Alpha Decay

Many types of heavy atomic nucleus spontaneously decay to produce daughter nucleii via the emission of $\alpha$-particles (i.e., helium nucleii) of some characteristic energy. This process is know as $\alpha$-decay. Let us investigate the $\alpha$-decay of a particular type of atomic nucleus of radius $R$, charge-number $Z$, and mass-number $A$. Such a nucleus thus decays to produce a daughter nucleus of charge-number $Z_1=Z-2$ and mass-number $A_1=A-4$, and an $\alpha$-particle of charge-number $Z_2=2$ and mass-number $A_2=4$. Let the characteristic energy of the $\alpha$-particle be $E$. Incidentally, nuclear radii are found to satisfy the empirical formula

$$R = 1.5\times 10^{-15} A^{1/3} {\rm m}=2.0\times 10^{-15} Z_1^{1/3} {\rm m}$$ (353)

for $Z\gg 1$.

In 1928, George Gamov proposed a very successful theory of $\alpha$-decay, according to which the $\alpha$-particle moves freely inside the nucleus, and is emitted after tunneling through the potential barrier between itself and the daughter nucleus. In other words, the $\alpha$-particle, whose energy is $E$, is trapped in a potential well of radius $R$ by the potential barrier

$$V(r) = \frac{Z_1 Z_2 e^2}{4\pi \epsilon_0 r}$$ (354)

for $r>R$.

Making use of the WKB approximation (and neglecting the fact that $r$ is a radial, rather than a Cartesian, coordinate), the probability of the $\alpha$-particle tunneling through the barrier is

$$\vert T\vert^{ 2} = \exp\left(-\frac{2\sqrt{2 m}}{\hbar}\int_{r_1}^{r_2} \sqrt{V(r)-E} dr\right),$$ (355)

where $r_1=R$ and $r_2 = Z_1 Z_2 e^2/(4\pi \epsilon_0 E)$. Here, $m=4 m_p$ is the $\alpha$-particle mass. The above expression reduces to

$$\vert T\vert^{ 2} = \exp\left(-2 \sqrt{2} \beta \int_{1}^{E_c/E}\left[\frac{1}{y}-\frac{E}{E_c}\right]^{1/2} dy\right),$$ (356)

where

$$\beta = \left(\frac{Z_1 Z_2 e^2 m R}{4\pi \epsilon_0 \hbar^2}\right)^{1/2} = 0.74 Z_1^{2/3}$$ (357)

is a dimensionless constant, and

$$E_c = \frac{Z_1 Z_2 e^2}{4\pi \epsilon_0 R} = 1.44 Z_1^{2/3} {\rm MeV}$$ (358)

is the characteristic energy the $\alpha$-particle would need in order to escape from the nucleus without tunneling. Of course, $E\ll E_c$. It is easily demonstrated that

$$\int_1^{1/\epsilon}\left[\frac{1}{y} - \epsilon\right]^{1/2} dy \simeq \frac{\pi}{2 \sqrt{\epsilon}}-2$$ (359)

when $\epsilon\ll 1$. Hence.

$$\vert T\vert^{ 2} \simeq \exp\left(-2 \sqrt{2} \beta\left[\frac{\pi}{2}\sqrt{\frac{E_c}{E}}-2\right]\right).$$ (360)

Now, the $\alpha$-particle moves inside the nucleus with the characteristic velocity $v= \sqrt{2 E/m}$. It follows that the particle bounces backward and forward within the nucleus at the frequency $\nu\simeq v/R$, giving

$$\nu\simeq 2\times 10^{28} {\rm yr}^{-1}$$ (361)

for a 1 MeV $\alpha$-particle trapped inside a typical heavy nucleus of radius $10^{-14}$m. Thus, the $\alpha$-particle effectively attempts to tunnel through the potential barrier $\nu$ times a second. If each of these attempts has a probability $\vert T\vert^{ 2}$ of succeeding, then the probability of decay per unit time is $\nu \vert T\vert^2$. Hence, if there are $N(t)\gg 1$ undecayed nuclii at time $t$ then there are only $N+dN$ at time $t+dt$, where

$$dN = - N \nu \vert T\vert^2 dt.$$ (362)

This expression can be integrated to give

$$N(t) = N(0) \exp(-\nu \vert T\vert^2 t).$$ (363)

Now, the half-life, $\tau$, is defined as the time which must elapse in order for half of the nuclii originally present to decay. It follows from the above formula that

$$\tau = \frac{\ln 2}{\nu \vert T\vert^2}.$$ (364)

Note that the half-life is independent of $N(0)$.

Finally, making use of the above results, we obtain

$$\log_{10}[\tau ({\rm yr})] = -C_1 - C_2 Z_1^{ 2/3} + C_3 \frac{Z_1}{\sqrt{E({\rm MeV})}},$$ (365)

where

$$C_1 = 28.5,$$ (366)

$$C_2 = 1.83,$$ (367)

$$C_3 = 1.73.$$ (368)

Figure 15: The experimentally determined half-life, $\tau _{ex}$, of various atomic nucleii which decay via $\alpha$ emission versus the best-fit theoretical half-life $\log_{10}(\tau_{th}) = -28.9 - 1.60 Z_1^{ 2/3} + 1.61 Z_1/\sqrt{E}$. Both half-lives are measured in years. Here, $Z_1=Z-2$, where $Z$ is the charge number of the nucleus, and $E$ the characteristic energy of the emitted $\alpha$-particle in MeV. In order of increasing half-life, the points correspond to the following nucleii: Rn 215, Po 214, Po 216, Po 197, Fm 250, Ac 225, U 230, U 232, U 234, Gd 150, U 236, U 238, Pt 190, Gd 152, Nd 144. Data obtained from IAEA Nuclear Data Centre.

The half-life, $\tau$, the daughter charge-number, $Z_1=Z-2$, and the $\alpha$-particle energy, $E$, for atomic nucleii which undergo $\alpha$-decay are indeed found to satisfy a relationship of the form (365). The best fit to the data (see Fig. 15) is obtained using

$$C_1 = 28.9,$$ (369)

$$C_2 = 1.60,$$ (370)

$$C_3 = 1.61.$$ (371)

Note that these values are remarkably similar to those calculated above.