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Gamma decay
An excited nucleus sheds energy as a photon. The photon carries away angular momentum L, so it leaves in a pattern set by L and its projection M, and the rate falls steeply with L: at 150 keV and A = 99 the single-particle estimate runs from femtoseconds for E1 to years for M4 (table below).
Rendering
What is drawn. Radiation pattern: the glowing surface is the polar plot of the photon intensity per solid angle, |XLM(θ)|², radius proportional to intensity, with the quantization axis z vertical; it is the same for E and M of the same L and M. The short white strokes on it show the direction of the electric field of the outgoing wave over one period (E ∝ n × XLM for EL, E ∝ XLM for ML: the strokes of E and M are turned by 90°). The shells are wavefronts, bright where the intensity is high. Photons leave in directions drawn at random from the pattern, each trailing a wiggle along its own field direction (wavelength and speed are not to scale). The nucleus: for EL its surface oscillates as Re YLM e−iωt, the oscillating charge multipole, amplitude exaggerated; for ML it keeps its shape and only a tint follows the same function, a reminder that magnetic radiation comes from currents and spins. Correlation: the first photon leaves along z (the beam), the surface is W(θ) of the second, the faint sphere is W = 1 (no correlation); the two detectors are at the angle θ of the slider. Materials, glow and floor are decoration.
Model, exactly
Rate of a transition of multipolarity σL between nuclear states, long-wavelength limit: λ(σL) = 8π(L+1) / (L[(2L+1)!!]²) · (1/ħ)(Eγ/ħc)2L+1 B(σL), with B(EL) in e²fm2L (e² = αħc) and B(ML) in μN²fm2L−2.
Weisskopf units, radius R = 1.2 A1/3 fm: BW(EL) = (1/4π)[3/(L+3)]² R2L e²fm2L, BW(ML) = (10/π)[3/(L+3)]² R2L−2 μN²fm2L−2 (BW(E2) = 0.059404 A4/3 e²fm⁴, BW(M1) = 1.790 μN²). This is the convention of the W.u. values ENSDF prints. ENSDF prints 184 such values for the gammas in the page's data, 159 as equalities. The page computes 150 of these 159 from the printed half-lives, intensities, conversion coefficients and mixing ratios: all within 2σ, 148 within 1σ; 4 of the 150 are bounds on the page (¹⁶O 6917 and 7117 keV levels), because another gamma of the level has an intensity limit, and the bound is within 0.01% of the value. The other 9 belong to levels where a gamma has no intensity (8) or no half-life is measured (1). Of the 25 limits and approximate values, 18 are computed with the same relation (<, >, ≈) and within 5%; the 7 others are the minor component of a mixed transition, whose limit depends on the range of δ, not propagated here. The B(E2) table's W.u. column uses the rounded unit 0.0594 A1.3333, below this one by 1.5×10−4 (A = 10) to 2.6×10−4 (A = 252).
The partial γ half-life of one gamma out of a level: tγ = T½ × Σj Iγ,j(1 + αj) / Iγ, divided by the isomeric-transition fraction when the level also decays otherwise. α is the total conversion coefficient as printed in ENSDF; this page computes none. Its source is per evaluation: BrIcc where the evaluation names it (⁶⁰Ni, ⁹⁹Tc and the isomer files of ¹⁸⁰Ta, ²²⁹Th, ²⁴²Am), measured for some E0-admixed transitions (¹⁵²Sm, ²³⁸U, stated in a comment and shown with the transition), and for ²²⁹ᵐTh derived from an assumed B(M1). Limits in ENSDF (a half-life or an intensity given as <, > or ≈) are carried through: B is then a limit, the partial half-life and the hindrance the opposite limit; a limit on another branch of the level makes the others bounds too, within that branch's share of the decays (shown). A limit on δ leaves the photon rate alone and bounds the two components in opposite directions. A pure E0 branch emits no photon: its (electron) intensity enters the branching, it gets no B. "M1,E2" in ENSDF means M1 or E2, not determined: no B. A mixed transition with mixing ratio δ splits as 1/(1+δ²) to the lower multipole and δ²/(1+δ²) to the higher. B in W.u. = λ / λW; the hindrance is its inverse, tγ/tW.
Radiation pattern of a pure multipole (L, M): |XLM|² = [M² YLM²/sin²θ + (∂YLM/∂θ)²] / L(L+1), normalised to 1 over the sphere, XLM = LYLM/√(L(L+1)) the vector spherical harmonic. A real source is an incoherent mixture of all M (a nucleus with no preferred axis): their sum is isotropic. An oriented nucleus (polarised or aligned) populates the M substates unequally and radiates the weighted sum of their patterns; detecting the first gamma of a cascade along z aligns the intermediate state in this way, which is where W(θ) comes from.
γγ angular correlation, pure multipoles, no perturbation of the intermediate state: W(θ) = Σk even Ak Pk(cos θ), Ak = Fk(L1L1JiJ) Fk(L2L2JfJ), Fk(LL′JfJi) = (−1)Jf+Ji−1[(2k+1)(2L+1)(2L′+1)(2Ji+1)]½ (L L′ k; 1 −1 0){L L′ k; Ji Ji Jf} (Frauenfelder and Steffen), 3j and 6j by Racah's formulas. An independent sum over magnetic substates (Clebsch-Gordan coefficients from diagonalising J², Wigner d from the matrix exponential) agrees to 10−13 for 16 cascades. k runs to 6 (L ≤ 3 on the page). Finite detector size, mixing ratios and hyperfine perturbation are left out.