The chance, $W_n\mathrm{}(0,t)\mathrm{}$, that $n$ atoms of a radioactive source will disintegrate in the interval ($0,t$) may be found if we know $f_r\mathrm{}\left(t\right)\mathrm{dt}$, the chance that a disintegration will occur in $\mathrm{dt}$ at $t$ after $r$ have occurred in ($0,t$). Bateman's differential equation is generalized to cover the case in which $f_r$ depends on $t$ and $r$. The solution is given for the case in which it depends only on $t$. A detector of efficiency $g$ subtends a solid angle $4\pi A$ at a source (decay constant $\lambda$), containing $N$ atoms at time zero. The probability of $n$ counts in the interval ($T_1,T_1+T_2$) is $P_n(T_1,T_1+T_2)=C_n^{}{}_{}{}^N{(e^{\lambda T_2}-1)}^n{e}^{-N\lambda (T_1+T_2)}\times {\mathrm{}\left(\mathrm{gA}\right)\mathrm{}}^n{[1+e^{\lambda T_2}(e^{\lambda T_1}-1)+(1-gA\left)\right(e^{\lambda T_2}-1\left)\right]}^{N-n}.$ Putting $g$ and $A$ equal to 1 we get the probability $W_n(T_1,T_1+T_2)$ of $n$ disintegrations in this interval. Bortkiewicz's formula $W_n\mathrm{}(0,t)=C_n^{}{}_{}{}^N{(e^{\lambda t}-1)}^n{e}^{-N\lambda t}$, is a special case. If we measure a great number, $r$, of intervals between the disintegrations of atoms in a single source, the fraction of the intervals that exceed $t$ has the "expected" value $\frac{(e^{-N\lambda t}-e^{-(N-r)\mathrm{}\lambda t})}{r(1-e^{\lambda t})}.$ The problem of determining fluctuations in the disintegration of a single substance from fluctuations in counting is solved for the case in which each disintegration gives a single ray capable of actuating the counter. Fluctuations in counts produced by two or more independent sources are considered. Since gamma-rays and secondary beta-rays are not emitted by every disintegrating atom, the distribution of counts due to such rays is discussed. The effect of the recovery time, $\tau$, of the counter is discussed, using at first the assumption that $\tau$ is the same for all counts, an approximation useful at low counting rates. With a source that would produce $f$ counts per sec. if $\tau$ were zero, the probability of $n$ counts in the interval ($0,t$) is obtained for two cases: (1) The counter is not clogged at time zero; (2) it is clogged. The second case has practical interest; the probability of an interval greater than $t$ between counts is 1 if $t\leqq r$ and $e^{-f(t-r)\mathrm{}}$ if $t\geqq r$. Differential-difference equations for the fluctuation-functions are derived. The fluctuations of counts due to a constant source, in a counter with variable recovery time, are obtained, using Skinner's formula for the frequency distribution of recovery times. For $t$ values greater than the maximum recovery time, the probability of an interval greater than $t$ is $e^{-f(t-{\tau }^{\prime })}$; ${\tau }^{\prime }$ is a constant. Formulas are derived for the stock fluctuations of each substance in a source containing several members of a radioactive series, subject to any desired initial conditions; recurrence equations governing the stock probabilities are given. The probability of a stock $n$ of the daughter of a constant parent which yields $f$ disintegrations per sec. is $S_n=\frac{{\left(\frac{f}{\lambda }\right)}^n{e}^{-\left(\frac{f}{\lambda }\right)}}{n!}$; here $\lambda$ is the decay constant of the daughter. General methods for finding the fluctuations in the emission of an entire radioactive series, or any part of a series, are given. The disintegration-fluctuations of the daughter of a constant parent obey the formula $W_n=\frac{{\mathrm{}\left(\mathrm{ft}\right)\mathrm{}}^n{e}^{-ft}}{n!}$ which applies also to the parent. However, fluctuations of parent and daughter are coupled, so that the Bateman type of formula does not apply to their combined emission. Instead, the probability of an interval greater than $t$ is $\exp [-ft-(\frac{f}{\lambda }\left)\right(1-e^{-\lambda t}\left)\right]$.

• Received 30 November 1935

DOI:https://doi.org/10.1103/PhysRev.49.355