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Physics LibreTexts

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=Z2 and mass-number A1=A4, 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=1.5×1015A1/3m=2.0×1015Z1/31m for Z1.

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(22mr2r1V(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(22βEc/E1[1yEEc]1/2dy), where β=(Z1Z2e2mR4πϵ02)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, EEc. It is easily demonstrated that 1/ϵ1(1yϵ)1/2dyπ2ϵ2 when ϵ1. Hence. |T|2exp(22β[π2EcE2]).

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 νv/R, giving ν2×1028yr1 for a 1 MeV α-particle trapped inside a typical heavy nucleus of radius 1014 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)]=C1C2Z2/31+C3Z1E(MeV), where C1=28.5,C2=1.83,C3=1.73.

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Figure 15: The experimentally determined half-life, τex of various atomic nucleii which decay via $\alpha $ emission versus the best-fit theoretical half-life log10(τth)=28.91.60Z2/31+1.61Z1/E. Both half-lives are measured in years. Here, Z1=Z2. Both half-lives are measured in years. Here, Z1=Z2, 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.

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=Z2, 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)


This page titled 4.5: Alpha Decay is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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