\label{eq.2.1.4}$$ that looks like (4.21) in [@Bogoliubov.1980] (with accuracy up to factor $(2\pi)^{2}$) and like (42.5) in [@Landau.v3.1989] (exactly, see § 42, p. 189).

Linear and circular polarizations of the photon emitted \[sec.2.2\]
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Rewrite vectors of *linear* polarization $\mathbf{e}^{(\alpha)}$ through *vectors of circular polarization* $\mathbf{\xi}_{\mu}$ with opposite directions of rotation (see ref. [@Eisenberg.1973], (2.39), p. 42): $$\begin{array}{ccc}
  \mathbf{\xi}_{-1} = \displaystyle\frac{1}{\sqrt{2}}\,
                      \bigl(\mathbf{e}^{(1)} - i\mathbf{e}^{(2)}\bigr), &
  \mathbf{\xi}_{+1} = -\displaystyle\frac{1}{\sqrt{2}}\,
                      \bigl(\mathbf{e}^{(1)} + i\mathbf{e}^{(2)}\bigr), &
  \mathbf{\xi}_{0} = \mathbf{e}^{(3)} = 0.
\end{array}
\label{eq.2.2.1}$$ Then $p\,(k_{i},k_{f})$ can be rewritten so: $$p\,(k_{i}, k_{f}) =
    \sum\limits_{\mu = -1, 1}  h_{\mu}\,\mathbf{\xi}^{*}_{\mu}
    \int
      \psi^{*}_{f}(\mathbf{r})\:
      e^{-i\mathbf{kr}} \displaystyle\frac{\partial}{\partial \mathbf{r}} \:
      \psi_{i}(\mathbf{r}) \;
    \mathbf{dr},
\label{eq.2.2.2}$$ $$\begin{array}{ccc}
  h_{\pm} = \mp \displaystyle\frac{1 \pm i}{\sqrt{2}}, &
  h_{-1} + h_{+1} = -i\sqrt{2}, &
  \sum\limits_{\alpha = 1,2} \mathbf{e}^{(\alpha),*} =
    h_{-1} \mathbf{\xi}_{-1}^{*} + h_{+1} \mathbf{\xi}_{+1}^{*}.
\end{array}
\label{eq.2.2.3}$$

Different expansions of the vector potential $\mathbf{A}$ \[sec.2.3\]
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In order to simplify calculations of the matrix element $p\,(k_{i}, k_{f})$ with taking into account angular correlation between vectors $\mathbf{k}$ and $\mathbf{r}$, function $e^{-i\mathbf{kr}}$ connected with the vector potential $\mathbf{A}$ of the electro-magnetic field of the daughter nucleus should be expanded into basis of functions. At present, only three different types of expansion have been used in this problem:

-   The expansion 