Phonon Decomposition

To describe inelastic thermal neutron scattering, the scattering law \(S(\alpha,\beta)\) must be obtained, which is performed using a phonon distribution. LEAPR performs a phonon decomposition, which means that a phonon distribution could consist of the following three contributions:

1. Continuous, solid-type spectrum

2. Translational, diffusive behavior

3. Discrete (Einstein) oscillators

The continuous spectrum (which requires a phonon distribution) is always required. The other two components are optional, and can be used to augment the existing phonon distribution to add very sharp peaks or fluid-like behaviour.

\[\rho(\beta) = \rho_{solid}(\beta) + \underbrace{\rho_{trans.}(\beta)}_{\text{optional}} + \underbrace{\sum_{n=1}^{N_{osc}}\omega_n\delta(\beta_n) }_{\text{optional}}\]

As \(\beta\) approaches zero, the solid-type spectrum varies as \(\beta^2\). Additionally, the solid-type spectrum must integrate to \(\omega_s\), which is the weight for the solid type law (this is a user provided value).

The translational spectrum integrates to the translational weight \(\omega_t\) (user provided) and can be either a free gas law or a diffusion type spectrum.

The sum of all weights must add to unity:

\[\omega_s + \omega_t + \sum_{n=1}^{N_{osc}}\omega_n = 1\]

Using the phonon decomposition method, the final scattering law can be calculated by recursively convolving individual scattering laws.

\[S(\alpha,\beta) = S^{(K)}(\alpha,\beta)\]

where

\[S^{(J)}(\alpha,\beta) = \int_{-\infty}^\infty S^{(J)}(\alpha,\beta')~S^{(J-1)}(\alpha,\beta-\beta')~d\beta'.\]

Debye-Waller Coefficient

The Debye-Waller coefficient is a temperature-dependent quantity that is related to the average displacement of an atom from its equilibrium position. The full Debye-Waller coefficient \(\lambda\) is the combination of the solid-type value \(\lambda_s\) and and discrete oscillator values \(\lambda_i\),

\[\lambda = \lambda_s + \sum_{i=1}^{N_{osc}}\lambda_i.\]

Note that the translational component is not represented in the Debye-Waller coefficient.

Effective Temperature

The effective temperature for all modes is defined as

\[\overline{T} = \omega_t~T + \omega_s~\overline{T}_s+\sum_{i=1}^{N_{osc}}\omega_i\frac{\beta_i}{2}~\coth(\beta_i/2)~T\]

where \(\omega_t,\omega_s,\) and \(\omega_i\) are the fractional weights for the translational piece, the solid-type piece, and each oscillator.

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Example

As an example, consider a phonon distribution that is comprised of a continuous component \([\rho_s(\beta)]\) and two discrete oscillators \([\rho_{d1}(\beta)\) and \(\rho_{d2}(\beta)]\). These three phonon contributions generate corresponding scattering law components \(S_s(\alpha,\beta),S_{d1}(\alpha,\beta),S_{d2}(\alpha,\beta)\).

The zeroth term consists solely of the continuous, solid-type contribution \(S_s(\alpha,\beta)\)

\[S^{(0)}(\alpha,\beta) = S_s(\alpha,\beta)\]

and we convolve the second scattering law \(S_{d1}(\alpha,\beta)\) with the zeroth term,

\[\begin{split}\begin{align} S^{(1)}(\alpha,\beta) &= \int_{-\infty}^\infty S_{d1}(\alpha,\beta')~S^{(0)}(\alpha,\beta-\beta')~d\beta'\\ &= \int_{-\infty}^\infty S_{d1}(\alpha,\beta')~S_s(\alpha,\beta-\beta')~d\beta'\\ \end{align}\end{split}\]

and the last scattering law \(S_{d2}(\alpha,\beta)\) is convolved with \(S^{(1)}(\alpha,\beta)\),

\[\begin{split}\begin{align} S^{(2)}(\alpha,\beta) &= \int_{-\infty}^\infty S_{d2}(\alpha,\beta')~S^{(1)}(\alpha,\beta-\beta')~d\beta'\\ \end{align}\end{split}\]

and \(S^{(2)}(\alpha,\beta)\) is the final scattering law that combines these three components.