Gamma decay, multipolarity and lifetimes

A gamma spectrum is a list of photon energies; a level scheme is the model that explains them. Excited states sit anywhere from tens of keV to several MeV above the ground state — 14.4 keV in ⁵⁷Fe, 661.7 keV in ¹³⁷Ba, 4.44 MeV in ¹²C — and each carries a spin and a parity, written Jπ. Gamma decay moves the nucleus between two of them and changes nothing else: Z and A are untouched, so no new element appears. The transition energy is very slightly above the photon energy, because the nucleus recoils: E_R = E_γ²/2Mc², which for a 1 MeV photon from A = 100 is 1²/(2 × 93 150) MeV = 5.4 eV. Against a germanium detector's 2 keV resolution that is invisible; against a Mössbauer linewidth of a few neV it is enormous, and the same formula decides the last section below. Many levels can decay to more than one lower level, not only to the ground state, so what you actually fit are branching ratios between competing transitions.

Expand the radiation field in multipoles. The EL term carries L units of angular momentum with parity (−1)^L; the ML term carries L units with parity (−1)^(L+1). Conservation then gives two hard rules. Spins: |Jᵢ − J_f| ≤ L ≤ Jᵢ + J_f. Parity: πᵢ π_f = +1 admits even-L electric and odd-L magnetic, while πᵢ π_f = −1 admits odd-L electric and even-L magnetic. And L ≥ 1 always, because a free photon's field has no monopole term — that exclusion is not a selection rule about nuclei, it is a statement about light. Work one: 5⁻ → 2⁺. Spins give L = 3, 4, 5, 6, 7; the parity flips, so the survivors are E3, M4, E5, M6, E7. The lowest order dominates, but the next one up is of the opposite character and can mix in — an M1 transition almost always carries some E2, and the mixing ratio δ is measured from angular correlations, never assumed.

To get a rate you need a matrix element, and Weisskopf's estimate supplies the crudest defensible one: a single proton moving between two shell-model orbitals inside a uniform sphere of radius R = 1.2A¹⁄³ fm, with unit radial overlap and no collective motion. With E in MeV and λ in s⁻¹ the results are λ(E1) = 1.0×10¹⁴ A²⁄³E³, λ(E2) = 7.3×10⁷ A⁴⁄³E⁵, λ(E3) = 34 A²E⁷ and λ(E4) = 1.1×10⁻⁵ A⁸⁄³E⁹, with magnetic partners λ(M1) = 5.6×10¹³ E³, λ(M2) = 3.5×10⁷ A²⁄³E⁵, λ(M3) = 16 A⁴⁄³E⁷ and λ(M4) = 4.5×10⁻⁶ A²E⁹. Every one of them goes as E^(2L+1). Nobody expects these to be right, and they are not: divide a measured rate by the estimate and you have the strength in Weisskopf units. E2 transitions in deformed rare earths reach 100 W.u., because whole nuclei rather than single protons are moving. E1 transitions typically land near 10⁻⁵ W.u., most of the dipole strength having been swallowed by the giant resonance at 15 MeV. M1 sits near 10⁻² W.u. Those deviations are the physics; the estimate is only the yardstick.

Take the ratio of two Weisskopf rates one unit of L apart and it is dominated by (kR)², with k = E_γ/ħc. For A = 137 and E_γ = 0.66 MeV, R = 6.19 fm and kR = 0.0207, so (kR)² = 4.3×10⁻⁴; the statistical coefficients contribute roughly another factor of 60, giving about five decades of half-life per extra unit. That is the whole mechanism of nuclear isomerism. ¹³⁷ᵐBa is the case to memorise: an 11/2⁻ state 661.7 keV above a 3/2⁺ ground state, hence M4, living 2.55 minutes where an E1 of the same energy would be gone in 10⁻¹⁵ s. The spins are no accident. Just below the magic numbers 50, 82 and 126 a high-j intruder orbital — g9/2, h11/2, i13/2 — sits beside low-j neighbours, so nuclei there routinely offer a level pair differing by four units of J. Those are the islands of isomerism. ⁹⁹ᵐTc, a 1/2⁻ p1/2 state above a 9/2⁺ g9/2 ground state, is the same physics, and its six-hour half-life is why it is in a hospital rather than a laboratory.

The nuclear multipole field does not only make photons. Where a bound atomic electron's wavefunction overlaps the nucleus, the field can couple to it directly and eject it with kinetic energy Tₑ = Eₜᵣ − Bₑ. This is internal conversion, a first-order process in its own right. Its strength is booked as α = λₑ/λ_γ, so a measured total rate is λₜₒₜₐₗ = λ_γ(1 + α); α grows roughly as Z³ and with L, falls steeply with transition energy, and reaches 10² and beyond for low-energy high-multipole transitions in heavy atoms. The extreme case is 0⁺ → 0⁺. The spins permit only L = 0, and there is no L = 0 photon, so the single-photon rate is exactly zero — yet the level still empties. ⁷²Ge's 691 keV 0⁺ state converts, with T½ = 444 ns. Above 2m_ec² = 1.022 MeV a second channel opens, internal pair creation, and ¹⁶O's 6.05 MeV 0⁺ state uses it, emptying in 67 ps almost entirely by emitting an electron-positron pair; a second-order two-photon branch exists but carries less than 10⁻³ of the decays.

A level's lifetime and its energy width are one measurement: Γτ = ħ, with ħ = 6.582×10⁻¹⁶ eV s. The timescale picks the technique. Above about a nanosecond, time the delayed coincidence between the feeding photon and the decaying one. Between roughly 1 ps and 1 ns, let the recoiling nucleus fly a measured distance to a stopper and compare Doppler-shifted with unshifted lines. From about 10 fs to 1 ps, use Doppler-shift attenuation as the recoil slows inside the stopper. Below that, abandon clocks and measure Γ directly — 1 fs corresponds to 0.66 eV. The sharp-line end becomes a tool. ⁵⁷Fe's 14.4 keV level has τ = 141 ns, so Γ = 4.7 neV, a fractional width of 3.2×10⁻¹³. But a free ⁵⁷Fe atom recoils with E_R = 1.96 meV, some 4×10⁵ linewidths, so emission and absorption lines miss each other entirely. Lock the nucleus into a crystal and a sizeable fraction of events leave the lattice in its original vibrational state; M in E_R = E_γ²/2Mc² becomes the mass of the whole crystal and the recoil vanishes. Pound and Rebka then used that line to measure a gravitational shift of 2.5×10⁻¹⁵ over 22.5 m.