# Incoherent inelastic neutron scattering -- Vanadium

## Experimental data

The following is a typical experiment I(Q,E) data for a Vanadium plate
measured at the SNS ARCS insturment.

## Scattering Cross Section

Incoherent inelastic neutron scattering spectra are easiest to
interpret.
The coherent neutron scattering length of Vanadium is far less than
its incoherent neutron scattering length, making it an ideal example
for illustrating incoherent inelastic scattering, and a standard
calibration sample for neutron experiments.

The one-phonon incoherent inelastic neutron scattering cross section
for a single-element material is given by

$$
\left( \frac{d^2\sigma}{d\Omega dE_f} \right) _{\pm 1} =
\frac{\sigma_{inc}}{4\pi} \frac{k_f}{k_i} N \; \frac{\hbar^2Q^2}{2M} \exp(-2W)
\frac{Z(E)}{E}
\left \{ \coth \left( \frac{E}{2k_B T} \right) \pm 1 \right \}/2
$$

where $k_i$, $k_f$ is the momentum of the incident and outgoing neutron,
respectively,
$Q$ is the momentum transfer, $E$ is the energy transfer,
$\left\{ \coth \left(\frac{E}{2k_B T} \right) \pm 1 \right\}/2$ is the thermal factor,
and
$Z(E)$ is the phonon density of states as a function of phonon
energy $E$.

### Q-dependency

One can see the (Q,E) distribution solly depends on
the phonon Density of States (DOS) of the sample.
This property makes the spectra easy to understand and interpret.
Let us take a more careful look at the various factors in this
formula.

The factor $2W$ is the Debye-Waller factor
$$
2W = \frac{\hbar^2 Q^2}{2M} \int \frac{1}{E}
\coth(\frac{1}{2}\beta E) Z(E) dE
$$

which can be rewritten as

$$
2W = C_{DW} Q^2
$$
and
$$
C_{DW} = \frac{\hbar^2}{2M} \int \frac{1}{E}
\coth(\frac{1}{2}\beta E) Z(E) dE
$$

The $Q$ dependency of the spectrum is solely determined by the term

$$ \frac{\hbar^2 Q^2}{2M} \exp(-C_{DW} Q^2) $$

The following is an interactive plot of I(Q)
in which you can play with the $C_{DW}$ factor.

### E-dependency

is governed by
$$
\frac{Z(E)}{E}
\left \{ \coth \left( \frac{E}{2k_B T} \right) \pm 1 \right \}/2
$$
Both
the thermal factor
$ \left \{ \coth \left( \frac{E}{2k_B T} \right)\pm 1 \right \}/2$
and density of states $Z(E)$ are easy to understand.
The additional $\frac{1}{E}$ factor comes from the amplitude of the
vibration:
$$
\hbar \omega \sim m \omega^2 \langle u^2 \rangle
$$
and
$$
\langle u^2 \rangle \sim 1/\omega \sim 1/E
$$
The following is an interactive plot of the thermal factor.
You can play with the temperature slider on the right to see how
detailed balance change with temperature.

One observation of this plot is that the thermal factor diverges
near $E=0$.
Indeed, it diverges by $1/E$ near $E=0$.
Considering the additional $1/E$ factor from the amplitude of
vibration,
the scattering intensity would have diverged by $1/E^2$
without considering the behavior of the density of states.
Actually, for 3D systems, the behavior of DOS near $E=0$ goes with
$$ Z(E) \sim E^2 $$
and the one-phonon scattering intensity should approach a constant at
$E=0$.

### Modeled IQE

As shown above, the $Q$ and $E$ dependencies of scattering intensities for incoherent
phonon scattering are separable, and it is easy to model the full
$I(Q,E)$ given Density of States:

### Dark angle

### Multi-phonon scattering

## MCViNE simulation results

MCViNE simulation automatically takes care of all the factors
presented above.
In addition, it also takes care of factors such as instrument resolution,
dynamical range, sample geometry, and multiple scattering.