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PDFFourierTransform v2¶

../_images/ImageNotFound.png — Enable screenshots using DOCS_SCREENSHOTS in CMake¶

Summary ¶

Fourier transform from S(Q) to G(r), which is paired distribution function (PDF). G(r) will be stored in another named workspace.

Properties ¶

Name	Direction	Type	Default	Description
InputWorkspace	Input	MatrixWorkspace	Mandatory	Input spectrum density or paired-distribution function
OutputWorkspace	Output	MatrixWorkspace	Mandatory	Result paired-distribution function or Input spectrum density
Direction	Input	string	Forward	The direction of the fourier transform. Allowed values: [‘Forward’, ‘Backward’]
rho0	Input	number	Optional	Average number density used for g(r) and RDF(r) conversions (optional)
Filter	Input	boolean	False	Set to apply Lorch function filter to the input
InputSofQType	Input	string	S(Q)	To identify spectral density function (deprecated). Allowed values: [‘S(Q)’, ‘S(Q)-1’, ‘Q[S(Q)-1]’]
SofQType	Input	string	S(Q)	To identify spectral density function. Allowed values: [‘S(Q)’, ‘S(Q)-1’, ‘Q[S(Q)-1]’]
DeltaQ	Input	number	Optional	Step size of Q of S(Q) to calculate. Default = \(\frac{\pi}{R_{max}}\).
Qmin	Input	number	Optional	Minimum Q in S(Q) to calculate in Fourier transform (optional).
Qmax	Input	number	Optional	Maximum Q in S(Q) to calculate in Fourier transform. (optional, defaults to 40 on backward transform.)
PDFType	Input	string	G(r)	Type of output PDF including G(r). Allowed values: [‘G(r)’, ‘g(r)’, ‘RDF(r)’, ‘G_k(r)’]
DeltaR	Input	number	Optional	Step size of r of G(r) to calculate. Default = \(\frac{\pi}{Q_{max}}\).
Rmin	Input	number	Optional	Minimum r for G(r) to calculate. (optional)
Rmax	Input	number	Optional	Maximum r for G(r) to calculate. (optional, defaults to 20 on forward transform.)

The algorithm transforms a single spectrum workspace containing spectral density \(S(Q)\), \(S(Q)-1\), or \(Q[S(Q)-1]\) (as a function of MomentumTransfer or dSpacing units) to a PDF (pair distribution function) as described below and also the reverse. The available PDF types are the reduced pair distribution function \(G(r)\), the pair distribution function \(g(r)\), the radial distribution function \(RDF(r)\), and the total radial distribution function \(G_k(r)\).

The output from this algorithm will have an x-range between 0.0 and the maximum parameter of the output, i.e. if converting from g(r) to S(Q) the output will be between 0.0 and Qmax.

The spectrum density should be in the Q-space (MomentumTransfer) units . (d-spacing is not supported any more. Contact development team to fix that and enable dSpacing again)

References ¶

B. H. Toby and T. Egami, Accuracy of Pair Distribution Functions Analysis Appliced to Crystalline and Non-Crystalline Materials, Acta Cryst. (1992) A 48, 336-346 doi: 10.1107/S0108767391011327
B.H. Toby and S. Billinge, Determination of Standard uncertainties in fits to pair distribution functions Acta Cryst. (2004) A 60, 315-317] doi: 10.1107/S0108767304011754

PDF Options ¶

g(r)¶

\(g(r) = \rho(r)/\rho_0 = 1+\frac{1}{2\pi^2\rho_0r} \int_{0}^{\infty} Q[S(Q)-1]\sin(Qr)dQ\)

and in this algorithm, it is implemented as

\(g(r)-1 = \frac{1}{2\pi \rho_0 r^3} \sum_{Q_{min}}^{Q_{max}} M(Q,Q_{max})(S(Q)-1)[\sin(Qr) - Qr\cos(Qr)]^{right bin}_{left bin}\)

where \(M(Q,Q_{max})\) is an optional filter function. If Filter property is set (true) then

\(M(Q,Q_{max}) = \frac{\sin(\pi Q/Q_{max})}{\pi Q/Q_{max}}\)

otherwise

\(M(Q,Q_{max}) = 1\,\)

# Simulates Load of a workspace with all necessary parameters
import numpy as np;
xx = np.array(range(0,100))*0.1
yy = np.exp(-(2.0 * xx)**2)
yy = np.delete(yy,-1) # need one less Y value than X value for histogram data
ws = CreateWorkspace(DataX=xx, DataY=yy, UnitX='MomentumTransfer')
Rt = PDFFourierTransform(ws, SofQType='S(Q)-1', PDFType='g(r)')

# Look at sample results:
print('part of S(Q)-1 and its correlation function')
for i in range(10):
   print('! {0:4.2f} ! {1:5f} ! {2:f} ! {3:5f} !'.format(xx[i], yy[i], Rt.readX(0)[i], Rt.readY(0)[i]))

Output:

part of S(Q)-1 and its correlation function
! 0.00 ! 1.000000 ! 0.317333 ! 1.003494 !
! 0.10 ! 0.960789 ! 0.634665 ! 1.003423 !
! 0.20 ! 0.852144 ! 0.951998 ! 1.003308 !
! 0.30 ! 0.697676 ! 1.269330 ! 1.003154 !
! 0.40 ! 0.527292 ! 1.586663 ! 1.002965 !
! 0.50 ! 0.367879 ! 1.903996 ! 1.002750 !
! 0.60 ! 0.236928 ! 2.221328 ! 1.002515 !
! 0.70 ! 0.140858 ! 2.538661 ! 1.002269 !
! 0.80 ! 0.077305 ! 2.855993 ! 1.002018 !
! 0.90 ! 0.039164 ! 3.173326 ! 1.001770 !

Categories: AlgorithmIndex | Diffraction\Utility