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Dynamical magnetic susceptibility

The dynamical magnetic susceptibility describes a system's linear response to a small inhomogeneous magnetic field with wave vector $\mathbf {Q}$, and energy $\hbar \omega$, consisting of real and imaginary parts represented as $\chi (\mathbf {Q} ,\omega )=\chi ^{\prime }(\mathbf {Q} ,\omega )+i\chi ^{\prime \prime }(\mathbf {Q} ,\omega )$.

In inelastic neutron scattering the imaginary part of the dynamical magnetic susceptibility is obtained from the differential cross section,

${\frac {d^{2}\sigma }{d\Omega dE_{f}}}={\frac {k_{f}}{k_{i}}}\left(1-e^{-\hbar \omega /k_{B}T}\right)^{-1}\chi ^{\prime \prime }(\mathbf {Q} ,\omega ),$

where $k_{f}$ and $k_{i}$ are the final and initial wave vectors, respectively. This is related to the dynamic spin correlation function $\chi ^{\prime \prime }(\mathbf {Q} ,\omega )=\left(1-e^{-\hbar \omega /k_{B}T}\right)S(\mathbf {Q} ,\omega )$, where $1-e^{-\hbar \omega /k_{B}T}$ is the Detailed balance factor.

Properties of the generalized susceptibility from the Kramers–Kronig relations:

$\chi (\mathbf {Q} ,-\omega )=\chi ^{\prime }(\mathbf {Q} ,\omega )-i\chi ^{\prime \prime }(\mathbf {Q} ,\omega )$

$\chi ^{\prime }(\mathbf {Q,0} )={\frac {1}{\pi }}\int _{-\infty }^{\infty }\chi ^{\prime \prime }(\mathbf {Q} ,\omega )\,{\frac {d\omega }{\omega }}$

This implies that one may extract the static susceptibility by integrating over the absorption spectrum in certain cases. For example, $\chi ^{\prime }(0,0)$ would be equivalent to the bulk susceptibility $\chi$, such as what would be measured with a SQUID magnetometer, in a paramagnetic system.

Expression for coherent one-phonon (acoustic) cross-section

The cross-section for an acoustic phonon can be written as

$\left({\frac {d^{2}\sigma }{d\Omega dE}}\right)_{coh}=N{\frac {k_{f}}{k_{i}}}\left|b_{ph}(q)\right|^{2}\delta (E-\varepsilon (q))$

The phonon scattering length is defined as

$\left|b_{ph}(q)\right|^{2}=2.09{\frac {q^{2}\cos ^{2}\beta }{M_{cell}\varepsilon (q)}}\left|F(q)\right|^{2}$

Resolution function (neutron scattering)

The resolution function $R(\mathbf {Q} ,\omega )$ is determined by the parameters related to instrument setup and is unique to each specific experimental configuration. The resolution function directly affects the width of a observed peak. Simple analytical techniques to estimate the resolution function are appropriate for simple double- or triple-axis instruments with steady-state neutron sources, but are impractical for more complicated configurations, such as time-of-flight instruments with choppers, curved monochromators, and/or pulsed neutron sources. For the more complex configurations, computational procedures, e.g. using a Monte Carlo technique or a matrix technique, are commonly employed.

The Gaussian approximation of the resolution function is given by

$R(\mathbf {X} )=R_{0}(2\pi )^{-3/2}(\det M)^{1/2}\mathrm {exp} \left(-{\frac {1}{2}}\mathbf {X} ^{T}M\mathbf {X} \right),$

where $M$ is a $4\times 4$ resolution matrix, $R_{0}$ is resolution volume (the optimal value of the resolution function), and $\mathbf {X} =\left[\mathbf {Q} ,\omega \right]$. Neutron scattering intensity as measured by a detector can be written as a convolution of the differential cross section and an instrument resolution function (neutron scattering).

$I(\mathbf {Q} ,\omega )=\int {{\frac {d^{2}\sigma }{d\Omega d\omega }}R(\mathbf {Q} ,\omega )\,d\mathbf {Q} d\omega }$

Addition for Dynamic structure factor

There is a useful sum rule for the dynamic structure factor, where by integrating over all energies in a single Brillouin Zone (BZ) one can obtain

$\int _{-\infty }^{\infty }{d\omega \,\int _{\mathrm {BZ} }{d\mathbf {Q} \,S(\mathbf {Q} ,\omega )}}=S\left(S+1\right),$

where $S=1/2$ is the spin, enabling quantitative analysis of the distribution of magnetic scattering.