Gamow Factor

The Gamow factor is the dominant semiclassical suppression for two positively charged particles to reach short separation through their repulsive Coulomb barrier. Its characteristic dependence,

$$\exp\left(-2\pi\eta_{\mathrm C}\right),$$

explains two otherwise startling observations: alpha-decay lifetimes can change by many orders of magnitude when the decay energy changes modestly, and low-energy nuclear fusion is concentrated in an energy window far above the thermal mean but far below the Coulomb-barrier height.

This page owns the Coulomb-specific WKB integral, its finite-radius correction, and those two consequences. Barrier Penetration and Tunneling owns the reusable smooth-barrier formula. Tunneling Applications: First Encounters gives the broader application map without repeating the derivation.

Conventions and Scope

Consider two particles with positive charge numbers $Z_1$ and $Z_2$. In the center-of-mass frame, their relative coordinate has reduced mass

$$\mu = \frac{m_1m_2}{m_1+m_2}$$

and positive relative energy $E$. Define

$$\begin{aligned} C &= \frac{Z_1Z_2e^2}{4\pi\varepsilon_0}, \\ V_{\mathrm C}(r) &= \frac{C}{r}, \\ v &= \sqrt{\frac{2E}{\mu}}. \end{aligned}$$

Thus $C\gt0$ for the repulsive problem. In units where the Coulomb potential is written differently, every result below remains valid after identifying the coefficient $C$ in $V_{\mathrm C}=C/r$.

The dimensionless Coulomb, or Sommerfeld, parameter is

$$\eta_{\mathrm C} = \frac{C}{\hbar v} = Z_1Z_2\alpha\frac{c}{v},$$

where

$$\alpha = \frac{e^2}{4\pi\varepsilon_0\hbar c}.$$

The subscript on $\eta_{\mathrm C}$ matters. Elsewhere in WKB, an $\eta$ is often used for an accumulated forbidden-region phase. Here $\eta_{\mathrm C}$ always means the Coulomb parameter.

We use the following terminology:

• $G$ is the dimensionless one-way barrier action.
• $e^{-G}$ is the leading amplitude suppression.
• $P_{\mathrm{WKB}}\approx e^{-2G}$ is the leading probability suppression.
• $P_{\mathrm G}=e^{-2\pi\eta_{\mathrm C}}$ is the point-Coulomb Gamow factor.

Some authors instead call $2\pi\eta_{\mathrm C}$ the Gamow factor or the Gamow exponent. Always inspect the author’s definition before comparing formulas.

Coulomb Barrier Geometry

The outer classical turning point satisfies

$$V_{\mathrm C}(b)=E,$$

so

$$b = \frac{C}{E}.$$

A short-range nuclear interaction becomes important at a radius $R$. In the simplest exterior-barrier model,

$$R\lt b,$$

and the interval $R\lt r\lt b$ is classically forbidden. The region $r\gt b$ is classically allowed.

Minimal radial model for charged-particle escape or approach. The pure-Coulomb calculation begins at the nuclear matching radius $R$ and ends at the outer turning point $b=C/E$. A realistic calculation replaces the sharp interior by a nuclear potential and determines the inner matching region dynamically.

For partial wave $\ell$, the exact reduced radial wavefunction obeys

$$\left[ - \frac{\hbar^2}{2\mu} \frac{d^2}{dr^2} + V_{\mathrm{eff}}(r) \right] u_\ell(r) = E u_\ell(r),$$

with

$$V_{\mathrm{eff}}(r) = V_{\mathrm N}(r) + \frac{C}{r} + \frac{\hbar^2\ell(\ell+1)} {2\mu r^2}.$$

The pure Gamow derivation keeps the exterior Coulomb term and takes $\ell=0$. The radius $R$ is then a matching cutoff, not generally a smooth turning point. This distinction becomes important when an accurate prefactor or decay rate is required.

Evaluating the WKB Exponent

In the exterior forbidden interval, define the positive under-barrier momentum

$$\kappa(r) = \sqrt{ 2\mu \left( \frac{C}{r}-E \right) }.$$

The dimensionless action is

$$G(R,E) = \frac{1}{\hbar} \int_R^b \sqrt{ 2\mu \left( \frac{C}{r}-E \right) } \,dr.$$

Introduce

$$x = \frac{R}{b} = \frac{RE}{C}, \qquad 0\lt x\lt1,$$

and use the substitution

$$r=b\sin^2\theta.$$

Since $C/b=E$,

$$\begin{aligned} \frac{C}{r}-E &= E\cot^2\theta, \\ dr &= 2b\sin\theta\cos\theta\,d\theta. \end{aligned}$$

The lower limit is

$$\theta_R = \arcsin\sqrt{x},$$

and $r=b$ corresponds to $\theta=\pi/2$. Therefore

$$I_R = \int_{\theta_R}^{\pi/2} \cos^2\theta\,d\theta.$$

The elementary integral can be written as

$$I_R = \frac12\mathcal F(x),$$

where

$$\begin{aligned} \mathcal F(x) ={}& \arccos\sqrt{x} \\ &- \sqrt{x(1-x)}. \end{aligned}$$

The action is therefore

$$G(R,E) = \frac{b\sqrt{2\mu E}}{\hbar} \mathcal F(x).$$

The prefactor simplifies because

$$\frac{b\sqrt{2\mu E}}{\hbar} = \frac{C\sqrt{2\mu}}{\hbar\sqrt E} = 2\eta_{\mathrm C}.$$

Hence the finite-radius Coulomb action is

$$G(R,E) = 2\eta_{\mathrm C} \mathcal F\left(\frac{R}{b}\right).$$

The corresponding leading penetrability is

$$P_{\mathrm{WKB}}(R,E) \approx \exp\left[-2G(R,E)\right].$$

This expression is the main result for a Coulomb tail cut off at radius $R$. It retains both the energy dependence and the large correction caused by the finite nuclear size.

Point-Coulomb Limit

If $R/b\to0$, then

$$\arccos\sqrt{\frac{R}{b}} \longrightarrow \frac{\pi}{2}$$

and the square-root term vanishes. Thus

$$G(0,E) = \pi\eta_{\mathrm C},$$

so

$$P_{\mathrm G}(E) = \exp\left(-2\pi\eta_{\mathrm C}\right).$$

This is the point-Coulomb Gamow factor under the convention used here.

It is often useful to define the Gamow energy

$$E_{\mathrm G} = 2\mu c^2 \left( \pi\alpha Z_1Z_2 \right)^2.$$

Then

$$2\pi\eta_{\mathrm C} = \sqrt{\frac{E_{\mathrm G}}{E}},$$

and

$$P_{\mathrm G}(E) = \exp\left[ - \sqrt{\frac{E_{\mathrm G}}{E}} \right].$$

The characteristic nonanalytic dependence $e^{-\text{constant}/\sqrt E}$ cannot be obtained from a finite-order expansion in the Coulomb interaction. It is a genuinely nonperturbative low-energy effect.

Energy sensitivity

Differentiating the point-Coulomb exponent gives

$$\frac{d\ln P_{\mathrm G}}{dE} = \frac{1}{2} \sqrt{\frac{E_{\mathrm G}}{E^3}} = \frac{\pi\eta_{\mathrm C}}{E}.$$

For a small energy change,

$$\delta\ln P_{\mathrm G} \approx \pi\eta_{\mathrm C} \frac{\delta E}{E}.$$

When $\eta_{\mathrm C}\gg1$, a percent-level change in energy can therefore produce an order-one or larger change in the logarithm of the penetrability.

Finite-Radius Correction

The point-Coulomb limit isolates the dominant low-energy dependence, but it is not usually an accurate absolute penetrability for a nucleus. For $x\ll1$, the Coulomb shape function has the expansion

$$\begin{aligned} \mathcal F(x) ={}& \frac{\pi}{2} - 2\sqrt{x} + \frac{x^{3/2}}{3} \\ & + \frac{x^{5/2}}{20} + O\left(x^{7/2}\right). \end{aligned}$$

It follows that

$$\begin{aligned} G(R,E) ={}& \pi\eta_{\mathrm C} - \frac{2\sqrt{2\mu C R}}{\hbar} \\ &+ \frac{2}{3} \eta_{\mathrm C} \left( \frac{RE}{C} \right)^{3/2} + \cdots. \end{aligned}$$

For fixed $C$, $\mu$, and $R$, the leading finite-radius term is independent of $E$. Therefore

$$2G(R,E) = 2\pi\eta_{\mathrm C} - \frac{4\sqrt{2\mu C R}}{\hbar} + O(E).$$

This explains a subtle but important fact:

• $e^{-2\pi\eta_{\mathrm C}}$ captures the leading low-energy slope.
• The finite radius can change the absolute penetrability enormously.
• Across a related family of nuclei, radius, charge, and structure dependence shifts the intercept and can also alter the slope beyond the leading approximation.

Calling every finite-radius correction a “prefactor” can be misleading. The leading correction is energy-independent in this expansion, but it still appears in an exponential and need not be numerically close to unity.

Illustrative heavy-nucleus scale

Take an illustrative alpha-like channel with

$$\begin{aligned} Z_1Z_2&=164, \\ E&=8.95\ \mathrm{MeV}, \\ \mu c^2&=3.66\times10^3\ \mathrm{MeV}, \\ R&=9.0\ \mathrm{fm}. \end{aligned}$$

Using

$$C \approx (1.43996\ \mathrm{MeV\,fm})Z_1Z_2$$

gives

$$\begin{aligned} b&\approx26.4\ \mathrm{fm}, \\ \eta_{\mathrm C}&\approx17.1, \\ 2\pi\eta_{\mathrm C}&\approx107.5. \end{aligned}$$

The point-Coulomb factor is then

$$P_{\mathrm G} \approx 2.1\times10^{-47}.$$

But $R/b\approx0.341$, and the finite-radius integral gives

$$\begin{aligned} 2G(R,E)&\approx32.4, \\ P_{\mathrm{WKB}}(R,E) &\approx 8.8\times10^{-15}. \end{aligned}$$

The example is not a precision decay calculation. It shows why the point-Coulomb factor should be read as the canonical energy-dependent suppression, not as permission to integrate an alpha-decay barrier all the way to $r=0$.

Exact Coulomb Cross-Check

The regular repulsive $s$-wave Coulomb solution contains the normalization factor

$$\mathcal C_{\eta_{\mathrm C}}^2 = \frac{ 2\pi\eta_{\mathrm C} }{ e^{2\pi\eta_{\mathrm C}}-1 }.$$

For $\eta_{\mathrm C}\gg1$,

$$\mathcal C_{\eta_{\mathrm C}}^2 \sim 2\pi\eta_{\mathrm C} e^{-2\pi\eta_{\mathrm C}}.$$

The exact result confirms the same dominant exponential and supplies an algebraic factor. It does not replace the finite-radius barrier calculation for escape from a nuclear interior, nor does it include the short-range reaction dynamics. The exact long-range phases and scattering conventions belong to Coulomb Scattering.

Alpha Decay and Nuclear Tunneling

In alpha decay, a parent nucleus produces an alpha particle and a daughter nucleus. The relative coordinate sees a short-range attractive nuclear region and a repulsive exterior Coulomb tail. For an alpha particle,

$$Z_1=2, \qquad Z_2=Z_{\mathrm d},$$

where $Z_{\mathrm d}$ is the daughter charge.

For a two-body decay to the daughter ground state, the relative kinetic energy is the decay energy $Q_\alpha$:

$$E_{\mathrm{rel}} \approx Q_\alpha.$$

The alpha particle’s laboratory kinetic energy is slightly smaller because the daughter recoils. The WKB barrier calculation uses the relative energy and reduced mass, not the alpha mass alone.

Rate model

A useful factorization of the decay rate is

$$\lambda \approx P_\alpha \nu P_{\mathrm{WKB}}.$$

Here:

• $P_\alpha$ is the probability that an alpha-like cluster is preformed in the parent;
• $\nu$ is an assault or attempt frequency associated with motion in the interior;
• $P_{\mathrm{WKB}}$ is the probability of penetrating the effective barrier on one attempt.

The decay width and half-life are

$$\Gamma=\hbar\lambda, \qquad T_{1/2} = \frac{\ln2}{\lambda}.$$

Only the penetrability is fixed by the elementary Gamow calculation. Predicting an absolute half-life also requires the nuclear interior, preformation, matching normalization, and angular-momentum channel.

Geiger–Nuttall behavior

In the point-Coulomb approximation,

$$2\pi\eta_{\mathrm C} = 2\pi \left( 2Z_{\mathrm d}\alpha \right) \sqrt{ \frac{\mu c^2}{2Q_\alpha} }.$$

If the other factors vary slowly within a related sequence of emitters, then

$$\log_{10}T_{1/2} \approx a \frac{Z_{\mathrm d}}{\sqrt{Q_\alpha}} + b.$$

This is the semiclassical origin of the Geiger–Nuttall trend. The symbols $a$ and $b$ are empirical family-dependent coefficients; they are not universal constants.

The finite-radius expansion explains why the trend is approximately linear rather than exact. Its leading correction changes the intercept, while variations in radius, reduced mass, preformation, angular momentum, shell structure, and daughter excitation produce additional deviations.

Angular momentum

If the emitted cluster carries orbital angular momentum, the barrier includes

$$\frac{\hbar^2\ell(\ell+1)} {2\mu r^2}.$$

This raises the action and suppresses the decay. In a careful radial WKB treatment, the Langer prescription replaces

$$\ell(\ell+1) \longrightarrow \left(\ell+\frac12\right)^2$$

inside the semiclassical momentum. The replacement repairs the behavior of radial WKB near the singular endpoint; it is not a change to the exact Hamiltonian.

Astrophysical Fusion Preview

For a nonresonant reaction between charged nuclei, the cross section is commonly written

$$\sigma(E) = \frac{S(E)}{E} \exp\left(-2\pi\eta_{\mathrm C}\right).$$

The astrophysical $S$-factor collects the short-range nuclear matrix element and the more slowly varying kinematic and Coulomb contributions. This factorization is useful only over an energy range where $S(E)$ is genuinely smoother than the Gamow exponential.

In a nondegenerate thermal plasma, the Maxwell–Boltzmann factor contributes $e^{-E/(k_{\mathrm B}T)}$. Ignoring slowly varying factors, the reaction-rate integrand contains

$$\exp\left[-\Phi(E)\right],$$

where

$$\Phi(E) = \frac{E}{k_{\mathrm B}T} + \sqrt{\frac{E_{\mathrm G}}{E}}.$$

The first term penalizes particles in the high-energy thermal tail. The second penalizes low-energy Coulomb penetration. Their competition creates the Gamow peak.

Setting $\Phi'(E_0)=0$ gives

$$E_0 = \left[ \frac{ E_{\mathrm G} \left(k_{\mathrm B}T\right)^2 }{4} \right]^{1/3}.$$

At the stationary point,

$$\Phi(E_0) = \frac{3E_0}{k_{\mathrm B}T}.$$

A quadratic expansion gives the conventional full $1/e$ width

$$\Delta = 4 \sqrt{ \frac{E_0k_{\mathrm B}T}{3} }.$$

The interval centered near $E_0$ with this scale is called the Gamow window. It is an asymptotic diagnostic, not a hard-edged interval.

Resonances, subthreshold states, laboratory electron screening, plasma screening, quantum-degenerate distributions, and rapidly varying $S(E)$ can all invalidate the simplest peak estimate. A reaction-rate calculation must inspect the actual integrand.

Historical Significance

In 1928, George Gamow applied wave mechanics to nuclear disintegration through barrier penetration. Ronald Gurney and Edward Condon independently developed the same physical explanation of alpha decay that year. The success was conceptually important: a quasi-bound nuclear state could have energy below the exterior barrier and still decay, with its lifetime controlled by a calculable exponential.

The historical argument did more than label alpha decay as “tunneling.” It connected the strong energy dependence of measured lifetimes to the shape of the Coulomb tail and helped establish barrier penetration as a quantitative prediction of wave mechanics.

Limitations and Domain of Validity

The Gamow factor is trustworthy as a dominant exponential only when its assumptions match the problem.

Semiclassical control

Away from the outer turning point, the local WKB condition is

$$\epsilon_{\mathrm{WKB}}(r) = \hbar \frac{ \left|\kappa'(r)\right| }{ \kappa^2(r) } \ll1.$$

It fails as $r\to b$ because $\kappa\to0$. A turning-point connection formula repairs the local divergence. The integrated exponent can remain accurate when the action is large, but the local WKB wavefunction is never valid exactly at $b$.

Model boundaries

An accurate nuclear calculation may need:

• a realistic nuclear plus Coulomb potential rather than a sharp radius;
• alpha or cluster preformation amplitudes;
• centrifugal and spin-coupling effects;
• daughter excitation and channel thresholds;
• deformation and orientation dependence;
• coupled channels or resonant dynamics;
• electron or plasma screening;
• a controlled matching prefactor.

For near-barrier energies, $R/b$ is not small and the point-Coulomb form should not be used. For $R\ge b$, there is no exterior Coulomb-forbidden interval in this model.

Sign of the interaction

The suppression derived here assumes

$$Z_1Z_2\gt0.$$

For an attractive Coulomb interaction, $\eta_{\mathrm C}\lt0$ under a signed convention and there is no repulsive Coulomb barrier of the form used in this derivation. Substituting a negative $\eta_{\mathrm C}$ into $e^{-2\pi\eta_{\mathrm C}}$ and calling the result a tunneling probability is physically wrong.

Common Mistakes

• Calling $e^{-\pi\eta_{\mathrm C}}$ a probability instead of an amplitude-scale suppression.
• Integrating the alpha-decay Coulomb tail to $r=0$ when the nuclear interaction takes over near $R$.
• Using the alpha-particle mass instead of the two-body reduced mass.
• Using the measured alpha laboratory energy where the relative decay energy is required, without accounting for recoil.
• Treating $e^{-2\pi\eta_{\mathrm C}}$ as a complete cross section or decay rate.
• Assuming the astrophysical $S$-factor is smooth across a resonance.
• Forgetting the centrifugal barrier for $\ell\ne0$.
• Applying the repulsive result to an attractive Coulomb problem.
• Trusting the local WKB form exactly at the outer turning point.

Reporting Checklist

A reproducible Gamow-factor estimate should state:

1. the charge product $Z_1Z_2$ and sign convention;
2. the reduced mass $\mu$ and relative energy $E$;
3. whether $R=0$ or a finite matching radius is used;
4. whether $G$ denotes the amplitude exponent or $2G$ the probability exponent;
5. the partial wave and any Langer correction;
6. which prefactor, preformation factor, or $S$-factor model is included;
7. the approximation regime and dominant omitted physics.

Exercises

1. Evaluate the Coulomb action

Starting from

$$G(R,E) = \frac1\hbar \int_R^{C/E} \sqrt{ 2\mu \left( \frac Cr-E \right) } \,dr,$$

use $r=(C/E)\sin^2\theta$ to derive the finite-radius result.

Solution

Let

$$\begin{aligned} b&=\frac CE, \\ x&=\frac Rb, \\ \theta_R&=\arcsin\sqrt x. \end{aligned}$$

Then

$$\frac Cr-E = E\cot^2\theta$$

and

$$dr = 2b\sin\theta\cos\theta\,d\theta.$$

Therefore

$$\begin{aligned} G &= \frac{ 2b\sqrt{2\mu E} }{\hbar} \int_{\theta_R}^{\pi/2} \cos^2\theta\,d\theta \\ &= \frac{ b\sqrt{2\mu E} }{\hbar} \left[ \frac{\pi}{2} - \theta_R - \frac12\sin(2\theta_R) \right]. \end{aligned}$$

Using

$$\frac{\pi}{2}-\theta_R = \arccos\sqrt x$$

and

$$\frac12\sin(2\theta_R) = \sqrt{x(1-x)},$$

while

$$\frac{ b\sqrt{2\mu E} }{\hbar} = 2\eta_{\mathrm C},$$

gives

$$G = 2\eta_{\mathrm C} \left[ \arccos\sqrt x - \sqrt{x(1-x)} \right].$$

2. Expand the finite-radius result

Show that the first correction to $\pi\eta_{\mathrm C}$ is independent of energy for fixed $C$, $\mu$, and $R$.

Solution

For $x\ll1$,

$$\begin{aligned} \mathcal F(x) ={}& \frac\pi2 - 2\sqrt x + \frac{x^{3/2}}3 \\ & + \cdots. \end{aligned}$$

Thus

$$G = \pi\eta_{\mathrm C} - 4\eta_{\mathrm C}\sqrt x + \frac23 \eta_{\mathrm C}x^{3/2} + \cdots.$$

Now

$$\eta_{\mathrm C}\sqrt x = \frac C{\hbar} \sqrt{\frac{\mu}{2E}} \sqrt{\frac{RE}{C}} = \frac1\hbar \sqrt{\frac{\mu C R}{2}}.$$

Hence

$$-4\eta_{\mathrm C}\sqrt x = - \frac{ 2\sqrt{2\mu C R} }{\hbar},$$

which has no $E$ dependence. The next term scales as

$$\eta_{\mathrm C}x^{3/2} \propto E.$$

3. Compare point and finite-radius estimates

Use

$$\begin{aligned} Z_1Z_2&=164, \\ E&=8.95\ \mathrm{MeV}, \\ \mu c^2&=3656\ \mathrm{MeV}, \\ R&=9.0\ \mathrm{fm}. \end{aligned}$$

and

$$\begin{aligned} \alpha^{-1} &= 137.036, \\ \frac{e^2}{4\pi\varepsilon_0} &= 1.43996\ \mathrm{MeV\,fm}. \end{aligned}$$

to estimate $b$, $2\pi\eta_{\mathrm C}$, and $2G(R,E)$.

Solution

First,

$$\begin{aligned} C &= 164 \left( 1.43996\ \mathrm{MeV\,fm} \right) \\ & \approx 236.2\ \mathrm{MeV\,fm}. \end{aligned}$$

Therefore

$$b = \frac CE \approx 26.4\ \mathrm{fm}, \qquad x = \frac Rb \approx 0.341.$$

The relative speed is

$$\frac vc = \sqrt{ \frac{2E}{\mu c^2} } \approx 0.0700,$$

so

$$\eta_{\mathrm C} = \frac{164}{137.036} \frac c v \approx 17.1$$

and

$$2\pi\eta_{\mathrm C} \approx 107.5.$$

For the finite radius,

$$\arccos\sqrt x - \sqrt{x(1-x)} \approx 0.473.$$

Hence

$$2G = 4\eta_{\mathrm C}(0.473) \approx 32.4.$$

The corresponding suppressions are approximately

$$e^{-107.5} \approx 2.1\times10^{-47}$$

and

$$e^{-32.4} \approx 8.8\times10^{-15}.$$

The finite nuclear radius is indispensable for an absolute alpha-decay estimate.

4. Recover the Geiger–Nuttall slope

Assume $P_\alpha\nu$ and the leading finite-radius correction vary slowly within an isotopic family. Show why $\log_{10}T_{1/2}$ is approximately linear in $Z_{\mathrm d}/\sqrt{Q_\alpha}$.

Solution

Since

$$T_{1/2} \approx \frac{\ln2}{P_\alpha\nu} e^{2G},$$

the dominant energy-dependent term in its logarithm is

$$2G \approx 2\pi\eta_{\mathrm C}.$$

For alpha decay,

$$2\pi\eta_{\mathrm C} = 4\pi Z_{\mathrm d}\alpha \sqrt{ \frac{\mu c^2}{2Q_\alpha} }.$$

Define the slowly varying slope coefficient

$$A = \frac{ 4\pi\alpha }{\ln10} \sqrt{ \frac{\mu c^2}{2} }.$$

Converting the natural logarithm to base ten then gives

$$\begin{aligned} \log_{10}T_{1/2} \approx{}& A \frac{Z_{\mathrm d}}{\sqrt{Q_\alpha}} \\ & + \text{slowly varying terms}. \end{aligned}$$

The slowly varying terms include the attempt frequency, preformation probability, finite-radius contribution, and the chosen time unit. They form the family-dependent intercept and corrections.

5. Locate the Gamow peak

For

$$\Phi(E) = \frac{E}{k_{\mathrm B}T} + \sqrt{\frac{E_{\mathrm G}}E},$$

find the stationary energy $E_0$ and show that $\Phi(E_0)=3E_0/(k_{\mathrm B}T)$.

Solution

Differentiate:

$$\Phi'(E) = \frac1{k_{\mathrm B}T} - \frac12 \sqrt{E_{\mathrm G}} E^{-3/2}.$$

The stationary condition is

$$E_0^{3/2} = \frac12 \sqrt{E_{\mathrm G}} k_{\mathrm B}T.$$

Therefore

$$E_0 = \left[ \frac{ E_{\mathrm G} \left(k_{\mathrm B}T\right)^2 }{4} \right]^{1/3}.$$

The stationary condition also implies

$$\sqrt{\frac{E_{\mathrm G}}{E_0}} = \frac{2E_0}{k_{\mathrm B}T}.$$

Thus

$$\Phi(E_0) = \frac{E_0}{k_{\mathrm B}T} + \frac{2E_0}{k_{\mathrm B}T} = \frac{3E_0}{k_{\mathrm B}T}.$$

6. Diagnose two invalid substitutions

Explain why neither of the following is valid:

1. use $e^{-2\pi\eta_{\mathrm C}}$ with signed $\eta_{\mathrm C}\lt0$ as an attractive-Coulomb tunneling probability;
2. use the $s$-wave Coulomb action unchanged for a decay that requires $\ell\gt0$.

Solution

For an attractive interaction there is no repulsive Coulomb-forbidden interval with outer turning point $b=C/E\gt0$. A negative signed $\eta_{\mathrm C}$ makes $e^{-2\pi\eta_{\mathrm C}}\gt1$, which already shows that the expression is not a probability in that problem. Attractive Coulomb waves exhibit enhancement rather than penetration through this barrier.

For $\ell\gt0$, the radial effective potential contains a positive centrifugal term,

$$\frac{\hbar^2\ell(\ell+1)} {2\mu r^2}.$$

It changes both the turning points and the action. A radial WKB calculation should also consider the Langer replacement $(\ell+1/2)^2$. Omitting the angular-momentum barrier systematically overestimates the penetrability.

References

1. G. Gamow, “Zur Quantentheorie des Atomkernes”, Zeitschrift für Physik 51, 204–212 (1928). Original barrier-penetration treatment of nuclear disintegration.
2. R. W. Gurney and E. U. Condon, “Wave Mechanics and Radioactive Disintegration”, Nature 122, 439 (1928). Independent wave-mechanical explanation of alpha decay.
3. H. Geiger and J. M. Nuttall, “The ranges of the alpha particles from various radioactive substances and a relation between range and period of transformation,” Philosophical Magazine 22, 613–621 (1911). Empirical lifetime–energy relation.
4. L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed. (Butterworth-Heinemann, 1981). Quasi-stationary states, barrier penetration, and alpha decay.
5. K. S. Krane, Introductory Nuclear Physics (Wiley, 1987). Alpha decay, preformation, angular momentum, and empirical systematics.
6. C. Iliadis, Nuclear Physics of Stars, 2nd ed. (Wiley-VCH, 2015). Charged-particle reaction rates, astrophysical $S$-factors, and Gamow windows.
7. E. G. Adelberger et al., “Solar fusion cross sections II”, Reviews of Modern Physics 83, 195–245 (2011). Evaluated low-energy charged-particle reactions and limitations of simple extrapolations.
8. M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics”, Reports on Progress in Physics 35, 315–397 (1972). WKB barriers, turning points, and semiclassical accuracy.