# Decay/Counts/Number of Nuclei

I'm writing up a lab report and have a question about the following formula

$$N = N_0e^{-\lambda t}$$

$N$ indicates the number of nuclei left after a time $t$ and $N_0$ indicates how much there was to begin with. In the experiment, we used a scintillation counter (consists of a crystal and a photomultiplier tube) to detect the gamma rays beginning emitted by a decaying metastable Ba-137 to Ba-137. After recording the "counts" $C$, we corrected for the background radiation counts $BG$ by calculating $C -BG$. Then we plotted $\ln(C-BG)$ vs $t$. We did this because from the above formula

$$C-BG = (C-BG)_0e^{-\lambda t}$$

Then,

$$\ln(C-BG) = \ln(C-BG)_0 - \lambda t$$

Thus a plot of $\ln(C-BG)$ vs $t$ yields a straight line with slope $-\lambda$. My question is, how does counts relate to $N$? I thought first formula above indicates the number of nuclei left. However, the detector is detecting gamma rays from decaying nuclei, i.e. counting how many nuclei are decay?

So why can I replace $N$ with counts?

• The count rate $C$ is directly proportional to the number of emitters $N$ remaining in the sample.
– rob
Commented Apr 27, 2014 at 22:44
• Note also that $N = N_0 e^{-\lambda t}$ is all decays (at all angles), the decays seen by your device will actually be $N = \ell \epsilon A (N_0 e^{-\lambda t})$ where $\ell$ is live time (very nearly 1.0), $\epsilon$ is the quantum efficiency of your detector (probably also near 1), and $A$ is acceptance (a number expressing what fraction of decay gammas actually hit the detector). But all those corrections are effectively constant, so they don't affect your measured lifetime. Commented Apr 27, 2014 at 23:17
• Is your measurement $C$ the total number of counts accumulated since the beginning of the experiment, or is it the number of counts in a short time interval (like counts per second, for example)? Commented Apr 28, 2014 at 0:25
• @DavidZ $C$ is the number of counts in a short time interval. We made like 900 different recordings. Each recording was like 4 seconds. Commented Apr 28, 2014 at 5:18
• @DWade64 I'd suggest editing the question to reflect that, as it's important information. Commented Apr 28, 2014 at 18:26

When you measure the number of decays in a short time interval, $C$, you're effectively measuring the rate $R$ at which decays happen, because $C - BG \approx R\Delta t$ for some fixed interval $\Delta t$. But the rate of decay is related to the number of nuclei remaining as follows:
$$R = -\frac{\mathrm{d}N}{\mathrm{d}t} = -\frac{\mathrm{d}}{\mathrm{d}t}N_0 e^{-\lambda t} = N_0\lambda e^{-\lambda t} = \lambda N$$
So since $R = \lambda N$, you can multiply both sides of $N = N_0e^{-\lambda t}$ by $\lambda$ and find that $R = R_0e^{-\lambda t}$. Then if you multiply both sides of that by your sampling time $\Delta t$, you get
$$C - BG \approx (C - BG)_0 e^{-\lambda t}$$
which justifies why you can plot $\ln(C - BG)$ versus $t$ and get a straight line.
Essentially, you're not really replacing $N$ with $C - BG$, but rather converting the equation for $N$ into an equation for $C - BG$ that happens to take the same form.