4.5: Alpha Decay
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Many types of heavy atomic nucleus spontaneously decay to produce daughter nucleii via the emission of α-particles (i.e., helium nucleii) of some characteristic energy. This process is know as α-decay. Let us investigate the α-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 Z1=Z−2 and mass-number A1=A−4, and an α-particle of charge-number Z2=2 and mass-number A2=4. Let the characteristic energy of the α-particle be E. Incidentally, nuclear radii are found to satisfy the empirical formula R=1.5×10−15A1/3m=2.0×10−15Z1/31m for Z≫1.
In 1928, George Gamow proposed a very successful theory of α-decay, according to which the α-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 α-particle, whose energy is E, is trapped in a potential well of radius R by the potential barrier V(r)=Z1Z2e24πϵ0r 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 α-particle tunneling through the barrier is |T|2=exp(−2√2mℏ∫r2r1√V(r)−Edr), where r1=R and r2=Z1Z2e2/(4πϵ0E). Here, m=4mp is the α-particle mass. The previous expression reduces to |T|2=exp(−2√2β∫Ec/E1[1y−EEc]1/2dy), where β=(Z1Z2e2mR4πϵ0ℏ2)1/2=0.74Z2/31 is a dimensionless constant, and Ec=Z1Z2e24πϵ0R=1.44Z2/31MeV is the characteristic energy the α-particle would need in order to escape from the nucleus without tunneling. Of course, E≪Ec. It is easily demonstrated that ∫1/ϵ1(1y−ϵ)1/2dy≃π2√ϵ−2 when ϵ≪1. Hence. |T|2≃exp(−2√2β[π2√EcE−2]).
Now, the α-particle moves inside the nucleus with the characteristic velocity v=√2E/m. It follows that the particle bounces backward and forward within the nucleus at the frequency ν≃v/R, giving ν≃2×1028yr−1 for a 1 MeV α-particle trapped inside a typical heavy nucleus of radius 10−14 m. Thus, the α-particle effectively attempts to tunnel through the potential barrier ν times a second. If each of these attempts has a probability |T|2 of succeeding then the probability of decay per unit time is ν|T|2. Hence, if there are N(t)≫1 undecayed nuclii at time t then there are only N+dN at time t+dt, where dN=−Nν|T|2dt. This expression can be integrated to give N(t)=N(0)exp(−ν|T|2t). Now, the half-life, τ, is defined as the time which must elapse in order for half of the nuclii originally present to decay. It follows from the previous formula that τ=ln2ν|T|2. Note that the half-life is independent of N(0).
Finally, making use of the previous results, we obtain log10[τ(yr)]=−C1−C2Z2/31+C3Z1√E(MeV), where C1=28.5,C2=1.83,C3=1.73.
Figure 15: The experimentally determined half-life, τex of various atomic nucleii which decay via emission versus the best-fit theoretical half-life log10(τth)=−28.9−1.60Z2/31+1.61Z1/√E. Both half-lives are measured in years. Here, Z1=Z−2. Both half-lives are measured in years. Here, Z1=Z−2, where Z is the charge number of the nucleus, and
the characteristic energy of the emitted
-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.
Equation ([e5.64]) is known as the Geiger-Nuttall formula, because it was discovered empirically by H. Geiger and J.M. Nuttall in 1911 .
The half-life, τ, the daughter charge-number, Z1=Z−2, and the α-particle energy, E, for atomic nucleii which undergo α-decay are indeed found to satisfy a relationship of the form ([e5.64]). The best fit to the data (see Figure [fal]) is obtained using C1=28.9,C2=1.60,C3=1.61. Note that these values are remarkably similar to those calculated previously.
Contributors and Attributions
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)