Gamow Factor and the Geiger-Nuttall Law

Derivation

Gamow Factor and the Geiger-Nuttall Law

D-269 Home PU-304 Threads energy · chance · matter Depends on wkb-tunnelling-probability, Coulomb Energy of a Uniformly Charged Nucleus

Statement

For alpha decay treated as a pre-formed alpha particle tunnelling through the Coulomb barrier of the daughter nucleus, the WKB transmission probability is \( T = e^{-G} \) with Gamow exponent \( G = \dfrac{2}{\hbar}\displaystyle\int_{R}^{b}\sqrt{2m\big(V(r)-Q\big)}\,dr \). Evaluating the integral for the Coulomb potential \( V(r)=\dfrac{zZe^{2}}{4\pi\varepsilon_{0}r} \) in the thick-barrier limit gives \( G \simeq \dfrac{\pi zZe^{2}}{4\pi\varepsilon_{0}\hbar}\sqrt{\dfrac{2m}{Q}} - \dfrac{4}{\hbar}\sqrt{\dfrac{2m\,zZe^{2}R}{4\pi\varepsilon_{0}}} \), so that \( \log_{10}t_{1/2} = \dfrac{a\,Z}{\sqrt{Q}} + b \) — the linear Geiger–Nuttall relation between the logarithm of the half-life and \( 1/\sqrt{Q} \).

Why it matters

Alpha half-lives span more than twenty-four orders of magnitude — from sub-microseconds for \( {}^{212}\mathrm{Po} \) to \( 10^{17}\,\mathrm{s} \) for \( {}^{238}\mathrm{U} \) — while the released energies \( Q \) vary by less than a factor of three. No classical model can bridge such a gap. Gamow, and independently Gurney and Condon, showed in 1928 that quantum tunnelling through the Coulomb barrier does exactly this: because \( G \) sits in an exponent and scales as \( Z/\sqrt{Q} \), a modest change in \( Q \) is amplified into an enormous change in lifetime.

Historically this was one of the first quantitative triumphs of quantum mechanics applied to the nucleus, turning the empirical Geiger–Nuttall plot of 1911 into a derived law. The same barrier-penetration factor reappears in thermonuclear reaction rates, in cluster radioactivity, and in cold fusion of heavy ions, making the Gamow factor a cornerstone of both nuclear astrophysics and nuclear structure.

Assumptions

The alpha particle exists as a pre-formed cluster inside the parent.If dropped, the decay constant must include an explicit spectroscopic (pre-formation) factor \( P_\alpha \lesssim 10^{-2} \); the exponential energy dependence survives but the absolute normalisation shifts by orders of magnitude.

The interaction outside the nuclear radius is the point-Coulomb potential \( V(r)=zZe^{2}/4\pi\varepsilon_{0}r \).If dropped (finite nuclear charge distribution, screening, or nuclear polarisation), the barrier shape changes near \( r\approx R \) and the constant offset \( b \) in the Geiger–Nuttall law is modified.

The WKB approximation holds: the de Broglie wavelength varies slowly across the barrier, \( \left|\dfrac{d\lambda}{dr}\right|\ll 1 \).If dropped, the simple \( e^{-G} \) form acquires a pre-exponential connection factor of order unity from the turning points; the leading exponential is unchanged but sub-order-of-magnitude accuracy is lost.

The energy is well below the barrier top, \( Q \ll V(R) \), so a single classically forbidden region spans \( R \) to \( b \).If dropped (near-barrier-top decay), the thick-barrier expansion \( f(x)\to\pi/2 \) fails and the full integral \( f(x)=\arccos\sqrt{x}-\sqrt{x(1-x)} \) must be retained.

Angular momentum carried off is zero (\( \ell=0 \)).If dropped, a centrifugal term \( \hbar^{2}\ell(\ell+1)/2mr^{2} \) adds to the barrier, raising \( G \) and further suppressing high-\( \ell \) transitions.

Derivation

\[ T = e^{-G}, \qquad G = \frac{2}{\hbar}\int_{R}^{b}\sqrt{2m\big(V(r)-Q\big)}\;dr \]

The WKB transmission probability through a single classically forbidden region between turning points is \( T=\exp\!\big(-2\!\int|k(r)|\,dr\big) \) with \( \hbar|k| = \sqrt{2m(V-Q)} \); imported from the WKB tunnelling result. A

\[ V(r)=\frac{zZe^{2}}{4\pi\varepsilon_{0}\,r}, \qquad V(b)=Q \;\Rightarrow\; b=\frac{zZe^{2}}{4\pi\varepsilon_{0}\,Q} \]

The alpha (charge \( z=2 \)) sees the daughter (charge \( Z \)) through the uniform-sphere Coulomb term, which reduces to the point form for \( r>R \). The outer classical turning point \( b \) is defined by \( V=Q \). A

\[ G = \frac{2\sqrt{2mQ}}{\hbar}\int_{R}^{b}\sqrt{\frac{b}{r}-1}\;dr \]

Substitute \( V(r)=Qb/r \) (using \( b=zZe^{2}/4\pi\varepsilon_{0}Q \)) into \( V-Q=Q(b/r-1) \) and pull the constant \( \sqrt{Q} \) outside. Pure algebraic rearrangement. A

\[ \int_{R}^{b}\sqrt{\frac{b}{r}-1}\;dr = b\Big[\arccos\sqrt{\tfrac{R}{b}} - \sqrt{\tfrac{R}{b}\big(1-\tfrac{R}{b}\big)}\Big] \]

Put \( r=b\sin^{2}\theta \), so \( \sqrt{b/r-1}=\cot\theta \) and \( dr=2b\sin\theta\cos\theta\,d\theta \); the integrand becomes \( b(1+\cos 2\theta)\,d\theta \), giving \( b[\theta+\sin\theta\cos\theta] \) between \( \theta_1=\arcsin\sqrt{R/b} \) and \( \pi/2 \). Exact closed form. B

\[ G = \frac{2\sqrt{2m}}{\hbar}\,\frac{zZe^{2}}{4\pi\varepsilon_{0}\sqrt{Q}}\;f(x), \qquad f(x)=\arccos\sqrt{x}-\sqrt{x(1-x)}, \quad x=\frac{R}{b} \]

Insert the integral of step 4 and use \( b\sqrt{Q}=zZe^{2}/(4\pi\varepsilon_{0}\sqrt{Q}) \) to collect the prefactor. The whole geometric dependence is now packed into the dimensionless barrier-thickness function \( f(x) \). B

\[ x\ll 1:\quad f(x) = \frac{\pi}{2} - 2\sqrt{x} + \frac{1}{3}x^{3/2} + \mathcal{O}(x^{5/2}) \]

Thick-barrier (deep-tunnelling) limit \( R\ll b \): expand \( \arccos\sqrt{x}=\pi/2-\sqrt{x}-\tfrac16 x^{3/2}-\dots \) and \( \sqrt{x(1-x)}=\sqrt{x}-\tfrac12 x^{3/2}-\dots \) and subtract. Retaining the constant and the \( \sqrt{x} \) term is sufficient. B

\[ G \simeq \underbrace{\frac{\pi zZe^{2}}{4\pi\varepsilon_{0}\hbar}\sqrt{\frac{2m}{Q}}}_{\propto\, Z/\sqrt{Q}} \;-\; \underbrace{\frac{4}{\hbar}\sqrt{\frac{2m\,zZe^{2}R}{4\pi\varepsilon_{0}}}}_{\text{constant in }Q} \]

Substitute the expansion of step 6, using \( \sqrt{x}=\sqrt{R/b}=\sqrt{4\pi\varepsilon_{0}RQ/zZe^{2}} \) so the \( Q \)-dependence cancels in the second term. The first (leading) term is the Sommerfeld-like barrier factor. A

\[ \lambda = f_{0}\,e^{-G}, \quad t_{1/2}=\frac{\ln 2}{\lambda} \;\Rightarrow\; \log_{10}t_{1/2} = \frac{G}{\ln 10} + \log_{10}\!\frac{\ln 2}{f_{0}} = \frac{a\,Z}{\sqrt{Q}} + b \]

The decay constant is the assault frequency \( f_{0} \) times the transmission probability. Take the base-10 logarithm; the leading term of \( G \) supplies the slope \( a \) against \( 1/\sqrt{Q} \) and the constant term plus \( \log(\ln2/f_{0}) \) supplies the intercept \( b \). B

Result

\[ T=e^{-G},\qquad G \simeq \frac{\pi zZe^{2}}{4\pi\varepsilon_{0}\hbar}\sqrt{\frac{2m}{Q}} \;=\; 2\pi zZ\,\alpha\,\sqrt{\frac{mc^{2}}{2Q}} \;\approx\; \frac{3.96\,Z}{\sqrt{Q\,[\mathrm{MeV}]}} \]

\[ \boxed{\;\log_{10}t_{1/2} = \frac{a\,Z}{\sqrt{Q}} + b, \qquad a\approx 1.72\ (Q\ \text{in MeV})\;} \]

Reading. The tunnelling suppression is exponential in the Gamow exponent \( G \), which grows linearly with the daughter charge \( Z \) and inversely with \( \sqrt{Q} \). Because \( G \) is an exponent, halving \( Q \) multiplies \( \sqrt{Q} \) in the denominator, adding tens to \( G \) and multiplying the half-life by many powers of ten. Plotting \( \log_{10}t_{1/2} \) against \( 1/\sqrt{Q} \) for an isotopic chain gives a straight line of slope \( \propto Z \) — the Geiger–Nuttall law.

Units check. \( \alpha \) is dimensionless, \( mc^{2} \) and \( Q \) are energies, so \( \sqrt{mc^{2}/Q} \) is dimensionless and \( G \) is a pure number, as an exponent must be. In the integral form, \( \sqrt{2m(V-Q)} \) has units of momentum \( \mathrm{kg\,m\,s^{-1}} \), times \( dr \) gives \( \mathrm{J\,s} \), divided by \( \hbar \) is dimensionless. In SI, using \( \hbar c=197.3\,\mathrm{MeV\,fm} \) and \( e^{2}/4\pi\varepsilon_{0}=1.440\,\mathrm{MeV\,fm} \), the coefficient evaluates to \( 3.96 \) with \( Q \) in MeV.

Limiting cases

Thick barrier \( x=R/b\to 0 \): \( f\to\pi/2 \), recovering the pure Sommerfeld factor \( G=2\pi zZ\alpha\,c/v \) with \( v=\sqrt{2Q/m} \).
Energy near barrier top \( Q\to V(R) \): \( x\to 1 \), \( f(x)\to 0 \), \( G\to 0 \) and \( T\to 1 \) — no suppression, the alpha essentially rolls over the barrier.
Gamow-energy form: writing \( G=\sqrt{E_G/Q} \) defines the Gamow energy \( E_G=(2\pi zZ\alpha)^{2}\,mc^{2}/2 \), a fixed energy scale set only by the charges and reduced mass.
Nonzero \( \ell \): the centrifugal barrier adds \( \Delta G\approx \hbar\ell(\ell+1)/\!\big(\sqrt{2mzZe^{2}R/4\pi\varepsilon_0}\big) \), hindering high-spin transitions.
Light emitters / small \( Z \): the slope \( a\propto z \) collapses, and the Geiger–Nuttall line flattens toward the phase-space-dominated regime.

Breaks when

The alpha is not pre-formed. For nuclei far from closed shells the pre-formation probability \( P_\alpha \) varies by orders of magnitude across the chart; the Gamow factor still sets the \( Q \)-dependence but absolute half-lives scatter about the Geiger–Nuttall line by factors of \( 10^{2}\!-\!10^{3} \).
Decay proceeds by a competing or exotic channel. Spontaneous fission, proton emission, or cluster radioactivity (e.g. \( {}^{14}\mathrm{C} \) emission) follow different barrier geometries and mass factors, so the single-slope alpha law no longer describes the total lifetime.
The energy approaches the barrier top. The thick-barrier expansion \( f(x)\approx\pi/2-2\sqrt{x} \) breaks; the full \( f(x) \) is needed and the log–\( 1/\sqrt{Q} \) plot curves.
The barrier is strongly non-Coulombic near \( R \). Deformation, nuclear-force tail, and finite charge radius reshape the inner turning region, shifting the intercept \( b \) and spoiling a universal fit across widely different \( A \).

Failure modes

Using the parent charge instead of the daughter. The alpha tunnels away from the daughter nucleus, so \( Z=Z_{\text{parent}}-2 \) must appear in \( G \), not \( Z_{\text{parent}} \).
Putting \( E \) = total alpha kinetic energy instead of \( Q \). The relevant energy is the disintegration energy \( Q \) (kinetic energy plus daughter recoil), \( Q=T_\alpha\,(A/(A-4)) \); using \( T_\alpha \) alone underestimates \( Q \) by a few percent.
Forgetting the factor of 2 in \( G=2\gamma \). The transmission probability is \( e^{-2\gamma} \), not \( e^{-\gamma} \); dropping it halves the exponent and gives wildly wrong lifetimes.
Ignoring the lower limit \( R \). Taking the integral from \( 0 \) makes it diverge or overcounts; the barrier begins at the nuclear surface \( r=R \).
Treating the assault frequency as the fitted quantity. \( f_{0}\sim v/2R\sim 10^{21}\,\mathrm{s^{-1}} \) only sets the intercept; the dramatic lifetime range comes almost entirely from the exponential, not from \( f_{0} \).
Mixing units of \( Q \). The numerical coefficients \( 3.96 \) and \( 1.72 \) assume \( Q \) in MeV; using keV or joules without rescaling corrupts the slope.

Discussion

The heart of the result is that a lifetime observable is controlled by an exponentiated action integral. The action \( S=\int_R^b|p|\,dr \) is large — of order \( 30\hbar \)–\( 90\hbar \) for real alpha emitters — so \( e^{-2S/\hbar} \) is astronomically small and astronomically sensitive to \( S \). This is the same semiclassical structure that governs any barrier problem, from field emission of electrons (the Fowler–Nordheim law) to the escape rate over a potential barrier (Arrhenius / Kramers). Alpha decay is simply the cleanest nuclear realisation because the potential outside the surface is the exactly known Coulomb tail.

The Geiger–Nuttall line is the empirical shadow of this exponential: because \( G\propto Z/\sqrt{Q} \), plotting \( \log t_{1/2} \) against \( 1/\sqrt{Q} \) collapses an isotopic chain onto a straight line whose slope is fixed by \( Z \) and whose small deviations diagnose nuclear structure — shell closures, deformation, and pre-formation. The very steepness that makes half-lives hard to measure also makes \( Q \) an exquisitely precise handle: a 1% error in \( Q \) can move the predicted lifetime by a factor of a few.

The identical Gamow factor, read in the opposite direction (fusion rather than fission), sets stellar burning rates. There the relevant quantity is the Gamow peak — the product of the Maxwell–Boltzmann tail \( e^{-Q/kT} \) and the tunnelling factor \( e^{-\sqrt{E_G/Q}} \) — whose sharp maximum picks out the narrow energy window in which nuclear reactions actually occur in a star. Thus the same integral that explains why uranium survives billions of years explains why the Sun burns slowly and steadily.

A subtler point is that the pre-exponential factor is not truly a "frequency" in any rigorous sense: the alpha particle does not exist as a well-defined object inside the nucleus, and \( f_0 \) absorbs both the pre-formation amplitude and the WKB normalisation at the inner turning point. Modern R-matrix and cluster-model treatments compute a reduced width \( \gamma_\alpha^2 \) in place of \( f_0 \), and the "single-particle" Gamow estimate is understood as the limit in which that width takes its maximal, structureless value. The remarkable success of the crude model reflects the fact that the exponent, being large, dominates all this structure — the physics is in the barrier, not in the poorly known prefactor.

Common misconceptions. The alpha does not "borrow energy" to climb the barrier; energy is conserved throughout and the wavefunction merely decays exponentially in the classically forbidden region. Nor is the decay "caused" by an internal collision at the assault frequency in any deterministic sense — the process is intrinsically probabilistic, and \( f_0 \) only calibrates how often the tunnelling attempt is, so to speak, refreshed.

Worked examples

Example 1 — Gamow exponent and transmission for \( {}^{210}\mathrm{Po}\to{}^{206}\mathrm{Pb}+\alpha \).

\[ b=\frac{zZe^{2}}{4\pi\varepsilon_{0}Q}=\frac{(2)(82)(1.440\,\mathrm{MeV\,fm})}{5.407\,\mathrm{MeV}} \]

Daughter charge \( Z=82 \), \( z=2 \), \( Q=5.407\,\mathrm{MeV} \). Outer turning point from \( V(b)=Q \). A

\[ b=43.68\,\mathrm{fm}, \qquad R=1.20\big(206^{1/3}+4^{1/3}\big)\,\mathrm{fm}=8.99\,\mathrm{fm}, \qquad x=\frac{R}{b}=0.206 \]

Touching radius of daughter and alpha with \( r_0=1.20\,\mathrm{fm} \). The barrier is far from thin: \( x \) is not negligible. A

\[ f(x)=\arccos\sqrt{0.206}-\sqrt{0.206(1-0.206)} = 1.100-0.404 = 0.695 \]

Full barrier-thickness function (not the thick-barrier limit \( \pi/2=1.571 \)), since \( x \) is sizeable. B

\[ G_{0}\equiv\frac{3.96\,Z}{\sqrt{Q}}=\frac{3.96\times 82}{\sqrt{5.407}}=139.6 \quad(\text{full-barrier }R\to0) \]

Leading coefficient \( 3.96=2\pi z\alpha\sqrt{mc^{2}/2}\, \) evaluated numerically; this is \( G \) when \( f=\pi/2 \). A

\[ G = G_{0}\,\frac{f(x)}{\pi/2}=139.6\times\frac{0.695}{1.571}=61.8 \]

Rescale the full-barrier value by the finite-\( R \) reduction factor \( f/(\pi/2)=0.443 \). The nuclear radius roughly halves the exponent. B

\[ T=e^{-G}=e^{-61.8}=10^{-26.8}=1.5\times10^{-27} \]

Transmission per barrier assault. A

\[ v=\sqrt{\frac{2Q}{m_\alpha}}=c\sqrt{\frac{2Q}{m_\alpha c^{2}}}, \qquad f_{0}=\frac{v}{2R}=\frac{1.62\times10^{7}\,\mathrm{m/s}}{2(8.99\times10^{-15}\,\mathrm{m})}=9.0\times10^{20}\,\mathrm{s^{-1}} \]

Assault frequency from the alpha speed set by \( Q \) and the well width \( 2R \). B

\[ \lambda=f_{0}T=1.3\times10^{-6}\,\mathrm{s^{-1}}, \qquad t_{1/2}=\frac{\ln 2}{\lambda}=5.3\times10^{5}\,\mathrm{s}\approx 6\ \text{days} \]

Combine. A

\[ G\approx 62,\qquad T\approx 1.5\times10^{-27},\qquad t_{1/2}\sim 10^{6}\,\mathrm{s} \]

Reading. The measured half-life is \( 138\ \mathrm{days}\approx1.2\times10^{7}\,\mathrm{s} \). The crude single-particle Gamow model lands within about one-and-a-half orders of magnitude — excellent given that the answer ranges over \( 10^{27} \) from the exponential alone. The residual gap is absorbed by the pre-formation factor and the schematic assault frequency, and is highly sensitive to the chosen \( R \).

Units check. \( b \): \( \mathrm{MeV\,fm/MeV=fm} \) ✓. \( G \) dimensionless ✓. \( \lambda \): \( \mathrm{s^{-1}}\times(\text{dimensionless})=\mathrm{s^{-1}} \) ✓, \( t_{1/2}=\ln2/\lambda \) in seconds ✓.

Example 2 — Why a factor \( \sim2 \) in \( Q \) spans \( \sim25 \) decades in half-life.

\[ G\simeq\frac{3.96\,Z}{\sqrt{Q}}, \qquad Z=86\ (\text{daughter of a radon-region emitter}) \]

Use the leading thick-barrier coefficient to isolate the \( Q \)-dependence, holding \( Z \) fixed. A

\[ G(Q=4.0)=\frac{3.96\times86}{\sqrt{4.0}}=\frac{340.6}{2.00}=170.3 \]

Low-energy emitter. A

\[ G(Q=9.0)=\frac{3.96\times86}{\sqrt{9.0}}=\frac{340.6}{3.00}=113.5 \]

High-energy emitter. A

\[ \Delta G = 170.3-113.5 = 56.8 \]

The exponent difference between the two energies. A

\[ \frac{t_{1/2}(4.0)}{t_{1/2}(9.0)}=e^{\Delta G}=e^{56.8}=10^{56.8/2.303}=10^{24.7} \]

Ratio of half-lives at fixed assault frequency and \( Z \); the prefactors cancel. B

\[ \frac{t_{1/2}(Q=4\,\mathrm{MeV})}{t_{1/2}(Q=9\,\mathrm{MeV})}\approx 10^{25} \]

Reading. A factor of \( 2.25 \) in \( Q \) produces a factor of \( 10^{25} \) in half-life — precisely the enormous dynamic range Geiger and Nuttall charted empirically, and the reason alpha half-lives run from microseconds to gigayears. The whole effect lives in the exponent \( G\propto1/\sqrt{Q} \).

Units check. \( G \) dimensionless, so \( e^{\Delta G} \) is a pure ratio; converting to base 10 uses \( \ln10=2.303 \) ✓.

Problems

Compute the outer classical turning point \( b \) for \( {}^{238}\mathrm{U}\to{}^{234}\mathrm{Th}+\alpha \), given \( Q=4.270\,\mathrm{MeV} \) and daughter \( Z=90 \).
Solution
\( b=\dfrac{zZe^{2}}{4\pi\varepsilon_0 Q}=\dfrac{(2)(90)(1.440\,\mathrm{MeV\,fm})}{4.270\,\mathrm{MeV}}=\dfrac{259.2}{4.270}=60.7\,\mathrm{fm} \). The alpha must tunnel from \( R\approx1.20(234^{1/3}+4^{1/3})=9.35\,\mathrm{fm} \) out to about \( 61\,\mathrm{fm} \) — a barrier some \( 50\,\mathrm{fm} \) thick, which is why the half-life is \( 4.5\times10^{9}\,\mathrm{yr} \).
Using the leading (thick-barrier) formula \( G=3.96\,Z/\sqrt{Q} \), estimate \( G \) and the transmission probability \( T \) for the \( {}^{238}\mathrm{U} \) decay of Problem 1.
Solution
\( G=\dfrac{3.96\times90}{\sqrt{4.270}}=\dfrac{356.4}{2.066}=172.5 \) (full-barrier). Applying the finite-\( R \) reduction with \( x=R/b=9.35/60.7=0.154 \), \( f(x)=\arccos\sqrt{0.154}-\sqrt{0.154\times0.846}=1.174-0.361=0.813 \), so \( G=172.5\times0.813/1.571=89.3 \). Then \( T=e^{-89.3}=10^{-38.8}\approx1.6\times10^{-39} \). The much larger \( G \) than \( {}^{210}\mathrm{Po} \) (62) — driven by the lower \( Q \) — accounts for the \( \sim10^{20} \) longer life.
Show that the leading Gamow exponent can be written \( G=\sqrt{E_G/Q} \) and evaluate the Gamow energy \( E_G \) for an alpha emitter with daughter \( Z=82 \).
Solution
From \( G=2\pi zZ\alpha\sqrt{mc^{2}/2Q} \), square: \( G^{2}=(2\pi zZ\alpha)^{2}\,mc^{2}/2Q\equiv E_G/Q \), so \( E_G=(2\pi zZ\alpha)^{2}mc^{2}/2 \). Numerically \( 2\pi zZ\alpha=2\pi(2)(82)/137.04=7.52 \), squared \( =56.5 \), times \( mc^{2}/2=3727.4/2=1863.7\,\mathrm{MeV} \) gives \( E_G=1.05\times10^{5}\,\mathrm{MeV}\approx105\,\mathrm{GeV} \). Check: at \( Q=5.407\,\mathrm{MeV} \), \( G=\sqrt{105360/5.407}=139.6 \), matching the full-barrier value of Example 1.
Two even–even isotopes of the same element decay with daughter charge \( Z=84 \), one at \( Q_1=6.00\,\mathrm{MeV} \) and one at \( Q_2=7.00\,\mathrm{MeV} \). Using only the leading term, find the ratio of their half-lives.
Solution
\( G_1=\dfrac{3.96\times84}{\sqrt{6.00}}=\dfrac{332.6}{2.449}=135.8 \); \( G_2=\dfrac{332.6}{\sqrt{7.00}}=\dfrac{332.6}{2.646}=125.7 \). \( \Delta G=10.1 \), so \( t_1/t_2=e^{10.1}=10^{4.39}\approx2.4\times10^{4} \). A single-MeV increase in \( Q \) shortens the life by more than four orders of magnitude — the local slope of the Geiger–Nuttall line.
For an \( \ell=2 \) alpha transition, estimate the extra suppression from the centrifugal barrier relative to \( \ell=0 \), for \( {}^{210}\mathrm{Po} \) (\( R=8.99\,\mathrm{fm} \), \( Q=5.407\,\mathrm{MeV} \)). Treat the centrifugal term as a small constant addition to the barrier evaluated near \( r\sim R \).
Solution
The centrifugal energy at the surface is \( E_\ell=\dfrac{\hbar^{2}\ell(\ell+1)}{2m R^{2}}=\dfrac{(\hbar c)^{2}\ell(\ell+1)}{2mc^{2}R^{2}}=\dfrac{(197.3)^{2}(6)}{2(3727.4)(8.99)^{2}}=\dfrac{2.335\times10^{5}}{6.02\times10^{5}}=0.388\,\mathrm{MeV} \). As a fractional barrier increase this is small compared with \( V(R)=zZe^{2}/4\pi\varepsilon_0 R=236.2/8.99=26.3\,\mathrm{MeV} \). Estimating the added exponent by scaling the leading term, \( \Delta G\approx \tfrac12 G\,(E_\ell/[V(R)-Q])=\tfrac12(62)(0.388/20.9)\approx0.6 \), giving an extra hindrance factor \( e^{0.6}\approx1.8 \). The centrifugal effect is genuine but modest for low \( \ell \), consistent with the observation that alpha transitions to low-lying rotational states are only mildly hindered.