Difference between revisions of "Integrated asymmetry"

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+
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=Integrated asymmetry calculation=
 
=Integrated asymmetry calculation=
 
<math>Asymm^{detA,detC} = \frac
 
<math>Asymm^{detA,detC} = \frac
 
  {\left(\frac{N_{up}^{D_2O}}{W_{up}^{D_2O}}-\frac{N_{up}^{H_2O}}{W_{up}^{H_2O}}\right)
 
  {\left(\frac{N_{up}^{D_2O}}{W_{up}^{D_2O}}-\frac{N_{up}^{H_2O}}{W_{up}^{H_2O}}\right)
- \left(\frac{N_{sd}^{D_2O}}{W_{sd}^{D_2O}}-\frac{N_{sd}^{H_2O}}{W_{sd}^{H_2O}}\right)}
+
- \left(\frac{N_{side}^{D_2O}}{W_{sided}^{D_2O}}-\frac{N_{side}^{H_2O}}{W_{sd}^{H_2O}}\right)}
 
  {\left(\frac{N_{up}^{D_2O}}{W_{up}^{D_2O}}-\frac{N_{up}^{H_2O}}{W_{up}^{H_2O}}\right)
 
  {\left(\frac{N_{up}^{D_2O}}{W_{up}^{D_2O}}-\frac{N_{up}^{H_2O}}{W_{up}^{H_2O}}\right)
+ \left(\frac{N_{sd}^{D_2O}}{W_{sd}^{D_2O}}-\frac{N_{sd}^{H_2O}}{W_{sd}^{H_2O}}\right)}
+
+ \left(\frac{N_{side}^{D_2O}}{W_{sd}^{D_2O}}-\frac{N_{sided}^{H_2O}}{W_{sd}^{H_2O}}\right)}
 
</math><br><br>
 
</math><br><br>
  
 
where<br>
 
where<br>
  <math>N_{up}^{D_2O}=\sum_{ii=min}^{max}{count(ii)};</math>  integrated number of counts for detector Up, D2O<br>
+
  <math>N_{k}^{m}=\sum_{ii=min}^{max}{count(ii)};</math>  total number of neutrons detected, k = [Up,Side], m = [D2O, H2O]<br>
<math>N_{up}^{H_2O}=\sum_{ii=min}^{max}{count(ii)};</math>  integrated number of counts for detector Up, H2O<br>
+
  <math>W_{k}^{m};</math>  weighted (NaI, Ref) coefficient, k = [Up,Side], m = [D2O, H2O] <br>
<math>N_{sd}^{D_2O}=\sum_{ii=min}^{max}{count(ii)};</math>  integrated number of counts for detector Side, D2O<br>
 
<math>N_{sd}^{H_2O}=\sum_{ii=min}^{max}{count(ii)};</math>  integrated number of counts for detector Side, H2O<br>  
 
  <math>W_{up}^{D_2O};</math>  weighted (NaI, Ref) coefficient for detector Up, D2O
 
<math>W_{up}^{H_2O};</math>  weighted (NaI, Ref) coefficient for detector Up, H2O
 
<math>W_{sd}^{D_2O};</math>  weighted (NaI, Ref) coefficient for detector Side, D2O
 
<math>W_{sd}^{H_2O};</math>  weighted (NaI, Ref) coefficient for detector Side, H2O
 
  
 
*For detector A summation is over [1000:1600] bin numbers<br>
 
*For detector A summation is over [1000:1600] bin numbers<br>
Line 24: Line 18:
 
=Error calculation=
 
=Error calculation=
 
<math>\delta (Asymm) =
 
<math>\delta (Asymm) =
  \sqrt{\left(\frac{\partial A}{\partial N_{up}^{D_2O}}\right)^2(\delta N_{up}^{D_2O})^2 +
+
  \sqrt{\sum_{k=up}^{side}\ \sum_{m=D_2O}^{H_2O}
\left(\frac{\partial A}{\partial W_{up}^{D_2O}}\right)^2(\delta W_{up}^{D_2O})^2 + ...}</math>
+
      \left[\left(\frac{\partial A}{\partial N_{k}^{m}}\delta N_{k}^{m}\right)^2 +
 +
            \left(\frac{\partial A}{\partial W_{k}^{m}}\delta W_{k}^{m}\right)^2\right]}</math>
  
If assume:<br>
+
What about
<math>\delta N_{up}^{D_2O} = \sqrt{N_{up}^{D_2O}}</math>
+
 
<math>\delta W_{up}^{D_2O} = \sqrt{N_{up}^{D_2O}};\quad...</math><br><br>
+
<math>\frac{\partial A}{\partial N_{k}^{m}}\frac{\partial A}{\partial W_{k}^{m}}</math>
  
 
<pre>
 
<pre>
Line 44: Line 39:
 
</pre>
 
</pre>
  
Then<br>
+
If we take:<br>
<math>\delta (Asymm) = \sqrt{\left(\frac{(+)-(-)}{(+)^2}\right)^2\left[\frac{N_{up}^{D_2O}}{W_{up}^{D_2O}^2}+\frac{N_{up}^{D_2O}^2}{W_{up}^{D_2O}}+\frac{N_{up}^{H_2O}}{W_{up}^{H_2O}^2}+\frac{N_{up}^{H_2O}^2}{W_{up}^{H_2O}}\right] +
+
<math>\delta N_{k}^{m} = \sqrt{N_{k}^{m}}</math><br>
                    \left(\frac{(+)+(-)}{(+)^2}\right)^2\left[\frac{sd}{Wsd^2}+\frac{sd^2}{Wsd}+\frac{sdbg}{Wsdbg^2}+\frac{sdbg^2}{Wsdbg}\right]}</math><br>
+
<math>\delta W_{k}^{m} = \sqrt{N_{k}^{m}}</math><br><br>
 +
 
 +
then<br>
 +
<math>\delta (Asymm) =
 +
\sqrt{
 +
\left(\frac{(+)-(-)}{(+)^2}\right)^2\left(
 +
\frac{N_{up}^{D_2O}}{(W_{up}^{D_2O})^2}+\frac{(N_{up}^{D_2O})^2}{W_{up}^{D_2O}}
 +
+\frac{N_{up}^{H_2O}}{(W_{up}^{H_2O})^2}+\frac{(N_{up}^{H_2O})^2}{W_{up}^{H_2O}}\right)+
 +
\left(\frac{(+)+(-)}{(+)^2}\right)^2\left(
 +
\frac{N_{sd}^{D_2O}}{(W_{sd}^{D_2O})^2}+\frac{(N_{sd}^{D_2O})^2}{W_{sd}^{D_2O}}
 +
+\frac{N_{sd}^{H_2O}}{(W_{sd}^{H_2O})^2}+\frac{(N_{sd}^{H_2O})^2}{W_{sd}^{H_2O}}\right)}</math><br><br>
 +
 
 
where<br>
 
where<br>
<math>(-)=\left[\left(\frac{up}{Wup}-\frac{upbg}{Wupbg}\right) - \left(\frac{sd}{Wsd}-\frac{sdbg}{Wsdbg}\right)\right]</math>
+
<math>(-)=\left[
<br><br>
+
\left(\frac{N_{up}^{D_2O}}{W_{up}^{D_2O}}-\frac{N_{up}^{H_2O}}{W_{up}^{H_2O}}\right)
<math>(+)=\left[\left(\frac{up}{Wup}-\frac{upbg}{Wupbg}\right) + \left(\frac{sd}{Wsd}-\frac{sdbg}{Wsdbg}\right)\right]</math>
+
-\left(\frac{N_{sd}^{D_2O}}{W_{sd}^{D_2O}}-\frac{N_{sd}^{H_2O}}{W_{sd}^{H_2O}}\right)\right]</math>
 
<br><br>
 
<br><br>
 +
<math>(-)=\left[
 +
\left(\frac{N_{up}^{D_2O}}{W_{up}^{D_2O}}-\frac{N_{up}^{H_2O}}{W_{up}^{H_2O}}\right)
 +
+\left(\frac{N_{sd}^{D_2O}}{W_{sd}^{D_2O}}-\frac{N_{sd}^{H_2O}}{W_{sd}^{H_2O}}\right)\right]</math><br><br>
  
 
=Cases was analysed=
 
=Cases was analysed=
 
Det A (was analized all possible combination):  
 
Det A (was analized all possible combination):  
  D2O Up,  files# [40,56,102,108,134,205,210,230];<br>
+
  D2O Up,  files# [40,56,102,108,205,210,230];<br>
 
  H2O Up,  files# [44];<br>
 
  H2O Up,  files# [44];<br>
  D2O Side, files# [48,74,78,82,86,90,94,146,180,190,225,235];<br>
+
  D2O Side, files# [48,74,78,82,86,90,94,146,190,225,235];<br>
 
  H2O Side, files# [52];<br>
 
  H2O Side, files# [52];<br>
 
Det C (was analized all possible combination):
 
Det C (was analized all possible combination):
  D2O Up,  files# [49,75,79,83,87,91,95,147,181,191,226,236];<br>
+
  D2O Up,  files# [49,75,79,83,87,91,95,147,191,226,236];<br>
 
  H2O Up,  files# [53];<br>
 
  H2O Up,  files# [53];<br>
  D2O Side, files# [41,57,103,107,135,206,211,231];<br>
+
  D2O Side, files# [41,57,103,107,206,211,231];<br>
 
  H2O Side, files# [45];<br><br>
 
  H2O Side, files# [45];<br><br>
  
Line 74: Line 83:
 
  Table 4: Det C, weighted with <math>{\color{Red}Ref \ detector}</math><br><br><br>
 
  Table 4: Det C, weighted with <math>{\color{Red}Ref \ detector}</math><br><br><br>
 
([[Image:Asymm_table.pdf]])<br><br>
 
([[Image:Asymm_table.pdf]])<br><br>
 
Change the X-axis to nanosecond or neutron energy (TF). A:Will do it.
 
  
 
=Example of bin by bin asymmetry=
 
=Example of bin by bin asymmetry=
[[Image:Asymm_RefDet_DetA_40,44,48(74,78),52.jpg]]
+
Change the X-axis to nanosecond or neutron energy (TF). A:Have done.
 +
[[Image:Asymm_RefDet_DetA_40,44,48(74,78),52.jpg]]<br><br>
 +
[[Image:Asymm_RefDet_DetC_49,53,41(57,103),45.jpg]]
  
  
 
[http://wiki.iac.isu.edu/index.php/PhotoFission_with_Polarized_Photons_from_HRRL Go Back]
 
[http://wiki.iac.isu.edu/index.php/PhotoFission_with_Polarized_Photons_from_HRRL Go Back]

Latest revision as of 19:02, 24 May 2012

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Integrated asymmetry calculation

[math]Asymm^{detA,detC} = \frac {\left(\frac{N_{up}^{D_2O}}{W_{up}^{D_2O}}-\frac{N_{up}^{H_2O}}{W_{up}^{H_2O}}\right) - \left(\frac{N_{side}^{D_2O}}{W_{sided}^{D_2O}}-\frac{N_{side}^{H_2O}}{W_{sd}^{H_2O}}\right)} {\left(\frac{N_{up}^{D_2O}}{W_{up}^{D_2O}}-\frac{N_{up}^{H_2O}}{W_{up}^{H_2O}}\right) + \left(\frac{N_{side}^{D_2O}}{W_{sd}^{D_2O}}-\frac{N_{sided}^{H_2O}}{W_{sd}^{H_2O}}\right)} [/math]

where

[math]N_{k}^{m}=\sum_{ii=min}^{max}{count(ii)};[/math]   total number of neutrons detected, k = [Up,Side], m = [D2O, H2O]
[math]W_{k}^{m};[/math] weighted (NaI, Ref) coefficient, k = [Up,Side], m = [D2O, H2O]
  • For detector A summation is over [1000:1600] bin numbers
  • For detector C summation is over [900:1600] bin numbers

Error calculation

[math]\delta (Asymm) = \sqrt{\sum_{k=up}^{side}\ \sum_{m=D_2O}^{H_2O} \left[\left(\frac{\partial A}{\partial N_{k}^{m}}\delta N_{k}^{m}\right)^2 + \left(\frac{\partial A}{\partial W_{k}^{m}}\delta W_{k}^{m}\right)^2\right]}[/math]

What about

[math]\frac{\partial A}{\partial N_{k}^{m}}\frac{\partial A}{\partial W_{k}^{m}}[/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>

If you are measuring total number of neutrons detected then you will have binomial if you 
break up the time spectrum into bins and don't integrate then probability distribution
is combination of detection probability and time measurement probability.

If we take:
[math]\delta N_{k}^{m} = \sqrt{N_{k}^{m}}[/math]
[math]\delta W_{k}^{m} = \sqrt{N_{k}^{m}}[/math]

then
[math]\delta (Asymm) = \sqrt{ \left(\frac{(+)-(-)}{(+)^2}\right)^2\left( \frac{N_{up}^{D_2O}}{(W_{up}^{D_2O})^2}+\frac{(N_{up}^{D_2O})^2}{W_{up}^{D_2O}} +\frac{N_{up}^{H_2O}}{(W_{up}^{H_2O})^2}+\frac{(N_{up}^{H_2O})^2}{W_{up}^{H_2O}}\right)+ \left(\frac{(+)+(-)}{(+)^2}\right)^2\left( \frac{N_{sd}^{D_2O}}{(W_{sd}^{D_2O})^2}+\frac{(N_{sd}^{D_2O})^2}{W_{sd}^{D_2O}} +\frac{N_{sd}^{H_2O}}{(W_{sd}^{H_2O})^2}+\frac{(N_{sd}^{H_2O})^2}{W_{sd}^{H_2O}}\right)}[/math]

where
[math](-)=\left[ \left(\frac{N_{up}^{D_2O}}{W_{up}^{D_2O}}-\frac{N_{up}^{H_2O}}{W_{up}^{H_2O}}\right) -\left(\frac{N_{sd}^{D_2O}}{W_{sd}^{D_2O}}-\frac{N_{sd}^{H_2O}}{W_{sd}^{H_2O}}\right)\right][/math]

[math](-)=\left[ \left(\frac{N_{up}^{D_2O}}{W_{up}^{D_2O}}-\frac{N_{up}^{H_2O}}{W_{up}^{H_2O}}\right) +\left(\frac{N_{sd}^{D_2O}}{W_{sd}^{D_2O}}-\frac{N_{sd}^{H_2O}}{W_{sd}^{H_2O}}\right)\right][/math]

Cases was analysed

Det A (was analized all possible combination):

D2O Up,   files# [40,56,102,108,205,210,230];
H2O Up, files# [44];
D2O Side, files# [48,74,78,82,86,90,94,146,190,225,235];
H2O Side, files# [52];

Det C (was analized all possible combination):

D2O Up,   files# [49,75,79,83,87,91,95,147,191,226,236];
H2O Up, files# [53];
D2O Side, files# [41,57,103,107,206,211,231];
H2O Side, files# [45];

Weighted coefficients was used

(File:Weight coeff.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)

Example of bin by bin asymmetry

Change the X-axis to nanosecond or neutron energy (TF). A:Have done.

Asymm RefDet DetA 40,44,48(74,78),52.jpg

Asymm RefDet DetC 49,53,41(57,103),45.jpg


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