diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzkvxu" "b/data_all_eng_slimpj/shuffled/split2/finalzzkvxu"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzkvxu"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\nEvery particle accelerator needs a vacuum system to transport its particle beams with low losses. With a~certain probability beam particles interact with residual gas molecules. The scattering leads to immediate particle losses from the beam, or to a deterioration of the beam quality. Both effects are not wanted and the beam losses can even lead to a problematic activation of accelerator components. For different situations, i.e. beam particle species and beam energies, the requirements for the quality of acceptable vacuum conditions can differ significantly. For example in a hadron accelerator with a section that is passed by the beam only once, a pressure of $10^{-5}\\,$mbar might be sufficient. On the other hand an~electron storage ring might need a base pressure of $10^{-10}\\,$mbar in the absence of beam. Consequently the~technology and physics of vacuum systems covers a broad range of effects and concepts, depending on the application. Vacuum systems for electron and proton storage rings are discussed for example in Ref. \\cite{benvenuti}, and a dedicated CERN school on accelerator vacuum is documented under Ref. \\cite{CERN}. Fundamentals of vacuum physics are discussed in Ref. \\cite{redhead}. \nFigure\\,\\ref{fig:p_overview} shows the range of pressure levels from ambient conditions down to the lowest pressures in accelerators. The pressure ranges for typical applications, the volume density of molecules and the mean free path of nitrogen molecules are indicated. The pressure range for beam vacuum is in technical language referred to as Ultra-High-Vacuum (UHV). One often speaks about dynamic vacuum, when the presence of an intense beam affects the residual gas pressure.\n\n\\begin{figure}\n\\centering\\includegraphics[width=0.85\\textwidth]{Fig_01_Vac.png}\n\\caption{Gas pressures ranging from ambient conditions down to cold accelerator vacuum cover 13 orders of magnitude.}\n\\label{fig:p_overview}\n\\end{figure}\n\n\n\\section{Pressure and Gas Equation}\nPressure is basically a force that gas molecules apply to a surface by mechanical momentum transfer, averaged over a huge number of collisions during a macrosopic time interval. Pressure is measured as a~force per area. A common unit is Pascal, 1\\,Pa~=~1\\,N\/m$^2$. Another common unit is 1\\,mbar~=~$100$\\,Pa. The~average velocity of molecules depends on the square root of the temperature as shown in Eq. (\\ref{eq:velo}). Using the molecule density $n_v$ the average velocity can be related to the rate of molecules impinging on the wall of a vessel per area and per time.\n\n\\begin{equation}\n\\overline{v} = \\sqrt{ \\frac{8 k_b}{\\pi m_0} ~ T },~~ \\d{N}{A\\,dt} = \\frac{1}{4} n_v \\overline{v}\n\\label{eq:velo}\n\\end{equation}\n\nHere $k_b = 1.38\\times 10^{-23}\\,$J\/K is the Boltzmann constant. The gas equation is a fundamental law relating pressure, volume, temperature and amount of a gas:\n\n\\begin{equation}\nPV = N k_b T = nRT.\n\\label{eq:pv}\n\\end{equation}\n\n$R = 8.314\\,$N\\,m\\,\/\\,mole\\,K is the gas constant. The majority of vacuum systems operate at room temperature and we can consider the temperature as being constant. Consequently the product of pressure and volume is a measure of the amount of gas, related to the number of molecules $N$ or the number of moles $n$. In practice a leak rate is often given in terms of mbar\\,l\/s, which is the amount of gas that is entering the considered recipient per unit of time.\nVacuum systems are equipped with pumps, devices that absorb gas molecules. A vacuum pump is characterised by its pumping speed $S = Q\/P$, quantified in l\/s at its interface. Here $Q$ is the gas load at the interface of the pump. The pumping speed varies for different gas species. Some chemically inactive gases like the inert gases He, Ar, and for example methane (CH$_4$), are pumped less efficiently by several types of pumps like Titanium sublimation pumps and NEG pumps. Turbo pumps and cryo pumps are not based on chemical binding reactions and are suited for all gas types. \nIn vacuum systems we can discriminate two types of flow regimes - viscous flow and molecular flow. Viscous flow occurs for higher gas densities. The higher the density of a gas the~shorter is the mean free path between two collisions of a gas molecule with others. The Knudsen number is the ratio between the mean free path and a typical dimension of the vacuum recipient. If the~mean free path is much larger than a typical dimension, we have molecular flow. With the cross section $\\sigma$ for collissions the mean free path $\\lambda$ is calculated as follows:\n\n\\begin{equation}\n\\lambda = \\frac{k_b T}{\\sqrt{2} \\sigma P}.\n\\end{equation}\n\nFor example Nitrogen residual gas at room temperature and a pressure of 10$^{-6}$\\,mbar has a mean free path of 60\\,m, which is much larger than the diameter of a typical vacuum pipe. Thus for particle accelerators we normally deal with molecular flow from a source of residual gas to a pump. We define the conductance of a vacuum component, e.g. an orifice or a piece of vacuum tube, as the ratio of the~molecular flow and the pressure drop across the element: \n\n\\begin{equation}\nC = \\frac{Q}{\\Delta P} . \n\\label{eq:cond}\n\\end{equation}\n\nThe conductance of an orifice of cross section $A$ can be estimated by:\n\\begin{equation}\nC = \\sqrt{\\frac{k_b T}{2\\pi M}} A\\, , ~~~\nC_\\mathrm{air} = 11.6 \\mathrm{[l\/s]} ~\nA \\mathrm{[cm^2]}.\n\\end{equation}\n\nOrifices may be used to realise a defined pumping speed for outgassing measurements. The conductance of a circular tube with diameter $d$ and length $l$ is given by:\n\\begin{equation}\nC = \\sqrt{\\frac{2 \\pi k_b T}{M}} \\frac{d^3}{l} \\, , ~\nC_\\mathrm{air} = 12.1 \\mathrm{[l\/s]} \\,\n\\frac{d^3 \\mathrm{[cm]}}{l \\mathrm{[cm]}}.\n\\end{equation}\n\nConductance varies with the type of gas, and in both formulas $M$ denotes the molecular mass of the~considered gas species. If two components are connected, the resulting conductance can be calculated from the individual conductances $C_1, C_2$. For a concatenation in series the resulting conductance is obtained by inverse addition: \n\n\\begin{equation}\nC_\\mathrm{total} = \\left( \\frac{1}{C_1} + \\frac{1}{C_2} \\right) ^{-1}.\n\\end{equation}\n\nAnd for a combination in parallel, simple addition has to be applied: \n\n\\begin{equation}\nC_\\mathrm{total} = C_1 + C_2. \n\\end{equation}\n\nIn a practical example an ion sputter pump of $400\\,$l\/s is connected to a recipient by a 30\\,cm long, $d=8\\,$cm tube. This results in an effective reduction of the pumping speed to $136\\,$l\/s. \n\nIn an accelerator vacuum system the operating pressure is established as a balance between release rate of free gas molecules and the removal rate that results from conductance and pumping. Gas molecules are released by different effects, such as thermal desorption from surfaces, beam induced desorption, diffusion of gas molecules from the bulk of material, permeation from ambient conditions through material, but also from leak rates through sealed connections between components. Among those effects thermal desorption is typically a significant contribution. Because of thermal energy, physical and chemical bindings of gas molecules to the material of the wall can be broken up and molecules are released. The sojourn time of a molecule at the wall is an exponential function of the temperature: $\\tau \\propto \\exp{\\left( E_d\\,\/\\,k_b T \\right) }$. For example a binding energy of 1\\,eV at room temperature leads to a sojourn time of 5 hours. The qualitative behaviour of the pressure in a recipient during pump-down, that results from the mentioned outgassing mechanisms, is shown in Fig.\\,\\ref{fig:pdown}. In the first phase of the pumpdown the~amount of gas removed from the recipient per unit time is proportional to the pressure, and thus one observes an exponential pressure decay. At some point a balance is reached between pumping speed and release of gas molecules from the surface of the walls into the volume of the recipient. In this phase the pressure decays inversely with time until the surface is emptied. Afterwards the outgassing is dominated by molecules that are transported in the bulk material to the surface by thermal diffusion. According to the~nature of diffusion this process scales as $1\/\\sqrt{t}$. It is possible to reduce this contribution to outgassing by a heat treatment of the material under vacuum conditions, thereby enhancing diffusion speed \\cite{calder}. Finally an equilibrium is reached when the remaining pressure is dominated by gas that permeates from the outside through the wall of the vacuum vessel. Naturally the permeation rate is very small in typical situations, and diffusion coefficients in metals are relevant only for light molecules like hydrogen.  \n\n\\begin{figure}\n\\centering\\includegraphics[width=0.90\\textwidth]{Fig_02_Vac.png}\n\\caption{Qualitative pump down behaviour of a recipient. Note the large logarithmic time scale on which $10^8$\\,s corresponds to three years.}\n\\label{fig:pdown}\n\\end{figure}\n\nSo far we have considered thermal outgassing of materials as the main source of gas load in a~vacuum recipient. In the presence of an intense particle beam the release of gas molecules from the~chamber walls can be drastically enhanced, by orders of magnitude, and those effects must be taken into account and quantified. A prominent effect is photo-desorption due to synchrotron radiation (SR), emitted by an electron or positron beam \\cite{fischer}. When a photon is absorbed on a metallic surface, the~photo effect may lead to the emission of a free electron. Absorption of the electron may then result in the~release of a gas molecule that was bound on the surface of the vacuum chamber. In a dipole magnet with bending radius $\\rho$ the number of photons emitted per unit length and time is given by:\n\n\\begin{equation}\n\\frac {dN_\\gamma}{dtds} = 1.28 \\cdot 10^{17} \\frac{I\\,[\\RM{mA}]\\,E\\,[\\RM{GeV}]}{\\rho\\,[\\RM{m}]}\\,.\n\\end{equation}\nHere $E$ is the beam energy and $I$ the beam current. The photons exhibit a statistical distribution of their energies and the given formula delivers the total number. More details on the properties of synchrotron radiation are given for example in Ref. \\cite{vacuumelectronic}. The SR induced desorption results in a specific outgassing rate per unit length of:\n\n\\begin{equation}\nq = \\eta \\, k_b\\, T\\, \\frac {dN_\\gamma}{dt\\,ds}\\,.\n\\end{equation}\n\nThe desorption yield $\\eta$ equals the number of gas molecules released per absorbed photon. It depends on the material of the chamber wall, the preparation of the material and foremost the conditioning of the surface under the bombardment with photons. For a surface cleaned with standard procedures one observes a decrease of the desorption yield inversely proportional to the total number of absorbed photons, e.g. Ref.\\,\\cite{billy}. The pressure obtained in a storage ring as a balance between beam induced desorption and installed pumping speed is called dynamic pressure as it depends on the beam intensity. In Fig.\\,\\ref{fig:petra} the~conditioning process of a NEG pumped vacuum chamber for the synchrotron light source PETRA-III is shown. The pressure rise per beam current is reduced by several orders of magnitude over the course of the conditioning process. In a well conditioned vacuum system the desorption yield $\\eta$ may reach values of $10^{-6}$ or lower. Rings with high intensity beams exhibit a high photon flux and one might be tempted to expect also high dynamic pressure for those. However, in this situation also the conditioning process advances faster and so the achieved pressure after a certain operating time is often similar for different storage rings. The typical gas composition is dominated by H$_2$ and a $\\sfrac{1}{4}$ fraction of CO.\n\n\\begin{figure}\n\\centering\\includegraphics[width=0.85\\textwidth]{Fig_03_Vac.png}\n\\caption{The pressure rise per beam current is called dynamic pressure. It is shown here for a test chamber of the~PETRA-III storage ring as a function of integrated beam current.}\n\\label{fig:petra}\n\\end{figure}\n\nBesides the discussed mechanism of photo desorption there are other mechanisms causing beam induced desorption. If ion beams are accelerated, small losses of energetic ions may lead to high desorption rates. For example there are situations where a single ion, impinging on the wall of the vacuum chamber, may release of the order of $10^5$ gas molecules \\cite{sen}. Ion induced desorption cannot easily be reduced by conditioning. \n\nAnother important effect is the electron cloud instability \\cite{ecloud}. A free electron is accelerated in the~electromagnetic field of the beam and releases more than one secondary electron when absorbed on the chamber wall. Due to the multiplicative behavior of the process, the intensity of these electrons grows exponentially, resulting in a proportional growth of gas release in the vacuum chamber. A key factor for this process is the secondary emission yield, that is the number of electrons released after the~absorption of one electron. It depends on the type of material and can be reduced by coating of the~surface, for example with TiN or C. Another countermeasure is the application of a small longitudinal magnetic field, by winding a coil around the beampipe, to force the low energy free electrons on spiral paths.\n\n\\section{Pressure Computation for One-Dimensional Systems}\n\nAccelerator vacuum systems are build for beam transport and are typically lengthy, that is the longitudinal dimension is much larger than the transverse dimensions. For a calculation of the pressure profile a one-dimensional calculation is then sufficient. We start the derivation of the corresponding diffusion equation from the previously discussed relation between gas flow $Q$ and conductance $C$ in Eq. (\\ref{eq:cond}). Note that we have to introduce at this point a minus sign that ensures positive gas flow towards smaller pressure. In the~limit of small differences we obtain a differential equation for the gas flow. Here a specific conductance $\\mathcal{C} = C \\Delta s$ is introduced, which is a property of the vacuum vessel cross section. \n\n\\begin{eqnarray}\nQ & \\propto & - \\d{P(s)}{s} \\nonumber \\\\\nQ(s) & = & -\\mathcal{C}\\cdot \\d{P(s)}{s} \n\\label{eq:flow}\n\\end{eqnarray}\n\nA second equation is the continuity equation for the gas flow, which expresses that the change in the amount of transported gas is given by the sum of outgassing from the wall minus the pumped gas of a considered section with infinitesimal length. Here we use the specific pumping speed $\\mathcal{S} = S\/\\Delta s$ and the specific outgassing rate $q$ per unit length in mbar\\,l\/m\\,s.\n\n\\begin{equation}\n\\d{Q(s)}{s} = q - \\mathcal{S}\\,P(s)\n\\label{eq:continuity}\n\\end{equation}\n\nThese two Eqs. (\\ref{eq:flow}) and (\\ref{eq:continuity}) can now be combined into a~second order diffusion equation for a~one-dimensional vacuum system with the independent coordinate $s$.\n\n\\begin{equation}\n\\d{}{s}\\,\\mathcal{C}\\,\\d{}{s}\\,P(s)-\\mathcal{S}P(s) + q = 0\n\\label{eq:diffeq}\n\\end{equation}\n\nDepending on the nature of the pumps two different types of solutions are obtained for Eq. (\\ref{eq:diffeq}). In the most common situation of systems with lumped pumps the pumping speed $\\mathcal{S}$ is non-zero only for short sections of the system. In-between these pumps the pressure distribution follows a quadratic function with a maximum half-way between a pair of pumps that are installed at a distance $l$:\n\n\\begin{equation}\nP(s) = \\frac{ql}{S}+ \\frac{q}{8\\mathcal{C}} \\left(l^2-4s^2\\right).\n\\label{eq:qprofile}\n\\end{equation}\n\nPeak pressure and average pressure for this situation are given by:\n\n\\begin{equation}\nP_\\RM{avg} = ql\\,\\left(\\frac{1}{S}+\\frac{l}{12\\,\\mathcal{C}}\\right), \\quad\nP_\\RM{max} = ql\\,\\left(\\frac{1}{S}+\\frac{l}{8\\,\\mathcal{C}}\\right).\n\\label{eq:aprofile}\n\\end{equation}\n\nThese functions have two terms. Even with infinite pumping speed (left term) the pressure is still limited by the conduction of the vacuum vessel. The economy of a technical solution is thus to be optimized w.r.t. density and size of the vacuum pumps, and the transverse dimensions of the vacuum vessel that determine the specific conductance.\nBesides lumped pumps, also distributed pumps are used in accelerator vacuum systems. Distributed pumps can be realised by NEG strips or NEG coating, and by distributed ion sputter pumps that utilize the magnetic field of accelerator magnets. For the case of non-zero pumping speed we obtain an exponential solution of Eq. (\\ref{eq:diffeq}). At some distance from neighboured sections the pressure in a long section with distributed pumps approaches a value of $P=q\/\\mathcal{S}$. Figure\\,\\ref{fig:v2} shows an exemplary calculation of a pressure distribution with lumped pumps in the right part and a long distributed pump in the left part of the graph.\n\n\\begin{figure}\n\\centering\\includegraphics[width=0.925\\textwidth]{Fig_04_Vac.png}\n\\caption{Simulated pressure profile in the upper part of the image. The lower part shows the pumping speed along the beamline. A $d=10\\,$cm beam pipe is assumed for this demonstration. While the left section contains a 20m long distributed pump, the  right section has 5 lumped pumps installed.}\n\\label{fig:v2}\n\\end{figure}\n\nPressure profiles can also be calculated in a numerical approach using matrix multiplications \\cite{ziemann}: \n\n\\begin{eqnarray}\n\\left(\\begin{array}{c} P(l) \\\\ Q(l) \\end{array} \\right) & = &\n\\left( \\begin{array}{cc}\n\\cosh(\\alpha l) & -\\frac{1}{c\\alpha} \\sinh(\\alpha l) \\\\\n-\\alpha c \\sinh( \\alpha l) & \\cosh(\\alpha l) \\end{array} \\right)\n\\left( \\begin{array}{c} P(0) \\\\ Q(0) \\end{array} \\right)\n+ \\frac{q}{\\alpha} \\left(\n\\begin{array}{c} \\frac{1-\\cosh(\\alpha l)}{\\alpha c} \\\\\n\\sinh(\\alpha l)  \\end{array} \\right). \\hphantom{8888} \n\\label{eq:matrix}\n\\end{eqnarray}\n\nHere the variable $\\alpha = \\sqrt{\\mathcal{S}\/\\mathcal{C}}$ was introduced. For sections without pumping, $\\mathcal{S}=0$, the elements containing $\\alpha$ in Eq. (\\ref{eq:matrix}) have to be taken in the limit $\\alpha\\rightarrow 0$, leading to the quadratic profile Eq. (\\ref{eq:qprofile}). So far we have considered the static situation of equilibrium pressure profiles. This is a special case of the general time dependent diffusion equation: \n\n\\begin{equation}\n\\mathcal{V} \\d{}{t} P(s,t) = \\d{}{s}\\,\\mathcal{C}\\,\\d{}{s}\\,P(s,t)\n-\\mathcal{S}P(s,t) + q.\n\\label{eq:timediff}\n\\end{equation}\n\nThe specific volume $\\mathcal{V}$ denotes the volume of the vessel per unit length. This equation describes also dynamic evolutions of the pressure profile, for example when He gas is injected for reasons of leak search. By comparison with a classical diffusion equation $\\d{}{t} f(x,t) = \\d{}{x}\\,\\mathcal{D}\\,\\d{}{x} f(x,t)$ that is known from the literature, we identify the~diffusion coefficient of our problem as:\n\n\\begin{equation}\n\\mathcal{D} = \\frac{<\\Delta x^2>}{<\\Delta t>} = \\frac{\\mathcal{C}}{\\mathcal{V}}.\n\\label{eq:diffcoeff}\n\\end{equation}\n\nUsing this parameter one can estimate that it takes 3 seconds for He gas to travel a 5\\,m distance in a 2\\,cm pipe. Figure\\,\\ref{fig:diff_He} shows the time evolution of the pressure distribution for a Delta-function like inlet of He gas in such a pipe.\n\n\\begin{figure}\n\\centering\\includegraphics[width=0.85\\textwidth]{Fig_05_Vac.png}\n\\caption{A pressure bump of He in this example diffuses in a 2\\,cm diameter tube at limited speed.}\n\\label{fig:diff_He}\n\\end{figure}\n\nThe shown analytic calculations allow one to estimate the pressure in vacuum systems and to calculate the required number of pumps and their distance. For complex geometries Monte Carlo methods for pressure calculations might be more accurate. The code MolFlow \\cite{molflow} follows individual gas molecules on their scattered path through a vacuum system, including sticking probabilities and sojourn times. Pressure and other variables are then calculated as statistical averages. For lepton storage rings the~code may be augmented by another code, SynRad \\cite{synrad}, which allows to simulated the photo desorption process.\n\n\\section{Requirements for Accelerator Vacuum Quality}\n\nIn order to assess the required vacuum quality in terms of pressure or density it is necessary to study the~different mechanisms of beam gas interaction. Scattering of beam particles may lead to immediate loss of beam particles, or to a degradation of the beam properties. The number of lost particles $\\Delta N_b$ from a beam passing through a section of length $\\Delta l$ can be estimated in the following way:\n\n\n\\begin{eqnarray}\n\\Delta N_b & = & -N_b \\times \\frac{\\mathrm{area\\,of\\,molecules}}{\\mathrm{total\\,area}} \\nonumber \\\\\n& = & -N_b \\times \\frac{n_v V \\sigma}{V\/\\Delta l} \\nonumber \\\\\n& = & -N_b n_v \\sigma \\Delta l \\nonumber \\\\\n& = & -N_b n_v \\sigma \\beta c \\Delta t \\, . \\nonumber\n\\label{eq:cross}\n\\end{eqnarray}\n\nHere $\\sigma$ is the cross section of a generic interaction process, $N_b$ is the number of beam particles and $n_v$ is the volume density of the residual gas. In case the considered scattering mechanism leads to the loss of the particle, and by converting this relation into a differential equation, the solution is given by an exponential decay of the beam intensity:\n\n\\begin{equation}\nN_b(t) = N_0 \\exp \\left( - \\sigma \\beta c n_v \\, t \\right), \n~\\tau = \\frac{1}{\\beta c \\, \\sigma \\, n_v}.\n\\label{eq:diffcoeff}\n\\end{equation}\n\nThe beam lifetime $\\tau$ is deduced from the exponential solution, and for most situations we can safely assume $\\beta=1$. In the following we consider the effect of different scattering mechanisms for electrons and protons.\\hfill\\\\\n\n\\begin{figure}\n\\centering\\includegraphics[width=0.85\\textwidth]{Fig_06_Vac.png}\n\\caption{A beam with $N_b$ particles passes through a volume with residual gas density $n_v$. The gas molecules exhibit an effective cross section $\\sigma$ that describes the interaction probability with the beam for specific processes.}\n\\label{fig:beam_gas}\n\\end{figure}\n\n\\subsection{Coulomb Scattering for Electrons}\n\n\\noindent\nThis process is described by the well known formula for Rutherford Scattering, which gives the differential cross section for the occurrence of a scattering angle $\\theta$:\n\n\\begin{equation}\n\\frac{d\\sigma_i}{d\\Omega} = \\frac{Z_i^2\\,r_e^2}{4\\gamma^2}~\n\\frac{1}{\\sin^4(\\theta\/2)}.\n\\label{eq:rutherford}\n\\end{equation}\n\nThe charge of the residual gas atom is $Z_i$ and $r_e=2.8\\,$fm the classical electron radius. If we integrate this differential cross section from the angle $\\theta_0$ above which particles are lost to $\\pi$ and use $\\theta_0 \\ll 1$, the total elastic scattering cross section for particle loss is:\n\n\\begin{equation}\n\\sigma_{i, \\RM{el}} = \\frac{2\\pi\\,Z^2_i\\,r_e^2}{\\gamma^2}~\n\\frac{1}{\\theta_0^2}.\n\\label{eq:elastic}\n\\end{equation}\n\nThe limiting angle $\\theta_0$ can be estimated from a typical value for the $\\beta$-function $\\overline{\\beta_y}$ and the minimum aperture $A_y$ of the accelerator: $\\theta_0 = A_y\/\\overline{\\beta_y}$. Using the parameters for the vertical plane is sufficient since the vertical aperture is usually smaller than the horizontal one in electron storage rings. Average values of the $\\beta$ functions at the locations of particle scattering and particle loss have to be used. Combining the~above relations and carrying out the sum over different atom species we obtain the following formula for the beam lifetime due to elastic scattering:\n\n\\begin{equation}\n\\tau^{-1}_\\RM{el} = \\frac{2\\pi r_e^2 c}{\\gamma^2}~\n\\frac{\\overline{\\beta_y}^2}{A_y^2} \\sum_i n_i \\sum_j k_{ij} Z_j^2.\n\\label{eq:telastic}\n\\end{equation}\n\nHere $k_{ij}$ is the number of atoms of type $j$ within the molecule of type $i$. By inserting numbers for the fundamental constants and expressing the gas density in terms of pressure at room temperature we obtain the following formula for the beam lifetime in electron rings due to elastic scattering:\n\n\\begin{equation}\n\\tau_\\RM{el}\\,[\\RM{h}] = 2839~\\frac{E^2\\,[\\RM{GeV}^2]\\,\\,A_y^2\\,[\\RM{mm}^2]}\n{\\overline{\\beta_y}^2\\,[\\RM{m}^2]}~\n\\left( \\sum_i P_i\\,[\\RM{pbar}] \\sum_j k_{ij} Z^2_j \\right)^{-1}.\n\\label{eq:handy}\n\\end{equation}\n\nNote that the quadratic function of $Z$ causes a sensitive dependence of the beam lifetime on the~presence of heavy gas species in the gas composition.\\hfill\\\\\n\n\\subsection{Bremsstrahlung} \\noindent\nDue to deceleration of a beam particle in the Coulomb field of a residual gas atom and the emission of a high energy photon, the particle may leave the energy acceptance of the accelerator. The important parameter in this context is the largest allowed relative energy deviation for the particles to stay confined within the beam: $\\delta_E = \\Delta E\/E_0$.  The cross section for the inelastic process is\n\n\\begin{equation}\n\\sigma_\\RM{inel} \\approx -\\frac{4}{3}\\,\\, \\frac{V_n}{N_A}\\,\\,\\frac{1}{X_0}\\,\\ln\\,\\delta_E.\n\\label{eq:inelastic}\n\\end{equation}\n\nFrom this the following lifetime $\\tau_\\RM{brems}$ is computed. For a gas mixture one has to sum up contributions of gas species with their partial pressures $P_i$ and corresponding radiation lengths $X_{0,i}$. \n\n\\begin{eqnarray}\n\\frac{1}{\\tau_\\RM{brems}} & = & -\\frac{4}{3}~\\frac{c}{P_n}\\ln(\\delta_E)~ \\sum_i \\frac{P_i}{X_{0,i}} \\nonumber \\\\\n\\tau_\\RM{brems}\\,[\\RM{h}] & = & \\frac{-0.695}{\\ln(\\delta_E)}\\,\\left(\\sum_i \\frac{P_i\\,[\\RM{pbar}]}{X_{0, i}\\,[\\RM{m}]}\\right)^{-1}.\n\\label{eq:brems}\n\\end{eqnarray}\n\n$N_A$ is the Avogadro constant and $V_n=22.4\\,$l\/mol, $P_n$ the molar volume and the pressure under standard conditions. The \\emph{radiation length} $X_0$ is the length over which a particles energy has dropped by a factor $1\/e$. $X_0$ scales roughly inversely proportional to the square of the nuclear charge of the~residual gas, and also inversely proportional to its density.  Radiation length values for common gases under normal conditions are tabulated for example in Ref. \\cite{pd}. In Table \\ref{tab:radlength} we list $X_0$ for the important gases of accelerator vacuum systems. With common energy acceptance and transverse acceptance of storage rings, the mechanism of Bremsstrahlung is the more severe mechanism for loss of particles from the~beam. \n\n\n\\subsection{Emittance Growth for Hadrons} \\noindent\nFor a proton or ion beam already the degradation of the beam emittance from elastic gas scattering at small angles is harmful. Due to the absence of radiation damping any decrease of the beam density over the storage time cannot be recovered. For the beam emittance a growth time can be defined:\n\n\\begin{equation}\n\\frac{1}{\\tau_\\varepsilon} = \\frac{1}{\\varepsilon_x}~\\frac{d\\varepsilon_x} {dt}.\n\\label{eq:emmlife}\n\\end{equation}\n\nThe scattering causes a diffusive growth of the mean squared angular deviation of the particles momentum vector which is linear in time.  The emittance growth is related to this angle as follows ($\\theta_0$ is the rms scattering angle projected on a transverse plane):\n\n\\begin{equation}\n\\frac{d\\varepsilon}{dt} = \\frac{1}{2}~\\overline{\\beta_y}~\\frac{d(\\theta_0^2)}{dt} = \n\\frac{1}{2}~\\overline{\\beta_y}~\\frac{(13.6)^2}{(cp)^2\\,[\\RM{MeV}^2]}~\\frac{c}{P_0}~\n\\sum_i \\frac{P_i}{X_{0,i}}.\n\\label{eq:emmgrowth}\n\\end{equation}\n\nUsing Eq. (\\ref{eq:emmlife}) the resulting emittance growth time for protons is:\n\n\\begin{equation}\n\\tau_\\varepsilon\\,[\\RM{h}] \\approx 34.2~\\frac{\\varepsilon_y\\,[\\RM{m\\,rad}]\\,\nE^2\\,[\\RM{GeV}^2]\\,T\\,[\\RM{K}]}{\\overline{\\beta_y}\\,[\\RM{m}]}~\\left(\n\\sum_i \\frac{P_i\\,[\\RM{pbar}]}{X_{0, i}\\,[\\RM{m}]}\\right)^{-1}.\n\\end{equation}\n\nThe temperature $T$ has been included since proton accelerators often use superconducting magnets and cold beam pipes. The described mechanism ignores other elastic scatting mechanisms besides Coulomb scattering. \\hfill\\\\\n\n\\subsection{Inelastic Scattering for Hadrons}\n\nAnother process is the complete removal of particles from the beam by an inelastic reaction. The beam lifetime for this effect can be computed using the inelastic interaction length $\\lambda_\\RM{inel}$ which is also tabulated in Table\\,\\ref{tab:radlength}:\n\n\\begin{equation}\n\\frac{1}{\\tau_\\RM{inel}} = \\frac{\\beta c}{P_0}~\n\\sum_i \\frac{P_i}{\\lambda_{\\RM{inel},i}}.\n\\label{eq:tauinel}\n\\end{equation}\n\nThe inelastic interaction length is related to the corresponding nuclear cross section via $\\lambda_\\RM{inel} = A\/\\rho\\,N_A\\,\\sigma_\\RM{inel}$, where $A$ is the molar mass and $\\rho$ the density. If we again include the gas temperature and take out all constants we obtain the following formula for the inelastic beam-gas lifetime:\n\n\\begin{equation}\n\\tau_\\RM{inel}\\,[\\RM{h}] = 3.2\\cdot10^{-3}~T\\,[\\RM{K}]\\,\\left(\n\\sum_i \\frac{P_i\\,[\\RM{pbar}]}{\\lambda_{\\RM{inel}, i}\\,[\\RM{m}]}\\right)^{-1}.\n\\end{equation}\n\n\\begin{table}\n\\begin{center}\n\\caption{Radiation length $X_0$ and inelastic interaction length $\\lambda_i$ \nfor different gases under atmospheric pressure and 20\\,$^\\circ$C \\cite{pd}.}\n\\label{tab:radlength}\n\\begin{tabular}{|l|cc|cc|cc|cc|cc|cc|cc|cc|cc|c|}\n\\hline\\hline\n&& \\textbf{H}$_\\textbf{2}$ && \\textbf{He} && \\textbf{CH}$_\\textbf{4}$ && \\textbf{H}$_\\textbf{2}$\\textbf{O} &&\\textbf{CO}&&\\textbf{N}$_\\textbf{2}$&&\\textbf{Ar}&&\\textbf{CO}$_\\textbf{2}$&&\\textbf{air} \\\\\n\\hline\nA && 2   && 4  && 16     && 18     &&28&& 28  &&40&&44    &&     \\\\\n$X_0$\\,[m] && 7530 && 5670 && 696 && 477 && 321&&326&&117&&196&& 304    \\\\\n$\\lambda_\\RM{inel}$\\,[m]&&6107&&3912&&1103&&1116&&763&&753&&704&&490&&747 \\\\\n\\hline\\hline\n\\end{tabular} \\end{center}\n\\end{table}\n\n\\section{Vacuum Technology for Accelerators}\n\\subsection{Pumping}\n\\label{pumping}\n\nAs discussed in the previous sections the average pressures required for accelerators range from $10^{-6}\\,$mbar down to $10^{-11}\\,$mbar. For facilities of large size, containing a huge number of components, reliability of the overall system becomes a critical aspect. Pumps without moving mechanical parts are thus advantageous. Figure\\,\\ref{fig:pumps} shows an overview of the most commonly used types of pumps in large accelerator systems. Initial pumpdown of accelerator vacuum systems is often done by turbo-molecular pumps in combination with rotary pumps that are installed on mobile pump carts. During normal operation these are disconnected while the required vacuum conditions are maintained by sputter ion pumps, titanium sublimation pumps (TSP) and non-evaporable getter (NEG) pumps. These types of pumps are almost exclusively used for routine operation in storage rings and large linear accelerators. \n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.9\\textwidth]{Fig_07_Vac.png}}\n\\caption{Overview of the most commonly used types of pumps and gauges in large accelerator vacuum systems including their operating range.} \n\\label{fig:pumps}\n\\end{figure} \n\n\\emph{Turbo-molecular pumps} are fully mechanical pumps, based on a mechanism to transfer physical momentum to residual gas molecules in a preferred direction. The momentum transfer is achieved through fast moving blades with rotation frequencies of $30\\dots60\\cdot 10^3\\,$ RPM (rounds per minute). At these rotation frequencies the blades reach a speed of $300\\dots600\\,$m\/s, which is to be compared with the speed of the residual gas molecules that should ideally not escape the blades. At room temperature heavier molecules like CO have a speed of $470\\,$m\/s, while hydrogen moves at $1800\\,$m\/s. Consequently the compression ratio (gas density ratio) achieved with a turbo-molecular pump may vary by several orders of magnitude for these gas species. Turbo-molecular pumps are always combined with mechanical roughing pumps that provide an intermittent vacuum level between UHV conditions and the normal atmospheric pressure.\n\n\\emph{Sputter ion pumps} are capable of pumping all gases and they can be operated at relatively high pressure. These pumps use penning cells with an applied high voltage of 3\\dots7\\,kV and a superimposed magnetic field. Permanent magnet blocks are used to generate the magnetic field. Residual gas molecules are ionized and accelerated in the electric field. The current drawn from the high voltage power supply is thus proportional to the residual gas pressure. This synergy is often used to measure the pressure in accelerators in a cost effective way. Hydrogen is pumped by diffusion into the bulk of the cathodes. All other reactive gases are chemisorbed by the cathode material. A commonly used material is titanium, which is sputtered onto the walls and anodes by the ions upon incident on the cathodes. Noble gases are physically buried by the sputtered cathode atoms. The pumping speed for noble gases is small, but it can be increased to values of $25-30\\,$\\% of that for N$_2$ replacing the cathode plate by heavy material such as tantalum. Such \\emph{noble diodes} are often used in systems with enhanced risk of helium leaks as in accelerator sections \nusing superconducting magnets or resonators cooled with liquid helium.\n\n\\begin{figure}\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig_08_Vac.png}\n\\caption{Penning cells are elements in ion sputter pumps to ionize residual gas molecules. In a combined electric and magnetic field electrons are accelerated on spiral paths, thereby enhancing their efficiency to ionize neutral gas molecules. Ions are then also accelerated and implanted in the cathod material or burried under sputtered metal on the anodes.}\n\\label{fig:penning}\n\\end{figure}\n\nA cost effective solution are so-called integrated sputter ion pumps inserted linearly into a channel parallel to the beam channel \\cite{cummings}, \\cite{koupt}. These pumps utilize the magnetic field of the bending magnets of the accelerator. This solution has been adopted at the electron ring of HERA, PETRA, PEP and TRISTAN reaching typical pump speeds of $25\\,$l\/s\/m. On the downside this couples the functions of the accelerator magnets and the vacuum system. Magnets must be powered to maintain pumping. In-between ramp cycles of storage rings these pumps are not active. Furthermore the magnetic field intensity is decreased when the particles are injected at lower energies, resulting in a reduced pumping speed. This becomes crucial for accelerators where the injection energy and thus the corresponding magnetic field are much lower than the nominal beam energy, such that the discharge in the sputter ion pumps extinguishes. Thus additional pumps are required to ensure good vacuum conditions. \n\n\\emph{Titanium sublimation pumps} are sorption pumps. In a metallic vessel titanium\nis evaporated by temporary electrical heating and deposited on the walls, forming a getter surface. These pumps exhibit a~high pumping speed for active gases but have limited pumping capacity since the thin titanium film saturates quickly. The pumping surface is renewed\nby deposition of fresh titanium from a heated filament by sublimation. A pump with $1000\\,$l\/s pumping speed is saturated after one hour at a pressure of $10^{-7}\\,$mbar. Titanium sublimation pumps cannot pump noble gases and are therefore used in combination with a low pumping speed ion sputter pump. Often a chicane is incorporated in the pumping port, to avoid that titanium atoms enter the beam chamber or contaminate surfaces, e.g. mirrors, ceramic insulators or instrumentation.  \n\n\\emph{Non-evaporable getter (NEG) pumps} are sorption pumps as well. The NEG material is made of special alloys which forms stable chemical compounds with the majority of active gas molecules. The~sorption of hydrogen is reversible by heating of the material. Also NEG pumps have a limited pumping capacity. The NEG material is activated by heating for times below one hour. Activation temperatures depend on the NEG material and range from $180^\\circ$C to $400^\\circ$C. During heating the gas molecules are not evaporated from the NEG material, but the molecules diffuse into the bulk material. Hydrogen is an exception, which is released again into the gas phase, thus requiring other pumps during the process of activation and reactivation. The heating produces fresh surface sites for further adsorption of active gases. The NEG material is a compound of different metals, for example Zircon, Vanadium, Iron, and is typically sintered in the form of a powder onto flexible strips. Such strips can be integrated in a side channel of the vacuum chamber design. The initial specific pumping speed provided by such schemes may exceed 1000\\,l\\,s$^{-1}$\\,m$^{-1}$. This technique has been developed for LEP \\cite{ben-NEG} and is now applied in many accelerators. At LEP about 24 km of the beam pipe have been equipped with 30\\,mm wide and 0.2\\,mm thick constantan ribbons coated with 0.1 mm thin NEG material on both sides. A newer application for the synchrotron light source PETRA-III is shown in Fig.\\,\\ref{fig:chamber}. The strips are installed onto a~rigid stainless steel carriage via insulating ceramics inside a separate pump channel. For electric heating the pumps are connected to current feedthroughs. NEG pumps are also available commercially as lumped pumps, cartridge units that can be connected to recipients by standard flange connections.\n\nAnother approach to the NEG pumping concept is the deposition of thin film coatings of TiZrV, sputtered onto a vacuum chamber. The activation temperature of these coatings is relatively low, a fact that is important to limit the thermal stress on the vacuum system during activation \\cite{Benvenuti-coating-1}, \\cite{Benvenuti-coating-2}. If the~design of the vacuum chamber allows that, the coating may cover the complete inner surface of the~beam vacuum system and thus the outgassing of the vacuum vessel itself is drastically reduced. It should be emphasised that even with low activation temperature the need for baking the entire vacuum system to ca. $200\\,^\\circ$C implies severe restrictions for the design of vacuum chambers, support structures and magnets. For accelerators in which the vacuum conditions are dominated by beam induced desorption, it is possible to reduce outgassing with NEG coated surfaces in the vicinity of the particle beam. In particular for light sources NEG coated surfaces are advantageous to reach good vacuum conditions without long conditioning times \\cite{chiggiato}. Modern light sources use complex multi-bend achromat lattice cells to achieve extremely small horizontal emittances. As a result the apertures of vacuum chambers are small, and the achievable pressure is conductance limited. Lumped pumping concepts are less efficient in this situation since a large number of pumps had to be installed per unit length. Today it is common for such light sources to use NEG coating for efficient pumping of the narrow beam chambers. Also hadron accelerators benefit from NEG coated chambers through the reduction of the secondary emission yield of electrons \\cite{wfischer}.\n\n\\begin{figure}\n\\centering\\includegraphics[width=0.85\\textwidth]{Fig_09_Vac.png}\n\\caption{Concept view of a vacuum chamber with integrated NEG pump for an electron beam storage ring. The~aluminum profile is extruded with integrated cooling channels. The synchrotron radiation is absorbed on the left side of the chamber, while the right side contains the NEG strip (blue) in a side channel, which is shielded from direct view to the beam to suppress capture of dust particles (courtesy DESY).}\n\\label{fig:chamber}\n\\end{figure}\n\n\\emph{Cryo pumps} use the effect of cryosorption to bind gas molecules on cold surfaces inside the pump vessel. With sufficiently low temperature these pumps can remove all gas species and they provide high pumping speed and capacitance. On the downside cryo pumps must be regenerated by regular warm-up cycles, for which the accelerator operation must be interrupted. During the regeneration cycle all pumped gases are released again, and are typically removed from the vacuum system by turbo molecular pumps. A concept sketch of a cryo pump is shown in Fig.\\,\\ref{fig:cryo}.\n\nFor performance reasons many accelerators make use of superconducting magnets or superconducting accelerating structures. With the beam pipe being integrated into the cryostat this leads to a \\emph{cold bore} vacuum system, which takes on the characteristics of a huge cryopump. Design and operation of such cold bore vacuum systems have many specific implications \\cite{grobner}.\n\n\\begin{figure}\n\\centering\\includegraphics[width=0.575\\textwidth]{Fig_10_Vac.png}\n\\caption{Concept view of a cryo pump with a large aperture flange connection on the top side (courtesy Lothar Schulz, PSI).}\n\\label{fig:cryo}\n\\end{figure}\n\n\\subsection{Instrumentation} \\label{instrumentation}\n\nAccelerator based research infrastructures are expensive installations, they use a lot of grid energy and their operation should be efficient and reliable. Also for vacuum systems it is therefore important that the integrity of the system is continuously monitored and problems like leaks or regions of unusual high pressure resulting from beam impact can be identified quickly to allow fast and targeted intervention. Using gate valves the beam vacuum system of accelerators is divided into several sections. In this way it is possible to exchange components without venting the entire facility. Fast shutters, capable to stop shock waves in milliseconds, are commonly used to avoid contamination of sensitive sections caused by a sudden break of the vacuum system.\n\nEach of the segments of a larger vacuum system should have at least one gauge to monitor the~integrated residual gas pressure. Total pressure gauges are included in Fig.\\,\\ref{fig:pumps} with their operating range. For practical operation it is usually not necessary to obtain precise absolute measurements, but to be able to diagnose relative changes over time and to compare sections with each other. Cold cathode gauges or Bayard-Alpert gauges are frequently used at low pressures. Current monitoring of sputter ion pumps is a~cost effective method, particularly for large facilities to monitor the residual gas pressure down to levels of $10^{-9}\\,$mbar. \n\nUsing quadrupole mass spectrometers the residual gas composition in a vacuum system can be analysed. The relative occurrence of molecule masses is determined, which allows to diagnose problems in an accelerator vacuum system. For example it can be determined whether a leak or a contamination is the reason for unusually high pressure. Such residual gas analysers (RGA) are typically not part of a standard installation, but these are temporarily connected to the recipient. Often the sensitive electronics is not compatible with the radiation environment of the stray magnetic fields of an accelerator in operation. \n\n\\subsection{Materials and Technology Choices for Accelerator Vacuum Systems}\n\nFor large facilities production cost can be optimized by a careful design and by utilization of industrial manufacturing processes. For example extrusion processes serve as a cost effective method to produce large lengths of beam pipe profiles. Over time a wide variety of best practices have been developed for accelerator vacuum systems and more details may be found for example in Ref. \\cite{CERN}.\n\nMaterials for the manufacturing of beam chambers and other beam vacuum components have to be selected carefully according to the specific requirements of each accelerator. A significant number of aspects must be addressed in parallel. The air pressure under normal conditions on a surface area of $10\\times10\\,\\RM{cm}^2$ results in a force equivalent to the weight of 100\\,kg. Consequently mechanical robustness is already one important criterion for a vacuum chamber, while the costly aperture of accelerator magnets might favour a thin wall solution. Vacuum chambers should be bake-able and thus stability must be guaranteed also at elevated temperatures. The magnetic guide field for the beam should not be disturbed by magnetic properties of the beam chamber. If synchrotron radiation is deposited on the chamber wall a high thermal conductivity is desirable. And the transport of beam image currents in the chamber walls requires good surface conductivity and smooth electrical connections across components. For machines susceptible to electron cloud instabilities the secondary electron emission yield of the material should be low. Last but not least the compatibility with UHV vacuum conditions and low outgassing rates are important. The requirement of UHV class pressure in combination with resistance to radiation and corrosive atmospheres demands all metal solutions for beam vacuum system.  \n\nStainless steel, copper or aluminum are the most common vacuum chamber materials. Electrical and thermal conductivity are by factors better for copper and aluminum surfaces compared to steel. On the other hand the mechanical strength of steel is outstanding. Copper and aluminum alloys exhibit better mechanical strength than the pure metals, at the expense of somewhat lower conductivity. A~few properties for common materials are listed in Table\\,\\ref{tab:materials}. Joining techniques for materials of vacuum systems include inert gas shielded arc welding, electron-beam welding, laser beam welding as well as brazing in a~furnace. Laser and electron beam welding can deposit large amounts of heating power to a~small volume in a short time. These technologies can be used advantageously for welding of sensitive components that allow only local heating while another part of the component must be kept at moderate temperature.\n\n\\begin{table}\n\\begin{center}\n\\caption{Properties of materials that are utilized for accelerator vacuum components.} \n\\label{tab:materials}\n\\begin{tabular}{|l|cc|cc|cc|cc|} \n\\hline\\hline\n                      && \\textbf{density}    && \\textbf{thermal}      && \\textbf{electrical}         && \\textbf{yield}      \\\\   \n                      &&            && \\textbf{conductivity} && \\textbf{conductivity}       && \\textbf{strength}   \\\\   \n\t                  && \\textbf{[g\/cm}$^3$] && \\textbf{[W\/K\/m]}      && \\textbf{[10}$^6\/ \\Omega$\\textbf{\/m]} && \\textbf{[N\/mm}$^2$] \\\\ \\hline \nstainless steel 316LN && 8.00  \t    && 16     \t    && 1.35 \t\t      && 205        \\\\ \naluminum pure         && 2.70\t    && 235     \t    && 37  \t\t          && 35         \\\\  \nAlMgSi0.5             && 2.70       && 200          && 30  \t\t          && 70-150     \\\\  \ncopper pure           && 8.95       && 394          && 58  \t\t          && 40-80      \\\\ \nCu\\,Sn$_2$            && 8.90       && 140          && 25  \t\t          && 150        \\\\ \n\\hline\\hline\n\\end{tabular} \\end{center}\n\\end{table}\n\nAluminum or copper are favoured for electron facilities due to their high thermal conductivity. The synchrotron radiation can be absorbed directly by the vacuum chamber if the power line-density is not exceeding values of $\\approx 100\\,$W\/m. With appropriate water cooling of the chamber a power load of several 10\\,kW\/m can be accepted. For intense synchrotron radiation fans with a shallow height the temperature profile should be simulated. The often quite elaborate beam pipe cross sections in combination with pumping and cooling ducts can be economically produced by continuous extrusion as shown in the example of Fig.\\,\\ref{fig:chamber}. More complex chambers are produced from solid blocks, which is associated with higher manufacturing cost. Neighboured chambers may be connected by aluminum flanges in combination with Helicoflex$^\\RM{TM}$ gaskets. Alternatively aluminum\/stainless steel transitions, for example made by explosion bonding, allow the usage of standard stainless steel Conflat$^\\RM{TM}$ flanges. Overall the sealing of aluminum systems is not as reliable as the standard stainless steel system with copper gaskets. \n\nIn some cases copper or copper alloy chambers are used to benefit from the even higher conductivity compared to Al. Examples include the accelerator facilities HERA-e, KEK-B and PEP. Brazing techniques are typically applied to join copper components and to connect stainless steel flanges. Massive water cooled copper blocks are often used as collimators or absorbers for synchrotron radiation at locations of high power density.   \n\nFor proton accelerators austenitic stainless steel has become the most widely used material. Beam pipes are often fabricated from seamless tubes with discrete pumps attached every few meters. For sealing usually welded stainless steel flanges and copper Conflat$^\\RM{TM}$ gaskets are used. Only metal sealed flange connections allow to achieve leak rates compatible with UHV conditions and radiation resistance at the same time. A concept sketch of the Conflat system is shown in Fig.\\,\\ref{fig:conflat}.\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=1.0\\textwidth]{Fig_11_Vac.png}}\n\\caption{In a Conflat$^\\RM{TM}$ metal sealed flange connection the copper seal is clamped between two stainless steel flanges that contain a knife edge.} \n\\label{fig:conflat}\n\\end{figure} \n\nFor accelerators with superconducting magnets the vacuum chamber is often also operated at low temperature, since there is no room for thermal insulation in the tight space between the magnet coils. Examples include the proton ring of HERA\/DESY, RHIC at Brookhaven and LHC at CERN. The~cool down over a large temperature range from room temperature to an operating temperature of, for example, $4.5\\,$K presents another challenge for the beam vacuum system. A system of bellows and space for expansion must be foreseen to handle the mechanical contraction. Usually the beam pipes are made from a type of stainless steel with the inner surface coated by copper to enhance the thermal conductivity. \n\nFor electrical feedthroughs into vacuum ceramics are used as insulators. In some cases ceramics are used for entire vacuum chambers. This is necessary if the chamber is to be placed in rapidly changing magnetic fields, and in a metallic chamber strong eddy currents would be induced. Applications include for example kicker magnets and rapid cycling synchrotrons \\cite{csns}. Beam chambers inside particle physics detectors should influence particles generated at the interaction point as less as possible. For this purpose very thin wall tubes are used, made from low Z material, such as aluminum, beryllium or carbon fiber materials. For exit windows of beams similar solutions are common.\n\nThe beam chamber carries an image current that acts back on the beam and may deteriorate beam quality properties like emittance or energy spread. To minimize such effects the chamber should carry the~image current with low resistance, avoiding geometric discontinuities. Variations of the vacuum chamber cross section should be applied gradually using tapered sections. Bellows and openings for pumping ports must be electrically shielded. In bellows often a set of flexible, sliding spring contacts of copper-beryllium alloys is installed around their circumference. Perforated electric screens with circular holes of some millimeters diameter or longitudinal slits are used for pump ports. Of course such electrical shielding measures result in a reduction of the effective pumping speed. Also for gate valves in the open state, gaps have to be covered by spring contacts that move with the gate mechanism. Inappropriate shielding of transverse openings will not only affect the beam quality, but may  also result in the excitation of trapped RF modes, leading to a strong heating effect for vacuum components.\n\nAccelerator components installed in the vicinity of the beam pipe should be radiation resistant and must withstand corrosive atmospheres produced by the primary radiation. This includes cables and electronics, which must either be properly chosen or shielded. For high energy electron\/positron storage rings and synchrotron radiation facilities the generated X-rays may cause problems, and often the beam chamber is wrapped into a lead shield to absorb as much radiation as possible. Figure\\,\\ref{fig:att} shows the~attenuation of X-rays in various materials as function of the photon energy. Using beam energy and bending radius the critical photon energy may be estimated with the expression: \n\n\\begin{equation}\nE_c [\\RM{keV}] \\approx 2.218 ~ \\frac{E^3 [\\RM{GeV}^3]}{\\rho [\\RM{m}]}.\n\\label{crit}\n\\end{equation}\n\n\\begin{figure}\n\\centering\\includegraphics[width=0.85\\textwidth]{Fig_12_Vac.png}\n\\caption{Attenuation length for X-rays of commonly used materials for vacuum vessels and shielding as a function of photon energy.}\n\\label{fig:att}\n\\end{figure}\n\n\\section{Summary}\nThe vacuum system presents a challenging technical aspect of each particle accelerator. To achieve required beam lifetimes and beam quality, and to minimize unwanted losses of high energy particles associated with radio-activation and damage to components, vacuum systems must be carefully designed to provide the adequate low gas densities.\n\nSteps for designing an accelerator vacuum system can be roughly categorized into three groups. The first step involves an evaluation of the {\\bf gas sources} in an accelerator. This includes outgassing rates for surfaces and beam induced outgassing dynamics, such as synchrotron radiation, electron cloud, heavy ion bombardment to name a few. Certain accelerator designs require quick handling of activated components thus involving relatively leaky inflating seals. In such cases leak rates must be considered. In practice a clean and baked stainless steel surface might exhibit an outgassing rate of $q_0 = 10^{-11}\\,$mbar\\,l\\,s$^{-1}$\\,cm$^{-2}$. In the presence of synchrotron radiation and after a reasonable time of conditioning the outgassing is still dominated by releasing at least $\\eta = 10^{-6}$ gas molecules per photon incident on the chamber wall.\n\nThe second step covers the definition of the {\\bf target residual gas pressure and composition} (more precisely the gas density). Physics effects of beam gas interaction and the resulting performance degradation must be considered for this purpose. To give a couple of examples the acceptable dynamic gas pressure in an electron storage ring is in the order of $10^{-8}\\,$mbar with a typical composition of $\\sfrac{3}{4}$ H$_2$ and $\\sfrac{1}{4}$ CO. A proton cyclotron as a single pass accelerator is more forgiving and a pressure of $10^{-6}$\\,mbar is sufficient.\n\nThe third step involves then to lay out the vacuum system in terms of geometry, installed {\\bf pumping speed} and perhaps more complex technical measures like surface coating to achieve the required vacuum quality under operating conditions. Types of pumps may include turbo pumps, ion sputter pumps, NEG coating or cryo pumps for large recipients. A typical electron storage ring may be equipped with $S = 100$\\,l\/s ion sputter pumps at a distance of 5\\,m. Today software can be used to compute the pressure profile using Monte Carlo methods or numerical solutions of the diffusing Eq. (\\ref{eq:diffeq}). \nIn addition to such conceptual considerations the design of vacuum systems requires a lot of engineering. Keywords include: UHV compatible materials and materials preparation, mechanical stability, thermo-mechanical problems under heat load, pumps, gauges, flange systems and valves.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}%\n\n\\par A vast range of space and astrophysical scenarios are driven by the rapid expansion of plasmas through space. Such examples include interplanetary coronal fast ejecta~\\cite{Burlaga2001}, the expansion of the stellar material from supernova remnants~\\cite{1990ApJ...356..549S}, and artificial magnetospheric releases of tracer ions~\\cite{Krimigis1982}. When these expanding plasmas encounter obstacles of magnetic nature, the resultant interaction leads to highly nonlinear and complex dynamics. In the solar system, the interaction between the plasma flow (\\textit{i.e.} the solar wind) and planetary-sized magnetic obstacles leads to the formation of magnetospheres~\\cite{Russell1991}.%\n\n\\par The effective size of the magnetic obstacles is determined by the equilibrium position between the kinetic pressure of the solar wind and the magnetic pressure exerted by the planetary magnetic fields~\\cite{Schield1969}. The region of equilibrium, called the magnetopause, can be described using the pressure balance derived from magnetohydrodynamics (MHD)%\n\\begin{equation}\n    n_dm_{i,d}v_0^2 = \\frac{B^2}{8\\pi} \\ ,\n    \\label{eq:pressure-equilibrium}\n\\end{equation}\nwhere $n_d$ is the density of the solar wind, $v_0$ is its flow velocity, $m_{i,d}$ is the mass of its ions, and $B$ is the total magnetic field at the magnetopause. The total magnetic field can be written as $B = B_0 + B_\\textrm{dip}$, where $B_0$ is the collective magnetic field and $B_\\textrm{dip} = M \/ L_0^3$ is the magnetic field of the obstacle, often well described by a dipolar profile of magnetic moment $M$. The distance $L_0$ between the center of the dipole and the magnetopause, often referred to as the plasma standoff distance, measures the effective size of the magnetic obstacle.%\n\n\\par For planetary-sized magnetospheres, the obstacle size is typically tens of thousands of kilometers. However, magnetospheres with a few hundreds of kilometers are also observed in space environments such as the lunar surface. When the magnetic obstacle size is smaller or of the order of the ion kinetic scales of the plasma, \\textit{i.e.} the ion skin depth or the ion gyroradius, the interaction with the solar wind results in ion-scale magnetospheres, or mini-magnetospheres.%\n\n\\par The study of mini-magnetospheres in past years was mainly motivated by the observation of crustal magnetic anomalies on the lunar surface~\\cite{Lin1998,Halekas2008,Kato2010,Wieser2010,Kramer2021}. Although the Moon does not have a global magnetic field like Earth, it does have small localized regions of crustal magnetic field, of 10-100 nT over distances of 100-1000 km~\\cite{Lin1998}, which are of the same order as the gyroradius of solar wind ions near the Moon's surface. As a result, when these regions of the lunar surface are exposed to the solar wind, mini-magnetospheres are formed. The deflection of charged particles off of lunar mini-magnetospheres commonly leads to the formation of ``lunar swirl'' structures~\\cite{Bamford2012}. Similar interactions between the solar wind and small-sized patches of magnetic field also occur in other planets and natural satellites without a planetary magnetosphere, such as Mars~\\cite{Lillis2013}, Mercury~\\cite{doi:10.1126\/science.1211001}, Ganymede~\\cite{https:\/\/doi.org\/10.1029\/97GL02201}, and comets and asteroids~\\cite{https:\/\/doi.org\/10.1029\/GL011i010p01022}.%\n\n\\par Multiple experiments have been performed in laboratory environments that replicate the interaction between plasma flows and magnetic obstacles. With a proper re-scaling of parameters~\\cite{Ryutov_2002}, these experiments represent highly controlled configurations where a large variety of diagnostics can be used to obtain more accurate measurements than those obtained from the direct probing of astrophysical events. In experimental studies, fast-moving plasma flows are usually driven resorting to high-intensity lasers focused onto solid targets of plastic or metal composition~\\cite{ablated,ablated2}. These laser-ablated plasmas can be mildly collisional or collisionless, replicating astrophysical conditions~\\cite{Niemann2014a,Bondarenko2017}. By adding dipole field sources against the plasma flow, previous experiments of mini magnetospheres studied possible applications for spacecrafts~\\cite{Winglee2007,Bamford2008,Bamford2014}, the formation of lunar swirls~\\cite{Bamford2012}, and the conditions for the formation of magnetosphere features~\\cite{Brady2009,Shaikhislamov2013, Shaikhislamov2014,Rigby2018}. Although these experiments achieved important breakthroughs in the study of ion-scale magnetospheric physics, they were limited to i) 1D measurements of the magnetic field and plasma density profiles and ii) fixed properties of the obstacle and plasma flow.%\n\n\\par Numerical simulations play a key role in interpreting and designing experiments. Early MHD simulations attempted to explain the formation and characteristics of lunar mini-magnetospheres and validate experimental and analytical models~\\cite{Harnett2000,Harnett2002,Shaikhislamov2013}. Hybrid simulations were used to study the role of ion kinetic effects, and obtain conditions for the formation of magnetospheres~\\cite{Blanco-Cano2004} and replicate previous experimental results~\\cite{Gargate2008}. However, these simulations do not resolve the electron scales and do not capture important kinetic effects on the magnetosphere's boundary, \\textit{e.g.} charge separation effects and nonthermal particle distributions. Particle-in-cell (PIC) simulations were used to fully resolve the micro-physics of these systems and study its role in the formation of lunar mini-magnetospheres~\\cite{Kallio2012,Deca2014,Deca2015,Zimmerman2015,Bamford2016}, the scaling of their properties with solar wind speed and magnetic field orientation~\\cite{Deca2021} and the conditions for the formation of collisionless shocks~\\cite{Cruz2017}.%\n\n\\par In this work, we use PIC simulations of ion-scale magnetospheres to interpret the results of recent experiments~\\cite{Schaeffer2021} performed at the LArge Plasma Device (LAPD), University of California, Los Angeles. In these experiments, fast collisionless plasma flows generated by high-repetition-rate lasers were collided with the magnetized ambient plasma provided by the LAPD and with a dipolar magnetic field obstacle, leading to the formation of ion-scale magnetospheres. Using motorized probes, high spatial and temporal resolution measurements of the magnetic field allowed characterization of 2D magnetic field and current density structures. Apart from validating the experimental results, the simulations presented in this work explore a set of upstream and magnetic parameter scans and configurations not accessible in the laboratory to determine the importance of each system parameter on the magnetospheric properties. The simulations show that the background ions, and then the driver ions, are responsible for the formation of the magnetopause observed in the experiments. They also show that a reflection of the downstream magnetic compression is observed for certain parameters of the driver plasma, and that the distance between the main current features is dependent on the dipolar and driver plasma parameters. \n\n\\par This paper is organized as follows. In Sec.~\\ref{sec:experiments}, we briefly review the LAPD experiments and their main results. In Sec.~\\ref{sec:simulations}, we present PIC simulations of ion-scale magnetospheres. In Sec.~\\ref{sec:numerical-methods}, we outline the standard configuration and parameters used for the simulations. In Sec.~\\ref{sec:additional}, we provide an overview of the temporal evolution of these systems and show that the simulations agree with the results of the LAPD experiments. We discuss the origin of the structures observed in current density and magnetic field synthetic diagnostics and use particle phase spaces to interpret them. In Sec.~\\ref{sec:length}, we present the results for different lengths of the plasma flow and define the conditions required to reproduce the features observed experimentally. The coupling between the laser-ablated driver and background plasmas is characterized in Sec.~\\ref{sec:density} with simulations with different driver densities. In Sec.~\\ref{sec:momentum}, different magnetic moments are considered, and we show that the main current density features are highlighted and more easily visible for weaker magnetic obstacles.\nIn Secs.~\\ref{sec:realistic} and~\\ref{sec:finite}, we discuss and illustrate the validity of the key simplifications and approximations used for the parameter scans presented in Secs.~\\ref{sec:additional}-\\ref{sec:momentum}. Finally, we outline the conclusions of this work in Sec.~\\ref{sec:conclusions}.%\n\n\\par This paper is the second part of a two part series. Detailed experimental results are presented in Part I~\\cite{Schaeffer2021}. \n\n\\section{LAPD Experiment}\n\\label{sec:experiments}\n\n\\par A new experimental platform has been developed on the LAPD to study mini-magnetospheres. The platform combines the large-scale magnetized ambient plasma generated by the LAPD, a fast laser-driven plasma, and a pulsed dipole magnet, all operating at high-repetition-rate ($\\sim1$ Hz). In the experiments, a supersonic plasma is ablated from a plastic target and then expands into the dipole magnetic field embedded in the ambient magnetized plasma. By measuring 2D planes of the magnetic field over thousands of shots, detailed maps of the magnetic field evolution are constructed.  Additional details on the platform and results can be found in Part I.\n\n\\par Example results are shown in Fig.~\\ref{fig:experiment} for the measured change in magnetic field $\\Delta B_z = B_{z}-B_{z,\\textrm{initial}}$ and the current density $J_x = \\partial \\Delta B_z\/\\partial y$.  Here, $B_{z}$ is the total magnetic field, $B_{z,\\textrm{initial}}=B_0 + B_\\textrm{dip}$ is the total initial magnetic field, $B_0$ is the background LAPD field, and $B_\\textrm{dip}$ is the dipole magnetic field.  These results are taken along $y$ at $x=0$ from the $z=0$ plane probed experimentally. In the experiments, the dipole is centered at $(x,y,z)=(0,0,0)$ and has a magnetic moment $M=475$ Am$^2$.\n\n\\par As seen in Fig.~\\ref{fig:experiment}(a), the expanding laser-driven plasma creates a leading magnetic field compression followed by a magnetic cavity.  The cavity reaches a peak position of $y\\approx-13$ cm, while the compression propagates closer to the dipole before being reflected back towards the target.  The current density in Fig.~\\ref{fig:experiment}(b) shows two prominent structures.  Following the expansion of the magnetic cavity is a diamagnetic current, which reaches a peak position of $y\\approx-15$ cm before stagnating for approximately 1 $\\mu$s and then dissipating.  Ahead of the diamagnetic current is the magnetopause current near $y\\approx-13.5$ cm, which lasts for about 0.5 $\\mu$s.\n\n\\par Here, we aim to qualitatively model these experiments in order to address key questions that aid in the interpretation of the experimental results. In particular, simulations can explain the role of each system component in the features observed, address which plasma component (ambient or laser-driven) is responsible for the features observed and which pressure balances are most relevant. \n\n\\par For convenience, the notations used in this paper are different for the ones used in Part I. Here, we use the CGS system, and the axis system is rotated from the used in Part I.\n\n\\begin{figure}[ht]\n    \\includegraphics[width=0.85\\linewidth]{plots\/experiment.pdf}\n    \\caption{\\label{fig:experiment} LAPD experimental results for the temporal evolution of a) the variation of the magnetic field $\\Delta B_z$ and b) the current density $J_x$ at $x=z=0$. The experimental results are discussed with more detail in Part I.}\n\\end{figure}\n\n\\section{PIC simulations}\n\\label{sec:simulations}\n\n\\subsection{Configuration of the simulations} \\label{sec:numerical-methods}\n\n\\par Motivated by the results of experiments described in Sec.~\\ref{sec:experiments}, we performed 2D simulations with OSIRIS, a massively parallel and fully relativistic PIC code~\\cite{Fonseca2002,Fonseca2013}. With PIC simulations, we can accurately resolve the plasma kinetic scales characteristic of mini-magnetospheres dynamics.%\n\n\\par The numerical simulations presented in this work stem from a simplified description of the LAPD experimental setup, represented in Fig.~\\ref{fig:config}. In these simulations, a driver plasma moves against a background plasma permeated by a uniform magnetic field $\\mathbf{B_0}$ and a dipolar magnetic field $\\mathbf{B_{dip}}$. $\\mathbf{B_0}$ and $\\mathbf{B_{dip}}$ are oriented along the $z$ direction and are transverse to the driver plasma flow. Since the most relevant dynamics of the simulations occurs at the ion kinetic scales, all the spatial scales are normalized to the ion skin depth of the background plasma $d_i=c\/\\omega_{pi}=\\sqrt{m_{i,0}c^2\/4\\pi n_0e^2}$, where $c$ is the speed of light in vacuum, $\\omega_{pi}$ is the ion plasma frequency, $m_{i,0}$ is the mass of the background plasma ions, $n_0$ is the background density, and $e$ is the electron charge. In turn, the temporal scales are normalized to $1\/\\omega_{ci}$, where $\\omega_{ci} = eB_0\/m_{i,0}c$ is the ion cyclotron frequency of the background. The simulation box is a 12 $d_i$ $\\times$ 12 $d_i$ area with open and periodic boundary conditions in the $x$ and $y$ directions, respectively. The flow is in the $x$ direction and the size of the simulation domain in the $y$ direction is large enough to avoid re-circulation of the particles through the whole interaction. The simulations considered 25 particles per cell per species. To resolve the dynamics of the electron kinetic scales, we used 10 grid cells per electron skin depth $d_e=d_i\\sqrt{m_e\/m_{i,0}}$ in both $x$ and $y$ directions, where $m_e$ is the electron mass.\n\n\\begin{figure}[ht]\n    \\includegraphics[width=0.85\\columnwidth]{plots\/config.pdf}\n    \\caption{\\label{fig:config} Schematic illustration of the initial setup of the 2D PIC simulations performed. The system considers a vacuum region at the left, a driver plasma (I) of density $n_d$ and length $L_x$, travelling to the right with flow velocity $v_0$, and a background plasma (II) with constant density $n_0$ and with an internal magnetic field $B_0$. A dipole is included at the center of the background region. Both the uniform and the dipolar magnetic fields are oriented in the $z$ direction. An illustration of the effective magnetic obstacle created by the dipole and of the magnetic field profile at $y=0$ are also shown in a dashed circumference and in a solid black line, respectively.}%\n\\end{figure}\n\n\\par The driver plasma, shown in region I in Fig.~\\ref{fig:config}, represents ideally the plasma ablated from the plastic target in the experiments. We assume that this driver has a length $L_x$ that is typically 2 $d_i$, and a width $L_y$ that is typically infinite. It has a constant density $n_{d}$, and it is initialized moving to the right side with initial flow velocity $v_0$. The driver is composed of an electron species and a single ion species, with ion mass $m_{i,d}$. Because the driver plasma is reflected during the interaction with the background, an empty region at the left of the driver was added to accommodate the reflecting particles.%\n\n\\par The background plasma is represented in region II. It is an 8 $d_i$ length and infinite width plasma and it has uniform density $n_0$. The initial interface between the driver and background plasma is located at $x_B=-4\\ d_i$. Like the driver plasma, it has an electron species and a single ion species, of mass $m_{i,0}$. The background plasma is magnetized with an internal uniform magnetic field $\\mathbf{B_0} = B_0 \\mathbf{\\hat{z}}$, and its magnitude is defined such that the Alfv\u00e9nic Mach number of the flow, $M_A \\equiv v_0\/v_A = v_0\\sqrt{4\\pi n_0m_{i,0}}\/B_0$ matches the peak experimental value $M_A=1.5$, where $v_A$ is the Alfv\u00e9n velocity.%\n\n\\par A dipolar magnetic field is externally imposed in our simulations (\\textit{i.e.}, it is added to the plasma self-consistent electromagnetic fields to advance particle momenta but is not included in Maxwell's equations to advance the fields). The dipole is centered at $(x,y)=(0,0)$ and its associated magnetic field is $\\mathbf{B_{dip}} = B_\\mathrm{dip} \\mathbf{\\hat{z}}$, with $B_\\mathrm{dip} = M \/ r^3$, where $M$ is the dipolar magnetic moment, $r = \\sqrt{x^2 + y^2+\\delta^2}$ is the distance to the origin of the dipole and $\\delta=0.25\\ d_i$ is a regularization parameter. For most simulations, the magnetic moment $M$ was chosen such that the expected standoff, obtained from Eq.~\\eqref{eq:pressure-equilibrium}, is similar to the experimental value $L_0=1.8\\ d_i$. For this particular magnetic moment, the total initial magnetic field $B_0+B_\\mathrm{dip}$ is $\\approx 3.0~B_0$ at the standoff distance. Near the interface between the driver and background plasmas, the magnetic field of the dipole is relatively small and the initial magnetic field is $\\approx 1.2~B_0$.%\n\n\\par In this work, we present simulations with different drivers and magnetic dipole moments. All the simulations presented here, and their respective parameter sets, are listed in Table~\\ref{tab:runs}. Simulations B-G are discussed through Sec.~\\ref{sec:simulations} on equally labeled subsections. Simulation B is used to discuss the overall dynamics of the system, while simulations C, D, and E illustrate the role of the driver length, the density ratio, and the magnetic moment, respectively. Simulations F show the results for more realistic choices of parameters and simulation G for a more realistic driver shape. The physical parameters of the simulations (\\textit{e.g.} $M_A$, $L_0\/d_i$) were adjusted to be similar to the LAPD experiments, whereas other parameters (\\textit{e.g.} $m_i\/m_e$, $v_0$, $v_{the}$) were chosen to make simulations computationally feasible. The experimental and numerical parameters are presented in Table~\\ref{tab:parameters} and compared with lunar mini-magnetospheres.%\n\n\\begin{table*}\n    \\caption{\\label{tab:runs} List of simulations performed and their parameters. $v_{the,x}$ and $v_{thi,x}$ represent the $x$ component of the electron and ion thermal velocities, respectively. All the runs considered $v_{th,x}=v_{th,y}=v_{th,z}$ for the electrons and ions.}%\n    \\begin{ruledtabular}\n        \\begin{tabular}{cccccccccc}\n            \\textrm{Name} & $v_{the,x}\/v_0$ & $v_{thi,x}\/v_0$ & $n_d\/n_0$ & $m_{i}\/m_e$ & $m_{i,0}\/m_e$ & $L_x\/d_i$ & $L_y\/d_i$ & $L_0\/d_i$\\\\ \\colrule\n            \\textrm{B\/D2\/E2} & 0.1 & 0.01 & 2 & 100 & 100 & 2 & $+\\infty$ & 1.8 \\\\\n            \\textrm{C1} & 0.1 & 0.01 & 2 & 100 & 100 & 1 & $+\\infty$ & 1.8 \\\\\n            \\textrm{C2} & 0.1 & 0.01 & 2 & 100 & 100 & 4 & $+\\infty$ & 1.8 \\\\\n            \\textrm{C3} & 0.1 & 0.01 & 2 & 100 & 100 & $+\\infty$ & $+\\infty$ & 1.8 \\\\\n            \\textrm{D1} & 0.1 & 0.01 & 1 & 100 & 100 & 2 & $+\\infty$ & 1.8 \\\\\n            \\textrm{D3} & 0.1 & 0.01 & 4 & 100 & 100 & 2 & $+\\infty$ & 1.8 \\\\\n            \\textrm{E1} & 0.1 & 0.01 & 2 & 100 & 100 & 2 & $+\\infty$ & 2.3 \\\\ \n            \\textrm{E3} & 0.1 & 0.01 & 2 & 100 & 100 & 2 & $+\\infty$ & 1.4 \\\\ \\colrule\n            \\textrm{F1} & 0.1 & 0.002 & 2 & 1836 & 1836 & 2 & $+\\infty$ & 1.8 \\\\\n            \\textrm{F2} & 2.5 & 0.033 & 2 & 1836 & 1836 & 2 & $+\\infty$ & 1.8 \\\\\n            \\textrm{F3} & 2.5 & 0.033 & 2 & 100 & 100 & 2 & $+\\infty$ & 1.8 \\\\\n            \\textrm{G} & 0.1 & 0.01 & 2 & 100 & 100 & 2 & 6 & 1.8 \\\\\n        \\end{tabular}\n    \\end{ruledtabular}\n\\end{table*}\n\n\\par In most simulations, we considered a reduced mass ratio $m_i \/ m_e = 100$, a flow velocity $v_0 \/ c = 0.1$, and cold plasmas to reduce the required computational resources, allow extended scans over the different parameters of the system, and simplify our analysis. The thermal effects are negligible for the main results, and the chosen ion-to-electron mass ratio is high enough to ensure sufficient separation between electron and ion spatial and temporal scales. We confirm the validity of our assumptions in Sec.\\ref{sec:realistic}.%\n\n\\par In most of the simulations presented in this work, we have assumed that ions and electrons are initially in thermal equilibrium, and thus used the electron thermal velocities $v_{the}$ shown in Table~\\ref{tab:runs}, to compute the ion thermal velocities $v_{thi}$. Because we aim to study the role of the hydrogen ions of the experimental driver in the interaction with the background plasma, these simulations considered equal ion masses for the driver and background plasmas, \\textit{i.e.} $m_{i,d}=m_{i,0}$.%\n\n\\begin{table*}\n    \\caption{\\label{tab:parameters} Typical parameters associated with lunar mini-magnetospheres~\\cite{Russell1991,Bamford2012,Lin1998}, the range of parameters of LAPD~\\cite{Schaeffer2021} and the canonical simulation B. The parameters are written in both physical and normalized units to facilitate the comparison between the space, the laboratory environments and the PIC simulations. The experimental parameters are presented in ranges of values computed with the possible LAPD values for the flow velocity $v_0$, the density $n_0$ and the temperature $T$. The plasma parameters shown for lunar mini-magnetospheres are relative to the solar wind, while for the experiments and the simulations, they are relative to the background plasma. The ion data shown corresponds to Hydrogen ions. The magnetic field $B_{\\textrm{std}}$ is calculated at the standoff position, \\textit{i.e.}, at a distance $L_0$ from the center of the obstacle.}%\n    \\begin{ruledtabular}\n        \\begin{tabular}{cccccc}\n            \\multirow{2}{*}{\\textrm{Parameters}} & \\multicolumn{2}{c}{Lunar mini-magnetospheres} & \\multicolumn{2}{c}{LAPD experiments} & PIC simulations \\\\ \n            & Physical units & Normalized units & Physical units & Normalized units & Normalized units\\\\ \\colrule \\\\ [-9 pt]\n            Flow velocity, $v_0$ & $400$ km\/s & $10^{-3}$ $c$ & 200-300 km\/s & 0.7-1.0$\\times10^{-3}$ $c$ & 0.1 $c$ \\\\\n            Density, $n_0$ & 5 cm$^{-3}$ & --- & $10^{12}$-$10^{13}$ cm$^{-3}$ & --- & ---\\\\\n            Mass ratio, $m_i\/m_e$ & --- & 1836 & --- & 1836 & 100 \\\\\n            Ion skin depth, $d_i$ & 100 km & --- & 7-23 cm & --- & --- \\\\\n            Electron skin depth, $d_e$ & 2 km & 2$\\times10^{-2}$ $d_i$ & 0.2-0.5 cm & 0.7-7.0$\\times10^{-2}$ $d_i$ & 0.1 $d_i$ \\\\\n            Magnetic obstacle size, $L_0$ & 300 km & 3 $d_i$ & 14-18 cm & 0.6-2.5 $d_i$ & 1.8 $d_i$\\\\\n            Internal magnetic field, $B_{0}$ & $10^{-4}$ G & $10^{-2}$ $m_ec^2\/ed_e$ & 300 G & 3-9$\\times10^{-2}$ $m_ec^2\/ed_e$ & 0.67 $m_ec^2\/ed_e$ \\\\\n            Ion gyroradius, $\\rho_i$ & 500 km & 5 $d_i$ & 7-10 cm & 0.3-1.5 $d_i$ & 1.5 $d_i$ \\\\\n            Electron gyroradius, $\\rho_e$ & 800 m & $8\\times10^{-3}$ $d_i$ & 4-6$\\times10^{-3}$ cm & 2-8$\\times10^{-4}$ $d_i$ & 0.15 $d_i$ \\\\\n            Ion gyroperiod, $\\omega_{ci}^{-1}$ & 1 s & --- & 230-520 ns & --- & --- \\\\\n            Alfv\u00e9n velocity, $v_A$ & 80 km\/s & $3\\times10^{-4}$ $c$ & 140-980 km\/s & 0.5-3.3$\\times10^{-3}$ $c$ & 0.067 $c$ \\\\\n            Alfv\u00e9nic Mach number, $M_A$ & --- & 5 & --- & 0.3-1.5 & 1.5 \\\\\n            Temperature, $T$ & 5 eV & --- & 1-10 eV & --- & --- \\\\\n            Electron thermal velocity, $v_{the}$ & 1500 km\/s & 4 $v_0$ & 730-2300 km\/s & 2.4-11.5 $v_0$ & 0.1 $v_0$ \\\\\n            Ram pressure, $n_0m_iv_0^2$ & 1 nPa & --- & 70-1500 Pa & --- & --- \\\\\n            Standoff magnetic field, $B_{\\textrm{std}}$ & $5\\times10^{-4}$ G & 0.07 $m_ec^2\/ed_e$ & 100-600 G & 0.02-0.2 $m_ec^2\/ed_e$ & 2.0 $m_ec^2\/ed_e$ \\\\\n        \\end{tabular}\n    \\end{ruledtabular}\n\\end{table*}\n\n\\subsection{\\label{sec:additional} Evolution and main features of the system}\n\n\\par To identify the main magnetospheric and kinetic-scale structures that arise from the initial configuration, simulation \\textrm{B} was performed. It considered a driver with length $L_x=2\\ d_i$ and density $n_d=2\\ n_0$ (twice the background density). Figs.~\\ref{fig:movie} a1-3) represent the total ion density $n_i=n_{i,d}+n_{i,0}$, for three different times, and Figs.~\\ref{fig:movie} b1-3) show the variation of the $z$ component of the magnetic field, from its initial value, $\\Delta B_z=B_z-B_{z,\\textrm{initial}}$.%\n\n\\begin{figure*}[t]\n    \\includegraphics[width=0.98\\linewidth]{plots\/movie.pdf\n    \\caption{\\label{fig:movie} Spatiotemporal evolution of a) the total ion density and b) the variation of the $z$ component of the magnetic field in simulation \\textrm{B} (see Table~\\ref{tab:runs} for a list of parameters). Columns 1-3 correspond to three different times in the simulation. The vertical and circular dashed lines mark the initial border between the driver and background plasma and the dipolar magnetic obstacle with radius $L_0$, respectively.}%\n\\end{figure*}\n\n\\par In Fig.~\\ref{fig:movie} a1), we see the total ion density for an early time ($t\\omega_{ci}=1.5$). Given the small distance propagated by the driver plasma at this time, the dipolar magnetic field does not significantly affect the interaction between the plasmas. For this reason, we can express the early system as a driver flowing against a uniform magnetized background plasma. \nIn Fig.~\\ref{fig:movie} b1), we observe that this interaction creates a region of compressed magnetic field in the downstream region, where the background plasma is located, and expels the magnetic field in the region of the driver, leading to a magnetic cavity in the upstream region with approximately null magnetic field~\\cite{Bondarenko2017}.%\n\n\\par In Figs.~\\ref{fig:movie} a2) and b2), we start to observe the effects of the dipolar magnetic field for a later time ($t\\omega_{ci}=3.0$). As the magnetic pressure exerted against the plasmas increases, a region of compressed background plasma forms in front of the dipole, as Fig.~\\ref{fig:movie} a2) shows. After the interaction between the background and the dipole, the magnetic field pressure becomes large enough to counterbalance the kinetic pressure of the driver, reflecting it upstream. This can be seen seen in Fig.~\\ref{fig:movie} a3) for a subsequent time ($t\\omega_{ci}=4.5$). After the reflection, there is no longer a plasma flow pushing the magnetic compression forward or holding the decompression by the left side of the background region, and as a result, the region near the dipole quickly decompresses --- see Fig.~\\ref{fig:movie} b3).%\n\n\\par To compare the numerical results with the experimental data shown in Fig.~\\ref{fig:experiment}, synthetic diagnostics were obtained from the simulations. In Fig.~\\ref{fig:standard}, the variation of the magnetic field $\\Delta B_z$ and the density current $J_y$ measured at the axis of symmetry $y=0$ and as a function of time are plotted for simulation \\textrm{B}. These diagnostics are important to comprehend the system dynamics, due to the importance of the $z$ direction of the magnetic field in the motion of the particles.%\n\n\\begin{figure}[ht]\n    \\includegraphics[width=0.85\\columnwidth]{plots\/standard.pdf}\n    \\caption{\\label{fig:standard} Temporal evolution of a) the variation of the magnetic field $B_z$ and b) current density $J_y$ at $y=0$ for the simulation \\textrm{B}. The driver has a 2 $d_i$ length and a density $n_d = 2\\ n_0$. The dashed lines have slopes that match the flow velocity $v_0$, the coupling velocity $v_c$ and the reflection velocity $v_r$.}%\n\\end{figure}\n\n\\par The main features of Fig.~\\ref{fig:standard} are consistent with the experimental results. In the magnetic field plot of Fig.~\\ref{fig:standard} a), both the upstream magnetic cavity and the downstream magnetic compression are present. Between $t\\omega_{ci}=0$ and $t\\omega_{ci}\\approx1.5$, the system behaves approximately as a driver piston moving against a uniform magnetized plasma. As the driver pushes the background plasma and magnetic field, the discontinuity that separates these two media travels at constant coupling velocity $v_c < v_0$, measured as $v_c\\approx0.49\\ v_0$ for this simulation. The leading edge of the compression of the magnetic field travels with a velocity close to $v_0$ for the runs considered.%\n\n\\par The driver experiences increasingly higher magnetic fields until the magnetic pressure is enough to reflect the driver near the expected standoff $x_0 = -L_0$, at $t\\omega_{ci}\\approx3$. The magnetic cavity and magnetic compression are also reflected, and the boundary between these two regions travels with a velocity $v_r$ after reflection. The background magnetic decompression is seen after $t\\omega_{ci}=5$.%\n\n\\par In the current density plot of Fig.~\\ref{fig:standard} b), we can observe the diamagnetic current that supports the magnetic field gradient between the driver and background plasmas and that identifies the leading edge of the magnetic cavity. During the driver reflection, this current branches into multiple components due to the multi-stream velocity distributions developed in the driver and background plasmas. We can also verify that this structure is reflected near the expected standoff $x_0 =-L_0$. Between $t\\omega_{ci}\\approx2$ and $t\\omega_{ci}\\approx3$, a second current structure is present in the background region. It is associated with the magnetopause of the system and the small decompressed field region that we see in Fig.~\\ref{fig:standard} a), and it arises from the interaction of the accelerated background ions with the dipole, as we show in Sec.~\\ref{sec:momentum}. The presence of these two current structures is consistent with the experimental results.%\n\n\\par In Fig.~\\ref{fig:standard} b) we can also see the formation of waves in the background plasma, near the dipole region. These waves are excited in regions of highly non-uniform density and magnetic field, and have periods and wavelengths between the ion and electron kinetic scales. We have verified that their properties change significantly for different ion thermal velocities. In particular, we have found these waves to be more clearly excited for lower ion temperatures, which may explain why these waves have not been observed in the experiments performed at the LAPD. A detailed characterization of these waves and the conditions for their formation is out of the scope of this paper, and shall be addressed in a future work.%\n\n\\par To better understand the particle motion during the events described, we show in Fig.~\\ref{fig:phase-space} the phase spaces of ions and electrons located near $y=0$. For the ions, the $x$ component of the velocity of the particles is presented, to illustrate their reflection and accumulation, while for the electrons, the $y$ component is shown instead, to show the formation of the currents. The magnetic field $B_z$ and the current density $J_y$ profiles for $y=0$ are also represented. Once again, we used the parameter set \\textrm{B} of Table~\\ref{tab:runs}.%\n\n\\begin{figure*}[t]\n    \\includegraphics[width=0.95\\linewidth]{plots\/phase-scan-standard.pdf\n    \\caption{\\label{fig:phase-space} Ion (a) and electron (b) phase spaces, and magnetic field $B_z$ and current density $J_y$ profiles at $y=0$, for simulation \\textrm{B} and for three different times (1-3). The particles shown were randomly selected in the region $-0.2\\ d_i < y < 0.2\\ d_i$. The frames labeled a1) to a3) show the $v_x$ velocity of the ions, while the frames labeled b1) to b3) show the $v_y$ velocity of the electrons. Blue\/orange markers correspond to background\/driver plasma particles. The green and purple lines correspond to the magnetic field $B_z$ and current density $J_y$, respectively. The left dashed line marks the initial border between the driver and the background plasmas, and the right dashed line marks the expected standoff $x_0=-L_0$.}%\n\\end{figure*}\n\n\\par Fig.~\\ref{fig:phase-space} a1) shows the $v_x$ velocity of the ions when the dipole field is still negligible. The ions initially move upstream with velocity $v_0$ until they interact with the background field. After reaching the background, they are mostly decelerated and reflected by electric field in the interface between the plasmas~\\cite{Bondarenko2017}, and end up with a flow velocity that is close to zero for the simulation considered. The reflection occurs near the boundary of the magnetic cavity, which moves with velocity $v_c$ through the background, as mentioned above. During this stage, the background ions accelerate from rest to velocities of average close to $v_c$.%\n\n\\par The driver and the accelerated background ions continue to approach the dipole until they are reflected. This can be seen in Fig.~\\ref{fig:phase-space} a2). During this interaction, two main current structures are visible in the $J_y$ profile. The first one (from the left) corresponds to the typical diamagnetic current, while the second one corresponds to the magnetopause. To the right of these two main current structures, we can see the background waves observed in Fig.~\\ref{fig:standard} b). In Fig.~\\ref{fig:phase-space} a3), the driver ions are totally reflected. The ions reflected by the dipole obtain a velocity close to $-v_0$, while the magnetic cavity moves back with velocity $v_r$.%\n\n\\par Because the simulation considers a cold plasma approximation, the ion thermal velocities remain small most of the time, except for the boundary between the two plasmas, where the velocity of the ions changes abruptly. The same does not occur for the electrons. We can see in the $v_y$ velocity of the electrons, represented in Figs.~\\ref{fig:phase-space} b1) to b3) that, although the electron thermal velocities are initially small, they rapidly increase considerably. At the boundary, the electrons can reach thermal velocities of $6\\ v_0$, much higher than the ion velocities. Because the electron and ion density profiles are very similar during the entire evolution of the system, the current density $J_y = e(n_iv_{iy}-n_ev_{ey})$ is then mainly transported by the electrons, where $n_j$ is the density and $v_{jy}$ the $y$ component of the velocity of the ions and electrons ($j=i,e$, respectively). This is also consistent with the observed spatial distribution of electrons during the reflection, which shows an excess of fast electrons around the standoff position.\n\n\\subsection{Driver length} \\label{sec:length}\n\n\\par To choose a driver length that best reproduces the experimental results shown in Fig.~\\ref{fig:experiment} and to understand its role on the magnetic field and current density structures, we performed simulations \\textrm{C1} to \\textrm{C3} (see Table~\\ref{tab:runs}) with varying driver length $L_x$. In Fig.~\\ref{fig:scan-length}, we show $\\Delta B_z$ and $J_y$ at $y=0$ for $L_x=1\\ d_i$ (C1), $L_x=4\\ d_i$ (C2) and for an infinite driver (C3). For these simulations, the properties of the background plasma and the width of the driver $L_y$ were kept unchanged. The density of the driver was $n_d=2\\ n_0$.%\n\n\\par In Figs.~\\ref{fig:scan-length} a1) and b1), we see the magnetic field and current density plots for the short driver length $L_x=1\\ d_i$. We observe most of the features of Fig.~\\ref{fig:standard}, namely the reflection of the compressed magnetic field in a1) and the diamagnetic and magnetopause currents in b1). For this length, however, the driver never fully interacts with the dipole. The closest that the diamagnetic current structure gets to the dipole is $x_r\\approx-3.0\\ d_i$, \\textit{i.e.}, much farther than the expected standoff $x_0 = -L_0=-1.8\\ d_i$. To replicate the experimental results and ensure that the driver can reach the dipole, we should thus use a sufficiently long driver such that $x_r>x_0$. Additionally, short drivers risk entering in a decoupling regime between the two plasmas~\\cite{Hewett2011}, which can compromise the observation of a magnetopause. The coupling effects on the results are discussed in detail in Sec.~\\ref{sec:density}.%\n\n\n\\par The position where the driver is fully reflected by the background can be estimated as $x_r\\approx x_B+L_xv_c\/(v_0-v_c)$, where $x_B$ is the initial boundary position between the two plasmas. This estimate is obtained by computing the volume of the background plasma required for the driver plasma to deposit its kinetic energy, \\textit{i.e.} $x_r - x_B$ corresponds to the magnetic stopping radius of the system~\\cite{rb}.\n\n\n\\par In the simulation with $L_x=4\\ d_i$, represented in Figs.~\\ref{fig:scan-length} a2) and b2), we observe once more the main features identified in Fig.~\\ref{fig:standard}, but unlike the $L_x=1\\ d_i$ case, the driver is long enough and ends up reflected by the dipole. We observe that the diamagnetic current reaches the expected standoff and has enough plasma to maintain it near the dipole for a time period ($t\\omega_{ci}\\approx3$ to $t\\omega_{ci}\\approx5$) longer than the $2\\ d_i$ case shown in Fig.~\\ref{fig:standard}. As a result, the magnetic decompression in the background region is delayed for longer drivers. However, because the full driver reflection also occurs later, longer drivers will result in short-lived reflections of the compression of the magnetic field.%\n\n\\par In Figs.~\\ref{fig:scan-length} a3) and b3), we show the results for a driver with infinite length ($L_x=+\\infty$). In this simulation, the driver plasma is only partially initialized inside the simulation domain, and a flow is continuously injected from the lower $x$ boundary. An infinite driver configuration allows us to understand the dynamics of the system in an asymptotic regime in which the driver plasma stays close to the dipole. As expected, until $t\\omega_{ci}=3$, the features observed are very similar to $L_x=2\\ d_i$ and $L_x=4\\ d_i$. After this time, the magnetic and the driver kinetic pressures balance each other near $x_0$, so the diamagnetic current remains stationary. Because the driver can hold for longer near the dipole, the decompression in the background region is much slower and is not visible for the time range of the plot. We can also observe that the background waves are only visible during a transient.%\n\n\\par In all the three simulations, the coupling velocity measured was always $v_c \\approx 0.49\\ v_0$. Given the results shown in Fig.~\\ref{fig:scan-length}, we chose a driver length of $2\\ d_i$ to reproduce the experimental results. This driven length is large enough to ensure that the driver arrives at the dipole and small enough to observe a significant reflection of the compression of the magnetic field as we see in the experiments.%\n\n\\begin{figure*}[t]\n    \\includegraphics[width=\\linewidth]{plots\/different-lengths-plot.pdf\n    \\caption{\\label{fig:scan-length} Temporal evolution of the variations of the magnetic field $\\Delta B_z$ and current density $J_y$ at $y=0$, for driver lengths of a) 1 $d_i$, b) 4 $d_i$ and for c) an infinite driver length (see Table~\\ref{tab:runs} for a full list of the parameters). The dashed lines represent the slopes of the flow velocity $v_0$, the coupling velocity $v_c$, and the reflection velocity $v_r$.}%\n\\end{figure*}\n\n\\subsection{Plasma coupling with density ratio} \\label{sec:density}\n\n\\par As expected from previous works, increasing the ratio between the driver and background plasma densities should improve the coupling between the two plasmas~\\cite{Bondarenko2017, Hewett2011}, meaning that, for denser drivers, the transfer of momentum and energy from the driver to the background plasma is more efficient. To better understand the role of the coupling mechanism, we performed simulations with different values of the driver density, namely $n_d=n_0$ (\\textrm{D1}), $n_d=2\\ n_0$ (\\textrm{D2}) and $n_d=4\\ n_0$ (\\textrm{D3}), while keeping a constant background density $n_0$ and a driver length $L_x = 2\\ d_i$. For each run, the magnetic moment was chosen such that the expected standoff obtained from Eq.~\\eqref{eq:pressure-equilibrium} was always $L_0=1.8\\ d_i$. The synthetic magnetic field and current density diagnostics were obtained for these simulations and are shown in Fig.~\\ref{fig:scan-density}.%\n\n\\begin{figure*}[t]\n    \\includegraphics[width=\\linewidth]{plots\/different-densities-plot.pdf\n    \\caption{\\label{fig:scan-density} Temporal evolution of the variations of the magnetic field $\\Delta B_z$ and current density $J_y$ at $y=0$, for different ratios between the driver and background densities $n_d \/ n_0$. The magnetic moment was chosen so that the expected standoff distance $L_0$, calculated from Eq.~\\eqref{eq:pressure-equilibrium}, was kept as 1.8 $d_i$ for all the simulations. Panels a-c) show results for $n_d = n_0$,  $n_d = 2\\ n_0$ and $n_d=4\\ n_0$, respectively.}%\n\\end{figure*}\n\n\\par In Figs.~\\ref{fig:scan-density} a1) and b1) we can see $\\Delta B_z$ and $J_y$ for the lowest driver density considered, $n_d = n_0$ (\\textit{i.e.}, background and driver with the same initial density). In this regime, the coupling is less efficient and, as a result, the coupling velocity $v_c \\approx 0.38\\ v_0$ is lower than obtained in the higher densities cases represented in Figs.~\\ref{fig:scan-density} b) and c). Due to the low coupling velocity, the driver plasma is reflected more quickly by the background than for denser drivers, and the expected position $x_r$ for the total reflection on the background is farther from the dipole than the expected standoff $x_0$, meaning $x_r<x_0$. As a result, Fig.~\\ref{fig:scan-density} a) shows similarities with the short driver length represented in Fig.~\\ref{fig:scan-length} a), because, in both simulations, the driver parameters do not ensure that the driver arrives near the dipole.%\n\n\\par In Figs.~\\ref{fig:scan-density} a2) and b2), we show the results for $n_d = 2\\ n_0$, which is the same run represented in Fig.~\\ref{fig:standard}. For this density, the coupling velocity, measured as $v_c\\approx0.49\\ v_0$, is high enough to secure a reflection of the driver by the dipole, as we observe at $t\\omega_{ci}\\approx3$. In Figs.~\\ref{fig:scan-density} a3) and b3) we show the case with the highest driver density $n_d=4\\ n_0$, which is similar to the $n_d=2\\ n_0$ case, because the measured coupling velocity for Fig.~\\ref{fig:scan-density} c) was $v_c \\approx 0.56\\ v_0$, \\textit{i.e.}, only slightly larger than the $v_c$ measured for Fig.~\\ref{fig:scan-density} b). In the high density case, we also see that the current density structures during the plasma reflection are filamented, due to analogous multi-stream velocity distributions discussed for Fig.~\\ref{fig:scan-length} b).%\n\n\\par To guarantee that the driver reaches the expected standoff, we thus require that $x_r>x_0$. In fact, the position where the driver is reflected $x_r$, for no dipole cases, increases with the driver length $L_x$ and the velocity ratio $v_c\/v_0$, and thus, both quantities must be large enough to guarantee that $x_r>x_0$. In turn, the ratio $v_c\/v_0$ increases with increasing driver density ratio $n_d\/n_0$, and so, the driver should be sufficiently long and dense to effectively couple to the background plasma. Our results (in particular Sec.~\\ref{sec:additional}) show that a driver with $L_x=2\\ d_i$ and $n_d=2\\ n_0$ qualitatively reproduces the experimental results.%\n\n\\par A separate study was also performed to analytically determine the properties of the driver-background plasma coupling. The results of this study will be presented in a future paper.\n\n\\subsection{Dependency of the magnetopause position with the magnetic moment} \\label{sec:momentum}\n\n\\par To confirm that the features previously associated with the magnetopause location change according with its expected position, we performed simulations with a 2 $d_i$ long driver with density $n_d = 2\\ n_0$ for three different magnetic moments. Considering the magnetic moment that results in the expected standoff $L_0=1.8\\ d_i$ as $M_0$ (simulation \\textrm{B\/E2} on Table~\\ref{tab:runs}), simulations with the magnetic moments $2\\ M_0$ (\\textrm{E1}) and $M_0\/2$ (\\textrm{E3}) were also performed, corresponding respectively to the expected standoffs $L_0\\approx2.3\\ d_i$ and $L_0\\approx1.4\\ d_i$. Fig.~\\ref{fig:scan-momentum} shows the $\\Delta B_z$ and $J_y$ synthetic diagnostics at $y=0$ for the three simulations.%\n\n\\begin{figure*}[t]\n    \\includegraphics[width=\\linewidth]{plots\/different-momentums-plot.pdf\n    \\caption{\\label{fig:scan-momentum} Temporal evolution of the variation of the magnetic field $\\Delta B_z$ and current density $J_y$ at $y=0$, for three different magnetic moments. The magnetic moments $M$ considered were a) $M=2\\ M_0$, b) $M=M_0$ and c) $M=M_0\/2$, where $M_0$ represents the magnetic moment that corresponds to a standoff $L_0=1.8\\ d_i$ for a driver density $n_d=2\\ n_0$. The corresponding standoffs for these simulations are a) $L_0\\approx2.3\\ d_i$, b) $L_0 = 1.8\\ d_i$ and c) $L_0 \\approx 1.4 \\ d_i$.}%\n\\end{figure*}\n\n\\par Figs.~\\ref{fig:scan-momentum} a1) and b1) show the results for the highest magnetic moment $M=2\\ M_0$. We see that the current structures associated with the magnetopause and the background waves are less evident than for the lower magnetic moments, as they are formed farther from the dipole. Figs.~\\ref{fig:scan-momentum} a2) and b2) correspond to the magnetic moment $M_0$ that leads to $L_0=1.8\\ d_i$ and are the same results shown in Fig.~\\ref{fig:standard}. As previously mentioned, there are two main observable current structure standoffs. The first one is associated to the diamagnetic current, which is reflected around $t\\omega_{ci}\\approx3$ near the expected value $x_0=-L_0=-1.8\\ d_i$. This standoff is related to the interaction between the driver ions and the dipole. The second standoff occurs between $t\\omega_{ci}\\approx2$ and $t\\omega_{ci}\\approx3$ and it is located in the background plasma region. This standoff also occurs near $x=-1.8\\ d_i$.%\n\n\\par In Figs.~\\ref{fig:scan-momentum} a3) and b3), we show the results obtained for the half magnetic moment $M=M_0\/2$. In this case, the magnetic pressure exerted by the dipole is lower, leading to a smaller $L_0$, and consequently, the diamagnetic current feature visible in b3) is closer to the dipole than in Figs.~\\ref{fig:scan-momentum} b1) and b2). The main changes, however, occur in the magnetopause current. Unlike what we observe for the other magnetic moments, the magnetopause current, pinpointed in the current density plot, lasts for a longer time (until $t\\omega_{ci}\\approx4$). This current is also more separated from the diamagnetic current standoff and is easier to identify. This is consistent with the experimental observations.%\n\n\\par To identify the pressure balances associated with the two observed standoffs, and because the magnetic and kinetic pressures vary over time, we studied the temporal evolution of the different plasma and magnetic pressure components of the system. In particular, we calculated the spatial profiles of the magnetic pressure $B^2\/8\\pi$, the ram pressure $n_jm_jv_{flj}^2$ and the thermal pressure $n_jm_jv_{thj}^2$ as a function of time for $y = 0$. In these expressions, $n_j$, $m_j$, $v_{flj}$ and $v_{thj}$ refer to the density, mass and flow and thermal velocities, respectively, of the ions ($j=i$) and electrons ($j=e$). The magnetic pressure was calculated from the magnetic field measured in each PIC grid cell located at $y=0$. The flow and thermal pressures, were calculated from averaged particle data. To ensure that the calculation of each kinetic pressure considered a sufficiently large number of particles, all the particles between $-0.1\\ d_i<y<0.1\\ d_i$ were binned into equal-sized bins of width $0.05\\ d_i$ over the $x$ direction. For each bin we computed: i) the average density of each species, ii) the flow velocity, corresponding to the average of the velocity of the particles, and iii) the thermal velocity, corresponding to the standard deviation of the velocity of the particles~\\cite{Liboff2003}. With these averaged quantities, the ram and thermal pressures were calculated in each bin for each species of ions and electrons and each component of the velocity $x$, $y$, and $z$. The $y$ and $z$ components of the pressures, however, are negligible.\n\n\\par These pressure profiles were obtained for simulation E3 with magnetic moment $M=M_0\/2$ and are plotted in Fig.~\\ref{fig:pressures} for times where a) the magnetopause and b) the diamagnetic current standoff can be observed. The kinetic pressures represented were calculated by adding all the components of the ram and thermal pressures of the ions and electrons for the background ($P_0$) and the driver ($P_\\mathrm{d}$) plasmas. The magnetic pressures represented were calculated by considering the total and the relative magnetic field pressures ($P_\\mathrm{mag} = B_z^2\/8\\pi$ and $P_\\mathrm{rel} = P_\\mathrm{mag}-B_0^2\/8\\pi$, respectively). The pressures were normalized to the initial ram pressure of the driver ions.%\n\n\\begin{figure*}[t]\n\\includegraphics[width=\\linewidth]{plots\/pressures_and_energy3.pdf\n\\caption{\\label{fig:pressures} Pressures profiles calculated for simulation \\textrm{E3} with a magnetic moment $M=M_0\/2$ (shown in Fig.~\\ref{fig:scan-momentum} c)), during the occurrence of a) the magnetopause and b) the standoff of the diamagnetic current. The magnetic pressures are $P_\\mathrm{mag} = B_z^2\/8\\pi$ and $P_\\mathrm{rel} = P_\\mathrm{mag}-B_0^2\/8\\pi$. The kinetic pressures $P_\\mathrm{d}$ and $P_0$, corresponding to the driver and background plasmas, respectively, consider both the ions and electrons and the flow and thermal components of the velocity. c) Temporal evolution of the variation of the total kinetic energies of the driver $\\Delta W_{\\mathrm{kin,d}}$ and background $\\Delta W_{\\mathrm{kin,0}}$ plasmas, the magnetic energy $\\Delta W_{\\mathrm{mag}}$, and the total energy of the simulation box $\\Delta W_{\\mathrm{tot}}$. The total energy is calculated by adding all the kinetic energies and the electric and magnetic energies. Since the background plasma is magnetized, the electric energy term is many orders of magnitude smaller than the magnetic energy term. The energies were normalized to the initial total energy of the driver ions $W_{\\textrm{d,ini}}$. The loss of energy conservation near $t\\omega_{ci}\\approx 4$ is caused by the escape of background plasma particles and magnetic field through the right hand side of the simulation box.}%\n\\end{figure*}\n\n\\par Fig.~\\ref{fig:pressures} a) shows the magnetic and kinetic pressures at time $t\\omega_{ci}\\approx2.33$ where we observed the magnetopause in Fig.~\\ref{fig:scan-momentum} b3). When the driver starts pushing the background, the pressure of the driver at the interface between the plasmas increases because the driver density and thermal velocities also increase. During the flow, the driver transfers energy and momentum to the background plasma, and as a result, the background develops a strong kinetic pressure. At the time represented in Fig.~\\ref{fig:pressures} a), the background plasma pressure equals the total and the dipolar magnetic pressures in $x \\approx -1.4\\ d_i$, near the location of the magnetopause current of Fig.~\\ref{fig:scan-momentum} b3). This observation supports the hypothesis that this current emerges from the standoff between the background and magnetic pressures. Fig.~\\ref{fig:pressures} b) shows the pressures for $t\\omega_{ci}=3$ where we see the beginning of the reflection of the driver. The driver pressure equals the magnetic and dipolar pressures near $x_0=-L_0\\approx-1.4\\ d_i$. After this time, the driver is incapable of moving any further into the background because the magnetic pressure exceeds its kinetic pressure.%\n\n\\par The energy variations integrated over the entire simulation domain can also help us understand the system dynamics. Fig.~\\ref{fig:pressures} c) shows the variation of the total driver and background kinetic energies, $\\Delta W_{\\mathrm{kin,d}}$ and $\\Delta W_{\\mathrm{kin,0}}$, respectively, as well as the variation of the magnetic energy $\\Delta W_{\\mathrm{mag}}$, and of the total energy $\\Delta W_{\\mathrm{tot}}$. The kinetic energies of the background and driver plasmas consider all ions and electrons. In early times $t\\omega_{ci}<3$, as the driver and background plasmas interact, the driver transfers its energy to the background plasma and the magnetic field. The total energy, given by the sum of the electromagnetic energy and the kinetic energies, remains constant during this period. After the driver is fully reflected by the dipole for $t\\omega_{ci}>3$, the magnetic field loses most of its energy to the background and driver plasmas leading to a drop of the magnetic energy. After $t\\omega_{ci}\\approx4$, the background ions start to leave the simulation box, and the total energy is no longer conserved. The background kinetic energy remains approximately constant because the background plasma loses energy to the sink at the right boundary of the simulation but gains energy from the magnetic field. For both driver and background plasma, the ions carry most of the energy.%\n\n\\par From Fig.~\\ref{fig:pressures}, we can identify the positions where multiple pressure balances occur, and therefore, develop an insight into the pressure equilibria that are behind the structures of the current density synthetic diagnostics. Using the previously calculated pressures, we obtained the equilibrium positions where certain pressure balances manifested and plotted them in Fig.~\\ref{fig:equilibrium} alongside $J_y$.%\n\n\\begin{figure}[ht]\n\\includegraphics[width=\\columnwidth]{plots\/pressure-equilibrium3.pdf}\n\\caption{\\label{fig:equilibrium} Temporal evolution of the current density $J_y$ at $y=0$, with the closest locations to the dipole of different pressure balances for multiple times. The represented locations of pressure balances are the equilibria between the driver kinetic pressure $P_{\\mathrm{d}}$ with the total magnetic field pressure $P_{\\mathrm{mag}}=B_z^2\/8\\pi$, represented by the solid line; the background kinetic pressure $P_0$ with the pressure exerted by the relative magnetic field $P_{\\mathrm{rel}}=P_{\\mathrm{mag}}-B_0^2\/8\\pi$, by the dotted line, and $P_{\\mathrm{d}}=P_{\\mathrm{rel}}$, by the dashed line. The results correspond to simulation \\textrm{E3} (see Table~\\ref{tab:runs}).}%\n\\end{figure}\n\n\\par This analysis shows that the system has, in general, two magnetopause structures: one driven by the background, and one by the driver plasma. The former structure is defined by the balance $P_{\\mathrm{0}}=P_{\\mathrm{rel}}$. For the latter structure to form, the driver needs to have almost enough energy to push the diamagnetic current up to the magnetopause, defined by Eq.~\\ref{eq:pressure-equilibrium}. This is illustrated in Fig.~\\ref{fig:equilibrium}, where we show the location of the pressure equilibrium between the driver kinetic pressure and the total magnetic pressure, $P_\\mathrm{d} = P_\\mathrm{mag}$. \n\n\\par As shown in Fig.~\\ref{fig:pressures}, the current associated with the background magnetopause seems to overlap with the region of background and magnetic pressure balance. Unlike the driver, the background plasma is magnetized. If we neglect the compression of the magnetic field in the downstream region, the pressure balance that describes this magnetopause can then be estimated by the equilibrium of the kinetic pressure of the background plasma with the relative magnetic pressure, $P_0 = P_\\mathrm{rel}$. In Fig.~\\ref{fig:equilibrium}, we show that this pressure balance, represented by the dotted line, describes well the position of the current feature identified as the magnetopause between times $t\\omega_{ci}\\approx2$ and $t\\omega_{ci}\\approx3$. \n\n\n\\par After $t\\omega_{ci} \\approx 3$, the magnetopause current is well described by the pressure balance $P_\\mathrm{d} = P_\\mathrm{rel}$, as illustrated by the dashed line in Fig.~\\ref{fig:equilibrium}. In fact, after inspecting the phase spaces in Figs.~\\ref{fig:phase-space} a3) and b3), we can observe that a combination of driver plasma particles (separated from the bulk distribution) and background ions pushes the dipolar field and sets the position of the magnetopause.\n\n\n\n\\par We stress that, because we are determining equilibria via MHD pressure balances but are checking the intersection between pressure curves with kinetic resolution, some caution must be made to ensure that we are observing the equilibrium between pressures and not merely the interface between the different regions of interest. To ensure that the pressure equilibria were correctly obtained, the corresponding pressure profiles were always carefully inspected with additional diagnostics.%\n\n\\subsection{Realistic parameters}\\label{sec:realistic}\n\n\\par Due to the need for more extensive scans (and thus using physically equivalent but computationally feasible parameters), the simulations shown so far considered reduced ion mass ratios, cold plasmas, and higher velocities than the ones used in the LAPD experiments - see Table~\\ref{tab:parameters}. To ensure that the main results presented in the previous sections are also valid with realistic parameters, we have performed a set of simulations with parameters similar to those expected experimentally. \n\n\\par Three simulations were performed, labeled as runs F1 to F3. Run F1 employs realistic mass ratios $m_{i,d}\/{m_e} = m_{i,0}\/{m_e} = 1836$. Additionally, run F2 also considers a ratio between the electron thermal and flow velocities close to the ones expected for the LAPD experiments, namely $v_{the,x}\/v_0=2.5$ and $v_{thi,x}\/v_0=0.033$, leading to higher temperatures than in the previous simulations, and thus allowing possible thermal effects on the system. Finally, run F3 considers the same electron thermal velocity ratios of F2 but the standard reduced mass ratios. \n\n\\par The $\\Delta B_z$ and $J_y$ plots for these simulations are shown in Fig.~\\ref{fig:realistic}. Note that, due to changes in $m_i\/m_e$, the spatial and temporal scales were recalculated for the new parameters. Once again, the magnetic dipole moment for the three simulations was adjusted to ensure that $L_0=1.8\\ d_i$.%\n\n\\begin{figure*}[t]\n    \\includegraphics[width=\\linewidth]{plots\/realistic2.pdf\n    \\caption{\\label{fig:realistic} Temporal evolution of a) the variation of the magnetic field $\\Delta B_z$ and b) the current density $J_y$ at $y=0$, for the simulations with similar parameters to the experiments. Run F1 considers realistic mass ratios for the driver and background plasmas and low ratios between the thermal and flow velocities; run F2 uses realistic mass ratios and thermal velocity ratios close to the ones expected in the experiments; run F3 uses the realistic thermal velocity ratios but reduced mass ratios.}%\n\\end{figure*}\n\n\\par As expected, these simulations show the same main structures discussed in the previous sections. We observe the typical reflection of the compression of the magnetic field and the current structures of the magnetopause and diamagnetic cavity. However, some differences are also visible. In Figs.~\\ref{fig:realistic} a1) and b1), \\textit{i.e.} for the realistic mass ratios but cold plasmas simulation, we observe a stronger filamentation of the plasma flow reflected off the dipole and a thinner diamagnetic current. This is because $d_e$ is the characteristic length scale of the current layer and we have lower $d_e\/d_i$ values for larger $m_i\/m_e$. Figs.~\\ref{fig:realistic} a2) and b2), for the simulation with higher temperatures, show no major differences with Figs.~\\ref{fig:realistic} a1) and b1), even though there is a significant increase in the thermal velocities. \n\n\\par In Figs.~\\ref{fig:realistic} a3) and b3), however, we observe significant differences for reduced mass ratios with realistic thermal velocity ratios. In particular, we observe in the current density plot smoother magnetic and current structures and less defined background waves between the magnetopause and the dipole. We also observed for increased ion thermal velocities, for example, $v_{thi}\/v_0 \\approx 0.25$, that the background waves are no longer visible. \n\n\\par Additionally, other simulations were performed to look for possible changes with realistic parameters. A simulation with a lower flow velocity $v_0=0.01\\ c$ and realistic thermal velocity ratios lead to no significant features observed, and the obtained synthetic diagnostics were very similar to the ones in Figs.~\\ref{fig:realistic} a3) and b3), meaning that the system scales well with $v_0$. Another simulation was performed to observe if the shape of the initial density profiles of the plasmas would affect the main results. Namely, the constant density profiles used on both the driver and background plasmas were replaced by Gaussian density profiles with a typical gradient scale $\\sigma=1\\ d_i$ on the edges of the plasmas. This simulation did not show meaningful differences, in agreement with previous plasma coupling works, which observed that the leading edge of the plasmas evolves similarly for different initial density profiles~\\cite{doi:10.1063\/1.1694472}.%\n\n\\subsection{Finite transverse size}\\label{sec:finite}\n\n\\par For simplicity, and because we were more interested in studying the system along the axis of symmetry $y=0$, the previous simulations only considered a driver with infinite width $L_y$ and a length of $L_x=2\\ d_i$. In the experiments, however, the drivers had a width comparable to their lengths and did not have the sharp boundaries used in the simulations. To investigate if and how our results are modified with a more complex-shaped driver, we performed a simulation with a finite width, semi-circular-shaped driver plasma. This driver is initially defined with the conditions $(x+7.25\\ d_i)^2+y^2<(3.25\\ d_i)^2$ and $x>-6\\ d_i$ and has length $L_x=2\\ d_i$ and width $L_y=6\\ d_i$. Fig.~\\ref{fig:finite} shows the results of this simulation and includes the initial shape of the driver in Fig.~\\ref{fig:finite} a).%\n\n\\begin{figure*}[t]\n    \\includegraphics[width=\\linewidth]{plots\/finite-driver.pdf}\n    \\caption{\\label{fig:finite} a) Total ion density at time $t\\omega_{ci}=3.0$, and temporal evolution of b) the variation of the magnetic field $\\Delta B_z$ and c) the current density $J_y$ at $y=0$, for simulation G with a finite width driver with a circular segment shape. The dashed lines at a) represent the initial position of the driver and the left border of the background plasma.}%\n\\end{figure*}\n\n\\par Due to the finite width of the new driver and its particular shape, we should expect to see significant differences in the regions of the simulation plane far from $y=0$. In the total ion density plot of Fig.~\\ref{fig:finite} a) for a time $t\\omega_{ci}=3$, when there is a strong interaction of the driver with the dipolar magnetic field, we observe the propagation of waves at the lower and upper sides of the dipole caused by the finite width of the driver, that was not present for infinite width drivers.%\n\n\\par In Figs.~\\ref{fig:finite} b) and c), we see the usual magnetic and current density plots at $y=0$ for this simulation. By shortening the driver plasma width, the background particles escape from the bottom and top regions of the simulation box, and the driver has more difficulty holding the magnetic decompression in the background region. The decompression, therefore, occurs quicker for finite drivers, as seen in Fig.~\\ref{fig:finite} b), leading to short reflections of the magnetic compression.\n\n\\par Although this complex-shaped driver gets us closer to the experimental configuration, the simulations did not include all the properties of the experimental driver, as for example, the non-uniform density, velocity profiles of the plasmas and the flow divergence. Additionally, 3D effects should also be considered. Future simulations are planned to study the effect of these properties in the results. However, we expect that these features will not change the main results of the simulations.%\n\n\\section{Conclusions} \n\\label{sec:conclusions}\n\n\\par In this work, we have performed PIC simulations of mini-magnetospheres in the interaction between a plasma flow and a magnetized background plasma. In particular, we have successfully reproduced results from recent experiments performed at the LAPD, validating the experimental platform to study mini-magnetospheres in the laboratory. We have also explored an extensive parameter space defining the interaction, allowing us to i) determine how the main properties of the system change with the parameters and ii) identify the required conditions for the creation of a mini-magnetosphere.%\n\n\\par Our simulations have shown that some system features are present across multiple regimes. The initial flow of the driver expels the magnetic field in the upstream region, leading to a magnetic cavity, and compresses the downstream magnetic field. The driver travels through the background until the magnetic field pressure is large enough to counterbalance the driver plasma pressure. A fast decompression of the background magnetic field then follows. If the background decompression occurs after the total reflection of the driver plasma, then we can observe the reflection of the compression of the magnetic field. To see this feature, the driver needs to be short enough to anticipate the driver reflection relative to the decompression but sufficiently long to ensure that it can get close to the dipole.%\n\n\\par For the super-Alfv\u00e9nic flows considered, the driver particles are reflected upstream during the interaction with the background plasma and the magnetic field. The coupling velocity (\\textit{i.e.}, the velocity at which the leading end of the driver travels through the background) is lower than the flow velocity and increases with the increase of the ratio between the driver and background densities. The coupling velocity and the length of the driver determine how far the driver can go through the background region without a dipole, for a uniform driver plasma. \n\n\\par The interaction of the plasmas with the dipole results in two magnetopauses. The first describes the balance between the kinetic pressure of the propelled background plasma plus the pressure of the plasma internal magnetic field and the total magnetic pressure. The seconds describes approximately the balance between the kinetic pressure of the driver plasma separated from the bulk distribution and the relative magnetic pressure. Using simulations with different dipole moments, we have shown that, for lower magnetic moments, the driver and background standoffs are closer to the center of the dipole, and the magnetopause current is more clearly identified than for higher magnetic moments. Furthermore, it is also easier to separate the magnetopause and diamagnetic currents for lower magnetic moments, consistent with experimental observations.%\n\n\\par In the simulations performed, we also observed the formation of waves in the background plasma region, between the magnetopause and the center of the dipole, where the magnetic field gradient was significant. These waves result from the excitation that always followed the formation of the magnetopause and were only observed for background plasmas with relative low ion thermal velocities. This condition may explain the absence of these waves in the experimental plots.%\n\n\\par Most of the simulations presented in this work were performed in idealized configurations. In particular, we used reduced ion-to-electron mass ratios, unrealistically high flow velocities, a simple flat-top driver density profile, and neglected thermal effects. In Sec.~\\ref{sec:realistic} and~\\ref{sec:finite}, we presented simulations that drop some of these simplifications. Replacing reduced ion mass ratios with realistic ones and considering high thermal velocities ratios close to the obtained in the experiments did not lead to significant changes in the results. The same occurred when considering smoothed density profiles. It was also possible to conclude that the main features of the system scaled as expected with the absolute value of the driver flow velocity. We also presented a simulation to study possible effects associated with the complexity of the experimental laser-ablated driver. A simple circular segment-shaped driver was considered and led to similar results in the axis of symmetry as the infinite width driver simulations. However, wave-like structures were observed on both the bottom and upper sides of the dipole. For future studies on the regions outside the axis of symmetry, the driver shape and complexity must be considered.%\n\n\\par Additionally, we also performed other parameter scans related to the complexity of the driver. For instance, we performed simulations where the driver ions were heavier than the background ions to simulate the small role of the carbon ions in the experimental driver. These studies showed no significant differences to the lighter ions simulations.\n\n\\par In conclusion, the simulations were consistent with the LAPD experimental results, and the multiple parameter scans performed dictated the formation conditions of the main features of mini-magnetospheres. For future works, we intend to exploit the features present in the sides of the dipole, exploit anti-parallel magnetic field configurations, perform 3D simulations, and consider even more realistic properties of the driver.\n\n\\begin{acknowledgments}\n\n\\par We acknowledge the support of the European Research Council (InPairs ERC-2015-AdG 695088), FCT (PD\/BD\/114307\/2016 and APPLAuSE PD\/00505\/2012), the NSF\/DOE Partnership in Basic Plasma Science and Engineering (Award Number PHY-2010248), and PRACE for awarding access to MareNostrum (Barcelona Supercomputing Center, Spain). The simulations presented in this work were performed at the IST cluster (Lisbon, Portugal) and at MareNostrum.\n\n\\end{acknowledgments}\n\n\\nocite{*}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nRadio continuum emission traces star formation on timescales of up to\n100 Myr \\citep{C92}. Two physical processes resulting from massive\nstar formation produce most of the radio continuum emission between 1\nand 100 GHz in star-forming galaxies: (1) nonthermal synchrotron\nemission from relativistic electrons accelerated by magnetic fields as\na result of recent supernovae and (2) thermal free-free emission from\ngas ionized by young massive stars \\citep{C92}. The nonthermal\nemission is closely tied to the number of supernova-generating massive\nstars produced in recent episodes of star formation, while the thermal\nemission gives a nearly direct measure of the current equivalent\nnumber of O stars via the ionizing flux in the sampled area. Since\neach component traces a physical process with a well-known timescale,\nwe can use measurements of the radio continuum to determine star\nformation rates and constrain the ages of recent episodes of star\nformation.\n\nRecent studies of nearby star-forming galaxies with interferometers\nhave emphasized resolving individual star-forming regions\n\\citep*[e.g.][]{B00,J03,JK03,J04,T06,R08a,J09,A11}. Since radio\ncontinuum emission is not affected by extinction, it can be used to\nobserve deeply embedded regions of current star formation that have\nnot yet shed their surrounding material and are thus invisible at\nshorter wavelengths. These studies have taken advantage of\ninterferometers' exceptional spatial resolution to probe very young\nstarbursts whose optical emission is obscured by dust. While these\nstudies have been invaluable for determining star formation properties\nin galaxies outside of our own, the high angular resolution and\nmissing short-spacing data of interferometers, especially at higher\nfrequencies, ``resolves out'' the diffuse radio continuum emission\noutside of compact star-forming regions. This effect disproportionally\nimpacts synchrotron emission, which tends to be much more diffuse than\nthe primarily thermal emission surrounding areas of ongoing massive\nstar formation \\citep{J09}. Unlike interferometers, single dish\ntelescopes are not plagued by missing short spacings. Therefore, these\ntelescopes provide a way to simultaneously measure the compact thermal\nand diffuse non-thermal components of a galaxy's radio continuum\nemission in order to characterize its \\emph{global} star formation\nproperties.\n\nDetermining the relative contributions of the thermal and nonthermal\ncomponents of the measured flux of entire galaxies can be challenging.\nFortunately, each component has a distinct behavior with respect to\nfrequency, and therefore we can model radio continuum emission with a\nsimple two-component fit. Radio continuum flux follows a power law\nrelation such that $S_{\\nu} \\propto \\nu^{\\alpha}$, where $\\alpha$ is\nthe spectral index that is characteristic of a source's\nemission. Optically thin thermal emission exhibits $\\alpha =$ -0.1,\nand nonthermal emission exhibits $\\alpha \\approx$ -0.8 \\citep{C92}. To\ndetermine a reliable fit to these parameters, observations sampling\nthe same physical area at multiple, widely-spaced frequencies are\nrequired. If only one frequency is observed, it is impossible to\ndetermine the relative contributions of each emission process without\nprevious knowledge of the source. Since single dish telescopes are\nsensitive to both compact thermal and diffuse nonthermal emission over\nlarge spatial extents, they are useful for constraining the\nlarge-scale properties of multiple components of a galaxy's radio\ncontinuum emission, and are thus powerful probes of star formation.\n\n\nThe goal of this paper is to characterize the \\emph{global} star\nformation properties of local galaxies. Our observations were taken in\nfour independent channels continuous in frequency across the full\n26-40 GHz span of Ka band. This range in frequencies is where a\ntypical star-forming galaxy's global radio continuum emission would be\nexpected to contain relatively equal amounts of flux from\nsteep-spectrum synchrotron and flat-spectrum thermal sources\n\\citep{C92}. Our observations are thus ideal for approximating the\nrelative contributions of each type of emission at this ``lever arm''\nfrequency range. Using new radio continuum observations centered at\n27.75 GHz, 31.25 GHz, 34.75 GHz, and 38.25 GHz, as well as archival\nNVSS 1.4 GHz and IRAS 60 $\\rm \\mu m$ and 100 $\\rm \\mu m$ data, we have\ndetermined these galaxies' thermal fractions and star formation\nrates. We have also explored the radio-far-infrared correlation in\nthese galaxies and its implications for their star formation\ntimescales. We will describe the galaxy sample and our observations\nand data reduction in Section 2, present our results and address the\nprocess of fitting spectral energy distributions to our data in\nSection 3, and finally conclude in Section 4.\n\n\\section{Data}\n\n\\subsection{Sample Selection}\n\nWe selected a heterogeneous group of 27 local (D $<$ 70 Mpc),\nwell-studied star-forming galaxies with known thermal radio continuum\nemission. Our sample contains galaxies spanning a variety of shapes,\nsizes, and environments, from blue compact dwarfs to grand-design\nspirals, including major and minor mergers, with members of compact\ngroups as well as more isolated galaxies (see Table\n\\ref{fig:obssummary} for galaxy types). The intention was to observe\nas many types of star-forming galaxies as possible to probe star\nformation in a diverse range of environments. See Table\n\\ref{fig:obssummary} and Figure \\ref{fig:sampleproperties} for sample\nproperties. For more information on each galaxy's previous radio\ncontinuum observations and discussions of their properties, see the\npapers named in Table \\ref{fig:obssummary}.\n\n\nThe galaxies in our sample span a range of distances (1-70 Mpc) and\nproperties. They all have previously detected radio emission and\nongoing star formation that covers three orders of magnitude in star\nformation rate. Thus, they are strong targets for a study of global\nradio continuum properties at a frequency range that probes both\nthermal free-free and nonthermal synchrotron star formation\nindicators. As seen in Figure \\ref{fig:sampleproperties}, these\ngalaxies are largely less massive and have higher star formation rates\nthan the Milky Way, and have subsolar metallicities. However, their\nproperties are not so similar that they can be considered as a single\nclass. It would not be surprising if their radio continuum properties\nalso encompassed a range of values. Our analysis is best understood as\nreflecting properties of nearby star-forming galaxies, though it is\nbeyond the scope of this paper to perform detailed analysis on each\ngalaxy individually.\n\n\\subsection{Observations and Data Reduction}\nWe observed the galaxies in our sample with the Caltech Continuum\nBackend (CCB) on the Robert C. Byrd Green Bank Telescope\n(GBT)\\footnotemark{} using single pointings. The CCB is designed for\nthe GBT's dual-beam Ka band receiver spanning the entire range of\nfrequencies from 26-40 GHz. The primary observing mode of the CCB is a\n70 second ``nod'', where each beam takes a turn as the on-source beam\nwhile the other beam is off-source. We observed 24 of the galaxies\nusing a single nod each, while we observed eight galaxies using\nmultiple nods, which we then averaged. Five galaxies in our sample\nwere observed on two different nights with both of these methods; we\ntreated these on a case-by-case basis and chose the observation(s)\nwith the best weather and elevation conditions. We used the standard\nNRAO primary flux calibrators 3C 147 and 3C 48 for flux calibration,\nas well as nearby pointing calibrators to ensure accurate\npointing. See Table \\ref{fig:obssummary} for a summary of the\nobservations.\n\n\\footnotetext{The National Radio Astronomy Observatory is a facility\n  of the National Science Foundation operated under cooperative\n  agreement by Associated Universities, Inc.}\n\n\nWe reduced our data using IDL reduction routines developed by B.\nMason \\citep[for details on the data reduction process,\n  see][]{M09}. Data with wind speeds over 5 $\\rm m \\ s^{-1}$ were\nexcluded due to the possibility of large pointing errors. We detected\n22 galaxies in all four channels. When a galaxy's flux was lower than\nthe 2$\\sigma$ level in one or more of the four channels, we combined\nthe channels' fluxes to produce one average flux across the entire\nband, centered at 33 GHz. One galaxy, Pox 4, was detected at the\n$5\\sigma$ level after averaging the four channels. Three additional\ngalaxies were marginally detected (between 2$\\sigma$ and 3$\\sigma$)\nusing this method. We report an upper limit 33 GHz flux for one\ngalaxy, Tol 35. The galaxies' observed fluxes are reported in Table\n\\ref{fig:uncorrectedflux}.\n\nSince the angular size of a telescope's beam is inversely proportional\nto the frequency observed, the beam size of the GBT varies appreciably\nacross the 26-40 GHz range of our observations ($\\sim27\\arcsec$ in the\nlowest-frequency sub-band versus $\\sim19\\arcsec$ in the\nhighest-frequency sub-band, see Figure ~\\ref{fig:beamsizes} for an\nillustration). We followed the procedure of \\citet{M10} to correct for\ndiffering beam sizes in each of the four sub-bands. First, we imaged\nan archival VLA radio continuum map of each galaxy (typically at\nfrequencies of 4-10 GHz) using the AIPS task IMAGR. For these maps, we\nselected the archival UV data from the NRAO science data\narchive\\footnotemark{} with the closest beamsize to our Ka-band\ndata. We explicitly imposed each of the four CCB beam sizes on these\nimages using BMIN and BMAJ (assuming a circular beam). We determined\ncorrection factors for each beam by normalizing the flux contained in\neach beam area in the archival map to the flux in the 34.75 GHz port's\n$\\sim21\\arcsec$ beam. This procedure partially adjusts for more flux\nbeing observed at lower frequencies due to these frequencies'\nintrinsically larger beam sizes. We then could approximate\n``beam-matched'' flux measurements to determine spectral indices\nbetween 26-40 GHz (see Figure \\ref{fig:histogram} for an illustration\nof the galaxies' spectral indices before and after applying the\ncorrections). NGC 1222 did not have available archival data, so we\napplied to it the average correction factors of all of the other\ngalaxies. We emphasize that these correction factors are only\napproximate. In many cases, they are based on resolved archival images\nthat may not contain all of the galaxies' radio flux. In particular,\nthese resolved images may contain most of the thermal emission, which\ntends to be compact, but underestimate the galaxies' nonthermal\nemission, which tends to be diffuse. This could bias the correction\nfactors to be closer to 1.0 than should be the case, especially in the\nhighest-frequency (and thus highest-resolution) channel. See Table\n\\ref{fig:beamcorrections} for the beam correction factors for each\ngalaxy. \n\nThe dominant sources of uncertainty in our beam corrections are\nsystematic errors due to the geometries of our sources. The smallest\ncorrections possible are for a galaxy whose most diffuse, extended\nflux is still contained within the smallest beam and is unresolved by\nthe lower-frequency interferometric observations. This type of source\nwould look identical to all four of the GBT beam sizes. In this case,\nthe correction factors would be 1.0 for each sub-band. For a source\nmuch more extended than the beam sizes, the maximum deviations from no\nbeam corrections in each sub-band are -36$\\%$, -19$\\%$, 0$\\%$, and\n+21$\\%$. Pointing offsets from the peak of radio emission can also be\nsources of systematic error, though the errors depend on whether the\nsource is compact or extended, and the magnitude of the pointing\noffset from the central peak of radio emission. These errors are\ntypically smaller than the maximum deviations discussed above. Since\nwe do not know how much diffuse emission is missing in the archival\ndata, we do not have enough information to quantify uncertainties in\nthe beam correction factors for each galaxy.\n\n\\footnotetext{https:\/\/archive.nrao.edu}\n\nMany of the galaxies that we observed were more extended than the\n$\\sim 23\\arcsec$ beam size of the GBT at 33 GHz. In these cases, radio\ncontinuum fluxes and star formation rates should only be interpreted\nas covering the inner $\\sim 23\\arcsec$ of the galaxies. The galaxies\nwith resolved lower-frequency archival data that was more extended\nthan the beam size are flagged with a ``1'' in Table\n\\ref{fig:correctedflux}.\n\nThe CCB has a beam separation of $78\\arcsec$ between the ``on'' and\n``off'' beams. M 51 and M 101 are more extended than that separation\nin both optical images (see Figure \\ref{fig:beamseparation}) and in\nmaps of their lower-frequency radio continuum emission\n\\citep{K84,G90}. In these cases, our flux measurements may be lower\nthan the true amount of flux contained within the beam. There is\nlikely to be radio continuum emission at the ``off'' positions, which\nwould cause an oversubtraction of flux in the reduction process.\n\n\\section{Results and Discussion}\n\n\\subsection{Fluxes}\nFor 22 of the 27 galaxies that we observed, the fluxes we report are\nthe first detections (either in all four sub-bands or averaged) at\n$\\sim$33 GHz. Four of the galaxies in our sample were previously\nobserved with the CCB by \\citet{M12}, three of which were detected (M\n101 was reported as an upper limit by \\citet{M12} and is a $2.6\\sigma$\nmarginal detection when the four sub-bands were averaged in our\nobservations). Only one galaxy in our sample, Tol 35, was not detected\nwhen averaging four sub-bands' fluxes. Its $3\\sigma$ upper-limit flux\nis 0.87 mJy. This galaxy was observed at a very low elevation\n($7.9^{\\circ}$), so it was observed with large atmospheric\nextinction. See Table \\ref{fig:uncorrectedflux} for the uncorrected\nfluxes, and Table \\ref{fig:correctedflux} for the fluxes corrected for\nthe differing beam sizes of each frequency.\n\n\\subsection{Spectral energy distribution fitting}\nWe fit a spectral energy distribution (SED) for each galaxy that was\ndetected in all four sub-bands using the four CCB fluxes and archival\nNRAO VLA Sky Survey (NVSS) 1.4 GHz fluxes (measured with a 45$\\arcsec$\naperture). We assumed a two-component fit of nonthermal emission with\na spectral index $\\alpha_{N}$ = -0.8 and thermal emission with a\nspectral index $\\alpha_{T}$ = -0.1. These fits are plotted in Figure\n\\ref{fig:sedpanel}. Though the spectral index of nonthermal emission\ncan vary (this phenomenon is described further in Section 3.2.1), we\nused this simple model because we only fit to five data points for\neach galaxy; our model did not include enough data to justify\nadditional free parameters. We do not see evidence of anomalous dust\nemission in the observed regions of these galaxies \\citep[for\n  explanations of anomalous dust, see][]{D98,M10}. Our observations\nare also at frequencies low enough to have negligible contributions\nfrom the low-frequency tail of the dust blackbody. Therefore, we did\nnot include any thermal dust emission in our fits. Our spectra also do\nnot show the inverted structure characteristic of self-absorption or\noptically thick thermal emission, so we did not include either of\nthese components. From these fits, we determined each galaxy's thermal\nand nonthermal fluxes at 33 GHz.\n\nNone of the galaxies have globally flat spectra indicative of purely\nthermal emission, nor the inverted spectra seen in some resolved\nobservations of very young, obscured thermal sources. Thermal emission\nwas the primary component at 33 GHz in some galaxies, while others had\nless prominent or even negligible thermal components in the observed\nregions. In contrast to radio continuum studies done at high spatial\nresolution, our single dish observations detect the diffuse\nsynchrotron emission produced by past supernovae in addition to the\nstrong compact thermal emission from H II regions, so the spectral\nindices that we derive are typically much steeper than those derived\nonly from detections of compact radio sources. Since our observations\ndo not spatially separate regions of thermal and nonthermal emission,\nwe cannot further distinguish the two components in that way.\n\n\\subsubsection{Galaxies with steep radio spectra}\nThe fitted spectra for eight of the 27 galaxies (Arp 217, NGC 4449,\nNGC 2903, Maffei II, NGC 4038, M 51, NGC 4490, and NGC 1741) are\nsignificantly steeper than can be fit by a combination of thermal\n($\\alpha_{T}$ = -0.1) and nonthermal ($\\alpha_{N}$ = -0.8) components\n(see Figure \\ref{fig:sedpanel}). When we could not fit a galaxy's SED\nwith both the thermal and nonthermal components at the $2\\sigma$\nlevel, we used only a single-component fit that assumed no thermal\nflux and a fixed nonthermal spectral index of $\\alpha_{N}$ = -0.8 for\nconsistency. The thermal fluxes and associated properties of this\ngroup of galaxies are reported as upper limits. We used the total flux\nin the 34.75 GHz channel plus 3$\\sigma$ as a conservative upper limit\nto the thermal flux in these cases.\n\nThere are two possible explanations for the steep spectra that we see\nin some galaxies. There could be technical considerations due to\nimperfect beam-matching in our data, or there could be physical\nprocesses taking place within these galaxies causing their spectra to\nsteepen at high frequencies. In order to have more accurate SED\nfits---and more precise star formation rates---we would need to have\nbeam-matched observations of the same regions at many different\nfrequencies. \n\nThe correction factors for differing beam sizes that are given in\nTable \\ref{fig:beamcorrections} are limited by being calculated using\nhigher-resolution data that could be missing extended emission. If\nextended emission is missing in the archival data, the correction\nfactors in Table \\ref{fig:beamcorrections} could be closer to 1.0 than\nis actually the case. While all of the correction factors calculated\nact to flatten the SED between 26 GHz and 40 GHz with respect to the\nuncorrected data, it is possible that they do not flatten the SED\nenough if they do not reflect contributions from extended emission (as\ndiscussed in Section 2.2). In addition, we did not correct for\nmismatched beams between the $\\sim 45\\arcsec$ NVSS data and the $\\sim\n23\\arcsec$ CCB data. This beam difference only affects resolved\ngalaxies (those marked with a ``1'' in Table \\ref{fig:correctedflux}),\nwhich comprise $33\\%$ of our sample. It is possible that synchrotron\nemission is more adversely affected by the differences in beam sizes\nthan thermal emission. More diffuse synchrotron emission could be\nundetected at higher frequencies (and thus smaller beam sizes) than\nwould be expected for a smooth flux distribution observed with two\napertures of different sizes. If this is the case in our observed\nregions, it could explain why some of our galaxies' spectra steepen at\nthe frequencies we observed. It is also possible that the choice of\nwhere the GBT beams were pointed within a galaxy could affect its\nfluxes in different beam sizes. If the beam is not centered on the\ngalaxy (in the case of unresolved galaxies) or is not centered on a\nbright knot of emission (in the case of resolved galaxies), the\nsmaller beams could contain even less flux than would be expected\nafter corretions for the beams' areas. NGC 1741 and NGC 4490 are\nlikely affected by pointing offsets, as seen by comparing the GBT\npointing in Table \\ref{fig:obssummary} to previous radio continuum\nmaps in Figure 2 of \\citet{B00} and Figure 4 of \\citet{A11}. As\ndescribed in Section 2.2, pointing offsets from the peak of radio\ncontinuum emission result in the need for larger beam correction\nfactors than derived from the archival radio continuum data, the lack\nof which result in steep spectra at the observed frequencies.\n\n\nIn addition to the technical issue of mismatched beam sizes, there are\npossible physical explanations for steep spectra in star-forming\ngalaxies. It is difficult to distinguish between a spectrum with a\nnonthermal component having $\\alpha_{N} \\approx -0.8$ coupled with a\nlow thermal fraction from a spectrum with a steeper nonthermal\ncomponent coupled with a relatively high thermal fraction\n\\citep{C92}. Though the spectral indices that we used are typical\nvalues \\citep{C92}, they can vary depending on the physical parameters\nof the observed regions. Thermal emission can have a positive spectral\nindex if the emission regions are optically thick, though we do not\nsee any evidence that this is occuring on the angular scale of our\nobservations. Nonthermal spectral indices can be positive at low\nfrequencies due to synchrotron self-absorption (which we do not\nobserve), or become more negative with increasing frequency and\nincreasing scale height from the disk due to aging cosmic ray\nelectrons losing energy as they propagate outward from their parent\nsupernovae \\citep{S85,CK91,H09}. \\citet{K11b} calculated the timescale\nfor synchrotron losses for cosmic ray electrons in NGC 4214 to be 44\nMyr at 1.4 GHz and 18 Myr at 8.5 GHz. There is also some evidence of\nsteepening spectra at higher frequencies ($\\gtrsim$ 10 GHz) for\nluminous and ultra-luminous infrared galaxies, as well as in the\npost-starburst galaxy NGC 1569 \\citep{I88,L04,C08,C10,L11}. These\nauthors hypothesize that winds or outflows may disperse synchrotron\nemission from its parent source more quickly than would be expected\nfor simple diffusion. This rapid dispersal could cause a dearth of\nsynchrotron emission at higher frequencies on shorter timescales than\nwould be predicted from the timescale of energy loss. \\citet{L11} also\nhypothesize that there could be a modified injection spectrum in\ngalaxies where this is the case. Our sample of galaxies does not\ncontain any LIRGs or ULIRGs, and we do not see steepening in our\nmeasurements of NGC 1569. We are only observing the inner region of\nNGC 1569, while the dispersed synchrotron emission resides in its\nouter halo, so it is not surprising that we do not observe a\nsteepening spectrum in this galaxy.\n\nWe suspect that the steep spectra seen in our sample are primarily a\nresult of imperfect beam matching as discussed above. This is\nespecially likely to be the case for the galaxies resolved by the GBT\nat 33 GHz, since these galaxies will have emission that is outside of\nthe view of the GBT beam but is included in the NVSS flux. As\ndiscussed earlier, the galaxies that appear unresolved in archival\nmaps could still have diffuse synchrotron emission that was not\ndetected in the archival data but that is more extended than the\n$23\\arcsec$ beam at 33 GHz. Five of the eight resolved galaxies that\nwere detected in all sub-bands had steep spectra (four out of those\nfive are classified as spiral galaxies), while only three of the\nfourteen unresolved galaxies had this feature. Two of these three are\nclassified as SABbc galaxies, while the third is classified as\npeculiar. It is possible that these steep-spectrum galaxies contain\nemission in their spiral arms that is extended with respect to the\nGBT's smaller beam but is observed in the NVSS data. In Figure\n\\ref{fig:sedpanel}, most of the galaxies with steep spectra (and thus\nsingle component fits) also showed the NVSS 1.4 GHz data point being\nlocated above the best-fit line expected for purely nonthermal\nemission. This could be a consequence of the larger beam at 1.4 GHz\nsampling a larger physical area of emission. Even so, the alternative\nphysical explanations merit consideration, especially in the case of\nthe unresolved galaxies. In Figure \\ref{fig:rad-IR}, which will be\ndiscussed further in Section 4.6, the three unresolved galaxies with\nsteep spectra (NGC 1741, NGC 2903, and Arp 217) have elevated 1.4 GHz\nfluxes with respect to what would be expected from the radio-far\ninfrared correlation. Since the 33 GHz fluxes of these galaxies are\nnot similarly elevated with respect to their far-infrared fluxes in\nFigure \\ref{fig:rad-IR}, their steep radio spectra may indicate an\ninternal physical process that strongly increases the amount of\nsynchrotron emission.\n\n\\subsection{Thermal fractions}\nThe average thermal fraction fit by two-component models at 33 GHz was\n54$\\%$, with a $1\\sigma$ scatter of 24$\\%$ and a range of\n10$\\%$-90$\\%$. The average is consistent, albeit with large scatter,\nwith the average global thermal fraction at 33 GHz in star-forming\ngalaxies without active galactic nuclei following the relation\n\\begin{equation}\n\\frac{S}{S_{T}} \\sim 1 + 10\\left(\\frac{\\nu}{GHz}\\right)^{0.1+\\alpha_{N}}\n\\end{equation}\n where $\\alpha_{N} = -0.8$ is the nonthermal spectral index, $S_{T}$\n is the thermal flux at a given frequency, and S is the total flux at\n that frequency \\citep{CY90}. When a two-component fit was not\n possible, we report the thermal flux as the corrected flux at 34.75\n GHz plus $3\\sigma$, which gives a very conservative upper limit. We\n expect from the galaxies' SEDs that their true thermal fractions are\n very low at 33 GHz, which we assume in the rest of our analysis.\n\n\\subsubsection{Implications for star formation timescales}\nThe large scatter in the thermal fraction is likely a consequence of\nour heterogeneous galaxy sample; these galaxies are at different\nstages of evolution and have different star formation rates, stellar\npopulations, and physical properties \\citep{B00}. Some of them may\nhave a very recent ($<$ 10 Myr) burst of star formation that produces\na large amount of free-free emission that dominates their spectra from\n1-100 GHz. Others may be in between episodes of very active star\nformation and instead be experiencing a more quiescent phase, which\nwould result in a relatively low thermal fraction and steepening\nnonthermal component at 33 GHz due to synchrotron energy losses at\nhigh frequencies.\n\nThermal emission traces very recent star formation, since it comes\nfrom ionized regions around short-lived, massive stars. For a single\nstarburst, a spectrum showing solely thermal emission requires that\ntoo few supernovae have yet occurred to detect their emission. This\nwould constrain the starburst to be less than $\\sim$ 6 Myr old (or\neven younger, depending on the mass and lifetime of the most massive O\nstars in the starburst; \\citet{M89} find the lifetime of a 120\n$M_{\\sun}$ star to be 3.4 Myr). A complete absence of thermal flux\nimplies the absence of enough massive O stars to have detectable\nfree-free emission for a long enough period of time that the emission\nhas dissipated from its parent region. If this was the case, the\nstarburst is likely at least 30 Myr old (the lifetime of the least\nmassive supernova progenitors). On the other hand, nonthermal emission\nprobes star formation on longer timescales (30 Myr $< \\tau <$ 100\nMyr). It is produced by recent supernovae of stars that can be less\nmassive and have longer lifetimes than the O stars that produce\nthermal emission \\citep[see Figure 9 of][]{C92}. The presence of\nnonthermal emission implies that the starburst is at least 6 Myr old\nbut younger than the timescale dictated by synchrotron energy loss for\nthe galaxy's magnetic field ($\\sim$ 100 Myr) \\citep{C92}.\n\nWe note that there are limits to the amount of each component that we\ncan detect, so the timescales quoted in the previous paragraph are\nonly approximate. To constrain how much nonthermal emission could be\npresent in a spectrum that appears purely thermal, we generated\nspectra with varying thermal fractions with fluxes at the same five\nfrequencies as those in our data set (1.4 GHz, 27.75 GHz, 31.25 GHz,\n34.75 GHz, and 38.25 GHz) and 10$\\%$ errors on the fluxes. When these\nspectra are fit with a two-component model assuming $\\alpha_{T} =\n-0.1$ and $\\alpha_{N} = -0.8$, nonthermal emission can only be\ndetected in the spectra for thermal fractions less than 97$\\%$. This\nmeans that the galaxy could have some nonthermal emission (up to $3\\%$\nfor $10\\%$ errors on the fluxes), but the emission would be\nundetectable and thus the starburst would appear younger than it\nis. Similarly, a spectrum could look like it contains no thermal\nemission while actually containing quite a bit. For the same spectra\nwith $10\\%$ errors on the fluxes, thermal fractions of up to $20\\%$\nresulted in undetectable thermal components. This means that a galaxy\ncould look like its massive star formation has ceased while still\nhaving a small thermal component.\n\nFor the galaxies in our sample, this picture could be more\ncomplicated. The quoted timescales in this section are for an isolated\nsingle starburst. Since our observations measure star formation\nproperties on large angular scales, the galaxies may have multiple\noverlapping generations of star formation that are not easily\nseparated in time. We are also sampling different structures and\nphysical scales in each galaxy. For some galaxies, we are only\nobserving the most central region. For these galaxies, we may be\nmissing the majority of the ongoing star formation happening in outer\nregions and spiral arms. For the more compact galaxies, however, we\nare likely measuring the entirety of the galaxy's star formation\nwithin the GBT beam, so our measurements characterize their global\nstar formation properties.\n\n\\subsection{O stars producing ionizing photons}\nFor those galaxies whose SEDs were fit with thermal components, we\nused their fluxes at 33 GHz to calculate their thermal\nluminosities. We then used those luminosities to calculate the number\nof ionizing photons responsible for the thermal fluxes seen within the\nGBT beam following Equation 2 in \\citet{C92}:\n\\begin{equation}\n\\left(\\frac{Q_{Lyc}}{s^{-1}}\\right) \\geq 6.3 \\times 10^{52}\n  \\left(\\frac{T_{e}}{10^{4}K}\\right)^{-0.45} \\left(\\frac{\\nu}{GHz}\\right)^{0.1}\n  \\left(\\frac{L_{T}}{10^{20}W Hz^{-1}}\\right),\n\\end{equation}\nwhere $Q_{Lyc}$ is the number of Lyman continuum photons emitted by\nthe region on thermal emission, $T_{e}$ is the electron temperature,\nand $L_{T}$ is the thermal luminosity. The resulting values are\ndetailed in Table \\ref{fig:sfrfractableunres} (unresolved galaxies)\nand Table \\ref{fig:sfrfractableres} (resolved galaxies). We used an\nelectron temperature of $10^{4}$K, as is typical for star-forming\nregions \\citep{C92}, and used $Q_{0} = 10^{49} s^{-1}$ as the number\nof Lyman continuum photons emitted by an O7.5V star from Table 5 in\n\\citet{Vacca96}. We report the total number of O7.5V stars in the\ngalaxies that are unresolved by the GBT at 33 GHz, and the number of\nO7.5V stars per square kiloparsec for the resolved galaxies in Tables\n\\ref{fig:sfrfractableunres} and \\ref{fig:sfrfractableres}. As seen in\nTable \\ref{fig:sfrfractableunres}, the number of O7.5V stars in each\nunresolved galaxy varies widely (log $\\#$ O7.5V stars is between 2.42\nand 4.66). This is likely due to the wide range in the unresolved\ngalaxies' overall star formation rates and physical areas observed.\n\n\\subsection{Supernova rates}\nSince we were able to fit nonthermal components for all of our\ngalaxies, we calculated supernova rates ($\\nu_{SN}$) for each of them\nfollowing Equation 18 in \\citet{C92}:\n\\begin{equation}\n\\left(\\frac{L_{N}}{10^{22}W Hz^{-1}}\\right) \\sim 13\\left(\\frac{\\nu}{GHz}\\right)^{-0.8} \\left(\\frac{\\nu_{SN}}{yr^{-1}}\\right),\n\\end{equation}\nwhere $L_{N}$ is the nonthermal luminosity. We report the total\nsupernova rate of the unresolved galaxies in Table\n\\ref{fig:sfrfractableunres}, while for the resolved galaxies we\nreport the supernova rate per square kiloparsec in Table\n\\ref{fig:sfrfractableres}. The supernova rates of the unresolved\ngalaxies vary by three orders of magnitude (log SNe rate between -3.72\nand -0.71), which is not surprising given the differences in star\nformation rates and physical areas sampled.\n\n\\subsection{Star formation rates}\nWe calculated massive star formation rates (SFRs) from thermal fluxes\nfor each galaxy whose SEDs have a thermal component and from\nnonthermal fluxes for all of our galaxies following Equations 21 and\n23 of \\citet{C92}:\n\\begin{equation}\n\\left(\\frac{L_{N}}{W Hz^{-1}}\\right) \\sim 5.3 \\times 10^{21}\n  \\left(\\frac{\\nu}{GHz}\\right)^{-0.8} \\left(\\frac{SFR_{N}\\left(M \\geq 5M_{\\sun}\\right)}{M_{\\sun} yr^{-1}}\\right)\n\\end{equation}\n\\begin{equation}\n\\left(\\frac{L_{T}}{W Hz^{-1}}\\right) \\sim 5.5 \\times 10^{20}\n  \\left(\\frac{\\nu}{GHz}\\right)^{-0.1} \\left(\\frac{SFR_{T}\\left(M \\geq 5M_{\\sun}\\right)}{M_{\\sun}yr^{-1}}\\right)\n\\end{equation}\nwhere $L_{T}$ and $L_{N}$ are thermal and nonthermal luminosities,\nrespectively, calculated from each galaxy's thermal and nonthermal\nfluxes, and $\\nu = 33$ GHz. These equations are derived from Equations\n2 and 18 of \\citet{C92} (reproduced as Equations 2 and 3 in this\npaper). Those equations were derived assuming (1) an extended\nMiller-Scalo IMF \\citep{MS79} with an exponent of $-2.5$, (2) that all\nstars with masses greater than $8 \\rm M_{\\sun}$ become supernovae, and\n(3) that dust absorption is negligible \\citep{C92}. We then scaled the\nmassive SFRs generated by each equation by a factor of 5.6 to\ntransform them to total SFRs (M $\\geq$ 0.1$M_{\\sun}$) calculated with\na Kroupa IMF \\citep{K01}. The galaxies' SFRs calculated from their\nthermal and nonthermal fluxes are shown in Table\n\\ref{fig:sfrfractableunres} (unresolved galaxies) and Table\n\\ref{fig:sfrfractableres} (resolved galaxies). We report the total\nmassive SFRs of the unresolved galaxies, while we report the massive\nSFR per square kiloparsec of the resolved galaxies. All of the\ngalaxies for which we calculated both thermal and nonthermal SFRs\nshowed agreement between the two to within an order of magnitude, but\nnot necessarily to within their margins of uncertainty. The\ndisagreement correlates with the thermal fractions of each galaxy:\ngalaxies with high thermal fractions were likely to have higher\nthermal SFRs than nonthermal SFRs, while galaxies with low thermal\nfractions showed the opposite relation. Like the differences in\nthermal fractions between galaxies in our sample, disagreement could\nbe due to the different star formation timescales traced by the\nthermal and nonthermal fluxes. Since these two emission components are\ncaused by physical processes that operate over differing lengths of\ntime (as discussed in Section 3.3.1), it is possible that the\ndiscrepancies between the star formation rates could be used to infer\nthe recent star formation histories of the observed regions.\n\nWe compared the radio continuum SFRs to monochromatic SFRs from 24$\\mu\nm$ fluxes as described in \\citet{Cal10}. The galaxies' SFRs (for the\nunresolved galaxies) and SFR densities (for the resolved galaxies)\nderived from 24$\\mu m$ fluxes are listed in Table\n\\ref{fig:sfrfractableunres} and Table \\ref{fig:sfrfractableres}. In\nFigure~\\ref{fig:comparesfrs}, we compare the SFRs derived from thermal\nand nonthermal radio continuum fluxes of the unresolved galaxies for\nwhich we fit two-component SEDs to SFRs derived from 24$\\mu m$\nfluxes. We find that most of the galaxies in our sample have higher\nradio continuum SFRs (both from thermal and nonthermal fluxes) than\nSFRs from 24$\\mu m$ data. One possible explanation for this is that\nextinction is lower at radio wavelengths than it is at 24$\\mu\nm$. Another possible explanation is that since radio continuum\nemission traces very young star formation while 24$\\mu m$ emission\ntraces less recent star formation, higher SFRs calculated from radio\ncontinuum observations than from 24$\\mu m$ data could be another\nindication that our sample of galaxies is undergoing recent star\nformation.\n\n\\subsection{Radio-far-infrared correlation}\nThere is a well-established tight correlation between far-infrared\n(FIR) and radio flux in star-forming galaxies\n\\citep[e.g.][]{H85,M06,M12}. When plotted on a log-log scale, the\nrelationship between radio continuum and FIR flux for star-forming\ngalaxies appears linear. This correlation has been well-studied at low\nfrequencies ($\\sim$ 1.4 GHz) where synchrotron emission is the\ndominant component of radio emission in a star-forming galaxy. We\ninvestigated whether this correlation could also be found on a global\nscale at 33 GHz, where synchrotron emission is weaker than it is at\n1.4 GHz and the relative contribution from thermal emission is more\nsignificant.\n\nWe limited our study of the radio-FIR correlation to the galaxies in\nour sample that are unresolved with the GBT beam at 33 GHz (as\ndiscussed in Section 2.2). We chose this limit to ensure that we were\nobserving both the total area of radio emission and total area of\nfar-infrared emission in each galaxy. This minimizes issues related to\nthe different beam sizes of the GBT and IRAS (objects are considered\npoint sources to IRAS if they are more compact than 1$\\arcmin$ at 60\n$\\rm \\mu m$ and 2$\\arcmin$ at 100 $\\rm \\mu m$). \n\nWe fit a power law to our 33 GHz flux as a function of total FIR\nflux. The total FIR flux was determined by a combination of archival\nIRAS 100 $\\rm \\mu m$ and 60 $\\rm \\mu m$ fluxes as described in\n\\citet{H88} ($S_{FIR} = 2.58 S_{60\\mu m} + S_{100 \\mu m}$). We chose\nto compute each galaxy's 33 GHz flux by taking the average of its\nfluxes in the four sub-bands. We used this measure (rather than the\nflux at 33 GHz inferred from the galaxies' SEDs) in order to eliminate\npossible uncertainties in the flux due to using assumed spectral\nindices in our fits. We found that the fluxes were related by $\\rm log\n\\ S_{33} = (0.88 \\pm 0.01) log \\ S_{FIR} + log \\ (5.3\\times 10^{-4}\n\\pm 6\\times 10^{-5})$. This correlation is relatively well-fit (the\nfractional errors of both fit parameters are small) even though our\nsample contains a wide range of thermal fractions. \\citet{M12} found a\nsimilar correlation between 33 GHz and $24 \\rm \\mu m$ fluxes for\nresolved nuclei and individual star-forming regions of galaxies. We\nfind that the radio-FIR correlation at 33 GHz can be extended to\nglobal measurements of galaxies' fluxes.\n\nAs a control of the tightness of the radio-FIR correlation in our\nsample, we also fit a relationship between the galaxies' NVSS 1.4 GHz\nfluxes and their total FIR fluxes. This relationship for the\nunresolved galaxies in our sample is $\\rm log \\ S_{1.4} = (0.85 \\pm\n0.01) log \\ S_{FIR} + log \\ (0.0047 \\pm 0.0006)$. The fractional\nuncertainties on the fit parameters are similar to those of the fit at\n33 GHz. We plot both correlations in Figure \\ref{fig:rad-IR}.\n\nAs discussed in Section 3.2, we have determined thermal fractions from\nSED fits assuming fixed thermal and nonthermal spectral indices. Due\nto the limited number of radio data points we have for each galaxy, we\ncannot more accurately constrain the thermal fractions at 33 GHz of\nthe galaxies in our sample at this time. Therefore, we do not have\nenough information to definitively isolate thermal and nonthermal\ncomponents to explore whether the radio-FIR correlation is equally\ntight for each. As an estimate, we have coded approximate thermal\nfractions in the plot. Even given these limitations, we are confident\nthat a correlation exists between the unresolved galaxies' total radio\nflux at 33 GHz and total FIR flux. \\citet{M12} found a similar\ncorrelation at 33 GHz for resolved nuclei and star-forming regions of\ngalaxies.\n\nTo further constrain the radio-FIR correlation at 33 GHz in our\nsample, we calculated $q_{\\nu}$ for each galaxy. $q_{\\nu}$ is a\nlogarithmic measure of the ratio of total far-infrared flux ($S_{FIR}$\nin Janskys) to radio continuum flux ($S_{\\nu}$) in units of $\\rm W\nm^{-2} Hz^{-1}$ at a given frequency.  It is defined in \\citet{H85} as\n\\begin{equation}\nq_{\\nu} = log\\left(\\frac{S_{FIR} \\cdot 1.26 \\times 10^{-14} W\n  m^{-2}}{3.75 \\times 10^{12} W m^{-2}}\\right) -\nlog\\left(\\frac{S_{\\nu}}{W m^{-2} Hz^{-1}}\\right).\n\\end{equation}\nThe average $q_{33}$ for our sample is $q_{33}=3.3$, with a 1$\\sigma$\nscatter of 0.3. \\citet{C92} reported that at 1.4 GHz, the average\nvalue of $q_{1.4}$ from a large sample of galaxies is $q_{1.4}=2.3 \\pm\n0.2$. The average value of $q_{\\nu}$ at 1.4 GHz for this set of\ngalaxies is $q_{1.4} = 2.4 \\pm 0.2$, consistent with the \\citet{C92}\nvalue. Since $q_{\\nu}$ is a function of the ratio of FIR flux to radio\nflux at a given frequency, it makes sense that $q_{\\nu}$ is larger\nusing 33 GHz fluxes than it is using 1.4 GHz fluxes (star-forming\ngalaxies are generally much brighter at 1.4 GHz than at 33 GHz). The\nscatter on $q_{\\nu}$ at 33 GHz is larger than that at 1.4 GHZ, which\nindicates that the radio-FIR correlation is not as tight at 33 GHz as\nat 1.4 GHz. This may be due to contamination from increased thermal\nflux at 33 GHz. If the correlation is solely between synchrotron and\nFIR emission, thermal flux at 33 GHz will increase the scatter in the\ncorrelation. However, due to our small sample size, we cannot rule out\nthe possibility that the correlation is just as strong at 33 GHz,\nwhere thermal fractions are higher, as it is at 1.4 GHz, where\nnonthermal emission is typically much stronger. We note that the\ngalaxies with the highest thermal fractions lie above the fitted\ncorrelation at 33 GHz, while the same is not true at 1.4 GHz, which\nsupports thermal emission being the cause of increased scatter.\n\nIn addition to plotting the radio-FIR correlation, we also plot the\nratio of 33 GHz flux to FIR flux, $q_{33}^{-1}$, against\n$\\alpha_{1.4-33}$ for our unresolved galaxies in Figure\n\\ref{fig:inverseqvsalpha}, similar to \\citet{M12}. The plot shows an\nincreasing $q_{33}^{-1}$ for flatter values of\n$\\alpha_{1.4-33}$. Flatter $\\alpha_{1.4-33}$ values are presumably\nindicative of a higher proportion of thermal flux to nonthermal flux,\nwhich is reflected in the highest thermal fractions in our sample also\nhaving the flattest $\\alpha_{1.4-33}$. A correlation between an\nelevated $q_{33}^{-1}$ and flat values of $\\alpha_{1.4-33}$ is not\nsurprising if the radio-FIR correlation is solely dependent on\nsynchrotron emission. If the radio-FIR correlation was independent of\nthe type of radio emission, $q_{33}^{-1}$ should be relatively\nconstant between galaxies and should not be affected by different\nspectral indices or thermal fractions. Our data support that the\nradio-FIR correlation is independent of a galaxy's thermal emission\nsince the addition of thermal emission results in elevated ratios of\n33 GHz flux to FIR flux.\n\n\\subsubsection{Implications for star formation timescales}\n\nWhen the timescales of the emission mechanisms for thermal,\nnonthermal, and FIR fluxes are taken into account, the observed\nrelationship between the ratio of 33 GHz and FIR fluxes and\n$\\alpha_{1.4-33}$ may be a way to age-date an episode of star\nformation. Since thermal flux is only produced by the shortest-lived\n($\\tau<10$ Myr) massive stars, its presence in large quantities\nrelative to synchrotron emission is indicative of very young star\nformation. Since in addition to massive stars, infrared emission also\ntraces less massive stars ($\\rm M > 5M_{\\sun}$) that live longer than\nthe $\\rm M > 8M_{\\sun}$ stars that produce thermal and nonthermal\nradio emission \\citep{D90}, FIR emission is a tracer of star formation\non longer timescales. Stars with these masses can live up to $\\sim$100\nMyr, while nonthermal radio emission traces stars with lifetimes of up\nto $\\sim$30 Myr and whose emission is detectable for up to 100 Myr\n\\citep[for an illustrative plot of stellar lifetimes, see Figure 3\n  of][]{R05}. In addition, infrared emission also contains a component\nfrom diffuse dust that is heated by lower-mass stars with lifetimes\nlonger than 100 Myr. These timescales could mean that the galaxies\nthat show both flat spectral indices and enhanced $q_{33}^{-1}$ also\nhost the youngest areas of ongoing star formation. This correlation\ncould then be a method of determining approximate ages for galaxies'\nglobal star formation. As a simple test, we used a Starburst 99 model\nof a single instantaneous burst using default inputs (solar\nmetallicity, a 2-component Kroupa IMF, and no effects of cosmic ray\naging, escape, or absorption taken into account) run for 100 Myr\n\\citep{L99,V05,L10}. This model, depicted in Figure \\ref{fig:sb99},\nshows the flattest spectral indices and highest thermal fractions at\nthe earliest times of the starburst. Similarly, the steepest spectral\nindices and lowest thermal fractions were seen as the lowest-mass\nstars that produce supernovae were dying (at $\\sim$40 Myr). The\nstarburst's ratio of 33 GHz luminosity to FIR luminosity was also high\nat early times (between 3 Myr and 40 Myr) while the lowest ratios of\n33 GHz luminosity to FIR luminosity were seen even later (after 40\nMyr). While modeling a more robust quantitative relationship between\nthis observed correlation and the age of each galaxy's star-forming\nepisode is beyond the scope of this work (the simple model we used\ndoes not take into account multiple co-existing generations of star\nformation), the apparent relationship between enhanced 33 GHz flux,\nflat spectral indices, and high thermal fractions is a promising\nmetric for future global radio and far-infrared photometric studies of\nstar-forming galaxies. Our simple model is not robust enough to\nconstrain the timescales' uncertainties, but is only meant to be\nillustrative of a correlation visible in our data.\n\n\n\\section{Conclusions}\n\nWe have observed 27 local, well-studied, star-forming galaxies between\n26-40 GHz with the GBT and obtained the first detections at this\nfrequency range for 22 of the galaxies. We determined the\ncontributions of thermal free-free and nonthermal synchrotron emission\nto the galaxies' total radio emission. We have used these measures to\nderive the number of massive, short-lived O stars and the number of\nrecent supernovae in the observed regions of each galaxy. In addition,\nwe have calculated SFRs for each galaxy using thermal and nonthermal\nfluxes and explored the radio-FIR correlation for the unresolved\ngalaxies. We found that\n\\begin{itemize}\n\\item None of the galaxies have spectral incides indicative of purely\n  thermal emission; eight galaxies show spectra that are too\n    steep to fit thermal components,\n\\item Thermal fractions range from 10$\\%$ to 90$\\%$, with a\n  median of $55\\%$,\n\\item The radio-far infrared correlation holds for the unresolved\n  galaxies at 1.4 GHz and 33 GHz, though the scatter at 33 GHz is\n  larger due to the increased influence of thermal emission at higher\n  frequencies, and\n\\item Galaxies with flat $\\alpha_{1.4-33}$ and high thermal fractions\n  have enhanced radio flux at 33 GHz with respect to far-infrared\n  flux, which identifies them as galaxies with recent star\n  formation. This is consistent with a simple model of a single\n  starburst.\n\\end{itemize}\n\nWe found that the observed regions of our galaxies had a diverse mix\nof radio continuum characteristics, with some galaxies' SEDs being\ndominated at 33 GHz by the thermal emission indicative of ongoing\nmassive star formation, while others have little or no detectable\nthermal emission. Even with this spread in the relative contributions\nof thermal and nonthermal emission, we saw that there is still a\ncorrelation between the global 33 GHz and far-infrared flux in the\nunresolved galaxies. The scatter in the correlation is larger than\nthat at 1.4 GHz, likely due to the increased influence of thermal\nemission at 33 GHz. We cannot, however, rule out that the radio-FIR\ncorrelation is not solely dependent on synchrotron emission. We also\nfound that higher ratios of 33 GHz emission to FIR emission correlated\nwith flatter spectral indices (and higher thermal fractions) for\nunresolved galaxies, which is consistent with younger ages in simple\nstarburst models. This correlation may be useful as a rough indicator\nof the age of the most recent episode of star formation. Future global\nstudies of more homogeneous galaxy populations or resolved studies of\nindividual star-forming regions will enable better modeling of star\nformation timescales using this metric.\n\nIn giving a broad measure of nearby galaxies' radio continuum\nemission, our observations complement previous studies done with\ninterferometers in which individual star-forming regions in local\ngalaxies were highly resolved. With the GBT, we can simultaneously\nobserve compact and diffuse thermal and nonthermal emission and\ndetermine their relative intensities, and in doing so estimate the\ntimescale for the current episode of star formation. Unfortunately, we\ncannot make stricter timescale estimates than those discussed in\nSection 3.3 at this time, as we do not have enough radio data points\nto robustly fit thermal and nonthermal flux components with varying\nspectral indices. Obtaining more unresolved radio fluxes at lower and\nhigher frequencies would help this effort.\n\nThis research has made use of the NASA\/IPAC Extragalactic Database\n(NED) which is operated by the Jet Propulsion Laboratory, California\nInstitute of Technology, under contract with the National Aeronautics\nand Space Administration. We acknowledge the use of NASA's SkyView\nfacility (http:\/\/skyview.gsfc.nasa.gov) located at NASA Goddard Space\nFlight Center. We thank the telescope operators and support staff at\nthe GBT for assistance with this project. K.R. acknowledges support\nfrom an NRAO student observing support award (GSSP10-0002). K.R. also\nthanks Brian Mason for his help with understanding the CCB observation\nand data reduction process.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Appendix A. <some heading here>}\n\n\\widetext\n\\section*{\\large Supplemental material for stacking-induced Chern insulator}\n\n\\section{I. Haldane model in bilayer honeycomb lattice}\n\nWe consider a HM bilayer with AB and AA stackings where the layers are assumed to have complex NNN phases $\\Phi_1=-\\Phi_2=\\frac{\\pi}2$, to drop the global energy shift ($a^0_{\\mathbf{k}}=0$ (Eq.~\\ref{al})) which does not affect the topology of the system.\nIn the basis of the four orbitals forming the unit cell ($A_1,B_1,A_2,B_2$) the corresponding Hamiltonians can be written as\n\\begin{eqnarray}\n H_\\text{AA-HM}(\\mathbf{k})=\n\\begin{pmatrix}\na_{\\mathbf{k}} +M_1 & f_{\\mathbf{k}} &2t_{\\perp}&0 \\\\\nf^{\\ast}_{\\mathbf{k}} & -a_{\\mathbf{k}} -M_1&0&2t_{\\perp}\\\\\n2t_{\\perp}&0&-a_{\\mathbf{k}} +M_2 & f_{\\mathbf{k}}\\\\\n0&2t_{\\perp}&f^{\\ast}_{\\mathbf{k}} & a_{\\mathbf{k} -M2}.\n\\end{pmatrix}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n H_\\text{AB-HM}(\\mathbf{k})=\n\\begin{pmatrix}\na_{\\mathbf{k}} +M_1 & f_{\\mathbf{k}} &0 &2t_{\\perp} \\\\\nf^{\\ast}_{\\mathbf{k}} & -a_{\\mathbf{k}} -M_1&0&0\\\\\n0&0&-a_{\\mathbf{k}} +M_2 & f_{\\mathbf{k}}\\\\\n2t_{\\perp}&0&f^{\\ast}_{\\mathbf{k}} & a_{\\mathbf{k} -M2}.\n\\end{pmatrix}\n\\end{eqnarray}\n\nThese Hamiltonians can be expressed, using the layer and the sublattice pseudospin matrices $\\boldsymbol{\\sigma}$ and $\\boldsymbol{\\tau}$, as\n\\begin{eqnarray}\nH_\\text{AA-HM}(\\mathbf{k})&=&\\left( b_{\\mathbf{k}}\\sigma_x+c_{\\mathbf{k}}\\sigma_y\\right)\\tau_0\n+2t_{\\perp}\\sigma0\\tau_x+ a_{\\mathbf{k}}\\sigma_z\\tau_z\n+\\frac 12 \\left(M_1+M_2\\right)\\sigma_z\\tau_0+\\frac 12 \\left(M_1-M_2\\right)\\sigma_z\\tau_z,\\\\\nH_\\text{AB-HM}(\\mathbf{k})&=&\\left( b_{\\mathbf{k}}\\sigma_x+c_{\\mathbf{k}}\\sigma_y\\right)\\tau_0\n+t_{\\perp}\\left(\\sigma_x\\tau_x-\\sigma_y\\tau_y\\right)+ a_{\\mathbf{k}}\\sigma_z\\tau_z\n+\\frac 12 \\left(M_1+M_2\\right)\\sigma_z\\tau_0+\\frac 12 \\left(M_1-M_2\\right)\\sigma_z\\tau_z.\n\\label{AB-HM-mass}\n\\end{eqnarray}\nwhere $a_{\\mathbf{k}}$ is given by Eq.~\\ref{al} in the main text.\\\n\n$H_\\text{AA-HM}$ ($H_\\text{AB-HM}$) breaks TRS $\\mathcal{T}=K$, the charge conjugation, represented by $\\mathcal{C}=\\sigma_z\\tau_zK$ ($\\mathcal{C}=\\sigma_z\\tau_0K$) with $\\mathcal{C}^2=\\mathds{1}$, and the chirality $\\mathcal{S}=\\tau_z\\sigma_z$ ($\\mathcal{S}=\\tau_0\\sigma_z$).\\\n\nIn the following, we will show based on numerical band structure calculations on bilayer ribbons, that coupling two HM with opposite chiralities ($C_1=-C_2$), resulting from oppositely broken TRS ($\\Phi_1=-\\Phi_2$), gives rise, as expected, to a trivial Chern insulator with $C=C_1+C_2=0$.\\\nWe will discuss the stacking order, the nature of the ribbon edges (zigzag or armchair) and the effect of the intralayer Semenoff masses $M_l$, where $l=1,2$ is the layer index. The case of AA stacking was discussed in Ref.~\\onlinecite{Dutta} for a fixed value of the mass term $M_l$.\\\n\nFigure~\\ref{band-HM-ZZ} shows the band structure of the AB bilayer HM on zigzag ribbons for $\\Phi_1=\\Phi_2= \\frac{\\pi}2$, $M_1=M_2=0$ and at different values of the interlayer hopping $t_{\\perp}$ . Starting from uncoupled ($t_{\\perp}=0$) chiral layers, with equal Chern number $C_1=C_2=\\pm 1$, the system turns, under the interlayer coupling, into a Chern insulator with a Chern number $C=\\pm 2$ characterized by a pair of chiral edge states propagating at the boundaries of each layer as shown in Fig.~\\ref{table-res} of the main text.\n \n\\begin{figure}[hpbt] \n\\begin{center}\n$\n\\begin{array}{ccc}\n\\includegraphics[width=0.3\\columnwidth]{HM-AB-ZZ-a.eps}\n\\includegraphics[width=0.3\\columnwidth]{HM-AB-ZZ-b.eps}\n\\includegraphics[width=0.3\\columnwidth]{HM-AB-ZZ-c.eps}\n\\end{array}\n$\n\\end{center}\n\\caption{Tight binding calculations of the electronic band structure of an AB bilayer HM on zigzag nanoribbons of a width $W = 60$ atoms. The interlayer hopping is (a) $t_{\\perp}=0$, \n(b) $t_{\\perp}=0.5t$ and (c) $t_{\\perp}=0.8t$.\nCalculations are done for $\\Phi_1=\\Phi_2= \\frac{\\pi}2$, $M_1=M_2=0$ and $t_2 = 0.1t$, where $t$ is the NN hopping integral.}\n\\label{band-HM-ZZ}\n\\end{figure}\nThe $C=\\pm 2$ Chern insulating phase occurs as far as the Semenoff mass $|M_l|<|M_{lc}|$, where\n\\begin{eqnarray}\nM_{lc}=3\\sqrt{3}t_2\\sin \\Phi_l.\n\\label{Mlc}\n\\end{eqnarray}\n$M_{lc}$ is the critical mass at which the transition from a topological phase ($C_l=\\pm 1$) to a trivial gapped phase ($C_l=0$), takes place in the monolayer HM~\\cite{Haldane}.\\\n\nThis feature is depicted in Fig.~\\ref{band-HM-mass}, showing that the pair of chiral edge states appears at the boundaries of the bilayer ribbons only if both layers have the same chirality~\\cite{Haldane}.\\\n\n\\begin{figure}[hpbt] \n$\n\\begin{array}{ccc}\n\\includegraphics[width=0.3\\columnwidth]{HM_AB_mass_a.eps}\n\\includegraphics[width=0.3\\columnwidth]{HM_AB_mass_b.eps}&\\includegraphics[width=0.3\\columnwidth]{HM_AB_mass_c.eps}\\\\\n\\includegraphics[width=0.3\\columnwidth]{HM_AB_mass_d.eps}\n\\includegraphics[width=0.3\\columnwidth]{HM_AB_mass_e.eps}&\n\\includegraphics[width=0.3\\columnwidth]{HM_AB_mass_f.eps}&\n\\end{array}\n$\n\\begin{center}\n\n\\end{center}\n\\caption{Electronic band structure of an AB bilayer HM on zigzag\nnanoribbons of a width $W = 60$ atoms. Calculations are done for $t_2 = 0.1t$, $\\Phi_1=\\Phi_2= \\frac{\\pi}2$, $t_{\\perp}=0.5t$ and for (a) $M_1=M_2=0$, (b) $M_1=M_2=\\sqrt{3}t_2$, (c) $M_1=-M_2=\\sqrt{3}t_2$,\n(d) $M_1=0, M_2=3\\sqrt{3}t_2$, (e) $M_1=0, M_2=5\\sqrt{3}t_2$, (f) $M_1=5\\sqrt{3}t_2, M_2=5\\sqrt{3}t_2$.}\n\\label{band-HM-mass}\n\\end{figure}\n\nThis behavior is independent of the nature (zigzag or armchair) of the ribbon boundaries as shown in Fig.~\\ref{band-HM-AC}.\n\n\\begin{figure}[hpbt] \n\\begin{center}\n$\n\\begin{array}{cc}\n\\includegraphics[width=0.3\\columnwidth]{HM-AB-ZZ-b.eps}\n\\includegraphics[width=0.3\\columnwidth]{HM_AC_AB_b.eps}\n\\end{array}\n$\n\\end{center}\n\\caption{Electronic band structure of an AB bilayer HM on (a) zigzag and (b) armchair nanoribbons of a width $W = 60$ atoms. \nCalculations are done for $t_2 = 0.1t$, $t_{\\perp} = 0.5t$, $\\Phi_1=\\Phi_2=\\frac{\\pi}2$ and\n$M_1=M_2=0$.}\n\\label{band-HM-AC}\n\\end{figure}\n\nRegardless of the stacking type (AB or AA), the bilayer HM is~\\cite{Dutta}:\n$(i)$ a trivial insulator, if the layers have opposite Chern numbers $C_1=-C_2$,\n$(ii)$ a topological chiral insulator with $C=\\pm 2$, if the layers have the same chirality ($C_1=C_2$),\n$(iii)$ and a Chern insulator with $C=\\pm1$ if one layer has a non-vanishing Chern number $C_1=\\pm 1$ and the other layer is a trivial insulator $C_2=0$, as depicted in Fig.~\\ref{band-HM-AA} showing the band structure of an AA bilayer HM on zigzag ribbons.\n\\begin{figure}[hpbt] \n\\begin{center}\n$\n\\begin{array}{cc}\n\\includegraphics[width=0.3\\columnwidth]{HM-AA-a.eps}\n\\includegraphics[width=0.3\\columnwidth]{HM-AA-b.eps}\\\\\n\\includegraphics[width=0.3\\columnwidth]{HM-AA-c.eps}\n\\includegraphics[width=0.3\\columnwidth]{HM-AA-d.eps}\n\\end{array}\n$\n\\end{center}\n\\caption{Electronic band structure of an AA Bilayer HM on zigzag\nnanoribbons of a width $W = 60$ atoms for $t_{\\perp} = 0.5t$,\n(a) $\\Phi_1=\\Phi_2=\\frac{\\pi}2$, $M_1=M_2=0$, (b) $\\Phi_1=\\Phi_2=-\\frac{\\pi}2$, $M_1=M_2=0$, (c) $\\Phi_1=\\Phi_2=\\frac{\\pi}2$, $M_1=0$, $M_2=5\\sqrt{3}t_2$ and\n(d) $\\Phi_1=\\frac{\\pi}2, \\Phi_2=0$,  $M_1=\\sqrt{3}t_2$, $M_2=0$. Calculations are done for\n$t_2=0.1t$ in (a), (b) and (d) and $t_2=0.2t$ in (c).}\n\\label{band-HM-AA}\n\\end{figure}\n\n\n\\section{ II. Modified Haldane model in AB stacked layers}\n\nTo derive the low energy Hamiltonian given by Eq.~\\ref{Heff} in the main text, we use the \nL\\\"owdin partitioning method~\\cite{Lowdin,McCann} in the case of bilayer graphene. \nFor simplicity, we consider the case $\\Phi_1=-\\Phi_2=\\frac{\\pi}2$ to remove the energy-shift terms $a^0_{l,\\mathbf{k}}$ (Eq.~\\ref{al}). We rewrite the full Hamiltonian (Eq.~\\ref{HBL}), in the basis ($A_2,B_1,A_1,B_2$) as\n\\begin{eqnarray}\n H_{B}(\\mathbf{k})=\n\\begin{pmatrix}\nH_{\\alpha\\alpha} & H_{\\alpha\\beta} \\\\\nH_{\\beta\\alpha} & H_{\\beta\\beta}\n\\end{pmatrix},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nH_{\\alpha\\alpha}=\n\\begin{pmatrix}\n -a_{\\mathbf{k}}+M_2& 0 \\\\\n0 & a_{\\mathbf{k}}-M1\n\\end{pmatrix},\\,\nH_{\\alpha\\beta}=H_{\\beta\\alpha}\n\\begin{pmatrix}\n0&f_{\\mathbf{k}} \\\\\nf^{\\ast}_{\\mathbf{k}} & 0\n\\end{pmatrix},\\,\nH_{\\beta\\beta}=\n\\begin{pmatrix}\n a_{\\mathbf{k}}+M_1& t_{\\perp} \\\\\n t_{\\perp}& -a_{\\mathbf{k}}-M_2\n\\end{pmatrix}.\n\\end{eqnarray}\nThe corresponding effective Hamiltonian is~\\cite{McCann}\n\n\\begin{eqnarray}\n H_\\text{eff}({\\mathbf{k}},E)=H_{\\alpha\\alpha}+H_{\\alpha\\beta}\\left( E-H_{\\beta\\beta}\\right)^{-1} H_{\\beta\\alpha},\n\\end{eqnarray}\nwhich reduces in the limit $M_l\\sim t_2\\ll t_{\\perp}$ ($l=1,2$), and for $E\\sim 0$ to\n\\begin{eqnarray}\nH_\\text{eff}({\\mathbf{k}},E=0)\\equiv H_\\text{eff}({\\mathbf{k}})\\sim H_{\\alpha\\alpha}-\\frac 1{X^2}H_{\\alpha\\beta}H_{\\beta\\beta} H_{\\beta\\alpha},\n\\end{eqnarray}\nwhere $X^2=\\left(a_{\\mathbf{k}}+M_1\\right)\\left(a_{\\mathbf{k}}+M_2\\right)+4t^2_{\\perp}$.\\\n\nAssuming $M_l\\frac{|f_{\\mathbf{k}}|^2}{X^2}\\ll M_{l^{\\prime}}\\sim t_2 \\, (l,l^{\\prime}=1,2)$, the corresponding effective Hamiltonian gives rise to Eq.~\\ref{Heff} of the main text.\n\n\\section{III. Modified Haldane model in AA stacked layers}\n\n\\begin{figure}[hpbt] \n$\n\\begin{array}{cc}\n\\includegraphics[width=0.3\\columnwidth]{mHM-AA-ZZ-a.eps}\n\\includegraphics[width=0.3\\columnwidth]{mHM-AA-AC-b.eps}\\\\\n\\includegraphics[width=0.3\\columnwidth]{mHM-AA-ZZ-c.eps}\n\\includegraphics[width=0.3\\columnwidth]{mHM-AA-ZZ-d.eps}\n\\end{array}\n$\n\\begin{center}\n\\end{center}\n\\caption{Band structure of the mHM on AA nanoribbons of a width of $W=60$ atoms with  (a,c,d) zigzag and (b) armchair boundaries. Calculations are done for $t_2 = 0.1t$, $t_{\\perp}=0.5t$, (a) and (b) $\\Phi_1=-\\Phi_2= \\frac{\\pi}2$, $M_1=M_2=0$, while in (c)  $\\Phi_1=-\\Phi_2= \\frac{\\pi}2$, $M_1=-M_2=\\sqrt{3}t_2$ and (d) $\\Phi_1=-\\Phi_2= \\frac{\\pi}2$, $M_1=0$, $M_2=\\sqrt{3}t_2$. }\n\\label{AA}\n\\end{figure}\nFigure~\\ref{AA} shows the band structure of the mHM in AA stacked ribbons with zigzag and armchair boundaries in the case of opposite complex phases $\\Phi_1=-\\Phi_2$.\nIn the absence of the Semenoff masses ($M_1=M_2=0$), the system remains gapless under the interlayer coupling. However, it turns to a trivial insulator if the layers have Semenoff mass terms. \\\n\nTherefore, in the absence of the Semenoff masses, the Fermi surface (Fig.~\\ref{Fig-intro}) of the mHM in AA stacked bilayer is,\ncontrary to the AB stacking, stable against the interlayer hopping which cannot induce a gap opening.\\\n\nTo understand the Fermi surface stability, we start by writing the corresponding Hamiltonian in the basis of the four orbitals forming the unit cell ($A_1,B_1,A_2,B_2$) and we consider, for simplicity, the case of opposite complex NNN phases $\\Phi_1=-\\Phi_2=\\frac{\\pi}2$ to have a vanishing global energy shift ($a^0_{\\mathbf{k}}=0$ (Eq.~\\ref{al}))\n\\begin{eqnarray}\n H_{AA-mHM}(\\mathbf{k})=\n\\begin{pmatrix}\na_{\\mathbf{k}} & f_{\\mathbf{k}} &2t_{\\perp}&0 \\\\\nf^{\\ast}_{\\mathbf{k}} & a_{\\mathbf{k}} &0&2t_{\\perp}\\\\\n2t_{\\perp}&0&-a_{\\mathbf{k}} & f_{\\mathbf{k}}\\\\\n0&2t_{\\perp}&f^{\\ast}_{\\mathbf{k}} & -a_{\\mathbf{k}}\n\\end{pmatrix}.\n\\label{AA-mHM}\n\\end{eqnarray}\nThis Hamiltonian can be written, using the layer and the sublattice pseudospin matrices $\\boldsymbol{\\sigma}$ and $\\boldsymbol{\\tau}$, as\n\\begin{eqnarray}\nH_\\text{AA-mHM}(\\mathbf{k})&=&\\left( b_{\\mathbf{k}}\\sigma_x+c_{\\mathbf{k}}\\sigma_y\\right)\\tau_0\n+2t_{\\perp}\\sigma_0\\tau_x+ a_{\\mathbf{k}}\\sigma_0\\tau_z,\n\\label{HBLAA}\n\\end{eqnarray}\nwhere $a_{\\mathbf{k}}$ is given by Eq.~\\ref{al} in the main text.\\\n\nThe Hamiltonian of Eq.~\\ref{HBLAA} breaks TRS, $\\mathcal{T}=K\\tau_x$, the charge conjugation, represented by $\\mathcal{C}=\\sigma_z\\tau_zK$ with $\\mathcal{C}^2=\\mathds{1}$, and the chirality $\\mathcal{S}=\\tau_z\\sigma_z$.\\\n\nThe gap separating the two bands, $E_{-,-}(\\mathbf{k})$ and $E_{+,-}(\\mathbf{k})$, around the zero energy is $\\Delta=\\mathrm{min}_{\\mathbf{k}}\\left(\\Delta_{\\mathbf{k}}\\right)$, where \n\\begin{eqnarray}\n \\Delta_{\\mathbf{k}}=2\\sqrt{A_{\\mathbf{k}}-B_{\\mathbf{k}}},\\;\n A_{\\mathbf{k}}=a^2_{\\mathbf{k}}+|f_{\\mathbf{k}}|^2+4t^2_{\\perp},\\; B_{\\mathbf{k}}=2|f_{\\mathbf{k}}|\\sqrt{a^2_{\\mathbf{k}}+4t^2_{\\perp}}\n\\end{eqnarray}\n$\\Delta_{\\mathbf{k}}=0$ leads to\n\\begin{eqnarray}\n|f_{\\mathbf{k}}|^2=a^2_{\\mathbf{k}}+4t^2_{\\perp},\n\\label{FL}\n\\end{eqnarray}\nwhich defines a closed Fermi line. \\\n\nFor $a_{\\mathbf{k}}=0$, Eq.~\\ref{FL} corresponds to the Fermi line of the AA graphene bilayer in the absence of NNN hopping terms. \\\n\nFor $t_{\\perp}=0$, Eq.~\\ref{FL} describes the mHM in AA bilayer with a particle-hole Fermi line obeying to $|f_{\\mathbf{k}}|=|a_{\\mathbf{k}}|$.\\\n\nBy turning on $t_{\\perp}$, this Fermi line is, simply, shifted but cannot be gapped (Eq.~\\ref{FL}).\nThe mHM on AA bilayer remains, then, metallic for vanishing Semenoff masses.\\\n\n\n\\section{IV. Modified Haldane model in AB stacked layers: effect of Semenoff masses}\n\nThe effect of Semenoff mass on the mHM in monolayer graphene nanoribbon is represented in Fig.~ \\ref{supp-mass-ML} showing that the mass term lifts the degeneracy of the antichiral edge modes which\nsurvive as far as $M<M_c\\equiv 3\\sqrt{3}t_2\\sin \\Phi$.\n\n\\begin{figure}[hpbt] \n\\begin{center}\n$\\begin{array}{cccc}\n\\includegraphics[width=0.2\\columnwidth]{mHM-ML-mass-a.eps}\n\\includegraphics[width=0.2\\columnwidth]{mHM-ML-mass-b.eps}\n\\includegraphics[width=0.2\\columnwidth]{mHM-ML-mass-c.eps}\n\\includegraphics[width=0.2\\columnwidth]{mHM-ML-mass-d.eps}\n\\end{array}$\n\\end{center}\n\\caption{Band structure of the mHM in monolayer graphene nanoribbon with zigzag boundaries and a width $W=60$ atoms. Calculations are done for $t_2=0.1t$, $\\Phi=\\frac{\\pi}2$, (a) $M=0$, (b) $M=0.1t_2$, (c) $M=M_c\\equiv 3\\sqrt{3}t_2$, and (d) $M_2=4\\sqrt{3}t_2$.}\n\\label{supp-mass-ML}\n\\end{figure}\n\nIn Fig.~\\ref{mass}, we plot the band structure of the mHM on AB bilayer honeycomb lattices with different choices of intralayer Semenoff masses.\n\n\\begin{figure}[hpbt] \n\\begin{center}\n$\\begin{array}{cccc}\n\\includegraphics[width=0.2\\columnwidth]{mHM-supp-mass-a.eps}\n\\includegraphics[width=0.2\\columnwidth]{mHM-supp-mass-b.eps}\n\\includegraphics[width=0.2\\columnwidth]{mHM-supp-mass-c.eps}\n\\includegraphics[width=0.2\\columnwidth]{mHM-supp-mass-d.eps}\\\\\n\\includegraphics[width=0.2\\columnwidth]{mHM-supp-mass-e.eps}\n\\includegraphics[width=0.2\\columnwidth]{mHM-supp-mass-f.eps}\n\\includegraphics[width=0.2\\columnwidth]{mHM-supp-mass-g.eps}\n\\includegraphics[width=0.2\\columnwidth]{mHM-supp-mass-h.eps}\n\\end{array}$\n\\end{center}\n\\caption{Band structure of the mHM in AB stacked bilayer zigzag ribbons of a width $W=60$ atoms with $t_2=0.1t$, $t_{\\perp}=0.5 t$, $\\Phi_1=-\\Phi_2=\\frac{\\pi}2$, (a) $M_1=M_2=0$, (b) $M_1=0$, $M_2=t_2$, (c) $M_1=0$, $M_2=3\\sqrt{3}t_2$ (d) $M_1=0$, $M_2=6\\sqrt{3}t_2$, (e) $M_1=M_2=\\sqrt{3}t_2$, (f) $M_1=-M_2=\\sqrt{3}t_2$, (g) $M_1=M_2=3\\sqrt{3}t_2$, (h) $M_1=M_2=4\\sqrt{3}t_2$.}\n\\label{mass}\n\\end{figure}\n\nFigure \\ref{mass} shows that, in AB bilayer mHM, the chiral edge states emerge as far as $M_l<M_{lc}$ (Eq.~\\ref{Mlc}), for which the nodal lines, originating from the overlap of the two layer bands, can occur, as discussed in the main text \\ref{Fig-intro}.\n\n\\section{ V. Modified Haldane model in AB stacked layers: effect of the NNN hopping complex phase}\n\nIt is noteworthy that the  induced Chern insulator in the AB bilayer mHM occurs as far as $\\Phi_1$ and $\\Phi_2$ have opposite signs and not only in the case where $\\Phi_1=-\\Phi_2=\\frac{\\pi}2$, which we considered to obtain simple analytical expressions. This feature is shown in Fig.~\\ref{phi}.\n\\begin{figure}[hpbt] \n\\begin{center}\n$\\begin{array}{c}\n\\includegraphics[width=0.3\\columnwidth]{mHM-phi-a.eps}\n\\includegraphics[width=0.3\\columnwidth]{mHM-phi-b.eps}\n\\includegraphics[width=0.3\\columnwidth]{mHM-phi-c.eps}\n\\end{array}$\n\\end{center}\n\\caption{Band structure of the mHM in AB stacked bilayer ribbons of a width $W=60$ atoms, for $t_2=0.1t$, $t_{\\perp}=0.5 t$, $M_1=M_2=0$ and (a) $\\Phi_1=-\\Phi_2=\\frac{\\pi}3$, $\\Phi_1=\\frac{\\pi}3$, $\\Phi_2=-\\frac{\\pi}4$ and (c) $\\Phi_1=0,\\Phi_2=-\\frac{\\pi}2$.}\n\\label{phi}\n\\end{figure}\n\\section{VI. Modified Haldane model in bilayer graphene: from AB to BA stacking \\label{AB-BA}}\n\nWe consider a generic Hamiltonian that, continuously, interpolates between AB $(\\theta=0)$ and BA $(\\theta=\\pi\/2)$ stacking of two single graphene layers described by the mHM. The interlayer hopping terms, for a given value of $\\theta$ between these limits, does not represent physical couplings.\nWe take, for simplicity, vanishing Semenoff masses and inplane complex phases $\\Phi_1=-\\Phi_2=\\frac{\\pi}2$. The Bloch Hamiltonian written in the basis ($A_1,B_1,A_2,B_2$) is\n\\begin{eqnarray}\n H_{B}(\\mathbf{k},\\theta)=\n\\begin{pmatrix}\na_{\\mathbf{k}}& f_{\\mathbf{k}} &0& 2t_{\\perp} c_{\\theta}\\\\\nf^{\\ast}_{\\mathbf{k}} & a_{\\mathbf{k}} & 2t_{\\perp} s_{\\theta} &0\\\\\n0&2t_{\\perp} s_{\\theta}&-a_{\\mathbf{k}}& f_{\\mathbf{k}}\\\\\n2t_{\\perp} c_{\\theta}&0&f^{\\ast}_{\\mathbf{k}} & -a_{\\mathbf{k}}\n\\end{pmatrix}\n\\label{HBLtheta}\n\\end{eqnarray}\nwhere $f_{\\mathbf{k}}=t\\sum_{i=1}^3 e^{i\\mathbf{k}\\cdot\\boldsymbol{\\delta}_i}$ and $a_{\\mathbf{k}}=-2t_2\\sum_{i=1}^3  \\sin\\left(\\mathbf{k}\\cdot\\mathbf{a}_i\\right)$. \\\n\nThe AB (BA) stacking corresponds to $c_{\\theta}\\equiv\\cos\\theta=1$ and $s_{\\theta}\\equiv\\sin\\theta=0$ ($c_{\\theta}=0$, $s_{\\theta}=1$). \\\n\nBy varying $\\theta$, the system can be interpolated between the two stacking configurations, without going through the AA stacking.\\\nUsing the intralayer and interlayer pseudo-spin matrices $\\boldsymbol{\\sigma}$ and $\\boldsymbol{\\tau}$, the Bloch Hamiltonian of Eq.~\\ref{HBLtheta} becomes\n\n\\begin{eqnarray}\nH_{B}(\\mathbf{k},\\theta)=\\left( b_{\\mathbf{k}}\\sigma_x+c_{\\mathbf{k}}\\sigma_y\\right)\\tau_0\n+2t_{\\perp}\\left[\\left(c_{\\theta}\\sigma_++s_{\\theta}\\sigma_-\\right)\\tau_+\n+2t_{\\perp}\\left(c_{\\theta}\\sigma_-+s_{\\theta}\\sigma_+\\right)\\tau_-\\right],\n\\label{HBL2theta}\n\\end{eqnarray}\nwhere $b_{\\mathbf{k}}$, $c_{\\mathbf{k}}$, $\\sigma_{\\pm}$ and $\\tau_{\\pm}$ are given in the main text.\\\n\nThe Hamiltonian of Eq.~\\ref{HBL2theta} breaks TRS, chirality but is invariant under charge-conjugation and inversion since $M_l=0$ and $\\Phi_1=-\\Phi_2=\\frac{\\pi}2$.\\\nThe corresponding energy spectrum is expressed, as in Eq.~\\ref{E+-} of the main text with\n\\begin{eqnarray}\n A_{\\mathbf{k}}&=&a^2_{\\mathbf{k}}+|f_{\\mathbf{k}}|^2+2t_{\\perp}^2,\\nonumber\\\\\n B_{\\mathbf{k}}&=&2\\sqrt{|f_{\\mathbf{k}}|^2\\left(a^2_{\\mathbf{k}}+t_{\\perp}^2\\right)+c^2_{2\\theta}t_{\\perp}^4+s_{2\\theta}t_{\\perp}^2\\left( b^2_{\\mathbf{k}}-c^2_{\\mathbf{k}}\\right)}\\nonumber\\\\\n \\label{ABktheta}\n\\end{eqnarray}\nThe gap separating the lowest energy band around zero energy is $\\Delta=\\min_{\\mathbf{k}}\\left(\\Delta_{\\mathbf{k}}\\right)=2\\sqrt{A_{\\mathbf{k}}-B_{\\mathbf{k}}}$. \\\n\n$\\Delta_{\\mathbf{k}}$ closes, for uncoupled layers ($t_{\\perp}=0$), for $A_{\\mathbf{k}}=B_{\\mathbf{k}}$ which defines, as we have seen in the main text, two non-intersecting closed Fermi lines.\nFor a non vanishing interlayer hopping $t_{\\perp}$, $\\Delta_{\\mathbf{k}}$ can be expressed as\n\n\\begin{eqnarray}\n\\Delta_{\\mathbf{k}}=2\\frac{\\sqrt{A^2_{\\mathbf{k}}-B^2_{\\mathbf{k}}}}{A_{\\mathbf{k}}+B_{\\mathbf{k}}}\n\\sim \\frac{2t_{\\perp}}{|f_{\\mathbf{k}}|}\\sqrt{\\left(1-s_{2\\theta}\\right)b^2_{\\mathbf{k}}+\\left(\\lambda^2_{\\mathbf{k}}-c^2_{2\\theta} \\right)t_{\\perp}^2},\n\\label{Delta}\n\\end{eqnarray} \nwhere we introduced the parameter $\\lambda_{\\mathbf{k}}$ defined as \n\\begin{eqnarray}\na^2_{\\mathbf{k}}=|f_{\\mathbf{k}}|^2+\\left(2\\lambda_{\\mathbf{k}}-1\\right)t_{\\perp}^2\n\\label{lambda}\n\\end{eqnarray}\nIn Eq.~\\ref{Delta}, the numerator is given by its exact expression, whereas the denominator has been approximated to the zeroth order in $t_{\\perp}$.\n\nAccording to Eq.~\\ref{Delta}, $\\Delta_{\\mathbf{k}}$ cannot vanish for $\\theta\\neq \\pi\/4,3\\pi\/4$. \\\n\nFor $\\theta=\\pi\/4 $, $\\Delta_{\\mathbf{k}}=0$ if $\\lambda_{\\mathbf{k}}=0$, which give rise, according to Eq.~\\ref{lambda}, to a closed loop defined by $a^2_{\\mathbf{k}}+t_{\\perp}^2=|f_{\\mathbf{k}}|^2$.\\\n\nFor $\\theta=3\\pi\/4$, the closing of the gap $\\Delta_{\\mathbf{k}}$ requiers $\\lambda_{\\mathbf{k}}=0$ and $b_{\\mathbf{k}}=0$.\\\n\nThe critical values $\\theta=\\pi\/4,3\\pi\/4 $ correspond to the semimetallic phase marking the transition from the topological Chern insulator phase $C=2$, occurring for $0\\le\\theta<\\pi\/4 $ to the Chern insulator phase $C=-2$, associated to $3\\pi\/4<\\theta\\le\\pi $.\nAt this topological phase transition, the gap closes at the four Dirac points of the bilayer system where, right after the transition, the signs of the gap flip, inducing a variation $\\Delta C=-4$ of the Chern number.\n\nThe flipping of the Chern number sign, at the crossing form the AB to BA stacking, could be understood from Fig.~\\ref{AB-AA}. Such crossing can be regarded as a sign change of the complex phases: since the inplane sublattices have opposite fluxes, the AB stacking corresponds to the dimer ($A_1$,$B_2$) for which the complex phases are $\\Phi_1=-\\Phi_2=\\pi\/2$ while the BA stacking is ascribed to the dimer ($B_1$,$A_2$) with $\\Phi_1=-\\Phi_2=-\\pi\/2$ (Fig.~\\ref{AB-AA} of the main text).\n\n\n\\end{document}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \nHigh energy heavy ion collisions at RHIC and the LHC have created a new form of nuclear matter comprised of deconfined, yet strongly interacting quarks and gluons. This matter exhibits strong collective and anisotropic flow in the transverse plane, which is well described by relativistic hydrodynamics~\\cite{Alver:2010dn,Gale:2012in,Heinz:2013th}. The magnitude of the anisotropic flow has been to found to be sensitive to the transport properties, as well as the space-momentum profile in the initial state. A central goal of current research is to understand the nature of various fluctuations in the initial state and how these fluctuations influence the hydrodynamic evolution of the matter in the final state.\n\nIn heavy ion collisions, the anisotropy of the particle distribution in azimuthal angle $\\phi$ is customarily characterized by a Fourier series:\n\\begin{equation}\n\\label{eq:flow0}\n\\frac{dN}{d\\phi}\\propto1+2\\sum_{n=1}^{\\infty}v_{n}\\cos n(\\phi-\\Phi_{n})\\;,\n\\end{equation}\nwhere $v_n$ and $\\Phi_n$ (event plane or EP) represent the magnitude and phase of the $n^{\\mathrm{th}}$-order harmonic flow. These flow harmonics have been associated to various shape components of the created matter~\\cite{Alver:2010gr}. The magnitude and direction of each shape component can be estimated via a simple Glauber model from the transverse positions (r, $\\phi$) of the participating nucleons~\\cite{Qin:2010pf,Teaney:2010vd}:\n\\begin{eqnarray}\n\\label{eq:ena}\n\\epsilon_n &=& \\frac{\\sqrt{\\langle r^m\\cos n\\phi\\rangle^2+\\langle r^m\\sin n\\phi\\rangle^2}}{\\langle r^m\\rangle},\\\\\\label{eq:glau} \n\\tan (n\\Phi_n^*+\\pi)&=&\\frac{\\langle r^m\\sin n\\phi\\rangle}{\\langle r^m\\cos n\\phi\\rangle},\\\\\\nonumber\n&&m=3\\; {\\rm if}\\; n=1, m=n\\; \\rm{ for}\\; n>1\n\\end{eqnarray} \nwith $\\epsilon_n$ and $\\Phi^*_n$ referred to as the eccentricity and participant plane (PP), respectively. Model calculations suggest that hydrodynamic response to the shape component is linear for the first few flow harmonics, i.e. $\\Phi_n\\approx \\Phi_n^*$ and $v_n\\propto \\epsilon_n$ for $n=$1--3~\\cite{Teaney:2010vd,Qiu:2011iv}. But these simple relations are violated for higher-order harmonics, due to strong mode-mixing effects intrinsic in the collective expansion~\\cite{Qiu:2011iv,Gardim:2011xv,Teaney:2012ke}.\n\nThe presence of large event-by-event (EbyE) fluctuations of the initial geometry suggests a general set of observables that involve correlations between $v_n$ and $\\Phi_n$:\n\\begin{equation}\n\\label{eq:flow}\np(v_n,v_m,...., \\Phi_n, \\Phi_m, ....)=\\frac{1}{N_{\\mathrm{evts}}}\\frac{dN_{\\mathrm{evts}}}{dv_ndv_m...d\\Phi_{n}d\\Phi_{m}...},\n\\end{equation}\nwith each variable being a function of $\\pT$, $\\eta$ etc~\\cite{Gardim:2012im}. Among these, the joint probability distribution of the EP angles:\n\\small{\n\\begin{eqnarray}\n\\nonumber\n\\frac{dN_{\\mathrm{evts}}}{d\\Phi_{1}d\\Phi_{2}...d\\Phi_{l}} &\\propto& \\sum_{c_n=-\\infty}^{\\infty} a_{c_1,c_2,...,c_l} \\cos(c_1\\Phi_1+c_2\\Phi_2...+c_l\\Phi_l),\\\\\\label{eq:ep}\na_{c_1,c_2,...,c_l}&=&\\left\\langle\\cos(c_1\\Phi_1+c_2\\Phi_2+...+c_l\\Phi_l)\\right\\rangle\n\\end{eqnarray}}\\normalsize\ncan be reduced to the following event-plane correlators required by symmetry~\\cite{Bhalerao:2011yg,Qin:2011uw,Jia:2012ma}:\n\\begin{eqnarray}\n\\label{eq:ep2}\n\\left\\langle\\cos(c_1\\Phi_1+2c_2\\Phi_2...+lc_l\\Phi_l)\\right\\rangle, c_1+2c_2...+lc_l=0.\n\\end{eqnarray}\nThese observables are sensitive to the fluctuations in the initial density profile and the final state hydrodynamics response~\\cite{Teaney:2012ke}.\n\n\nEarlier flow measurements were aimed at studying the individual $v_n$ coefficients for $n=$1--6 averaged over many events~\\cite{Adare:2011tg,star:2013wf,Aamodt:2011by,Aad:2012bu,CMS:2012wg}. Recently, the LHC experiments exploited the EbyE observables defined in Eq.~\\ref{eq:flow} by performing the first measurement of $p(v_n)$~\\cite{Aad:2013xma} for $n=2-4$ and fourteen correlators involving two or three event planes~\\cite{Jia:2012sa,ALICE:2011ab}. The measured event-plane correlators are reproduced by EbyE hydrodynamics~\\cite{Qiu:2012uy,Teaney:2012gu} and AMPT transport model~\\cite{Bhalerao:2013ina} calculations. The EP correlation measurement provides detailed insights on the non-linear hydrodynamic response, for example the correlators $\\left\\langle\\cos 4(\\Phi_{2}-\\Phi_{4})\\right\\rangle$ and $\\left\\langle\\cos 6(\\Phi_{3}-\\Phi_{6})\\right\\rangle$ mainly arise from the non-linear effects, which couple $v_4$ to $(v_2)^2$ and $v_6$ to $(v_3)^2$. Similarly, the correlator $\\left\\langle\\cos (2\\Phi_{2}+3\\Phi_{3}-5\\Phi_5)\\right\\rangle$ is driven by the coupling between $v_5$ and $v_2v_3$~\\cite{Gardim:2011xv,Teaney:2012ke}. \n\n\nThis paper focuses on two subsets of the observables defined by Eq.~\\ref{eq:flow}: $p(v_n,v_m)$ and $p(v_n, \\Phi_m, \\Phi_l, ....)$, which can provide further insights on the linear and non-linear effects in the hydrodynamics response. The correlation $p(v_n,v_m)$  quantifies directly the coupling between $v_m$ and $v_n$, while $p(v_n, \\Phi_m, \\Phi_l, ...)$ allows us to study how the event-plane correlations couples to a specific flow harmonics $v_n$. The probability distributions of these correlations are difficult to measure directly, instead we explore them systematically using the recently proposed event shape selection method~\\cite{Schukraft:2012ah} (also investigated in Ref.~\\cite{Petersen:2013vca,Lacey:2013eia}): Events in a given centrality interval are first classified according to the observed $v_n$ signal in certain $\\eta$ range, and the $p(v_m)$ and $p(\\Phi_m, \\Phi_l, ...)$ are then measured in other $\\eta$ range for each class. The event shape observables should be those that correlate well with the $\\epsilon_n$ of the initial geometry, such as the observed $v_1$ (dipolar flow), $v_2$ and $v_3$. The roles of these selection variables are similar to the event centrality, except that they further divide events within the same centrality class.\n\nThe event shape selection method also provides a unique opportunity to investigate the longitudinal dynamics of the collective flow. For example, events selected with large $v_2$ in one pseudorapidity window, in addition to having bigger $\\epsilon_2$, may also have stronger density fluctuations, larger initial flow or smaller viscous correction~\\cite{Pang:2012he}. Studying how the $v_n$ values or EP correlations vary with the $\\eta$ separation from the selection window may provide better insights on the longitudinal dynamics in the initial and the final states. Earlier efforts in this front can be found in Refs.~\\cite{Petersen:2011fp,Pang:2012he,Xiao:2012uw}.\n\nIn this paper, we apply the event shape selection technique to events generated by the AMPT model, to investigate the $p(v_n,v_m)$, $p(v_n, \\Phi_m, \\Phi_l, ....)$, and the longitudinal flow fluctuations. These correlations are studied for events binned according to the observed $v_2\/v_3$ signal, which are then compared with results for events binned directly in $\\epsilon_2\/\\epsilon_3$. This comparison helps to elucidate whether the changes in the correlation are driven mostly by the selection of the initial geometry or due to additional dynamics in the final state. This study also help to develop and validate the analysis method to be used in the actual data analysis.\n\nThe structure of the paper is as follows: Section~\\ref{sec:1} introduces the observables and method of the event shape selection in the AMPT model. Section~\\ref{sec:2} studies how the correlations among the eccentricities and PP angles vary with event shape selection. Section~\\ref{sec:3} presents a study of the rapidity fluctuations of flow. Section~\\ref{sec:4} studies how the correlations among the $v_n$'s and $\\Phi_n$'s vary with event shape selection. Section~\\ref{sec:5} gives a discussion and summary of the results.\n\n\n\n\\section{The method}\n\\label{sec:1}\nA Muti-Phase Transport model (AMPT)~\\cite{Lin:2004en} has been used frequently to study the higher-order $v_n$ associated with $\\epsilon_n$ in the initial geometry~\\cite{Xu:2011jm,Xu:2011fe,Ma:2010dv}. It combines the initial fluctuating geometry based on Glauber model from HIJING with the final state interaction via a parton and hadron transport model. The collective flow in this model is driven mainly by the parton transport. The AMPT simulation in this paper is performed with string-melting mode with a total partonic cross-section of 1.5 mb and strong coupling constant of $\\alpha_s=0.33$~\\cite{Xu:2011fe}, which has been shown to reproduce reasonably the $\\pT$ spectra and $v_n$ data at RHIC and LHC~\\cite{Xu:2011fe,Xu:2011fi}. The initial condition of the AMPT model with string melting has been shown to contain significant longitudinal fluctuations that can influence the collective dynamics~\\cite{Pang:2012he,Pang:2012uw}.\n\nThe AMPT sample used in this study is generated for $b=8$~fm Pb+Pb collisions at LHC energy of $\\sqrt{s_{NN}}=2.76$ TeV, corresponding to $\\sim 30\\%$ centrality. The particles in each event are divided into various subevents along $\\eta$, one example division scheme is shown in Fig.~\\ref{fig:m1}. Four subevents labelled as S, A, B, C, with at least 1 unit $\\eta$ gap between any pair except between S and A, are used in the analysis. Note that particles in $-6<\\eta<-2$ are divided randomly into two equal halves, labelled as S and A, respectively. The particles in subevent S are used only for the event shape selection purpose, and they are excluded for $v_n$ and event-plane correlation analysis. This choice of subevents and analysis scheme ensure that the event shape selection does not introduce non-physical correlations between S and A, B or C.\n\n\nThe flow vector in each subevent is calculated as:\n\\begin{eqnarray}\n\\nonumber\n&&\\overrightharp{q}_n =(q_{x,n},q_{y,n}) = \\frac{1}{\\Sigma_i w_i}\\left(\\textstyle\\Sigma_i (w_i\\cos n\\phi_i), \\Sigma_i (w_i\\sin n\\phi_i)\\right)\\;, \\\\\\label{eq:me1}\n&&\\tan n\\Psi_n = \\frac{q_{y,n}}{q_{x,n}}\\;,\n\\end{eqnarray}\nwhere the weight $w_i$ is chosen as the $\\pT$ of $i$-th particle and $\\Psi_n$ is the measured event plane. Due to finite number effects, $\\Psi_n$  smears around the true event-plane angle $\\Phi_n$. Hence $q_n$ represents the weighted raw flow coefficients $v_n^{\\mathrm{obs}}$, $q_n=\\Sigma_i \\left(w_i (v_n^{\\mathrm{obs}})_i\\right)\/\\Sigma_i w_i$. In this study, each subevent in Fig.~\\ref{fig:m1} has 1400-3000 particles, so $q_n$ is expected to follow closely the true $v_n$.\n\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{dets}\n\\caption{\\label{fig:m1} (Color online) The $\\eta$ range of the subevent for the event shape selection (S) and three other subevents for correlation analysis (A, B and C). Note that the particles in $-6<\\eta<-2$ are divided randomly and equally into subevents A and S.}\n\\end{figure}\n\nFor each generated event, the following quantities are calculated for $n=$1--6: $(\\epsilon_n, \\Phi^*_n)$ from initial state, $(q_n^{\\mathrm{A}}, \\Psi^{\\mathrm{A}}_n)$ for subevent A, $(q_n^{\\mathrm{B}}, \\Psi^{\\mathrm{B}}_n)$ for subevent B, $(q_n^{\\mathrm{C}}, \\Psi^{\\mathrm{C}}_n)$ for subevent C, and $(q_n^{\\mathrm{S}}, \\Psi^{\\mathrm{S}}_n)$ for subevent S, a total of 60 quantities. The event shape selection is performed by dividing the generated events into 10 bins in $q_2^{\\mathrm{S}}$ or $q_3^{\\mathrm{S}}$ with equal statistics. Similar event shape selection procedure is also performed by slicing the values of $\\epsilon_2$ or $\\epsilon_3$ directly, with the aim of studying how well the physics for events selected in the final state correlates with those selected purely on the initial geometry.\n\nFigure~\\ref{fig:m2} shows the performance of the event shape selection on $q_2^{\\mathrm{S}}$ and $q_3^{\\mathrm{S}}$. Strong positive correlations between $\\epsilon_n$ and $q_n^{\\mathrm{S}}$ seen in the top panels reflect the fact that collective response is linear for $n=2$ and 3~\\cite{Qiu:2012uy}. The bottom panels show that events selected with top 10\\% of the $q_2^{\\mathrm{S}}$ have a $\\left\\langle\\epsilon_2\\right\\rangle$ value that is nearly 3 times that for events with the lower 10\\% of $q_2^{\\mathrm{S}}$. For $n=3$ the difference in $\\epsilon_3$ in the two event classes is about a factor of 2. These results suggest that the ellipticity and triangularity of the initial geometry can be selected precisely by slicing the flow vector in the final state. \n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=1\\linewidth]{cqe3paper_cent8_m1}\n\\caption{\\label{fig:m2} (Color online) Correlations between $\\epsilon_n$ and magnitude of flow vector $q_n^{\\mathrm{S}}$ calculated using half of the particles in $-6<\\eta<-2$ (top panels), the 10 bins in $q_n^{\\mathrm{S}}$ with equal statistics (middle panels) and the corresponding distributions of $\\epsilon_n$ for events in the top 10\\%, bottom 10\\% and total of $q_n^{\\mathrm{S}}$ (bottom panels). The results are calculated for $b=8$~fm for $n=2$ and $n=3$, and are shown in the left and right column, respectively.}\n\\end{figure}\n\nIn the event shape selection method, $p(v_m,v_n)$ is not directly calculated. Instead, the calculated correlation is:\n\\begin{eqnarray}\n\\label{eq:m2}\np(q_m^{\\mathrm{S}},v_n^{\\mathrm{obs}}) =  p(q_m^{\\mathrm{S}})\\times p(v_n^{\\mathrm{obs}})_{q_m^{\\mathrm{S}}},\\;\\; m=2,3\n\\end{eqnarray}\nwhere conditional probability $p(v_n^{\\mathrm{obs}})_{q_m^{\\mathrm{S}}}$ represents the distribution of $v_n^{\\mathrm{obs}}$ for events selected with given $q_m^{\\mathrm{S}}$ value. To minimize non-flow effects, the $v_n^{\\mathrm{obs}}$ is calculated for particles separated in $\\eta$ from the subevent that provides the event plane. To minimize non-flow effects, a $\\eta$ gap from the corresponding event plane in each case is required. The probability $p(v_n)_{q_m^{\\mathrm{S}}}$ can be obtained from $p(v_n^{\\mathrm{obs}})_{q_m^{\\mathrm{S}}}$ via the unfolding technique~\\cite{Aad:2013xma,Jia:2013tja}, or if one is interested in the event-averaged $v_n$ values, the standard method~\\cite{Poskanzer:1998yz} can be applied for each $q_m^{\\mathrm{S}}$ bin:\n\\begin{eqnarray}\n\\label{eq:m3}\nv_n(\\pT,\\eta)_{q_m^{\\mathrm{S}}} = \\left[\\frac{v_n^{\\mathrm{obs}}(\\pT,\\eta)}{\\mathrm{Res}\\{ n\\Psi_n \\} }\\right]_{q_m^{\\mathrm{S}}},\n\\end{eqnarray}\nwhere the event-plane resolution factor $\\mathrm{Res}\\{ n\\Psi_n \\}$ is calculated separately for A, B, and C via the three-subevent method, providing three independent $v_n$ estimates~\\cite{Poskanzer:1998yz}. Since the magnitude and direction of the flow vector are uncorrelated, the event shape selection is not expected to introduce biases to the resolution correction. One special case of Eq.~\\ref{eq:m3} is $n=m$, which probes into the rapidity fluctuation of the $v_n$ itself (see Section~\\ref{sec:3}).\n\nTo calculate the event-plane correlation for each $q_m^{\\mathrm{S}}$ bin, the standard method introduced by the ATLAS collaboration based on event-plane correlation~\\cite{Jia:2012sa,Jia:2012ma}, and the method based on scalar products in Refs.~\\cite{Luzum:2012da,Bhalerao:2013ina} are adopted:\n\\begin{eqnarray} \n\\nonumber\n\\langle\\cos (\\Sigma \\Phi)\\rangle &=& \\frac{\\langle\\cos (\\Sigma \\Psi)\\rangle} {\\mathrm{Res}\\{c_1\\Psi_1\\}\\mathrm{Res}\\{c_22\\Psi_2\\}...\\mathrm{Res}\\{c_ll\\Psi_l\\}}\\\\\\label{eq:m4a}\n\\\\\\nonumber\n\\langle\\cos (\\Sigma \\Phi)\\rangle_w &=& \\frac{\\langle q_1^{c_1} q_2^{c_2}... q_l^{c_l}\\cos (\\Sigma \\Psi)\\rangle} {\\mathrm{Res}\\{c_1\\Psi_1\\}_w\\mathrm{Res}\\{c_22\\Psi_2\\}_w...\\mathrm{Res}\\{c_ll\\Psi_l\\}_w}\\\\\\label{eq:m4}\n\\end{eqnarray}\nwhere shorthand notions $\\Sigma \\Phi = c_1\\Phi_{1}+2c_2\\Phi_{2}+...+lc_l\\Phi_{l}$ and $\\Sigma \\Psi = c_1\\Psi_{1}+2c_2\\Psi_{2}+...+lc_l\\Psi_{l}$ are used. They are referred to as the EP method (Eq.~\\ref{eq:m4a}) and the SP method (Eq.~\\ref{eq:m4}) for the rest of this paper. The resolution factors $\\mathrm{Res}\\{c_nn\\Psi_n\\}$ and $\\mathrm{Res}\\{c_nn\\Psi_n\\}_w$ are calculated via three-subevent method involving subevents A, B and C:\n\\small{\n \\begin{eqnarray}\n \\label{eq:m5a}\n &&{\\mathrm{Res}}\\{jn\\Psi^{\\mathrm A}_{n}\\}=\\sqrt{\\frac{\\left\\langle {\\cos\\Delta\\Psi^{AB}_n} \\right\\rangle\\left\\langle {\\cos \\Delta\\Psi^{AC}_n}\\right\\rangle}{\\left\\langle {\\cos \\Delta\\Psi^{BC}_n} \\right\\rangle}}.\\\\\\nonumber\n&&{\\mathrm{Res}}\\{jn\\Psi^{\\mathrm A}_{n}\\}_w= \\sqrt{\\frac{\\left\\langle { (q_n^{\\mathrm A}q_n^{\\mathrm B})^j \\cos \\Delta\\Psi^{AB}_n} \\right\\rangle \\left\\langle { (q_n^{\\mathrm A}q_n^{\\mathrm C})^j\\cos \\Delta\\Psi^{AC}_n }\\right\\rangle}{\\left\\langle { (q_n^{\\mathrm B}q_n^{\\mathrm C})^j\\cos \\Delta\\Psi^{BC}_n} \\right\\rangle}}.\\\\\\label{eq:m5}\n \\end{eqnarray}}\\normalsize\nwhere $\\Delta\\Psi^{AB}_n = jn \\left(\\Psi_n^{\\mathrm A} - \\Psi_n^{\\mathrm B}\\right)$ etc. Each $\\Psi_n$ angle in Eq.~\\ref{eq:m4} is calculated in a separate subevent to avoid auto-correlations. The two subevents involved in two-plane correlation are chosen as A and C in Fig.~\\ref{fig:m1}, while the three subevents in three-plane correlation are chosen as A, B, and C in Fig.~\\ref{fig:m1}. Note that selecting on $q_n^{\\mathrm{S}}$ explicitly breaks the symmetry between subevents A and C even though they still have symmetric $\\eta$ acceptance. Thus their resolution factors are different and need to be calculated separately.\n\n\\section{Correlations in the initial state}\n\\label{sec:2}\nBefore discussing correlations in the final state, it is instructive to look first at how the initial geometry variables $\\epsilon_n, \\Phi^*_n$ and their correlations vary with the event shape selection. Figure~\\ref{fig:i1} shows the correlations between pairs of $\\epsilon_n$ for $n\\leq4$ for the generated AMPT events. Significant correlations are observed between $\\epsilon_2$ and $\\epsilon_3$~\\cite{Lacey:2013eia,ATLAS2014-022}, $\\epsilon_1$ and $\\epsilon_3$. The correlations between $\\epsilon_1$ and $\\epsilon_2$ are weak for this impact parameter but become more significant for $b=10$~fm (see Appendix~\\ref{sec:7}). Since the hydrodynamic response is nearly linear for $n=1-3$~\\cite{Qiu:2011iv}, these correlations are expected to survive into correlations between $v_n$ of respective order. The $\\epsilon_2$ and $\\epsilon_4$ correlation is also significant, especially for large $\\epsilon_2$ values, this correlation may survive to the final state but it competes with non-linear effects expected for $v_4$~\\cite{Gardim:2011xv}. \n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{ceepaperb_cent8}\n\\caption{\\label{fig:i1} (Color online)  Selected correlations between $\\epsilon_n$ of different order for Pb+Pb events at $b=8$~fm. More examples are given in Appendix~\\ref{sec:7}. The $x$- and $y$-profiles of the 2D correlations are represented by the solid symbols.}\n\\end{figure}\n\nFigure~\\ref{fig:i2} shows selected correlations between $\\Phi_n^*$ of different order for events binned in $\\epsilon_2$ (boxes) or $q_2^{\\mathrm{S}}$ (circles)~\\cite{Jia:2012ma,Jia:2012ju}. It is clear that the correlation signal varies dramatically with $\\epsilon_2$, implying that the correlations between $\\Phi_n^*$'s can vary a lot for events with the same impact parameter. Figure~\\ref{fig:i2} also shows that events with different correlations in the initial geometry can be selected with nearly the same precision between using $q_2^{\\mathrm{S}}$ and using $\\epsilon_2$.\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{cephilonga_cent8}\n\\caption{\\label{fig:i2} (Color online) Dependence of the participant-plane correlations on $\\epsilon_2$ (boxes) or $q_2^{\\mathrm{S}}$ (circles), calculated for Pb+Pb events with $b=8$~fm. Results for three two-plane, three three-plane and two four-plane correlators proposed in Refs.~\\cite{Jia:2012ma,Jia:2012ju} are shown.}\n\\end{figure}\n\n\n\n\\section{The correlation between $v_n$ and $v_m$ and longitudinal fluctuations}\n\\label{sec:3}\nFigure~\\ref{fig:eta1} shows the $v_2(\\eta)$ values for events selected for lower 10\\% (top panels) and upper 10\\% (bottom panels) of the values of either $q_2^{\\mathrm{S}}$ (left panels) or $\\epsilon_2$ (right panels). They are calculated via Eqs.~\\ref{eq:m3} and \\ref{eq:m5a} using all final state particles with $0.1<\\pT<5$ GeV, excluding those particles used in the event shape selection (i.e. subevent S). The event-plane angles are calculated separately for the three subevents A, B, and C, and a minimum 1--2 unit of $\\eta$ gap is required between $v_n(\\eta)$ and the subevent used to calculate the event plane. Specifically, the $v_2$ values in $-6<\\eta<0$ are obtained using the EP angle in subevent C covering $2<\\eta<6$ (open boxes), the $v_2$ values in $0<\\eta<6$ are obtained using the EP angle in subevent A covering $-6<\\eta<-2$ (open circles), and the $v_2$ values in $|\\eta|>2$ are also obtained using the EP angle in subevent B covering $|\\eta|<1$ (solid circles).\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=1\\linewidth]{cvnbpaper_cent8_pt0_har1_m1}\n\\caption{\\label{fig:eta1} (Color online) $v_2(\\eta)$ for events selected with lower 10\\% (top panels) and upper 10\\% (bottom panels) of the values of either $q_2^{\\mathrm{S}}$ (left panels) or $\\epsilon_2$ (right panels) for AMPT Pb+Pb events with $b=8$~fm. In each case, the integral $v_2$ calculated for particles in $0.1<\\pT<5$ GeV relative to the event plane of subevent A, B and C (Their $\\eta$ coverages are indicated in the legend) with a minimum $\\eta$ gap of 1 unit are shown.}\n\\end{figure}\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=1\\linewidth]{cvncombpaper_cent8_pt0_m1}\n\\caption{\\label{fig:eta2} (Color online) $v_n(\\eta)$ for events selected with lower 10\\% (open symbols) and upper 10\\% (solid symbols) of the values of $q_2^{\\mathrm{S}}$ for AMPT Pb+Pb events with $b=8$~fm. Results are shown for $v_2(\\eta)$, $v_3(\\eta)$,..., and $v_6(\\eta)$ from left panel to the right panel. The ratios of $v_n(\\eta)$ between events with $q_2^{\\mathrm{S}}$ selection to the inclusive events are shown in the bottom panels.}\n\\end{figure*}\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=1\\linewidth]{cvnQdeppaper_cent8_pt0_eta1_m1}\n\\caption{\\label{fig:eta3} (Color online)  Dependence of the $v_n$ on $q_2^{\\mathrm{S}}$ in a forward ($-5<\\eta<-4$) and a backward ($4<\\eta<5$) pseudorapidity ranges.}\n\\end{figure*}\n\nThere are several interesting features in the observed $\\eta$ dependence of $v_2$. The $v_2(\\eta)$ values for events selected with lower 10\\% values of $q_2^{\\mathrm{S}}$ or $\\epsilon_2$ are significantly lower (by a factor of 3) than for events selected with upper 10\\%, indicating that the $v_2$ signal correlates well with both the $q_2^{\\mathrm{S}}$ and the $\\epsilon_2$. Furthermore, a significant forward\/backward asymmetry of $v_2(\\eta)$ is observed for events selected on $q_2^{\\mathrm{S}}$ but not $\\epsilon_2$. This asymmetry is already observed outside the $\\eta$ range covered by subevent S, but is bigger towards larger $|\\eta|$. This asymmetry may reflect the dynamical fluctuations exposed by the $q_2^{\\mathrm{S}}$ selection. Additional cross-checks performed by choosing subevent S in a more restricted $\\eta$ range show similar asymmetry (see Fig.~\\ref{fig:a0} in the Appendix).\n\n\nBased on the good agreement between the three $v_2$ estimations in Fig.~\\ref{fig:eta1}, they are combined into a single $v_2(\\eta)$ result. Good agreement is also observed for higher harmonics, hence they are combined in the same way. The resulting $v_2(\\eta)$--$v_6(\\eta)$ are shown in Fig.~\\ref{fig:eta2} for events with lower 10\\% and upper 10\\% of the values of $q_2^{\\mathrm{S}}$. The asymmetry of $v_n$ in $\\eta$ is much weaker for the higher-order harmonics. The values of $v_n(\\eta)$ for $n>3$ are also seen to be positively correlated with $v_2$, $i.e.$ events with large $q_2^{\\mathrm{S}}$ also have bigger $v_n(\\eta)$. On the other hand, $v_3$ values are observed to decrease with increasing $q_2^{\\mathrm{S}}$. This decrease reflects the anti-correlation between $\\epsilon_2$ and $\\epsilon_3$ in Fig.~\\ref{fig:i1} (also confirm by ATLAS data~\\cite{ATLAS2014-022}). Figure~\\ref{fig:eta3} quantifies the forward\/backward asymmetry of $v_2$--$v_6$ in two $\\eta$ ranges: $-5<\\eta<-4$ and $4<\\eta<5$. Clear asymmetry can be seen for $v_2$, $v_4$ and $v_5$, but not for $v_3$. This behavior re-enforces our earlier conclusion that the correlation between $v_2$ and $v_3$ in the AMPT model is mostly geometrical, $i.e.$ reflecting correlation between $\\epsilon_2$ and $\\epsilon_3$.\n\nAn identical analysis is also performed for events selected on $q_3^{\\mathrm{S}}$ or $\\epsilon_3$. The $v_n(\\eta)$ for events with the upper 10\\% and lower 10\\% values of $q_3^{\\mathrm{S}}$ or $\\epsilon_3$ are shown in Fig.~\\ref{fig:eta1b}. A strong $\\eta$ asymmetry is observed as a result of $q_3^{\\mathrm{S}}$ selection, but not for $\\epsilon_3$ selection. Nevertheless, the overall magnitude of the $v_3$ is similar between the two selections. In the $-6<\\eta<-2$ range where $q_3^{\\mathrm{S}}$ is calculated, $v_3(\\eta)$ values for events with the lower 10\\% of $q_3^{\\mathrm{S}}$ drop to below zero. This implies that the $\\Phi_3$ angle for large negative $\\eta$ region become out of phase with the $\\Phi_3$ angle in the large positive range. This $\\Phi_3$ angle decorrelation is also observed for events selected with lower 10\\% of $\\epsilon_3$ values as shown in the top-right panel of Fig.~\\ref{fig:eta1b}. This behavior suggests that in the AMPT model, rapidity decorrelation of $v_3$ is stronger for events with small $\\epsilon_3$ and grows towards large $|\\eta|$ (negative $v_3$ implies its phase is opposite to that in the $\\eta$ region used to obtain the event plane). An earlier study~\\cite{Xiao:2012uw} has show evidences of $\\eta$ decorrelation of $v_3$ in the AMPT; Our later studies published in separate papers trace this decorrelation to the independent fluctuations of the $\\epsilon_n$ for the projectile nucleus and the $\\epsilon_n$ for the target nucleus~\\cite{Jia:2014vja,Jia:2014ysa}.\n\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=1\\linewidth]{Q3cvnbpaper_cent8_pt0_har2_m1}\n\\caption{\\label{fig:eta1b} (Color online) The $v_3(\\eta)$ for events selected for lower 10\\% (top panels) and upper 10\\% (bottom panels) of the values of either $q_3^{\\mathrm{S}}$ (left panels) or $\\epsilon_3$ (right panels) for AMPT Pb+Pb events with $b=8$~fm. In each case, the integral $v_3$ calculated for particles in $0.1<\\pT<5$ GeV relative to the event plane of subevent A, B and C (Their $\\eta$ coverages are indicated in the legend) with a minimum $\\eta$ gap of 1 unit are shown. }\n\\end{figure}\n\n Figure~\\ref{fig:eta3b} quantifies the rapidity asymmetry of $v_n$ between $-5<\\eta<-4$ and  $4<\\eta<5$ as a function of $q_3^{\\mathrm{S}}$ and $\\epsilon_3$. The even harmonics $v_2$ and $v_4$ show little asymmetry and are nearly independent of $q_3^{\\mathrm{S}}$. In contrast, the $v_5$ values show a strong $\\eta$-asymmetry similar to that for $v_3$. \n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=1\\linewidth]{Q3cvnQdeppaperb_cent8_pt0_eta1_m1}\n\\caption{\\label{fig:eta3b} (Color online)  Dependence of the $v_n$ on $q_3^{\\mathrm{S}}$ in a forward ($-5<\\eta<-4$) and a backward ($4<\\eta<5$) pseudorapidity ranges.}\n\\end{figure}\n\nFigure~\\ref{fig:eta5} shows the particle multiplicity distributions $dN\/d\\eta$ for events selected on $q_2^{\\mathrm{S}}$ (left) or $q_3^{\\mathrm{S}}$ (right). The distributions remain largely symmetric in $\\eta$ and the overall magnitude is nearly independent of the event selection. We also verified explicitly that the number of participating nucleons for the projectile and target are nearly equal for all $q_2^{\\mathrm{S}}$ or $q_3^{\\mathrm{S}}$ bins. This suggests that the underlying mechanism is not due to the EbyE fluctuations of the $dN\/d\\eta$ distribution.\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.5\\linewidth]{cvnNpaper_cent8_pt0_m0}\\includegraphics[width=0.5\\linewidth]{Q3cvnNpaper_cent8_pt0_m0}\n\\caption{\\label{fig:eta5} (Color online) The $dN\/d\\eta$ distributions of all particles for events selected on $q_2^{\\mathrm{S}}$ (left) and $q_3^{\\mathrm{S}}$ (right).}\n\\end{figure}\n\n\n\\section{event-plane correlations}\n\\label{sec:4}\nThe AMPT model has been shown to reproduce~\\cite{Bhalerao:2013ina} the centrality dependence of various two-plane and three-plane correlations measured by the ATLAS Collaboration~\\cite{Jia:2012sa}. Here we use AMPT model to study how these correlators change with $q_n^{\\mathrm{S}}$ or $\\epsilon_n$. In this analysis, the two-plane correlators $\\langle\\cos k(\\Phi_{n}-\\Phi_{m})\\rangle$ are calculated by correlating the EP angles from subevent A and subevent C. Each subevent provides its own estimation of the EPs, leading to two statistically independent estimates of the correlator: Type1 $\\langle\\cos k(\\Phi_{n}^{\\mathrm{A}}-\\Phi_{m}^{\\mathrm{C}})\\rangle$ and Type2 $\\langle\\cos k(\\Phi_{n}^{\\mathrm{C}}-\\Phi_{m}^{\\mathrm{A}})\\rangle$. The two estimates are identical for events selected on $\\epsilon_2$, and hence they are averaged to obtain the final result. But for events selected based on $q_2^{\\mathrm{S}}$, the two estimates can differ quite significantly. \n\nFigure~\\ref{fig:ep1} shows the values of four two-plane correlators in bins of $q_2^{\\mathrm{S}}$ or $\\epsilon_2$. The values of the correlators are observed to increase strongly with increasing $q_2^{\\mathrm{S}}$ or $\\epsilon_2$. The two estimates based on $q_2^{\\mathrm{S}}$ selection differ significantly, reflecting the influence of longitudinal flow fluctuations exposed by the $q_2^{\\mathrm{S}}$ selection. Interestingly, the correlators whose $\\Phi_2$ angle is calculated in subevent C agree very well with those based on $\\epsilon_2$ event shape selection, such as $\\langle\\cos 4(\\Phi_{2}^{\\mathrm{C}}-\\Phi_{4}^{\\mathrm{A}})\\rangle$. This is because $\\Phi_{2}^{\\mathrm{C}}$ is expected to be less dependent on the $q_2^{\\mathrm{S}}$ selection than $\\Phi_{2}^{\\mathrm{A}}$ (see Fig.~\\ref{fig:eta3}(a)). These observations suggest that the dependence of $\\langle\\cos 4(\\Phi_{2}^{\\mathrm{C}}-\\Phi_{4}^{\\mathrm{A}})\\rangle$ and $\\langle\\cos 6(\\Phi_{2}^{\\mathrm{C}}-\\Phi_{6}^{\\mathrm{A}})\\rangle$ on $q_2^{\\mathrm{S}}$ reflects mainly the change in the initial geometry and the ensuing non-linear effects in the final state. Note that the last bin in each panel represents the value obtained without event shape selection, which agrees between the three calculations by construction.\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=1\\linewidth]{c2pcomp2paperw_cent8_type_m1}\n\\caption{\\label{fig:ep1} (Color online) The four two-plane correlations as a function of bins of $q_2^{\\mathrm{S}}$ or $\\epsilon_2$ for AMPT Pb+Pb events with $b=8$~fm. The two event planes $\\Phi_n$ and $\\Phi_m$ are measured by subevent A and subevent C. The $q_2^{\\mathrm{S}}$-binned results are presented separately for the two combinations: $\\langle\\cos k(\\Phi_{n}^{\\mathrm{A}}-\\Phi_{m}^{\\mathrm{C}})\\rangle$ (open circles) and $\\langle\\cos k(\\Phi_{n}^{\\mathrm{C}}-\\Phi_{m}^{\\mathrm{A}})\\rangle$ (solid circles).}\n\\end{figure}\n\nFigure~\\ref{fig:ep2} compares various two-plane correlators calculated via the EP method and the SP method given by Eqs.~\\ref{eq:m4a}-\\ref{eq:m5}. The SP method is observed to give systematically higher values for Type1 correlators where the first angle is measured by subevent A, while it gives consistent or slightly lower values for Type2 correlators. The last bin in each panel shows the result obtained without event shape selection, where the values from the SP method are always higher, as expected~\\cite{Bhalerao:2013ina}. \n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=1\\linewidth]{c2pcomp3paper_cent8_type_m1}\n\\caption{\\label{fig:ep2} (Color online) The four two-plane correlators in bins of $q_2^{\\mathrm{S}}$ for AMPT Pb+Pb events with $b=8$~fm, shown for two different combinations of the event-plane angles (solid symbols), and compared with the correlations calculated via the scalar product method (open symbols).}\n\\end{figure}\n\nFigure~\\ref{fig:ep1b} shows $\\langle\\cos 6(\\Phi_{2}-\\Phi_{3})\\rangle$ and $\\langle\\cos 6(\\Phi_{3}-\\Phi_{6})\\rangle$ in bins of $q_3^{\\mathrm{S}}$ or $\\epsilon_3$. The first correlator shows little dependence on $q_3^{\\mathrm{S}}$ or $\\epsilon_3$, while the second correlator does. This is in sharp contrast to the results seen in Fig.~\\ref{fig:ep1}, where both correlators show strong but opposite dependence on $q_2^{\\mathrm{S}}$ or $\\epsilon_2$. This behavior is consistent with a strong coupling between $v_6$ and $v_2$, $v_6$ and $v_3$, but weak coupling between $v_2$ and $v_3$.\n\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=1\\linewidth]{Q3c2pcomp2paperw_cent8_type_m1}\n\\caption{\\label{fig:ep1b} (Color online) Two two-plane correlators as a function of either $q_3^{\\mathrm{S}}$ or $\\epsilon_3$ for AMPT Pb+Pb events with $b=8$~fm. The event planes $\\Phi_n$ and $\\Phi_m$ are measured by subevent A and subevent C. The $q_3^{\\mathrm{S}}$-binned results are presented separately for the two estimates: $\\langle\\cos k(\\Phi_{n}^{\\mathrm{A}}-\\Phi_{m}^{\\mathrm{C}})\\rangle$ (open circles) and $\\langle\\cos k(\\Phi_{n}^{\\mathrm{C}}-\\Phi_{m}^{\\mathrm{A}})\\rangle$ (solid circles).}\n\\end{figure}\n\nTo calculate three-plane correlations, subevents A, B and C are used. Each subevent provides its own estimation of the three EP angles, and hence there are $3!=6$ independent ways of estimating a given three-plane correlator. For $c_nn\\Phi_{n}+c_mm\\Phi_{m}+c_ll\\Phi_{l}$ with $n<m<l$, these six estimates are labelled as the following:\n\\begin{itemize}\n\\item Type 1a: $c_nn\\Phi_{n}^{\\mathrm{B}}+c_mm\\Phi_{m}^{\\mathrm{A}}+c_ll\\Phi_{l}^{\\mathrm{C}}$\n\\item Type 1b: $c_nn\\Phi_{n}^{\\mathrm{B}}+c_mm\\Phi_{m}^{\\mathrm{C}}+c_ll\\Phi_{l}^{\\mathrm{A}}$\n\\item Type 2a: $c_nn\\Phi_{n}^{\\mathrm{A}}+c_mm\\Phi_{m}^{\\mathrm{B}}+c_ll\\Phi_{l}^{\\mathrm{C}}$\n\\item Type 2b: $c_nn\\Phi_{n}^{\\mathrm{C}}+c_mm\\Phi_{m}^{\\mathrm{B}}+c_ll\\Phi_{l}^{\\mathrm{A}}$\n\\item Type 3a: $c_nn\\Phi_{n}^{\\mathrm{A}}+c_mm\\Phi_{m}^{\\mathrm{C}}+c_ll\\Phi_{l}^{\\mathrm{B}}$\n\\item Type 3b: $c_nn\\Phi_{n}^{\\mathrm{C}}+c_mm\\Phi_{m}^{\\mathrm{A}}+c_ll\\Phi_{l}^{\\mathrm{B}}$\n\\end{itemize}\nFor events selected on $\\epsilon_n$, the symmetry of the $\\eta$-coverage between A and C reduces them into three equivalent pairs of estimates. However for events selected on $q_n^{\\mathrm{S}}$, all six estimates can be different.\n\nFigure~\\ref{fig:ep3} summarizes the results for five three-plane correlators, one for each column, calculated for events classified by $q_2^{\\mathrm{S}}$ or $\\epsilon_2$~\\footnote{We also calculate these correlators using the SP method (a.la. $q_n$-weights), the dependences on $q_2^{\\mathrm{S}}$ are found to be qualitatively the same, though the overall magnitudes may differ (see Fig.~\\ref{fig:a3}).}, including a previously unnoticed strong correlator $\\left\\langle\\cos (2\\Phi_{2}-8\\Phi_{4}+6\\Phi_6)\\right\\rangle$. The six estimates of each correlator are grouped into three pairs and are shown in the three rows. Many of these correlators exhibit a breaking of the symmetry between subevent A and C, see for example the Type1 and Type2 for the first two correlators, as well as the Type1 and Type3 for the third and fourth correlators. Some correlators even show opposite dependence on $q_2^{\\mathrm{S}}$, such as the Type1 and Type3 for the fourth correlator ($\\left\\langle\\cos (2\\Phi_{2}-6\\Phi_{3}+4\\Phi_4)\\right\\rangle$). In most cases, however, the overall dependences on $\\epsilon_2$ bin (open circles) are reasonably captured by the dependence on $q_2^{\\mathrm{S}}$ (open boxes), implying that these correlations reflect mainly the intrinsic hydrodynamic response to the change in the selected initial geometry. \n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=1\\linewidth]{c3pcomp2paperw_cent8_m1}\n\\caption{\\label{fig:ep3} (Color online) \nThe five three-plane correlators (from left to right) as a function of either $q_2^{\\mathrm{S}}$ (solid symbols) or $\\epsilon_2$ (open symbols) for AMPT Pb+Pb events with $b=8$~fm. The results for the three type groups are shown in different row: Type1a and Type1b (top row), Type2a and Type2b (middle row), Type2a and Type2b (bottom row). The last bin in each panel shows the inclusive result.}\n\\end{figure*}\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=1\\linewidth]{Q3c3pcomp2paperw_cent8_m1}\n\\caption{\\label{fig:ep3b} (Color online) \nThe five three-plane correlators (from left to right) as a function of either $q_3^{\\mathrm{S}}$ (solid symbols) or $\\epsilon_3$ (open symbols) for AMPT Pb+Pb events with $b=8$~fm. The results for the three type groups are shown in different row: Type1a and Type1b (top row), Type2a and Type2b (middle row), Type2a and Type2b (bottom row). The last bin in each panel shows the inclusive result.}\n\\end{figure*}\n\nFigure~\\ref{fig:ep3b} summarizes the results for the same five three-plane correlators, but calculated for events classified by $q_3^{\\mathrm{S}}$ or $\\epsilon_3$. Comparing with Fig.~\\ref{fig:ep3}, we find that the values of $\\left\\langle\\cos (2\\Phi_{2}-3\\Phi_{3}+5\\Phi_5)\\right\\rangle$ are more sensitive to $q_3^{\\mathrm{S}}$ or $\\epsilon_3$ than to $q_2^{\\mathrm{S}}$ or $\\epsilon_2$ for small bin numbers, possibly reflecting stronger rapidity decorrelation effects associated with $\\Phi_3$ seen for small $q_3^{\\mathrm{S}}$ or $\\epsilon_3$ in Fig.~\\ref{fig:eta1b}. The correlators $\\left\\langle\\cos (2\\Phi_{2}-6\\Phi_{3}+4\\Phi_4)\\right\\rangle$ and $\\left\\langle\\cos (2\\Phi_{2}+6\\Phi_{3}-8\\Phi_4)\\right\\rangle$ show better agreement between the six types than when they are selected by $q_2^{\\mathrm{S}}$ or $\\epsilon_2$ as shown in Fig.~\\ref{fig:ep3}, again reflecting the weak correlation between $v_3$ and $v_2$, and between $v_3$ and $v_4$.\n\n\n\n\n\n\\section{Discussion and Summary}\n\\label{sec:5}\n\nThis paper studies two sets of flow observables involving correlations between harmonic flow coefficients $v_n$ and their phases $\\Phi_n$, utilizing the recently proposed event shape selection technique~\\cite{Schukraft:2012ah}. The shape of the collision geometry is selected by cutting on $q_n$ (the magnitude of the flow vector in a subevent) and eccentricity $\\epsilon_n$. The $p(v_m)$ or the differential distribution of event-averaged $v_m$, and the event-plane correlations $p(\\Phi_m, \\Phi_l, ....)$ are then be studied in each $q_n$ or $\\epsilon_n$ bin. A special case of interest is the $v_n(\\eta)$ for events selected on $q_n$, which is sensitive to rapidity fluctuations of collective flow. The feasibility of measuring these new observables is investigated using the AMPT model. This model combines the Glauber initial fluctuating geometry with collective flow generated by partonic transport, and hence allows one to correlate the initial geometry information, $e.g.$ ($\\epsilon_n$, $\\Phi_n^*$), with ($v_n$,$\\Phi_n$) on a EbyE basis. Since the AMPT model describes reasonably the experimentally measured $\\pT$ spectra, $v_n$~\\cite{Xu:2011fe,Xu:2011fi}, and event-plane correlations~\\cite{Bhalerao:2013ina}, it should provide a good benchmark for the performance of event shape selection, as well as provide qualitative understanding of the physics behind these new observables.\n\n\nTo summarize, our study has been performed for Pb+Pb collisions at $\\sqrt{s_{NN}}=2.76$ TeV with fixed impact parameter $b=8$~fm, which corresponds to $\\sim30\\%$ centrality. The event shape selection is performed on the $q_2^{\\mathrm{S}}$ and $q_3^{\\mathrm{S}}$ calculated using half of the particles in $-6<\\eta<-2$, as well as on $\\epsilon_2$ and $\\epsilon_3$. The EP angles used to calculate the EP correlators are calculated in subevent A (the other half particles in $-6<\\eta<-2$), subevent B covering $-1<\\eta<1$ and subevent C covering $2<\\eta<6$. Our main findings can be summarized below:\n\\begin{itemize}\n\\item\nThe eccentricity $\\epsilon_n$ distribution is very broad for events with fixed impact parameter and correlates well with $q_n$ for $n=2$ and $3$, hence events with different ellipticity or triangularity can be selected precisely using $q_2$ and $q_3$, respectively.\n\\item\nThe participant plane correlations, such as $\\langle\\cos4(\\Phi_{2}^*-\\Phi_{4}^*)\\rangle$, $\\langle\\cos6(\\Phi_{2}^*-\\Phi_{6}^*)\\rangle$ and $\\langle\\cos6(\\Phi_{3}^*-\\Phi_{6}^*)\\rangle$, are observed to vary strongly with $\\epsilon_2$ or $\\epsilon_3$, and much of these dependences are preserved when selecting on $q_2$ or $q_3$. \n\\item\nSignificant correlations or anti-correlations are observed among the $\\epsilon_n$, such as $p(\\epsilon_2, \\epsilon_3)$, $p(\\epsilon_1, \\epsilon_3)$ and $p(\\epsilon_2, \\epsilon_4)$. These correlations can be easily probed by the event shape selection technique, some of these correlations, $e.g.$ $p(\\epsilon_1, \\epsilon_3)$ and $p(\\epsilon_2, \\epsilon_3)$, are expected to survive to the final state. Indeed, the correlation between $\\epsilon_2$ and $\\epsilon_3$ seems to be captured by the observed correlation between $v_2$ and $v_3$ in the AMPT model. \n\\item\nThe overall $v_n$ values in each $q_n^{\\mathrm{S}}$ bin are similar to that in the corresponding $\\epsilon_n$ bin ($n=2$ and 3). However a strong forward\/backward asymmetry of $v_n(\\eta)$ is observed in $q_n^{\\mathrm{S}}$ selected events (also $v_4$, $v_5$ for $q_2^{\\mathrm{S}}$ and $v_5$ for $q_3^{\\mathrm{S}}$), reflecting the dynamical fluctuations either in the initial state~\\cite{Pang:2012he,Pang:2013pma} or during the collective expansion present in the AMPT model. These dynamical fluctuations are exposed by the $q_n^{\\mathrm{S}}$ selection, probably because they are either short range in $\\eta$ or have strong $\\eta$ dependence. These dynamical effects also contribute strongly to event-plane decorrelations. We also observe that the $dN\/d\\eta$ distribution is not affected much by the event shape selection.\n\\item\nThe $v_n$ values are always positively correlated with $q_n^{\\mathrm{S}}$, however other harmonics $v_m$ for $m\\neq n$ show non-trivial dependence on $q_n^{\\mathrm{S}}$. For $q_2^{\\mathrm{S}}$ selection, $v_4$, $v_5$ and $v_6$ show positive correlation, while $v_3$ shows negative correlation consistent with the behavior of $p(\\epsilon_2, \\epsilon_3)$. For $q_3^{\\mathrm{S}}$ selection, $v_5$ shows positive correlation, while the $v_2$ and $v_4$ show slight negative correlation. These correlations are qualitatively consistent with couplings between different harmonics expected for collective flow~\\cite{Teaney:2012ke}.\n\\item\nThe strengths of several event-plane correlators vary strongly with $q_n^{\\mathrm{S}}$ and $\\epsilon_n$. These variations are observed to be similar between the EP method and the SP method. In many cases, they are also found to be similar between $q_n^{\\mathrm{S}}$ selection and $\\epsilon_n$ selection, namely the $\\langle\\cos 4(\\Phi_{2}^{\\mathrm{C}}-\\Phi_{4}^{\\mathrm{A}})\\rangle$, $\\langle\\cos 6(\\Phi_{2}^{\\mathrm{C}}-\\Phi_{6}^{\\mathrm{A}})\\rangle$, and many of $\\langle\\cos (2\\Phi_{2}+3\\Phi_3-5\\Phi_{5})\\rangle$, $\\langle\\cos (2\\Phi_{2}+4\\Phi_4-6\\Phi_{6})\\rangle$ and $\\langle\\cos (2\\Phi_{2}-8\\Phi_4+6\\Phi_{6})\\rangle$. For these correlators, the change of signal reflects mainly the selection on the collision geometry. In some other cases, results differ significant between $q_n^{\\mathrm{S}}$ and $\\epsilon_n$ selections, such as $\\langle\\cos 4(\\Phi_{2}^{\\mathrm{A}}-\\Phi_{4}^{\\mathrm{C}})\\rangle$, $\\langle\\cos 6(\\Phi_{2}^{\\mathrm{A}}-\\Phi_{6}^{\\mathrm{C}})\\rangle$ and $\\langle\\cos (2\\Phi_{2}^{\\mathrm{B}}+3\\Phi_3^{\\mathrm{A}}-5\\Phi_{5}^{\\mathrm{C}})\\rangle$ between $q_2^{\\mathrm{S}}$ and $\\epsilon_2$, and $\\langle\\cos (2\\Phi_{2}^{\\mathrm{B}}+3\\Phi_3^{\\mathrm{A}}-5\\Phi_{5}^{\\mathrm{C}})\\rangle$ and $\\langle\\cos (2\\Phi_{2}^{\\mathrm{A}}+3\\Phi_3^{\\mathrm{B}}-5\\Phi_{5}^{\\mathrm{C}})\\rangle$ between $q_3^{\\mathrm{S}}$ and $\\epsilon_3$. For these correlators, the change of signal is probably also affected by the dynamical fluctuations present in the AMPT model.\n\\item\nFor some event-plane correlators, the signal for different types show very different dependence on $q_2^{\\mathrm{S}}$. For example $\\langle\\cos (2\\Phi_{2}^{\\mathrm{A}}-6\\Phi_3^{\\mathrm{C}}+4\\Phi_{4}^{\\mathrm{B}})\\rangle$ and $\\langle\\cos (2\\Phi_{2}^{\\mathrm{C}}-6\\Phi_3^{\\mathrm{A}}+4\\Phi_{4}^{\\mathrm{B}})\\rangle$ have opposite dependence, and both differ from the dependence on $\\epsilon_2$. This may be a result of the rapidity fluctuations of $v_3(\\eta)$, as shown in Fig.~\\ref{fig:eta1b}.\n\\end{itemize}\n\n\n\n\nHeavy ion collisions at RHIC and LHC are a finite number system. A typical Pb+Pb collision involves a few hundred colliding nucleons and produces on the order of 10,000 particles in the final state. Both the collision geometry defined by the nucleons and the collective expansion of produced particles can fluctuate strongly event-by-event. It remains a challenge to disentangle the fluctuations at the initial state and dynamical fluctuations generated in the collective expansion, a prerequisite for understanding the physics behind these fluctuations. Future progress requires a detailed and systematic study of the correlations between all $v_n$ and their phases $\\Phi_n$: $p(v_n,v_m,..., \\Phi_n, \\Phi_m,...)$, in order to disentangle different sources of fluctuations. The study presented in this paper attempts to establish the methodology for accessing these correlations and provides the initial guidance on how these correlations are connected to the initial geometry and dynamics in the collective expansion. Much more theoretical efforts, especially those based on the application of event shape selection technique in EbyE hydrodynamics, will provide more realistic and detailed insights on the nature of the fluctuations and non-linear dynamics in various stages of the heavy ion collisions.\n\nThis research is supported by NSF under grant number PHY-1305037 and by DOE through BNL under grant number DE-AC02-98CH10886.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}