n}_{r}),
\label{eq.2.4.6}$$ we obtain: $$\displaystyle\frac{\partial}{\partial \mathbf{r}}\: \psi_{i}(\mathbf{r}) =
  -\,\displaystyle\frac{d\,\varphi_{i}(r)}{dr}\: \mathbf{T}_{01,0}(\mathbf{n}^{i}_{r}),
\label{eq.2.4.7}$$ and then we calculate the matrix components: $$\begin{array}{lcl}
  p_{l_{ph}}^{M} & = & -\; I_{M}(l_{f},l_{ph}, l_{ph}) \cdot J(l,l), \\
  p_{l_{ph}}^{E} & = &
    -\sqrt{\displaystyle\frac{l_{ph}+1}{2l_{ph}+1}}\; I_{E}(l_{f},l_{ph},l_{ph}-1) \cdot J(l_{f},l_{ph}-1)\; +
    \sqrt{\displaystyle\frac{l_{ph}}{2l_{ph}+1}}\; I_{E}(l_{f},l_{ph},l_{ph}+1) \cdot J(l_{f},l_{ph}+1),
\end{array}
\label{eq.2.4.8}$$ where $$\begin{array}{lcl}
  J(l_{f},n) & = &
  \displaystyle\int\limits^{+\infty}_{0}
    \varphi^{*}_{f}(l,r)\, \displaystyle\frac{d\varphi_{i}(r)}{dr}\:
    j_{n}(kr)\; r^{2} dr, \\

  I_{M}(l_{f}, l_{ph}, n) & = &
    \displaystyle\sum\limits_{\mu = \pm 1}
    \mu\, h_{\mu}
    \displaystyle\int
    Y_{l_{f}m}^{*}({\mathbf n}_{r}^{f}) \:
    \mathbf{T}_{01,0}(\mathbf{n}^{i}_{r})
    \: \mathbf{T}_{l_{ph} n,\mu}^{*}({\mathbf n}_{ph}) \: d\Omega, \\

  I_{E}(l_{f}, l_{ph}, n) & = &
    \displaystyle\sum\limits_{\mu = \pm 1}
    h_{\mu}
    \displaystyle\int
    Y_{l_{f}m}^{*}({\mathbf n}_{r}^{f}) \:
    \mathbf{T}_{01,0}(\mathbf{n}^{i}_{r})
    \: \mathbf{T}_{l_{ph} n,\mu}^{*}({\mathbf n}_{ph}) \: d\Omega.
\end{array}
\label{eq.2.4.9}$$

Using the following value of the Clebsh-Gordon coefficient (see Appendix A): $$(110\,|1, -1, 0) = (110\,|-1, 1, 0) = \sqrt{\displaystyle\frac{1}{3}},
\label{eq.2.4.10}$$ from (\[eq.2.3.5\]) and (\[eq.2.4.7\]) we obtain: $$\begin{array}{c}
  \mathbf{T}_{01,0}(\mathbf{n}^{i}_{r}) =
    \displaystyle\sum\limits_{\mu = \pm 1} (110\,|-\mu\mu 0) \:
      Y_{1,-\mu}(\mathbf{n}^{i}_{r}) \: \mathbf{\xi}_{\mu} =
    \sqrt{\displaystyle\frac{1}{3}}
      \displaystyle\sum\limits_{\mu = \pm 1}
      Y_{1,-\mu}(\mathbf{n}^{i}_{r}) \: \mathbf{\xi}_{\mu}, \\

  \displaystyle\frac{\partial}{\partial \mathbf{r}}
    \psi_{i}(\mathbf{r}) =
  -\,\sqrt{\displaystyle\frac{1}{3}}\:
    \displaystyle\frac{d\varphi_{i}(r)}{dr}
    \displaystyle\sum\limits_{\mu = -1, 1}
    Y_{1,-\mu}(\mathbf{n}^{i}_{r}) \: \mathbf{\xi}_{\mu}
\end{array}
\label{eq.2.4.11}$$ and for the angular integrals for transition into the superposition of all possible final $f$-states with different $m_{f}$ at the same $l_{f}$ from eq. (\[eq.2.4.9\]) we obtain: $$\begin{array}{lcl}
  I_{M}(l_{f},l_{ph},n) & 