Integrated asymmetry

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Integrated asymmetry calculation

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

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

where

[math]up=\sum{up(ii)}[/math] - D2O, detector Up
Wup - weighted coefficient (Nai or Ref)

[math]upbg=\sum{upbg(ii)}[/math] - H2O, detector Up
Wup - weighted coefficient (Nai or Ref)

[math]sd=\sum{sd(ii)}[/math] - D2O, detector Side
Wup - weighted coefficient (Nai or Ref)

[math]sdbg=\sum{sdbg(ii)}[/math] - H2O, detector Side
Wup - weighted coefficient (Nai or Ref)

• 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]

what is [math]\sqrt{up}[/math] ?  Does [math]\sqrt{up}[/math] = number of counts in in the detector up spectrum as suggested above?  Why not use [math]N_{up}[/math] as variable.  If distribution is Binomial (detector yes/no) then width of distribution is [math]1.265\sqrt{N}[/math] = 2 times e error.  Relative error = [math]\frac{error}{Ave} = \frac{0.6325 \sqrt{N}}{N/2}[/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 (was analized all possible combination):

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 (was analized all possible combination):

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];

Weighted coefficients was used

(File:Asymm table .pdf)

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). A:Will do it.
Only one "s" in Asymmetry. A:Will do it.

Example of bin by bin asymmetry

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