that it gives us the most reliable approach to describe the angular emission of photon during the $\alpha$-decay. Matrix-function $D_{\nu\mu}^{l,*}(\varphi,\theta,0)$ defines direction of vector $\mathbf{k}$ relatively axis $z$ in the frame system for $\mathbf{r}$: angles $\varphi$ and $ \theta$ point to direction of vector $\mathbf{k}$, but not the vector $\mathbf{r}$. The functions $\mathbf{T}_{ll',\nu}(\mathbf{n})$ have the following form (${\mathbf \xi}_{0} = 0$, see ref.[@Eisenberg.1973], p. 45): $$\mathbf{T}_{jl,m} (\mathbf{n}) =
  \sum\limits_{\mu = \pm 1} (l, 1, j \,\big| \,m-\mu, \mu, m) \; Y_{l,m-\mu}(\mathbf{n}) \; \mathbf{\xi}_{\mu},
\label{eq.2.3.5}$$ where $(l, 1, j \,\bigl| \, m-\mu, \mu, m)$ are *Clebsh-Gordon coefficients* and $Y_{lm}(\theta, \varphi)$ are *spherical functions* defined, according to [@Landau.v3.1989] (see p. 119, (28,7)–(28,8)).

Approximation of the spherically symmetric $\alpha$-decay \[sec.2.4\]
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Now we shall study the $\alpha$-decay in the spherically symmetric approximation. We orientate the frame system so that axis $z$ will be parallel to the vector $\mathbf{k}$ and $$D_{\nu\mu}^{l}(\varphi,\theta,0) = \delta_{\mu\nu}.
\label{eq.2.4.1}$$ In the spherically symmetric approximation, wave functions of the decaying system in the initial and final states are separated into the radial and angular components, and these states are characterized by quantum numbers $l$ and $m$. We shall be interesting in such photon emission when the system transits to superposition 