Theory of Dynamical Diffraction

Bloch wave

$$\begin{equation} I_{\bb G}(z_{thick}) = \left| \bigl(\bb C\bb e^{2i\pi\boldsymbol{\gamma_j}z_{thick}}\bb C^{-1} \bigr)_{\bb G} \right|^2 \end{equation}$$

additional details

Assumptions :

• Elastic scattering : $E$ is constant at the incident electron energy.
• High energy approximation of collimated incident beam : $k_0\approx|\bb k_j|\gg|\bb G|$ hence ignoring backscattering.
• Continuity at the interface for individual beams : $\bb k_j=(k_0+\gamma_j)\bb e_z$.
• Small angle approximation : $\gamma_j^2\ll 2\gamma_j(\bb k_0+\bb G)\cdot \bb e_z$

Eigen value equation in $\gamma_j$ : $$\begin{eqnarray} 2k_0S_{\bb G}C_{j,\bb G} + \sum_{\bb G^{'}}V_{\bb G-\bb G^{'}}C_{j,\bb G^{'}} &=& 2k_0\gamma_jC_{j,\bb G} \\ S_{\bb G}C_{j,\bb G} + \sum_{\bb G^{'}}\frac{V_{\bb G-\bb G^{'}}}{2k_0} C_{j,\bb G^{'}} &=& \gamma_jC_{j,\bb G} \end{eqnarray}$$

where $2k_0S_{\bb G}=k_0^2-|\bb k_0+\bb G|^2=-2\bb k_0\cdot\bb G-|\bb G|^2\approx -2k_0G_z$ is the excitation error and $C_{j,\bb G}$ is the component of eigen vector $j$ related to the contribution of reciprocal lattice vector (or beam) $\bb G$.

Using the boundary conditions that the incident wave is a plane wave solely along the $\bb G=\bb 0$ direction enables to compute the wave function at the other boundary through a transfer matrix formalism : $$\begin{equation} \Psi(\bb r_{xy},T) = e^{2i\pi k_0T}\sum_{\bb G}\bb S_{\bb G,0}e^{2i\pi G_z T}e^{2i\pi\bb G_{xy}\cdot\bb r_{xy}} \end{equation}$$

where $T$ is the thickness, $\bb S=\bb C\bb e^{2i\pi\boldsymbol{\gamma_j}T}\bb C^{-1}$ is the scattering matrix. The far field diffraction pattern is then obtained from standard Fourier transform since it corresponds to propagation in free space. As a result $S_{\bb G0}^2$ can readily be identified as the diffraction intensity of beam $\bb G$.

Geometry documentation

Vectors can be expressed in various referential :

• $lab$ : laboratory cartesian frame where $x$ along rotation axis and $-z$ along beam
• $car$ : crystal cartesian frame where $x$ along $\bb a_1$ or $\bb b_1$ and $(x,y)$ is the same plane as $(\bb a_1,\bb a_2)$ or $(\bb b_1,\bb b_2)$.
• $lat$ : lattice frame where $(\bb a_1,\bb a_2,\bb a_3)$, may be non orthogonal
• $rec$ : reciprocal lattice frame where $(\bb b_1,\bb b_2,\bb b_3)$, may be non orthogonal

Relationship :

• $\bb u_{lab}=\bb A_{car}\bb u_{car}$
• $\bb u_{rec}=\bb A_{rec}^{-1}\bb u_{car}=(h,k,l)$
• $\bb u_{car}=\bb A_{lat} (u,v,w)$

Pets provides $\alpha$,$\beta$ angles and the orientation matrix $\bb A_{UB}$ so that $\bb R=\bb R_x(\alpha)\cdot\bb R_y(\beta)\cdot\bb A_{UB}$ and $\bb g_{lab}=\bb R\cdot\bb (h,k,l)$.

Pets provides the zone axis beam orientation as $\bb K_{lat} = (u,v,w)$ so that in the laboratory frame we should have $\bb K_{lab}=\bb R\cdot\bb A^{-1}_{rec}\bb A_{lat}\cdot\bb K_{lat}=(0,0,-1)$

Applications

2-beam configuration

In Bloch theory using 2 beams approximation gives the intensity as : $$\begin{equation} I_{dyn-2}(w_g;t,\xi_{\bb g}) = \left(\frac{\pi t}{\xi_{\bb g}}\right)^2 sinc^2\left(\frac{t}{\xi_{\bb g}}\sqrt{1+w_{\bb g}^2}\right) \end{equation}$$ where :

• $\xi_{\bb g}=k_0/U_{\bb g}$ is the Pendellosung thickness (in $A$) with $k_0=1/\lambda$ being the wave number.
• $w_{\bb g}=S_g\xi_g$ is the excitation error scaled by $\xi_g$.
• $U_{\bb g}=2me/h^2 v_{\bb g}=F_{\bb g}/\pi\cc V_{cell}$ is the form factor Fourier component (in $A^{-2}$).

kinematic approximation

The intensity of a Bragg spot can be established as : $$\begin{equation} I_{kin}(S_g;t,v_g) = \left(\sigma v_g t\right)^2 sinc^2\left(S_gt\right) \end{equation}$$ where :

• $S_g$ is the excitation error.
• $t$ is the thickness.
• $\sigma$ the interaction parameter (in $rad/kVA$).
• $v_g$ the Fourier components (in $V$).

Comparison 2-beam vs kinematic

Since $\sigma v_g=\pi/\xi_g$ one can see that for $S_G=0$ the 2-beam intensity reduces to kinematic limit case for small thickness : $$\begin{equation} I(0;t,\xi_g) = \left(\frac{\pi t}{\xi_g}\right)^2 sinc^2\left(\frac{t}{\xi_g}\right) \underset{t\ll\xi_g}{\rightarrow} \left(\frac{\pi t}{\xi_g}\right)^2 \end{equation}$$

Taking the arbitrary values $K=100A^{-1}$, $U_g=0.1A^{-2}$ gives a Pendullosung thickness $\xi_g=1000 A$ with patterns illustrated below for $t=100 A$ and $t=500A$.

1	2

1) I(0) kin vs dyn 2-beam $\xi_g=1000 A$ and 2) I(w) kin vs dyn 2-beam.

1	2

1) 2-beam rocking curves as function of thickness and 2) 2-beam(red) and kinematic(blue) integrated rocking curves as function of thickness. $\xi_g=500A$

Collision approach

The probabilities of an electron to undergo $m$ elastic collisions and $n$ inelastic collisions of mean free path $l_i$ after going through a specimen of length $z$ follows the Poisson distribution : $$\begin{equation} P_{mn}(z) = \frac{1}{m!}\left(\frac{z}{l_e}\right)^me^{-z/l_e} \frac{1}{n!}\left(\frac{z}{l_i}\right)^ne^{-z/l_i} \end{equation}$$ where

• $l_e=1/\sigma_e\rho$ is the average elastic collision mean free path, $\sigma_e=|f^{(e)}_a|^2$ being the interaction cross section and $f^{(e)}_a$ the atomic elastic scattering factor.
• $l_i=1/\sigma_i\rho$ is the average inelastic collision mean free path.
• $\rho$ is the number of atoms per volume area.

Latychevskaiaabrahams2019 followed a similar approach where the order of the collisions is considered to determine which sequence of events contribute to Bragg spots. A simplification of her equations considering only 0,1 and more than 1 elastic collisions reads : $$\begin{eqnarray} P_{coh}(z+dz) &=& (1-P_e(dz))P_{coh}(z) \\ P_{kin}(z+dz) &=& (1-P_e(dz))P_{kin}+P_e(dz)P_{coh}(z) \\ P_{dyn}(z+dz) &=& P_{dyn}(z) + P_e(z)P_{kin} \\ \end{eqnarray}$$

which analytical solutions are the Poisson distribution above. The program nearBragg should follow this statistics. Taking for protein $\rho=106$ atoms per $nm^3$ and $\sigma_e=0.001-0.005A^2$ which covers beam energies $E=100-1000keV$ and protein average scattering powers give mean free paths ranging $l_e=200-1000nm$.

small $\sigma_e$	medium $\sigma_e$	large $\sigma_e$