states with different $m_{f}$ at the same $l_{f}$.

We shall find the matrix element at the first values of $l_{f}$ and $l_{ph}$. We have $l_{f} = 1$, $l_{ph} = 1$. Calculating coefficients $C_{11 n}^{m \mu^{\prime}}$ and functions $f_{1 n}^{m \mu^{\prime}}(\theta)$ (see Appendix \[app.4\] and \[app.5\]), from eq. (\[eq.2.5.4\]) we obtain: $$\begin{array}{lcl}
  \displaystyle\frac{d \,\tilde{p}_{1}^{M}}{\sin{\theta}\,d\theta} & = &
    - \displaystyle\frac{3}{8} \: \sqrt{\displaystyle\frac{1}{\pi}} \cdot
    J(1,1) \cdot
    \sin^{2}{\theta} \cos{\theta}, \\
  \displaystyle\frac{d \,\tilde{p}_{1}^{E}}{\sin{\theta}\,d\theta} & = &
    i \: \displaystyle\frac{1}{8} \: \sqrt{\displaystyle\frac{2}{\pi}} \cdot  J(1,0) \cdot  \sin^{2}{\theta} \: + \:
    i \: \displaystyle\frac{1}{8} \: \sqrt{\displaystyle\frac{1}{\pi}} \cdot  J(1,2) \cdot
    \sin^{2}{\theta} \: \Bigl( 1 - 3 \sin^{2}{\theta} \Bigr).
\end{array}
\label{eq.2.6.1}$$ Integrating these expressions over angle $\theta$, we find the integral matrix elements: $$\begin{array}{lcllcl}
  \tilde{p}_{1}^{M} & = & 0, &
  \tilde{p}_{1}^{E} & = &
    i \: \displaystyle\frac{1}{6} \, \sqrt{\displaystyle\frac{2}{\pi}} \cdot
    \Bigl\{ J(1,0) - \displaystyle\frac{7}{10} \, \sqrt{2} \cdot J(1,2) \Bigr\}.
\end{array}
\label{eq.2.6.2}$$

Dipole approximation and Fermi’s Golden rule \[sec.2.7\]
--------------------------------------------------------

### Vector potential $\mathbf{A}$ and integrated matrix element \[sec.2.7.1\]

The matrix element $p\,(k_{i},k_{f})$ in the dipole approximation of the vector potential $\mathbf{A}$ has a form: $$p\,(k_{i}, k_{f}) =
    \sum\limits_{\mu = -1, 1}  h_{\mu}\,\mathbf{\xi}^{*}_{\mu}
    \int
      \psi^{*}_{f}(\mathbf{r})\:
      \displaystyle\frac{\partial}{\partial \mathbf{r}}\: \psi_{i}(\mathbf{r})\; \mathbf{dr}.
\label{eq.2.7.1.1}$$ Applying the following transformation at $l_{i}=0$ introduced in [@Papenbrock.1998.PRLTA] (see new 