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+{"text":"\\section{Introduction}\nThe study of quantum chaotic systems has gained extensive attention in the last few years. In classical mechanics, the chaos of a system is described as the sensitivity of the particle trajectory to the initial conditions. Or, in other words, the sensitivity of the position of a system in the phase space to the initial conditions. For chaotic systems, the trajectory can deviate exponentially due to small changes in the initial conditions, which is called the butterfly effect. However, quantum chaos is still difficult to be well understood. One important reason is that the concept of trajectory becomes ill-defined after considering quantum effects. One has difficulty in finding a physical observable in quantum systems to measure the extent of chaos. Despite the difficulties in measuring quantum chaos, great progress has been made in the last few years in understanding quantum chaos. To date, many methods have been developed to probe quantum chaos.\n\nIt is worth noting that the study of holography has greatly advanced the understanding of quantum chaotic systems. One of the most exciting and promising examples is the study of operator growth. In a quantum chaotic system, a simple operator becomes extraordinarily complicated after the Heisenberg time evolution. Measuring how an operator becomes more and more \"complex\" from a \"simple\" can be an important indicator in chaotic systems. Similar to the classical case, we expect the measurement to be sensitive enough to the initial conditions, and the deviation caused by different initial conditions (or rather, the complexity of the operator) grows exponentially with time.\n\nThe out-of-time-order correlator (OTOC) method has gained a lot of attention in the study of operator growth \\cite{Rozenbaum:2016mmv,Hashimoto:2017oit,Nahum:2017yvy,Shen:2017kez,Fan:2016ean,Khemani:2017nda,Rozenbaum:2018sns,Pilatowsky-Cameo:2019qxt,Xu:2019lhc,Rozenbaum:2019nwn,Hashimoto:2020xfr}. One reason is that OTOC can reproduce the exponential growth in classical chaotic models. Using the OTOC method, we can generalize the Lyapunov exponent to quantum systems, i.e., the quantum Lyapunov exponent. Another important reason is that, in the context of AdS\/CFT correspondence or quantum gravity, it can be an indicator of possible gravity dual\\cite{Shenker_2014_1,Shenker_2014_2,Maldacena_2016}. Gedanken experiment about shock waves in the black hole background can lead to a maximum bound of Lyapunov exponent\\cite{Maldacena_2016}. This Lyapunov bound is saturated in the SachdevYe-Kitaev (SYK) model \\cite{Sachdev:1992fk}, which implies that the gravity duality of the SYK model may describe a quantum black hole. In addition, the operator size has been argued to correspond to the radial momentum of the particle in the gravity side\\cite{https:\/\/doi.org\/10.48550\/arxiv.1802.01198}. The radial momentum is defined by the maximal volume of the Cauchy surface in the bulk. We will see that this has a very close relation with another important concept in quantum chaotic systems, circuit complexity.\n\nIn parallel developments, circuit complexity (or computational complexity) is another physical observable that has been given significant importance in the study of AdS\/CFT and quantum chaos\\cite{https:\/\/doi.org\/10.48550\/arxiv.1403.5695,https:\/\/doi.org\/10.48550\/arxiv.0804.3401,Stanford_2014,https:\/\/doi.org\/10.48550\/arxiv.1411.0690,Brown_2016,PhysRevLett.116.191301}. In 2014, Susskind and Stanford proposed that the circuit complexity in the boundary state is dual to the maximal volume of the Cauchy surface in bulk. Since then several proposals for the holographic complexity have been proposed, including the CA correspondence \\cite{Brown_2016,PhysRevLett.116.191301} and CV2.0 correspondence \\cite{Couch_2017}. To be more specific, in the study of the eternal AdS-Schwarzchild black hole, it was found that the volume of the wormhole connecting two boundaries grows linearly with time. And an important feature of computational complexity is it grows linearly in chaotic systems until the Heisenberg time $t \\sim e^S$. The conjecture is the growth of the wormhole in bulk is dual to the growth of the complexity on the boundary.\n\nAlthough both the studies of operator growth and circuit complexity have made exciting progress in their respective directions. However, the relation between these two similar but parallel directions has been ambiguous. An emerging concept in recent years, Krylov complexity (or K-complexity in short) \\cite{Parker_2019,Barb_n_2019,Rabinovici_2021,Jian:2020qpp}, seems to bridge the gap between these two fields. It is not only a well-defined physical measure of operator growth but also has the same growth pattern as circuit complexity. In other words, there are possible geometric duals for K-complexity. We will focus on this concept in this paper. \n\nKrylov complexity was first put forward in \\cite{Parker_2019} to measure the universal features of operator growth. Precisely, it measures the extent how an operator spread in the Krylov basis, which is constructed from the Baker-Campbell-Hausdorff formula for the Heisenberg time evolution of chosen operator. The process of constructing the Krylov basis is known as the Lanczos algorithm. When implementing the Lanczos algorithm, we obtain the so-called Lanczos coefficients $b_n$, which compactly contain the information in the Krylov basis. In the chaotic system, Lanczos coefficients are expected to show a linear growth pattern at the beginning and then go into saturation. Also, Lanczos coefficients are argued to have the fastest rate of growth in chaotic systems. In addition, the slope of the linear growth $b_n \\sim a n$ of Lanczos coefficients is conjectured to give the upper bound $\\lambda < 2a$ of the quantum Lyapunov exponent. This upper bound is saturated in the SYK model. The hypothesis itself is even valid beyond the semiclassical system where the Lyapunov exponent is ill-defined. From this point of view, Krylov complexity is important for finding a well-defined physical quantity to measure operator growth in quantum systems.\n\nK-complexity has a growth pattern similar to that of circuit complexity, i.e., exponential growth at the beginning, followed by linear growth, and finally saturation. In the AdS\/CFT correspondence, some possible gravitational duals of K-complexity have been investigated, e.g., the SYK model and JT gravity \\cite{Jian:2020qpp}. Besides, one of the most important drawbacks of circuit complexity is that it is extremely hard to calculate circuit complexity in strongly interacting quantum field theory. While K-complexity seems to have a greater potential to be computed in quantum field theory\\cite{Caputa:2021ori,Caputa:2021sib,Dymarsky:2021bjq}. It is for these reasons that calculating K-complexity in strongly interacting QFT becomes an interesting problem. In this paper, we will show the calculation of K-complexity in the SU(2) Yang-Mills theory. \n\nSU(2) Yang-Mills theory was shown to be chaotic in \\cite{matinyan1981stochasticity}. It is expected to have a gravity dual in the large N and strong coupling limit. So, this model can be a starting point to investigate the relation between K-complexity and AdS\/CFT correspondence. From another point of view, it can be recognized as a rather simple version of the Quantum chromodynamics (QCD). this model has attracted a lot of attention over the years. For these reasons, the SU(2) Yang-Mills theory and related models have attracted attention for many years.\n\nThis paper is organized as follows. In section \\ref{section2}, we review how the SU(2) Yang-Mills theory can be reduced to a coupled harmonic oscillator(CHO) model by dimensional reduction, and analyze the chaotic nature of this model from both classical and quantum perspectives. In section \\ref{section3}, we review the concepts of Krylov complexity and Lanczos coefficients, then summarized our approach to numerically implement the Lanczos algorithm. In section \\ref{section4}, we numerically compute the Krylov complexity and Lanczos coefficients in the CHO model for different coupling constants. We found that Krylov complexity in the CHO model shows a quadratic growth in the early time stage, then enters a linear growth. By analyzing the Lanczos coefficients, we find an upper bound on the quantum Lyapunov exponent. In section \\ref{section5}, we discuss the effect of different energy modes on K-complexity, and how different temperatures and energy affect Lanczos coefficients. We conclude this paper with a brief discussion in section \\ref{section6}.\n\n\\section{Classical and quantum description of coupled harmonic oscillators}\\label{section2}\nIn this section, we review how SU(2) Yang-Mills theory can be reduced to a coupled harmonic model (CHO) by dimensional reduction, following the work of \\cite{Akutagawa_2020,matinyan1981stochasticity}. And then in section \\ref{2.2}, we analyze the CHO model classically. In section \\ref{2.3}, we review the OTOC method and quantum Lyapunov exponent in the CHO model.\n\n\\subsection{Reduction of SU(2) Yang-Mills theory}\\label{2.1}\nThe four-dimensional SU(2) Yang-Mills theory has the following Lagrangian\n\\begin{equation}\n    \\mathcal{L} = (D_\\mu \\phi)^\\dagger (D^\\mu \\phi) - V(\\phi) - \\frac{1}{4} F^a_{\\mu \\nu}F^{a \\mu \\nu}\n\\end{equation}\nwhere the field strength $F^a_{\\mu \\nu} = \\partial_\\mu A^a_\\nu -\\partial_nu A^a_{\\mu} + g \\epsilon^{abc} A_{\\mu}^b A_{\\nu}^{c}$, and $D_\\mu = \\partial_\\mu -i g A^a_\\mu T^a$, $T^a = \\sigma^a\/2$ ($\\sigma^a$ is the Pauli matrix ). Here, the index $a= 1,2,3$ and $\\mu, \\nu = {0,1,2,3}$. The scalar potential $V(\\phi)$ is given by\n\\begin{equation}\n    V(\\phi) = \\mu^2 |\\phi|^2 +\\lambda |\\phi|^4\n\\end{equation}\nThe gauge condition we chose is $A^a_0 = 0$. Furthermore, we assume the fields satisfy the spatial homogeneousness condition.\n\\begin{equation}\n    \\partial_i A^a_j =0, \\ \\ \\ \\partial_i \\phi =0\n\\end{equation}\nThen the remaining degrees of freedom are $A^1_1, A^2_2$. Let $A^1_1=\\sqrt{2} x(t), A^2_2= \\sqrt{2}y(t)$, then the Hamiltonian becomes \n\\begin{equation}\n    H = \\dot{x}^2+\\dot{y}^2 + \\frac{\\omega^2}{4}(x^2+y^2) + g^2 x^2 y^2\n\\end{equation}\nwhere we have introduced the frequency $\\omega = g \\langle \\phi \\rangle_{vac}$.\n\nThis Hamiltonian can also be recognized as a toy model of the Quantum\nchromodynamics (QCD) with the colors $N_C = 2$. Then the coupling constant $g$ controls the non-linear interaction between the colors \\cite{Matinyan:1981dj}, and $\\omega$ denotes the mass, which is finite in the presence of the Higgs mechanism.\n\nIn classical systems, Poincar\u00e9 sections and Lyapunov exponents are often used to measure the chaotic nature of a system. While in quantum mechanics, OTOC is the physical quantity that is most often used to measure the properties of chaotic systems. We will review these methods in the following sections.\n\n\\subsection{Classical analysis of coupled harmonic oscillators} \\label{2.2}\nN-dimensional Coupled harmonic oscillators (CHO) is a quantum mechanical model whose Hamiltonian is the following.\n\\begin{equation}\n    H = \\sum_{i=1}^{N}\\left( p_i^2 + \\frac{\\omega^2}{4} q_i^2 \\right) +\\frac{1}{2}\\sum_{i,j=1}^N g_{ij} q_i^2 q_j^2 \n\\end{equation}\nwhere $\\omega$ is the frequency of the harmonic oscillators. And $g_{ij}$ are the coupling constants, which determine the strength of the nonlinear interaction between different harmonic oscillators. \n\nWhen $g_{ij}$ is zero, the whole system is simplified to N decoupled harmonic oscillators, which is an integrable system. As $g_{ij}$ slowly increases, the non-linear term $g_{ij} q_i^2 q_j^2$ starts to dominate and the model becomes chaotic.\n\nThe Poincar\u00e9 section is an intuitive way to qualitatively measure the chaotic nature of a classical system. We plotted the Poincar\u00e9 sections with energy $E = 1$ for different coupling constants in Figure \\ref{fig120}. It can be seen that when the coupling constant $g_0 = 0.1$, the corresponding Poincar\u00e9 sections show clear orbits, which indicates the system is in a regular phase. The chaotic property of the system starts to appear when $g_0 = 0.3$. Randomly scattered points around the fixed points gradually replace the continuous trajectories. When $g_0 = 0.7$, only a small part of the Poincar\u00e9 section has continuous trajectories, while most of the figure is covered by scattered points. And when the coupling constant $g_0 =1$, the Poincar\u00e9 section is almost completely covered by scattered points, so the system is in the chaos phase. \n\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[$g_0 =0.1$]{\n\\includegraphics[width=5.5cm]{Poincare section\/g0=0.1,E=1.png}\n}\n\\quad\n\\subfigure[$g_0 =0.3$]{\n\\includegraphics[width=5.5cm]{Poincare section\/g0=0.3,E=1.png}\n}\n\\quad\n\\subfigure[$g_0 =0.7$]{\n\\includegraphics[width=5.5cm]{Poincare section\/g0=0.7,E=1.png}\n}\n\\quad\n\\subfigure[$g_0 =1$]{\n\\includegraphics[width=5.5cm]{Poincare section\/g0=1,E=1.png}\n}\n\\caption{Poincar\u00e9 sections of the CHO model for four different coupling constants $g_0 = 0.1,0.3,0.7,1$. As the coupling constant rises, the clear orbits with strong periodicity gradually become randomly scattered points.}\n\\label{fig120}\n\\end{figure}\n\nOn the other hand, energy is also an important factor that affects the chaos in the system. We show in figure \\ref{fig147} the Poincar\u00e9 section at different energy $E = 1, 3, 5, 15$. We can see the transition from an integrable system to a chaotic system. At $E = 1$, the system is in regular phase. When the energy becomes $E = 3, 5$, the system transforms into a mixed phase. Finally, when $E$ rises to $15$, the system enters the chaos phase.\n\n\\begin{figure}[tbp]\n\\centering\n\\subfigure[E=3]{\n\\includegraphics[width=5.5cm]{Poincare section\/g0=0.1,E=3.png}\n}\n\\quad\n\\subfigure[E=5]{\n\\includegraphics[width=5.5cm]{Poincare section\/g0=0.1,E=5.png}\n}\n\\quad\n\\subfigure[E=7]{\n\\includegraphics[width=5.5cm]{Poincare section\/g0=0.1,E=7.png}\n}\n\\quad\n\\subfigure[E=15]{\n\\includegraphics[width=5.5cm]{Poincare section\/g0=0.1,E=15.png}\n}\n\\caption{Poincar\u00e9 sections in the CHO Hamiltonian system for four different energy $E=3, E=5, E=7, E=15$. When the energy is low, the Poincar\u00e9 section shows the characteristics of integrable systems, i.e., clear periodic orbits. As the energy increases, the system gradually enters the chaotic phase.}\n\\label{fig147}\n\\end{figure}\n\nAnother parameter that affects the chaotic property of the system is time. If the coupling constant $g_0<1$, according to the perturbative theory, when $t$ is small, the nonlinear term in the Hamiltonian is suppressed. So, the system shows the characteristics of an integrable system in the early time stage. After that, chaotic characteristics start to emerge. We analyze in section 4 the time scale of this transition.\n\nThrough the discussion in this section, we find, classically, the chaotic property of the CHO model is governed by three parameters coupling constant $g_0$, energy scale $E$, and the evolution time t. We will further discuss their effects on quantum chaotic systems in sections \\ref{section4} and \\ref{section5}.\n\n\\subsection{OTOC method for coupled harmonic oscillators}\\label{2.3}\nIn classical chaotic systems, the Lyapunov exponent $\\lambda_{cl}$ is defined by the exponential growth of the orbit deflection\n\\begin{equation}\n    \\Delta x(t) \\simeq \\Delta x(0)e^{\\lambda_{cl} t}\n\\end{equation}\nwhere $\\Delta x(0)$ means the small differences in initial conditions, which can cause huge deviations for the orbit afterward. The corresponding exponent is defined the be the classical Lyapunov exponent $\\lambda_{cl}$. This concept can be generalized to quantum systems. The quantum Lyapunov exponent is related to the out-of-time-order correlator (OTOC). In a thermal system, the OTOC is defined by \n\\begin{equation}\n    C_\\beta(t) = -\\langle [\\hat{x}(t),\\hat{p}]^2 \\rangle_{\\beta}\n\\end{equation}\nwhere $\\beta$ is the inverse temperature, $\\langle \\rangle_\\beta$ denotes the thermal expectation. Here we have chosen $\\hat{p}$ as the reference operator. The OTOC is expected to grow exponentially in chaotic system $C_\\beta(t) \\simeq C_\\beta(0)e^{\\lambda t}$. And the quantum Lyapunov exponent is defined to be the exponent $\\lambda$.\n\nQuantum Lyapunov exponent is an important observable to measure the chaotic structure of a system. It was conjectured that the linear growth of Lanczos coefficients gives a maximum bound on quantum Lyapunov exponent \\cite{Parker_2019}. We will analyze how the Lanczos coefficients in the CHO model bound the quantum Lyapunov exponent in section \\ref{section5}.\n\n\\section{Krylov complexity in coupled harmonic oscillators}\\label{section3}\n\\subsection{Brief review of Krylov complexity}\\label{3.1}\n\nIn this subsection, we briefly review the concept of operator growth and Krylov complexity. In a quantum system, the time evolution of an operator is given by\n\\begin{equation}\n    \\mathcal{O}(t) = e^{iHt}\\mathcal{O}e^{-iHt}\n\\end{equation}\nwhere $\\mathcal{O}$ is the operator when $t=0$. We can expand it via the Baker-Campbell-Hausdorff formula\n\\begin{equation}\n    \\mathcal{O}(t) = \\sum_{n=0}^{\\infty} \\frac{(i t)^n}{n!} \\bar{\\mathcal{O}}_n\n\\end{equation}\nwhere $\\bar{\\mathcal{O}}_n$ denotes the n-th commutator of $\\mathcal{O}$ with Hamiltonian\n\\begin{equation}\n    \\bar{\\mathcal{O}}_0 = \\mathcal{O}, \\ \\ \\ \\ \\bar{\\mathcal{O}}_1 = [H,\\mathcal{O}], \\ \\ \\ \\ \\bar{\\mathcal{O}}_2 = [H,[H,\\mathcal{O}]],...\n\\end{equation}\nAlternatively, we can write in another formalism, using the well-known Liouvillian super operator. The Liouvillian super operator acts on the operator $\\mathcal{O}$ by\n\\begin{equation}\n    \\mathcal{L} \\mathcal{O} = [H, \\mathcal{O}]\n\\end{equation}\nIn this way, we can write the time evolution of the operator as\n\\begin{equation}\n    \\mathcal{O}(t) = e^{i\\mathcal{L} t}\\mathcal{O}\n\\end{equation}\n\nInspired by this, we can define a set of orthonormal bases in the operator space, which is called Krylov basis. Krylov basis is constructed by imposing an iterative Gram-Schmidt orthogonalization for $\\mathcal{O}_n$. This process is known as the Lanczos algorithm. However, before introducing the Lanczos algorithm, we need to determine the inner product used in the operator space. following the work of \\cite{Jian:2020qpp}, we take the Wightman inner product\n\\begin{equation}\\label{Wightman}\n    (A|B) = \\langle e^{H \\beta\/2} A^{\\dagger} e^{-H \\beta\/2} B \\rangle_\\beta\n\\end{equation}\nwhere the notation $\\langle ... \\rangle_\\beta$ represents the thermal expectation of operators\n\\begin{equation}\\label{191}\n    \\langle A \\rangle_\\beta =\\frac{ \\Tr (e^{-\\beta H} A)}{Z},\\ \\ \\ \\  Z = \\Tr(e^{-\\beta H})\n\\end{equation}\nWith the definition of inner product, we can start building the orthonormal basis. The Lanczos algorithm process is as follows\n\n\\begin{enumerate}\n\\item Setting the starting conditions:$| \\mathcal{O}_0 ) =  |\\bar{\\mathcal{O}}_0 )\/ \\Vert \\bar{\\mathcal{O}}_0 \\Vert $, \\ \\ \\ \n$| \\mathcal{O}_1 ) =  b_1^{-1}\\mathcal{L}|\\mathcal{O}_0 )$,\\ \\ \\ $b_1 = (\\mathcal{O}_0\\mathcal{L} |\\mathcal{L}\\mathcal{O}_0 )$\n\\item For $n \\geq 2$, operator $A_n$ are constructed iteratively by:\n\\begin{equation}\\label{151}\n|A_n) = \\mathcal{L} |\\mathcal{O}_{n-1} ) - b_{n-1} |\\mathcal{O}_{n-2} )\n\\end{equation}\nThe normalization constant $b_n$ is given by $b_n = (A_n|A_n)^{\\frac{1}{2}}$. Then the basis element is constructed as follows:\n\\begin{equation}\\label{155}\n|\\mathcal{O}_n) = b_n^{-1}|A_n)\n\\end{equation}\n\\item If $b_n = 0$, stop the algorithm.\n\\end{enumerate}\n\nThe normalization constant $b_n$ is known as the Lanczos coefficients. If we consider a system with finite-dimensional Hilbert space. Then the dimension $K$ of the space spanned by the Krylov basis is the value of n when $b_n = 0$. It was proven that Krylov space dimension satisfies $K \\leq D^2-D+1$ \\cite{Rabinovici_2021}, where $D$ is the dimension of Hilbert space. We will see later that this bound of the Krylov space is the reason for the saturation of Krylov complexity. However, if we consider a physical system with an infinite dimensional Hilbert space, where the Krylov space is infinite-dimensional, the condition $b_n = 0$ will never be satisfied. Accordingly, the Krylov complexity will keep growing forever. The CHO model we studied in this paper belongs to the latter one.\n\nOnce we obtained the Krylov basis, we can use it to expand the time-dependent operator $\\mathcal{O}(t)$\n\\begin{equation}\\label{159}\n    |\\mathcal{O}(t)) = \\sum_n i^n \\varphi_n(t) |\\mathcal{O}_n)\n\\end{equation}\nThe Krylov amplitude $\\varphi_n(t)$ is a real number and it indicates the extent to which the operator spreads on the Krylov chain. If we multiply both sides of eq(\\ref{159}) by $(\\mathcal{O}_n|$, we get an explicit expression for $\\varphi_n(t)$.\n\\begin{equation}\\label{amplitude}\n    \\varphi_n(t) = i^{-n} (\\mathcal{O}_n|\\mathcal{O}(t))\n\\end{equation}\nThe sum of the Krylov amplitude $\\varphi_n(t)$ is conserved over time and satisfies the normalization condition. \n\\begin{equation}\\label{171}\n    \\sum_n |\\varphi_n(t)|^2 = 1\n\\end{equation}\n\nSince the CHO model has an infinite dimensional Hilbert space, we must make a truncation $n \\leq N_{cut}$ on the Krylov basis $|\\mathcal{O}_n )$ when implementing the numerical calculation. To ensure the accuracy of the numerical result, we will always keep the difference between eq(\\ref{171}) and 1 no greater than $10^{-5}$. \n\nIf we plug eq(\\ref{159}) in the Heisenberg equation, $\\partial_t \\mathcal{O} = i[H,\\mathcal{O}]$, we can obtain the following recursion relation for $\\varphi_n(t)$\n\\begin{equation}\n    \\partial_t \\varphi_n(t) = b_n \\varphi_{n-1}(t) - b_{n+1} \\varphi_{n+1}(t)\n\\end{equation}\nFrom this, we can see that the time evolution of the Krylov amplitude $\\varphi_n(t)$ is completely determined by the Lanczos coefficient $b_n$. This can also be seen from the matrix representation of the Liouvillian operator $(\\mathcal{O}_n|\\mathcal{L}|\\mathcal{O}_m) = L_{nm}$\n\\begin{equation}\nL = \n    \\begin{pmatrix}\n    0 & b_1 & 0 & ... & 0 \\\\\n    b_1 & 0 & b_2 & ... & 0 \\\\\n    0 & b_2 & 0 & ... & 0 \\\\\n    \\vdots &  & \\vdots & \\vdots & \\vdots\\\\\n     & ... & & ... & b_{K-1}\\\\\n    0 & 0 & ... & b_{K-1} & 0\n    \\end{pmatrix}\n\\end{equation}\n\nWith the previous knowledge laid out, we can next introduce the definition of the Krylov complexity\n\\begin{equation}\\label{K-complexity}\n    C_K = \\sum_n n |\\varphi_n(t)|^2\n\\end{equation}\nwhich can be recognized as the average expectation of the first-order moment of $\\varphi_n(t)$ over the Krylov chain.\n\nKrylov complexity is considered to be an important indicator of chaotic systems. For general integrable systems, it is observed that the Lanczos coefficient shows the following sub-linear growth pattern at $n\\ll \\ln{S}$ (S is the entropy of the system)\n\\begin{equation}\n    b_n \\sim a n^\\delta\n\\end{equation}\nwhere $a$ is a constant coefficient and $\\delta$ satisfies $0<\\delta<1$. Correspondingly, Krylov complexity satisfies polynomial growth in the early time\n\\begin{equation}\\label{241}\n    C_K(t) \\sim (a t)^{\\frac{1}{1-\\delta}}\n\\end{equation}\nFor chaotic systems, Lanczos coefficients satisfy linear growth when n is small.\n\\begin{equation}\n    b_n \\sim a n\n\\end{equation}\nAfter ending the linear growth, Lanczos coefficients enter the saturation plateau. Correspondingly, K-complexity satisfies the exponential growth in the early time stage.\n\\begin{equation}\\label{250}\n   C_K(t) \\sim e^{2 a t}\n\\end{equation}\nAt this point, the coefficient $2 a$ on the exponent is considered to provide an above bound for the quantum Lyapunov exponent $\\lambda$ \\cite{Parker_2019}.\n\\begin{equation}\\label{bound}\n    \\lambda \\leq 2 a\n\\end{equation}\n\nThe exponential growth of the Krylov complexity continues until $t \\sim \\ln(S)$. From then on, the Krylov complexity becomes to grows linearly $C_K(t) \\propto t$. \n\nThe behavior of K-complexity can be traced back to the saturation of Lanczos coefficients $b_n$. For chaotic systems, the Lanczos coefficients $b_n$ reach saturation after $n \\sim \\ln(S)$, which leads to the linear growth of K-complexity.\n\nFor systems with finite-dimensional Hilbert space, the growth of K-complexity will continue until the Heisenberg time $t \\sim e^{S}$, after which it enters the saturation plateau. The Lanczos coefficient $b_n$ starts to decrease rapidly after $n \\sim e^{S}$ until it reaches zero. This operator growth behavior is conjectured to apply to all local operators in chaotic systems \\cite{Parker_2019}. In the case we consider, the K-complexity will never saturate since the Hilbert space is infinite-dimensional. \n\n\\subsection{Numerical computation of Krylov complexity}\nIn this subsection, we describe our approach to numerically calculate the K-complexity in CHO model. The CHO model has the following Schr\u00f6dinger equation.\n\\begin{equation} \\label{Schr\u00f6dinger equation}\n    -\\left(\\frac{\\partial^2}{\\partial x^2}+\\frac{\\partial^2}{\\partial y^2}\\right)\\psi_n(x,y)+\\left[ \\frac{\\omega^2}{4}(x^2+y^2) + g_0 x^2y^2 \\right]\\psi_n(x,y) = E_n \\psi_n(x,y)\n\\end{equation}\n\nBy numerical calculation we can obtain a series of eigenvalues $\\{E_n\\}$ and eigenfunctions $\\{\\psi_n(x,y)\\}$ of the Schr\u00f6dinger equation (\\ref{Schr\u00f6dinger equation}). So, to implement the numerical calculation, all we have to do is to represent the Wightman inner product as a function of $ \\{ E_n \\}$ and $\\{\\psi_n(x,y)\\}$.\n\nFollowing the work of \\cite{Akutagawa_2020}. We use $\\hat{x}(t)$ as our local operator, i.e., $\\mathcal{O}(t) = \\hat{x}(t)$. Notice that the CHO model has the following commutation relation\n\\begin{equation}\n  \\begin{aligned}\n    [H,\\hat{x}] = -2ip_x, \\ \\ \\ \\ [H,\\hat{y}] = -2ip_y\n  \\end{aligned}    \n\\end{equation}\n\nTherefore, all local operators in the CHO model can be represented as functions of $\\hat{x}$ and $H$. It is worth noting that the commutation relation is independent of how many oscillators are in the model. So, this computational method can be used in an arbitrary dimensional CHO model. However, in this paper, we will only focus on the two-dimensional CHO model.\n\nIn the Lanczos algorithm, we will only do the inner product among the Krylov basis $\\{\\mathcal{O}_n\\}$. So we just need to express the inner product of the Krylov basis $\\{\\mathcal{O}_n\\}$ as a function of $ \\{ E_n \\}$ and $\\{\\psi_n(x,y)\\}$. According to (\\ref{151}) and (\\ref{155}), $\\mathcal{O}_n$ can be expressed as a polynomial of $\\hat{x}$ and $H$, where the order of $\\hat{x}$ in each term is 1 and the order of $H$ is at most $n$.\n\nFrom (\\ref{Wightman}), the Wightman inner product can be expressed as\n\\begin{equation}\n    (A|B) = \\frac{1}{Z} \\Tr \\left( e^{-H \\beta\/2} A^\\dagger e^{-H \\beta\/2} B \\right)\n\\end{equation}\nAfter inserting the completeness condition $\\sum_l |l\\rangle \\langle l|=1$, we get\n\\begin{equation}\n    (A|B) = \\frac{1}{Z} \\sum_{m,l} e^{-\\frac{\\beta}{2}(E_m+E_l)}A^\\dagger_{ml} B_{lm}\n\\end{equation}\nwhere we introduced $A_{ml} = \\langle m | A | l \\rangle$. Since we only need to consider the Wightman inner product among $\\{|\\mathcal{O}_n)\\}$, and $|\\mathcal{O}_n)$ can be expressed as a polynomial of $\\hat{x}$ and $H$. Therefore, we have\n\\begin{equation}\\label{234}\n\\begin{aligned}\n    (\\mathcal{O}_n)_{lm} &= \\sum_{k=1}^n \\sum_{a+b=k} C_{ab} E_l^a E_m^b \\langle l | \\hat{x} | m \\rangle\\\\\n    (\\mathcal{O}_n)^{\\dagger}_{ml} &= \\sum_{k=1}^n \\sum_{a+b=k} C^{\\prime}_{ab} E_m^a E_l^a \\langle m | \\hat{x} | l \\rangle\n\\end{aligned}\n\\end{equation}\nwhere $C_{ab}$ and $C^{\\prime}_{ab}$ are constant coefficients. According to eq(\\ref{234}), we only need to numerically integrate $\\langle m| x | l \\rangle$ to get the value of the Wightman inner product. Through this method, we can numerically compute the K-complexity and Lanczos coefficients.\n\nOur numerical approach can be summarized as follows\n\n\\begin{enumerate}\n    \\item Numerically solve the Schr\u00f6dinger equation and obtain the corresponding energy eigenvalues $\\{E_n\\}$ and eigenfunctions $\\{\\psi_n(x,y)\\}$.\n    \\item For different $m$ and $l$, compute $\\langle m | \\hat{x} | l \\rangle = \\int dx dy \\ \\psi_n^{\\dagger}(x,y) x \\psi_n(x,y)$ numerically.\n    \\item Implement the Lanczos algorithm by writing the Wightman inner product involved in each step as a function of $\\langle m | \\hat{x} | l \\rangle$, then plug in $\\langle n | \\hat{x} | m \\rangle$ obtained by step 2.\n    \\item Through the Lanczos algorithm, obtain Krylov basis $|O_n)$ and Lancaos coefficient $|b_n)$.\n    \\item Calculate the Krylov amplitude $\\varphi_n$ by (\\ref{amplitude}). The corresponding K-complexity is obtained by plugging $\\varphi_n$ into (\\ref{K-complexity}).\n\\end{enumerate}\n\n\\subsection{Energy spectrum and spectral form factor}\nBefore moving to the calculation of the K-complexity, we show in this section the energy spectrum as well as the spectral form factor of the CHO model. Figure \\ref{energyspectrum} shows the energy spectrum of the CHO model for different coupling constants from the numerical calculation of the Schr\u00f6dinger equation (\\ref{Schr\u00f6dinger equation}).\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=7cm]{energyspectrum.png}\n\\caption{}\n\\label{energyspectrum}\n\\end{figure}\n\nAnother important observable in chaotic systems is the spectral form factor $g(\\beta,t)$. Its behavior can show many characteristics of chaotic systems. And in recent developments it has been related to the information loss problem and AdS\/CFT correspondence \\cite{Dyer:2016pou,Cotler:2016fpe,Cotler:2017jue}. The spectral form factor $g(\\beta,t)$ is defined by\n\\begin{equation}\n    g(\\beta,t) \\equiv \\frac{|Z(\\beta + it)|^2}{Z^2(\\beta)}\n\\end{equation}\nwhere we have analytically continued partition function $Z(\\beta)$ in eq(\\ref{191}) into the complexity plane. In figure \\ref{spectralformfactor} we show the spectral form factor evolution with $t$ for coupling constant $g=0.1$ under different temperatures $T = 1, T=2, T=10$.\n\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[]{\n\\includegraphics[width=5.5cm]{spectralformfactor.png}\n}\n\\quad\n\\subfigure[]{\n\\includegraphics[width=5.5cm]{spectralformfactor2.png}\n}\n\\caption{Spectral form factor $g(\\beta, t)$ as a function of time, for $g_0 = 0.1$ and three different temperatures $T=1,T=2,t=10$. In the early stage, the spectral form factor holds 1. After that, it drops to a dip and then bounces back to a plateau with erratic fluctuations. The fluctuation in the spectral form factor reflects the discrete eigenvalues of the system.}\n\\label{spectralformfactor}\n\\end{figure}\n\nAt low temperatures, $g(\\beta,t)$ seems to first decrease to the dip, and then bounce back to a certain value with erratic fluctuations. At high temperatures, $g(\\beta,t)$ first drops and then rises to a plateau with a relatively small value. We find the spectral form factor is very much reminiscent of that from the random matrix theory \\cite{Dyer:2016pou}, which implies that there may be a random matrix description for the SU(2) Yang-Mills theory.\n\n\n\\section{Krylov Complexity in chaotic transition}\\label{section4}\nIn this section, we numerically compute the behavior of Krylov Complexity and Lanczos coefficient in the CHO model using the method we presented in the previous section. In section \\ref{4.1}, we analyze the time scale for the system's chaotic property to start to emerge as well as the time scale for the K-complexity to grow linearly. In section \\ref{4.2}, we show the growth of Lanczos coefficients with different coupling constants. We found that the Lanczos coefficient grows linearly at the beginning and then goes into saturation. In section \\ref{4.3}, we present the result for the time evolution of Krylov complexity. In this section, we will fix the inverse temperature $\\beta =1$.\n\n\\subsection{Time scale for chaotic transition}\\label{4.1}\nIn the CHO model, K-complexity has two different growth patterns in the early and late time evolution. This is because, according to the BCH formula, at a very short time t, we can treat the nonlinear interaction $g_0 x^2 y^2$ in the Hamiltonian as a perturbative term. And the corresponding operator evolution can be approximated as\n\\begin{equation}\ne^{i H t} \\hat{x} e^{i H t} \\sim (1+ i g_0 \\hat{x}^2 \\hat{y}^2 t) e^{i H_0 t} \\hat{x} e^{i H_0 t} (1- i g_0 \\hat{x}^2 \\hat{y}^2 t) \n\\end{equation}\nwhere $H_0$ is the non-perturbative Hamiltonian. We want to find a time scale when the contribution from the non-linear term becomes important, i.e., the chaotic property of the system begins to emerge.\n\nWe assume that if the correction of the non-linear term for energy exceeds $10\\%$, the effect of the coupling term becomes non-negligible. In other words, when the expectation value of $g_0 \\hat{x}^2 \\hat{y}^2 t$ under the ground state $| \\Omega \\rangle = \\mathcal{N} e^{- \\beta E_n} |n \\rangle$ reaches $10\\% E$ (E is the energy of the system and $\\mathcal{N}$ is the normalization constant), the system starts to become chaotic.\n\nBy calculation, we can find that the expectation of the non-linear term $g_0 \\hat{x}^2 \\hat{y}^2 t$ is roughly\n\n\\begin{equation}\n    \\langle \\Omega | g_0 \\hat{x}^2 \\hat{y}^2 t |\\Omega \\rangle \\sim g_0 t\n\\end{equation}\n\nIf we choose $g_0 = 0.1$, then our chaotic transition time is roughly $t_0=1$. As we will show in the next sections, our estimate here matches our numerical results.\n\nAnother different time scale is the scrambling time $t_*$ at which the K-complexity starts to grow linearly. As stated in early sections, $t_* \\sim \\ln(S)$. The Shannon entropy $S_H$ of our system can be computed by\n\\begin{equation}\n    S_H = -\\sum_n \\mathcal{N}^2 e^{-2 E_n} \\ln(\\mathcal{N}^2 e^{-2 E_n})\n\\end{equation}\nThrough numerical calculation, we find the scrambling time $t_*$ is roughly around $t\\sim1$. We find that this is very close to the chaotic transition time $t_0$ we calculated. We will see in the following sections because the non-linear term of the system is suppressed in the early time, K-complexity in early growth satisfies not exponential growth but power-law growth. And, around the chaotic transition time $t_0$, K-complexity shifts directly from power-law growth to linear growth.\n\n\\subsection{Lanczos coefficients and upper bound for quantum Lyapunov exponent}\\label{4.2}\nLanczos coefficients are an important indicator in chaotic systems. First, the growth of Krylov Complexity is captured by the Lanczos coefficients. The linear growth of Lanczos coefficients corresponds to the exponential growth of K-complexity in the early time. Generally, in chaotic systems, the Lanczos coefficients grow linearly at the beginning, which is called Lanczos ascent. After the linear growth, it enters the saturation plateau. The K-complexity keeps growing linearly while the Lanczos coefficients are on the plateau. Besides, the linear growth of the Lanczos coefficients $b_n \\sim a n $ is expected to give a maximum bound $ \\lambda < 2 a$ for the quantum Lyapunov exponent.\n\nWe show in figure \\ref{fig335} the Lanczos coefficients $b_n$ for different coupling constants. From the figure, we can see that the Lanczos coefficients rise approximately linearly when n is small. Around $n \\sim 12$, the Lanczos coefficients go into saturation, which ensures an asymptotic linear growth of K-complexity. The system we consider has a Hilbert space with infinite dimension, so the Lanczos coefficient stays on the plateau and will not drop to zero. The effect of different coupling constants on the Lanczos coefficient is mainly to change the saturation value of Lanczos coefficients. The greater the coupling constant, the larger the saturation value of Lanczos coefficients. It can be translated to the behavior of K-complexity, i.e., the larger coupling constant will lead to a steeper linear growth for K-complexity.\n\n\\begin{figure}[htbp] \n\\centering\n\\includegraphics[width=0.7\\textwidth]{Lanczo.png} \n\\caption{Lanczos coefficients for different coupling constants $g_0 = 0.1, 0.3, 0.7$ at temperature $\\beta =1$. In the beginning, the Lanczos coefficients $b_n$ increase linearly with $n$. And after about n = 12, Lanczos coefficients enter the saturation plateau and fluctuate around a constant value.} \n\\label{fig335} \n\\end{figure}\n\nNotice that there is an oscillation phenomenon in the Lanczos coefficients. One possible explanation for this oscillation is that the Schr\u00f6dinger equation of Kyrlov amplitude allows the following solution\\cite{Yates:2020lin,Yates:2020hhj,Bhattacharjee:2022vlt}\n\\begin{equation}\n    b_n = f(n)+ (-1)^{n} \\Tilde{f}(n)\n\\end{equation}\nThe asymptotic behavior of the oscillatory function $\\Tilde{f}(n)$ can be derived from the auto-correlation function $C(t) = \\varphi_0(t)$. In general, if the auto-correlation function is power-law decaying $C(t) \\sim t^{-b}$, then the oscillation of the corresponding Lanczos coefficient satisfies a logarithmically decay $\\Tilde{f}(n) \\sim \\ln (n)^{-b}$\\cite{Bhattacharjee:2022vlt}.\n\nAccording to (\\ref{bound}), by computing the slope of the linear growth of Lanczos coefficients $b_n$, we can obtain the above bound of the quantum Lyapunov exponent. \n\n\\begin{equation}\n    \\lambda_{0.1} \\leq 1.57, \\ \\ \\\n    \\lambda_{0.3} \\leq 1.79, \\ \\ \\ \n    \\lambda_{0.7} \\leq 2.09\n\\end{equation}\nThe subscript of $\\lambda$ is the corresponding coupling constant.\n\n\\subsection{K-complexity growth in the chaotic transition}\\label{4.3}\nThrough the perturbative analysis in section \\ref{4.1}, we know that the non-linear term of the Hamiltonian can be neglected when $t$ is small enough. Therefore, the system can be considered as two decoupled harmonic oscillators, which is a system with the Heisenberg-Weyl symmetry. According to \\cite {Caputa:2021sib}, a system with the Heisenberg-Weyl symmetry has a quadratic growing K-complexity. \n\n\\begin{equation}\n    K \\sim t^2\n\\end{equation}\n\nThus, the K-complexity is expected to grow quadratically at the beginning. We show the early growth of K-complexity under different coupling constants in figure \\ref{K-complexity growth}. As expected, in the early time stage, the K-complexity shows a quadratic growth behavior. The dash lines in the figure are fitting functions $C_K(t) = \\gamma t^{2}$. The coefficients $\\gamma$ are respectively $\\gamma_{0.1} = 1.39$, $\\gamma_{0.3} = 1.76$, $\\gamma_{0.7} = 2.23$. Then in the period $0.1<t<1$, K-complexity starts to deviate from quadratic growth and turn into linear growth. The linear growth can be numerically fitted by $C_K(t) = \\eta t + \\xi$. The slop $\\eta$ for different coupling constants are $\\eta_{0.1} = 1.21$, $\\eta_{0.3} = 1.51$, $\\eta_{0.7} = 1.85$. From the analysis of Lanczos coefficients in section \\ref{4.2}, this linear growth will last forever. \n\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[]{\n\\includegraphics[width=5.5cm]{K-complexity_growth1.pdf}\n}\n\\quad\n\\subfigure[]{\n\\includegraphics[width=5.5cm]{K-complexity_growth2.pdf}\n}\n\\quad\n\\subfigure[]{\n\\includegraphics[width=5.5cm]{K-complexity_growth3.pdf}\n}\n\\caption{K-complexity evolution in different time stages. When t<0.1, the K-complexity shows a quadratic growth. During $0.1<t<1$, K-complexity deviates from the quadratic growth and then grows linearly with time.}\n\\label{K-complexity growth}\n\\end{figure}\n\nAs stated in section \\ref{3.1}, the power-law growth of K-complexity corresponds to integrable systems, while the linear growth corresponds to chaotic systems. Therefore, it can be seen clearly that in the CHO model, there is a transformation from integrability to chaos. The transition time $0.1 < t_{0} < 1$. The exponential growth of K-complexity (\\ref{250}) is also suppressed by this effect.\n\n\\section{Microcanonical K-complexity}\\label{section5}\nAs we saw in the classical analysis, the energy of the system will affect its chaotic nature. In the thermal state, high-modes show more chaotic properties, while low-modes show more integrable properties. In this section, we will analyze the effect of different energies as well as different temperatures on the K-complexity and Lanczos coefficients.\n\\subsection{Contribution from different energy}\nTo investigate the contribution from different energy modes to the K-complexity. we change the Wightman inner product as follows\n\\begin{equation}\\label{379}\n    (A|B)_E \\equiv \\frac{1}{Z} \\sum_{E_n = E} \\langle n | e^{H \\beta \/2} A^\\dagger e^{-H \\beta \/2} B | n \\rangle\n\\end{equation}\nwhere the summation means summing over all the states with degeneracy $E_n = E$. In this way, the original Wightman inner product can be expressed as\n\\begin{equation}\n    (A|B) = \\int dE \\ \\rho(E) e^{-\\beta E} (A|B)_E\n\\end{equation}\nwhere we defined $\\rho(E) = \\sum_{n}(E_n - E) $. Keeping the Krylov basis fixed, according to eq(\\ref{amplitude}), our Krylov amplitude becomes\n\\begin{equation}\n    \\varphi_n(t,E) = i^{-n} (\\mathcal{O}_n|\\mathcal{O}(t))_E\n\\end{equation}\nThis is related to the original Krylov amplitude by\n\\begin{equation}\n    \\varphi_n(t) =\\int dE \\ \\rho(E) e^{-\\beta E} \\varphi_n(t,E)\n\\end{equation}\nWith this definition, we can extract the contribution from different energy sectors to the Krylov amplitude $\\varphi_n(t)$.\n\nWe show in figure \\ref{fig421} the evolution of $|\\varphi_n(t,E)|$ for different energies $E$ in the CHO model with $g_0=0.1$ and $n=10$. It can be seen that after lifting the suppression of the Boltzmann factor $e^{-\\beta E}$, the high-energy modes have a significantly larger contribution to $\\varphi_n(t)$. Notice that when E is small, $\\varphi_n(t,E)$ has a distinct periodicity, which is a characteristic of the integrable system. In other words, the effect of the non-linear term in the CHO model is more on the higher energy modes. Therefore, when the energy of the system is elevated, it deviates from the integrable system and shows stronger chaotic features.\n\n\\begin{figure}[htbp] \n\\centering \n\\includegraphics[width=0.6\\textwidth]{energysector.pdf} \n\\caption{The norm of energy-dependent Krylov amplitude $\\varphi_n(t,E)$ vs evolution time $t$. Here we choose $n=10$ and five different values of energy $E=3.44,E=5.00,E=9.81,E=12.90,E=19.12$. At low energy sectors, $|\\varphi_n(t,E)|$ has a strong periodicity. While for high-energy, the periodicity of $|\\varphi_n(t,E)|$ is broken and the oscillations become random and erratic. on the other hand, high-modes of $|\\varphi_n(t,E)|$ have larger values than low-modes, which means it can contribute more to the K-complexity.} \n\\label{fig421} \n\\end{figure}\n\n\\subsection{Energy and temperature dependence of Lanczos coefficients}\n\nIn the classical limit, the Lanczos coefficients growth rate $a$ can be calculated by the singularity of the analytically continued auto-correlation function $C(t)$. Consider a classical orbit $E = H(x,y,\\frac{\\partial x}{\\partial t},\\frac{\\partial y}{\\partial t})$. Assume the equations of motion have the solution $x(t)=F_E(t)$, and inversely, $t(x)=F_E^{-1}(x)$. $x$ is analytic for all $t\\in \\mathbb{R}$. So the singularity in $x(t)$ has the imagination $\\sigma_* = -i F_E^{-1}(\\infty)$. So, $x(t)$ is analytic in the strip $\\{t:\\Im(t) \\leq \\sigma_* \\}$. Thus, the analytic region of auto-correlation function $C(t) = \\langle x(t) x(0) \\rangle$ can be determined by the singularity in $x(t)$. If we replace the ensemble average by the time average $C(t) = \\frac{1}{T}\\int x(s+t\/2)x(s-t\/2)ds$, then $C(t)$ is analytic in the strip $\\{t:\\Im(t) \\leq  2 \\sigma_* \\}$. According to \\cite{Avdoshkin:2019trj}, $2\\sigma_*$ is related to the Lanczos coefficients growth rate $a$ by $a = \\frac{\\pi}{4 \\sigma_*}$. \n\nIn the high energy limit, the harmonic potential $\\frac{\\omega^2}{4}(x^2+y^2)$ can be ignored. So, the remaining energy is $E = p_x^2+p_y^2+g_0 x^2 y^2$, which allows the scaling transformation, $x\\to \\Lambda x, y\\to \\Lambda y, t\\to \\Lambda^{-1} t, E\\to \\Lambda^4 E$. Plugging the scaling relation to $\\sigma_*$, it turns out that $\\sigma_*$ scales as $\\sigma_* \\propto E^{-1\/4}$. Therefore, the growth rate $a$ scales as $a= \\frac{\\pi}{4 \\sigma_*} \\propto E^{1\/4}$. In the low energy limit, the harmonic potential dominates, so $\\sigma_*$ doesn't scale with the energy. \n\nIn the thermal K-complexity, the contribution from different energy regions is suppressed by the Boltzmann factor $e^{-\\beta E_n}$. This means that increasing temperature will enhance the contribution from higher energy modes. We show in figure \\ref{fig431} the upper bound of quantum Lyapunov exponent $\\lambda_{max} = 2a$ for different temperatures when the coupling constant $g_0=0.1$.\n\n\\begin{figure}[htbp]\n\\centering \n\\includegraphics[width=0.6\\textwidth]{LanczosTemperature.pdf} \n\\caption{Numerical result for the maximum bound of quantum Lyapunov exponent $\\lambda_{max} <2a$ in different temperature. The horizontal coordinate denotes the inverse temperature $\\beta$.} \n\\label{fig431} \n\\end{figure}\n\n\\section{Conclusion and discussion}\\label{section6}\n\nIn this paper, we investigated the Krylov complexity and associated Lanczos coefficients growth in the SU(2) Yang-Mills theory, which can be reduced to a nonlinearly coupled harmonic oscillators model. We developed a method to numerically calculate Krylov complexity in this chaotic model. We computed the growth patterns of K-complexity and Lanczos coefficients for different coupling constants. We observed a transition from integrability to chaos in the CHO model. The K-complexity grows quadratically in the early stage, which is characteristic of integrable systems. After the transition time $t_0$, K-complexity grows linearly, which is characteristic of chaotic systems.\n\nWhile for Lanczos coefficients, we found that it linearly increases with $n$ when $n$ is small and then enters the saturation plateau, which is consistent with the universal hypothesis in \\cite{Parker_2019}. It was conjectured that the quantum Lyapunov exponent $\\lambda$ is upper bounded by the slope of the linear growth of the Lanczos coefficients $a$, i.e.,  $\\lambda < 2a$. Using this relation, we found the upper bound of the quantum Lyapunov exponent in the CHO model at different temperatures and coupling constants.\n\nBy analyzing the microcanonical K-complexity, we extracted the contributions from different energy sectors to the K-complexity and Lanczos coefficients. We found that higher energy modes have a greater contribution to the growth of K-complexity, which means that the higher the energy, the more the system shows the nature of quantum chaos. This is consistent with our classical analysis. Classically, by analyzing the singularity of the auto-correlation function $C(t)$, we obtained that the bound for Lyapunov exponent $\\lambda_{max}$ is proportional to $E^{1\/4}$ in the high-energy limit. In the thermal state, the energy of the system is closely related to the temperature. We calculated the effect of different temperatures on Lanczos coefficients. In addition, we also numerically computed the spectral form factor in the CHO model. We found a behavior similar to that of the random matrix theory, which indicates there might be a random matrix description of SU(2) Yang-Mills theory.\n\nIn conclusion, in SU(2) Yang-Mills theory, Krylov complexity is an interesting observable which encodes many features of quantum chaos. This may have implications for a good definition of quantum chaos. Also, the application of Krylov complexity to strongly interacting field theories may advance the search for its gravity dual. Although our study focuses on the two-dimensional CHO model, our approach can be generalized to higher dimensions. For future research, it will be interesting to see whether Krylov complexity can be computed in more general quantum chaotic systems and quantum field theories.\n\n\n\n\\bibliographystyle{JHEP}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nStars form in dense cores of dust and gas within molecular clouds, but remain cocooned in the material of their natal envelopes. Though young stars and protostars can be extremely luminous, the enveloping dust absorbs nearly all of the ultraviolet and optical starlight and reradiates it at longer wavelengths. In the submillimetre regime, the dust emission is generally optically thin. Submillimetre observations thus sample all the dust emission within the cloud along the line of sight, and can be used to determine the column density and mass of material present \\citep{hild83}. Interpretation of the observations, however, depends on the assumed dust grain temperature, composition and size distribution. The effects of composition and grain size on the dust emission are usually combined into a dust opacity, $\\kappa_\\nu$, which can be reasonably fit empirically with a power law dependence on frequency in the submillimetre.\n\n\\begin{table*}\n\\centering\n\\begin{minipage}{106mm}\n\\caption{Positions and peak brightnesses for observed objects}\n\\label{positions}\n\\begin{tabular}{cccccc}\n\\hline\nSource & $\\alpha_{J2000}$ & $\\delta_{J2000}$ & $I_{450}$ & $I_{850}$ & Distance\\\\\n--- & \\fh\\,\\fm\\,\\fs & \\degr\\,\\,\\,\\arcmin\\,\\,\\arcsec & Jy\/8.5\\arcsec~beam & Jy\/15\\arcsec~beam & kpc\\\\\n\\hline\nG10.47 & 18 08 38.2 & $-$19 51 50 & 260 $\\pm$ 65 & 39.0 $\\pm$ 5.9 & 5.8\\\\\nG31.41 & 18 47 34.5 & $-$01 12 43 & 130 $\\pm$ 32 & 24.2 $\\pm$ 3.6 & 7.9\\\\\nG12.21 & 18 12 39.7 & $-$18 24 20 & 121 $\\pm$ 30 & 13.0 $\\pm$ 2.0 & 13.5\\\\\n\\hline\n\\end{tabular}\n\n\\medskip\nThe positions, peak brightnesses at 450\\,\\micron~and 850\\,\\micron, and distances to each of the hot molecular cores observed in this study, ordered by $I_{850}$. Positions and distances are taken from \\citet{hatchell00}, while peak brightnesses were calculated from the archived SCUBA data.\n\\end{minipage}\n\\end{table*}\n\nAt submillimetre wavelengths, dust is generally assumed to radiate as a blackbody at some average temperature $T_d$ modified by the optical depth, $\\tau_\\nu \\propto \\kappa_\\nu$. For optically thin emission, we then find $S_{\\nu} \\propto \\kappa_\\nu \\, B_{\\nu}(T_d)$. Assuming a power law dependence on frequency, the dust opacity $\\kappa_\\nu$ can be parameterized for a given opacity $\\kappa_0$ at frequency $\\nu_0$ as $\\kappa_\\nu=\\kappa_0 (\\nu \/ \\nu_0)^\\beta$, where $\\beta$ is the spectral index of the dust emission. For optically thin emission, the dust continuum emission then follows the power law:\n\\begin{equation}\nS_{\\nu} \\propto \\kappa_{0}(\\frac{\\nu}{\\nu_0})^{\\beta}B_{\\nu}(T_d)\n\\label{eqn:dust_law}\n\\end{equation}\nThrough mathematical modelling, $\\kappa_\\nu$ and $\\beta$ have been calculated for various dust grain compositions and size distributions \\citep{draine84,ossen94,poll94}, which can be compared with astronomical observations. Often, however, various combinations of the dust parameters can reproduce existing observational data with similar accuracy, while dust temperatures and densities can vary dramatically within star forming environments. Calculations often assume $\\beta\\simeq2$ at these wavelengths, based on work by \\citet{hild83}. In practice, however, theoretical dust grain models find spectral indices anywhere between zero and three (see the summary by Goldsmith, Bergin \\& Lis (1997))\\nocite{gold97}, depending on the physical dust properties of the model, while observations have found a similar range of values in different states of the interstellar medium.\n\nIn high mass star forming regions and the cores containing warm protostars, \\citet{will04} find the average frequency dependence of dust emission from more than $60$ high mass protostellar objects to be characterized by $\\beta=0.9\\pm0.4$. This is comparable to values observed in less evolved, colder objects \\citep{gold97,visser98,hoger00,beuther04}, but is systematically lower than spectral indices determined through submillimetre observations of hot molecular cores (HMC) and ultra-compact H\\,{\\small II} (UC\\,H\\,{\\small II}) regions, which are likely the next phases in the evolution of high mass protostars. In these more evolved cores, $\\beta$ values have been determined to be near to \\citep{osorio99} or slightly higher than \\citep{hunter98} $\\sim2$. \n\nThe changing values of $\\beta$ in these studies are indicative of the evolution of the dust itself in the changing physical environments associated with the process of star formation. The growth of dust grains, the formation of icy mantles, and changes in grain composition are likely all significant factors in the variation of the frequency dependence of the dust emission in these regions. In order to understand the properties and evolution of dust in star forming regions, observational methods are required that can discern between temperature and density effects and actual changes in the dust composition and opacity. \n\nTo determine $\\kappa_0$, observations must be calibrated against known dust column densities, which generally requires comparison with observations at shorter wavelengths. In contrast, the spectral index $\\beta$ of the dust opacity can be calculated solely in the submillimetre by comparing observations at two widely separated submillimetre wavelengths. This method provides great leverage on $\\beta$, but is complicated by calibration issues from observing in the different passbands, and has the disadvantage that the value determined for $\\beta$ is dependent on the assumed dust temperature. The Planck function at 450\\,\\micron~is far from the Rayleigh-Jeans limit for dust temperatures less than $\\sim40$\\,K, and even at the higher temperatures expected in high mass star forming regions, the change in emission between passbands could be due to either small changes in temperature or large changes in the dust emissivity. The uncertainties in the determination of $\\beta$ are thus large enough that the subsequently derived physical properties of dusty systems, such as the mass estimates of the material in dense cores in molecular clouds, may range over factors of a few. For this reason, it is beneficial to observe in one passband only.\n\nOne instrument that is well-suited to this problem is a Fourier Transform Spectrometer (FTS). FTS have several advantages and disadvantages compared with other spectrometer designs. The highest spectral resolution of an FTS is determined by the maximum optical path difference between the two beams of the interferometer, which is set by the length of the translation stage. In practice, space limitations require that most submillimetre FTS provide low to intermediate spectral resolution (R\\,$\\sim200$ - R\\,$\\sim2000-3000$ at 345-375\\,GHz), excellent for observations of sources with wide, bright spectral lines and continuum emission. For these studies FTS have a distinct advantage over high spectral resolution but narrow bandwidth heterodyne receivers. The instantaneous bandwidth of an FTS is inherently broad, and in the submillimetre is limited only by the atmospheric transmission windows, enabling a wider spectral coverage within these windows than is possible with the heterodyne receivers currently in use. In the past decade, two groups have successfully used FTS to study the submillimetre emission from several astronomical sources, such as the Orion molecular cloud \\citep{serabyn}, the Sun \\citep{naylor00}, and the atmospheres of planets \\citep{naylor94,davis97}.\n\nA submillimetre FTS operated at low resolution can sample the continuum dust emission from a source with a large enough bandwidth that the change in dust emissivity with frequency, and thus $\\beta$, can be determined solely within one passband. At very low temperatures, it is still difficult to separate the temperature component of the emission from the dust power-law emissivity function, but single bandpass observations are able to place limits on the combined dust temperature and $\\beta$. \n\nIn this study, we present low resolution 350\\,GHz continuum observations of four hot molecular cores performed with a submillimetre FTS of the Mach-Zehnder design (MZ FTS) developed for use at the JCMT by the FTS group at the University of Lethbridge, AB, Canada (described in detail by \\citet{naylor03}). \n\nWe describe in the following the observational procedure and analysis of the data. We compare the dust spectral index values determined for our sources with those calculated from previous submillimetre measurements, and discuss the detrimental effects of the variable submillimetre atmosphere on our sensitivity. Finally, we comment on the continuum measurement capabilities of an imaging FTS currently in preparation for use at the JCMT with SCUBA-2.\n\n\\section{Observations}\n\n\\subsection{Source selection}\n\nThe sources for this pilot study were chosen from a sample of HMC from work by \\citet{hatchell00} for their large submillimetre fluxes and lack of extended structure as shown by \\citet{hatchell00} and \\citet{walsh03}. The sources, G10.47, G12.21 and G31.41, have been previously studied in the submillimetre regime with the Submillimetre Common User Bolometer Array (SCUBA) at the JCMT \\citep{hatchell00, walsh03} and in the radio continuum (\\citet{wc89}, among others). All three have also been detected in numerous molecular lines and maser emission (see the detailed source descriptions in \\citet{hatchell00}). Table \\ref{positions} gives source positions and distances from \\citet{hatchell00}, and 450\\,\\micron~(660\\,GHz) and 850\\,\\micron~(350\\,GHz) peak brightnesses calculated using the archived SCUBA data\\footnote{Guest User, Canadian Astronomy Data Centre, which is operated by the Dominion Astrophysical Observatory for the National Research Council of Canada's Herzberg Institute of Astrophysics}. Each source is associated with one or more UC\\,H\\,{\\small II}~regions near the peak of the submillimetre emission. The hot core in G10.47 contains three UC\\,H\\,{\\small II}~regions, G10.47-0.03A, B and C, while the other sources are coincident with one UC\\,H\\,{\\small II}~region each, G12.21-0.10 and G31.41+0.31. These HMC are very peaked in the 850\\,\\micron~emission, although G31.41 does contain some more extended emission, and the hot cores themselves of these sources are much smaller than the JCMT beam with diameters of only 1-2\\arcsec. Core mass estimates are on the order of a few thousand M$_\\odot$ \\citep{hatchell00}. A fourth HMC, G43.89, was observed by us to only S\/N\\,$\\sim1.5$, too low to analyse with any degree of certainty, and is therefore omitted from further discussion. Observations of Mars were also performed for flux calibration.\n\n\\subsection{Observational method}\nThe observations were performed during 2003 April 21 - 28, using the MZ FTS mounted on the right Nasmyth platform of the JCMT on Mauna Kea, Hawaii. The April 2003 weather was good to excellent with the zenith opacity measured at 225\\,GHz~(1.3\\,mm), $\\tau_{225}$, $\\sim0.07$ for most of the observations, giving precipitable water vapour (pwv) levels averaging $\\sim1.1$\\,mm. At 350\\,GHz, the FTS bandwidth is $\\sim30$\\,GHz. The FTS was operated at low spectral resolution (1.5\\,GHz), resulting in $\\sim\\,20$ resolution elements across the band. The full width half maximum (FWHM) of the 350\\,GHz~beam at the JCMT is $\\sim15\\arcsec$, which was matched by a cold (4\\,K) aperture located at the field stop of the detector system \\citep{naylor99}. The atmospheric emission and transmission window at  350\\,GHz is shown in Figure \\ref{fig:ULTRAM}. \n\nSpectra were obtained in groups, with five spectra taken of a molecular core immediately followed by five spectra of the background sky, +1425\\arcsec~in right ascension from the source. The offset in right ascension was calculated to compensate for the change in airmass of the source due to the Earth's rotation during the time taken to obtain five spectra, allowing the sky measurements to be taken at the same airmass as the original source observations. This is crucial for accurate background sky subtraction, as the brightness and variability of the atmosphere at 350\\,GHz is easily the largest source of uncertainty in our final results. Five scans were obtained approximately every 90 seconds. Each target source was observed for between three and four hours, with approximately half of that time spent on source and half spent on background observations. Due to its brightness, Mars was observed for a half hour. A pointing check performed approximately every 45 minutes resulted in small ($\\sim1-2\\arcsec$) corrections to the pointing. \n\n\\subsection{Data reduction}\n\n\\begin{figure}\n\\includegraphics[width=84mm]{Fig1.ps}\n\\caption{The atmospheric radiance (solid line) in units of Jy\/15\\arcsec~beam and transmission (dash-dotted line) calculated for precipitable water vapour levels of 1.0mm.}\n\\label{fig:ULTRAM}\n\\end{figure}\n\nPreliminary data reduction was performed using processing pipeline software written for the FTS \\citep{naylor03}. The data were first manually screened for transients, such as cosmic rays, which occur approximately once in every ten interferograms and produce large intensity spikes at one or two data points in the interferograms. These spikes are easily detected and can be removed within the pipeline. Some spikes in the interferograms were spread over multiple data points; these interferograms were omitted from the final dataset. Only a handful of interferograms were rejected for this reason. Since the FTS optics and detector have negligible dispersion over the range of interest, a linear phase correction was applied to each interferogram. The individual interferograms were then Fourier transformed. The spectra were then coadded into the on- and off-source groups of five, and the coadded off-source background scans were subtracted from the coadded on-source scans. The coadded, sky-subtracted spectra of G10.47, G12.21, G31.41 and Mars are shown in Figure \\ref{fig:mean_spc}. \n\n\\begin{figure*}\n\\begin{minipage}{170mm}\n\\begin{center}\n\\includegraphics[width=84mm]{Fig2_1.ps}\n\\hfill\n\\includegraphics[width=84mm]{Fig2_2.ps}\\\\\n\\includegraphics[width=84mm]{Fig2_3.ps}\n\\hfill\n\\includegraphics[width=84mm]{Fig2_4.ps}\n\\caption{The raw, coadded, sky-subtracted spectra of G10.47, G12.21, G31.41 and Mars before calibration or atmospheric transmission correction. Error bars are at the 1\\,$\\sigma$ level. Note that the error bars for Mars lie within the line thickness.}\n\\label{fig:mean_spc}\n\\end{center}\n\\end{minipage}\n\\end{figure*}\n\n\nEach background-subtracted spectrum was then corrected for atmospheric transmittance within the band using the technique described by \\citet{davis97}. Updated atmospheric opacities were calculated using the University of Lethbridge Transmission and Radiance Atmospheric Model (ULTRAM; \\citet{chapman}). ULTRAM is a radiative transfer atmospheric model which calculates the radiance and transmission of the atmosphere through a line-by-line, layer-by-layer analysis, and was written with the specific goal of accurately modelling the atmosphere above Mauna Kea. The model parameterizes the atmospheric transmittance in terms of the pwv and airmass, and includes H$_2$O, O$_3$ and O$_2$ as atmospheric absorbers. The airmass of each scan was recorded with each interferogram. The opacity at 225\\,GHz, $\\tau_{225}$, was recorded by a dipping radiometer operated by the Caltech Submillimeter Observatory (CSO). The use of this opacity value in the data analysis introduces some uncertainty into the final results, as the CSO radiometer operates at a fixed azimuth which rarely corresponds with the azimuth of the JCMT observations, and only updates approximately every 10-15 minutes, while the pwv levels in the atmosphere may vary significantly on shorter timescales. This can be seen in Figure \\ref{fig:pwv_vs_cso}, where $\\tau_{225}$ from the CSO is plotted against $\\tau_{225}$ determined from measurements of the pwv recorded by the JCMT's water vapour monitor (which is pointed slightly off the main JCMT beam) during the observations of G10.47. The CSO $\\tau_{225}$ value is often offset from that recorded by the water vapour monitor, and does not reflect the short timescale of the opacity variability. The water vapour monitor, while operating during our observations, was new to the telescope and thus was not incorporated into the data reduction pipeline. In future, the use of the water vapour monitor will provide more accurate atmospheric pwv measurements.\n\n\\begin{figure}\n\\includegraphics[width=84mm]{Fig3.ps}\n\\caption{The opacity at 225\\,GHz measured from the CSO and determined through measurements of the precipitable water vapour by the JCMT's water vapour monitor (WVM) during observations of G10.47. The solid line is the CSO-measured opacity, corrected to match the JCMT observational airmass, which operates at fixed azimuth and updates approximately every 10-15 minutes. The WVM is pointed slightly off the main JCMT beam, and updates every 6 seconds.}\n\\label{fig:pwv_vs_cso}\n\\end{figure}\n\n\nMars observations were then used to calibrate the final, atmospheric transmission corrected, coadded spectra.\n\n\\section{Analysis}\n\n\\subsection{Source signal analysis}\n\nWe first discuss sky subtraction and instrumental contributions to the FTS signal. We then detail calibration issues, which include atmospheric variation and spectral line contamination of the continuum emission. \n\nThe measured signal $S$ as a function of frequency $\\nu$ from an astronomical source through the FTS can be expressed as\n\\begin{eqnarray}\nS(\\nu) &=& R(\\nu)\\,G\\,[\\eta_a\\,J_s(\\nu)\\,\\mbox{e}^{-\\tau(\\nu)} + \\eta_a\\,J_{sky}(\\nu) (1-\\mbox{e}^{-\\tau(\\nu)}) \\nonumber \\\\\n&+& (1-\\eta_a)\\,J_{amb}(\\nu)-J^\\prime(\\nu)] \n\\end{eqnarray}\nwhere the contribution from any additional background has been neglected. Here, $\\tau(\\nu)$ is the atmospheric optical depth as a function of frequency $\\nu$, the telescope aperture efficiency is given by $\\eta_a$,  $R(\\nu)$ is the responsivity or response function of the FTS, and $G$ is the detector gain when observing an astronomical source. For all observations, the gain was set to $G=10^4$. $J_s(\\nu)$ is the true spectrum of the source, $J_{sky}(\\nu)$ is the true spectrum of the atmosphere, $J_{amb}(\\nu)$ is the power received from ambient temperature surfaces of the telescope, and $J^\\prime(\\nu)$ is the power received through the $2^{nd}$ port of the FTS, which is differenced in the interferometric measurement \\citep{naylor00}. The first term therefore represents the power received from the astronomical source itself through the atmosphere and instrument, the second term represents the power received from the atmosphere and the third term represents the power received from the surfaces of the telescope, which are at ambient temperature.\n\nObservations at the background position produce solely the intensity received from the sky, $S_{sky}(\\nu)$, and ambient surfaces:\n\\begin{eqnarray}\nS_{sky}(\\nu) &=& G\\,R(\\nu)\\,[\\eta_a\\,J_{sky}(\\nu)(1-\\mbox{e}^{-\\tau(\\nu)}) \\nonumber \\\\\n&+& (1-\\eta_a)J_{amb}(\\nu)-J^\\prime(\\nu)]\n\\end{eqnarray}\nIf the atmospheric temperature and opacity are assumed to remain unchanged during each pair of source and background scans, the difference of the on- and off-source scans gives the observed power of the source itself, $S_s(\\nu)$:\n\\begin{eqnarray}\nS_s(\\nu)&=& S(\\nu) - S_{sky}(\\nu) = R(\\nu)\\,G\\,\\eta_a\\,J_s(\\nu)\\,\\mbox{e}^{-\\tau_s(\\nu)}\n\\end{eqnarray}\nSky brightnesses at 345\\,GHz~for the typical pwv levels during our observations of $\\sim1.1$\\,mm are $\\sim800$\\,Jy\/beam, an order of magnitude larger than our brightest source, making accurate background subtraction essential. During the observations, measurements of the precipitable water vapour, from the CSO radiometer, and observational airmass were recorded. With this information, the atmospheric model described in Section 2 was used to determine the transmission of the atmosphere in the direction of the source, and thus the contribution of the term $\\mbox{e}^{-\\tau(\\nu)}$ to the observed source intensity $S_s(\\nu)$. Typical values of $\\tau_{225}$ during the observations were $\\sim0.07$, leading to atmospheric transmission levels at 345\\,GHz~of $\\sim75$\\%.\n\nObservations of Mars and a brightness temperature model were used for calibration of the spectra obtained by the instrument. At the time of our observations, Mars had an angular diameter of 8.97\\arcsec. The Mars observations were done in the same on- and off-source pattern as the HMC observations, and the intensity $S_m(\\nu)$ received from Mars after sky subtraction is thus\n\\begin{eqnarray}\nS_m(\\nu)&=& S(\\nu) - S_{sky}(\\nu) = R(\\nu)\\,G\\,\\eta_a\\,J_m(\\nu)\\,\\mbox{e}^{-\\tau_m(\\nu)}\n\\end{eqnarray}\n\nThe brightness temperature of Mars, $T_{B,m}$, on the date of our observations was calculated using the STARLINK FLUXES program \\citep{starlink}. The Mars flux, $J_m(\\nu)$, at the telescope can then be calculated:\n\\begin{equation}\nJ_m(\\nu) = \\frac{2\\nu^2k}{c^2T_{B,m}}\n\\label{eqn:flux}\n\\end{equation}\n\nThe true spectrum of the hot core sources, $J_s(\\nu)$, can then be determined: \n\\begin{equation}\nJ_s(\\nu)=\\frac{S_s(\\nu)\\,\\mbox{e}^{-\\tau_m(\\nu)}}{S_m(\\nu)\\,\\mbox{e}^{-\\tau_s(\\nu)}} J_m(\\nu)\n\\end{equation}\n\nThe final 350\\,GHz spectrum of each source can be seen in Figure \\ref{fig:slopes} in units of Jy\\,\/beam. The derived intensity near the edges of the band has larger error due to the strong water vapour transitions seen in Figure \\ref{fig:ULTRAM}, which define the 350\\,GHz spectral window. Variations in the column abundance of water vapour between source and background measurements result in greater noise at the steepest part of the water vapour line profile.\n\n\\begin{figure}\n\\includegraphics[width=84mm]{Fig4_1.ps}\\\\\n\\includegraphics[width=84mm]{Fig4_2.ps}\\\\\n\\includegraphics[width=84mm]{Fig4_3.ps}\n\\caption{Final calibrated spectra of the hot core sources (top to bottom: G10.47, G31.41, G12.21) with 1$\\sigma$ error bars at 1.5\\,GHz~resolution. The dash-dotted line is the best fit linear slope. The dashed lines indicate the fitting window. Arrows indicate the positions of five molecular emission lines, listed in the text, expected to be strong in HMC. With the exception of the CO(3-2) line at 345.6\\,GHz, no spectral features are coincident with the expected bright line positions.} \\ \\ \\ \\ \\\n\\label{fig:slopes}\n\\end{figure}\n\n\\subsection{Signal-to-noise and atmospheric effects}\n\nThe instrumental signal-to-noise (S\/N) ratio was calculated from the MZ FTS characteristics in \\citet{naylor03} for the spectral resolution and specific observing times for each source, and is solely due to the detector noise of the FTS. The S\/N ratios of the final spectra are compared with the instrumental S\/N ratios in Table \\ref{SN}, with the average S\/N achieved across the band a factor of order $\\sim10$ lower than the instrumental value for all sources. With all detector effects accounted for in the instrumental S\/N ratio, the sensitivity degradation is solely due to the atmospheric variance during our observations. \n\n\\begin{table}\n\\caption{Instrumental and observed S\/N values}\n\\label{SN}\n\\begin{tabular}{cccc}\n\\hline\nObject & Time on source & S\/N$_{Ins}$ & S\/N$_{Obs}$ \\\\\n--- & (hours) & --- & --- \\\\\n\\hline\nG10.47 & $3.5$ & $147$ & $8.5$\\\\\nG31.41 & $4.0$ & $99$ & $8.0$ \\\\\nG12.21 & $4.0$ & $73$ & $7.4$ \\\\\n\\hline\n\\end{tabular}\n\n\\medskip\nThe instrumental and observed signal-to-noise ratio for each source.\n\\end{table}\n\nAt submillimetre wavelengths, the brightness and variability of the sky is an overwhelming obstacle to accurate flux calibration, and observations generally require some form of chopping between source and background measurements in order to remove the sky emission. Ideally, the time between source and background measurements is as short as possible to limit the level of sky variance between observations \\citep{arch02}. While our observational method was designed to minimize the effects of the ever-changing atmosphere on the data by switching between source and background observations every 90\\,s, the atmosphere still contributed substantially to our uncertainties for several reasons. First, the opacity varies on timescales shorter than 90\\,s, as shown in Figure \\ref{fig:pwv_vs_cso}. Second, delays in slewing the telescope to the background position resulted in slight differences between the airmass of the background and source scans, introducing a small systematic effect by effectively offsetting the opacity in the background scan relative to the on-source scan. \n\nAs a quick test, we can think of a small increase in opacity as a small increase in the precipitable water vapour (with the caveat that there are other minor contributions to the opacity). Using atmospheric emissivities calculated for various pwv levels with ULTRAM, we have calculated that a modest jump in pwv of 0.05\\,mm, very reasonable when combining small deviations between source and background airmass with actual changes in the atmospheric pwv levels, increases the atmospheric emission by $\\sim35$\\,Jy\/beam at 350\\,GHz. The pwv values calculated from the CSO $\\tau_{225}$ for the observations contain jumps of this size and a few even larger. For this reason, stable weather is more crucial to this study than particularly good weather. A change in atmospheric emission of 35\\,Jy is comparable to the flux of our brightest source, G10.47, and brighter than the others (see Table \\ref{positions}). As a consequence, some of the subtractions of the source and background spectra produced unphysical, negative flux measurements, while others resulted in impossibly high source brightnesses. When the spectra are coadded, however, on the whole these overly positive and negative scans will average out unless there is a large systematic offset between source and background observations or the pwv levels in the atmosphere are systematically increasing or decreasing. \n\nWhile there was a small systematic offset in the airmass between the on- and off-source scans, it was not large enough to produce significant changes in the sky emission except in the case of G31.41, which was observed while setting. For this source, the offset may have decreased the observed flux by at most 1-2\\,Jy\/beam. During observations of G10.47, the recorded atmospheric opacity steadily increased, although it remained low. After atmospheric transmission corrections, the recorded intensity of this source near the start of the observations was greater by $\\sim10$\\% on average than that recorded later. Again, this may have decreased the observed flux by a few Jy\/beam at most, and is the largest contributor to the uncertainty per data point in the spectrum. There do not appear to be any systematic trends in the atmospheric opacity for the G12.21 observations.\n\n\\begin{figure}\n\\includegraphics[width=84mm]{Fig5_1.ps}\\\\\n\\includegraphics[width=84mm]{Fig5_2.ps}\n\\caption{Sky-subtracted spectra, divided into two groups corresponding to observations made in the first and second halves of each night, then coadded and ratioed for Mars and G10.47. The dash-dotted line indicates the best-fit slope of the ratioed data for G10.47, showing that while the variable atmosphere may introduce offsets into our final observed intensity of the sources, it has not significantly affected the slope of the continuum emission.}\n\\label{fig:source_vs_source}\n\\end{figure}\n\nTo estimate the effects of the varying atmosphere on our spectra, and to determine the level of any systematic effects which may have been introduced into the data, we divided the sky-subtracted spectra into two groups for each source corresponding to observations made in the first and second halves of each night. We then coadded the spectra in these groups, and ratioed the coadded spectrum of the first half with that of the second. Figure \\ref{fig:source_vs_source} shows this ratio for Mars, our calibrator, and G10.47. It is clear that while there is a small shift in overall flux between the early and later observations, within the band there is little overall change in the continuum slope. The fainter sources showed similar results, and we thus expect that while the variable atmosphere may introduce offsets into our final observed intensity of the sources, it has not significantly affected the slope of the continuum emission. \n\n\\subsection{Signal strength}\n\nThe final brightness values recorded by the FTS for each source are listed in Table \\ref{fluxes} with calculated uncertainties. The uncertainty in the final flux value comes from the uncertainty in the observed flux of the source, $S_s(\\nu)$, the observed flux of Mars, $S_m(\\nu)$, the uncertainty in the Mars brightness temperature model, and the uncertainty in the atmospheric transmission, $\\mbox{e}^{-\\tau(\\nu)}$. From the S\/N of the observations, the uncertainty in $S_s(\\nu)$ is $\\sim11-12$\\% for each source. We take an uncertainty in the atmospheric transmission of 10\\%, while the estimated uncertainty for the Mars model is of order 5\\%. The uncertainty in $S_m(\\nu)$ is negligible compared to the other factors due to Mars' brightness. Added in quadrature, this gives an uncertainty in the calibrated source flux of 16\\%. Additionally, the error lobe of the 850\\,\\micron~(350\\,GHz) JCMT beam may contain up to $\\sim5$\\% of the source flux, making the final uncertainties of order 21\\%. \n\n\\begin{table}\n\\caption{Comparison of source brightnesses from SCUBA and the FTS}\n\\label{fluxes}\n\\begin{tabular}{ccc}\n\\hline\nObject & $I_\\nu$ (SCUBA) & $I_\\nu$ (FTS) \\\\\n--- & Jy\/15\\arcsec~beam & Jy\/15\\arcsec~beam \\\\\n\\hline\nG10.47 & $39.0\\pm5.9$ & $44.0\\pm8.8$ \\\\\nG31.41 & $24.2\\pm3.6$ & $24.6\\pm4.9$ \\\\\nG12.21 & $13.0\\pm2.0$ & $27.2\\pm5.4$ \\\\\n\\hline\n\\end{tabular}\n\n\\medskip\nA comparison of the 850\\,\\micron~source brightnesses determined using the FTS with the peak brightnesses recorded by the Submillimetre Common User Bolometer Array. The uncertainties in the SCUBA fluxes are 15\\%, while the uncertainties in the FTS fluxes are 21\\%.\n\\end{table}\n\nThrough calibration with Mars, the observed fluxes of G10.47 and G31.41 agree very well with those recorded by SCUBA; however, G12.21 is much brighter in our observations than the submillimetre continuum measurements indicate, and the values are not within uncertainties. The peak fluxes from the archived SCUBA data for G12.21, which are from two separate observations, agree within uncertainties, suggesting that a variation of the source brightness is unlikely. G12.21 is the faintest source by the SCUBA measurements, and is thus more susceptible to atmospheric subtraction problems than G10.47 and G31.41. Since the largest effect of the variable atmosphere on our observations is to introduce an offset into the overall continuum, and because this offset is cumulative, it is possible that the final flux for this source has been inflated by the bright, variable atmosphere.\n\n\\subsection{Molecular line contamination}\n\nThe sources observed in this study are hot molecular cores. The moderate temperatures of these objects ($T>100$\\,K) cause molecules and elements frozen onto the surface of dust grains during the colder collapse phase to evaporate, releasing these species into the gas phase. The resulting emission lines are often strong in the 350\\,GHz spectral window for HMC, and can therefore contaminate measurements of the submillimetre continuum. The sources in this study have been detected in many molecular lines \\citep{hatchell98} from multiple species present in the 350\\,GHz atmospheric window.\n\nAt our low spectral resolution only very bright, relatively broad lines would contribute noticeably to the dust continuum spectrum. Our spectra show little evidence of bright line contamination, as shown in Figure \\ref{fig:slopes}, where we have indicated the expected positions of $^{13}$CO (3-2), CS (7-6), CO (3-2), HCN (4-3), and HCO$^+$ (4-3) emission lines. These are the strongest, broadest lines observed in the 350\\,GHz window of the hot core G5.89-0.39 \\citep{thompson}, which we expect to be similar to the cores studied here. There is some evidence of small peaks in all the spectra, one of which appears to be at the CO(3-2) transition frequency (345.6\\,GHz); however the data do not allow a conclusive identification. None of the other observed peaks in the spectra are coincident with expected bright line positions. Overall the peaks are small and do not significantly influence the continuum slope, and the contamination of the continuum spectrum by single bright emission lines is therefore not a large problem for this analysis. The concern for this study is then the contamination of the continuum spectrum by a significant number of low-level emission lines that may be present and are not immediately visible, but which may influence the slope of the continuum.\n\nThe frequency coverage of molecular line studies for these sources is much less than that covered by SCUBA and the FTS, so no absolute estimates of the level of contamination have been made \\citep{hatchell00}. Observing at 350\\,GHz with the JCMT, the conversion factor from brightness temperature $T_b$ in Kelvin to flux $S$ in Janskys is approximately $S\\,\\mbox{(Jy)} \\simeq 13 T_b\\,\\mbox{(K)}$ \\citep{hatchell98}. A brightness temperature of 40\\,K thus corresponds to $\\sim 500$\\,Jy.  The line width of strong molecular features, however, is seldom greater than 15\\,km\\,s$^{-1}$ (at least for the bright core of the line) while the spectral resolution of the FTS for this study is $\\sim 1500$\\,km\\,s$^{-1}$, so that individual lines are diluted by at least a factor of 100. Strong molecular lines, with peak brightness $\\sim 40$\\,K, will thus add approximately 5\\,Jy to the measured flux in a given spectral bin. Therefore, we do not expect contamination of the spectra by individual lines by more than approximately 10-20\\%.\n\nThe contamination of the submillimetre continuum of hot core sources by molecular lines can range between 10\\% and (in extreme cases) 60\\% of the total integrated flux \\citep{groes94}. \\citet{johns03} find that the 350\\,GHz continuum emission from their observed protostellar sources, however, is never dominated by molecules other than CO (although line emission from HCN, HNC, CN and CH$_3$OH can contribute a substantial fraction of the contamination for more energetic sources) and line contamination in general is typically less than $\\sim10\\%$ at 350\\,GHz, even for photon-dominated regions. The hot core regions of the sources observed in this study amount to a small fraction of the 350\\,GHz beam (1-2\\arcsec~in diameter compared with 15\\arcsec~FWHM), whereas the more extended emission is from cooler material in which fewer molecules will be excited. Overall, we argue that molecular line contamination of our low resolution 350\\,GHz continuum is minimal, and does not significantly affect the analysis. We recognize, however, that it is important to consider the influence of these low-level molecular lines when making any determination of the slope of the dust continuum emission using low spectral resolution submillimetre observations. \n\n\\section{Spectral index results}\n\n\\subsection{Calculation of the spectral index}\n\nFrom Equation \\ref{eqn:dust_law}, the dust emission $S(\\nu) \\propto \\nu^\\gamma$, where $\\gamma=\\beta+\\alpha$ is the sum of the dust spectral index $\\beta$ and the frequency dependence $\\alpha$ from the Planck function. For small ranges in frequency $\\Delta\\,\\nu$ around a central frequency $\\nu_0$, we can expand this relationship:\n\\begin{eqnarray} \nS(\\nu_0+\\Delta\\,\\nu) &\\propto& (\\nu_0+\\Delta\\,\\nu)^\\gamma \\nonumber \\\\*\n&\\propto& \\nu_0^\\gamma \\,\\biggl(1+\\gamma\\frac{\\Delta\\,\\nu}{\\nu_0}\\biggr)\n\\end{eqnarray}\nThus, for small $\\Delta\\,\\nu\\,\/\\nu_0$, the dust emission is expected to increase linearly with frequency. The FTS passband at 350\\,GHz is $\\simeq30$\\,GHz~wide, or $\\simeq10$\\% of the observing frequency, making the approximation valid. In practice, this window was made slightly smaller due to a decrease in S\/N caused by increased levels of contaminating atmospheric flux near the edges of the band, leaving $\\Delta\\,\\nu\/\\nu\\simeq0.08$ on average. A $\\chi^2$ minimizing linear fit routine was used to determine the slope of the continuum emission inside the passband. All results quoted here have been determined from the unapodized data to retain the highest spectral resolution. The results varied slightly depending on the fraction of the band used (due to increasing error bars at the band edges when using more of the band, and conversely due to a smaller lever arm when using less of the band); however any variations were within uncertainties. \n\nColumn 2 of Table \\ref{tab:betas} shows the best fit continuum emission slope $\\gamma$ for each source with uncertainties determined through the $\\chi^2$ fitting routine, and Figure \\ref{fig:slopes} shows the final spectrum for each source overlaid with the best linear fit. The average reduced $\\chi^2$ value for the fits was $\\sim0.6$, indicating a good fit to the data within the band. We expect, for a good linear fit, that the reduced $\\chi^2$ should be close to unity for a sufficiently large number of data points. The small values of the reduced $\\chi^2$ in our fits suggest that the uncertainty per data point has been overestimated by $\\sim20$\\% for each source. Since the uncertainty in the calculated slope varies as the uncertainty per data point, this implies that the uncertainties in our slopes may be overestimated by $\\sim20$\\%. \n\n\\begin{table}\n\\caption{Spectral indices}\n\\label{tab:betas}\n\\begin{tabular}{cccc}\n\\hline\nSource & $\\gamma_{FTS}$ & $\\beta$ & $\\gamma_{SCUBA}$ \\\\\n\\hline\nG10.47 & $3.6\\pm1.2$  & $1.6\\pm1.2$  & $3.4\\pm0.6$  \\\\\nG31.41 & $4.1\\pm1.2$  & $2.1\\pm1.2$  & $3.3\\pm0.6$  \\\\\nG12.21 & $3.3\\pm1.1$  & $1.3\\pm1.1$  & $4.1\\pm0.6$  \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nFor emission in the Rayleigh-Jeans limit, $\\alpha=2$ and thus the dust spectral index $\\beta=\\gamma-2$. In order for the R-J approximation to apply at 350\\,GHz, $h\\nu\/kT_d\\ll 1$, so $T_d\\gg h\\nu\/k\\sim17$\\,K. Although HMC are warm, they are surrounded by a shell of cold dust. The observed emission is integrated over the line of sight and the beam, which at 15\\arcsec~FWHM is much larger than the hot cores. There is therefore substantial contribution to the observed emission from the colder outer regions along the line of sight. Models of the emission from these sources which take the source temperature and density profiles into account, however, appear to produce spectra which are in the R-J limit at 350\\,GHz \\citep{hatchell00,hatchell03}. When considering only the peak flux, a greater proportion of the emission from the warm inner regions is contained within the single beam, and the R-J approximation still applies. Column 3 of Table \\ref{tab:betas} lists the dust spectral index $\\beta$ calculated for each source assuming the 350\\,GHz emission is in the R-J limit. We find $\\beta$ in the range of $1.3 - 2.1$ with a mean of 1.6.\n\n\n\\subsection{Comparison with SCUBA data}\n\nAs discussed in Section 1, the spectral index of dust emission can also be calculated using observations at two widely separated wavelengths. From Equation \\ref{eqn:dust_law}, we see that we can solve for $\\gamma=\\beta+\\alpha$ with a ratio of the flux at two different wavelengths:\n\\begin{equation}\n\\gamma = \\frac{\\ln(S_{2}\\,\/\\,S_{1})}{\\ln(\\nu_2\\,\/\\,\\nu_1)} = \\beta + \\alpha\n\\label{eqn:scuba_calc}\n\\end{equation}\nFor our sources, archived SCUBA data at 450\\,$\\mu$m (660\\,GHz) and 850\\,$\\mu$m (350\\,GHz) are publicly available. In order to properly compare the flux at the two wavelengths, the 8.5\\arcsec~resolution 450\\,\\micron~data were first convolved to the 15\\arcsec~resolution of the 850\\,\\micron~data. The ratio of the peak brightnesses was then used to calculate $\\gamma$ from Equation \\ref{eqn:scuba_calc}. Column 4 of Table \\ref{tab:betas} gives the $\\gamma$ value, $\\gamma_{SCUBA}$, calculated from the ratio of the 450\\,\\micron~and 850\\,\\micron~fluxes for each source. Within uncertainties, $\\gamma_{SCUBA}$ is consistent with $\\gamma_{FTS}$. The uncertainties in this calculation are dominated by the uncertainty in the source flux at both wavelengths.\n\nIf the R-J approximation applies at both frequencies $\\nu_1$ and $\\nu_2$, then $\\alpha=2$ and $\\beta=\\gamma_{SCUBA}-2$. As shown previously, the emission at 350\\,GHz is likely in the R-J limit, but for the R-J approximation to apply at 660\\,GHz, $T_d\\gg h\\nu\/k\\sim32$\\,K.  For the average dust temperatures likely associated with these hot molecular core sources, the 660\\,GHz emission cannot be assumed to be in the R-J limit. In this case, less power is contributed to the dust emission from the Planck function, and $\\alpha<2$. Using this information, we can use the 350\\,GHz (850\\,\\micron) slope determined using the FTS and $\\gamma_{SCUBA}$ to estimate the temperatures of the observed HMC. \n\nThe spectral indices calculated for G10.47 agree well with $\\alpha\\simeq2$, suggesting that the average dust temperature for this source is high enough ($T_d>60$\\,K) for the emission at both wavelengths to be in the R-J limit. G12.21 is also likely warm, while the results for G31.41 indicate a lower average dust temperature. The uncertainties in these temperature estimates are very large due to the large uncertainties in the evaluated slopes; it is clear, however, that with smaller uncertainties in the value of $\\beta$ determined from FTS observations, it will be possible to independently determine both $T_d$ and $\\beta$ of the dust opacity in star forming regions. \n\n\\subsection{Interpretation}\n\nOur results are consistent with the values of $\\beta$ found in embedded UC\\,H\\,{\\small II}~regions and other HMC. Fitting models of HMC to observational data have found $\\beta=1.6-2$ \\citep{osorio99,church90}. In a study of 17 UC\\,H\\,{\\small II}~regions, \\citet{hunter98} finds an average spectral index of $\\beta=2.0\\pm0.25$ through greybody modelling of IRAS, submillimetre and millimetre observations. Observations and modelling of six compact HII regions, one of which was G10.47, by \\citet{hoare91} find an average $\\beta\\simeq1.5$. They note, however, that the frequency dependence is somewhat affected by the assumed density distribution of the model. The dense ridge in Orion has $\\beta\\simeq1.9$ near embedded infrared sources \\citep{gold97}. Overall, the spectral indices of these warmer, likely more evolved star forming regions tend to be higher than those found in cooler, less evolved dense cores, indicating that dust properties have changed through some mechanism which appears to correlate with protostellar age. \n\nSeveral dust models predict a dust emissivity frequency dependence similar to that found here, but our observations are not sensitive enough to determine a most likely dust composition. Higher values of $\\beta$ have been associated with dust grains covered in thick ice mantles \\citep{aanestad,lismenten}, while many dust grain models with combinations of silicate and graphite compositions find a spectral index of $\\beta\\sim1.5-2$. \\citet{ossen94} tabulate the dust opacity properties for nine different dust distributions, including bare grains versus grains with ice mantles (both thick and thin), and various degrees of dust coagulation. The emissivity results presented in this paper are unable to definitively rule out any of the models. The best fit, however, is for bare grains in relatively dense environments, where coagulation is more pronounced (column 2 of Table 1 in \\citet{ossen94}).\n\nWhile we had hoped to determine $\\gamma$ to an accuracy of $\\pm\\,0.1$, the variable atmosphere at 350\\,GHz degraded the achieved signal-to-noise ratio by a factor $\\ga10$ compared with the instrumental S\/N, leading to uncertainties in our calculation of $\\beta$ of approximately $\\pm1.2$. Even with the atmospheric complications, however, the uncertainties determined here are of the same order as those found when calculating the dust spectral index through multiple wavelength observations of single sources. These results show that with better sky subtraction techniques, it should be possible to determine the spectral index of bright astronomical sources with high accuracy using an FTS. Future instruments with greater sensitivity and precise sky subtraction will enable the measurement of spectral indices of much fainter sources. These results show that careful sky subtraction is \\textit{essential} to future dust continuum emission studies with an FTS.\n\n\\subsection{An imaging FTS for SCUBA-2}\n\n\\begin{table*}\n\\begin{minipage}{135mm}\n\\caption{Source brightnesses required to determine $\\gamma$ in 12 hours of observing}\n\\label{tab:ifts}\n\\begin{tabular}{cccccc}\n\\hline\nWavelength & Frequency & Resolution & 12 hrs 1$\\sigma$ $\\Delta$T & Source brightness & Source brightness \\\\\n\\micron & \\,GHz & \\,GHz & mK & $\\gamma \\pm 0.1$ (Jy\/beam) & $\\gamma \\pm 0.3$ (Jy\/beam) \\\\\n\\hline\n450 & 660 & 3.0 & 1.44 & 4.7 & 1.6 \\\\\n850 & 350 & 3.0 & 0.2 & 0.40 & 0.14 \\\\\n\\hline\n\\end{tabular}\n\n\\medskip\nSource brightnesses required to determine the dust spectral index $\\gamma$ in 12 hours of observing using an IFTS on SCUBA-2.\n\\end{minipage}\n\\end{table*}\n\nSCUBA-2 is the next generation submillimetre instrument currently in development to replace SCUBA in 2006 \\citep{holl03}. An imaging FTS (IFTS) is being developed as an ancillary instrument to SCUBA-2 \\citep{naylor04}. Based on detector noise calculations alone, the IFTS will be at least 10 times more sensitive per pixel than the single bolometer FTS used in these observations \\citep{gom04}. The extremely accurate sky subtraction possible with the IFTS will provide an even more substantial increase in the observational precision. When also considering the variable atmosphere noise, the per-pixel sensitivity of the IFTS increases to at least 100 times greater than in the observations presented in this paper. Using the detector sensitivities in \\citet{gom04}, we have calculated that it will be possible to determine in 12 hours of observing with the IFTS the spectral index $\\gamma=\\beta+\\alpha$ of the submillimetre dust emission at 850\\,\\micron~ to $\\pm0.1$ for sources of only 400\\,mJy\/beam in brightness, and to $\\pm0.3$ for sources only $\\sim140$\\,mJy\/beam in brightness. Similar calculations for observations at 450\\,\\micron~are listed with those at 850\\,\\micron~in Table \\ref{tab:ifts}.\n\n\\section{Summary}\n\nWe have determined the spectral index of the dust emission of three hot molecular cores solely within the 350\\,GHz (850\\,\\micron) passband using an FTS on the JCMT. We find an average dust spectral index $\\beta\\simeq1.6$, in agreement with the spectral indices calculated using archived SCUBA data at 450\\,\\micron~and 850\\,\\micron. These results for $\\beta$ are consistent with measurements of the spectral index in other HMC. The uncertainties in $\\beta$ from these observations are dominated by the difficulties in subtracting the very bright and variable submillimetre atmosphere from the data. With better sky subtraction techniques, as will be possible with an imaging FTS on SCUBA-2, these uncertainties will be greatly reduced. The per-pixel sensitivity of the imaging FTS planned for SCUBA-2 will be greater by a factor of $\\sim100$ than that of the single pixel detector used in this study, and will allow the determination of the spectral index of dust emission of sources with brightnesses of only a few hundred mJy\/JCMT beam. \n\n\\section*{Acknowledgments}\nWe thank the referee, Jenny Hatchell, for her suggestions which improved this paper. We also thank all TSS involved in FTS runs for their invaluable assistance, as well as G. J. Tompkins, B. Gom and I. Chapman for much FTS-related support. Additional thanks to G. Fuller for providing the source list. RKF also thanks B. Hesman for helpful advice and discussion. This work was funded in part by the University of Victoria (RKF) and by grants from NSERC Canada (DJ, DAN and GRD). The JCMT is operated by the Joint Astronomy Centre in Hilo, Hawaii on behalf of the parent organizations Particle Physics and Astronomy Research Council in the United Kingdom, the National Research Council of Canada and The Netherlands Organization for Scientific Research. SCUBA-2 is a jointly funded project through the JCMT Development Fund with substantial additional contributions from the UK Office of Science and Technology and the Canada Foundation for Innovation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe X complex in the Tholen taxonomy comprises the E, M and P classes which\nhave very different inferred mineralogies but which are spectrally similar \nto each other and featureless in the visible wavelengths (Tholen, 1984).  \nBy convention X-types with measured geometric albedos are classified into Tholen E-type\n(high albedo, $p_{v} > 0.3$), M-type (medium albedo, $0.1 < p_{v} < 0.3$), or P-type\n(low albedo, $ p{v} < 0.1$) asteroids (Tholen\\&Barucci, 1989). The X-type\nasteroid indicates an \"EMP\" asteroid for which we do not have albedo\ninformation. \n\n\nThe X-type asteroids are distributed throughout the main belt (Moth\\'e-Diniz et al. 2003) but\ntend to be concentrated around 3.0 AU, and in the Hungaria region (Warner et al. 2009). The X complex includes bodies with very different mineralogies. \nM asteroids may be composed of metals such as iron and \nnickel and may be the progenitors of differentiated iron-nickel\nmeteorites (Gaffey, 1976; Cloutis et al, 1990,\nGaffey et al. 1993).  Enstatite chondrites have also been proposed\nas M-type asteroid analogs (Gaffey 1976; Vernazza et al. 2009; Fornasier et al. 2010; Ockert-Bell et al. 2010). \nA metallic iron interpretation requires that M asteroids are\nfrom differentiated parent bodies that were heated to at least 2000 $^{\\circ} C$ to\nproduce iron meteorites (Taylor, 1992). \n\nE-type asteroids have high albedo values and their surface  compositions \nare consistent with iron-free or iron-poor  silicates  such as enstatite,\nforsterite or feldspar. E-types are thought to be the parent bodies of the\naubrite (enstatite achondrite) meteorites (Gaffey et al. 1989, Gaffey et al.,\n1992, Zellner et al., 1977, Clark et al. 2004a, Fornasier et al. 2008).  \nP-type asteroids have low albedo values, and featureless red spectra. \nP-types are located mainly in the outer region of the main belt (Dahlgren \\& Lagerkvist 1995, \t\nDahlgren et al. 1997, Gil-Hutton \\& Licandro 2010) and in the Jupiter Trojan\nclouds (Fornasier et al. 2004, 2007a). They are probably primitive bodies, however\nthey have no clear meteorite analogues.  P-type asteroids\nare presumed to be similar to carbonaceous chondrites, but with a higher organic content to explain the strong red spectral slopes (Gaffey et al. 1989, Vilas et al. 1994).\n\nIn the last 15 years, several near-infrared spectral surveys have been devoted to the study of the X group and in particular the M and E-type asteroids, revealing that these bodies \nshow great diversity in the infrared, even within the same class. In addition, polarimetric measurements (Lupishko \\&\nBelskaya 1989) and radar observations (Ostro et al., 1991, 2000; Margot and Brown, 2003; Magri et al.,\n2007; Shepard et al., 2008, 2010)  of selected M-type asteroids have revealed surface\nproperties that are in some cases inconsistent with metallic iron. \nMoreover, spectra of M-types are not uniformly featureless as \ninitially believed; \nseveral spectral absorption bands have been detected in the visible and near\ninfrared ranges. Faint absorption bands near 0.9 and 1.9\n$\\mu$m have been identified on the surfaces of some M-types (Hardersen \net al., 2005; Clark et al., 2004b; Ockert-Bell et al., 2008; Fornasier et al., 2010), and of some E-type asteroids (Clark et al. 2004a; Fornasier et al. 2008) and\nattributed to orthopyroxene. A very peculiar absorption band, centered around\n0.49-0.50 $\\mu$m, extending from about 0.41 to 0.55 $\\mu$m, was found on some E-type spectra (Burbine et al, 1998; Fornasier \\& Lazzarin, 2001; Clark et al. 2004a,b;\nFornasier et al., 2007b, 2008), and attributed to the presence of sulfides such as\ntroilite or oldhamite (Burbine et al., 1998, 2002a). Busarev (1998) detected weak\nfeatures tentatively attributed to pyroxenes (at 0.51 $\\mu$m) and oxidized or\naqueously altered mafic silicates (at 0.62 and 0.7 $\\mu$m) in the spectra of 2\nM-type asteroids. \\\\\n\nOf note is the evidence for hydrated minerals on the surfaces of some  E and M-type asteroids (Jones et al, 1990; Rivkin et al, 1995,\n2000, 2002), inferred from  the identification of an absorption feature around 3\n$\\mu$m.  This feature is typically attributed \nto the first overtone of H$_{2}$O and to OH vibrational\nfundamentals in hydrated silicates. If the origin of the 3 $\\mu$m band on E and \nM-type asteroids is actually due to hydrated minerals, then these objects \nare not all igneous as previously believed and the thermal scenario\nin the inner main belt would need revision.\nGaffey et al. (2002) propose alternative explanations for the 3 $\\mu$m band: materials normally considered to be anhydrous containing structural OH; troilite, an anhydrous mineral that shows a feature at 3 $\\mu$m, \nor xenolithic hydrous meteorite components on asteroid surfaces from impacts \nand solar wind implanted H. Rivkin et al. (2002) provide arguments\nrefuting the Gaffey suggestions, and provide evidence in support of the\nhydrated mineral interpretation of the observations.\\\\\n\nThe X complex is consistently identified in the two taxonomies \nrecently developed based on CCD spectra acquired in the visible (Bus \\& Binzel 2002) or in the visible and near infrared range (DeMeo et al., 2009), even though there is not a direct one-to-one correspondence between the Bus and Tholen classes. For instance 92 Undina (Tholen M-type), and 65 Cybele (Tholen P-type) are both  classified as Xc-type in the Bus taxonomy. A description of the X complex classes (X, Xk, Xe, Xc) in the Bus \\& Binzel (2002) classification is summarized in Clark et al. (2004b), while in section 6 we describe the X complex classes in the Bus-DeMeo taxonomy.\n\nAiming to constrain and understand the composition of the\nasteroids belonging to the mysterious X group, we have carried out a\nspectroscopic survey of these bodies in the visible and near infrared range at\nthe 3.5m Telescopio Nazionale Galileo (TNG), in La Palma, Spain, at the 3.5m\nNew Technology Telescope (NTT) of the European Southern \nObservatory, in La Silla,\nChile, and at the Mauna Kea Observatory  3.0 m NASA Infrared Telescope Facility\n(IRTF) in Hawaii, USA. \nResults on asteroids classified as E and M-type following the Tholen \ntaxonomy are presented elsewhere (Fornasier et al. 2008, 2010).\nIn this paper we present new VIS-NIR spectra of 24 asteroids belonging to\nthe X type as defined by Tholen \\& Barucci (1989), that is an \"E--M--P\" type\nasteroid for which albedo information was not available at the time of their\nclassification. To constrain their compositions, we conduct a search for\nmeteorite analogues using the RELAB database, and we model the asteroid\nsurface composition with geographical mixtures of selected minerals when a\nmeteorite match is not satisfactory. \nIn addition, we present an analysis of X complex spectral \nslope values and class distributions in the asteroid main belt, \nwhere we include previously published observations of M and E-type\nasteroids obtained during the same survey (Fornasier et al. 2008, 2010). \n\n\n\\section{Observations and Data Reduction}  \n\n\n[HERE TABLE 1 ]\n\nThe data presented in this work were mainly obtained  during 2 runs \n(February and November 2004) at the Italian Telescopio Nazionale Galileo \nTNG of the European Northern Observatory (ENO) in La Palma, Spain, and 2 runs (August 2005, and January 2007) at the New Technology Telescope (NTT)\nof the European Southern Observatory (ESO), in Chile. Two asteroids (92 Undina, 275 Sapientia),\ninvestigated in the visible range at the NTT and TNG telescopes, were separately\nobserved in the near infrared at the Mauna Kea Observatory  3.0 m NASA Infrared\nTelescope Facility (IRTF) in Hawaii during 2 runs in July 2004 and September 2005 (Table~\\ref{tab1}).\nWe had a total of 11 observing nights. Here\nwe present the results of our observations of 24 X-type asteroids: for 16\nobjects we obtained new visible and near-infrared wavelength spectra (VIS-NIR),\nfor one object we obtained only a near-infrared spectrum (NIR), \nand for 7 objects we obtained only visible range spectra (VIS).\n\nThe instrument setups are the same as used \nin Fornasier et al. (2008, 2010). We refer the reader to those papers\nfor complete\ndescriptions of observations and data reduction techniques.\nThe spectra of the observed asteroids, all normalized at 0.55 $\\mu$m, are shown\nin Figs.~\\ref{fig1}-- \\ref{fig4}, and the observational conditions are reported in Table~\\ref{tab1}.\n\n[Here Figs 1,2,3, and 4]\n\nTo analyze the data, spectral slope values were calculated with linear\nfits to different wavelength regions: $S_{cont}$ is the spectral slope in the whole\nrange observed for each asteroid, $S_{UV}$  in the 0.49-0.55 $\\mu$m range, $S_{VIS}$ is the slope in the 0.55-0.80 $\\mu$m\nrange, S$_{NIR1}$ is the slope in the 1.1-1.6 $\\mu$m range, and S$_{NIR2}$ is\nthe slope in the 1.7-2.4 $\\mu$m range. Values are reported in\nTable~\\ref{slope}. \nBand centers and depths were calculated for each asteroid showing an \nabsorption feature, following the Gaffey et al. (1993) method. \nFirst, a linear continuum was fitted at the edges of the band, \nthat is at the points on the spectrum outside the absorption \nband being characterized. Then the asteroid spectrum was divided \nby the linear continuum and the region of the band was fitted with a \npolynomial of order 2 or more. The band center was then calculated \nas the position where the minimum of the spectral reflectance \ncurve occurs, and the band depth as the minimum in the ratio of the \nspectral reflectance curve to the fitted continuum (see Table~\\ref{band}).\n\n\n[HERE TABLE 2]\n\n\n[HERE TABLE 3]\n\n\\section{Spectral Analysis and Absorption Features}\n\nThe observed asteroids show very different spectral behaviors\n(Figs~\\ref{fig1}-- \\ref{fig4}). Eleven asteroids show at least one absorption\nfeature (Table~\\ref{band}). \n\n50 Virginia (see Fig. 1 and Table 3) shows spectral features centered at 0.43, 0.69, and 0.87 $\\mu$m (Table~\\ref{band}) associated with aqueous alteration products (Vilas et al. 1994, Fornasier et al. 1999), and seems to be a typical 'hydrated' C-type asteroid. \n\nFive asteroids (50 Virginia, 283 Emma, 337 Devosa, 517 Edith, and 1355 Magoeba) show a \nweak band at 0.43 $\\mu$m. For 50 Virginia, 283 Emma, and 517 Edith, \nthe band might be associated with an Fe$^{3+}$  spin-forbidden \ntransition in the iron sulfate jarosite, as suggested by  \nVilas et al. (1993) for low-albedo asteroids. For 337 Devosa and 1355 Magoeba, \nthe band might be associated with chlorites and Mg-rich serpentines,  \nas suggested by King \\& Clark (1989) for enstatite chondrites, \nor to clinopyroxenes such us pigeonite or augite as\nsuggested by Busarev (1998) for M-asteroids. \n\nThe 1355 Magoeba spectrum also shows a band \ncentered at $\\sim$ 0.49 $\\mu$m that\nresembles the absorption seen on some E-type (subclass EII) asteroids (Fornasier \\& Lazzarin 2001; Fornasier et al. 2007b; 2008, Clark et al. 2004a), where it is attributed to sulfides such\nas oldhamite and\/or troilite (Burbine et al. 1998, 2002a, 2002b). Nevertheless, the estimated albedo of 1355 Magoeba has a moderate value (0.267$\\pm$0.095, Gil-Hutton et al. 2007) more consistent with a Tholen M-type classification.\n\n92 Undina has a very weak band centered at $\\sim$ 0.51 $\\mu$m that is similar to the Fe$^{2+}$\nspin-forbidden crystal field transitions in terrestrial and lunar pyroxenes (Burns et\nal. 1973, Adams 1975, Hazen et al. 1978). This band has been previously detected\nby Busarev (1998) in the spectra of two M asteroids, 75 Eurydike and 201\nPenelope. \n\nAn absorption feature has been identified  in the 0.9 $\\mu$m\nregion in the spectra of 4 X-type asteroids that\nhave albedos placing them in the Tholen M-class (92 Undina,\n337 Devosa, 417 Suevia, and 1124 Stroobantia). \nThe band center ranges from 0.86 $\\mu$m to 0.92 $\\mu$m with a\nband depth (as compared to the continuum) of 3--6 \\%.\nAlso 522 Helga, having a low albedo (4\\%)\nconsistent with the Tholen P-type classification, exhibits a faint band centered at 0.94 $\\mu$m.  \nThis band was previously \nreported by several authors in the spectra of Tholen M, E and X-type\nasteroids (Hardersen et al. 2005; Clark et al. 2004a, 2004b, Fornasier\net al. 2008, 2010) and is attributed to low-Fe, low-Ca orthopyroxene\nminerals. \n\nA peculiar feature centered at 1.6 $\\mu$m was found on the spectrum of 758 Mancunia. This band resembles one seen on 755 Quintilia (Fornasier et al., 2010, Feiber-Beyer et al. 2006) whose interpretation is still unknown. No bands due to silicates in the 0.9 and 1.9 $\\mu$m regions have been identified in our spectra, except for a change in slope at around 0.77 $\\mu$m.\n\nFinally, 1122 Nieth shows a strong 0.96 $\\mu$m absorption band (depth of 24\\%) and a steep slope in the NIR region, typical of olivine-rich bodies. \\\\\n\nSeveral asteroids of this survey were observed previously by other authors.\nFive X-type asteroids (50 Virginia, 283 Emma, 337 Devosa, 517 Edith, and 909 Ulla) of the Clark et al. (2004b) survey of the X complex were \nalso studied in this work. The spectral behavior for these objects looks quite similar,\nexcept for 283 Emma (our spectrum is flat in the NIR, while the Clark et\nal. spectrum has a concave shape with a higher spectral gradient),  and 909 Ulla\n(our spectrum has a higher spectral gradient \nas compared to the Birlan et al. (2007) and the Clark et al. (2004b) spectra).\nOckert-Bell et al. (2008, 2010) presented near-infrared spectra of 77 Frigga and 758 Mancunia,\nthat look slightly different compared to ours. In particular, they detect faint spectral bands in the 0.9 and 1.9 $\\mu$m region for 758 Mancunia, and in the 0.9 $\\mu$m region for 77 Frigga, bands that are not detected in our spectra.\n1355 Magoeba and 3447 Burckhalter were observed in the visible\nrange by Carvano et al. (2001), and their data look very similar to ours.\\\\ \nIn sum, comparing our spectra with those in the existing literature, we suggest\nthat asteroids 77 Frigga, 283 Emma, 909 Ulla, and 758 Mancunia may display surface variability as they show different spectral behaviors throughout independent observations. For 758 Mancunia this variability is confirmed also by radar measurements that show the radar cross-section and the polarization ratio to\nvary considerably with rotation phase (Shepard et al. 2008). For the other 3 asteroids, we cannot exclude that some spectral differences may be linked with unresolved observational differences (such as\nbackground stars, or undetected troubles in removing the atmosphere). \n\n \n\n\\section{Tholen Taxonomic Classification}\n\nOur targets were classified as belonging to the Tholen X class on the basis of\ntheir spectra from 0.4 to 1 $\\mu$m and because their albedos were not known\nat the time of classification.\nSince the original X-type classification in the Tholen taxonomy, \nthe albedo value has become available for most of the asteroids \nwe observed (Tedesco et al., 2002). Taking into account this important \ninformation, together with the visible and near infrared (when available) \nspectral behavior,  we suggest a re-classification of the X-type \nasteroids in our sample following the Tholen-classification scheme. \nThe suggested new classifications are reported in Table ~\\ref{slope}.\n\n1122 Nieth (Fig.~\\ref{fig4}), initially classified as X-type in the Tholen taxonomy,\nhas atypical spectral properties that diverge from all other asteroids.\nThe near infrared spectrum of Nieth\nclearly shows a strong 0.96 $\\mu$m absorption band \nand a steep slope in the NIR\nregion. We infer that this asteroid belongs \nto the olivine-rich A-types (used in\nthe Tholen, Bus, and Bus-DeMeo taxonomic systems).\n\n1328 Devota has a very steep featureless red spectrum. Considering its low albedo value (4\\%), \nDevota falls within the D class in the Tholen taxonomy (Fig.~\\ref{fig4}).\n\nWe suggest that the ten asteroids (77 Frigga, 92 Undina, \n184 Dejopeja, 337 Devosa, 417 Suevia, 741 Botolphia, \n758 Mancunia, 1124 Stroobantia, 1146 Biarmia, and 1355 Magoeba) with \nmoderate albedo values (0.1--0.3) are M-type asteroids while those with albedos lower than 10\\% fall in the C\/P classes. \nTo discriminate between C and P-type asteroids, we \nanalyzed spectral slope values in the visible \nrange ($S_{VIS}$ from 0.55 to 0.80 $\\mu$m) and searched for \nthe UV drop-off below $\\sim 0.5 ~\\mu$m that is\ntypical of C-type asteroids. Comparing the $S_{VIS}$ slopes \nwith the $S_{UV}$ slopes (calculated from 0.49 to 0.55 $\\mu$m) \nof the low albedo asteroids, we find two distinct distributions \ncorrelated with the C and P-classes. The 6 asteroids \n(50, 220, 223, 283, 536, and 517) having the UV absorption and\n$S_{vis} < 2 \\%\/10^{3}$\\AA\\ belong to the C-type, while the \n4 asteroids (275, 463, 522, and 909) having \n$3 <S_{vis} < 6 \\%\/10^{3}$\\AA\\ belong to the P-type. \n1902 Shaposhnikov, observed only in the near infrared \nrange, is probably also a P-type, while 3447 Burckhalter, for which the albedo value is 0.336$\\pm$0.164 \n (Gil-Hutton et al. 2007) could possibly be an E-type.\n\n\n\n\\section{Meteorite Spectral Matches and Mixing Models}\n\n\\subsection{Meteorite Analogues}\n\nTo constrain the possible mineralogies of our asteroids, we conducted  \na search for meteorite and\/or mineral spectral matches.\nWe sought matches only for objects observed across the entire \nVIS-NIR wavelength range.  A complete description of \nthe search methodology can be found in Clark et al. (2010) and\nFornasier et al. (2010).  We used the  \npublicly available RELAB spectrum library (Pieters 1983), which  \nconsisted of nearly 15,000 spectra in November of 2008.  RELAB spectra were \nnormalized to 1.0 at 0.55 $\\mu$m, and then a Chi-squared value \nwas calculated for each RELAB spectrum relative to  \nthe normalized input asteroid spectrum. The RELAB spectra were sorted according  \nto Chi-squared values, and then visually examined for dynamic weighting of  \nspectral features by the spectroscopist.  \nGiven similar Chi-squared  \nvalues, a match that mimicked spectral features was preferred over a  \nmatch that did not.  We visually examined the top ~200 Chi-squared  \nmatches for each asteroid and we constrain the search for \nanalogues to those laboratory spectra with\nbrightness (reflectance at 0.55 $\\mu$m) roughly comparable to the albedo of\nthe asteroid.  \nIt must be noted that the asteroid data are disk integrated, \nwhich could mask spectral variations due to differences in \nparticle size, mineralogy, or abundance. However the meteorite \nand mineral spectra are well characterized powders measured under \nwell defined laboratory conditions. It should also be noted that\n an asteroid's albedo \nis defined at phase angle zero,\nwhereas meteorites' reflectance is taken at phase angle  $> 5^{o}$.  \nFor the asteroid albedo value, we used the IRAS  \nalbedo published by Tedesco et al. (2002).  For the darker asteroids  \n(below 10\\% albedo), this is a fairly well-constrained value.  The  \nuncertainty for brighter asteroid albedos (above 10\\% albedo) is  \nrelatively larger. To account for this, we filtered out lab spectra if\nbrightness differed by more than $\\pm$ 3\\% in absolute albedo for the darker  \nasteroids (albedos less than 10\\%), and we filtered out lab spectra if  \nbrightness differed by more than $\\pm$ 5\\% in absolute albedo for the  \nbrighter asteroids  (albedos greater than 10\\%).  For example, if the asteroid' s albedo was 10\\%, we searched over all lab spectra with  \n0.55 $\\mu$m reflectance between 5\\% and 15\\%.  Once the brightness filter was  \napplied, all materials were normalized before comparison by least- \nsquares. \n\nOur search techniques effectively emphasized the  \nspectral characteristics of brightness and shape, and de-emphasized  \nminor absorption bands and other parametric characteristics.  As such,  \nwe suggest that our methods are complementary to band parameter  \nstudies.\n\n[HERE TABLE 4]\n\n[HERE FIGURES 5 and 6]\n\n\nThe best matches between the observed X-type asteroids and meteorites\nfrom the RELAB database are reported\nin Table~\\ref{tab}. We matched only the 15 asteroids \nobserved both in the visible and near infrared range that\nhave published albedo values. Some meteorite matches are quite good, while for some asteroids the best meteorite \nmatch we found does not satisfactorily reproduce both the visual and near infrared\nspectral behavior. In particular our attempt to find a meteorite or mineral match failed for the 3 asteroids 1122 Nieth (A-type), 1328 Devota (D-type), and 1355 Magoeba (M-type).\n\nThe best matches we found \nfor 7 moderate albedo M-type asteroids are presented in Figure~\\ref{fig5met}.  Two asteroids, \n741 Botolphia and 1146 Biarmia were not compared with RELAB spectra\ndue to the lack of near-infrared spectral data. \nWe find that iron or pallasite meteorites are the best matches both for the featureless M-type asteroids and for those having minor absorption bands. This result supports the link between M-type asteroids and iron or pallasite meteorites suggested in the literature by several authors. In this small sample, no enstatite chondrites, which have also been suggested to be linked with M-type asteroids, have been found as the best meteorite analogue.\n\nThe strongest analogy \nwe found is between 758 Mancunia and a sample of the iron meteorite Landes which has a large number of silicate inclusions in a nickel-iron matrix.\nIt must be noted that no specific grain-size sampling\nfilter was applied, and of concern is the fact that Mancunia's best match is with a spectrum obtained of the cut\nslab surface of the Landes iron meteorite. \n\n758 Mancunia has a radar albedo of 0.55$\\pm$0.14 \n(Shepard et al. 2008) one of the highest values measured for the \nasteroids, suggesting a very high metal content. Shepard et al. (2008) also found variations in the radar cross-section and polarization ratio with Mancunia's rotational phase, variations that are not related to shape but to the regolith depth, porosity, or near-surface roughness. The Ockert-Bell et al. (2010) Mancunia spectrum shows absorption bands in the 0.9 and 1.9 $\\mu$m region, and is found to be similar to the ordinary chondrite Paragould. Our spectrum does not show these features, but a change of slope at 0.77 $\\mu$m and a faint (1.3\\% depth) feature centered at 1.6 $\\mu$m, similar to that observed on 755 Quintilia (Fornasier et al. 2010, Fieber-Beyer et al. 2006), whose origin is not yet understood.  Considering both radar and spectral observations, 758 Mancunia could present an heterogeneous surface composition, with a high metal content but also the presence of some silicates that are not uniformly distributed on the surface.\n\nThe Landes iron meteorite sample was found to be the best spectral match also for 337 Devosa, even if the meteorite does not fully reproduced the 0.88 $\\mu$m absorption band of the asteroid. A similar problem occurred for 417 Suevia and 1124 Stroobantia, where different grain sized samples of the iron meteorite DRP78007 are found to be the best spectral matches, but they did not reproduce the asteroids absorption bands in the 0.9 $\\mu$m region. \nWe note that grain size effects can play an important role in the spectral behavior of iron-nickel meteorites. In fact Britt and Pieters (1988) found that\nthe spectra of M-type asteroids show good agreement with those of iron\nmeteorites with surface features in the range of 10 $\\mu$m to 1 mm, that is\nlarger that the wavelength of incident light.  Meteorites with these roughness\nvalues are diffuse reflectors and show the classic red slope continuum of iron,\nwith practically no geometric dependence on reflection. On the other hand, a  \ndecreasing of the meteorite surface roughness changes the reflectance\ncharacteristics:  complex scattering behavior is seen for roughness in the\n0.7-10 $\\mu$m range, while for roughness values $<$ 0.7 $\\mu$m the reflectance\nis characterized by two distinct components, the specular one which is bright and\nred sloped, and the nonspecular one which is dark and flat.  \\\\\n\nFor 77 Frigga we were not able to find any meteoritic\/mineral \nanalog within the medium albedo range that could\nreproduce Frigga's red visible slope and flat near-infrared slope. \nThe best spectral match is shown in Fig.~\\ref{fig5met}.\nThe iron meteorite Chulafinnee\nmodels the spectral behavior below 0.55 $\\mu$m very well,\nand the albedo values are similar, \nhowever the change in spectral slope around 0.9 $\\mu$m is not\nreproduced.\n\nFor 92 Undina we propose two different meteorites: \nthe best spectral match is with a metal-rich powder sample of the pallasite meteorite \nEsquel, whose albedo of 0.14 is lower than the 0.25 value\nof the asteroid. The iron meteorite Babb's Mill\nwas a better numerical fit (minimum chi squared fit)\nand had a more comparable albedo, but was not favored because \nBabb's Mill has a near  infrared spectrum that is \nmuch flatter than that of Undina. \n\n\n\nIn Figure~\\ref{fig6met} we present the best matches \nproposed for six low albedo asteroids. \nIt has long been noticed that reflectance spectra of carbonaceous\nchondrites are similar to those of the low-albedo asteroids\n(Gaffey \\& McCord 1978; Hiroi et al. 1996). In particular, CM meteorites \nshow features due to aqueous alteration processes and are linked\nto low albedo aqueous altered asteroids like the C and G-types (see Vilas et al. 1994, Burbine 1998, Fornasier et al. 1999).   \nIndeed, all the observed low albedo asteroids are best fit with CM meteorites, either unaltered (50 Virginia and 283 Emma) \nor altered by heating episodes (275 Sapientia and 517 Edith) \nor laser irradiation (522 Helga and 909 Ulla) (see Fig.~\\ref{fig6met}). \nThe CM2 MET00639 meteorite matches very well the visible range spectrum of 50 Virginia and in particular the 0.7 $\\mu$m band attributed to $Fe^{2+}\\rightarrow Fe^{3+}$ charge transfer absorptions in phyllosilicate minerals. Nevertheless, the meteorite does not well reproduce the asteroid's near-infrared spectrum. The same meteorite was found as a best match for the near infrared region of 283 Emma, but it does not well reproduce the spectrum in the visible range, where the asteroid is featureless. The absence of the 0.7 $\\mu$m band in the asteroid spectrum led us to discard this as a spectral match.\n\nThe P-type asteroids 275 Sapientia and the C-type 517 Edith have as best match two different samples of the Murchison carbonaceous chondrite heated at 600$^{o}$C (grain size $<$ 63 $\\mu$m) and 700$^{o}$C (63 $\\mu$m $<$  grain size $<$ 125 $\\mu$m), respectively. It must be noted that at these temperatures the phyllosilicates are dehydrated and transformed into olivine and pyroxene and that the 3 $\\mu$m hydratation band vanishes (Hiroi et al., 1996).  \nOur spectral match may indicate that Sapientia and Edith have no hydrated silicates on their surfaces. If hydrated silicates were originally present, the asteroids may have experienced important thermal episodes that dehydrated them.\n\nThe P-type asteroids 522 Helga and 909 Ulla are reproduced by a sample of the CM meteorite Migei after laser irradiation. This meteorite is composed of a black matrix with olivine-rich chondrules, olivine aggregates and individual grains, carbonates, and sulfides (Moroz et al. 2004). Its unaltered spectrum is dark and shows absorption features related to hydrated silicates such as the 2.7-3 $\\mu$m band, the UV-falloff and a 0.75 $\\mu$m band. Once irradiated, the meteorite is dehydrated, the absorption bands are significantly weakened (Moroz et al. 2004), and the material remains dark -- most probably due to abundant submicron inclusions of Fe-rich phases finely dispersed in the glassy mesostasis (Shingareva et al. 2004). \nBecause laser irradiation is a laboratory technique for simulating micrometeorite bombardment, the spectral match between Helga and Ulla and a laser irradiated sample of CM Migei may indicate that the surfaces of these outer belt asteroids, if composed of CM carbonaceous-like materials, may be spectrally reddish due to micrometeoritic bombardment.\n\n\n\\subsection{Mixing Models}\n[HERE FIGURE 7]\n\n[HERE TABLE 5]\n\nTo constrain the surface compositions of the investigated \nX-type asteroids and the materials needed to reproduce the weak 0.9 $\\mu$m band seen on some spectra\nwe considered geographical (spatially segregated) mixtures of \nseveral terrestrial and meteoritic materials in different grain \nsizes: in particular we took into account all the samples \nincluded in the US Geological Digital Spectral Library \n($http:\/\/speclab.cr.usgs.gov\/spectral-lib.html$), in \nthe RELAB database, and in the ASTER spectral library ($http:\/\/speclib.jpl.nasa.gov$),\ntogether with \norganics solids (e.g. kerogens by Clark et al. 1993 and Khare et al. 1991;\nTitan tholins from Khare et al. 1984; and\nTriton tholins from McDonald et al. 1994) and \namorphous carbon (by Zubko et al. 1996).\nThe synthetic spectra  were compared with \nthe asteroid spectra, using the known IRAS albedo, and \nthe VIS-NIR spectral behavior and continuum slopes. \nWe present these matches as non-unique examples of how\ngeographical mixtures can be used to explain some of the\nvariety found in X-type asteroid spectra.  We do not intend\nto indicate that these are unique derivations of the composition\nof these asteroids.  Such an inversion requires much more\ninformation than we currently possess (grain sizes, optical\nconstants, endmembers present, etc.).\nNevertheless, the models we present can\nprovide a first order understanding of\n possible surface composition.\n\n\nIn Figure~\\ref{modelli} and Table~\\ref{models} we present \nmixing models \nfor the 4 X-type asteroids showing the 0.9 $\\mu$m band (this band is\nattributed to low-Fe, \nlow-Ca orthopyroxene and is not reproduced by the meteorite analogues proposed \nin Table~\\ref{tab}). We also present\nthe low albedo asteroids 517 Edith (for which \nthe spectral match with a heated CM \nmeteorite is poor in the 1--2 $\\mu$m region), and 1328 Devota (for which we \ncannot find any satisfactory meteorite analogue).\n\nAs shown in Fig.~\\ref{fig5met} and discussed in Section 5.2 we did not find \na meteorite with a similar albedo value\nthat could match the whole VIS-NIR spectral behavior of \n92 Undina.\nHowever, we were able to produce a model of\nthe surface of this M-type asteroid\nwith a geographical mixture of 99\\% of pallasite Esquel and 1\\% \northopyroxene.\nThis mixture model reproduces the spectral behavior of Undina,\nhowever its albedo is much lower (0.14) than that of the asteroid\n(0.25).\n\nFor 337 Devosa the spectral behavior has been reproduced \nby two different mixtures: one composed of 99\\% pallasite\nEsquel and 1\\% orthopyroxene (for an albedo of 0.14) \n(red line in Fig.~\\ref{modelli}), and one \ncomposed of 98\\% pallasite Esquel and 2\\% goethite (for an albedo of 0.14).\n337 Devosa has spectral properties in the VIS-NIR \nspectral range that are very similar\nto the M-type asteroid 22 Kalliope (see Fornasier et al. 2010). \nFor these asteroids, \nthe 0.9 $\\mu$m band is consistent with a small\namount of anhydrous silicate (such as orthopyroxene), or with \na small amount of goethite (an aqueous alteration product).\n\nFor 417 Suevia the proposed meteoritic analogue (the iron meteorite \nDRP78007, Fig.~\\ref{fig5met}) does not fit the observed 0.9 $\\mu$m band. \nWe found that a geographical mixture of 97\\% IM DRP78007 (RELAB file cdmb47) \nand 3\\% orthopyroxene  models the asteroid spectral behaviour. \nThe albedo of the mixture is 0.16, a bit lower than the value estimated for \nSuevia (0.20).\nFor 1124 Stroobantia the proposed meteorite analogue\n(metallic meteorite DRP78007) also does not fit the faint 0.9\n$\\mu$m band.  For this asteroid, we propose two mixing models. One is a \ngeographical mixture of 98\\% IM MET101A Odessa and 2\\% orthopyroxene, with \nan albedo of 0.13 (blu line in Fig.~\\ref{modelli}). \nThis model reproduces the asteroid spectrum below 1.4 $\\mu$m \nbut not the longer wavelengths.\nAlternatively, a mixture of 96\\% iron meteorite DRP78007 and 4\\% \nolivine (albedo 0.16, red line in Fig.~\\ref{modelli}) fits \nStroobantia \nbeyond 0.6 $\\mu$m, but does not match the spectral region below 0.6 $\\mu$m.  \nAlthough none of the modeled mixtures reproduces all spectral features (albedo, \nspectral slopes, band depth, and band center) of the asteroid, our best inferrence from the modeling is to suggest that the surface composition of Stroobantia is probably consistent with metallic meteorites enriched in silicates.\n\nThese results show that a few percent (less than 3\\%) \nof orthopyroxenes or goethite added to iron or pallasite meteorites \ncan reproduce the weak spectral features around 0.9 $\\mu$m seen on some asteroids \nbelonging to the X-complex. \n\n517 Edith shows a flat and featureless spectrum. The observed spectral behavior appears \nsimilar to the spectrum of pure  amorphous carbon (from Zubko et al., 1996), although we \ncannot exclude the possibility that a small amount \nof silicates may be present and masked by the dark and opaque \nmaterials.\n \n Considering the low albedo value and the featureless and reddish spectral behavior, we classify  1328 Devota as a D-type. Since the Tagish Lake meteorite is usually \nconsidered the best meteorite analog for \nD-type asteroids (Hiroi et al. 2001), we\ncompared the spectrum of Devota with laboratory spectra of\nseveral samples of this meteorite. \nThe spectrum of Devota appears to be redder than that of Tagish Lake. \nThis could be due to the different ages of the asteroids and meteorites.\nThe surface of Devota could be older than\nthe meteorite, and the reddening of the asteroid spectrum could be the result\nof space weathering processes. \nAlternatively, the spectral behavior of Devota can be reproduced with \na geographical mixture of 94\\% Tagish Lake meteorite (RELAB file c1mt11)\nand 6\\% \nTriton tholins (from McDonald et al. 1994), resulting in an albedo of 0.02.\nOur observations of Devota do not cover the wavelengths necessary for detection of the  tholin spectral feature at about 3 $\\mu$m, so\nalthough we cannot constrain the presence of \ntholins with the data in hand, \nwe suggest that some tholin or tholin-like\ncomponent could be the reddening agent producing the observed \nspectral slope between 0.5 and 2.5 $\\mu$m.\n \n\\section{The X complex: overview and discussion}\n\n[HERE TABLE 6 and 7]\n\n[HERE FIGURES 8 and 9]\n\n\nOur survey devoted to the X-complex asteroids as defined by the \nTholen (1984) taxonomy is composed of 78 objects, including the \nE and M-type asteroids already published in Fornasier et al. (2008, 2010).\nOn the basis of their spectral behavior and albedo values,\nwe classified 22 E-types and distributed them \ninto 3 subclasses I, II, and III, \n(see Fornasier et al. 2008 and references therein). \nOne of the objects (3447) was originally classified as X-type and \nwas tentatively attributed to the E(I) class in this work, due to its moderately high albedo value but with a high uncertainty. \nOur survey includes 38 M-types, ten of which (77, 92, 184, 337, 417, 741, 758, 1124, \n1146) were originally classified as X-types, 7 C-types (498 Tokio was \noriginally classified as an M-type, and all the others as X-type),\n1 D (originally classified as X-type), 5 P-type (originally classified as X-type), 3 A-types (2577, 7579 and 1122 which were originally classified as E-types,\nand an  X-type, respectively), and 2 S-types (5806 and 516, which were originally classified as an\nE and M-type, respectively).\nIn Fig.~\\ref{slopea} we show the spectral slope value $S_{VIS}$ calculated \nin the visible wavelength range versus the semimajor axis for all  the E, \nM and C\/P-types observed. As expected, high albedo E-type asteroids populate \nmainly the Hungaria region and the inner part of the main belt, while low \nalbedo C and P-types are located mainly in the outer part of the main belt \nor beyond it, and M-type are located between 2.4 and 3.2 AU. \nThe mean visible spectral slope for the different E, M and P classes are \nvery similar, as expected (Table~\\ref{slopes}). In the NIR range, M-type asteroids having a band in the 0.9 $\\mu$m region have mean spectral slope S$_{NIR1}$ and S$_{NIR2}$ values similar to P-type asteroids. These values are higher than those of the  M-type asteroids without the 0.9 $\\mu$m band. For the E-type asteroids, observations in the infrared range are available only for a very few objects. We give the NIR  mean slope values only for the subtype III (4 asteroids observed in the NIR range: 44 Nysa, 214 Aschera, 317 Roxane, and 437 Rhodia), and the subtype II (64 Angelina, 2867 Steins, and 4660 Nereus), that show the lower S$_{NIR1}$ mean value within the X complex (no data are available in the NIR range for the subtype I).\n\nOne M-type asteroid, 77 Frigga, show a very peculiar spectrum, with a near infrared spectrum similar to that of other M-type bodies, but with a very high $S_{VIS}$ value comparable with that of the D-type Devosa. Nevertheless its moderate albedo value allows us to exclude any possible link with dark P or D-types. Also A or S type asteroids have $S_{VIS}$ value similar to that of Frigga, but the absence of any absorption band due to olivine and pyroxene lets us exclude any link with these asteroid classes. As discussed in section 3, Frigga may show surface variability, and the 3 $\\mu$m band, usually associated with hydrated silicates, has been detected on its surface (Rivkin et al., 2000). Polarimetric data reveals that Frigga has a large inversion angle (Gil-Hutton, 2007), implying that the surface is composed of small particles (comparable to the wavelength), and\/or of a mixture of particles with high contrast in albedo, like refractory inclusions seen on some carbonaceous chondrites (see Belskaya et al. 2010, and references therein).  Our attempt to find a meteorite analogue was unsuccessful as the best match found (the iron meteorite Chulafinnee) does not satisfactory reproduce the asteroid spectrum. No radar observations are available for this body. Additional observations of 77 Frigga will be very important to confirm its surface heterogeneities and possibly to constrain its surface composition. \\\\\nIn our sample 43 objects were observed in the complete \nV+NIR spectral range.\nAmong them 37 have spectral features and albedo values compatible with\nthe X-complex (5 E-types, 29 M-types, and 3 P-types).\nFor all objects we investigated the spectral matches  with\nmeteorites\/minerals or geographical mixture models. \nFor the 5 E-type asteroids (44, 64, 214, 317 and 437) \nwe found good spectral matches with the enstatite\nachondrite meteorites, in several cases enriched with troilite, oldhamite or\northopyroxenes (Fornasier et al. 2008).\nThe 3 P-type asteroids presented in this paper (275, 522, and 909) \nexhibit spectral behaviors and albedo values compatible with carbonaceous chondrite meteorites. \nMost of the M-type asteroids have spectral features and albedo values well represented \nby iron meteorites, pallasites, and enstatite chondrites -- in several cases \nenriched with orthopyroxenes, olivines, or goethite.\n\nOur new spectral observations enhance the available physical information for the observed \nasteroids and allow us to apply the  Bus-DeMeo classification recently published\n(DeMeo et al., 2009). The Bus-DeMeo system is based on the\nasteroids' spectral characteristics over the wavelength range 0.45 to 2.45 $\\mu$m\nwithout taking the albedo into consideration. \nIn the Bus-DeMeo taxonomy, the X complex comprises \n4 types (X, Xk, Xe, Xc) as in the  Bus \\& Binzel (2002) system: \nan asteroid belongs to the Xe-type if the 0.49 $\\mu$m feature is present, \nto the  Xk-type if a feature is present in the 0.8--1.0 $\\mu$m range, \nto the Xc-type if the spectrum is red and featureless with slight \nconcave-down curvature, and to the X-type if the spectrum is straight and\nfeatureless.\nThe Tholen X-type asteroids we observed show different spectral behaviors in the near infrared range. \nWe therefore re-classified 16 of the asteroids presented in this paper and observed both in the VIS and NIR range according to the Bus-DeMeo taxonomy. The corresponding classes are summarized in Table\n~\\ref{slope}. All 5 asteroids showing the faint 0.9 $\\mu$m band \nfall in the Xk-type (337 Devosa also has the 0.43 $\\mu$m band), \nwhile 1122 Nieth, which shows a broad and deep 0.96 $\\mu$m band \nand a steep infrared spectrum, is classified as an A-type.\nThe 5 asteroids having the 0.43 $\\mu$m absorption band (note that this region is outside the wavelength limits where the Bus-DeMeo taxonomy is defined) belong \nto 5 different classes: 337 Devosa is classified as Xk; \n50 Virginia, showing bands due to hydrated materials, is a Ch; \n283 Emma is a C-type; 517 Edith is an Xc; and 1355 Magoeba, \nhaving also the peculiar band at 0.49 $\\mu$m, is classified \nas Xe-type in the Bus-DeMeo taxonomy. \nThe featureless objects belong to the X-type (77, 758, and 909), \nD-type (1328, and 1902), and to the C-type (275).\n\nIn Table~\\ref{busDeMeotx} we report Bus-DeMeo taxonomic classifications\ntogether with the Tholen classifications (existing or proposed by us) \nfor all the asteroids observed during our survey devoted to the X-complex asteroids. \nIn Fig~\\ref{busalbedo} we show the albedo value versus the semimajor axis \nfor the observed asteroids classified with the Bus-DeMeo taxonomy. We aim \nto investigate if possible correlations between the classes of this taxonomy \nand the albedo may exist, although the albedo is not a parameter taken into \naccount in the Bus-DeMeo classification system.  \nWe dispose of the albedo value for 7 out of the 10 Xe asteroids, that show the 0.49$\\mu$m band typical of this class and attributed to sulfides. Most of them \ncorrespond to the high albedo subgroup II of the Tholen E-class, with only two asteroids (132 \nAertha and 1355 Magoeba) being classified as M-type in the Tholen taxonomy. So this peculiar band, attributed to the presence of sulfides like oldhamite, seems not to be exclusively associated with high albedo E-type asteroids. \\\\\nThe Bus-DeMeo Xk-class asteroids show two distinct albedo distributions: \na high albedo group (albedo $>$ 0.4), corresponding to the subgroup III of \nthe Tholen E-class, and a medium albedo group (0.1 $<$ albedo $<$ 0.3) which \nincludes all the Tholen M-types showing a faint feature in the 0.9 $\\mu$m \nregion. The Xk-class includes also a low albedo object, 522 Helga, that we classified here as a P-type in the Tholen taxonomy. \nIron bearing pyroxenes such as orthopyroxene are suggested to cause the feature around 0.9 $\\mu$m, characteristic of the Xk class, and seem to be present on asteroids with very different albedo values. \\\\\nMost of the Bus-DeMeo X-class asteroids have 0.1 $<$ albedo $<$ 0.2, with \nthe exception of 504 Cora (albedo = 0.34, therefore classified as E[I] \nasteroid) and 909 Ulla (albedo=0.034, therefore classified as Tholen P-type).\nAll the C-types have low albedo values, but the Xc and D-types span low \nand moderate albedo values. In particular the asteroid 849 Ara has a steep spectral slope and falls in the D class according to the Bus-DeMeo taxonomy (Fornasier et al. 2010), but its albedo is very high (0.27), excluding a surface composition of organic-rich silicates, carbon, and anhydrous silicates as commonly expected on low-albedo Tholen D-type asteroids.\n\nFrom this comparison, it is evident that the Bus-DeMeo taxonomy is very helpful in constraining the asteroids' surface compostion (for example Xk-type means presence of orthopyroxene), in particular when some absorption bands are present on the spectra. But it is clear that the asteroid albedo is also a very important parameter for constraining the surface properties and also their evolution in time (space weathering effects). We strongly encourage the development of a next-stage Bus-DeMeo taxonomy that includes the important albedo parameter for asteroid composition\/classification.\n\n\\section{Summary}\n\nWe present new visible and near infrared spectra of 24 asteroids belonging to\nthe X-type as defined by Tholen \\& Barucci (1989), that is an \"E--M--P\" type\nasteroid for which albedo information was not available at the time of their\nclassification. The X complex in the Tholen taxonomy is comprised of\nthe E, M and P classes which\nhave very different mineralogies but which are spectrally similar with\nfeatureless spectra in broadband visible wavelengths. \nOur observations reveal a large variety of spectral behaviors within the \nX class, and we identify  \nweak absorption bands on 11 asteroids. We combine our spectra with the albedo \nvalues available since 2002 for the observed bodies to \nsuggest new Tholen-like classifications.  We find:  1 A-type (1122), 1 D-type \n(1328), 1 E-type (possibly, 3447 Burckhalter), 10 M-types (77, 92, 184, 337, 417, \n741, 758, 1124, 1146, and 1355), 5 P-types (275, 463, 522, 909, 1902), and 6 C-types (50, 220, 223, 283, 517, and 536).\nFour new M-type asteroids (92 Undina, 337 Devosa, 417 Suevia, and 1124 \nStroobantia) show a faint band in the 0.9 $\\mu$m region, attributed to low \ncalcium, low iron orthopyroxene. Indeed, several works based on spectral and \nradar observations show that not all the M-type asteroids have a pure metallic \ncomposition (Fornasier et al. 2010 and reference therein).\\\\\nThree low albedo asteroids (50 Virginia, 283 Emma, and 517 Edith) show a weak \nband centered at 0.43 $\\mu$m  that we interpreted as due to Fe$^{3+}$  \nspin-forbidden transition in hydrated minerals (hematite, goethite). \nAlso the medium albedo bodies \n337 Devosa and 1355 Magoeba have the same absorption.  In this case the band \nmay be associated with chlorites and Mg-rich serpentines or pyroxene minerals \nsuch us pigeonite or augite. 50 Virginia shows also \ntwo absorptions centered at $\\sim$ 0.69 and 0.87 $\\mu$m which are typical of \nhydrated silicates. \\\\\nWe performed a search for meteorite and\/or\nmineral spectral matches between the asteroids observed in the visible and \nnear infrared range (with published albedo values) and the RELAB \ndatabase. \\\\\nThe best matches found for all the M-types of our sample are iron or pallasite meteorites as suggested in the literature, however we note that these meteorites often do not reproduce the faint absorption band features (such as the possible orthopyroxene absorption bands at 0.9 $\\mu$m) detected on the asteroids.\nWe tried to constraint the asteroids' surface compositions using geographical mixing models for new M-types having the 0.9 $\\mu$m feature. We found \ngood spectral matches by enriching the iron or pallasite meteorites with small \namounts ($<3\\%$) of orthopyroxene or goethite.   \nFor the low albedo asteroids we found as the best match CM carbonaceous \nchondrites, either unaltered or altered \n(submitted to heating episodes or laser irradiation).\nOur sample includes also two objects whose spectra diverge completely from \nother X complex asteroids: 1328 Devota, a low albedo body with \na red featureless spectrum, suggested to belong to the D-type, and 1122 Nieth, \na medium albedo asteroid with a broad and deep 0.9 $\\mu$m band and a steep \ninfrared spectrum, suggested to belong to the A-type. A \nsynthetic model made with the Tagish \nLake meteorite, considered as the best meteorite analogue of D-type asteroids \n(Hiroi et al 2001), and a reddening agent (Triton tholin) reproduces the \nspectral behavior of 1328 Devota.\n\nThe whole sample of asteroids included in our work is 72 X-type\nobjects (we exclude the A, D and S\/Sq-type asteroids), partly published here and partly  \nalready published in Fornasier et al. (2008) and Fornasier et al. (2010). The analysis of this complete sample \nhas clearly shown that, although\nthe mean visible spectral slopes of M-, E- and P-type asteroids\nare very similar to each other, the differences in albedo indicate major differences in mineralogy and composition.\n\n\\bigskip\n\n{\\bf Acknowledgment} \\\\\nThe authors thank Dr. M. Ockert-Bell, Dr. M. Shepard, and \nDr. A. Migliorini for their help with observations, and A. W. Rivkin and an anonymous referee for their \ncomments and suggestions.\nBEC thanks Jonathan Joseph for computer programming assistance.\nSF thanks Dr. F. DeMeo for her help in the classification \nof the observed asteroids using her taxonomy. \nTaxonomic type results presented in this work were determined, in whole or in part, using a Bus-DeMeo Taxonomy Classification Web tool by Stephen M. Slivan, developed at MIT with the support of National Science Foundation Grant 0506716 and NASA Grant NAG5-12355. \nThis research utilizes spectra acquired with\nthe NASA RELAB facility at Brown University.\n\n\\bigskip\n\n{\\bf References} \\\\\n\nAdams, J. B. 1975. Interpretation of visible and near-infrared reflectance\nspectra of pyroxenes other rock-forming minerals. In Infrared and Raman\nSpectroscopy of Lunar and Terrestrial Minerals (C. Karr, Jr., Ed.), pp. 90--116.\nAcademic Press, New York.\n\nBelskaya, I. N., Fornasier, S., Krugly, Yu. N., Shevchenko, V. G., Gaftonyuk, N. M., Barucci, M. A., Fulchignoni, M., Gil-Hutton, R, 2010. Puzzling asteroid 21 Lutetia: our knowledge prior to the Rosetta fly-by. Astron. and Astroph. 515, A29\n\nBirlan, M., Vernazza, P., Nedelcu, D. A., 2007. Spectral properties of nine\nM-type asteroids. Astron. Astroph. 475, 747--754\n\nBritt, D.T.,Pieters, C. M., 1988. Bidirectional reflectance properties of\niron-nickel meteorites. In: Lunar and Planetary Science Conference, 503--512\n\nBurbine, T.H., 1998. Could G-class asteroids be the parent bodies of the CM chondrites? Meteoritics \\& Plan. Sci. 33, 253--258\n\nBurbine, T. H., Cloutis, E. A., Bus, S. J., Meibom, A., Binzel, R. P., 1998. The detection of troilite (FeS) on the surfaces of E-class asteroids.  Bull. Am. Astron. Soc. 30, 711 \n \n Burbine, T. H., McCoy, T. J., Nittler, L., Benedix, G., Cloutis, E., Dickenson, T. 2002a. Spectra of extremely reduced assemblages: Implications for Mercury. Meteorit. Planet. Sci. 37, 1233--1244\n\n Burbine, T.H., McCoy, T.J., Meibom, A., 2002b. Meteorite parent bodies. In\nAsteroids III (Bottke, W. et al. editors), pp 653--664, Univ. of Arizona Press, Tucson \n\nBurns, R. G., D. J. Vaughan, R. M. Abu-Eid, M. Witner, and A. Morawski 1973.\nSpectral evidence for Cr$^{3+}$, Ti$^{3+}$, and Fe$^{2+}$ rather than Cr$^{2+}$,\nand Fe31 in lunar ferromagnesian silicates. In Proc. 4th Lunar Sci. Conf.\n983--994\n\n  Bus, S. J., Binzel, R. P.,  2002. Phase II of the Small Main-Belt Asteroid Spectroscopic SurveyA Feature-Based Taxonomy. Icarus 158, 146--177\n  \n  Busarev, V. V., 1998. Spectral Features of M-Asteroids: 75 Eurydike and 201 Penelope. Icarus 131, 32--40\n\nCarvano, J. M., Lazzaro, D., Moth\\'e-Diniz, T., Angeli, C. A., Florczak, M., 2001. Spectroscopic Survey of the Hungaria and Phocaea Dynamical Groups. Icarus 149, 173--189\n\nClark, R.N., Swayze G.A., Gallagher, A.J., King, T.V.V., Calvin, W.M., 1993.\nU.S. Geological Survey Open File Report 93-592, http:\/\/speclab.cr.usgs.gov \n\nClark, B.E., Bus, S.J., Rivkin, A.S., McConnochie, T.,\nSander, J., Shah, S., Hiroi, T., Shepard, M., 2004a. E-Type asteroid\nspectroscopy and compositional modeling, JGR 109, 1010--1029\n\nClark, B.E., Bus, S.J., Rivkin, A.S., Shepard, M., Shah, S., 2004b. Spectroscopy\nof X-type asteroids. Astron. J. 128, 3070--3081\n\nClark, B.E., J. Ziffer, D. Nesvorny, H. Campins, A. S. Rivkin, T. Hiroi, M.A. Barucci, M. Fulchignoni, R. P. Binzel, S. Fornasier, F. DeMeo, M. E. Ockert-Bell, J. Licandro, T. Moth\u00e9-Diniz, 2010. Spectroscopy of B-Type Asteroids: Subgroups and Meteorite Analogs.  Journal of Geophysical Research, 115, E06005\n\nCloutis, E. A., Gaffey, M. J., Smith, D. G. W., Lambert, R. St. J., 1990. Metal Silicate Mixtures: Spectral Properties and Applications to Asteroid Taxonomy. J. Geophys. Res., 95, 281, 8323--8338\n\nDahlgren, M., Lagerkvist, C. I., 1995. A study of Hilda asteroids. I. CCD spectroscopy of Hilda asteroids. Astron. Astrophys. 302, 907--914\n\nDahlgren, M., Lagerkvist, C.I., Fitzsimmons, A., Williams, I. P., Gordon, M., 1997. A study of Hilda asteroids. II. Compositional implications from optical spectroscopy. Astron. Astrophys. 323, 606--619\n\nDeMeo, F. E., Binzel, R P., Slivan, S. M., Bus, S. J., 2009. An extension of the Bus asteroid taxonomy into the near-infrared. Icarus 202, 160--180\n\nFieber-Beyer, S. K., Gaffey, M. J., Hardersen, P. S., 2006. Near-Infrared\nSpectroscopic Analysis of Mainbelt M-Asteroid 755 Quintilla. 37th Annual Lunar\nand Planetary Science Conference, March 13-17, 2006, League City, Texas,\nabstract no.1315\n\nFornasier, S., Lazzarin, M., Barbieri, C., Barucci, M.A., 1999. Spectroscopic\ncomparison of aqueous altered asteroids with CM2 carbonaceous chondrite\nmeteorites. Astron. Astrophys. 135, 65--73\n\nFornasier, S., \\& Lazzarin, M., 2001. E-Type asteroids: Spectroscopic investigation on the $0.5 \\mu$m absorption band. Icarus 152, 127--133\n\nFornasier, S., Dotto, E., Marzari, F., Barucci, M.A., Boehnhardt, H., Hainaut, O.,\nde Bergh, C., 2004. Visible spectroscopic and photometric survey of L5 Trojans\n: investigation of dynamical families. Icarus 172,  221--232\n\nFornasier, S., Dotto, E., Hainaut, O., Marzari, F., Boehnhardt, H., de Luise, F., Barucci, M. A., 2007a.\nVisible spectroscopic and photometric survey of Jupiter Trojans: Final results on dynamical families. Icarus 190, 622--642\n\nFornasier, S., Marzari, F., Dotto, E., Barucci, M. A., Migliorini, A., 2007b.\nAre the E-type asteroids (2867) Steins, a target of the Rosetta mission,\nand NEA (3103) Eger remnants of an old asteroid family? Astron. Astroph. 474, 29--32\n\nFornasier, S., Migliorini, A., Dotto, E., Barucci, M. A., 2008. Visible and near infrared spectroscopic investigation of E-type asteroids, including 2867 Steins, a target of the Rosetta mission. Icarus 196, 119--134\n\nFornasier, S., Clark, B.E., Dotto, E., Migliorini, Ockert-Bell, M., Barucci, M. A., 2010. Spectroscopic survey of  M--type asteroids. Icarus 210, 655--673\n\nGaffey,  M. J., 1976. Spectral reflectance characteristics of the meteorite classes. J. Geophys. Res. 81, 905--920\n\nGaffey, M. J., McCord, T. B., 1978. Asteroid surface materials - Mineralogical characterizations from reflectance spectra.\nSpace Sci. Reviews 21, 555--628\n\n  Gaffey, M. J., J. F. Bell, and D. P. Cruikshank 1989. Reflectance spectroscopy and aster\noids surface mineralogy. In Asteroids II (R.P. Binzel, T. Gehrels, and M. S. Matthew\ns, Eds.),  pp. 98--127, Univ. of Arizona Press, Tucson. \n\n  Gaffey, M. J., K. L. Reed, and  M. S. Kelley 1992. Relationship of E-type Apollo asteroi\nds 3103 (1982 BB) to the enstatite achondrite meteorites and Hungaria asteroids. Icarus 100, 95--109 \n\nGaffey, M. J., Burbine, T. H., Piatek, J. L., Reed, K. L., Chaky, D. A., Bell, J. F., Brown, R. H., 1993. Mineralogical variations within the S-type asteroids class. Icarus 106, 573--602\n\n Gaffey, M. J., Cloutis, E.A., Kelley, M. S., Reed, K. L., 2002. Mineralogy of Asteroids. In\nAsteroids III (Bottke W. et al. editors), pp. 183-204, Univ. of Arizona Press, Tucson.\n\nGil-Hutton, R., Lazzaro, D., Benavidez, P., 2007. Polarimetric observations of Hungaria asteroids. Astron. Astrophys. 468, 1109--1114\n\nGil-Hutton, R., 2007. Polarimetry of M-type asteroids. Astron. Astrophys. 468, 1127--1132\n\nGil-Hutton, R., Licandro, J., 2010. Taxonomy of asteroids in the Cybele region from the analysis of the Sloan Digital Sky Survey colors. Icarus 206, 729--734\n\n\nHardersen, P. S., Gaffey, M. J., Abell, P. A., 2005. Near-IR spectral evidence\nfor the presence of iron-poor orthopyroxenes on the surfaces of six M-type\nasteroids. Icarus 175, 141--158\n\nHazen, R. M., P. M. Bell, and H. K. Mao 1978. Effects of compositional variation\non absorption spectra of lunar pyroxenes. In Proc. 9th Lunar Planet. Sci. Conf.,\n2919--2934\n\n\nHiroi, T., Zolensky, M. E., Pieters, C. M., 1996. Thermal metamorphism of the C, G, B, and F asteroids seen from the 0.7 micron, 3 micron and UV absorption strengths in comparison with carbonaceous chondrites. Meteoritics and Planet. Sci. 31, 321--327\n\nHiroi, T., Zolensky, M. E., Pieters, C.M., 2001. The Tagish Lake Meteorite: A Possible Sample from a D-Type Asteroid. Science 293, 2234--2236\n\nJones, T. D., Lebofsky, L. A., Lewis, J. S.,  Marley, M. S., 1990. \nThe composition and origin of the C, P, and D asteroids: Water\nas a tracer of thermal evolution in the outer belt. Icarus, 88, 172--192.\n\nKhare, B. N., Sagan, C., Arakawa, E. T., Suits, F., Callcott, T. A.,\nWilliams, M. W., 1984. Optical constants of organic tholins produced in a \nsimulated Titanian atmosphere - From soft X-ray to microwave frequencies. \nIcarus 60, 127--137.\n\nKhare, B.N., Thompson, W.R., Sagan, C., Arakawa, E. T., Meisse, C.,\nGilmour, I. 1991. Optical Constants of Kerogen from 0.15 to 40 micron:\nComparison with Meteoritic Organics. In Origin and Evolution\nof Interplanetary Dust, IAU Colloq. 126, eds. A.C. Levasseur-Regours and H.\nHasegawa, Kluwer Academic Publishers, ASSL 173, 99.\n\nKing, T.V.V., Clark, R.N., 1989. Spectral characteristics of chlorites and\nMgserpentines using high resolution reflectance spectroscopy. J. Geophys. Res. 94,\n13997--14008.\n\nLupisko, D. F., Belskaya, I. N., 1989. On the surface composition of the M-type asteroids. Icarus 78, 395--401.\n\n\nMagri, C. Nolan, M. C.; Ostro, S. J., Giorgini, J. D., 2007. A radar survey of\nmain-belt asteroids: Arecibo observations of 55 objects during 1999 2003. Icarus\n186, 126--151.\n\nMargot, J.-L., Brown, M.E., 2003. A low-density M-type asteroid in the mainbelt.\nScience 300, 1939--1942.\n\nMcDonald, G.D., Thompson, W.R., Heinrich, M., Khare, B.N., Sagan, C. 1994.\nChemical investigation of Titan and Triton tholins. Icarus 108, 137--145.\n\nMoroz, L. V., Hiroi, T., Shingareva, T. V., Basilevsky, A. T., Fisenko, A. V., Semjonova, L. F., Pieters, C. M., 2004.\nReflectance Spectra of CM2 Chondrite Mighei Irradiated with Pulsed Laser and Implications for Low-Albedo Asteroids and Martian Moons. Lunar Planet. Sci. Conf. 35, abstract 1279\n\n\nMoth\\'e-Diniz, T., Carvano, J. M., Lazzaro, D. 2003. Distribution of taxonomic classes in the main belt of asteroids. Icarus, 162, 10--21.\n\nOckert-Bell, M.E., Clark, B.E., Shepard, M.K., Rivkin, A.S., Binzel, R.P.,\nThomas, C.A., DeMeo, F.E., Bus, S.J., Shah, S., 2008. Observations of X\/M\nasteroids across multiple wavelengths. Icarus 195, 206--219.\n\nOckert-Bell, M.E., Clark, B.E., Shepard, M.K.,Isaacs, R.A, Cloutis, E.A., Fornasier, S., Bus, J.S., 2010.\nThe composition of M-type asteroids:synthesis of spectroscopic and radar observations. Icarus 210, 674--692.\n\nOstro, S. J., Campbell, D. B., Chandler, J. F., Hine, A. A., Hudson,\nR. S., Rosema, K. D., Shapiro, I. I., 1991. Asteroid 1986\nDA: Radar evidence for a metallic composition. Science, 252,\n1399--1404.\n\nOstro, S. J., Hudson, R. S., Nolan, M. C., Margot, J. L., Scheeres,\nD. J., Campbell, D. B., Magri, C., Giorgini, J. D., Yeomans,\nD. K., 2000. Radar observations of asteroid 216 Kleopatra.\nScience, 288, 836--839.\n\nPieters, C. 1983. Strength of mineral absorption features in the transmitted\ncomponent of near-infrared reflected light: First results from RELAB, J.\nGeophys. Res., 88, 9534--9544.\n\nRivkin, A. S., Howell, E. S., Britt, D. T., Lebofsky, L. A., Nolan,\nM. C., Branston, D. D., 1995. 3-micron spectrophotometric\nsurvey of M- and E-class asteroids. Icarus, 117, 90--100.\n\nRivkin, A.S., Howell, E.S., Lebofsky, L.A., Clark, B.E., Britt, D.T., 2000.\nThe nature of M-class asteroids from 3-\u03bcm observations. Icarus 145, 351--\n368.\n\nRivkin, A. S., Howell, E. S., Vilas, F., Lebofsky L. A., 2002.\nHydrated minerals on asteroids: The astronomical record. In\nAsteroids III (Bottke W. et al. editors), pp 235--253, Univ. of Arizona Press, Tucson.\n\nShepard, M. K., Clark, B. E., Nolan, M. C., Howell, E. S., Magri, C., Giorgini,\nJ. D., Benner, L. A. M., Ostro, S. J., Harris, A. W., Warner, B., et al., 2008.\nA radar survey of M- and X-class asteroids. Icarus 195, 184--205.\t\n\nShepard, M. K., Clark, B. E., Ockert-Bell, M., Nolan, M. C., Howell, E. S., Magri, C., Giorgini,\nJ. D., Benner, L. A. M., Ostro, S. J., Harris, A. W., Warner, B., Stephens, R. D., Mueller, M., 2010. A radar survey of M- and X-class asteroids II. Summary and synthesis. Icarus 208, 221--237. \n\nShingareva, T. V., Basilevsky, A. T., Fisenko, A. V., Semjonova, L. F., Korotaeva, N.N.., 2004. \nMineralogy and Petrology of Laser Irradiated Carbonaceous Chondrite Mighei. Lunar Planet. Sci. Conf. 35, abstract 1137\n\nTaylor, S. R., 1992.  Solar system evolution: a new perspective. an inquiry into the chemical composition, origin, and evolution of the solar system. In Solar System evolution: A new Perspective, Cambridge Univ.Press. \n\nTedesco, E.F., P.V. Noah, M. Moah, and S.D. Price, 2002. The supplemental IRAS\nminor planet survey. The Astronomical Journal 123, 10565--10585.\n\nTholen, D.J., 1984. Asteroid taxonomy from cluster analysis of photometry.\nPh.D. dissertation, University of Arizona, Tucson.\n\nTholen, D.J., Barucci, M.A., 1989. Asteroids taxonomy. In: Binzel, R.P.,\nGehrels, T., Matthews, M.S. (Eds.), Asteroids II. Univ. of Arizona Press,\nTucson, pp. 298--315. \n\nVernazza, P., Brunetto, R., Binzel, R. P., Perron, C., Fulvio, D., Strazzulla,\nG., Fulchignoni, M., 2009. Plausible parent bodies for enstatite chondrites and\nmesosiderites: Implications for Lutetia's fly-by. Icarus 202, 477-486\n\nVilas, F., Hatch, E.C., Larson, S.M., Sawyer, S.R., Gaffey,\nM.J., 1993. Ferric iron in primitive asteroids - A 0.43-$\\mu$m absorption feature. Icarus 102, 225--231\n\nVilas, F., Jarvis, K.S., Gaffey, M.J., 1994. Iron alteration minerals in the visible and near-infrared spectra of low-albedo asteroids. Icarus 109, 274--283 \n\nWarner, B. D., Harris, A. W., Vokrouhlick\u00fd, D., Nesvorn\u00fd, D., Bottke, W. F., 2009. Analysis of the Hungaria asteroid population.\nIcarus 204, 172--182\n\nZellner, B., M. Leake, J. G. Williams,  Morrison, D., 1977. The E asteroids \nand the origin of the enstatite achondrites. Geochim. Cosmochim. Acta 41, 1759--1767.\n\nZubko, V. G., Mennella, V., Colangeli, L., Bussoletti, E., 1996. Optical \nconstants of cosmic carbon analogue grains - I. Simulation of clustering \nby a modified continuous distribution of ellipsoids. MNRAS 282, 1321--1329.\n\n\n\\newpage\n\n\n{\\bf Tables}\n{\\scriptsize\n       \\begin{center}\n     \\begin{longtable} {|l|l|c|c|c|c|c|c|l|}  \n\\caption[]{Observational circumstances for the observed X type asteroids. Solar analog stars named \"hip\" come from the Hipparcos catalogue, \"la\" from the\nLandolt photometric standard stars catalogue, and \"HD\" from the Henry Draper\ncatalogue}. \n        \\label{tab1} \\\\\n\\hline \\multicolumn{1}{|c|} {\\textbf{Object    }} & \\multicolumn{1}{c|}\n{\\textbf{Night}} & \\multicolumn{1}{c|} {\\textbf{UT$_{start}$}} &\n\\multicolumn{1}{c|} {\\textbf{T$_{exp}$}} &     \\multicolumn{1}{c|}\n{\\textbf{Tel.}} & \\multicolumn{1}{c|} {\\textbf{Instr.}} & \\multicolumn{1}{c|}\n{\\textbf{Grism}} & \\multicolumn{1}{c|} {\\textbf{airm.}} & \\multicolumn{1}{c|}\n{\\textbf{Solar Analog (airm.)}} \\\\  \\hline \n  \n\\endfirsthead\n\\multicolumn{9}{c}%\n{{\\bfseries \\tablename\\ \\thetable{} -- continued from previous page}} \\\\ \\hline \n\\endfoot\n\\hline \\multicolumn{1}{|c|} {\\textbf{Object    }} & \\multicolumn{1}{c|}\n{\\textbf{Night}} & \\multicolumn{1}{c|} {\\textbf{UT$_{start}$}} &\n\\multicolumn{1}{c|} {\\textbf{T$_{exp}$}} &     \\multicolumn{1}{c|}\n{\\textbf{Tel.}} & \\multicolumn{1}{c|} {\\textbf{Instr.}} & \\multicolumn{1}{c|}\n{\\textbf{Grism}} & \\multicolumn{1}{c|} {\\textbf{airm.}} & \\multicolumn{1}{c|}\n{\\textbf{Solar Analog (airm.)}} \\\\  \\hline \n\\endhead\n\\hline \\multicolumn{9}{r}{{Continued on next page}} \\\\ \n\\endfoot\n\\hline \\hline\n\\endlastfoot\nObject  &  Night &  UT$_{start}$  & T$_{exp}$ & Tel. & Instr. & Grism & airm. & Solar Analog (airm.) \\\\\n & & (hh:mm) & (s) & & &   &   \\\\ \\hline\n50 Virginia & 18 Nov. 04 & 04:45 & 120 &  TNG & Dolores & LR-R & 1.09 & hyades64 (1.4) \\\\ \n50 Virginia & 18 Nov. 04 & 04:49 & 180 &  TNG & Dolores & MR-B & 1.10 & hyades64 (1.4) \\\\ \n50 Virginia & 19 Nov. 04 & 04:23 & 360 & TNG & NICS & Amici   & 1.07  & la98-978 (1.17) \\\\\n77 Frigga  & 29 Feb. 04 & 05:12 & 90 & TNG & Dolores & LR-R    & 1.33 & la107684 (1.20)   \\\\\n77 Frigga  & 29 Feb. 04 & 05:15 & 90 & TNG & Dolores & LR-B    & 1.34 & la107684 (1.20)   \\\\\n77 Frigga  & 1 Mar. 04 & 05:27 & 120 & TNG & NICS & Amici   & 1.38  & la102-1081 (1.46) \\\\\n92 Undina     &  20 Jan 07 & 04:29 & 40 & NTT & EMMI &\tGR1 & 1.69 & la98-978 (1.35) \\\\\n92 Undina     & 17 Sep. 05& 11:37 & 2560  & IRTF & SPEX &\tPrism &\t1.15 &\nhyades64 (1.00),la115-270(1.10), \\\\\n& & & & & & & & la93-101(1.10), la112-1333(1.10) \\\\\n184 Dejopeja  &13 Aug 05&06:13&   300& NTT & EMMI &\tGR1 &\t1.07   & HD1835 (1.07) \\\\        \t\n184 Dejopeja  &12 Aug 05 & 06:59 & 360 & NTT & SOFI & GB  &1.11  & hip103579  (1.06) \\\\\n184 Dejopeja  &12 Aug 05 & 07:04 & 360 & NTT & SOFI & GR  &1.12  & hip103579  (1.06) \\\\\n220 Stefania & 21 Nov. 04 & 00:13 & 180 & TNG & Dolores & LR-R  & 1.12 & hyades64 (1.05)  \\\\\n220 Stefania & 21 Nov. 04 & 00:17 & 180 & TNG & Dolores & MR-B  & 1.12 & hyades64 (1.05)  \\\\\n223 Rosa & 20 Nov. 04 & 20:58 & 360 & TNG & Dolores & LR-R  & 1.38 & la115-271 (1.16)      \\\\\n223 Rosa & 20 Nov. 04 & 21:05 & 420 & TNG & Dolores & MR-B  & 1.40 & la115-271 (1.16)   \\\\\n275 Sapientia &13 Aug 05&06:44\t&480& NTT & EMMI &\tGR1 &\t1.09  & HD1835 (1.07) \\\\    \n275 Sapientia & 12 Jul. 04 & 08:50 & 1650 &\tIRTF &\tSPEX &\tPrism &\t  1.30 &\n16CyB(1.20), la107-684(1.10)\\\\\n & & & & & & & &  la110-361(1.10) \\\\\n283 Emma & 16 Nov. 04 & 04:08 & 180 & TNG & Dolores & LR-R & 1.08 & hyades64 (1.03) \\\\\n283 Emma & 16 Nov. 04 & 04:13 &  180 & TNG & Dolores & MR-B & 1.09 & hyades64 (1.03) \\\\\n283 Emma  & 19 Nov. 04 & 01:45 & 240 & TNG & NICS & Amici   & 1.01  & Hyades64 (1.01) \\\\\n337 Devosa    &13 Aug 05&01:31\t&300& NTT & EMMI &\tGR1 &\t1.08 & HD144585 (1.22) \\\\         \t\n337 Devosa    &12 Aug 05 & 03:16 & 280 & NTT & SOFI & GB &1.38  & hip083805  (1.23) \\\\ \n417 Suevia    &13 Aug 05 &08:12\t&600& NTT & EMMI &\tGR1 &\t1.31  & HD1835 (1.07) \\\\       \t\n417 Suevia    &14 Aug 05 &06:04 &480& NTT & SOFI &      GB &   1.15  & la115-271 (1.15) \\\\\n417 Suevia    &14 Aug 05 &06:20 &720& NTT & SOFI &      GR &   1.16  & la115-271 (1.15) \\\\\t \t\n463 Lola & 20 Nov. 04 & 23:48 & 180 & TNG & Dolores & LR-R  & 1.10 & hyades64 (1.05)      \\\\\n463 Lola & 20 Nov. 04 & 23:52 & 240 & TNG & Dolores & LR-R  & 1.11 & hyades64 (1.05)      \\\\\n517 Edith & 16 Nov. 04 & 02:42 & 180 &  TNG & Dolores & LR-R & 1.01 & hyades64 (1.03) \\\\\n517 Edith & 16 Nov. 04 & 02:46 & 180 &  TNG & Dolores & MR-B & 1.01 & hyades64 (1.03) \\\\\n517 Edith &  19 Nov. 04 & 03:50 & 360 & TNG & NICS & Amici   & 1.02  &  Hyades64 (1.01) \\\\\n522 Helga     &14 Aug 05 &07:20 &600& NTT & SOFI &      GB &   1.13  & la115-271 (1.15) \\\\\n522 Helga     &14 Aug 05 &07:38 &900& NTT & SOFI &      GR &   1.12  & la115-271 (1.15) \\\\\n522 Helga     &  20 Jan 07 & 02:49 & 600 & NTT & EMMI &\tGR1 & 1.67 & Hyades64 (1.45) \\\\\n\n536 Merapi & 20 Nov. 04 & 22:23 & 150 & TNG & Dolores & LR-R  & 1.34 & la115-271 (1.28)      \\\\\n536 Merapi & 20 Nov. 04 & 22:27 & 150 & TNG & Dolores & MR-B  & 1.34 & la115-271 (1.28)      \\\\\n741 Botolpia  & 20 Jan 07 & 04:44 & 300 & NTT & EMMI &\tGR1 & 1.70 & la98-978 (1.35) \\\\\n758 Mancunia  &13 Aug 05 &06:05 &180 & NTT & EMMI &\tGR1 &\t1.03  & HD1835 (1.07) \\\\       \t\n758 Mancunia  &12 Aug 05 &06:40 &320 & NTT & SOFI &     GB &   1.05  & hip103572 (1.06) \\\\\n758 Mancunia  &12 Aug 05 &06:46 &320 & NTT & SOFI &     GR &   1.05  & hip103572 (1.06) \\\\\n909 Ulla       & 20 Nov. 04 & 04:41 & 180 & TNG & Dolores & LR-R  & 1.11 &  hyades64 (1.10)  \\\\\n909 Ulla       & 20 Nov. 04 & 04:45 & 240 & TNG & Dolores & MR-B  & 1.11 &  hyades64 (1.10)  \\\\\n909 Ulla       & 21 Nov. 04 & 05:31 & 300 & TNG & NICS & Amici   & 1.17  & la98-978 (1.17)  \\\\\n1122 Neith & 20 Nov. 04 & 03:28 & 120 & TNG & Dolores & LR-R  & 1.14 &  hyades64 (1.10)  \\\\\n1122 Neith  & 20 Nov. 04 & 03:31 & 120 & TNG & Dolores & MR-B  & 1.15 &  hyades64 (1.10)  \\\\\n1122 Neith  & 21 Nov. 04 & 04:19 & 160 & TNG & NICS & Amici   & 1.32  & Hyades64 (1.23) \\\\\n1124 Stroobantia  & 20 Nov. 04 & 03:50 & 300 & TNG & Dolores & LR-R  & 1.02 &  hyades64 (1.10)  \\\\\n1124 Stroobantia  & 20 Nov. 04 & 03:57 & 360 & TNG & Dolores & MR-B  & 1.02 &  hyades64 (1.10)  \\\\\n1124 Stroobantia  & 21 Nov. 04 & 04:40  & 480 & TNG & NICS & Amici   & 1.19  & la98-978 (1.19) \\\\ \n1146 Biarmia & 20 Nov. 04 & 20:10 & 300 & TNG & Dolores & LR-R  & 1.31 &  la115-271 (1.16)  \\\\\n1146 Biarmia & 20 Nov. 04 & 20:19 & 300 & TNG & Dolores & MR-B  & 1.34 &  la115-271 (1.16)  \\\\\n1328 Devota  & 20 Nov. 04 & 02:58 & 300 & TNG & Dolores & LR-R  & 1.01 &  hyades64 (1.02)  \\\\\n1328 Devota  & 20 Nov. 04 & 03:04 & 300 & TNG & Dolores & MR-B  & 1.02 &  hyades64 (1.02)  \\\\\n1328 Devota  & 21 Nov. 04 & 03:45 & 480 & TNG & NICS & Amici   & 1.03  & la98-978 (1.17) \\\\\n1355 Magoeba & 16 Nov. 04 & 04:28 & 300 & TNG & Dolores & LR-R & 1.30 & hyades64 (1.03) \\\\\n1355 Magoeba & 16 Nov. 04 & 04:34 &  360 & TNG & Dolores & MR-B & 1.31 & hyades64 (1.03) \\\\\n1355 Magoeba &  19 Nov. 04 & 03:31 & 480 & TNG & NICS & Amici   & 1.17  & Hyades64 (1.21) \\\\\n1902 Shaposhnikov & 14 Aug 05 & 08:03 & 480 & NTT & SOFI &    GB &   1.20  & hip113948 (1.17) \\\\ \n1902 Shaposhnikov &14 Aug 05 & 08:12 & 480 & NTT & SOFI &    GR &   1.25  & hip113948 (1.17) \\\\ \n3447  Burckhalter      & 20 Nov. 04 & 02:22 & 300 & TNG & Dolores & LR-R  & 1.42 &  hyades64 (1.02)  \\\\\n3447  Burckhalter      & 20 Nov. 04 & 02:29 & 300 & TNG & Dolores & MR-B  & 1.46 &  hyades64 (1.02)  \\\\ \n\\hline\n\\hline\n\\end{longtable}\n\\end{center}\n}\n\\begin{list}{}{}\n\\item \n\\end{list}\n\n\n\n       \\begin{sidewaystable}\n       \\caption{\nPhysical and orbital parameters of the  X-type asteroids\nobserved and selected on the basis of the Tholen taxonomy (Tholen, 1984). The Bus and Bus-DeMeo \nclassifications are also reported, together with the new Tholen-like classification proposed here, \nconsidering the available albedo values (Tedesco et al. 2002; the albedos marked by $^{a}$ come from Gil-Hutton et al. 2007) and the spectral behaviors. The spectral slopes values\nare also given (S$_{UV}$ calculated in the 0.49--0.55 $\\mu$m wavelength range,\nS$_{VIS}$ in the 0.55-0.8 $\\mu$m range, S$_{NIR1}$ in the 1.1-1.6 $\\mu$m range, S$_{NIR2}$ in the 1.7-2.4 $\\mu$m\nrange, S$_{cont}$ in the 0.43-2.40 $\\mu$m for asteroids with VIS and NIR spectra, \nand 0.43-0.92 $\\mu$m for those observed only in the visible range (they are marked with an $^{*}$).  }\n        \\label{slope}\n        \\scriptsize{\n\\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|c|c|c|} \\hline\n\\hline\nAsteroid & Bus &  Bus-DeMeo & NEW TX & albedo &  D (km)  & a\n(AU) & e &    i$^{o}$ & S$_{UV}$ & S$_{Vis}$ & S$_{NIR1}$ & S$_{NIR2}$ & $S_{cont}$\\\\  \\hline\n50 Virginia\t  & Ch &Ch  & C &0.04  &99.82   & 2.6518 & 0.2838 &2.8  & 1.57$\\pm$0.74  & -0.40$\\pm$0.54&  1.39$\\pm$0.84& -0.16$\\pm$0.77 &  1.05$\\pm$0.71 \\\\\n77 Frigga\t  & X  &X   & M &0.14  &69.25   & 2.6674 & 0.1328 &2.4  & 18.23$\\pm$1.32 &  10.16$\\pm$0.57&  2.85$\\pm$0.87&  1.08$\\pm$0.83 &  5.37$\\pm$0.78 \\\\\n92 Undina\t  & Xc &Xk  & M &0.25  &126.42  & 3.1895 & 0.1001 &9.9  &  6.56$\\pm$0.31 & 3.15$\\pm$0.54&  2.30$\\pm$0.76&  1.63$\\pm$0.73 &  2.46$\\pm$0.71 \\\\\n184 Dejopeja\t  & X  &X   & M &0.19  &66.47   & 3.1804 & 0.0763 &1.1  & 5.05$\\pm$1.02  & 3.98$\\pm$0.55&  2.72$\\pm$0.75&  1.38$\\pm$0.75 &  2.57$\\pm$0.71 \\\\\n220 Stephania\t  & C  &$-$ & C &0.07  &31.12   & 2.3482 & 0.2585 &7.5  & 2.14$\\pm$0.32  & 0.36$\\pm$0.55&  --&  -- &  0.74$\\pm$0.72$^{*}$ \\\\\n223 Rosa\t  & C  &$-$ & C &0.03  &87.61   & 3.0920 & 0.1238 &1.9  & 3.99$\\pm$0.61  & 1.76$\\pm$0.57&  --&  -- &  1.86$\\pm$0.73$^{*}$ \\\\\n275 Sapientia\t  &  C &C   & P &0.04   &103.00  & 2.7721 & 0.1615 &4.7  & 7.49$\\pm$0.69  & 4.11$\\pm$0.57&  2.21$\\pm$0.75&  2.13$\\pm$0.73 &  2.38$\\pm$0.72 \\\\\n283 Emma\t  & $-$ &C  & C &0.03  &148.06  & 3.0447 & 0.1499 &7.9  & 2.89$\\pm$0.77  & 1.87$\\pm$0.54&  0.38$\\pm$0.81&  0.24$\\pm$0.79 &  0.73$\\pm$0.72 \\\\\n337 Devosa\t  & X  &Xk  & M &0.16  &59.11   & 2.3835 & 0.1379 &7.8  & 5.57$\\pm$1.28  & 3.95$\\pm$0.57&  3.64$\\pm$0.77&  -- &  3.58$\\pm$0.02 \\\\\n417 Suevia\t  & Xk &Xk  & M &0.20  &40.69   & 2.8006 & 0.1331 &6.6  & 5.88$\\pm$2.33  & 3.31$\\pm$0.76&  5.94$\\pm$0.76&  2.89$\\pm$0.81 &  4.06$\\pm$0.72 \\\\\n463 Lola\t  & $-$ &$-$& P &0.08  &19.97   & 2.3988 & 0.2202 &13.4 & 4.50$\\pm$0.96  & 4.11$\\pm$0.55&  --&  -- &  4.08$\\pm$0.73$^{*}$ \\\\\n517 Edith\t  & C  &Xc  & C &0.04  &91.12   & 3.1558 & 0.1819 &3.1  & 2.61$\\pm$0.78  & 1.88$\\pm$0.54&  1.52$\\pm$0.81&  0.89$\\pm$0.81 &  1.59$\\pm$0.71 \\\\\n522 Helga\t  & X  &Xk  & P &0.04  &101.22  & 3.6287 & 0.0755 &4.4  & 5.64$\\pm$1.09  & 3.79$\\pm$0.55&  3.82$\\pm$0.78&  3.38$\\pm$0.76 &  3.59$\\pm$0.71 \\\\\n536 Merapi\t  & C  &$-$ & C &0.04  &151.42  & 3.5016 & 0.0819 &19.4 & 1.72$\\pm$0.85  & 1.16$\\pm$0.55&  --&  -- &  1.13$\\pm$0.72$^{*}$ \\\\\n741 Botolphia\t  & X  &$-$ & M &0.14  &29.64   & 2.7198 & 0.0678 &8.4  & 6.37$\\pm$0.88  & 3.42$\\pm$0.54&  --&  -- &  3.22$\\pm$0.74$^{*}$ \\\\\n758 Mancunia\t  & $-$ &X  & M &0.13  &85.48   & 3.1861 & 0.1518 &5.6  & 6.79$\\pm$1.05  & 4.06$\\pm$0.56&  2.53$\\pm$0.75&  1.36$\\pm$0.73 &  2.19$\\pm$0.71 \\\\\n909 Ulla\t  & $-$ &X  & P  &0.03  &116.44 & 3.5550 & 0.0959&18.8 &  5.12$\\pm$1.54  & 3.36$\\pm$0.61&  3.20$\\pm$0.89&  1.36$\\pm$0.85 &  3.64$\\pm$0.72 \\\\\n1122 Nieth\t  & $-$ &A  & A  & 0.45 & 12.01 & 2.6068 & 0.2562 &4.7  & 15.42$\\pm$0.58  & 8.03$\\pm$0.58& 15.31$\\pm$1.10&  6.18$\\pm$0.92 &  8.57$\\pm$0.76 \\\\\n1124 Stroobantia  & $-$ &Xk & M  &0.16  &24.65  & 2.9253 & 0.0344 &7.7 & 2.19$\\pm$1.23   & 2.01$\\pm$0.63&  4.73$\\pm$0.88&  3.29$\\pm$0.87 &  3.49$\\pm$0.73 \\\\\n1146 Biarmia\t  & X  &$-$ & M &0.22  &31.14   & 3.0511 & 0.2508 &17.0 & 6.08$\\pm$0.49  & 1.92$\\pm$0.56&  --&  -- &  1.78$\\pm$0.73$^{*}$ \\\\\n1328 Devota\t  & $-$ &D  & D &0.04  &57.11   & 3.5093 & 0.1407 &5.7   & 10.67$\\pm$0.96  & 11.90$\\pm$0.55& 11.70$\\pm$1.10&  7.06$\\pm$1.01 & 11.01$\\pm$0.72 \\\\\n1355 Magoeba\t  & Xe &Xe  & M & 0.27$^{a}$  &12.90   & 1.8535 & 0.0450 &22.8  & 13.42$\\pm$0.98  &  5.63$\\pm$0.55&  2.23$\\pm$0.81& -0.80$\\pm$0.88 &  3.61$\\pm$0.74 \\\\\n1902 Shaposhnikov  & $-$ &D  & P &0.03  &96.86  & 3.9727 & 0.2242 &12.5 & --  & --&  4.08$\\pm$0.77&  2.29$\\pm$0.77 &  -- \\\\\n3447 Burckhalter   & Xc &$-$ & E ? & 0.34$^{a}$  &15.60  & 1.9907 & 0.0285 &20.7 & 16.02$\\pm$1.32  & 4.66$\\pm$0.60&  --&  -- &  6.17$\\pm$0.75$^{*}$ \\\\ \\hline\n\\end{tabular}\n}\n\\end{sidewaystable}\n\n\n\n\n\n       \\begin{table}\n       \\begin{center}\n       \\caption{\nBand center, depth and width for the features detected in the\nasteroid spectra.}\n        \\label{band}\n\\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \\hline\n\\hline\nAsteroid & Band Center ($\\mu$m) & Depth (\\%)  & Width ($\\mu$m) \\\\ \\hline\n50 Virginia\t  &0.6920$\\pm$0.0070  &2.2  &0.540--0.839 \\\\\n50 Virginia\t  &0.4310$\\pm$0.0040  &1.1  &0.416--0.445 \\\\\n50 Virginia\t  &0.8660$\\pm$0.0080  &1.2  &0.823--0.911 \\\\\n92 Undina\t  &0.9050$\\pm$0.0080  &2.9  &0.742--1.027 \\\\\n92 Undina\t  &0.5103$\\pm$0.0050  &0.7  &0.487--0.531 \\\\\n283 Emma\t  &0.4310$\\pm$0.0040  &1.4  &0.417--0.443 \\\\\n337 Devosa\t  &0.8820$\\pm$0.0080  &2.8  &0.763--1.017\\\\\n337 Devosa\t  &0.4280$\\pm$0.0040  &3.0  &0.410--0.450 \\\\\n417 Suevia\t  &0.8593$\\pm$0.0100  &5.2  &0.730--1.007 \\\\\n517 Edith\t  &0.4300$\\pm$0.0040  &1.5  &0.416--0.444 \\\\\n522 Helga\t  &0.9400$\\pm$0.0070  &2.3  &0.874--1.004\\\\\n758 Mancunia      &1.5960$\\pm$0.0060  &1.3  &1.520-1.705 \\\\ \n1122 Nieth\t  &0.9650$\\pm$0.0080  &24.3 &0.756--1.564 \\\\\n1124 Stroobantia  &0.9170$\\pm$0.0100  &5.9  &0.807--1.100\\\\\n1355 Magoeba\t  &0.4910$\\pm$0.0070  &3.0  &0.443--0.543\\\\\n1355 Magoeba\t  &0.4300$\\pm$0.0040  &1.9  &0.417--0.445\\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}\n       \\begin{center}\n       \\caption{RELAB Matches (Exp means experimental).}\n        \\label{tab}\n         \\scriptsize{\n\\begin{tabular}{|l|c|c|c|c|l|l|} \\hline\nASTEROID & ALBEDO & BEST FIT & MET    & MET  & NAME & GRAIN SIZE \\\\\n          &       &          &  CLASS & REFL  &  & \\\\ \\hline\n50 Virginia & 0.04 & PH-D2M-032\t  & CM & 0.03 & MET00639    & $<$75$\\mu$m \\\\\n77 Frigga   & 0.14 & MR-MJG-082   & IM & 0.18 & Chulafinnee &  \\\\\n92 Undina   & 0.25 & MR-MJG-083   & IM & 0.23 & Babb's Mill & \\\\\n92 Undina   & 0.25 & MB-TXH-043-H & Pall&0.14 & Esquel      & $<$63 $\\mu$m \\\\\n184 Dejopeja& 0.19 & MB-TXH-043-H & Pall&0.14 & Esquel      & $<$63 $\\mu$m \\\\\n275 Sapientia& 0.04 & MB-TXH-064-C   & CM Exp. & 0.03 & Murchison heated to 600$^{\\circ}$C   & $<$63 $\\mu$m\\\\\n283 Emma    & 0.03 & PH-D2M-032   & CM & 0.03 & MET00639    & $<$75$\\mu$m\\\\\n337 Devosa  & 0.16 & MB-TXH-046   & IM & 0.16 & Landes      & slab \\\\\n417 Suevia  & 0.20 & MB-TXH-047-D & IM & 0.15 & DRP78007    & 75-125$\\mu$m \\\\\n517 Edith   & 0.04 & MB-TXH-064-D & CM-Exp& 0.03 & CM Murchison  heated to 700$^{\\circ}$C & 63--125$\\mu$m \\\\ \n522 Helga   & 0.04 & MA-ATB-068   & CM Exp& 0.05 & CM Migei Laser Irrad. & $<$45$\\mu$m \\\\ \n758 Mancunia& 0.13 & MB-TXH-046   & IM & 0.16 & Landes      & slab \\\\\n909 Ulla    & 0.03 & MA-ATB-068   & CM Exp& 0.05 & CM Migei Laser Irrad. & $<$45$\\mu$m \\\\ \n1124 Stroobantia& 0.16 & MB-TXH-047-A & IM & 0.12 & DRP78007 &  $<$25 \\\\\n\\hline \n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\n\n\\begin{table}\n       \\begin{center}\n       \\caption{Geographical mixing models. $^1$ blue line in Fig.~\\ref{modelli},\n$^2$ red line in Fig.~\\ref{modelli}}\n        \\label{models}\n         \\scriptsize{\n\\begin{tabular}{|l|c|l|c|c|} \\hline\nASTEROID & ALBEDO & GEOGRAPHICAL MODEL & FILE SOURCE & MODEL ALBEDO \\\\ \\hline\n92 Undina   & 0.25 &  99\\% pallasite Esquel & RELAB ckmb43 & 0.14 \\\\\n            &      &   1\\% orthopyroxene  & RELAB cbpe40 & \\\\ \n337 Devosa$^1$  & 0.16 &  98\\% pallasite Esquel & RELAB ckmb43 & 0.14 \\\\\n            &      &   2\\% goethite  & ASTER & \\\\\n337 Devosa$^2$  & 0.16 &  98\\% pallasite Esquel & RELAB ckmb43 & 0.14 \\\\\n             &      &   1\\% orthopyroxene  & RELAB cbpe40 & \\\\ \n417 Suevia  & 0.20 &   97\\% iron met. DRP78007 & RELAB cdmb47 & 0.16 \\\\\n            &      &   3\\% orthopyroxene  & RELAB cbpe40 & \\\\ \n517 Edith   & 0.04 & 100\\% amorphous carbon &  Zubko et al., 1996 & \\\\\n1124 Stroobantia$^1$ & 0.16 & 98\\% iron met. MET101A & RELAB c1sc99 & 0.13 \\\\\n            &     & 2\\% orthopyroxene & RELAB  cbpe40 & \\\\\n1124 Stroobantia$^2$ & 0.16 & 96\\% iron met. DRP78007 & RELAB cdmb47  & 0.16 \\\\\n            &     & 4\\% olivine & USGS & \\\\            \n1328 Devota & 0.04 & 94\\% carb. chond. Tagish Lake & RELAB c8mt11 & 0.03 \\\\\n            &      & 6\\% Trit. Tholin & McDonald et al. 1994  & \\\\ \\hline\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\n  \\begin{table}\n       \\begin{center}\n       \\caption{Mean spectral slopes, for the different classes of asteroids with the associated standard deviation value. We discard from the following analysis a few asteroids initially classified as\nE\/M\/X in the Tholen taxonomy but whose spectral properties diverge completely\nfrom these classes and who have been since  reclassified as A, S or D-type\n(A: 1122, 2577, and 7579; S\/Sq: 516, and 5806; D: 1328).}\n        \\label{slopes}\n\\begin{tabular}{|l|l|l|l|} \\hline\n\\hline\nAsteroid Class & Mean S$_{VIS}$ & Mean S$_{NIR1}$& Mean S$_{NIR2}$ \\\\ \n               & (\\%\/($10^{3}$ \\AA) & (\\%\/($10^{3}$ \\AA) & (\\%\/($10^{3}$ \\AA) \\\\ \\hline\nE (all)                       & 3.92$\\pm$1.73  & -- & --\\\\\nE[I]                          & 4.60$\\pm$1.18  & -- & -- \\\\\nE[II]                         & 5.15$\\pm$1.31  & 0.85$\\pm$0.30 &  0.32$\\pm$0.15 \\\\\nE[III]                        & 2.43$\\pm$1.12  & 1.07$\\pm$0.81 &  1.09$\\pm$0.87    \\\\\nM with the 0.9 $\\mu$m band    & 3.54$\\pm$0.90  & 3.14$\\pm$1.51 &  1.82$\\pm$1.42   \\\\\nM without the 0.9 $\\mu$m band & 4.00$\\pm$1.97  & 1.73$\\pm$2.07 &  0.81$\\pm$1.21 \\\\\nP                             & 3.34$\\pm$1.69  & 2.66$\\pm$1.65  & 1.83$\\pm$1.25 \\\\\nC                             & 1.77$\\pm$1.94  & 0.48$\\pm$0.68  & 0.17$\\pm$0.35     \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n  \\begin{table}\n       \\begin{center}\n       \\caption{The Tholen (existing or proposed by this work) and the Bus--DeMeo classification for all the asteroids observed withing our spectroscopical survey of the X complex. 1 = this work, 2 = Fornasier et al. 2010, 3 = Fornasier et al. 2008 and reference therein.}\n        \\label{busDeMeotx}\n        \\scriptsize{\n\\begin{tabular}{|l|l|l|l|l|l|l|l|} \\hline\nAsteroid & Tholen & Bus-DeMeo & reference & Asteroid & Tholen & Bus-DeMeo & reference\\\\\n16   & M  & Xk & 2       &       498  & C & Xc & 2          \\\\\n22   & M  & Xk & 2       &   \t 504  & E (I) & -- & 3      \\\\\n44   & E (III) & Xk & 3  &\t 516  & S & Sq & 2          \\\\\n50   & C & Ch & 1        & \t 517  & C & Xc & 1          \\\\\n55   & M  & Xk & 2       &\t 522  & P  & Xk & 1         \\\\\n64   & E (II) & Xe & 3   &\t 536  & C  & C & 1         \\\\\n69   & M  & Xk & 2       &\t 558  & M  & Xk & 2        \\\\\n77   & M  & X  & 1       &\t 620  & E (III) & Xk & 3   \\\\\n92   & M  & Xk & 1       &\t 741  & M  & X  & 1        \\\\\n97   & M  & Xc & 2       &\t 755  & M  & Xk & 2        \\\\\n110  & M  & Xk & 2       &\t 758  & M  & X  & 1        \\\\\n125  & M  & -- & 2\t &\t 785  & M  & X  & 2        \\\\\n129  & M  & Xc & 2\t &\t 849  & M &  D  & 2        \\\\\n132 & E (II) & Xe & 2 \t &\t 860  & M  & X  & 2        \\\\\n135  & M  & Xk & 2\t &\t 872  & M  & Xk & 2        \\\\\n161  & M & Xc & 2\t &\t 909  & P  & X  & 1          \\\\\n184  & M  & X  & 1\t &\t 1025 & E (I) & -- & 3       \\\\\n201  & M  & X  & 2\t &\t 1103 & E (III) & Xk & 3     \\\\\n214  & E (III) & Xk & 3\t &\t 1122 & A & A & 1            \\\\\n216  & M  & Xk & 2\t &\t 1124 & M  & Xk & 1          \\\\\n220  & C  & C & 1        &\t 1146 & M  & X  & 1         \\\\\n223  & C  & C & 1\t &\t 1251 & E (III) & Xk & 3    \\\\\n224  & M  & Xc & 2\t &\t 1328 & D & D & 1           \\\\\n250  & M  & Xk & 2\t &\t 1355 & M  & Xe & 1     \\\\\n275  & P  & C  & 1\t &\t 1902 & P      & D & 1      \\\\\n283  & C  & C  & 1\t &\t 2035 & E (II) & Xe & 3     \\\\\n317  & E (III) & Xk & 3\t &\t 2048 & E (II) & Xe & 3     \\\\\n325  & M  & X  & 2\t &\t 2449 & E (I) & -- & 3      \\\\\n337  & M  & Xk & 1\t &\t 2577 &  A & A &3           \\\\\n338  & M  & Xk & 2\t &\t 2867 & E (II) & Xe & 3     \\\\\n347  & M  & Xk & 2       &\t 3050 & E (III) & Xk & 3     \\\\\n369  & M  & Xk & 2\t &\t 3103 & E (II) & Xe & 3      \\\\\n382  & M  & -- & 2\t &\t 3447 & E (I) ?  & Xc & 1      \\\\\n417  & M  & Xk & 1\t &\t 4660 & E (II) & Xe & 3      \\\\\n418  & M  & X  & 2\t &\t 5806 & S      & S & 3       \\\\\n434  & E (II) & Xe & 3\t &\t 6435 & E (I) & -- & 3      \\\\\n437  & E (III) & Xk & 3\t &\t 6911 & E (II) & Xe & 3     \\\\\n441  & M  & X  & 2\t &\t 7579 & A      & A & 3      \\\\\n463  & P  & -- & 1\t &\t 144898 & E (I) & -- & 3    \\\\ \\hline\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\n\n\\newpage\n\n{\\bf Figure captions}\n\n\\vspace{1cm}\n\nFig. 1 - Visible and near infrared spectra  of X-type asteroids.\n\nFig. 2 - Visible and near infrared spectra of X-type asteroids.\n\nFig. 3 - Visible and near infrared spectra of X-type asteroids.\n\nFig. 4 - Visible and near infrared spectra of X-type asteroids.\n\nFig. 5 - Best spectral matches between the observed medium albedo \nasteroids and meteorites from the RELAB database \n(see Table~\\ref{tab} for details of the meteorite samples). \nAll these asteroids are re-classified as M-type \nfollowing the Tholen classification scheme. For 92 Undina \n2 meteorites are proposed: in red the IM Babb's Mill \nand in blue the IM Esquel.\n\nFig. 6 - Best spectral matches between the observed low-albedo asteroids and meteorites from the RELAB database (see Table~\\ref{tab} for details of the meteorite samples). \n\nFig. 7 - Geographical mixing model for the asteroids 92 Undina, 337 Devosa, 417 Suevia, 517 Edith, 1124 Stroobantia, and 1328 Devota (see details of each model in Table~\\ref{models}). \n\nFig. 8 - Spectral slope value ($S_{VIS}$) versus the semimajor axis for the different asteroids observed, classified following the Tholen taxonomy. The size of\nthe symbols is proportional to the asteroids' diameter.\n\nFig. 9 - Albedo versus semimajor axis for the different asteroids observed, classified following the Bus-DeMeo taxonomy. The size of the symbols is proportional to the asteroids' diameter.\n\n\n\\newpage\n\n\n{\\bf Figures}\n\n\\begin{figure*}[b]\n\\includegraphics[width=14cm]{I11726_f1.eps}\n\\caption{}\n\\label{fig1}\n\\end{figure*}\n\\begin{figure*}\n\\includegraphics[width=14cm]{I11726_f2.eps}\n\\caption{}\n\\label{fig2}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=14cm]{I11726_f3.eps}\n\\caption{}\n\\label{fig3}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=14cm]{I11726_f4.eps}\n\\caption{}\n\\label{fig4}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=14cm]{I11726_f5.eps}\n \\caption{}\n   \\label{fig5met}\n \\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=14cm]{I11726_f6.eps}\n \\caption{}\n   \\label{fig6met}\n \\end{figure*}\n\n\n\\begin{figure*}\n\\includegraphics[width=14cm]{I11726_f7.eps}\n \\caption{}\n   \\label{modelli}\n \\end{figure*}\n\n\n\\begin{figure*}\n\\includegraphics[width=14cm]{I11726_f8.ps}\n\\caption{}\n\\label{slopea}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\includegraphics[width=14cm]{I11726_f9.ps}\n\\caption{}\n\\label{busalbedo}\n\\end{figure*}\n\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nKitaev quantum spin liquids have an exactly solvable ground state in which the excitations are Majorana fermions \\cite{Kitaev2006,Jackeli2009}, and the $J_{\\textrm{eff}}$ = $\\frac{1}{2}$ layered honeycomb magnet $\\alpha$-RuCl$_3$ has emerged as one of the prominent material candidates\\cite{Plumb2014,Trebst2017,Takagi2019}. In the absence of an applied magnetic field, the ground state has zig-zag antiferromagnetic order ($T_\\textrm{N}$ $\\approx$ 7 K), but upon applying an in-plane magnetic field the ordered state is suppressed ($H_{\\text{C2}} \\approx$ 7 T) and a quantum paramagnetic regime is accessed \\cite{Johnson2015,Kubota2015,Sears2015}. In this field-induced state, a thermal Hall conductivity close to the half-quantized value was reported and discussed as a response of a topological edge state of Majorana fermions \\cite{Kasahara2018a,Nasu2017}. Although that result and interpretation are the subject of ongoing debate \\cite{Kasahara2018a,Yamashita2020,Yokoi2021,Yamashita2020,Czajka2021a,Lefrancois2021,Bruin2022,Czajka2022}, the field-induced state clearly exhibits exotic behavior.\n\nAnother unusual property of the field-induced state was reported recently by Czajka \\textit{et al.}\\cite{Czajka2021a}, who observed a sequence of minima in the magnetic field dependence of thermal conductivity $\\kappa$ and interpreted them as quantum oscillations originating from a Fermi surface of neutral quasiparticles. The amplitudes of these oscillations were shown to grow upon cooling down to $\\sim$1 K, in line with the expected Lifshitz-Kosevich behavior. The present authors, however, have pointed out that two of the reported oscillation minima coincide with known magnetic phase transitions, the sequential collapse of zig-zag order at  $H_{\\text{C1}}$ and $H_{\\text{C2}}$, and that amplitudes of these oscillations collapse upon cooling below 1 K\\cite{Bruin2022}. These observations contradict expectation for conventional quantum oscillations. The origin of the oscillatory features remains to be examined carefully by more comprehensive experiments over a wide range of temperatures and fields. We note that, while the measurements by Czajka \\textit{et al.} were conducted using single crystals grown by a chemical vapor transport (CVT) method, the measurements by the present authors were done on a Bridgman-grown sample (the same batch of single crystals as those used in the original report of half-quantization \\cite{Kasahara2018a}). The possibility of a sample dependence must also be considered when addressing the controversial issue of oscillations.\n\nIn this study we focus on the question of the origin of the oscillatory structures in the magnetic field dependence of thermal conductivity $\\kappa(H)$ of $\\alpha$-RuCl$_3$.  We present detailed measurements of magnetothermal conductivity along the $a$-axis and magnetization with the in-plane field oriented along either the crystal $a$ or $b$ axes, down to the low temperature of 100 mK for single crystals grown by both the Bridgman and the CVT methods. All the experimental results point to magnetic phase transitions as the origin of the oscillatory structures, independent of the crystal growth method.\n\n\\section{Methods}\n\nTwo single crystal pieces grown by a CVT- and a Bridgman technique were measured. The CVT-grown crystal was sourced from the same batch as reported in Ref.~\\onlinecite{Suzuki2021}. The Bridgman-grown crystal was the same as reported in Ref.~\\onlinecite{Bruin2022}. They were both plate-like and similar in size, with approximate dimensions 2 mm x 1 mm x 0.02 mm (length x width x thickness). The crystal structures and their orientations were determined by single-crystal x-ray diffraction at room temperature using a SMART-APEX-II CCD X-ray diffractometer (Bruker AXS, Karlsruhe, Germany) with graphite monochromated MoK$\\alpha$ radiation. The Bridgman sample was best fitted to a C2\/m monoclinic unit cell with lattice parameters a = 6.041(4) \\AA, b = 10.416(8) \\AA, c = 6.088(4) \\AA, $\\beta$ = 108.54(2)$^{\\circ}$, and the CVT sample was determined to be trigonal P3$_1$12 with lattice parameters a = b = 6.012(3) \\AA, c = 17.27(1) \\AA. Lattice parameters were refined with the SAINT subprogram in the Bruker Suite software package\\cite{Bruker2019}, using 448 and 776 reflections, respectively. Despite the differences in the room temperature structures of these two samples, we will see that their low temperature conductivities and magnetic properties are similar to each other.\n\nThermal conductivity was measured using a one heater- two thermometer technique on a dilution refrigerator. RuO$_2$ chip thermometers, calibrated in-situ against a field-calibrated RuO$_2$ reference, were used to determine the temperature gradients. To measure the magnetic field dependence of thermal conductivity, the sample was allowed to thermalize fully at each field before every measurement so that true steady state conditions were achieved. Complementary continuous field sweeps were performed in some cases to examine the detailed field dependence, and these data were cross-checked against steady-state field sweeps to verify consistency. All measurements of $\\kappa$ were performed with heat applied along the crystal $a$-axis. The Bridgman sample $\\kappa$ data with $H \\parallel a$ plotted in Figs. \\ref{fig1}b,c and \\ref{fig2}b,d are reproduced from Ref.~\\onlinecite{Bruin2022}. For the CVT crystal, magnetization data before and after mechanical bending are presented, and all thermal conductivity data for that crystal were measured before it was bent.\n\nThe crystal $a$ and $b$ axes are defined as being perpendicular and parallel to the Ru-Ru bond directions, respectively. In the case of the monoclinic C2\/m structure of the Bridgman crystal, the $a$-axis is uniquely determined as the direction of the layer-to-layer Ru honeycomb shift. In a monoclinic unit cell, the honeycomb axes perpendicular and parallel to Ru-Ru bonds other than the $a$ and $b$ axes are distinct, and we denote these as $a$*,$a$** and $b$*,$b$**, respectively (Fig.~\\ref{fig4}a). In the case of the trigonal CVT crystal, $a$ was defined as the direction of the longest sample dimension. Magnetization measurements were performed on both samples using a Physical Properties Measurement System vibrating sample magnetometer option (Quantum Design, USA) along all 6 named crystal axes.\n\n\\section{Results and discussion}\n\nThe magnetothermal conductivity displays rich structures in both the Bridgman and the CVT crystals, for magnetic field applied along both $a$ and $b$ axes as shown in Figs.~\\ref{fig1}a, \\ref{fig1}b and \\ref{fig1}d. In all the studied crystals and field configurations, the observed minima or kinks in $\\kappa(H)$ coincide with the magnetic fields where the field derivative of magnetization d$M$\/d$H$ has a peak or shoulder, including at the magnetic phase transitions associated with the collapse of zig-zag order at $H_{\\text{C1}}$ and $H_{\\text{C2}}$, which already suggests a link between magnetic transitions and the structures in $\\kappa(H)$. In Fig.~\\ref{fig1}c, the magnetothermal conductivity with $H \\parallel a$ at 2 K for the Bridgman and the CVT crystals are compared together with the data from Ref.~\\onlinecite{Czajka2021a} at 1.75 K. The absolute magnitude of the conductivity of our CVT-grown sample, as well as the sample in Ref.~\\onlinecite{Czajka2021a}, is lower than that of the Bridgman-grown sample, which suggests more disorder in the former. However, the structures are more prominent for the CVT-grown samples, so the degree of disorder correlates with the magnitude of oscillatory features, which goes against expectation for canonical quantum oscillations in metals; the cleaner the sample is, the larger the amplitude of oscillations. The positions of minima occur at nearly the same magnetic fields for all three crystals: at $H_{\\text{1}}\\sim$1.5 T, $H_{\\text{2}}\\sim$4.2 T, $H_{\\text{3}}\\sim$4.8 T, $H_{\\text{C1}}\\sim$5.8 T, $H_{\\text{C2}}\\sim$7 T, $H_{\\text{4}}\\sim$8.3 T, $H_{\\text{5}}\\sim$9 T and $H_{\\text{6}}\\sim$10.5 T (Figs.~\\ref{fig1}a,b), except that the distinction between $H_{\\text{4}}$ and $H_{\\text{5}}$ is unresolved for the Bridgman sample. For the Bridgman sample, the features are clearly less prominent in both $\\kappa(H)$ and d$M$\/d$H$ than in the CVT sample, in particular $H_{\\text{2}}$ and $H_{\\text{3}}$ are very weak in $\\kappa(H)$ and not resolved in d$M$\/d$H$. For magnetic field applied along the $b$-axis we resolve fewer features, and their positions are different than for $H \\parallel a$ (Fig.~\\ref{fig1}d), but the coincidence between structures in $\\kappa(H)$ and maxima or shoulders in d$M$\/d$H$ persists.\n\n\\begin{figure*}\n\\includegraphics[width=0.8\\textwidth, angle=0]{Figure1.jpg}\n\\caption{\\label{fig1} (a) Isotherms of thermal conductivity $\\kappa$ (red) and d$M$\/d$H$ (black) with $H \\parallel a$ at $T$=2 K for the same CVT-grown sample. Open and closed black markers denote d$M$\/d$H$ before and after the sample was lightly bent with tweezers, respectively. Several prominent peaks are enhanced by bending. Dotted lines indicate the locations of minima or kinks in $\\kappa(H)$ and arrows indicate the coincident peaks or shoulder features in d$M$\/d$H$. (b) Isotherms of thermal conductivity $\\kappa$ (blue) and d$M$\/d$H$ (black) with $H \\parallel a$ at $T$=2 K for the same Bridgman-grown sample. Dotted lines indicate the same fields as in panel a. Underlying $\\kappa(H)$ data were reproduced from Ref.~\\onlinecite{Bruin2022} (c) Isotherms of thermal conductivity around 2 K with $H \\parallel a$, comparing the Bridgman (blue) and CVT-grown (red) samples, as well as the result from Ref.~\\onlinecite{Czajka2021a} (green). All principal features are reproduced, and the similarity of the two CVT-grown samples is especially striking. (d) Isotherms of thermal conductivity $\\kappa$ (blue) and d$M$\/d$H$ (black) with $H \\parallel b$ for the Bridgman sample. Dotted lines indicate the locations of minima or kinks in $\\kappa(H)$ and arrows indicate the coincident peaks or shoulder features in d$M$\/d$H$.}\n\\end{figure*}\n\nFor both the Bridgman and the CVT crystals, the amplitudes of the oscillatory structures initially grow upon cooling, but then turn down sharply at lowest temperatures below 1 K. We demonstrate this by first subtracting a background (non-oscillatory) magnetic field dependence $\\kappa_{\\text{bg}}$ at each temperature by fitting a spline through the locations of greatest field derivative in $\\kappa$. By plotting the isotherms of the normalized oscillatory content $(\\kappa-\\kappa_{\\text{bg}})\/\\kappa_{\\text{bg}}$ for $H \\parallel a$ in Figs.~\\ref{fig2}a and \\ref{fig2}b, it is apparent that the amplitudes do not increase monotonically upon cooling, but instead decrease for lowest temperature, especially those of the highest field features. The temperature dependences of the amplitudes at the minima fields (Figs.~\\ref{fig2}c and \\ref{fig2}d) show this turnover between 500 mK - 1 K, which is observed in both samples. We note that the present data are consistent with those reported by Czajka \\textit{et al.} \\cite{Czajka2021a} which were taken above $\\sim$1 K, and we only observe the decrease by further lowering temperature. Our observation speaks against Lifshitz-Kosevich behavior and a quantum oscillation interpretation.\n\n\\begin{figure*}\n\\includegraphics[width=0.8\\textwidth, angle=0]{Figure2.jpg}\n\\caption{\\label{fig2} High-field oscillatory component of $\\kappa(H)$ expressed as $(\\kappa-\\kappa_{\\text{bg}})\/\\kappa_{\\text{bg}}$. (a,b) Field dependences for the CVT and Bridgman-grown samples for temperatures ranging from 0.1 K (purple) to 2.5 K (red). Vertical arrows indicate the well-defined minima at $H_{\\text{C1}}$ = 5.79 T (green) and $H_{\\text{C2}}$ = 7.05 T (orange) as well as two higher field minima arising from transitions at $H_{\\text{4\/5}}$ $\\approx$ 8.52 T and $H_{\\text{6}}$ = 10.47 T (red, purple), which all coincide with peaks or shoulders in d$M$\/d$H$. Whilst the structure at $H_{\\text{4\/5}}$ appears to consist of two minima in the CVT sample, only one broad minimum can be identified in the Bridgman sample. Underlying $\\kappa(H)$ data for the Bridgman sample were reproduced from Ref.~\\onlinecite{Bruin2022}. (c,d) Temperature dependences of the amplitudes of the minima in $(\\kappa-\\kappa_{\\text{bg}})\/\\kappa_{\\text{bg}}$ for the CVT and Bridgman samples for all fields where minima can be identified. Upon cooling from 2.5 K to $\\sim$1 K, the amplitudes grow, but they rapidly collapse upon further cooling below 1 K.}\n\\end{figure*}\n\nWhat is then the origin of the structures in the magnetothermal conductivity? We previously argued\\cite{Bruin2022} that the high-field features observed in the Bridgman sample may be due to magnetic phase transitions or crossovers, as two minima in $\\kappa(H)$ coincide with the magnetic phase transitions at $H_{\\text{C1}}$ and $H_{\\text{C2}}$. The minima would then be caused by soft magnetic scattering of phonons, which dominate the thermal conductivity in $\\alpha$-RuCl$_3$ \\cite{Hentrich2020}. We argue that the high-field transitions at $H_{\\text{4}}$ and $H_{\\text{5}}$ are connected to weak magnetic transitions which have higher transition temperatures than the 7 K main phase. In zero field, signatures of transitions around 13 K \\cite{Johnson2015,He2018,Mi2021} and 10 K \\cite{Sears2015,Kubota2015,He2018,Mi2021} were reported previously in the magnetic susceptibility, the specific heat and the dielectric constant, which were discussed to originate from secondary phases associated with stacking faults. As shown in Figs.~\\ref{fig4}a and \\ref{fig4}b, the 7 K, 10 K and 13 K transitions can be identified as sequential kink-like structures in the temperature dependence of magnetization at a low magnetic field of 1 T. \n\n\\begin{figure*}\n\\includegraphics[width=1\\textwidth, angle=0]{Figure3.jpg}\n\\caption{\\label{fig3} Tracking phase transitions in features of the temperature and field derivatives of magnetization of both the CVT and Bridgman samples. (a-d) d$M$\/d$T$ at constant magnetic fields between 0.1 T and 10 T for $H \\parallel a$ and $H \\parallel b$. Colored triangles indicate the features tracking the 7 K (blue), 10 K (red) and 13 K (brown) transitions. Peaks of unknown origin are marked by black triangles. Dashed lines (Bridgman sample) show additional data taken at half-tesla intervals. (e-h) d$M$\/d$H$ isotherms between 2 K and 10 K for $H \\parallel a$ and $H \\parallel b$ (shifted vertically for clarity). Blue triangles indicate the features tracking the main phase transitions at $H_{\\text{C1}}$ and $H_{\\text{C2}}$, additional colored triangles point at secondary features.}\n\\end{figure*}\n\nThe magnetic transitions in $M(T)$ can be identified more clearly as peaks in the temperature derivative d$M$\/d$T$ as shown in Figs.~\\ref{fig3}a-d, from which we can trace the magnetic field dependence of the phase boundaries. For the 7 K transition (dark blue triangles), the peak in d$M$\/d$T$ becomes hard to define for fields above 6 T due to the rapid decrease of the transition temperature. Instead, well-defined peaks in d$M$\/d$H$ appear at low temperature. They can be traced up to about 7 K (Figs.~\\ref{fig3}e-h) and overlap with the transition temperature determined by the anomalies in d$M$\/d$T$. Taken together, the peaks in d$M$\/d$T$ and d$M$\/d$H$ can be used to construct the entire phase diagram for a given direction of the in-plane magnetic field, including for features independent of the 7 K transition, as is shown in Figs.~\\ref{fig4}c-f. We find that for both $H \\parallel a$ and $H \\parallel b$ the 10 K transition connects to $H_{\\text{4}}$ for the CVT sample and the 13 K transition connects to $H_{\\text{4\/5}}$ for the Bridgman sample. The structures at the 13 K transition for the CVT sample and the 10 K transition for the Bridgman sample are difficult to track below $\\sim$10 K but appear to connect to the $H_{\\text{4\/5}}$ features, although detailed behavior is complex and not fully traceable.\n\n\\begin{figure*}\n\\includegraphics[width=1\\textwidth, angle=0]{Figure4.jpg}\n\\caption{\\label{fig4} (a,b) Temperature dependence of $M\/H$ at $B=1\\textrm{T}$ for the CVT and Bridgman crystals, respectively. The six different magnetic field orientations, separated by $30^{\\circ}$ rotations in the honeycomb plane are sketched in the inset of (a). Kink-like features at the three phase transitions at 7 K, 10 K and 13 K are indicated with arrows. (c-f) Phase diagrams for the CVT and Bridgman crystals with magnetic field along the $a$ and $b$-axes, constructed from the extracted maxima in d$M$\/d$H$ (horizontal error bars) and d$M$\/d$T$ (vertical error bars) (see: Fig.~\\ref{fig3}). The sizes of the error bars represent the widths of the measured features. Solid black lines at low temperature trace the measured locations of minima in $\\kappa(H)$ (see: Fig.~\\ref{fig1}). Dashed lines are guides to the eye tracing possible phase transition lines. (g,h) $M\/H$ at $T=2\\textrm{K}$ as a function of in-plane-field angle $\\phi$ at different magnetic fields, showing the gradual suppression of two-fold modulation upon increasing field.}\n\\end{figure*}\n\nWe conclude that the dip structures in $\\kappa(H)$ at $H_{\\text{4}}$ and $H_{\\text{5}}$ indicated by the black lines at low temperatures in Figs.~\\ref{fig4}c-f coincide reasonably with the phase boundaries connecting to the 10 K and 13 K transitions in zero field, just like the connection between $H_{\\text{C1}}$ and $H_{\\text{C2}}$ and the main 7 K transition, and therefore originate from the high temperature magnetic transitions. The 13 K and 10 K transitions have been discussed to originate from secondary phases closely related to stacking faults sensitive to mechanical deformation \\cite{Cao2016,He2018,Mi2021,Banerjee2016}. The crystals used in this study have a soft, foil-like morphology and are easily deformed. Mechanical bending of the CVT crystal indeed enhances the structure at $H_{\\text{4}}$ and $H_{\\text{5}}$ in d$M$\/d$H$ as shown in Fig.~\\ref{fig1}a, which can be ascribed to the increase of volume of the secondary 10 K and 13 K phases by bending and support these phases as the origin of structure in $\\kappa(H)$ and $M(H)$.\n\n\\begin{figure*}\n\\includegraphics[width=0.8\\textwidth, angle=0]{Figure5.jpg}\n\\caption{\\label{fig5} Locations of minima in $\\kappa(H)$ of the CVT sample at $H_{\\text{C1}}$, $H_{\\text{C2}}$ and $H_{\\text{6}}$ on the $H$-$T$ plane. Background colors indicate regions of negative d$\\kappa(H)$\/d$H$ (pink) and positive d$\\kappa(H)$\/d$H$ (blue) which define the locations of the minima in $\\kappa(H)$ at their boundary. Field scales are expanded to emphasize the convex nature and inverse melting displayed by the transition lines of $H_{\\text{C2}}$ and $H_{\\text{6}}$.}\n\\end{figure*}\n\nIn Fig.~\\ref{fig1}a, we see a bending-induced enhancement of the structures at $H_{\\text{3}}$ and $H_{\\text{6}}$ similar to those at $H_{\\text{4}}$ and $H_{\\text{5}}$. It strongly suggests that the structures at $H_{\\text{3}}$ and $H_{\\text{6}}$ are also linked to the phase transitions of bending-induced secondary phases. As the features in d$M$\/d$H$ at $H_{\\text{3}}$ and $H_{\\text{6}}$ are weakly resolved at 2 K and rapidly disappear upon heating above 4 K (Figs.~\\ref{fig3}e,f), we cannot trace them to corresponding phase transitions at low fields. Instead, we find that additional evidence for the phase transition origin of all the high-field features above $H_{\\text{C1}}$ comes from the small but appreciable temperature dependence of positions of high-field dips in the magnetothermal conductivity. These are not expected for canonical (single frequency) quantum oscillations and support magnetic phase transitions as the origin of the structures in $\\kappa(H)$. The phase transition lines of $H_{\\text{C1}}$, $H_{\\text{C2}}$ and $H_{\\text{6}}$ are traced Fig.~\\ref{fig5}. We also observed temperature dependences for $H_{\\text{4}}$ and $H_{\\text{5}}$ (see: Figs.~\\ref{fig2}a and \\ref{fig2}b) but due to the flatness of the associated minimum in $\\kappa(H)$ the assignment of exact fields to those two features is less certain. The transition line of $H_{\\text{C2}}$ (7 K main phase) as defined by minima in $\\kappa(H)$ has a convex shape below 2.5 K with a field width of $\\sim$70 mT, which agrees very well with the shape of that phase transition determined by a recent measurement of the Gr\\\"uneisen parameter, where it was discussed as an \\textit{inverse melting} due to the higher entropy of the low-field ordered phase at very low temperatures \\cite{Bachus2021}. The similarity in behavior of the features at $H_{\\text{C2}}$ and $H_{\\text{6}}$ confirms that the latter is also a thermodynamic phase transition with an appreciable entropy effect even below 1 K. Suetsugu \\textit{et al.}\\cite{Suetsugu2022} recently reported a rapid increase in $\\kappa(H)$ around 11 T, where we observe the $H_{\\text{6}}$ anomaly, and discussed that it represents a first-order topological transition from a quantum spin liquid with a half-quantized thermal Hall effect to a topologically trivial phase. Although the scenario proposed by Suetsugu \\textit{et al.} is attractive and cannot be excluded by the present data, we note that the similarity between the features at $H_{\\text{6}}$ and $H_{\\text{C2}}$ in $\\kappa(H)$, d$M$\/d$H$ and in their inverse melting behavior means that the origin of $H_{\\text{6}}$ is likely a magnetic phase transition in a secondary phase related to stacking faults. \n\nFinally, we suggest that the feature at $H_{\\text{1}}$, and possibly also that at $H_{\\text{2}}$, might originate from pseudo-spin reorientations within the zig-zag ordered phase, which is supported by the evolution of the in-plane field angle ($\\phi$) dependence of magnetization in this field range. At 1 T (below $H_{\\text{1}}$) $M\/H$ in Figs.~\\ref{fig4}a and \\ref{fig4}b shows a clear two-fold anisotropy implying an easy-axis fixed along $a$, consistent with the reported $ac$-plane orientation of pseudospins in the zig-zag phase \\cite{Cao2016}. The observed easy-axis anisotropy along $a$ is consistent with the monoclinic structure of the Bridgman sample, but in the trigonal CVT sample the origin of the anisotropy is unclear, perhaps a weak easy-axis anisotropy could be induced by strain, or alternatively the low temperature crystal structure may have a lowered symmetry due to the reported structural phase transition around 150 K \\cite{He2018,Mi2021,Park2016}. The antiferromagnetic easy axis will rotate to perpendicular to the field direction when the field is strong enough to overcome the lattice pinning, resulting in a spin-flop transition. Such behavior was previously reported in studies of the magneto-optical dichroism \\cite{Wagner2022} and terahertz spectroscopy \\cite{Wu2018} at $\\sim$1.5 T, where we observe the anomaly at $H_{\\text{1}}$. Increasing the in-plane field from 1 T to 5 T suppresses the easy-axis anisotropy fully in the CVT crystal and substantially in the Bridgman crystal, as can be seen in Figs.~\\ref{fig4}g and \\ref{fig4}h. It is therefore likely that spin-flop or reorientation transitions have happened in between.\n\n\\section{Conclusion}\nIn summary, we show that by measuring the low temperature features in the thermal conductivity and magnetization of two $\\alpha$-RuCl$_3$ single crystals grown with different sample growth techniques, a consistent picture emerges to explain high-field features. Minima in $\\kappa(H)$ coincide with peaks in d$M$\/d$H$, tracing out the magnetic phase transitions of both the 7 K main phase and other likely coexistent secondary phases, including those which have transition temperatures of 10 K and 13 K. These phases are enhanced by sample manipulation and exist in both types of samples. It is interesting to consider that whilst we treat them as essentially independent of the 7 K main phase and coexistent with it, it is clear that they strongly affect the bulk thermal transport, and their existence was shown to suppress the thermal Hall effect rapidly at low temperatures \\cite{Bruin2022}. It would be highly desirable to isolate samples with only one of the 7 K, 10 K or 13 K phases present, but whether this is feasible in bulk crystals is still an open question.\n\n\\begin{acknowledgments}\nWe thank Y. Kasahara, Y. Matsuda, S. Suetsugu and H. Suzuki for insightful discussions and M. Dueller and K. Pflaum for technical assistance. The work done in Germany has been supported in part by the Alexander von Humboldt foundation. H.TAN and N.K. have been supported by JSPS KAKENHI Grant Number JP17H01142 and JP19K03711, respectively. H.TAK has been supported in part by JSPS KAKENHI, Grant Numbers JP22H01180 and JP17H01140. S.L. thanks the Science and Engineering Research Board (SERB), Government of India, for the award of a Ramanujan Fellowship (RJF\/2021\/000050).\n\\end{acknowledgments}\n\n\\section*{Author Declarations}\n\\subsection*{Conflict of Interest}\nThe authors have no conflicts to disclose.\n\n\\section*{Data Availability}\nThe data that support the findings of this study are available from the corresponding author upon reasonable request.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{intro}\n\nOver the past several decades, astrophysicists have pushed the ``high-redshift frontier,\" where the most distant known galaxies and quasars reside, farther and farther back -- now encompassing star-forming galaxies at $z \\sim 7$ \\cite{bouwens08} and billion-solar mass black holes shining as quasars at $z \\sim 6.5$ \\cite{fan06}.  Two strategies can extend these efforts even farther over the next decade. The first, direct observations with large ground- and space-based near-infrared telescopes, will teach us about processes internal to these objects.  But a second method promises a beautiful complement to these direct probes:  low-frequency radio arrays using the 21 cm hyperfine line of neutral hydrogen to explore this era through the galaxies' indirect effects on the intergalactic medium (IGM).  Here we will describe the key astrophysical questions that this unique tool can address:  {\\bf What were the properties of high$-z$ galaxies? How did they affect the Universe around them?}\n\n\\section{Scientific Context} \\label{basic}\n\n\\begin{figure}\n\\includegraphics[width=6in]{steve.eps}\n\\caption{Maps ($\\approx 0.6^{\\circ}$ or $94$ comoving Mpc across) of ionized (white) and neutral (black) gas during reionization, showing the features visible in 21 cm maps.  From left to right, the panels assume that increasingly massive galaxies drive reionization (all have $x_{\\rm HI} \\approx 0.5$ and take $z \\approx 7.7$).  From \\cite{mcquinn07}.}\n\\label{fig:pics}\n\\end{figure}\n\nThe 21 cm brightness temperature of an IGM gas parcel at a redshift $z$, relative to the cosmic microwave background (CMB), is \\cite{madau97, furl06-review}\n\\begin{equation}\n\\delta T_b \\approx 25\\;x_{\\rm HI}(1+\\delta) \\, \\left( { 1+z \\over 10} \\right)^{1\/2}\\, \\left[1-\\frac{T_{\\rm CMB}(z)}{T_S}\\right] \\, \\left[ \\frac{H(z)\/(1+z)}{{\\rm d} v_\\parallel\/{\\rm d} r_\\parallel} \\right] \\mbox{ mK},\n\\label{eq:dtb}\n\\end{equation}\nwhere $x_{\\rm HI}$ is the neutral fraction, $\\delta = \\rho\/\\VEV{\\rho}-1$ is the fractional IGM overdensity in units of the mean, $T_{\\rm CMB}$ is the CMB temperature, $T_S$ is the spin (or excitation) temperature of this transition, $H(z)$ is the Hubble constant, and ${\\rm d} v_\\parallel\/{\\rm d} r_\\parallel$ is the line-of-sight velocity gradient.\n\nAll four of these factors contain unique astrophysical information.  The dependence on $\\delta$ traces the development of the cosmic web \\cite{scott90}, while the velocity factor sources line-of-sight ``redshift-space distortions\" that separate aspects of the cosmological and astrophysical signals \\cite{barkana05-vel}.  The other two factors depend strongly on the ambient radiation fields in the early Universe:  the ionizing background for $x_{\\rm HI}$ and a combination of the ultraviolet background (which mixes the 21 cm level populations through the Wouthuysen-Field effect \\cite{wouthuysen52, field58}) and the X-ray background (which heats the gas \\cite{chen04}) for $T_S$.  Fig.~\\ref{fig:pics} shows how $x_{\\rm HI}$ influences the 21 cm signal:  ionized regions appear as ``holes\" in the $\\sim 20 \\mbox{ mK}$ background of neutral gas.\n\n\\begin{SCfigure}\n\\centering\n\\includegraphics[scale=0.35]{f1.eps}\n\\caption{Fiducial histories of the sky-averaged 21 cm brightness temperature, $\\bar{\\delta T}_b$.  The solid blue curve uses a typical Population II star formation history, while the dashed red curve uses only very massive Population III stars.  Both fix reionization to end at $z_r \\approx 7$. From \\cite{furl06-glob}.}\n\\label{fig:global}\n\\end{SCfigure}\n\nFig.~\\ref{fig:global} shows two example scenarios for the sky-averaged 21 cm brightness temperature (see also \\cite{gnedin04}).  In the solid curve (a very typical model), we assume that high-$z$ star formation has similar properties to that at lower redshifts.  When the first galaxies form (here at $z \\sim 25$), the Wouthuysen-Field effect triggers strong 21 cm absorption \\cite{madau97}.  Somewhat later, X-rays -- produced by the first supernovae or black holes -- heat the entire IGM above $T_{\\rm CMB}$, turning this absorption into emission -- which fades and eventually vanishes as ionizing photons from the maturing galaxies destroy the intergalactic HI.\n\nThe red dashed curve shows a scenario in which very massive Population III stars dominate.  Their reduced efficiency for Wouthuysen-Field coupling (relative to their ionizing emissivity) decreases the strength of the absorption era \\cite{furl06-glob}, leading to a markedly different history.  The 21 cm line is clearly sensitive to the basic parameters of high-$z$ star formation.\n\nWe have so far illustrated the 21 cm signal through its all-sky background, but of course it actually fluctuates strongly as individual IGM regions grow through gravitational instability and are heated and ionized by luminous sources.  These fluctuations can ideally be imaged, but over the short term statistical measurements are likely to be more powerful  (see \\S \\ref{mile} below).  Fortunately, as we discuss next (also see Fig.~\\ref{fig:pk}), these statistical fluctuations contain an enormous amount of information about this era.\n\n\\section{Key Questions} \\label{reion}\n\nThe rich physics of the 21 cm line allows us to study the over-arching questions of {\\bf how the first galaxies evolved and affected the Universe around them}.  In particular, we can approach these questions from the following specific directions.  (Note that related cosmological questions, and more detailed IGM physics, are addressed in the companion white paper ``Cosmology from the Highly-Redshifted 21 cm Transition.\")\n\n\\underline{\\it When did reionization occur?}  One of the most dramatic events in the IGM's history was the reionization of intergalactic hydrogen, most likely via ultraviolet photons from star-forming galaxies.  This event marked the point when the small fraction of matter inside galaxies completely changed the landscape of the diffuse IGM gas (and hence future generations of galaxies), rendering it transparent to high-energy photons and heating it substantially.  The 21 cm background provides the \\emph{ideal} probe of reionization.  Its weak oscillator strength (in comparison to Ly$\\alpha$) allows us to penetrate even extremely high redshifts.  We can also image it across the entire sky -- instead of only rare, isolated Ly$\\alpha$  forest lines of sight.  Moreover, unlike the CMB, it is a spectral line measurement, and we can distinguish different redshift slices and study the full history of the ``dark ages\" -- extremely difficult even with a ``perfect\" CMB measurement \\cite{zald08-cmbpol}.  Finally, it directly samples the $ 95\\%$ (or more) of the baryons that reside in the IGM, requiring no difficult inferences about this material from the properties of the rare luminous galaxies. \n\nObservations have provided tantalizing hints about reionization, but even more unanswered questions \\citep{fan06-review}.  For example, CMB observations imply that reionization completed by $z \\sim 10$ \\citep{dunkley08}, but quasar absorption spectra suggest that it may have continued until $z \\sim 6$ \\cite{fan06}, albeit both with substantial uncertainties.  Both the sky-averaged $\\bar{\\delta T}_b$ and the 21 cm power spectrum yield much more precise measures of $\\bar{x}_{\\rm HI}(z)$ (see Fig.~\\ref{fig:global} and the left panel of Fig.~\\ref{fig:pk}).  The fluctuations briefly fade as galaxies ionize their dense surroundings in the first stage of reionization, then increase as large ionized bubbles form, finally fading again as the gas is ionized.  The principal goal of first-generation experiments (now under construction) is to constrain this time evolution \\cite{lidz08-constraint}.\n\n\\underline{\\it What sources were responsible for reionization?}  The 21 cm sky contains much more information about reionization, because the properties of the ionizing sources strongly affect its topology and have quantifiable impacts on the power spectrum.  Most fundamentally, stars produce well-defined ionized regions, while ``miniquasars\" (small accreting black holes) produce much more diffuse features \\cite{zaroubi05}. But the 21 cm background can even distinguish between different stellar reionization scenarios \\cite{lidz08-constraint}.  For example, more massive galaxies produce larger ionized regions \\cite{furl04-bub, mcquinn07}, as shown by the four panels in Fig.~\\ref{fig:pics}.  Photoheating (accompanied by a suppression of galaxy formation due to the increased IGM pressure) also slows the growth of bubbles \\cite{iliev07-selfreg}.  We can thus indirectly constrain the sources of ionization, including even extremely faint galaxies (which probably dominate the stellar mass budget), and gauge the importance of exotic, very massive Population III stars.  \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.35]{f2a.eps}\n\\includegraphics[scale=0.35]{f2b.eps}\n\\caption{\\emph{Left panel:}  Evolution of the spherically-averaged 21 cm power spectrum during reionization, from a numerical simulation of that process.  Here the fluctuations are presented in units of $\\bar{\\delta T}_b^2$ for a fully neutral universe (see eq.~\\ref{eq:dtb}); $\\VEV{x_i}$ is the ionized fraction during reionization.  From \\cite{lidz08-constraint}.  \\emph{Right panels:}   Evolution of the spherically-averaged power spectrum \\emph{before} reionization begins in earnest, including X-ray heating and Wouthuysen-Field coupling fluctuations.  From \\cite{pritchard07}.  In all panels, we use models with parameters similar to the solid curve in Fig.~\\ref{fig:global}.  }\n\\label{fig:pk}\n\\end{center}\n\\end{figure}\n\nOf course, exploring this era is a primary goal of many other forthcoming experiments.  Comparison with, e.g., galaxy surveys will provide a clear and complete picture connecting the small-scale physics of galaxies with landmark events encompassing larger scales and illuminate the relation between galaxies and the IGM, if appropriately wide and deep near-IR surveys can be done \\cite{lidz09}.  Targeted near-IR follow-up of individual HII regions identified in 21 cm will also identify how reionization proceeds on smaller scales.\n\n\\underline{\\it How did the cosmic web evolve?}  Once they leave their sources, ionizing photons must propagate through the IGM, where they can be absorbed by dense pockets of neutral gas around galaxies.  These affect the topology of the ionized bubbles during the tail end of reionization, when the pattern of well-defined ionized bubbles disintegrates into the ``cosmic web\" that we see in the Ly$\\alpha$ forest \\cite{furl05-rec, choudhury08}.  This process also imprints distinctive features on the 21 cm signal and allows us to measure directly the evolution of the IGM's structure, as well as the interactions of ionizing sources with the IGM.\n\n\\underline{\\it Where did the first generations of quasars form, and what were their properties?}  Bright quasars form enormous HII regions in the IGM ($> 40 \\mbox{ Mpc}$ across), visible in the 21 cm sky even after the quasar becomes dormant, because the recombination time is relatively long \\cite{furl08-fossil}.  Studying these regions in detail will constrain the quasar emission mechanism and their lifetimes, luminosity function, and redshift evolution \\cite{wyithe05-qso}.\n\n\\underline{\\it When did the first galaxies form, and what were their properties?}  As mysterious as they are, the galaxies responsible for reionizing the Universe were probably relatively mature.  The 21 cm transition also probes the very first structures to appear in the Universe -- an era most likely inaccessible even to \\emph{JWST}.  The right panel of Fig.~\\ref{fig:pk} shows the 21 cm power spectrum during this early era in one typical model.  The first stars produced a ``soft\" UV background (below the HI ionization edge) that triggered an absorption feature in the 21 cm background (at $z \\approx 20$ in Fig.~\\ref{fig:global}).  The background is strongest near these galaxies, so the absorption fluctuations trace their locations, masses, UV spectra, luminosity function and redshift evolution \\cite{barkana05-ts}.  The $z=19$ curve in Fig.~\\ref{fig:pk} illustrates this era; the fluctuation amplitude can be \\emph{stronger} than that during reionization.\n\n\\underline{\\it When did the first black holes form, and what were their properties?}  Somewhat later, X-rays from the first black holes (and supernovae) heated the Universe (at $z \\approx 15$ in Fig.~\\ref{fig:global}).  Again, the heating was concentrated around the galaxies hosting these sources, so the observable fluctuations trace their redshift evolution, abundance, masses, and spectra \\cite{pritchard07}.  The $z=17$ and 15 curves in Fig.~\\ref{fig:pk} show that the sharp contrast between hot and cold regions produces distinctive troughs in the power spectrum, as well as especially large fluctuation amplitudes.  Thus we can securely identify the era in which the first X-ray sources heated the IGM and then follow their importance throughout the reionization era \\cite{oh01, ricotti05}.\n\n\\underline{\\it How does radiative feedback affect high-z galaxy formation?}  Both the UV and X-ray backgrounds are crucial feedback mechanisms in early galaxy formation.  For example, they strongly affect H$_2$, the primary coolant in primordial gas, and therefore regulate both the possible locations of star formation and its end products \\cite{haiman00, ahn08}.  Moreover, X-ray heating also affects structure formation and clumping in the IGM \\cite{oh03}.  The 21 cm background is the only method to measure  these mechanisms \\emph{directly}.  \n\n\\section{Milestones} \\label{mile}\n\nThe ultimate goal of studying the 21 cm background is to make detailed maps of the IGM throughout the ``dark ages\" and reionization, as in Fig.~\\ref{fig:pics}.  The top axis of Fig.~\\ref{fig:global} shows the observed frequency range for these measurements: well within the low-frequency radio regime.  Unfortunately, this is an extremely challenging band, because of terrestrial interference, ionospheric refraction, and (especially) other astrophysical sources (see \\cite{furl06-review}).  In particular, the polarized Galactic synchrotron foreground has $T_{\\rm sky} \\sim 180 (\\nu\/180 \\mbox{ MHz})^{-2.6} \\mbox{ K}$, at least four orders of magnitude larger than the signal.  For an interferometer, the noise per resolution element (with an angular diameter $\\Delta \\theta$ and spanning a bandwidth $\\Delta \\nu$) is then \\cite{furl06-review}\n\\begin{equation}\n\\Delta T_{\\rm noise} \\sim 20  \\mbox{ mK} \\ \\left( \\frac{10^4 \\mbox{ m$^2$}}{A_{\\rm eff}} \\right) \\, \\left( \\frac{10'}{\\Delta \\theta} \\right)^2 \\, \\left( \\frac{1+z}{10} \\right)^{4.6} \\,  \\left( \\frac{{\\rm MHz}}{\\Delta \\nu} \\, \\frac{100 \\mbox{ hr}}{t_{\\rm int}} \\right)^{1\/2},\n\\label{eq:if-sens}\n\\end{equation}\nwhere $A_{\\rm eff}$ is the effective collecting area and $t_{\\rm int}$ is the integration time.  These angular and frequency scales correspond to $\\sim 30 \\mbox{ Mpc}$.  Here we outline the steps required to explore this era in detail, given the challenges implicit in this huge noise.\n\n{\\bf The all-sky signal:}  The global background illustrated in Fig.~\\ref{fig:global} contains an extraordinary amount of information about the gross properties of galaxies.  These measurements can easily beat down the foreground noise with only a single dipole, so they may provide our first constraints at very high redshifts.  The challenge lies in calibration that is precise enough to extract the signal from instrumental artifacts and the bright foregrounds.  The EDGES experiment \\cite{bowman08} has already set upper limits and hopes to measure this signal to $z \\sim 20$ over the next decade.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.35]{f3a.eps}\n\\includegraphics[scale=0.35]{f3b.eps}\n\\caption{\\emph{Left panels:}  Sensitivity of some fiducial arrays to the spherically-averaged power spectrum at $z=10$ and $z=15$, (conservatively) assuming a fully neutral emitting IGM.  The solid and dashed curves take $A_{\\rm eff}=0.1$ and 1 km$^2$, respectively; in both cases we take  4000 hr of observing split over four fields, a total bandwidth $B=8 \\mbox{ MHz}$, $T_{\\rm sky} = (471,\\,1247) \\mbox{ K}$ at $z=(10,15)$, and $N=5000$ stations centered on a filled core and with an envelope out to $R_{\\rm max}=3$~km.  The black squares show the location of the independent $k$ bins in each measurement.  The dotted curve shows the projected MWA sensitivity at $z=10$.  The vertical dotted line corresponds to the bandwidth; modes with smaller wavenumbers are compromised by foreground removal.   \\emph{Right panels:} Sensitivity of the same arrays to angular components of the power spectrum at $z=10$.}\n\\label{fig:sensitivity}\n\\end{center}\n\\end{figure}\n\n{\\bf First-Generation Arrays:}  The arrays now operating or under construction, including the Murchison Widefield Array (MWA), the Precision Array to Probe the Epoch of Reionization (PAPER), LOFAR, the 21 Centimeter Array, and the Giant Metrewave Radio Telescope, have $A_{\\rm eff} \\sim 10^4 \\mbox{ m$^2$}$ and so are limited to imaging only the most extreme ionized regions (such as those surrounding bright quasars).  Nevertheless,  these arrays have sufficiently large fields of view ($> 400^{\\circ2}$) to make reasonably good statistical measurements \\cite{mcquinn06-param, bowman07}.  Fig.~\\ref{fig:sensitivity} shows the projected errors for the MWA at $z=10$.  It will be able to detect fluctuations over a limited spatial dynamic range (thanks to foreground removal and thermal noise) and only at $z < 12$, constraining the timing of reionization and some of the source physics.\n\n{\\bf Second-Generation Arrays:}  Fig.~\\ref{fig:sensitivity} shows that larger telescopes, with $A_{\\rm eff} \\sim 10^5 \\mbox{ m$^2$}$ (and large fields of view), will clearly be needed for precise measurements, and especially to identify the distinctive features in the power spectrum (such as the flattening at $k > 1 \\mbox{ Mpc$^{-1}$}$ during reionization and the troughs during X-ray heating; see Fig.~\\ref{fig:pk}).  Instruments in this class will also be able to measure some more advanced statistics.  This is illustrated in the right panels in Fig.~\\ref{fig:sensitivity}, which consider the components of the redshift-space distortions induced by velocity fluctuations.  These are extremely useful for breaking degeneracies in the signal \\cite{barkana05-vel} (for more information, see the companion white paper ``Cosmology from the Highly-Redshifted 21 cm Line\") but lie beyond the reach of first-generation experiments.\n\n{\\bf Imaging Arrays:}  At $A_{\\rm eff} \\sim 10^6 \\mbox{ m$^2$}$, imaging on moderate scales becomes possible, and statistical constraints become exquisite even at high redshifts  (provided that the large field of view, not strictly necessary for imaging, is maintained; see Fig.~\\ref{fig:sensitivity}).  \n\nPlans for these later generations will evolve as we learn more about ``dark age\" physics  and the experimental challenges ahead; for example, the Long Wavelength Array and other lower-frequency instruments will study the ionospheric calibration required to explore the high-$z$ regime ($z > 12$, or $\\nu < 110 \\mbox{ MHz}$) and help determine the relative utility of a terrestrial Square Kilometer Array or a far-side Lunar Radio Array.  At the same time, we must explore whether innovative new telescope designs more closely aligned with the observables, such as the FFT Telescope \\cite{tegmark09}, can provide cost-effective improvements.\n\nThis roadmap, with accompanying efforts to improve theoretical modeling of the first galaxies and data analysis techniques (such as specialized statistical measures) will ideally position the community to explore the major science questions of high-$z$ galaxy formation.\n\n\\newpage\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}