---
title: Fine Structure, Angular Momentum, and Hindrance Factors
module: Alpha Decay
moduleNumber: 5
lessonNumber: 2
order: 502
summary: >
  A single parent emits several alpha groups of slightly different energy, each feeding
  a distinct level of the daughter, so the alpha spectrum maps the daughter's low-lying
  states. Emission with orbital angular momentum L raises the barrier by a centrifugal
  term and is allowed only when angular-momentum and parity selection rules permit.
  Comparing the measured partial half-life to the Gamow estimate defines a hindrance
  factor near unity for even-even ground-state transitions and large for odd-A decays
  that must rearrange the unpaired nucleon.
topics: [Alpha Decay]
sources:
  - book: Krane
    ref: "Ch. 8 — Alpha Decay; §8.5 Angular Momentum and Parity in Alpha Decay, §8.6 Alpha Decay Spectroscopy"
  - book: Wong
    ref: "Ch. 4 — Nuclear Collective Motion; §4-2 Alpha-Particle Decay"
draft: false
---

The Gamow theory of the [previous lesson](/nuclear-physics/alpha-decay/alpha-decay-gamow-theory)
treats alpha decay as a single transition to the daughter ground state. Measured with a
magnetic spectrometer, the alpha particles from one parent instead appear as several
discrete groups a few tens to a few hundreds of keV apart. Each group feeds a different
state of the daughter, and their energies and intensities read out the daughter's
low-lying level scheme together with the angular-momentum structure of the parent. Two
new ingredients govern which transitions occur and how fast: the centrifugal barrier that
accompanies emission with orbital angular momentum, and the overlap between parent and
daughter wavefunctions that the one-dimensional barrier model omits.[^krane-am]

## Fine structure and the daughter level scheme

The disintegration energy is fixed by the masses, but the alpha need not leave the
daughter in its ground state. If it feeds an excited level at energy $E_x$, the available
kinetic energy is reduced,

$$
T_\alpha = (Q_\alpha - E_x)\,\frac{A-4}{A},
$$

so the alpha group is shifted down in energy by $E_x$ (times the recoil factor). A parent
therefore produces one alpha group per populated daughter level, and the spacing of the
groups reproduces the spacing of those levels. Because the penetrability falls steeply
with decreasing energy, ground-state and low-lying transitions dominate the intensity; the
higher the daughter level, the smaller its $Q$ and the weaker its branch.

The even-even alpha emitters give the cleanest example, because their daughters are
deformed rotors with a ground-state band $0^+, 2^+, 4^+, \dots$ at energies
$E_I = \tfrac{\hbar^2}{2\mathcal{I}}I(I+1)$. The decay of ${}^{228}\mathrm{Th}$ to the
rotational band of ${}^{224}\mathrm{Ra}$ ($Q_\alpha = 5.520\ \mathrm{MeV}$,
$t_{1/2} = 1.91\ \mathrm{yr}$) feeds the $0^+$ ground state ($72.2\%$, $E_\alpha =
5.423\ \mathrm{MeV}$), the $2^+$ level at $84.4\ \mathrm{keV}$ ($26.4\%$), and the $4^+$
level at $250.8\ \mathrm{keV}$ ($0.4\%$), with progressively weaker feeding of the higher
band members.[^nndc] The alpha spectrum is a direct picture of a rotational band.

$$
% caption: Alpha groups from thorium-228 feed the rotational band of the radium-224
% daughter; each branch loses the level excitation energy from its kinetic energy, and the
% intensities fall steeply as the daughter level rises.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % parent level
  \draw[very thick] (0.2,4.4) -- (2.4,4.4);
  \node[anchor=south, black] at (1.3,4.45) {${}^{228}$Th};
  % daughter band levels (rising)
  \draw[black!70, thick] (4.6,0.4) -- (7.6,0.4);
  \node[anchor=west, black!70] at (7.65,0.4) {$0^+$, $0$};
  \draw[black!70, thick] (4.6,1.15) -- (7.6,1.15);
  \node[anchor=west, black!70] at (7.65,1.15) {$2^+$, $84$};
  \draw[black!70, thick] (4.6,2.0) -- (7.6,2.0);
  \node[anchor=west, black!70] at (7.65,2.0) {$4^+$, $251$};
  \node[anchor=north, black, font=\scriptsize] at (6.1,0.32) {${}^{224}$Ra band};
  % alpha branches from parent to each level (weight by line width)
  \draw[acc, very thick, ->] (2.4,4.4) .. controls (3.6,3.0) and (4.0,1.2) .. (4.6,0.4);
  \node[acc, font=\scriptsize] at (3.05,2.2) {72\%};
  \draw[black, thick, ->] (2.4,4.3) .. controls (3.8,3.2) and (4.2,1.7) .. (4.6,1.15);
  \node[black, font=\scriptsize] at (3.75,2.75) {26\%};
  \draw[black, ->] (2.4,4.2) .. controls (4.0,3.6) and (4.3,2.5) .. (4.6,2.0);
  \node[black, font=\scriptsize] at (4.25,3.05) {0.4\%};
\end{tikzpicture}
$$

## Angular momentum and parity selection rules

The alpha particle has spin zero and positive intrinsic parity. Conservation of angular
momentum in the decay therefore couples the parent spin $I_i$ and daughter spin $I_f$
entirely through the orbital angular momentum $L$ carried off by the alpha,

$$
\vec I_i = \vec I_f + \vec L, \qquad |I_i - I_f| \le L \le I_i + I_f.
$$

Because the alpha carries no intrinsic parity change, the parity of the final state relative
to the initial is set by the orbital parity $(-1)^L$,

$$
\pi_i = \pi_f\,(-1)^L.
$$

The two rules together restrict $L$ to a subset of the triangle range: if parent and
daughter have the same parity, only even $L$ contribute; if opposite, only odd $L$. A
$0^+ \to 0^+$ transition forces $L = 0$ uniquely. A $0^+ \to 2^+$ transition (as in the
band feeding above) requires $L = 2$. Transitions that would demand a parity change with
$L = 0$, such as $0^+ \to 0^-$, are strictly forbidden.

$$
% caption: The alpha carries orbital angular momentum L that vector-couples the parent spin
% to the daughter spin, with the allowed range bounded by the triangle inequality and the
% parity of the two states fixing L to be even or odd.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % vectors from a common origin forming triangle
  \coordinate (O) at (0.6,0.4);
  \draw[black, very thick, ->] (O) -- (4.4,0.4);
  \node[black, anchor=north] at (2.5,0.32) {$I_i$};
  \draw[black, very thick, ->] (4.4,0.4) -- (2.9,2.4);
  \node[black!70, anchor=west] at (3.75,1.5) {$L$};
  \draw[black!70, very thick, ->] (O) -- (2.9,2.4);
  \node[black!70, anchor=east] at (1.6,1.5) {$I_f$};
  \node[black, anchor=west, font=\scriptsize] at (5.0,1.9) {triangle rule bounds $L$};
  \node[black, anchor=west, font=\scriptsize] at (5.0,1.3) {even $L$ if parities agree};
  \node[black, anchor=west, font=\scriptsize] at (5.0,0.7) {odd $L$ if parities disagree};
\end{tikzpicture}
$$

## The centrifugal barrier

Emission with $L > 0$ adds a centrifugal term to the potential outside the nucleus, so the
alpha tunnels through an effective barrier

$$
V_{\text{eff}}(r) = \frac{2(Z-2)e^2}{4\pi\epsilon_0\,r} + \frac{\hbar^2 L(L+1)}{2m r^2}.
$$

The centrifugal term is small next to the Coulomb barrier but not negligible. Evaluated at
the nuclear surface $R \approx 9.3\ \mathrm{fm}$,

$$
\frac{\hbar^2}{2m R^2} = \frac{(\hbar c)^2}{2mc^2 R^2}
= \frac{(197.3)^2}{2(3727)(9.3)^2} = 0.060\ \mathrm{MeV},
$$

so the added height is $0.36\ \mathrm{MeV}$ for $L = 2$ and $1.8\ \mathrm{MeV}$ for
$L = 5$, on top of a Coulomb barrier near $26\ \mathrm{MeV}$. A higher and slightly thicker
barrier enlarges the Gamow integral and lowers the penetrability. The reduction is modest,
a factor of a few per unit of $L$ for a typical actinide, and it works alongside the much
larger structural hindrance discussed below.

$$
% caption: The centrifugal term raises the effective barrier for emission with nonzero
% orbital angular momentum; higher L pushes the barrier up and slightly outward, enlarging
% the tunneling integral and suppressing the penetrability.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.0,0) node[right, black!70] {$r$};
  \draw[->, black] (0,0) -- (0,4.0) node[above, black!70] {$V(r)$};
  % L=0 pure Coulomb barrier
  \draw[acc, very thick]
    (1.4,2.7) .. controls (2.2,1.9) and (3.2,1.25) .. (4.2,0.95)
    .. controls (5.4,0.6) and (6.6,0.42) .. (7.8,0.32);
  % L=4 raised barrier
  \draw[black, very thick, dashed]
    (1.4,3.55) .. controls (2.3,2.55) and (3.3,1.7) .. (4.4,1.25)
    .. controls (5.6,0.8) and (6.7,0.55) .. (7.8,0.4);
  % inner wall
  \draw[black] (1.4,0) -- (1.4,3.55);
  \node[anchor=north, black!70] at (1.4,-0.05) {$R$};
  \node[acc, anchor=west, font=\scriptsize] at (2.3,1.65) {$L = 0$};
  \node[black, anchor=west, font=\scriptsize] at (2.4,2.85) {$L = 4$};
\end{tikzpicture}
$$

## Favored and hindered transitions

The Gamow formula predicts a partial half-life from the barrier alone. Dividing that
prediction into the measured partial half-life for a given branch defines the **hindrance
factor**,

$$
\mathrm{HF} = \frac{t_{1/2}^{\text{obs, partial}}}{t_{1/2}^{\text{theory}}},
$$

where the theoretical value assumes an unhindered transition with the appropriate $L$. The
hindrance factor isolates everything the one-body barrier model leaves out, above all the
probability that an alpha is preformed at the surface with the daughter left in the target
state. It plays the role of an inverse reduced width.

> **Definition (Favored transition).** An alpha transition is _favored_ (unhindered) when
> $\mathrm{HF} \approx 1$–$4$, meaning the observed rate matches the barrier-penetration
> estimate. The even-even ground-state decays $0^+ \to 0^+$ are the archetype: the parent
> is a paired $0^+$ state and the alpha forms from two paired protons and two paired
> neutrons at the surface, leaving the daughter in its own paired $0^+$ ground state with
> no rearrangement.

For even-even parents the ground-state transition and the transitions to the members of
the daughter's ground-state rotational band are all favored, because the alpha decay does
not disturb the underlying pairing structure. This is why the ${}^{228}\mathrm{Th}$ band
feeding above follows the Gamow ordering so cleanly, its branch ratios set almost entirely
by penetrability and the $L$-dependent centrifugal factor.

Odd-$A$ and odd-odd parents are different. The unpaired nucleon occupies a specific
Nilsson orbital, and forming an alpha from the paired nucleons leaves that odd nucleon
behind. The transition is favored only if it feeds a daughter state with the same
quantum numbers as the parent's odd nucleon; every other branch requires the odd nucleon
to change orbital and is hindered, often by factors of $10$ to $10^3$. In
${}^{241}\mathrm{Am} \to {}^{237}\mathrm{Np}$ ($Q_\alpha = 5.638\ \mathrm{MeV}$,
$t_{1/2} = 432.6\ \mathrm{yr}$), the strongest branch feeds an excited level at
$59.5\ \mathrm{keV}$ ($84.5\%$) rather than the ground state ($13.0\%$), precisely because
the $59.5\ \mathrm{keV}$ level matches the parent's odd-proton configuration while the
ground-state transition is hindered.[^krane-spec]

$$
% caption: Hindrance factors for the ground-state transitions along a decay series; the
% even-even parents sit near unity while the odd-mass and odd-odd parents are hindered by
% one to three orders of magnitude by the rearrangement of the unpaired nucleon.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.0,0) node[right, black!70] {parent along series};
  \draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {$\log_{10}\mathrm{HF}$};
  % gridline at HF=1
  \draw[black, dashed] (0,0.5) -- (7.7,0.5);
  \node[anchor=east, black, font=\scriptsize] at (-0.05,0.5) {$0$};
  \foreach \y/\lab in {1.5/1,2.5/2,3.5/3} {\draw[black] (-0.07,\y) -- (0.07,\y); \node[anchor=east, black, font=\scriptsize] at (-0.05,\y) {\lab};}
  % even-even points near HF~1
  \fill[acc] (1.0,0.62) circle (1.8pt);
  \fill[acc] (2.4,0.75) circle (1.8pt);
  \fill[acc] (3.9,0.58) circle (1.8pt);
  \fill[acc] (5.4,0.7) circle (1.8pt);
  \node[acc, anchor=south west, font=\scriptsize] at (0.8,0.9) {even-even};
  % odd-A points, hindered
  \draw[black, thick] (1.7,2.1) circle (2.0pt);
  \draw[black, thick] (3.1,2.9) circle (2.0pt);
  \draw[black, thick] (4.6,2.4) circle (2.0pt);
  \draw[black, thick] (6.1,3.2) circle (2.0pt);
  \node[black!70, anchor=west, font=\scriptsize] at (4.4,3.55) {odd $A$ / odd-odd};
\end{tikzpicture}
$$

## Alpha spectroscopy as a structure probe

The measured spectrum inverts into nuclear-structure information. The alpha-group energies
give the daughter level energies through $E_x = Q_\alpha - T_\alpha\,A/(A-4)$; the relative
intensities, once corrected for penetrability and the centrifugal factor, give the reduced
widths and hence the degree of hindrance; and the pattern of favored versus hindered branches
identifies the parent's single-particle configuration. A rotational band shows up as a
sequence of groups whose energies follow $I(I+1)$ and whose intensities fall smoothly with
$L$; an odd-$A$ decay shows up as anomalously strong feeding of an excited state.

$$
% caption: A magnetic-spectrometer alpha spectrum resolves discrete groups at energies set
% by the daughter levels; the tallest peak feeds the state matching the parent structure,
% and the spacing reproduces the daughter level scheme.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.0,0) node[right, black!70] {$E$};
  \draw[->, black] (0,0) -- (0,4.2) node[above, black!70] {intensity};
  % three peaks (groups) as narrow filled spikes with decreasing height toward lower E
  \fill[acc!25, draw=acc, thick] (6.0,0) rectangle (6.24,3.6);
  \draw[thick] (4.4,0) rectangle (4.64,1.9);
  \draw[thick] (2.7,0) rectangle (2.94,0.55);
  % labels for the daughter levels each group feeds
  \node[acc, anchor=south, font=\scriptsize] at (6.12,3.62) {$0^+$};
  \node[black, anchor=south, font=\scriptsize] at (4.52,1.92) {$2^+$};
  \node[black, anchor=south, font=\scriptsize] at (2.82,0.57) {$4^+$};
  \node[black, anchor=north, font=\scriptsize] at (6.12,-0.05) {ground};
  \node[black, anchor=north, font=\scriptsize] at (4.52,-0.05) {84 keV};
  \node[black, anchor=north, font=\scriptsize] at (2.82,-0.05) {251 keV};
\end{tikzpicture}
$$

The energies and lifetimes that drive this analysis are read from the same nuclide
databases used throughout the module.[^nndc] Alpha spectroscopy dovetails with the
[shell](/nuclear-physics/nuclear-models/shell-model-single-particle) and
[collective](/nuclear-physics/nuclear-models/collective-model-rotations-vibrations) models:
the favored transitions map the ground-state configurations, while the fine-structure
groups trace the rotational and vibrational bands the collective model predicts. The next
module turns to beta decay, where a continuous rather than discrete spectrum forced the
introduction of the neutrino.

[^krane-am]: **Krane**, _Introductory Nuclear Physics_, §8.5 (Angular Momentum and Parity in
Alpha Decay). The spin-zero, positive-parity alpha, the coupling $\vec I_i = \vec I_f + \vec
L$, the parity rule $\pi_i = \pi_f(-1)^L$, and the centrifugal barrier
$\hbar^2 L(L+1)/2mr^2$.

[^krane-spec]: **Krane**, §8.6 (Alpha Decay Spectroscopy). Favored even-even ground-state
transitions ($\mathrm{HF} \approx 1$), hindrance factors of $10$–$10^3$ for odd-$A$ decays,
and the ${}^{241}\mathrm{Am} \to {}^{237}\mathrm{Np}$ example in which the excited-state
branch dominates the ground-state branch. The hindrance factor is defined relative to the
barrier-penetration half-life of the [Gamow
theory](/nuclear-physics/alpha-decay/alpha-decay-gamow-theory).

[^nndc]: Alpha-group energies, intensities, daughter level schemes, and half-lives are from
the NNDC evaluated nuclear data, [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/). The
${}^{228}\mathrm{Th} \to {}^{224}\mathrm{Ra}$ band feeding ($0^+$ $72.2\%$, $2^+$ $26.4\%$,
$4^+$ $0.4\%$) and the ${}^{241}\mathrm{Am}$ branchings are the evaluated values.