see that integration of such functions over the solid angle $\Omega$ or over the angle $\theta$ (with needed limits) gives the integrated matrix element $p_{\rm dip}$ exactly. Comparing eqs. (\[eq.2.7.2.3\]) and (\[eq.2.7.1.10\]), we obtain connection between the differential and integral components of the matrix element in the dipole approximation: $$\begin{array}{lclcl}
  \displaystyle\frac{d\, p_{\rm dip}}{d\Omega} & = &
  2\pi \cdot \displaystyle\frac{d\, p_{\rm dip}}{\sin{\theta}\,d\theta} & = &
  p_{\rm dip} \cdot
    \displaystyle\frac{3}{8\pi}\;
    \sin^{2}{\theta}.
\end{array}
\label{eq.2.7.2.4}$$ The differential matrix element characterizes the angular distribution of emitted $\alpha$-particles, which becomes anisotropic in result of emission of photons.

Angular probability of emission of photon with impulse $\mathbf{k}$ and polarization $\mathbf{e}^{(\alpha)}$ \[sec.2.8\]
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I define the probability of transition of the system for time unit from the initial $i$-state into the final $f$-states, being in the given interval $d \nu_{f}$, with emission of photon with possible impulses inside the given interval $d \nu_{ph}$, so (see ref.[@Landau.v3.1989], (42,5) § 42, p. 189; ref.[@Berestetsky.1989], § 44, p. 191): $$\begin{array}{lll}
  d W = \displaystyle\frac{|a_{fi}|^{2}}{T} \cdot d\nu =
    2\pi \:|F_{fi}|^{2} \: \delta (w_{f} - w_{i} + w) \cdot d\nu, &
  d \nu = d\nu_{f} \cdot d\nu_{ph}, &
  d \nu_{ph} = \displaystyle\frac{d^{3} k}{(2\pi)^{3}} =
          \displaystyle\frac{w^{2} \, dw \,d\Omega_{ph}}{(2\pi c)^{3}},
\end{array}
\label{eq.2.8.1}$$ where $d\nu_{ph}$ 