diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzdks" "b/data_all_eng_slimpj/shuffled/split2/finalzdks" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzdks" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nNon-orthogonal multiple access (NOMA) is a promising multiple access technique for 5G networks\nwhich has the potential to address the\nissues associated with the exponential growth of data traffic\nsuch as spectrum scarcity and massive connectivity \\cite{6730679,6868214,7273963,7676258,7263349}.\nIn contrast to conventional multiple access schemes,\nNOMA allows different users to efficiently share the same resources (i.e., time, frequency and code)\nat different power levels so that the user with lower channel gain is served with a higher power and vice versa.\nIn this technique, a successive interference cancellation (SIC) approach is employed at receivers to separate multi-user\nsignals, which significantly enhances\nthe overall spectral efficiency.\nIn other words,\nNOMA has the capability to control the\ninterference by sharing resources\nwhile increasing system throughput\nwith a reasonable additional complexity \\cite{7263349}.\n\nRecently, a significant amount of research has focused in studying\nseveral practical issues in NOMA scheme.\nIn particular, beamforming designs for multiple antenna NOMA networks have received \na great deal of interest in the research community due to their additional degrees of freedom and diversity gains \\cite{7433470,7095538,7277111}.\nA general framework for a multiple-input multiple-output (MIMO) NOMA system has been developed for both the downlink\nand the uplink in \\cite{7433470} whereas\nthe throughput maximization problem was studied for a two-user MIMO NOMA system in \\cite{7095538}.\nThe sum rate maximization problem for a multiple-input single-output (MISO) NOMA has been investigated in \\cite{7277111}\nthrough the minorization maximization algorithm.\nIn most of the existing work, beamforming designs have been proposed \nfor NOMA schemes with the assumption of perfect channel state information (CSI) at the transmitter \\cite{7433470,7095538,7277111}.\nHowever,\nthis assumption might not be always valid for practical scenarios\ndue to channel estimation and quantization errors \\cite{4700145,4895658,5073854,5074429,5665898,6967767,7047328}. On the other hand, channel uncertainties significantly influence the performance of the\nSIC based receivers as the\ndecoding order of the received multi-user signals is determined with respect to the users' effective channel gains.\nTherefore, it is important to take into account the channel\nuncertainties especially in the beamforming design for NOMA networks.\nMotivated by this practical constraint, we\nfocus on robust beamforming design based on the worst-case performance optimization framework\nto tackle the norm-bounded channel uncertainties \\cite{5755208,6156468,7177089,7547398}.\nIn \\cite{5755208}, the robust beamforming design has been developed\nfor providing secure communication in wireless networks with \nimperfect CSI.\n\nBy incorporating the\nbounded channel uncertainties, the robust sum power minimization\nproblem\nis investigated in \\cite{6156468}\nfor a downlink multicell network with the worst-case signal-to-interference-plus-noise-ratio (SINR) constraints\nwhereas the robust weighted sum-rate maximization was studied for multicell downlink MISO systems in \\cite{7177089}.\nIn \\cite{7547398}, a robust minimum mean square error based beamforming technique is\nproposed for multi-antenna relay channels with imperfect CSI between the relay and the users.\nIn the literature, there are two types of NOMA schemes considered:\nI) clustering NOMA \\cite{6735800,7015589,7442902}, II) non-clustering NOMA \\cite{5755208,6156468,7177089,7547398}.\nIn the clustering NOMA scheme, all the users in a cell are grouped into $N$ clusters with two users\nin each cluster, for which a transmit beamforming vector is designed to support those two users\nthrough conventional multiuser beamforming designs. The users in each cluster are supported by a NOMA\nbeamforming scheme. However, in the non-clustering NOMA scheme, there is no clustering and each\nuser is supported by its own NOMA based beamforming vector.\nIn \\cite{7442902}, the authors studied a robust NOMA scheme for the MISO channel\nto maximize the worst-case achievable sum rate with a\ntotal transmit power constraint.\n\nIn this letter, we follow the second class of research where NOMA scheme applied between all users and there is the spectrum\nsharing between all users in cell. Then, we propose a robust beamforming design for NOMA-based MISO\ndownlink systems.\nIn particular, the robust power minimization problem is solved\nbased on worst-case optimization framework to provide the required quality of service at each user\nregardless of the associated channel uncertainties.\nBy exploiting \\mbox{S-Procedure,} the original \\mbox{non-convex} problem is\nconverted into a convex one by recasting the non-convex constraints into linear matrix inequality (LMI) forms.\nSimulation results are provided\nto validate the \\mbox{effectiveness}\nof the robust design by comparing the performance of the robust scheme with that of the non-robust approach.\n{The work in \\cite{7442902} also studied the worst-case based robust scheme\nfor MISO NOMA system, however, there are main differences between our proposed scheme and the work in \\cite{7442902}.\nA clustering NOMA scheme is developed in \\cite{7442902} by grouping\nusers in each cluster.\nIn this scheme, a single beamformer is designed to transmit the signals for all users\nin the same cluster whereas, in this letter, the signal for each user is transmitted with a dedicated beamformer.\nIn addition, both beamforming designs are completely different as the work in \\cite{7442902}\nproposes robust sum-rate maximization based design whereas this letter solves robust power minimization problem\nwith rate constraint on each user. In terms of solutions, the work in \\cite{7442902}\nexploits the relationship between MSE and achievable rate and derives an equivalent non-convex problem, which\nis decoupled into four sub-problems and those problems are iteratively solved to realize the solution of the original problem.\nIn this letter, the robust power minimization problem is formulated by deriving the worst-case achievable rate.\nThe original problem formulation turns out to be non-convex and we exploit S-Procedure and semidefinite\nrelaxation to convert it to a convex one. Hence, the work in \\cite{7442902} and the proposed work in\nthis letter are different including problem formulation and the solution approaches.}\n\n\\vspace{-0.2cm}\n\\section{System Model and Problem Formulation}\nWe consider NOMA-based downlink transmission where a\nbase station (BS) sends information to $K$ users\n$U_1,U_2, \\ldots, U_K$.\nIt is assumed that the BS is equipped with $M$ antennas whereas each user consists of a single antenna.\nThe channel coefficient vector between the BS and the $k^{th}$ user $U_k$ is denoted by $\\mathbf {h}_k \\in \\mathbb{C}^{M \\times 1}~ (k = 1,\\ldots,K)$\nand $\\mathbf{w}_{k}\\in \\mathbb{C}^{M \\times 1}$ represents the corresponding beamforming vector of the $k^{th}$ user $U_{k}$.\nThe received signal at $U_k$ is given by\n\\begin{equation}\\label{signal}\n y_k = \\mathbf {h}_k^H \\mathbf{w}_k s_k + \\sum_{m\\neq k}\\mathbf{h}_k^H \\mathbf {w}_m s_m + n_k, \\qquad \\forall k,\n\\end{equation}\n\n\\vspace{-0.3cm}\n\\noindent where $s_{k}$ denotes the symbol intended for $U_{k}$\nand \\mbox{$n_k \\sim \\mathcal{CN }(0, \\sigma_k^2)$} represents a zero-mean additive white Gaussian noise with variance $\\sigma_k^{2}$.\nThe power of the symbol $s_{k}$ is assumed to be unity, i.e., $\\mathbb{E}(|s_k|^2)=1$. In practical scenarios, it is difficult to provide perfect CSI at the transmitter due to \nchannel estimation and quantization errors. Therefore, we consider a robust beamforming design to overcome these channel uncertainties.\nIn particular, we incorporate norm-bounded channel uncertainties in the design as\n\n\\vspace{-0.5cm}\n\\begin{align} \\label{delta}\n& \\mathbf{{h}}_k=\\mathbf{\\hat{h}}_k +\\Delta\\mathbf{\\hat{h}}_k,\\quad \\|\\Delta\\mathbf{\\hat{h}}_k\\|_2=\\|\\mathbf{{h}}_k-\\mathbf{\\hat{h}}_k\\|_2 \\leq \\epsilon,\n\\end{align}\n\n\\vspace{-0.25cm}\n\\noindent where $\\mathbf{\\hat{h}}_k$, $\\Delta\\mathbf{\\hat{h}}_k$ and\n$\\epsilon \\geq 0$ denote the estimate of $\\mathbf{{h}}_k$, the norm-bounded channel estimation error and the channel estimation error bound, respectively.\n\nIn the NOMA scheme, user multiplexing is performed in the power domain\nand the SIC approach is employed at receivers to separate signals between different users.\nIn this scheme, users are sorted\nbased on the norm of their channels,\ni.e., $\\|\\mathbf{h}_{1}\\|_2\\leq\\|\\mathbf{h}_{2}\\|_2\\leq\\ldots\\leq\\|\\mathbf{h}_{K}\\|_2$.\nFor example, the $k^{th}$ user decodes the signals intended for the users from $U_{1}$ to $U_{k-1}$\nusing the SIC approach\nwhereas the signals intended for the rest of the users (i.e., $U_{k+1},\\ldots,U_{K}$) are treated as interference at the $k^{th}$ user.\nBased on this SIC approach, the $l^{th}$ user can detect and remove\nthe $k^{th}$ user's signals for $ 1 \\leq k < l $ \\cite{7277111}.\nHence, the signal at the $l^{th}$ user after removing the first $k-1$ users' signals to detect the $k^{th}$ user is represented as\n\n\\vspace{-0.6cm}\n\\begin{align}\\nonumber\ny_l^k=\\mathbf {h}_l^H \\mathbf{w}_k s_k+\\sum_{m=1}^{k-1} \\Delta\\mathbf{\\hat{h}}_l \\mathbf{w}_m s_m+\\sum_{m=k+1}^{K}\\mathbf{h}_l^H \\mathbf {w}_m s_m+n_l,&&\n\\end{align}\n\n\\vspace{-0.7cm}\n\\begin{eqnarray} \\label{signal2}\n \\qquad\\qquad \\qquad\\qquad\\quad\\quad\\forall k, \\, l\\in\\{k,k+1,\\ldots,K\\},&&\n\\end{eqnarray}\n\n\\vspace{-0.3cm}\n\\noindent\nwhere the first term is the desired signal to detect $s_k$ and\nthe second term is due to imperfect CSI at the receivers during\nthe SIC process.\nDue to the channel uncertainties, the signals intended for the users $U_{1},\\ldots,U_{k-1}$ cannot be completely removed by the $l^{th}$ user. The third term is the interference\nintroduced by the signals intended to the users $U_{k+1},\\ldots,U_{K}$.\nAccording to the SIC based NOMA scheme, the $l^{th}$ user should be able to detect\nall $k^{th}\\:(k =\nN_f (246.7 MeV)^3$ and $f_{\\pi} = 93 MeV$. For the remainder of this paper we\nwill use the numerical solution\n\\cite{Be,BeHaMe}\n\\begin{eqnarray}\nM = 335.1 MeV \\qquad \\Lambda = 630.9 MeV \\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\n\\tilde{G}\\Lambda^2 = 2.196,\n\\label{e2.24}\n\\end{eqnarray}\nwhich assumes a current quark mass $m_u = m_d = 4 MeV$.\n\n\\section{POLYAKOV LOOP POTENTIAL}\n\\label{s3}\nWe denote $tr_c {\\cal P}$ by $\\phi$. Near the deconfinement temperature $T_D$\nthe potential of Polyakov loops in pure QCD can be approximated by a polynomial\npotential of the form \\cite{TrWi}\n\\begin{eqnarray}\nU(\\phi ,T) = \\lambda \\left( \\phi - {\\phi^2 \\over \\phi_D} \\right)^2 -\n{T - T_D \\over T_D L^{-1}} \\left( {\\phi \\over \\phi_D} \\right)^3\n\\label{e3.1}\n\\end{eqnarray}\nThe potential is parametrized such that for a pure gauge theory the phase\ntransition occurs at $T_D$ with a latent heat $L$ and the Polyakov loop jumping\nfrom $0$ to $\\phi_D$.\n\nThis choice for $U$ is phenomenological. Although perturbative QCD does yield a\nquartic polynomial potential for the $A_0$ field, the potential so obtained\ndoes not display critical behavior, and it is only valid at high temperatures\n\\cite{We}.\nHowever, it does suggest that the parameter $\\lambda$ should be taken to be on\nthe order of $T_D^4$. We have chosen $\\lambda = T_D^4$ and $L = 2T_D^4$ for the\nnumerical calculations below, neglecting any possible temperature corrections\nor other dependencies in $\\lambda$, $T_D$, $\\phi_D$ and $L$. The direct\nassociation of $T_D$, $\\phi_D$ and $L$ with measureable quantities holds only\nfor very heavy quarks. For light quarks, these parameters must be determined\nby fitting to the observed behavior.\n\nWe can now couple the chiral symmetry and deconfinement phase transitions. Let\n\\begin{eqnarray}\n{\\cal V}(M,\\phi,T) = V(M,\\phi,T) + U(\\phi,T),\n\\label{e3.3}\n\\end{eqnarray}\nwhere $V(M,\\phi,T)$ is given in Eq.~(\\ref{e2.20}) with ${\\rm tr_c}{\\cal P} =\n\\phi$. To determine the critical behavior of the coupled effective potential,\nthe absolute minimum of the potential ${\\cal V}(M,\\phi,T)$ as a function of $M$\nand $\\phi$ must be found as $T$ is varied. A satisfactory determination of the\nentire phase diagram requires numerical investigation.\n\n\\section{RESULTS}\n\\label{s4}\nThe two flavor Nambu-Jona-Lasinio model has a second-order chiral phase\ntransition for massless quarks. For the parameter set used here, this\ntransition occurs at $T_{\\chi} = 194.6$ MeV, as determined numerically from\n$V(M,\\phi \\equiv 1,T)$. The order of the transition is consistent with Monte\nCarlo simulations of two-flavor QCD, and the critical temperature is plausible.\nHowever, the finite temperature quark determinant is not consistent with the\nknown behavior of QCD unless the effects of a non-trivial Polyakov loop are\nincluded. As Eq.~\\ref{e2.16} makes clear, the effects of finite temperature in\nthe quark determinant are suppressed by the small expected value of the\nPolyakov loop at low temperatures. Without the Polyakov loop effects, the\nconventional Nambu-Jona-Lasinio model displays the $T^4$ behavior of a free\nquark gas at arbitrarily low temperatures.\n\n\\begin{figure}[htb]\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\special{psfile=Fig1.ps angle=0 hoffset=30 voffset=-28 hscale=35 vscale=35}\n\\caption {Schematic behavior of $\\phi$ and $M$ in the three regions of the\n$(T_D, \\phi_D)$ plane as temperature increases.}\n\\label{f1}\n\\end{figure}\n\nNumerical investigation shows that the critical behavior of\n${\\cal V}(M,\\phi,T)$ is most sensitive to $T_D$ and $\\phi_D$. Figure~\\ref{f1}\nillustrates the three distinct types of critical behavior observed as $T_D$ and\n$\\phi_D$ are varied with the other parameters held fixed and the current mass\n$m$ set to $0$. In Region I of the $(T_D, \\phi_D)$ plane the effective quark\nmass changes continuously from its $T = 0$ value to a mass of $0$ MeV as the\ntemperature increases, signaling a second-order chiral phase transition.\nSimultaneously, the expectation value of the trace of the Polyakov loop\nincreases smoothly with temperature, exhibiting a large, but continuous,\nincrease in the vicinity of the chiral transition. This region exhibits\nbehavior most similar to that observed in simulations of two-flavor QCD. In\nRegion II the Polyakov loop experiences a first-order jump at some $T_c$. At\nthe same $T_c$ the constituent quark mass undergoes a sudden drop, but chiral\nsymmetry is {\\it not} restored. As the temperature increases beyond $T_c$, the\nconstituent mass moves continuously to zero. Thus, there are two distinct phase\ntransitions in Region II, a first-order transition driven by the dynamics of\ndeconfinement, and a later second-order chiral symmetry restoring transition.\nIn Region III the Polyakov loop experiences a first-order jump at some $T_c$,\nand simultaneously the effective quark mass drops suddenly to $0$ MeV,\nindicating a single first-order transition which restores chiral symmetry.\n\nFigure~\\ref{f2} shows the location of the three regions in the $(T_D, \\phi_D)$\nplane. Figure~\\ref{f3} plots the behavior of the Polyakov loop, $\\phi$, and the\nconstituent quark mass, $M$, as a function of temperature at the point\n$(T_D = 220 {\\rm MeV}, \\phi_D = 1.2)$ in Region II. There is a clear separation\nof the first and second-order transitions.\n\n\\begin{figure}[htb]\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\special{psfile=Fig2.ps angle=0 hoffset=12 voffset=-45 hscale=30 vscale=25}\n\\caption{Phase diagram in the $(\\phi_D, T_D)$ plane.}\n\\label{f2}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\special{psfile=Fig3.ps angle=0 hoffset=9 voffset=-60 hscale=25 vscale=25}\n\\caption{Plot of $\\phi$ and $M$ as a function of temperature at the point\n$(T_D = 220 {\\rm MeV}, \\phi_D = 1.2)$ in Region II.}\n\\label{f3}\n\\end{figure}\n\n\\section{LATTICE VERSION OF THE EFFECTIVE POTENTIAL}\n\\label{s5}\nA lattice version of the effective potential has the advantage that many\nparameters can be set from QCD simulations. There are two related issues in\nextending our work to the lattice: the choice of a lattice fermion formalism\nand setting other parameters in the effective action.\n\nA lattice version of the NJL model can be straightforwardly implemented using\nnaive fermions \\cite{BiVr}. Species doubling can be handled by a formal\nreplacement of $N_c$ with $N_c \/ 16$. Using the single quark current mass,\nchiral condensate, and constituent mass of the last section as inputs, we find\n\\begin{eqnarray}\na^{-1} = 389.0 MeV \\qquad \\tilde{G}a^{-2} = 0.8343.\n\\label{e5.8}\n\\end{eqnarray}\nHence, $f_{\\pi} = 94.09$ $MeV$. Also note that $M_{T=0}a$ is of order 1.\nThe temperature is given by $T = 1 \/ (N_t a)$ where $N_t$ is the temporal\nextent of the lattice. This places a problematic upper limit on a discrete\nrange of available temperatures. Given our choice of parameters,\n$T_{max}$ is only $194.5$ $MeV$.\n\nThe temperature may be changed continuously by an asymmetric rescaling of the\nlattice in which the lattice spacing $a_t$ in the temporal direction varies\nindependently of the spatial lattice spacing $a_s$ \\cite{Ka}. Defining\n$a_s = a$ and $\\xi = a_t \/ a_s$, we now have in physical units\n$T = 1 \/ (N_t a \\xi)$. The effective potential is\n\\begin{eqnarray}\nV(M, \\phi, \\xi) = \\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\n- {N_f N_c \\over 4 \\xi} \\int_{-\\pi}^{\\pi} {d^3 \\vec{k} \\over (2 \\pi)^3}\n\\rm{ln} \\left[ \\sqrt{1 + {\\xi}^2 A^2(\\vec{k})} + \\xi A(\\vec{k})\n\\right] \\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\n- {N_f N_c \\over 2 N_t \\xi} \\int_{-\\pi}^{\\pi} {d^3 \\vec{k} \\over (2 \\pi)^3}\n\\rm{ln} \\Bigl\\{ 1 + \\Bigl[ \\sqrt{1 + {\\xi}^2 A^2(\\vec{k})} \\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\n- \\xi A(\\vec{k}) \\Bigr]^{N_t} {tr_c(\\phi) \\over N_c} \\Bigr\\}\n+ {(M - m)^2 \\over 4 \\tilde{G}}\n\\label{e5.11}\n\\end{eqnarray}\nwhere $A(\\vec{k}) = \\sqrt{M^2 + \\sum_j {\\sin}^2(p_j)}$.\nHowever, using the same inputs as the symmetric lattice case, as well as\n$f_{\\pi} = 93$ $MeV$, we find the unique answer\n\\begin{eqnarray}\na^{-1} = 477.2 MeV \\qquad \\tilde{G}a^{-2} = 1.255\n\\label{e5.15}\n\\end{eqnarray}\nwith $\\xi = 2.002$.\n\nAn alternative to asymmetric lattices might be variant or improved actions for\nwhich $a^{-1}$ is larger.\n\n\\section{CONCLUSION}\n\\label{s6}\nPolyakov loop effects have a strong impact on the possible critical behavior of\nthe NJL model. The universality argument \\cite{PiWi} which predicts a\nsecond-order chiral transition for two flavors and a first-order transition for\nthree or more flavors may fail when this additional order parameter is\nincluded.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe existence of gravitational waves is predicted in general relativity\nand many modified gravity theories. By observing gravitational waves (GWs),\nwe can not only test the general relativity and modified gravity\ntheories in strong gravitational fields, but also reveal the merger\nprocess of compact objects through the history of the universe and\nphysics in the early universe such as inflation ({\\it e.g.}\n\\cite{2011PhRvL.107u1301G}). In particular, GWs which\noriginate from the cosmological inflation have almost scale-invariant\nspectrum and propagate freely since their generation, and thus the\ndetection of such scale-invariant GWs can be\nconsidered as a direct proof of the existence of cosmological inflation.\nIn order to prove the primordial GWs to be\nscale-invariant, we have to observe GWs with a wide frequency range \nand high sensitivity.\n\nPrecise measurements of primordial GWs at high\nfrequencies will tell us about the thermal history of the early\nuniverse, which could not be reached in other ways. For example, it has\nbeen suggested that the amplitude of GWs at the\nfrequency range $f \\gtrsim 10^{-2.5}$ Hz can be used to infer the reheating\ntemperature \\cite{2008PhRvD..77l4001N}, and the epoch of quark-gluon\nphase transition and neutrino decoupling at lower frequency range \n$10^{-10} \\lesssim f \\lesssim 10^{-8}$ Hz \\cite{2009PhRvD..79j3501K}. \nIf the universe underwent a strongly\ndecelerating phase after inflation realized in quintessential inflation\nmodels \\cite{1999PhRvD..59f3505P},\ninflationary GWs have a blue spectrum at higher frequency $f\\gtrsim 10^{-2}$ Hz\n\\cite{1999CQGra..16.2905G,2004CQGra..21.1761T}. Therefore, while the\nfirst detection of primordial GWs is expected through\nthe CMB experiments which probe the waves at smallest frequencies\n({\\it e.g.} \\cite{2000cils.book.....L})\nit is significant to explore gravitational\nwaves with a wide range of frequency.\n\nIn addition, measurements of GWs at high frequency will\nopen a new window on the black hole merger events in the universe.\nMatsubayashi {\\it et al.} showed that the frequency of gravitational\nwaves that originate from mergers of intermediate mass black holes\n(IMBHs) falls inside the range of $10^{-5}\\lesssim f \\lesssim 10^{1}$ Hz, and one\ncan discover or set constraints on these merger events within several\ngigapersec from the Earth in future gravitational wave observation\nprojects \\cite{2004ApJ...614..864M}.\n\nIn order to observe GWs, many kinds of gravitational wave\ndetectors have been proposed and developed. The Laser Interferometer\nGravitational Wave Observatory (LIGO) has already set a strong\nconstraint on the strain amplitude $h_{\\rm c}<4\\times 10^{-24}$ at the\nfrequency of GWs around $10^{2}$ Hz\n\\cite{2009RPPh...72g6901A,2013PrPNP..68....1R}. However, seismic\noscillations interfere ground-based observations of gravitational\nwaves at lower frequencies as $f\\lesssim 10$ Hz ({\\it see}\n\\cite{2001PhyU...44R...1G}). Aiming at this frequency range, Torsion\nbar detectors such as TOBA have been developed. Ishidoshiro {\\it et al.}\nset a constraint on the strain amplitude of the continuous component of\nGWs as $h_{\\rm c}< 2\\times 10^{-9}$ at $f\\sim 0.2$Hz\n\\cite{2011PhRvL.106p1101I}. \n\n\n\nAt slightly lower frequency range, with some planetary explorers,\nconstraints such as $h_{\\rm c}<1\\times 10^{-15}$ at $f=10^{-2}$ Hz \\cite{1995A&A...296...13B}\nand $h_{\\rm c}<2\\times 10^{-15}$ at $f=3\\times 10^{-4}$ Hz \\cite{2003ApJ...599..806A}\nhave been set by the Doppler tracking method.\nHowever it is difficult to distinguish\nsignals of GWs as the frequency of targeting\nGWs increases higher than $10^{-2}$ Hz due to the noise\nin the electric circuits of which are installed in the receiver in this method. \n\nIn this work, we consider the modulation of frequency of radio\nwaves from GPS satellites by stationary\/prompt GWs.\nGPS satellites emit precious, stable and high intensity radio waves \nin order to provide positional accuracy for the GPS in navigation. \nIt is possible because the oscillators of GPS satellites are designed to \nbe synchronized with atomic clocks which are loaded with the satellites.\nThese oscillators are so stable that \nthe fractional error of the frequency of their emitting radio waves \nis suppressed to $\\Delta \\nu \\slash \\nu<10^{-15}$\n\\cite{2008.gps.standard}.\n\nThe GPS method is an application of the Doppler tracking method which has two\nadvantages. First, GPS method can probe in the frequency range \n$f\\gtrsim 10^{-2}$ Hz where the conventional Doppler tracking methods can not\nreach because GPS satellites are much closer than the planetary\nexplorers, while the distance is large enough for the noise due to\nseismic oscillations to be negligible as shown in the next section.\nSecondly, the radio waves from GPS \nsatellites can be detected everywhere and everytime on the ground.\nThis condition will be suitable to make cross correlation study and\ncrucial to detect GWs from prompt events. \n\nIn this study, we set a constraint on the amplitude of GWs\nfor $0.01 \\le f \\le 1$ Hz by using the Doppler tracking method with\nGPS disciplined oscillators (GPSDOs). The GPSDO is the stable oscillator with\na quartz oscillator whose output is controlled to agree with the signals\nfrom GPS satellites \\cite{2008.gps.standard,2005ptti.conf..677L}. To\nhave GPSDOs operating with high stability, the amplitude of the\ncontinuous components of GWs should be small. Recently,\nthe frequency stability of GPSDOs for a short time interval about from\nseconds to a few hundred seconds has been reached to the level of\n${\\Delta \\nu}\\slash{\\nu} \\simeq 10^{-12}$ \\cite{2008.gps.standard,2005ptti.conf..677L}. \nThis short time stability of the\nfrequency of GPSDOs enables us to set a constraint on the strain\namplitude of the continuous GWs.\n\nThis paper is organized as follows. We estimate the effects of\nGWs on the \nmeasurements of the radio waves from GPS satellites and derive the \nconstraint in \\S 2. In our\nformulation, we adopt TT gauge to describe the GWs. In\n\\S 3, we discuss implications of the result to the\nmerger events of IMBHs and inflationary GWs. We conclude\nthis paper in \\S 4. Throughout \nthis paper, $\\nu$ and $f$ mean the frequency of the electro-magnetic\nwaves and the GWs, respectively. The speed of light is\ndenoted by $c$. A dot represents a partial derivative respective to the\nphysical time, {\\it i.e.} $\\dot{x}\\equiv {\\partial x}\\slash{\\partial\nt}$.\n\n\n\\section{Effects of GWs on GPS measurements}\nIn this section, we consider the effect of gravitational wave\nbackgrounds (GWBs) on the radio waves emitted by GPS satellites. Here\nwe assume that GWBs are expected to be isotropic and stationary ({\\it\nsee.} \\cite{2000PhR...331..283M}). For GWs whose\nwavelength are longer than the typical distance between GPS satellites\nand detectors on the ground, the frequency of radio waves emitted by the\nsatellites is modulated as \n\\begin{equation} \n\\dfrac{\\Delta \\nu}{\\nu}=\n\\dfrac{l}{2c}\\dot{h}(t)\\sin^{2}\\theta ~,\\label{fractional01}\n\\end{equation}\nwhere $\\dot{h}(t)$, $\\theta $ and $l$ are the time derivative of the\namplitude of GWs at time $t$, the angle between the \ndirections of propagation of radio waves and gravitational\nwaves, and the distance between the GPS satellite and the observer,\nrespectively.\n\nBy assuming GWs are monotonic and plane waves\nparallel to the $z$-axis, the amplitude of GWs $h(t)$ and its time derivative $\\dot{h}(t)$\ncan be written with the strain $h_{\\rm c}$ as\n\\begin{eqnarray} \nh(t)&=& h_{\\rm c}\\sin \\left(2\\pi f \\left(t-\\dfrac{z}{c}\\right)+\\phi \\right)~,\\\\\n\\dot{h}(t)&=& 2\\pi f h_{\\rm c}\\cos \\left(2\\pi f \\left(t-\\dfrac{z}{c}\\right)+\\phi \\right)~,\\label{differential}\n\\end{eqnarray}\nwhere $\\phi $ is the phase of GWs at $z=t=0$.\n\nFrom Eq. (\\ref{fractional01}), signals of GWs generate\nan additional shift of frequency of radio waves. When one considers the\ncase where $\\theta =\\pi \\slash 2$, the effect of GWs on\nthe frequency modulation of electro-magnetic waves can be estimated as\n\\begin{equation} \n\\dfrac{\\Delta \\nu}{\\nu}=\n\\dfrac{\\pi fl}{c}h_{\\rm c}~.\n\\label{fractional02}\n\\end{equation}\nIn reality, the signal sourced by GWs is buried within \nthe noise. Therefore, if one can receive \nthe radio waves which is emitted at a distance of $l$ \nwith a time variance of the frequency fluctuations $\\sigma$\n\\cite{baran2010modeling}, \none can set the upper bound of the strain amplitude of gravitational\nwaves as, \n\\begin{equation} \nh_{\\rm c}<\\dfrac{c}{\\pi lf}\\sigma ~.\\label{hc}\n\\end{equation}\n\nFor the GPS, the distance $l$ is approximately $2\\times 10^{7}$ m and \nthe standard variation of frequency from GPS satellites converges to \n$\\sigma \\simeq 1\\times 10^{-12}$ \nby integrating the signal from GPS satellites for the period \nfrom one second to one hundred seconds \n\\cite{2005ptti.conf..677L,2008.gps.standard}.\nBy receiving the signal from the GPS for a period $t_{\\rm i}$, \nthe frequency range of the GWs which one can probe is limited\nas $f\\ge t_{\\rm i}^{-1}$. By combining it with the condition \nthat the wavelength of GWs is longer than \nthe distance between the GPS satellites and the observer, \nthe probed range of frequency with the GPS can be written as\n\\begin{equation} \nt_{\\rm i}^{-1}\\le f \\le \\dfrac{l}{c}~.\n\\end{equation}\n\nFrom Eq. (\\ref{fractional01}) \nbecause the amplitude of\nthe modulation is proportional to the frequency of GWs\n$f$, the strain amplitude $h_{\\rm c}$ is constrained tighter for higher\nfrequency. By substituting the numbers into Eq. (\\ref{hc}), we can set\na constraint on the strain of GWBs as\n\\begin{equation} \nh_{\\rm c}<4.8\\times 10^{-12}\\left(\\dfrac{1 {\\rm Hz}}{f} \\right)\n~.\\label{constraintGPS}\n\\end{equation}\nIn figure \\ref{fig.1}, we plot the constraint on the strain amplitude of\nthe continuous component of GWs and compare the result\nfrom the torsion bar detector \\cite{2011PhRvL.106p1101I}. We find that\nthe GPS gives a tighter constraint on the relevant frequency range.\n\nThe GPS method does not suffer seismic oscillations that disturb\nground-based observations. The reason can be given as follows.\nIn the GPS method, the effect of GWs can be seen \nas changes of observed distances between the satellites and observers.\nWhen plane and monotonic GWs come to an observer, \nthe observed distance $l$ between the satellite and the observer on the\nground can be written in terms of the proper distance $l_0$ as \n\\begin{equation} \nl=l_{0}\\left(1+h \\right)+x_{\\rm s}~,\\label{equivalentEq}\n\\end{equation}\nwhere $h$ is the amplitude of GWs and $x_{\\rm s}$ is the\namplitude of seismic oscillations. Thus the effect of seismic\noscillations relative to the strain amplitude of GWs can\nbe characterized by $x_{\\rm s}\\slash l_{0}$. Shoemaker {\\it et al.}\nreported a typical power spectrum of seismic oscillations as\n\\cite{1988PhRvD..38..423S}\n\\begin{equation} \nx_{\\rm s}\\simeq 3\\times 10^{-7} \\left(f\\slash 1~{\\rm Hz}\n\t\t\t\t\\right)^{-2}~{\\rm [m]}~. \n\\end{equation}\nBecause $l_{0}\\simeq 2.0\\times 10^{7}$ m, the effect of seismic\noscillations is suppressed as,\n\\begin{equation} \n\\dfrac{x_{\\rm s}}{l_{0}}\\simeq \n1.5\\times 10^{-14}\\left(f\\slash 1~{\\rm Hz}\n\t\t\t\t\\right)^{-2}~.\n\\end{equation}\nThis is smaller than the upper limit of our constraint, Eq\n(\\ref{constraintGPS}). \n\nIn addition, because GPS satellites fly much closer to the ground\nthan the planetary explorers which have been used in the Doppler\ntracking method, we can set a constraint on the strain amplitude of GWBs\nin the higher frequency region than those of ULYSSES and Cassini (\n\\cite{1995A&A...296...13B,2003ApJ...599..806A}).\n\\begin{figure}\n \\begin{center}\n \\includegraphics[width=85mm]{gw}\n \\end{center}\n \\caption{The upper limit on the strain amplitude of GWBs from GPS\n satellites (solid line). The dashed line represents the constraint\n from the torsion bar detector \\cite{2011PhRvL.106p1101I}. The dotted\n line represents the constraint from the Doppler tracking\n method\\cite{1995A&A...296...13B}. } \\label{fig.1}\n\\end{figure}\n\nThe intensity of GWBs can be characterized by the dimensionless\ncosmological density parameter $\\Omega_{\\rm gw}(f)$. The parameter is\ndefined as\n\\begin{equation} \n\\Omega_{\\rm gw}(f)=\\dfrac{10\\pi^{2}}{3H_{0}^{2}}(fh_{\\rm c})^{2}~,\n\\end{equation}\nwhere $H_{0}$ is the Hubble constant, and the Planck collaboration\nreported as $H_{0}=67.11~{\\rm km\/sec\/Mpc}=2.208\\times 10^{-18}~{\\rm\nsec}^{-1}$\\cite{2013arXiv1303.5076P}. Then the constraint Eq.~(\\ref{constraintGPS}) can be written as\n\\begin{equation} \n\\Omega_{\\rm gw}(f)<1.7\\times 10^{14} ~~{\\rm for}~10^{-2} \\lesssim f \\lesssim 10^{0}[{\\rm Hz}]~.\n\\end{equation}\nIn figure \\ref{fig.2}, we compare our constraint with the previous ones.\nIt can be seen that the Doppler tracking method with the GPS is setting\na constraint on the amplitude of continuous component of gravitational\nwaves in the frequency range of $10^{-2} \\lesssim f \\lesssim 10^{0}$ Hz, a window between\nthe constraints from the Torsion bar and the planetary explores.\n\n\\begin{figure}\n \\begin{center}\n \\includegraphics[width=85mm]{gwOmega2}\n \\end{center}\n \\caption{Summary of the constraints on GW background in terms of\n$\\Omega_{\\rm gw}(f)$, which includes {\\tt COBE} \\cite{2000PhR...331..283M}, {\\tt CMB\nhomogeneity} and {\\tt BBN} \\cite{2006PhRvL..97b1301S}, {\\tt Pulsar\ntiming} \\cite{2006ApJ...653.1571J}, {\\tt LLR}\n\\cite{1981ApJ...246..569M,2009IJMPD..18.1129W}, {\\tt Cassini}\n\\cite{2003ApJ...599..806A}, {\\tt ULYSSES} \\cite{1995A&A...296...13B},\n{\\tt Lunar orbiter} \\cite{2001Icar..150....1K},\n{\\tt Torsion bar} \\cite{2011PhRvL.106p1101I}, {\\tt LIGO}\n\\cite{2009RPPh...72g6901A,2013PrPNP..68....1R}, and {\\tt GPS}.} \\label{fig.2}\n\\end{figure}\n\n\n\\section{Discussion}\n\n\nIn this work, we show that satellites with atomic clocks are available\nfor setting constraints on the strain amplitude of GWs\nat $10^{-2}\\lesssim f \\lesssim 10^{0}$ Hz. The constraints based on the Doppler tracking\nmethod with planetary explorer such as ULYSSES and Cassini are mainly\nlimited from the stability of the hydrogen maser clock on the ground\n$\\sim 10^{-15}$ at the frequency range $10^{-6}\\lesssim f\\lesssim 10^{-2}$ Hz. \nThe stability of the optical lattice clock is expected\nto reach $10^{-18}$ \\cite{2003PhRvL..91q3005K}. If future explorers \nhave the optical lattice clock on board, they will become useful\ninstruments for detection\/setting constraints on the strain amplitude\nof GWs ({\\it see also }\\cite{2006LRR.....9....1A}).\n\nDetermination accuracy of frequency fluctuations of received radio waves \nfrom GPS satellites depends mostly on the time resolution of \nthe received electric-signal in the A\\slash D converter\nof the GPS receiver $\\sigma_{\\rm r}$. It is reported that $\\sigma_{\\rm r}\n\\ge 2\\times 10^{-13}\\slash t_{\\rm i}$ and the largest error comes from\nthe frequency transfer \\cite{2005ptti.conf..149F}. It is difficult to\nimprove the GPS constraints on the strain amplitude until the \nresolution of the quantization in the A\\slash D converter is much improved.\n\nHere let us apply our result to setting a constraint on the\nmerger events of compact objects. In particular, it is predicted that\nmergers of the binary of IMBHs emit large GWs in the\nrelevant frequency range, while it is difficult to detect the events\nthrough X-rays or radio waves if the systems do not have accretion\ndisks. Therefore observations or constraints of GWs\nenable one to estimate the number density of the binary of IMBHs and the\nmerger events. \n\nDuring the quasi-normal (QNM) phase of the IMBHs mergers, in which\nmerging IMBHs are expected to emit the largest amplitude of \nGWs, the amplitude $h_{\\rm QNM}$, typical\nfrequency $f_{\\rm QNM}$ and the duration time of the event $t_{\\rm QNM}$\nare given by \\cite{2004PThPS.155..415M},\n\\begin{eqnarray} \nf_{\\rm QNM}&\\simeq &4\\times 10^{-2}\\left(\\dfrac{M}{10^{6} M_{\\odot}} \\right)^{-1}~[{\\rm Hz}],\\label{f.insp}\\\\\nh_{\\rm QNM}&\\simeq &2\\times 10^{-12}\\left(\\dfrac{M}{10^{6} M_{\\odot}} \\right)\n\\left(\\dfrac{\\varepsilon }{10^{-2}} \\right)^{\\frac{1}{2}}\n\\left(\\dfrac{R}{4 {\\rm kpc}}\\right)^{-1}\n~,\\label{h.insp} \\\\\nt_{\\rm QNM}&\\simeq &30\\times \n\\left(\\dfrac{M}{10^{6} M_{\\odot}} \\right)~[{\\rm sec}].\\label{t.insp}\n\\end{eqnarray}\nHere $\\varepsilon$ is the eccentricity of the orbit, $R$ is the distance\nto the binary, and we assumed that two IMBHs have the same mass $M$.\nFrom Eqs. (\\ref{constraintGPS}), (\\ref{f.insp}) and (\\ref{h.insp}), \none can rule out merger events of IMBHs from the GPS constraint as\n\\begin{eqnarray} \nR&\\gtrsim&0.1\\left(\\dfrac{\\varepsilon }{0.01} \\right)^{-1\\slash 2} [{\\rm kpc}]~\\\\\n& &{\\rm for}~4\\times 10^{4}M_{\\odot} \\le M \\le 4\\times 10^{6}M_{\\odot}~. \\notag\n\\end{eqnarray}\nThe minimum and maximum masses are \ndetermined from the frequency range we can probe in this method, i.e. Eq.(4).\n\nFrequency modulation signals induced by GWs may be\ndisturbed by the plasma effect in the ionosphere and the atmosphere. \nFluctuations of the column density of free electrons in the plasma,\ncalled the dispersion measure (DM), induce those in frequency of GPS\nradio waves as\n({\\it e.g.} \\cite{2004tra..book.....R,Jackson})\n\\begin{equation} \n\\dfrac{\\Delta \\nu}{\\nu}=\\dfrac{e^{2}\\nu^{-2}}{2\\pi m_{\\rm e}cr}{\\rm DM}~,\n\\end{equation}\nwhere ${\\rm DM}=\\int_{0}^{r}n_{\\rm e}ds$ is the column density of\nelectrons, $m_{\\rm e}$ and $e$ are the mass and the electric charge of\nan electron, respectively, and $r$ is the distance between the GPS\nsatellite and the observer. By comparing the above equation with\nEq. (\\ref{fractional02}), one can see that the frequency dependences are\ndifferent between the modulations induced by GWs and the\nplasma effect. Therefore the modulation originated from the plasma can\nin principle be removed by using multiple frequencies. Some of the\nGPSDOs observe the two bands of GPS radio waves (L1 \\& L2) \\footnote{The\nfrequency of the band L1 and L2 are 1.57542 GHz and 1.2276 GHz,\nrespectively.} and take into account this modulation.\n\nHowever, even when GPSDOs give the best performance, the distance at\nwhich one can detect the merger of IMBHs with GPSDOs is limited only to\n0.1 kilopersec from the Earth. In order to detect mergers of IMBHs and compact\nobjects and the GW background from inflation with realistic amplitude,\nspace-based gravitational wave detectors are needed. \nAs future gravitational wave detectors, Laser Interferometer Space Antenna (LISA)\nand Deci-hertz Interferometer Gravitational wave Observatory (DECIGO)\nare in progress. LISA and DECIGO are expected to reach $h_{\\rm c}\\simeq 3\\times\n10^{-21}$ at $f\\simeq 6 \\times 10^{-3}$ Hz and $h_{\\rm c}\\simeq 2\\times 10^{-24}$ at\n$f\\simeq 0.3$ Hz, respectively\n\\cite{2013arXiv1305.5720C,2009JPhCS.154a2040S}. These sensitivities\nwill enable us to detect almost all the merger events of IMBHs with mass\n$M\\sim 10^{3}M_{\\odot}$ within the current horizon \\footnote{The current\nhorizon scale is approximately equal to the distance to the last\nscattering surface of CMB $d_{\\rm A(CMB)}$. Jarosik {\\it et al.}\nreports that $d_{\\rm A}=14.116^{+0.160}_{-0.163}$ Gpc\n\\cite{2011ApJS..192...14J}.}, and to reveal the properties of the strong\ngravity field and the cosmological inflation\n\\cite{2004PThPS.155..415M}.\n\n Finally let us discuss another constraint that can be obtained from the lunar\norbitting explorers in a similar way to the GPS constraint. \nIn the Apollo mission, in order\nto study the gravity field of the moon,\nlunar explorers such as Apollo 15 and 16 measured \nthe change in distance between the\nexplorers and the Earth precisely \nvia S-band transponders \\cite{2001Icar..150....1K}.\nThe typical distance between the explorers and the ground is $l\\approx\n3.8\\times 10^{8}$ m, which is much longer than that of the GPS case.\nIt was reported that anomalous oscillating motions had never been\nfound in their every ten-second sampling data with $1\\times\n10^{-4}~$m\/sec accuracy \\cite{2001Icar..150....1K}.\nFrom Eqs. \\eqref{differential} and \\eqref{equivalentEq}, this \nresult is translated to\nthe strain amplitude of GWs as $h_{\\rm c}<2.6\\times 10^{-13}$ at\nthe frequency range $f < c\/l \\simeq 0.8$Hz \\footnote{ To be more\nprecise, the distance between the moon and the earth is so long that\nthis constraint should be corrected at frequencies above 0.1Hz as\n$h_{\\rm c}<2.6\\times 10^{-13}\\sqrt{1+\\left(f\/0.16~{\\rm\n[Hz]}\\right)^{2}}$.\nAt $f=1$ Hz, we obtain the constraint $h_{\\rm c}<1.6\\times 10^{-12}$.\n}.\nThe constraint is also depicted in Fig. \\ref{fig.2}.\nRecent lunar explorers such as Kaguya \\cite{2009Sci...323..900N} \nand GRAIL \\cite{2013Sci...339..668Z,2010EGUGA..1213921Z} may improve this upper bound. \nIn addition, future\nmeasurements with lunar surface transponders \nsuch as those proposed by Gregnanin {\\it et al.} \n\\cite{2012P&SS...74..194G} will be available for setting \nconstraints on $h_{\\rm c}$.\n\n\\section{Conclusion}\nWe set a constraint on the strain amplitude of the continuous component\nof GWs as $h_{\\rm c }<4.8\\times 10^{-12}\\left(1[{\\rm\nHz}]\\slash f \\right)$ at the frequency range $10^{-2}\\lesssim f \\lesssim 1$ Hz \nwith the radio waves emitted by GPS\nsatellites in operation. \nBecause the distance between GPS satellites and observers\nis order $10^{7}$ m, seismic oscillations do not affect the\nconstraints on the strain amplitude. The sensitivity to the\nGWs is limited to that of the A\\slash D converter on the\nGPS receiver at the frequency range.\n\n\\section{Acknowledgments}\n{\nWe thank the referee, Peter L. Bender, for making us aware of the fact that past and future lunar orbitar experiments can be used to set better constraints on the amplitude of continuous GWs. Our thanks also go to Jun'ichi Miyao for discussion on the characteristics of the GPS, and Seiji Kawamura, Yoichi Aso on the noise in the electric circuits installed on the gravitational wave detectors.}\nThis work is supported in part by scientific research expense for Research Fellow of the Japan Society for the Promotion of Science from JSPS Nos. 24009838\n(SA) and 24340048 (KI).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe Auger spectroscopy involves the creation of two localized holes at\nor close to the same atom, hence giving access to local electronic\nproperties.\nDirect information on the local density of valence states is brought by\ncore-core-valence transitions; in the case of core-valence-valence (CVV)\nones, which will be investigated here, one can in addition access the\nscreened Coulomb repulsion amongst the two valence holes in the final\nstate which is relevant to a wide class of phenomena, and study its\neffects.\n\nFrom the theoretical point of view, a large amount of work has been\ndevoted to the calculation of the Auger spectra of solids during the\nlast three decades.\\cite{Verdozzi:2001:AugerRev} A general formulation\nof the dynamical Auger decay, where the creation of the initial core\nhole and the Auger decay are considered as coherent processes, was\ngiven by Gunnarsson and\nSch\\\"{o}nhammer\\cite{Gunnarsson:1980:dynTheoryAuger} but is of hard\npractical implementation.\nIn solids with (almost) closed valence bands, where no dynamical\ncore-hole screening can occur before the Auger decay, one can employ a\nsimpler two-step approximation and consider the above events as\nindependent.\nUnder this assumption, Cini\\cite{Cini:1977:CSM} and\nSawatzky\\cite{Sawatzky:1977:CSM} (CS) proposed a simple model\nproviding the Green's function describing the two valence holes left\nafter the Auger decay.\nGood agreement with experiments was achieved using fitting parameters\nfor the screened Coulomb interaction, giving a quantitative\nunderstanding of the Auger spectra of transition metals located at the\nbeginning and the end of the row, such as Ti,\\cite{Cini:1995:AugerTi}\nAg,\\cite{Cole:1994:AgAugerOffSite} and\nAu.\\cite{Verdozzi:1995:AuAugerOffSite} These results confirmed the\nusefulness of including explicitly on-site Hubbard terms to one-body\nHamiltonian, and prompted an extension to nearest-neighbour\ninteractions.\\cite{Verdozzi:1995:OffSiteRev}\n\nThese studies determined the relevant physical parameters by\nreproducing experimental findings within a semi-empirical approach.\nOf particular interest is the parameter governing the interaction\namongst the two holes in the final state, which has an analogue in the\npopular LDA+U description for correlated\nsystems.\\cite{Anisimov:1991:LDA+U, Anisimov:1993:LDA+U} Even if\nmethods for its {\\em ab initio} evaluation have been proposed, also\nthis quantity is often determined by phenomenological arguments.\n\nThe possibility of evaluating CVV spectra from first-principles, rather\nthan from a model with parameters fitted to experiments, would then be\nvery desirable as it would allow predicting different situations (e.g.,\ninvestigate the effect of a given chemical environment on the Auger\ncurrent) and a deeper interpretation of experimental findings.\n\nThis paper addresses such possibility, by proposing a\nmethod to compute the parameters entering the CS model by {\\em ab\ninitio} simulations. In this step towards a first-principle\ndescription of CVV Auger spectra in systems where the interaction of\nthe final-state holes cannot be neglected, we aim at highlighting the\nmost important contributions to the spectrum, which one should\nfocus to in forthcoming improvements.\nThe method is based on Density Functional Theory (DFT) simulations in\nthe Kohn-Sham (KS) framework, with constrained occupations. We make\nuse of comparison with reference atomic calculations to extrapolate\nthe electronic properties of the sample when they are more difficult\nto evaluate directly. Results are presented for the\n$L_{23}M_{45}M_{45}$ Auger line of Cu and Zn metals, which have been\nchosen as benchmark systems with closed $3d$ bands: the former being\nmore challenging for the proposed procedure, and the latter bearing\nmore resemblance with the atomic case.\n\nThe paper is organized as follows. In section~\\ref{sec:methods} we\ndescribe our method to evaluate the Auger spectra by first-principles\ncalculations. Section~\\ref{sec:results} presents our theoretical\nresults for Cu and Zn metals, comparing them with experimental results\nin the literature. In section~\\ref{sec:discussion} we analyze the\nweight of various contributions and discuss improvements. Finally,\nsection~\\ref{sec:conclusions} is devoted to conclusions.\n\n\n\\section{\\label{sec:methods}Theoretical methods}\n\n\\subsection{Model Hamiltonian and the Cini-Savatzky solution}\n\nWe describe the electron system in the hole representation, by an\nHubbard-like~\\cite{Hubbard:1963:HubbardH} model Hamiltonian\n\\begin{equation}\n\\label{eq:HHubbard}\nH=\\epsilon_c c_c^\\dagger c_c\n +\\sum_v\\epsilon_v c_v^\\dagger c_v\n +\\frac{1}{2}\\sum_{\\varphi_1 \\varphi_2 \\varphi_3 \\varphi_4} U_{\\varphi_1 \\varphi_2 \\varphi_3 \\varphi_4}\nc^\\dagger_{\\varphi_1}c^\\dagger_{\\varphi_2} c_{\\varphi_4}c_{\\varphi_3},\n\\end{equation}\nwhere $c$ and $v$ label the core state involved in the transition and\nthe valence states of the system, respectively, including the spin\nquantum number. In bulk materials, $v$ is a continuous index. The\nlast term is the hole-hole interaction Hamiltonian, parametrized by\nthe screened repulsion $U$, and is for simplicity restricted here to a\nfinite set of wavefunctions, $\\varphi$, centered at the emitting atom\n(hence neglecting interatomic interactions).\nIn closed shell systems, $\\epsilon_c$ and $\\epsilon_v$ yield the core\nand valence photoemission energies, since the two-body term has no\ncontribution on the one-hole final state or on the zero-hole initial\none.\n\nA two-step model is adopted to represent the Auger process, assuming\nthat the initial ionization and the following Auger decay of the core\nhole can be treated as two independent events. In other terms, we assume\nthat the Auger transition we are interested in follows a fully relaxed\nionization of a core shell.\nIf the ground state energy of the neutral $N$-electron system is chosen\nas a reference, the energy of the initial state is simply given by\n$\\epsilon_c$.\nThe total spectrum for electrons emitted with kinetic energy $\\omega$ is\nproportional to\n\\begin{equation}\n\\label{eq:Spectrum}\nS(\\omega)=\\sum_{XY} A^{*}_X D_{XY}(\\epsilon_c-\\omega) A_Y,\n\\end{equation}\nwhere $X$ and $Y$ are the final-state quantum numbers, $A_X$ is the\nAuger matrix element corresponding to the final state $X$, and\n$D_{XY}$ represents the two-hole density of states.\nNotice that Eq.~(\\ref{eq:Spectrum}) coincides with the Fermi golden rule\nif the states $X$, $Y$ are eigenstates of the Hamiltonian, so that\n$D_{XY}$ is diagonal.\nThe presence of the transition matrix elements effectively reduces the\nset of states contributing to Eq.~(\\ref{eq:Spectrum}) to those with a\nsignificant weight close to the emitting atom.\nThis motivates the approximation to restrict $X$ and $Y$ to two-hole\nstates based on wavefunctions centered at the emitting atom, such as\nthe set $\\{\\varphi\\}$ previously introduced.\nTherefore, the CVV spectrum is a measure of the two-hole local density\nof states (2hLDOS), with modifications due to the matrix elements.\nThe 2hLDOS could in principle be determined as the imaginary part of\nthe two-hole Green's function, $G_{XY}$, solution of\nEq.~(\\ref{eq:HHubbard}). However, because of the presence of the\nhole-hole interaction term, evaluating $G_{XY}$ is in general a\nformidable task.\n\nFor systems with filled valence bands, the two holes are created in a\nno-hole vacuum and one is left with a two-body problem. A solution in\nthis special case has been proposed by Cini\\cite{Cini:1977:CSM} and\nSawatzky,\\cite{Sawatzky:1977:CSM} and is briefly reviewed here (see\nRef.~\\onlinecite{Verdozzi:2001:AugerRev} for an extended review).\nThe two-holes interacting Green's function, $G$, is found as the\nsolution to a Dyson equation with kernel $U$, which reads:\n\\begin{equation}\n\\label{eq:G}\nG(\\omega)=G^{(0)}(\\omega) \\big(1-UG^{(0)}(\\omega)\\big)^{-1}.\n\\end{equation}\nHere, $G^{(0)}$ is the non-interacting Green's function which can be\ncomputed from the non-interacting 2hLDOS, $D^{(0)}$, via Hilbert\ntransform. Such 2hLDOS results from the self-convolution of the\none-hole local density of states (1hLDOS),\n$D^{(0)}{\\equiv}d{*}d$.\n\nThe quantum numbers $LSJM_J$ (intermediate coupling scheme) are the\nmost convenient choice to label the two-hole states, allowing for the\nstraightforward inclusion of the spin-orbit interaction in the final\nstate by adding to the Hamiltonian the usual diagonal term,\nproportional to $[J(J+1)-L(L+1)-S(S+1)]$.\nFinally, the CVV lineshape is:\n\\begin{equation}\n\\label{eq:PCiniLSJMJ}\nS(\\omega)=-\\frac{1}{\\pi}\n\\sum_{LSJM_J \\atop L'S'J'M_{J'}'} A^{*}_{LSJ} A_{L'S'J'} \\text{Im}\n\\left[\n\\frac{\n G^{(0)} ( \\epsilon_c - \\omega )\n}{\n 1 - U G^{(0)}( \\epsilon_c - \\omega )\n}\n\\right]_{LSJM_J \\atop L'S'J'M_{J'}'}.\n\\end{equation}\nFor comparison with experimental results, this is to be convoluted\nwith a Voigt profile to account for core-hole lifetime and\nexperimental resolution.\n\n\nIt is customary to isolate two limiting regimes:\n(i) When $U$ is small with respect to the valence band width $W$\n(broad, band-like spectra) the 2hLDOS is well represented by\n$D^{(0)}(\\omega)$. However, in such a case it might be even\nqualitatively important to account for a dependence of the matrix\nelements on the Auger energy $\\omega$. As a consequence, accurate\ncalculations of the lineshape require the simultaneous evaluation of\nthe matrix elements and the DOS.\n(ii) For $U$ larger than $W$, narrow atomic-like peaks dominate the\nspectrum, each peak from an $LSJ$ component. Hence, to the first\napproximation the spectrum is described by a sum of\n$\\delta$-functions, weighted by matrix elements whose dependence on\nthe Auger energy may be neglected. If we take the matrix $U$ diagonal\nin the $LSJ$ representation, and indicate by $E^{(0)}_{LSJ}$ the\nweighted average of $D^{(0)}_{LSJ}(\\omega)$, one obtains:\n\\begin{equation}\n\\label{eq:PDeltaLSJ}\nS(\\omega)\\approx\\sum_{LSJ}(2J+1)|A_{LSJ}|^2\n\\delta\\big( (\\epsilon_c - E^{(0)}_{LSJ} - U_{LSJ}) - \\omega \\big).\n\\end{equation}\nAtomic matrix elements can be taken as a first approximation, often\nsatisfactory, and can be evaluated as shown in\nRef.~\\onlinecite{Antonides:1977:LMM_Cu_Zn_Ga_Ge_I}.\nAn approach which could bridge between these two limiting regimes,\nconsidering both finite values of $U$ and the energy dependence of the\nmatrix elements, is still missing to our knowledge.\n\nIn the present work we adopt Eq.~(\\ref{eq:PCiniLSJMJ}) in order to\nsimulate the spectrum. Accordingly, one has to determine the quantities\n$A$, $U$, $D^{(0)}(\\omega)$, and $\\epsilon_c$.\nIn this paper we make use of a $U$ matrix which does not include the\nspin-orbit interaction, and is diagonal on the $LS$ basis.\nWe take atomic results in the literature for the matrix elements $A$,\nwhich are assumed independent of $J$\ntoo.\\cite{Antonides:1977:LMM_Cu_Zn_Ga_Ge_I}\nThe other quantities are computed by DFT simulations, as detailed in the\nfollowing Section.\n\n\n\n\\subsection{{\\em Ab initio} determination of the relevant parameters}\n\nTo evaluate {\\em ab initio} the photoemission energies we use a method\nclosely related to Slater's transition-state theory, while the\nparameter $U$ is computed following a general procedure first proposed\nin Ref.~\\onlinecite{Gunnarsson:1989:DFTcalcAnderson} and then adopted\nby several authors.\n\n\nOne extrapolates total energies for the system with $N$, $N-1$ and $N-2$\nelectrons by DFT calculations with constrained occupations for $N-q$\nelectrons, with $q$ small (typically, up to $0.05$), so that ionized\natoms in otherwise periodic systems can be treated in rather small\nsupercells. We make the approximation that the total energy of the\nsystem with $q_i$ electrons removed from the level $i$ is given by a\npower expansion in $q_i$ up to third order:\n\\begin{equation}\n\\label{eq:EtotQ}\nE(N-q_i) = E(N) + A_i q_i + B_i q_i^2 + C_i q_i^3.\n\\end{equation}\nIn the following we shall assume that this can be extended to finite\nvalues of $q_i$.\nThe introduction of the cubic term $C_i q_i^3$ allows for a\n$q$-dependence of the screening properties of the system.\nThe coefficients $A_i$, $B_i$, and $C_i$, where $i$ labels core and\nvalence states involved in the transition, are in this framework all\nwhat is needed to compute the Auger electron energy. They can be\nevaluated in two equivalent ways, whichever is most convenient: by\ntaking the first, second, and third derivatives of the total energy\n$E(N-q_i)$ for $q_i\\rightarrow0$; by using Janak's\ntheorem\\cite{Janak:1978:theorem} and computing the KS eigenvalue of\nlevel $i$ and its first and second derivatives:\n\\begin{equation}\n\\label{eq:JanakQ}\n-\\epsilon^\\text{KS}_i(N-q_i) = A_i + 2 B_i q_i + 3 C_i q_i^2.\n\\end{equation}\nIn particular, $A_i$ is given by (minus) the KS eigenvalue in the\nneutral system.\n\nThe binding energy of a photoemitted electron, $\\epsilon_i\n{\\equiv}E^\\text{XPS}_i=E(N-1_i)-E(N)$, to be used in\nEq.~(\\ref{eq:HHubbard}), is given by Eq.~(\\ref{eq:EtotQ}) as:\n\\begin{equation}\n\\label{eq:XPSABC}\n \\epsilon_i = A_i + B_i + C_i.\n\\end{equation}\nThis is very close to the well-known Slater's transition-state\napproach, in which the XPS energy equals the (minus) eigenvalue at\nhalf filling. The latter amounts to $A_i+B_i+\\frac{3}{4}C_i$ when\napproximating the total energy by a cubic expansion as in\nEq.~(\\ref{eq:EtotQ}). In other terms, it differs from the result of\nEq.~(\\ref{eq:XPSABC}) only by $\\frac{1}{4}C_i$, with $C_i\\lesssim1$~eV\nin the cases considered here (see below).\nIt is worth noticing that the term $B_i+C_i$ acts like a correction to\nthe (minus) KS eigenvalue $A_i$, accounting for dynamical relaxation\neffects even though all terms are evaluated within KS-DFT.\n\nThe evaluation of the $A$, $B$, and $C$ coefficients for localized\nstates poses no additional difficulty. Instead, care must be taken\nwhen determining those corresponding to the delocalized valence shells\nof bulk materials ($A_v$, $B_v$, and $C_v$), for which we propose the\nfollowing method.\nAs for $A_v$, this is a continuous function of the quantum number $v$\nand, by taking advantage of Janak's theorem, it is the KS band energy\nwith reversed sign.\nTo estimate $B_v$ and $C_v$, we neglect their dependence on $v$ and\nassume that a single value can be taken across the valence band,\nacting as a rigid shift of the band. Hence, the 1hLDOS is obtained\nfrom the KS LDOS, $d^\\text{KS}(\\omega)$, as:\n\\begin{equation}\n\\label{eq:1hLDOS}\nd(\\omega)=d^\\text{KS}(-\\omega+B_v+C_v).\n\\end{equation}\nFor sake of the forthcoming discussion, one can also define a single\nvalue of $A_v$ in the solid by taking the KS valence band average.\n\nWe expect the above approximation to be a good one as long as the\nvalence band is sufficiently narrow and deep (since eventually the\ncorrection should approach zero at the Fermi level). Still under this\nsimplification, the direct evaluation of $B_v$ and $C_v$ would ask for\nconstraining the occupations for fairly delocalized states, which is a\nfeasible but uneasy task.\nAs an alternative route, we suggest a simpler approach based on the\nworking hypothesis that the environment contribution to the screening\nof the positive charge $q_i$ in Eq.~(\\ref{eq:EtotQ}) does not depend\nstrongly on the shape of the charge distribution.\nPractically, we take the neutral isolated atom as a reference\nconfiguration in Eq.~(\\ref{eq:EtotQ}), and evaluate the coefficients\n$B_i^a$ and $C_i^a$ for this system. The two quantities\n${\\Delta}B=B_i-B_i^a$ and ${\\Delta}C=C_i-C_i^a$ can be easily computed\nfor core levels.\nSuch bulk-atom corrections are reported in Table~\\ref{tab:deltaBC} for\nCu and Zn, which demonstrates that they are almost independent of the\ncore level. This supports our working hypothesis, and enables us to\nextrapolate to the valence shell. Accordingly, $B_v$ and $C_v$ are given\nby:\n\\begin{eqnarray}\n\\label{eq:B=Ba+DeltaB}\nB_v=B_v^a+{\\Delta}B,\\\\\n\\label{eq:C=Ca+DeltaC}\nC_v=C_v^a+{\\Delta}C.\n\\end{eqnarray}\n\n\nWe remark that by choosing the neutral atom as the reference system\nsome degree of arbitrariness is introduced. In principle, one could\nevaluate the atomic coefficients in a configuration which is\nclosest to the one of the atom in the solid, depending on its chemical\nenvironment. However, such arbitrariness has limited effect on the\nfinal value of $B_v$ (similar discussion applies for $C_v$), owing to\ncancellations between $B_c^a$ and $B_v^a$ in\nEq.~(\\ref{eq:B=Ba+DeltaB}), as will be demonstrated in the following.\n\n\\begin{table}\n\\begin{tabular}{|c|c|rrrrr|r|}\n\\hline\n\\hline\n &\n & \\multicolumn{1}{c}{$1s$}\n & \\multicolumn{1}{c}{$2s$}\n & \\multicolumn{1}{c}{$2p$}\n & \\multicolumn{1}{c}{$3s$}\n & \\multicolumn{1}{c|}{$3p$}\n & \\multicolumn{1}{c|}{average} \\\\\n\\hline\nCu\n & $\\Delta{}B$\n & $-4.77$\n & $-4.92$\n & $-4.90$\n & $-4.81$\n & $-4.76$\n & $-4.85\\pm0.07$ \\\\\n & $\\Delta{}C$\n & $-0.88$\n & $-0.80$\n & $-0.82$\n & $-0.73$\n & $-0.72$\n & $-0.81\\pm0.07$ \\\\\n\\hline\nZn\n & $\\Delta{}B$\n & $-4.21$\n & $-4.26$\n & $-4.27$\n & $-4.19$\n & $-4.17$\n & $-4.23\\pm0.04$ \\\\\n & $\\Delta{}C$\n & $-0.49$\n & $-0.33$\n & $-0.26$\n & $-0.32$\n & $-0.35$\n & $-0.35\\pm0.09$ \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{\\label{tab:deltaBC}\nDifferences amongst the values of $B$ and $C$ in the bulk and the atom,\n$\\Delta{}B=B-B^a$ and $\\Delta{}C=C-C^a$, for core levels of Cu and Zn.\nThe last column reports the average and standard deviation across the\ncore levels.\nAll values in eV.\n}\n\\end{table}\n\nRegarding the interaction energy $U$ for the two holes in a valence\nlevel, defined by $[E(N-2)-E(N)]-2[E(N-1)-E(N)]$, let us consider the\ncase of spherically symmetric holes (non-spherical contributions,\ngiving rise to multiplet splitting, will then be added). Such\nspherical interaction, denoted by $U_\\text{sph}$, can be determined\nvia Eq.~(\\ref{eq:EtotQ}), resulting in\n\\begin{equation}\nU_\\text{sph} = 2B_v + 6C_v.\n\\end{equation}\nThis amounts to the second derivative of the DFT energy as a function\nof the band occupation,\n$U(q)= \\partial^2 E(N-q) \/ \\partial q^2 $,\nas originally suggested by Gunnarsson and\ncoworkers,\\cite{Gunnarsson:1989:DFTcalcAnderson} here evaluated for\nthe $N-1$-electron system rather than for the neutral one.\nDifferently, the interaction energy commonly used in LDA+U\ncalculations of the ground state is defined as $E(N+1)+E(N-1)-2E(N)$\nand hence evaluated by the second derivative in $q=0$, resulting in\n$2B_v$ only.\nNotice here that the role of the cubic term in Eq.~(\\ref{eq:EtotQ}) is\nto introduce a dependence of the interaction energy on the particle\nnumber, following the one of the screening properties of the system.\nFinally, non-spherical contributions, which give rise to multiplet\nsplitting, are added to $U_\\text{sph}$. It has been\ndemonstrated\\cite{Antonides:1977:LMM_Cu_Zn_Ga_Ge_I} for a number of\nmaterials, including Cu and Zn, that these terms are well reproduced\nby a sum of atomic Slater integrals,\\cite{Slater:1960:atoms}\n$a_2F^2+a_4F^4$, where the coefficients $a_2$ and $a_4$ depend on the\nmultiplet configuration and\n\\begin{equation}\nF^k=\\int_0^\\infty r_1^2 dr \\int_0^\\infty r_2^2 dr_2\n \\frac{r^k_<}{r^{k+1}_>}\n \\left[\\varphi^a(r_1)\\varphi^a(r_2)\\right]^2.\n\\end{equation}\nHere $\\varphi^a(r)$ is the atomic radial wave function relevant to the\nprocess under investigation (e.g., the $3d$ one for a $CM_{45}M_{45}$\nAuger transition), and $r_<$ ($r_>$) is the smaller (larger) of $r_1$\nand $r_2$.\nNotice that the spherical Slater integral $F^0$ is implicit into\n$U_\\text{sph}$, which has the meaning of a screened Coulomb\nintegral.\\cite{Anisimov:1991:F0effDFT}\n\nSummarizing, one has:\n\\begin{equation}\n\\label{eq:U}\nU=2B_v^a+6C_v^a+2{\\Delta}B+6{\\Delta}C+a_2F^2+a_4F^4.\n\\end{equation}\nIt is customary to write $U=F-R$, where $F=F^0+a_2F^2+a_4F^4$, and $R$\nis the ``relaxation energy''.\\cite{Shirley:1973:KLL_relaxation} This can\nbe further decomposed into an atomic and an extra-atomic contribution,\n$R=R_a+R_e$. From Eq.~(\\ref{eq:U}), one identifies\n$R_a=F^0-2B_v^a-6C_v^a$ and $R_e=-2{\\Delta}B-6{\\Delta}C$. Notice that by\nour approach we compute $F^0-R_a$ as a single term, so that it is not\npossible to separate the two contributions.\n\n\nIn other formulations,\\cite{Pickett:1998:LDA+U, Cococcioni:2005:LDA+U}\nthe derivative of the energy with respect to the occupation number of\na broad band is computed by shifting the band with respect to the\nFermi level. This adds a non-interacting contribution to the\ncurvature of the energy, since the level whose occupation is varied is\nitself a function of the band occupancy. Such non-interacting term\nhas to be subtracted when computing $U$ by these\napproaches.\\cite{Cococcioni:2005:LDA+U} Our formulation is\nconceptually more similar to scaling the occupation of all valence\natomic levels in a uniform way, and the non-interacting term is\nvanishing.\n\n\n\n\\subsection{Computational details}\n\nThe results presented in this paper have been obtained by DFT\ncalculations with the Perdew-Burke-Ernzerhof\\cite{Perdew:1996:PBE}\ngeneralized gradient approximation for the exchange and correlation\nfunctional.\nWe used an all-electron linearized augmented-plane-wave code to perform\nthe simulations with constrained core occupations. Periodically repeated\nsupercells at the experimental lattice constants were adopted to\ndescribe the solids.\nOne atom was ionized in a unit cell containing four and eight atoms\nfor Cu and Zn, respectively. In both cases the ionized atom has no\nionized nearest neighbours.\nCell neutrality is preserved by increasing the number of the valence\nelectrons, simulating the screening of the core hole by the solid.\nThe spin-orbit splitting in core states as well as in the final state\nwith two holes was taken into account by adopting DFT energy shifts\nfor free atoms,~\\cite{NIST::DFTdata} and is here assumed independent\non the fractional charge $q$ (we verified that the latter approximation\naffects our final results by no more than $0.2$~eV).\nAs for the coefficients $A$, $B$, and $C$ in Eq.~(\\ref{eq:EtotQ}), we\nfound values numerically more stable, with respect to convergence\nparameters, by performing a second order expansion of the eigenvalues\nrather than a third order expansion of the total energy. Therefore,\nwe made use of Janak's theorem and Eq.~(\\ref{eq:JanakQ}), with\neigenvalues relative to the Fermi level in the solid (hence, resulting\nXPS and Auger energies are given with respect to the same reference).\nFulfillment of Janak's theorem and coincidence of results of\nEq.~(\\ref{eq:EtotQ}) and (\\ref{eq:JanakQ}) were numerically verified\nto high accuracy in a few selected cases. The values of $q$ ranged\nfrom $0$ to $0.05$ at intervals of $0.01$.\nComparison with denser and more extended meshes for the free atom case\nshowed that results are not dependent on the chosen mesh. Matrix\nelements and Slater integrals $F^2$ and $F^4$ are taken from\nRef.~\\onlinecite{Antonides:1977:LMM_Cu_Zn_Ga_Ge_I}, and core hole\nlifetimes from Ref.~\\onlinecite{Yin:1973:widthL2L3}.\n\n\\section{\\label{sec:results}Results}\n\nIn this Section we report our results for the $L_{23}M_{45}M_{45}$\nAuger lineshape of Cu and Zn. The core and the valence indices, $c$\nand $v$ in the previous Section, are specialized to the $2p$ and $3d$\nlevel of such elements, respectively.\n\nAs an example of our procedure to extract the parameters $A$, $B$, and\n$C$ [see Eq.~(\\ref{eq:EtotQ})], we report the case for the $2p$ level\nof Cu metal in Fig.~\\ref{fig:sampleFit} (the following considerations\nare also valid in the other cases).\nWe remove the fractional number of electrons $q$ from the $2p$ level\nof a Cu atom, and plot its (minus) KS $2p$ eigenvalue in\nFig.~\\ref{fig:sampleFit}a.\nSuch a curve is fitted by the expression in Eq.~(\\ref{eq:JanakQ}).\nIt is apparent from Fig.~\\ref{fig:sampleFit}a that a linear fit\nalready reproduces the KS eigenvalue in this range of $q$ to high\naccuracy. However, since results are to be extracted up to $q=1$ or\n$2$, the quadratic term in the expansion is also of interest. This is\nshown in Fig.~\\ref{fig:sampleFit}b, where the linear contribution\n($A+2Bq$) has been subtracted. The parabola accurately fits the\nnumerical results, with residuals of the order of $10-50$~$\\mu$eV.\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig1_sampleFit.eps}\n\\caption{\\label{fig:sampleFit}Example of fitting the Kohn-Sham\neigenvalue to extract the coefficients $A$, $B$, and $C$. Results are\nshown for the $2p$ level of metal Cu. Panel (a) plots the KS\neigenvalue (relative to the Fermi energy) with reversed sign (circles)\nand the fitted parabola from Eq.~(\\ref{eq:JanakQ}) (line), as function\nof the number of electrons removed from the $2p$ level, $q$. Panel (b)\nreports the same quantities after subtracting the linear term\n$A+2Bq$.}\n\\end{figure}\n\nTable~\\ref{tab:ABC_CuZn} collects our results for the coefficients\n$A$, $B$, and $C$, needed for the determination of the\n$L_{23}M_{45}M_{45}$ lineshape of Cu and Zn.\nThe values of $B$ and $C$ for the $L_2$ and $L_3$ cases are identical,\nfollowing the assumption that spin-orbit splitting is independent of\nthe fractional charge.\nThe coefficients $B_{M_{45}}$ and $C_{M_{45}}$ in the solid have been\nobtained by comparing results for core levels in the bulk and in the\nfree neutral atom according to\nEqs.~(\\ref{eq:B=Ba+DeltaB}-\\ref{eq:C=Ca+DeltaC}), with the values of\n$\\Delta{}B$ and $\\Delta{}C$ averaged across the core levels as\nreported in Table~\\ref{tab:deltaBC}.\nTheir negative sign indicates that the interaction amongst the two\nholes is more effectively screened in the solid.\n\n\n\n\\begin{table}\n\\begin{tabular}{|c|c|rrr|rrr|rr|}\n\\hline\n\\hline\n\\multicolumn{2}{|c|}{Level} & \\multicolumn{1}{c}{$A^a$} & \\multicolumn{1}{c}{$B^a$}\n& \\multicolumn{1}{c|}{$C^a$} &\n \\multicolumn{1}{c}{$A$} & \\multicolumn{1}{c}{$B$}\n& \\multicolumn{1}{c|}{$C$} &\n \\multicolumn{1}{c}{$E^\\text{XPS}$} & \\multicolumn{1}{c|}{Exp.}\n\\\\\n\\hline\n &$L_2$ & $ 930.17$ & $27.74$ & $1.25$ & $ 928.40$ & $22.84$ & $0.43$ & $ 951.67$ & $ 952.0$ \\\\\nCu &$L_3$ & $ 909.81$ & $27.74$ & $1.25$ & $ 908.04$ & $22.84$ & $0.43$ & $ 931.31$ & $ 932.2$ \\\\\n &$M_{45}$ & $ 5.04$ & $ 5.72$ & $0.91$ & $ 2.86$ & $ 0.87$ & $0.10$ & $ 3.84$ & $ 3.1$ \\\\\n\\hline\n &$L_2$ & $1019.40$ & $30.44$ & $1.08$ & $1016.98$ & $26.17$ & $0.82$ & $1043.97$ & $1044.0$ \\\\\nZn &$L_3$ & $ 995.69$ & $30.44$ & $1.08$ & $ 993.27$ & $26.17$ & $0.82$ & $1020.26$ & $1020.9$ \\\\\n &$M_{45}$ & $ 10.14$ & $ 7.06$ & $0.77$ & $ 7.53$ & $ 2.82$ & $0.42$ & $ 10.78$ & $ 9.9$ \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{\\label{tab:ABC_CuZn}\nCoefficients for the expansion of the total energy as a function of\nthe number of electrons, $E(N-q)$, for atomic ($A^a$, $B^a$, $C^a$)\nand bulk ($A$, $B$, $C$) Cu and Zn.\nAs for the $M_{45}$ values: by $A_{M_{45}}$ we indicate (minus) the\nweighted average of the $3d$ KS band; $B_{M_{45}}$ and $C_{M_{45}}$\nare obtained according to\nEqs.~(\\ref{eq:B=Ba+DeltaB}-\\ref{eq:C=Ca+DeltaC}). Theoretical XPS\nenergies are given by Eq.(\\ref{eq:XPSABC}); experimental data are\ntaken from Ref.~\\onlinecite{Antonides:1977:LMM_Cu_Zn_Ga_Ge_I}.\nValues in eV.\n}\n\\end{table}\n\n\nLet us now consider the dependence of our results on the particular\nchoice of the reference atomic configuration.\nFor comparsion, the Cu$^+$ and Zn$^+$ ions (with one electron removed\nfrom the $4s$ shell) have been used as a starting point for the\nevaluation of the atomic coefficients instead of the neutral one.\nWe find similar modifications, canceling each other in\nEq.~(\\ref{eq:B=Ba+DeltaB}), for core and valence $B_i^a$ atomic\ncoefficients (larger by about $1.5$~eV in Cu and $1.3$~eV in Zn). The\nsame is found for the $C_i^a$ coefficients (lower by $0.2$~eV in Cu\nand $0.1$~eV in Zn).\nAs a consequence, the values for $B_{M_{45}}$ differ by less than\n$0.2$~eV, and those for $C_{M_{45}}$ are identical within $0.01$~eV,\nwith data reported in Table~\\ref{tab:ABC_CuZn}.\nHence, as anticipated in the previous section, the choice of the\nreference atomic configuration does not affect significantly the\nevaluated XPS and Auger energies.\n\nRecall now that $A+B+C$ is our estimate for the XPS excitation\nenergies [see Eq.~(\\ref{eq:XPSABC})], which are reported in\nTable~\\ref{tab:ABC_CuZn}, and compared with experimental\nvalues.\\cite{Antonides:1977:LMM_Cu_Zn_Ga_Ge_I}\nNotice that bare KS excitations energies can be $30$~eV smaller than\nthe experimental value, but the addition of $B$ and, to a smaller\nextent, of $C$, properly accounts for the missing relaxation energy,\nthe left discrepancy being smaller that $1$~eV.\n\n\nWe report next our results for the $3d$ component of the 1hLDOS,\n$d(\\omega)$, for Cu and Zn in Fig.~\\ref{fig:1hLDOS_CuZn}. We remind\nthat such quantity is obtained by converting the KS density of states\ninto the hole picture, and by translating the result by $B_v+C_v$ to\naccount for relaxation effects [see Eq.~(\\ref{eq:1hLDOS})].\nThe total $d$ 1hLDOS, $\\bar{d}(\\omega)$, is shown as a shaded area\ntogether with the components on the different irreducible\nrepresentations over which the $d$ matrix is diagonal. For both\nmetals, the various components differ among themselves in the detailed\nenergy dependence, but their extrema are very similar.\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig2a_1hdosCu.eps}\n\\includegraphics[width=8.5cm]{fig2b_1hdosZn.eps}\n\\caption{\\label{fig:1hLDOS_CuZn}One-hole local density of states,\n$d(\\omega)$, for Cu (top) and Zn (bottom), relative to the Fermi\nenergy and normalized to unity. The shaded area is the total $d$ DOS,\n$\\bar{d}$.}\n\\end{figure}\n\nAs a final ingredient, Table~\\ref{tab:U} lists the values of $U$ for\nthe five $LS$ components of the multiplet, computed by\nEq.~(\\ref{eq:U}). Notice that the inclusion of a $q$ dependence in\n$U$ (via a cubic term in the expansion of the total energy with\nrespect to a fractional charge) proves to be quite important.\nIndeed, in evaluating $U$ the $C$ coefficient is counted six times,\nhence bringing a larger contribution than in the XPS energies\npreviously discussed.\nSuch inclusion gives an estimate of $U=U(q=1)$ which is $0.60$~eV and\n$2.52$~eV larger for Cu and Zn, respectively, than the corresponding\nvalues obtained as $U(q=0)$.\n\n\\begin{table}\n\\begin{tabular}{|c|r|rrrrr|}\n\\hline\\hline\n & \\multicolumn{1}{c|}{$U_\\text{sph}$}\n & \\multicolumn{1}{c}{$^1S$}\n & \\multicolumn{1}{c}{$^1G$}\n & \\multicolumn{1}{c}{$^3P$}\n & \\multicolumn{1}{c}{$^1D$}\n & \\multicolumn{1}{c|}{$^3F$} \\\\\n\\hline\nCu & 2.38\n&7.76\n&3.34\n&2.67\n&2.25\n&0.33\n\\\\\n\\hline\nZn & 8.16\n&14.31\n&9.26\n&8.49\n&8.01\n&5.82\n\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{\\label{tab:U} Values of $U$ resulting from the application of\nEq.~(\\ref{eq:U}), in eV. Slater's integrals from\nRef.~\\onlinecite{Antonides:1977:LMM_Cu_Zn_Ga_Ge_I}.}\n\\end{table}\n\n\nWe then compute the $L_{23}M_{45}M_{45}$ Auger spectrum following\nEq.~(\\ref{eq:PCiniLSJMJ}).\nThe outcome has been convoluted with a core hole lifetime of $0.49$\nand $0.27$~eV ($0.42$ and $0.33$~eV)\\cite{Yin:1973:widthL2L3} for the\n$L_2$ and $L_3$ lines of Cu (Zn), respectively, and results in a\nmultiplet of generally narrow atomic-like peaks, shown in\nFig.~\\ref{fig:LVV}.\n\n\nTo analyze these results, let us focus on the principal peak ($^1G$) in\nthe spectrum, which can be associated with the absolute position of the\nmultiplet. (The internal structure of the multiplet in our description\nonly depends on the values of $F^2$ and $F^4$ which, as previously\nspecified, were taken from the literature.)\nThe experimental energy of the (most intense) $^1G$\ntransition\\cite{Antonides:1977:LMM_Cu_Zn_Ga_Ge_I} is marked by a\nvertical line in Fig.~\\ref{fig:LVV}. The agreement of our results is\nrather good considering the absence of adjustable parameters in the\ntheory: focusing on the $L_3VV$ line, the $^1G$ peak position\n($918.0$~eV from experiments) is overestimated by $1.6$~eV, while the\none for Zn ($991.5$~eV) is underestimated by $1.9$~eV.\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig3_spectrum.eps}\n\\caption{\\label{fig:LVV}Simulated $L_{23}M_{45}M_{45}$ spectrum for Cu\n(top) and Zn (bottom) metals. The vertical lines mark the position of\nthe principal ($^1G$) peaks from\nexperiments.\\cite{Antonides:1977:LMM_Cu_Zn_Ga_Ge_I}}\n\\end{figure}\n\n\\section{\\label{sec:discussion}Discussion}\n\nIt is interesting to compare these results with those obtained by an\nexpression commonly adopted for Auger energies, i.e., $\\omega_{LS}\n\\approx \\epsilon_c - 2\\epsilon_v - U_{LS}$. This is an excellent\napproximation when $U$ is larger than $W$: for example in Zn, where\n$W\\approx1.5$~eV and $U_{^1G}=9.26$~eV, its application to the\ncomputed parameters yields a value which is only $0.10$~eV larger than\nthe $^1G$ peak position derived from Eq.~(\\ref{eq:PCiniLSJMJ}).\nHowever, when $U$ is of order of $W$, significant deviations can be\nobserved: e.g., for Cu ($W\\approx3.5$~eV and $U_{^1G}=3.34$~eV) the\n$^1G$ position is overestimated by $0.66$~eV. For smaller values of\n$U$, the quasi-atomic peak is lost for a broad band-like structure.\nThis is the case for the $^3F$ component of Cu (the rightmost shoulder\nin the spectrum) for which we obtain $U_{^3F}=0.33$~eV. However, this\nis an artifact of our underestimate of $U_\\text{sph}$ in Cu:\nexperimentally, the $^3F$ peak is resolved as well.\n\nBesides these observations, the expression $\\omega_{LS} \\approx\n\\epsilon_c - 2\\epsilon_v - U_{LS}$ is accurate enough for the $^1G$\npeak to discuss the discrepancy of our results with respect to the\nexperimental ones.\nLet us focus on the $L_3VV$ part of the spectrum, and decompose the\nAuger kinetic energy $\\omega$ into its contributions (see\nTable~\\ref{tab:decomposeEkin}).\nDespite the fact that the overall agreement is similar in magnitude for\nCu and Zn, it is important to remark that this finding has different\norigins.\nIn both metals, we underestimate slightly the core photoemission\nenergy and overestimate the valence photoemission energy by a similar\namount. Both effects contribute underestimating the kinetic energy.\nIn Zn, where our value of $U$ is excellent, the error in $\\omega$ stems\nfrom the errors in the photoemission energies.\nIn Cu, instead, $U$ is seriously underestimated. This overcompensates\nthe error in the photoemission energies, resulting in a fortuitous\noverall similar accuracy.\n\n\\begin{table}\n\\begin{tabular}{|c|c|rrr|r|}\n\\hline\n\\hline\n\n &\n & \\multicolumn{1}{c}{$\\epsilon_c$}\n & \\multicolumn{1}{c}{$-2\\epsilon_v$}\n & \\multicolumn{1}{c|}{$-U_{^1G}$}\n & \\multicolumn{1}{c|}{$\\omega_{^1G}$} \\\\\n\\hline\n & Theory & $931.31$ & $-7.67$ & $-3.34$ & $920.29$ \\\\\nCu & Exp. & $932.2$ & $-6.2$ & $-8.0$ & $918.0$ \\\\\n & Diff. & $-0.9$ & $-1.5$ & $4.7$ & $2.3$ \\\\\n\\hline\n & Theory & $1020.26$ & $-21.55$ & $-9.26$ & $989.45$ \\\\\nZn & Exp. & $1020.9$ & $-19.8$ & $-9.5$ & $991.5$ \\\\\n & Diff. & $-0.6$ & $-1.8$ & $0.2$ & $-2.1$ \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{\\label{tab:decomposeEkin}\nDecomposition of the $^1G$ $L_3M_{45}M_{45}$ Auger kinetic energy into\nits contributions, according to the simple approximation\n$\\omega=\\epsilon_c-2\\epsilon_v-U$. Theoretical values of\n$\\epsilon_c$, $\\epsilon_v$, and $U$ from tables \\ref{tab:ABC_CuZn} and\n\\ref{tab:U}; experimental data from\nRef.~\\onlinecite{Antonides:1977:LMM_Cu_Zn_Ga_Ge_I}.\nValues in eV.\n}\n\\end{table}\n\nWe now examine the relative weight of two different ingredients of our\nmethod. First, the role of the spin-orbit interaction in the final\nstate, which will be analyzed by comparing with results where such\nterm is neglected; second, the resolution of the 2hLDOS in its angular\ncomponents. To this respect, we notice that the matrix expression for\nthe spectrum given in Eq.~(\\ref{eq:PCiniLSJMJ}) can be significantly\nsimplified under the assumption that the 2hLDOS is spherically\nsymmetric and the spin-orbit contribution can be neglected. In this\ncase, we can just take the spherically averaged, i.e., the total $d$\n1hLDOS, and compute its self-convolution,\n$\\bar{D}^{(0)}\\equiv\\bar{d}*\\bar{d}$.\nAn averaged Green's function, $\\bar{G}^{(0)}$, is then defined as the\nHilbert transform of $\\bar{D}^{(0)}$. By replacing\n$G^{(0)}_{LSJM_J,L'S'J'M_{J'}'}$ in Eq.~(\\ref{eq:PCiniLSJMJ}) with the\ndiagonal matrix $\\delta_{LSJM_J,L'S'J'M_{J'}'}\\bar{G}^{(0)}$, we\nobtain the simple scalar equation\n\\begin{equation}\n\\label{eq:PCini0}\nS(\\omega)=-\\frac{1}{\\pi}\\sum_{LSJ} (2J+1)|A_{LS}|^2 \\text{Im}\n\\left[\n\\frac{\n \\bar{G}^{(0)} ( \\epsilon_c - \\omega )\n}{\n 1 - U_{LS} \\bar{G}^{(0)}( \\epsilon_c - \\omega )\n}\n\\right],\n\\end{equation}\nwhere the dependence on $LS$ quantum numbers is only via the matrix\nelements and the interaction matrix $U$, and each $LS$ component of\nthe spectrum is decoupled from the others.\n\nThe Auger spectra simulated neglecting the spin-orbit interaction and\ncalculated with the simplified expression of Eq.~(\\ref{eq:PCini0}) are\nplotted in Fig.~\\ref{fig:SOvsALL} as dashed and dotted line,\nrespectively, to be compared with the result of the full calculation\n[Eq.~(\\ref{eq:PCiniLSJMJ})], solid line.\nFor simplicity, we limit the discussion to the $L_3VV$ line of Zn. The\nresemblance of the three results is remarkable. Indeed, in Cu and Zn\nthe spin-orbit splitting for $3d$ levels is relatively small, $0.27$\nand $0.36$~eV, respectively.\\cite{NIST::DFTdata} Furthermore, despite\nthe differences which characterize the angular components of the\n1hLDOS (see Fig.~\\ref{fig:1hLDOS_CuZn}), the convoluted 2hLDOS are\nonly mildly different from $\\bar{D}^{(0)}$, as reported in\nFig.~\\ref{fig:2hLDOS_CuZn}.\nNow, in systems with a large $U\/W$ ratio, fine details of the 2hLDOS\nare not relevant for the position of quasi-atomic peaks, which only\ndepend on the weighted averages $E^{(0)}_{LS}$ as in\nEq.~(\\ref{eq:PDeltaLSJ}). In our case, the values of $E^{(0)}_{LS}$\nlie within $0.1$~eV from those corresponding to the averaged\n2hLDOS. Hence the practically equivalent results obtained by\nEq.~(\\ref{eq:PCiniLSJMJ}) and Eq.~(\\ref{eq:PCini0}).\n\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig4_comparison.eps}\n\\caption{\\label{fig:SOvsALL}Simulated spectrum for the\n$L_3M_{45}M_{45}$ line of Zn. Solid line: full treatment of\nEq.~(\\ref{eq:PCiniLSJMJ}), as presented in this paper. Dashed line:\nneglecting the spin-orbit interaction in the two-hole final\nstate. Dotted line: adopting the spherically averaged 2hLDOS and the\nscalar formulation, Eq.~(\\ref{eq:PCini0}). The origin of the vertical\naxis is shifted for improved clarity.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig5_2pdos.eps}\n\\caption{\\label{fig:2hLDOS_CuZn}Non-interacting two-holes density of\nstates, $D^{(0)}$, for Cu (left) and Zn (right). The solid curve\nrepresents $\\bar{D}^{(0)}\\equiv\\bar{d}*\\bar{d}$, the self-convolution\nof the total $d$ 1hLDOS; the shaded area indicates the largest\ndeviations from this result found amongst the angular resolved\n2hLDOSs.}\n\\end{figure}\n\nThis analysis shows that, for a wide class of systems with strong\nhole-hole interaction, weak spin-orbit interaction, and a spherical\nsymmetry to some extent, the simple formulation presented in\nEq.~(\\ref{eq:PCini0}) is practically as accurate as the expression in\nEq.~(\\ref{eq:PCiniLSJMJ}). One should instead adopt the full treatment\nfor, e.g., heavier elements, or systems with low dimensionality.\nThis remark is independent of the methodology to determine the\nparameters entering the model, either fully {\\em ab initio} as in the\npresent approach, or by phenomenological arguments.\n\nOur method provides an agreement with experimental photoemission\nenergies of the order of $1$~eV and of $2$~eV for Auger energies.\nWe consider this to be a rather good result, as a starting point,\nconsidering the absence of adjustable parameters in the model which is\nthe new feature of our approach for CVV transitions in correlated\nsystems.\nWe trust that our simple method could already yield qualitative\ninformation on the variations of the spectrum to be expected following\nto modifications in the sample, e.g., when the emitting atom is\nlocated in different environments.\nOf course, a much better agreement would be obtained by inserting\nphenomenological parameters, but at the cost of loosing predictive\npower.\n\nThe internal structure of the lineshape, i.e., the multiplet\nsplitting, is given very precisely. However, the first-principles\ntreatment of the latter is not a new aspect of our approach, which is\nindeed based in this respect on atomic results available in the\nliterature since decades.\nLet us instead focus again on the estimated position of the multiplet,\nwhich crucially depends on the parameters evaluated by our {\\em ab\ninitio} method.\nEven though the agreement with the experiment is about as good as for\nZn as for Cu, actually the results for Zn are much better.\nIn Zn, the $3d$ band is deep and narrow, and electronic states bear\nmostly atomic character. Their hybridization with the states closer to\nthe Fermi level, which mainly contributes to screening, is small,\nsomehow in an analogous way as for core states. Consequently, the\napproximations to neglect the energy dependence of $B_v$ and $C_v$,\nand to transfer the values of $\\Delta{}B$ and $\\Delta{}C$ from the\ncore states to the valence ones, produce very good results.\nIn Cu, instead, the $3d$ band is higher and broader, and hybridizes\nsignificantly with the $s$-like wavefunctions. Our approximations\nturn out to be less adequate: the resulting $U$ is about half the one\nderived from experiments.\n\n\nPart of this discrepancy might also have a deeper physical origin,\nsince the Hamiltonian adopted, Eq.~(\\ref{eq:HHubbard}), does not allow\nfor the interaction amongst two holes located at different atomic\nsites. The CS model has been extended to consider the role of\ninteratomic (``off-site'') correlation effects which mainly produce an\nenergy shift of the Auger line to lower\nenergies.\\cite{Verdozzi:1995:OffSiteRev} Studies based on\nphenomenological parameters suggest that such an energy shift could be\nof about $2.5$~eV in Cu\\cite{Ugenti:2008:spinAPECS} (smaller values\nare expected in Zn where holes are more localized and screening is\nmore effective). For sake of simplicity, the off-site term has not\nbeen considered here and is left for future investigations. The\nparameters entering this term could be determined by {\\em ab initio}\nmethods in analogy to the procedure shown here for evaluating $U$. It\nis however important to notice that adding the off-site term would not\nfix all the discrepancies observed in Cu, where also the lineshape, in\naddition to the peak position, is not satisfactory owing to small\nvalues for the on-site interaction $U$ (e.g., the non-resolved $^3F$\npeak).\n\nEnhancing the accuracy of the values of $U$ seems therefore the most\nimportant improvement for the method presented here, especially for\nsystems with broad valence bands. As a possibility, it would be\ninteresting to use approaches which are capable to compute the total\nenergy in presence of holes in the valence state. The methodology\npresented in Ref.~\\onlinecite{Cococcioni:2005:LDA+U}, in which the\nvalence occupation is changed by means of Lagrange multipliers\nassociated with the KS eigenvalues, could be particularly effective.\nOne should pay attention as some arbitrariness is anyway introduced.\nNamely, the value of $U$ does depend on the chosen form of the valence\nwavefunctions. Such an arbitrariness is compensated in LDA+U\ncalculations performed self-consistently.\\cite{Cococcioni:2005:LDA+U}\nFurthermore, to apply this method to systems with closed band lying\nwell below the Fermi energy, large shifts of the KS eigenvalues would\nbe needed to alter the occupation of the band to an appreciable\namount.\n\nAnother possible improvement concerns the photoemission energies.\nCalculations by the $GW$ approach\\cite{Aryasetiawan:1998:GW} of the\n1hLDOS could be used to account for relaxation energies, rather than\nadopting Eq.~(\\ref{eq:XPSABC}). Results available in the literature\n(e.g., for Cu~\\cite{Marini:2002:GWCu}) are very promising in that\nsense.\nIt is interesting to notice that the factor $B+C$ plays the role of a\nself-energy expectation value, and that the use of a single value of\n$B+C$ to shift rigidly the band is formally analogous to the ``scissor\noperator'' often introduced to avoid expensive self-energy calculations.\nThe accuracy of such rigid shifts for valence-band photoemission in Cu\nis discussed in Ref.~\\onlinecite{Marini:2002:GWCu}.\n\nSystems with larger band width or smaller hole-hole interaction would\nrequire to extend the approach to treat the dependence of the matrix\nelements on energy together with the interaction in the final state.\nReleasing the assumption that matrix elements equal the atomic ones,\nas in the current treatment, or that particles are non-interacting in\nformulations accounting for such an energy dependence (like, e.g., the\none in Ref.~\\onlinecite{Bonini:2003:MDS}), would allow switching\ncontinously between systems with band-like and atomic-like spectra.\nThis possibility is currently under investigation.\n\nFinally, let us recall the basic assumption considered here that the\nvalence shell is closed, which is crucial to the CS model in its\noriginal form. Efforts have been devoted towards releasing this\nassumption, resulting in a formulation by more complicated three-hole\nGreen's functions,\\cite{Marini:1999:3bodyCVVAuger} for which no {\\em\nab initio} treatment is nowadays available to our knowledge.\n\n\n\\section{\\label{sec:conclusions}Conclusions}\n\nWe have presented an {\\em ab initio} method for computing CVV Auger\nspectra for systems with filled valence bands, based on the\nCini-Sawatzky model.\nOnly standard DFT calculations are required, resulting in a very\nsimple method which allows working out the spectrum with no adjustable\nparameters. The accuracy on the absolute position of the Auger\nfeatures is estimated to a few eV, as we have demonstrated by the\nanalysis of Cu and Zn metals.\nWe have shown that in these systems further simplifications like\nneglecting spin-orbit interaction for the two valence holes, or the\nnon sphericity of the emitting atom, give results practically\nequivalent to the full treatment.\nAttention should be paid to the problematic parameter $U$. We\nobtained such a term with a good accuracy for the more localized,\natomic-like valence bands in Zn, while it results underestimated in\nCu.\nIts occupation number dependence, included via a cubic term in the\nexpansion of the total energy, has been considered, and shown to play\nan important role.\n\nThis step towards a first-principles description of CVV spectroscopy\nin closed-shell correlated systems enables identifying improvements\nwhich future investigations could focus on. In particular, one would\nbenefit from detailed calculations of the single-particle densities of\nstates (e.g., by the $GW$ method), from truly varying the valence-band\noccupation to obtain the $U$ parameter, and from including off-site\nterms in the Hamiltonian. Prospectively, it would be desirable to take\ninto account the energy dependence of the transition matrix elements.\n\n\\section{Acknowledgment}\n\nThis work was supported by the MIUR of Italy (Grant\nNo. 2005021433-003) and the EU Network of Excellence NANOQUANTA (Grant\nNo. NMP4-CT-2004-500198). Computational resources were made available\nalso by CINECA through INFM grants. EP is financially supported by\nFondazione Cariplo (n. Prot. 0018524).\n\n\n\n\\bibliographystyle{apsrev}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec:intro}\n\nWhen a black hole is viewed against a backdrop of light sources,\nthe observer sees a black disc in the sky which is known as the\n\\emph{shadow} of the black hole. Points inside this black disc\ncorrespond to past-oriented light rays that go from the observer\ntowards the horizon of the black hole, while points outside this\nblack disc correspond to past-oriented light rays that are more\nor less deflected by the black hole and then meet one of the \nlight sources. For a Schwarzschild black hole, which is non-rotating\nthe shadow is circular and its boundary corresponds to light rays \nthat asymptotically spiral towards circular photon orbits\nthat fill the so-called photon sphere at 1.5 Schwarzschild radii\naround the black hole. For a Kerr black hole, which is rotating, \nthe shadow is flattened on one side and its boundary corresponds \nto light rays that spiral towards spherical photon orbits\nthat fill a 3-dimensional photon region around the black hole.\nFor the supermassive black hole that is assumed to sit at the\ncentre of our Galaxy, the predicted angular diameter of the shadow is \nabout 53 microarcseconds which is within reach of VLBI observations.\nThere is an ongoing effort to actually observe this shadow, and \nalso the one of the second-best black-hole candidate at the centre\nof M87, see {\\tt http:\/\/www.eventhorizontelescope.org}.\n\nWhen calculating the shadow one usually considers an \\emph{eternal}\nblack hole, i.e., a black hole that is static or stationary and exists\nfor all time. For a Schwarzschild black hole, there is a simple\nanalytical formula for the angular radius of the shadow which goes\nback to Synge \\cite{Synge1966}. (Synge did not use the word ``shadow''\nwhich was introduced much later. He calculated what he called the\n\\emph{escape cones} of light. The opening angle of the escape \ncone is the complement of the angular radius of the shadow.) For\na Kerr black hole, the shape of the shadow was \ncalculated for an observer at infinity by Bardeen \\cite{Bardeen1973}.\nMore generally, an analytical formula for the boundary curve\nof the shadow was given, for an observer anywhere\nin the domain of outer communication of a Pleba{\\'n}ski-Demia{\\'n}ski\nblack hole, by Grenzebach et al. \n\\cite{GrenzebachPerlickLaemmerzahl2014,GrenzebachPerlickLaemmerzahl2015}.\nFor the Kerr case, this formula was further evaluated by Tsupko\n\\cite{Tsupko2017}.\nThese analytical results are complemented by ambitious numerical\nstudies, performing ray tracing in black hole spacetimes with various\noptical effects taken into account. We mention in particular a \npaper by Falcke et al. \\cite{FalckeMeliaAgol2000} where the perspectives\nof actually observing black-hole shadows were numerically investigated\ntaking the presence of emission regions and scattering into account, and\na more recent article by James et al. \n\\cite{JamesTunzelmannFranklinThorne2015}\nwhich focusses on the numerical work that was done for the\nmovie \\emph{Interstellar} but also reviews earlier work.\n\nAs we have already emphasised, in all these analytical and numerical\nworks an eternal black hole is considered. Actually, we believe that\nblack holes are not eternal: They have come into existence some finite\ntime ago by gravitational collapse (and are then possibly growing by \naccretion or mergers with other black holes). This brings us to the \nquestion of how an observer who is watching the collapse would see \nthe shadow coming about in the course of time. This is the question \nwe want to investigate in this paper.\n\nThe visual appearance of a star undergoing gravitational collapse\nhas been studied in several papers, beginning with the pioneering\nwork of Ames and Thorne \\cite{AmesThorne1968}. In this work, and \nin follow-up papers e.g. by Jaffe \\cite{Jaffe1969}, Lake and Roeder\n\\cite{LakeRoeder1979} and Frolov et al. \\cite{FrolovKimLee2007},\nthe emphasis is on the frequency shift of light coming from the \nsurface of the collapsing star. More recent papers by Kong et al.\n\\cite{KongMalafarinaBambi2014,KongMalafarinaBambi2015} and\nby Ortiz et al. \n\\cite{OrtizSarbachZannias2015a,OrtizSarbachZannias2015b}\ninvestigated the frequency shift of light passing through a \ncollapsing transparent star, thereby contrasting the collapse\nto a black hole with the collapse to a naked singularity. In \ncontrast to all these earlier articles, here we consider\na \\emph{dark} and \\emph{non-transparent} collapsing star which\nis seen as a black disc when viewed against a backdrop of light \nsources and we ask how this black disc changes \nin the course of time. \n\nFor the collapsing star we use a particularly simple model: We\nassume that the star is spherically symmetric and that it begins\nto collapse, at some instant of time, in free fall like a ball of\ndust until it ends up in a point singularity at the centre. \nThe metric inside such a collapsing ball of dust was\nfound in a classical paper by Oppenheimer and Snyder\n\\cite{OppenheimerSnyder1939}. However, for our purpose, as \nwe assume the collapsing star to be non-transparent, we do\nnot need this interior metric. All we need to know is\nthat a point on the surface follows \na timelike geodesic in the ambient Schwarzschild spacetime. \nWe will demonstrate that in this situation the time \ndependence of the shadow can be given analytically. We do\nthis first for a static observer who is watching the collapse from a\ncertain distance, and then also for an observer who is falling\ntowards the centre and ending up in the point singularity \nafter it has formed. The latter situation is (hopefully) not\nof relevance for practical astronomical observations but\nwe believe that the calculation is quite instructive from\na conceptual point of view.\n\nThe paper is organised as follows. In Section \\ref{sec:PG}\nwe review some basic facts on the Schwarzschild solution \nin Painlev{\\'e}-Gullstrand coordinates. These coordinates \nare particularly well suited for our purpose because they\nare regular at the horizon, so they allow to consider worldlines\nof observers or light signals that cross the horizon without the\nneed of patching different coordinate charts together. In \nSection \\ref{sec:shbh} we rederive in Painlev{\\'e}-Gullstrand\ncoordinates the equations for the shadow of an eternal \nSchwarzschild black hole. We do this both for a static and\nfor an infalling observer. The results of this section will then\nbe used in the following two sections for calculating the shadow \nof a collapsing star. We do this first for a static observer\nin Section \\ref{sec:shcoll1} and then for an infalling observer\nin Section \\ref{sec:shcoll2}. We summarise our results in \nSection \\ref{sec:conclusions}. \n\n\\section{Schwarzschild metric in Painlev{\\'e}-Gullstrand coordinates}\\label{sec:PG}\n\nThroughout this paper, we work with the Schwarzschild metric \nin Painlev{\\'e}-Gullstrand coordinates \\cite{Painleve1921,Gullstrand1922},\n\n\\begin{equation}\\label{eq:gPG}\ng_{\\mu \\nu}dx^{\\mu} dx^{\\nu} \n= - \\left( 1 - \\dfrac{2m}{r} \\right) c^2 d T ^2 + 2 \\sqrt{\\dfrac{2m}{r}} \\, c \\, d T \\, dr\n+ dr^2 +\nr^2 \\big( d \\vartheta ^2 + \\mathrm{sin} ^2 \\vartheta \\, \nd \\varphi ^2 \\big) \\, .\n\\end{equation}\nHere\n\n\\begin{equation}\\label{eq:m}\nm=\\dfrac{GM}{c^2}\n\\end{equation}\nis the mass parameter with the dimension of a length; $M$ is \nthe mass of the central object in SI units, $G$ is Newton's \ngravitational constant and $c$ is the vacuum speed of light.\n\nThe Painlev{\\'e}-Gullstrand coordinates $(T,r,\\vartheta, \\varphi )$\nare related to the standard text-book Schwarzschild coordinates\n$(t,r,\\vartheta , \\varphi )$ by\n\n\\begin{equation}\\label{eq:Tt}\nc \\, dT= c \\, dt + \n\\sqrt{\\dfrac{2m}{r}} \\, \\dfrac{dr}{\\Big( 1- \\dfrac{2m}{r} \\Big)}\n\\, .\n\\end{equation} \nAs a historical side remark, we mention that both Painlev{\\'e}\n\\cite{Painleve1921} and Gullstrand \\cite{Gullstrand1922} believed\nthat they had found a new solution to Einstein's vacuum field\nequation before Lema{\\^\\i}tre \\cite{Lemaitre1933} demonstrated \nthat it is just the Schwarzschild solution in other coordinates. \nWhereas in the standard Schwarzschild coordinates the metric\nhas a coordinate singularity at the horizon at $r=2m$, in the\nPainlev{\\'e}-Gullstrand coordinates the metric is regular\non the entire domain $0 < r < \\infty$.\n\nOn the domain $2m2m)$, proper time $\\tau$ is related \nto the Painlev{\\'e}-Gullstrand time coordinate $T$ by \n\\begin{equation}\\label{eq:taustatic}\n\\dfrac{dT}{d \\tau} = \\dfrac{1}{\\sqrt{1- \\dfrac{2m}{r}}}\n\\, .\n\\end{equation}\n\nIn the Painlev{\\'e}-Gullstrand coordinates, the geodesics \nin the Schwarzschild spacetime are the solutions to the\nEuler-Lagrange equations of the Lagrangian \n \n\\begin{equation}\\label{eq:LagPG}\n\\mathcal{L} \\big( x , \\dot{x} \\big) \n= \\dfrac{1}{2} \\left(\n- \\left( 1 - \\dfrac{2m}{r} \\right) c^2 \\dot{T}{} ^2 + \n2 \\sqrt{\\dfrac{2m}{r}} \\, c \\, \\dot{T} \\, \\dot{r}\n+ \\dot{r}{}^2 +\nr^2 \\, \\Big( \\mathrm{sin}^2 \\vartheta \\, \\dot{\\varphi}{} ^2 \n+ \\dot{\\vartheta}{} \\Big) \\right) \n\\, .\n\\end{equation}\nHere the overdot means derivative with respect to an affine \nparameter.\n\nThe $T$ and $\\varphi$ components of the Euler-Lagrange\nequations give us the familiar constants of motion \n$E$ and $L$ in Painlev{\\'e}-Gullstrand coordinates,\n\n\\begin{equation}\\label{eq:E}\nE = - \\dfrac{\\partial \\mathcal{L}}{\\partial \\dot{T}} \n= \\left( 1 - \\dfrac{2m}{r} \\right) c^2 \\dot{T} \n- \\sqrt{\\dfrac{2m}{r}} \\, c \\, \\dot{r} \n\\end{equation}\nand\n\n\\begin{equation}\\label{eq:L}\nL = \n\\dfrac{\\partial \\mathcal{L}}{\\partial \\dot{\\varphi}} \n= r^2 \\mathrm{sin} ^2 \\vartheta \\, \\dot{\\varphi} \\, .\n\\end{equation}\nFor the purpose of this paper we will need the \nradial timelike geodesics and the lightlike geodesics\nin the equatorial plane. \n\n\\subsection{Radial timelike geodesics}\\label{sec:radial}\n\nWe consider massive objects in radial free \nfall, i.e., radial geodesics ($\\dot{\\varphi}=0$ and \n$\\dot{\\vartheta} =0$) which are timelike. Then we may \nchoose the affine parameter equal to proper time $\\tau$, \n\n\\begin{equation}\\label{eq:Lagrad}\n- \\, c^2 =\n - \\left( 1 - \\dfrac{2m}{r} \\right) c^2 \\Big( \\dfrac{dT}{d \\tau} \\Big) ^2 \n+ 2 \\sqrt{\\dfrac{2m}{r}} \\, c \\, \\dfrac{dT}{d \\tau} \\, \\dfrac{dr}{d \\tau}\n+ \\Big( \\dfrac{dr}{d \\tau} \\Big) ^2 \\, .\n\\end{equation}\nIn this notation (\\ref{eq:E}) can be rewritten as\n\n\\begin{equation}\\label{eq:Erad}\n\\varepsilon := \\dfrac{E}{c^2} = \n\\left( 1 - \\dfrac{2m}{r} \\right) \\dfrac{dT}{d \\tau} \n- \\sqrt{\\dfrac{2m}{r}} \\, \\dfrac{1}{c} \\, \\dfrac{dr}{d \\tau} \n\\end{equation}\nwhereas (\\ref{eq:L}) requires $L=0$. \n\nIn the following we restrict to the case that the parametrisation by proper\ntime is future oriented with respect to the Painlev{\\'e}-Gullstrand\ntime coordinate, $dT\/d \\tau >0$, and we consider only ingoing motion,\n$dr\/d \\tau <0$, that starts in the domain $r>2m$. Then $\\varepsilon >0$ \nand (\\ref{eq:Erad}) and (\\ref{eq:Lagrad}) imply\n\n\\begin{equation}\\label{eq:rTrad}\n\\dfrac{dr}{d \\tau} = \n- \\, c \\, \\sqrt{\\varepsilon ^2- 1+ \\dfrac{2m}{r}} \n\\, , \\quad\n\\dfrac{dT}{d \\tau} =\n\\dfrac{\n\\varepsilon - \\sqrt{\\dfrac{2m}{r}} \n\\sqrt{\\varepsilon ^2- 1 + \\dfrac{2m}{r} }\n }{\n1 - \\dfrac{2m}{r} \n}\n\\end{equation} \nWe distinguish three cases, see Fig.~\\ref{fig:radial}:\n\n(a) $0<\\varepsilon < 1$: Then $dr\/d \\tau =0$ at a radius coordinate \n$r_i$ given by $\\varepsilon ^2=1-2m\/r_i$, i.e., this case \ndescribes free fall from rest at $r_i$. Clearly, the \npossible values of $r_i$ are $2m1$ (dashed). For the four \nworldlines with $\\varepsilon \\ge 1$ we have chosen the initial conditions \nsuch that they arrive simultaneously at $r=0$. For the three worldlines \nwith $\\varepsilon <1$ we have assumed that they start from rest at the \nsame Painlev{\\'e}-Gullstrand time; then these worldlines do not arrive \nsimultaneously at $r=0$. As an alternative, one could have considered \nworldlines that start from rest at the same Schwarzschild time; they \nwould arrive simultaneously at $r=0$.\n\\label{fig:radial}}\n\\end{figure}\n\nChoosing a value of $\\varepsilon >0$ defines a family of infalling observers.\nWe associate with this family the tetrad \n\n\\begin{gather}\n\\nonumber\n\\tilde{e}{}_0 =\n\\dfrac{\n\\varepsilon -\\sqrt{\\dfrac{2m}{r}} \\sqrt{\\varepsilon ^2-1+\\dfrac{2m}{r}}\n}{ \n\\Big( 1 - \\dfrac{2m}{r} \\Big) \\, c\n}\n\\, \\dfrac{\\partial}{\\partial T} \n-\n\\sqrt{\\varepsilon ^2-1+\\dfrac{2m}{r}} \\, \\dfrac{\\partial }{\\partial r} \\, ,\n\\\\\n\\nonumber\n\\tilde{e}{}_1 = \\varepsilon \\, \\dfrac{\\partial}{\\partial r} \n+\n\\dfrac{\n\\varepsilon \\sqrt{\\dfrac{2m}{r}} -\\sqrt{\\varepsilon ^2-1+\\dfrac{2m}{r}}\n}{\n\\Big( 1- \\dfrac{2m}{r} \\Big) \\, c\n}\n\\dfrac{\\partial}{\\partial T} \\, ,\n\\\\\n\\tilde{e}{}_2 = \\dfrac{1}{r }\n\\, \\dfrac{\\partial}{\\partial \\vartheta} \\, , \\quad\n\\tilde{e}{}_3 = \\dfrac{1}{r \\, \\mathrm{sin} \\, \\vartheta}\n\\, \\dfrac{\\partial}{\\partial \\varphi} \\, .\n\\label{eq:temu}\n\\end{gather}\nFor $\\varepsilon \\ge 1$ this tetrad is well-defined and orthonormal\non the entire domain $0 < r < \\infty$. For $0 < \\varepsilon < 1$ the tetrad\nis restricted to the domain $0 < r < r_i = 2m\/(1- \\varepsilon ^2)$.\n\nThe relative velocity $v$ of a radially infalling observer \nwith respect to the static observer at the same event can \nbe calculated from the special relativistic formula\n\n\\begin{equation}\\label{eq:v1}\ng_{\\mu \\nu} \\, e_0^{\\mu} \\, \\tilde{e}{}_0^{\\nu} \n= \\dfrac{-1}{\\sqrt{1-\\dfrac{v^2}{c^2}}} \\, .\n\\end{equation}\nThis results in\n\n\\begin{equation}\\label{eq:vrad}\n\\dfrac{v}{c} = \n\\dfrac{1}{\\varepsilon} \\, \\sqrt{\\varepsilon ^2- 1+ \\dfrac{2m}{r} }\n\\, .\n\\end{equation}\nClearly, this formula makes sense only on the domain on which \nboth families of observers are defined. For $\\varepsilon \\ge 1$ this\nis true on the domain $2m2m$, in the last expression the second term is \nbigger than the first. Therefore, in this domain the \nupper sign has to be chosen if $dT\/dr > 0$ and the \nlower sign has to be chosen if $dT\/dr <0$.\n\nBy differentiating (\\ref{eq:orbit}) with respect to $\\varphi$ \nwe find \n\n\\begin{equation}\\label{eq:d2rdphi}\n\\dfrac{d^2r}{d \\varphi ^2} =\n\\dfrac{4E^2r^3}{c^2L^2} - 2r +2m \\, .\n\\end{equation}\nIf along a lightlike geodesic the radius coordinate goes \nthrough an extremum at value $r_m$, (\\ref{eq:orbit}) and \n(\\ref{eq:d2rdphi}) imply\n\n\\begin{equation}\\label{eq:Erm}\n0 = \n\\dfrac{E^2r_m^3}{c^2L^2} - r_m +2m \\, ,\n\\end{equation}\n\n\\begin{equation}\\label{eq:Erm2}\n\\dfrac{d^2 r}{d \\varphi ^2} \\Big| _{r=r_m} = \nr_m-3m \\, .\n\\end{equation}\nThis demonstrates that only local minima may occur in the domain $r>3m$ \nand only local maxima may occur in the domain $r<3m$. The sphere at\n$r=3m$ is filled with circular photon orbits that are unstable with respect \nto radial perturbations. These well-known facts will be crucial for the \nfollowing analysis.\n\nFor a lightlike geodesic with an extremum of the radius coordinate at $r_m$,\n(\\ref{eq:Erm}) may be used for expressing $E^2\/L^2$ in (\\ref{eq:orbit}) and \n(\\ref{eq:trtime}) in terms of $r_m$. This results in\n\n\\begin{equation}\\label{eq:orbit2}\n\\dfrac{dr}{d \\varphi} = \\pm \\sqrt{\\dfrac{(r_m-2m)r^4}{r_m^3} - r^2 +2mr}\n\\end{equation}\nand\n\n\\begin{equation}\\label{eq:trtime2}\nc \\, \\dfrac{dT}{d r} = \n\\dfrac{\\sqrt{2mr}}{(r-2m)} \\pm \n\\dfrac{\\sqrt{(r_m-2m)r^5}}{(r-2m) \\sqrt{(r_m-2m)r^3-(r-2m)r_m^3}}\n\\end{equation}\n\n\\vspace{0.5cm}\n\n\\section{The shadow of an eternal Schwarzschild black hole}\\label{sec:shbh}\nIn this section we rederive the formulas for the angular radius\nof the shadow of an eternal Schwarzschild black hole, both for\na static and for an infalling observer, using Painlev{\\'e}-Gullstrand\ncoordinates. The results of this section will then be used for calculating \nthe shadow of a collapsing star in the following sections. \n\nWe consider a lightlike geodesic $\\big( T(s),r(s),\\varphi (s) \\big)$ \nin the equatorial plane, where $s$ is an affine parameter. As before,\nwe denote the derivative with respect to $s$ by an overdot. We may\nthen expand the tangent vector of the lightlike geodesic with respect \nto the static tetrad (\\ref{eq:emu}) and also, as an\nalternative, with respect to the infalling tetrad \n(\\ref{eq:temu}) for some chosen $\\varepsilon >0$. Of course, \nthe resulting equations are restricted to the domain where\nthe respective tetrad is well-defined and orthonormal. As the \ntangent vector is lightlike, these expansions may be written in \nterms of two angles $\\alpha$ and $\\tilde{\\alpha}$,\n\\begin{gather}\n\\nonumber\n\\dot{T} \\dfrac{\\partial}{\\partial T}\n+\n\\dot{r} \\dfrac{\\partial}{\\partial r}\n+\n\\dot{\\varphi} \\dfrac{\\partial}{\\partial \\varphi}\n=\n\\chi \\Big( \ne_0 + \\mathrm{cos} \\, \\alpha \\, e_1 -\n\\mathrm{sin} \\, \\alpha \\, e_3 \\Big) \n\\\\\n=\n\\tilde{\\chi} \\Big( \n\\tilde{e}{}_0 + \\mathrm{cos} \\, \\tilde{\\alpha} \\, \\tilde{e}{}_1 -\n\\mathrm{sin} \\, \\tilde{\\alpha} \\, \\tilde{e}{}_3 \\Big) \\, .\n\\label{eq:tangent1}\n\\end{gather}\nIf the parametrisation of the lightlike geodesic is future-oriented\nwith respect to the $T$ coordinate, the scalar factors $\\chi$ and \n$\\tilde{\\chi}$ are positive; otherwise they are negative.\n$\\alpha$ is the angle between the lightlike geodesic and \nthe radial direction in the rest system of the static observer, \nwhereas $\\tilde{\\alpha}$ is the analogously defined angle in \nthe rest system of the infalling observer. \n$\\alpha$ and $\\tilde{\\alpha}$ may take all values between 0 and $\\pi$.\nOf course, $\\alpha$ is well-defined on the domain where the \nstatic observer exists (i.e., for $2m < r < \\infty$) whereas\n$\\tilde{\\alpha}$ is well-defined on the domain where the \ninfalling observer exists (i.e., for $0 < r < r_i = 2m\/(1-\\varepsilon ^2)$ \nif $0< \\varepsilon < 1$, and for $0 < r < \\infty$ if $1 \\le \\varepsilon < \\infty$).\n\nComparing coefficients of $\\partial\/ \\partial r$\nand $\\partial\/\\partial \\varphi$ in (\\ref{eq:tangent1}) yields\n\n\\begin{equation}\\label{eq:cc2}\n\\dot{r} \n= \\, \\chi \\, \\sqrt{1-\\dfrac{2m}{r}} \\, \\mathrm{cos} \\, \\alpha \n= \\tilde{\\chi} \\left( \\varepsilon \\, \\mathrm{cos} \\, \\tilde{\\alpha} \n- \\sqrt{\\varepsilon ^2 -1+\\dfrac{2m}{r}} \\right) \\, ,\n\\end{equation}\n\n\\begin{equation}\\label{eq:cc3}\n\\dot{\\varphi} \n= - \\chi \\, \\dfrac{\\mathrm{sin} \\, \\alpha}{r} \n= - \\tilde{\\chi} \\, \\dfrac{\\mathrm{sin} \\, \\tilde{\\alpha}}{r} \\, .\n\\end{equation}\n\nFrom (\\ref{eq:cc2}) and (\\ref{eq:cc3}) we find\n\n\\begin{equation}\\label{eq:alpha1}\n\\dfrac{\\dot{\\varphi}}{\\dot{r}} \n=\n\\dfrac{\n- \\mathrm{sin} \\, \\alpha\n}{\nr \\sqrt{1-\\dfrac{2m}{r}} \\mathrm{cos} \\, \\alpha\n}\n=\n\\dfrac{\n- \\mathrm{sin} \\, \\tilde{\\alpha}\n}{\nr \\left( \\varepsilon \\, \\mathrm{cos} \\, \\tilde{\\alpha} - \n\\sqrt{\\varepsilon ^2 -1+\\dfrac{2m}{r}} \\right)\n} \\, .\n\\end{equation}\nNow we apply these results to the case of a lightlike geodesic that goes through an extremum\nof the radius coordinate at some value $r_m$. If we evaluate (\\ref{eq:alpha1})\nat a radius value $r>3m$ this extremum is necessarily a local minimum, whereas it is necessarily \na local maximum if we evaluate (\\ref{eq:alpha1}) at a radius value $r<3m$. \nIn either case (\\ref{eq:orbit2}) implies that the angles $\\alpha$ and $\\tilde{\\alpha}$ \nat $r$ satisfy\n\n\\begin{equation}\\label{eq:alpha2}\n\\dfrac{1}{\\dfrac{r^2(r_m-2m)}{r_m^3}-1 + \\dfrac{2m}{r}} \n=\n\\dfrac{ \\mathrm{sin} ^2 \\alpha}{ \\left( 1-\\dfrac{2m}{r} \\right) \n\\mathrm{cos} ^2 \\alpha}\n=\n\\dfrac{\n\\mathrm{sin} ^2 \\tilde{\\alpha}\n}{\n\\left( \\varepsilon \\, \\mathrm{cos} \\, \\tilde{\\alpha} \n- \\sqrt{\\varepsilon ^2 -1+\\dfrac{2m}{r}} \\right) ^2\n} \\, .\n\\end{equation}\nFrom the second equality sign in (\\ref{eq:alpha2}) we find\n\n\\begin{equation}\\label{eq:aberr2}\n\\mathrm{sin} \\, \\alpha =\n\\dfrac{\n\\sqrt{1-\\dfrac{2m}{r}} \\, \\dfrac{1}{\\varepsilon} \\, \\mathrm{sin} \\, \\tilde{\\alpha}\n}{\n1- \\dfrac{1}{\\varepsilon} \\sqrt{\\varepsilon ^2 -1+\\dfrac{2m}{r}} \\, \n\\mathrm{cos} \\, \\tilde{\\alpha}\n} \\, .\n\\end{equation}\nBy (\\ref{eq:vrad}), this just demonstrates that $\\alpha$ and $\\tilde{\\alpha}$ are \nrelated by the standard aberration formula. \n\nFrom the first equality sign in (\\ref{eq:alpha2}) we find\n\n\\begin{equation}\\label{eq:alpha4}\n\\mathrm{sin} \\, \\alpha =\n\\sqrt{\\dfrac{r_m^3(r-2m)}{r^3(r_m-2m)}}\n\\end{equation}\nand equating the first to the third expression in (\\ref{eq:alpha2}) yields\n\n\\begin{equation}\\label{eq:alpha5}\n\\mathrm{sin} \\, \\tilde{\\alpha} =\n\\dfrac{\n\\sqrt{1 -\\dfrac{2m}{r}} \\, \n\\sqrt{\\dfrac{(r-2m)r_m^3}{(r_m-2m) r^3}} \n}{\n\\varepsilon \\pm \\, \n\\sqrt{\\varepsilon ^2 -1+\\dfrac{2m}{r}} \\, \n\\sqrt{1-\\dfrac{(r-2m)r_m^3}{(r_m-2m) r^3}}\n} \\, .\n\\end{equation}\nIn (\\ref{eq:alpha5}) the upper sign is valid if $dr\/d \\varphi >0$ \nand the lower sign is valid if $dr\/d \\varphi <0$ at $r$. \n\n\n\\begin{figure}[h]\n \\psfrag{x}{$r_O$}\n \\psfrag{y}{$\\,$ \\hspace{-0.3cm} $\\alpha _{\\mathrm{sh}}$}\n \\psfrag{a}{\\small $\\,$ \\hspace{-0.5cm} $2m$}\n \\psfrag{b}{\\small $\\,$ \\hspace{-0.42cm} $3m$}\n \\psfrag{c}{\\small $\\,$ \\hspace{-0.5cm} $\\dfrac{\\pi}{2}$}\n \\psfrag{d}{\\small $\\,$ \\hspace{-0.5cm} $\\pi$}\n\\centerline{\\epsfig{figure=alphabh.eps, width=11cm}}\n\\caption{Angular radius $\\alpha _{\\mathrm{sh}}$ of the shadow of \na Schwarzschild black hole for a static observer. What we have \nplotted here is Synge's formula (\\protect{\\ref{eq:alpha6}}). For \nan observer at $r_O=3m$, we have $\\alpha = \\pi\/2$, i.e., half\nof the sky is dark. For $r_O \\to 2m$, we have $\\alpha \\to \\pi$, \ni.e., in this limit the entire sky becomes dark. \\label{fig:alphabh}}\n\\end{figure}\n\nFrom (\\ref{eq:alpha4}) and (\\ref{eq:alpha5}) we can now easily determine \nthe angular radius of the shadow. The latter is defined in the following way:\nConsider an observer at radius coordinate $r=r_O >3m$. Then a lightlike geodesic \nissuing from the observer position into the past may either go to infinity,\npossibly after passing through a minimum of the radius coordinate\nat some $r_m>3m$, or it may go to the horizon. Similarly, for an observer position \n$2m 1$ \n(dashed). In any case $\\tilde{\\alpha}{}_{\\mathrm{sh}}$ approaches $\\pi\/2$, i.e., half of \nthe sky becomes dark, for $r_O \\to 0$. For the sake of comparison the \nangular radius $\\alpha _{\\mathrm{sh}}$ for a static observer is again\nplotted in this diagram as a thick (blue) line, cf. \nFigure~\\protect{\\ref{fig:alphabh}}. \\label{fig:talphabh}}\n\\end{figure}\nTherefore, we get the angular radius of the shadow for an observer\nat $r=r_O$ if we send $r_m \\to 3m$ in (\\ref{eq:alpha4}) and \n(\\ref{eq:alpha5}). This results in \n\n\\begin{equation}\\label{eq:alpha6}\n\\mathrm{sin} \\, \\alpha _{\\mathrm{sh}}=\n\\dfrac{\\sqrt{27} m}{r_O} \\sqrt{ 1- \\dfrac{2m}{r_O}}\n\\end{equation}\nand\n\n\\begin{equation}\\label{eq:alpha7}\n\\mathrm{sin} \\, \\tilde{\\alpha} {}_{\\mathrm{sh}} =\n\\dfrac{\n\\dfrac{\\sqrt{27} m}{r_O} \\, \n\\left(\n\\varepsilon \\mp \n\\sqrt{\\varepsilon ^2 -1+\\dfrac{2m}{r_O}}\n\\, \\sqrt{1- \\dfrac{27 m^2}{r_O^2} \n\\Big( 1 - \\dfrac{2m}{r_O} \\Big)} \n\\right)\n}{\n1 + 27 \\, \\dfrac{m^2}{r_O^2} \\, \n\\left( \\varepsilon ^2 -1+\\dfrac{2m}{r_O} \\right)\n} \\, ,\n\\end{equation}\nrespectively. (\\ref{eq:alpha6}) gives us the angular radius \n$\\alpha _{\\mathrm{sh}}$ as it is seen by a static observer \nat $r_O$, see Figure~\\ref{fig:alphabh}. \nThis formula is known since Synge \\cite{Synge1966}. It is \nmeaningful only for observer positions $2m < r_O < \\infty$ \nbecause the static observers exist on this domain only. \nBy contrast, (\\ref{eq:alpha7}) gives us the angular radius \n$\\tilde{\\alpha}{}_{\\mathrm{sh}}$ of the shadow as it is \nseen by an infalling observer at momentary radius coordinate \n$r_O$, see Figure~\\ref{fig:talphabh}. A similar formula was\nderived by Bakala et al. \\cite{BakalaEtAl2007}, even for \nthe more general case of a Schwarzschild-deSitter (Kottler) \nblack hole. (\\ref{eq:alpha7}) is meaningful for \n$00$ the worldline of an\nobserver on the surface of the star is then given by one of the dotted lines\nin Figure~\\ref{fig:radial}. From (\\ref{eq:rTrad}) with $\\varepsilon ^2=1-2m\/r_i$\nwe find that for $T_S>0$\n\n\\begin{equation}\\label{eq:surface}\nc T_S = \n\\bigintss _{r_S} ^{r_i} \n\\dfrac{ \\sqrt{r_i-2m} \\, \\sqrt{r^3} \\, dr}{(r-2m) \\sqrt{2m(r_i-r)}}\n- \\bigintss _{r_S}^{r_i} \\dfrac{\\sqrt{2mr} \\, dr}{(r-2m)}\n\\end{equation}\nEquating $r_S$ to zero gives the collapse time, $T_S^{\\mathrm{coll}}$, i.e., the\nPainlev{\\'e}-Gullstrand time when the star has collapsed to a point singularity \nat the centre, see Figure~\\ref{fig:collstat},\n\n\\begin{equation}\\label{eq:Tcoll}\nc T_S^{\\mathrm{coll}} = \n\\bigintss _{0} ^{r_i} \n\\dfrac{ \\sqrt{r_i-2m} \\, \\sqrt{r^3} \\, dr}{(r-2m) \\sqrt{2m(r_i-r)}}\n- \\bigintss _{0}^{r_i} \\dfrac{\\sqrt{2mr} \\, dr}{(r-2m)} \\, .\n\\end{equation}\nNote that necessarily $2m 3m$. \nFor calculating the shadow we have to consider lighlike geodesics\nthat graze the surface of the collapsing star. If such a lightlike\ngeodesic passes through a minimum radius value $r_m$, we may\ndetermine $r_m$ by equating the first and the last expression in \n(\\ref{eq:alpha2}) with $r=r_S$, $\\tilde{\\alpha} = \\pi\/2$ and $\\varepsilon ^2 = \n1- 2m\/r_i$. This results in\n\n\\begin{equation}\\label{eq:rmcoll2}\n\\dfrac{r_m^3}{r_m-2m} = \\dfrac{r_i r_S^2}{r_i-2m} \\, . \n\\end{equation}\nRecall that a minimum value is possible only for $r_m>3m$,\ni.e., in (\\ref{eq:rmcoll2}) $r_S$ must satisfy the inequality $r_S>r_S^{(2)}$\nwhere \n\n\\begin{equation}\\label{eq:rS2}\nr_S ^{(2)} = 3 m \\sqrt{3-\\dfrac{6m}{r_i}} \\, .\n\\end{equation}\nAs $r_i>3m$, (\\ref{eq:rS2}) implies that \n\n\\begin{equation}\\label{eq:rS2in}\n3mT_S$. With (\\ref{eq:surface}) this results in\n\n\\begin{gather}\\label{eq:Tcollstat}\nc T _O = \n\\bigintss _{r_S} ^{r_i} \n\\dfrac{ \\sqrt{r_i-2m} \\, \\sqrt{r^3} \\, dr}{(r-2m) \\sqrt{2m(r_i-r)}}\n+ \\bigintss _{r_i} ^{r_O} \n\\dfrac{\\sqrt{2mr} \\, dr}{(r-2m)}\n\\\\\n\\nonumber\n+\n \\bigintss _{r_S}^{r_O} \\dfrac{\\sqrt{r_i-2m} \\, \\sqrt{r^5} \\, dr}{(r-2m) \\sqrt{(r_i-2m)r^3-(r-2m)r_ir_S^2}}\n\\, .\n\\end{gather}\n\nIf $r_i$ and $r_O$ are given, with $3m T_O^{(2)}$, the angular radius of \nthe shadow is given by Synge's formula (\\ref{eq:alpha6}). Past-oriented light \nrays grazing the surface of the star cannot escape to infinity anymore,\ni.e., they do not give the boundary of the shadow; the latter is determined by light\nrays that spiral asymptotically to $r=3m$.\n\n\\begin{figure}[h]\n \\psfrag{x}{$r_O$}\n \\psfrag{y}{$\\,$ \\hspace{-1.3cm} $cT_O^{(2)}-cT_O^{(1)}$}\n \\psfrag{a}{\\small $\\,$ \\hspace{-0.5cm} $25m$}\n \\psfrag{b}{\\small $\\,$ \\hspace{-0.5cm} $50m$}\n \\psfrag{c}{\\small $\\,$ \\hspace{-0.5cm} $75m$}\n \\psfrag{d}{\\small $\\,$ \\hspace{-0.5cm} $100m$}\n \\psfrag{e}{\\small $\\,$ \\hspace{-0.9cm} $25m$}\n \\psfrag{f}{\\small $\\,$ \\hspace{-0.9cm} $50m$}\n \\psfrag{g}{\\small $\\,$ \\hspace{-0.9cm} $75m$}\n\\centerline{\\epsfig{figure=colltime.eps, width=10.1cm}}\n\\caption{Time $T_O^{(2)}-T_O^{(1)}$ over which the static observer sees the \nstar collapse, plotted against the observer position $r_O$. The star is\ncollapsing from an initial radius $r_i=5m$ (dotted), $r_i=10m$ (solid)\nor $r_i=15m$ (dashed), respectively. \\label{fig:colltime}}\n\\end{figure}\n\nWe summarise our analysis in the following way. In the first phase, which lasts \nfrom $T_O = - \\infty$ to $T_O = T_O^{(1)}$ given by (\\ref{eq:TO1}), the observer\nsees a shadow of constant angular radius given by (\\ref{eq:shcollstat1}). In the\nsecond phase, which lasts from $T_O= T_O^{(1)}$ until $T_O=T_O^{(2)}$ given by\n(\\ref{eq:TO2}), the observer sees a shrinking shadow whose angular radius\nas a function of observer time $T_O$ is given in parametric form by (\\ref{eq:Tcollstat})\nand (\\ref{eq:shcollstat2}). The parameter $r_S$ runs down from $r_S^{(1)}=r_i$ to\n$r_S^{(2)}= 3m \\sqrt{3-6m\/r_i}$. The third phase lasts from $T_O = T_O^{(2)}$ to\n$T_O=\\infty$. In this period the observer sees a shadow of constant angular\nradius given by Synge's formula (\\ref{eq:alpha6}). The angular radius of the\nshadow is plotted against $T_O$, over all three periods, for $r_i=5m$\nand $r_O=10m$ in Figure \\ref{fig:alphacoll1}. \n\nIn Fig. \\ref{fig:colltime} we plot the time $T_O^{(2)}-T_O^{(1)}$ over which \nthe observer sees the star collapse against the observer position $r_O$. We\nsee that this time is largely independent of $r_O$, unless the observer is\nvery close to the star. For a star collapsing from an initial radius of 5\nSchwarzschild radii, $r_i=10m$, we see that $T_O^{(2)}-T_O^{(1)} \\approx\n34 \\, m\/c$ for a sufficiently distant observer. For a stellar black hole, a typical\nvalue would be $m \\approx 15 \\, \\mathrm{km}$, resulting in $T_O^{(2)}-T_O^{(1)}\n\\approx 0.001 \\, \\mathrm{sec}$, so such a collapse would happen quite quickly.\nEven for a supermassive black hole of $m \\approx 10^6 \\, \\mathrm{km}$, the\nobserver would see the collapse happen in less than 2 minutes. For the case \nof a collapsing cluster of galaxies the formation of the shadow would take longer, \nbut in this case it is more reasonable to model the collapsing object as transparent.\nNote that on the worldline of a distant static observer Painlev{\\'e}-Gullstrand \ntime $T_O$ is practically the same as proper time $\\tau _O$ because, \nby (\\ref{eq:taustatic}), \n \n\\begin{equation}\\label{eq:tauTO}\n \\tau _O = \\sqrt{1- \\dfrac{2m}{r_O}} \\, T_O + \\mathrm{constant}.\n\\end{equation}\n\n\\begin{figure}[h]\n \\psfrag{x}{$c T _O$}\n \\psfrag{y}{$\\,$ \\hspace{-0.3cm} $\\alpha _{\\mathrm{sh}}$}\n \\psfrag{a}{\\small $\\,$ \\hspace{-0.62cm} $\\begin{matrix} \\, \\\\[-0.12cm] 4 m\\end{matrix}$}\n \\psfrag{b}{\\small $\\,$ \\hspace{-0.58cm} $\\begin{matrix} \\, \\\\[-0.12cm] 8 m\\end{matrix}$}\n \\psfrag{c}{\\small $\\,$ \\hspace{-0.62cm} $\\begin{matrix} \\, \\\\[-0.12cm] 12 m\\end{matrix}$}\n \\psfrag{d}{\\small $\\,$ \\hspace{-0.58cm} $\\begin{matrix} \\, \\\\[-0.12cm] 16 m\\end{matrix}$}\n \\psfrag{p}{\\small $\\,$ \\hspace{-0.56cm} $\\begin{matrix} \\, \\\\[-0.2cm] cT_O^{(0)} \\end{matrix}$}\n \\psfrag{q}{\\small $\\,$ \\hspace{-0.64cm} $\\begin{matrix} \\, \\\\[-0.2cm] cT_O^{(2)} \\end{matrix}$}\n \\psfrag{e}{\\small $\\,$ \\hspace{-0.5cm} $\\dfrac{\\pi}{4}$}\n \\psfrag{f}{\\small $\\,$ \\hspace{-0.5cm} $\\dfrac{\\pi}{2}$}\n \\psfrag{r}{\\small $\\,$ \\hspace{-0.85cm} $\\alpha _{\\mathrm{sh}}^{(0)}$}\n\\centerline{\\epsfig{figure=alphacoll2.eps, width=11.7cm}}\n\\caption{Angular radius $\\alpha _{\\mathrm{sh}}$ of the shadow of a collapsing \ndark star for a static observer. The observer is at $r_O=4.5 m$, the \nsurface of the star is collapsing from $r_i = 5 m$.\nThe observation begins at Painlev{\\'e}-Gullstrand time $T_O^{(0)}$ \nwhen the surface of the star passes through the radius value $r_O$. \nAt this moment the angular radius of the shadow takes a value $\\alpha _{\\mathrm{sh}}^{(0)}$ \nwhich is smaller than $\\pi \/2$, because of aberration.\nAs a function of the Painlev{\\'e}-Gullstrand time coordinate $T _O$, the \nangular radius $\\alpha _{\\mathrm{sh}}$ then decreases until it becomes a \nconstant at time $T_O^{(2)}$. This constant value, which is again shown as a dotted\nline, is given by Synge's formula (\\protect{\\ref{eq:alpha6}}). \\label{fig:alphacoll2}}\n\\end{figure}\n\nA very similar analysis applies to case (b). The only difference is that then in \nthe beginning the observer is inside the star. The observation can begin only at \nthe time when the surface of the star passes through the radius value $r_O$\nwhich, by assumption, is bigger than $r_S^{(2)}$. From that time on, the angular \nradius of the shadow is given by the same equations as before for the second and \nthe third phase. A plot of the angular radius of the shadow against $T_O$ is\nshown in Figure~\\ref{fig:alphacoll2} for $r_O=4.5 m$ and $r_i=5m$.\n\nIn case (c) the observer is initially inside the star, as in case (b). The\ndifference is in the fact that now the radius of the star is smaller than $r_S^{(2)}$ \nat the moment when the observation begins. Therefore, the shadow is never\ndetermined by light rays that graze the surface of the star; it is always \ndetermined by light rays that spiral towards $r=3m$, i.e., the angular radius \nof the shadow is constant from the beginning of the observation and given by \nSynge's formula. \n \n\\section{The shadow of a collapsing star for an infalling observer}\\label{sec:shcoll2}\n\nWe consider the same collapsing dark star as in the preceding section,\nbut now we want to calculate the shadow as it is seen by an infalling\nobserver. The relation between the coordinates $r_O$ and $T_O$ of the \ninfalling observer can be found by integrating (\\ref{eq:rTrad}),\n\\begin{equation}\\label{eq:TOrO}\nc T_O = \\bigintss_{r_O} ^{r_O^*} \n\\dfrac{\n\\Big( \\varepsilon \\sqrt{r^3}- \\sqrt{2mr} \\sqrt{ \\varepsilon ^2 r -r+2m}\\Big) dr\n}{\n( r -2m ) \\sqrt{\\varepsilon ^2r -r+2m} \n} \\, .\n\\end{equation}\nHere $r_O^*$ is an integration constant that gives the position of the observer at $T=0$ which\nis the time when the star begins to collapse.\nWe assume that $r_i>3m$, hence $3m < r_S^{(2)}