View source for Se170063 Thin Window Analysis

In order to try to pin down the ratio of pure selenium to the selenium in the soil, try using a 2 channel window to find the signal. The error in the signal will be found by integrating the fit gaussian function over the window and "widening" it by half its standard deviation. Since the max value is fixed, it does not appear in the stats box, but I have included it in the table. It would seem that widening the gaussian function by half of its standard deviation yields results that agree up to 3 digits after the decimal with the result of widening the gaussian by its standard deviation. This would imply not much has changed between these methods.



{| border="3"  cellpadding="5" cellspacing="0"
|| || 400 <t< 640 sec || 1100 < t < 1360 sec || 1875 < t < 2150 || 2650 < t < 2930 sec || 3400 < t < 3690 sec || 4120 < t < 4400 sec || 4840 < t < 5130 sec 
|- 
||Thin Window||[[File:170063 PureSe ThinWindow 400 640Sec.png|100px]] || [[File:170063 PureSe 1100 1360Sec ThinWindow.png|100px]]  || [[File:170063 PureSeSpec 1875 2150Sec ThinWindow.png|100px]] || [[File:170063 PureSeSpec 2650 2930Sec ThinWindow.png|100px]] || [[File:170063 PureSeSpec 3400 3690Sec ThinWindow.png|100px]] || [[File:170063 PureSeSpec 4120 4400Sec ThinWindow.png|100px]] || [[File:170063 PureSeSpec 4840 5130Sec ThinWindow.png|100px]]
|-
||Original Window ||[[File:170063 PureSe 400 640Sec OGWindow.png|100px]] || [[File:170063 PureSe 1100 1360 Seconds OGWindow.png|100px]] || [[File:170063 PureSeSpec 1875 2150Sec OGWindow.png|100px]] || [[File:170063 PureSeSpec 2650 2930Sec OGWindow.png|100px]] || [[File:170063 PureSeSpec 3400 3690Sec OGWindow.png|100px]] || [[File:170063 PureSeSpec 4120 4400Sec OGWindow.png|100px]] || [[File:170063 PureSeSpec 4840 5130 OGWindow.png|100px]] 
|-
||Expanded Window || [[File:170063 PureSe 400 640Sec ExpWindow.png|100px]] || [[File:170063 PureSeSpectrum 1100 1360Sec ExpWindow.png|100px]] || [[File:170063 PureSeSpec 1875 2150 ExpWindow.png|100px]] || [[File:170063 PureSeSpec 2650 2930Sec ExpWindow.png|100px]] ||[[File:170063 PureSeSpec 3400 3690Sec ExpWindow.png|100px]] || [[File:170063 PureSeSpec 4120 4400Sec ExpWindow.png|100px]] || [[File:170063 PureSeSpec 4840 5130Sec ExpWindow.png|100px]]
|-
||Maximum of Histogram|| 260980 || 241756 || 220348 || 191476 || 165549 || 139528 || 130930
|-
||Original Background || 10560 +/- 51.58 || 10560 +/- 51.41 || 9496 +/- 48.9 || 8898 +/- 47.1 || 9418 +/- 48.2 || 7485 +/- 43 || 7032 +/- 31.1
|-
||Expanded Background || 10750 +/- 42.42 || 10630 +/- 42.09 || 9674 +/- 40.2 || 9029 +/- 38.7 || 9248 +/- 39.1 || 7634 +/- 35.5 || 7161 +/- 34.2
|-
||Signal Background (Take Larger Error) || 10560 +/- 51.58 || 10560 +/- 51.41 || 9496 +/- 48.9 || 8898 +/- 47.1 || 9418 +/- 48.2 || 7485 +/- 43 || 7161 +/- 34.2
|-
||Integrated Background || 21120 +/- 103.16 || 21120 +/- 102.82 || 18992 +/- 97.8 || 17796 +/- 94.2 || 18836 +/- 96.4 || 14970 +/- 86 || 14322 +/- 68.4
|-
||Signal in Thin Window || 466700 +/- 100640 || 444100 +/- 107283 || 412800 +/- 105315 || 356800 +/- 90071 || 304100 +/- 70533 || 258700 +/- 63093 || 233500 +/- 52318
|-
||Background Subtracted Signal || 445580 +/- 100640 || 422980 +/- 107283 || 393808 +/- 105315 || 339004 +/- 90071 || 285264 +/- 70533 || 243730 +/- 63093 || 219178 +/- 52318
|-
||Time Correction Factor (Sec) || 400 || 370  || 395 || 400 || 350 || 345 || 360 
|-
||Corrected Counts || 483015.53 +/- 109095.5248 || 455750 +/- 115595.0622 || 426463.63 +/- 114048.1804 || 367485.52 +/- 97638.49218 || 306127.92 +/- 75691.81615 || 261292.55 +/- 67639.40782 || 235683.31 +/- 56257.91145
|-
||
|-
||.dat file entry || 7.607165162 +/- 0.2258633895 || 7.469018062 +/- 0.2536369988 || 7.346511269 +/- 0.2676838585 || 7.179649592 +/- 0.2656934406 || 6.96187741 +/- 0.2472555138 || 6.838606337 +/- 0.2588646627 || 6.700363353 +/- 0.2387012956
|-
|}
Below is the table for the Se-Ash Mixture



{| border="3"  cellpadding="5" cellspacing="0"
|| || 0 <t< 300 sec || 730 < t < 1020 sec || 1480 < t < 1775 || 2250 < t < 2550 sec ||  3050< t < 3300 sec || 3775 < t < 4050 sec || 4480 < t < 4770 sec 
|- 
||Thin Window|| [[File:170063 MixSpec 0 300Sec ThinWindow.png|100px]] || [[File:170063 MixSpec 730 1020Sec ThinWindow.png|100px]] || [[File:170063 MixSpec 1480 1775Sec ThinWindow.png|100px]] || [[File:170063 MixSpec 2250 2550Sec ThinWindow.png|100px]] || [[File:170063 MixSpec 3050 3300Sec ThinWindow.png|100px]] || [[File:170063 MixSpec 3775 4050Sec ThinWindow.png|100px]] || [[File:170063 MixSpec 4480 4770Sec ThinWindow.png|100px]]
|-
||Original Window || [[File:170063 MixSpec 0 300 OGWindow.png|100px]] || [[File:170063 MixSpec 730 1020Sec OGWindow.png|100px]] || [[File:170063 MixSpec 1480 1775Sec OGWindow.png|100px]] || [[File:170063 MixSpec 2250 2550Sec OGWindow.png|100px]] || [[File:170063 MixSpec 3050 3300Sec OGWindow.png|100px]] || [[File:170063 MixSpec 3775 4050Sec OGWindow.png|100px]] || [[File:170063 MixSpec 4480 4770Sec OGWindow.png|100px]]
|-
||Expanded Window || [[File:170063 MixSpec 0 300Sec ExpWindow.png|100px]] || [[File:170063 MixSpec 730 1020Sec ExpWindow.png|100px]] || [[File:170063 MixSpec 1480 1775Sec ExpWindow.png|100px]] || [[File:170063 MixSpec 2250 2550Sec ExpWindow.png|100px]] || [[File:170063 MixSpec 3050 3300Sec ExpWindow.png|100px]] || [[File:170063 MixSpec 3775 4050Sec ExpWindow.png|100px]] ||[[File:170063 MixSpec 4480 4770Sec ExpWindow.png|100px]]
|-
||Maximum of Histogram|| 173289 || 155770 || 135740 || 124656 || 88582 || 80993 || 77781
|-
||Original Background || 29440 +/- 84.93 || 20580 +/- 70.31 || 16420 +/- 62.52 || 14460 +/- 58.47 ||10950   +/- 50.65 || 10530 +/- 49.90 || 9809 +/- 48.2
|-
||Expanded Background || 28960 +/- 69.12 || 20580 +/- 57.79 || 16500 +/- 51.60 || 14500 +/- 48.2 || 10630 +/- 41.29 || 10460 +/- 41.01 || 9814 +/- 39.7
|-
||Signal Background (Take Larger Error) || 29440 +/- 84.93 || 20580 +/- 70.31 || 16420 +/- 62.52 || 14460 +/- 58.47 || 10950 +/- 50.65 || 10530 +/- 49.90 || 9809 +/- 48.2  
|-
||Integrated Background || 58880 +/- 169.86 || 41160 +/- 140.62 || 32840 +/- 131.04 || 28920 +/- 116.94 || 21900 +/- 101.3 || 21060 +/- 99.8 || 19618 +/- 96.4
|-
||Signal in Thin Window || 342200 +/- 92880 || 300100 +/- 78390 || 267200 +/- 74626 || 238600 +/- 63463 || 170100 +/- 46436 || 153200 +/- 39780 || 147900 +/- 38766
|-
||Background Subtracted Signal || 283320 +/- 92880 || 258940 +/- 78390 || 234360 +/- 74626 || 209680 +/- 63463 || 148200 +/- 46436.11 || 132140 +/- 39780.13 || 128282 +/- 37866.12
|-
||.dat file entry ||6.850549805 +/- 0.3278271919 || 6.794470731 +/- 0.3027342241 || 6.677638317 +/- 0.3181179382 || 6.549555363 +/- 0.3026659672  || 6.384857074 +/- 0.3184246458 || 6.174846148 +/- 0.3010453307 || 6.092105322 +/- 0.2951787468 
|-
|}


Below is a plot of the ratio of the activities where the pure selenium sample has been time corrected back to match the run time of the mixture.

[[File:170063 MixVsPureRatio ThinWindowMethod.png|200px]]

Note that this does give the expected result that the ratio between the two samples is statistically the same as 0.5, which is the ratio of the masses. The only problem here is that the error is quite large, The average of the uncertainty divided by the mean is 39.6%.

Below are the half life plots for the Pure Selenium sample and the ash mixture


[[File:170063 MixHLPlot ThinWindowMethod.png|200px]]

[[File:170063 PureSeHLPlot ThinWindowMethod.png|200px]]


The slope of the mixture plot gives a half life of 63.48 +/- 26.85 minutes. This does overlap, but the error is extremely large.

The slope of the pure selenium sample plot gives a half life of 56.08 +/- 16.33 minutes. Again this does overlap, but the error is still quite large.

The ratio of the initial activities is 0.49 +/- 0.13. Now this does fall within 1 standard deviation of 0.5, which is the value we want.


==Dead Time Corrected==
I have shown that by tracing the activity back in time for the pure selenium sample that the activities are the same here [[LB Thesis Thin Window Analysis]], but now I must make sure that this will actually give the ratio that is expected. 

Begin with the third measurement that was taken for the pure selenium sample

=Pure Se Analysis=

==Third Pure Se Measurement==
Below is the histogram of interest

[[File:170063 PureSeSpec 1875 2150Sec ThinWindow.png|200px]]

From this we can see that 

<math> I \pm \sigma_I = 412800 \pm 642.50 Counts </math> 

and the integrated background is given by 

<math> B \pm \sigma_B = 18992 \pm 97.8 Counts</math>

Now we can find the background subtracted signal 

<math> N \pm \sigma_N = 393808 \pm 649.90 Counts</math> 

Now convert this into an activity by dividing by the runtime 

<math> A_{Measure} \pm \sigma_{A_{Measure}} = \frac{393808}{275} \pm \frac{649.90}{275} = 1432.03 \pm 2.36 Hz </math>

Now we can find the true value using the integrated measurement

<math> A_{True}  = \frac{e^{\lambda \times 275} \times 1432.03 \times 275 \times \lambda}{e^{\lambda \times 275}-1} = 1472.11 Hz </math>

<math> \sigma_{A_{True}} = \frac{e^{\lambda \times 275} \times 2.36 \times 275 \times \lambda}{e^{\lambda \times 275}-1} = 2.43 Hz </math>

So now the true value for the activity is 

<math> A_{True} \pm \sigma_{A_{True}} = 1472.11 \pm 2.43 Hz </math>

Now we must correct for the dead time. The third run had an average count rate of 1387.06 Hz, which corresponds to a dead time of 2.53 +/- 0.3

<math> A_{True}^{DT} = A_{True} \times \frac{1}{1-DT} = 1472.11 \times \frac{1}{0.9747} = 1510.32 Hz </math> 

<math> \sigma_{DTC} = \frac{\sigma_{DT}}{1-DT^2} = 0.003 </math> 

<math> \sigma_{A_{True}^{DT}} = \sqrt{\frac{\sigma_{A_{True}^{DT}}^2}{(1-DT)^2} + \frac{A_{True}^2 \sigma_{DT}^2}{(1-DT)^4}} </math>

So the real activity for the third measurement of the Pure Se sample is 

<math> A_{True}^{DT} \pm \sigma_{A_{True}^{DT}} = 1510.32 \pm 5.27 </math> 

Below is a table representing the numbers that will be put into a data file to be linearly fit (Also using the values from the thesis portion)

==Fourth Pure Se Measurement==
Now do the analysis for the 4th measurement taken on the pure selenium sample.

Below is the histogram of interest

[[File:170063 PureSeSpec 2650 2930Sec ThinWindow.png|200px]]

From this we can see that 

<math> I \pm \sigma_I = 356800  \pm 597.33 Counts </math> 

and the integrated background is given by 

<math> B \pm \sigma_B = 17796 \pm 94.2 Counts</math>

Now we can find the background subtracted signal 

<math> N \pm \sigma_N = 339004 \pm 604.71 Counts</math> 

Now convert this into an activity by dividing by the runtime 

<math> A_{Measure} \pm \sigma_{A_{Measure}} = \frac{339004}{280} \pm \frac{604.71}{280} = 1210.73 \pm 2.16 Hz </math>

Now we can find the true value using the integrated measurement

<math> A_{True} = \frac{e^{\lambda \times 280} \times 1210.73 \times 280 \times \lambda}{e^{\lambda \times 280}-1} =1245.24  Hz </math>

<math> \sigma_{A_{True}} = \frac{e^{\lambda \times 280} \times 2.16 \times 280 \times \lambda}{e^{\lambda \times 280}-1} = 2.22 Hz </math>

So now the true value for the activity is 

<math> A_{True} \pm \sigma_{A_{True}} = 1245.24 \pm 2.22 Hz </math>

Now we must correct for the dead time. The 4th run had an average count rate of 1274.32 Hz, which corr5esponds to a dead time of 2.26 +/- 0.33

<math> A_{True}^{DT} = A_{True} \times \frac{1}{1-DT} = 1245.24 \times \frac{1}{0.9774} = 1274.03 Hz </math> 

<math> \sigma_{DTC} = \frac{\sigma_{DT}}{1-DT^2} = 0.004 </math> 

<math> \sigma_{A_{True}^{DT}} = \sqrt{\frac{\sigma_{A_{True}^{DT}}^2}{(1-DT)^2} + \frac{A_{True}^2 \sigma_{DT}^2}{(1-DT)^4}} </math>

So the real activity for the third measurement of the Pure Se sample is 

<math> A_{True}^{DT} \pm \sigma_{A_{True}^{DT}} = 1274.03 \pm 5.47 </math>

==Fifth Pure Se Measurement== 

Now do the analysis for the 5th measurement taken on the pure selenium sample.

Below is the histogram of interest

[[File:170063 PureSeSpec 3400 3690Sec ThinWindow.png|200px]]

From this we can see that 

<math> I \pm \sigma_I = 304100  \pm 551.45 Counts </math> 

and the integrated background is given by 

<math> B \pm \sigma_B = 18836 \pm 96.4 Counts</math>

Now we can find the background subtracted signal 

<math> N \pm \sigma_N = 285264 \pm 559.81 Counts</math> 

Now convert this into an activity by dividing by the runtime 

<math> A_{Measure} \pm \sigma_{A_{Measure}} = \frac{285264}{290} \pm \frac{559.81}{290} = 983.67 \pm 1.93 Hz </math>

Now we can find the true value using the integrated measurement

<math> A_{True}  = \frac{e^{\lambda \times 290} \times 983.67 \times 290 \times \lambda}{e^{\lambda \times 290}-1} = 1012.72 Hz </math>

<math> \sigma_{A_{True}} = \frac{e^{\lambda \times 290} \times 1.93 \times 290 \times \lambda}{e^{\lambda \times 290}-1} = 1.99 Hz </math>

So now the true value for the activity is 

<math> A_{True} \pm \sigma_{A_{True}} = 1012.72 \pm 1.99 Hz </math>

Now we must correct for the dead time. The 5th run had an average count rate of 1168.06 Hz, which corresponds to a dead time of 2.26 +/- 0.33

<math> A_{True}^{DT} = A_{True} \times \frac{1}{1-DT} = 1012.72 \times \frac{1}{0.9774} = 1036.14 Hz </math> 

<math> \sigma_{DTC} = \frac{\sigma_{DT}}{1-DT^2} = 0.003 </math> 
<math> \sigma_{A_{True}^{DT}} = \sqrt{\frac{\sigma_{A_{True}^{DT}}^2}{(1-DT)^2} + \frac{A_{True}^2 \sigma_{DT}^2}{(1-DT)^4}} </math>

So the real activity for the third measurement of the Pure Se sample is 

<math> A_{True}^{DT} \pm \sigma_{A_{True}^{DT}} = 1036.14 \pm 3.65 </math>

==Sixth Pure Se Measurement==
Now do the analysis for the 6th measurement taken on the pure selenium sample.

Below is the histogram of interest

[[File:170063 PureSeSpec 4120 4400Sec ThinWindow.png|200px]]

From this we can see that 

<math> I \pm \sigma_I = 258700  \pm 508.63 Counts </math> 

and the integrated background is given by 

<math> B \pm \sigma_B = 14970 \pm 86 Counts</math>

Now we can find the background subtracted signal 

<math> N \pm \sigma_N = 243730 \pm 515.85 Counts</math> 

Now convert this into an activity by dividing by the runtime 

<math> A_{Measure} \pm \sigma_{A_{Measure}} = \frac{243730}{280} \pm \frac{515.85}{280} = 870.46 \pm 1.84 Hz </math>

Now we can find the true value using the integrated measurement

<math> A_{True}  = \frac{e^{\lambda \times 280} \times 870.46 \times 280 \times \lambda}{e^{\lambda \times 280}-1} = 895.27 Hz </math>

<math> \sigma_{A_{True}} = \frac{e^{\lambda \times 280} \times 1.84 \times 280 \times \lambda}{e^{\lambda \times 280}-1} = 1.89 Hz </math>

So now the true value for the activity is 

<math> A_{True} \pm \sigma_{A_{True}} = 895.27 \pm 1.89 Hz </math>

Now we must correct for the dead time. The 6th run had an average count rate of 1168.06 Hz, which corresponds to a dead time of 1.96 +/- 0.26

<math> A_{True}^{DT} = A_{True} \times \frac{1}{1-DT} = 895.27 \times \frac{1}{0.9804} = 913.17 Hz </math> 

<math> \sigma_{DTC} = \frac{\sigma_{DT}}{1-DT^2} = 0.003 </math> 
<math> \sigma_{A_{True}^{DT}} = \sqrt{\frac{\sigma_{A_{True}^{DT}}^2}{(1-DT)^2} + \frac{A_{True}^2 \sigma_{DT}^2}{(1-DT)^4}} </math>

So the real activity for the 6th measurement of the Pure Se sample is 

<math> A_{True}^{DT} \pm \sigma_{A_{True}^{DT}} = 913.17 \pm 3.28 </math>

== Seventh Pure Se Measurement==

Now do the analysis for the 7th measurement taken on the pure selenium sample.

Below is the histogram of interest

[[File:170063 PureSeSpec 4840 5130Sec ThinWindow.png|200px]]

From this we can see that 

<math> I \pm \sigma_I = 233500  \pm 483.22 Counts </math> 

and the integrated background is given by 

<math> B \pm \sigma_B = 14322 \pm 68.4 Counts</math>

Now we can find the background subtracted signal 

<math> N \pm \sigma_N = 219178 \pm 488.04 Counts</math> 

Now convert this into an activity by dividing by the runtime 

<math> A_{Measure} \pm \sigma_{A_{Measure}} = \frac{219178}{290} \pm \frac{488.04}{290} = 755.79 \pm 1.68 Hz </math>

Now we can find the true value using the integrated measurement

<math> A_{True}  = \frac{e^{\lambda \times 290} \times 755.79 \times 290 \times \lambda}{e^{\lambda \times 290}-1} = 778.11 Hz </math>

<math> \sigma_{A_{True}} = \frac{e^{\lambda \times 290} \times 1.68 \times 290 \times \lambda}{e^{\lambda \times 290}-1} = 1.73 Hz </math>

So now the true value for the activity is 

<math> A_{True} \pm \sigma_{A_{True}} = 778.11 \pm 1.73 Hz </math>

Now we must correct for the dead time. The 7th run had an average count rate of 1012 Hz, which corresponds to a dead time of 1.96 +/- 0.26

<math> A_{True}^{DT} = A_{True} \times \frac{1}{1-DT} = 778.11 \times \frac{1}{0.9804} = 793.67 Hz </math> 

<math> \sigma_{DTC} = \frac{\sigma_{DT}}{1-DT^2} = 0.003 </math> 
<math> \sigma_{A_{True}^{DT}} = \sqrt{\frac{\sigma_{A_{True}^{DT}}^2}{(1-DT)^2} + \frac{A_{True}^2 \sigma_{DT}^2}{(1-DT)^4}} </math>

So the real activity for the 7th measurement of the Pure Se sample is 

<math> A_{True}^{DT} \pm \sigma_{A_{True}^{DT}} = 793.67 \pm 2.92 </math>

{| border="3"  cellpadding="5" cellspacing="0"
|| || 400 <t< 640 sec || 1100 < t < 1360 sec || 1875 < t < 2150 || 2650 < t < 2930 sec || 3400 < t < 3690 sec || 4120 < t < 4400 sec || 4840 < t < 5130 sec 
|- 
||.dat file entry ||7.58 +/- 0.003 || 7.45 +/- 0.003 || 7.32 +/- 0/003 || 7.15 +/- 0.004 || 6.94 +/- 0.004 || 6.82 +/- 0.004 || 6.68 +/- 0.004 
|-
|}

=Se-Sage Mix Analysis=
==Activity of the First Measurement==

Below is the histogram of interest

[[File:170063 MixSpec 0 300Sec ThinWindow.png|200px]]

From this we can see that 

<math> I \pm \sigma_I = 342200  \pm 584.98 Counts </math> 

and the integrated background is given by 

<math> B \pm \sigma_B = 58880 \pm 169.86 Counts</math>

Now we can find the background subtracted signal 

<math> N \pm \sigma_N = 283320 \pm 609.14 Counts</math> 

Now convert this into an activity by dividing by the runtime 

<math> A_{Measure} \pm \sigma_{A_{Measure}} = \frac{283320}{300} \pm \frac{609.14}{300} = 944.4 \pm 2.03 Hz </math>

Now we can find the true value using the integrated measurement

<math> A_{True}  = \frac{e^{\lambda \times 300} \times 944.4 \times 300 \times \lambda}{e^{\lambda \times 300}-1} = 973.26 Hz </math>

<math> \sigma_{A_{True}} = \sqrt{(\frac{t\times e^{\lambda t} \times \lambda \times \sigma_{A_{Measure}}}{e^{\lambda t}-1})^2 + (\frac{e^{\lambda t}(A t e^{\lambda t} - \lambda^2 - A t + A t \lambda)}{(e^{\lambda t}-1)^2})^2 \times \sigma_{\lambda}^2} = 2.09 Hz </math>

So now the true value for the activity is 

<math> A_{True} \pm \sigma_{A_{True}} =973.26  \pm 2.09 Hz </math>

Now we must correct for the dead time. The 1st run had an average count rate of 2434.44 Hz, which corresponds to a dead time of 5.06 +/- 0.39

<math> A_{True}^{DT} = A_{True} \times \frac{1}{1-DT} = 944.4 \times \frac{1}{0.9494} = 994.73 Hz </math> 

<math> \sigma_{A_{True}^{DT}} = \sqrt{\frac{\sigma_{A_{True}^{DT}}^2}{(1-DT)^2} + \frac{A_{True}^2 \sigma_{DT}^2}{(1-DT)^4}} </math>

So the real activity for the 1st measurement of the Mixed sample is 

<math> A_{True}^{DT} \pm \sigma_{A_{True}^{DT}} = 994.73  \pm 4.75 </math>

==Activity of the Second Measurement==

Below is the histogram of interest

[[File:170063 MixSpec 730 1020Sec ThinWindow.png|200px]]

From this we can see that 

<math> I \pm \sigma_I =  300100 \pm 547.81 Counts </math> 

and the integrated background is given by 

<math> B \pm \sigma_B = 41160 \pm 140.62 Counts</math>

Now we can find the background subtracted signal 

<math> N \pm \sigma_N = 258940 \pm 565.57 Counts</math> 

Now convert this into an activity by dividing by the runtime 

<math> A_{Measure} \pm \sigma_{A_{Measure}} = \frac{258940}{290} \pm \frac{547.81}{290} = 892.90 \pm 1.89 Hz </math>

Now we can find the true value using the integrated measurement

<math> A_{True}  = \frac{e^{\lambda \times 290} \times 892.90 \times 290 \times \lambda}{e^{\lambda \times 300}-1} = 919.27 Hz </math>

<math> \sigma_{A_{True}} = \frac{e^{\lambda \times 290} \times 1.89 \times 290 \times \lambda}{e^{\lambda \times 290}-1} = 1.95 Hz </math>

So now the true value for the activity is 

<math> A_{True} \pm \sigma_{A_{True}} = 919.27 \pm 1.95 Hz </math>

Now we must correct for the dead time. The 1st run had an average count rate of 1756.30 Hz, which corresponds to a dead time of 3.06 +/- 0.30

<math> A_{True}^{DT} = A_{True} \times \frac{1}{1-DT} = 919.27 \times \frac{1}{0.9694} = 948.29 Hz </math> 

<math> \sigma_{DTC} = \frac{\sigma_{DT}}{1-DT^2} = 0.0.003 </math> 
<math> \sigma_{A_{True}^{DT}} = \sqrt{\frac{\sigma_{A_{True}^{DT}}^2}{(1-DT)^2} + \frac{A_{True}^2 \sigma_{DT}^2}{(1-DT)^4}} </math>

So the real activity for the 2nd measurement of the Mixed sample is 

<math> A_{True}^{DT} \pm \sigma_{A_{True}^{DT}} = 948.20  \pm 3.56 </math>

==Activity of the Third Measurement==

Below is the histogram of interest

[[File:170063 MixSpec 1480 1775Sec ThinWindow.png|200px]]

From this we can see that 

<math> I \pm \sigma_I =  267200 \pm 516.91 Counts </math> 

and the integrated background is given by 

<math> B \pm \sigma_B = 32840 \pm 131.04 Counts</math>

Now we can find the background subtracted signal 

<math> N \pm \sigma_N = 234360 \pm 533.26 Counts</math> 

Now convert this into an activity by dividing by the runtime 

<math> A_{Measure} \pm \sigma_{A_{Measure}} = \frac{234360}{295} \pm \frac{533.26}{295} = 794.44 \pm 1.81 Hz </math>

Now we can find the true value using the integrated measurement

<math> A_{True}  = \frac{e^{\lambda \times 295} \times 794.44 \times 295 \times \lambda}{e^{\lambda \times 295}-1} = 818.31 Hz </math>

<math> \sigma_{A_{True}} = \frac{e^{\lambda \times 295} \times 1.81 \times 295 \times \lambda}{e^{\lambda \times 295}-1} = 1.86 Hz </math>

So now the true value for the activity is 

<math> A_{True} \pm \sigma_{A_{True}} = 818.31 \pm 1.86 Hz </math>

Now we must correct for the dead time. The 1st run had an average count rate of 1401.98 Hz, which corresponds to a dead time of 2.53 +/- 0.31

<math> A_{True}^{DT} = A_{True} \times \frac{1}{1-DT} = 818.31 \times \frac{1}{0.9747} = 839.55 Hz </math> 

<math> \sigma_{DTC} = \frac{\sigma_{DT}}{1-DT^2} = 0.003 </math> 
<math> \sigma_{A_{True}^{DT}} = \sqrt{\frac{\sigma_{A_{True}^{DT}}^2}{(1-DT)^2} + \frac{A_{True}^2 \sigma_{DT}^2}{(1-DT)^4}} </math>

So the real activity for the 2nd measurement of the Mixed sample is 

<math> A_{True}^{DT} \pm \sigma_{A_{True}^{DT}} = 839.55  \pm 3.28 </math>

==Fourth Mix Sample Measurement==

Below is the histogram of interest

[[File:170063 MixSpec 2250 2550Sec ThinWindow.png|200px]]

From this we can see that 

<math> I \pm \sigma_I =  238600 \pm 488.47 Counts </math> 

and the integrated background is given by 

<math> B \pm \sigma_B = 28920 \pm 116.94 Counts</math>

Now we can find the background subtracted signal 

<math> N \pm \sigma_N = 209680 \pm 502.27 Counts</math> 

Now convert this into an activity by dividing by the runtime 

<math> A_{Measure} \pm \sigma_{A_{Measure}} = \frac{209680}{300} \pm \frac{502.27}{300} = 698.93 \pm 1.67 Hz </math>

Now we can find the true value using the integrated measurement

<math> A_{True}  = \frac{e^{\lambda \times 300} \times 698.93 \times 300 \times \lambda}{e^{\lambda \times 300}-1} = 720.29 Hz </math>

<math> \sigma_{A_{True}} = \frac{e^{\lambda \times 300} \times 1.67 \times 300 \times \lambda}{e^{\lambda \times 300}-1} = 1.72 Hz </math>

So now the true value for the activity is 

<math> A_{True} \pm \sigma_{A_{True}} = 720.29 \pm 1.72 Hz </math>

Now we must correct for the dead time. The 1st run had an average count rate of 1401.98 Hz, which corresponds to a dead time of 2.53 +/- 0.31

<math> A_{True}^{DT} = A_{True} \times \frac{1}{1-DT} = 720.29 \times \frac{1}{0.9747} = 738.99 Hz </math> 

<math> \sigma_{DTC} = \frac{\sigma_{DT}}{1-DT^2} = 0.003 </math> 

<math> \sigma_{A_{True}^{DT}} = \sqrt{\frac{\sigma_{A_{True}^{DT}}^2}{(1-DT)^2} + \frac{A_{True}^2 \sigma_{DT}^2}{(1-DT)^4}} </math>>

So the real activity for the 2nd measurement of the Mixed sample is 

<math> A_{True}^{DT} \pm \sigma_{A_{True}^{DT}} = 738.99  \pm 2.94 </math>

==Activity of the Fifth Measurement==

Below is the histogram of interest

[[File:170063 MixSpec 3050 3300Sec ThinWindow.png|200px]]

From this we can see that 

<math> I \pm \sigma_I =  170100 \pm 412.43 Counts </math> 

and the integrated background is given by 

<math> B \pm \sigma_B = 21900 \pm 101.3 Counts</math>

Now we can find the background subtracted signal 

<math> N \pm \sigma_N = 148200 \pm 424.69 Counts</math> 

Now convert this into an activity by dividing by the runtime 

<math> A_{Measure} \pm \sigma_{A_{Measure}} = \frac{148200}{250} \pm \frac{424.69}{250} = 592.8 \pm 1.70 Hz </math>

Now we can find the true value using the integrated measurement

<math> A_{True}  = \frac{e^{\lambda \times 250} \times 592.8 \times 250 \times \lambda}{e^{\lambda \times 250}-1} = 607.87 Hz </math>

<math> \sigma_{A_{True}} = \frac{e^{\lambda \times 250} \times 1.70 \times 250 \times \lambda}{e^{\lambda \times 250}-1} = 1.74 Hz </math>

So now the true value for the activity is 

<math> A_{True} \pm \sigma_{A_{True}} = 607.87 \pm 1.74 Hz </math>

Now we must correct for the dead time. The 1st run had an average count rate of 1025.13 Hz, which corresponds to a dead time of 1.92 +/- 0.26

<math> A_{True}^{DT} = A_{True} \times \frac{1}{1-DT} = 607.8 \times \frac{1}{0.9808} = 619.70 Hz </math> 

<math> \sigma_{DTC} = \frac{\sigma_{DT}}{1-DT^2} = 0.003 </math> 

<math> \sigma_{A_{True}^{DT}} = \sqrt{\frac{\sigma_{A_{True}^{DT}}^2}{(1-DT)^2} + \frac{A_{True}^2 \sigma_{DT}^2}{(1-DT)^4}} </math>

So the real activity for the 2nd measurement of the Mixed sample is 

<math> A_{True}^{DT} \pm \sigma_{A_{True}^{DT}} = 619.70  \pm 2.42 </math>

==Sixth Mix Sample Measurement==

Below is the histogram of interest

[[File:170063 MixSpec 3775 4050Sec ThinWindow.png|200px]]

From this we can see that 

<math> I \pm \sigma_I =  153200 \pm 391.41 Counts </math> 

and the integrated background is given by 

<math> B \pm \sigma_B = 21060 \pm 99.8 Counts</math>

Now we can find the background subtracted signal 

<math> N \pm \sigma_N = 132140 \pm 403.93 Counts</math> 

Now convert this into an activity by dividing by the runtime 

<math> A_{Measure} \pm \sigma_{A_{Measure}} = \frac{132140}{275} \pm \frac{403.93}{275} = 480.51 \pm 1.47 Hz </math>

Now we can find the true value using the integrated measurement

<math> A_{True}  = \frac{e^{\lambda \times 275} \times 480.51 \times 275 \times \lambda}{e^{\lambda \times 275}-1} = 493.96 Hz </math>

<math> \sigma_{A_{True}} = \frac{e^{\lambda \times 275} \times 1.47 \times 275 \times \lambda}{e^{\lambda \times 275}-1} = 1.51 Hz </math>

So now the true value for the activity is 

<math> A_{True} \pm \sigma_{A_{True}} = 493.96 \pm 1.51 Hz </math>

Now we must correct for the dead time. The 1st run had an average count rate of 898.19 Hz, which corresponds to a dead time of 1.65 +/- 0.34

<math> A_{True}^{DT} = A_{True} \times \frac{1}{1-DT} = 493.96 \times \frac{1}{0.9835} = 502.25 Hz </math> 

<math> \sigma_{DTC} = \frac{\sigma_{DT}}{1-DT^2} = 0.004 </math> 

<math> \sigma_{A_{True}^{DT}} = \sqrt{\frac{\sigma_{A_{True}^{DT}}^2}{(1-DT)^2} + \frac{A_{True}^2 \sigma_{DT}^2}{(1-DT)^4}} </math>

So the real activity for the 2nd measurement of the Mixed sample is 

<math> A_{True}^{DT} \pm \sigma_{A_{True}^{DT}} = 502.25  \pm 2.32 </math>

==Activity of the 7th Measurement==

Below is the histogram of interest

[[File:170063 MixSpec 4480 4770Sec ThinWindow.png|200px]]

From this we can see that 

<math> I \pm \sigma_I =  147900 \pm 384.58 Counts </math> 

and the integrated background is given by 

<math> B \pm \sigma_B = 19618 \pm 96.4 Counts</math>

Now we can find the background subtracted signal 

<math> N \pm \sigma_N = 128282 \pm 396.48 Counts</math> 

Now convert this into an activity by dividing by the runtime 

<math> A_{Measure} \pm \sigma_{A_{Measure}} = \frac{128282}{290} \pm \frac{396.48}{290} = 442.35 \pm 1.38 Hz </math>

Now we can find the true value using the integrated measurement

<math> A_{True}  = \frac{e^{\lambda \times 290} \times 442.35 \times 290 \times \lambda}{e^{\lambda \times 290}-1} = 455.41 Hz </math>

<math> \sigma_{A_{True}} = \frac{e^{\lambda \times 290} \times 1.38 \times 290 \times \lambda}{e^{\lambda \times 290}-1} = 1.42 Hz </math>

So now the true value for the activity is 

<math> A_{True} \pm \sigma_{A_{True}} = 455.41 \pm 1.42 Hz </math>

Now we must correct for the dead time. The 1st run had an average count rate of 846.40 Hz, which corresponds to a dead time of 1.65 +/- 0.34

<math> A_{True}^{DT} = A_{True} \times \frac{1}{1-DT} = 455.41 \times \frac{1}{0.9835} = 463.05 Hz </math> 

<math> \sigma_{DTC} = \frac{\sigma_{DT}}{1-DT^2} = 0.004 </math> <math> \sigma_{A_{True}^{DT}} = \sqrt{\frac{\sigma_{A_{True}^{DT}}^2}{(1-DT)^2} + \frac{A_{True}^2 \sigma_{DT}^2}{(1-DT)^4}} </math>

So the real activity for the 2nd measurement of the Mixed sample is 

<math> A_{True}^{DT} \pm \sigma_{A_{True}^{DT}} =  463.05 \pm 2.16 </math>

{| border="3"  cellpadding="5" cellspacing="0"
|| || 0<t<300 || 730 <t< 1020 || 1480<t<1775 || 2250<t<2550 || 3050<t<3300 || 3775<t<4050 || 4480<t<4770
|-
||Data File Entry || 6.90 +/- 0.004 || 6.85 +/- 0.004 ||6.73 +/- 0.004 || 6.61 +/- 0.004 || 6.43 +/- 0.004 || 6.23 +/- 0.005 || 6.14 +/- 0.005
|-
|}

==Plots==

Below are the time vs. frequency plots

[[File:170063 MixHL DTCorrected.png|200px]]
[[File:170063 PureSeHL DTCorrected.png|200px]]

To check consistency I would time correct measurements and take a ratio to compare, for example measurement 1 was taken and a 2nd measurement was taken some time later. So correct the rate of the second measurement back to the first measurement by using the radioactive decay equation, then take a ratio of these 2 numbers.

[[File:170063 MixtureRateComparison.png|200px]]
[[File:170063 PureSe RateComparison.png|200px]]

Another check to do is to take the ratio of the mixture and the pure selenium sample (time corrected to the same time for each measurement) and see if the ratio here is relatively constant

[[File:170063 CrossSample Ratio.png|200px]]
 
By exponentiating we find that the initial activity of the Mixture is 1056.48 +/- 2.91 Hz, while the pure sample has an initial activity of 2162.90 +/- 5.16 Hz which gives a ratio of 

<math> 0.488 \pm 0.002 </math> 

We can also check the ratio of the nickel with the selenium mixture. The rate of the nickel foil was <math> 1.127931 \times 10^6 \pm 7.258 \times 10^3 Hz </math> 

So using the efficiency corrected rate for the soil, we find that the ratio is

<math> 0.134 \pm 0.001 </math> 


The measurement was made 20cm away from the detector face, which has an efficiency of 0.70 +/- 0.0011, so the measurements with the efficiency taken into account are 

150925.7143 +/- 392.5923 Hz for the mixture and 

308985.7143 +/- 882.6876 Hz for the pure selenium sample