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+{"text":"\\section{Introduction}\n\nThe knowledge of phase diagrams is of great importance for many research areas in physics, chemistry, and engineering. A phase diagram is a map used to show in which conditions of pressure, temperature, volume, etc, one can find each specific phase of a system and the boundaries between these various phases.\n\nThe determination of phase diagrams by computer simulation requires the computation of the free-energy of the various phases of the system, this task is far from trivial. The difficulties stem from the fact that entropy and free-energies depend on the volume in phase space available to the system, in contrast to other thermodynamic properties, such as internal energy, enthalpy, temperature, pressure, etc., that can be easily computed as simple averages of functions of the phase space coordinates.\n\nThe determination of phase diagrams by computer simulation can be achieved by employing free-energy evaluation techniques such as thermodynamic integration (TI) \\cite{Kirkwood1935}, which consists in the construction of a sequence of equilibrium states along a path between two thermodynamic states of interest.\nThe calculated free-energy difference between these two states is essentially exact, if the free-energy of one of them is known, which we call reference system. From that the absolute free-energy of the other system, which we call system of interest, is readily obtained. In practice, since the integration along the thermodynamic path is performed numerically, it requires the calculation of several ensemble averages, which is computationally very demanding.\n\nIn recent years, several studies\\cite{Freitas2016,Leite2016,Leite2019,Cajahuaringa2018,Cajahuaringa2019} have been proving the efficiency of the non-equilibrium (NE) techniques, making free-energy  calculations much more accessible\\cite{Freitas2016,Leite2019}, which were made available in the \\texttt{LAMMPS} molecular dynamics (MD) package\\cite{LAMMPS}. In contrast to the equilibrium TI method, the NE approaches estimate the desired free-energy difference by traversing the thermodynamic path between the systems of interest and reference in an explicitly time-dependent process, which have been shown to give accurate results using only a few relatively short non-equilibrium simulations. \n\nHowever, several NE calculations are required for each coexistence condition between the phases of the system to map correctly the phase boundaries. In order to optimize the calculation of phase boundaries, it was proposed by de Koning et al.\\cite{deKoning2001} a method for the integration of the Clausius-Clapeyron equation using non-equilibrium simulations. This method allows the calculation of the phase boundary using, in principle, a single simulation, whose length is comparable to a regular equilibrium simulation.\n\nIn this paper, we demonstrate the potential of the dynamic Clausius-Clapeyron integration (dCCI) method by using it to obtain, entirely from atomistic simulations employing a realistic interatomic potential, the phase diagram in a wide range of temperatures and pressures, focusing on the implementation in the widely used \\texttt{LAMMPS} MD package. We show how the NE methods can be used to compute the Gibbs free-energy of different phases, in a broad range of temperature and\/or pressure, in order to determine an initial coexistence condition and, from that, how the dCCI method can be used to accurately and efficiently compute the phase boundaries of atomistic systems. We provide excerpts from the used \\texttt{LAMMPS} scripts to exemplify the practical details of the calculations. Complete \\texttt{LAMMPS} scripts, source codes and post-processing tools have also been made available in the following github account. As an illustration we performed non-equilibrium free-energy calculations of the phase-boundaries of silicon, described by the Stillinger-Weber\\cite{Stillinger1985} potential.\n\nThe paper is organized as follows. In Sec.~\\ref{sec:NFEM} we review the relevant aspects of NE free-energy methods to calculate the Gibbs free-energy along isothermal and isobaric paths. Sec. \\ref{sec:DCCI} provides the details of the dCCI methodology, in which from a given initial coexistence condition one is able to determine the entire phase boundary from a single non-equilibrium simulation. Sec. \\ref{sec:applications} describes the application of dCCI to compute the phase diagram of Si. We end with a summary and conclusions in Sec. \\ref{sec:conclusions}.\n\\section{Non-equilibrium free-energy methods}\n\\label{sec:NFEM}\nThe TI method\\cite{Kirkwood1935} has been widely used to compute Helmholtz free-energy differences by using the calculation of reversible work. It consists in the construction of a path connecting two thermodynamic states by defining a parametrized Hamiltonian $H(\\lambda)$, where $\\lambda$ is a coupling parameter describing the interpolation between the two ends of the thermodynamic path.\n\nIt is straightforward to show, starting from the definition of the partition function, that the Helmholtz free-energy difference between the two equilibrium states corresponding to $H(\\lambda_{i})$ and $H(\\lambda_{f})$ is given by the reversible work $W_{rev}$ done along the quasi-static process that connects these states, i.e.\n\\begin{equation}\n\\Delta F = F(\\lambda_{f})-F(\\lambda_{i})=\\int_{\\lambda_{i}}^{\\lambda_{f}}\\left \\langle\\frac{\\partial H}{\\partial \\lambda}\\right \\rangle d\\lambda\\equiv W_{rev},\n\\label{eqn:TI}\n\\end{equation}\nwhere $\\left\\langle\\cdots \\right\\rangle_{\\lambda}$ is the canonical ensemble average at a particular value of parameter and $\\frac{\\partial H}{\\partial \\lambda}$ is the so-called driving-force. In this approach, the integral is discretized on a grid of $\\lambda$-values between $\\lambda_{i}$ and $\\lambda_{f}$ and a separate equilibrium simulation is performed for each value of lambda.\n\nIn the NE approach, the adiabatic switching\\cite{Watanabe1990} (AS) method estimates the integral in Eq.~\\ref{eqn:TI} by replacing the integration over equilibrium ensemble averages, by an integration over instantaneous values of driving force along a single simulation in which $\\lambda=\\lambda(t)$ changes continuously throughout the simulation \n\\begin{equation}\nW_{dyn}=\\int_{0}^{t_{s}}\\frac{d\\lambda}{dt}\\frac{\\partial H(\\lambda)}{\\partial \\lambda}\\Bigg|_{\\lambda(t)} dt,\n\\label{eqn:Wdyn}\n\\end{equation}\nwhere $t_{s}$ is the total time of the switching process and $W_{dyn}$ is the dynamical work done. Due to the irreversible nature of the process, entropy is produced, causing $W_{dyn}$ to be a stochastic variable whose mean value, by the second law of thermodynamics, differs from $W_{rev}$ according to the relation\n\\begin{equation}\n\\Delta F=W_{rev}=\\overline{W}_{dyn}-\\overline{Q}_{diss},\n\\end{equation}\nwhere the overbar means an average over an ensemble of realizations of the AS process and $\\overline{Q}_{diss}\\geqslant 0$ is the average dissipated heat, which is zero only in the quasi-static limit ($t_{s}\\to \\infty$). However, provided that the non-equilibrium process is sufficiently ''close'' to the ideal quasi-static process, hence the linear response theory is valid, it can be shown that the systematic error can be eliminated by combining the results of the processes realized in both directions (forward and backward). Accordingly, one has\n\\begin{equation}\n\\Delta F= \\frac{1}{2}[\\overline{W}_{dyn}^{i\\to f}-\\overline{W}_{dyn}^{f\\to i}],\n\\label{eqn:dF}\n\\end{equation}\nwhere we have used the fact that within the linear response theory the heat dissipated in both processes are identical $\\overline{Q}_{diss}^{i\\to f}=\\overline{Q}_{diss}^{f\\to i}$. Similarly, the dissipated heat can be estimated by\n\\begin{equation}\n\\overline{Q}_{diss}^{i\\to f}=\\overline{Q}_{diss}^{f\\to i}=\\frac{1}{2}[\\overline{W}_{dyn}^{i\\to f}+\\overline{W}_{dyn}^{f\\to i}],\n\\label{eqn:Qdiss}\n\\end{equation}\nEqs.~\\ref{eqn:dF} and \\ref{eqn:Qdiss} can be used for monitoring of the convergence of $\\Delta F$ and the heat dissipation as a function of the switching time $t_{s}$. The AS method can lead to significant efficiency gains when compared to standard equilibrium methods, providing accurate estimates of $\\Delta F$ using only a few relatively short non-equilibrium simulations.\n\nNext we discuss the application of three specific thermodynamic paths for $H(\\lambda)$ that allow the calculation of, (i) the Helmholtz free-energy difference between two systems described by different Hamiltonians, (ii) the temperature-dependence of the Gibbs free-energy for a given system Hamiltonian along an isobaric path, and (iii) the pressure-dependence of the Gibbs free-energy for a given system Hamiltonian along an isothermal path.\n\\subsection{Helmholtz free-energy difference between two systems: Hamiltonian interpolation\nmethod}\\label{subsec:AS}\nSuppose we wish to compute the Helmholtz free-energy of some system of interest described by a Hamiltonian $H_{int}$ of the form\n\\begin{equation}\n    H_{int}=K+U_{int}(\\textbf{r}),\n\\end{equation}\nwhere $K$ is the kinetic energy and $U_{int}(\\textbf{r})$ is the system of interest's potential energy, where $\\textbf{r}$ stands for the set of particles coordinates. A second system, in the same thermodynamic phase of the system of interest, can be used as a reference system if its Helmholtz free-energy is known (analytically or numerically), described by the Hamiltonian $H_{ref}$, given by\n\\begin{equation}\n    H_{ref}=K+U_{ref}(\\textbf{r}_{i}),\n\\end{equation}\nwith $U_{ref}(\\textbf{r})$ being its potential energy.\n\nA typical functional form of $H(\\lambda)$ that couples the systems of interest and reference is given by a linear interpolation as follow\n\\begin{equation}\n    H_{\\lambda}=\\lambda H_{int}+(1-\\lambda)H_{ref},\n\\end{equation}\nnote that this form allows a continuous switching between $H_{int}$ and $H_{ref}$ by varying the coupling parameter $\\lambda$ between $\\lambda_{i}=1$ and $\\lambda_{f}=0$. According to Eq. \\ref{eqn:TI}, the reversible work is given by \n\\begin{equation}\nW_{rev}=\\int_{\\lambda_{i}}^{\\lambda_{f}}\\langle U_{int}-U_{ref}\\rangle d\\lambda\n\\label{eqn:Wrev_HI}\n\\end{equation}\nand its corresponding dynamical work estimator for the forward process is given by\n\\begin{equation}\nW_{dyn}^{i\\to f}=\\int_{0}^{t_{s}}\\frac{d\\lambda}{dt}(U_{int}-U_{ref})dt,\n\\end{equation}\nnote that for the backward process the time derivative $\\frac{d\\lambda}{dt}$ has the reversed sign of the forward process. By combining the average results obtained from a number of independent realizations of both switching processes, the desired Helmholtz free-energy is estimated as\n\\begin{equation}\nF_{int}=F_{ref}+\\frac{1}{2}[\\overline{W}_{dyn}^{i\\to f}-\\overline{W}_{dyn}^{f\\to i}].\n\\label{eqn:dA}\n\\end{equation}\nFor solids systems, either crystalline or amorphous, the choice of the reference system is straightforward, the Einstein crystal\\cite{Frenkel,Freitas2016,deKoning1997} whose Helmholtz free-energy is analytically known. For fluid-phase systems recently it was proposed that the Uhlenbeck-Ford (UF) model\\cite{Leite2016,Leite2019}, which is a robust reference system that provides an accurate fluid-phase Helmholtz free-energy.\n\\subsection{Gibbs free-energy as a function of the temperature: the Reversible Scaling method}\nSuppose the Gibbs free-energy $G_{int}(P,T_{0})$ of a system of interest is known at some temperature $T_{0}$ and fixed pressure $P$, and we now wish to determine $G_{int}(P,T)$ for another temperature $T$ along an isobaric path. That can be achieved in a single simulation by using the reversible scaling (RS) technique\\cite{deKoning1997}. The RS method in the NPT ensemble\\cite{deKoning2001} is based on the following parametric Hamiltonian\n\\begin{equation}\nH_{RS}(\\lambda)=K+\\lambda U_{int}(\\textbf{r})+P_{S}(\\lambda)V,\n\\label{eqn:H_RS}\n\\end{equation}\nin this case, this Hamiltonian represents the enthalpy of the scaled system, in which the potential energy is scaled by the coupling parameter and $P_{S}$ the external pressure on the scaled system. The configurational and volume contributions to the partition function in the NPT ensemble $Z_{RS}(P_{S}(\\lambda),\\lambda)$ for the RS Hamiltonian at temperature $T_{0}$ and pressure $P_{s}$ is given by\n\\begin{eqnarray}\nZ_{RS}(P_{S}(\\lambda),\\lambda) &=& \\int dV\\exp[-P_{S}(\\lambda)V\/k_{B}T_{0}]\\int_{V}d^{3N}\\mathbf{r}\\exp[-\\lambda U_{int}(\\textbf{r})\/k_{B}T_{0}] \\nonumber \\\\\n&=& Z_{int}(P,T),\n\\label{eqn:Z_RS}\n\\end{eqnarray}\nwhich is equal to the partition function of the system of interest at temperature $T$ and pressure $P$ by using the following scaling relations:\n\\begin{equation}\nT=\\frac{T_{0}}{\\lambda}\n\\label{eqn:T_sc}\n\\end{equation}\nand\n\\begin{equation}\nP_{S}(\\lambda)=\\lambda P.\n\\label{eqn:P_sc}\n\\end{equation}\nOn the basis of Eqs. \\ref{eqn:Z_RS}, \\ref{eqn:T_sc}, and \\ref{eqn:P_sc}, the Gibbs free energies of the system of interest $G_{int}$ and the scaled system $G_{RS}$, are related according to\n\\begin{equation}\nG_{int}(P,T)=\\frac{G_{RS}(P_{S}(\\lambda),T_{0};\\lambda)}{\\lambda}+\\frac{3}{2}Nk_{B}T_{0}\\frac{\\ln\\lambda}{\\lambda}\n\\label{eq:GRS}\n\\end{equation}\nwhere $N$ is the number of atoms. Eq.~\\ref{eq:GRS}, implies that each value of $\\lambda$ in the scaled Hamiltonian $H_{RS}$ at a fixed temperature $T_{0}$ and scaled pressure $P_{S}(\\lambda)$ corresponds to the system of interest described by $H_{int}$ at a temperature $T$ and pressure $P$. Thus, the task of calculating the free-energy of the physical system as a function of temperature $T$ at fixed pressure $P$ can be accomplished from $H_{RS}(\\lambda)$ by varying the scaling parameter $\\lambda$ and the scaling pressure $P_{S}(\\lambda)$ at fixed temperature $T_{0}$. In this case one can identify  the driving force as\n\\begin{equation}\n    \\frac{\\partial H_{RS}(\\lambda)}{\\partial \\lambda}=U_{int}(\\textbf{r})+\\frac{dP_{S}(\\lambda)}{d\\lambda}V.\n    \\label{eqn:dH_RS}\n\\end{equation}\nThe reversible work in the RS method is given by\n\\begin{equation}\n    W_{rev}(\\lambda)=\\int_{1}^{\\lambda}\\Bigg(\\langle U_{int}\\rangle+\\frac{dP_{S}(\\lambda')}{d\\lambda'}\\langle V\\rangle\\Bigg)d\\lambda',\n    \\label{eqn:Wrev_RS}\n\\end{equation}\ncan be obtained within very good accuracy through the AS method, which is used to estimate the forward dynamical work, with $\\lambda(t)$ varying from $\\lambda(0)=1$ to $\\lambda(t_{s})=\\lambda_{f}$\n\\begin{equation}\n    W_{dyn}^{i\\to f}=\\int_{0}^{t_{s}}\\frac{d\\lambda}{dt}\\Bigg(U_{int}+\\frac{dP_{S}(\\lambda)}{d\\lambda}\\Bigg|_{\\lambda(t)}\\Bigg)dt.    \n    \\label{eqn:Wdyn_RS}\n\\end{equation}\nBy carrying out the scaling process in the opposite direction and using Eq.~\\ref{eqn:dF}, $G_{int}(P,T)$ is given by\n\\begin{equation}\nG_{int}(P,T)=\\frac{G_{int}(P,T_{0})}{\\lambda}+\\frac{3}{2}Nk_{B}T_{0}\\frac{\\ln\\lambda}{\\lambda}+\\frac{1}{2\\lambda}[\\overline{W}_{dyn}^{1\\to \\lambda}-\\overline{W}_{dyn}^{\\lambda\\to 1}],\n\\label{eqn:Gibbs_RS}\n\\end{equation}\nwhere $\\lambda$ varies between $1$ and $\\lambda_{f}$, and $G_{int}(P,T_{0}) = G_{RS}(P,T_{0})$.\n\nIt is important to note that the application of this approach requires the knowledge of the absolute Gibbs free-energy of the system of interest at a temperature $T_{0}$ and pressure $P$, which can be computed by knowing the Helmholtz free-energy at temperature $T_{0}$ and the average volume of the system $V$, which corresponds to the external pressure $P$, through the thermodynamic relation\n\\begin{equation}\nG_{int}(P,T_{0})=F_{int}(V,T_{0})+PV\n\\label{eqn:G_and_A}\n\\end{equation}\nwhere the $F(T_{0},V)$ can be determined by employing the AS method, detailed in \\ref{subsec:AS}.\n\\subsection{Gibbs free-energy as a function of pressure: Adiabatic Switching method}\nSuppose the Gibbs free-energy $G_{int}(P_{0},T)$ of a system of interest is known at some pressure $P_{0}$ and fixed temperature $T$, and we now wish to determine $G_{int}(P,T)$ for other pressure $P$ along an isothermal path, where the Hamiltonian of the interest system is \n\\begin{equation}\nH_{int}=K+U_{int}(\\textbf{r})+PV,    \n\\end{equation}\nif we take the external pressure as coupling parameter, the work associated with a reversible process along a path connecting the physical system at the reference pressure $P_{0}$ to the system at a pressure of interest $P$ is\n\\begin{equation}\nG_{int}(P,T)-G_{int}(P_{0},T)=\\int_{P_{0}}^{P}\\left \\langle\\frac{\\partial H}{\\partial P'}\\right \\rangle dP'=\\int_{P_{0}}^{P}\\langle V\\rangle dP'\\equiv W_{rev},\n\\label{eqn:Wmech}\n\\end{equation}\nwhere the brackets in Eq.~\\ref{eqn:Wmech} indicate the equilibrium average of the volume in the isothermal-isobaric ensemble. An efficient alternative to performing several equilibrium simulations can be achieved by using the AS method\\cite{Cajahuaringa2018}, the integral can be evaluated along a single non-equilibrium simulation during which the magnitude of the external pressure $P(t)$ changes dynamically, in such a way that, at the beginning of the simulation $P(0)=P_{0}$ and at the end $P(t_{s})=P$. In this case the reversible work will be estimated in terms of the dynamical work \n\\begin{equation}\nW_{dyn}=\\int_{0}^{t_{s}}\\frac{dP(t)}{dt}\\bigg|_{t}V(t)dt.\n\\label{eqn:Wmech_dyn}\n\\end{equation}\nIn this switching process not only the initial and final points on the trajectory correspond to physical systems, as it is usual in AS simulations, but the information gathered at the intermediate states of the path also has physical meaning. As a consequence, one obtains the Gibbs free-energy of a system in a wide interval of pressure, provided that the Gibbs free-energy of the system of interest in a reference state is known.\n\\begin{equation}\nG_{int}(P,T)=G_{int}(P_{0},T)+\\frac{1}{2}[\\overline{W}_{dyn}^{i\\to f}-\\overline{W}_{dyn}^{f\\to i}].\n\\label{eqn:Gibbs_AS}\n\\end{equation}\n\\section{Dynamic Clausius-Clapeyron integration}\\label{sec:DCCI}\nFirst-order phase transitions for pure substances, are usually represented as lines in a PT diagram, which indicate the phase boundaries between the phases and its slope can be described by the Clausius\u2013Clapeyron equation (CCE), which uses thermodynamic properties of the both coexisting phases. The CCE in the PT diagram can be written in several forms, the most common one is,\n\\begin{equation}\n    \\frac{dP}{dT}=\\frac{\\Delta H}{T\\Delta V},\n    \\label{eqn:CCE}\n\\end{equation}\nwhere $\\Delta H$ and $\\Delta V$ are the molar enthalpy and molar volume differences between the two phases, respectively. The CCE is remarkable because if a point of the coexistence curve is known, one can obtain the whole phase boundary from Eq. \\ref{eqn:CCE}.\n\nIn the nineties Kofke\\cite{kofke1993} proposed a method that employs the CCE combined with results from computer simulations to calculate phase boundaries. The method is very effective, however, it requires a series of independent equilibrium simulations of both phases of interest, which is quite demanding in terms of computing time.\n\nIn order to avoid performing a series of independent equilibrium simulations for each phase required to map the phase boundaries, de Koning et al.\\cite{deKoning2001} based on a similar idea to the RS method, proposed a method in which the integration of Clausius-Clapeyron equation is performed dynamically. Starting at a single point of the coexistence curve, this technique allows the exploration of the coexistence curve over a wide range of states from two non-equilibrium simulations, one for each phase, running simultaneously.\n\nLet us start from a known condition of phase coexistence of two phases $I$ and $II$, that occurs when the Gibbs free energies of both phases are equal $G_{s,I}=G_{s,II}$, if reversible perturbations $d\\lambda$ and $d\\lambda(dP_{S}\/d\\lambda)$ are applied, the Gibbs free energies of the scaled systems will change according to\n\\begin{equation}\ndG_{S,I}=dW_{rev,I}=d\\lambda\\Bigg[\\left \\langle U_{int}\\right \\rangle_{I}+\\frac{dP_{S}}{d\\lambda}\\left \\langle V\\right \\rangle_{I}\\Bigg]\n\\end{equation}\nand\n\\begin{equation}\ndG_{S,II}=dW_{rev,II}=d\\lambda\\Bigg[\\left \\langle U_{int}\\right \\rangle_{II}+\\frac{dP_{S}}{d\\lambda}\\left \\langle V\\right \\rangle_{II}\\Bigg],\n\\end{equation}\nwhere all ensemble averages are evaluated at temperature $T_{0}$, scaling parameter $\\lambda$, and scaled pressure $P_{S}(\\lambda)$. In order to maintain phase-coexistence upon the application of this disturbance, one must have that $dG_{S,I}= dG_{S,II}$, from this condition we obtain the following relationship between pressure and temperature of the scaled systems,\n\\begin{equation}\n\\frac{dP_{S}}{d\\lambda}=-\\Bigg(\\frac{\\left \\langle U_{int}\\right \\rangle_{I}-\\left \\langle U_{int}\\right \\rangle_{II}}{\\left \\langle V\\right \\rangle_{I}-\\left \\langle V\\right \\rangle_{II}}\\Bigg),\n\\label{eqn:RS_CCE}\n\\end{equation}\nthis equation is the CCE for the scaled systems, where the temperature is represented by the coupling parameter $\\lambda$.\n\nSimilarly to what is done to estimate the free-energy from non-equilibrium simulations, we can modify Eq.~\\ref{eqn:RS_CCE} in a way that the parameter $\\lambda$ is varied dynamically, transforming it into the dynamic Clausius-Clapeyron equation \n\\begin{equation}\n\\frac{dP_{S}}{dt}=-\\frac{d\\lambda}{dt}\\Bigg(\\frac{U_{int,I}(t)-U_{int,II}(t)}{V_{I}(t)-V_{II}(t)}\\Bigg),\n\\end{equation}\nwhere the ensemble averages of the potential energies and volumes have been replaced by instantaneous values along the time-dependent process. Provided that the dynamic process is ideally reversible, the coexistence curve is given by\n\\begin{equation}\nP_{S,coex}(\\lambda(t_{s}))=P_{S,coex}(\\lambda=1)-\\int_{0}^{t_{s}}\\frac{d\\lambda}{dt}\\Bigg(\\frac{U_{int,I}(t)-U_{int,II}(t)}{V_{I}(t)-V_{II}(t)}\\Bigg)dt.\n\\label{eqn:Ps_integration}\n\\end{equation}\n\nGiven an initial coexistence condition, the integration of this equation then provides a dynamic estimator for the entire coexistence curve from two non-equilibrium simulations, one for each phase, which are performed simultaneously. In practice, one ought to consider finite changes of the parameter $\\lambda$ in Eq.~\\ref{eqn:Ps_integration}, in such a way we obtain\n\\begin{equation}\nP_{S,coex}(\\lambda_{k+1})=P_{S,coex}(\\lambda_{k})-(\\lambda_{k+1}-\\lambda_{k})\\Bigg(\\frac{\\Delta U_{I-II,k}}{\\Delta V_{I-II,k}}\\Bigg)\n\\label{eqn:Ps_coex_discreted}\n\\end{equation}\nwhere $\\Delta U_{I-II,k}=U_{int,I}(\\textbf{z}_{k})-U_{int,II}(\\textbf{z}_{k})$ and $\\Delta V_{I-II,k}=V_{I,k}-V_{II,k}$, in which $\\textbf{z}_k$ stands for the phase space coordinates of the $k$-th microstate. The initial value of $\\lambda$ is always set to be equal to 1, and it is varied according to Eq.~\\ref{eqn:T_sc} in order to attain the desired temperature $T$, depending on the rate of change of $\\lambda$ a given scaled coexistence pressure $P_{S,coex}(\\lambda)$ can be attained for different values of $\\lambda$ from Eq.~\\ref{eqn:Ps_coex_discreted}. Using the scaling relation given by Eq.~\\ref{eqn:P_sc}, we obtain the coexistence pressure $P_{coex}$, in other words, we obtain the coexistence curve as the pressure coexistence, $P_{coex}(T)$, for a wide interval of the temperature.\n\nIf the coexistence curve has a small slope (in absolute value) i.e. the pressure derivative with respect the temperature has a small value, which is common in solid-solid coexistence lines, it is more convenient use the $\\lambda$ parameter as control parameter and the Eq.~\\ref{eqn:Ps_coex_discreted}. If, on the other hand, the coexistence curve is expected to have a large slope (in absolute value) i.e. the pressure derivative with respect the temperature has a large value, which is common in solid-fluid and fluid-fluid phase boundaries, in these case is more convenient use the pressure $P$ as control parameter, which can be obtained by inverting the relationship between $\\lambda$ and $P_{S,coex}$ in Eq.~\\ref{eqn:Ps_coex_discreted}\n\\begin{equation}\n\\lambda_{k+1}=\\lambda_{k}\\Bigg(\\frac{1+P_{k}\\Delta V_{I-II,k}\/\\Delta U_{I-II,k}}{1+P_{k+1}\\Delta V_{I-II,k}\/\\Delta U_{I-II,k}}\\Bigg)\n\\label{eqn:l_coex_discreted}\n\\end{equation}\nand by using the scaling relations Eqs.~\\ref{eqn:P_sc} and ~\\ref{eqn:T_sc}, we obtain the  coexistence curve as the coexistence temperature in a wide interval of pressure $T_{coex}(P)$.\n\nEvidently, the dynamic estimators of $P_{coex}(T_{k})$ or $T_{coex}(P_{k})$ are subject to errors associated with the irreversible nature of the time-dependent scaling process, however those effects decrease quickly with increasing number of simulation steps, allowing an accurate determination of the entire phase boundary from a couple of single, relatively short dynamic scaling simulations.\n\\section{Application: results and discussion}\\label{sec:applications}\nIn this section we describe the implementation of the dCCI method in the widely used Molecular Dynamics code \\texttt{LAMMPS}. We demonstrate the use of the NE methods to calculate coexistence conditions by computing the Gibbs free-energy of the phases in a wide interval of temperature and pressure. Through the use of the dCCI method we compute the silicon phase boundaries, described by the Stillinger-Weber (SW)\\cite{Stillinger1985} potential. Complete \\texttt{LAMMPS} scripts, source code and auxiliary files are available on the following github project \\cite{Samuelgithub}.\n\nInitially we need to determine the phase coexistence conditions: (i) the melting temperature of silicon in the diamond phase (Si-cd) at 0.0 GPa. Next, we compute (ii) the melting temperature of silicon in the $\\beta$-tin phase (Si-$\\beta$-tin) at 15.0 GPa. Then, we compute (iii) the coexistence pressure between Si-cd and Si-$\\beta$-tin at 400 K. Finally, using these coexistence conditions (iv) we apply the dCCI method to calculate the phase diagram of silicon and determine the triple point predicted by the SW potential.\n\\subsection{Melting points of silicon}\nFirst, we determine the melting points of Si-cd phase at zero pressure and Si-$\\beta$-tin at $15.0$ GPa. To this end, we need compute the Gibbs free-energy curves $G(P,T)$ for the solids and liquid phases in order to determine the melting temperature $T_{m}$ at their crossing points. To achieve this we follow Refs. \\citenum{Freitas2016} and \\citenum{Leite2019}, which provides details of the methodologies for calculating the free-energy of the solid and liquid phases using the \\texttt{LAMMPS} package.\n\nWe use computational cells containing 8000 atoms in the calculations of the Si-cd and liquid phases, and in the case of Si-$\\beta$-tin it was used a cell of 7448 atoms. All systems were subject to periodic boundary conditions. Pressure and temperature control was attained using a Parrinello-Rahman\\cite{Parrinello1981} barostat and a Langevin\\cite{Schneider1978} thermostat. The corresponding equations of motion were integrated using the velocity-Verlet algorithm with a time step of $\\Delta t= 1$ fs.\n\nIn order to compute the Gibbs free-energy of Si-cd as a function of the temperature at zero-pressure, first we determine the equilibrium volume at $T_{0}=400$ K, using an initial equilibration of 0.2 ns and then averaging the volume over a time interval of 1.0 ns. Next, we compute the Gibbs free-energy at the reference temperature of 400 K, using as reference system the Einstein crystal with a spring constant of 6.113 eV\/\\AA$^{2}$. The system was then equilibrated during 0.1 ns prior to the switching runs. The unbiased dynamical-work estimators were obtained from 10 independent forward and backward realizations using a switching time of 0.2 ns to converge the results. Finally, we use the RS method to compute the Gibbs free-energy curve of Si-cd as a function of temperature at 0.0 GPa. The RS path was chosen in such a way that the scaling parameter $\\lambda$ was varied between $\\lambda=1$ to $\\lambda=\\lambda_{f}=400\/2000$, thus, covering a temperature range between $T_{0}=400$ K and $T_{f}=2000$ K, within a switching time of $t_{s}=0.5$ ns. The dynamical work was determined as the average over 10 independent realizations of the RS process in the forward and backward directions. Full details of the methodology to compute the Gibbs free-energy curve as a function of temperature at zero pressure are described in Ref. \\citenum{Freitas2016}.\n\\begin{figure}[tbp]\n    \\centering\n         \\includegraphics[scale=0.44]{Tm_diamond_liquid.eps}\n        \n     \\caption{\\label{fig:Tm1} Gibbs free-energy per atom of Si-cd and liquid phases at zero pressure. The crossing of the curves indicates the melting temperature at zero pressure.}\n\\end{figure}\n\nNow we compute the absolute Gibbs free-energy of liquid phase at zero-pressure as a function of the temperature. We start by determining the equilibrium volume at $T_{2}=2200$ K. After an initial equilibration of 0.2 ns, the average value of the volume is determined over a time interval of 1.0 ns. Next, we compute the Gibbs free-energy at a reference temperature 2200 K, using as reference system the UF model\\cite{Leite2016}, with the following parameters: p=25 and $\\sigma=2.0$. The system was equilibrated during 0.1 ns prior to the switching runs. The unbiased dynamical-work estimators were obtained from 10 independent forward and backward realizations using a switching time of 0.1 ns to converge the results. Finally, we use the RS method to compute the Gibbs free-energy curve of the liquid phase as a function of temperature at 0.0 GPa. In this case, we used a RS path in which the scaling parameter $\\lambda$ was varied between $\\lambda=1$ to $\\lambda=\\lambda_{f}=2200\/800$, thus covering a temperature range between $T_{0}=2200$ K and $T_{f}=800$ K, within a switching time of $t_{s}=0.5$ ns. The dynamical work was determined as the average over 10 independent realizations of the RS process along the forward and backward directions. Full details of the calculation of the Gibbs free-energy curve of the liquid phase as a function of temperature at zero pressure are described in Ref.~\\citenum{Leite2019}. Fig. \\ref{fig:Tm1} shows the melting temperature of Si-cd at 0.0 GPa at the value of $T_{m}=1689.2(3)$ K. This result is in agreement with that previously reported in Ref. \\citenum{Ryu2008}.\n\n\\begin{figure}[tbp]\n    \\centering\n         \\includegraphics[scale=0.44]{as-pressure-liquid.eps}\n        \n     \\caption{\\label{Fig:AS1}(\\textbf{Upper panel}) Gibbs free-energy of liquid Si as a function of pressure at 2200~K. \\textbf{Lower panel} volume per atom as a function of the dynamical pressure along the forward and backward processes}\n\\end{figure}\n\nAfter computing the melting point at zero pressure, we now determine the second melting of silicon at 15.0 GPa by the crossing of the absolute Gibbs free-energy curves of the Si-$\\beta$-tin and liquid phases. This is accomplished using a similar procedure used for Si-cd. To compute the absolute Gibbs free-energy as a function of temperature for the Si-$\\beta$-tin phase, we determine the equilibrium volume at $T_{1}=400$ K, using an initial equilibration of 0.2 ns and then the average value of the volume is determined over a time interval of 1.0 ns. Next, we compute the Helmholtz free-energy at the temperature of 400 K (Eq.~\\ref{eqn:dA}), using as reference system the Einstein crystal with a spring constant of 1.362 eV\\AA$^{2}$, the system was equilibrated during 0.1 ns prior to the switching runs. The unbiased dynamical-work estimators were obtained from 10 independent forward and backward realizations using an switching time of 0.5 ns. From these calculations, the Gibbs free-energy is obtained from the Eq.~\\ref{eqn:G_and_A}. Finally, we use the RS method to compute the Gibbs free-energy curve as a function of temperature at 15.0 GPa. In this case, during the RS path, aside from the scaling of the force done by the \\texttt{fix adapt} command in LAMMPS, we need to scale the external pressure of the barostat as well, in order to take into account the effect of volume change of the system. To this end, we modified the \\texttt{fix npt (fix nph)} to include a new tag \\texttt{ners}, which multiply the external pressure by the scaling parameter $\\lambda$. This is achieved using the following fix commands in the \\texttt{LAMMPS} script:\n\n\\texttt{variable lambda equal 1\/(1+(elapsed\/\\$\\{t\\_s\\})*(1\/\\$\\{lf\\}-1))}\n\n\\texttt{fix f1 all nph aniso 15000.0 15000.0 1.0 ners v$\\_$lambda}\n\nThe RS path chosen was such that the scaling parameter $\\lambda$ is varied between $\\lambda=1$ to $\\lambda=\\lambda_{f}=400\/1600$. Thus, covering the temperature range between $T_{0}=400$ K and $T_{f}=1600$ K. The temperature dependence of the Gibbs free-energy is then computed using Eq.~\\ref{eqn:Gibbs_RS}, with a switching time of $t_{s}=0.5$ ns. The dynamical work was determined as the average over 10 independent realizations of the RS process along the forward and backward directions.\n\n\\begin{figure}[tbp]\n    \\centering\n         \\includegraphics[scale=0.44]{Tm_beta_tin_liquid.eps}\n        \n     \\caption{\\label{fig:Tm2} Gibbs free-energy per atom of Si-$\\beta$-tin and liquid phases at 15.0 GPa. The crossing of the curves indicates the melting temperature at 15.0 GPa.}\n\\end{figure}\n\nTo compute the absolute Gibbs free-energy as a function of the temperature of the liquid phase at 15.0 GPa, we follow a different procedure. First, we determine the Gibbs free-energy as a function of pressure at 2200 K using the AS method (Eq.~\\ref{eqn:Gibbs_AS}), in which the external pressure $P$ is changed dynamically from 0.0 GPa to 15.0 GPa. The system was equilibrated during 0.1 ns prior to the switching runs, and the unbiased dynamical work (Eq.~\\ref{eqn:Wmech_dyn}) was obtained from 10 independent forward and backward realizations using switching times between 0.01 ns to 0.2 ns. We observed a very quick convergence of the dynamical work for switching times of 0.05 ns. The Gibbs free-energy of the liquid phase at 2200~K as a function of pressure, as well as the evolution of the atomic volume with pressure are shown in Fig.~\\ref{Fig:AS1}. Finally, we use the RS method to compute the Gibbs free-energy curve for the liquid phase as a function of temperature at 15.0 GPa using an identical procedure to that used in the case of Si-$\\beta$-tin. This is achieved using the following fix commands in the \\texttt{LAMMPS} script:\n\n\\texttt{variable lambda equal 1\/(1+(elapsed\/\\$\\{t\\_s\\})*(1\/\\$\\{lf\\}-1))}\n\n\\texttt{fix f1 all nph iso 15000.0 15000.0 1.0 ners v$\\_$lambda}\n\nThe RS path chosen was such the scaling parameter $\\lambda$ is varied between $\\lambda=1$ to $\\lambda=\\lambda_{f}=2200\/1000$. Thus, covering a temperature range between $T_{0}=2200$ K and $T_{f}=1000$ K. The temperature dependence of the Gibbs free-energy is then computed using Eq.~\\ref{eqn:Gibbs_RS}, with a switching time of $t_{s}=0.5$ ns. The dynamical work was determined as the average over 10 independent realizations of the RS process along the forward and backward directions. Fig. \\ref{fig:Tm2} shows the melting temperature of silicon  at 15.0 GPa at $T_{m}=1365.3(3)$ K\n\n\\subsection{Pressure coexistence point between solids phases}\nTo determine the pressure coexistence of the solid phases at 400 K, we compute the Gibbs free-energy curves $G(P, T)$ for both solids as a function of pressure along an isothermal path by applying the AS method (Eq.~\\ref{eqn:Gibbs_AS}). From the knowledge of the Gibbs free-energy for Si-cd at 400 K and zero pressure, we compute the dynamical work required to change the pressure dynamically between 0.0 GPa to 15.0 GPa, in order to determine the Gibbs free-energy of Si-cd at 15.0 GPa and 400 K. The system at both pressures was equilibrated during 0.1 ns prior to the switching runs. The unbiased dynamical estimators of the mechanical work were obtained from 10 independent forward and backward realizations using switching times between 0.01 ns to 0.2 ns. Also in this case, we observed a very quick convergence of the dynamical work for switching times of 0.05 ns. For Si-$\\beta$-tin, the pressure is varied dynamically between 15.0 GPa to 5.0 GPa, using the same protocol used for Si-cd. Again is observed a very quick convergence of dynamical work for switching times of 0.05 ns. Fig. \\ref{fig:Pcoex} shows the coexistence pressure at 400K of the solid phases at the value of $P_{coex}=13.17346(5)$ GPa, which is in agreement with the value previously reported by Romano et al. \\cite{Romano2014}.\n\\begin{figure}[htbp]\n    \\centering\n         \\includegraphics[scale=0.44]{Pcoex_diamond_beta_tin.eps}\n        \n     \\caption{\\label{fig:Pcoex} Gibbs free-energy per atom of Si-cd and Si-$\\beta$-tin phases at 400 K. The crossing indicates the pressure coexistence at 400 K.}\n\\end{figure}\n\\subsection{Phase diagram of silicon:}\nFirst, we determine the phase boundary between the Si-cd and liquid phases using the coexistence point determined at 0.0 GPa, as an initial condition to apply the dCCI method. In this case, the external pressure was chosen to be the independent variable in the integration process and varied linearly between the boundaries $P=0.0$ GPa and $P=10.2$ GPa for scaling times between 1 ps to 200 ps for testing the convergence of the calculation. Each process was started from equilibrated configurations in both cells corresponding to the known coexistence condition at $P=0.0$ GPa.\n\nThese calculations are accomplished using the following commands, \\texttt{fix adapt\/dcci} and \\texttt{dcci} together, the command  \\texttt{fix adapt\/dcci} has a similar functionality of \\texttt{fix adapt}, but in this case the scaling parameter is controlled by the \\texttt{dcci} command, which must run using one or more processors per each phase. The usage of these commands in the \\texttt{LAMMPS} script is:\n\n\\texttt{fix f1 all nph iso \\$\\{Pcoex\\} \\$\\{Pcoex\\} \\$\\{Pdamp\\}}\n\n\\texttt{fix f2 all langevin \\$\\{Tcoex\\} \\$\\{Tcoex\\} \\$\\{Tdamp\\} 1234 zero yes}\n\n\\texttt{fix f3 all adapt\/dcci \\$\\{lambda\\} pair sw fscale * *}\n\n\\texttt{dcci \\$\\{Tcoex\\} \\$\\{Pcoex\\} \\$\\{lambda\\} f3 f1 \\$\\{t\\_s\\} press \\$\\{Pi\\} \\$\\{Pf\\}}\n\nThe \\texttt{dcci} command performs the time-dependent calculation of the coexistence curve,  \\texttt{\\$\\{Tcoex\\}} and \\texttt{\\$\\{Pcoex\\}} define the initial coexistence condition, \\texttt{\\$\\{lambda\\}} is the scaling parameter and must be defined as well in the command \\texttt{fix adapt\/dcci}, which is identified by fix id \\texttt{f3}. Also the command \\texttt{dcci} needs to control the external pressure of the barostat, which is identified by the command fix id \\texttt{f1}, \\texttt{\\$\\{t\\_s\\}} is the scaling time. Finally the syntax \\texttt{press \\$\\{Pi\\} \\$\\{Pf\\}}, which indicates the initial and final pressures along the coexistence the curve, is used to obtain the coexistence temperature as a function of the pressure i.e. $T_{coex}(P)$. \n\\begin{figure}[tbp]\n    \\centering\n         \\includegraphics[scale=0.44]{dcci_diamon_liquid.eps}\n        \n     \\caption{\\label{fig:dcci1} Melting curves of Si-cd for different scaling times. The points represent the melting temperature at zero pressure and 9.4 GPa determined through the NE methods}\n\\end{figure}\n\nIn Fig. \\ref{fig:dcci1} it is shown the phase boundaries between Si-cd and liquid for different scaling times. We observe the convergence of the results for scaling times above 50.0 ps. In order to estimate the effects associated with the irreversible nature of dCCI, we investigated the convergence of the coexistence temperatures as a function of the scaling time $t_{s}$, compared with the melting point of Si-cd at $P=9.4$ GPa calculated by the AS and RS methods (see Supplementary material Figs. S1-S4). \n\n\nFig. \\ref{fig:dcci2} shows the convergence of the coexistence temperature at the pressure P=9.4 GPa determined from dCCI runs. The coexistence temperature converges to the RS result in approximately 50 ps per process, reaching a plateau above 100 ps.\n\n\\begin{figure}[tbp]\n    \\centering\n         \\includegraphics[scale=0.44]{Tm_convergence_dcci.eps}\n        \n     \\caption{\\label{fig:dcci2} Points indicate the melting temperature of Si-cd at P=9.4 GPa as a function of the scaling time determined by the dCCI method and the dashed red line corresponds to the value obtained by the RS method.}\n\\end{figure}\n\nWe now turn to the other phase boundary between the Si-$\\beta$-tin and liquid phase. By using the coexistence point determined at 15.0 GPa, as an initial condition to apply the dCCI method, the pressure is varied linearly between the boundaries P=15.0 GPa and P=9.4 GPa for an scaling time of 100 ps. The procedure is practically identical to the previous case, but with an import difference, in the case of Si-cd and liquid phases, where we used isotropic volume fluctuations, because both phases present isotropic symmetry \\texttt{fix f1 all nph iso}, in this case, the Si-$\\beta$-tin phase presents anisotropic symmetry, to include the correct volume fluctuations for each system, i.e. isotropic for liquid \\texttt{fix nph iso} and anisotropic \\texttt{fix nph aniso} for the Si-$\\beta$-tin, we need to define a different style of barostat for each phase in the \\texttt{LAMMPS} script, this is achieved using the following commands:\n\n\\texttt{variable barostat word iso aniso}\n\n\\texttt{fix f1 all nph \\$\\{barostat\\} \\$\\{Pcoex\\} \\$\\{Pcoex\\} \\$\\{Pdamp\\}}\n\n\\texttt{fix f2 all langevin \\$\\{Tcoex\\} \\$\\{Tcoex\\} \\$\\{Tdamp\\} 1234 zero yes}\n\n\\texttt{fix f3 all adapt\/dcci \\$\\{lambda\\} pair sw fscale * *}\n\n\\texttt{dcci \\$\\{Tcoex\\} \\$\\{Pcoex\\} \\$\\{lambda\\} f3 f1 \\$\\{t\\_s\\} press \\$\\{Pi\\} \\$\\{Pf\\}}\n\n\\begin{figure}[t]\n    \\centering\n         \\includegraphics[scale=0.44]{phase_diagram.eps}\n     \\caption{\\label{fig:dcci3} Phase boundaries between silicon phases by the dCCI method. The dashed lines correspond to experimental results (the symbols {\\color{blue}$\\square$} Ref. \\citenum{Jayaraman1963}, {\\color{red}$\\square$} Ref. \\citenum{Bundy1964} and {\\color{black}$\\square$} Ref. \\citenum{Voronin2003}). Symbols {\\color{black}$\\square$} to the left indicate the pressure in which for a given temperature the Si-$\\beta$-tin phase begins to appear, whereas symbols {\\color{black}$\\square$} to the right indicate the pressure in which for that given temperature the Si-cd phase disappears completely.  \\textbf{Inset}: triple point is indicated by the crossing of coexistence curves.}\n\\end{figure}\nNext, we determine the coexistence curve between the solid phases, by using the coexistence point determined at 400 K, as an initial  condition to apply the dCCI method. In this case is more convenient to control the temperature, because the coexistence curve between the solid phases exhibit a very small change in pressure, whereas temperature varies in a large interval. Therefore, temperature is varied linearly between 400 K to 1250 K in a scaling time of 100 ps, each process was started from equilibrated configurations in both cells corresponding to the known coexistence condition at 400 K, with isotropic and anisotropic volume fluctuations for the Si-cd and Si-$\\beta$-tin, respectively. This is achieved using the following commands:\n\n\\texttt{variable barostat word iso aniso}\n\n\\texttt{fix f1 all nph \\$\\{barostat\\} \\$\\{Pcoex\\} \\$\\{Pcoex\\} \\$\\{Pdamp\\}}\n\n\\texttt{fix f2 all langevin \\$\\{Tcoex\\} \\$\\{Tcoex\\} \\$\\{Tdamp\\} 1234 zero yes}\n\n\\texttt{fix f3 all adapt\/dcci \\$\\{lambda\\} pair sw fscale * *}\n\n\\texttt{dcci \\$\\{Tcoex\\} \\$\\{Pcoex\\} \\$\\{lambda\\} f3 f1 \\$\\{t\\_s\\} temp \\$\\{Ti\\} \\$\\{Tf\\}}\nthe syntax is similar to that described previously, but in this case we need to indicate the control of the temperature in the \\texttt{dcci} command, using the following command \\texttt{temp \\$\\{Ti\\} \\$\\{Tf\\}}, which indicates the initial and final temperature along the coexistence the curve to obtain the coexistence pressure as a function of temperature ($P_{coex}(T)$). \n\nFinally, in Fig. \\ref{fig:dcci3} the phase diagram of silicon modeled by the SW potential is shown for the three phases considered in this work. The found triple point is $T_{p}$=(1210(2) K, 9.90(3) GPa), in agreement with previous values reported by Romano et al. \\cite{Romano2014}. We have also compared our results with previous experimental results \\cite{Jayaraman1963,Bundy1964,Voronin2003}. We note discrepancies between the phase diagram calculated using non-equilibrium methods and experimental results, can be attributed to the potential model chosen to describe the silicon phases, since originally the Stillinger-Weber potential was developed to study silicon in the diamond and liquid phase at low pressure, as can be observed by the agreement between our results and the experimental findings.\n\nDespite the quantitative discrepancies between the computed and experimental phase diagrams, qualitative agreement between them is observed: such as the negative slope of the fusion curve of the silicon diamond phase, the positive slope of the fusion curve of the silicon $\\beta$-tin phase, and the negative slope of the phase boundaries between the solids phases. Furthermore, it is remarkable that the crossing of the calculated coexistence lines define a very small region, spanning few degrees and hundredths of GPa, whose size is similar to that defined by the error bars in temperature and pressure. Thus, clearly defining the triple point for this model of Si. However, considering that the SW potential was designed to provide the experimental melting temperature of the Si-cd phase at low pressure and the description of other solid phases, such as Si-$\\beta$-tin, were not taken into account, the agreement between our results and those from experiments is quite reasonable.\n\\section{Summary}\\label{sec:conclusions}\nThis paper provides a guide for computing the Gibbs free-energy in a wide interval of the temperatures and pressures, which is used to determine the phase boundaries through the dCCI method within the \\texttt{LAMMPS} MD simulation package. In addition, it describes implementation details in \\texttt{LAMMPS} and makes available the computational tools in the form of full source code, scripts and auxiliary files.\n\nAs an illustrative example, the phase diagram of silicon model by SW potential was determined. Initially, the coexistence states between the phases of silicon were determined by the use of AS and RS methods, that allowed to efficiently calculate the Gibbs free-energy of the phases in a wide interval of temperatures and pressures. After that, with the knowledge the coexistence states between phases, the dCCI method was applied to determine the phase boundaries of Silicon using single non-equilibrium simulations. \n\nThe techniques described in this paper, together with the supplied source code,  scripts  and  post-processing  files,  provide  a  platform  for  computing the phase boundaries of atomistic systems, which can be easily and efficiently determined using the \\texttt{LAMMPS} software.\n\n\\section*{Acknowledgments} \nS.C. acknowledges the Brazilian agency FAPESP for the Post-Doctorate Scholarship under Grant No. 21\/03224-3. \n\nA.A. gratefully acknowledges support from the Brazilian agencies CNPq and FAPESP under Grants No. 2010\/16970-0, No. 2013\/08293-7, No. 2015\/26434-2, No. 2016\/23891-6, No. 2017\/26105-4, and No. 2019\/26088-8. The calculations were performed at CCJDR-IFGW-UNICAMP and at CENAPAD-SP in Brazil.\n\n\\section*{Appendix A. Supplementary data} \nThe following are the Supplementary data to this article can be found online at\n\\url{https:\/\/drive.google.com\/file\/d\/1wcnm0V9Uun0ZDzjEfcHtn_D5eGmo2LZ9\/view?usp=sharing}.\n\n\\bibliographystyle{elsarticle-num}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Introduction}\n\"How did the Milky Way form?\" is likely the most fundamental question facing the field of Galactic archaeology. When posed in a cosmological context, the $\\Lambda$-CDM model predicts that the Galaxy formed in great measure via the process of hierarchical mass assembly. In this scenario, nearby satellite galaxies are consumed by the Milky Way due to them being attracted to its deeper gravitational potential, and as a result merge with the Galaxy. In such cases, these merger events shape the formation and evolution of the Milky Way. Therefore, an understanding of the assembly history of the Milky Way in the context of $\\Lambda$-CDM depends critically on the determination of the properties of the systems accreted during the Galaxy's history, including their masses and chemical compositions. Moreover, the merger history of the Galaxy has a direct impact on its resulting stellar populations at present time, and plays a vital role in shaping its components.\n\n Since the seminal work by \\citet{Searle1978}, many studies have aimed at characterising the stellar populations of the Milky Way, linking them to either an \"\\emph{in situ}\" or accreted origin. Although detection of substructure in phase space has worked extremely well for the identification of on-going and\/or recent accretion events (\\citealp[e.g. Sagittarius dSph,][]{Ibata1994}; \\citealp[Helmi stream,][]{Helmi1999}), the identification of accretion events early in the life of the Milky Way has proven difficult due to phase-mixing. A possible solution to this conundrum resides in the use of additional information, typically in the form of detailed chemistry and\/or ages (\\citealp[e.g.,][]{Nissen2010,Hawkins2015,Hayes2018,Haywood2018,Mackereth2019b,Das2020,Montalban2020,Horta2021, Hasselquist2021, Buder2022,Carrillo2022}).\n\nThe advent of large spectroscopic surveys such as APOGEE \\citep{Majewski2017}, GALAH \\citep{DeSilva2015}, SEGUE \\citep{Segue2009}, RAVE \\citep{Steinmetz2020}, LAMOST \\citep{Zhao2012}, H3 \\citep{Conroy2019}, amongst others, in combination with the outstanding astrometric data supplied by the \\emph{Gaia} satellite \\citep{Gaia2018,Gaiaedr3}, revolutionised the field of Galactic archaeology, shedding new light into the mass assembly history of the Galaxy.\n\nThe core of the Sagittarius dSph system and its still forming tidal stream \\citep[][]{Ibata1994} have long served as an archetype for dwarf galaxy mergers in the Milky Way. Moreover, in the past few years, several phase-space substructures have been identified in the field of the Galactic stellar halo that are believed to be the debris of satellite accretion events, including the \\emph{Gaia}-Enceladus\/Sausage (\\citealp[GE\/S,][]{Helmi2018,Belokurov2018,Haywood2018,Mackereth2019b}), Heracles \\citep{Horta2021}, Sequoia (\\citealp[][]{barba2019,Matsuno2019,Myeong2019}), Thamnos 1 and 2 \\citep{Koppelman2019b}, Nyx \\citep[][]{Necib2020}, the substructures identified using the H3 survey: namely Aleph, Arjuna, I'itoi \\citep[][]{Naidu2020}, LMS-1 \\citep[][]{Yuan2020}\\footnote{This structure also goes by the name of Wukong \\citep[][]{Naidu2020}.}, Icarus \\citep[][]{Refiorentin2021}, Cetus \\citep[][]{Newberg2009}, and Pontus \\citep[][]{Malhan2022}. While the identification of these substructures is helping constrain our understanding of the mass assembly history of the Milky Way, their association with any particular accretion event still needs to be clarified. Along those lines, predictions from numerical simulations suggest that a single accretion event can lead to multiple substructures in phase space \\citep[e.g.,][]{Jean2017, Koppelman2020}. Therefore, in order to ascertain the reality and\/or distinction of these accretion events, one must combine phase-space information with detailed chemical compositions for large samples.\n\nRecent work modelling the stellar halo suggests that accreted populations constitute over two thirds of all of the halo stellar mass within $\\sim$25 kpc from the Galactic centre \\citep[e.g.,][]{Mackereth2020}, of which approximately 30$\\%$-50$\\%$ ($\\sim$3$\\times$10$^{8}$M$\\odot$), is associated with GE\/S. This result is in agreement with recent work by \\citet{Naidu2020}, which find that >~65$\\%$ of stellar halo populations in the inner $\\sim$20 kpc have an accreted origin, with the majority being from the GE\/S progenitor. Other studies have estimated an even higher stellar mass to this accretion event (namely, $\\sim$0.5-$2 \\times 10^{9}$M$\\odot$), suggesting that most (if not all) of the mass of the inner $\\sim$20 kpc of the Galactic stellar halo is comprised of a single accretion event (\\citealp[e.g.,][]{Helmi2018,Vincenzo2019,Das2020}). Another important source of halo stellar mass not accounted for in previous studies is Heracles, whose progenitor mass was estimated by \\citet{Horta2021} to be of the order of $\\sim$5$\\times$10$^{8}$M$\\odot$. \n\n\nThe combined estimated mass from accreted populations within $\\sim$20kpc, when added to the estimated mass from dissolved and\/or evaporated globular clusters (\\citealp[GCs, namely, $\\sim$1-2$\\times$10$^{8}$M$\\odot$][]{Schiavon2017_nrich,Horta2021_nrich}), outweighs earlier estimates of the mass of the Galactic stellar halo, namely $\\sim$4-7$\\times$10$^{8}$M$\\odot$ \\citep[see][and references therein]{Bland2016}. When compared to more recent estimates, the sum of the mass arising from accreted populations and dissolved and\/or evaporated globular clusters yields a total mass that is approximately equivalent to the total estimated mass of the stellar halo, namely $\\sim$1-1.3$\\times$10$^{9}$M$\\odot$ (\\citealp[e.g.,][]{Deason2019,Mackereth2020}), therefore suggesting a very small contribution from \\emph{in situ} halo populations at low [Fe\/H].\n\nAside from assessing the mass contributed from accreted halo populations, understanding the chemo-dynamical properties of each substructure enables the characterisation of each accretion event. By comparing the chemo-dynamical properties of each substructure with $in$ $situ$ populations, it is possible to infer properties of the debris' progenitor, helping shed light into the Galaxy's past life and environment. The properties of these disrupted satellite galaxies also provide a window for near-field cosmology, and the study of lower-mass galaxies in unrivalled detail.\n\nIn this work we set out to combine the latest data releases from the APOGEE and \\emph{Gaia} surveys in order to dynamically determine and chemically characterise previously identified halo substructures in the Milky Way. We attempt, where possible, to define the halo substructures using kinematic information only, so that the distributions of stellar populations in various chemical planes can be studied in an unbiased fashion. This allows us to understand in more detail the reality and nature of these identified halo substructures, as chemical abundances encode more pristine fossilised records of the formation environment of stellar populations in the Galaxy.\n\n\nThe paper is organised as follows: Section \\ref{data} presents our selection of the parent sample used in this work. Section~\\ref{method} includes a detailed description of how we identified the halo substructures. In Section~\\ref{kinematics}, we discuss the resulting orbital distributions of the identified structures in the orbital energy and angular momentum plane. Section \\ref{chemistry} presents an in-depth chemical analysis of the identified halo substructures in different chemical abundance planes, where we show the $\\alpha$ elements in Section~\\ref{sec_alphas}, the iron-peak elements in Section~\\ref{sec_ironpeak}, the odd-Z elements in Section~\\ref{sec_oddZ}, the carbon and nitrogen abundances in Section~\\ref{sec_cn}, a neutron capture element (namely, Ce) in Section~\\ref{sec_neutron_capture}, and the [Mg\/Mn]-[Al\/Fe] chemical composition plane in Section~\\ref{sec_other_chem}. The chemical abundances for each halo substructure are then quantitatively compared in Section~\\ref{sec_abundances}. We then discuss our results in the context of previous work in Section \\ref{discussion}, and finalise this work by presenting our conclusions in Section~\\ref{conclusion}.\n\n\n\\section{Data and sample} \n\\label{data} \nThis paper combines the latest data release \\citep[DR17,][]{sdss2021} of the SDSS-III\/IV (\\citealp{Eisenstein2011,Blanton2017}) and APOGEE survey (\\citealp[][]{Majewski2017}) with distances and astrometry determined from the early third data release from the \\textit{Gaia} survey \\citep[EDR3,][]{Gaia2020}. The celestial coordinates and radial velocities supplied by APOGEE \\citep[][Holtzman et al, in prep]{Nidever2015}, when combined with the proper motions and inferred distances \\citep{Leung2019} based on \\textit{Gaia} data, provide complete 6-D phase space information for approximately $\\sim$730,000 stars in the Milky Way, for most of which exquisite abundances for up to $\\sim$20 different elements have been determined.\n\nAll data supplied by APOGEE are based on observations collected by (almost) twin high-resolution multi-fibre spectrographs \\citep{Wilson2019} attached to the 2.5m Sloan telescope at Apache Point Observatory \\citep{Gunn2006}\nand the du Pont 2.5~m telescope at Las Campanas Observatory\n\\citep{BowenVaughan1973}. Elemental abundances are derived from\nautomatic analysis of stellar spectra using the ASPCAP\npipeline \\citep{Perez2015} based on the FERRE\\footnote{github.com\/callendeprieto\/ferre} code \\citep[][]{Prieto2006} and the line lists from \\citet{Cunha2017} and \\citet[][]{Smith2021}. The spectra themselves were\nreduced by a customized pipeline \\citep{Nidever2015}. For details on\ntarget selection criteria, see \\citet{Zasowski2013} for APOGEE and \\citet{Zasowski2017} for APOGEE-2, and \\citet[][]{Beaton2021} for APOGEE north, and \\citet[][]{Santana2021} for APOGEE south.\n\nWe make use of the distances for the APOGEE DR17 catalogue generated by \\citet{Leung2019b}, using the \\texttt{astroNN} python package \\citep[for a full description, see][]{Leung2019}. These distances are determined using a re-trained \\texttt{astroNN} neural-network software, which predicts stellar luminosity from spectra using a training set comprised of stars with both APOGEE DR17 spectra and \\textit{Gaia} EDR3 parallax measurements \\citep{Gaia2020}. The\nmodel is able to simultaneously predict distances and account for\nthe parallax offset present in \\textit{Gaia}-EDR3, producing high precision,\naccurate distance estimates for APOGEE stars, which match well\nwith external catalogues \\citep[][]{Hogg2019} and standard candles like red clump stars \\citep[][]{Bovy2014}. We note that the systematic bias in distance measurements at large distances for APOGEE DR16 as described in \\citet[][]{Bovy2019} have been reduced drastically in APOGEE DR17. Therefore, we are confident that this bias will not lead to unforeseen issues during the calculation of the orbital parameters. Our samples are contained within a distance range of $\\sim$20 kpc and have a mean $d_{\\mathrm{err}}$\/$d$ $\\sim$0.13 (except for the Sagittarius dSph, which extends up to $\\sim$30 kpc and has a mean $d_{\\mathrm{err}}$\/$d$ $\\sim$0.16).\n\nWe use the 6-D phase space information\\footnote{The positions, proper motions, and distances are taken\/derived from \\textit{Gaia} EDR3 data, whilst the radial velocities are taken from APOGEE DR17.} and convert between astrometric parameters and Galactocentric cylindrical coordinates, assuming a solar velocity combining the proper motion from Sgr~A$^{*}$ \\citep{Reid2020} with the determination of the local standard of rest of \\citet{Schonrich2010}. This adjustment leads to a 3D velocity of the Sun equal to [U$_{\\odot}$, V$_{\\odot}$, W$_{\\odot}$] = [--11.1, 248.0, 8.5] km s$^{-1}$. We assume the distance between the Sun and the Galactic Centre to be R$_{0}$ = 8.178~kpc \\citep{Gravity2019}, and the vertical height of the Sun above the midplane $z_{0}$ = 0.02~kpc \\citep{Bennett2019}. Orbital parameters were then determined using the publicly available code \\texttt{galpy}\\footnote{\\href{https:\/docs.galpy.org\/en\/v1.6.0\/}{https:\/docs.galpy.org\/en\/v1.6.0\/.}} (\\citealp[][]{Galpy2015,Galpy2018}), adopting a \\citet{McMillan2017} potential and using the St\\\"ackel approximation of \\citet[][]{Binney2012}.\n\nThe parent sample employed in this work is comprised of stars that satisfy the following selection criteria:\n\\begin{itemize}\n    \\item APOGEE-determined atmospheric parameters: $3500<\\mathrm{T}_{\\rm eff}<5500$~K and $\\log{g}< 3.6$,\n    \\item APOGEE spectral S\/N $>$ 70,\n    \\item APOGEE \\texttt{STARFLAG} = 0,\n    \\item  astroNN distance accuracy of $d_{\\odot, \\rm err}\/d_{\\odot}<0.2$,\n\\end{itemize}\nwhere $d_{\\odot}$ and $d_{\\odot\\rm err}$ are heliocentric distance and its uncertainty, respectively. The S\/N criterion was implemented to maximise the quality of the elemental abundances. The T$_{\\rm eff}$ and $\\log{g}$ criteria aimed to minimise systematic effects at high\/low temperatures, and to minimise contamination by dwarf stars. We also removed stars with \\texttt{STARFLAG} flags set, in order to not include any stars with issues in their stellar parameters. \nA further 7,750 globular cluster stars were also removed from consideration using the APOGEE Value Added Catalogue of globular cluster candidate members from Schiavon et al. (2022, in prep.), \\citep[building on the method from][using primarily radial velocity and proper motion information]{Horta2020}. Finally, stars belonging to the Large and Small Magellanic clouds were also excluded using the sample from \\citet[][]{Hasselquist2021} (removing 3,748 and 1,002 stars, respectively). The resulting parent sample contains 199,077 stars.\n\nIn the following subsection we describe the motivation behind the selection criteria employed to select each substructure in the stellar halo of the Milky Way. The criteria are largely built on selections employed in previous works and are summarised in Table \\ref{tab:selection} and in Figure~\\ref{fig:hr}.\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{plots_submit\/hr_icarus_pontus_new.png}\n\\caption{Distribution of the identifed substructures in the Kiel diagram. The parent sample is plotted as a 2D histogram, where white\/black signifies high\/low density regions. The coloured markers illustrate the different halo substructures studied in this work. For the bottom right panel, green points correspond to Pontus stars, whereas the purple point is associated with Icarus. Additionally, in the Aleph and Nyx panels, we also highlight with purple edges those stars that overlap between the APOGEE DR17 sample and the samples determined in \\citet[][]{Naidu2020} and \\citet[][]{Necib2020}, respectively, for these halo substructures.}\n\\label{fig:hr}\n\\end{figure*}\n\n\\subsection{Identification of substructures in the stellar halo} \n\\label{method}\n\nWe now describe the method employed for identifying known substructures in the stellar halo of the Milky Way. We set out to select star members belonging to the various halo substructures.\n\nWe strive to identify substructures in the stellar halo by employing solely orbital parameter and phase-space information where possible, with the aim of obtaining star candidates for each substructure population that are unbiased by any chemical composition selection. \n\nWe take a handcrafted approach and select substructures based on simple and reproducible selection criteria that are physically motivated by the data and\/or are used in previous works, instead of resorting to clustering software algorithms, which we find cluster the $n$-dimensional space into too many fragments. In the following subsections we describe the selection procedure for identifying each substructure independently.\n\n We note that our samples for the various substructures are defined by a strict application to the APOGEE survey data of the criteria defined by other groups, often on the basis of different data sets.  The latter were per force collected as part of a different observational effort, based on specific target selection criteria.  It is not immediately clear whether or how differences between the APOGEE selection function and those of other catalogues may imprint dissimilarities between our samples and those of the original studies.  We nevertheless do not expect such effects to influence our conclusions in an important way.\n\n\\subsubsection{Sagittarius}\n\\label{sec_sgr} \nSince its discovery \\citep{Ibata1994}, many studies have sought to characterise the nature of the Sagittarius dwarf spheroidal (\\citealp[hereafter Sgr dSph; e.g.,][]{Ibata2001,Majewski2003,Johnston2005,Belokurov2006,Yanny2009,Koposov2012,Carlin2018,Vasiliev2020}), as well as interpret its effect on the Galaxy using numerical simulations (\\citealp[e.g.,][]{Johnston1995,Ibata1997,Law2005,Law2010,Purcell2011,Gomez2013}). More recently, the Sgr~dSph has been the subject of comprehensive studies on the basis of APOGEE data. This has enabled a detailed examination of its chemical compositions, both in the satellite's core and in its tidal tails (\\citealp[e.g.,][]{Hasselquist2017,Hasselquist2019,Hayes2020}). Moreover, in a more recent study, \\cite{Hasselquist2021} adopted chemical evolution models to infer the history of star formation and chemical evolution of the Sgr dSph.  Therefore, in this paper the Sgr dSph is considered simply as a template massive satellite whose chemical properties can be contrasted to those of the  halo substructures that are the focus of our study. \n \n \nWhile it is possible to select high confidence Sgr~dSph star candidates using Galactocentric positions and velocities \\citep{Majewski2003}, \\citet{Hayes2020} showed it is possible to make an even more careful selection by considering the motion of stars in a well-defined Sgr orbital plane.  We identify Sgr star members by following the method from \\citet{Hayes2020}. Although the method is fully described in their work, we summarise the key steps for clarity and completeness. We take the Galactocentric positions and velocities of stars in our parent sample and rotate them into the Sgr orbital plane according to the transformations described in \\citet{Majewski2003}. This yields a set of position and velocity coordinates relative to the Sgr orbital plane, but still centered on the Galactic Centre. As pointed out in \\citet{Hayes2020}, Sgr star members should stand out with respect to other halo populations in different Sgr orbital planes. Using this orbital plane transformation, we select from the parent sample Sgr star members if they satisfy the following selection criteria:\n\n\\begin{itemize}\n\\item |$\\beta_{\\mathrm{GC}}$| < 30 ($^{\\circ}$),\n\\item 18 < L$_{z,\\mathrm{Sgr}}$ < 14 ($\\times$10$^{3}$ kpc kms$^{-1}$),\n\\item --150 < $\\mathrm{V}_{z,\\mathrm{Sgr}}$ < 80 (kms$^{-1}$),\n\\item  X$_{\\mathrm{Sgr}}$ > 0 or X$_{\\mathrm{Sgr}}$ < --15 (kpc), \n\\item Y$_{\\mathrm{Sgr}}$ > --5 (kpc) or Y$_{\\mathrm{Sgr}}$ < --20 (kpc),\n\\item  Z$_{\\mathrm{Sgr}}$ > --10 (kpc),\n\\item pm$_{\\alpha}$ > --4 (mas),\n\\item $d_{\\odot}$ > 10 (kpc),\n\\end{itemize}\nwhere $\\beta_{\\mathrm{GC}}$ is the angle subtended between the Galactic Centre and the Sgr dSph, L$_{\\mathrm{z,Sgr}}$ is the azimuthal component of the angular momentum in the Sgr plane, $\\mathrm{V}_{\\mathrm{z,Sgr}}$ is the vertical component of the velocity in the Sgr plane, (X$_{\\mathrm{Sgr}}$, Y$_{\\mathrm{Sgr}}$, Z$_{\\mathrm{Sgr}}$) are the cartesian coordinates centred on the Sgr dSph plane, pm$_{\\alpha}$ is the right-ascension proper motion, and $d_{\\odot}$ is the heliocentric distance, which for Sgr has been shown to be $\\sim$ 23 kpc \\citep{Vasiliev2020}. Our selection yields a sample of 266 Sgr star members, illustrated in the L$_{\\mathrm{z,Sgr}}$ vs $\\mathrm{V}_{\\mathrm{z,Sgr}}$ plane in Fig~\\ref{sag_selec}.\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{plots_dr17\/sgr_vz_lz2_v2.pdf}\n\\caption{Parent sample used in this work in the L$_{\\mathrm{z,Sgr}}$ vs $\\mathrm{V}_{\\mathrm{z,Sgr}}$ plane (see Section~\\ref{sec_sgr} for details). Here, Sgr stars clearly depart from the parent sample, and are easily distinguishable by applying the selection criteria from \\citet[][]{Hayes2020}, demarked in this illustration by the orange markers.}\n    \\label{sag_selec}\n\\end{figure}\n\n\n\\subsubsection{Heracles}\nHeracles is a recently discovered metal-poor substructure located in the heart of the Galaxy \\citep{Horta2021}. It is characterised by stars on eccentric and low energy orbits. Due to its position in the inner few kpc of the Galaxy, it is highly obscured by dust extinction and vastly outnumbered by its more abundant metal-rich (\\textit{in situ}) co-spatial counterpart populations. Only with the aid of chemical compositions has it been possible to unveil this metal-poor substructure, which is discernible in the [Mg\/Mn]-[Al\/Fe] plane. It is important at this stage that we mention a couple of recent studies which, based chiefly on the properties of the Galactic globular cluster system, proposed the occurrence of an early accretion event whose remnants should have  similar properties to those of Heracles \\citep[named Kraken and Koala, by][respectively]{Kruijssen2020,Forbes2020}. In the absence of a detailed comparison of the dynamical properties and detailed chemical compositions of Heracles with these putative systems, a definitive association is impossible at the current time.\n\nIn this work we define Heracles candidate star members following the work by \\citet{Horta2021}, and select stars from our parent sample that satisfy the following selection criteria:  \n\\begin{itemize}\n\\item $e$ > 0.6,\n\\item --2.6 < E < --2 ($\\times$10$^{5}$ km$^{2}$s$^{-2}$), \n\\item $[\\mathrm{Al\/Fe}]$ < --0.07 $\\&$ [Mg\/Mn] $\\geqslant$ 0.25, \\item $[\\mathrm{Al\/Fe}]$ $\\geqslant$ --0.07 $\\&$ [Mg\/Mn] $\\geqslant$ 4.25$\\times$[Al\/Fe] + 0.5475.\n\\end{itemize}\nMoreover, we impose a [Fe\/H] > --1.7 cut to select Heracles candidate star members in order to select stars from our parent sample that have reliable Mn abundances in APOGEE DR17. Our selection yields a resulting sample of 300 Heracles star members.\n\n\n\\subsubsection{Gaia-Enceladus\/Sausage}\nRecent studies have shown that there is an abundant population of stars in the nearby stellar halo (namely, R$_{\\mathrm{GC}} \\lesssim$ 20-25 kpc) belonging to the remnant of an accretion event dubbed the $Gaia$-Enceladus\/Sausage (\\citealp[GES, e.g.,][]{Belokurov2018,Haywood2018,Helmi2018,Mackereth2019b}). This population is characterised by stars on highly radial\/eccentric orbits, which also appear to follow a lower distribution in the $\\alpha$-Fe plane, presenting lower [$\\alpha$\/Fe] values for fixed metallicity than $in$ $situ$ populations.\n\nFor this paper, we select GES candidate star members by employing a set of orbital information cuts. Specifically, GES members were selected adopting the following criteria:\n\\begin{itemize}\n\\item |L$_{z}$| < 0.5 ($\\times$10$^{3}$ kpc km$^{-1}$),\n\\item --1.6 < E < --1.1 ($\\times$10$^{5}$ km$^{2}$s$^{-2}$).\n\\end{itemize}\nThis selection is employed in order to select the clump that becomes apparent in the E-L$_{z}$ plane at higher orbital energies and roughly L$_{z}$ $\\sim$ 0 (see Fig~\\ref{elz}), and to minimise the contamination from high-$\\alpha$ disc stars on eccentric orbits (\\citealp[namely, the \"Splash\"][]{Bonaca2017,Belokurov2020}), which sit approximately at E$\\sim$--1.8$\\times$10$^{5}$ km$^{2}$s$^{-2}$ (see Kisku et al, in prep). The angular momentum restriction ensures we are not including stars on more prograde\/retrograde orbits. We find that by selecting the GES substructure in this manner, we obtain a sample of stars with highly radial (J$_{R}$ $\\sim$1x10$^{3}$ kpc kms$^{-1}$) and therefore highly eccentric ($e$$\\sim$0.9) orbits, in agreement with selections employed in previous studies to identify this halo substructure (\\citealp[e.g.,][]{Mackereth2020,Naidu2020,Feuillet2021,Buder2022}). The final GES sample is comprised of 2,353 stars.\n\nWe note that in a recent paper, \\citet[][]{Hasselquist2021} undertook a thorough investigation into the chemical properties of this halo substructure and compared it to other massive satellites of the Milky Way (namely, the Magellanic Clouds, Sagittarius dSph, and Fornax). Although their selection criteria differs slightly from the one employed in this study, we find that their sample is largely similar to the one employed here, as both studies employed APOGEE DR17 data.\n\n\n\\subsubsection{Retrograde halo: Sequoia, Thamnos, Arjuna, and I'Itoi}\nA number of substructures have been identified in the retrograde halo. The first to be discovered was Sequoia (\\citealp{barba2019,Matsuno2019,Myeong2019}), which was suggested to be the remnant of an accreted dwarf galaxy. The Sequoia was identified given the retrograde nature of the orbits of its stars, which appear to form an arch in the retrograde wing of the Toomre diagram. Separately, an interesting study by \\citet{Koppelman2019b} showed that the retrograde halo can be further divided into two components, separated by their orbital energy values in the E-L$_{z}$ plane. They suggest that the high energy component corresponds to Sequoia, whilst the lower energy population would be linked to a separate accretion event, dubbed Thamnos. In addition, \\cite{Naidu2020} proposed the existence of additional retrograde substructure overlapping with Sequoia, characterised by different metallicities, which they named Arjuna and I'Itoi.\n\nAs the aim of this paper is to perform a comprehensive study of the chemical abundances of substructures identified in the halo, we utilise all the selection methods used in prior work and select the same postulated substructures in multiple ways, in order to compare their abundances later. Specifically, we build on previous works (\\citealp[e.g.,][]{Myeong2019,Koppelman2019b,Naidu2020}) that have aimed to characterise the retrograde halo and select the substructures following a similar selection criteria.\n\nFor reference, throughout this work we will refer to the different selection criteria of substructures in the retrograde halo as the \"\\textit{GC}\", \"\\textit{field}\", and \"\\textit{H3}\" selections, in reference to the method\/survey employed to determine the Sequoia substructure in \\citet{Myeong2019}, \\citet{Koppelman_thamnos}, and \\citet{Naidu2020}, respectively. We will now go through the details of each selection method independently.\n\nThe \\textit{GC} method (used in \\citealt{Myeong2019}) selects Sequoia star candidates by identifying stars that satisfy the following conditions:\n\\begin{itemize}\n\\item E > --1.5 ($\\times$10$^{5}$ km$^{2}$s$^{-2}$),\n\\item J$_{\\phi}$\/J$_{\\mathrm{tot}}$ < --0.5,\n\\item J$_{\\mathrm{(J_{z}-J_{R})}}$\/J$_{\\mathrm{tot}}$ <0.1.\n\\end{itemize}\nHere, J$_{\\phi}$, J$_{R}$, and J$_{z}$ are the azimuthal, radial, and vertical actions, and J$_{\\mathrm{tot}}$ is the quadrature sum of those components. This selection yields a total of 116 Sequoia star candidates. \n\nThe \\textit{field} method (used in \\citealt{Koppelman_thamnos}) identifies Sequoia star members based on the following selection criteria:\n\\begin{itemize}\n\\item 0.4 < $\\eta$ < 0.65,\n\\item --1.35 < E < --1 ($\\times$10$^{5}$ km$^{2}$s$^{-2}$),\n\\item L$_{z}$ < 0 (kpc kms$^{-1}$),\n\\end{itemize}\nwhere $\\eta$ is the circularity and is defined as $\\eta$ = $\\sqrt{1 - e^{2}}$ \\citep{Wetzel2011}. These selection criteria yield a total of 95 Sequoia stars.\n\nLastly, we select the Sequoia based on the \\textit{H3} selection criteria (used in \\citealt{Naidu2020}) as follows:\n\\begin{itemize}\n\\item $\\eta$ > 0.15,\n\\item E > --1.6 ($\\times$10$^{5}$ km$^{2}$s$^{-2}$),\n\\item L$_{z}$ < --0.7 (10$^{3}$ kpc kms$^{-1}$).\n\\end{itemize}\nThis selection produces a preliminary sample comprised of 236 Sequoia stars. However, we note that \\citet{Naidu2020} use this selection to define not only Sequoia, but all the substructures in the high-energy retrograde halo (including the Arjuna and I'itoi substructures). In order to distinguish  Sequoia, I'itoi, and Arjuna, \\citet{Naidu2020} suggest performing a metallicity cut, motivated by the observed peaks in the metallicity distribution function (MDF) of their retrograde sample. Thus, we follow this procedure and further refine our Sequoia, Arjuna and I'itoi samples by requiring [Fe\/H] > --1.6 cut for Arjuna, --2 < [Fe\/H] < --1.6 for Sequoia, and [Fe\/H] < --2 for I'itoi, based on the distribution of our initial sample in the MDF (see Fig~\\ref{mdf_highe}). This further division yields an \\textit{H3} Sequoia sample comprised of 65 stars, an Arjuna sample constituted by 143 stars, and an I'itoi sample comprised of 22 stars.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{plots_dr17\/mdf_highe.pdf}\n\\caption{Metallicity distribution function of the high-energy retrograde sample determined using the selection criteria from \\citet{Naidu2020}. The dashed black vertical lines define the division of this sample used by \\citet{Naidu2020} to divide the three high-energy retrograde substructures: Arjuna, Sequoia, and I'itoi. This MDF dissection is based both on the values used in \\citet[][]{Naidu2020}, and the distinguishable peaks in this plane (we do not use a replica value of the [Fe\/H] used in \\citet{Naidu2020} in order to account for any possible metallicity offsets between the APOGEE and H3 surveys).}\n    \\label{mdf_highe}\n\\end{figure}\n\n\nFollowing our selection of substructures in the high-energy retrograde halo, we set out to identify stars belonging to the intermediate-energy and retrograde Thamnos 1 and 2 substructures. \\citet{Koppelman_thamnos} state that Thamnos 1 and 2 are separate debris from the same progenitor galaxies. For this work we consider Thamnos as one overall structure, given the similarity noted by \\citet{Koppelman_thamnos} between the two smaller individual populations in chemistry and kinematic planes. Stars from our parent sample were considered as Thamnos candidate members if they satisfied the following selection criteria:\n\\begin{itemize}\n\\item --1.8 < E < --1.6 ($\\times$10$^{5}$ km$^{2}$s$^{-2}$),\n\\item L$_{z}$ < 0 (kpc kms$^{-1}$),\n\\item $e$ < 0.7,\n\\end{itemize}\nThese selection cuts are performed in order to select stars in our parent sample with intermediate orbital energies and retrograde orbits (see Fig~\\ref{elz} for the position of Thamnos in the E-L$_{z}$), motivated by the distribution of the Thamnos substructure in the E-L$_{z}$ plane illustrated by \\citet{Koppelman_thamnos}. This selection yields a Thamnos sample comprised of 121 stars.\n\n\n\\subsubsection{Helmi stream}\nThe Helmi stream was initially identified as a substructure in orbital space due to its high V$_{\\mathrm{z}}$ velocities \\citep{Helmi1999}. More recent work by \\citet{Koppelman2019} characterised the Helmi stream in \\emph{Gaia} DR2, and found that this stellar population is best defined by adopting a combination of cuts in different angular momentum planes. Specifically, by picking stellar halo stars based on the azimuthal component of the angular momentum ($\\mathrm{L}_{z}$), and its perpendicular counterpart ($\\mathrm{L}_{\\bot}$ = $\\sqrt{\\mathrm{L}_{x}^{2}+\\mathrm{L}_{y}^{2}}$), the authors were able to select a better defined sample of Helmi stream star candidates.\nWe build on the selection criteria from \\citet{Koppelman2019} and define our Helmi stream sample by including stars from our parent population that satisfy the following selection criteria:\n\\begin{itemize}\n\\item 0.75 < $\\mathrm{L}_{z}$ < 1.7 ($\\times$10$^{3}$ kpc kms$^{-1}$),\n\\item 1.6 < $\\mathrm{L}_{\\bot}$ < 3.2 ($\\times$10$^{3}$ kpc kms$^{-1}$).\n\\end{itemize}\n\nOur final sample is comprised of 85 Helmi stream stars members.\n\n\n\\subsubsection{Aleph}\nAleph is a newly discovered substructure presented in a detailed study of the Galactic stellar halo by \\cite{Naidu2020} on the basis of the H3 survey \\citep{Conroy2019}. It was initially identified as a sequence below the high $\\alpha$-disc in the $\\alpha$-Fe plane. It is comprised by stars on very circular prograde orbits. For this paper, we follow the method described in \\citet{Naidu2020} and define Aleph star candidates as any star in our parent sample which satisfies the following selection criteria:\n\\begin{itemize}\n\\item 175 < $\\mathrm{V}_{\\phi}$ < 300 (kms$^{-1}$),\n\\item  |V$_{R}$| < 75  (kms$^{-1}$),\n\\item $[\\mathrm{Fe\/H}]$ > --0.8,\n\\item $[\\mathrm{Mg\/Fe}]$ < 0.27,\n\\end{itemize}\nwhere $\\mathrm{V}_{\\phi}$ and V$_{R}$ are the azimuthal and radial components of the velocity vector (in Galactocentric cylindrical coordinates), and we use Mg as our $\\alpha$ tracer element. The selection criteria yield a sample comprised of 128,578 stars. We find that the initial selection criteria determine a preliminary Aleph sample that is dominated by \\textit{in situ} disc stars, likely obtained due to the prograde nature of the velocity cuts employed as well as the chemical cuts. Thus, in order to remove disc contamination and select $true$ Aleph star members, we employ two further cuts in vertical height above the plane (namely, |$z$| > 3 kpc) and in vertical action (i.e., 170 < J$_{z}$ < 210 kpc kms$^{-1}$), which are motivated by the distribution of Aleph in these coordinates (see Section 3.2.2 from \\citealt{Naidu2020}). We also note that the vertical height cut was employed in order to mimic the H3 survey selection function. After including these two further cuts, we obtain a final sample of Aleph stars comprised of 28 star members.\n\n\n\\subsubsection{LMS-1}\nLMS-1 is a newly identified substructure discovered by \\citet{Yuan2020}. It is characterised by metal poor stars that form an overdensity at the foot of the omnipresent GES in the E-L$_{z}$ plane. This substructure was later also studied by \\citet[][]{Naidu2020}, who referred to it as Wukong. We identify stars belonging to this substructure adopting a similar selection as \\citet{Naidu2020}, however adopting different orbital energy criteria to adjust for the fact that we adopt the \\citet{McMillan2017} Galactic potential (see Fig 23 from Appendix B in that study). Stars from our parent sample were deemed LMS-1 members if they satisfied the following selection criteria:\n\\begin{itemize}\n\\item 0.2 < $\\mathrm{L}_{z}$ < 1 ($\\times$10$^{3}$ kpc kms$^{-1}$),\n\\item  --1.7 < E < --1.2 ($\\times$10$^{5}$ km$^{-2}$s$^{-2}$),\n\\item $[\\mathrm{Fe\/H}]$ < --1.45,\n\\item 0.4 < $e$ < 0.7,\n\\item |$z$| > 3 (kpc).\n\\end{itemize}\nWe note that the $e$ and $z$ cuts were added to the selection criteria listed by \\citet{Naidu2020}. This is because we conjectured that instead of eliminating GES star members from our selection (as \\citet[][]{Naidu2020} do), it is more natural in principle to find additional criteria that distinguishes these two overlapping substructures. Thus, we select stars on less eccentric orbits than those belonging to GES, but still more eccentric than most of the Galactic disc (i.e., 0.4 < $e$ < 0.7). Furthermore, in order to ensure we are observing stars at the same distances above the Galactic plane as in \\citet{Naidu2020} (defined by the selection function of the H3 survey), we add a vertical height cut of |$z$| > 3 (kpc). Our selection identifies 31 stars belonging to the LMS-1 substructure.\n\n\n\\subsubsection{Nyx}\nNyx has recently been put forward by \\cite{Necib2020}, who identified a stellar stream in the solar neighbourhood, that they suggest to be the remnant of an accreted dwarf galaxy \\citep{Necib2020}. Similar to Aleph, it is characterised by stars on very prograde orbits, at relatively small mid-plane distances (|Z| < 2 kpc) and close to the solar neighbourhood (i.e., |Y| < 2 kpc and |X| < 3 kpc). The Nyx structure is also particularly metal-rich (i.e., [Fe\/H] $\\sim$ --0.5). Based on the selection criteria used in \\citet{Necib2020}, we select Nyx star candidates employing the following selection criteria:\n\\begin{itemize}\n\\item 110 < V$_{R}$ < 205 (kms$^{-1}$),\n\\item 90 < V$_{\\phi}$ < 195 (kms$^{-1}$),\n\\item |X| < 3 (kpc), |Y| < 2 (kpc), |Z| < 2 (kpc).\n\\end{itemize}\nThe above selection criteria yield a sample comprising of 589 Nyx stars.\n\n\\subsubsection{Icarus}\n\\label{sec_icarus}\nIcarus is a substructure identified in the solar vicinity, comprised by stars that are significantly metal-poor ([Fe\/H] $\\sim$ --1.45) with circular (disc-like) orbits \\citep{Refiorentin2021}. In this work, we select Icarus star members using the mean values reported by those authors and adopting a two sigma uncertainty cut around the mean. The selection used is listed as follows:\n\\begin{itemize}\n    \\item $\\mathrm{[Fe\/H]}$ < --1.05,\n    \\item $\\mathrm{[Mg\/Fe]}$ < 0.2,\n    \\item 1.54 < $\\mathrm{L}_{z}$ < 2.21 ($\\times$10$^{3}$ kpc kms$^{-1}$),\n    \\item L$_{\\bot}$ < 450 (kpc kms$^{-1}$),\n    \\item $e$ < 0.2,\n    \\item $z_{\\mathrm{max}}$ < 1.5 (kpc).\n\\end{itemize}\nThese selection criteria yield an Icarus sample comprised of one star. As we have only been able to identify one star associated with this substructure, we remove it from the main body of this work and focus on discerning why our selection method only identifies 1 star in Appendix~\\ref{appen_icarus}. Furthermore, we combine the one Icarus star found in APOGEE DR17 with 41 stars found by \\citet[][]{Refiorentin2021} in APOGEE DR16, in order to study the nature of this substructure in further detail. Our results are discussed in Appendix~\\ref{appen_icarus}.  \n\n\\subsubsection{Pontus}\n\n Pontus is a halo substructure  recently proposed by \\cite{Malhan2022}, on the basis of an analysis of {\\it Gaia} EDR3 data for a large sample of Galactic globular clusters and stellar streams.  These authors identified a large number of groupings in action space, associated with known substructures.  \\cite{Malhan2022} propose the existence of a previously unknown susbtructure they call {\\it Pontus}, characterised by retrograde orbits and intermediate orbital energy.  Pontus is located just below {\\it Gaia}-Enceladus\/Sausage in the E-Lz plane, but displays less radial orbits (Pontus has an average radial action of J$_{R}$$\\sim$500 kpc kms$^{-1}$, whereas \\textit{Gaia}-Enceladus\/Sausage displays a mean value of J$_{R}$$\\sim$1,250 kpc kms$^{-1}$). In this work, we utilise the values listed in Section 4.6 from \\citet[][]{Malhan2022} to identify Pontus candidate members in our sample. We note that because both that study and ours use the \\citet[][]{McMillan2017} potential to compute the IoM, the orbital energy values will be on the same scale. Our selection criteria for Pontus are the following:\n\n\\begin{itemize}\n    \\item  --1.72 < E < --1.56 ($\\times$10$^{5}$ km$^{-2}$s$^{-2}$),\n    \\item --470 < $\\mathrm{L}_{z}$ < 5 ($\\times$10$^{3}$ kpc kms$^{-1}$),\n    \\item 245 < $\\mathrm{J}_{R}$ < 725 (kpc kms$^{-1}$),\n    \\item 115 < $\\mathrm{J}_{z}$ < 545 (kpc kms$^{-1}$),\n    \\item 390 < $\\mathrm{L}_{\\perp}$ < 865 (kpc kms$^{-1}$),\n    \\item 0.5 < $e$ < 0.8,\n    \\item 1 < $R_{\\mathrm{peri}}$ < 3 (kpc),\n    \\item 8 < $R_{\\mathrm{apo}}$ < 13 (kpc),\n    \\item $\\mathrm{[Fe\/H]}$ < --1.3,\n\\end{itemize}\n \n\\noindent where $R_{\\mathrm{peri}}$ and $R_{\\mathrm{apo}}$ are the perigalacticon and apogalacticon radii, respectively. Using these selection criteria, we identify two Pontus candidate members in our parent sample. As two stars comprise a sample too small to perform any statistical comparison, we refrain from comparing the Pontus stars in the main body of this work. Instead, we display and discuss the chemistry of these two Pontus stars in Appendix~\\ref{appen_pontus}, for completeness.\n\n\\subsubsection{Cetus}\n\nAs a closing remark, we note that we attempted to identify candidate members belonging to the Cetus \\citep{Newberg2009} stream. Using the selection criteria defined in Table 3 from \\citet[][]{Malhan2022}, we found no stars associated with this halo substructure that satisfied the selection criteria of our parent sample. This is likely due to a combination of two factors: {\\it (i)} Cetus is a diffuse stream orbiting at large heliocentric distances ($d_{\\odot}$$\\gtrsim$30 kpc, \\citealp[][]{Newberg2009}), which APOGEE does not cover well; {\\it (ii)} it occupies a region of the sky around the southern polar cap, where APOGEE does not have many field pointings, at approximately $l$$\\sim$143$^{\\circ}$ and $b$$\\sim$--70$^{\\circ}$ \\citep[][]{Newberg2009}.\n\n\n\\setlength{\\tabcolsep}{25pt}\n\\begin{table*}\n\\centering\n\\begin{tabular}{ |p{4.5cm}|p{8.75cm}|p{0.3cm}}\n\\hline\n Name & Selection criteria & N$_{\\mathrm{stars}}$\\\\\n\\hline\n\\hline\nHeracles & $e$ > 0.6; --2.6 < E < --2 ($\\times$10$^{5}$ km$^{2}$s$^{-2}$); $[\\mathrm{Al\/Fe}]$ < --0.07 $\\&$ [Mg\/Mn] $\\geqslant$ 0.25; $[\\mathrm{Al\/Fe}]$ $\\geqslant$ --0.07; [Mg\/Mn] $\\geqslant$ 4.25$\\times$[Al\/Fe] + 0.5475; [Fe\/H] > --1.7 & 300 \\\\\n\\hline\n\\textit{Gaia}-Enceladus\/Sausage &  |L$_{z}$| < 0.5 ($\\times$10$^{3}$ kpc km s$^{-1}$) ; --1.6 < E < --1.1 ($\\times$10$^{5}$ km$^{2}$s$^{-2}$) & 2353  \\\\\n\\hline\nSagittarius & |$\\beta_{\\mathrm{GC}}$|\n< 30 ($^{\\circ}$); 1.8 < L$_{z,\\mathrm{Sgr}}$ < 14 ($\\times$10$^{3}$ kpc kms$^{-1}$); --150 < $\\mathrm{V}_{z,\\mathrm{Sgr}}$ < 80 (kms$^{-1}$);  X$_{\\mathrm{Sgr}}$ > 0 (kpc) or X$_{\\mathrm{Sgr}}$ <--15 0 (kpc); Y$_{\\mathrm{Sgr}}$ > --5 (kpc) or Y$_{\\mathrm{Sgr}}$ < --20 (kpc); Z$_{\\mathrm{Sgr}}$ > --10 (kpc); pm$_{\\alpha}$ > --4 (mas); $d_{\\odot}$ > 10 (kpc) & 266\\\\\n\\hline\nHelmi stream & 0.75 < $\\mathrm{L}_{z}$ < 1.7 ($\\times$10$^{3}$ kpc kms$^{-1}$); 1.6 < $\\mathrm{L}_{\\bot}$ < 3.2 ($\\times$10$^{3}$ kpc kms$^{-1}$)& 85\\\\\n\\hline\n(\\textit{GC}) Sequoia \\citep[][]{Myeong2019} & E > --1.5 ($\\times$10$^{5}$ km$^{2}$s$^{-2}$); J$_{\\phi}$\/J$_{\\mathrm{tot}}$<--0.5; J$_{\\mathrm{(J}_{z}-\\mathrm{J}_{R})}$\/J$_{\\mathrm{tot}}$<0.1 & 116\\\\\n\\hline\n(\\textit{field}) Sequoia \\citep[][]{Koppelman_thamnos}& 0.4<$\\eta$<0.65; \n--1.35<E<--1 ($\\times$10$^{5}$ km$^{2}$s$^{-2}$); L$_{z}$<0 (kpc kms$^{-1}$) &95\\\\\n\\hline\n(\\textit{H3}) Sequoia \\citep[][]{Naidu2020}& $\\eta$>0.15; E>--1.6 ($\\times$10$^{5}$ km$^{2}$s$^{-2}$); L$_{z}$<--0.7 ($\\times$10$^{3}$ kpc kms$^{-1}$); --2 < [Fe\/H] < --1.6 & 65\\\\\n\\hline\nThamnos & --1.8 < E < --1.6 ($\\times$10$^{5}$ km$^{2}$s$^{-2}$); L$_{z}$ < 0 (kpc kms$^{-1}$); $e$ < 0.7 & 121\\\\\n\\hline\nAleph & 175 < $\\mathrm{V}_{\\phi}$ < 300 (kms$^{-1}$); |V$_{R}$| < 75  (kms$^{-1}$); Fe\/H > --0.8; Mg\/Fe < 0.27; |z| > 3 (kpc); 170 < J$_{z}$ < 210 (kpc kms$^{-1}$) & 28\\\\\n\\hline\nLMS-1 &  0.2 < $\\mathrm{L}_{z}$ < 1 ($\\times$10$^{3}$ kpc kms$^{-1}$);  --1.7 < E < --1.2 ($\\times$10$^{5}$ km$^{-2}$s$^{-2}$); [Fe\/H] < --1.45; 0.4 < $e$ < 0.7; |$z$| > 3 (kpc)& 31\\\\\n\\hline\nArjuna & $\\eta$ > 0.15; E > --1.6 ($\\times$10$^{5}$ km$^{2}$s$^{-2}$); L$_{z}$ < --0.7 ($\\times$10$^{3}$ kpc kms$^{-1}$); [Fe\/H] > --1.6 & 143\\\\\n\\hline\nI'itoi & $\\eta$ > 0.15; E > --1.6 ($\\times$10$^{5}$ km$^{2}$s$^{-2}$); L$_{z}$ < --0.7 ($\\times$10$^{3}$ kpc kms$^{-1}$); [Fe\/H] < --2 & 22\\\\ \n\\hline\nNyx &  110 < V$_{R}$ < 205 (kms$^{-1}$); 90 V$_{\\phi}$ < 195 (kms$^{-1}$); |X| < 3 (kpc), |Y| < 2 (kpc), |Z| < 2 (kpc)& 589\\\\\n\\hline\nIcarus & $\\mathrm{[Fe\/H]}$ < --1.45; $\\mathrm{[Mg\/Fe]}$ < 0.2; 1.54 < $\\mathrm{L}_{z}$ < 2.21 ($\\times$10$^{3}$ kpc kms$^{-1}$); L$_{\\bot}$ < 450 (kpc kms$^{-1}$); $e$ < 0.2; $z_{\\mathrm{max}}$ < 1.5 & 1\\\\\n\\hline\nPontus &  --1.72 < E < --1.56 ($\\times$10$^{5}$ km$^{-2}$s$^{-2}$); --470 < $\\mathrm{L}_{z}$ < 5 ($\\times$10$^{3}$ kpc kms$^{-1}$); 245 < $\\mathrm{J}_{R}$ < 725 (kpc kms$^{-1}$); 115 < $\\mathrm{J}_{z}$ < 545 (kpc kms$^{-1}$); 390 < $\\mathrm{L}_{\\perp}$ < 865 (kpc kms$^{-1}$); 0.5 < $e$ < 0.8; 1 < $R_{\\mathrm{peri}}$ < 3 (kpc); 8 < $R_{\\mathrm{apo}}$ < 13 (kpc); [Fe\/H] < --1.3 & 2\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{Summary of the selection criteria employed to identify the halo substructures, and the number of stars obtained for each sample. For a more thorough description of the selection criteria used in this work, see Section~\\ref{method}. We note that all the orbital energy values used are obtained adopting a \\citet[][]{McMillan2017} potential.}\n\\label{tab:selection}\n\\end{table*}\n\n\\section{Kinematic properties}\n\\label{kinematics}\n\nIn this Section, we present the resulting distributions of the identified halo substructures in the orbital energy (E) versus the azimuthal component of the angular momentum (L$_{z}$) plane in Fig~\\ref{elz}. The parent sample is illustrated as a density distribution and the halo substructures are shown using the same colour markers as in Fig~\\ref{fig:hr}. By construction, each substructure occupies a different locus in this plane. However, we do notice some small overlap between some of the substructures (for example, between GES and Sequoia, or GES and LMS-1), given their similar selection criteria. More specifically, we find that  Heracles dominates the low energy region (E < --2$\\times$10$^{5}$ km$^{-2}$s$^{-2}$), whereas all the other substructures are characterised by higher energies. As shown before (\\citealp[e.g.,][]{Koppelman_thamnos,Horta2021}), we find that GES occupies a locus at low L$_{z}$ and relatively high E, which corresponds to very radial\/eccentric orbits. We find the retrograde region (i.e., L$_{z}$ < 0 $\\times$10$^{3}$ kpc kms$^{-1}$) to be dominated by Thamnos at intermediate energies (E $\\sim$ --1.7$\\times$10$^{5}$ km$^{-2}$s$^{-2}$), and by Sequoia, Arjuna and I'itoi at higher energies (E > --1.4$\\times$10$^{5}$ km$^{-2}$s$^{-2}$); on the other hand, in the prograde region (L$_\\mathrm{z}$ > 0 $\\times$10$^{3}$ kpc kms$^{-1}$), we find the LMS-1 and Helmi stream structures, which occupy a locus at approximately E $\\sim$ --1.5$\\times$10$^{5}$ km$^{-2}$s$^{-2}$ and L$_\\mathrm{z}$ $\\sim$ 500 kpc kms$^{-1}$, and E $\\sim$ --1.4$\\times$10$^{5}$ km$^{-2}$s$^{-2}$ and L$_\\mathrm{z}$ $\\sim$ 1,000 kpc kms$^{-1}$, respectively. Furthemore, the loci occupied by the Aleph and Nyx substructures closely mimic the region defined by disc orbits. Lastly, sitting above all other structures we find the Sagittarius dSph, which occupies a position at high energies and spans a range of angular momentum between 0 < L$_\\mathrm{z}$ < 2,000 kpc kms$^{-1}$.\n\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{plots_dr17\/E_Lz_split_paper.png}\n\\caption{Distribution of the identified halo substructures in the orbital energy (E) versus angular momentum w.r.t. the Galactic disc (L$_{z}$) plane. The parent sample is plotted as a 2D histogram, where white\/black signifies high\/low density regions. The coloured markers illustrate the different structures studied in this work, as denoted by the arrows (we do not display Pontus(Icarus) as we only identify 2(1) stars, respectively). The figure is split into two panels for clarity.}\n    \\label{elz}\n\\end{figure*}\n\n\n\\section{Chemical Compositions} \n\\label{chemistry}\nIn this Section we turn our attention to the main focus of this work: a chemical abundance study of substructures in the stellar halo of the Milky Way. In this Section, we seek to first characterise these substructures qualitatively in multiple chemical abundance planes that probe different nucleosynthetic pathways. In Section~\\ref{sec_abundances} we then compare mean chemical compositions across various substructures in a quantitative fashion. Our aim is to utilise the chemistry to further unravel the nature and properties of these halo substructures, and in turn place constraints on their star formation and chemical enrichment histories. We also aim to compare their chemical properties with those from {\\it in situ} populations (see Fig~\\ref{discs} for how we determine \\textit{in situ} populations). By studying the halo substructures using different elemental species we aim to develop an understanding of their chemical evolution contributed by different nucleostynthetic pathways, contributed either by core-collapse supernovae (SNII), supernovae type Ia (SNIa), and\/or Asymptotic Giant Stars (AGBs). Furthermore, as our method for identifying these substructures relies mainly on phase space and orbital information, our analysis is not affected by chemical composition biases (except for the case of particular elements in the Heracles, Aleph, LMS-1, Arjuna, I'itoi, and (H3) Sequoia sample). \n\nOur results are presented as follows. In Section \\ref{sec_alphas} we present the distribution of the halo substructures in the $\\alpha$-Fe plane, using Mg as our $\\alpha$ element tracer. In Section \\ref{sec_ironpeak}, we show the distribution of these substructures in the Ni-Fe abundance plane, which provides insight into the chemical evolution of the iron-peak elements. Section \\ref{sec_oddZ} displays the distribution of the halo substructures in an odd-Z-Fe plane, where we use Al as our tracer element. Furthermore, we also show the C and N abundance distributions in Section \\ref{sec_cn}, the Ce abundances (namely, an $s$-process neutron capture element) in Section~\\ref{sec_neutron_capture}, and the distribution of the halo substructures in the [Mg\/Mn]-[Al\/Fe] plane in Section \\ref{sec_other_chem} \\footnote{For each chemical plane, we impose a further set of cuts of \\texttt{X$_{-}$FE$_{-}$FLAG=0} and [X\/Fe]$_{\\mathrm{error}}$<0.15 to ensure there are no unforeseen issues when determining the abundances for these halo substructures in \\texttt{ASPCAP}.}. This last chemical composition plane is interesting to study as it has recently been shown to help distinguish stellar populations from \"\\textit{in situ}\" and accreted origins (\\citealp[e.g.,][]{Hawkins2015,Das2020,Horta2021}). Upon studying the distribution of the substructures in different chemical composition planes, we finalise our chemical composition study in Section \\ref{sec_abundances} by performing a quantitative comparison between the substructures studied in this work for all the (reliable) elemental abundances available in APOGEE. For simplicity, we henceforth refer to the [X\/Fe]-[Fe\/H] plane as just the X-Fe plane.\n\nAs mentioned in Section~\\ref{data}, we exclude the Pontus and Icarus substructures from our quantitative chemical comparisons as the number of candidate members of these substructures in the APOGEE catalogue is too small.  The properties of Icarus and Pontus are briefly discussed in Appendices~\\ref{appen_icarus} and \\ref{appen_pontus}, respectively.\n\n\\subsection{$\\alpha$-elements}\n\\label{sec_alphas}\nWe first turn our attention to the distribution of the substructures in the $\\alpha$-Fe plane. This is possibly the most interesting chemical composition plane to study, as it can provide great insight into the star formation history and chemical enrichment processes of each subtructure (\\citealp[e.g.,][]{Matteucci1986,Wheeler1989,McWilliam1997,Tolstoy2009,Nissen2010,Bensby2014}). Specifically, we seek to identify the presence of the $\\alpha$-Fe knee. For this work, we resort to magnesium as our primary $\\alpha$ element, as this has been shown to be the most reliable $\\alpha$ element in previous APOGEE data releases \\citep[e.g., DR16;][]{Jonsson2020}. For the distributions of the remaining $\\alpha$ elements determined by ASPCAP (namely, O, Si, Ca, S, and Ti), we refer the reader to Fig~\\ref{sife}-Fig.~\\ref{tife} in Appendix~\\ref{app_alphas}.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{plots_dr17\/discs.png}\n\\caption{Parent sample in the Mg-Fe plane. The solid red lines indicate cut employed to select the high- and low-$\\alpha$ disc samples that we use in our $\\chi^{2}$ analysis, where the diagonal dividing line is defined as [Mg\/Fe] > --0.167[Fe\/H] + 0.15.}\n    \\label{discs}\n\\end{figure}\n\nFigure \\ref{mgfes} shows the distribution of each substructure in the Mg-Fe plane (coloured markers) compared to the parent sample (2D density histogram). We find that all the substructures --except for Aleph and Nyx-- occupy a locus in this plane which is typical of low mass satellite galaxies and accreted populations of the Milky Way (\\citealp[e.g.,][]{Tolstoy2009,Hayes2018,Mackereth2019b}), characterised by low metallicity and lower [Mg\/Fe] at fixed [Fe\/H] than {\\it in situ} populations. Moreover, we find that different substructures display distinct [Mg\/Fe] values, implying certain differences despite their overlap in [Fe\/H]. However, we also note that at the lowest metallicities ([Fe\/H] < --1.8), the overlap between different halo substructures increases. \n\nNext we discuss the distribution  on the Mg-Fe plane of stars in our sample belonging to each halo substructure.\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{plots_submit\/mgfe.png}\n\\caption{The resulting parent sample and identified structures from Fig~\\ref{elz} in the Mg-Fe plane. The mean uncertainties in the abundance measurements for halo substructures (colour) and the parent sample (black) are shown in the bottom left corner. Colour coding and marker styles are the same as Fig~\\ref{elz}. For the Aleph and Nyx substructures, we also highlight with purple edges stars from our APOGEE DR17 data that are also contained in the Aleph and Nyx samples from the \\citet[][]{Naidu2020} and \\citet[][]{Necib2020} samples, respectively.}\n    \\label{mgfes}\n\\end{figure*}\n\n\\begin{itemize}\n\n\\item $Sagittarius$: As shown in \\citet{Hasselquist2017}, \\citet{Hasselquist2019} and \\citet{Hayes2020}, the stellar populations of the Sgr dSph galaxy are characterised by substantially lower [Mg\/Fe] than even the low-$\\alpha$ disc at the same [Fe\/H] and traces a tail towards higher [Mg\/Fe] with decreasing [Fe\/H]. Conversely, at the higher [Fe\/H] end, we find that Sgr stops decreasing in [Mg\/Fe] and shows an upside-down \"\\textit{knee}\", likely caused by a burst in SF at late times that may be due to its interaction with the Milky Way (\\citealp[e.g.,][]{Hasselquist2017,Hasselquist2021}). Such a burst of star formation intensifies the incidence of SNe II, with a consequent boost in the ISM enrichment in its nucleosynthetic by-products, such as Mg, over a short timescale ($\\sim10^7$~yr).  Because Fe is produced predominantly by SN Ia over a considerably longer timescale ($\\sim10^8-10^9$~yr), Fe enrichment lags behind, causing a sudden increase in [Mg\/Fe]. \n\n\n\\item $Heracles$: The [Mg\/Fe] abundances of the Heracles structure occupy a higher locus than that of other halo substructures of similar metallicity, with the exception of Thamnos. As discussed in \\citet{Horta2021}, the distribution of Heracles in the $\\alpha$-Fe plane is peculiar, differing from that of GES and other systems by the absence of the above-mentioned $\\alpha$-knee. As conjectured in \\citet{Horta2021}, we suggest that this distribution results from an early quenching of star formation, taking place before SN~Ia could contribute substantially to the enrichment of the interstellar medium (ISM). \n\n\\item $Gaia$-$Enceladus\/Sausage$: Taking into account the small yet clear contamination from the high-$\\alpha$ disc at higher [Fe\/H] (see Fig~\\ref{mgfe_GES} for details), the distribution of GES dominates the metal-poor and $\\alpha$-poor populations of the Mg-Fe plane (as pointed out in previous studies \\citealp[e.g.,][]{Helmi2018,Hayes2018,Mackereth2019b}), making it easily distinguishable from the high-\/low-$\\alpha$ discs. We find that GES reaches almost solar metallicities, displaying the standard distribution in the Mg-Fe plane with a change of slope --the so-called ``$\\alpha$-knee''-- occurring at approximately [Fe\/H]$\\sim$--1.2 \\citep{Mackereth2019b}. The metallicity of the ``knee'' has long been thought to be an indicator of the mass of the system \\cite[e.g.,][]{Tolstoy2009}, and indeed it occurs at [Fe\/H]$>$--1 for both the high- and low-$\\alpha$ discs.  As a result, GES stars in the $shin$ part of the  $\\alpha$-$knee$ are characterised by lower [Mg\/Fe] at constant metallicity than disc stars. Interestingly, even at the plateau ([Fe\/H]$<$--1.2), GES seems to present lower [Mg\/Fe] than the high-$\\alpha$ disc, although this needs to be better quantified. Furthermore, we note the presence of a minor population of [Mg\/Fe] < 0 stars at --1.8 < [Fe\/H] < --1.2, which could be contamination from a separate halo substructure, possibly even a satellite of the GES progenitor (see Fig~\\ref{mgfe_GES} for details).\n\nBased purely on the distribution of its stellar populations on the Mg-Fe plane, one would expect the progenitor of GES to be a relatively massive system \\citep[see][ for details]{Mackereth2019b}. The fact that the distribution of its stellar populations in the Mg-Fe plane covers a wide range in metallicity, bracketing the knee and extending from [Fe\/H]<--2 all the way to [Fe\/H]$\\sim$--0.5 suggests a substantially prolonged history of star formation.\n\n\n\\item $Sequoia$: The distribution of the \\textit{GC}, \\textit{field}, and \\textit{H3} selected samples (selected on the criteria described in \\citet{Myeong2019}, \\citet{Koppelman_thamnos}, and \\citet{Naidu2020}, respectively) occupy similar [Mg\/Fe] values ([Mg\/Fe]$\\sim$0.1) at lower metallicities ([Fe\/H] $\\lesssim$--1). More specifically, we find that all three Sequoia samples occupy a similar position in the Mg-Fe plane, one that overlaps with that of GES and other substructures at similar [Fe\/H] values. Along those lines, we find that the \\textit{field} and \\textit{GC} Sequoia samples seem to connect with the \\textit{H3} Sequoia sample, where the \\textit{H3} sample comprises the lower metallicity component of the \\textit{field}\/\\textit{GC} samples. We examine in more detail the distribution of the Sequoia stars in the $\\alpha$-Fe plane in Section~\\ref{sec:Sequoia}.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{plots_dr17\/mgfe_ges.pdf}\n\\caption{\\textit{Gaia}-Enceladus\/Sausage (GES) sample in the Mg-Fe plane, colour coded by the [Al\/Fe] abundance values. The low [Al\/Fe] stars are true GES star candidates, which display the expected low [Al\/Fe] abundances observed in accreted populations (see Section~\\ref{sec_oddZ} for details). Conversely, the high [Al\/Fe] stars are clear contamination from the high-$\\alpha$ disc, likely associated with disc stars on very eccentric and high energy orbits (\\citealp{Bonaca2017,Belokurov2020}). A striking feature becomes apparent in this plane: at --1.8 < [Fe\/H] < --0.8, there is a population of very [Mg\/Fe]-poor stars (i.e., [Mg\/Fe] below $\\sim$0), that could possibly be contamination from a separate halo substructure (although these could also be due to unforeseen problems in their abundance determination).}.\n    \\label{mgfe_GES}\n\\end{figure}\n\n\n\n\\item $Helmi$ $stream$: Despite the low number of members associated to this substructure, its chemical composition in the Mg-Fe plane appears to follow a single sequence, and is confined to low metallicities ([Fe\/H]<--1.2) and intermediate magnesium values ([Mg\/Fe]$\\sim$0.2). However, we do note that the stars identified for this substructure appear to be scattered across a wide range of [Fe\/H] values. Specifically, the Helmi stream occupies a locus that overlaps with the GES for fixed [Fe\/H]. In Fig~\\ref{fig:knees_subs} we show that the best-fitting piece-wise linear model prefers a knee that is \"inverted\" similar to, although less extreme than, the case of Sgr dSph. We discuss this in more detail in Section~\\ref{sec:helmi}.\n\n\\item $Thamnos$: The magnesium abundances of Thamnos suggest that this structure is clearly different from other substructures in the retrograde halo (namely, Sequoia, Arjuna, and I'itoi). It presents a much higher mean [Mg\/Fe] for fixed metallicity than the other retrograde substructures, and appears to follow the Mg-Fe relation of the high-$\\alpha$ disc. We find that Thamnos presents no $\\alpha$-knee feature, and occupies a similar locus in the Mg-Fe plane to that of Heracles. The distribution of this substructure in this plane with the absence of an $\\alpha$-Fe \"knee\" suggests that this substructure likely quenched star formation before the onset of SN~Ia. \n\n\\item $Aleph$: By construction, Aleph occupies a locus in the Mg-Fe plane that overlaps with the metal-poor component of the low-$\\alpha$ disc. Given the distribution of this substructure in this chemical composition plane, and its very disc-like orbits, we suggest it is possible that Aleph is constituted by warped\/flared disc populations. Because the data upon which our work and that by \\cite{Naidu2020} are based come from different surveys, it is important however to ascertain that selection function differences between APOGEE and H3 are not responsible for our samples to have very different properties, even though they are selected adopting the same kinematic criteria.  In an attempt to rule out that hypothesis we cross-matched the \\citet[][]{Naidu2020} catalogue with that of APOGEE DR17 to look for Aleph stars in common to the two surveys. We find only two such stars\\footnote{We find that these two stars have a \\texttt{STARFLAG} set with \\texttt{PERSIST$_{-}$LOW} and \\texttt{BRIGHT$_{-}$NEIGHBOUR}, and thus do not survive our initial parent selection criteria. However, these warnings are not critical, and should not have an effect on their abundance determinations.}, which we highlight in Fig~\\ref{mgfes} with purple edges. While this is a very small number, the two stars seem to be representative of the chemical composition of the APOGEE sample of Aleph stars, which is encouraging.\n\n\\item $LMS-1$: Our results from Fig~\\ref{mgfes} show that the LMS-1 occupies a locus in the Mg-Fe plane which appears to form a single sequence with the GES at the lower metallicity end. Based on its Mg and Fe abundances, and the overlap in kinematic planes of LMS-1 and GES, we suggest it is possible that these two substructures could be linked, where LMS-1 constitutes the more metal-poor component of the GES. We investigate this possible association further in Section~\\ref{sec_abundances}.\n\n\\item $Arjuna$: This substructure occupies a distribution in the Mg-Fe plane that follows that of the GES. Despite the sample being lower in numbers than that for GES, we still find that across --1.5 < [Fe\/H] < --0.8 this halo substructure overlaps in the Mg-Fe plane with that of the GES substructure. Given the strong overlap between these two systems, as well as their proximity in the E-L$_{z}$ plane (see Fig~\\ref{elz}), we suggest it is possible that Arjuna could be part of the GES substructure, and further investigate this possible association in Section~\\ref{sec_abundances}.\n\n\\item $I'itoi$: Despite the small sample size, we find I'itoi presents high [Mg\/Fe] and low [Fe\/H] values (the latter by construction), and occupies a locus in the Mg-Fe plane that appears to follow a single sequence with the Sequoia (all three samples) and the GES sample. \n\n\\item $Nyx$: The position of this substructure in the Mg-Fe strongly overlaps with that of the high-$\\alpha$ disc. Given this result, and the disc-like orbits of stars comprising this substructure, we conjecture that Nyx is constituted by high-$\\alpha$ disc populations, and further investigate this association in Section~\\ref{sec_abundances}. Furthermore, in a similar fashion as done for the Aleph substructure, we highlight in Fig~\\ref{mgfes} with purple edges those stars in APOGEE DR17 that are also contained in the Nyx sample from \\citet[][]{Necib2020}, in order to ensure that our results are not biased by the APOGEE selection function. We find that the overlapping stars occupy a locus in this plane that overlaps with the Nyx sample determined in this study, and the high-$\\alpha$ disc.\n\n\\end{itemize}\n\n\n\\subsection{Iron-peak elements}\n\\label{sec_ironpeak}\n\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{plots_submit\/nife.png}\n\\caption{The same illustration as in Fig~\\ref{mgfes} in the Ni-Fe space. We note that the grid limit of appears clearly in this plane at the lowest [Fe\/H] values. }\n    \\label{nifes}\n\\end{figure*}\n\n\n\nFollowing our analysis of the various substructures in the Mg-Fe plane, we now focus on studying their distributions in chemical abundance planes that probe nucleosynthetic pathways contributed importantly by Type Ia supernovae. We focus on nickel (Ni), which is the Fe-peak element that is determined the most reliably by ASPCAP, besides Fe itself.  For the distribution of the structures in other iron-peak element planes traced by ASPCAP (e.g., Mn, Co, and Cr), we refer the reader to Fig.~\\ref{mnfe}-- Fig.~\\ref{crfe} in Appendix~\\ref{app_ironpeak}. \n\nThe distributions of the halo substructures in the Ni-Fe plane are shown in Fig.~\\ref{nifes}. We find that the distributions of GES, Sgr dSph, the Helmi stream, Arjuna, LMS-1, and the three Sequoia samples occupy a locus in this plane that is characteristic of low mass satellite galaxies and\/or accreted populations of the Milky Way, displaying lower [Ni\/Fe] abundances than the low- and high-$\\alpha$ disc populations (\\citealp[e.g.,][Shetrone et al. 2022, in prep.]{Shetrone2003,Mackereth2019b,Horta2021}). In contrast, the data for Heracles and Thamnos display a slight correlation between [Ni\/Fe] and [Fe\/H], connecting with the high-$\\alpha$ disc at  ${\\rm [Fe\/H]}\\sim-1$ (despite the differences of these substructures in the other chemical composition planes with \\textit{in situ} populations). Conversely, we find that the Aleph and Nyx structures clearly overlap with \\textit{in situ} disc populations at higher [Fe\/H] values, agreeing with our result for these substructures on the Mg-Fe plane. The distribution of I'itoi shows a spread in [Ni\/Fe] for a small range in [Fe\/H], that is likely due to observational error at such low metallicities. \n\nInterpretation of these results depends crucially on an understanding of the sources of nickel enrichment.  Like other Fe-peak elements, nickel is contributed by a combination of SNIa and SNe~II \\cite[e.g.,][]{Weinberg2019,Kobayashi2020}.  The disc populations display a bimodal distribution in Figure~\\ref{nifes}, which is far less pronounced than in the case of Mg.  This result suggests that the contribution by SNe~II to nickel enrichment may be more important than previously thought (but see below).  It is thus possible that the relatively low [Ni\/Fe] observed in MW satellites and halo substructures has the same physical reason as their low [$\\alpha$\/Fe] ratio, namely, a low star formation rate \\cite[e.g.,][]{Hasselquist2021}. This hypothesis can be checked by examining the locus occupied by halo substructures in a chemical plane involving an Fe-peak element with a smaller contribution by SNe~II, such as manganese \\cite[e.g.,][]{Kobayashi2020}.  If indeed the [Ni\/Fe] depression is caused by a decreased contribution by SNe~II, one would expect [Mn\/Fe] to display a different behaviour.  Figure~\\ref{mnfe} confirms that expectation, with substructures falling on the same locus as disc populations on the Mn-Fe plane.  \n\nAnother possible interpretation of the reduced [Ni\/Fe] towards the low metallicity characteristic of halo substructures is a metallicity dependence of nickel yields \\citep{Weinberg2021}.  We may need to entertain this hypothesis since, in contrast to the results presented in Figure~\\ref{nifes}, no [Ni\/Fe] bimodality is present in the solar neighbourhood disc sample studied by \\cite{Bensby2014}, which may call into question our conclusion that SNe~II contribute relevantly to nickel enrichment.  It is not clear whether the apparent discrepancy between the data for nickel in \\cite{Bensby2014} and this work is due to lower precision in the former, sample differences, or systematics in the APOGEE data.\n\nGiven the distribution of the substructures in the Ni-Fe plane, our results suggest that: {\\it i)} Sgr dSph, GES, Sequoia (all three samples), and the Helmi stream substructures show a slightly lower mean [Ni\/Fe] than \\textit{in situ} populations at fixed [Fe\/H], as expected for accreted populations in the Milky Way on the basis of previous work \\citep[e.g.,][]{Nissen1997,Shetrone2003}; {\\it ii)} Heracles and Thamnos fall on the same locus on the Ni-Fe plane, presenting a slight correlation between [Ni\/Fe] and [Fe\/H]; {\\it iii)} as in the case of the Mg-Fe plane, Arjuna and LMS-1 occupy a similar locus in the Ni-Fe plane to that of GE\/S, further supporting the suggestion that these substructures may be associated; {\\it iv)} Aleph\/Nyx mimic the behaviour of {in situ} low-\/high-$\\alpha$ disc populations, respectively. \n\n\\subsection{Odd-Z elements}\n\\label{sec_oddZ}\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{plots_submit\/alfe.png}\n\\caption{The same illustration as in Fig~\\ref{mgfes} in Al-Fe space. We note that the grid limit appears clearly in this plane at the lowest [Fe\/H] values.}\n    \\label{alfes}\n\\end{figure*}\n\nAside from $\\alpha$ and iron-peak elements, other chemical abundances provided by ASPCAP\/APOGEE that are interesting to study are the odd-Z elements. These elements have been shown in recent work to be depleted in satellite galaxies of the MW and accreted systems relative to populations formed \\textit{in situ} (\\citealp[e.g.,][]{Hawkins2015,Das2020,Horta2021,Hasselquist2021}). For this paper, we primarily focus on the most reliable odd-Z element delivered by ASPCAP: aluminium. For the distribution of the structures in other odd-Z chemical abundance planes yielded by APOGEE (namely, Na, and K), we refer the reader to Fig.~\\ref{nafe} and Fig.~\\ref{kfe} in Appendix~\\ref{app_oddZ}.\n\nFig.~\\ref{alfes} displays the distribution of the substructures and parent sample in the Al-Fe plane, using the same symbol convention as adopted in Fig.~\\ref{mgfes}. We note that the parent sample shows a high density region at higher metallicities, displaying a bimodality at approximately [Fe\/H] $\\sim$ --0.5, where the high-\/low-[Al\/Fe] sequences correspond to the high-\/low-$\\alpha$ discs, respectively. In addition, there is a sizeable population of aluminium-poor stars with ${\\rm -0.5\\mathrel{\\copy\\simlessbox}[Al\/Fe]\\simless0}$ ranging from the most metal-poor stars in the sample all the way to ${\\rm [Fe\/H]\\sim-0.5}$\\footnote{The clump located at {$\\rm [Al\/Fe]\\sim-0.1$} and [Fe\/H] > 0 is not real, but rather an artifact due to systematics in the abundance analysis which does not affect the bulk of the data.}.  This is the locus occupied by MW satellites and most accreted substructures, with the exception of Aleph and Nyx.  Note also that the upper limit of the distribution of the Heracles population on this plane is determined by the definition of our sample \\cite[see][for details]{Horta2021}.\n\n\nThe majority of the substructures studied occupy a similar locus in this plane, which agrees qualitatively with the region where the populations from MW satellites are usually found \\cite[e.g.,][]{Hasselquist2021}.  There is strong overlap between stars associated with the GES, the Helmi stream, Arjuna, Sequoia (all three samples), and LMS-1 substructures. More specifically, we find that GES dominates the parent population sample at [Fe\/H] < --1, being located at approximately [Al\/Fe] $\\sim$ --0.3. At a slightly higher value of [Al\/Fe] $\\sim$ --0.15 and similar metallicities, we find Heracles and Thamnos. In contrast, Sgr dSph is characterised by an overall  lower [Al\/Fe] $\\sim$ --0.5 value, which extends below the parent disc population towards higher [Fe\/H], reaching almost solar metallicity.  Within the ${\\rm -2 \\mathrel{\\copy\\simlessbox} [Fe\/H] \\mathrel{\\copy\\simlessbox} -1}$ interval, Heracles, GES, Helmi streams, Thamnos, Nyx, and, to a lesser extent, Sequoia, show some degree of correlation between [Al\/Fe] and [Fe\/H].   Towards the metal-poor end, we find the LMS-1 located at [Al\/Fe]$\\sim$--0.3, which is consistent with the value found for I'itoi, although the sample of aluminium abundances for this latter structure is very small and close to the detection limit. As in the case of magnesium and nickel, all three Sequoia samples occupy the same locus in the Al-Fe plane as Arjuna, which strongly overlap with GES.  Again in the case of the Al-Fe plane, we find that the case for Nyx and Aleph follow closely the trend established by {\\it in situ} disc populations.\n\n\\subsection{Carbon and Nitrogen}\n\\label{sec_cn}\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{plots_submit\/cfe.png}\n\\caption{The same illustration as Fig.~\\ref{mgfes} in the C-Fe plane. For this chemical plane we restrict our sample to a surface gravity range of 1 < log$g$ < 2 in order to minimise the effect of internal mixing in red giant stars.} \n    \\label{cfe}\n    \n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{plots_submit\/nfe.png}\n\\caption{The same illustration as Fig.~\\ref{mgfes} in the N-Fe plane. As done in Fig~\\ref{cfe}, for this chemical plane we restrict our sample to a surface gravity range of 1 < log$g$ < 2 in order to minimise the effect of internal mixing in red giant stars. We note that the grid limit appears clearly in this plane at the lowest [Fe\/H] values.}\n    \\label{nfe}\n    \n\\end{figure*}\n\nIn this subsection, we examine the distribution of stars belonging to various substructures in the C-Fe and N-Fe abundance planes, shown in Figures~\\ref{cfe} and ~\\ref{nfe}, respectively. We note that in these chemical planes, we impose an additional surface gravity constraint of 1 < log$g$ < 2 in order to minimise the effect of internal mixing along the giant branch.\n\n In the C-Fe plane, most substructures are characterised by sub-solar [C\/Fe], displaying a clear correlation between that abundance ratio and metallicity.  The exceptions, as in all previous cases, are Aleph and Nyx, which again follow the same trends as {\\it in situ} populations. Interestingly, the Sgr dSph presents the lowest values of [C\/Fe] at fixed [Fe\/H], tracing a tight sequence at approximately [C\/Fe]$\\sim$--0.5, spanning from --1.4 < [Fe\/H] < --0.2, approximately $\\sim$0.5 dex below that of the Galactic disc. In the case of I'itoi, due to the low numbers of stars in this sample we are unable to draw any conclusions.\n\n The distribution of substructures in the N-Fe plane follows a different behaviour than seen in all other chemical planes.  Again, except for Aleph and Nyx, all systems display a trend of increasing [N\/Fe] towards lower metallicities, starting at ${\\rm [Fe\/H]\\mathrel{\\copy\\simlessbox}-1}$.  This trend cannot be ascribed to systematics in the ASPCAP abundances or evolutionary effects, as the abundances are corrected for variations with $\\log g$.  Nitrogen abundances are notoriously uncertain, particularly in the low metallicity regime.  The compilation by \\cite{Kobayashi2020} shows that the [N\/Fe] trend at low metallicity is strongly dependent on the analysis methods.  Discerning the source of systematics in the ASPCAP abundances at ${\\rm [Fe\/H]\\mathrel{\\copy\\simlessbox}-1}$ is beyond the scope of this paper, which focuses on a strictly differential analysis of the data, within a metallicity regime where ASPCAP elemental abundances attain exceedingly high precision (Section~\\ref{sec_abundances}).\n\nFor completeness, data covering the whole range of $\\log g$ for all substructures are displayed in the (C+N)-Fe plane in Fig~\\ref{cnfe} in Appendix~\\ref{app_carbon_nitrogen}. By combining carbon and nitrogen abundances, we minimise the effect of CNO mixing along the giant branch. In this plane, MW satellites and accreted populations typically display a lower [(C+N)\/Fe] chemical composition than their \\textit{in situ} counterparts \\citep[e.g.,][]{Horta2021,Hasselquist2021}. This is in fact what we observe for all the structures identified, again with the exception of Aleph and Nyx, whose locus overlaps with that of \\textit{in situ} disc populations.\n\n\\subsection{Cerium}\n\\label{sec_neutron_capture}\n\nCerium is a neutron capture element of the $s$-process family, with a large enrichment contribution from AGB stars \\citep{Sneden2008,Jonsson2020,Kobayashi2020}. In Fig.~\\ref{cefes}, the disc sample at [Fe\/H]$>-1$ has a roughly horizontal locus at [Ce\/Fe]$\\approx-0.1$ dominated by stars in the {\\it high}-$\\alpha$ population and an upward-pointing triangular locus dominated by stars in the {\\it low}-$\\alpha$ sequence, reaching [C\/Fe]$\\approx+0.4$ at [Fe\/H]$\\approx-0.2$.  The scatter within each of these components is large and may be partly observational.  The presence of substantial Ce in high-$\\alpha$ stars suggests that massive stars with short lifetimes make a significant, prompt contribution.  The rising-then-falling trend in the low-$\\alpha$ population is expected from the metallicity-dependent yield of intermediate mass AGB nucleosynthesis: at low [Fe\/H] the number of seeds available for neutron capture increases with increasing metallicity, but at high [Fe\/H] the number of neutrons per seed becomes to low to produce the heavier $s$-process elements \\citep{Gallino1998}. See \\cite{Weinberg2021} for plots of [Ce\/Mg] vs. [Mg\/H] and further discussion of the disc trends.\n\nIn this chemical composition plane, we find that all the identified substructures, with the exception of Aleph and Nyx, present [Ce\/Fe] abundances that follow the mean trend with [Fe\/H] of the parent population until [Fe\/H]~$\\sim-1$.  Aleph and Nyx have higher [Fe\/H] stars that generally lie within the broad disc locus.  Interestingly, the Sgr stars with [Fe\/H]~$>-1$ show a rising [Ce\/Fe] trend that tracks the behaviour of the low-$\\alpha$ disc population.  This trend is not obvious in the other substructures, though with the exception of Aleph and Nyx they have few stars at [Fe\/H]~$>-1$.  We interpret this upturn in both Sgr and the low-$\\alpha$ disc as the signature of an AGB contribution with a metallicity dependent yield.\n\nAt [Fe\/H]~$<-1$, the parent halo population and most substructure stars exhibit mildly sub-solar [Ce\/Fe] with substantial scatter, which may have a significant observational component.  The [Ce\/Fe] is similar to that of typical high-$\\alpha$ disc stars at [Fe\/H]~$>-1$.  However, these stars have lower [$\\alpha$\/Fe] than the high-$\\alpha$ disc, the signature of Fe enrichment from Type Ia SNe, so if the Ce in these populations is a prompt contribution from massive stars one might have naively expected them to have depressed [Ce\/Fe].  It is difficult to disentangle the effects of metallicity-dependent Ce yields, differences in the relative contributions of high-mass and intermediate-mass stars, and the impact of Type Ia SN enrichment on the Fe abundance; further observational investigation and theoretical modeling will be needed to do so. The [Ce\/Fe] locus of substructure stars at $-2 < {\\rm [Fe\/H]} < -1$ is similar to that in the dwarf satellites studied by \\cite{Hasselquist2021}.\n\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{plots_submit\/cefe.png}\n\\caption{The same illustration as Fig.~\\ref{mgfes} in the Ce-Fe plane. We note that the grid limit appears clearly in this plane at the lowest [Fe\/H] values. }\n    \\label{cefes}\n\\end{figure*}\n\n\n\\subsection{ The [Al\/Fe] vs [Mg\/Mn] plane}\n\\label{sec_other_chem}\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{plots_submit\/mgmn.png}\n\\caption{The same illustration as Fig.~\\ref{mgfes} in the [Mg\/Mn]-[Al\/Fe] plane. }\n    \\label{mgmn}\n    \n\\end{figure*}\n\nHaving studied the distribution of the identified structures in chemical abundance planes that aimed to give us an insight into the different nucleosynthetic pathways, contributed either by core-collapse, type Ia supernovae, and AGB stars, we now focus our attention on analysing the distribution of substructures in the stellar halo in an abundance plane that lends insights into the accreted or \\textit{in situ} nature of Galactic stellar populations: namely, the [Mg\/Mn]-[Al\/Fe] plane. \n\n\nFig.~\\ref{mgmn} shows the resulting distribution of the various structures in the [Mg\/Mn]-[Al\/Fe] plane. This chemical plane has been proposed by \\cite{Das2020} as a means to distinguishing accreted populations from those formed {\\it in situ}.  \\cite{Horta2021} showed that {\\it in situ} stellar populations with a small degree of chemical evolution occupy the same locus in that plane as accreted populations. By construction, the Heracles substructure falls in the accreted locus of the diagram (see Fig~1 in \\citet{Horta2021} for reference). However, our results show that all the other structures, except for Aleph and Nyx, also occupy the accreted locus of this plane. Interestingly, we find that although the GES, Sgr dSph, the Helmi stream, Sequoia (all three samples), Thamnos, LMS-1, Arjuna, and I'itoi substructures occupy the same locus, they appear to show some small differences. Specifically, we find that Sgr dSph occupies a locus in this plane positioned at lower mean [Mg\/Mn] than the other structures. This is likely due to Sgr being more recently accreted by the Milky Way, and thus had more time to develop stellar populations with enriched Mn abundances that have been contributed on a longer timescale by type Ia supernovae. This feature is also seen to a lesser extent for I'itoi. Conversely, at higher [Mg\/Mn] values (but still low [Al\/Fe]) we find GES, Heracles (by construction), Thamnos, LMS-1, the Helmi stream, Sequoia (all three samples), and Arjuna. The distribution of these substructures in this chemical plane reinforces the hypothesis of these halo substructures arising from an accreted origin. \n\nIn a similar fashion to the other chemical composition planes, we find that Aleph and Nyx overlap with \\textit{in situ} (disc) populations at higher [Al\/Fe], suggestive that Aleph and Nyx are likely substructures comprised of \\textit{in situ} disc populations.\n\n\n\n\\section{A quantitative comparison between halo substructure abundances}\n\\label{sec_abundances}\nAfter qualitatively examining the chemical compositions of the previously identified halo substructures in a range of chemical abundance planes, we now focus on comparing the abundances in a quantitative fashion using a $\\chi^{2}$ method. To do so, we compare the mean value of thirteen different elemental abundances, manufactured in different nucleosynthetic channels, for each substructure at a fixed metallicity that is well covered by the data. The set of elemental abundances chosen to run this quantitative test was determined based on the distribution of the parent sample in the respective chemical composition plane, where we only chose those elements that did not display a large scatter towards low metallicity due to increased abundance uncertainties (on the order of $\\sigma$ $\\sim$0.15 dex). Out of the initial 20 elemental abundances available in ASPCAP (excluding Fe), we utilise the following thirteen elements: C, N, O, Mg, Al, Si, S, K, Ca, Ti, Mn, Ni, and Ce. We note that Na, P, V, Cr, and Co were removed due to the large scatter at low metallicity, whereas Cu and Nd were not considered due to \\texttt{ASPCAP} not being able to determine abundances for these elements in APOGEE DR17. \n\nFor our quantitative comparison, we proceeded as follows:\n\ni) We select a high- and low-$\\alpha$ disc population based on Fig~\\ref{discs} for reference, and utilise these samples as representative disc samples for any comparison between \\textit{in situ} populations and halo substructures. In order to account for any distance selection function effects, we restrict our high-\/low-$\\alpha$ samples to stars within $d < 2$ kpc, and also determine an \"inner high-$\\alpha$ disc\" sample (restricted to R$_{\\mathrm{GC}} < 4$ kpc which we will use to compare to Heracles (which has a spatial distribution that is largely contained within $\\sim$4 kpc from the Galactic Centre).\n\nii) Before performing any chemical composition comparisons, we correct the abundances for systematic trends with surface gravity. Systematic abundance variations trends with $\\log g$ can be caused, one one hand, by real physical chemical composition variations as a function of evolutionary stage and\/or, on the other, by systematic errors in elemental abundances as a function of stellar parameters.  The former chiefly impact elements such as C, N, and O, whose atmospheric abundances are altered by mixing during evolution along the giant branch.  The latter impact various elements in distinct, though more subtle, ways \\citep[see discussion in][]{Weinberg2021}.  Surface gravity distributions of various substructures differ in important ways (Figure~\\ref{fig:hr}), so that such systematic abundance trends with $\\log g$ can induce spurious artificial chemical composition differences between substructures.  We thus follow a procedure similar to that outlined by \\cite{Weinberg2021} to correct each elemental abundance using the full parent sample. As systematic trends with $\\log g$ are more important towards the low and high ends of the $\\log g$ distribution, we restrict our sample to stars within the $1 < \\log g  < 2$ range. We then fit a second order polynomial to the [X\/H]-$\\log g$ relation, and calculate the difference between that fit and the overall [X\/H] median. The difference between these two quantities for any given $\\log g$ is then added to the original [X\/H] values so as to produce a flat relation between [X\/H]$_{\\rm corrected}$ and $\\log g$. We then use these values to determine corrected [X\/Fe] abundances. In a recent study, \\cite{Eilers2021} pointed out that simple corrections for abundance trends as a function of $\\log g$ could erase real differences associated with abundance gradients within the Galaxy.  That is because a magnitude limited survey may cause an artificial dependence of $\\log g$ on distance. Our study aims at contrasting the chemical compositions of substructures that are in principle associated with spatially self-contained progenitors.  Thus, systematic differences linked to spatial abundance variations within each structure are irrelevant for our purposes, so a straightforward correction for abundance variations with $\\log g$ are perfectly acceptable for our goals.\n\niii) Upon obtaining corrected abundances for every halo substructure, we determined the uncertainties in the abundances using a bootstrapping resampling with replacement method (utilising the \\texttt{astropy.stats.bootstrap} routine by \\citealp[][]{astropy:2018}).  We generated 1,000 realisations of the X-Fe chemical composition planes for every element and every halo substructure sample in order to assess the scatter in the abundance distribution. For example, we generated 1,000 realisations of the C-Fe distribution for GES by drawing 2,353 values from the observed distribution, with replacement (where 2,353 is the size of our GES sample).\n\niv) For each one of the 1,000 bootstrapped realisations of a chemical composition plane of a halo substructure, we determine the [X\/Fe] value at a given metallicity ([Fe\/H]$_{\\mathrm{comp}}$) that is covered by both halo substructures being compared by taking a 0.05 dex slice in [Fe\/H] around [Fe\/H]$_{\\mathrm{comp}}$ and determining the median value for stars in that [Fe\/H] interval. This yields 1,000 [X\/Fe] median values for each of the thirteen elements studied for every halo substructure (and disc sample) compared.\n\nv) We take the mean and standard deviation of the medians distribution for every [X\/Fe] as our representative [X\/Fe] and uncertainty value, respectively, and use these to quantitatively compare the chemical abundances between two populations. This sample median from the mean of the bootstrap medians is always close to the full sample median itself.\n\nvi) Upon obtaining the mean and uncertainty chemical abundance values for every halo substructure at [Fe\/H]$_{\\mathrm{comp}}$ (for a range of thirteen reliable elemental abundances in APOGEE), we quantitatively compare the chemical compositions across halo substructures adopting a $\\chi^{2}$ statistic to assess the chemical similarities between different substructures. This quantity was computed by using the following relation:\n\\begin{equation}\n    \\chi^{2} =  \\sum_{i} \\frac{\\Big(\\mathrm{[X\/Fe]}_{i,\\mathrm{sub}} - \\mathrm{[X\/Fe]}_{i,\\mathrm{ref}}\\Big)^{2}}{\\Big(\\sigma_{\\mathrm{[X\/Fe]}_{i,\\mathrm{sub}}}^{2} + \\sigma_{\\mathrm{[X\/Fe]}_{i,\\mathrm{ref}}}^{2}\\Big)},\n    \\label{eq:chi2}\n\\end{equation}\nwhere [X\/Fe]$_{\\mathrm{sub}}$ and [X\/Fe]$_{\\mathrm{ref}}$ are the abundances of the halo substructure and the compared reference stellar population, respectively, and $\\sigma_{\\mathrm{[X\/Fe],sub}}$ and $\\sigma_{\\mathrm{[X\/Fe],ref}}$ are the corresponding uncertainties to those abundance values. Since GES is the halo substructure for which our sample is the largest, we use this substructure as our main reference against which all other substructures, as well as \\textit{in situ} disc populations are contrasted. \n\nvii) Lastly, in order to infer if two stellar populations present consistent chemical abundances in a statistical manner, we determine the probability value of the $\\chi^{2}$ result for twelve degrees of freedom using the \\texttt{scipy} \\citep[][]{Scipy2020} \\texttt{stats.chi2.cdf} routine. In addition to the $\\chi^{2}$ value, we also compute a metric of separation (defined as $\\Sigma_{\\mathrm{[X\/Fe]}}$), that is calculated by setting the denominator of Eq~\\ref{eq:chi2} equal to 1. This separation metric provides an additional way to quantify how similar the chemical compositions of two halo substructures are that is unaffected by the sample size (as smaller halo substructure samples will have larger uncertainties on their mean abundance values).\n\nviii) For the case of Heracles and Aleph, as the selection of these substructures relies heavily on the use of [Al\/Fe] and [Mg\/Fe], respectively, we remove these elements when comparing these substructures, and reduce the numbers of degrees of freedom to eleven when calculating the $\\chi^{2}$ probability value.\\\\\n\n\n In order to develop a more clear notion of the meaning of the resulting $\\chi^{2}$ values resulting from the above comparisons we perform an additional exercise aimed at gauging the expected $\\chi^2$ values for the cases where two samples are identical to each other, or very different.  To accomplish this, we draw, for each substructure, three $N_{\\rm sub}$-sized random samples, two from the high-$\\alpha$ and one from the low-$\\alpha$ disc samples, where $N_{\\rm sub}$ is the size of the sample of that substructure.  We then calculate, for each substructure, two $\\chi^2$ values, one resulting from the comparison of the high-$\\alpha$ disc against itself, and the other from the comparison of the high-$\\alpha$ disc against the low-$\\alpha$ disc samples. As the high- and low-$\\alpha$ disc both cover a similar range in [Fe\/H], and their abundances vary with [Fe\/H], we select a narrow bin in [Fe\/H] from which to draw our high- and low-$\\alpha$ disc samples (namely, between --0.45 < [Fe\/H] < --0.35). This enables us to obtain random samples of high- and low-$\\alpha$ disc populations at the same [Fe\/H], and allows us to directly compare the mean and scatter values of the chemical abundances using the $\\chi^{2}$ method.\n\nThe reader may inspect the resulting $\\chi^{2}$ and probability ($p_{\\chi^{2}}$) values obtained in Table~\\ref{tab:chi2}. Here, a $p_{\\chi^{2}}$ $\\sim$ 1 signifies that two populations are statistically equal, and $p_{\\chi^{2}}$ $\\sim$ 0 means that they are statistically different. For the quantitative comparisons, we employ a threshold of $p_{\\chi^{2}}$ = 0.1 as our benchmark, where we will deem two substructures to be statistically similar if their associated probability value is higher than $p_{\\chi^{2}}$ > 0.1, and different if below $p_{\\chi^{2}}$ < 0.1.\n\nThe resulting quantitative comparison between the halo substructures indicates that approximately half of the halo substructures are statistically equal with regards to their chemical abundances, whilst the other half are statistically different. For example, the $\\chi^{2}$ comparison between the GES and the three Sequoia samples imply that these four substructures are statistically the same, as we find that the \\textit{GC}, \\textit{field}, and \\textit{H3} Sequoia samples all have a high probability of being statistically similar to the GES substructure (0.2 < $p_{\\chi^{2}}$ < 0.85). We find this also to be the case for the LMS-1 substructure, for which we obtain a probability value of 0.46. Similarly, an ever closer match is found for the GES-Arjuna comparison, yielding a probability value of 0.88. Along similar lines, we find that when comparing the Nyx to the high-$\\alpha$ disc, we obtain a probability value of 0.8, reinforcing our initial hypothesis that the Nyx is not an accreted substructure, but instead is a stellar population constituted of high-$\\alpha$ disc stars. Moreover, we find that when comparing Heracles, Sgr dSph, and Thamnos with the GES that the probability of these substructures being statistically equal is $\\sim$0. This result is not entirely surprising, as all these substructures are postulated to be debris from separate accretion events, and thus should present differences in their chemical abundances. Interestingly, we find two surprising results: i) despite the Aleph substructure presenting qualitatively the same chemistry as the low-$\\alpha$ disc, its $\\chi^{2}$ value yields a probability of 0.05, suggesting that these are not as similar as initially hypothesised; ii) although Heracles occupies a position in several chemical composition planes that appears to follow a single sequence with the high-$\\alpha$ disc, the $\\chi^{2}$ yielded when comparing this substructure to the high-$\\alpha$ disc indicate that these are statistically different (with a probability value of $p_{\\chi^{2}}$=0).\n\nFor the complete resulting probability values obtained when comparing the chemistry between every halo substructure, as well as the high- and low-$\\alpha$ disc, we refer the reader to Fig~\\ref{confusion_matrix}.\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{plots_dr17\/ges_mcs.png}\n\\caption{$\\Delta$[X\/Fe] differences between the resulting mean values obtained using the procedure outlined in Section~\\ref{sec_abundances} (at [Fe\/H]$_{\\mathrm{comp}}$) for the \\textit{Gaia}-Enceladus\/Sausage substructure and the high-$\\alpha$ disc stars (top) and for the Large and Small Magellanic Cloud (LMC\/SMC) samples from \\citet[][]{Hasselquist2021} (bottom) in thirteen different chemical abundance planes, grouped by their nucleosynthetic source channel. The shaded regions illustrate the uncertainty on this $\\Delta$[X\/Fe] value. Also illustrated in the top right\/left are the $\\chi^{2}$\/$p_{\\chi^{2}}$\/[Fe\/H]$_{\\mathrm{comp}}$ values for the comparison between these two populations. As can be seen from the abundance values, the $\\chi^{2}$ value, and the $p_{\\chi^{2}}$ value, it is evident that the GES\/high-$\\alpha$ disc and the LMC\/SMC are quantitatively different given their chemical compositions.}\n    \\label{ges_disc}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{plots_dr17\/comb_abun1.png}\n\\caption{The same mean and mean error abundance values as shown in Fig~\\ref{ges_disc} but comparing the Heracles, Sagittarius dSph, Helmi stream, Arjuna, and Thamnos substructures with the \\textit{Gaia}-Enceladus\/Sausage substructure. We note that those substructures with fewer stars present larger uncertainties in their $\\Delta$[X\/Fe] value.}\n    \\label{ges_combined1}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{plots_dr17\/comb_abun2.png}\n\\caption{The same as Fig~\\ref{ges_combined1} but comparing the three Sequoia samples, LMS-1, and I'itoi substructures with the \\textit{Gaia}-Enceladus\/Sausage substructure.}    \\label{ges_combined2}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{plots_dr17\/comb_abun3.png}\n\\caption{The same as Fig~\\ref{ges_disc} but comparing the Nyx substructures with the \\textit{Gaia}-Enceladus\/Sausage substructure and the high-$\\alpha$ discs, as well as a comparison between the Heracles and inner high-$\\alpha$ disc, Heracles and Thamnos, and Aleph and the low-$\\alpha$ disc.}    \\label{ges_combined3}\n\\end{figure*}\n\n\n\n\\setlength{\\tabcolsep}{8pt}\n\\begin{table*}\n\\centering\n\\begin{tabular}{ p{3.5cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{2cm}|p{2cm}}\n\\hline\nCompared samples & [Fe\/H]$_{\\mathrm{comp}}$& $\\chi^{2}$& $p_{\\chi^{2}}$ & $\\Sigma \\Delta_{\\mathrm{[X\/Fe]}}$ & high$\\alpha$-high$\\alpha$ $\\chi^{2}$ & high$\\alpha$-low$\\alpha$ $\\chi^{2}$ \\\\\n\\hline\n\\hline\nHigh$\\alpha$ disc-GES & --0.9&  1753.4 & 0.00 & 0.50 & 9.78& 4622.96 \\\\ \n\\hline\nLMC-SMC & --1.1&  53.1 & 0.00 & 0.04& 10.05 & 1517.67 \\\\ \n\\hline\nGES-Heracles & --1.3& 359.0& 0.00& 0.06& 10.74&507.43  \\\\ \n\\hline\nGES-Sgr dSph& --1.0 & 150.0& 0.00& 0.11& 8.55  & 542.27\\\\\n\\hline\nGES-Helmi stream& --1.2 & 22.1 & 0.04 & 0.08& 6.61 &153.93 \\\\ \n\\hline\nGES-Sequoia (\\textit{GC})& --1.2 & 7.1& 0.85& 0.02& 3.40& 207.94 \\\\\n\\hline\nGES-Sequoia (\\textit{field})& --1.3 & 15.9& 0.20& 0.16& 4.04&210.68\\\\ \n\\hline\nGES-Sequoia (\\textit{H3})& --1.9 & 12.7& 0.39& 0.24& 3.42&229.83 \\\\\n\\hline\nGES-Thamnos& --1.4 & 74.2& 0.00& 0.18& 8.82 &162.81\\\\\n\\hline\nGES-LMS-1& --2.1& 11.8& 0.46& 0.23& 6.27&371.65 \\\\\n\\hline\nGES-Arjuna& --1.3 &6.6& 0.88& $<$0.01&5.48 &230.82\\\\\n\\hline\nGES-I'itoi& --2.1& 13.9&0.30& 0.23& 6.75& 123.99\\\\\n\\hline\nGES-Nyx& --0.6& 62.7&0.00& 0.34& 6.45&1246.71  \\\\\n\\hline\nhigh$\\alpha$ disc-Nyx& --0.6 & 7.9 & 0.80& $<$0.01& 6.45&1246.71\\\\\n\\hline\nlow$\\alpha$ disc-Aleph& --0.6& 19.5& 0.05& 0.04& 8.82&195.69\\\\\n\\hline\nInner high-$\\alpha$ disc-Heracles& --1&53.1&0.00& 0.04 & 10.74&507.43\\\\\n\\hline\nHeracles-Thamnos&--1.3 & 33.7& 0.0& 0.08& 8.82 &162.81 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{From left to right: compared halo substructures, [Fe\/H] value used to compare the two compared substructures, resulting $\\chi^{2}$ value from the comparison between the listed halo substructures, the probability value the $\\chi^{2}$ result falls upon for a $\\chi^{2}$ test with twelve (or eleven for the case of Heracles and Aleph) degrees of freedom, the metric separation $\\Sigma \\Delta_{\\mathrm{[X\/Fe]}}$, $\\chi^{2}$ value between two randomly chosen high-$\\alpha$ disc samples of the same size as the smallest substructure compared, $\\chi^{2}$ value between a randomly chosen high-$\\alpha$ and low-$\\alpha$ disc sample of the same size as the smallest substructure compared. The LMC\/SMC samples were taken from \\citet[][]{Hasselquist2021}.}\n\\label{tab:chi2}\n\\end{table*}\n\n\\begin{figure*}\n\\includegraphics[width=0.7\\textwidth]{plots_dr17\/confusion_prob.png}\n\\caption{Confusion matrix of the probability values (estimated using the $\\chi^{2}$ calculated using Eq~\\ref{eq:chi2}) obtained when comparing the chemical compositions of all the halo substructures with each other and with a high-\/low-$\\alpha$ discs. Here, each substructure is compared with its counterpart using a [Fe\/H] value that is well covered by the data (see Fig~\\ref{confused_matrix_feh} in Appendix~\\ref{appen_confusion} for further details), where red(blue) signifies a high(low) probability of two systems being statistically equal given their chemical compositions. Comparisons with blank values are due to the two substructures being compared not having any overlap in [Fe\/H].}    \\label{confusion_matrix}\n\\end{figure*}\n\n\n\\section{Discussion}\n\\label{discussion}\n\n\n\\subsection{Summary of substructure in the stellar halo}\n\\label{discussion_subs}\n\nHaving qualitatively and quantitatively compared the chemical abundances of all the halo substructures under study, in this Section we discuss the results obtained for each identified substructure in the context of previous work. \n\n\\subsubsection{Heracles}\n\nStars from this substructure follow low energy, often eccentric orbits with low L$_{z}$, being largely phase-mixed in velocity and action space. All these features are to be expected in the scenario where Heracles was a massive system that merged early in the history of the Milky Way.  Under this hypothesis,  dynamical friction would have driven the system quickly into low energy orbits, sinking it into the heart of the Galaxy (\\citealp{Horta2021,Pfeffer2021})\n\nThe chemical compositions of the stars associated with Heracles are in broad agreement with this scenario. The distribution of Heracles in the $\\alpha$-Fe plane does not display the $\\alpha$-Fe knee or shin components of chemically evolved systems \\citep{McWilliam1997}, which suggests that its star formation ceased before the contribution of supernovae type Ia became substantial \\citep[][]{Horta2021}. In \\cite{Horta2021} we checked that this result was not an effect of the chemical composition criteria adopted in the selection of Heracles stars by comparing them with a similarly selected GES sample, which did display a clear $\\alpha$-Fe knee signature. Furthermore,  Heracles occupies a locus in different chemical planes that resembles that of low mass galaxies of the Milky Way and\/or accreted populations. In particular \\cite{Horta2021} show that the stars associated with Heracles make up a clump in the [Mg\/Mn]-[Al\/Fe] plane in the region occupied by accreted and\/or chemically unevolved populations, characterised by low [Al\/Fe] and high [Mg\/Mn].\n\nRecent work by \\citet[][]{Lane2021} suggests that the concentration associated with Heracles in E-L$_z$ space could be an artifact of the APOGEE selection function, so the authors caution that the reality of this halo substructure should be further tested. As discussed extensively in \\cite{Horta2021}, phase mixing makes it especially hard to discriminate accreted systems from their {\\it in situ} counterparts co-located in the inner few kpc of Milky Way on the sole basis of kinematics.  Nonetheless, theoretical predictions predict their existence \\cite[e.g.][Horta et al., 2022, in prep.]{Fragkoudi2020,Kruijssen2020,Pfeffer2021}.  Detailed chemistry is thus crucial to tease out the remnants of accreted systems from the maze of {\\it in situ} populations overlapping in the inner few kpc of the Galaxy.\n\nFor that reason, we examine closely the comparison between the abundance patterns of Heracles data and the inner high-$\\alpha$ disc at the same [Fe\/H] (Fig.~\\ref{ges_combined3}). Our $\\chi^2$ analysis shows that the two populations differ chemically with high statistical significance ($\\chi^2=53.1$, $p_{\\chi^{2}}$=0, similar to the value obtained when comparing the LMC to the SMC, see Table~\\ref{tab:chi2}). To check whether this result is sensitive to the choice of [Fe\/H]$_{\\mathrm{comp}}$, we reran the analysis adopting [Fe\/H]$_{\\mathrm{comp}}$=--0.9 and [Fe\/H]$_{\\mathrm{comp}}$=--0.95 obtaining $\\chi^{2}$ value of 36.28 and 43.40, respectively, which corresponds to a probability value of $p_{\\chi^{2}}$=0 in both cases. \n\nExamining more closely the contrast between the abundance patterns of Heracles and the inner high-$\\alpha$ disc we find that they differ in interesting ways.  By far the elements displaying the largest differences are oxygen, magnesium, and silicon, whose abundances are lower in Heracles than in the inner high-$\\alpha$ disc. At face value, this difference implies less SNII enrichment in Heracles than in the inner high-$\\alpha$ disc, possibly reflecting a lower star formation rate. A similar, but smaller, difference is seen in carbon, nitrogen, and potassium. Interestingly (as in the case of GES, see Section~\\ref{sec:GES}), we find [Ce\/Fe]$\\sim$0, suggesting a small contribution to chemical enrichment from the $s$-process nucleosynthesis channel. \n\nWe point out that it is possible that the chemical properties ascribed to Heracles may be to some extent influenced by our selection method, which is partly based on chemistry. That selection, however, is far from arbitrary. It is rather informed by the fact that the distribution of inner Galaxy stellar populations form a clear clump in the accreted\/chemically unevolved region of the [Mg\/Mn]-[Al\/Fe] plane \\citep[see Figure 1 of][]{Horta2021}.\nOur $\\chi^2$ analysis shows that Heracles presents an almost unique abundance pattern, differing in a (sometimes small, yet) statistically significant way from most stellar populations under study. For instance, we find that the abundance patterns of Heracles and GES are different (i.e., $p_{\\chi^{2}}$ = 0). This is also the case when comparing Heracles to the Sequoia (all three samples), Arjuna, Thamnos, and Nyx. This result reinforces the hypothesis from \\citet[][]{Horta2021} that Heracles is likely the remnant of a separate early\/massive accretion event. \n\n\n We conclude by stating that our data are consistent with Heracles being the remnant of a satellite that merged with the Milky Way in its early history.  Further studies based on an expanded set of elemental abundances for a larger sample, as well as detailed modelling, based both on cosmological numerical simulations and standard chemical evolution prescriptions, are required to definitively establish the origin of Heracles. \n\n\n\n\\subsubsection{\\textit{Gaia}-Enceladus\/Sausage} \\label{sec:GES}\n\nSince its discovery (\\citealp[][]{Belokurov2018,Helmi2018}), the \\textit{Gaia}-Enceladus\/Sausage (GES) substructure has been extensively studied, both from an orbital and chemical compositions perspective (\\citealp[e.g.,][]{Hayes2018,Mackereth2019b,Koppelman2019b,Vincenzo2019,Aguado2020,Feuillet2020,Simpson2020,Deokkeun2021,Horta2021,Hasselquist2021,Buder2022, Carrillo2022}). In this work, we have identified a large sample of GES stars, and have shown that stars belonging to this population are characterized by intermediate-to-high orbital energies and high eccentricity, displaying no significant systemic disc-like rotation. \n\nThe chemical compositions of the GES substructure are characterised by lower [$\\alpha$\/Fe] at [Fe\/H]~$\\mathrel{\\copy\\simgreatbox}-1.6$, than high-$\\alpha$ disc for most $\\alpha$ elements (namely Mg, O, Si, Ca, S), in agreement with previous work (\\citealp[e.g.,][]{Hayes2018,Haywood2018,Helmi2018,Mackereth2019b,Horta2021,Buder2022}), and displays an $\\alpha$-knee at [Fe\/H]$\\sim$--1.1 (see Fig~\\ref{fig:knees_subs}). The stellar populations of GES are also characterised by lower Al, C, and Ni than \\textit{in situ} populations, resembling the abundance patterns of stars from satellites of the Milky Way \\citep{Horta2021,Hasselquist2021}. They also occupy the accreted\/unevolved region of the [Mg\/Mn]-[Al\/Fe] plane. In summary, the chemical compositions of GES confirm the results from previous studies, and reinforce the idea that this halo substructure is the remnant of an accreted satellite whose debris dominate the local\/inner regions of the stellar halo. \n\n\\subsubsection{Sequoia} \n\\label{sec:Sequoia}\nThe Sequoia substructure was initially discovered due to the highly unbound and retrograde orbits of its constituent stars and associated globular clusters (\\citealp[e.g.,][]{barba2019,Matsuno2019,Myeong2019}), which made it easily distinguishable from \\textit{in situ} populations. Since its discovery, several groups have sought to identify it using different surveys. It has also been examined by \\citet{Koppelman2020} on the basis of $N$-body simulations by \\citet{Villalobos2008},  suggesting that, rather than a separate system, Sequoia may constitute a fringe population of stars in low eccentricity retrograde orbits left over after the GES merger. In this work, we have selected the Sequoia substructure by employing three independent definitions, derived from the \\citet[][]{Myeong2019}, \\citet{Koppelman_thamnos}, and \\citet{Naidu2020} works, respectively. \n\nWhen inspecting the chemical distribution of these three independent Sequoia samples in multiple chemical composition planes, we have found that the Sequoia samples defined by \\citet[][]{Myeong2019}, \\citet[][]{Koppelman_thamnos}, and \\citet[][]{Naidu2020} occupy a similar locus in all chemical composition planes, that resembles that of low mass satelite galaxies of the MW and\/or accreted populations, displaying low Mg, Al, C, and Ni, that overlap with the GES substructure. Furthermore, when running a quantitative $\\chi^{2}$ comparison between the three Sequoia samples, we find that their chemical compositions are indistinguishable from each other, yielding a high probability value of $p_{\\chi^{2}}$=0.93 for the comparison between the \\citet[][]{Koppelman2020} samples and \\citet[][]{Naidu2020} samples, $p_{\\chi^{2}}$=0.95 between the \\citet[][]{Koppelman2020} and \\citet[][]{Myeong2019} samples, and $p_{\\chi^{2}}$=0.99 for that between the \\citet[][]{Myeong2019} and \\citet[][]{Naidu2020} samples. Therefore we conclude, reassuringly, that the chemical composition we obtain for the Sequoia system does not depend on the selection criterion adopted.\n\nAs in the case of the systems discussed above, Sequoia occupies the accreted\/unevolved region of the [Mg\/Mn]-[Al\/Fe] plane, which is encouraging given that two of those samples were selected purely on the basis of orbital parameters. Interestingly, we note that the \\citet[][]{Naidu2020} sample appears to be simply the metal-poor tail of the \\citet[][]{Myeong2019} and \\citet[][]{Koppelman_thamnos} samples. \n\nA quantitative comparison of these Sequoia samples with GES shows that all four populations have consistent and similar chemistry. More specifically, we find that the \\citet[][]{Myeong2019}, \\citet[][]{Koppelman_thamnos}, and \\citet[][]{Naidu2020} Sequoia samples yield a probability value of $p_{\\chi^{2}}$=0.85, $p_{\\chi^{2}}$=0.2, and $p_{\\chi^{2}}$=0.39, respectively. At face value, this chemical similarity corroborates the hypothesis raised by \\citet[][]{Koppelman2020}, that stars with low eccentricity and retrograde orbits with relatively high energies could have resulted from the GES merger.\n\nThe hypothesis advanced by \\cite{Koppelman2020} embodies a falsifiable prediction of a metallicity difference between the two systems.  Such a difference would be expected had Sequoia been originally associated with the stellar populations located in the outskirts of GES.  They argue that, for a GES system of $M_{\\star}\\sim10^{9.6}$M$_{\\odot}$, the metallicity gradient expected between the outer regions stripped first (i.e., Sequoia) and the main body (i.e., GES) would be of the order of $\\sim$0.3 dex.  We exclude from that test the sample selected by \\cite{Naidu2020}, since it adopts a metallicity cut which would bias our results.  The mean metallicities inferred when adopting either the \\citet[][]{Myeong2019} or the \\citet[][]{Koppelman_thamnos} Sequoia samples are ${\\rm \\langle[Fe\/H]\\rangle = -1.41}$ and --1.31, respectively.  In contrast, we obtain ${\\rm \\langle[Fe\/H]\\rangle = -1.19}$ for our GES sample, suggesting a difference of $\\sim$0.1-0.2~dex, which is  roughly consistent with the prediction by \\cite{Koppelman2020}.  It is interesting, however, that no difference in abundance pattern exists between Sequoia and GES at fixed [Fe\/H], which under \\cite{Koppelman2020}'s hypothesis would suggest the absence of abundance ratio gradients in the GES progenitor.  At face value, this result is at odds with the observations of existing satellites of the Milky Way.  For instance, an [$\\alpha$\/Fe] gradient is known to be present in  the Sgr dSph \\citep[e.g.,][]{Hayes2020,Hasselquist2021}.  The latter difference may however be explained away as resulting from the occurrence of recent episodes of star formation in the Sgr dSph, which would have a strong impact on [$\\alpha$\/Fe] of the younger populations \\citep[e.g.,][]{Hasselquist2021}.  It is thus conceivable that the absence of a detectable [$\\alpha$\/Fe] difference between Sequoia and GES results from an early quenching of star formation. \n\n Our samples for GES and Sequoia cover in a statistically meaningful way a wide range of metallicities, so that they lend themselves nicely to a more detailed comparison of the distributions of those two systems in the $\\alpha$-Fe plane. A crucial diagnostic is the metallicity of the ``$\\alpha$-knee'', [Fe\/H]$_{\\rm knee}$ (Section~\\ref{sec_alphas}), which is strongly sensitive to the details of the star formation history of the system. In recent studies, \\cite{Matsuno2019},  \\cite{Monty2020}, and \\cite{Aguado2020} suggest that Sequoia is characterised by a substantially lower [Fe\/H]$_{\\rm knee}$ than GES. To test that hypothesis, we perform a piece-wise linear fit to our GES and Sequoia samples in order to accurately determine the position of the [Fe\/H]$_{\\rm knee}$ in those two systems, in a manner similar to the approach followed by \\cite{Mackereth2019b}.\n\nThe resulting fits (solid lines) and 1-$\\sigma$ dispersions (shaded regions) are displayed along with respective samples for GES (blue), Sequoia \\citep[green and red for the][samples, respectively]{Myeong2019,Koppelman_thamnos} in Figure~\\ref{fig:knees_subs}. The piece-wise function was determined using the \\texttt{PiecewiseLinFit} function included as part of the \\texttt{pwlf} package \\citep[][]{pwlf}. Due to its low metallicity upper limit, the Sequoia sample from \\citet[][]{Naidu2020} is excluded from the comparison. The [Fe\/H]$_{\\rm knee}$ values for GES and the two Sequoia samples are within $\\sim$0.1 dex from each other. The largest difference in fact is that between the \\cite{Myeong2019} and \\cite{Koppelman_thamnos}, with the latter resulting in a larger [Fe\/H]$_{\\rm knee}$.  This result suggests the absence of significant differences in the star formation histories of the two systems, despite the slightly different mean metallicities discussed above.  Since the [Fe\/H]$_{\\rm knee}$ parameter is considerably less sensitive to uncertainties due to selection effects, we conclude that the precursors of GES and Sequoia underwent similar star formation histories, which  supports the hypothesis that they were once different components of the same system.\n\nIn a recent paper, \\cite{Naidu2021} propose that Sequoia, rather than constituting the outskirts of GES, was instead one of its  satellites.  That notion is predicated on their estimated ratio between the masses of the two systems (roughly 1\/10) and their chemical composition differences.  \\cite{Naidu2021} call particular attention to the large metallicity gradient implied by an association between Sequoia and GES, referring as well to the difference in [Fe\/H]$_{\\rm knee}$ from the literature.  Our results call into question the existence of an important difference in [Fe\/H]$_{\\rm knee}$ and mean metallicity, thus possibly accommodating comfortably the possibility of an association between the two systems.\n\nAlong those lines, in a recent work \\citet[][]{Matsuno2021} determined the abundances of 12 Sequoia stars on the basis of high-resolution spectra obtained with the Subaru High Dispersion Spectrograph. Their Sequoia sample was selected following the criteria by \\citet[][]{Koppelman_thamnos}.\n\\cite{Matsuno2021} compared their chemical compositions to those of GES members selected from \\citet[][]{Nissen2010} and \\citet[][]{Reggiani2017}. The authors found that the abundances of Na, Mg, and Ca differed from GES for 8 out of the 10 stars in the cleaned Sequoia sample, at the 2$\\sigma$ level, which is at odds with the findings presented in this work.  Thus, we decided to double-check our results by running \\cite{Matsuno2021}'s methodology on our data.  We fit a quadratic polynomial to our GES data and estimated the residuals of the Sequoia sample stars relative to the polynomial value at their measured [Fe\/H]. Because APOGEE does not yield reliable sodium abundances at low metallicities, we replaced that element by aluminium to run this test, as it is the APOGEE element sharing a nucleosynthetic source that is closest to that of Na. Out of the 45 Sequoia stars (selected as in \\citet[][]{Myeong2019}) that fall within the --1.8 < [Fe\/H] < --1.4 metallicity range adopted by \\citet[][]{Matsuno2021}, we find that 42\/40\/43 stars fall \\textit{within} 2$\\sigma$ of the quadratic polynomial fit to the GES sample when examining the Mg\/Al\/Ca abundance planes. When the \\citet[][]{Koppelman_thamnos} Sequoia sample is considered, we find that 22\/22\/23 out of the total 26 stars match the abundances of Mg\/Al\/Ca. In conclusion, when replicating the procedure performed by \\citet[][]{Matsuno2021} on the basis of APOGEE data, we find that the majority of our sample (>90$\\%$) falls within 2$\\sigma$ of the [X\/Fe]-[Fe\/H] quadratic polynomial fit to the GES sample.  This test confirms our finding that Sequoia and GES are very similar in the chemical space sampled by APOGEE.  The reason for the discrepancy between our results and those by \\citet[][]{Matsuno2021} is unclear.\n\nIn summary, we conclude that the chemical composition data are possibly consistent with a common origin between Sequoia and GES, although further data and modeling are required to fully clarify the matter.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{plots_dr17\/ges_seq_arjuna_hs_knees.png}\n\\caption{Piece-wise polynomial fit (solid line) and 1-$\\sigma$ dispersion (shaded region) for  GES, Sequoia, Arjuna, and Helmi stream samples. Data and fits are displaced vertically for clarity. The resulting [Fe\/H]$_{\\rm knee}$ values are shown. The Mg-Fe knee of GES and Sequoia are within 0.1~dex from each other, with the largest difference being found between the two Sequoia samples.  By the same token, [Fe\/H]$_{\\rm knee}$ for Arjuna differs from that GES by only 0.05~dex. The star formation efficiencies of these systems, as indicated by [Fe\/H]$_{\\rm knee}$, seem not to have been substantially different. Conversely, for the Helmi stream we find an \"inverted\" knee, that occurs at [Fe\/H]$_{\\rm knee}$$\\sim$--1.7, suggestive of a very different star formation history when compared to GES, Sequoia, and Arjuna.}\n    \\label{fig:knees_subs}\n\\end{figure}\n\n\n\\subsubsection{Helmi stream} \n\\label{sec:helmi}\nThe Helmi stream is a halo substructure that appears to jut out of the Galactic disc. It is characterised by stars on highly perpendicular (i.e., high L$_{\\perp}$ and v$_{\\mathrm{z}}$) and prograde orbits (\\citealp{Helmi1999,Koppelman2019}), which appear to form a pillar at high orbital energies in the prograde wing of the E-L$_{z}$ plane (see Fig~\\ref{elz}). Despite this substructure being discovered decades ago, its chemical compositions have not been studied in great detail, largely due to the difficulty of obtaining a high confidence sample with reliable chemical composition information.\n\nOur results on the chemical compositions of the Helmi stream imply that, unsurprisingly, this substructure presents chemistry that is typical of accreted populations and\/or dwarf satellites (i.e., low Mg, Al, C, and Ni). We also find that, despite it being selected purely on a kinematic and position basis, it occupies the accreted\/unevolved region of the [Mg\/Mn]-[Al\/Fe] which, combined with its orbital properties, confirms its accreted nature. Furthermore, we find that, when comparing this halo substructure with the others studied in this work, the Helmi stream differs statistically from all other halo substructures.  In Figure~\\ref{fig:knees_subs} we display the data for the Helmi stream alongside a piecewise polynomial fit performed in the same way as described for GES and the Sequoia samples.  Interestingly, the best fit for the Helmi stream indicate the occurrence of an ``inverted knee'', whereby the slope of the relation between Mg-Fe becomes less negative.  As discussed above for the case of the Sgr dSph, this is the signature of a burst of star formation. This is a very interesting result, which merits further investigation on the basis of a larger sample.\n\n\\subsubsection{Arjuna}\nThe existence of this substructure was proposed by \\cite{Naidu2020} as part of the H3 survey \\citep{Conroy2019}.  \\cite{Naidu2020} show that the MDF of the retrograde component of the halo displays three peaks, which they ascribe to Arjuna (the most metal-rich), Sequoia, and I'itoi (the most metal-poor, see Fig~\\ref{mdf_highe}). \\cite{Naidu2021} argue that Arjuna corresponds to the outer parts of the GES progenitor, which, according to their fiducial numerical simulation was stripped early in the accretion process, thus preserving the highly retrograde nature of the GES approaching orbit. In additional support to that proposition, \\cite{Naidu2021} point out that the peak [Fe\/H] and mean [$\\alpha$\/Fe] of Arjuna are in excellent agreement with those of GES, which in turn should be consistent with a much lower metallicity gradient in the GES progenitor than suggested by \\cite{Koppelman2020}.\n\nThe detailed quantitative comparison of the chemical compositions of Arjuna and GES (Figure~\\ref{ges_combined1}) shows that the similarity of these two systems indeed encompasses a broader range of elemental abundances, leading to a $p_{\\chi^{2}}$ = 0.88. In addition, the distributions of Arjuna and GES stars in the $\\alpha$-Fe plane are also very similar, with the two values for [Fe\/H]$_{\\rm knee}$ agreeing within 0.05~dex (see Fig~\\ref{fig:knees_subs}).\n\nTherefore, our results are at face value in agreement with the suggestion by \\cite{Naidu2021} that the stars associated with the Arjuna substructure were originally part of GES. That association predicts a very low metallicity gradient for GES at the time of the merger with the Milky Way. Further theoretical and observational work is required to ascertain the reality of that prediction \\cite[see discussion in, e.g.,][]{Horta2021,Naidu2021}.\n\n\n\\subsubsection{I'itoi}\n\nSimilarly to Arjuna, the I'itoi substructure is a high-energy retrograde substructure identified by \\citet{Naidu2020}. However, it is comprised by more metal-poor stars (see Fig~\\ref{mdf_highe}) than its high-energy retrograde counterparts, Arjuna and Sequoia.  \\cite{Naidu2021} propose that I'itoi was in fact a satellite of GES, based on its low metallicity and high energy retrograde orbit.  Our detailed comparison of the chemistry of GES and I'itoi suggests that their abundance patterns are consistent with a $p_{\\chi^{2}}$ = 0.3 (Figure~\\ref{ges_combined2}). We point out, however, that this result is highly uncertain, given the relatively small size of our I'itoi sample and its low metallicity ([Fe\/H]$_{\\rm comp}=-2.1$), which places its stars in a regime where ASPCAP abundances are relatively uncertain. The matter needs revisiting on the basis of more detailed chemical composition studies applied to a larger sample.\n\n\n\\subsubsection{Thamnos}\n\nInitially conjectured by its discoverers to be the amalgamation of two smaller systems \\citep{Koppelman_thamnos}, Thamnos is a substructure that occupies a locus in the retrograde wing of the velocity and IoM planes. It is comprised of stars with intermediate orbital energy (i.e., E$\\sim$--1.8$\\times$10$^{5}$ km$^{2}$ s$^{-2}$) and fairly eccentric orbits ($e$$\\sim$0.5), that occupies a position at the foot of GES in the Toomre diagram. \n\nAs in the case of most orbital substructures in this study, we find that the locus occupied by Thamnos in the Ni-Fe, C-Fe, and [Mg\/Mn]-[Al\/Fe] chemical  planes resembles that of low-mass satellite galaxies and accreted populations of the Milky Way.  However, Thamnos distinguishes itself from other substructures by showing a relatively high [$\\alpha$\/Fe] ratio, although not as high as Heracles. In fact, Thamnos does not match the abundance pattern of any other substructure in this study.\n\n\\subsubsection{Aleph}\n\nAleph was identified in a study of the stellar halo based on the H3 survey \\citep{Naidu2020}. In this work we identify Aleph members by selecting from our parent sample stars that satisfy the selection criteria outlined in \\citet{Naidu2020}. We have also searched for stars that are included in both the sample by \\citet[][]{Naidu2020} and in APOGEE DR17 (highlighted in the chemical abundance figures with purple edges).\n As described in their work, this substructure is comprised of stars on very strongly prograde orbits with low eccentricity, which appear to follow the same distribution as the Galactic disc component, although at higher J$_{z}$ values. \n\n We find that the locus occupied by Aleph in various chemical planes is placed somewhere in between low- and high-$\\alpha$ disc stars, while sitting squarely within the \\textit{in situ} region of the [Mg\/Mn]-[Al\/Fe] plane. This is also the case for the two stars contained in both the sample by \\citet[][]{Naidu2020} and in the APOGEE DR17 data. Our quantitative comparison of the Aleph chemistry with that of the low-$\\alpha$ disc suggests a statistically different, albeit on the borderline ($\\chi^{2}$ = 19.47 and $p_{\\chi^{2}}$ = 0.05). This is in fact not surprising, seeing as the distribution of Aleph stars on the Mg-Fe plane straddles both the high and low-$\\alpha$ discs (Figure~\\ref{mgfes}). These results suggest that Aleph may be a stellar population comprised of a mix of warped\/flared low-$\\alpha$ disc, and high-$\\alpha$ disc stars, which also explains its location within the locus of {\\it in situ} stellar populations in the [Mg\/Mn]-[Al\/Fe] plane. \n \n\n\\subsubsection{LMS-1}\n\nLMS-1 is a metal-poor substructure comprised of stars on mildly prograde orbits at intermediate\/high orbital energies. \\citet{Yuan2020} identified this substructure by applying a clustering algorithm to SDSS and LAMOST data in the E-L$_{z}$ plane. Although it presents great overlap with the GES in IoM space, it is suggested to be an independent substructure based on the detection of a metallicity peak in the MDF of the stars included within the E-L$_{z}$ box defining this system \\citep{Naidu2020}. \\cite{Yuan2020} also note that there are potentially several globular clusters with similar metallicity and orbital properties. In this work, we identified LMS-1 candidates adopting the same selection criteria as \\citet{Naidu2020}'s, obtaining a relatively small sample of only 31 stars. \n\nWe examine the distribution of LMS-1 stars in various chemical planes, concluding that its chemistry is consistent with those of other accreted systems, with all its stars falling in the \"accreted\/unevolved\" region of the [Mg\/Mn]-[Al\/Fe] plane. Furthermore, the $\\sim$0.5 probability value obtained when comparing this halo substructure with GES suggests that {\\it at face value} LMS-1 could be part of either of the omnipresent GES substructure. However, these comparisons are made at [Fe\/H]$_{\\rm comp}=-2.1$, where our samples are small and ASPCAP abundances are relatively more uncertain and {\\it in situ} and accreted structures tend to converge towards the same locus in the regions of chemical space sampled by APOGEE \\citep[e.g.,][]{Horta2021}.  Moreover, given its location in IoM space, it is possible that our LMS-1 sample is importantly contaminated by GES stars.\n\nAs mentioned in Section~\\ref{sec:helmi}, \\citet[][]{Jean2017} show that a single accretion event can lead to multiple overdensities in orbital space \\citep[see also][]{Koppelman2020}. Due to the chemical similarity between LMS-1 and GES, and the close proximity between these two halo substructures in orbital planes (see Fig~\\ref{elz}), an association between these two systems seems tempting. However, we defer a firmer conclusion to a future when more elemental abundances are obtained for a larger sample of both LMS-1 and GES candidates.\n\n\n\\subsubsection{Nyx}\nThe Nyx substructure is conjectured to be a stellar stream in the solar vicinity of the Milky Way \\citep{Necib2020}. Given the chemical compositions obtained for this substructure in this work and its strong overlap with the high-$\\alpha$ disc in all the chemical planes studied, we suggest that the Nyx is likely comprised by \\textit{in situ} high-$\\alpha$ disc populations.  Our quantitative estimate of the similarity between Nyx and the stars from the high-$\\alpha$ disc yields a very low $\\chi^{2}$ with associated likelihood probability of 0.8.  We note that the stars in common between our sample and that of \\citet{Necib2020} seem to boldly confirm this result. Along these lines, we note that our result is in agreement with a recent study focused on studying the chemical compositions of the Nyx substructure \\citep[][]{Zucker2021}, who also conjecture Nyx to be comprised of Galactic (high-$\\alpha$) disc populations. \n\n\n\\section{Conclusions}\n\\label{conclusion}\nThe unequivocal association with accretion events of halo substructures identified on the basis of orbital information alone is extremely difficult, as demonstrated by recent numerical simulations \\citep[e.g.,][]{Jean2017,Koppelman2020,Naidu2021}. Substructure in integrals of motion space can also be influenced or even artificially created by survey selection effects \\citep[][]{Lane2021}. Because the chemical compositions of halo substructures contain a fossilised record of the  evolutionary histories of their parent galaxies, abundance pattern information can help linking substructure in orbital space to their progenitor systems.\n\nIn this work we have utilised a cross-matched catalogue of the latest APOGEE (DR17) and \\textit{Gaia} (EDR3) data releases in order to study the chemo-dynamic properties of substructures previously identified in the stellar halo of the Milky Way. We have successfully distinguished stars in the APOGEE DR17 catalogue that are likely associated with the following substructures: \\textit{Gaia}-Enceladus\/Sausage, Sagittarius dSph, Heracles, Helmi stream, Sequoia, Thamnos, Aleph, LMS-1, Arjuna, I'itoi, Nyx, Icarus, and Pontus. Using the wealth of chemical composition information provided by APOGEE, we have examined the distributions of the stellar populations associated with these substructures in a range of abundance planes, probing different nucleosynthetic channels. We performed a quantitative comparison of the abundance patterns of all the substructures studied in order to evaluate their mutual associations, or lack thereof.  Below we summarise our main conclusions:\n\n\\begin{itemize}\n\n\\item We show that the chemical compositions of the majority of the halo substructures so far identified in the Galactic stellar halo (namely, \\textit{Gaia}-Enceladus\/Sausage, Heracles, Sagittarius dSph, Helmi stream, Sequoia, Thamnos, LMS-1, Arjuna, I'itoi) present chemical compositions which resemble those of low-mass satellites of the MW and\/or accreted populations. There are however a couple of exceptions, namely Nyx and Aleph, that do not follow this pattern and instead present chemical properties similar to those of populations formed \\textit{in situ}. Furthermore, in Appendix~\\ref{appen_icarus} we discuss the nature of Icarus, concluding that this substructure is likely comprised of stars formed \\textit{in situ}, for which the ASPCAP abundances are unreliable. \n\n\\item The chemical properties of the inner-Galaxy Heracles substructure differ from those of its co-located low-metallicity high-$\\alpha$ disc counterparts in a statistically significant way. Abundances of $\\alpha$ elements oxygen, magnesium, and silicon are lower in Heracles than in its co-spatial high-$\\alpha$ disc counterparts, suggesting that the ratio of SNII\/SNIa enrichment in Heracles has been lower in that system than in the early disc.  By the same token, Heracles is found to have higher [$\\alpha$\/Fe] ratios than GES which, as suggested by \\cite{Horta2021}, is an indication of an early truncation of star formation and the resulting absence of an $\\alpha$-knee in the former system. The abundance pattern of Heracles is indeed found to differ from all of the other substructures studied in this work. Further studies based on additional elemental abundances for larger samples, as well as detailed numerical modelling, are required to definitively ascertain the existence of this system and more thoroughly characterise its properties.\n\n\n\\item We show that a large fraction of the substructures studied (namely, Sequoia, Arjuna, LMS-1, I'itoi) present chemistry indistinguishable from that of the omnipresent \\textit{Gaia}-Enceladus\/Sausage. These findings bring into question the independence of these substructures, which are at least partially overlapping with GES in kinematic planes (see Fig~\\ref{elz}). In view of these similarities, claims in the literature about the nature of Sequoia as being originally a higher angular momentum component located in the outskirts of GES \\citep[][]{Koppelman2020}, a satellite of GES \\citep{Naidu2021}, or an altogether unrelated system \\citep{Myeong2019} may need to be reexamined. The possibility that Sequoia was a detached, but much less massive galaxy than GES is likely challenged by their chemical similarity.  On the other hand, confirmation that it, or Arjuna, might be the remains of populations originally located in the outskirts of GES depends crucially on the magnitude of chemical composition gradients one should expect for dwarf galaxies at $z\\approx2$, and on whether that is a sufficiently discriminating criterion.\n\n\\item We found that the halo hosts substructures which differ from GES in a statistically significant way. Three among those susbstructures display chemistry that is genuinely suggestive of an accreted nature, namely Heracles, Thamnos, and the Helmi stream (although for the latter this conclusion is not firm due to uncertainties in the chemistry and small sample size). \n\n\\item Conversely, the chemistry of the two remaining substructures, Nyx and Aleph, is indistinguishable from that of {\\it in situ} populations. We conjecture that Nyx forms part of the high-$\\alpha$ disc. For the case of Aleph, we suggest that it is likely comprised of stars both from the low-$\\alpha$ (flared\/warped) disc as well as stars from the high-$\\alpha$ disc.\n\n\\item Our results suggest that the local\/inner (R$_{GC}$ $\\lesssim$ 20kpc) halo is comprised of the debris from at least three massive accretion events (Heracles, GES, and Sagittarius) and two lower mass debris (Thamnos and Helmi stream). Upcoming large spectroscopic surveys probing deeper into the outer regions of the stellar halo (beyond R$_{GC}$ $\\sim$20kpc) will likely uncover the debris from predicted lower-mass\/more recent accretions (\\citealp[e.g.,][Horta et al. 2022, in prep]{Bosch2016}), and will enable the full characterisation of those already known. Conversely the precise contribution of massive accretion to the stellar populations content of the inner few kpc of the halo is still to be fully gauged. Heracles is likely the result of a real accretion event, likely the most massive to have ever happened in the history of the Milky Way, but we are still scratching its surface.\n\n\\end{itemize}\n\nThis paper presents a chemical composition study of substructures identified (primarily) on the basis of phase-space and orbital information in the stellar halo of the Milky Way. Current and upcoming surveys will continue to map the stellar halo and will provide further clues to the nature and reality of phase-space substructures discovered in recent years. For the inner regions of the Galaxy and the local halo, the Milky Way Mapper \\citep[][]{sdssv} and the Galactic component of the MOONS GTO survey \\citep{moons} will provide revolutionising chemo-dynamical information for massive samples. For the outer regions of the stellar halo, the WEAVE \\citep[][]{Dalton2012}, 4MOST \\citep[][]{DeJong2019}, and DESI \\citep[][]{DESI2016} surveys will provide spectroscopic data for millions of stars in both high and low resolution. In addition, in this work we have only studied the chemistry of phase-mixed accretion events. However, there is a plethora of halo substructures in the form of stellar streams that has not been studied in this work and also require to be fully examined (see \\citealt[][]{Li2021} for a recent example). The advent of surveys like $S^{5}$ \\citep[][]{Li2019} will aid in such endeavours. All this information, when coupled with the exquisite astrometry and upcoming spectroscopic information from the \\textit{Gaia} satellite, will provide the necessary information for further discoveries and examinations of substructure in the stellar halo of the Milky Way.\n\n\n\\section*{Acknowledgements}\nThe authors thank Rohan Naidu and Charlie Conroy for making available the H3 catalogue in digital format, and Paola Re Fiorentin and Alessandro Spagna for sharing their Icarus star's APOGEE IDs. DH thanks Melissa Ness, Holger Baumgardt, and Cullan Howlett for helpful scientific discussions, and Sue, Alex and Debra for their constant support. D.G. gratefully acknowledges financial support from the Direcci\\'on de Investigaci\\'on y Desarrollo de la Universidad de La Serena through the Programa de Incentivo a la Investigaci\\'on de Acad\\'emicos (PIA-DIDULS). Funding for the Sloan Digital Sky Survey IV has been provided by\nthe Alfred P. Sloan Foundation, the U.S. Department of Energy Office\nof Science, and the Participating Institutions. SDSS acknowledges\nsupport and resources from the Center for High-Performance Computing\nat the University of Utah. The SDSS web site is www.sdss.org. SDSS\nis managed by the Astrophysical Research Consortium for the\nParticipating Institutions of the SDSS Collaboration including the\nBrazilian Participation Group, the Carnegie Institution for Science,\nCarnegie Mellon University, the chilean Participation Group, the\nFrench Participation Group, Harvard-Smithsonian Center for Astrophysics,\nInstituto de Astrof\\'{i}sica de Canarias, The Johns Hopkins University,\nKavli Institute for the Physics and Mathematics of the Universe\n(IPMU) \/ University of Tokyo, the Korean Participation Group,\nLawrence Berkeley National Laboratory, Leibniz Institut f\\\"{u}r Astrophysik\nPotsdam (AIP), Max-Planck-Institut f\\\"{u}r Astronomie (MPIA Heidelberg),\nMax-Planck-Institut f\\\"{u}r Astrophysik (MPA Garching), Max-Planck-Institut\nf\\\"{u}r Extraterrestrische Physik (MPE), National Astronomical Observatories\nof china, New Mexico State University, New York University, University\nof Notre Dame, Observat\u00f3rio Nacional \/ MCTI, The Ohio State University,\nPennsylvania State University, Shanghai Astronomical Observatory,\nUnited Kingdom Participation Group, Universidad Nacional Aut\u00f3noma\nde M\u00e9xico, University of Arizona, University of Colorado Boulder,\nUniversity of Oxford, University of Portsmouth, University of Utah,\nUniversity of Virginia, University of Washington, University of\nWisconsin, Vanderbilt University, and Yale University.\n\nThis work presents results from the European Space Agency (ESA) space mission Gaia. Gaia data are being processed by the Gaia Data Processing and Analysis Consortium (DPAC). Funding for the DPAC is provided by national institutions, in particular the institutions participating in the Gaia MultiLateral Agreement (MLA). The Gaia mission website is \\href{https:\/\/www.cosmos.esa.int\/gaia}{https:\/\/www.cosmos.esa.int\/gaia}. The Gaia archive website is \\href{https:\/\/archives.esac.esa.int\/gaia}{https:\/\/archives.esac.esa.int\/gaia}.\n\n{\\it Software:} Astropy \\citep{astropy:2013,astropy:2018}, SciPy\n\\citep{Scipy2020}, NumPy \\citep{NumPy}, Matplotlib \\citep{Hunter:2007},\nGalpy \\citep{Galpy2015,Galpy2018}, TOPCAT \\citep{Taylor2005}.\n\n{\\it Facilities:} Sloan Foundation 2.5m Telescope of Apache Point Observatory (APOGEE-North), Ir\\'en\\'ee du Pont 2.5m Telescope of Las Campanas Observatory (APOGEE-South), \\textit{Gaia} satellite\/European Space Agency (\\textit{Gaia}).\n\n\n\\section*{Data availability}\nAll APOGEE DR17 data used in this study is publicly available and can be found at: https: \/www.sdss.org\/dr17\/\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCore-collapse supernovae are among the most energetic transients that occur in the Universe, originating from the death of very massive stars~\\cite{Janka:2016fox,Janka:2012wk}. Despite the remarkable progress we have seen in our understanding of the core-collapse physics over the last decade, we are still far from fully grasping the physical processes that underlie the supernova (SN) engine and, in particular, the role that neutrinos play in powering it~\\cite{Janka:2016fox,Mirizzi:2015eza,Chakraborty:2016yeg}. A high-statistics detection of neutrinos from the next Galactic SN explosion in detectors that operate with different technologies will shed light on both the stellar engine and the properties of neutrinos. Neutrino flavour discrimination will be crucial to investigate neutrino oscillation physics and scenarios with non-standard neutrino properties~\\cite{Mirizzi:2015eza,Esmaili:2014gya,Wu:2014kaa,EstebanPretel:2007yu,Stapleford:2016jgz,Hidaka:2007se}.\nOn the other hand, the detection of all six neutrino flavours will be essential to reconstruct global emission properties, such as the total explosion energy emitted into neutrinos~\\cite{Drukier:1983gj,Beacom:2002hs,Horowitz:2003cz}. \n\nAt present, several neutrino detectors are ready for the next Galactic SN explosion, while others are under construction or being planned~\\cite{Mirizzi:2015eza,Scholberg:2012id}. Among these experiments, the most promising technologies include Cherenkov telescopes and liquid scintillators, as used in or proposed for IceCube~\\cite{Abbasi:2011ss}, Super-Kamiokande~\\cite{Abe:2010hy,Abe:2016waf}, IceCube-Gen2~\\cite{Aartsen:2014njl}, Hyper-Kamiokande~\\cite{Abe:2011ts}, LVD~\\cite{Agafonova:2014leu}, Borexino~\\cite{Cadonati:2000kq}, JUNO~\\cite{An:2015jdp}, RENO-50~\\cite{Kim:2014rfa}, and KamLAND~\\cite{Barger:2000hy,Asakura:2015bga}. Both of these technologies will be able to probe $\\bar{\\nu}_e$ neutrinos with high accuracy. In contrast, the planned liquid argon detector within the DUNE facility~\\cite{Acciarri:2015uup} will accurately probe the $\\nu_e$ channel. There are also proposals to study the $\\nu_e$ properties with Cherenkov telescopes or liquid scintillators~\\cite{Laha:2014yua,Laha:2013hva,Jia-Shu2016} or with experiments that use lead or iron targets~\\cite{Kolbe:2000np,Volpe:2001gy,Shantz:2010th}. Together, these experiments will accurately measure the $\\bar{\\nu}_e$ and $\\nu_e$ fluxes from the next Galactic SN explosion~\\cite{Scholberg:2012id}.\n\nThe elastic scattering of neutrinos on protons~\\cite{Beacom:2002hs,Dasgupta:2011wg} and on nuclei~\\cite{Drukier:1983gj} are alternative tools to detect astrophysical neutrinos. Neutrino-nucleus scattering is especially attractive because, at low energies, the scattering cross-section is coherently enhanced by the square of the nucleus's neutron number~\\cite{Freedman:1977xn}. Supernova neutrinos with energies of $\\mathcal{O}(10)$~MeV induce $\\mathcal{O}(1)$~keV nuclear recoils through coherent elastic neutrino-nucleus scattering ($\\mathrm{CE}\\nu\\mathrm{NS}$). Although recoils in this energy range are too small to be detected by conventional neutrino detectors, it is precisely this energy range for which direct detection dark matter experiments are optimized~\\cite{Undagoitia:2015gya}. The primary purpose of these experiments is to search for nuclear recoils induced by Galactic dark matter particles. Yet, sufficiently large experiments ($\\gtrsim$ tonne of target material) are also sensitive to $\\mathrm{CE}\\nu\\mathrm{NS}$\\ from SN neutrinos~\\cite{Horowitz:2003cz,Monroe:2007xp,Strigari:2009bq,XMASS:2016cmy}.\n\nMediated by $Z$-boson exchange, $\\mathrm{CE}\\nu\\mathrm{NS}$\\ is especially intriguing because it is equally sensitive to all neutrino flavours. Detectors that observe $\\mathrm{CE}\\nu\\mathrm{NS}$\\ are therefore sensitive to the $\\bar{\\nu}_{\\mu}$, $\\nu_{\\mu}$, $\\bar{\\nu}_{\\tau}$ and $\\nu_{\\tau}$ (otherwise dubbed~$\\nu_x$) neutrinos within their main detection channel, in addition to the $\\bar{\\nu}_e$ and $\\nu_e$ neutrinos~\\cite{Drukier:1983gj}. This feature carries numerous implications for experiments that detect $\\mathrm{CE}\\nu\\mathrm{NS}$. For instance, the neutrino light curve could be reconstructed without the uncertainties that arise from neutrino oscillation in the stellar envelope~\\cite{Chakraborty:2013zua}, the total energy emitted into all neutrino species could be measured, or, by assuming adequate reconstruction of the $\\bar{\\nu}_e$ and $\\nu_e$ emission properties with other detectors,\n $\\mathrm{CE}\\nu\\mathrm{NS}$\\  detectors provide a way to reconstruct the $\\nu_x$ emission properties. \n\nIn this paper, we revisit the possibility of detecting $\\mathrm{CE}\\nu\\mathrm{NS}$\\ from SN neutrinos in the context of XENON1T~\\cite{Aprile:2015uzo} and larger forthcoming direct detection dark matter experiments that employ a xenon target, such as XENONnT~\\cite{Aprile:2015uzo}, LZ~\\cite{Akerib:2015cja}, and DARWIN~\\cite{Aalbers:2016jon}. Among the various technologies used in direct detection experiments, dual-phase xenon experiments have many advantages: the large neutron number of the xenon nucleus enhances the $\\mathrm{CE}\\nu\\mathrm{NS}$\\ rate compared to nuclei used in other direct detection experiments; they are sensitive to sub-keV nuclear recoils; the deployment of XENON1T heralds the era of tonne-scale experiments, which are relatively straightforward to scale to even larger masses; despite their large size, the background rates are very low; and, finally, they have excellent timing resolution, $\\mathcal{O}(100)$~\\!$\\mu$s, in the data analysis mode discussed here. As we demonstrate in section~\\ref{sec:results}, these factors mean that XENON1T is already able to detect neutrinos from a SN up to 25~kpc at a significance of more than~$5\\sigma$.\n\nTo forecast the signal that is expected from SN neutrinos in forthcoming xenon detectors, we adopt inputs of four hydrodynamical SN simulations from the Garching group~\\cite{Mirizzi:2015eza,Huedepohl:2013} that differ in the progenitor's mass and nuclear equation of state in such a way as to provide a reasonable estimate of the signal band. The neutrino properties for the adopted progenitor models are introduced in section~\\ref{sec:SNmodels}. In section~\\ref{sec:xenondet}, for the first time, we accurately simulate the expected signal in terms of the measured quantities in dual-phase xenon experiments (scintillation photons and ionization electrons). In section~\\ref{sec:S2only}, we discuss the advantages of a dual-phase xenon detector in the observation of SN neutrinos (the low-energy sensitivity of the proportional-scintillation-signal analysis mode) as well as the expected backgrounds and achievable threshold. Section~\\ref{sec:results} contains our main physics results. We discuss the detection significance of SN neutrinos with future-generation xenon detectors, the reconstruction of the SN neutrino light curve as well as the average neutrino energy and the total explosion energy. Uncertainties related to the detector modeling are outlined in section~\\ref{sec:discussion}. Finally, in section~\\ref{sec:conclusions}, we present our conclusions.\n\n\\section{Supernova neutrino emission} \\label{sec:SNmodels}\n\nIn order to forecast the expected recoil signal in a xenon detector, we require the differential flux of the $\\nu_e$, $\\bar{\\nu}_e$ and $\\nu_x$ neutrinos as a function of time and energy. The differential flux for each neutrino flavour~$\\nu_{\\beta}$ at a time~$t_{\\rm pb}$ after the SN core bounce for a SN at a distance~$d$ is parametrized by\n\\begin{equation}\n\\label{eq:nuflux}\nf_{\\nu_\\beta}^0(E,t_{\\rm pb})= \\frac{L_{\\nu_\\beta}(t_{\\rm pb})}{4 \\pi d^2}\\,\\frac{\\varphi_{\\nu_\\beta}(E,t_{\\rm pb})}{\\langle E_{\\nu_\\beta}(t_{\\rm pb}) \\rangle}\\ ,\n\\end{equation}\nwhere $L_{\\nu_\\beta}(t_{\\rm pb})$ is the $\\nu_\\beta$ luminosity, $\\langle E_{\\nu_\\beta}(t_{\\rm pb}) \\rangle$ is the mean energy, and $\\varphi_{\\nu_\\beta}(E,t_{\\rm pb})$ is the neutrino energy distribution. The neutrino energy distribution is defined in~\\cite{Keil:2002in,Tamborra:2012ac} as\n\\begin{equation}\n\\begin{split}\n\\label{alphafit}\n\\varphi_{\\nu_\\beta}(E,t_{\\rm pb})&=\\xi_\\beta(t_{\\rm pb}) \\left(\\frac{E}{\\langle E_{\\nu_\\beta}(t_{\\rm pb}) \\rangle}\\right)^{\\alpha_\\beta(t_{\\rm pb})}\\\\\n&\\quad\\times \\exp\\left[-\\frac{(\\alpha_\\beta(t_{\\rm pb})+1) E}{\\langle E_{\\nu_\\beta}(t_{\\rm pb}) \\rangle}\\right] .\n\\end{split}\n\\end{equation}\nThe fit parameter $\\alpha_\\beta(t_{\\rm pb})$ satisfies the relation\n\\begin{equation}\n\\label{eq:earelate}\n\\frac{\\langle E_{\\nu_\\beta}(t_{\\rm pb})^2 \\rangle}{\\langle E_{\\nu_\\beta}(t_{\\rm pb}) \\rangle^2} =\n\\frac{2+\\alpha_\\beta(t_{\\rm pb})}{1+\\alpha_\\beta(t_{\\rm pb})}\\ ,\n\\end{equation}\nwhile $\\xi_\\beta(t_{\\rm pb})$ is a normalization factor defined such that $\\int dE \\,\\varphi_{\\nu_\\beta}(E,t_{\\rm pb})=1$. In the following, we show results for a benchmark distance of $d=10$~\\!kpc for the Galactic~SN.\n\n\\subsection{Supernova neutrino emission properties}\n\nThe neutrino emission properties that we adopt are from the one-dimensional (1D) spherically symmetric SN hydrodynamical simulations by the Garching group~\\cite{Mirizzi:2015eza,Huedepohl:2013,snarchive}. More recent three-dimensional (3D) SN simulations exhibit hydrodynamical instabilities such as large-scale convective overturns and the standing accretion shock instability (SASI) that are responsible for characteristic modulations in the neutrino signal not observable in 1D SN simulations~\\cite{Lund:2010kh,Tamborra:2013laa,Tamborra:2014hga,Tamborra:2014aua,Janka:2016fox}. However, as in this paper we are interested in the general qualitative behaviour of the SN neutrino event rate in a xenon detector, we can safely neglect these effects and adopt the outputs from 1D spherically symmetric SN simulations.\n\nTo investigate the variability of the expected signal as a function of the progenitor mass, we use the neutrino emission properties from the hydrodynamical simulations of two SN progenitors with masses of $11.2~\\!\\mathrm{M}_{\\odot}$ and $27~\\!\\mathrm{M}_{\\odot}$. We also consider the dependence of the expected event rates on the nuclear equation of state (EoS) by adopting, for each SN progenitor, simulations obtained from the Lattimer and Swesty EoS~\\cite{Lattimer:1991nc} with a nuclear incompressibility modulus of $K = 220$~\\!MeV (LS220 EoS) and the Shen EoS~\\cite{Shen:1998gq}. These four progenitors provide a gauge of the astrophysical variability of the expected recoil signal.\n\n\\begin{figure*}[!htbp]\n\\centering\n\\includegraphics[width=1.9\\columnwidth]{luminosity_energy_panel.pdf}\n\\caption{The upper and lower panels show the neutrino luminosity~$L_{\\nu_{\\beta}}$ and mean energy $\\langle E_{\\nu_\\beta} \\rangle$, respectively, as a function of the post-bounce time~$t_{\\rm{pb}}$ for the $11~\\!\\mathrm{M}_{\\odot}$ (in blue) and $27~\\!\\mathrm{M}_{\\odot}$ (in red) SN progenitors with the LS220 EoS for $\\nu_e$ (continuous lines), $\\bar{\\nu}_e$ (dashed lines) and $\\nu_x$ (dot-dashed lines). The panels on the left show the neutrino properties during the neutronization burst phase, the middle panes refer to the accretion phase, and panels on the right describe the Kelvin-Helmholtz cooling phase. The differences in the neutrino properties from different progenitors during the neutronization burst are small, but they become considerable at later times. The variation of the neutrino properties owing to a different nuclear EoS is smaller than the differences from a different progenitor mass, so for clarity, the progenitors with the Shen EoS are not shown here.}\n\\label{fig:LEpanels}\n\\end{figure*}\n\nFigure~\\ref{fig:LEpanels} displays the neutrino luminosities (top panels) and mean energies (bottom panels) of all neutrino flavours ($\\nu_e$, $\\bar{\\nu}_e$ and $\\nu_x$) as a function of the post-bounce time in the observer frame for the $27~\\!\\mathrm{M}_{\\odot}$ and $11~\\mathrm{M}_{\\odot}$ SN progenitors with the LS220 EoS. The variation of the neutrino properties owing to a different nuclear EoS is smaller than the differences shown here from the different progenitor mass. The neutrino signal emitted from a SN explosion lasts for more than 10~\\!s but, as Fig.~\\ref{fig:LEpanels} demonstrates, the luminosity drops considerably after a few seconds. Therefore, in this paper, we focus on the initial 7~\\!s of the neutrino signal after the core bounce.\n\nThe left, middle, and right panels of Fig.~\\ref{fig:LEpanels} show the three main phases of the SN neutrino signal: the neutronization burst, the accretion phase, and the Kelvin-Helmholtz cooling phase, respectively. The neutronization burst originates while the shock wave is moving outwards through the iron core. Free protons and neutrons are released as the shock wave dissociates iron nuclei. Consequently, rapid electron capture by nuclei and free protons produces a large~$\\nu_e$ burst. As evident from the top left panel in Fig.~\\ref{fig:LEpanels}, the width and amplitude of the~$\\nu_e$ luminosity during the neutronization burst are approximately independent of the SN progenitor mass and EoS~\\cite{Kachelriess:2004ds,Serpico:2011ir}. Generally, the~$\\nu_x$ luminosity rises more quickly than that of~$\\bar{\\nu}_e$ during the first 10--20~ms of the signal due to the high abundance of $\\nu_e$ and electrons, which suppress the rapid production of $\\bar{\\nu}_e$.\n\nThe accretion phase is shown in the middle panels of Fig.~\\ref{fig:LEpanels}. During this phase, the SN shock loses energy while moving outward and dissociating iron nuclei until it stalls at a radius of about 100--200~\\!km. According to the delayed-neutrino SN explosion mechanism~\\cite{Bethe:1984ux,Bethe:1990mw}, neutrinos provide additional energy to the shock to revive it after tens to hundreds of milliseconds and finally trigger the explosion. As the in-falling material accretes onto the core, it is heated, and the subsequent $e^+e^-$ annihilation produces neutrinos of all flavours. Due to the high abundance of $\\nu_e$ during the neutronization burst, the production of $\\bar{\\nu}_e$ and $\\nu_x$ is initially suppressed. The production of $\\bar{\\nu}_e$ increases as the capture of electrons and positrons on free nucleons starts to become more efficient. The non-electron neutrinos remain less abundant as they can only be produced via neutral-current interactions. \n\nThe explosion of 1D SN simulations may require an artificial initiation, especially for more massive progenitors. In the simulation  shown in Fig.~\\ref{fig:LEpanels}, the explosion was triggered at $t_{\\rm{pb}}\\simeq0.5$~\\!s. After this, the Kelvin-Helmholtz cooling phase of the newly born neutron star begins. As shown in the right panel of Fig.~\\ref{fig:LEpanels}, the neutrino luminosities gradually decrease as the proto-neutron star cools and de-leptonizes. As the explosion is artificially triggered in these simulations, the exact transition time from the accretion to the Kelvin-Helmholtz cooling phase should be taken with caution. The neutrino signal during this phase is sensitive to the progenitor mass and the EoS. In fact, while the differences among the neutrino properties from different progenitors during the neutronization burst are small, at later times they become considerable. \n\n\\subsection{Neutrino flavour conversion}\n\nThe neutrino transport in SN hydrodynamical simulations is solved within the weak-interaction basis for all three neutrino flavours. Neutrinos oscillate while they are propagating through the stellar envelope as well as on their way to Earth. This affects the neutrino flavour distribution detected on Earth. In particular, neutrinos undergo the Mikheev-Smirnov-Wolfenstein (MSW) effect~\\cite{wolf,Wolfenstein:1977ue,Mikheev:1986if}, which affects the survival probability of each neutrino flavour according to the adiabaticity of the matter profile. The MSW effect could be modified by turbulence or significant stochastic fluctuations in the stellar matter density (see e.g.~\\cite{Borriello:2013tha,Sawyer:1990tw,Fogli:2006xy,Friedland:2006ta}). In addition, neutrino--neutrino interactions are believed to be important and can affect the neutrino flavour evolution and therefore the expected energy distribution~\\cite{Duan:2010bg,Mirizzi:2015eza,Chakraborty:2016yeg}.\n\nFor our purpose, however, the details of the oscillation physics are not important. This is because $\\mathrm{CE}\\nu\\mathrm{NS}$\\ is sensitive to all neutrino flavours and the total neutrino flux is conserved. Hence, the same total flux produced at the SN core will reach the detector on Earth.\n\nNon-standard physics may lead to situations where the total flux is not conserved, such as a scenario with light sterile neutrinos~\\cite{Esmaili:2014gya,Wu:2014kaa,Tamborra:2011is,Pllumbi:2014saa}, non-standard neutrino interactions~\\cite{EstebanPretel:2007yu,Nunokawa:1996tg,Stapleford:2016jgz}, or light dark matter particles~\\cite{Fuller:2009zz,Hidaka:2007se,Davis:2016dqh}. All of these cases affect the heating of the star, implying that the total neutrino flux reaching the Earth could be different from the total neutrino flux at the neutrinosphere. In this paper, we do not consider these scenarios further but focus on the Standard Model scenario.\n\n\\section{Supernova neutrino scattering with dual-phase xenon detectors} \\label{sec:xenondet}\n\nWith the launch of the XENON1T experiment~\\cite{Aprile:2015uzo}, which contains two tonnes of instrumented xenon, direct detection dark matter searches have entered the era of tonne-scale targets. The detection principle of this experiment is similar to smaller predecessors, including LUX~\\cite{Akerib:2012ys}, PandaX~\\cite{Cao:2014jsa}, XENON100~\\cite{Aprile:2011dd}, XENON10~\\cite{Aprile:2010bt}, and the three ZEPLIN experiments~\\cite{Alner:2005pa, Alner:2007ja,Akimov:2006qw}. Future experiments using the same technology include XENONnT~\\cite{Aprile:2014zvw} and LZ~\\cite{Akerib:2015cja} with each planning for approximately seven tonnes of instrumented xenon. The DARWIN consortium~\\cite{Baudis:2012bc,Schumann:2015cpa,Aalbers:2016jon} is investigating an even larger experiment to succeed XENONnT and LZ with approximately 40~tonnes of instrumented xenon. In principle, the technology can be extended to even larger detectors at comparatively modest cost.\n\nThese experiments consist of a dual-phase cylindrical time projection chamber (TPC) filled primarily with liquid xenon and a gaseous xenon phase on top. The energy deposited by an incident particle in the instrumented volume produces two measurable signals, called the~S1 and~S2 signals, respectively, from which the energy deposition can be reconstructed. An energy deposition in the liquid xenon creates excited and ionized xenon atoms, and the prompt de-excitation of excited molecular states yields the S1 (or prompt scintillation) signal. An electric drift field of size $\\mathcal{O} (1)$~\\!kV\/cm draws the ionization electrons to the liquid-gas interface. A second electric field of size $\\mathcal{O} (10)$~\\!kV\/cm extracts the ionization electrons from the liquid to the gas. Within the gas phase, these extracted electrons collide with xenon atoms to produce the S2 (or proportional scintillation) signal. The S1 and S2 signals are observed with two arrays of photomultiplier tubes (PMTs) situated at the top and bottom of the TPC. A measurement of both the S1 and S2 signals allows for a full 3D reconstruction of the position of the energy deposition in the TPC. In typical dark matter searches, only an inner volume of the xenon target is used to search for dark matter (the ``fiducial volume\"), but the background rate for the duration of the SN signal is sufficiently small such that all of the instrumented xenon can be used to search for SN neutrino scattering (see section~\\ref{sec:S2only} for further discussion). In the following, we will thus always refer to the instrumented volume.\n\nThe general expression for the differential scattering rate $dR$ in terms of the observable~S1 and~S2 signals for a perfectly efficient detector is\n\\begin{equation}\n\\label{eq:rateS1S2}\n\\frac{d^2R}{d \\mathrm{S1} d \\mathrm{S2}}= \\int\\! dt_{\\mathrm{pb}} d E_{\\mathrm{R}} \\,\\mathrm{pdf} \\left( \\mathrm{S1},\\mathrm{S2} | E_{\\mathrm{R}} \\right) \\frac{d^2R}{d E_{\\mathrm{R}} {d t_{\\mathrm{pb}}}}.\n\\end{equation}\nThe differential rate is an integral over the time-period of the SN neutrino signal, expressed in terms of the post-bounce time~$t_{\\mathrm{pb}}$, and an integral over the recoil energy~$E_{\\mathrm{R}}$ of the xenon nucleus. The differential scattering rate in terms of~$E_{\\mathrm{R}}$ is convolved with the probability density function $(\\mathrm{pdf})$ to obtain S1 and S2 signals for a given energy deposition~$E_{\\mathrm{R}}$. In subsections~\\ref{sec:subEr} and~\\ref{sec:S1S2}, we describe the procedure to calculate~$d^2R\/dE_{\\mathrm{R}} d t_{\\mathrm{pb}}$ and~$\\mathrm{pdf}(\\mathrm{S1},\\mathrm{S2} | E_{\\mathrm{R}})$ respectively. Subsequently, in subsection~\\ref{sec:obrates}, we present the expected neutrino-induced scattering rates in terms of the~S1 and~S2 observable quantities.\n\nBefore moving on, we briefly comment on single-phase xenon experiments, such as XMASS~\\cite{Abe:2013tc}, which only have the liquid phase. The absence of the gas-phase implies that there is no S2 signal, so any generated ionization only adds to the S1 signal. Thus, the instrument is more sensitive in S1 but lacks the inherent amplification of the S2 signal using proportional scintillation. Ultimately, due to quantum efficiencies of photon detection and some sources of background, single-phase detectors have a higher energy threshold compared to dual-phase detectors. As we demonstrate in the next subsection, the recoil spectrum increases rapidly at low energies; so, dual-phase experiments are significantly more sensitive to SN neutrinos. For this reason, we do not consider single-phase detectors and refer the reader to the literature for further discussion~\\cite{XMASS:2016cmy}.\n\n\\subsection{Scattering rates in terms of recoil energy \\label{sec:subEr}}\n\nThe interaction of a SN neutrino with a xenon nucleus through $\\mathrm{CE}\\nu\\mathrm{NS}$\\ causes the nucleus to recoil with energy~$E_{\\mathrm{R}}$. The differential scattering rate in terms of~$E_{\\mathrm{R}}$ is given by\n\\begin{equation}\n\\label{eq:d2RdEdt}\n\\frac{d^2R}{d E_{\\mathrm{R}} d t_{\\rm{pb}}}=\\sum_{\\nu_{\\beta}}N_\\text{Xe}\\int_{E_{\\nu}^{\\rm{min}}} dE_{\\nu}\\, f_{\\nu_\\beta}^0(E_{\\nu},t_{\\rm pb}) \\frac{d\\sigma}{d E_{\\mathrm{R}}} \\ ,\n\\end{equation}\nwhere the sum is over all six neutrino flavours, $N_{\\rm{Xe}}\\simeq4.60\\times10^{27}$ is the number of xenon nuclei per tonne of liquid xenon, $E_\\nu^{\\rm{min}}\\simeq\\sqrt{m_{\\mathrm{N}} E_{\\mathrm{R}}\/2}$ is the minimum neutrino energy required to induce a xenon recoil with energy~$E_{\\mathrm{R}}$, $m_{\\mathrm{N}}$~is the mass of the xenon nucleus, and $f_{\\nu_\\beta}^0(E_{\\nu},t_{\\rm pb})$~is as defined in Eq.~\\eqref{eq:nuflux}. Finally, $d\\sigma\/ d E_{\\mathrm{R}}$ is the coherent elastic neutrino-nucleus scattering cross-section~\\cite{Drukier:1983gj},\n\\begin{equation}\n\\frac{d\\sigma}{d E_{\\mathrm{R}}}= \\frac{G_F^2 m_{\\mathrm{N}}}{4\\pi}Q_W^2\\left(1-\\frac{m_{\\mathrm{N}} E_{\\mathrm{R}}}{2 E_\\nu^2}\\right)F^2(E_{\\mathrm{R}}) \\ ,\n\\label{eq:Xsection}\n\\end{equation}\nwhere $G_F$ is the Fermi constant, $Q_W=N-(1-4\\sin^2\\theta_W)Z$ is the weak nuclear hypercharge of a nucleus with $N$ neutrons and $Z$ protons, $\\sin^2\\theta_W\\simeq0.2386$ is the weak mixing angle at small momentum transfer~\\cite{Erler:2004in}, and $F(E_{\\mathrm{R}})$ is the nuclear form factor. For xenon, the Helm form factor provides an excellent parametrization for the small values of $E_{\\mathrm{R}}$ induced by $\\mathrm{CE}\\nu\\mathrm{NS}$\\ with which we are concerned~\\cite{Vietze:2014vsa},\n\\begin{equation}\nF(E_{\\mathrm{R}})=\\frac{3 j_1 (q r_n)}{qr_n}\\exp\\left(-\\frac{(qs)^2}{2} \\right) \\, ,\n\\end{equation}\nwhere $q^2=2m_{\\mathrm{N}} E_{\\mathrm{R}}$ is the squared momentum transfer, $s=0.9$~fm is the nuclear skin thickness, $r^2_n=c^2+\\frac{7}{3}\\pi^2 a^2-5s^2$ is the nuclear radius parameter, $c=1.23A^{1\/3}-0.60$~\\!fm, $a=0.52$~\\!fm, $A$ is the atomic number of xenon, and~$j_1(q r_n)$ is the spherical Bessel function. \n\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[width=0.975\\columnwidth]{diffrecspec.pdf}\n\\includegraphics[width=0.975\\columnwidth]{lightcurves.pdf}\n\\includegraphics[width=0.975\\columnwidth]{interecspec.pdf}\n\\caption{The upper panel shows the expected differential recoil spectrum~$dR\/dE_{\\mathrm{R}}$ as a function of the recoil energy~$E_{\\mathrm{R}}$. The differential rate~$dR\/dt_{\\rm{pb}}$ as a function of the post-bounce time~$t_{\\rm{pb}}$ is plotted in the middle panel. The lower panel represents the number of observable events~$R$ as a function of the detector's energy threshold~$E_{\\rm{th}}$. All panels show results for $11~\\!\\mathrm{M}_{\\odot}$ and $27~\\!\\mathrm{M}_{\\odot}$ progenitors with LS220 and Shen EoSs for a SN at 10~\\!kpc. In the upper and lower panels, the neutrino flux is integrated over $[0,7]$~\\!s after the core bounce, while the middle panel assumes~$E_{\\rm{th}}=0$~\\!keV. All panels show that the event rate is larger for the $27~\\!\\mathrm{M}_{\\odot}$ SN progenitors while the LS220 EoS results in an $\\mathcal{O}(25\\%)$ larger rate than the Shen EoS. Note that these rates are not directly observable since it is S1 and S2 that is measured, rather than $E_{\\mathrm{R}}$.}\n\\label{fig:DiffRecSpectra}\n\\end{figure}\n\nThe differential scattering rates~$dR\/dE_{\\mathrm{R}}$ as a function of the xenon recoil energy~$E_{\\mathrm{R}}$ for the four progenitor models are shown in the upper panel of Fig.~\\ref{fig:DiffRecSpectra} for $t_{\\rm{pb}}$ integrated over $[0,7]$~\\!s. As evident in all of this figure's panels, the event rate is larger for the $27~\\!\\mathrm{M}_{\\odot}$ SN progenitors, while there is a smaller difference owing to the different equations of state, with the LS220 EoS resulting in a slightly larger predicted event rate. \n\nThe middle panel of Fig.~\\ref{fig:DiffRecSpectra} shows the differential scattering rate~$dR\/dt_{\\rm{pb}}$ as a function of the post-bounce time~$t_{\\rm{tb}}$ for the four progenitor models. In this figure, we have integrated over the recoil energy, assuming an idealistic threshold energy $E_{\\rm{th}}=0$~\\!keV. The qualitative behaviour is similar for other threshold energies although the rate is smaller. The differential scattering rates among the different SN progenitors are comparable for $t_{\\rm pb}\\lesssim 10^{-2}$~\\!s, reflecting the similarities of the neutrino emission properties during the neutronization burst (cf.~left panels of Fig.~\\ref{fig:LEpanels}). As the post-bounce time increases, the differences among the differential rates become larger. Most of the scattering events occur for~$t_{\\rm{pb}}\\lesssim1$~\\!s. \n\nThe total number of events observed by an experiment is determined by integrating the differential scattering rate above a given energy threshold $E_{\\rm{th}}$ over the full time period of the SN burst. These integrated spectra are shown in the lower panel of Fig.~\\ref{fig:DiffRecSpectra}. Again, we see that the number of signal events is about twice as large for the $27~\\!\\mathrm{M}_{\\odot}$ SN progenitors, while there is $\\mathcal{O}(25\\%)$ difference owing to the different equation of state. The total number of events drops quickly as $E_{\\rm{th}}$ increases, demonstrating the importance of pushing $E_{\\rm{th}}$ as low as possible. However, as $E_{\\mathrm{R}}$ is not directly measurable, these rates are not directly observable. Therefore, a careful treatment is needed to discuss the rates in terms of~S1 and~S2 signals instead.\n\nBesides scattering off xenon nuclei, neutrinos can also scatter off electrons in the xenon atom. We neglect the latter interaction as the rate of electron recoils is very small, approximately $10^{-5}~\\mathrm{counts}\/\\mathrm{tonne}$, compared to the rate of approximately $10~\\mathrm{counts}\/\\mathrm{tonne}$ for recoils with a xenon nucleus. \n\n\\subsection{Generation of the observable S1 and S2 signals \\label{sec:S1S2}}\n\nTo convert the nuclear recoil energy~$E_{\\mathrm{R}}$ induced by a SN neutrino into the S1 and S2 signals, we perform a Monte Carlo simulation of a xenon TPC following the method employed by the XENON1T collaboration~\\cite{Aprile:2015uzo}. In this subsection, we discuss the technical details of our Monte Carlo simulation.\n\nThe S1 and S2 signals are directly proportional to the number of scintillation photons $N_{\\rm{ph}}$ and ionization electrons $N_{\\rm{el}}$, respectively. The mean numbers of photons and electrons are modeled as\n\\begin{align}\n\\langle N_{\\rm{ph}} \\rangle&=E_{\\mathrm{R}}\\ L_{y}(E_{\\mathrm{R}}) \\ , \\\\\n\\langle N_{\\rm{el}} \\rangle&=E_{\\mathrm{R}}\\  Q_y(E_{\\mathrm{R}}) \\label{eq:QYdef}\\ ,\n\\end{align}\nwhere both the photon yield, $L_{y}$, and electron yield, $Q_y$, are functions of~$E_{\\mathrm{R}}$. We use the emission model developed by the LUX collaboration with data from an {\\it in situ} nuclear recoil calibration~\\cite{Akerib:2015rjg}. We ignore the small effects that may arise from having different drift fields~\\cite{Szydagis:2011tk,Mock:2013ila,Lenardo:2014cva} across the various detectors. The quantities~$Q_y$ and~$L_{y}$ have been directly measured down to an energy of 0.7~\\!keV and 1.1~\\!keV, respectively~\\cite{Akerib:2015rjg}. Unless otherwise stated, we assume that~$Q_y$ and~$L_{y}$ are zero below 0.7~\\!keV. Hence, our rate predictions tend to be conservative, and we discuss the impact of this assumption in section~\\ref{sec:discussion}.\n\nIn a realistic detector, we must account for quantum and statistical fluctuations. Part of the nuclear recoil energy is lost to heat dissipation. Therefore, the number of scintillation photons and ionization electrons that are produced by the xenon target, $N_{\\rm{Q}}^{\\rm{NR}}= N_{\\rm{ph}} + N_{\\rm{el}}$, is only a fraction of the total number of quanta produced by an electronic recoil at the same energy. In electronic recoils, the energy lost to heat is negligible and $\\langle N_{\\rm{Q}} \\rangle =E_{\\mathrm{R}}\/13.7$~\\!eV is the total number of quanta available~\\cite{Szydagis:20111006,Szydagis:2013sih}. We model the intrinsic fluctuation in $N_{\\rm{Q}}^{\\rm{NR}}$ with a Binomial distribution characterized by a trial factor $\\langle N_{\\rm{Q}} \\rangle$ and probability $f_{\\rm{NR}}=\\langle N_{\\rm{Q}}^{\\rm{NR}} \\rangle\/ \\langle N_{\\rm{Q}} \\rangle$,\n\\begin{equation}\nN_{\\rm{Q}}^{\\rm{NR}}=\\text{Binomial}(\\langle N_{\\rm{Q}}\\rangle,f_{\\rm{NR}}) \\ .\n\\end{equation}\nIn addition to the intrinsic fluctuation in $N_{\\rm{Q}}^{\\rm{NR}}$, the fraction of quanta that is emitted as scintillation photons also fluctuates. We model this with a second Binomial distribution with trial factor $N_{\\rm{Q}}^{\\rm{NR}}$ and probability $f_{\\rm{ph}}=\\langle N_{\\rm{ph}} \\rangle\/\\langle N_{\\rm{Q}}^{\\rm{NR}} \\rangle$:\n\\begin{equation}\nN_{\\rm{ph}}=\\text{Binomial}(N_{\\rm{Q}}^{\\rm{NR}},f_{\\rm{ph}}) \\ .\n\\end{equation}\nBy conservation of the number of quanta, we have that the number of electrons is simply $N_{\\rm{el}}=N_{\\rm{Q}}^{\\rm{NR}}-N_{\\rm{ph}}$.\n\nNext, we consider detector-specific fluctuations as we convert the number of generated scintillation photons and ionization electrons into observed S1 and S2 signals. Both S1 and S2 are measured in photoelectrons (PE). For the S1 signal, the number of detected photoelectrons~is\n\\begin{equation}\nN_{\\rm{PE}}=\\text{Binomial}(N_{\\rm{ph}},f_{\\rm{PE}}) \\ ,\n\\end{equation}\nwhere $f_{\\rm{PE}}$ is the photon detection efficiency (also referred to as $g_1$ or $\\epsilon_1$ in other studies~\\cite{Schumann:2015cpa,Akerib:2015rjg}). LUX calibration measurements indicate that $f_{\\rm{PE}}\\simeq0.12$~\\cite{Akerib:2015rjg} and simulations of the XENON1T detector predict a similar value~\\cite{ Aprile:2015uzo}; we assume that $f_{\\rm{PE}}\\simeq0.12$ for all other detectors, as well. This efficiency may be optimistic for detectors that are much larger than XENON1T since the geometry of larger detectors generally means that~$f_{\\rm{PE}}$ decreases. However, as we discuss in section~\\ref{sec:S2only}, the~S1 signal is less important than the~S2 signal such that this assumption does not affect our conclusions.\n\nFinally, for the S1 signal, we must account for the response of a PMT, which is modeled with a Gaussian distribution\n\\begin{equation}\n\\mathrm{S1}=\\text{Gauss}(N_{\\rm{PE}},0.4\\sqrt{N_{\\rm{PE}}}) \\ .\n\\end{equation}\n\n\\begin{figure*}[!htbp]\n\\centering\n\\includegraphics[width=0.97\\columnwidth]{s1_rates.pdf} \\hspace{1mm}\n\\includegraphics[width=0.97\\columnwidth]{s2_rates.pdf}\n\\caption{The left and right panels show the differential rates of the four SN progenitors in terms of the observable~S1 and~S2 signals, respectively. We have integrated the neutrino flux over the first 7~\\!s after the core bounce and assumed that the SN burst occurs 10~\\!kpc from Earth. Note that the axes in the right panel have units of 100~PE compared to~PE in the left panel meaning that the~S2 signal is generally larger than the~S1 signal. The light and charge yields,~$L_y$ and~$Q_y,$ respectively, have been set to zero below recoil energies of 0.7~\\!keV. Integrated rates are given in Table~\\ref{table:S1S2numbers}.}\n\\label{fig:S2S1space}\n\\end{figure*}\n\nTo obtain the S2 signal, we must account for the loss of ionization electrons due to electronegative impurities in the liquid as they drift toward the liquid-gas interface. This attenuation is represented by the electron survival probability\n\\begin{equation}\np_{\\rm{sur}}=\\exp\\left[-\\frac{\\Delta z}{v_{\\rm{d}}\\tau} \\right] \\ ,\n\\end{equation}\nwhere $\\Delta z$ is the distance an electron traverses, $\\tau$ is the so-called electron lifetime in the liquid and~$v_{\\rm{d}}$ is the electron drift velocity. The value measured in XENON100 was $v_{\\rm{d}}\\simeq1.7$~\\!mm\/$\\mu$s~\\cite{Aprile:2012vw} and we assume this value for all other detectors (cf.~\\cite{Yoo:2015yza}). The electron lifetime in the liquid is in general a detector-dependent parameter. For the purpose of this study, we therefore assume that $\\Delta z\/\\tau$ is distributed uniformly over $[0,2\/3]$~\\!mm\/$\\mu$s and that it holds for all of the detector sizes that we consider. This condition means that, as detectors become larger, their purity increases proportionally to meet or exceed this requirement. For XENON1T (LZ), the maximum drift length $\\Delta z_\\text{\\rm{max}}\\simeq967~\\! (1300)$~\\!mm implies that the electron lifetime is at least $\\tau\\simeq1450~\\! (1950)$~\\!$\\mu$s, which is straightforward to achieve~\\cite{Akerib:2015cja}.\n\nThe number of ionization electrons that reach the liquid-gas interface is\n\\begin{equation}\n\\tilde{N}_{\\rm{el}}=\\text{Binomial}(N_{\\rm{el}},p_{\\rm{sur}}) \\ ,\n\\end{equation}\nand we assume that the extraction efficiency from the liquid into gas is 100\\%. Finally, the extracted electrons are accelerated by a strong electric field in the gas phase and induce an S2 signal that is modeled with a Gaussian distribution:\n\\begin{equation}\n\\label{eq:S2gas}\n\\mathrm{S2}=\\text{Gauss}(20\\ \\tilde{N}_{\\rm{el}},7\\ \\sqrt{\\tilde{N}_{\\rm{el}}}) \\ .\n\\end{equation}\nWe have conservatively assumed that the average yield for each extracted electron is 20~PE, but the yield could be higher. For example, the LZ design goal is 50~PE per electron~\\cite{Akerib:2015cja}. \n\n\n\n\\subsection{Observable scattering rates \\label{sec:obrates}}\n\nNow that we have given expressions for $dR\/dE_{\\rm{R}}$ and detailed our procedure for generating the~S1 and~S2 signals, it is straightforward to use Eq.~\\eqref{eq:rateS1S2} to calculate the differential rates $dR\/d\\mathrm{S1}$ and~$dR\/d\\mathrm{S2}$. These rates are shown in the left and right panels of Fig.~\\ref{fig:S2S1space}, respectively, where we have integrated $t_{\\rm{pb}}$ over $[0,7]$~\\!s and assumed that~$L_y$ and~$Q_y$ are zero below 0.7~\\!keV, as discussed in the previous subsection. In the left panel, we have integrated over all~S2 values by assuming an~S2 threshold of zero, while the~S1 signal has been integrated with an~S1 threshold of zero in the right panel. Similar to the previous figures, the differential rates are highest for the $27~\\!\\mathrm{M}_{\\odot}$ SN progenitors and for the LS220~EoS. Comparing the two panels, it is apparent that the~S2 signal is generally larger than the~S1 signal (note that the axes in the right panel have units 100~PE compared to~PE in the left panel). The left panel shows that the differential rate has peaks at integer multiples of 1~PE. A similar behaviour is also present in the right panel, although the effect is smaller and the peaks appear at multiples of 20~PE, the average number of photoelectrons generated for each extracted electron, cf.~Eq.~\\eqref{eq:S2gas} (see also Fig.~\\ref{fig:QyLyvary4} where the effect is more apparent). The roll-off in~$dR\/d\\mathrm{S2}$ below approximately $100$~PE (right panel) is a result of the assumption that~$Q_y$ is zero below 0.7~\\!keV. In section~\\ref{sec:discussion}, we show that this assumption does not have a significant impact on our results.\n\n\n\\begin{table}\n\\caption{Expected number of SN neutrino events per tonne of xenon target above various~S1 and~S2 thresholds. The SN burst occurs at 10~\\!kpc from Earth and the neutrino flux has been integrated over the first 7~\\!s after the core bounce. The light and charge yields,~$L_y$ and~$Q_y$, respectively, have been set to zero below recoil energies of 0.7~\\!keV. The number of events, for the case in which the threshold includes $0$~PE (`$\\geq0$') and when it does not (`$>0$'), have been separated to show that many of the events have an~S1 or~S2 signal that is exactly zero. The symbol $(\\star)$ indicates the most likely threshold values (see discussion in sections~\\ref{sec:S2only} and~\\ref{sec:discussion} for details). An S2-only search for $\\mathrm{CE}\\nu\\mathrm{NS}$\\ from SN neutrinos is optimal as it results in a higher number of detected events. \\label{table:S1S2numbers}}\n\\centering\n\\begin{ruledtabular}\n\\begin{tabular}{@{}c c c c c c @{}} \n& & \\multicolumn{2}{c}{$27\\,\\mathrm{M}_{\\odot}$} & \\multicolumn{2}{c}{$11\\,\\mathrm{M}_{\\odot}$} \\\\ \\cline{3-4}\\cline{5-6}\n& & LS220 & Shen & LS220 & Shen \\\\ \\colrule\n$\\mathrm{S1}_{\\mathrm{th}}$ [PE] & $\\langle N_{\\rm{ph}} \\rangle$ & & & \\\\\\colrule\n$\\geq 0$ & 0 & 26.9 & 21.4 & 15.1 & 12.3 \\\\\n$> 0$ & 0 & 13.3 & 9.8 & 6.9 & 5.2 \\\\\n1 & 8.3 & 11.0 & 8.0 & 5.6 & 4.1 \\\\\n2 & 16.7 & 7.3 & 5.1 & 3.6 & 2.6 \\\\\n3 $(\\star)$  &25  & 5.2 & 3.5 & 2.4 & 1.7 \\\\ \\colrule\n$\\mathrm{S2}_{\\mathrm{th}}$ [PE] & $\\langle N_{\\rm{el}} \\rangle$ & & & \\\\ \\colrule\n$\\geq 0$ & 0 & 26.9 & 21.4 & 15.1 & 12.3 \\\\\n$> 0$ & 0 &  18.5 & 14.0 & 9.9 & 7.6 \\\\\n20 & 1.2 & 18.4 & 14.0 & 9.8 & 7.6 \\\\\n40 & 2.4 & 18.1 & 13.7 & 9.7 & 7.4 \\\\\n60 $(\\star)$ & 3.6 & 17.6 & 13.3 & 9.4 & 7.2 \\\\\n80 & 4.8 & 17.0 & 12.8 & 9.0 & 6.9 \\\\\n100 & 6.0 &16.3 & 12.2 & 8.6 & 6.5 \\\\ \n\\end{tabular} \n\\end{ruledtabular}\n\\end{table}\n\nTable~\\ref{table:S1S2numbers} lists the total number of expected events per tonne of xenon target for various values of the~S1 and~S2 thresholds. For the listed S1 thresholds, we have integrated over all~S2 values, and vice versa for the listed S2 thresholds. The S2 thresholds are given as multiples of 20~PE, the average number of detected photoelectrons for each extracted electron. The second column lists the mean number of primary photons and electrons required to produce an~S1 and~S2 signal for the listed thresholds, calculated with the relations $\\langle \\mathrm{S1} \\rangle= f_{\\rm{PE}} \\langle N_{\\rm{ph}} \\rangle$ and $\\langle \\mathrm{S2} \\rangle= 20 \\langle p_{\\rm{sur}} \\rangle \\langle N_{\\rm{el}} \\rangle$ (we quote the raw~S2 value, rather than the position corrected value). The number of events is reported for our four SN progenitor models located 10~kpc from Earth and for~$t_{\\rm pb}$ integrated in the range $[0,7]$~\\!s. We have separated the number of events for the case in which the threshold includes 0~PE and when it does not to show that approximately~50\\% and~30\\% of the events have an~S1 or~S2 signal that is exactly zero, respectively, and are therefore not observable even in an ideal detector. Generally, the number of~S2 events is much higher than the number of~S1 events, and the event rate drops more slowly as the~S2 threshold is increased, compared to an increase in the~S1 threshold. This trend reflects the fact that the~S2 signal from low-energy depositions is easier to detect in a dual-phase xenon TPC due to the amplification that is inherent to the process of proportional scintillation. For example, the mean~S1 signal of a 1~\\!keV energy deposition is $\\langle \\mathrm{S1} \\rangle\\simeq0.5$~PE, while the mean number of electrons and mean~S2 signal are $\\langle \\mathrm{N_{\\rm{el}}} \\rangle\\simeq7.4$ and $\\langle \\mathrm{S2} \\rangle\\simeq150$~PE, respectively. Since dual-phase xenon detectors are sensitive to single electrons~\\cite{Edwards:2007nj,Aprile:2013blg}, even very small energy depositions result in detectable~S2 signals.\n\nOn the basis of these preliminary results, we show in the next section that an S2-only analysis is the optimal channel for detecting $\\mathrm{CE}\\nu\\mathrm{NS}$\\ from SN neutrinos. We discuss realistic values of the S2~threshold and show that an S2-only search is not limited by background events. In section~\\ref{sec:discussion}, we also show the signal uncertainty is not a limitation.\n\n\\section{S2-only analysis} \\label{sec:S2only}\n\n\n\nThe canonical dark matter search in a dual-phase xenon experiment requires the presence of both an~S1 and an~S2 signal. This stipulation reduces the background rate by two primary means. Firstly, measuring both S1 and S2 enables discriminated between the dominant electronic recoil backgrounds and the expected nuclear recoil signal, based on the ratio S2\/S1 at a given value of S1. Secondly, the S1 and S2 signals allow for a 3D reconstruction of the interaction vertex, based on the time difference between the S1 and S2 signal events and the PMT hit pattern. The latter means that events can be selected from the central region of the detector, where the background rate is lowest. In these canonical dark matter searches, which utilize data collected over $\\mathcal{O}(100)$~days, the~S1 threshold is typically 2~PE or 3~PE, while the S2 threshold is typically $\\sim150$~PE (see e.g.~\\cite{Aprile:2012nq,Akerib:2015rjg,Tan:2016diz}). \n\nFor SN neutrinos though, the brevity of the $\\mathcal{O}(10)$~\\!s burst enables the signal to be discrimination from background based on the timing information rather than the charge-to-light ratio. Although the requirement of detecting both an~S1 and an~S2 signal has the effect of further reducing the background rate, it also significantly reduces the signal rate, especially for processes such as SN neutrino scattering where the nuclear recoil energy is small~\\cite{Hagmann:2004uv,Angle:2011th,Frandsen:2013cna,Essig:2011nj,Santos:2011ju,Essig:2012yx}. For example, for $\\mathrm{S2}_{\\rm{th}}=60$~PE and any value of~S1 (including no S1 signal), the number of SN neutrino events for the $27~\\!\\mathrm{M}_{\\odot}$ SN progenitor with the LS220 EoS is 17.6~events\/tonne. However, when additionally requiring an~S1 signal with $\\mathrm{S1}_{\\rm{th}}=2$~PE, the number of events drops to only 7.2~events\/tonne. Requiring both an~S1 and an~S2 signal therefore significantly reduces the rate of $\\mathrm{CE}\\nu\\mathrm{NS}$\\ compared to an S2-only analysis.\n\nWe now show that, for a SN burst, the expected background rate in a tonne-scale detector is small enough such that an S2-only analysis does not require the additional discrimination capabilities otherwise afforded by the S1 signal. Although the low-energy S2 background in dual-phase xenon experiments is not yet fully understood, the dominant contribution is believed to arise from photoionization of impurities in the liquid xenon and the metal surfaces in the TPC~\\cite{Aprile:2013blg}, caused by the relatively high energy of the 7-\\!eV xenon scintillation photons. Another background contribution may be from delayed extraction of electrons from the liquid to gas-phase~\\cite{Edwards:2007nj}. Such processes create clusters of single-electron~S2 signals and, occasionally, these single-electron signals overlap and appear as a single~S2 signal from multiple electrons. The resultant low-energy background~S2 signals are very similar to those expected in the case of a SN neutrino interaction. The background rate for these lone-S2 events has been characterized by XENON10~\\cite{Angle:2011th,Essig:2012yx} and XENON100~\\cite{Aprile:2016wwo}, which found background rates of approximately $2.3 \\times 10^{-2}$ and $1.4 \\times 10^{-2}$~events\/tonne\/s, respectively. These rates are consistent with the general expectation that the S2-only background rate is independent of the detector size. Based on these measurements, we therefore assume that the average background rate in XENON1T and future detectors will lie in the range $(1.4-2.3) \\times 10^{-2}$~events\/tonne\/s. This background rate corresponds to $0.1-0.2$~events\/tonne during the initial~7~\\!s of the~SN signal, which is at least a factor of~40 smaller than the signal rate from the $11~\\!\\mathrm{M}_{\\odot}$ with Shen EoS progenitor, the smallest rate in Table~\\ref{table:S1S2numbers} (assuming $\\mathrm{S2}_{\\rm{th}}=60$~PE). Additionally, it is worth recalling that the background signal grows linearly in time, whereas the SN neutrino signal does not, resulting in an even better signal-to-background ratio in the early times of the SN burst.\n\nFinally, we motivate an appropriate choice of the~S2 threshold. This threshold is largely determined by two factors. The first is the `trigger-efficiency' for an experiment to detect an S2 signal. For XENON10, the trigger-efficiency was 50\\% for $\\mathrm{S2}\\simeq20$~PE and reached 100\\% for $\\mathrm{S2}\\simeq30$~PE~\\cite{Essig:2012yx}, while for XENON100, it was 50\\% for $\\mathrm{S2}\\simeq60$~PE and reached 100\\% for $\\mathrm{S2}\\simeq140$~PE~\\cite{Aprile:2012vw}. Values have not yet been reported for LUX. The trigger system for XENON1T has been significantly upgraded relative to XENON100 and is expected to lead to an improvement in the trigger-efficiency. Therefore, while the trigger-efficiency does vary between different experiments, here we assume a benchmark value of $\\mathrm{S2}_{\\rm{th}}=60$~PE and make the simplifying assumption that the trigger-efficiency is 100\\% above this value. This benchmark value is consistent with the threshold in the sensitivity studies of LZ, where it was assumed that the S2-only threshold is 2.5 extracted electrons~\\cite{Akerib:2015cja}, corresponding to $\\mathrm{S2}_{\\rm{th}}=50$~PE with an average of 20~PE per extracted electron (and ignoring the small loss owing to the finite electron lifetime).\n\nThe second consideration when deciding $\\mathrm{S2}_{\\rm{th}}$ is the signal uncertainty induced by the choice of the electron yield $Q_y$ (cf.~Eq.~\\eqref{eq:QYdef} for where it enters our analysis). We postpone a full discussion of this uncertainty until section~\\ref{sec:discussion} and, for now, simply state that the signal uncertainty from $Q_y$ is smaller than~10\\% when $\\mathrm{S2}_{\\rm{th}}=60$~PE. This is appreciably smaller than the $\\sim25\\%$ variation for the LS220 and Shen EoS for the same progenitor mass as well as the approximate factor-of-two difference for different progenitor masses; so, this uncertainty should only have a small effect on our results.\n\nFor all of the reasons outlined above, our main results have been obtained by adopting an S2-only analysis with $\\mathrm{S2}_{\\rm{th}}=60$~PE. Figure~\\ref{fig:Nevents} displays the expected number of SN neutrino events from an S2-only analysis with this threshold for the three detectors and four SN progenitors that we consider in this study. Finally, since the background rate is significantly smaller than the signal rate, we ignore it in section~\\ref{sec:results} unless stated otherwise.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.95\\columnwidth]{exp_events.pdf}\n\\caption{The expected number of SN neutrino events from an S2-only analysis with a threshold of 60~PE for a SN burst at 10~\\!kpc from Earth. The different colours refer to XENON1T (red), XENONnT and LZ (blue), and DARWIN (green) detectors. For each detector, the number of events is shown for the four SN progenitors that we consider in this study.}\n\\label{fig:Nevents}\n\\end{figure}\n\n\n\\section{Supernova neutrino detection} \\label{sec:results}\n\nIn this section, we calculate the discovery potential of an S2-only search for SN neutrinos as a function of the SN distance and discuss the discrimination power of xenon detectors with respect to the SN progenitor. We then show that it is possible to reconstruct the SN neutrino light curve and, therefore, to discriminate among the different phases of the neutrino signal. Furthermore, we demonstrate that xenon detectors can reconstruct both the neutrino differential spectrum and the total energy emitted by the SN into all flavours of neutrinos. Finally, we present a concise comparison of the performance of xenon detectors with dedicated neutrino detectors.\n\n\\subsection{Detection significance}\nWe first investigate the sensitivity of present and upcoming xenon detectors to a SN burst as a function of the SN distance from Earth. Figure~\\ref{fig:milkyway} shows the SN burst detection significance as a function of the SN distance from Earth for the 27~\\!$\\mathrm{M}_{\\odot}$ progenitor with LS220 EoS. We see that XENON1T will be able to detect this SN burst at more than~$5\\sigma$ significance up to 25~kpc from Earth, while XENONnT and LZ will make at least a~$5\\sigma$ discovery anywhere in the Milky Way. DARWIN's much larger target mass will extend the sensitivity to a 5$\\sigma$ discovery past the Large Magellanic Cloud (LMC) and the Small Magellanic Cloud~(SMC).\n\n\\begin{figure}[!tbp]\n\\centering\n\\includegraphics[width=0.95\\columnwidth]{significance.pdf}\n\\caption{The detection significance is given as a function of the SN distance for a 27~\\!$\\mathrm{M}_{\\odot}$ progenitor with LS220 EoS. The SN signal has been integrated over $[0,7]$~\\!s. The different bands refer to XENON1T (red), XENONnT and LZ (blue), and DARWIN (green). The band width reflects uncertainties from our estimates for the background rate, discussed in section~\\ref{sec:S2only}. The vertical dotted lines mark the centre and edge of the Milky Way as well as the Large and Small Magellanic Clouds (LMC and SMC, respectively). For this SN progenitor, XENONnT\/LZ could make at least a~$5\\sigma$ discovery of the neutrinos from a SN explosion anywhere in the Milky Way. DARWIN extends the sensitivity beyond the SMC.}\n\\label{fig:milkyway}\n\\end{figure}\n\n\nIn this figure, the SN signal has been integrated over the first 7 seconds after the core bounce. We calculate the detection significance following the likelihood-based test for the discovery of a positive signal described in~\\cite{Cowan:2010js}. Our null hypothesis is that the observed events are only due to the background processes described in section~\\ref{sec:S2only}, while our alternative hypothesis is that the observed events are due to both the background processes and from SN neutrino scattering. A detection significance of~$5\\sigma$ means that we reject the background-only hypothesis at this significance, which we therefore regard as a~$5\\sigma$ discovery of the SN neutrino signal. The bands in Fig.~\\ref{fig:milkyway} show the detection significance for a background rate spanning the range $(1.4-2.3) \\times 10^{-2}$~events\/tonne\/s, our assumption for the background rate discussed in section~\\ref{sec:S2only}, based on the measured rates in XENON10 and XENON100. \n\nFigure~\\ref{fig:milkyway} shows the detection significance for the 27~\\!$\\mathrm{M}_{\\odot}$ LS220 EoS progenitor, which gives the highest event rate among the four progenitors that we consider. However, from this figure and Table~\\ref{table:S1S2numbers}, it is straightforward to calculate the detection significance for the other progenitors. The expected number of events simply scales with the inverse square of the SN distance, which implies that the distance $d_2|_{n \\sigma}$ for an $n\\sigma$ detection of an alternative SN progenitor is related to the distance $d_{27,\\mathrm{LS220}}|_{n \\sigma}$ for an $n\\sigma$ detection of the 27~\\!$\\mathrm{M}_{\\odot}$ LS220 EoS progenitor by $d_2|_{n \\sigma}=d_{{27,\\mathrm{LS220}}}|_{n \\sigma} \\sqrt{\\mathrm{events}_2\/\\mathrm{events}_{27,\\mathrm{LS220}}}$. Here `events' is simply the number of events calculated from the $\\mathrm{S2}_{\\rm{th}}=60$~PE row in Table~\\ref{table:S1S2numbers} (which gives the number of events per tonne and thus must be multiplied by the detector size). With this formula, we estimate that the SN burst from the 11~\\!$\\mathrm{M}_{\\odot}$, Shen EoS progenitor can be detected at $5\\sigma$ significance at 16~kpc, 26~kpc and 44~kpc from Earth for XENON1T, XENONnT\/LZ and DARWIN, respectively.\n\n\\subsection{Distinguishing between supernova progenitors}\n\nBesides spotting a SN burst, we are also interested in investigating whether dual-phase xenon detectors could help us to constrain the SN progenitor physics and the neutrino properties. Given the sensitivity of xenon detectors to SN neutrinos and the expected insignificant background, detection should allow the progenitor mass to be discerned. With the neutrino flux from only four progenitor models, we cannot perform a detailed study of the precision with which the progenitor mass could be reconstructed. However, we can make some general statements on the performance of the different xenon experiments.\n\nFor a SN at 10~kpc, the expected numbers of events in XENON1T, XENONnT\/LZ and DARWIN for the 27~\\!$\\mathrm{M}_{\\odot}$ LS220 EoS progenitor are 35, 123 and 704, respectively, which are $3.8\\sigma$, $7.1\\sigma$ and $16.9\\sigma$ higher than the 11~\\!$\\mathrm{M}_{\\odot}$ LS220 EoS progenitor, where the expectations are 19, 66 and 376 events. This demonstrates that when the SN distance is well known, DARWIN will be able to discern between these progenitor masses with a high degree of certainty, while even XENON1T's ability will be reasonably good. \n\n\\subsection{Reconstructing the supernova neutrino light curve}\n\n\\begin{figure*}[!tbp]\n\\centering\n\\includegraphics[width=0.99\\columnwidth]{light7.pdf}\n\\includegraphics[width=0.99\\columnwidth]{light1.pdf}\n\\caption{Event rate from an S2-only analysis as a function of the post-bounce time for a SN burst at 10~kpc. The event rate is shown for a $27~\\!\\mathrm{M}_{\\odot}$ SN progenitor with LS220 EoS for three target masses: 2,~7, and~40 tonnes in red, blue, and green respectively. The left panel covers the full time evolution with 500~\\!ms time bins, including the Kelvin-Helmholtz cooling phase. The right panel shows the early evolution with 100~\\!ms time bins and focuses on the neutronization and accretion phases.}\n\\label{fig:light}\n\\end{figure*}\n\nWe now discuss the reconstruction of the SN neutrino light curve from a Galactic SN burst. Figure~\\ref{fig:light} shows the neutrino event rate for the most optimistic of the four SN progenitors ($27~\\!\\mathrm{M}_{\\odot}$ with LS220 EoS) as a function of the time after the core bounce. The rate has been obtained for a SN at 10~kpc from Earth by adopting an S2-only analysis with a benchmark threshold of 60~PE for XENON1T, XENONnT\/LZ and DARWIN. In this analysis, we neglect the small background rate.\n\nThe left panel of Fig.~\\ref{fig:light} shows the light curve during the full time evolution of the SN burst with 500~\\!ms bins. For a Galactic SN, a detector the size of DARWIN clearly shows the characteristic behaviour of the Kelvin-Helmholtz cooling phase where the event rate slowly decreases between 1 to 7~\\!s, following the same neutrino luminosity trend (cf.\\ Figs.~\\ref{fig:LEpanels} and~\\ref{fig:DiffRecSpectra}). This behaviour is also partially distinguishable with XENONnT\/LZ, albeit with a smaller significance, and essentially unobservable with XENON1T where the number of events per bin is too low.\n\nThe right panel of Fig.~\\ref{fig:light} focuses on the early time evolution of the SN signal during the neutronization and the accretion phases of the burst. In this panel, 100~\\!ms time binning is used. A detector the size of DARWIN will be able to discern the neutronization peak in the neutrino signal shown in Fig.~\\ref{fig:LEpanels}. However, the neutronization peak cannot be distinguished at a high level of significance in a seven-tonne or two-tonne detector for a SN at 10~kpc. We thus conclude that it will be necessary to have a xenon detector with $\\mathcal{O}(40)$~tonnes of xenon in order to constrain the SN light curve with high precision. Such an experiment will be competitive with existing neutrino telescopes. We stress that the results shown here are for a SN exploding 10~kpc from Earth. The DARWIN results shown in Fig.~\\ref{fig:light} for a SN at 10~kpc are equivalent to the XENON1T or XENONnT\/LZ results for a SN at 2.2~kpc and 4.2~kpc respectively. \n\nAs shown in Figs.~\\ref{fig:LEpanels} and~\\ref{fig:DiffRecSpectra}, the neutrino signal is almost independent of the progenitor mass and the nuclear EoS for $t_{\\rm{pb}} \\lesssim 10$~ms, while, for later times, it depends on the progenitor properties. Therefore, an accurate measurement of the later-time light curve will tell us about properties of the SN progenitor, such as its mass and~EoS. \n\nThe single-flavour light curve, which depends on the neutrino mass ordering and on flavour oscillation physics~\\cite{Mirizzi:2015eza}, should be accurately reconstructed with traditional neutrino detectors. The fact that xenon-based direct detection dark matter experiments are flavour insensitive will allow for the possibility of combining the all-flavour light curve with results from detectors sensitive to a single-flavour light curve. This complementarity between xenon detectors and traditional neutrino experiments should, therefore, allow for tests of oscillation physics as well as the possible existence of non-standard physics scenarios~(e.g.~\\cite{Keranen:2004rg}). We leave a detailed study of this feature for future work.\n\n\n\\subsection{Neutrino differential flux}\\label{sec:Erec}\n\n\\begin{figure*}[!tbp]\n\\centering\n\\includegraphics[width=0.9\\columnwidth]{acc_nuflux_ae.pdf}\\hspace{2mm}\n\\includegraphics[width=0.9\\columnwidth]{acc_nuflux.pdf}\n\\caption{The left panel shows the reconstructed average neutrino energy $\\langle E_T \\rangle$ and amplitude parameter $A_T$ (green dot-circle) compared to the true value for the $27~\\!\\mathrm{M}_{\\odot}$ LS220 EoS progenitor at 10~kpc from Earth integrated between 0.1~\\!s and 1~\\!s (black triangle). Also shown are the $1\\sigma$ contours from 2-, 7- and 40-tonne mock experiments following our maximum likelihood~(ML) analysis. The right panel shows the reconstructed neutrino flux as a function of the neutrino energy. The dashed green line represents the differential flux obtained with the best fit ML estimators and is compared with the true flux, shown by the dashed black line. Also shown are the $1\\sigma$ intervals from our 2-, 7- and 40-tonne mock experiments.}\n\\label{fig:S2Reconstruction2}\n\\end{figure*}\n\nUp to this point, we have extracted information using the event rate integrated over the S2 range for a given threshold $\\mathrm{S2}_{\\rm{th}}$. However, xenon detectors are also able to accurately measure the S2 value of an individual event. For the first time, we investigate the physics that can be extracted from this spectral information. In particular, in this subsection, we demonstrate that xenon detectors can reconstruct the all-flavour neutrino differential flux as a function of the energy and, in the next subsection, that the total SN energy emitted into all flavours of neutrinos can be reconstructed.\n\nThe neutrino differential flux, $f_{\\nu_\\beta}^0(E_{\\nu},t_{\\rm pb})$, defined in Eq.~\\eqref{eq:nuflux}, enters the calculation for the rate of events in a xenon detector in Eq.~\\eqref{eq:d2RdEdt}. From this equation, we see that it is the time-integrated differential flux summed over all neutrino flavours that determines the number of SN neutrino scattering events in a detector. This flux is typically dominated by the $\\nu_x$ flavours, simply because it contributes four flavours ($\\bar{\\nu}_{\\mu}$, $\\nu_{\\mu}$, $\\bar{\\nu}_{\\tau}$ and $\\nu_{\\tau}$) out of the six flavours that comprise the total flux. We now show that this flux may be reconstructed. It depends on the SN progenitor and the time window of the observation and we would like to reconstruct it by making as few assumptions as possible about the initial SN progenitor. We thus make the following ansatz for the time-integrated differential flux summed over all neutrino flavours:\n\\begin{equation}\n\\label{eq:ATETdef}\n\\begin{split}\n&\\sum_{\\nu_{\\beta}} \\int_{t_1}^{t_2} dt_{\\rm{pb}}\\,f_{\\nu_\\beta}^0(E_{\\nu},t_{\\rm pb}) \\\\\n&\\qquad\\equiv A_T\\, \\xi_T\\left(\\frac{E_{\\nu}}{\\langle E_T \\rangle} \\right)^{\\alpha_T} \\exp{\\left(\\frac{-(1+\\alpha_T)E_{\\nu}}{\\langle E_T \\rangle} \\right)}\\;.\n\\end{split}\n\\end{equation}\nWith this ansatz, we assume that the time-integrated differential flux can be parametrized with three free parameters: an amplitude~$A_T$, an average energy~$\\langle E_T \\rangle$, and a shape parameter~$\\alpha_T$. Here,~$ \\xi_T$ is a normalization parameter defined such that\n\\begin{equation}\n\\int d E_{\\nu}\\,\\xi_T\\left(\\frac{E_{\\nu}}{\\langle E_T \\rangle} \\right)^{\\alpha_T} \\exp{\\left(\\frac{-(1+\\alpha_T)E_{\\nu}}{\\langle E_T \\rangle} \\right)}=1\\;.\n\\end{equation}\nWith these definitions, $A_T$ has units of area$^{-1}$, $\\alpha_T$ is dimensionless, and $\\langle E_T \\rangle$ has units of energy. As suggested by our notation, $\\langle E_T \\rangle$ is the average neutrino energy of the time-integrated flux summed over all flavours.\n\nIn practice, however, the shape parameter $\\alpha_T$ is difficult to constrain experimentally since it is degenerate with $\\langle E_T \\rangle$, which also controls the shape of the observed recoil spectrum i.e.\\ $d R\/d\\mathrm{S2}$. We therefore make a simplifying assumption motivated by the observation from SN simulations that the differential neutrino flux can be approximated by a Fermi-Dirac distribution with zero chemical potential~\\cite{1989A&A...224...49J,Keil:2002in,Tamborra:2012ac}. For this distribution, the relation $\\langle E^2 \\rangle\/\\langle E \\rangle^2\\simeq1.3$ holds~\\cite{Keil:2002in}, and from Eq.~\\eqref{eq:earelate}, implies $\\alpha_T\\simeq2.3$. This value of~$\\alpha_T$ is fixed in the subsequent results. This should be a reasonably good approximation everywhere, except during the very short neutronization burst phase $(t_{\\rm{pb}}\\lesssim10~\\!\\mathrm{ms})$, where the spectrum is significantly pinched with respect to a Fermi-Dirac distribution~\\cite{Tamborra:2012ac}.\n\nWe first consider the reconstruction of the time-integrated differential flux summed over all neutrino flavours in the time window $[t_1,t_2]=[0.1,1]$~\\!s (cf.\\ Eq.~\\eqref{eq:ATETdef}) for the $27~\\!\\mathrm{M}_{\\odot}$ LS220 EoS progenitor at 10~kpc from Earth. This window corresponds to the accretion phase of the~SN burst. To reconstruct $A_T$ and $\\langle E_T \\rangle$, we perform a maximum likelihood (ML) analysis and maximize the extended likelihood function\n\\begin{equation}\n\\ln \\mathcal{L}(A_T,\\langle E_T \\rangle)= N \\ln \\mu -\\mu +\\sum_{i=1}^{N}\\ln f \\left( \\mathrm{S2}_i;\\langle E_T \\rangle \\right) \\;,\n\\end{equation}\nwhere $\\mu=\\mu(A_T,\\langle E_T \\rangle)$ is the mean number of expected events, $N$ is the observed number of events, and $f(\\mathrm{S2}_i;\\langle E_T\\rangle)$ is the probability density function evaluated at the~S2 value of the $i$th event.\n\nFigure~\\ref{fig:S2Reconstruction2} shows the ML estimators for~$A_T$ and~$\\langle E_T \\rangle$ for XENON1T, XENONnT\/LZ and DARWIN mock experiments, where, by construction, each mock experiment has the same ML estimators for~$A_T$ and~$\\langle E_T \\rangle$. The~$N$ observed events are randomly drawn from the~$d R\/d\\mathrm{S2}$ spectrum of the $27~\\!\\mathrm{M}_{\\odot}$ LS220 EoS progenitor at 10~kpc, integrated from 0.1~\\!s to 1~\\!s. We consider all events in the~S2 range from $\\mathrm{S2}_{\\rm{th}}=60$~PE to $\\mathrm{S2}_{\\rm{max}}=2000$~PE. For this progenitor and this time window, the mean number of expected events is 7.0~events\/tonne, so $N$ is drawn from Poisson distributions with means of 14, 49 and 280 events for XENON1T, XENONnT\/LZ and DARWIN, respectively.\n\nThe left panel of Fig.~\\ref{fig:S2Reconstruction2} shows the best fit ML estimators (green dot-circle) together with the $1\\sigma$ contours for XENON1T, XENONnT\/LZ and DARWIN. These contours are obtained from the ML by $\\ln \\mathcal{L}=\\ln \\mathcal{L}_{\\rm{max}}-2.3\/2$. The black triangle shows the values of these parameters from our input SN progenitor. The DARWIN reconstruction of the parameters is excellent, while the XENON1T reconstruction has a significantly larger uncertainty. Besides reconstructing $A_T$ and $\\langle E_T \\rangle$, an estimation of the expected $\\nu_x$ average energy should be also possible analogously to what was proposed for neutrino--proton elastic scattering~\\cite{Beacom:2002hs,Dasgupta:2011wg}.\n\nThe dashed green line in the right panel of Fig.~\\ref{fig:S2Reconstruction2} shows the differential flux obtained with the best-fit ML estimators substituted into Eq.~\\eqref{eq:ATETdef}. This can be compared with the true flux from the $27~\\!\\mathrm{M}_{\\odot}$ LS220 EoS progenitor, which is shown by the dashed black line. The ML reconstruction is in very good agreement with the true flux. Also shown are the $1\\sigma$ intervals. At each value of the neutrino energy, the intervals were obtained by propagating all points in the $1\\sigma$ regions for~$A_T$ and~$\\langle E_T \\rangle$ through Eq.~\\eqref{eq:ATETdef} and selecting the maximum and minimum values of the neutrino flux. The right panel demonstrates that a DARWIN-sized experiment will be capable of accurately reconstructing the neutrino flux. The errors from XENON1T however are substantial, owing to the fact that with XENON1T one would observe only~14 events during this time window, compared to~280 with DARWIN.\n\n\\subsection{Total energy emitted into neutrinos}\n\nFinally, we show that it is possible to reconstruct the total energy emitted by neutrinos, which is simply the luminosity integrated over the duration of the SN burst (here taken as the first 7~\\!s) and summed over all neutrino flavours. This is related to the free parameters in our ansatz by (see Eqs.~\\eqref{eq:nuflux} and \\eqref{eq:ATETdef})\n\\begin{align}\nE_{\\rm{tot}}=\\sum_{\\nu_{\\beta}} \\int_{0~\\!\\!s}^{7~\\!\\!s} d t_{\\rm{pb}}\\, L_{\\nu_{\\beta}}(t_{\\rm{pb}})\n =4 \\pi d^2 A_T \\langle E_T \\rangle \\label{eq:Etot}\\;.\n\\end{align}\nThis relation follows from noting that \n\\begin{eqnarray}\nA_T \\langle E_T \\rangle=\\int d E_{\\nu} \\,E_{\\nu} \\sum_{\\nu_{\\beta}} \\int d t_{\\rm{pb}} f_{\\nu_\\beta}^0(E_{\\nu},t_{\\rm pb})\\ ,\n\\end{eqnarray}\nand using Eq.~\\eqref{eq:nuflux} to express $f_{\\nu_\\beta}^0(E_{\\nu},t_{\\rm pb})$ in terms of~$L_{\\nu_{\\beta}}(t_{\\rm{pb}})$.\n\n\\begin{figure}[!tbp]\n\\centering\n\\includegraphics[width=0.9\\columnwidth]{e27ls220f.pdf}\n\\caption{The reconstructed $1\\sigma$ band of the total energy emitted into neutrinos in~30 mock experiments for each of XENON1T (red), XENONnT\/LZ (blue) and DARWIN (green). The true value for the $27~\\!\\mathrm{M}_{\\odot}$ LS220 EoS progenitor integrated over the total time of the SN burst (taken as the first 7~\\!s) is shown by the dashed vertical line.}\n\\label{fig:EReconstruction}\n\\end{figure}\n\nFigure~\\ref{fig:EReconstruction} shows the~$1\\sigma$ range of the reconstructed total energy emitted into neutrinos in~30 mock experiments for each of XENON1T, XENONnT\/LZ and DARWIN. As in the previous subsection, we use the ML method to find the estimators of the parameters~$A_T$ and~$\\langle E_T \\rangle$ for the signal integrated over the first 7~\\!s of a $27~\\!\\mathrm{M}_{\\odot}$ LS220 EoS progenitor at 10~kpc from Earth. Then, we calculate~$E_{\\rm{tot}}$ from Eq.~\\eqref{eq:Etot} and the $1\\sigma$ range using the propagation of errors as described in~\\cite{Agashe:2014kda}. We do not include any uncertainty on the distance~$d$ in our reconstruction.\n\n\\begin{table}[t!]\n\\caption{The typical precision of the reconstructed total energy emitted in neutrinos over the first 7~\\!s, assuming our four SN progenitors situated 10~kpc from Earth, for different current and upcoming detectors.\\label{table:Ttot}}\n\\centering\n \\begin{ruledtabular}\n\\begin{tabular}{@{}c c c c c @{}} \n& \\multicolumn{2}{c}{$27\\,\\mathrm{M}_{\\odot}$} & \\multicolumn{2}{c}{$11\\,\\mathrm{M}_{\\odot}$} \\\\ \\cline{2-3}\\cline{4-5}\n & LS220 & Shen & LS220 & Shen \\\\ \\colrule\nXENON1T (2t) & $20\\%$ & $25\\%$ & $30\\%$ & $36\\%$ \\\\\nXENONnT\/LZ (7t) & $11\\%$ & $13\\%$ & $16\\%$ & $20\\%$ \\\\\nDARWIN (40t) & $5\\%$ & $6\\%$ & $7\\%$ & $9\\%$ \\\\\n\\end{tabular} \n \\end{ruledtabular}\n\\end{table}\n\nThe dashed vertical line shows the total energy from the SN simulation of the $27~\\!\\mathrm{M}_{\\odot}$ LS220 EoS progenitor. As we would expect, each mock experiment results in a different mean and variance with the property that the~$1\\sigma$ region covers the true value in approximately 68\\% of the mock experiments. The typical uncertainty on the reconstructed energy for all four SN progenitors is given in Table~\\ref{table:Ttot}. This number is the average of the ratio of the $1\\sigma$ error over the mean for~250 mock experiments. The uncertainty is smallest for the $27~\\!\\mathrm{M}_{\\odot}$ LS220 EoS progenitor since it results in the highest number of events, and is largest for the $11~\\!\\mathrm{M}_{\\odot}$ Shen EoS progenitor, which gives the lowest number of events. Unsurprisingly, the errors decrease substantially as the target mass is increased from~2~tonnes in XENON1T to~40~tonnes in DARWIN. However, even XENON1T can give a reasonably precise estimate of the total energy emitted into neutrinos for a~SN at 10~\\!kpc.\n\n\\subsection{Comparison with dedicated neutrino detectors}\n\nWe briefly compare the expected number of events of the forthcoming xenon detectors with existing or future neutrino detectors (see also Table~1 of~\\cite{Mirizzi:2015eza} for an overview). For a SN burst at 10~\\!kpc, XENON1T and XENONnT\/LZ will measure approximately~35 and~120 events in total. This is similar to the projected number of events from neutrino-proton elastic scattering at scintillator detectors~\\cite{Dasgupta:2011wg}. However, it is one order of magnitude less than DUNE, which is expected to measure approximately $\\mathcal{O}(10^3)$ events mostly in the~$\\nu_e$ channel with a 40-tonne liquid argon detector (see Fig.~5.5 of~\\cite{Acciarri:2015uup}). In the $\\bar{\\nu}_e$ channel, larger event rates are expected from IceCube, which should see approximately $10^6$ events (see Fig.~52 of~\\cite{Abe:2011ts}), Hyper-Kamiokande, which is expected to measure approximately 10$^5$ events (see Fig.~54 of~\\cite{Abe:2011ts}), and JUNO, which should detect about 6000 events (see Figs.~4-7 of~\\cite{An:2015jdp}). The proposed DARWIN direct detection dark matter detector, with 40~tonnes of liquid xenon, will measure approximately 700~events for all six flavours, and is thus starting to be competitive in terms of the event rate with these dedicated neutrino detectors. Of course, the quoted numbers depend on different assumptions for the adopted SN model and therefore have to be viewed only as rough estimations of the expected number of events. \n\nFor what concerns the reconstruction of the SN neutrino light curve, IceCube, Hyper-Kamiokande and JUNO will all measure many more events [$\\mathcal{O}(10^4-10^5)$~events\/s] compared to DARWIN, which will see approximately 330 events during the first second and 370 events in the remainder of the SN burst. Even though the number of events is smaller for DARWIN, it is important to remember that it is sensitive to all six neutrino flavours, while the existing and planned neutrino detectors are primarily sensitive to a single flavour. Moreover, as discussed in previous sections, dual-phase xenon detectors will provide us with all-flavour information about the energetics of the explosion that should be compared with the flavour-dependent energy spectra possibly reconstructed, e.g., in JUNO or  Hyper-Kamiokande with high resolution. In this sense, in the event of a SN burst, a global analysis of the burst with events from all experiments will benefit from the inclusion of DARWIN data to better constrain the properties of neutrinos and the SN progenitor. \n\n\\section{Experimental factors}\\label{sec:discussion}\n\nIn this section, we discuss the uncertainties related to~$Q_y$, the detector performance during calibration periods, and the eventual pile-up of events that could prevent a clean identification of individual~S2 signals if a SN burst occurred too close to the Earth.\n\n\\subsection{Signal uncertainty from $Q_y$}\\label{subsec:signalQy}\n\nAn accurate prediction of the~S2 signal relies on knowledge of~$Q_y$, the charge yield in liquid xenon, at sub-keV energies. The LUX collaboration has provided the most accurate measurement of~$Q_y$ and has measured it down to a nuclear recoil energy of 0.7~\\!keV. The LUX data points from~\\cite{Akerib:2015rjg} are reproduced in the inset of Fig.~\\ref{fig:QyLyvary4}. The Lindhard model~\\cite{Lindhard:1961zz} provides a good fit to the measurements and, following the LUX collaboration, is the default parametrization that we have assumed for energies above 0.7~\\!keV. For energies below this value, we have conservatively assumed that~$Q_y=0$. In this subsection, we investigate the uncertainty that this assumption introduces on the number of events observed with a dual-phase xenon experiment.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.97\\columnwidth]{qyvar.pdf}\n\\caption{Variations in the $dR\/d\\mathrm{S2}$ differential spectrum under different assumptions for $Q_y$ for the $27\\,\\mathrm{M}_{\\odot}$ LS220 EoS progenitor at~10~\\!kpc and integrated over the first 7~\\!s. The quantity $Q_{y,\\rm{min}}$ is the energy below which $Q_y=0$ and the solid, dashed and dotted lines correspond to $Q_{y,\\rm{min}}$ values of 0.1~keV, 0.4~keV and 0.7 keV. The inset shows the Lindhard and Bezrukov $Q_y$ models together with the LUX measurements. The differences in~$dR\/d\\mathrm{S2}$ between the Lindhard and Bezrukov models are reasonably small compared to the larger differences from varying $Q_{y,\\rm{min}}$.}\n\\label{fig:QyLyvary4}\n\\end{figure}\n\nIn order to extrapolate $Q_y$ to the lowest energies, we use either the Lindhard model or the alternative model by Bezrukov~et.~al.~\\cite{Bezrukov:2010qa}. As can be seen in the inset of Fig.~\\ref{fig:QyLyvary4}, both the Lindhard and Bezrukov models fit the data well. We then set $Q_y$ to zero below various values~$Q_{y,\\rm{min}}$. The case of $Q_{y,\\rm{min}}=0.7$~keV can be seen as the minimum predicted signal. At approximately 0.1~\\!keV or below, the Lindhard model is expected to break down due to atomic effects~\\cite{Sorensen:2014sla}. We thus also test $Q_{y,\\rm{min}}=0.1$~keV and 0.4~\\!keV, as an intermediate example.\n\nThe main panel of Fig.~\\ref{fig:QyLyvary4} shows different realizations of the $dR\/d\\mathrm{S2}$ spectrum for the $27~\\!\\mathrm{M}_{\\odot}$ LS220 EoS progenitor at 10~kpc integrated over the first 7~\\!s. The spectra are obtained for the Lindhard and Bezrukov models of $Q_y$ with three values of~$Q_{y,\\rm{min}}$. As expected, the lower the assumed $Q_{y,\\rm{min}}$ value, the greater the number of signal electrons that can be detected from low-energy nuclear recoils. The differences between the Lindhard and Bezrukov models for $Q_y$ are much smaller than the differences from varying $Q_{y,\\rm{min}}$. For a given $Q_{y,\\rm{min}}$, the Lindhard model gives a signal that is shifted to lower S2 values, which follows from the lower energy yield for given recoil energy, as seen in the inset of Fig.~\\ref{fig:QyLyvary4}.\n\n\\begin{table}[!t]\n\\caption{The expected number of neutrino events per tonne for various~S2 thresholds under different assumptions for~$Q_y$. We compare the Lindhard and Bezrukov models and assume that $Q_y=0$ for energies below~$Q_{y,\\rm{min}}$. The results are for the $27\\,\\mathrm{M}_{\\odot}$ LS220 EoS progenitor at~10~\\!kpc and integrated over the first 7~\\!s. Similar results hold for other progenitor models. The signal uncertainty in each row is $(\\mathrm{S2}_{\\rm{max}}-\\mathrm{S2}_{\\rm{min}})\/(\\mathrm{S2}_{\\rm{max}}+\\mathrm{S2}_{\\rm{min}})$. The Lindhard model with $Q_{y,\\rm{min}}=0.7$~keV gives the smallest number of events per tonne and is the benchmark assumption that we have made in this paper. \n\\label{table:QyLyvary2}}\n\\centering\n \\begin{ruledtabular}\n\\begin{tabular}{@{}c c c c c c @{}} \n& \\multicolumn{5}{c}{$27\\,\\mathrm{M}_{\\odot}$ LS220 EoS} \\\\ \\cline{2-6}\n& \\multicolumn{2}{c}{Lindhard $Q_{y,\\rm{min}}$} & \\multicolumn{2}{c}{Bezrukov $Q_{y,\\rm{min}}$} & Signal \\\\ \\cline{2-3}\\cline{4-5}\n$\\mathrm{S2}_{\\mathrm{th}}$ [PE] & 0.1~keV & 0.7~keV &0.1~keV & 0.7~keV & uncertainty\\\\ \\colrule\n20 & 22.9 & 18.4 & 23.8 & 18.5 & 13\\% \\\\\n40 & 21.0 & 18.1 & 22.2 & 18.3 &10\\% \\\\\n60 $(\\star)$ & 19.4 & 17.6 & 20.6 & 17.9 &8\\% \\\\\n80 & 18.1 & 17.0 & 19.2 & 17.5 &6\\%\\\\\n100 & 16.9 & 16.3 & 17.9 & 16.9 &5\\% \\\\\n\\end{tabular} \n \\end{ruledtabular}\n\\end{table}\n\nTable~\\ref{table:QyLyvary2} shows the total number of expected events per tonne of xenon target in the various $Q_y$ scenarios considered. The number of events corresponds to the $27\\,\\mathrm{M}_{\\odot}$ LS220 EoS SN progenitor at 10~kpc and the neutrino signal is integrated over 7~\\!s. The final column in Table~\\ref{table:QyLyvary2} gives an estimate of the signal uncertainty for each~S2 threshold, calculated in each row as $(\\mathrm{S2}_{\\rm{max}}-\\mathrm{S2}_{\\rm{min}})\/(\\mathrm{S2}_{\\rm{max}}+\\mathrm{S2}_{\\rm{min}})$. In all cases, the minimum number of events per tonne is found for the Lindhard model with $Q_{y,\\rm{min}}=0.7$~keV, which is the benchmark assumption that we have made in all calculations reported in this paper. The highest number of events is found for the Bezrukov model with $Q_{y,\\rm{min}}=0.1$~keV. For the $27\\,\\mathrm{M}_{\\odot}$ LS220 EoS progenitor and our benchmark value $\\mathrm{S2}_{\\rm{th}}=60$~PE, the uncertainty from the $Q_y$ parametrization is around 8\\%. The signal uncertainties with this~S2 threshold for the other SN progenitors are similar, with an uncertainty of $9\\%$ ($9\\%$, $10\\%$) for the $27\\,\\mathrm{M}_{\\odot}$ Shen EoS ($11\\,\\mathrm{M}_{\\odot}$ LS220 EoS, $11\\,\\mathrm{M}_{\\odot}$ Shen EoS) progenitor. \n\nThe neutrino flux amplitude and mean energy reconstruction analyses in section~\\ref{sec:Erec} may be more adversely affected by the uncertainty in the charge yield $Q_y$, since they also take into account the shape of the recoil spectrum. The $Q_y$ modeling uncertainty could be straightforwardly incorporated into a ML analysis (as the $L_y$, $Q_y$ and Milky Way halo uncertainties are routinely incorporated into dark matter studies). Here, we simply test a higher~S2 threshold, $\\mathrm{S2}_{\\rm{th}}=120$~PE, to reduce the $Q_y$ modeling uncertainty by repeating our analysis that led to Fig.~\\ref{fig:S2Reconstruction2}. In this case, we find similar results as in Fig.~\\ref{fig:S2Reconstruction2}. The number of events is reduced from 7.0~events\/tonne to 6.3~events\/tonne, which leads to an increase in the $1\\sigma$ regions of the mean energy and amplitude by only 13\\% and 20\\% respectively. Thus, the quantitative conclusions drawn from this analysis are only slightly affected by the present uncertainty in $Q_y$. Clearly, it would be most desirable to further reduce the $Q_y$ uncertainty by having other low-energy measurements of this quantity.\n\n\\subsection{Sources of increased background rates}\n\nThe low background rates discussed in section~\\ref{sec:S2only} are applicable when the detector is in dark matter search mode. However, in contrast to dedicated SN neutrino detectors, direct detection dark matter detectors can in some cases spend half of their time taking calibration data~\\cite{Aprile:2015ibr}. Various calibration sources are utilized, from external Compton or neutron calibrations to radioactive isotopes that are dissolved directly in the liquid target. The particular background rate in the S2-only channel discussed previously can vary significantly during calibration and may depend on the particular calibration source employed. However, even with an event rate during calibration two orders of magnitude above the rates during a dark matter search, the background is still smaller than the expected signal rates from a Galactic SN.\n\nAnother potential source of increased background to SN signals comes from photoionization on impurities in the liquid xenon. During the commissioning of a detector, the purity may be low, and thus the background rate may be increased. Furthermore, the diminished electron survival probability from their drift would in effect raise the S2-based energy threshold, possibly rendering the detector blind to SN events. Since such initial commissioning times are supposed to be short, we do not discuss them further here.\n\n\\subsection{Sensitivity limitation from event pile-up}\n\nIn a xenon TPC, a single SN neutrino scattering event produces a number of ionization electrons that are drifted to the gas phase, where the~S2 signal is produced from proportional scintillation. At a drift velocity of order 2~mm\/$\\mu$s~\\cite{Yoo:2015yza}, a~1~\\!m high TPC is expected to smear the arrival times of the electrons by about 250~$\\mu$s. This aspect limits the timing resolution of this detection channel.\n\nTo get a better estimate of the maximum number of SN neutrino events ($N_{\\rm{pile-up}}$) before pile-up becomes an issue, we perform a Monte Carlo simulation of the events in the TPC. Once the electrons are extracted from the liquid, the observed~S2 signal is a pulse with a width of~$\\mathcal{O}(1)~\\!\\mu$s~\\cite{Aprile:2012vw}. To resolve individual events without the need to use information from the PMT hit pattern, one~S2 pulse should not overlap another S2~pulse. This will limit the sensitivity of the detector once pile-up becomes significant. Motivated by Fig.~8 in~\\cite{Aprile:2012vw}, we define events to be well separated if the spacing from the start of one~S2 pulse to the start of the next pulse is more than $10~\\!\\mu$s. We randomly distribute events in time according to the differential time distribution $dR\/d t_{\\mathrm{pb}}$. We test both the 27~\\!$\\mathrm{M}_{\\odot}$ LS220 EoS and 11~\\!$\\mathrm{M}_{\\odot}$ Shen EoS progenitors to get an idea about the impact of these models on our conclusions. As the event rate is highest at the start of the SN burst (cf.~Figs.~\\ref{fig:LEpanels} and~\\ref{fig:DiffRecSpectra}), we focus on the first second after the explosion. We then distribute the events uniformly throughout the TPC and take into account the time delay as the ionization electrons drift from the interaction site to the liquid-gas interface, assuming the XENON100 drift velocity~$v_{\\mathrm{d}}=1.7~\\!\\mathrm{mm}\/\\mu\\mathrm{s}$~\\cite{Aprile:2012vw}. In each mock (and real) experiment, the vertex sites, the number, and the time distribution of the events vary. We thus use a statistical procedure and define $N_{\\rm{pile-up}}$ to be the number of events at which~90\\% of mock experiments observe at least~5\\% of events with a spacing of less than~$10~\\!\\mu$s.\n\nWe have performed our calculation for three TPC sizes, 967~\\!mm, 1450~\\!mm and 2600~\\!mm, corresponding to the expected sizes of the XENON1T~\\cite{Aprile:2015uzo}, XENONnT\/LZ~\\cite{Akerib:2015cja} and DARWIN TPCs~\\cite{Schumann:2015cpa}. We find that~$N_{\\rm{pile-up}}$ is approximately independent of these three TPC sizes, varying by less than 0.2\\%. For the 27~\\!$\\mathrm{M}_{\\odot}$ LS220 EoS and 11~\\!$\\mathrm{M}_{\\odot}$ Shen EoS progenitors, $N_{\\rm{pile-up}}=4810$~events and $N_{\\rm{pile-up}}=4780$~events respectively, consistent with our simple estimate of approximately 250~$\\mu$s for the timing resolution.\n\nThe maximum number of SN neutrino events, $N_{\\rm{pile-up}}$, can be converted into a minimum progenitor distance from Earth so that pile-up is not an issue. Given that 8.3~events per tonne and 4.1~events per tonne are expected during the first second for the 27~\\!$\\mathrm{M}_{\\odot}$ LS220 EoS and 11~\\!$\\mathrm{M}_{\\odot}$ Shen EoS progenitors at 10~\\!kpc, we find minimum distances of $\\{0.6,1.1,2.6\\}$~\\!kpc and $\\{0.4,0.8,1.8\\}$~\\!kpc for \\{XENON1T, XENONnT\/LZ, DARWIN\\} for the 27~\\!$\\mathrm{M}_{\\odot}$ LS220 EoS and 11~\\!$\\mathrm{M}_{\\odot}$ Shen EoS progenitors, respectively. A SN explosion that is much closer than these distances will still be detected by a xenon detector, but precision studies of the SN neutrino light curve or neutrino flux parameters will become degraded as it becomes difficult to distinguish between individual events.\n\n\\section{Conclusions} \\label{sec:conclusions}\n\nWith the launch of XENON1T with 2~tonnes of xenon target, and given the plans for larger experiments employing the same technology such as XENONnT and LZ with 7~tonnes and DARWIN with 40~tonnes, we here revisited the possibility of detecting a Galactic supernova (SN) through coherent elastic neutrino-nucleus scattering~($\\mathrm{CE}\\nu\\mathrm{NS}$) with such dual-phase xenon direct detection dark matter experiments. In order to gauge the astrophysical variability of the expected signal, we studied the neutrino signal from four hydrodynamical SN simulations, differing in the progenitor mass and nuclear equation of state. For the first time, we have performed a realistic detector simulation of SN neutrino scattering, expressing the scattering rates in terms of the observed signals S1 (prompt scintillation) and S2 (proportional scintillation).\n\nWe have shown that focusing on the S2 channel maximizes the number of events that can be detected, thanks to the lower energy threshold. We have discussed appropriate values of the S2 threshold and proved that the background rate is negligible compared to the expected signal. Hence, high-significance discoveries can be expected. As a concrete example, we have shown that for a~$27~\\!\\mathrm{M}_{\\odot}$ SN progenitor, the XENON1T experiment will be able to detect a SN burst with more than $5\\sigma$ significance up to 25~\\!kpc from Earth. Furthermore, the XENONnT and LZ experiments will extend this sensitivity beyond the edge of the Milky Way, and the DARWIN experiment will be sensitive to SN bursts in the Large and Small Magellanic Clouds. Due to the low background rate, these experiments should even be able to actively contribute to the Supernova Early Warning System (SNEWS)~\\cite{Antonioli:2004zb,Scholberg:2008fa}.\n\nFor a~SN burst at 10~\\!kpc, features of the neutrino signal such as the neutronization burst, accretion phase, and Kelvin-Helmholtz cooling phase will be distinguishable with the DARWIN experiment. In addition, with DARWIN it will be possible to make a high-precision reconstruction of the average neutrino energy and differential neutrino flux. Since $\\mathrm{CE}\\nu\\mathrm{NS}$\\ is insensitive to the neutrino flavour, the signal in dual-phase xenon detectors is unaffected by uncertainties from neutrino oscillation physics. A high-precision measurement of $\\mathrm{CE}\\nu\\mathrm{NS}$\\ from~SN neutrinos will therefore offer a unique way of testing our understanding of the~SN explosion mechanism. The sensitivity to all neutrino flavours also means that it is straightforward to reconstruct the total energy emitted into neutrinos. We have shown that even XENON1T could provide a reasonably good reconstruction of this energy.\n\nIt has already been discussed that a large multi-tonne xenon detector such as DARWIN would be able to measure solar neutrino physics~\\cite{Baudis:2013qla} and exploit novel dark matter signals~\\cite{Baudis:2013bba,Schumann:2015cpa,McCabe:2015eia}. Here, we have illustrated that DARWIN will also be able to reconstruct many properties of~SN progenitors and their neutrinos with high precision. Large dual-phase xenon detectors are expected to be less expensive and more compact than future-generation dedicated neutrino telescopes, encouraging the construction of liquid xenon experiments as~SN neutrino detectors. Specifically, DARWIN will allow for all-flavour event statistics that are competitive with next-generation liquid argon or scintillation neutrino detectors~\\cite{Acciarri:2015uup,An:2015jdp}, which are sensitive to only some of the neutrino flavours. At the same time, being flavour blind, dual-phase xenon detectors will provide complementary information on the SN neutrino signal that is not obtainable with existing or planned neutrino telescopes.\n\n\\acknowledgments\nWe thank John Beacom and Sebastian Liem for discussions as well as Alec Habig and Georg Raffelt for comments on the manuscript. RFL and SR are supported by Grant No.~\\#PHYS-1412965 from the National Science Foundation (NSF). CM acknowledges support from the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientific Research (NWO). MS thanks the Istituto Nazionale di Fisica Nucleare (INFN). IT~acknowledges support from the Knud H\\o jgaard Foundation and from the Danish National Research Foundation (DNRF91).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\n\tMolecules consisting of one alkali-metal atom and one alkaline-earth atom receive rising interest for their prospective value in the field of ultracold quantum gases because their ground state  \\Xstate with its electric and a magnetic dipole moment offers advantageous properties (see e.g. references \\citeText{tscherbul_controlling_2006,pasquiou_quantum_2013,roy_photoassociative_2016}).\n\t\\ce{RbSr}, \\ce{RbYb} and \\ce{LiYb} have been studied in weakly bound states via Feshbach-resonance spectroscopy (see references \\citeText{pasquiou_quantum_2013,bruni_observation_2016,roy_photoassociative_2016}) and \\ce{LiBa} and \\ce{LiCa}  in deeply bound states via conventional spectroscopy in references \\citeText{dincan_electronic_1994} and \\citeText{stein_spectroscopic_2013}.\n\n\tFor many molecules of this class ab initio studies for potential energy curves (PECs), transition properties and electric dipole moments of the molecules exist, e.g. references \\citeText{guerout_ground_2010,augustovicova_ab_2012,gopakumar_dipole_2014,pototschnig_vibronic_2017,shao_ground_2017}.\n\tAdditional ab initio calculations for \\ce{LiSr} have been performed by other groups\\cite{gopakumar_ab_2011,kotochigova_ab_2011,gopakumar_ab_2013}. \t\\figref{fig:PotentialCurves} shows a part of the potential energy scheme of \\ce{LiSr} for the atom pair asymptotes Li(2s) + Sr(5s$^2$) and Li(2p) + Sr(5s$^2$) which is almost degenerate with Li(2s) + Sr(5s5p $^3$P). The latter atom pair leads also to molecular quartet states.\n\t\n\t\\begin{figure}\n\t\t\\includegraphics[width = \\columnwidth]{PotCurves.eps}\n\t\t\\caption{\\label{fig:PotentialCurves} Potential curves of \\ce{LiSr} from ab initio calculations\\cite{gopakumar_ab_2013}. Thick curves represent RKR PECs from this work.}\n\t\\end{figure}\n\n\t \n\tIn this work, we present the first spectroscopic observation of \\LiSr and its analysis from which the bottoms of the potential energy curves of the ground state \\Xstate and the first excited state \\bstate of \\LiSr are derived. This was achieved by creating the molecular gas in a heatpipe oven and observing the thermal emission spectrum in the near infrared region, which was expected from the ab initio results shown in \\figref{fig:PotentialCurves}. The spectrum gave no indication of more than one isotopologue. Therefore, \\ce{^7Li^{88}Sr}, formed by the most abundant isotopes of \\ce{Li} and \\ce{Sr}, was most likely observed. Laser excitations of the molecule were performed and were essential for an unambiguous assignment of the dense spectrum.\n\tFor the observed $2 ^2\\Sigma^+ \\leftrightarrow 1 ^2\\Sigma^+$ system, molecular parameters are derived. \\figref{fig:PotentialCurves} shows an avoided crossing between the excited states \\astate and $2^2\\Sigma^+$. Thus perturbations are expected in the spectrum and  estimations for the \\astate state will be derived from the observed coupling to the \\bstate state. A comparison of  results from ab initio studies with experimental findings of this work is presented.\n\n\n\\section{Experimental scheme}\n\n\tA sample of \\ce{LiSr} was prepared in a heatpipe and its thermal emission spectrum recorded with a Fourier Transform spectrometer (FTS). In a second experimental step, selected transitions of the molecule were excited by  a tunable external cavity diode laser and the resulting fluorescence was resolved through the FTS.\n\n\tA heatpipe of \\SI{88}{\\cm} in length and \\SI{3}{\\cm} in diameter was filled with about \\SI{25}{\\g} of \\ce{Sr} and about \\SI{2}{\\g} of \\ce{Li}. As a buffer gas, \\SI{30}{\\milli\\bar} of argon was used.  A mesh was installed in the heatpipe to enable reflow of material condensing in the cold areas to the ends. Both ends were closed with BK7 windows tilted by few degrees against the optical axis. One end was used for imaging the emitted light to the spectrometer, the other end opened to a beam dump for the laser beam. A \\SI{40}{\\cm} long region in the center of the heatpipe was heated to \\SI{915}{\\celsius} while the ends were kept at room temperature. The thermal emission spectrum of \\ce{LiSr} appears at about \\SI{870}{\\celsius}. \n\t\n\tThe spectrum was recorded with a Bruker FTS (IFS 120HR). Beam splitters for the near infrared  and an IR-enhanced Silicon avalanche photodiode (S11519-30, Hamamatsu) were installed. An optical low-pass filter (FGL850S, Thorlabs GmbH) and electronic filters in the detection circuit were applied to restrict the spectrum to the region of interest. It should be noted that the response of the detector vanishes around \\SI{8500}{\\kay}, which conveniently suppresses the detection of the rise of the thermal emission for a temperature around \\SI{900}{\\celsius}.\n\t\n\t\n\t\\begin{figure}\n\t\t\\includegraphics[width = \\columnwidth]{LiSr_1cm_Spectrum.eps}\n\t\t\\caption{\\label{fig:emSpectrum} Thermal emission spectrum of \\ce{LiSr} at \\SI{915}{\\celsius} with a resolution of \\SI{1}{\\kay}.}\n\t\\end{figure}\n\t\n\t\n\tThe recorded spectrum ranges from \\SI{8000}{\\kay} to \\SI{12500}{\\kay}, an example is given  in \\figref{fig:emSpectrum}. In the range from \\numrange{9000}{10000} \\si{\\kay}, \\ce{LiSr} dominates the spectral structure. Three prominent bands can be seen from \\SI{9100}{\\kay} to \\SI{9600}{\\kay}, tentatively assigned to the \\bands{$v^{\\prime}$}{$v^{\\prime\\prime}$}=\\bands{0}{0}, \\bands{0}{1} and \\bands{1}{0} bands. Beyond the \\bands{1}{0} band there are weaker structures seemingly following a band pattern. They can not be assigned so far. Starting around \\SI{10500}{\\kay}, a spectrum without obvious band structure can be seen. This structure coincides roughly with the expected $3^2\\Sigma^{+} \\leftrightarrow 1^2\\Sigma^{+}$ electronic system from the ab inito calculations \\cite{gopakumar_ab_2011}, but it is overlapped by the \\ce{Li_2} spectrum.\n\t\n\t\\begin{figure}\n\t\t\\includegraphics[width = \\columnwidth]{LiSr_rotational_Spectra.eps}\n\t\t\\caption{\\label{fig:rotSpectrum} Rotational structure near the \\bands{0}{0} bandhead at \\SI{915}{\\celsius} with a resolution of \\SI{0.03}{\\kay}(a) and superimposed with laser light at \\SI{9386.932}{\\kay} and a fluorescence line with a resolution of \\SI{0.05}{\\kay} (b). The \\fone \\ and \\ftwo \\ lines make two distinct bandheads.}\n\t\\end{figure}\n\t\n\t\n\t\\figref{fig:rotSpectrum} (a) shows a part of the recorded emission spectrum of \\ce{LiSr} in detail. To resolve the rotational structure, the thermal emission spectrum was recorded with \\SI{0.03}{\\kay}. It is given in the supplementary material. The Doppler width is expected to be \\SI{0.023}{\\kay} for \\LiSr at \\SI{915}{\\celsius} which justifies the selected resolution of the FTS. After averaging 1080 scans, a signal-to-noise ratio of about \\num{250} can be achieved.\n\t\n\tIn \\figref{fig:rotSpectrum} (a) the band heads for \\fone \\ and \\ftwo \\  of the \\bands{0}{0} band can be seen on the right side. The R branch turns at $N^{\\prime\\prime} \\approx 12$. Most of the higher peaks are due to more than one transition line. The decreasing intensity towards \\SI{9380}{\\kay} is due to a perturbation in the R branch. The same perturbation is reflected in the P branch around \\SI{9350}{\\kay}. See sections \\ref{sec:lineAss} and \\ref{sec:pert} for details on the perturbation. Starting around \\SI{9388}{\\kay} towards lower wavenumber, the R branch of the \\bands{1}{1} band starts to become visible from the background of the \\bands{0}{0} emission.\n\t\n\tTo observe laser induced fluorescence (LIF), the gas was excited with laser powers up to \\SI{100}{\\mW}. A diode laser in a Littrow configuration stabilized by a wavemeter (WS-U, High Finesse GmbH) was used to access the range from \\SI{9200}{\\kay} to \\SI{10600}{\\kay} with an accuracy of \\SI{20}{\\MHz}. The beam was collimated to a diameter of \\SI{2}{\\mm} and aligned with the heatpipe axis. LIF in the center of the heatpipe was imaged into the FTS minimizing stray light in the detection as the filters could not sufficiently block the backscattered laser light. The LIF spectra were recorded with a resolution of \\SI{0.05}{\\kay} and an example is shown in \\figref{fig:rotSpectrum}~(b).  The fluorescence line is greatly enhanced compared to the thermal emission lines and has a signal-to-noise ratio of ca. \\num{100} by averaging only 10 scans. In the example, the corresponding emission line in the pure thermal spectrum is an overlap of the lines \\fone \\ R18, \\ftwo \\ R7 and \\ftwo \\ R14. The LIF spectrum relates undoubtedly P or R lines for an excited rotational level. This is very important for the unambiguous assignment given in section \\ref{sec:lineAss}.\n\t\n\tThe observed LIF shows PR-doublets in bands \\bands{$v^{\\prime}$}{$(v^{\\prime\\prime}\\pm 1)$} directly neighbouring the excited band \\bands{$v^{\\prime}$}{$v^{\\prime\\prime}$}. Rotational satellites from collisional relaxation were sometimes recorded but never spanned more than about five rotational levels. Long vibrational progressions were not observed as expected from the PECs in \\figref{fig:PotentialCurves}. Laser excitations in the structure  from \\SI{10500}{\\kay} to \\SI{12000}{\\kay} have so far revealed PR-doublets that could be attributed to \\ce{Li2}\\cite{coxon_application_2006}.\n\t\n\tThe line position was determined by fitting one or multiple Gaussian curves to a spectral line. The average frequency uncertainty given by the fit is close to the Doppler width. Where this method was not successful due to too many overlapping lines, the center of the spectral line was used as frequency and the FWHM was used as uncertainty. In the further analysis, the uncertainty was adjusted to be at least \\SI{0.02}{\\kay}.\n\n\t\n\t\n\\section{Line assignment}\n\\label{sec:lineAss}\n\tBased on the ab initio calculations from reference \\citeText{gopakumar_ab_2013}, the observed band structure can be expected to be composed of $2^2\\Sigma^{+} \\leftrightarrow 1^2\\Sigma^{+}$ transitions (see \\figref{fig:PotentialCurves}). LIF experiments with the laser tuned to lines of the most intense band showed associated P and R lines only in the next visible band of lower frequency. Therefore, these bands can be tentatively assigned to be the \\bands{0}{0} and \\bands{0}{1} bands of this electronic system. \n\t\n\tA $^2\\Sigma^{+}$ state can be adequately described in Hund's coupling case (b) with basis vector $\\ket{\\Lambda, (N,S)J}$, where $\\Lambda$ is the quantum number of the projection on the molecular axis of the orbital angular momentum (here zero), $\\hat{N}$ is the total angular momentum without spins and $\\hat{S}$ is the electron spin.  $\\hat{J} = \\hat{N} + \\hat{S}$ is the total angular momentum of the molecule excluding nuclear spins. The energies of the rovibrational states can be expressed with the conventional Dunham expansion \\cite{herzberg_spectra_1950}\n\t\n\t\\begin{equation}\n\tE(v,N) = \\sum_{m,n} \\mathrm{Y}_{m,n}(v+\\nicefrac{1}{2})^m[N(N+1)]^n.\n\t\\label{eq:DunhamExpansion}\n\t\\end{equation}\n\t\n\tFor a doublet state, levels with $J=N+1\/2$ or $J=N-1\/2$ are labeled by \\fone\\ or \\ftwo, respectively. The energy differences between the \\fone \\ and \\ftwo \\ components are then attributed to the spin-rotation coupling, given by the Hamiltonian $\\gamma\\mathbf{\\hat{S}}\\cdot\\mathbf{\\hat{N}}$\\cite{herzberg_spectra_1950}, which is added to the rovibrational energies.  $\\gamma$ is the coupling constant. The energy of a $J$ level thus evaluates to\n\n\n\t\\begin{subequations}\n\t\\begin{align}\n\t\tE_1(v,J)&= E(v,N) + \\nicefrac{\\gamma}{2} \\times N  &\\quad\\mathrm{for\\; F}_1 \\\\\n\t\tE_2(v,J)&= E(v,N) - \\nicefrac{\\gamma}{2} \\times (N+1) &\\quad\\mathrm{for\\; F}_2\n\t\\end{align}\n\t\\label{eq:SRcoupling}\n\t\\end{subequations}\n\n\tfor the vibrational and rotational quantum numbers $v$ and $N$ and $E(v,N)$ given by the Dunham expression. The strength of the spin-rotation coupling can change with the internuclear separation and thus a slight dependence on $v$ and $N$ was observed. For mnemonic reasons, this dependence is modeled in analogy to the Dunham expansion:\n\n\t\\begin{equation}\n\t\t\\gamma(v,N) = \\sum_{m,n} \\gamma_{m,n}(v+\\nicefrac{1}{2})^m[N(N+1)]^n.\n\t\t\\label{eq:gammaExpansion}\n\t\\end{equation}\n\t\n\tMany lines of the \\bands{0}{0} band could be assigned to rotational transitions using frequency differences from fluorescence PR-doublets near the band head and the value for the rotational constant given in reference \\citeText{gopakumar_ab_2013} as a first approximation.\n\tThe corresponding fluorescence lines in the \\bands{0}{1} band were assigned accordingly. Lines of both spin components \\fone \\ and \\ftwo \\ were assigned up to $N^{\\prime\\prime} =104$. \\figref{fig:assignedLines} summarizes the levels of the state \\bstate that were addressed in the LIF experiments. Because the experimental procedure gives no information about the sign of the spin-rotation constant $\\gamma$,  a definite assignment of spectral lines to \\fone \\ or \\ftwo \\ could not be made. Inverting \\fone \\ and \\ftwo \\ will give the same result with opposite sign of $\\gamma$. \n\t\t\n\t\n\t\\begin{figure}\n\t\t\\includegraphics[width = \\columnwidth]{AdressedLevels.eps}\n\t\t\\caption{ Rovibrational levels of the \\bstate state which were addressed with a laser excitation from $v^{\\prime\\prime} = 0,1,2$ in the \\Xstate state}\n       \\label{fig:assignedLines}\n\t\\end{figure}\n\t\n\tWhile the ground state levels obtained  from the fluorescence progressions could be described with the formulas \\eqref{eq:SRcoupling} and \\eqref{eq:gammaExpansion}, the Dunham model proved insufficient to describe the energies of some rotational levels in the $v^\\prime = 0$ manifold. These levels show  systematic deviations from the energies given by the Dunham series as displayed in \\figref{fig:deviation}, suggesting a perturbation. For a range of about 20 rotational levels, centered around $N^\\prime \\approx 40$, a model including the spin-orbit coupling to \\astate is developed (section \\ref{sec:pert} below). For $N^\\prime>68$, the perturbations become complicated and this part of the rotational manifold was therefore not taken into account for this work.\n\n\t\\begin{figure}\n\t\t\\includegraphics[width = \\columnwidth]{ObsCal.eps}\t\t\n\t\t\\caption{ (a) Deviation of actual transition frequencies in the \\bands{0}{0,1} bands from frequencies calculated with the Dunham model. Circles represent transitions in the \\bands{0}{0} band and triangles represent transitions in the \\bands{0}{1} band. (b) Deviations considering coupling between electronic states.  The grey area depicts the experimental  uncertainty. }\n\t\t\\label{fig:deviation}\n\t\\end{figure}\n\n\tAnother feature of the perturbation is that the lines with perturbed energy levels have reduced intensity. This was seen in the thermal emission and LIF spectra. \\figref{fig:Intensities} shows the intensities from the emission spectrum for the perturbed range of quantum numbers identified above. For values of $N$ around 40 with large perturbation, the intensities reduce significantly and result in reduced or vanishing fluorescence from the most perturbed levels. For this reason, the lines from perturbing states, the so called extra lines, could not be identified because the LIF experiments yielded no identifiable response. This observation is in agreement with the expectation that the perturbation comes from the coupling to the \\astate state, which has a low electronic transition moment to the ground state and unfavorable Franck-Condon factors, according to ab initio calculations \\cite{gopakumar_ab_2013} (see also \\figref{fig:PotentialCurves}).\n\n\t\\begin{figure}\n\t\t\\includegraphics[width = \\columnwidth]{Intensity.eps}\t\t\n\t\t\\caption{ Intensities of thermal emission lines of the \\bands{0}{0} band. Line intensities were always determined on a local scale subtracting the background. Overlapping lines were left out if their respective intensities could not be determined. Thermal populations calculated with $E(v^\\prime=0,N^\\prime) \\approx \\mathrm{Y}^\\prime_{01}~\\times~N^\\prime(N^\\prime+1)$ are shown for comparison (black circles). }\n\t\t\\label{fig:Intensities}\n\t\\end{figure}\n\t\n\tAlso by LIF, a system of three connected bands was discovered which are identified as the \\bands{1}{0}, \\bands{1}{1} and \\bands{1}{2} bands inserting the vibrational spacing of the ground state obtained from the fluorescence progressions. LIF experiments in the less intense spectral structure seen between the \\bands{1}{0} band and \\SI{10000}{\\kay} in \\figref{fig:emSpectrum} were unsuccessful, possibly due to insufficient laser intensity. For the \\bands{1}{0} band, lines in the range of $N^{\\prime\\prime} = $ \\numrange{40}{60} were assigned using the already derived rotational energies of the ground state (see upper part in \\figref{fig:assignedLines}) but the excited state could not be described by the Dunham model. For this reason, only the $v'=0$ state with rotational levels up to $N^\\prime=68$ is considered in this work.\n\t\n\t468 measured transition frequencies and 821 frequency differences of the PR-doublets along with their assigned quantum numbers were used for a linear fit of the Dunham coefficients for both $^2\\Sigma^+$ states as given in \\eqref{eq:DunhamExpansion} and  spin-rotation parameters  in \\eqref{eq:gammaExpansion}. The transitions associated with an upper level that was recognized as perturbed were excluded from this fit.\n\t\n\t\n\t\n\n\\section{Perturbation}\n\\label{sec:pert}\n\t\n\tA substantial number of observed transition lines suggest perturbations in the \\bstate state (see \\figref{fig:deviation} (a)). For the lines with significant deviation from the Dunham model, the LIF experiments proved highly important in ascertaining the quantum number assignment since the PR-differences are governed by the involved (unperturbed) \\Xstate levels, exclusively.\n\t\n\tAccording to reference \\citeText{gopakumar_ab_2013}, the $1^2\\Pi_{1\/2}$ and $1^2\\Pi_{3\/2}$ states are energetically closest to the state under study.  $^2\\Sigma^+$ and $^2\\Pi$ states are coupled by the spin-orbit and rotational interaction.  Following reference \\citeText{lefebvre-brion_perturbations_1986}, the matrix representation of the Hamiltonian in Hund's coupling case (a) with state vector $\\ket{\\Lambda,S,\\Sigma,J}$ for a total angular momentum $J$ is derived, where $\\Sigma$ is the quantum number of the projection of $\\hat{S}$  and $\\Omega = \\Lambda + \\Sigma$ is the quantum number of the projection of $\\hat{J}$ to the molecular axis.  Hyperfine interaction has not been considered. No effects of the hyperfine structure by additional splitting or on  line shapes were observed and hence it can be assumed that its effects on line positions are negligible. The matrix is given in \\tabref{tab:intMatrix}. The upper and lower signs correspond to the \\fone \\ and \\ftwo \\ states of a given total angular momentum $J$.\n\t\n\t\n\t\\begin{table*}\\caption{\\label{tab:intMatrix} Interaction between the  $2^2\\Sigma^{+}$ and  $1^2\\Pi$ states in Hund's coupling case (a) for a set of vibrational levels $v_\\Sigma, v_\\Pi$. $\\mathrm{E}_\\mathrm{Dun} $ is the energy calculated from Dunham parameters, $\\gamma$ is the spin-rotation constant, $A$ is the spin-orbit coupling constant, $B_v$ is the rotational constant for a given vibrational state. $T_\\Pi$ is the electronic energy and $G_\\Pi(v)$ the vibrational energy of $1^2\\Pi$. The factor $p$ represents the expectation value $\\braket{v_\\Sigma |\\mathbf{\\hat{L}^{\\pm}} |v_\\Pi}$. Subscripts on the constants indicate a value for the $\\Sigma$ or $\\Pi$ state or a mixture thereof. Upper signs are for \\fone and lower signs for \\ftwo . }\n\t\t\n\t\t\t\\centering\n\t\t\t\\begin{tabular}{l|*3{c}} \n\t\t\t\t& $\\ket{^2\\Sigma^{+}_{1\/2}}$ & $\\ket{^2\\Pi_{1\/2}}$ & $\\ket{^2\\Pi_{3\/2}}$  \\\\\n\t\t\t\t\\hline\n\t\t\t\t$\\bra{^2\\Sigma^{+}_{1\/2}}$ & \\begin{tabular}{@{}c} $\\mathrm{E}_\\mathrm{Dun}(v,N=J\\mp \\nicefrac{1}{2}) $ \\\\$ -\\nicefrac{\\gamma_\\Sigma}{2} \\times \\left [1 \\mp (J+\\nicefrac{1}{2}) \\right ] $ \\end{tabular} & \\begin{tabular}{@{}c} $\\nicefrac{p}{2} \\times \\big[A_{\\Sigma\\Pi} - \\gamma_{\\Sigma\\Pi} \\, + $ \\\\ $2 \\, B_{\\Sigma\\Pi} ( 1 \\mp [J+\\nicefrac{1}{2}] ) \\big] $ \\end{tabular}& \\begin{tabular}{@{}c}$-p \\times B_{\\Sigma\\Pi} \\times $\\\\ $ \\sqrt{J(J+1) -\\nicefrac{3}{4}  }$\\end{tabular}\\\\[10pt]\n\t\t\t\t$\\bra{^2\\Pi_{1\/2}}$ & \\begin{tabular}{@{}c} $\\nicefrac{p}{2} \\times\\big[A_{\\Sigma\\Pi} - \\gamma_{\\Sigma\\Pi} \\, + $ \\\\ $2\\, B_{\\Sigma\\Pi} ( 1 \\mp [J+\\nicefrac{1}{2}] ) \\big] $ \\end{tabular}  & \\begin{tabular}{@{}c} $\\mathrm{T}_{\\Pi}+B_{\\Pi,v} [J(J+1)+\\nicefrac{1}{4}]+ $ \\\\ $G_\\Pi(v)- (A_\\Pi + \\gamma_\\Pi)\/2$ \\end{tabular} & \\begin{tabular}{@{}c} $ (\\nicefrac{\\gamma_\\Pi}{2} - B_\\Pi ) \\, \\times $\\\\ $\\sqrt{J(J+1) -\\nicefrac{3}{4}  }$ \\end{tabular} \\\\[10pt]\n\t\t\t\t$\\bra{^2\\Pi_{3\/2}}$ & \\begin{tabular}{@{}c}$-p \\times B_{\\Sigma\\Pi} \\times $\\\\ $ \\sqrt{J(J+1) -\\nicefrac{3}{4}  }$\\end{tabular}  & \\begin{tabular}{@{}c} $ (\\nicefrac{\\gamma_\\Pi}{2} - B_\\Pi ) \\, \\times $ \\\\ $ \\sqrt{J(J+1) -\\nicefrac{3}{4}  }$ \\end{tabular} &  \\begin{tabular}{@{}c} $\\mathrm{T}_{\\Pi}+B_{\\Pi,v} [J(J+1)-\\nicefrac{7}{4}]+ $ \\\\ $G_\\Pi(v) + (A_\\Pi - \\gamma_\\Pi)\/2$ \\end{tabular}\n\t\t\t\\end{tabular}\n\t\t}\n\t\\end{table*}\n\t\n\n\tThe diagonal entries describe the energies of the $^2\\Sigma^+$ and $^2\\Pi$ states without coupling. $\\mathrm{E}_\\mathrm{Dun}$ for the \\bstate state is the rovibrational energy according to \\equref{eq:DunhamExpansion}. For the $^2\\Pi$ state, the energy levels are given by the electronic energy $\\mathrm{T}_\\Pi$, vibrational energy $G_\\Pi(v)$ and rotational constant for a vibrational level $B_{\\Pi,v}$. Three additional parameters appear in the matrix: the coupling constants of the spin-rotation interaction and the spin-orbit interaction, $\\gamma$ and $A$, and the factor $p = \\braket{v_\\Sigma |\\mathbf{\\hat{L}^{\\pm}} |v_\\Pi}$, which will be approximated as the product of an overlap integral $\\braket{v_\\Sigma |v_\\Pi}$ of the coupled vibrational states and the expectation value $\\braket{\\Pi |\\mathbf{\\hat{L}^{+}}| \\Sigma}$over the electronic space. Assuming the electronic states belong to $L = 1$, $\\braket{\\mathbf{\\hat{L}^{\\pm}} }$ evaluates to $\\sqrt{2}$. For all parameters, a subscript indicates the corresponding electronic states. The non-diagonal terms for $\\Delta \\Omega =0$ come from spin-orbit interaction and those for $\\Delta \\Omega =\\pm1$ are rotational interactions, the later ones also couple $^2\\Sigma^+_{-1\/2}$ and $^2\\Pi_{+1\/2}$.\n\t\n\tTo keep the number of fit parameters low, some simplifications had to be made because we only have data for state $2^2\\Sigma^+$, which couples strongly through spin-orbit interaction to the component $\\Omega=1\/2$ of \\astate but only weakly by rotation to $\\Omega=3\/2$. Thus we reduce the $3\\times3$-matrix to the $2\\times2$ case. The coupling constants $A_{\\Sigma\\Pi}$ and $\\gamma_{\\Sigma\\Pi}$ cannot be separate in fitting experimental data. They are combined into the constant $d_{\\Sigma\\Pi}$. For the same reason we combine  $A_\\Pi$ and $\\gamma_\\Pi$ to the effective constant A. From the difference of the $1^2\\Pi_{1\/2}$ and $1^2\\Pi_{3\/2}$ PECs given in the supplement of reference \\citeText{gopakumar_ab_2013}, $A_\\Pi$ can be estimated to be \\SI{118}{\\kay}.  The approximate interaction matrix is shown in \\tabref{tab:redIntMatrix}.\n\t\n\t\n\t\\begin{table}\\caption{\\label{tab:redIntMatrix} Reduced interaction matrix between the  $2^2\\Sigma^{+}$ and  $2^2\\Pi_{1\/2}$ state. $d_{\\Sigma\\Pi} = A_{\\Sigma\\Pi} - \\gamma_{\\Sigma\\Pi}$ and $A = A_\\Pi \\pm \\gamma_{\\Pi} \\approx A_\\Pi$. The $^2\\Pi_{3\/2}$ state is ignored.}\n\t\t\n\t\t\t\\centering\n\t\t\t\\begin{tabular}{l|*2{c}} \n\t\t\t\t& $\\ket{^2\\Sigma^{+}_{1\/2}}$ & $\\ket{^2\\Pi_{1\/2}}$  \\\\\t\n\t\t\t\t\\hline\n\t\t\t\t$\\bra{^2\\Sigma^{+}_{1\/2}}$ & \\begin{tabular}{@{}c} $\\mathrm{E}_\\mathrm{Dun}(v,N=J\\mp \\nicefrac{1}{2}) $ \\\\$ -\\nicefrac{\\gamma_\\Sigma}{2} \\times \\left [1 \\mp (J+\\nicefrac{1}{2}) \\right ] $ \\end{tabular} & \\begin{tabular}{@{}c} $\\nicefrac{p}{2} \\big[d_{\\Sigma\\Pi} \\, + $ \\\\ $2\\, B_{\\Sigma\\Pi} ( 1 \\mp [J+\\nicefrac{1}{2}] ) \\big] $ \\end{tabular}\\\\[10pt]\n\t\t\t\t$\\bra{^2\\Pi_{1\/2}}$ & \\begin{tabular}{@{}c} $\\nicefrac{p}{2} \\big[d_{\\Sigma\\Pi} \\, + $ \\\\ $2\\, B_{\\Sigma\\Pi} ( 1 \\mp [J+\\nicefrac{1}{2}] ) \\big] $ \\end{tabular}  & \\begin{tabular}{@{}c} $\\mathrm{T}_{\\Pi} -\\nicefrac{A}{2} + G_\\Pi(v)    $ \\\\ $+B_{\\Pi,v} [J(J+1)-1]$ \\end{tabular}\n\t\t\t\\end{tabular}\n\t\t}\n\t\\end{table}\n\t\n\t\n\t\\begin{figure}\n\t\t\\includegraphics[width = \\columnwidth]{2Sigma_1Pi_Ladder.eps}\t\t\n\t\t\\caption{ Rotational energies of the $v^\\prime = 0$ level of the $2^2\\Sigma^{+}$ state and the three closest vibrational levels of the $1^2\\Pi_{1\/2}$  state.}\n       \\label{fig:energyLadders}\n\t\\end{figure}\n\t\n\tTo characterize the perturbation seen in \\figref{fig:deviation} (a), knowledge about the crossing of the rotational states of the \\bstate and  \\astate states and the various coupling strengths are required. For the \\astate state we start with parameters taken from reference \\citeText{gopakumar_ab_2013}. In order to come close to the observed resonant perturbation, $T_\\Pi$ and $B_\\Pi$ were adjusted to move the crossing points of the rotational ladders of one vibrational level of $1^2\\Pi_{1\/2}$ and $v^\\prime_{\\Sigma}= 0$ into the range of maximal deviation (see \\figref{fig:energyLadders}). The variation of the sign of the deviation shows that the rotational constant of the perturbing state must be larger than that of state \\bstate to obtain  repelling levels in the observed direction.\n\tTaking this initial choice of parameters for the $1^2\\Pi_{1\/2}$ state, only $p$, $d_{\\Sigma\\Pi}$ and $B_{\\Sigma\\Pi}$ are unknown for a fit.\n\n\tSince the \\astate state is not known well enough to assign $v_\\Pi$ unambiguously and thus  to calculate the desired overlap integral with the \\bstate state with satisfactory reliability, we incorporate the parameter $p$ into $d_{\\Sigma\\Pi}$ and set $B_{\\Sigma\\Pi}$ initially to zero.\n\n\t\n\t\\begin{table*}\n\t\t\\caption{\\label{tab:DunPar}Dunham and spin-rotation parameters for the first two $^2\\Sigma^{+}_{1\/2}$ states and the first  $^2\\Pi_{1\/2}$ state of \\LiSr. The parameters give an accurate description for levels with $N<105, v= 0,1$ and $40 \\leq N \\leq 60, v=2$ in the \\Xstate state, $N < 69, v=0$ in the \\bstate state and for $N<69, v \\approx 15$ in the \\astate state.  All values given in \\si{\\kay}.}\n\t\t\n\t\t\n\t\t\\begin{ruledtabular}\n\t\t\t\\begin{tabular}{*{5}lr}\n\t\t\t\t\\noalign{\\vskip 5pt} \n\t\t\t\t\\multicolumn{6}{c}{$1^2\\Sigma^{+}_{1\/2}$} \\\\\n\t\t\t\t$\\mathrm{Y}_{0n}$ & $\\mathrm{Y}_{1n}$ & $\\mathrm{Y}_{2n}$& $\\gamma_{0n}$ & $\\gamma_{1n}$ & $n$ \\\\\n\t\t\t\t\\num{0.0} & \\num{182.9305 +- 0.0054} & \\num{-3.0263 +-  0.0026}& \\num{8.88 +-  0.46 e-3} & \\num{-5.28 +-  0.16 e-4} & 0\\\\\n\t\t\t\t\\num{2.072284 +-   0.000079 e-1} & \\num{-3.3395 +-  0.0023 e-3} & \\num{-1.0525 +-  0.0092 e-4} &-&-& 1\\\\\n\t\t\t\t\\num{-1.0297 +-  0.0017 e-6} & \\num{-3.662 +-  0.042 e-8} & - &-&- & 2\\\\\n\t\t\t\t\\num{ -5.89 +-  0.10 e-12} & \\num{-2.187 +-  0.032 e-12} & - &-&- &3\\\\\n\t\t\t\t\\hline\n\t\t\t\t\\noalign{\\vskip 5pt} \n\t\t\t\t\\multicolumn{6}{c}{$2^2\\Sigma^{+}_{1\/2}$} \\\\\n\t\t\t\t\\num{9389.2125 +-  0.0026} & \\num{186.94} \\footnote{Value from reference \\citeText{gopakumar_ab_2013}} &- &\\num{4.651 +-  0.046 e-2} &- & 0 \\\\\n\t\t\t\t\\num{1.890807 +-  0.000081 e-1} & - &- & - &- & 1 \\\\\n\t\t\t\t\\num{-7.922  +- 0.018 e-7} & - & -&-&- & 2 \\\\\n\t\t\t\t\\num{3.77  +- 0.14 e-12} & - &- &-&- &3 \\\\\n\t\t\t\t\\hline\n\t\t\t\t\\noalign{\\vskip 5pt} \n\t\t\t\t\\multicolumn{6}{c}{$1^2\\Pi_{1\/2}$} \\\\\n\t\t\t\t\\num{5403.7 +-  2.4}\\footnote{Disregarding an offset of $A\/2$} & \\num{285.634} $^{\\text{a}}$ & \\num{-1.91289} $^{\\text{a}}$ &- &- & 0 \\\\\n\t\t\t\t\\num{2.837 +-  0.014 e-1} & \\num{-2.01 e-3} $^{\\text{a}}$& - &- &- & 1\\\\\n\t\t\t\t\\hline\n\t\t\t\t\\noalign{\\vskip 5pt} \n\t\t\t\t\\multicolumn{4}{r}{$p\/2 \\times d_{\\Sigma\\Pi}$:} & \\multicolumn{1}{l}{\\num{2.644 +- 0.043} } & \n\t\t\t\\end{tabular}\n\t\t\t\n\t\t\\end{ruledtabular}\n\t\t\n\t\t\n\t\\end{table*}\n\n\t\n\tThe rovibronic parameters $T_\\Pi$ and $B_\\Pi$ and $d_{\\Sigma\\Pi}$ were varied in order to minimize the deviation for all data points shown in \\figref{fig:deviation} by a non-linear least-squares fit using the energies obtained after matrix diagonalization.\n\tFrom this fit, unperturbed levels of the \\bstate state and corresponding transition frequencies were constructed and then applied in the linear fit with energies represented by \\equref{eq:SRcoupling} for improved Dunham coefficients for the \\Xstate and \\bstate states. For this, 1534 observations (transition frequencies from the thermal emission spectrum, fluorescence lines along with frequency differences from LIF spectra) were used. Such procedure was cycled and after three iterations the Dunham coefficients changed by less than the estimated standard deviation from the linear fit. Thus we obtained convergence of the iterative fitting procedure. Finally, the stability of this solution was checked by fitting the parameters for the \\bstate and $1^2\\Pi^+_{1\/2}$ states along with $d_{\\Sigma\\Pi}$ simultaneously in the non-linear fit step. Additionally, we allowed the variation of the J-dependent off-diagonal term by $B_{\\Sigma\\Pi}$. It turned out that this contribution is insignificant in the range of observations.\n\tThe final coefficients are listed in \\tabref{tab:DunPar} together with estimated standard deviations from the linear fit. The standard deviation of the fit was 0.492.\n\t\t\n\t\\figref{fig:deviation} (b) shows that the perturbation is well described because most deviations lie within the gray area, which represents the experimental uncertainty. For the \\fone \\ states closest to the perturbation, no ideal description could be achieved. For improving the modeling we would like to have data of the $1^2\\Pi$, especially the observation of the extralines expected around the perturbed lines of $2^2\\Sigma^+$, but we were so far unsuccessful in our efforts.\n\t\n\tAdditionally, a local deviation around $N^\\prime = 53$ for \\fone \\ and  $N^\\prime = 57$ for \\ftwo can be seen in \\figref{fig:deviation}. These might indicate a crossing of \\bstate with $1^2\\Pi_{3\/2}$. That state was ignored in the simplified interaction model and therefore the aforementioned lines were ignored in the fitting process. A manual adjustment of the spin-orbit coupling parameter $A_\\Pi$ to shift the $1^2\\Pi_{3\/2}$ state to the corresponding energies gives a value of $A_\\Pi = $\\SI{88 +- 2}{\\kay}, which is still close to the ab initio value, but we believe that this single observation is not yet conclusive.\n\t\n\t\n\\section{results and discussion}\n\tThe deeply bound rovibrational levels of the \\Xstate and \\bstate states of \\LiSr were modeled using the thermal emission and LIF spectra. \n\tThe $v^{\\prime\\prime} =0,1$ levels of the ground state \\Xstate could be described up to $N^{\\prime\\prime} = 105$ and the $v^{\\prime\\prime} = 2$ level with $N^{\\prime\\prime}$ ranging from \\numrange{41}{64} by the Dunham parameters including spin-rotation. \n\tTransitions in this system were found to take place mainly between states with the same or a directly neighbouring vibrational quantum number. This restricts the study of the molecule via LIF experiments and becomes a time consuming work because only short progressions were observed. To investigate higher vibrational states for a more complete ground state potential, other studies including higher lying electronic states need to be employed.\n\t\n\tWith the perturbation model from section \\ref{sec:pert}, molecular parameters for the $v^\\prime = 0$ level of the \\bstate state with $N^\\prime < 69$ were derived. Including higher rotational states in the evaluation was not yet successful due to the complex perturbation structure. The isolated perturbation of the rovibrational levels for $N^\\prime<69$ gave an opportunity to gauge the strength of the \\astate -- \\bstate coupling and gain first insights to the \\astate state. Since the \\astate state could not be observed directly with the applied method, this knowledge will be the initial ingredient when incorporating transition lines of higher \\bstate levels into the analysis of the spectrum. \n\t\n\tThe derived molecular parameters are given in \\tabref{tab:DunPar}. Since only a finite number of parameters were fitted, the given parameters are not the true Dunham parameters; they are significantly affected by the truncation of the power expansion. For example, $\\mathrm{Y}_{01}$ of \\bstate is the rotational constant $B_0$ of the evaluated vibrational level. Utilizing the observed perturbation, a  parametrization of the energy levels of the $1^2\\Pi_{1\/2}$ state in the neighbourhood of $v_\\Sigma =0$ of the \\bstate state was possible. The effective coupling parameter, $p\/2 \\times d_{\\Sigma\\Pi} = \\SI{2.644}{\\kay}$ is fairly low compared to the estimated spin-orbit parameter around \\SI{100}{\\kay}, indicating the weak overlap of the involved vibrational levels of the states \\astate and $2^2\\Sigma^+$. A fit with the $J$-dependent coupling parameter $B_{\\Sigma\\Pi}$ (see \\tabref{tab:redIntMatrix}) gave no significant improvement and shows variations in the value of $\\gamma_{00}$ of \\bstate of less than one percent. Thus the effective spin-rotation interaction of \\bstate does not originate from the investigated local perturbation but will be the integrated effect of the full manifold of \\astate -- \\bstate coupling and\/or couplings to other $^2\\Pi$ states. \n\tFor \\LiSr it is possible to observe the \\astate state indirectly due to the long $N$ interval of interaction with the \\bstate state. This is not guaranteed for all molecules and has not yet been observed in \\ce{LiCa}\\cite{ivanova_x_2011, stein_spectroscopic_2013}.\n\t\n\t\n\t\\begin{table*}\n\t\t\\caption{\\label{tab:constComp}Comparison of measured spectroscopic constants of \\LiSr with results of various ab initio works. All values given in \\si{\\kay}, except $R_e$ which is given in \\AA. }\n\t\t\\begin{ruledtabular}\n\t\t\t\\begin{tabular}{llccccccr}\n\t\t\t\t&Method&$R_e$ &$D_e$ &$\\omega_e \\approx \\mathrm{Y}_{10}$&$\\omega_e x_e \\approx -\\mathrm{Y}_{20}$ &$B_e \\approx \\mathrm{Y}_{01}$ &$T_e $ & Ref.\\\\\n\t\t\t\t\\hline\n\t\t\t\t\\noalign{\\vskip 5pt} \n\t\t\t\t$1^2\\Sigma^{+}_{1\/2}$&UCCSD(T) & 3.55\\phantom{0} & 2367\\phantom{00} & 182.2\\phantom{0} & - & -& 0 & \\citeText{kotochigova_ab_2011}$\\phantom{^{\\text{a}}}$\\\\\n\t\t\t\t&CCSD(T) & 3.531 & 2226.4\\phantom{0} & 182.1\\phantom{0} & 4.29 & 0.203$\\phantom{00^{\\text{a}}}$ & 0 & \\citeText{gopakumar_ab_2011}$\\phantom{^{\\text{a}}}$\\\\\n\t\t\t\t&SO-MS-CASPT2 & 3.579 & 2075.26 & 168.62 & - & 0.2036$\\phantom{0^{\\text{a}}}$ & 0 & \\citeText{gopakumar_ab_2013}\\footnote{Values have been converted to \\LiSr.}\\\\\n\t\t\t\t&MRCI & 3.574 & 2483\\phantom{.00} & 179.1\\phantom{0} & 3.22 & - & 0 & \\citeText{pototschnig_vibronic_2017}$\\phantom{^{\\text{a}}}$\\\\\n\t\t\t\t&spectroscopy & \\phantom{$^\\text{a}$}3.545\\footnote{From RKR calculation} & - & 182.93 & 3.03 & 0.207$\\phantom{00^{\\text{a}}}$ & 0 & this work$\\phantom{^{\\text{a}}}$\\\\\n\t\t\t\t\\hline\n\t\t\t\t\\noalign{\\vskip 5pt}    \n\t\t\t\t$2^2\\Sigma^{+}_{1\/2}$&SO-MS-CASPT2 & 3.785 & 6860.54 & 186.94 & - & 0.1711$\\phantom{0^{\\text{a}}}$ & 9488.63 & \\citeText{gopakumar_ab_2013}$^{\\text{a}}$\\\\\n\t\t\t\t&MRCI & 3.728 & 7811\\phantom{.00} & 183.0\\phantom{0} & 1.07 & - & 9469\\phantom{.00} & \\citeText{pototschnig_vibronic_2017}$\\phantom{^{\\text{a}}}$\\\\\n\t\t\t\t&spectroscopy & $\\phantom{^{\\text{a,b}}}$3.712$^\\text{b,}$\\footnote{$R_0 = \\SI{3.708}{\\AA}$} & - & $\\phantom{^{\\text{a}}}(186.94)\\footnote{RKR potentials were calculated with $\\omega_e$ taken from reference \\citeText{gopakumar_ab_2013}.}\\phantom{0}$ & - & 0.18908\\footnote{$B_0$, not $B_e$, because only one vibrational state was involved.} & $\\phantom{^{\\text{b,f}}}$9388.31$^\\text{b,}$\\footnote{$T_0=\\SI{9482.683}{\\kay}$}& this work$\\phantom{^{\\text{a}}}$\\\\\n\t\t\t\\end{tabular}\n\t\t\\end{ruledtabular}\n\t\\end{table*}\n\t\n\n\tUsing the Dunham parameters, PECs for the lowest vibrational states were calculated using the RKR-Method (see e.g. reference \\citeText{telle_fcfrkr_1982} and references therein). They are indicated by thick, blue lines in \\figref{fig:PotentialCurves} and show essential agreement with the ab initio PECs but energy displacements are in the order of \\SI{100}{\\kay} within the thick lines. Deriving Franck-Condon-factors\\cite{herzberg_spectra_1950} from these PECs, we get the confirmation by the values why the emission spectrum of $2 ^2\\Sigma^+ \\leftrightarrow 1 ^2\\Sigma^+$ is concentrated on few vibrational levels and why no long vibrational progressions are observed from laser excitations. The authors of reference \\citeText{pototschnig_vibronic_2017} found this to be generally expected for alkali-alkaline earth dimers. \n\t\n\t\n\t\n\t\\tabref{tab:constComp}  shows a comparison of some spectroscopic constants derived in this work with several ab initio calculations. Overall, the rotational constant $B_e\\approx Y_{01}$, the vibrational constant $\\omega_e\\approx Y_{10}$ and the electronic energy $T_e$ were found to be smaller than the ab initio values. The value of $\\omega_e$ varies by about \\SI{10}{\\percent} between different studies, two of which differ by less than \\SI{1}{\\percent} from the experimentally found value for the \\Xstate state. The equilibrium internuclear distances along with $B_e$ values deviate by only a few percent. The values for $T_e$ of the \\bstate state agree within \\SI{100}{\\kay}. The potential depth from the ab initio calculations disagree between each other by a few hundred \\si{\\kay} for the ground state but by up to \\SI{1000}{\\kay} for the \\bstate state. An experimental result for this parameter for the $^2\\Sigma^+$ states could not be achieved here. A comparison for the \\astate state is not yet possible because of the single perturbation level, which would be about $v_\\Pi=15$ applying the ab initio results from ref. \\citeText{gopakumar_ab_2013}.\n\t\n\tFurther work on \\ce{LiSr} will be done to describe the transition lines with $N^\\prime > 68$ in a more extensive model of the perturbation based on the findings presented here.\n\n\tTogether with data from different publications for \\ce{^7Li^{40}Ca}\\cite{ivanova_x_2011, stein_spectroscopic_2013} and \\ce{^7Li^{138}Ba} \\cite{dincan_electronic_1994}, a trend in the molecular constants seems to emerge. For the $\\Sigma$ states, the product of the reduced mass and the rotational constant decreases with increasing reduced mass while the product of the reduced mass and the spin-rotation coupling increases with the reduced mass. This finding relates nicely to the increase of the spin-orbit interaction from Ca via Sr to Ba.\n\t\n\tThe spectrum of \\ce{LiSr} is fairly dense and thus difficult to analyze due to the many overlapping lines. Thus alkali-alkaline earth dimers with a larger reduced mass should have denser spectra and the LIF method would be immensely advantageous to obtain simplified spectra to help in the assignment of quantum numbers. Work on \\ce{KCa} is in progress in our lab and confirms this expectation.\n\t\n\t\n\n\\section*{supplementary material}\n\n\tSee supplementary material for the full recorded thermal emission spectrum and a list of assigned emission lines for $N^\\prime < 69$ of the \\bands{0}{0} and \\bands{0}{1} bands.\n\n\\section*{acknowledgments}\n\tThis work received financial support from the Deutsche Forschungsgemeinschaft (DFG).\n\n\n\\section*{references}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nDue to a combination of strong spin-orbit coupling, large g-factor, and readily-induced proximity superconductivity the InAs based two-dimensional electron gas (2DEG) has gained traction recently as a promising platform for topological quantum computing \\citep{sarma:2005,alicea:2011,shabani:2016,kjaergaard:2016,kjaergaard:2017,suominen:2017,nichele:2017}.\nA structure composed of a shallow InAs quantum well can be engineered to have proximity induced superconductivity with an in-situ epitaxial Al top layer with high transparency \\citep{shabani:2016}.\nThis system has been demonstrated to contain Andreev bound states that coalesce into Majorana zero modes \\citep{suominen:2017,nichele:2017}.\nHowever, a pressing limitation is the quality of the 2DEG.\n\n\n\n\nWe investigate the limitations of 2DEG mobility in the InAs on InP substrate system.\nLow temperature transport measurements are performed on gated Hall bars using symmetric In$_{0.75}$Ga$_{0.25}$As\/InAs\/In$_{0.75}$Ga$_{0.25}$As quantum wells grown on (100) InP where we vary the width of the flanking InGaAs layers, the depth of the quantum well from the surface, and the width of the InAs layer. \nWhile InAs has a 3.3$\\%$ lattice mismatch to InP, the superior insulating property of Fe-doped InP substrates presents a crucial advantage for the measurement of high impedance devices necessary for the exploration of Majorana physics.\nOur results demonstrate record charge carrier mobility in excess of $1 \\times 10^{6}\\,$cm$^{2}\/$Vs for this system and that our mobility appears to be limited by unintentional background charge impurities.\nWe extract the Rashba parameter and spin-orbit length from the beating pattern of Shubnikov de-Haas oscillations.\nThese results may be leveraged to improve the quality of InAs 2DEG structures used for topological quantum computing.\n\n\n\\begin{table}[b]\n  \\begin{tabular}{ | l | c | c | c | }\n    \\hline\n    \\, & \\,\\,In$_{0.75}$Al$_{0.25}$As\\,\\, & \\,\\,In$_{0.75}$Ga$_{0.25}$As\\,\\, & \\,\\,InAs\\,\\, \\\\ \\hline\n    \\,\\,Sample A\\,\\, & $b=120\\,$nm & $d=5\\,$nm & $w=4\\,$nm \\\\ \\hline\n    \\,\\,Sample B\\,\\, & $b=120\\,$nm & $d=10.5\\,$nm & $w=4\\,$nm \\\\ \\hline\n    \\,\\,Sample C\\,\\, & $b=120\\,$nm & $d=15\\,$nm & $w=4\\,$nm  \\\\ \\hline\n    \\,\\,Sample D\\,\\, & $b=120\\,$nm & $d=10.5\\,$nm & $w=6\\,$nm \\\\ \\hline\n    \\,\\,Sample E\\,\\, & $b=180\\,$nm & $d=10.5\\,$nm & $w=4\\,$nm \\\\ \n    \\hline\n  \\end{tabular}\n  \\caption{Sample details for dimensions of the top In$_{0.75}$Al$_{0.25}$As barrier layer,  the symmetric In$_{0.75}$Ga$_{0.25}$As layers, and the InAs quantum well width.}\n\\label{table}\n\\end{table}\n\nOur samples are grown using molecular beam epitaxy (MBE); see Ref.\\,\\citep{gardner:2016} for greater detail on how our MBE has been set up and maintained.\nThe InAs structures are grown on semi-insulating InP (100) substrates that have been desorbed at $525^{\\degree}$C until a $(2\\times3)$ to $(2\\times4)$ surface phase transition is observed.\nFirst, a 100 nm thick In$_{0.52}$Al$_{0.48}$As lattice matched smoothing layer upon which a In$_{0.52}$Al$_{0.48}$As\/In$_{0.52}$Ga$_{0.48}$As  $2.5\\,$nm five period superlattice is grown at $480^{\\degree}$C \\citep{heyn:2003}.\nDue to a native lattice mismatch of 3.3$\\%$ between InAs and InP we grow a step graded buffer of In$_{x}$Al$_{1-x}$As where $x=0.52$ to $0.84$ using 18, $50\\,$nm wide each, followed by a linearly ramped reverse step from $x=0.84$ to $0.75$ to relieve any residual strain.  \nThe graded buffer layer and reverse step are grown at $360^{\\degree}$C.\n\nThe active region comprised of the composite quantum well plus barriers is then grown. The substrate temperature is increased to $480^{\\degree}$C to grow a $25\\,$nm In$_{0.75}$Al$_{0.25}$As bottom barrier and active region composed of a strained $w=4\\,$nm ($w=6\\,$nm for Sample E only) InAs layer flanked on either side by symmetric In$_{0.75}$Ga$_{0.25}$As layers to promote higher mobility \\citep{sexl:1997,wallart:2005}.\nFor Samples A, B, and C, we vary only the width of the In$_{0.75}$Ga$_{0.25}$As layers to be $d=5, 10.5,$ and $15\\,$nm, respectively.\nThe sample growth is completed with a $b=120\\,$nm ($b=180\\,$nm for Sample D only) In$_{0.75}$Al$_{0.25}$As top barrier to remove the active region from the surface and minimize anisotropy effects that can become apparent when the active region of the quantum well is to near the surface \\citep{lohr:2003}.\nLastly, we do not include an InGaAs capping layer to avoid formation of a parallel conduction channel \\citep{shabani:2014} or intentional doping.\nIn the inset of Fig.\\,\\ref{fig1} we schematically depict the layer stack for the active region.  \nA summary of the five samples discussed here are presented in Table\\,\\ref{table}.\n\n\n\n\\begin{figure}[t]\n\\vspace{-0.1 in}\n\\includegraphics[width=.48\\textwidth]{fig1_v7.JPG}\n\\vspace{-0.25 in}\n\\caption{(Color online)\nLongitudinal, $\\rho_{xx}$, and Hall, $\\rho_{xy}$ in units of inverse filling factor, resistivities vs magnetic field, $B$, for density $n=6.2\\times 10^{11}\\,$cm$^{-2}$, left and right axis respectively, for Sample B.\nInset: Schematic representation of the layer stack for the the active region of the quantum well, see text for greater detail.\n}\n\\vspace{-0.1 in}\n\\label{fig1}\n\\end{figure}\nOur samples are processed with standard wet etching techniques to define both straight and L-shaped (aligned along the $[1\\bar{1}0]$ and $[110]$ directions) Hall bars of width $w=150\\,\\mu$m. \nAfter etching we deposit Ti\/Au ohmic contacts of thickness $80\/250\\,$nm, a 40 nm Al$_{2}$O$_{3}$ dielectric using thermal atomic layer deposition, and a $20\/150\\,$nm Ti\/Au gate.\nAll samples have a zero gate voltage, $V_{\\mathrm{G}}=0$, density of  $n = 5.3-5.6 \\times 10^{11}\\,$cm$^{-2}$ with $\\Delta n$ versus $V_{\\mathrm{G}}$ in good agreement with a simple capacitance model. \nThe samples were measured in a $^{3}$He system at a base temperature of $T=300\\,$mK using standard low frequency lockin techniques with excitation current of $0.5 \\mu$A.\n\nInAs quantum wells based on GaSb substrates with Al$_{0.37}$Ga$_{0.67}$Sb barriers \\citep{shojaei:2016} have recently been shown to achieve mobilities of $\\mu=2.4 \\times 10^{6}\\,$cm$^{2}\/$Vs  at $n \\sim 1 \\times 10^{12}\\,$cm$^{-2} $\\citep{tschirky:2017}.\nThe sample structures investigated here are instead grown on lattice mismatched InP substrates that have superior insulating properties, a requirement when operating mesoscopic devices in high resistance configurations.\nTo our knowledge, the highest reported mobility for such a structure is $\\mu= 0.6 \\times 10^{6}\\,$cm$^{2}\/$Vs achieved at $n \\sim 5 \\times 10^{11}\\,$cm$^{-2}$ \\citep{shabani:2014}, but supporting transport data was not provided.\n\n\nWe begin our discussion with Sample B, which yielded the highest mobility.\nIn Fig.\\,\\ref{fig1} we present longitudinal ($\\rho_{xx}$, left axis) and Hall ($\\rho_{xy}$ in units of inverse filling factor, right axis) resistivities versus magnetic field ($B$) for $n = 6.2 \\times 10^{11}\\,$cm$^{-2}$.\nWe observe the absence of a parasitic parallel conduction channel from the linear low field $\\rho_{xy}$.\nThere is also good agreement between the extracted density from both the Hall slope and the period of Shubnikov de-Haas oscillations (SdHOs). \nWith increasing $B$ we observe a spin splitting onset at filling factor $\\nu=nh\/eB=19$, where $h$ is the Planck constant and $e$ the electron charge, as marked in Fig.\\,\\ref{fig1}, with well developed integer quantum Hall states, $\\rho_{xx} = 0$ and $\\rho_{xy} =N h\/e^{2}\\nu$, where $N$ is an integer. \n\n\n\\begin{figure}[t]\n\\vspace{-0.1 in}\n\\includegraphics[width=.48\\textwidth]{fig2_v7.jpg}\n\\vspace{-0.25 in}\n\\caption{(Color online)\nMobility vs density, $\\mu$ vs $n$:\n(a) Sample B on a straight Hall bar oriented along the $[1\\bar{1}0]$ direction demonstrating a peak mobility of $\\mu=1.1 \\times 10^{6}\\,$cm$^{2}\/$Vs at $n = 6.2 \\times 10^{11}\\,$cm$^{-2}$.\n(b) Sample B obtained from an L-shaped Hall bar along the two main crystallographic directions and a fit line to $\\mu \\propto n^{\\alpha}$ where $\\alpha=0.5$.\n}\n\\vspace{-0.1 in}\n\\label{fig2}\n\\end{figure}\nWe continue with Sample B with gating to obtain $\\mu$ versus $n$, shown in Fig.\\,\\ref{fig2}\\,(a) for a straight Hall bar aligned along the $[1\\bar{1}0]$ direction.\nA maximum mobility of $\\mu=1.1 \\times 10^{6}\\,$cm$^{2}\/$Vs occurs at $n = 6.2 \\times 10^{11}\\,$cm$^{-2}$.\nTo our knowledge this is the largest reported $\\mu$ for an InGaAs\/InAs\/InGaAs quantum well. \nOn a device processed during a different fabrication, from the same wafer as Sample B, we plot $\\mu$ versus $n$ where measurements were performed on an L-shaped Hall bar in Fig.\\,\\ref{fig2}\\,(b).\nThe gating dependence shows that there is minimal anisotropy between the $[1\\bar{1}0]$ and $[110]$ directions with less than $5\\%$ difference. \nThus we compare samples only along the $[1\\bar{1}0]$ direction for the remainder of the manuscript.\n\n\n\n\nTo determine what limits mobility in our structure we assume the $\\mu$ vs $n$ dependence can be described by a simple power law, $\\mu\\propto n^{\\alpha}$, and extract the exponent $\\alpha$, using a log-log plot (not shown) giving equal weight to all data, in the restricted density range of $n>1.5 \\times 10^{11}\\,$cm$^{-2}$.\nIn Fig.\\,\\ref{fig2}\\,(b) the extracted fit, dashed line, for $\\alpha = 0.5$ fits well over the density range of interest and is roughly equivalent for all samples measured in this study.\nIn Ref.\\,\\citep{shabani:2014_MIT} a similar structure was investigated that contained $8\\,$nm In$_{0.75}$Ga$_{0.25}$As layers where it is was observed that $\\alpha \\sim 0.8$.\nThese $\\alpha$-values indicate that the mobility is limited by unintentional background impurities \\citep{sarma:2013}.\nTheoretically, in the strong screening, $q_{\\mathrm{TF}} \\gg k_{\\mathrm{F}}$ where $q_{\\mathrm{TF}}$ is the Thomas-Fermi wave vector \\citep{stern:1967}, and high density limit $\\alpha \\rightarrow 1\/2$, however, in the case of remote two-dimensional impurities $\\alpha \\rightarrow 3\/2$.\nFor comparison, Ref.\\,\\citep{shabani:2014_MIT} investigated a sample with an additional $10\\,$nm In$_{0.75}$Ga$_{0.25}$As capping layer and observed $\\alpha = 1.35$, where the increase in $\\alpha$ was attributed to an unintentional parallel surface channel. \nIt is plausible that the introduction of this capping layer enhanced a remote layer that was competing with the background impurities favoring an increase in $\\alpha$.\nThe difference between $\\alpha=0.5$ and $\\alpha=0.8$ could also be due to unintentional background impurities within the well, as evidenced by the difference in $\\mu$ over the same $n$ range \\citep{sarma:2013}.\nAt present the exact nature of the charged impurities in our samples cannot be specified, but since a 2DEG is formed in the absence of modulation doping it is reasonable to assume that ionized donor-like defects exist in the lattice.  \nIdentification of the precise location and density of such defects requires further investigation beyond the scope of this paper.\n\n\n\\begin{figure}[t]\n\\vspace{-0.1 in}\n\\includegraphics[width=.48\\textwidth]{fig3_v7.jpg}\n\\vspace{-0.25 in}\n\\caption{(Color online)\n$\\mu$ vs $n$:\n(a) Comparison from Samples A-C where the width, $d$, of the InGaAs layer is varied.\n(b) Comparison between Sample B and D where the width, $w$, of the InAs quantum well is varied.\n(c) Comparison between Sample B and E where the width, $b$, of the top InAlAs barrier is varied.\n} \n\\vspace{-0.1 in}\n\\label{fig3}\n\\end{figure}\nComparison of the quality of our samples is evaluated using zero-field mobility as the metric.\nWe next investigate perturbations to Sample B beginning with the well width dependence of the In$_{0.75}$Ga$_{0.25}$As layers.\nIn Fig.\\,\\ref{fig2}\\,(c) we plot $\\mu$ versus $n$ for Samples A-C where the In$_{0.75}$Ga$_{0.25}$As layer widths are $d=5,10.5,\\,$and $15\\,$nm, respectively.\nFor Sample C, $d=15\\,$nm, there is a nonmonotonic $\\mu$ versus $n$ where $\\mu$ begins to decrease for $n > 4.25 \\times 10^{11}\\,$cm$^{-2}$.\nThis nonmonotonic $n$-dependence is due to occupation of the second subband.\nEstimation of the onset density of the second subband becoming populated occurs at $n \\sim 7.5, 6,$ and $5 \\times 10^{11}\\,$cm$^{-2}$ for Samples A-C, respectively, from self consistent calculations performed with Nextnano$^{3}$ \\citep{nextnano}.\n\n\nAt fixed $n$ there is a nonmonotonic dependence of $\\mu$ versus $d$.\nAt $n\\sim 4 \\times 10^{11}\\,$cm$^{-2}$, for example, $\\mu= 0.83 \\times 10^{6}\\,$cm$^{2}\/$Vs for $d=10.5\\,$nm that decreases to $\\mu=0.73 \\times 10^{6}\\,$cm$^{2}\/$Vs and $\\mu=0.59 \\times 10^{6}\\,$cm$^{2}\/$Vs for $d=15$ and $5\\,$nm, respectively.\nWith a large overlap in sample structure between the three samples we do not expect changes in scattering from background impurities, remote impurities, or charged dislocations due to the lattice mismatch to give reasonable explanation to the observed width dependence.\nIncreasing $d$ results in a spread of the charge distribution such that there is an increase of the amount of charge that resides in the In$_{0.75}$Ga$_{0.25}$As layers.\nAn increase of the amount of wavefunction extension into the In$_{0.75}$Ga$_{0.25}$As layer will decrease the mobility due to an increase in the amount of alloy scattering.\nThe small decrease of $\\mu$ of $\\sim 12\\%$ is due to a $\\sim 1\\%$ transfer of charge from the pure InAs to the In$_{0.75}$Ga$_{0.25}$As layer implies a strong dependence on alloy scattering.\nA more dramatic reduction in the mobility occurs when there is a decrease of $d$, which can come from two sources 1) alloy scattering and 2) interface scattering.\nThe charge distribution in the effective $14\\,$nm well of Sample C will penetrate into the In$_{0.75}$Al$_{0.25}$As barriers giving an increased amount of alloy scattering, as observed in Nextnano$^{3}$ simulations.\nAdditionally, the increased confinement of the charge results in an increase in scattering at the In$_{0.75}$Ga$_{0.25}$As\/InAs interface.\nComparison of the integrated charge density of the wells of Sample B and C in a restricted region of $0.5\\,$nm to either side of the InGaAs\/InAs interface shows that the amount of charge in the region of the interface increases giving rise to increased interface scattering.\n\n\nIn Fig.\\,\\ref{fig3}\\,(b) $\\mu$ versus $n$ for Sample B and D where the width of the InAs quantum well is increased from $w=4$ to $6\\,$nm is plotted.\nA large reduction in $\\mu$ throughout the entire $n$-range for $w=6\\,$nm is observed.\nNaively one might expect $\\mu$ to increase with an increase in $w$ as a larger percentage of the charge density would reside in the InAs part of the well resulting in a decrease in alloy scattering from the In$_{0.75}$Ga$_{0.25}$As layers.\nHowever, the severe reduction in $\\mu$ implies that the critical thickness, $w_{\\mathrm{c}}$, of the InAs has been exceeded which introduce misfit dislocations to the quantum well \\citep{capotondi:2005,shabani:2014}.\nWe estimate $w_{\\mathrm{c}}\\sim 5.5\\,$nm, for this In concentration.\n\n\nIn Fig.\\,\\ref{fig3}\\,(c) we plot $\\mu$ versus $n$ for increase of the In$_{0.75}$Al$_{0.25}$As barrier from $b=120$ to $180\\,$nm, Samples B and E respectively.\nAgain there is an overall decrease in $\\mu$.\nAs previously discussed $\\mu$ is limited by background charged impurities which suggests that while $b$ is increased in Sample E to reduce surface effects the possible gain is compensated by the increased level of charged impurities introduced by the additional In$_{0.75}$Al$_{0.25}$As layers resulting in decreased $\\mu$.\n\n\n\n\\begin{figure}[t]\n\\vspace{-0.1 in}\n\\includegraphics[width=.5\\textwidth]{fig4_v7.jpg}\n\\vspace{-0.25 in}\n\\caption{(Color online)\n(a) Low field magnetoresistivity with removal of a smoothly varying background, $\\Delta \\rho_{xx}$ vs $B$, for $n=5.1 \\times 10^{11}\\,$cm$^{2}$ on Sample B.\n(c) Amplitude of a FFT of $\\rho_{xx}$ from inverse $B$.\n(c) Rashba parameter, $\\alpha_{\\mathrm{r}}$ squares, and spin orbit length, $\\ell_{\\mathrm{SO}}$ circles, vs $n$, left and right axis respectively.\n}\n\\vspace{-0.1 in}\n\\label{fig4}\n\\end{figure}\nWe preformed further measurements of Sample B at low $B$ to investigate the spin-orbit coupling.\nIn Fig.\\,\\ref{fig4}\\,(a) we plot the oscillatory correction to the magnetoresistivity, $\\Delta\\rho_{\\mathrm{xx}}$, for $n=5.1 \\times 10^{11}\\,$cm$^{-2}$ after removal of a slowly varying background.\nWith increasing $B$ we observe the onset of SdHOs at $B\\sim 0.2\\,$T, this low $B$ value is another indication of the high quality of the 2DEG.\nThe amplitude of the SdHOs increase with increasing $B$ but demonstrate a beating pattern with a node at $B \\sim 0.3\\,$T.\nBy restricting our analysis to densities below occupation of the second subband, this beating pattern can be ascribed to two oscillation periods that are nearly equal and has been demonstrated in these structures to arise from zero field spin splitting between slightly different spin up and spin down densities \\citep{datta:1990,kim:2010,lee:2011}.\n\n\nIn Fig.\\,\\ref{fig4}\\,(b) we present the amplitude of the Fast Fourier Transform (FFT) versus frequency of the magnetotransport after conversion to inverse magnetic field.\nThis FFT split peak can be assigned to two spin-split subbands with densities $n_{+}$ and $n_{-}$, which can be calculated from $n_{\\pm}=ef\/h$.\nFrom this assignment, the estimated total density $n_{\\mathrm{T}}=n_{+}+n_{-}$ is in good agreement with that obtained from the Hall slope and the SdHO minima period.\n\nIn systems that lack inversion symmetry the dominant source of spin-orbit interaction is due to the Rashba effect, which arises from an electric field perpendicular to the plane of the 2DEG.\nThis electric field can be a result of an inversion asymmetry built into the system based on 2DEG design or from an applied field from a gate \\citep{nitta:1997}.\nFrom the SdHO beating pattern we extract the Rashba parameter $\\alpha_{\\mathrm{r}}=\\frac{\\Delta n\\hbar^{2}}{m^{*}}\\sqrt{\\frac{\\pi}{2(n_{\\mathrm{T}}-\\Delta n)}}$, where $\\Delta n = n_{+}-n_{-}$ and we assume $m^{*}=0.03$ \\citep{shabani:2014_MIT}.\nWe perform FFTs at different $V_{\\mathrm{G}}$ and extract $\\alpha_{\\mathrm{r}}$ versus $n$ in Fig.\\,\\ref{fig4}\\,(c), left axis.\nThe Rashba effect is due to an asymmetry in the azimuthal direction and is proportional to the electric field, $\\alpha_{\\mathrm{r}} = \\alpha_{\\mathrm{0}}\\langle E_{\\mathrm{z}}\\rangle$ where $\\alpha_{\\mathrm{0}}$ is a material specific parameter.\nOur gating density dependence is very nearly linear and follows from the simple capacitance model where we observe a linear change to $n$ so we expect a linear increase of $\\alpha_{\\mathrm{r}}$ with decreasing $n$, corresponding to an increase in $E_{\\mathrm{Z}}$.\nThe values we obtain for $\\alpha_{\\mathrm{r}}$ are of the same order as those obtained from InAs systems with symmetric Si doping \\citep{kim:2010}, built in In$_{0.53}$Ga$_{0.47}$As layer asymmetry \\citep{lee:2011,park:2013}, or those reported with AlSb barriers \\citep{shojaei:2016}.\n\n\nTo eliminate effective mass dependence we recast $\\alpha_{\\mathrm{r}}$ as the spin-orbit length, $\\ell_{\\mathrm{SO}}=\\frac{1}{\\Delta n}\\sqrt{\\frac{n_{\\mathrm{T}}-\\Delta n}{2\\pi}}$, versus $n$ and plot the result in Fig.\\,\\ref{fig4}\\,(c), right axis.\nPhysically, the spin-orbit length gives a measure of the average distance traversed by an electron before a spin flip occurs.\nIn the case of weak spin-orbit interaction, the high $n$ (low $V_{\\mathrm{G}}$) case, the spin will travel further through the system, larger $\\ell_{\\mathrm{SO}}$, before its spin orientation will become essentially randomized.\nWith increase of the spin-orbit interaction under applied gate voltage the electron traverses decreasing distance before its spin is randomized.\n\n\n\nThis research supported by Microsoft Station Q.\n\n\n\n\\bibliographystyle{apsrev}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}