Advanced computing in electron microscopy

Chap5 Thin specimens

Phase grating approximation

• The specimen potential is very small compare to the accelerating potential : $k_s\approx k_z(1+\sigma\nu_z(x,y))$ where $k_z=2\pi/\lambda_0$ is the wave number of the accelerating potential and $\sigma=2\pi me\lambda/h^2$ the interaction parameter.
• The specimen is thin enough so that the effect of the potential can be summarized as an accumulated projected potential along the beam direction $\nu_z(x,y)=\int V(x,y,z)dz$.

weak phase object

Since the interaction parameter $\sigma\approx 0.9$ the phase shift due to the projected potential of light atoms is usually quite small $\approx 0.4rad$ for Carbone at $r\approx 0.1A$. As a result they may be considered as weak phase objects. However the potential exhibits a singularity in real space due to the atomic nucleus so the phase shift may be large if the incident beam passes very close to the nucleus.

electron density $r^2\rho(r)$	3D potential $V(r)$	projected potential $\nu_z(r)$

The atomic scattering factor can be calculated within the Born approximation which will be obtained from a simple Fourier transform. The projected potential can therefore conveniently be deduced using the Fourier projection theorem.

However, a proper calculation based on Moliere shows that the atomic scattering factor within the Born approximation makes significant errors in the phase of the scattered electrons especially at large scattering angles where the effect of the nucleus dominates which is actually the region where the neglected effect of bonding is most relevant. Applying the Born approximation for the atomic scattering amplitudes assumes that the atoms are weak phase objects.

Born approximation for C,Si,Cu,Au,U	Moliere vs Born approximation for Gold

Image of thin specimens

Projected atomic potentials	Image intensities

Chap6 Theory

Electron scattering is accounted for by solving Schrodinger's equation : $$\begin{equation} \Big\{ \frac{\hbar^2}{2m_0}\grad^2 + V(\bb r) \Big\}\Psi(\bb r) = E\Psi(\bb r) \end{equation}$$

which is only an approximation of Dirac's equation the electron spin is neglected and relativistic effects not considered (accurate up to $1MeV$ electron beam if the energy/wavelength of the electron already takes into account relativistic effects). The magnetic field due to the lenses can be ignored since it affects the electron path over a $1mm$ distance despite being $\approx 1T$.

ED modelling history

Date	Name	Advance
1928	Davisson,Bethe	Electron diffraction exp,theory
1957	Cowley,Moodie	Multislice approach
1959,1962	Fujimoto,Sturkey	Scattering matrix Bloch
1961	Howie,Whelan	Dynamical theory non crystal based
1972,1979	O,Keefe	Multislice:First computer implementation SHRLI
1978	Glaeser,van Dyck	Path integral formulation
1978,1987	Cowley,Kirkland	Multi-slice CBED, ADF-STEM
1983	Kilaas	Real space multislice

Notes

Prop	Multislice	Bloch
Number of beams $N$	Fourier components at the FFT stage	Fourier components of the potential
Storage scales as	$N$	$N^2$
Computer time as	$N\log_2(N)$	$N^3$

Bloch(up) vs mutlislice(down)	CPU time

Bloch waves

see Bloch wave.

Fast electron wave equation

Using constant high energy collimated electrons, i.e. $\Psi_f=\Psi(x,y,z)e^{ik_0z}$ and slow varying envelope, i.e. $\dP^2_{z}\ll 2ik_0\dP_z$ assumptions, Schrodinger's equation reduces to:

$$\begin{equation} \frac{\dP\Psi(x,y,z)}{\dP_z} = \Big\{\frac{i\lambda}{4\pi}\grad^2_{xy} + i\sigma V(x,y,z)\Big\} \Psi(x,y,z) \end{equation}$$

where $\sigma=2\pi me\lambda/h^2$.

Howie Whelan

Assuming period potential and looking for solutions $\Psi(\bb r)=\sum_{\bb G}\phi_{\bb G}(z)e^{i\bb G\cdot\bb r}$ reduces to the system of differential equations : $$\begin{equation} \frac{\dP\phi_{\bb G}(x)}{\dP_z} = 2\pi S_{\bb G}\phi_{\bb G}(z) + i\sigma\sum_{\bb G^{'}}V_{\bb G-\bb G^{'}}\phi_{\bb G^{'}}(z) \end{equation}$$

Multislice

See multislice

Chap7 Applications

see multislice

Chap8 The programs

atompot+mulslice	autoslic Si[100]

atompot and mulslic

These codes are essentially described in temsim

autoslic

Used for non periodic specimen :

• The slices are computed automatically from the input file .xyz.
• The projected potential is directly computed in real space.