Angular Correlations and the Mössbauer Effect

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.¹

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.

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

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

set by the intermediate spin $I$ and the two multipole orders $L_1,L_2$. The coefficients $a_2,a_4,\dots$ 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 is¹

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.

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, 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

Whether the lines still overlap depends on their width. The natural linewidth is set by the excited-state lifetime through $\Gamma =\hslash /\tau$. 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.

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 $⟨x^2⟩$ of the emitter,

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

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 $\mathrm{mm}{\mathrm{s}}^{-1}$ 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 $\delta ⟨r^2⟩$ 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.

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.