# Na-22 Branching Ratio

I tried to calculate the branching ratio for $^{22}Na$ from experimental data using a Germanium detector. My derivation is as follows:

The formula for the branching ratio $BR$ for electron capture $EC$ is $$BR(EC)=\frac{1}{1+\frac{\beta^+}{EC}}$$ where $\beta^+,\;EC$ are the probability densities / rates / theoretical spectrum intensities for each process (all of these are equivalent in what follows). To calculate this ratio de facto, we get $$\frac{\beta^+}{EC}=\frac{I_{\beta^+}/\varepsilon_{\beta^+}}{2I_{EC}/\varepsilon_{EC}}$$ where $I$ is the measured intensity in the detector, and $\varepsilon$ is the relative efficiency of the detector (or the intrinsic photopeak efficiency). The $1/2$ factor is due to the fact that $\beta^+$ emits 2 photons thus doubling the probability for a photon to enter the detector.

Folowing the above, I calculated the relative efficiency using $^{152}Eu$ and interpolated the results. I then measured the spectrum for $^{22}Na$ and calculated $BR(EC)$. Theoretical results are about ~$0.1$, but I got ~$0.6$ using three different methods of photopeak assessment: maximum photopeak values, summing FWHM of photopeak gaussian, and fitting the data to a gaussian and numerically integrating to get the area under its curve.

I haven't found any errors in my code (yet), so I'm thinking maybe the above derivation is incomplete, or the value ~$0.1$ is not what I interpret it to be? Is there underlying physics I haven't taken into account? Or should I just keep looking for errors in my code, or deeper uncertainty analysis etc.?

• Difficult to separate these processes. What gamma energies are you looking at?
– user137289
Jan 9 '17 at 14:08
• For sodium it's 511keV for the beta decay, and for EC it's 1274keV. My Europium measurement had nice enough peaks relevant for interpolation at ~ 340, 780 keV and at ~ 1110, 1400keV.
– Yoni
Jan 9 '17 at 14:40
• The 1274 keV peak is caused by both processes nucleardata.nuclear.lu.se/toi/nuclide.asp?iZA=110022 Could that be a cause of the discrepancy?
– user137289
Jan 9 '17 at 16:55
• Yes, it most definitely could. Can you please elaborate on that? I didn't really understand the data presented in your reference. Thank you!
– Yoni
Jan 9 '17 at 18:20