$$

HeliumAnalyserEfficiency v1

Summary

Calculates the efficiency of a He3 analyser.

Properties

Name	Direction	Type	Default	Description
InputWorkspace	Input	WorkspaceGroup	Mandatory	Input group workspace to use for polarization calculation
SpinStates	Input	string	11,10,01,00	Order of individual flipper configurations in the input group workspace, e.g. “01,11,00,10”
PXD	Input	number	12	Gas pressure in bar multiplied by cell length in metres
PXDError	Input	number	0	Error in gas pressure multiplied by cell length
StartX	Input	number	1.75	Lower boundary of wavelength range to use for fitting
EndX	Input	number	8	Upper boundary of wavelength range to use for fitting
IgnoreFitQualityError	Input	boolean	False	Whether the algorithm should ignore a poor chi-squared (fit cost value) of greater than 1 and therefore not throw an error
OutputWorkspace	Output	MatrixWorkspace	Mandatory	Helium analyzer efficiency as a function of wavelength
OutputFitCurves	Output	MatrixWorkspace		The name of the matrix workspace containing the calculated fit curve, the original data, and the difference between the two.
OutputFitParameters	Output	TableWorkspace		The name of the table workspace containing the fit parameter results.

Description

Takes a normalised group workspace and combines this with the cell path length multiplied by the gas pressure (PXD) to calculate the efficiency of the analyser. The input workspace must be a group of four single spectrum histogram workspaces, each representing the transmission of a known spin state as specified by the SpinStates parameter.

If the transmission of the wanted spin state is $T_{para}$, and the transmission of the unwanted spin state is $T_{anti}$, then the polarization of of an unpolarised incoming beam after the analyser cell, $P_{cell}$ is given by [1]

$$P_{cell}=\frac{T_{para}-T_{anti}}{T_{para}+T_{anti}}=\tanh (0.0733pd\lambda p_{He})$$

The efficiency of the analyser cell is given by

$${ϵ}_{cell}=\frac{1+P_{cell}}{2}=\frac{T_{para}}{T_{para}+T_{anti}}=\frac{1+\tanh (0.0733pd\lambda p_{He})}{2}$$

If our four periods are $T_{00},T_{01},T_{10},T_{11}$, with the subscript denoting the spin configuration, then $T_{para}=T_{00}+T_{11}$ and $T_{anti}=T_{01}+T_{10}$, and we can calculate ${ϵ}_{cell}$ from the above equation. The value of $pd$ is given by the input PXD, and $\lambda$ is the wavelength of each bin, so we fit $\frac{1+\tanh (0.0733pd\lambda p_{He})}{2}$ to our calculated ${ϵ}_{cell}$ to give us $p_{He}$ and hence the theoretical efficiency curve.

To calculate the error, ${\sigma }_{{ϵ}_{cell}}$, we need the error on $p_{He}$, ${\sigma }_{p_{He}}$, and the error of $pd$, ${\sigma }_{pd}$, given by PXDError. The covariance between $pd$ and $p_{He}$ is assumed to be zero. Then the error ${\sigma }_{{ϵ}_{cell}}$ is given by

$$\frac{{\sigma }_{{ϵ}_{cell}}}{t_{crit}}=\sqrt{{\left(\frac{\partial {ϵ}_{cell}}{\partial p_{He}}{\sigma }_{p_{He}}\right)}^2+{\left(\frac{\partial {ϵ}_{cell}}{\partial mu}{\sigma }_{mu}\right)}^2+{\left(\frac{\partial {ϵ}_{cell}}{\partial \lambda }{\sigma }_{\lambda }\right)}^2}$$

where $mu=0.0733\ast pd$ and ${\sigma }_{\lambda }$ is the width of the wavelength bin.

If $n_b$ is the number of histogram bins used in the fit, then define $n:=n_b-1$, since we are fitting one parameter ($p_{He}$). Then the factor $t_{crit}$ follows a $t$ distribution with $n$ degrees of freedom, and probability density function $f_t(x,n)$. For a standard 68.3% (1-sigma) error the factor $t_{crit}$ is given by the solution to

$$\frac{1}{2}(1+\mathrm{erf}\left(\frac{1}{\sqrt{2}}\right))=P(X<t_{crit})={\int }_0^{t_{crit}}{f}_t(x;n)dx$$

As the number of histogram bins used in the fit increases, $t_{crit}\to 1$.

Usage

Example - Calculate Analyser Efficiency

wsPara = CreateSampleWorkspace('Histogram', Function='User Defined', UserDefinedFunction='name=UserFunction,Formula=0.5*exp(-0.0733*12*x*(1-0.9))',XUnit='Wavelength', xMin='1',XMax='8', BinWidth='1', NumBanks='1', BankPixelWidth='1')
wsPara1 = CloneWorkspace(wsPara)
wsAnti = CreateSampleWorkspace('Histogram', Function='User Defined', UserDefinedFunction='name=UserFunction,Formula=0.5*exp(-0.0733*12*x*(1+0.9))',XUnit='Wavelength', xMin='1',XMax='8', BinWidth='1', NumBanks='1', BankPixelWidth='1')
wsAnti1 = CloneWorkspace(wsAnti)

grp = GroupWorkspaces([wsPara,wsAnti,wsPara1,wsAnti1])
eff = HeliumAnalyserEfficiency(grp, SpinStates='11,01,00,10')

print('Efficiency at ' + str(mtd['eff'].dataX(0)[0]) + ' Å = ' + str(mtd['eff'].dataY(0)[0]))
print('Error in efficiency at ' + str(mtd['eff'].dataX(0)[0]) + ' Å = ' + str(mtd['eff'].dataE(0)[0]))

Output:

Efficiency at 1.0 Å = ...
Error in efficiency at 1.0 Å = ...

References

[1]

Polarization-analyzed small-angle neutron scattering. I. Polarized data reduction using Pol-Corr, Kathryn Krycka et al, Journal of Applied Crystallography, 45 (2012), 546-553 doi: 10.1107/S0021889812003445

Categories: AlgorithmIndex | SANS\PolarizationCorrections

Source

C++ header: HeliumAnalyserEfficiency.h

C++ source: HeliumAnalyserEfficiency.cpp