Coherent Nuclear Inelastic Scattering - 1 Phonon processes

The coherent inelastic nuclear scattering (by phonons) is given by

$$S_{\rm nuc}^{\rm inel}(\mathbf Q,\omega) = \frac{1}{2\pi\hbar}\int_{-\infty}^{+\infty}dt e^{i\omega t} \frac{1}{N}\sum_{nn'} b_n b_{n'} e^{-W_n(Q)- W_{n'}(Q)}$$ (249)

$$\times e^{-i\mathbf Q \cdot (\mathbf R_n-\mathbf R_{n'})} (\langle \math... ...\hat {\mathbf u}^{n}(t) {\mathbf Q} \cdot \hat{\mathbf u}^{n'}(0) \rangle_{T,H}$$

- here $b$ denotes the nuclear coherent scattering length and $\hat {\mathbf u}$ the displacement. If we split the index $n$ into basis and lattice part $n=(\ensuremath{\boldsymbol\ell},s)$ and compare equation (238), we see that the nucler scattering function depends on the correlation function between the inner product of scattering vector and displacement operator $\mathbf Q \cdot \hat \mathbf u$, which is the observable in the case of coherent nuclear inelastic neutron scattering ( $\mathcal O \leftrightarrow b_s e^{-W_s(Q)} \mathbf Q \cdot\hat \mathbf u^s$).

$$S_{\rm nuc}^{\rm inel}(\mathbf Q,\omega) = \sum_{ss'} \frac{{\Sigma^{ss'}}({\mathbf Q},\omega)}{2\pi \hbar N_b}$$ (250)

$W_s(Q)$ is the Debye-Waller factor of the atom number $s$ in the unit cell. $N_b$ denotes the number of magnetic atoms in the magnetic unit cell. Therefore, if the generalised eigenvalue problem (235) for the dynamical matrix has been solved, the nuclear neutron scattering function can be evaluated with the help of equations (239) and (247):

$$S_{\rm nuc}^{\rm inel}(\mathbf Q,\omega) = \sum_{r,ss'} \frac{(\sqrt{\Gamma_{\rm nuc}^s(\mathbf Q)})^\ast\sq... ..._{\rm nuc}^{s'}(\mathbf Q)}} {N_b(1-e^{-\hbar{\omega^r}(\mathbf Q)/kT})} \times$$ (251)

$$\times \mathcal V^s_{{\rm nuc},\alpha1}(\mathbf Q) { \mathcal T^{... ...cal T^{rs'\dag }}(\mathbf Q) \mathcal V^{s'\dag }_{{\rm nuc},1\beta}(\mathbf Q)$$

Once the eigenvectors $\underline{\mathcal T}$ of the system have been determined, this expression can be evaluated. mcdisp evaluates for every mode the expression (264) with exception of the $\delta$-function and multiplies it by $k'/k$ in order to get the nuclear Intensity $I_{\rm nuc}$ in barns/meV formula unit.