Application to Neutron scattering

We now apply the theory outlined above to the inelastic scattering of neutrons in a crystalline solid. The observables measured here are the atomic displacement (for phonons) and the Fourier transform of the magnetisation (=magnetic moment density) operator of a magnetic ion ( $\hat \mathbf M(\mathbf Q)$). The magnetisation operator consists of a spin and an orbital contribution and thus its Fourier transform may be written as a sum $\hat \mathbf M(\mathbf Q)=\hat \mathbf M_S(\mathbf Q)+\hat \mathbf M_L(\mathbf Q)$. For consistency with ref.[30] please note that we write the following formulas in terms of the magnetisation operator instead of the scattering operator $\hat \mathcal Q_{\alpha} \equiv -\hat M_{\alpha}(\mathbf Q)/(2\mu_B)$ (where $\alpha=x,y,z$ and $\mu_B$ is the Bohr magneton).

The double differential neutron scattering cross section is usually being evaluated base on the master equation (see e.g. [30])

  $$\frac{d^2\sigma}{d\Omega dE'}=\frac{k'}{k}\left( \frac{m_n}{2\pi ... ... Q)\vert f\rangle\vert s_n'\rangle \right \vert^2 \delta(\hbar \omega+E_i-E_f)$$ (252)

Here $m_n$ is the neutron mass, $\mathbf k$ and $\mathbf k'$ the incoming and scattered neutron wave vector, $\vert s_n\rangle$ and $\vert s_n'\rangle$ the spin state of the incoming and scattered neutron, $P_{s_n}$ the polarisation of the incoming neutron beam (i.e. the probability for the neutron spin state $\vert s_n>$ in the incoming neutron beam). $\vert i\rangle$ and $\vert f\rangle$ denote the initial and final states of the target, $E_i$ and $E_f$ the corresponding energies, $P_i$ the population number of the target state $\vert i\rangle$. Finally $H_{int}(\mathbf Q)$ is the interaction operator which consists of neutron spin dependent and spin independent parts

  $$H_{int}(\mathbf Q)=\hat \beta (\mathbf Q) + \hat {\mathbf s}_n \cdot \hat {\boldsymbol \alpha } (\mathbf Q)$$ (253)

Here $\hat \beta$ contains the nuclear spin independent scattering and $\hat {\boldsymbol \alpha }$ the nuclear spin dependent and the electronic magnetic scattering operators. The accurate evaluation of (268) involves the computation of terms $\hat \beta^{\dagger} \dots \hat \beta$ (nuclear scattering, phonons), $\hat \alpha^{\dagger} \dots \hat \alpha$ (electronic magnetic scattering and nuclear magnetic scattering) and mixed terms $\alpha^{\dagger} \dots \beta$ (interference terms, not included in mcdisp currently, maybe nonzero only for polarized beams). We do not take into account any preferred occupancy of nuclear spin states in the sample, either. Then the neutron intensity may be separated into nuclear (phonon) and magnetic intensity. The corresponding expression for the double differential scattering cross section for unpolarised neutrons has been given frequently in literature (see e.g. [30]):

	$$\frac{d^2\sigma}{d\Omega dE'}$$	$=$	$$N\frac{k'}{k}S_{\rm nuc}(\mathbf Q,\omega) + N\frac{k'}{k} Tr\{S_{\rm mag\perp}(\mathbf Q,\omega)\}$$
	$$S_{\rm nuc}$$	$=$	$$S_{\rm nuc}^{\rm el}+S_{\rm nuc}^{\rm inel}$$	(254)
	$$S_{\rm mag}$$	$=$	$$S_{\rm mag}^{\rm el}+S_{\rm mag}^{\rm inel}$$	(255)

In (270) $N$ denotes the number of magnetic atoms in the sample, $\mathbf k$ and $\mathbf k'$ the wave vector of the incoming and scattered neutron, respectively. $\hbar \omega=E-E'$ and $\mathbf Q=\mathbf k-\mathbf k'$ denote the energy and momentum transfer. The van Hove scattering functions $S_{\rm mag}$ and $S_{\rm nuc}$ may be split into an elastic and an inelastic part. We will discuss here only the inelastic scattering.