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ucph>Tommy at 21:01, 16 July 2019

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In this section we will specialize the master equation for magnetic scattering to describe spin waves. We start with repeating [[Magnetic neutron scattering#label-eq:magnetic_master_final|the magnetic scattering master equation]] from the [[Magnetic neutron scattering]] page:

\begin{align}\label{dummy1597252741}
\dfrac{d^2\sigma}{d\Omega dE_{\rm f}} &=
\left(\gamma r_0 \right)^2 \dfrac{k_{\rm f}}{k_{\rm i}} \left[ \dfrac{g}{2} F(q)\right]^2
\displaystyle\sum_{\alpha \beta} \left( \delta_{\alpha\beta}-\hat{ {\mathbf q}}_\alpha\hat{ {\mathbf q}}_\beta\right) \\
&\quad \times \displaystyle\sum_{\lambda_{\rm i} \lambda_{\rm f}} p_{\lambda_{\rm i}}
\displaystyle\sum_{j,j'}
\left\langle \lambda_{\rm i}| \exp(-i {\mathbf q} \cdot {\mathbf r}_{j}) {\mathbf s}_{j}^\alpha
| \lambda_{\rm f}\right\rangle
\left\langle \lambda_{\rm f}\left| \exp(i {\mathbf q} \cdot {\mathbf r}_{j'}) {\mathbf s}_{j'}^\beta
\right| \lambda_{\rm i}\right\rangle \nonumber\\
&\quad \times
\delta\left( \hbar\omega + E_{\lambda_{\rm i}}-E_{\lambda_{\rm f}}\right) . \nonumber
\end{align}

The next step closely follows the treatment of phonon scattering. First, we rewrite the energy-conserving \(\delta\)-function in terms of a time integral. To repeat [[Scattering from lattice vibrations#label-eq:delta_hw|the equation]] from the [[Scattering from lattice vibrations]] page:

\begin{equation}\label{eq:delta_transform}
\delta(\hbar\omega+E_{\lambda_{\rm i}}-E_{\lambda_{\rm f}})
= \dfrac{1}{2\pi\hbar} \displaystyle\int_{-\infty}^{\infty} dt
\exp\left(-\dfrac{i(\hbar\omega+E_{\lambda_{\rm i}}-E_{\lambda_{\rm f}})t}{\hbar}\right) .
\end{equation}

Subsequently, we transform the stationary operators, here denoted \(\hat{A}\), to time-dependent Heisenberg operators, \(\hat{A}(t)\), through

\begin{align}\label{dummy1091740695}
&\biggr\langle \lambda_{\rm i} \biggr| \exp\biggr( \dfrac{i E_{\lambda_{\rm i}} t}{\hbar} \biggr)
\hat{A} \exp\biggr( -\dfrac{i E_{\lambda_{\rm f}} t}{\hbar} \biggr) \biggr| \lambda_{\rm f} \biggr\rangle \\
&\quad= \biggr\langle \lambda_{\rm i} \biggr| \exp\biggr( \dfrac{i \hat{H}_\lambda t}{\hbar} \biggr)
\hat{A} \exp\biggr( -\dfrac{i \hat{H}_\lambda t}{\hbar} \biggr) \biggr| \lambda_{\rm f} \biggr\rangle \nonumber\\
&\quad= \langle \lambda_{\rm i} | \hat{A}(t) | \lambda_{\rm f} \rangle .\nonumber
\end{align}

Making this transformation and performing the closure sum over the final states, (\(\sum_{\lambda_{\rm f}}|\lambda_{\rm f}\rangle\langle\lambda_{\rm f}| = 1\)), we reach the equation for the magnetic scattering cross section in the Heisenberg picture<ref name="marshall82">W. Marshall and S.W. Lovesey. ''Theory of Thermal Neutron Scattering'' (Oxford, 1971), equation (8.2)</ref>:

\begin{align}\label{dummy72416411}
\dfrac{d^2\sigma}{d\Omega dE_{\rm f}} &=
\left(\gamma r_0 \right)^2 \dfrac{k_{\rm f}}{k_{\rm i}} \left[ \dfrac{g}{2} F(q)\right]^2
\displaystyle\sum_{\alpha \beta} \left( \delta_{\alpha\beta}-\hat{q}_\alpha\hat{q}_\beta\right)
\\
&\quad\times \dfrac{1}{2\pi\hbar} \displaystyle\sum_{j,j'} \displaystyle\int_{-\infty}^{\infty} dt
\exp(-i \omega t) \nonumber\\
&\quad\times \left\langle \exp(-i {\mathbf q} \cdot {\mathbf r}_{j}(0)) {\mathbf s}_{j}^\alpha(0)
\exp(i {\mathbf q} \cdot {\mathbf r}_{j'}(t)) {\mathbf s}_{j'}^\beta(t) \right\rangle
, \nonumber
\end{align}

where \(\langle \hat{A} \rangle\) means the thermal average of the expectation value of an operator \(\hat{A}\), in the same way as the calculations leading to the [[Scattering from lattice vibrations#label-eq:sum_lambda_f|phonon expression]] on the [[Scattering from lattice vibrations]] page.

The nuclear positions \({\bf r}(t)\) and the atomic spin \({\bf s}(t)\) are both operators. Each of these give rise to both an elastic and an inelastic contribution; in total the cross section becomes a sum of four terms. The one that is elastic in both channels is the magnetic diffraction signal, while the contribution that is elastic in the nuclear positions (the phonon channel) and inelastic in the spins (the magnetic channel) is the true magnetic inelastic signal. The two other contributions involve creation/annihilation of phonons via the magnetic interactions, and are not discussed further here; we refer to Ref. <ref name="marshall810">W. Marshall and S.W. Lovesey. ''Theory of Thermal Neutron Scattering'' (Oxford, 1971), equation (8.10)</ref>.

The two contributions that are elastic in the phonon channel are reached in analogy with the page [[Scattering from lattice vibrations]]. We replace the general nuclear position operators \({\bf r}_j\) with the nuclear equilibrium positions in a lattice and utilize the periodicity of the lattice. Furthermore, we multiply with the Debye-Waller factor, \(\exp(-2W)\)<ref name="marshall813825">W. Marshall and S.W. Lovesey. ''Theory of Thermal Neutron Scattering'' (Oxford, 1971), equation (8.13) and (8.25)</ref>:

\begin{align} \label{eq:magn_cross_master}
\left(\dfrac{d^2\sigma}{d\Omega dE_{\rm f}}\right)_{\rm magn.} &=
\left(\gamma r_0 \right)^2 \dfrac{k_{\rm f}}{k_{\rm i}} \left[ \dfrac{g}{2} F(q)\right]^2
\exp(-2W)
\displaystyle\sum_{\alpha \beta} \left( \delta_{\alpha\beta}-\hat{ {\mathbf q}}_\alpha\hat{ {\mathbf q}}_\beta\right)
\\
&\quad\times
N \dfrac{1}{2\pi\hbar} \displaystyle\int_{-\infty}^{\infty} dt \exp(-i \omega t)
\displaystyle\sum_{j'} \exp(i {\mathbf q} \cdot ({\mathbf r}_j))
\left\langle {\mathbf s}_{0}^\alpha(0) {\mathbf s}_{j'}^\beta(t) \right\rangle . \nonumber
\end{align}

This equation should be understood as the space and time Fourier transform of the spin-spin correlation function, and it is the starting point for most calculations of magnetic scattering cross sections.

<references/>

Inelastic magnetic neutron scattering - Revision history

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ucph>Tommy at 21:01, 16 July 2019