User:Pseudocubic/sandbox

= 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.

= References =