Euphonic Calculations: Overtones and Combinations (incoherent semi-analytic)

[ajjackson]

To expand a little on what this is, what requirements it creates and where it might fit in:

This refers to the "semi-empirical" method implemented in a-CLIMAX and AbINS, as well as o-CLIMAX. While o-CLIMAX also provides some other techniques, this is the only method currently implemented in AbINS, presumably as replacing a-CLIMAX was the initial top priority.

The method applies to incoherent scattering in a powder of semi-isotropic oscillators. This combination of approximations allows an analytical solution to be applied to inexpensively obtain the scattering function at a given q (or ω-q relationship) from a set of frequencies and eigenvectors.

It is also possible to analytically obtain contributions to S from higher-quantum-order events: overtones and combination modes. As the expressions would become increasingly complicated at higher orders, additional approximations seem to be applied in practice:

• the second-order term in a-CLIMAX and AbINS makes an isotropic approximation in the Debye-Waller factor;
• the third- and fourth-order terms seem to also make an isotropic approximation in the calculation of S from eigenvectors, but I have not yet found an explicit derivation/justification of this or a statement of exactly what is being assumed. As the method is built on an incoherent approximation, contributions are computed for each atom individually and then summed to create a spectrum at each quantum order.

An empirical energy-dependent broadening relationship is then applied in order to obtain a simulated spectrum for an indirect instrument.

So in principle there are two main functions to implement (with supporting machinery):

• a function which takes energies and eigenvectors as input and returns atom-resolved S(q,ω) for each supported quantum order
• a function that takes S(q,ω) and an empirical σ(ω) relationship and returns a broadened S(q,ω) I think these can be separate objectives and separate code modules.

An important distinction between a typical a-CLIMAX/Abins calculation for TOSCA and the general model in PACE is that the data is not really a 4-D S(q,ω); it is just S(ω) with a (scalar) q(ω) relationship determined by the instrument geometry. So ideally the data is passed around as a pair of 1D (frequency, intensity) where appropriate; requiring a corresponding 1D q vector would not be too onerous, but it should be possible to treat q as a scalar as we have no directional information.

[ajjackson]

It is not clear at this point whether broadening data on a regular grid is in Euphonic's scope; I believe this is currently handled in Horace? If it is decided that Euphonic does not deal with regular grids, then it does not make sense to include the second function (and broadening machinery) in Euphonic.

In that case the objectives of this Issue become:

• [ ] #67 Create a function/class which takes energies and eigenvectors as input, with options determining the extent of overtone/combination calculations, and implements the analytic powder-averaged "almost-anisotropic" incoherent spectrum as implemented in Abins/a-climax
• [ ] #68 Refactor Abins/Mantid to include a copy of Euphonic and access this code rather than duplicate the functionality. (This is not strictly a blocker for elements of Euphonic that rely on this feature. Nonetheless it could be a good idea to treat it as one, in order to avoid diverging codebases.)

Broadening-related work can be added as seperate issues if/when determined to be in-scope.