Magnetic inelastic Scattering

Similar, the magnetic inelastic scattering is given by

	$$S_{\rm mag\perp}^{\rm inel,\alpha\beta}(\mathbf Q,\omega)$$	$=$	$$\left( \frac{ \gamma r_0}{2 \mu_B} \right)^2 \frac{1}{2\pi\hbar}\... ...-\infty}^{+\infty}dt e^{i\omega t} \frac{1}{N}\sum_{nn'} e^{-W_n(Q)- W_{n'}(Q)}$$	(259)
		$\times$	$$e^{-i\mathbf Q \cdot (\mathbf R_n-\mathbf R_{n'})} (\langle \hat ... ...thbf Q)\rangle_{T,H} \langle \hat M^{n'}_{\perp\beta}(\mathbf Q) \rangle_{T,H})$$

The total magnetic cross section is $4\pi (\gamma r_0)^2=4\pi\left(\frac{\hbar \gamma e^2}{mc^2}\right)^2 =3.65$ barn. In (275) the first term in the bracket corresponds to the total and the second term to the elastic scattering. If we split the index $n$ into basis and lattice part $n=(\ensuremath{\boldsymbol\ell},s)$ and compare equation (248), we see that the scattering function depends on the correlation function between the magnetisation operator $\hat \mathbf M_{\perp}(\mathbf Q)=\hat \mathbf M(\mathbf Q)-\mathbf Q (\hat \mathbf M_{\perp}(\mathbf Q) \cdot \mathbf Q)/Q^2$, which is the observable in the case of magnetic neutron scattering ( $\mathcal O \leftrightarrow \frac{ \gamma r_0}{2 \mu_B} e^{-W(Q)} \hat \mathbf M_{\perp}(\mathbf Q)$).

  $$S_{\rm mag\perp}^{\rm inel,\alpha\beta}(\mathbf Q,\omega)= \sum_{ss'} \frac{{\Sigma^{ss'}_{\alpha\beta}}({\mathbf Q},\omega)}{2\pi \hbar N_b}$$ (260)

$W_s(Q)$ is the Debye-Waller factor of the atom number $s$ in the magnetic unit cell²⁸. $N_b$ denotes the number of magnetic atoms in the magnetic unit cell. Therefore, if the generalised eigenvalue problem (245) for the dynamical matrix has been solved, the magnetic neutron scattering function can be evaluated with the help of equations (249) and (257):

	$$S_{\rm mag\perp}^{\rm inel,\alpha\beta}(\mathbf Q,\omega)$$	$=$	$$\sum_{r,ss'} \frac{(\sqrt{\Gamma_{\rm mag}^s(\mathbf Q)})^\ast\sq... ...{\rm mag}^{s'}(\mathbf Q)} } {N_b(1-e^{-\hbar{\omega^r}(\mathbf Q)/kT})} \times$$	(261)
			$$\times \mathcal V^s_{{\rm mag},\alpha1}(\mathbf Q) { \mathcal T^{... ...cal T^{rs'\dag }}(\mathbf Q) \mathcal V^{s'\dag }_{{\rm mag},1\beta}(\mathbf Q)$$

Once the eigenvectors $\underline{\mathcal T}$ of the system have been determined, this expression can be evaluated. mcdisp evaluates for every mode the expression (277) 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. In addition, the components of the magnetic scattering function $S_{\rm mag}^{\rm inel,\alpha\beta}(\mathbf Q,\omega)$ can be output (to be used to interpret polarised magnetic neutron scattering).$\alpha,\beta$ then refer to either the xyz coordinate system ($y\vert\vert b$, $z\vert\vert(a \times b)$ and $x$ perpendicular to $y$ and $z$) or the uvw coordinate system ( $\mathbf u\vert\vert\mathbf Q=\mathbf k- \mathbf k'$,$\mathbf w$ perpendicular to the scattering plane (as determined by the cross product of subsequent vectors in the input q-vector list of mcdisp) and $\mathbf v$ perpendicular to $\mathbf u$ and $\mathbf w$, such that uvw for a righthanded system).

Form factor effects on the scattering intensity are due to the $\mathbf Q$-dependence of the magnetisation operator, which means that the transformation matrices $\overline{\mathcal V}^s$ and the eigenvalues $\Gamma^s$ are also $\mathbf Q$-dependent. These quantities have to be calculated by evaluating the transition matrix elements²⁹ of $\hat \mathbf M_{\perp}(\mathbf Q)$ for every scattering vector $\mathbf Q$ and diagonalising the matrix (253) with $\mathcal O \leftrightarrow \hat \mathbf M_{\perp}(\mathbf Q)$. For small $Q$ this procedure can be simplified by using the dipole approximation, which is described below.