**Appendix A**

Time-of-Flight neutrons instruments are a class of spectrometers which allows measuring inelastic neutron scattering, providing insights into the dynamics of matter. In **Figure 14**, we show, as an example, BRISP, a direct geometry Brillouin spectrometer once installed in the reactor hall of the High Flux Reactor of ILL. The neutrons scattered by the sample are collected by a highly pixeled detector covering a certain angular range. The scattering angle 2Θ is defined as the angle between the direct beam axis and the direction of the scattered neutrons (**Figure 15**). A Fermi chopper device splits the continuous beam coming from the monochromator into 10 *μs* bursts of neutrons and fixes for each burst an initial reference time. The wavevector **k0**, hence the energy *E*<sup>0</sup> of the neutrons impinging on the sample, are known and so is the time *t*<sup>0</sup> such incident neutrons take to fly from the reference initial time to the sample position. The detector electronics suitably synchronized with the chopper provides us a measure of the total time-of-flight *ttof* from the reference time to a

### **Figure 14.**

*Sketch of the Brillouin spectrometer BRISP. A monochromatized continuous beam, severely collimated is converted in a pulsed beam by a Fermi chopper which labels each pulse with an initial reference time as the starter in a running race. Once the neutrons interact with the sample they are scattered and finally collected by 2D-pixeled detector. Reproduced from ref. [28].*

### **Figure 15.**

*Sketch of the kinematic scattering triangle. Incident neutrons characterized by a wavevector* **k0** *are scattered by a sample with a wavevector* **k1***. The angle* 2Θ *between the incident and the scattered radiation is the scattering angle. The vector* **Q***=***k0***-***k1** *is the transferred momentum in the scattering process.*

*Bayesian Inference as a Tool to Optimize Spectral Acquisition in Scattering Experiments DOI: http://dx.doi.org/10.5772/intechopen.103850*

specific pixel and of course the position and hence the distance traveled by each scattered neutron. If we call *t*<sup>1</sup> the time the scattered neutron takes to fly from an interacting atom in the sample to a specific detector pixel we have that:

$$t\_{\text{tof}} = t\_0 + t\_1 = \frac{L\_0}{\nu\_0} + \frac{L\_1}{\nu\_1} \tag{10}$$

where *L*0, *L*<sup>1</sup> are the distances between the chopper and the sample and between the sample and the detector, respectively, and *v*<sup>0</sup> and *v*<sup>1</sup> are the initial and final velocities of the incident and scattered neutron.

From Eq. (10) it is straightforward to obtain the energy *E*<sup>1</sup> with which the neutron reaches the detector and then the energy *E* ¼ ℏ*ω* transferred from the probe particle to the sample. It is, in fact:

$$E\_1 = \frac{1}{2} m v\_1^2 = \frac{1}{2} m \left(\frac{L\_1}{t\_{\text{tof}} - \frac{L\_0}{t\_0}}\right)^2 = \frac{1}{2} m \left(\frac{L\_1}{t\_{\text{tof}} - L\_0 \sqrt{\frac{m}{2E\_0}}}\right)^2 \tag{11}$$

and

$$E = E\_1 - E\_0 = \hbar \alpha \tag{12}$$
