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PDFFourierTransform v1¶

../_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	S(Q), S(Q)-1, or Q[S(Q)-1]
OutputWorkspace	Output	MatrixWorkspace	Mandatory	Result paired-distribution function
InputSofQType	Input	string	S(Q)	To identify whether input function. Allowed values: [‘S(Q)’, ‘S(Q)-1’, ‘Q[S(Q)-1]’]
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)
Filter	Input	boolean	False	Set to apply Lorch function filter to the input
PDFType	Input	string	G(r)	Type of output PDF including G(r). Allowed values: [‘G(r)’, ‘g(r)’, ‘RDF(r)’]
DeltaR	Input	number	Optional	Step size of r of G(r) to calculate. Default = \(\frac{\pi}{Q_{max}}\).
Rmax	Input	number	20	Maximum r for G(r) to calculate.
rho0	Input	number	Optional	Average number density used for g(r) and RDF(r) conversions (optional)

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. The available output types are the reduced pair distribution function \(G(r)\), the pair distribution function \(g(r)\), and the radial distribution function \(RDF(r)\).

The input Workspace spectrum 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

Output Options ¶

G(r)¶

\(G(r) = 4\pi r[\rho(r)-\rho_0] = \frac{2}{\pi} \int_{0}^{\infty} Q[S(Q)-1]\sin(Qr)dQ\)

and in this algorithm, it is implemented as

\(G(r) = \frac{2}{\pi} \sum_{Q_{min}}^{Q_{max}} Q[S(Q)-1]\sin(Qr) M(Q,Q_{max}) \Delta Q\)

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

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

g(r)¶

\(G(r) = 4 \pi \rho_0 r [g(r)-1]\)

transforms to

\(g(r) = \frac{G(r)}{4 \pi \rho_0 r} + 1\)

RDF(r)¶

\(RDF(r) = 4 \pi \rho_0 r^2 g(r)\)

transforms to

\(RDF(r) = r G(r) + 4 \pi \rho_0 r^2\)

Note: All output forms except \(G(r)\) are calculated by transforming \(G(r)\).

Usage ¶

Example - PDF transformation examples:

# 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)
ws = CreateWorkspace(DataX=xx, DataY=yy, UnitX='MomentumTransfer')
Rt = PDFFourierTransform(ws, InputSofQType='S(Q)', PDFType='g(r)', Version=1)

# Look at sample results:
print('part of S(Q) 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) and its correlation function
! 0.00 ! 1.000000 ! 0.317333 ! -3.977330 !
! 0.10 ! 0.960789 ! 0.634665 ! 2.247452 !
! 0.20 ! 0.852144 ! 0.951998 ! 0.449677 !
! 0.30 ! 0.697676 ! 1.269330 ! 1.313403 !
! 0.40 ! 0.527292 ! 1.586663 ! 0.803594 !
! 0.50 ! 0.367879 ! 1.903996 ! 1.140167 !
! 0.60 ! 0.236928 ! 2.221328 ! 0.900836 !
! 0.70 ! 0.140858 ! 2.538661 ! 1.079278 !
! 0.80 ! 0.077305 ! 2.855993 ! 0.940616 !
! 0.90 ! 0.039164 ! 3.173326 ! 1.050882 !

Categories: AlgorithmIndex | Diffraction\Utility