Index
Bethe potential

The number of reflections to include in a dynamical calculations by the Bloch-wave method in order to get reliable results can be figure out looking a the convergence of the eigenvalues of 1s core state Bloch-waves. Since the calculation time grows as power 3 of the number of reflections n (~n3), it is necessary to somehow filter out the weak reflections.

The Bethe potential approximation method allows to reduce the size of the secular matrix, whose eigenvalues and eigenvectors make up the Bloch waves, by selecting only the strongest reflections. All the other reflections are introduced as perturbations of the strong reflections.

For example using 63 strong reflections and 28 weak reflections included as perturbations of the 63 strong ones, the eigenvalues of the 1s Te and Ze Bloch-waves are already close to the values of the full 91 reflections calculation (Table 1).

How good is the Bethe potential approximation?

The 3 next figures show the calculations conditions (ZnTe [0,1,1] 300kV). The 1s Bloch-wawes are located on the Te and Zn atomic columns respectively. The eigenvalues of the first 2 Bloch-waves are shown in Table 1.

63 0-72.51-22.87
6328-78.31-23.21
91 0-78.76-24.47
StrongWeakTe 1sZn 1s

Table 1 Eigenvalues of the first 2 Bloch-waves.

Figure1

Figure2

Figure3

Figure1a

Figure2a

Figure3a

Figure 1 ZnTe [0,1,1] calculation with 63 strong reflections.

Figure 2 ZnTe [0,1,1] 91 reflections, 63 strong reflections and 28 weak reflections.

Figure 3 ZnTe [0,1,1] 91 reflections.

The Bethe correction underestimates slightly the eigenvalues of the 1s Bloch-waves (the crystal projected potential is weaker). Repeating the calculation with an increasing number of reflections (no Bethe perturbation) the 1s core states converges slowly (Table 2). A sufficient number of reflections would be close to 163.

910-78.76-24.47
1190-81.17-24.92
1630-82.85-25.18
2220-83.59-25.26
2800-83.81-25.28
StrongWeakTe 1sZn 1s

Table 2 Eigenvalues of the 2 1s Bloch-waves as a function of the number of reflections.

Keep in mind that the calculation time grows as the cube of the number of reflections. The 91 reflections calculation is more than 10 times faster than the 222 reflections one. Very important speed increase indeed for Convergent Beam Electron Diffraction calculations, for example to determine crystal polarity (Figures 4, 5).

Figure4

Figure5

Figure 4 CBED calculation, ZnTe [0,1,1] (119 reflections).

Figure 5 CBED calculation, ZnTe [0,1,1] (200) reflection.