Continuous, Solid-type Spectrum¶

For a solid-type spectrum, the scattering law is defined as

\[S_s(\alpha,\beta) = \frac{1}{2\pi}\int_{-\infty}^\infty\mathrm{e}^{i\beta t}~\mathrm{e}^{\gamma(t)-\gamma(0)}~dt\]

\[\gamma(t)=\alpha\int_{-\infty}^\infty P(\beta) ~\mathrm{e}^{-i\beta t}~\mathrm{e}^{-\beta/2}~d\beta\]

where

\[P(\beta) = \frac{\rho(\beta)}{2\beta\sinh(\beta/2)}\]

and the phonon distribution \(\rho(\beta)\) is and even funcion and is normalized to the solid-type distribution weight \(\omega_s\),

\[\int_0^\infty\rho(\beta)~d\beta=\omega_s.\]

The phonon distribution can be input without normalization - it will be normalized automatically by LEAPR.

Recall that while \(\beta\) is defined as unitless energy change \((E'-E)/k_bT\), the input phonon distribution must be provided energy exchange \(E'-E\) in units of eV.

The Debye-Waller coefficient is defined as

\[\lambda_s = \int_{-\infty}^\infty P(\beta)~\mathrm{e}^{-\beta/2}~d\beta\]

which simplifies the scattering law \(S_s(\alpha,\beta)\) to be

\[S_s(\alpha,\beta) = \frac{1}{2\pi}\mathbf{e}^{-\alpha\lambda_s}\int_{-\infty}^\infty\mathrm{e}^{i\beta t}~\mathrm{e}^{\gamma(t)}~dt\]

Phonon Expansion

The exponential of \(\gamma(t)\) is a complex and highly oscillatory.

\[\begin{split}\mathrm{e}^{\gamma(t)}=\mathrm{exp}\left[ \alpha\int_{-\infty}^\infty P(\beta)~\mathrm{e}^{-i\beta t}~\mathrm{e}^{-\beta/2}~d\beta \right]\\\end{split}\]

To ease the burden of calculating the scattering law, LEAPR uses the phonon expanion method, which involves expanding the \(\gamma(t)\) exponential as a Taylor series,

\[\mathrm{e}^{\gamma(t)} =\sum_{n=0}^\infty\frac{1}{n!}\left[ \alpha\int_{-\infty}^\infty P(\beta)~\mathrm{e}^{-i\beta t}~\mathrm{e}^{-\beta/2}~d\beta \right]^n\]

which, when used in the solid-type scattering law definition, results in

\[\begin{split}\begin{align} S_s(\alpha,\beta) =~&\mathbf{e}^{-\alpha\lambda_s} \sum_{n=0}^\infty \frac{1}{n!}\alpha^n \\ \times~&\frac{1}{2\pi} \int_{-\infty}^\infty\mathrm{e}^{i\beta t}~\left[ \int_{-\infty}^\infty P(\beta)~\mathrm{e}^{-i\beta t}~\mathrm{e}^{-\beta/2}~d\beta \right]^n~dt \end{align}\end{split}\]

(note that the order of summation and integral have been swapped). Now, the second line of the above equation is redefined as \(\lambda_s^n\mathcal{T}_n(\beta)\). This allows for the solid-type scattering law to be redefined as

\[S_s(\alpha,\beta) = \mathrm{e}^{-\alpha\lambda_s}~\sum_{n=0}^\infty\frac{1}{n!}\Big[\alpha\lambda_s\Big]^n\mathcal{T}_n(\beta)\]

where

\[\mathcal{T}_0(\beta)=\frac{1}{2\pi}\int_{-\infty}^\infty\mathrm{e}^{i\beta t}~dt = \delta(\beta)\]

and

\[\begin{split}\begin{align} \mathcal{T}_1(\beta)&=\frac{1}{\lambda_s}\int_{-\infty}^\infty P(\beta')~\mathrm{e}^{-\beta'/2}~\left[\frac{1}{2\pi}\int_{-\infty}^\infty\mathrm{e}^{i(\beta-\beta')t}~dt\right]~d\beta'\\ &= \frac{1}{\lambda_s}P(\beta')~\mathrm{e}^{-\beta/2}. \end{align}\end{split}\]

In general subsequent \(\mathcal{T}_n(\beta)\) terms can be obtained by convolving the first term with the previous one,

\[\mathcal{T}_n(\beta) = \int_{-\infty}^\infty \mathcal{T}_1(\beta')~\mathcal{T}_{n-1}(\beta-\beta')~d\beta'.\]

These \(\mathcal{T}_n(\beta)\) follow the relationship

\[\mathcal{T}_n(\beta) = \mathrm{e}^{-\beta}~\mathcal{T}_n(-\beta)\]

and are normalized to unity,

\[\int_{-\infty}^\infty \mathcal{T}_n(\beta)~d\beta = 1.\]

This method is called the phonon expansion method because the \(n^{th}\) term of this sum corresponds to the creation/destructin of \(n\) phonons. Note that the \(n=0\) term (which has that \(\mathcal{T}_0(\beta)=\delta(\beta)\)) is carried forward separately. This is to ensure that if translational behavior is considered, then the elastic \(\beta=0\) behavior will appear there. Thus, the continuous distribution contribution is approximated as

\[S_s(\alpha,\beta) \approx \mathrm{e}^{-\alpha\lambda_s}~\sum_{n=1}^N\frac{1}{n!}\Big[\alpha\lambda_s\Big]^n\mathcal{T}_n(\beta)\]

tiny and the zero-phonon term,

\[\mathrm{e}^{-\alpha\lambda_s}~\delta(\beta)\]

is carried forward separately and discussed in Incoherent Elastic Scattering.

In LEAPR, the \(\mathcal{T}_n(-\beta)\) functions are precomputed on the user-requested \(\beta\) grid for \(n\) extending up to some maximum value (default value of 100). These are used to obtain the \(S_s(\alpha,-\beta)\), which is converted to \(S_s(\alpha,\beta)\) by multiplying by \(\mathrm{exp}(-\beta)\).

The short-collision time approximation is used to obtain an effective temperature,

\[\overline{T}_s=\frac{T}{2\omega_s}~\int_{-\infty}^\infty\beta^2~P(\beta')~\mathrm{e}^{-\beta}~d\beta\]

where \(\omega_s\) is the solid-type distribution weight.