---
title: Alpha Decay and the Gamow Theory of Tunneling
module: Alpha Decay
moduleNumber: 5
lessonNumber: 1
order: 501
summary: >
  The alpha Q-value turns positive above mass number 150 because the emitted
  helium-4 is exceptionally tightly bound. Emission proceeds by quantum tunneling
  through the Coulomb barrier: a WKB integral from the nuclear surface to the outer
  turning point gives the Gamow factor, and multiplying its penetrability by the
  assault frequency yields half-lives spanning more than twenty orders of magnitude.
  The leading term reproduces the Geiger-Nuttall relation, log t½ proportional to
  the daughter charge over the square root of Q.
topics: [Alpha Decay]
sources:
  - book: Krane
    ref: "Ch. 8 — Alpha Decay; §8.1 Why Alpha Decay Occurs, §8.2 Basic Alpha Decay Processes, §8.3 Alpha Decay Systematics, §8.4 Theory of Alpha Emission"
  - book: Wong
    ref: "Ch. 4 — Nuclear Collective Motion; §4-2 Alpha-Particle Decay"
draft: false
---

Alpha decay is the emission of a helium-4 nucleus from a heavier parent,

$$
{}^{A}_{Z}\mathrm{X}_{N} \longrightarrow {}^{A-4}_{Z-2}\mathrm{X}'_{N-2} + {}^{4}_{2}\mathrm{He}_{2}.
$$

Two facts about it demand explanation. The energy released is only a few MeV, yet
the process is universal among the heaviest nuclei; and the half-life is fantastically
sensitive to that energy, changing by twenty-four orders of magnitude while the decay
energy changes by a factor of two. Both follow from a single mechanism: the alpha
particle is bound inside a potential well but must tunnel through a Coulomb barrier
that classically it has nowhere near enough energy to cross.[^krane-why]

## The alpha Q-value and why heavy nuclei decay

The energy release is the difference in rest energies of the initial and final systems.
Written in nuclear masses,

$$
Q_\alpha = \bigl[m(A,Z) - m(A-4,Z-2) - m_\alpha\bigr]c^2,
$$

and the electron masses and binding cancel to a good approximation when atomic masses
are substituted. Expressing each mass through its binding energy $m c^2 = Z m_p c^2 +
N m_n c^2 - B$ collapses the nucleon rest energies and leaves a difference of binding
energies,

$$
Q_\alpha = B(A-4,Z-2) + B_\alpha - B(A,Z), \qquad B_\alpha = 28.30\ \mathrm{MeV}.
$$

Decay is energetically allowed when $Q_\alpha > 0$, that is when the binding lost by
removing four nucleons from the parent is less than the $28.3\ \mathrm{MeV}$ recovered
by assembling them into an alpha particle. The alpha is exceptionally tightly bound
because it is doubly magic, $N = Z = 2$, with a binding energy per nucleon of
$7.07\ \mathrm{MeV}$ far above its neighbors. No other light fragment is competitive:
the corresponding $Q$ for emitting a proton, a deuteron, or a triton is negative across
the same region of the chart, so the alpha is the fragment that heavy nuclei actually
release.

> **Definition (Alpha disintegration energy).** The alpha disintegration energy
> $Q_\alpha$ is the total kinetic energy shared by the daughter and the alpha particle
> in the rest frame of the parent. Momentum conservation, $p_D = p_\alpha$, splits it as
> $$
> T_\alpha = \frac{Q_\alpha}{1 + m_\alpha/m_D} \approx Q_\alpha\,\frac{A-4}{A},
> \qquad T_D = Q_\alpha\,\frac{4}{A},
> $$
> so the alpha carries away the fraction $(A-4)/A$, about $98\%$ for a heavy parent, and
> the daughter recoils with the small remainder.

Evaluating $Q_\alpha$ from the semi-empirical mass formula shows where the process
switches on. The Coulomb term grows as $Z^2/A^{1/3}$ while the volume and surface terms
scale more slowly, so the binding per nucleon falls beyond the iron peak, and
$Q_\alpha$ crosses zero near $A \approx 150$. Above that mass almost every nuclide is
unstable to alpha emission; below it the decay is energetically forbidden. The rare-earth
alpha emitters near $A = 150$ (for example ${}^{147}\mathrm{Sm}$, with $Q_\alpha =
2.31\ \mathrm{MeV}$) sit right at the threshold and have half-lives comparable to the age
of the universe.[^krane-syst]

$$
% caption: The alpha disintegration energy computed from the mass formula rises through
% zero near mass number 150 and climbs steadily through the actinides, with the observed
% emitters clustered where Q exceeds a few MeV.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.0,0) node[right, black!70] {$A$};
  \draw[->, black] (0,-1.3) -- (0,3.7) node[above, black!70] {$Q$ (MeV)};
  % zero line
  \draw[black, dashed] (0,0.9) -- (7.7,0.9);
  \node[anchor=east, black, font=\scriptsize] at (-0.05,0.9) {$0$};
  % x ticks
  \foreach \x/\lab in {1.2/120,2.9/160,4.6/200,6.3/240} \node[anchor=north, black, font=\scriptsize] at (\x,0.82) {\lab};
  % Q curve: negative below 150, crossing near A=150, rising through actinides
  \draw[acc, very thick]
    (0.4,-0.7)
    .. controls (1.4,-0.2) and (2.0,0.55) .. (2.55,0.9)
    .. controls (3.4,1.45) and (4.4,2.0) .. (5.4,2.55)
    .. controls (6.2,2.95) and (6.9,3.25) .. (7.4,3.35);
  % crossing marker
  \fill[black] (2.55,0.9) circle (1.6pt);
  \draw[black, dashed] (2.55,0) -- (2.55,0.9);
  \node[anchor=north, black!70, font=\scriptsize] at (2.55,-0.02) {$A = 150$};
  \node[black, anchor=east, font=\scriptsize] at (2.0,0.15) {forbidden};
  \node[black, anchor=west, font=\scriptsize] at (5.2,2.75) {actinides};
\end{tikzpicture}
$$

## The barrier and the tunneling picture

Model the alpha as a preformed particle moving in the field of the daughter. Inside the
nuclear radius $R$ the strong force holds it in a flat attractive well of depth $V_0$;
outside $R$ the only interaction is the Coulomb repulsion between the alpha charge $2e$
and the daughter charge $(Z-2)e$,

$$
V(r) = \begin{cases}
-V_0, & r < R,\\[2pt]
\dfrac{2(Z-2)e^2}{4\pi\epsilon_0\,r}, & r > R.
\end{cases}
$$

The barrier reaches its maximum at contact,

$$
B = \frac{2(Z-2)e^2}{4\pi\epsilon_0\,R},
$$

which for a uranium daughter ($Z-2 = 90$, $R \approx 9.3\ \mathrm{fm}$) is about
$28\ \mathrm{MeV}$, roughly six times the alpha's kinetic energy of $4$–$5\ \mathrm{MeV}$.
Classically the particle is trapped: it lacks the energy to reach $r = R$ from outside,
or to cross the barrier from inside. The alpha nonetheless escapes because its wavefunction
does not vanish in the classically forbidden region. It leaks through, emerging at the
outer turning point $b$ where the Coulomb potential has fallen back to the alpha energy,

$$
V(b) = Q_\alpha \quad\Longrightarrow\quad b = \frac{2(Z-2)e^2}{4\pi\epsilon_0\,Q_\alpha}.
$$

$$
% caption: Inside the radius R the alpha sits in an attractive well; outside, the Coulomb
% repulsion forms a barrier peaking at B. The alpha at energy Q must tunnel through the
% shaded region between the inner edge R and the outer turning point b.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % axes
  \draw[->, black] (0,-1.7) -- (0,3.7) node[above, black!70] {$V(r)$};
  \draw[->, black] (0,0) -- (8.2,0) node[right, black!70] {$r$};
  % well bottom
  \draw[very thick] (0.0,-1.5) -- (1.5,-1.5);
  \draw[very thick] (1.5,-1.5) -- (1.5,3.2);
  % coulomb barrier decaying like 1/r
  \draw[very thick]
    (1.5,3.2)
    .. controls (2.1,2.2) and (2.7,1.55) .. (3.4,1.2)
    .. controls (4.3,0.75) and (5.6,0.5) .. (7.9,0.28);
  % Q level line
  \draw[black, thick] (0,1.2) -- (5.0,1.2);
  \node[anchor=east, black!70] at (-0.05,1.2) {$Q$};
  % shaded tunneling band under curve above Q, between R and b
  \fill[acc!12] (1.5,1.2) -- (1.5,3.2)
    .. controls (2.1,2.2) and (2.7,1.55) .. (3.4,1.2) -- cycle;
  % turning points
  \draw[black, dashed] (1.5,0) -- (1.5,-1.5);
  \node[anchor=north, black!70] at (1.5,-0.05) {$R$};
  \draw[black, dashed] (3.4,0) -- (3.4,1.2);
  \node[anchor=north, black!70] at (3.4,-0.05) {$b$};
  % barrier height label
  \node[anchor=west, black!70] at (1.62,3.2) {$B$};
  \node[anchor=east, black!70] at (0.75,-1.5) {$V_0$};
  \node[acc, anchor=south west, font=\scriptsize] at (2.05,1.9) {tunneling};
  \node[black, anchor=west, font=\scriptsize] at (4.5,0.7) {Coulomb tail};
\end{tikzpicture}
$$

## The Gamow factor

The tunneling probability follows from the WKB approximation, in which the wavefunction
decays across the forbidden region as $\exp\!\bigl(-\!\int|k|\,\d r\bigr)$ with the local
wavenumber $|k(r)| = \sqrt{2m\,[V(r) - Q_\alpha]}/\hbar$. The transmission coefficient is
$P = e^{-2G}$, where the **Gamow factor** is the barrier integral

$$
G = \frac{1}{\hbar}\int_R^b \sqrt{2m\,[V(r) - Q_\alpha]}\ \d r
= \frac{\sqrt{2mQ_\alpha}}{\hbar}\int_R^b \sqrt{\frac{b}{r} - 1}\ \d r,
$$

using $V(r) = Q_\alpha\,b/r$ on the Coulomb tail. Here $m$ is the alpha mass, or more
precisely the reduced mass of the alpha-daughter pair, which for a heavy daughter is
within a percent of $m_\alpha$. The integral is elementary. With $x = R/b$,

$$
\int_R^b \sqrt{\frac{b}{r} - 1}\ \d r
= b\Bigl[\arccos\sqrt{x} - \sqrt{x(1-x)}\Bigr].
$$

$$
% caption: The Gamow factor is the area under the square root of V(r) minus Q between the
% turning points R and b; a thicker or higher barrier enlarges this area and suppresses
% the penetrability exponentially.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.6,0) node[right, black!70] {$r$};
  \draw[->, black] (0,0) -- (0,3.6) node[above, black!70] {barrier integrand};
  % integrand: zero at b (outer), rises toward R (inner), peaks near R
  \fill[acc!12]
    (1.4,0)
    -- (1.4,2.9)
    .. controls (2.3,2.3) and (3.2,1.5) .. (4.2,0.85)
    .. controls (4.9,0.42) and (5.4,0.12) .. (5.8,0)
    -- cycle;
  \draw[very thick]
    (1.4,2.9)
    .. controls (2.3,2.3) and (3.2,1.5) .. (4.2,0.85)
    .. controls (4.9,0.42) and (5.4,0.12) .. (5.8,0);
  % turning points
  \draw[black, dashed] (1.4,0) -- (1.4,2.9);
  \node[anchor=north, black!70] at (1.4,-0.05) {$R$};
  \node[anchor=north, black!70] at (5.8,-0.05) {$b$};
  \node[acc, anchor=west, font=\scriptsize] at (2.9,1.55) {area $= G$};
\end{tikzpicture}
$$

For a real actinide the barrier is thick, $R \ll b$, so $x$ is small. Expanding
$\arccos\sqrt{x} \approx \tfrac{\pi}{2} - \sqrt{x}$ and $\sqrt{x(1-x)} \approx \sqrt{x}$,
the bracket becomes $b\bigl(\tfrac{\pi}{2} - 2\sqrt{x}\bigr)$, and

$$
G \approx \frac{\pi (Z-2)e^2}{4\pi\epsilon_0\,\hbar}\sqrt{\frac{2m}{Q_\alpha}}
- \frac{4}{\hbar}\sqrt{\frac{2m(Z-2)e^2 R}{4\pi\epsilon_0}}.
$$

The first term, proportional to $(Z-2)\,Q_\alpha^{-1/2}$, dominates and carries the entire
sensitivity to the decay energy. The second, proportional to $\sqrt{(Z-2)R}$, is a slowly
varying correction that depends on the radius. The steep $Q^{-1/2}$ dependence in the
exponent is the origin of the enormous spread in half-lives: a small increase in
$Q_\alpha$ lowers $b$, thins the barrier, and multiplies the penetrability by a large
factor.

## From penetrability to half-life

The decay constant is the penetrability times the rate at which the alpha strikes the
barrier. Treating the preformed alpha as bouncing inside the well with velocity
$v_{\text{in}} = \sqrt{2(Q_\alpha + V_0)/m}$, it presents itself at the wall with an
**assault frequency**

$$
f = \frac{v_{\text{in}}}{2R} \sim 10^{21}\ \mathrm{s^{-1}},
$$

and the decay constant and half-life are

$$
\lambda = f\,P = f\,e^{-2G}, \qquad t_{1/2} = \frac{\ln 2}{\lambda}.
$$

The prefactor $f$ varies by less than an order of magnitude across the alpha emitters,
while $e^{-2G}$ ranges over more than forty powers of ten. The half-life is therefore
controlled almost entirely by the Gamow exponent, and a crude estimate of the
preformation probability and the assault frequency still lands within a couple of orders
of magnitude of the measured lifetime.

> **Worked example.** For ${}^{238}\mathrm{U} \to {}^{234}\mathrm{Th} + \alpha$, the decay
> energy is $Q_\alpha = 4.27\ \mathrm{MeV}$ and the daughter charge is $Z-2 = 90$. With
> $mc^2 = 3727\ \mathrm{MeV}$, $\hbar c = 197.3\ \mathrm{MeV\,fm}$, and $e^2/4\pi\epsilon_0
> = 1.44\ \mathrm{MeV\,fm}$:
> $$
> b = \frac{2(90)(1.44)}{4.27} = 60.7\ \mathrm{fm}, \qquad R \approx 9.3\ \mathrm{fm},
> \qquad x = \frac{R}{b} = 0.153.
> $$
> The bracket $\arccos\sqrt{x} - \sqrt{x(1-x)} = 1.168 - 0.360 = 0.808$, and
> $$
> G = \frac{\sqrt{2mc^2 Q_\alpha}}{\hbar c}\,b\,(0.808)
> = \frac{178.4}{197.3}\,(60.7)(0.808) = 44.3,
> $$
> so the penetrability is $P = e^{-88.6} \approx 3\times10^{-39}$. Taking $f \approx
> 8\times10^{20}\ \mathrm{s^{-1}}$ gives $\lambda \approx 2\times10^{-18}\ \mathrm{s^{-1}}$
> and $t_{1/2} \approx 3\times10^{17}\ \mathrm{s} \approx 10^{10}\ \mathrm{yr}$, against the
> measured $4.5\times10^{9}\ \mathrm{yr}$. A one-parameter barrier model reproduces a
> ten-billion-year lifetime to within an order of magnitude.

## The Geiger-Nuttall relation

Keeping only the dominant term of $G$ and absorbing the slowly varying prefactor and
radius correction into a constant, the logarithm of the half-life is linear in
$(Z-2)\,Q_\alpha^{-1/2}$:

$$
\log_{10} t_{1/2} = a\,\frac{Z-2}{\sqrt{Q_\alpha}} + c.
$$

The theoretical slope comes straight from $2G_1/\ln 10$:

$$
a = \frac{2\pi}{\ln 10}\,\frac{e^2}{4\pi\epsilon_0\,\hbar c}\sqrt{2mc^2}
= 1.72\ \mathrm{MeV^{1/2}},
$$

with $Q_\alpha$ in MeV. This is the **Geiger-Nuttall relation**, observed empirically in
1911 and explained by Gamow, and independently Condon and Gurney, in 1928 as the first
application of quantum tunneling.[^krane-theory] Plotting $\log_{10} t_{1/2}$ against
$Q_\alpha^{-1/2}$ for the isotopes of a single element gives a straight line, and the
lines for different elements are nearly parallel, their spacing set by the daughter charge
$Z-2$. Across the natural emitters the relation compresses a range from
$10^{-7}\ \mathrm{s}$ to $10^{17}\ \mathrm{s}$ onto one line.

$$
% caption: For a fixed element the base-ten logarithm of the half-life is linear in the
% inverse square root of the decay energy; the isotopes of thorium fall on a straight
% Geiger-Nuttall line covering more than twenty orders of magnitude in lifetime.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.0,0) node[right, black!70] {inverse root of $Q$};
  \draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {log half-life (s)};
  % y ticks
  \foreach \y/\lab in {0.4/{-6},1.4/0,2.4/8,3.4/14,4.2/18} {\draw[black] (-0.07,\y) -- (0.07,\y); \node[anchor=east, black, font=\scriptsize] at (-0.05,\y) {\lab};}
  % x ticks
  \foreach \x/\lab in {1.4/0.34,3.4/0.42,5.4/0.50,7.0/0.56} \node[anchor=north, black, font=\scriptsize] at (\x,-0.05) {\lab};
  % Geiger-Nuttall straight line
  \draw[acc, very thick] (1.0,0.35) -- (7.4,4.35);
  % data points along the line (thorium isotopes)
  \fill[acc] (1.5,0.66) circle (1.7pt);
  \fill[acc] (2.6,1.35) circle (1.7pt);
  \fill[acc] (3.8,2.10) circle (1.7pt);
  \fill[acc] (5.0,2.85) circle (1.7pt);
  \fill[acc] (6.2,3.60) circle (1.7pt);
  \node[acc, anchor=north west, font=\scriptsize] at (4.7,2.6) {Th isotopes};
  \node[black, anchor=west, font=\scriptsize] at (1.4,0.42) {short-lived, high $Q$};
  \node[black, anchor=east, font=\scriptsize] at (6.9,3.95) {long-lived, low $Q$};
\end{tikzpicture}
$$

The same steep dependence explains a systematic feature of the chart: within an isotopic
chain the alpha half-life grows rapidly as $Q_\alpha$ falls toward the stability line, and
the shortest-lived, highest-$Q$ emitters lie farthest from stability. The relation is
quantitative enough to predict an unknown half-life from a measured $Q$ to within about a
factor of ten, and conversely a measured lifetime fixes $Q$, which is why alpha
spectroscopy became an early tool for mapping the actinide masses.

> **Result.** The alpha half-life is set by a Coulomb-barrier tunneling exponent whose
> leading term is $2G_1 \propto (Z-2)\,Q_\alpha^{-1/2}$. A change in $Q_\alpha$ from
> $4$ to $9\ \mathrm{MeV}$ shortens the half-life by more than twenty orders of magnitude,
> and the linear Geiger-Nuttall plot of $\log t_{1/2}$ against $Q_\alpha^{-1/2}$ is the
> direct experimental signature of that exponent.

The one-dimensional barrier model treats every transition as feeding the daughter ground
state with the alpha carrying zero orbital angular momentum. Real spectra show several
alpha groups of slightly different energy, feeding excited states of the daughter, and
some transitions run far slower than the Gamow estimate. Those departures, governed by
angular-momentum and parity selection rules and by the overlap of the parent and daughter
wavefunctions, are the subject of the next lesson on
[fine structure and hindrance](/nuclear-physics/alpha-decay/alpha-fine-structure-hindrance).

[^krane-why]: **Krane**, _Introductory Nuclear Physics_, §8.1–8.2. The $Q$-value in binding
energies, the doubly magic stability of the alpha ($B_\alpha = 28.30\ \mathrm{MeV}$), and
the recoil split $T_\alpha = Q_\alpha(A-4)/A$. Alpha-decay energies and half-lives are
tabulated by the NNDC, [https://www.nndc.bnl.gov/](https://www.nndc.bnl.gov/), and the IAEA
Nuclear Data Services, [https://www-nds.iaea.org/](https://www-nds.iaea.org/).

[^krane-syst]: **Krane**, §8.3 (Alpha Decay Systematics): $Q_\alpha$ from the semi-empirical
mass formula crossing zero near $A \approx 150$, and the near-threshold rare-earth emitters.

[^krane-theory]: **Krane**, §8.4 (Theory of Alpha Emission), and **Wong**, _Introductory
Nuclear Physics_, §4-2. The WKB Gamow factor, the assault-frequency prefactor $f \approx
v/2R$, and the Geiger-Nuttall slope $a = 1.72\ \mathrm{MeV^{1/2}}$ from the leading barrier
term. The tunneling explanation is due to Gamow (1928) and, independently, Condon and Gurney.