Neutron
scattering Dynamic structure factor
The iFit infrastructure comes with a set of dedicated objects
and methods to read and analyze S(q,w) dynamic structure
factors, aka scattering laws, 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] (here for a monoatomic material):
d^{2}σ/dΩdEf = Nσ_{b} /4π
kf/ki S(q,w)
where N σ
_{b} kf and ki are the number of scattering
units, their bound scattering cross section [14], the final and
initial neutron wavevector, 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.
It is also possible to generate 2D S(q,w) models from analytical
expressions, or a density of states using the socalled
incoherent Gaussian approximation.
The iFit objects which are handled below are:
 the generic iData data set
 the S(q,w) 2D iData_Sqw2D
flavour, data set
 the S(alpha,beta) 2D iData_Sab
for nuclear data sets (thermal scattering laws)
 the g(w) 1D iData_vDOS
representing a vibrational density of states
 the generic iFunc model
 the S(q,w) 2D model iFunc_Sqw2D for e.g. liquids and
powders
 the S(q,w) 4D model iFunc_Sqw4D
from e.g. phonons and spinwaves in singlecrystals
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 Angs1), 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 (timeofflight or tripleaxis). Once the raw
data has been reduced, using e.g. LAMP
at the ILL:
 apply scripts normalise, vnorm, corr_tof, t2e, and
finally sqw_rebin) ;
 you get a S(q,w) data set as a workspace ;
 export it into a HDF/NeXus file or as text (write_lamp,
format='hdf').
Similar data sets can be measured using Xray spectrometers.
The file can then be read by iFit, using any of
 Sqw=load(iData, 'file')
 Sqw=iData('file')
which can read HDF, (for instance HDF/NeXus from Mantid)
text, NetCDF, and many other formats (see Loaders for a full description).
It is then advisable to convert the generic iData object into
a specific 2D S(q,w) flavour iData_Sqw2D,
so that the methods below are applicable. This can be done
automatically when importing a file, as shown in the 2nd
example
 Sqw=iData_Sqw2D(Sqw)
 Sqw=iData_Sqw2D('file')
 Sqw=iData_Sqw2D('SQW_coh_lGe.nc')
Common formats that can be read include McStas Isotropic Sqw,
ISIS SPE and SQW, ILL ToF experiments, ENDF (see below).
NOTE: it is required that the 2D data set axes allow
to identify their type, e.g. the labels match
'angle','time','momentum','energy', ...
From an
experiment, raw data
Direct experimental data can be automatically converted to
S(q,w) from any of (e.g. from ToF experiment):
 S(φ,channel) where φ is the radial
scattering angle, and channel is the channel id
(integer, step=1). This conversion requires to locate the
experimental neutron wavelength (searched as 'lambda' or 'wavelength'
in the data set), the sample to detector distance (searched
as 'distance', as a scalar for
radial geometry, or a vector for more complex geometries),
and the time channel width (searched as 'ChannelWidth', scalar or vector).
The time channel width can be given in [s] (preferred),
[ms], or [us].
 S(φ,t) where φ is the radial scattering
angle, and t is the detection time counted from the
sample location. This conversion requires to locate the
experimental neutron wavelength (searched as 'lambda' or 'wavelength'
in the data set), the sample to detector distance (searched
as 'distance', as a scalar for
radial geometry, or a vector for more complex geometries).
The time can be given in [s] (preferred), [ms], or [us].
 S(φ,w) where φ is the radial scattering
angle and w is the energy transfer (in [meV]). This
conversion requires to locate the experimental neutron
wavelength (searched as 'lambda'
or 'wavelength' in the data
set).
The conversion takes place when calling
iData_Sqw2D:
 Sqw=iData('file') % it may be a (phi,
ToF) data set
 Sqw = iData_Sqw2D(Sqw); %
when needed the data is converted to (q,w) space
NOTE: it is required that the 2D data set axes allow to
identify their type, e.g. the variable names or labels match
'angle','time','momentum','energy', ...
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:
 Sqw=iData(q,w,S)
 Sqw=iData_Sqw2D(Sqw)
NOTE: it is required that the 2D data set axes allow to
identify their type, e.g. the labels match
'angle','time','momentum','energy', ...
From a Model
(iFunc)
If S is a 2D iFunc Model (analytical, such as
obtained from the sqw_phonons),
then simply evaluate the model into an iData object, with
default Model parameters [], or specified ones. Axes must be
specified as momentum and energy values in [Angs1] and [meV]
resp.
 Sqw=iData(S, parameters, q,
w)
 Sqw=iData_Sqw2D(Sqw)
It is also possible to obtain a 2D data set from a 4D
iFunc_Sqw4D model by performing e.g. a powder average (see
powder method below), and then an
evaluation.
 model4d = sqw_cubic_monoatomic
 model4d = sqw_phonons('defaults') % using ASE/PhonoPy
and DFT calculator
 model4d = sqw_spinw('defaults') % PSI SpinW S. Toth
In order to apply the powder method, the model must be converted
to an iFunc_Sqw4D flavour (which may not be needed if already
generated as such)
 model4d = iFunc_Sqw4D(model4d)
Then apply the powder average:
 model2d = powder(model4d)
 Sqw = iData(model2d,
parameters, q, w, ...)
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
symmetrize method, as
detailed below.
You may as well apply the following iData operators:
 resize: change binning
in q,w, fast rebin.
 reshape: change
the dimensions of the Sqw data set, the number of elements
must not change.
 fill:
replaces missing data (NaN's, gaps) by interpolating
 smooth:
smooth the data set
 meshgrid:
creates a square 2D distribution, using interpolation if
necessary.
 powder:
converts a 4D S(hkl,w) model into a 2D S(q,w) one (see
below).
 xlim: the 'x' axis is the
momemtum/wavevector 'Q' e.g. in [Angs1]. It corresponds
with the axis of rank 2, in the way Matlab switches xy axes
for plots. To change the Q axis use xlim(Sqw, [min max]).
 ylim: the 'y' axis is the Energy transfer
'w' e.g. in [meV]. It corresponds with the axis of rank 1,
in the way Matlab switches xy axes for plots. To change the
w axis use ylim(Sqw, [min max]).
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 nonsymmetric (
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 iData_Sqw2D conversion/check, 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) format (change of axes, taking into
account the Jacobian).
Conventions:
 w = EiEf = energy lost by the neutron in meV.
 w > 0, neutron looses energy, material/sample gains
energy, w can not be higher than the incident
neutron energy (Stokes)
 w < 0, neutron gains energy, material/sample looses
energy (antiStokes)
 q = KiKf is the momentum exchange (or wavevector) in
Angs1.
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)
Sqw=iData_Sqw2D(file)

file

iData or iData_Sqw2D

Import any 2D S(q,w) as generated by e.g.
LAMP, Mantid and nMoldyn/MDANSE. Possibly convert from
experimental data S(φ,channel), S(φ,t) or S(φ,w).

Sqw=iData_Sqw2D(iData(q,w,S)) 
[q,w,S]

iData_Sqw2D 
Import data set S as a function of axes q
and w.

Sqw=iData_Sqw2D(sab) 
iData_Sab
S*(alpha,beta) 
iData_Sqw2D 
Import an S(alpha,beta)

symmetrize(sqw2d)

iData_Sqw2D
S*(q,w>0)

S*(q,w)=S*(q,w)

Extend the
S*(q,w) in both energy sides. The initial data set
should better be 'classical'.

Bosify(sqw2d,T)

iData_Sqw2D
S*(q,w) 
S(q,w) 
Convert a
symmetric scattering law (aka classical, for instance
from Molecular Dynamics) into a 'quantum' scattering law
(non symmetric, for instance from neutron and Xray
inelastic scattering experiment)

iData_Sqw2D(sqw2d)
iData_Sqw2D('file') 
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). 
deBosify(sqw2d,T)

iData_Sqw2D
S(q,w) 
S*(q,w) 
Convert a 'quantum' scattering law (non
symmetric, for instance from neutron and Xray inelastic
scattering experiment) into a symmetric scattering law
(aka classical, for instance from Molecular Dynamics)

trapz(sqw2d)
structure_factor(sqw2d)

iData_Sqw2D
S(q,w) 
S(q)

Compute the
static structure factor. The S(q) is about the same when
computed from the 'classical' and the 'quantum'
scattering law.

dynamic_range(sqw2d,Ei)

iData_Sqw2D
S(q,w) or S*(q,w) 
measurable S(q,w) for Ei

Restrict
the dynamic structure factor to the measurable one for
an incoming energy Ei [meV]. The detection angular range
can be set as 3rd argument.

moments(sqw2d,M)

iData_Sqw2D
S(q,w) or S*(q,w) 
S(q), Er, characteristic frequencies

Compute
the energy moments, and provide some of the
characteristic frequencies (recoil, isothermal,
longitudinal/harmonic, mean energy transfer)

dos(sqw2d)

iData_Sqw2D
S(q,w) or S*(q,w)or S(q,w)

g(w)

Compute the vibrational density of states
(DOS) using Bredov or Carpenter methods 
scattering_cross_section(sqw2d,Ei)

iData_Sqw2D
S(q,w) 
∫∫S(q,w)q dq dw /2Ki^{2}

Compute
the total thermal neutron scattering cross section which
is the integral of S(q,w) over the dynamic range vs.
incident neutron energy.

Sab(sqw2d, M)

iData_Sqw2D
S*(q,w) 
iData_Sab
S*(alpha,beta)

Convert a
dynamic structure factor to its unitless representation
S(α,β) 
powder(sqw4d model)

iFunc_Sqw4D
S(hkl,w) 4D model

iFunc_Sqw2D model
S(q,w)

Convert a
4D S(hkl,w) model into a powder averaged S(q,w) model.
This can be used e.g. after calling sqw_phonons and
other 4D S(q,w).

max(Sqw4d model)

iFunc_Sqw4D
S(hkl,w) 4D model 
energy maximum

Computes the maximum energy of modes in a
4D S(h,k,l,w) model. Also returns a fast evaluation of
the vDOS.

powder(sqw4d) 
iData_Sqw4D
S(hkl,w) 4D 
iData_Sqw2D
S(q,w) 
Convert a 4D S(h,k,l,w) data set into
S(q,w). 
band_structure(sqw4d
model)

iFunc_Sqw4D
S(hkl,w) 4D 
dispersion curves

Compute the dispersion curves and neutron
intensity along the main crystal axes.

publish(sqw4d
model)

iFunc_Sqw4D
S(hkl,w) 4D 
HTML page

Write a full document about a 4D
S(h,k,l,w) model.

thermochemistry(sqw4d
or g, T)

iFunc_Sqw4D
S(hkl,w) 4D
or
iData_vDOS g(w)

U,F,S,Cv

Compute the energy, entropy, heat
capacity, and also returns the DOS.

iData_vDOS(g)

iData g(w) 1D

iData_vDOS 
Convert a 1D data set assumed to be a
vDOS

incoherent(g)

iData_vDOS
g(w)

iData_Sqw2D array

Compute the incoherent S(q,w) in the
Gaussian/multiphonon approximation.

multi_phonons(g)

iData_vDOS
g(w) 
iData_vDOS 1D array

Compute the effective neutron weighted
density of states separate terms, which contain
multiphonon terms. 
gdos(g)

iData_vDOS
g(w) 
iData_vDOS

Compute the effective neutron weighted
density of states sum, which contains multiphonon
terms.

Dynamic
structure factor from Molecular Dynamics (MD)
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
abinitio Molecular
Dynamics (AIMD) with VASP, for 200 Ge atoms (GGAPW91 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:
 Sqw=iData_Sqw2D('SQW_coh_lGe.nc')
Molecular Dynamics trajectories are usually computed in the NVE
microcanonical
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.
It is often named 'classical'.
We plot the scattering law S*:
 plot(log10(Sqw), 'view3');
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.
 plot(log10(Sqw2), 'view3');
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 socalled '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
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 semiclassical 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./(1exp(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.
 SqwT = Bosify(Sqw2, 1250)
 plot(log10(SqwT));
When 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 downscattering are about the same (the
quantum correction remains close to 1). For materials at low
temperatures, the two sides are highly disymmetric.
The
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
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) = (Ki^{2} + Kf^{2}  q^{2}) /
(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*Ki^{2} =
81.8042/λ^{2} with Ki in [Angs1] and λ in [Angs]
We can determine the dynamic range accessible to neutrons with a
given incident energy, say Ei=25 meV.
 Sqw25 = dynamic_range(SqwT,
25)
 plot(log10(Sqw25));
The integrated scattering cross section is the integral of the
dynamic range, for a given incident neutron energy (see below).
The structure factor S(q) is
the integral of S(q,w) over the energy. We use the usual '
trapz' integrator, or
structure_factor 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". Also, on a real experiment, the integration
over momentum and angle is not identical, raising additional
corrections.
 plot(structure_factor([SqwT,
Sqw25]));
Characteristic
frequencies (moments)
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
nth
order frequency moment as:
<w
^{n} S> = ∫ w
^{n}
S(q,w) dw
and then derive:
 S(q) = ∫S(q,w) dq = <S> is the 0th frequency moment
 Er(q) = h^{2}q^{2}/2M = <w S> is the
recoil energy, 1st frequency moment
 Wc(q) = √[ 2kT*Er(q)/S(q) ] is the collective/isothermal
dispersion
 Wl(q) =√[ <w^{3}S>/Er(q) ] is the
harmonic/longitudinal excitation
 Wq(q) = 2q √[ kT / m.S(q) ] is the mean energy transfer,
or half width from normalized 2nd frequency moment
These frequencies are computed with:
 moments=moments(SqwT, 72.64)
 plot(moments(1:5))
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, vibrational 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
autocorrelation 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 timeofflight
experiment (such as done on IN4, IN5 and IN6 at the ILL). The
concept of the
generalised density of states has been
defined as [Carpenter/Bellissent 12,13,15]:
gDOS(q,w) = S(q,w) w /q^2/[1 + n(w)] ~
S(q,w) w^{2}/q^{2}
which is valid in the case of an incoherent scatterer. Here,
n(w) is the Bose factor. Then, the energy dependent density of
states is the lowwavevector 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. However, the Carpenter
method is satisfactory for incoherent materials, and results
from Molecular Dynamics where the q range reaches low q values.
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.
The Bredov/Oskotskii method [16] is also valid for coherent
scatterers, with better statistics:
gDOS(q,w) = w q S(q,w) e^{2W(q)}^{
}/[Q_{max}4  Q_{min}^{4}]/(1+n(w))
gDOS(w) = ∫ gDOS(q,w) dq
where Q
_{min} and Q
_{max} are the minimal and
maximal momentum values reachable in the dynamic range of the
neutron spectrometer. The Carpenter and Bredov methods usually
provide different results.
The syntax to compute the density of states is usually:
g = dos(sqw2d) %
uses Bredov method by default
g = dos(sqw2d, 'Carpenter')
g = dos(s, 'method', method, 'T', T, 'DW',
dw)
We import the heavy water NAMD simulation (3900 molecules, 290K,
E. Pellegrini) coherent dynamic structure factor from
here
[9].
Then we can compute the generalised density of states with the
dos procedure, from both the raw, and
the symmetrized, 'Bosified' (with quantum correction) data set.
 SqwT = Bosify(symmetrize(Sqw), 290);
 plot(log(SqwT))
 g=[ dos(Sqw, 'Carpenter')) dos(SqwT, 'Carpenter'))
];
 plot(g); xlim([0 Inf]);
It is also possible to evaluate the density of states from a
4D S(q,w) model, using:
g = dos(model4d)
The powder
average
When 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):
Q_{cart} = B*Q_{rlu}
The the average is obtained using e.g. the
powder method
>> s=sqw_phonons([ ifitpath 'Data/POSCAR_Al'],'metal','EMT');
>> pow=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.
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
deBosify procedure, specifying the
temperature if it is not defined in the data set. Then
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.
Dynamic
structure factor from a density of states
It is rather usual to derive a socalled density of states from
a neutron or Xray inelastic scattering experiment. Such
quantity can also be estimated from e.g. abinitio lattice
dynamics (see our
sqw_phonon
model), perturbation theory, molecular dynamics (use e.g.
MDANSE), ...
As
defined above, the vDOS
measures the density of vibrational states in [w, w+dw]. This
quantity is central in the determination of thermochemistry
quantities such as U, S, F, and Cv.
However, it can also be used to estimate the incoherent
scattering law S(q,w) [Sjölander theory, Gaussian approximation]
as well as a further estimate of the coherent scattering law
[Sköld approximation].
In this way, we suggest that, from an inelastic neutron
scattering experiment where the total intensity is measured, the
data is corrected and reduced to extract the gDOS. Then, the
incoherent approximation is used (see below), which can be
subtracted from the total intensity to reveal the coherent part
better and help in the 'background' subtraction.
Incoherent
scattering law estimate: monoatomic material
The estimate of the incoherent scattering law is based on the
socalled
incoherent approximation, which states that
any angle or momentum integrated quantity from both the
incoherent and the coherent scattering processes are equal. This
can be demonstrated in the case of a monoatomic cubic lattice
material, and has been extended to other materials within a 20%
difference in general.
In practice, this can be checked from any molecular dynamics
modelling, computing first the theoretical density of states
from the velocity autocorrelation function, as well as the
incoherent and coherent scattering laws. Then, we can estimate
from each S(q,w) the generalised density of states as
defined above,
and check that these two estimates are comparable.
As a consequence, the density of states can be estimated
similarly from incoherent, coherent and even total scattering
integrals [17].
The socalled
Gaussian approximation (aka multiphonons
expansion) starts from the general expression of the scattering
law. It is usual to write it:
d^{2}σ/dΩdEf = kf/ki S(q,w)
where S contains the bound scattering cross section σ and the
number of scatterers N (e.g. the mass or material density). In a
very general expression, S contains the single excitation term,
as well as further multiexcitation terms which are often
omitted, assumed negligible.
In a polyatomic material, the expression of S contains
'partial' terms as well as 'interference' (cross) terms:
S(q,w) = ∑ b_{i}b^{*}_{j}
S_{ij}(q,w)
where b
_{i} is the scattering length for atom 'i'. In
practice we find that the 'partial' terms per atomic species
contain a coherent (e.g. periodic features such as structure and
vibrations) and an incoherent part (e.g. from disorder).
Sjölander [18] has derived an extensive theoretical development
of the dynamic structure factor, including multiexcitations.
However, this methodology can only be applied to the incoherent
part. It may then seem odd to talk about multiphonons in a
incoherent scattering law which does not contain any phononic
part. For a monoatomic isotropic material, it is possible to
write:
S(Q,w) = N σ
_{inc} /4π exp[ hQ
^{2}/2m
f(0) ] ∫ e
^{iwt} exp[ hQ
^{2}/2m f(t) ] dt
where the function 'f' is computed from the
density of states:
f(t) = ∫ e^{iwt} g(w)/w (n(w)+1)
dw
The second exponential term is then expanded in Taylor
series, to draw:
S(Q,w) = N σ
_{inc} /4π exp[ h
^{2}Q
^{2}/2m
f(0) ] ∑
_{p} 1/p! ∫ e
^{iwt} [ h
^{2}Q
^{2}/2m
f(t) ]
^{p} dt
hence the naming as 'phonon expansion'. This rather complex
expression can be computed by introducing the time Fourier
tranform of f(t)
^{p}
T_{p}(w) = ∫ e^{iwt }f(t)^{p}
dt
which satisfies:
T_{0}(w) = δ(w)
T_{1}(w) = g(w)/w (n(w)+1)
T_{p} = T_{1} * T_{p1} with
'*' being the convolution operator
These function T are normalized as:
T_{1} = f(0) = ∫ T_{1}(w) dw
T_{p} = f(0)^{p}
from which we can estimate the DebyeWaller factor
W(Q) = h^{2}Q^{2}/2m
f(0)
The p=0 term is the Elastic Incoherent Structure Factor (mostly
a Dirac peak with a decreasing DebyeWaller amplitude)
S(Q,w)_{[p=0]} = (1/4π) e^{2W(Q)}
δ(w)
The p=1 term is the 'onephonon' response
S(Q,w)_{[p=1]} = Nσ_{inc}/8πm
Q^{2} e^{2W(Q)} g(w)/w (n(w)+1)
The following terms are obtained by iterative autoconvolution
by the vDOS
S(q,w)_{[p]} = Nσ_{inc}/4π/!p
e^{2W(Q)} [2W(q)]^{p} T_{p}(w)
All this mechanics allows to estimate the S(q,w) as a sum of
decreasing terms as 1/m
^{p} where m is the mass of the
atom. So for most materials, the additional terms are negligible
after e.g. p> 5. However, this methodology allows, only
starting from the density of states, to estimate the incoherent
scattering law. And it proves very efficient (fast to compute)
and reasonably accurate.
This strategy is the one used in the NJOY/LEAPR module, known as
"phonon expansion" to estimate the thermal scattering law in the
incoherent approximation.
The use with iFit is simple. If one has a density of states g,
the incoherent S(q,w) is obtained as:
If the atom mass (in [g/mol]) and Temperature (in [K]) need to
be defined, the syntax becomes:
 Sinc = incoherent(g, 'm', m, 'T',
T)
The final 2D S
_{inc}(q,w) must be multiplied by the
bound incoherent scattering cross section of the material.
As an example, we can get the density of states from an
incoherent S(q,w) obtained by MD, then compute back the S(q,w)
estimate. To estimate the DOS, we use the Carpenter method which
is well suited to MD results containing low 'q' values such as
here. The incoherent method is then used, specifying the
momentum axis 'q' to use as the one from the MD data set, and
the DW coefficient is set to a low value, as the MD does not
simulate the DW attenuation along Q in S(q,w). Last, the
temperature effect (Bose factor) is part of the incoherent
estimate. To compare with the initial S(q,w) from MD which is
'classical' (symmetric), we 'deBosify' the incoherent estimate.
The plot then restricts the energy range to only the positive
one.
 Sinc0 = iData_Sqw2D('D2O_liq_290_inc.sqw.zip')
 gDOS = dos(Sinc0, 'Carpenter'); % well suited for
MD results
 Sinc1 = incoherent(gDOS, 'DW', 0.0001,'q',Sinc{2});
% this is an array of 'p' terms, to be added with 'plus'
 subplot(log10([ Sinc0 deBosify(ylim(plus(Sinc1), [0
inf])) ]))
The 2 data sets
Sinc0 and
Sinc1 are not equal,
but they still show some similarities. Remember that
Sinc1
is only obtained from an estimate of the DOS.
Incoherent
scattering law estimate: polyatomic material
The methodology above is in principle only valid for a
monoatomic material, isotropic in density (liquid, powder, gas,
amorphous).
For a polyatomic material, two procedures can be used.
When the partial density of modes per atom are known, each
contribution will provide an incoherent scattering law, which
should be weighted with their respective bound scattering cross
sections of the atom, and finally added. This is usually the
case from molecular dynamics and lattice dynamics.
When the partial density of states are not known, the
computation of the incoherent S(q,w) with a total density of
states should be performed with
the weighted mass and incoherent cross sections:
m = [∑_{i} C_{i} σ_{i}]/[∑_{i}
C_{i} σ_{i}/m_{i}]
σ = ∑_{i} C_{i} σ_{i}
where the C
_{i} are the respective proportions of atoms
in the compound, and σ
_{i}, m
_{i} are their
incoherent cross section and mass.
Coherent
scattering law estimate
A phenomenological expression has been proposed by Sköld [19] to
derive an estimate of the coherent scattering law, given the
incoherent one and the structure factor S(q).
S_{coh}(q,w) ~ S_{inc}(q/√S(q),
w) S(q)
This methodology forces the structure into the incoherent
S(q,w), and should never be considered as exact. It may be
considered as a first guess, especially when very limited
information is available from the material. The 0th (structure)
and 2nd moment (mean energy transfer) of the scattering law are
correctly estimated. The way the structure is inserted into the
dynamics does not allow to reconstruct the lowq dynamics (e.g.
phonons), nor the socalled De Gennes narrowing which appears as
a sharpening of the dynamics close to structure peaks.
The iFit syntax to apply this approximation is:
where 'inc' is a 2D incoherent S(q,w) [iData_Sqw2D], and 'sq's
is a 1D data set [iData] holding the structure factor.
As an example, we can go beyond the example above by adding the
estimate of the coherent scattering law from the estimate of the
incoherent (let's be crazy):
 Sinc0 = iData_Sqw2D('D2O_liq_290_inc.sqw.zip')
 Scoh0 = iData_Sqw2D('D2O_liq_290_coh.sqw.zip')
 gDOS = dos(Sinc0, 'Carpenter'); % well suited for
MD results
 Sinc1 = incoherent(gDOS, 'DW', 0.0001,'q',Sinc{2});
% this is an array of 'p' terms, to be added with 'plus'
 sq = structure_factor(Scoh0);
 Scoh1 = coherent(Sinc1, sq)
% VERY approximate as we use the incoherent estimate
References:
 H. Schober, Journal of Neutron Research 17
(2014) 109–357. DOI 10.3233/JNR140016
(see esp. pages 328331)
References
 P. Schofield. Phys. Rev. Lett., 4, 239 (1960).
 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).
 L. Frommhold. Collisioninduced absorption in gases, 1 st
ed., Cambridge Monographs on Atomic, Molecular, and Chemical
Physics, Vol. 2, Cambridge Univ. Press: London (1993).
 B. Hehr, http://www.lib.ncsu.edu/resolver/1840.16/7422
PhD manuscript (2010).
 Helmut Schober, Journal of Neutron Research 17
(2014) pp. 109357<http://dx.doi.org/10.3233/JNR140016>
 G.L. Squires, Introduction to the Theory of Thermal
Neutron Scattering, Dover Publications Inc.(1997)
 Hugouvieux V, Farhi E, Johnson MR, et al., PRB 75
(2007) 104208
 E. Farhi, V. Hugouvieux, M.R. Johnson, W. Kob, Journal
of Computational Physics 228 (2009) 5251
 E. Farhi et al, J. Nucl. Sci. Tech. 52
(2015) 844; DOI: 10.1080/00223131.2014.984002
 JP.Hansen and I.R.McDonald, Theory of simple liquids
Academic Press New York 2006.
 NIST Neutron scattering lengths and cross sections <https://www.ncnr.nist.gov/resources/nlengths/>;
See also Sears, Neut. News 3 (1992) 26, and
the ILL
Neutron Data Bluebook.
 Price J. et al, Non Cryst Sol 92 (1987)
153
 BellisentFunel et al, J. Mol. Struct. 250
(1991) 213
 Sears, Neutron News 3 (1992) 26
 Suck et al, Journal of Alloys and Compounds 342
(2002) 314
 Bredov et al., Sov. Phys. Solid State 9,
214 (1967)
 V.S. Oskotskii, Sov. Phys. Solid State 9
(1967), 420.
 A. Sjolander, Arkiv for Fysik 14 (1958),
315.
 K. Skold, Phys. Rev. Lett. 19, 1023
(1967).
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/ENDFBVIIthermal/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 aptaddrepository 'deb http://packages.mccode.org/debian stable main'
% sudo aptget update
% sudo aptget install pyne
Total
scattering cross section, energy dependent
The energy dependent scattering cross section is computed in
the coldthermal range as the integral on the dynamic range
vs. the incident energy:
σ(Ei) = ∫∫ d
^{2}σ/dΩdEf dΩdEf = ∫∫
Nσ
_{b} /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) = σ
_{b}∫∫S(q,w)q dq dw /2ki
^{2}
where σ
_{b} is the
tabulated bound neutron cross section [Sears,
Neutron News
3 (1992) 26] for thermal energies. The integral can be
computed with the
scattering_cross_section
procedure, for a given energy range, which should then be
multiplied by the bound thermal scattering cross section σ
_{b}.
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):
 Sqw=iData_Sqw2D('D2O_liq_290_coh.sqw');
 SqwT = Bosify(symmetrize(Sqw), 290);
 Ei = logspace(3,3,50);%
from Ei=10^{3} to 10^{2} meV.
 XS = 15.4*scattering_cross_section(SqwT,Ei);
 plot(XS); set(gca, 'YScale','log','XScale','log');
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 σ
_{b}.
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) → σ_{b}∫∫ q dq dw /2ki^{2}
→ σ_{b}∫ q^{2}/2 dw /2ki^{2} up to
q=2ki
→ σ_{b}
However, for energies above the thermalhot range (e.g. above
14 eV), the total cross section should converge towards the
free neutron cross section σ(Ei) → σ
_{free}. The free
cross section, with A being the element mass, is defined as:
σ_{free} ~ A^{2}/(A+1)^{2}
σ_{b}
As most S(q,w) scattering law models in nuclear engineering
include an e
^{WEi }decaying factor (a kind of very
weak/slow DebyeWaller 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
DebyeWaller factor exp(<u2>Q2) !
The scatterer mass can be input as 3rd argument to
scattering_cross_section For a
polyatomic material, the effective mass is obtained by
weighting the bound cross sections with the A
^{2}/(A+1)
^{2}
factors, e.g.:
r^{2} = ∑ (A/(A+1))^{2} *
σ_{b}(A)) / ∑ σ_{b}(A)
M = r/(1r)
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 σ
_{b}=80.2) and one O atom (A=16 σ
_{b}=4.2),
we compute
r
^{2} =
(2*(1/(1+1))^2*80.2+(18/(18+1))^2*4.2)/(2*80.2+4.2) = 0.26
M = r/(1r) =
1.06 i.e. scattering is mostly the
hydrogen one.
In the case of heavy water we have two D
atoms (A=2 σ
_{b}=7.64) and one O atom (A=16 σ
_{b}=4.2),
we compute
r^{2} =
(2*(2/(2+1))^2*7.64+(18/(18+1))^2*4.2)/(2*7.64+4.2) =
0.54
M = r/(1r) =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 socalled 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 unitless momentum and energy variables
defined as:
S*(α,β) = kT S*(q,w) = kT exp(beta/2) S(q,w)
α= h^{2}q^{2}/2MkT = [Ei+Ef2 μ √(Ei*Ef)]/A.kT
β = hw/kT = (EfEi)/kT
A = M/m
μ = cos(θ) = (Ki^{2} + Kf^{2}  q^{2})
/ (2Ki.Kf)
The 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:
 M. Mattes and J. Keinert, IAEA INDC(NDS)0470, 2005. [link]
 R. E. MacFarlane, LA12639MS (ENDF356), 1994. [link]
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.
In 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.
 Sqw=iData_Sqw2D('D2O_liq_290_coh.sqw');
 SqwT = Bosify(Sqw_symmetrize(Sqw), 290);
 DDCS = dynamic_range(SqwT,
154, [5 15])
 plot(trapz(DDCS,2));
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:
 O. K. Harling, “Slow Neutron Inelastic Scattering and the
Dynamics of Heavy Water”,
Nucl. Sci. Eng., 33, 41 (1968). Provides heavy
water data at Beta=1.5, Ei=38.6, meV T=299 K, and Ei=101 meV
theta=60 deg. [link]
 F.G. Bischoff, M.L. Yeater, W.E. Moore, Nucl. Sci.
Eng. 48 (1972) 266. Provides heavy water ToF
data Ei=233 meV at θ=25 deg. [link]
 F.G. Bischoff et al., “Low Energy Neutron Inelastic
Scattering”, RPI32887 (1967). Provides light water data
Ei=154, 231, 631 meV at θ=60 deg.
 R. E. MacFarlane, LA12639MS (ENDF356), 1994. [link].
Indicates light water rotational vibrations at 205 and 408
meV
 E. Liu (RPI) <http://homepages.rpi.edu/~danony/Papers/RND2011/ND_Sym_Liu_4_27_11.pdf>
presents light water data at Ei=600 and Ei=154 meV, and
compares to older data from Bischoff and Esch.
 J. I. Márquez Damian et al, Annals of Nuclear Energy
65 (2014) 280–289. [link]
Gathers most of this data.
 Esch et al, Nuclear Science and Engineering: 46,
223235, 1971. Provides light water data Ei=632 meV and 154
meV theta=10 14 25 40 60 150 deg. [link]
Neutron
diffraction
The iFit package includes a few features relevant to
diffraction.
Creating
structure data files for McStas
The cif2hkl
tool is derived from the CrysFML
library (Juan RodriguezCarvajal and Javier GonzalezPlatas).
Its mai purpose is to read/convert material structure data
files into HKL F^{2} lists, mainly for use with
McStas.
It can read:
 CIF
 FullProf PCR/CFL
 ShelX SHX/INS/RES
The generated file can be tuned for powders or singlecrystal
diffraction.
The syntax from iFit is:
output = cif2hkl('input', 'output',
wavelength, 'mode')
where:
 input: data file to
read/convert (CIF, PCR, CFL, SHX, INS, RES). It can also be
given as a chemical formula with syntax 'cod: formula'.
 output: name/path of file for
output. If not given or empty, an extension is added to the
input file.
 wavelength: the wavelength used for generation of HKL
peaks, in Angs. Default is 0.5 Angs. This affects the
maximal reachable HKL indices.
 mode: conversion mode 'powder' or 'p'=powder,
'xtal' or
'x'=single crystal
The generated file name is returned after completion. It can
e.g. be used with McStas components PowderN,
Single_crystal,
Isotropic_Sqw.
The input material structure can also be entered as a
chemical formula in Hill notation (C, H, other atoms in alpha
order) with syntax 'cod: formula'
(access crystallography,net,
requires a valid internet connexion).
NOTE: This requires proxy settings to be
set (when behind a firewall)
 ProxyHost='proxy.ill.fr'; % Proxy
address if you are behind a proxy [e.g.
myproxy.mycompany.com or empty]
 ProxyPort=8888;
% Proxy port if you are behind a proxy [8888 or 0 or
empty]
 java.lang.System.setProperty('http.proxyHost',
ProxyHost);
 com.mathworks.mlwidgets.html.HTMLPrefs.setUseProxy(true);
 com.mathworks.mlwidgets.html.HTMLPrefs.setProxyHost(ProxyHost);
 java.lang.System.setProperty('http.proxyPort',
num2str(ProxyPort));
 com.mathworks.mlwidgets.html.HTMLPrefs.setProxyPort(num2str(ProxyPort));
Examples:
 lmo = cif2hkl('LaMnO3.cif')
 lmo = cif2hkl('cod: La Mn O3')
Pure
MonteCarlo Rietveld refinement
A Rietveld structure refinement model has been created. It is
documented in Models
Rietveld. It is very slow compared to usual analytical
procedures such as FullProf, but
it is based on a full McStas instrument model and goes much
beyond the Caglioti formalism.
Neutron
TripleAxis Spectrometers
The ResLibCal application
allows to model a conventional neutron scattering TripleAxis
Spectrometer (TAS). It also allows to convolve a 4D S(q,w)
dynamic structure factor model with the resolution function
and perform fits. Read more in the dedicated miFit example for a
phonon data set.
Neutron
Scattering Instruments with McStas
It is possible to control McStas
instrument models from within iFit. By doing so, you will be
able to perform parameter scans, optimisations, display the
instrument code as well as its geometry, and mode. Read more
at the McStas page.