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PDFFourierTransform v2

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Summary

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

See Also

FFT

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.)

Description

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

  1. 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

  2. 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

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

G(r)

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

RDF(r)

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

G_k(r)

\(G_k(r) = \frac{0.01 * \langle b_{coh} \rangle ^2 G^{PDF}(r)}{(4 \pi)^2 \rho_0 r} = 0.01 * \langle b_{coh} \rangle ^2 [g(r)-1]\)

Note: All output forms are calculated by transforming \(g(r)-1\).

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)
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

Source

C++ header: PDFFourierTransform2.h

C++ source: PDFFourierTransform2.cpp