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%Title of paper

\title{Anharmonic lattice dynamics of Ag$_2$O studied by inelastic neutron scattering and first principles molecular dynamics simulations\\
Supplemental Material}

% repeat the \author .. \affiliation  etc. as needed
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\author{Tian Lan} 
\email[]{tianlan@caltech.edu}
\affiliation{\mbox{Department of Applied Physics and Materials Science, \nolinebreak California Institute of Technology, Pasadena, California 91125, USA}}
\author{Chen W. Li}
\affiliation{Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA}
\author{J. L. Niedziela}
\affiliation{Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA}
\author{Hillary Smith}
\affiliation{\mbox{Department of Applied Physics and Materials Science, \nolinebreak California Institute of Technology, Pasadena, California 91125, USA}}
\author{Douglas L. Abernathy}
\affiliation{Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA}
\author{George R. Rossman}
\affiliation{Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California 91125, USA} 
\author{Brent Fultz}
\affiliation{\mbox{Department of Applied Physics and Materials Science, \nolinebreak California Institute of Technology, Pasadena, California 91125, USA}}





\date{\today}


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%\pacs{78.70.Nx, 78.30.-j, 63.20.kg, 63.20.Ry}

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To separate the effects of quasiharmonicity and explicit anharmonicity,
the mode frequency $\omegaup_j=\omegaup_j(V,T)$ is expressed as a function 
of volume and temperature
\begin{equation}
\label{eq:weight}
\left(\frac{\partial \ln \omegaup_j}{\partial T}\right)_P = - \gammaup_j\betaup +
\left(\frac{\partial \ln \omegaup_j}{\partial T}\right)_V
\end{equation}
where $j$ is the phonon mode index, $\betaup$ is the volume thermal expansivity 
and $\gammaup_j$ is the mode Gr\"{u}neisen parameter. 
The left-hand side %, as obtained in our Raman spectra shown in Fig. \ref{fig:Raman}, 
is the temperature-dependent  frequency shift at constant pressure, 
and includes contributions from both  quasiharmonicity and explicit anharmonicity. 
The first term on the right-hand side, the isothermal frequency shift as a function of pressure, 
is the quasiharmonic contribution to the frequency shift. 
The second term on the right is the pure temperature 
contribution to the frequency shift from the explicit anharmonicity.
From the difference of the isobaric and isothermal frequency shifts, 
the explicit anharmonicity can be determined.

In a molecular dynamics simulation, the quasiharmonic contribution can be evaluated explicitly by turning off the temperature-dependent anharmonicity. In principle, this method is equivalent to the QHA method implemented with  self-consistent lattice dynamics,\cite{ScF3} 
and in practice we have found this to be true.
For example, we performed simulations at 40\,K for volumes characteristic of 400\,K, 
which produced a pressure of 0.45\,GPa. This calculation therefore removed the temperature effect while preserving the quasiharmonic volume effect at 400 K. The corresponding phonon DOS curves from MD calculations are shown in Fig.~\ref{DOSSM}. By comparing the phonon spectrum of a simulation at 40 K and 0.45 GPa with a simulation at 400 K, the pure temperature
dependence is identified.
From the spectra of Fig.~\ref{DOSSM}, it is found that the explicit anharmonicity dominates
the softening and broadening of the phonon spectra. 
All features in the phonon spectra with the QHA 
showed little change with temperature, except for small stiffenings at high energies. 
%Of course,  the thermal lifetime broadening of phonon linewidths cannot be  
%predicted with the QHA because it assumes noninteracting normal modes.

\begin{figure}[b]
\includegraphics[width=0.8\columnwidth]{DOS1SM}
\caption{Neutron weighted phonon DOS of Ag$_2$O with the cuprite structure from MD simulations. 
The green spectrum is the MD simulated phonon DOS at 40\,K and 0.45\,GPa. Vertical dashed lines are aligned to the major peak centers at 40\,K and labeled by numbers. 
The incident energy was 100\,meV for panel (a), and 30\,meV for panel (b). The spectra were convoluted with the resolution function characteristic of ARCS for the different energies of the incident neutron beam. }
\label{DOSSM}
\end{figure} 
  




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