Integrated asymmetry

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Integrated asymmetry calculation

[math]Asymm^{detA} = \frac{\sum \left[\left(\frac{up(ii)^{detA}}{Wup^{detA}}-\frac{upbg(ii)^{detA}}{Wupbg^{detA}}\right) - \left(\frac{sd(ii)^{detA}}{Wsd^{detA}}-\frac{sdbg(ii)^{detA}}{Wsdbg^{detA}}\right)\right]} {\sum \left[\left(\frac{up(ii)^{detA}}{Wup^{detA}}-\frac{upbg(ii)^{detA}}{Wupbg^{detA}}\right) + \left(\frac{sd(ii)^{detA}}{Wsd^{detA}}-\frac{sdbg(ii)^{detA}}{Wsdbg^{detA}}\right)\right]}[/math]

[math]Asymm^{detC} = \frac{\sum \left[\left(\frac{up(ii)^{detC}}{Wup^{detC}}-\frac{upbg(ii)^{detC}}{Wupbg^{detC}}\right) - \left(\frac{sd(ii)^{detC}}{Wsd^{detC}}-\frac{sdbg(ii)^{detC}}{Wsdbg^{detC}}\right)\right]} {\sum \left[\left(\frac{up(ii)^{detC}}{Wup^{detC}}-\frac{upbg(ii)^{detC}}{Wupbg^{detC}}\right) + \left(\frac{sd(ii)^{detA}}{Wsd^{detC}}-\frac{sdbg(ii)^{detC}}{Wsdbg^{detC}}\right)\right]}[/math]

where

up(ii)   - D2O, detector Up,   Wup - weighted coefficient for up(ii)

upbg(ii) - H2O, detector Up,   Wup - weighted coefficient for upbg(ii)

sd(ii)   - D2O, detector Side, Wup - weighted coefficient for sd(ii)

sdbg(ii) - H2O, detector Side, Wup - weighted coefficient for sdbg(ii)

For detector A summation is over [1000:1600] bin numbers
For detector C summation is over [900:1600] bin numbers

Error calculation

[math]Error = \sqrt{\left(\frac{\boldsymbol\partial A}{\boldsymbol\partial up}\right)^2(\boldsymbol\delta up)^2 + \left(\frac{\boldsymbol\partial A}{\boldsymbol\partial Wup}\right)^2(\boldsymbol\delta Wup)^2 + ...}[/math]

If assume:
[math]\boldsymbol\delta up = \sqrt{up};\quad\boldsymbol\delta Wup = \sqrt{Wup};\quad...[/math]

Then
[math]Error = \sqrt{\left(\frac{(+)-(-)}{(+)^2}\right)^2\left[\frac{up}{Wup^2}+\frac{up^2}{Wup}+\frac{upbg}{Wupbg^2}+\frac{upbg^2}{Wupbg}\right] + \left(\frac{(+)+(-)}{(+)^2}\right)^2\left[\frac{sd}{Wsd^2}+\frac{sd^2}{Wsd}+\frac{sdbg}{Wsdbg^2}+\frac{sdbg^2}{Wsdbg}\right]}[/math]
where
[math](-)=\left[\left(\frac{up}{Wup}-\frac{upbg}{Wupbg}\right) - \left(\frac{sd}{Wsd}-\frac{sdbg}{Wsdbg}\right)\right][/math]
[math](+)=\left[\left(\frac{up}{Wup}-\frac{upbg}{Wupbg}\right) + \left(\frac{sd}{Wsd}-\frac{sdbg}{Wsdbg}\right)\right][/math]

Cases was analysed

Det A:

D2O Up,   files# [40,56,102,108,134,205,210,230];

H2O Up,   files# [44];

D2O Side, files# [48,74,78,82,86,90,94,146,180,190,225,235];

H2O Side, files# [52];

Det C:

D2O Up,   files# [49,75,79,83,87,91,95,147,181,191,226,236];

H2O Up,   files# [53];

D2O Side, files# [41,57,103,107,135,206,211,231];

H2O Side, files# [45];

Results

Table 1: Det A, weighted with [math]{\color{Red}NaI \ detector}[/math]

Table 2: Det C, weighted with [math]{\color{Red}NaI \ detector}[/math]

Table 3: Det A, weighted with [math]{\color{Red}Ref \ detector}[/math]

Table 4: Det C, weighted with [math]{\color{Red}Ref \ detector}[/math]

(File:Asymm table .pdf)

Change the X-axis to nanosecond or neutron energy (TF)

Only one "s" in Asymmetry

Example of bin by bin asymmetry

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