====== Raw Data Handling ======

Figure 1 shows the pulse height spectra (PHS) for Na22 with no absorber.  The full energy peaks for both the 512 keV and 1.27 MeV gammas are indicated.  The blue and green shaded portions of the PHS show the Region of Integration (ROI) I choose for each peak.  When an ROI is selected the USX software calculates the Gross counts and Net counts.  Gross counts are the sum of all the counts in each channel within the ROI.  Net counts are the number of counts that are considered to be above the background.  The software determines the background by connecting the end points of the ROI with a straight line which I will refer to as the Extrapolated Background Boundary (EBB).  The software calculates the sum of the counts in the ROI which are beneath the EBB.  These counts are assumed to be a background upon which the full energy peak resides.  The Net counts are simply the Gross minus the Background.  The EBB for each peak is shown as a dashed line in Figure 1, though it is more apparent for the 512 keV feature.

{FIXME ${/download/attachments/226427033/Na22_spectrum.PNG?version=1&modificationDate=1570655263000&api=v2}$
Fig 1.  Screen capture of pulse height spectrum of Na22.

Figure 2 shows the corresponding PHS for Ba133 with no absorber.

{FIXME ${/download/attachments/226427033/Ba133_spectrum.PNG?version=1&modificationDate=1570656812000&api=v2}$
Fig 2.  Screen capture of pulse height spectrum of Ba133 with no absorber.

For each peak the ROI was carefully chosen so that the EBB would give the best approximation of the background underling the corresponding full energy peak.  The ROI for each peak is the same for all absorber thicknesses.

====== Uncertainties ======

Four sources of uncertainty were taken into account.  

===== Geometry =====

As discussed in the manual the positioning of the absorbers relative to the source and detector can produce systematic biases due to scattering of gammas outside the acceptance cone into the detector.  Due to time constrains I did not investigate this effect and simply followed the advice in the wiki to place the absorbers midway between the source and detector.

===== Statistical and systematic uncertainties in the number of counts in the full energy peaks =====

Since the Na22 course emits radiation at random intervals, the total number of counts observed should obey normal counting statistics.  So we treat the statistical uncertainty in the number of measured counts //N// as  {FIXME $\sqrt{N}$ .  Since we use the net counts, which are calculated by subtracting the estimated background from the Gross counts, we compute the total statistical uncertainty of the Net as follows:
| [Math Processing Error]NN=NG−NB    {FIXME $N_{N} = N_{G} - N_{B}$  |

Since both the Gross count and the Background count determination depends on the ROI selected, I treat the uncertainties as correlated and add the uncertainties in the Gross and Background counts to get the uncertainty in the Net.

| [Math Processing Error]δNN=δNg+δNB    {FIXME $\delta N_{N} = \delta N_{g} + \delta N_{B}$  |

There exists the possibility that a poorly chosen ROI could introduce a systematic bias in the background estimation.  If time had allowed this effect could have been investigated by choosing different ROI's and seeing how much it affects the final results.  Since this was not possible I chose ROI's conservatively so that the resulting Net count estimations would be well within the boundaries of the full energy peaks.  For example see the ROI chosen for the 302keV peak in the Ba133 PHS.

The two higher energy peaks in the Ba133 spectrum are actually each comprised of two separate energy gammas which are close enough in energy that their full energy peaks overlap.  What I identify as the 302keV peak is actually a combination of 276keV and 302 keV gammas where the 302keV is roughly twice as intense as the 276keV so I expect this data to be biased to a cross section somewhere between the two energies.  The other peak is actually composed of gammas with energies 356keV and 382keV, however the 356keV line is ten times the intensity of the 382keV which I expect will cause negligible bias.

===== Uncertainty in the time of data collection =====

I will be expressing the intensities as rates where the rate is the Net counts divided by the PHA Live Time.  The Live Time (LT) is dead time corrected time over which a PHS was collected.  There was no opportunity to test the accuracy of the LT reported by the USX software, though we note in the documentation the LT is given to a precision of 0.01s.  My data collection runs varied from ~300s to ~3000s.  So I record all times in seconds and assume negligible uncertainty at this time scale.  

===== Absorber thickness =====

The thickness of each absorber was measured with a pair of calipers with a resolution of 0.05mm and was considered negligible.

To maintain consistency across all measurements the pmt HV and source-absorber-detector positions were all held constant.  Care was taken to make sure that no additional radioactive sources were within 3 feet of the apparatus which taking data.  As noted earlier, the same ROI was used for each gamma energy across all absorber thickness measurements.

====== Linear Attenuation Coefficients ======

Table 1 shows the rates with uncertainties extracted from the PHS.

| Absorber Thickness (cm)  | 31keV  | 81keV  | 302keV  | 356keV  | 512keV  | 1.27MeV  |
| 0  | 809.9 ± 0.8  | 206.1 ± 0.04  | 17.1 ± 0.1  | 129.59 ± 0.71  | 173.4 ± 0.92  | 26.36 ± 0.37  |
| 0.05  | 688.4 ± 1.3  | 194.1 ± 0.07  | 16.8 ± 0.2  | 127.10 ± 1.3  |
| 1  | 610.4 ± 1.1  | 187.2 ± 0.06  | 16.2 ± 0.2  | 122.4 ± 1.4  |
| 2  | 485.0 ± 0.9  | 187.9 ± 0.05  | 16.9 ± 0.2  | 124.7 ± 1.1  |
| 4  | 298.0 ± 0.9  | 170.4 ± 0.6  | 17.3 ± 0.2  | 120.1 ± 1.3  | 160.9 ± 0.90  | 25.8 ± 0.44  |
| 16  | 45.7 ± 0.3  | 93.1 ± 0.4  | 9.1 ± 0.1  | 84.5 ± 0.83  |
| 32  | 30.3 ± 0.1  | 44.5 ± 0.2  | 6.8 ± 0.07  | 57.9 ± 0.49  | 91.9 ± 0.82  | 16.79 ± 0.34  |
| 48  |
| 68.43 ± 0.50  | 13.25 ± 0.21  |
| 63  | 15.5 ± 0.1  | 15.5 ± 0.1  | 2.8 ± 0.03  | 26.9 ± 0.22  |
| 63.5  |
| 52.46 ± 0.35  | 10.51 ± 0.29  |

Table 1 showing data used for plotting and fitting.

For each energy the data are fitted using python's optimize function to the functional form  {FIXME $A + B\cdot e^{(\lambda \cdot x)}$  where _A _accounts for an assumed linear background, //B// is the intensity of the radiation and //λ// is the linear attenuation coefficient in units of cm-1.   The following plots show the absorption data for each gamma energy plotted as circles and the best fit shown as a solid line.  Note that error bars are included, but are often smaller that the point size.  Additionally, note that the X axis is mistakenly labeled in units of //mm// when it should be units of //cm//.  
| {FIXME ${/download/attachments/226427033/31kev.png?version=3&modificationDate=1570808094000&api=v2}$    //χ2// = 2455  | {FIXME ${/download/attachments/226427033/81keV.png?version=3&modificationDate=1570808108000&api=v2}$    //χ2// = 4976  |
| {FIXME ${/download/attachments/226427033/302keV.png?version=2&modificationDate=1570808122000&api=v2}$    //χ2// = 70  | {FIXME ${/download/attachments/226427033/356keV.png?version=2&modificationDate=1570808133000&api=v2}$    //χ2// = 4  |
| {FIXME ${/download/attachments/226427033/512keV.png?version=1&modificationDate=1570808144000&api=v2}$    //χ2// = 0.5  | {FIXME ${/download/attachments/226427033/1270keV.png?version=1&modificationDate=1570808156000&api=v2}$    //χ2// = 1.2  |

===== Linear attenuation coefficients =====

The best fit linear attenuation coefficients are given in Table 2 along with literature values for each gamma energy.  Note that the literature obtained by eye from the graph (_Harshaw Radiation Detectors Catalog) _provided in the experiment wiki.  The associated uncertainties in the literature values are my estimate of the accuracy with which these values could be determined by eye.

| Energy (keV)  | Measured    λ(cm-1)  | Literature    Total    λ(cm-1)  | Literature    Photoelectric    λ(cm-1)  | Literature    Compton    λ(cm-1)  |
| 31  | -2.51 ± 0.01  | -2.6 ± 0.1  |<WRAP> 
  **-2.2 ± 0.1** 
</WRAP> |  -0.46 ± 0.02  |
| 81  | -0.51 ± 0.01  | -0.55 ± 0.2  |<WRAP> 
  **-0.16 ± 0.02** 
</WRAP> | <WRAP> 
  **-0.5 ± 0.2** 
</WRAP> | 
| 302  | -0.35 ± 0.01  | -0.28 ± 0.02  | -0.007 ± 0.002  |<WRAP> 
  **-0.28 ± 0.02** 
</WRAP> | 
| 356  | -0.26 ± 0.01  | -0.26 ± 0.02  | -0.003 ± 0.002  |<WRAP> 
  **-0.26 ± 0.02** 
</WRAP> | 
| 512  | -0.22 ± 0.01  | -0.22 ± 0.02  | Negligible  |<WRAP> 
  **-0.22 ± 0.02** 
</WRAP> | 
| 1270  | -0.13 ± 0.03  | -0.15 ± 0.02  | Negligible  |<WRAP> 
  **-0.15 ± 0.02** 
</WRAP> | 

Table 2.  Linear attenuation coefficients obtained from fits to data compared with literature values.

====== Conclusions ======

  * As shown in Table 2, other than at 302keV the measured linear attenuation coefficients are in good agreement with the literature values.  The main reason for the discrepancy at 302keV is likely due to the fact that the measured full energy peak is a mixture of the 302keV and 276keV lines which overlap as a result of the detector energy resolution. 
  * Although the data are obviously well fitted, the reduced chi square values for most of the fits indicate that the variation in the data from the best fit are too large to be due to random fluctuations and therefor the uncertainties in the best fit parameters are under estimates. 
  * There are certainly significant systematic effects at play, some of which were discussed earlier.  Most likely there are biases due to background subtraction. 
  * Note that the data from the Na22 source is consistent with statistical fluctuations while the Ba133 data are not.  This seems reasonable in so far as the two full energy peaks in the Na22 spectrum are well separated and lie atop what appears to be a fairly linear background.  The Ba133 spectrum however consists of a more complicated collection of overlapping spectral features from the multiple energies present.  For the Ba133 source one would expect the background to be more complicated at the lowest energies where there is contribution from all of the higher energy spectra and this is reflected in the reduced chi square values.
  * From Table 2 it is clear that at 31keV attenuation is dominated by photoelectric effect, at 81keV Compton scattering is the dominant mechanism though photoelectric effect is still a factor.  At energies above 302keV Compton scattering is only significant process.  Note that even at 1270keV the probability of pair production is negligible so those values are not included in the table.