iFit: Data Treatment procedures


  1. Neutron scattering Dynamic structure factor
    1. Loading/generating S(q,w) data sets
      1. From an experiment, processed data
      2. From an experiment,raw data
      3. From Matlab variables
      4. From a Model (iFunc)
      5. Shaping the Sqw data set
    2. Isotropic dynamic structure factors S(|q|,w) in liquids, powders, gas, polymers and other amorphous materials
      1. Example: dynamic structure factor from molecular dynamics
        1. From classical to quantum scattering law (quantum correction, detailed balance, Bose factor)
        2. Dynamic range accessible for a given neutron incident energy, structure factor, inelasticity correction
        3. Characteristic frequencies
        4. The density of states (aka phonon or frequency spectrum)
        5. The powder average
      2. Example: dynamic structure factor from a neutron scattering experiment
      3. References
  2. Neutron scattering cross sections for nuclear engineering
    1. Reading ENDF data sets
    2. Total scattering cross section, energy dependent
    3. Dynamic structure factor S(alpha, beta) for e.g. ENDF MF7 and ACE files
    4. Double differential cross section at a given angle and incident neutron energy

Commands we use in this page: iData, load, save, Math, Models, methods

Neutron scattering Dynamic structure factor

The iFit infrastructure comes with a set of dedicated procedures to read and analyze S(q,w) dynamic structure factors, aka scattering law, in the case of neutron scattering [5,6].

The dynamic structure factor is defined from the double differential scattering cross section per unit solid angle and final neutron energy as [5,6]:

d2σ/dΩdEf = Nσb /4π kf/ki S(q,w)

where N σb kf and ki are the number of scattering units, their bound scattering cross section [14], the final and initial neutron wave-vector, respectively.

You may as well generate 4D S(q,w) models using e.g. the sqw_phonons and other specialized models. Then you can compute the 2D powder average, as shown below.

Loading/generating S(q,w) data sets

In this section, we describe how one can load existing data sets, or generate ones.

A 2D S(|q|,w) consists in a momentum axis q (wavevector, usually in Angs-1), an energy axis w (usually in meV), and a dynamic structure factor 2D matrix S(q,w).

From an experiment, processed data

Such data sets can be measured on neutron scattering spectrometers (time-of-flight or triple-axis). Once the raw data has been reduced, using e.g. LAMP at the ILL:

  1. apply scripts normalise, vnorm, corr_tof, t2e, and finally sqw_rebin) ;
  2. you get a S(q,w) data set as a workspace ;
  3. export it into an HDF/NeXus file or as text (write_lamp, format='hdf').

Similar data sets can be measured using X-ray spectrometers.

The file can then be read by iFit, using any of

which can read HDF, text, NetCDF, and many other formats (see Loaders for a full description).

Common formats that can be read include McStas Isotropic Sqw, ISIS SPE, ILL ToF experiments, ENDF (see below).

From an experiment, raw data

Direct experimental data can be automatically converted to S(q,w) from any of (e.g. from ToF experiment):

The conversion takes place when calling the Sqw_check routine below.

From Matlab variables

If you managed to get a matrix S and two vectors q,w as Matlab variables, create an S(q,w) data set with:

From a Model (iFunc)

If S is an iFunc Model (analytical, such as obtained from the sqw_phonons), then simply evaluate the model into an iData object, with default Model parameters:

Shaping the Sqw data set

The data set may extend on both negative and positive energy transfers, or only on one side.

If it has only positive or negative energy values, then create the other side using the Sqw_symmetrize method, as detailed below.

You may as well apply the following iData operators:
and you should of course plot it, e.g. with:

Isotropic dynamic structure factors S(|q|,w) in liquids, powders, gas, polymers and other amorphous materials

The following procedures apply to 2D S(q,w), i.e. those that are obtained by averaging the S(q,w) over |q| in isotropic density materials.
In the following, S*(q,w) is a symmetric (classical) dynamic structure factor, whereas S(q,w) denotes the non-symmetric (quantum) dynamic structure factor which contains the population factor (e.g. Bose for phonons). See below for more details on this.

All routines below call the Sqw_check routine, which makes sure the S(q,w) data satisfies the format definition. This routine also makes, when needed, the conversion from an experimental to the S(q,w) formating.

Conventions:
If the 'positive' energy transfer side does not corresponds with the neutron loss side (sample gains energy from the neutron), then you should revert the energy axis with e.g.:
  • setaxis(Sqw, 1, -Sqw{1}) or equally Sqw{1} = - Sqw{1};
All routines assume the wavevector 'q' is given in [Angs-1] and the energy is given in [meV].

1 meV = 241.8 GHz = 11.604 K = 0.0965 kJ/mol = 8.0657 cm-1
1 Angs-1 = 10 nm-1

Procedure
Input
Output
Description
Sqw=load(iData,file)
Sqw=iData(file)
file
iData
Import any S(q,w) as generated by e.g. LAMP and nMoldyn/MDANSE.
Sqw_symmetrize(Sqw)
S*(|q|,w>0)
S*(|q|,w)=S*(|q|,-w)
Treatment_Sqw_symmetrizeExtend the S*(|q|,w) in both energy sides. The initial data set should better be 'classical'.
Sqw_Bosify(Sqw,T)
S*(|q|,w) S(|q|,w) Treatment_Sqw_Bosify.pngConvert a symmetric scattering law (aka classical, for instance from Molecular Dynamics) into a 'quantum' scattering law (non symmetric, for instance from neutron and X-ray inelastic scattering experiment)
Sqw_check(Sqw)
S(|q|,w) or S*(|q|,w) S(|q|,w) or S*(|q|,w) Check the S(q,w) data set. Possibly convert from experimental data S(φ,channel), S(φ,t) or S(φ,w).
Sqw_deBosify(Sqw,T)
S(|q|,w) S*(|q|,w) Convert a 'quantum' scattering law (non symmetric, for instance from neutron and X-ray inelastic scattering experiment) into a symmetric scattering law (aka classical, for instance from Molecular Dynamics)
trapz(Sqw)
Sqw_Sq(Sqw)
S(|q|,w) S(|q|)
Treatment_Sqw_Sq.pngCompute the static structure factor. The S(q) is the same when computed from the 'classical' and the 'quantum' scattering law.
Sqw_dynamic_range(Sqw,Ei)
S(|q|,w) or S*(|q|,w) measurable S(|q|,w) for Ei
Treatment_Sqw_dynrange.pngRestrict the dynamic structure factor to the measurable one for an incoming energy Ei [meV]. The detection angular range can be set as 3rd argument.
Sqw_moments(Sqw,M)
S(|q|,w) or S*(|q|,w) S(q), Er, characteristic frequencies
Treatment_Sqw_momentsCompute the energy moments, and provide some of the characteristic frequencies (recoil, isothermal, longitudinal/harmonic, mean energy transfer)
Sqw_gDOS(Sqw)
S(|q|,w) or S*(|q|,w) g(w)
Compute the generalized density of states (gDOS)Treatment_Sqw_gdos.png
Sqw_scatt_xs(Sqw,Ei) S(|q|,w) ∫∫S(q,w)q dq dw /2Ki2

Treatment_Sqw_XS.pngCompute the total thermal neutron scattering cross section which is the integral of S(q,w) over the dynamic range vs. incident neutron energy.
Sqw_Sab(Sqw, M)
S*(|q|,w) S*(alpha,beta)
Treatment_Sqw_SabConvert a dynamic structure factor to its unit-less representation S(α,β)
sqw_powder(Sqw)
S(hkl,w) 4D
S(|q|,w)
Treatment_Sqw_powder.pngConvert a 4D S(hkl,w) model or data set into a powder averaged S(|q|,w). This can be used e.g. after calling sqw_phonons.

Example: dynamic structure factor from molecular dynamics

In the following, we demonstrate the different treatment procedures, starting from an example data set obtained from a molecular dynamics simulation.

We import the data, obtained from an ab-initio Molecular Dynamics (AIMD) with VASP, for 200 Ge atoms (GGA-PW91 potential) at T=1350K, from [7]. The data set is part of iFit, as well as in McStas. You can also try liquid Rubidium from here [8] and heavy water from here [9]. This is the coherent symmetric scattering law:
Molecular Dynamics trajectories are usually computed in the NVE micro-canonical ensemble. The integrator used along the trajectory is the Verlet, which is a central difference, energy conserving integrator. As a consequence, the time evolution is symmetric (i.e. trajectories can be read in positive or negative time steps without changing the system behaviour). This implies that the derived intermediate scattering function F(q,t) is real, symmetric in time, and as a consequence S(q,w) is also real, symmetric in energy. We denote this scattering law S* to label it as symmetric.Treatment_Sqw_load It is often named 'classical'.

We plot the scattering law S*:
Then we need to symmetrize the data set, as it comes defined only in w>=0. This procedure applies to a 'classical' symmetric S*(q,w). If the scattering law you have already extends on both energy sides, you can skip this step (e.g. nMoldyn4/MDANSE).
Then we can plot the symmetrized scattering law, which extends equally in both w<0 and w>0.Treatment_Sqw_symmetrize
We can check here that:
S*(q,-w) = S*(q,w).

Note: if the initial S(q,w) is already normalised so that lim S(q → Inf) = 1 , then the resulting symmetrized data set should be divided by a factor 2.

From classical to quantum scattering law (quantum correction, detailed balance, Bose factor)
The next step is to get the temperature in.

A warning is in place here
. There is only one S(q,w), but there are infinite ways to derive a symmetrized S*(q,w). So, having obtained S* from Molecular Dynamics, we have to choose how to 'desymmetrize' it in order to obey the so-called 'detailed balance' accounting for the population of vibrational modes [5,6]:

S(q,w)  = exp(hw/kT) S(q,-w)

This step is performed by the Sqw_Bosify procedure, which can use 3 different 'quantum corrections' to go from a classical/symmetric S*(q,w) data set to a 'real/quantum' scattering law.

S(q,w)  = Q(w) S*(q,w) with S*=classical limit

The semi-classical correction, Q, aka 'quantum' correction factor,  can be selected as [1,2,34]:

Q = exp(hw/kT/2) 'Schofield' or 'Boltzmann' [1]
Q = hw_kT./(1-exp(-hw/kT))
'harmonic'  or 'Bader' [2]
Q = 2./(1+exp(-hw/kT))
'standard'  or 'Frommhold' [3]. This is the default.

where hw is the energy (in meV), and k is the Boltzmann constant.

Even though the 'Boltzmann' correction is the most natural correction, as it directly derives from the detailed balance, it is not recommended as it leads to a divergence of the S(q,w) for large energies e.g. above few 100 meV. The 'harmonic' correction is weaker but does not fully avoid the divergence at large energies. The 'standard' correction is probably the best when dealing with large neutron energies. In the 'classical' limit where hw << kT (and kT ~ 30 meV a 300K), all corrections are equivalent. All these corrections satisfy the detailed balance equally.

So, from the symmetric 'classical' S* we can now determine the 'true' S(q,w) scattering law, at a given temperature, using the 'standard' quantum correction.
Treatment_Sqw_dynrange.pngWhen not given, the temperature will be searched in the Sqw data set. A 3rd argument allows to specify the quantum correction to apply ('standard' is default). Of course, in this example, for such a high temperature, the up- and down-scattering are about the same (the quantum correction remains close to 1). For materials at low temperatures, the two sides are highly disymmetric.

The Sqw_deBosify procedure performs the opposite operation, that is compute a symmetric S*(q,w) from e.g. a 'quantum' experimental S(q,w). It corresponds with Sqw_Bosify(Sqw, -T).

Dynamic range accessible for a given neutron incident energy, structure factor, inelasticity correction
The dynamic range is computed by applying the momentum and energy conservation rules to the S(q,w):

Ef = Ei - w is positive
cos(θ) = (Ki2 + Kf2 - q2) / (2 Ki.Kf)  is within [-1:1]

where Ei and Ef are the incident and final neutron energies, θ is the scattering angle, Ki and Kf are the incident and final neutron wavevectors, defined as:

Ei = 2.0721*Ki2 = 81.8042/λ2 with Ki in [Angs-1] and λ in [Angs]

We can determine the dynamic range accessible to neutrons with a given incident energy, say Ei=25 meV.
The integrated scattering cross section is the integral of the dynamic range, for a given incident neutron energy (see below).

Treatment_Sqw_Sq.pngThe structure factor S(q) is the integral of S(q,w) over the energy. We use the usual 'trapz' integrator, or Sqw_Sq which does the same.

S(q) = ∫S(q,w) dq

In addition, the structure factor should converge to 1 at high momentum values:

lim S(q → Inf) = 1
 
We then recommend to plot the structure factor, and possibly adapt the S(q,w) value so that the high wavevector S(q) limit should tend to 1. This is simply accomplished by dividing the dynamic structure factor S(q,w) by the structure factor limit S(q → Inf).

It is here possible to compare the structure factor as measured with Ei=25 meV (i.e. restricted to the corresponding dynamic range) to the ideal one. The ratio is called the "inelasticity correction".
Treatment_Sqw_momentsCharacteristic frequencies
From the dynamic structure factor, one can compute the energy moments, which provide some of the characteristic frequencies [5,6,10], specifying the molar weight m of the scattering units (here 72.64 g/mol for Ge).

Using the 'quantum' scattering law S(q,w), we can define n-th order frequency moment as:

<wn S> = ∫ wn S(q,w) dw

and then derive:
These frequencies are computed with:
The moments can equally be computed from a 'classical' symmetric S*(q,w) and a 'quantum' experimental S(q,w).
The density of states (aka phonon or frequency spectrum)
The scattering law, from its definition [5,6], is a correlation function of the density operator, which itself depends on the position of the particles.
The vibrational density of states is defined as the velocity auto-correlation function (VACF) of the particles. It is not a function of the positions. So, in principle, it is NOT possible to compute the vibrational density of states from the scattering law.

However, a few attempts have been made to estimate the density of states from e.g. a neutron scattering time-of-flight experiment (such as done on IN4, IN5 and IN6 at the ILL). The concept of the generalised density of states has been defined as [12,13]:
gDOS(q,w) ~ S(q,w) w2/q2

which is valid in the case of an incoherent scatterer. Then, the energy dependent density of states is the low-wavevector limit of the generalised density of states, which is a normalized quantity:

g(w) ~ gDOS(q → 0,w)

∫g(w) dw = 1

From an experimental S(q,w) data set, the g(w) is obtained from the intensity in the low angle detector bank vs. the energy transfer.

As such, it should only be considered as an approximation of the vibrational density of states, especially for most materials which have coherent scattering. The hydrogenated materials and e.g Vanadium are good examples of materials which are mostly incoherent, and the approximation is then good. For other materials, great care should be taken.

We import the heavy water NAMD simulation (3900 molecules, 290K) coherent dynamic structure factor from here [9].
Then we can compute the generalised density of states with the Sqw_gDOS procedure, from both the raw, and the symmetrized, 'Bosified' (with quantum correction) data set.
Treatment_Sqw_Bosify.pngTreatment_Sqw_gdos.png
The powder average
Treatment_Sqw_powder.pngWhen a 4D S(q,w) model is created, using e.g. the sqw_phonons and other specialized models, one  can compute the 2D powder average, which is the projection of the 4D data set onto the (|q|,w). space.

However, this projection must take care of the crystal structure. When the reciprocal lattice information is found in the object (matrix B=[a* b* c*]), the conversion from 'rlu' (e.g. 2pi/a units) to 'Angs-1' (cartesian) is automatically done (and this information is stored in the Model upon creation):

Qcart = B*Qrlu

The the average is obtained using e.g.
	>> s=sqw_phonons([ ifitpath 'Data/POSCAR_Al'],'metal','EMT'); 
>> pow=sqw_powder(s); plot(log(pow));
Here we have generated a 4D data set from the sqw_phonons iFunc Model, and then reduced it to 2D.
The same procedure can be applied on an evaluated iData object out of the model.

Example: dynamic structure factor from a neutron scattering experiment

The symmetrization step is not needed if you import experimental S(q,w) as measured on inelastic neutron scattering spectrometers such as IN5@ILL. This type of data set already 'contains' the Bose factor accounting for the population of modes.

Then the sequence of operations is first to load the data set, then apply the Sqw_deBosify procedure, specifying the temperature if it is not defined in the data set. Then Sqw_symmetrize can be applied to derive a symmetric data set which extends in both w>0and w<0. The data close to the elastic w=0 line, which can be measured in up- and down- scattering, is computed as the mean of both sides.

Then, the rest of the data treatment is the same as above.

References

  1. P. Schofield. Phys. Rev. Lett., 4, 239 (1960).
  2. J. S. Bader and B. J. Berne. J. Chem. Phys., 100, 8359 (1994) ; see also T. D. Hone and G. A. Voth. J. Chem. Phys., 121, 6412 (2004).
  3. L. Frommhold. Collision-induced absorption in gases, 1 st ed., Cambridge Monographs on Atomic, Molecular, and Chemical Physics, Vol. 2, Cambridge Univ. Press: London (1993).
  4. B. Hehr, http://www.lib.ncsu.edu/resolver/1840.16/7422 PhD manuscript (2010).
  5. Helmut Schober, Journal of Neutron Research 17 (2014) pp. 109-357<http://dx.doi.org/10.3233/JNR-140016>
  6. G.L. Squires, Introduction to the Theory of Thermal Neutron Scattering, Dover Publications Inc.(1997)
  7. Hugouvieux V, Farhi E, Johnson MR, et al., PRB 75 (2007) 104208
  8. E. Farhi, V. Hugouvieux, M.R. Johnson, W. Kob, Journal of Computational Physics 228 (2009) 5251
  9. E. Farhi et al, J. Nucl. Sci. Tech. 52 (2015) 844; DOI: 10.1080/00223131.2014.984002
  10. J-P.Hansen and I.R.McDonald, Theory of simple liquids Academic Press New York 2006.
  11. NIST Neutron scattering lengths and cross sections <https://www.ncnr.nist.gov/resources/n-lengths/>; See also Sears, Neut. News 3 (1992) 26.
  12. Price J. et al, Non Cryst Sol 92 (1987) 153
  13. Bellisent-Funel et al, J. Mol. Struct. 250 (1991) 213
  14. Sears, Neutron News 3 (1992) 26

Neutron scattering cross sections for nuclear engineering

Reading ENDF data sets

iFit can read ENDF Thermal Scattering data sections (MF=7 MT=2 and 4), as well as ACE files (requires PyNE 0.5).

>> endf = iData('http://t2.lanl.gov/nis/data/data/ENDFB-VII-thermal/HinH2O')
endf = array [10 1] iData (methods,doc,plot,plot(log),more...) object:

Index [Tag] [Dimension] [Title] [Last command] [Label/DisplayName]
1 iD72088 [1 1] 'HinH2O.txt ENDF "Data Signal"' iD72088=load(iData,'... HinH2O.txt
2 iD72089 [274 187] 'H(H2O) T=293.6 [K] Thermal ... "S(alp..."' iD72089=load(iData,'... H(H2O) T=293.6 ...
3 iD72090 [274 187] 'H(H2O) T=350 [K] Thermal ne... "S(alp..."' iD72090=load(iData,'... H(H2O) T=350 [K...
4 iD72091 [274 187] 'H(H2O) T=400 [K] Thermal ne... "S(alp..."' iD72091=load(iData,'... H(H2O) T=400 [K...
5 iD72092 [274 187] 'H(H2O) T=450 [K] Thermal ne... "S(alp..."' iD72092=load(iData,'... H(H2O) T=450 [K...
6 iD72093 [274 187] 'H(H2O) T=500 [K] Thermal ne... "S(alp..."' iD72093=load(iData,'... H(H2O) T=500 [K...
7 iD72094 [274 187] 'H(H2O) T=550 [K] Thermal ne... "S(alp..."' iD72094=load(iData,'... H(H2O) T=550 [K...
8 iD72095 [274 187] 'H(H2O) T=600 [K] Thermal ne... "S(alp..."' iD72095=load(iData,'... H(H2O) T=600 [K...
9 iD72096 [274 187] 'H(H2O) T=650 [K] Thermal ne... "S(alp..."' iD72096=load(iData,'... H(H2O) T=650 [K...
10 iD72097 [274 187] 'H(H2O) T=800 [K] Thermal ne... "S(alp..."' iD72097=load(iData,'... H(H2O) T=800 [K...
>> subplot(log(endf(2:end)),'view2 tight')
The resulting data sets are S(alpha,beta) for each temperature. The general information section (MF1/MT451) is stored in all S(alpha,beta) data sets.

We recommend to make use of PyNE in order to read ENDF and ACE files.
PyNE can be installed under Ubuntu with commands:
% sudo apt-add-repository 'deb http://packages.mccode.org/debian stable main'
% sudo apt-get update
% sudo apt-get install pyne

Total scattering cross section, energy dependent

The energy dependent scattering cross section is computed in the cold-thermal range as the integral on the dynamic range vs. the incident energy:

σ(Ei) = ∫∫ d2σ/dΩdEf dΩdEf = ∫∫ Nσb /4π kf/ki S(q,w)  dq dw

where the integral is carried out on the available dynamic range, i.e. the (q,w) values which satisfy the conservation rules. By applying the Jacobian of the transformation (dΩdEf) → (dq,dw) we find that dΩ = q/(2π ki kf) dq and

σ(Ei) = σb∫∫S(q,w)q dq dw /2ki2

Treatment_Sqw_XS.pngwhere σb is the tabulated bound neutron cross section [Sears, Neutron News 3 (1992) 26] for thermal energies. The integral can be computed with the Sqw_scatt_xs procedure, for a given energy range, which should then be multiplied by the bound thermal scattering cross section σb.

In the following example, we compute the coherent scattering cross section, with the bound cross section such as 15.4 barns for the coherent scattering in D2O [10] (data here):
This quantity is mostly used to estimate the scattering power of any material, considered as an incoherent scatterer, such as in the neutron transport codes MCNP, GEANT4, TRIPOLI, FLUKA, OpenMC, TART, ...

The above equation asymptotically converges to σb. Indeed, S(q) = ∫S(q,w) dq → 1 for large q values, and the dynamic range for large energies Ei becomes rectangular (the neutron velocity is large so that the dynamic range slope at q=0 and q=2ki is nearly vertical) and extends from q=0 to q=2ki. Then the total scattering cross section simplifies as:

σ(Ei) → σb∫∫ q dq dw /2ki2
→ σb∫ q2/2 dw /2ki2 up to q=2ki
→ σb

However, for energies above the thermal-hot range (e.g. above 1-4 eV), the total cross section should converge towards the free neutron cross section σ(Ei) → σfree. The free cross section, with A being the element mass, is defined as:

σfree ~ A2/(A+1)2 σb

As most S(q,w) scattering law models in nuclear engineering include an e-WEi decaying factor (a kind of very weak/slow Debye-Waller damping, see NJOY/LEAPR), we may compute the constant W so that the total cross section converges to the free cross section at e.g. Ei=1eV (used in e.g. OpenMC), then we find

W  = 2/1000*(log(A)-log(A+1))
and the corrected total cross section is then
σ(Ei) e-WEi

This correction is applied below the 1eV threshold, and is kept constant above it.
Warning: this factor is NOT the Debye-Waller factor exp(-<u2>Q2) !

The scatterer mass can be input as 3rd argument to Sqw_scatt_xs. For a poly-atomic material, the effective mass is obtained by weighting the bound cross sections with the A2/(A+1)2 factors, e.g.:

r2 = ∑ (A/(A+1))2 * σb(A)) / ∑ σb(A)
M = r/(1-r)

where the sums are done for each atom of mass A in the material.

For instance, in the case of light water we have two H atoms (A=1 σb=80.2) and one O atom (A=16 σb=4.2), we compute
r2 = (2*(1/(1+1))^2*80.2+(18/(18+1))^2*4.2)/(2*80.2+4.2) = 0.26
M = r/(1-r) = 1.06 i.e. scattering is mostly the hydrogen one.

In the case of heavy water we have two D atoms (A=2 σb=7.64) and one O atom (A=16 σb=4.2), we compute
r2 = (2*(2/(2+1))^2*7.64+(18/(18+1))^2*4.2)/(2*7.64+4.2) = 0.54
M = r/(1-r) =2.79 i.e. scattering is mostly from deuterium, but with an oxygen contribution.

Dynamic structure factor S(alpha, beta) for e.g. ENDF MF7 and ACE files

A dynamic structure factor as seen above can be expressed in the (q,w) space, and then the dynamical features are not highly sensitive to the temperature, except close to phase transitions. However, for historical reasons, most neutron transport codes such as MCNP, use so-called symmetric S*(α,β) which are stored in section MF=7 of e.g. ENDF and related neutron data bases. These data sets are highly sensitive to the temperature, as the axes alpha and beta are.

The S*(α,β) is a representation of the dynamic structure factor using unit-less momentum and energy variables defined as:

S*(α,β) = kT S*(q,w) = kT exp(beta/2) S(q,w)
α= h2q2/2MkT = [Ei+Ef-2 μ √(Ei*Ef)]/A.kT
β = -hw/kT    = (Ef-Ei)/kT
A    = M/m
μ   = cos(θ) = (Ki2 + Kf2 - q2) / (2Ki.Kf)

Treatment_Sqw_SabThe variable change (q,w) → (alpha,beta) is done with the Sqw_Sab procedure, which takes as input the mass of the scatterer A and the Temperature. When not given, these quantities are searched in the original data set. The input data S*(q,w) set must be 'classical' that is either obtained from a molecular dynamics (may have to symmetrize it), or from an experiment after deBosify and symmetrization:
The S*(α,β) may then be exported to e.g. ENDF and ACE files. A scaling factor may be needed, depending on how the S(q,w) data set is normalised.

References:

Double differential cross section at a given angle and incident neutron energy

It is usual, in the neutron cross section data bases, to compare an evaluated double differential cross section (measured intensity) at a given angle, with an actual measurement.

This procedure is trivial once the S(q,w) has been obtained, by applying the Sqw_dynamic_range procedure for the selected angle. A scaling factor may be needed, depending on how the S(q,w) data set is normalised.

Treatment_Sqw_DDCSIn the following example, we use Molecular Dynamics data, symmetrize it, then apply the detailed balance to go from the 'classical' to the 'quantum' S(q,w). Finally, we extract the S(q,w) for a given incident energy Ei=154 meV and around 10 deg detector angle. And we plot the integral along momentum (axis 1) to show a spectrum.
The energy axis must then be reversed and the incident energy be added, here 154 meV, to cope with the definition of the energy transfer used in nuclear engineering.

References for water data:



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