$$\begin{array}{ccl}
  \displaystyle\frac{d W(\theta_{f})} {d\,\Omega_{ph} \: d\cos{\theta_{f}}} & = &
  \displaystyle\frac{Z_{eff}^{2}\, \hbar\, e^{2}}{2\pi\, c^{3}}\: \displaystyle\frac{w_{fi}}{m^{2}} \;
    \biggl\{p\,(k_{i},k_{f}) \displaystyle\frac{d\, p^{*}(k_{i},k_{f}, \theta_{f})}{d\cos{\theta_{f}}} + {\rm h. e.} \biggr\}.
\end{array}
\label{eq.2.8.3}$$ This probability is inversely proportional to normalized volume $V$. With a purpose to have the probability independent on $V$, I divide eq. (\[eq.2.8.3\]) on flux $j$ of outgoing $\alpha$-particles, which is inversely proportional to this volume $V$ also. Using quantum field theory approach (where $v(\mathbf{p}) = |\mathbf{p}| / p_{0}$ at $c=1$, see [@Bogoliubov.1980], § 21.4, p. 174): $$\begin{array}{cc}
  j = n_{i}\, v(\mathbf{p}_{i}), &
  v_{i} = |\mathbf{v}_{i}| = \displaystyle\frac{c^{2}\,|\mathbf{p}_{i}|} {E_{i}} =
          \displaystyle\frac{\hbar\,c^{2}\,k_{i}} {E_{i}},
\end{array}
\label{eq.2.8.4}$$ where $n_{i}$ is average number of particles in time unit before photon emission (we have $n_{i}=1$ for the normalized wave function in the initial $i$-state), $v(\mathbf{p}_{i})$ is module of velocity of outgoing particle in the frame system where colliding center is not moved, I obtain the *differential absolute probability* (while let’s name $dW$ as the *relative probability*): $$\begin{array}{ccl}
  \displaystyle\frac{d\,P (\varphi_{f}, \theta_{f})}{d\Omega_{ph}\, d\cos{\theta_{f}}} & = &
  \displaystyle\frac{d\,W (\varphi_{f}, \theta_{f})}{d\Omega_{ph}\, d\cos{\theta_{f}}} \cdot
    \displaystyle\frac{E_{i}} {\hbar\, c^{2}\, k_{i}} =
    \displaystyle\frac{Z_{eff}^{2} \,e^{2}}{2\pi\,c^{5}}\:
      \displaystyle\frac{w_{ph}\,E_{i}}{m^{2}\,k_{i}} \;
      \biggl\{p\,(k_{i},k_{f}) \displaystyle\frac{d\, p^{*}(k_{i},k_{f}, \Omega_{f})}{d\,\cos{\theta_{f}}} + {\rm h. e.} \biggr\}.
\end{array}
\label{eq.2.8.5}$$

Multipolar approach \[sec.2.9\]
-------------------------------

Let us find the bremsstrahlung probability 