eq 1
$$F(hkl) = \sum_j \, f_j {\rm e}^{-M_j} \, {\rm e}^{2\pi\,i(hx_j + ky_j + lz_j)}$$

eq 2
$$n_r=n+ik=\sqrt{\epsilon}=1-\delta-i\beta=1-{r_0\over2\pi} \lambda^2 \sum_j \,n_jf_j$$

eq 3
$${\rm Re}(f)=f_0+f^\prime +f_{\rm NT}~,\quad f^\prime =f_1+f_{\rm rel}-Z$$

eq 4
$$f_0(q)=4\pi \int_0^{\infty}~ \frac{\rho(r)\sin(qr)r^2~{\rm d}r}{qr} \quad .$$

eq 5
$${\rm Im}(f)=f^{\prime\prime} (E)=f_2(E)=\frac{E\mu_{\rm PE}(E)}{2hc\,r_{\rm e}}$$

eq 6
$$f^\prime(E,Z)=f^\prime(\infty)-{2\over\pi}P \int_0^{\infty} \frac{\epsilon^\prime f^{\prime\prime}(\epsilon^\prime)}{(\hbar\omega)^2-(\epsilon^\prime)^2} ~{\rm d}\epsilon^\prime \quad .$$

eq 7
$$f^\prime(E)=f^\prime(\infty)-{2\over\pi} P \int_0^{\infty} \frac{(\epsilon^+-\epsilon_1) f^{\prime\prime}(\epsilon^+-\epsilon_1)} {(\hbar\omega)^2-(\epsilon^+-\epsilon_1)^2}~{\rm d}\epsilon^+ $$

eq 8
\begin{eqnarray*} f^+ &=&\left(\int_0^\infty \frac{\sigma(\epsilon^+ -\epsilon_1) (\epsilon^+ -\epsilon_1)^2 -\sigma(\hbar\omega)(\hbar\omega)^2} {(\hbar\omega)^2 - (\epsilon^+ -\epsilon_1)^2}~ {\rm d}\epsilon^+ \right. \\ ~&~& \left.+ P \int_0^\infty \frac{\sigma(\hbar\omega)(\hbar\omega)^2} {(\hbar\omega)^2 - (\epsilon^+ -\epsilon_1)^2}~ {\rm d}\epsilon^+\right) /2\pi^2 \alpha \end{eqnarray*}

eq 9
$$ x = \left\{ \begin{array}{l}{l} \frac{|\epsilon_1|}{\epsilon^+ -\epsilon_1} ~ , & \hbar\omega \simeq 2|\epsilon_1|\\ \left[ \frac{|\epsilon_1|}{\epsilon^+ -\epsilon_1}\right]^2 ~ , & \hbar\omega \leq \sqrt{2}|\epsilon_1|\\ \left[\frac{|\epsilon_1|}{\epsilon^+ -\epsilon_1}\right]^{1/2} ~ , & \hbar\omega \gg 2|\epsilon_1| \end{array} \right. \quad . $$

eq 10
$$ f^+ = \frac{1}{2\pi^2 \alpha} \int_0^1 \sigma_i {\rm d}x - \frac{\hbar\omega}{4\pi^2\alpha} ~\sigma(\hbar\omega) {\rm ln} \left( \frac{\hbar\omega-\epsilon_1}{\hbar\omega+\epsilon_1} \right)~,\quad |\epsilon_1|<\hbar\omega~;$$

eq 11
$$\sigma_i = \left\{ \begin{array}{l}{l} \sigma_2 ~, & \hbar\omega\gg 2|\epsilon_1|\\ \sigma_0 ~, & \hbar\omega\simeq 2|\epsilon_1|\\ \sigma_4 ~, & \hbar\omega\leq \sqrt{2} |\epsilon_1| \quad ;\end{array} \right. $$

eq 12
$$\sigma_2 = \left\{ \begin{array}{l}{l} -2\sigma(|\epsilon_1|/x^2) \epsilon_1/x^3 ~ , & |\epsilon_1|/x^2 = \hbar\omega\\ 2\left( \frac{\sigma(|\epsilon_1|/x^2) \epsilon_1^3/x^4 - \epsilon_1(\hbar\omega)^2 \sigma(\hbar\omega)} {x^3 (\hbar\omega)^2 - \epsilon_1^2 /x} \right) ~ , & {\rm otherwise} \quad ; \end{array} \right. $$

eq 13
$$\sigma_0 = \left\{ \begin{array}{l}{l} -\sigma(|\epsilon_1|/x) \epsilon_1/x^2 ~ , & |\epsilon_1|/x= \hbar\omega\\ \left(\frac{\sigma(|\epsilon_1|/x^2) \epsilon_1^3/x^2 - \epsilon_1(\hbar\omega)^2 \sigma(\hbar\omega)} {x^2 (\hbar\omega)^2 - \epsilon_1^2 } \right) ~ , & {\rm otherwise} \quad ; \right. \end{array} \right. $$

eq 14
$$\sigma_4 = \left\{ \begin{array}{l}{l} -0.5 \sigma(|\epsilon_1|/x^{1/2}) \epsilon_1/x ~ , & |\epsilon_1|/x^{1/2} = \hbar\omega\\ 0.5 \left( \frac{\sigma(|\epsilon_1|/x^{1/2}) \epsilon_1^3/x - \epsilon_1(\hbar\omega)^2 \sigma(\hbar\omega)}{\sqrt{x} \left[x(\hbar\omega)^2 - \epsilon_1^2 \right]} \right) ~ , & {\rm otherwise} \quad ; \end{array} \right. $$

eq 15
$$ f^+ = \frac{1}{2\pi^2 \alpha} \int_0^1 \sigma_i {\rm d}x + \frac{\epsilon_1^2 \sigma(|\epsilon_1|)}{4\pi^2 \alpha \hbar\omega} ~{\rm ln} \left(\frac{|\epsilon_1|-\hbar\omega}{|\epsilon_1|+\hbar\omega}\right) \quad ;$$

eq 16
$$\sigma_i= \left\{ \begin{array}{l}{l} sigma_3 ~ , & \hbar\omega \simeq 2|\epsilon_1|\\ sigma_5 ~ , & \hbar\omega \ll 2|\epsilon_1| \quad ; \end{array} \right. $$

eq 17
$$\sigma_3=\epsilon_1^3 \left( \frac{\sigma(|\epsilon_1|/x)/x^2 - \sigma(|\epsilon_1|)} {(x\hbar\omega)^2 - \epsilon_1^2}\right) \quad ; $$

eq 18
$$\sigma_5 = 0.5 \epsilon_1^3 \left( \frac{\sigma(|\epsilon_1|/x^{1/2})/x - \sigma(|\epsilon_1|)} {\sqrt{x}\left[x(\hbar\omega)^2 - \epsilon_1^2\right]}\right) \quad ; $$