Time-Reversal Invariance and Beta-Gamma Angular Correlation

The angular correlation functions between beta and gamma rays from oriented nuclei (with or without observing circular polarization) are given for use in testing the invariance of the beta interactions under time-reversal. If the beta interactions are noninvariant under time-reversal, the angular correlation functions have asymmetries, $W(\Theta , \theta , \varphi )\ne W(\Theta , \theta , -\varphi )$ and $W(\Theta , \theta , \varphi , P)\ne W(\Theta , \theta , -\varphi , P)$, where the axis of the nuclear orientation is chosen as the $z$-axis, and the beta and gamma rays are assumed to be emitted in the directions with polar angles $\Theta , \Phi \equiv \mathrm{zero}, \mathrm{and} \theta , \varphi$, respectively. $P$ indicates the circular polarization of the gamma rays, and $P=+1\mathrm{}\mathrm{}(-1\mathrm{})\mathrm{}$ for left (right) circular polarization. These asymmetries are of the order of $\left(\frac{p}{W}\right) \mathrm{Im}({C_T}^{*}{C_S}^{\prime }+{C_T}^{\prime *}{C}_S-{C_A}^{*}{C_V}^{\prime }-{C_A}^{\prime *}{C}_V)$.