Multislice 2D

2-beam dynamical diffraction

Setup

The structure is simple primitive square lattice with unique atom $Z_a=2$. The crystal is oriented along the [10 1] direction as shown in fig1-a which requires simulating a 10x10 super cell. For this case, the accelerated voltage has been voluntary chosen $E=3.75keV$ to allow for more Ewald sphere curvature. The configuration of the Ewald sphere is shown in fig1-b where the blue dots correspond to the reciprocal lattice rotated by $0.1rad$ from the zone axis [1 0] and the black dots correspond to the reciprocal lattice of the super cell setup. The beam intensities for $g=1$ and $g=2$ as a function of crystal thickness is shown in fig1-c.

Potential	Ewald configuration	Diffraction pattern	Beams intensities

Extinction distance

Below, the same simulation is run for different potential strengths.
Beam $g_1$ is not at the exact Bragg condition and its intensity with crystal thickness is mostly due to Ewald sphere curvature. Indeed, the oscillation period is independent of the potential strength and mostly depends on excitation error.

On the other hand, the beam $g_2$ being close to the Bragg condition, its extinction distance is sensitive to the strength of the potential. For this strongly excited beam, dynamical diffraction is present at all potential strengths but the kinematic regime is extended to larger crystal thickness at weaker potential.

$g_1$	$g_2$	$\zeta_g$

Rocking curve

Rocking curves are simulated by running simulations with various beam tilt angles ranging from 0 to 0.08 degrees.

The actual exact Bragg condition is satisfied for the $\theta_c=0.0385^{\circ}$. At this tilt angle, the Pendullosung thickness can be measured on the $I_b(z)$ giving $\zeta_g=293nm$. The analytical approach would give $\zeta_g=\pi/\sigma f_v(\theta_i,Z_a)$.

The rocking curves around $\theta_c$ are characteristic of 2-beam theory and shown for $z=[0.25,0.5,0.75,1,1.25,1.5]\zeta_g$.

Rocking curves	$I_g(z)$ at $\theta_c$

Padding

The purpose of this paragraph is to evaluate the effect of padding with zeros the potential of the sample in order to model finite size effects. The padding is expressed number of unit cells on each side of the sample.

Increasing padding for one periodic unit cell

$n_{pad}$	0	5	10	30
$T$
$I$

In this set of figures there is only one unit cell. In this case, the amount of padding has the same effect as increasing the size of the unit cell.

Increasing padding for two periodic unit cells

$n_{pad}$	0	5	10	30
$T$
$I$

In this set, 2 periodic unit cells are used.

Large padding with increasing number of unit cells

$n_{cells}$	1	2	5	10
$T$
$I$

In this set, a padding of 20 unit cells is applied while the number of periodic unit cells is increased.

Real space convolution

Performing the convolution by direct integration, using the small angle approximation : $$\begin{equation} \Psi(x,z_0+\Delta z) = \int_{-\delta x}^{\delta x} p(X,\Delta z) t(x-X,z_0)\Psi(x-X,z_0) dX \end{equation}$$

where $\delta x=\Delta z\theta$ with $\theta=0.1rad$ small angle and the Fresnel propagator $p(x,y,\Delta z)=\frac{e^{i\pi/4}}{i\lambda\Delta z}e^{ik_0\frac{x^2}{2\Delta z}}$, with $k_0=200keV$ and $\Delta z=5A$.

Transmission	Propagator	$\Psi(x,z_0+\Delta z)$

Below the real space transmission function, propagator and convolution product using the above integral and the FFT based multislice approach.