# Semiclassical interpretation of angular anisotropy in nuclear decay angular correlation studies?

Sometime ago a number of $\gamma-\gamma$, $\beta-\gamma$ and $\alpha-\gamma$ angular correlation studies were carried out to infer spin-parity assignments for cascades of decays from excited nuclear states (e.g., this one). A typical study would have a fixed detector and a moveable detector, with an angle $\vartheta$ subtended between them in relation to a radioactive source. Two $\gamma$ photons were recorded in coincidence, and the path of the first one, detected in the fixed detector, was chosen as the $z$-axis. The intensity of the correlation was measured as a function of $\vartheta$ as the moveable detector was moved around the source. The anisotropy in the intensity that was found as a function of $\vartheta$ was then used to infer things like the multipole order of the second $\gamma$ that had been emitted.

Suppose in a specific case an E1 (electric dipole) transition was determined (or perhaps an M1+E2 transition). Is there a semiclassical explanation for the anisotropy in the coincidence measurement around the radioactive source, or do we only understand the mathematics? Is there an intuitive explanation for the fact that the second $\gamma$ photon is preferentially emitted at specific angles in relation to the first $\gamma$?

The anisotropies are fitted by a function consisting of a sum of spherical harmonics, $Y_{\ell m}$. This is so that the multipolarity of the $\gamma$ can be inferred. If an E2 multipole (electric quadrupole) is assigned, this infers a deformed spheroidal distribution, either static (deformed ground state) or dynamic (deformed rotational or vibrational state). The "amount" of deformation is usually characterized by a parameter, $\beta_2$. If $\beta_2>0$, the deformation of the spheroid will be prolate (like rugby ball); if negative, it will be oblate, like a basketball being squeezed on opposite sides. A formula for describing the shape is $$R(\theta)=R_A\left(1+\beta_2 Y_{20} \right)$$ where, typically, $-0.2 < \beta_2 < 0.2$ and $R_A=R_oA^{1/3}$ with $R_o\simeq$1.2 - 1.44 fm.
As charge distributions change in the nucleus during gamma emissions, the radiation will be a linear combination of multipoles. Depending on the multipole mixture, the intensity of radiation will depend on the angle of emission relative to some defined axis. The detection of the first $\gamma$ defines the axis of measurement. The angular distribution of intensity of the second $\gamma$ is analyzed to find the multipole mixture of that second $\gamma$.