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RefinePowderInstrumentParameters v3

../_images/RefinePowderInstrumentParameters-v3_dlg.png

RefinePowderInstrumentParameters dialog.

Summary

Parameters include Dtt1, Dtt1t, Dtt2t, Zero, Zerot.

See Also

RefinePowderDiffProfileSeq

Properties

Name

Direction

Type

Default

Description

InputPeakPositionWorkspace

Input

Workspace2D

Mandatory

Data workspace containing workspace positions in TOF agains dSpacing.

WorkspaceIndex

Input

number

0

Workspace Index of the peak positions in PeakPositionWorkspace.

OutputPeakPositionWorkspace

Output

Workspace2D

Output data workspace containing refined workspace positions in TOF agains dSpacing.

InputInstrumentParameterWorkspace

Input

TableWorkspace

Mandatory

INput tableWorkspace containg instrument’s parameters.

OutputInstrumentParameterWorkspace

Output

TableWorkspace

Output tableworkspace containing instrument’s fitted parameters.

RefinementAlgorithm

Input

string

MonteCarlo

Algorithm to refine the instrument parameters. Allowed values: [‘OneStepFit’, ‘MonteCarlo’]

RandomWalkSteps

Input

number

10000

Number of Monte Carlo random walk steps.

MonteCarloRandomSeed

Input

number

0

Random seed for Monte Carlo simulation.

StandardError

Input

string

ConstantValue

Algorithm to calculate the standard error of peak positions. Allowed values: [‘ConstantValue’, ‘UseInputValue’]

Damping

Input

number

1

Damping factor for (1) minimizer ‘Damped Gauss-Newton’. (2) Monte Carlo.

AnnealingTemperature

Input

number

1

Starting annealing temperature.

MonteCarloIterations

Input

number

100

Number of iterations in Monte Carlo random walk.

ChiSquare

Output

number

Description

This algorithm refines the instrumental geometry parameters for powder diffractomers. The parameters that can be refined are Dtt1, Zero, Dtt1t, Dtt2t, Zerot, Width and Tcross.

It serves as the second step to fit/refine instrumental parameters that will be introduced in Le Bail Fit. It uses the outcome from FitPowderDiffPeaks algorithm.

Introduction

In order to do Rietveld refinement to experimental data, the diffractometer’s profile should be calibrated by the standards, such as LaB6 or Ni, with known crystal structure and lattice parameters.

For POWGEN and NOMAD, the type of the instrument profile is back-to-back exponential function convoluted with pseudo voigt of thermal neutron and epithermal neutron. It means that each diffraction peak is a back-to-back exponential,

\[I\frac{AB}{2(A+B)}\left[ \exp \left( \frac{A[AS^2+2(x-X0)]}{2}\right) \mbox{erfc}\left( \frac{AS^2+(x-X0)}{S\sqrt{2}} \right) + \exp \left( \frac{B[BS^2-2(x-X0)]}{2} \right) \mbox{erfc} \left( \frac{[BS^2-(x-X0)]}{S\sqrt{2}} \right) \right].\]

with peak parameter \(A\), \(B\), \(X_0\) and \(S\)

And their corresponding peak parameters are functions described as:

\[ \begin{align}\begin{aligned}n_{cross} = \frac{1}{2} \text{erfc}(Width(xcross\cdot d^{-1}))\\TOF_e = Zero + Dtt1\cdot d\\TOF_t = Zerot + Dtt1t\cdot d - Dtt2t \cdot d^{-1}\end{aligned}\end{align} \]

Final Time-of-flight is calculated as:

\[TOF = n_{cross} TOF_e + (1-n_{cross}) TOF_t\]

Formula for calculating \(A(d)\), \(B(d)\), \(\sigma(d)\) and \(\gamma(d)\)

\(\alpha(d)\):

\[ \begin{align}\begin{aligned}\alpha^e(d) = \alpha_0^e + \alpha_1^e d_h\\\alpha^t(d) = \alpha_0^t - \frac{\alpha_1^t}{d_h}\\\alpha(d) = \frac{1}{n\alpha^e + (1-n)\alpha^t}\end{aligned}\end{align} \]

\(\beta(d)\):

\[ \begin{align}\begin{aligned}\beta^e(d) = \beta_0^e + \beta_1^e d_h\\\beta^t(d) = \beta_0^t - \frac{\beta_1^t}{d_h}\\\beta(d) = \frac{1}{n\alpha^e + (1-n)\beta^t}\end{aligned}\end{align} \]

For \(\sigma_G\) and \(\gamma_L\), which represent the standard deviation for pseudo-voigt

\[ \begin{align}\begin{aligned}\sigma_G^2(d_h) = \sigma_0^2 + (\sigma_1^2 + DST2(1-\zeta)^2)d_h^2 + (\sigma_2^2 + Gsize)d_h^4\\\gamma_L(d_h) = \gamma_0 + (\gamma_1 + \zeta\sqrt{8\ln2DST2})d_h + (\gamma_2+F(SZ))d_h^2\end{aligned}\end{align} \]

The analysis formula for the convoluted peak at \(d_h\)

\[\Omega(TOF(d_h)) = (1-\eta(d_h))N\{e^u\text{erfc}(y)+e^v\text{erfc}(z)\} - \frac{2N\eta}{\pi}\{\Im[e^pE_1(p)]+\Im[e^qE_1(q)]\}\]

where

\[ \begin{align}\begin{aligned}\text{erfc}(x) = 1-\text{erf}(x) = 1-\frac{2}{\sqrt{\pi}}\int_0^xe^{-u^2}du\\E_1(z) = \int_z^{\infty}\frac{e^{-t}}{t}dt\\u = \frac{1}{2}\alpha(d_h)(\alpha(d_h)\sigma^2(d_h)+2x)\\y = \frac{\alpha(d_h)\sigma^2(d_h)+x}{\sqrt{2\sigma^2(d_h)}}\\p = \alpha(d_h)x + \frac{i\alpha(d_h)H(d_h)}{2}\\v = \frac{1}{2}\beta(d_h)(\beta(d_h)\sigma^2(d_h)-2x)\\z = \frac{\beta(d_h)\sigma^2(d_h)-x}{\sqrt{2\sigma^2(d_h)}}\\q = -\beta(d_h)x + \frac{i\beta(d_h)H(d_h)}{2}\end{aligned}\end{align} \]

\(\text{erfc}(x)\) and \(E_1(z)\) will be calculated numerically.

Break down the problem

If we can do the single peak fitting on each single diffraction peak in a certain range, then we can divide the optimization problem into 4 sub problems for \(X_0\), \(A\), \(B\) and \(S\), with the constraint on \(n\), the ratio between thermal and epi thermal neutrons.

The function to fit is

\(X_0\):

\[TOF\_h = n(Zero + Dtt1\cdot d) + (1-n)(Zerot + Dtt1t\cdot d + Dtt2t/d)\]

\(A\):

\[\alpha(d) = \frac{1}{n\alpha^e + (1-n)\alpha^t}\]

\(B\):

\[\beta(d) = \frac{1}{n\alpha^e + (1-n)\beta^t}\]

\(S\):

\[\sigma_G^2(d_h) = \sigma_0^2 + (\sigma_1^2 + DST2(1-\zeta)^2)d_h^2 + (\sigma_2^2 + Gsize)d_h^4\]

with constraint:

\[n = 1/2 \text{erfc}(W\cdot (1-Tcross/d))\]

The coefficients in this function are strongly correlated to each other.

Current Implementation

Only the parameters of the function for \(X_0\) are fitted in present implementation.

Refinement Algorithm

Two refinement algorithms, DirectFit and MonteCarlo, are provided.

DirectFit

This is a simple one step fitting. If there is one parameter to fit, Levenberg Marquart minimizer is chosen. As its coefficients are strongly correlated to each other, Simplex minimizer is used if there are more than 1 parameter to fit.

MonteCarlo

This adopts the concept of Monte Carlo random walk in the parameter space. In each MC step, one parameter will be chosen, and a new value is proposed for it. A constraint fitting by Simplex minimizer is used to fit the coefficients in new configuration.

Simulated annealing will be tried as soon as it is implemented in Mantid.

Constraint

In future, constaint will be considered.

How to use algorithm with other algorithms

This algorithm is designed to work with other algorithms to do Le Bail fit. The introduction can be found in the wiki page of LeBailFit v1.

Usage

Categories: AlgorithmIndex | Diffraction\Fitting

Source

C++ header: RefinePowderInstrumentParameters3.h

C++ source: RefinePowderInstrumentParameters3.cpp