---
title: Angular Correlations and the Mössbauer Effect
module: Gamma Decay
moduleNumber: 7
lessonNumber: 3
order: 703
summary: >
  Two gammas emitted in cascade are not independent in direction: detecting the
  first selects magnetic substates of the intermediate level and makes the second
  anisotropic, so the correlation function fixes the intermediate spin. The same
  nuclear resonance that recoil normally destroys is recovered when the emitter
  is locked in a lattice, giving the Mössbauer effect and its part-in-a-trillion
  resolution of isomer shifts and hyperfine fields.
topics: [Gamma Decay]
draft: false
sources:
  - book: Krane
    ref: "Ch. 10 — Gamma Decay; §10.8 Angular Correlations, §10.9 Nuclear Resonance and the Mössbauer Effect"
  - book: Wong
    ref: "Ch. 6 — Nuclear Collective Motion; §6-5 Electromagnetic Transitions"
---

The gammas from an unoriented sample are emitted isotropically: with the nuclear
spins randomly oriented, every magnetic substate is equally populated and no
direction is special. Two experimental facts break that isotropy and turn gamma
radiation into a precision probe. First, when a nucleus emits two photons in
**cascade**, detecting the first fixes a direction and selects a nonuniform set of
substates for the second, so the two emission directions are **correlated**.
Second, the recoil that normally spoils resonant absorption of a gamma vanishes
when the emitting nucleus is bound in a solid, recovering a resonance so sharp it
resolves the nucleus's electromagnetic environment.[^krane-corr]

## Gamma-gamma angular correlations

Consider a cascade $I_1 \to I \to I_2$ through an intermediate state of spin $I$,
emitting $\gamma_1$ then $\gamma_2$. Fixing the direction of $\gamma_1$ with a
detector picks out a subset of the $2I+1$ magnetic substates $m$ of the
intermediate level: the substates whose radiation pattern points $\gamma_1$ at the
detector are preferentially populated. Because those substates then emit
$\gamma_2$ with their own anisotropic patterns, the probability of detecting
$\gamma_2$ depends on the angle $\theta$ between the two photon directions.

$$
% caption: A cascade emits gamma-one into a fixed detector and gamma-two at
% angle theta; fixing the first direction aligns the intermediate-state
% substates, so the second photon's yield varies with theta.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[thick] (0,0) circle (0.34);
  \node[black, anchor=north] at (0,-0.5) {source};
  % gamma 1 fixed
  \draw[->, acc, very thick] (0.25,0) -- (3.2,0);
  \node[acc, anchor=south, font=\scriptsize] at (1.8,0.05) {gamma one};
  \draw[black, thick] (3.3,-0.45) rectangle (3.9,0.45);
  \node[black, anchor=west, font=\scriptsize] at (4.0,0) {detector 1};
  % gamma 2 at angle
  \draw[->, black, very thick] (0.2,0.25) -- (2.35,2.55);
  \node[black, anchor=south east, font=\scriptsize] at (2.35,2.55) {gamma two};
  % angle arc
  \draw[black] (1.0,0) arc (0:47:1.0);
  \node[black, font=\scriptsize] at (1.35,0.55) {theta};
\end{tikzpicture}
$$

The correlation function $W(\theta)$, the relative coincidence yield as a function
of the opening angle, is an expansion in even-order Legendre polynomials,

$$
W(\theta) = \sum_{k=0,2,4,\ldots}^{k_{\max}} A_k\, P_k(\cos\theta)
= 1 + a_2 P_2(\cos\theta) + a_4 P_4(\cos\theta) + \cdots.
$$

Only **even** orders appear, because parity conservation makes $W(\theta)$
symmetric under $\theta \to \pi - \theta$; odd Legendre terms would break that
symmetry. The series terminates at

$$
k_{\max} = \min(2I,\, 2L_1,\, 2L_2),
$$

set by the intermediate spin $I$ and the two multipole orders $L_1, L_2$. The
coefficients $a_2, a_4, \ldots$ are fixed geometrically by the spin sequence and
the multipolarities, so measuring $W(\theta)$ and matching the coefficients to
tabulated values determines the intermediate spin $I$ once the multipolarities are
known.

The standard case is the pure quadrupole cascade $0 \to 2 \to 0$, in which both
photons are $E2$. Here $k_{\max} = 4$, and the correlation is[^krane-corr]

$$
W(\theta) = 1 + \tfrac{1}{8}\,P_2(\cos\theta) + \tfrac{1}{24}\,P_4(\cos\theta),
$$

which is peaked at $\theta = 0$ and $\theta = \pi$ and flattest near
$\theta = \pi/2$. The presence of a nonzero $a_4$ term is itself information: it
requires $I \ge 2$ and both multipoles of order at least $2$, so its mere
observation rules out a dipole cascade.

$$
% caption: The 0-2-0 quadrupole cascade correlation W(theta) is largest at 0 and
% 180 degrees and smallest near 90 degrees; the anisotropy fixes the
% intermediate spin.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (6.6,0) node[right, black!70] {angle theta};
  \draw[->, black] (0,0) -- (0,3.6) node[above, black!70] {W};
  \foreach \x/\l in {0/0,1.5/45,3.0/90,4.5/135,6.0/180}{
    \node[black, anchor=north, font=\scriptsize] at (\x,0) {\l};
  }
  % W(theta): high at ends, min at 90
  \draw[acc, very thick]
    (0,3.0) .. controls (1.4,2.4) and (2.2,1.55) .. (3.0,1.5)
    .. controls (3.8,1.55) and (4.6,2.4) .. (6.0,3.0);
  \draw[black, dashed] (3.0,0) -- (3.0,1.5);
  \node[black, anchor=north, font=\scriptsize] at (3.0,1.45) {minimum};
\end{tikzpicture}
$$

If the intermediate state lives long enough (nanoseconds or more), a magnetic or
electric field at the nucleus makes its substates precess before $\gamma_2$ is
emitted, rotating the correlation pattern by an angle that grows with the delay.
These **perturbed angular correlations** turn the same cascade into a measurement
of the nuclear magnetic moment or the electric field gradient at the nuclear site,
a widely used condensed-matter probe.

## The recoil problem in resonant absorption

Optical resonance, in which an atom absorbs a photon its own species emitted, is
routine. The nuclear analog, resonant absorption of a gamma by a nucleus of the
same transition, is normally impossible, and the reason is recoil. From the
[energetics](/nuclear-physics/gamma-decay/gamma-multipole-radiation), an emitter
at rest produces a photon of energy $E_0 - E_R$ rather than the full level
spacing $E_0$, while an absorber at rest needs $E_0 + E_R$ to reach the excited
level, because it too must recoil. The emission and absorption lines are displaced
from each other by

$$
2E_R = \frac{E_\gamma^2}{Mc^2}.
$$

Whether the lines still overlap depends on their width. The natural linewidth is
set by the excited-state lifetime through $\Gamma = \hbar/\tau$. For atomic
transitions $E_R$ is far smaller than $\Gamma$ and the lines overlap fully; for
nuclear gammas the opposite holds by a wide margin.

> **Worked example (Recoil versus linewidth for $^{57}\mathrm{Fe}$).**
> The $14.4\ \mathrm{keV}$ first excited state of $^{57}\mathrm{Fe}$ has a mean
> life $\tau = 141\ \mathrm{ns}$. The recoil energy is
> $$
> E_R = \frac{E_\gamma^2}{2Mc^2}
> = \frac{(14.4\times10^{-3}\ \mathrm{MeV})^2}{2(57)(931.5\ \mathrm{MeV})}
> \approx 1.9\times10^{-3}\ \mathrm{eV}.
> $$
> The natural width is
> $$
> \Gamma = \frac{\hbar}{\tau}
> = \frac{6.58\times10^{-16}\ \mathrm{eV\,s}}{1.41\times10^{-7}\ \mathrm{s}}
> \approx 4.7\times10^{-9}\ \mathrm{eV}.
> $$
> The recoil shift exceeds the linewidth by a factor of about $4\times10^{5}$, so
> the emission and absorption lines miss each other completely and no resonance
> occurs for free nuclei. Thermal Doppler broadening, of order
> $\sqrt{E_R\,k_B T} \sim 10^{-2}\ \mathrm{eV}$ at room temperature, spreads the
> lines but does not restore useful overlap.

$$
% caption: For a free nucleus the emission line sits at E-zero minus the recoil
% and the absorption line at E-zero plus the recoil, separated by twice the
% recoil energy, far more than the natural linewidth, so they do not overlap.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.0,0) node[right, black!70] {photon energy};
  \draw[->, black] (0,0) -- (0,3.4) node[above, black!70] {intensity};
  % emission line (left of center)
  \draw[acc, very thick] (1.4,0.1) .. controls (2.0,0.1) and (2.1,2.8) .. (2.4,2.8)
    .. controls (2.7,2.8) and (2.8,0.1) .. (3.4,0.1);
  \node[acc, anchor=south, font=\scriptsize] at (2.4,2.8) {emission};
  % absorption line (right of center)
  \draw[black, very thick, dashed] (4.0,0.1) .. controls (4.6,0.1) and (4.7,2.8) .. (5.0,2.8)
    .. controls (5.3,2.8) and (5.4,0.1) .. (6.0,0.1);
  \node[black, anchor=south, font=\scriptsize] at (5.0,2.8) {absorption};
  % center line E0
  \draw[black, dashed] (3.7,0) -- (3.7,3.0);
  \node[black, anchor=south, font=\scriptsize] at (3.7,3.0) {$E_0$};
  % separation bracket
  \draw[<->, black] (2.4,-0.35) -- (5.0,-0.35);
  \node[black, anchor=north, font=\scriptsize] at (3.7,-0.4) {separation $2E_R$};
\end{tikzpicture}
$$

## The Mössbauer effect

Rudolf Mössbauer found in 1958 that when the emitting nucleus is bound in a
crystal, a fraction $f$ of the emissions occur with **no** nuclear recoil at all:
the momentum is taken up by the entire lattice, whose mass is of order Avogadro's
number times $M$, so the effective $E_R \to 0$ and the photon carries the full
energy $E_0$ at the natural linewidth. The **recoilless fraction** $f$, the
Lamb-Mössbauer factor, is governed by the mean-square vibrational amplitude
$\langle x^2 \rangle$ of the emitter,

$$
f = \exp\!\left(-k^2 \langle x^2 \rangle\right),
\qquad k = \frac{E_\gamma}{\hbar c},
$$

the same Debye-Waller factor that governs X-ray diffraction. A large $f$ requires
a low transition energy, a stiff lattice with a high Debye temperature, and a low
sample temperature. For recoilless emitter and absorber the emission and
absorption lines coincide at $E_0$ with width $\Gamma$, and resonance is restored
with a fractional resolution

$$
\frac{\Gamma}{E_0} \sim \frac{4.7\times10^{-9}\ \mathrm{eV}}{14.4\times10^{3}\ \mathrm{eV}}
\approx 3\times10^{-13}
$$

for $^{57}\mathrm{Fe}$, among the sharpest resonances available in the laboratory.

The resonance is scanned by moving the source at a small velocity $v$ relative to
the absorber, Doppler-shifting the emitted energy by $\delta E = (v/c)E_0$.
Velocities of a few $\mathrm{mm\,s^{-1}}$ shift the line by microelectronvolts,
enough to tune across the natural width many times over; the transmission dips
whenever the shifted emission line overlaps an absorber level.

## Mössbauer spectroscopy

Because the linewidth is a part in $10^{13}$, the spectrum resolves the tiny
shifts and splittings the nuclear levels acquire from their electromagnetic
environment.

- **Isomer (chemical) shift.** A difference in the s-electron density at the
  nucleus between source and absorber, combined with the change in mean-square
  charge radius $\delta\langle r^2 \rangle$ between ground and excited nuclear
  states, shifts the whole resonance. It measures oxidation state and chemical
  bonding.
- **Quadrupole splitting.** An electric field gradient at the nucleus couples to
  the nuclear quadrupole moment and splits a level of spin $I > 1/2$, producing a
  resonance doublet. It measures the local charge symmetry.
- **Magnetic hyperfine splitting.** A magnetic field at the nucleus Zeeman-splits
  the levels into their $m$ substates; for the $3/2 \to 1/2$ transition of
  $^{57}\mathrm{Fe}$ the allowed dipole transitions give a characteristic
  six-line pattern whose spacing measures the internal field, of order tens of
  teslas in metallic iron.

$$
% caption: Mössbauer transmission versus source velocity: a single line for an
% unsplit nucleus, a doublet from quadrupole splitting, and a six-line pattern
% from magnetic hyperfine splitting in iron.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.4,0) node[right, black!70] {source velocity};
  \draw[->, black] (0,1.6) -- (0,3.0) node[above, black!70, font=\scriptsize] {transmission};
  % single line (upper baseline) with one central dip
  \draw[black, very thick] (0.3,2.6) -- (3.55,2.6) -- (3.7,1.9) -- (3.85,2.6) -- (7.0,2.6);
  \node[black, anchor=west, font=\scriptsize] at (5.6,2.9) {single line};
  % sextet (lower baseline) with six dips
  \draw[black, very thick] (0.3,0.9) -- (7.0,0.9);
  \foreach \x in {0.9,1.7,2.5,4.9,5.7,6.5}{
    \draw[black, very thick] (\x-0.12,0.9) -- (\x,0.35) -- (\x+0.12,0.9);
  }
  \node[black, anchor=west, font=\scriptsize] at (0.2,1.3) {magnetic sextet};
\end{tikzpicture}
$$

The angular correlation and the Mössbauer resonance are two readings of the same
level scheme: the correlation fixes the spin of an intermediate state from the
geometry of the emitted radiation, and the Mössbauer line fixes the shifts and
splittings of the levels from the environment of the emitter. Together they close
the account of how a nucleus radiates its excitation energy and what that
radiation reveals about nuclear and atomic structure.

[^krane-corr]: **Krane**, _Introductory Nuclear Physics_, Ch. 10, §10.8 (Angular Correlations) for the Legendre expansion of $W(\theta)$, the termination rule $k_{\max} = \min(2I, 2L_1, 2L_2)$, and the $0 \to 2 \to 0$ cascade coefficients; §10.9 (Nuclear Resonance and the Mössbauer Effect) for the recoil obstruction, the recoilless fraction, and hyperfine Mössbauer spectroscopy. The $^{57}\mathrm{Fe}$ $14.4\ \mathrm{keV}$ level parameters are from the National Nuclear Data Center, [nndc.bnl.gov](https://www.nndc.bnl.gov/).