14. Simulation

There are a variety of simulation tools available in Horace, so that if you have a theoretical model to describe your data you can simulate the results for the specific data points that you measured.

Note

When simulating an S(Q,w) model (see sqw_eval below), bear in mind the difference between what is calculated for equivalent dnd and sqw datasets. See FAQ.

14.1. func_eval

This evaluates a user-supplied function at the x, y, z, … values of an n-dimensional dataset, or array of datasets. The syntax is as follows:

wout = func_eval(win, @myfunc, p);
wout = func_eval(win, @myfunc, p, 'all');

win is the input dataset or array of datasets (sqw or dnd type) for which you wish to perform a simulation.
myfunc is the name of a user-defined function to calculate the intensity at the points in the dataset(s).

Warning

The function must be of the form

y = myfunc(x1, x2, .., xn, p)

e.g.

y = gauss2d(x1, x2, [amplitude, centre1, centre2, width1, width2, background])

and accept equal sized arrays that contain the x1, x2, … values.

p is a row-vector containing the parameters needed by the function.
'all' is used if you wish to calculate the intensity from the function over the whole domain covered by the input dataset. In other words, your dataset may contain gaps due to the trajectory of the detectors through reciprocal space, but you may wish to simulate the scattering even in the gaps to get a complete image. This option only applies in the case of dnd, but not sqw objects.

14.2. sqw_eval

wout = sqw_eval(win, @my_sqw_func, p);
wout = sqw_eval(win, @my_sqw_func, p, 'all');

The syntax for sqw_eval is almost identical to that of func_eval. The only difference is the form of the required function my_sqw_func,

Warning

The function supplied to sqw_eval must be of the form:

weight = my_sqw_func(qh, qk, ql, en, p)

where qh, qk, ql, en are arrays that contain the co-ordinates in \(h, k, l\) and energy of each point in the dataset.

In the case of a dnd, bin centres are used. Where dimensions might be integrated away, the centres the range of integration are used.

p is a row-vector containing the parameters required by the function.

Note

These could be the values of exchange constants, intensity scale factor, or temperature, for example.

One would generally use sqw_eval in preference to func_eval if, for example, one had a model of the spin-wave cross-section for magnetic scattering.

14.3. dispersion

Calculate dispersion relation for a dataset or array of datasets.

[wdisp,weight] = dispersion(win, dispreln, p)   % dispersion and spectral weight

The output dataset (or array of data sets), wdisp, will retain only the Q axes, and the signal array(s) will contain the values of energy at the relevant Q-points. If the dispersion relation returns the spectral weight, this will be placed in the error array (actually the square of the spectral weight is put in the error array).

Note

The dispersion relation is calculated at the bin centres, i.e. the individual pixel information in an sqw object is not used.

Inputs:

win - Dataset (or array of datasets) that provides the axes and points for the calculation. If one of the plot axes is energy transfer, then the output dataset will have dimensionality one less than the input dataset.
dispreln - Handle to function that calculates the dispersion relation w(Q) and spectral weight, S(Q).
p Arguments needed by the function that calculates the dispersion relation(s). Most commonly, a vector of parameter values e.g. [A, js, gam] as intensity, exchange, lifetime. If a more general set of parameters is required by the dispersion relation function, then package these into a cell array {p, c1, c2, ...}.

Warning

The function dispreln must be of the form:

[w, s] = dispreln(qh, qk, ql, p, c1, c2, ..)

where the inputs are:

qh, qk, ql - Arrays containing the coordinates of a set of points in reciprocal lattice units
p - Vector or cell array of parameters needed by dispersion function, e.g. [A, js, gam] as intensity, exchange, lifetime
c1, c2, ... [Optional] - Other constant parameters e.g. file name for look-up table

and the outputs are:

w - Array of corresponding energies, or, if more than one dispersion relation, a cell array of arrays.
s [Optional] - Array of corresponding spectral weights, or, if more than one dispersion relation, a cell array of arrays.

Outputs:

wdisp Output dataset or array of datasets. Output is always a dnd. The output dataset (or array of data sets) will retain only the Q axes, the the signal array(s) will contain the values of energy at the Q points, and the error array will contain the square of the spectral weight.

Warning

If the function that calculates dispersion relations produces more than one branch, then in the case of a single input dataset the output will be an array of datasets, one for each branch.

If the input is an array of datasets, then only the first dispersion branch will be returned, so there is one output dataset per input dataset.

weight Mirror output: the signal is the spectral weight, and the error array contains the square of the frequency.

Note

If win is a 2D dataset with Q and E axes, then wdisp is a 1D dataset with just the Q axis

Note

In the case when the dispersion has been calculated on a plane in momentum space (i.e. wdisp is an IX_dataset_2d object) then the plot function ds2 (draw surface from two arrays)

ds2(wdisp)

will plot a surface with the z axis as energy, coloured according to the spectral weight.

If you wish to overplot a dispersion relation on top of, for example, a Q-E slice from your data, then you would use:

plot(my_qe_slice)

ploc(wdisp)       % for plot line on current

Warning

In the above there must not be a keep_figure command between plotting the Q-E slice and plotting the dispersion, since the ploc command works on the current figure.

14.4. disp2sqw_eval

Similar to sqw_eval, but takes as the input function a routine that calculates both the dispersion and the spectral weight, and only requires as its inputs \(h, k, l\) and some model parameters.

wout = disp2sqw_eval(win, @dispreln, pars, fwhh, 'all', 'ave')

win - the input dataset (sqw or dnd) or array of datasets
dispreln - Handle to function that calculates the dispersion relation w(Q) and spectral weight, S(Q).
p - Vector of parameters needed by dispersion function, e.g. [A, js, gam] as intensity, exchange, lifetime
fwhh - the full-width half-height of Gaussian broadening applied to dispersion relation.

Warning

The function dispreln must be of the form as specified above

The optional inputs are:

'all' [Optional] - Requests that the calculated sqw be returned over the whole of the domain of the input dataset. If not given, then the function will be returned only at those points of the dataset that contain data. Applies only to input with no pixel information - it is ignored if full sqw object.
'ave' [Optional] - Requests that the calculated sqw be computed for the average values of \(h, k, l\) of the pixels in a bin, not for each pixel individually. Reduces cost of expensive calculations. Applies only to the case of sqw object with pixel information - it is ignored if dnd object.

The output is:

wout - Output dataset or array of datasets

14.5. dispersion_plot

Plot dispersion relation or array of dispersion relations along a path in reciprocal space. It can be called in the following ways, with or without outputs, as below:

[wdisp, weight] = dispersion_plot(lattice, rlp, dispreln, pars, 'dispersion', 'weight' ...
                                  'labels', {'G', 'X', ..}, 'ndiv', n, 'noplot')

The inputs are as follows:

lattice [Optional] - Lattice parameters \([a,b,c,\alpha,\beta,\gamma]\) (Angstrom, degrees). Default is \([2\pi,2\pi,2\pi,90,90,90]\)

rlp - Array of reciprocal lattice points, e.g.

[0, 0,0;
 0, 0,1;
 0,-1,1;
 1,-1,1;
 1, 0,1;
 1, 0,0];

dispreln - Handle to function that calculates the dispersion relation w(Q) and spectral weight, S(Q).

Warning

The function dispreln must be of the form as specified above

The keyword options are:

'dispersion' [Optional] - Only plot the dispersion relations. The default is to plot and/or return dispersion, and weight if available
'weight' [Optional] - Only plot the spectral weights. The default is to plot and/or return dispersion, and weight if available
'labels' [Optional] - Tick labels to place at the positions of the Q points in argument rlp. e.g. {'G', 'X', 'M', 'R'}. By default the labels are character representations of rlp, e.g. {0, 0, 0; 0.5, 0, 0; 0.5, 0.5, 0; 0.5, 0.5, 0.5} becomes {'0, 0, 0', '0.5, 0, 0', '0.5, 0.5, 0', '0.5, 0.5, 0.5'}
'ndiv', N [Optional] - Number of points into which to divide the interval between two r.l.p. (default=100)
'noplot' [Optional] - Do not plot, just return the output IX_dataset_1d

The outputs are as follows

wdisp [Optional] - Array of IX_dataset_1d containing dispersion, one per dispersion relation. The x-axis is the distance in Ang^-1 along the path described
weight [Optional] - Array of IX_dataset_1d with corresponding spectral weight, one per dispersion relation

14.6. disp2sqw_plot

Generate an Q-E intensity plot for a dispersion relation along a path in reciprocal space. The function is very closely related to dispersion_plot, and most of the input arguments and options are the same for the two functions.

weight = disp2sqw_plot(lattice, rlp, dispreln, pars, ebins, fwhh, 'labels', {'G', 'X', ..}, 'noplot')

The inputs are as follows:

lattice [Optional] - Lattice parameters \([a, b, c, \alpha, \beta, \gamma]\) (Angstrom, degrees). Default is \([2\pi, 2\pi, 2\pi, 90, 90, 90]\)

rlp - Array of reciprocal lattice points, e.g.

[0, 0,0;
 0, 0,1;
 0,-1,1;
 1,-1,1;
 1, 0,1;
 1, 0,0];

dispreln - Handle to function that calculates the dispersion relation w(Q) and spectral weight, S(Q).

Warning

The function dispreln must be of the form as specified above

ebins - Defines the energy bin centres: a three-vector [ecentre_lo, bin_width, ecentre_hi]
fwhh - Full width half height of broadening applied to the dispersion to produce the intensity map

The keyword options (which can be abbreviated to single letter) are:

'labels' [Optional] - Tick labels to place at the positions of the Q points in argument rlp. e.g. {'G', 'X', 'M', 'R'}. By default the labels are character representations of rlp, e.g. {0, 0, 0; 0.5, 0, 0; 0.5, 0.5, 0; 0.5, 0.5, 0.5} becomes {'0, 0, 0', '0.5, 0, 0', '0.5, 0.5, 0', '0.5, 0.5, 0.5'}
'ndiv', N [Optional] - Number of points into which to divide the interval between two r.l.p. (default=100)
'noplot' [Optional] - Do not plot, just return the output IX_dataset_1d (see below)

The output is as follows:

weight [Optional] - IX_dataset_2d containing the spectral weights. The x-axis is the distance in Ang^-1 along the path described.

The image intensity, as a function of Q along the r.l.p path along the x-axis and the energy transfer along y-axis is determined by the equation:

\[weight(\mathbf{Q}, E) = \frac{S}{\sigma\sqrt{2\pi}} \exp\left[ \frac{ - \left( w(\mathbf{Q}, \{p\}) - E \right)^{2}}{2\sigma{}^{2}} \right]\]

where \(w\) is the dispersion relation function dispreln, \(\{p\}\) are the parameters given in p, \(E\) is the energy and

\[\sigma{} = \frac{\textrm{fwhh}}{\sqrt{\log(256)}}\]