ADR: DetailedBalance

[henrikjacobsenfys]

Detailed balancing is discussed in detail in easyscience/dynamics#6

Some of the information is pasted here for simplicity.

General

The core is simple: it is easier to create excitations than to absorb them at low temperature, and therefore, the scattering intensity is vanishing on the neutron energy gain/sample energy loss (E<0) side of the spectrum.
We define:

• $\omega$ is the energy transfer,
• $\beta = 1 / (k_B T)$ is the inverse temperature,
• $n(\omega) = \frac{1}{e^{\beta \omega}-1}$ is the Bose occupation factor.
• $n(\omega)+1 = \frac{1}{1 - e^{-\beta \omega}}$ is a useful relation to know

I have set $\hbar=1$ for simplicity.

Mathematically, detailed balancing is captured by the following relation of the scattering function (also known as the dynamical structure factor)

$$S(-Q,\omega)=\exp(-\beta \omega) S (Q,\omega).$$

It is related to the generalized susceptibiity $\chi''(\omega)$ by

$$S(\omega) = \frac{1}{\pi} \cdot (n(\omega)+1) \cdot \chi''(\omega).$$

Another way to write this important relation is (notice the difference between $\chi'$ and $\chi''$ )

$$S(\omega) = \omega \cdot (n(\omega)+1) \cdot \chi'(Q,\omega=0)\cdot F(\omega),$$

where $F(\omega)$ is the spectral weight function, which has area 1. More on that in a moment. Importantly, $\chi'(Q,\omega=0)$ does not depend on $\omega$.

$\chi'$ and $\chi''$ is typically what is calculated theoretically, while $S$ is typically what is measured. When fitting, we usually allow fitting of the area of the peak in the case DBF=1 (usually in arbitrary units), and is proportional to $\chi'(Q,\omega=0)$.

In other words, the fit function, if the data is $S(Q,\omega)$, is

$$I=DBF\cdot A \cdot F(\omega),$$

where
$DBF=\omega*(n(\omega)+1)$ is called the Detailed Balance Factor
$A$ is the "area" of the peak (or would be, if DBF=1)
$F$ is the shape of the peak, usually a Lorentzian, Gaussian, pairs of Lorentzians or Gaussians, or Damped Harmonic Oscillator

It should be noted that detailed balancing often isn't included, especially in analysis of diffusion. This is sometimes because the measurements are at high enough temperatures that it doesn't matter. Sometimes, it is not included because detailed balancing is, by definition, an equilibrium property, and scattering related to diffusion is not necessarily at equilibrium.

Implementation

Some users normalize the detailed balancing factor by temperature. This is supported in the implementation.

If users provide temperature in their SampleModel then detailed balancing will be applied as described above. The users will therefore for most purposes not need to interact with detailed balancing directly.

Care has been taken to ensure the correct behaviour at low and hig temperature, using the appropriate limiting forms of the equations.