Before decay, the nucleus is in state with total angular momentum J and symmetry axis quantization M :


Say, the emitted radiation (can be EM wave or particle ) carries angular momentum  l and axis quantization m, its wavefunction is:


then the daughter nucleus has angular momentum j and m_j, the wave function is

\Psi_{j m_j}

their relation is:

\Phi_{JM} = \sum_{m, m_j}{\phi_{lm} \Psi_{j m_j} \left< l m j m_j | JM \right>}

where the $Latex \left< l m j m_j | JM \right> $ is Clebsch-Gordan coefficient.

The wave function of the emitted radiation from a central interaction takes the form:

\phi_{lm} = A_0 u_{nl}(r) Y_l^m(\theta,\phi)

The angular distribution is:

\int |\phi_{lm}|^2 dx^3 =A_0^2 \int |u_{nl}|^2 r^2 dr |Y_l^m|^2

for a fixed distance detector, the radial part is a constant. Moreover, not every spherical harmonic contribute the same weight, there is weighting factor due to Clebsch-Gordan coefficient.   Thus, the angular distribution is proportional to:

W(\theta) \propto \sum_{m_j=M-m}{|Y_l^m|^2 |\left<l m j m_j |JM\right>|^2 }

For example, JM=00, The possible (l, j) are (0,0), (1,1), (2,2) and so on, the m=-m_j . The C-G coefficient are

$Latex \left<0 0 0 0 |0 0\right> = 1 $

$Latex \left<1 m 1 -m |0 0\right> = \frac{1}{sqrt{3} $

$Latex \left<l m l -m |0 0\right> = \frac{1}{sqrt{2l+1} $


Y_0^0 = \frac{1}{4\pi}

\sum_{m}{|Y_l^m|^2 |\left<l m l -m |0 0\right>|^2 } =  \sum_m|Y_l^m|^2 \frac{1}{2l+1}}= constant

Thus, the angular distribution is isotropic.