Gamow Tunneling and Alpha Decay

Derivation

Gamow Tunneling and Alpha Decay

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Statement

For an alpha particle of mass \(m\) and energy \(E\) confined behind the Coulomb barrier of a daughter nucleus of charge \(Z_d e\), the WKB transmission probability is \(T \approx e^{-G}\) with the Gamow exponent \(G = \dfrac{2}{\hbar}\displaystyle\int_{R}^{b}\sqrt{2m\!\left(V(r)-E\right)}\,dr\), whose closed form is \(G = \dfrac{2k}{\hbar}\sqrt{\dfrac{2m}{E}}\Big[\arccos\!\sqrt{R/b}-\sqrt{(R/b)(1-R/b)}\Big]\), reducing in the thick-barrier limit to \(G \to 2\pi\eta\) with the Sommerfeld parameter \(\eta = \dfrac{2Z_d e^2}{4\pi\epsilon_0\hbar v}\); the decay rate is \(\lambda \approx f\,e^{-G}\) and hence \(\log_{10} t_{1/2} \propto Z_d/\sqrt{E}\).

Why it matters

This was one of the first quantitative triumphs of quantum mechanics: Gamow (and independently Condon and Gurney) in 1928 explained why an alpha particle with only a few MeV escapes a nucleus whose Coulomb barrier is over 20 MeV tall. Classically forbidden, the escape is pure tunnelling, and the exponential sensitivity of \(G\) to \(E\) reproduces the empirical Geiger–Nuttall law spanning more than twenty orders of magnitude in half-life.

The very same barrier-penetration integral, run in reverse, governs thermonuclear fusion rates and the Gamow peak of stellar nucleosynthesis. Mastering this one integral therefore unlocks both radioactive decay and how stars burn.

Assumptions

The alpha particle exists preformed inside the nucleus.If clustering is weak, the true rate carries an extra preformation factor \(P \ll 1\), and the absolute rate is overestimated (though the energy scaling survives).

The barrier is a pure point-Coulomb potential \(V(r)=2Z_d e^2/4\pi\epsilon_0 r\) for \(r>R\).Drop it and the turning point \(b\) and the integrand change; nuclear-surface diffuseness and polarization shift \(G\) by a few percent, enough to move \(t_{1/2}\) by orders of magnitude.

Zero orbital angular momentum (\(\ell=0\)).For \(\ell\neq0\) a centrifugal term \(\hbar^2\ell(\ell+1)/2mr^2\) adds to \(V\), raising \(G\) and suppressing the rate; ignoring it overestimates high-spin decays.

The WKB semiclassical condition holds across the barrier interior.Near the turning points \(|d\lambda_{\mathrm{dB}}/dr|\to\infty\); the connection formulas (assumed prior result) patch this, but if \(E\) approaches the barrier top the two turning points merge and the treatment collapses.

The decay is a stationary, weakly-coupled single channel with \(\lambda\) small compared with the assault frequency.If the level is broad (\(\Gamma\) comparable to level spacing) the exponential-decay/quasi-stationary picture and the factorization \(\lambda=f\,e^{-G}\) break down.

Derivation

\[ T \approx \exp\!\left(-2\int_{r_1}^{r_2}\kappa(r)\,dr\right),\qquad \kappa(r)=\frac{\sqrt{2m\big(V(r)-E\big)}}{\hbar} \]

The WKB connection formulas (prior result) give the transmission of a smooth barrier as the exponential of the classically forbidden action; the factor \(2\) is the intensity (probability) exponent, not the amplitude one. A

\[ V(r)=\frac{2Z_d e^2}{4\pi\epsilon_0\,r}\equiv\frac{k}{r},\qquad r_1=R,\quad r_2=b\ \text{with}\ \frac{k}{b}=E \;\Rightarrow\; b=\frac{k}{E} \]

Outside the nuclear surface \(R\) the only force is the Coulomb repulsion between the alpha (\(2e\)) and daughter (\(Z_d e\)); the outer turning point \(b\) is where \(V=E\). We define the barrier strength \(k\equiv 2Z_d e^2/4\pi\epsilon_0\). A

\[ G=2\int_{R}^{b}\frac{\sqrt{2m(k/r-E)}}{\hbar}\,dr=\frac{2\sqrt{2mE}}{\hbar}\int_{R}^{b}\sqrt{\frac{b}{r}-1}\,\,dr \]

Substitute \(V=k/r\) and factor \(\sqrt{2mE}\) out of the radical using \(k/r-E=E(b/r-1)\); this isolates a dimensionless integral in \(r\). Symbols only, no numbers yet. B

\[ r=b\cos^2\theta \;\Rightarrow\; \frac{b}{r}-1=\tan^2\theta,\quad dr=-2b\cos\theta\sin\theta\,d\theta \]

\[ \int_{R}^{b}\sqrt{\frac{b}{r}-1}\,dr=\int_{0}^{\theta_1}2b\sin^2\theta\,d\theta=b\big[\theta-\sin\theta\cos\theta\big]_0^{\theta_1} \]

The trigonometric substitution rationalizes \(\sqrt{b/r-1}\to\tan\theta\); the sign flips because \(r=b\) maps to \(\theta=0\) and \(r=R\) to \(\theta_1=\arccos\sqrt{R/b}\), and reordering the limits restores a positive integrand. C

\[ \cos\theta_1=\sqrt{R/b},\ \ \sin\theta_1=\sqrt{1-R/b}\ \Rightarrow\ \int_R^b\!\sqrt{\tfrac{b}{r}-1}\,dr=b\!\left[\arccos\!\sqrt{\tfrac{R}{b}}-\sqrt{\tfrac{R}{b}\!\left(1-\tfrac{R}{b}\right)}\right] \]

\[ \boxed{\,G=\frac{2\sqrt{2mE}}{\hbar}\,b\left[\arccos\!\sqrt{\tfrac{R}{b}}-\sqrt{\tfrac{R}{b}\!\left(1-\tfrac{R}{b}\right)}\right]=\frac{2k}{\hbar}\sqrt{\frac{2m}{E}}\left[\arccos\!\sqrt{\tfrac{R}{b}}-\sqrt{\tfrac{R}{b}\!\left(1-\tfrac{R}{b}\right)}\right]\,}\]

Evaluate the bracket at the limits, then use \(\sqrt{2mE}\,b=\sqrt{2mE}\,(k/E)=k\sqrt{2m/E}\) to trade \(b\) for the barrier strength \(k\). This is the exact WKB result. B

\[ R\ll b:\quad \arccos\!\sqrt{R/b}\approx\frac{\pi}{2}-\sqrt{R/b},\qquad \sqrt{\tfrac{R}{b}(1-\tfrac{R}{b})}\approx\sqrt{R/b} \]

\[ G\approx\frac{2k}{\hbar}\sqrt{\frac{2m}{E}}\left(\frac{\pi}{2}-2\sqrt{\frac{R}{b}}\right) \]

Expand for a thick barrier (nuclear radius far inside the outer turning point). The leading \(\pi/2\) is the point-charge penetrability; the \(-2\sqrt{R/b}\) term is the finite-size correction that lets the alpha start at the surface rather than at \(r=0\). B

\[ G_0\equiv\frac{\pi k}{\hbar}\sqrt{\frac{2m}{E}}=\frac{\pi}{\hbar}\frac{2Z_d e^2}{4\pi\epsilon_0}\sqrt{\frac{2m}{E}}=2\pi\eta,\qquad \eta=\frac{2Z_d e^2}{4\pi\epsilon_0\hbar v},\ \ v=\sqrt{\tfrac{2E}{m}} \]

Collect the leading term. With \(v=\sqrt{2E/m}\) one has \(\sqrt{2m/E}=2/v\), so \(G_0=\frac{\pi k}{\hbar}\cdot\frac{2}{v}=2\pi\,\frac{k}{\hbar v}=2\pi\eta\) with \(\eta\equiv k/(\hbar v)\). Using \(e^2/4\pi\epsilon_0=\alpha\hbar c\) gives the compact \(\eta=2Z_d\alpha\sqrt{m c^2/2E}\). A

\[ \lambda \approx f\,e^{-G},\qquad f=\frac{v_{\mathrm{in}}}{2R},\qquad t_{1/2}=\frac{\ln 2}{\lambda} \]

\[ \log_{10} t_{1/2}\approx \frac{G_0}{\ln 10}-\log_{10}f = a\,\frac{Z_d}{\sqrt{E}}+b' \]

The rate factorizes into an assault frequency \(f\) (how often the alpha strikes the wall, with \(v_{\mathrm{in}}\) its internal speed) times the tunnelling probability \(e^{-G}\). Because \(G_0\propto Z_d/\sqrt{E}\), the logarithm of the half-life is linear in \(Z_d/\sqrt{E}\): the Geiger–Nuttall law. B

Result

\[ T\approx e^{-G},\quad G=\frac{2k}{\hbar}\sqrt{\frac{2m}{E}}\Big[\arccos\!\sqrt{\tfrac{R}{b}}-\sqrt{\tfrac{R}{b}(1-\tfrac{R}{b})}\Big]\ \xrightarrow{R\ll b}\ 2\pi\eta,\quad \lambda\approx\frac{v_{\mathrm{in}}}{2R}\,e^{-G} \]

Reading. The escape probability per wall-collision is the exponential of a negative action accumulated across the classically forbidden region between the nuclear surface \(R\) and the Coulomb outer turning point \(b=2Z_d e^2/4\pi\epsilon_0 E\). Because that action is huge (\(G\sim30\)–\(90\)) and enters exponentially, a factor-of-two change in \(E\) swings the half-life by dozens of decades. The leading piece \(2\pi\eta\) depends only on daughter charge and alpha speed; the negative correction rewards a wider nucleus (larger \(R\)), which thins the barrier it must cross.

Units check. \(G\) must be dimensionless. \(\sqrt{2mE}\) is a momentum (kg·m·s\(^{-1}\)); times a length \(b\) gives an action (kg·m\(^2\)·s\(^{-1}\)); divided by \(\hbar\) (J·s = kg·m\(^2\)·s\(^{-1}\)) it is pure number. The bracket is dimensionless (arccos and ratios of lengths), and \(\eta=(\text{energy·length})/(\text{action·velocity})\) is likewise dimensionless. \(\lambda\) has units \((\mathrm{m/s})/\mathrm{m}=\mathrm{s}^{-1}\), correct for a rate.

Limiting cases

Thick barrier \(R\ll b\): \(G\to 2\pi\eta-\dfrac{8}{\hbar}\sqrt{2mE\,R\,b}\big/2\)… i.e. \(G\to 2\pi\eta-\dfrac{4k}{\hbar}\sqrt{\tfrac{2m}{E}}\sqrt{\tfrac{R}{b}}\), the point-charge Gamow factor with a finite-size reduction.
Low energy \(E\to0\): \(b\to\infty\), \(\eta\propto1/\sqrt{E}\to\infty\), \(T\to0\): long-lived nuclei, the ultra-slow tail of Geiger–Nuttall.
\(E\) near barrier top \(E\to V(R)\): \(b\to R\), the bracket \(\to0\), \(G\to0\), \(T\to1\); no barrier remains and the state is effectively unbound.
Neutral emission (\(Z_d\to0\) or short range): \(k\to0\), \(G\to0\); no Coulomb barrier, so this Gamow suppression is specific to charged-particle emission.
Heavy alpha / classical limit \(\hbar\to0\): \(G\to\infty\), \(T\to0\): tunnelling vanishes, recovering classical confinement.

Breaks when

The energy approaches the barrier top. When \(E\!\to\!V(R)\) the inner and outer turning points coalesce; the WKB connection formulas (which assume well-separated linear turning points) fail, and the closed form with \(\arccos\) loses meaning.
The barrier is thin or low. If \(R\) is not \(\ll b\) the \(2\pi\eta\) approximation is invalid, and even the exact WKB bracket degrades because the semiclassical condition \(|d\lambda_{\mathrm{dB}}/dr|\ll1\) is violated over a large fraction of the barrier.
Non-spherical or strongly-clustered nuclei. Deformation, \(\ell\neq0\) centrifugal barriers, and a preformation probability \(P\neq1\) all rescale the absolute rate; the bare \(\lambda=f e^{-G}\) with \(P=1\) can be off by 1–3 orders of magnitude.
Broad resonances / non-exponential decay. For very short-lived, wide states the quasi-stationary factorization into assault-frequency times penetrability breaks; a full R-matrix or complex-energy (Gamow-state) treatment is required.

Failure modes

Using the parent charge \(Z\) instead of the daughter \(Z_d=Z-2\). The tunnelling alpha feels the residual nucleus, not the original; using \(Z\) inflates \(k\) and \(G\).
Dropping the factor of 2 in \(T=e^{-2\gamma}\). Confusing the amplitude action \(\gamma\) with the probability exponent halves \(G\) and mis-predicts \(t_{1/2}\) by many decades.
Feeding in the \(Q\)-value instead of the alpha kinetic energy \(E\). Recoil of the daughter takes a share: \(E=Q\,A_d/(A_d+4)\). Using \(Q\) makes \(v\), \(b\), and \(G\) all slightly wrong in a steeply sensitive formula.
Setting the lower limit to \(r=0\). The Coulomb form only applies outside the nuclear surface; integrating from \(0\) double-counts a region where \(V\) is not \(k/r\) and diverges spuriously.
Confusing \(b\) with the nuclear radius \(R\). \(b\) is the outer classical turning point (tens of fm), \(R\) the surface (\(\sim9\) fm); swapping them inverts the barrier width.
Plugging the nucleon mass rather than the reduced/alpha mass into \(m\). Use \(m\approx m_\alpha\) (or the reduced mass \(m_\alpha M_d/(m_\alpha+M_d)\)); a factor-4 mass error moves \(G\) substantially through \(\sqrt{m}\).

Discussion

The physical picture is a three-factor rate: a preformation probability that an alpha cluster exists at the surface, an assault frequency \(f\) counting how often it batters the wall, and a tunnelling probability \(e^{-G}\). Only the last carries the violent energy dependence, and it does so exponentially. The engine of Geiger–Nuttall is thus not any nuclear subtlety but the humble geometry of a \(1/r\) barrier: the action integral \(\int\sqrt{k/r-E}\,dr\) scales as \(k/\sqrt{E}\), so \(\ln t_{1/2}\) rides linearly on \(Z_d/\sqrt{E}\). This is why alpha half-lives fan out across twenty-four orders of magnitude while the alpha energies vary by barely a factor of two.

The same barrier integral, evaluated for an incoming particle, is the astrophysical S-factor's penetrability. In a stellar plasma the reaction rate is the convolution of the Maxwell–Boltzmann tail (rising with \(E\)) against \(e^{-2\pi\eta}\) (falling with \(E\)); their product is the sharply peaked Gamow window that sets which fusion reactions ignite at a given temperature. Alpha decay and stellar burning are two readings of one equation, forward and backward in time.

Historically this was decisive: in 1928 tunnelling was brand new, and the fact that a crude one-line estimate reproduced lifetimes from microseconds to gigayears convinced physicists that the Schrödinger equation governed the nucleus. The residual disagreements (factors of a few to a few hundred) are exactly the preformation and structure physics that modern nuclear theory refines.

Rigorously, the WKB penetrability is only the leading semiclassical term of the exact Coulomb penetration factor \(P_\ell=\rho/(F_\ell^2+G_\ell^2)\) built from the regular and irregular Coulomb functions evaluated at the channel radius \(\rho=k_{\infty}R\). R-matrix theory expresses the decay width as \(\Gamma=2P_\ell\gamma_\lambda^2\), separating the external penetrability from an internal reduced width \(\gamma_\lambda^2\) that encodes the many-body clustering. In this language the Gamow \(e^{-G}\) is the barrier factor \(P_0\) in the extreme sub-barrier limit, the assault frequency and preformation collapse into \(\gamma_\lambda^2\), and the whole heuristic \(\lambda=fe^{-G}\) is recovered as the stationary-phase evaluation of the exact expression. The finite width of the state is then the imaginary part of a complex (Gamow-state) eigenvalue with purely outgoing boundary conditions.

Common misconceptions. Tunnelling does not mean the alpha "borrows" energy to hop over the barrier — its energy is fixed at \(E\); it is the wavefunction that penetrates the forbidden region with an exponentially small tail. Nor does a higher barrier always mean a longer life: what matters is the barrier area (width × height), so a more energetic alpha in a higher-\(Z\) nucleus can still escape faster because its outer turning point \(b\) has moved inward, thinning the barrier.

Worked examples

Constants used: \(\hbar c=197.33\) MeV·fm, \(\alpha=1/137.036\), \(\alpha\hbar c=1.440\) MeV·fm, \(m_\alpha c^2=3727.4\) MeV, \(r_0=1.2\) fm, touching-sphere radius \(R=r_0(A_d^{1/3}+4^{1/3})\), and \(k=2Z_d\,\alpha\hbar c\).

Example 1 — \({}^{238}\mathrm{U}\to{}^{234}\mathrm{Th}+\alpha\). Daughter Th: \(Z_d=90\), \(A_d=234\); alpha kinetic energy \(E\approx4.20\) MeV.

\[ k=2Z_d\,\alpha\hbar c=2(90)(1.440)=259.2\ \text{MeV·fm},\qquad b=\frac{k}{E}=\frac{259.2}{4.20}=61.7\ \text{fm} \]

Barrier strength and outer turning point. A

\[ R=1.2\big(234^{1/3}+4^{1/3}\big)=1.2(6.16+1.587)=9.30\ \text{fm},\quad \frac{R}{b}=0.1507,\ \sqrt{R/b}=0.388 \]

Nuclear-surface start point. A

\[ \arccos(0.388)=1.172,\quad \sqrt{0.1507(0.8493)}=0.358,\quad \text{bracket}=1.172-0.358=0.814 \]

Evaluate the exact WKB bracket (radians). B

\[ \eta=2Z_d\alpha\sqrt{\tfrac{m c^2}{2E}}=\frac{180}{137.036}\sqrt{\frac{3727.4}{8.40}}=1.314(21.06)=27.66,\quad G_0=2\pi\eta=173.8 \]

\[ G=G_0\,\frac{\text{bracket}}{\pi/2}=173.8\times\frac{0.814}{1.5708}=90.1 \]

Leading factor times the exact-bracket ratio (the point-charge value would be \(G_0\)). B

\[ T=e^{-90.1}=10^{-90.1/\ln10}=10^{-39.1}\approx8\times10^{-40} \]

\[ v=c\sqrt{\tfrac{2E}{mc^2}}=c\sqrt{\tfrac{8.40}{3727.4}}=0.0475c=1.42\times10^{7}\ \text{m/s},\ \ f=\frac{v}{2R}=\frac{1.42\times10^{7}}{1.86\times10^{-14}}=7.6\times10^{20}\,\text{s}^{-1} \]

Penetrability and assault frequency. B

\[ \lambda=fT\approx6\times10^{-19}\,\text{s}^{-1},\quad t_{1/2}=\frac{\ln2}{\lambda}\approx1.1\times10^{18}\,\text{s}\approx4\times10^{10}\ \text{yr} \]

Reading. The crude model gives \(t_{1/2}\sim3\times10^{10}\) yr against the measured \(4.5\times10^9\) yr — agreement to within a factor \(\sim8\), which for a quantity spanning \(10^{40}\) is extraordinary. The residual factor is preformation and radius uncertainty.

Units check. \(v/2R\) is s\(^{-1}\); \(T\) dimensionless; \(\lambda\) in s\(^{-1}\); \(\ln2/\lambda\) in s. Consistent.

Example 2 — \({}^{212}\mathrm{Po}\to{}^{208}\mathrm{Pb}+\alpha\). Daughter Pb: \(Z_d=82\), \(A_d=208\); alpha kinetic energy \(E\approx8.78\) MeV. This high-energy decay contrasts sharply with Example 1.

\[ k=2(82)(1.440)=236.2\ \text{MeV·fm},\quad b=\frac{236.2}{8.78}=26.9\ \text{fm} \]

Higher \(E\) pulls the outer turning point in from 62 fm to 27 fm. A

\[ R=1.2\big(208^{1/3}+4^{1/3}\big)=1.2(5.925+1.587)=9.01\ \text{fm},\quad \frac{R}{b}=0.335,\ \sqrt{R/b}=0.579 \]

Barrier is far thinner: the alpha starts a third of the way to \(b\). A

\[ \arccos(0.579)=0.953,\ \sqrt{0.335(0.665)}=0.472,\ \text{bracket}=0.481 \]

\[ \eta=\frac{164}{137.036}\sqrt{\frac{3727.4}{17.56}}=1.197(14.57)=17.44,\quad G_0=2\pi\eta=109.6 \]

Exact bracket and leading factor. B

\[ G=109.6\times\frac{0.481}{1.5708}=33.5,\quad T=e^{-33.5}=10^{-14.55}\approx2.8\times10^{-15} \]

\[ v=c\sqrt{\tfrac{17.56}{3727.4}}=0.0686c=2.06\times10^{7}\,\text{m/s},\ f=\frac{2.06\times10^{7}}{1.80\times10^{-14}}=1.14\times10^{21}\,\text{s}^{-1} \]

Because both \(G_0\) and the bracket shrink, \(G\) falls from 90 to 34. B

\[ \lambda=fT\approx3.2\times10^{6}\,\text{s}^{-1},\quad t_{1/2}=\frac{\ln2}{\lambda}\approx2.2\times10^{-7}\ \text{s}\approx0.22\ \mu\text{s} \]

Reading. Predicted \(0.22\ \mu\)s versus measured \(0.3\ \mu\)s — near-perfect. Doubling the alpha energy from 4.2 to 8.8 MeV collapsed the half-life from \(\sim10^{10}\) yr to sub-microsecond: twenty-four decades from a factor-two energy change, the Geiger–Nuttall law in action.

Units check. Same chain as Example 1; \(t_{1/2}\) in seconds. Consistent.

Problems

(Easy — turning point.) For \({}^{226}\mathrm{Ra}\to{}^{222}\mathrm{Rn}+\alpha\) the daughter is Rn (\(Z_d=86\)) and the alpha kinetic energy is \(E=4.78\) MeV. Compute the barrier strength \(k\) and the outer turning point \(b\).
Solution
\(k=2Z_d\alpha\hbar c=2(86)(1.440)=247.7\) MeV·fm. \(b=k/E=247.7/4.78=51.8\) fm. (For reference the barrier height at the surface \(R\approx9.2\) fm is \(V(R)=k/R\approx27\) MeV, far above \(E\): a genuinely classically forbidden escape.)
(Easy–medium — Sommerfeld parameter.) For the same \({}^{226}\mathrm{Ra}\) decay compute the Sommerfeld parameter \(\eta=2Z_d\alpha\sqrt{m c^2/2E}\) and the point-charge Gamow factor \(G_0=2\pi\eta\).
Solution
\(\sqrt{m c^2/2E}=\sqrt{3727.4/9.56}=\sqrt{389.9}=19.75\). \(\eta=(2\times86/137.036)\times19.75=1.255\times19.75=24.79\). \(G_0=2\pi\eta=155.8\). Taken alone this would give \(T=e^{-155.8}\sim10^{-68}\); the finite-size correction of the next problem is essential.
(Medium — exact WKB factor.) Take \(R=1.2(222^{1/3}+4^{1/3})\) fm for \({}^{226}\mathrm{Ra}\). Using \(b=51.8\) fm from Problem 1, evaluate the exact bracket and hence \(G\).
Solution
\(R=1.2(6.055+1.587)=9.17\) fm. \(R/b=9.17/51.8=0.177\), \(\sqrt{R/b}=0.421\). \(\arccos(0.421)=1.136\) rad; \(\sqrt{0.177(0.823)}=\sqrt{0.1457}=0.382\); bracket \(=1.136-0.382=0.754\). \(G=G_0(\text{bracket})/(\pi/2)=155.8\times0.754/1.5708=155.8\times0.480=74.8\). The finite-size term has cut the exponent nearly in half.
(Medium–hard — half-life.) With \(G=74.8\) from Problem 3, estimate \(\lambda\) and \(t_{1/2}\) for \({}^{226}\mathrm{Ra}\) (measured: 1600 yr). Use \(v=c\sqrt{2E/m c^2}\) and \(f=v/2R\).
Solution
\(T=e^{-74.8}=10^{-32.49}=3.2\times10^{-33}\). \(v=c\sqrt{9.56/3727.4}=0.0507c=1.52\times10^{7}\) m/s. \(f=1.52\times10^{7}/(2\times9.17\times10^{-15})=8.3\times10^{20}\) s\(^{-1}\). \(\lambda=fT=8.3\times10^{20}\times3.2\times10^{-33}=2.7\times10^{-12}\) s\(^{-1}\). \(t_{1/2}=\ln2/\lambda=0.693/2.7\times10^{-12}=2.6\times10^{11}\) s \(\approx8300\) yr, versus 1600 yr measured — right order of magnitude (factor \(\sim5\)) for a two-parameter model of a \(10^{11}\)-second lifetime.
(Hard — Geiger–Nuttall coefficient and sensitivity.) (a) From \(G_0=2\pi\eta\) show \(\log_{10}t_{1/2}\approx a\,Z_d/\sqrt{E}+b'\) and evaluate the coefficient \(a\) (with \(E\) in MeV), ignoring the slowly-varying \(\log_{10}f\) and the finite-size term. (b) Using the leading term only, by what factor does \({}^{226}\mathrm{Ra}\)'s half-life change if \(E\) rises 10% to \(5.26\) MeV?
Solution
(a) \(G_0=2\pi\eta=4\pi\alpha\sqrt{m c^2/2}\,Z_d/\sqrt{E}\). Then \(\log_{10}t_{1/2}\approx G_0/\ln10+\text{const}\), so \(a=\dfrac{4\pi\alpha\sqrt{m c^2/2}}{\ln10}\). Numerically \(4\pi\alpha=0.09169\), \(\sqrt{m c^2/2}=\sqrt{1863.7}=43.17\) MeV\(^{1/2}\), product \(=3.96\) MeV\(^{1/2}\); divide by \(\ln10=2.303\): \(a\approx1.72\) MeV\(^{1/2}\). So \(\log_{10}t_{1/2}\approx1.72\,Z_d/\sqrt{E}+b'\), the empirical Geiger–Nuttall slope (the finite-size term lowers the effective slope toward the observed \(\sim1.6\)). (b) \(\Delta\log_{10}t_{1/2}=a Z_d\big(E_1^{-1/2}-E_2^{-1/2}\big)=1.72\times86\,(4.78^{-1/2}-5.26^{-1/2})=147.9\,(0.4574-0.4360)=147.9\times0.0214=3.16\). The half-life drops by \(10^{3.16}\approx1.4\times10^{3}\), i.e. a factor of about 1400 from a mere 10% energy increase — the hallmark exponential sensitivity of Gamow tunnelling.