Comparison of Two Simulation Methods in Electron Crystallography: BW Method and a Modified Direct Product Method of Scattering Matrix

Abstract

Abstract:

In this paper, a modified direct product method of scattering matrix (DPSM) was presented and the calculation formula was derived as follows:

? ｚ = ∑ n 1 n ! ( M ｚ ) n ? ( 0 ) ?and? ? ｚ + ε j = ∑ n 1 n ! ( M ε j ) n ? ( ｚ ) ,

where M is the scattering matrix of which the dimension can be reduced by ‘Bethe potential method’ drastically and therefore the calculation speed can be increased tremendously without losing accuracy very much. The results calculated with the DPSM method are in almost exact agreement with those calculated with BW method. However, the calculation speed for the modified DPSM method is approximately three times faster than that for the BW method. Furthermore, the DPSM is suitable for computing all types of matrices without requiring symmetry or conjugate symmetry.

Key words: Electron crystallography, Image simulation, Bethe potential method, Matrix diagonalization

Yang Yi,Cai* Canying,Yang Qibin. Comparison of Two Simulation Methods in Electron Crystallography: BW Method and a Modified Direct Product Method of Scattering Matrix[J]. J. Mater. Sci. Technol., 2017, 33(2): 210-214.

Figures/Tables 7

Table 1 Comparison of calculated time by the DPSM and BW methods for Kx?=?Ky?=?0.0, Kz?=?398.406?nm-1

No. of diffraction	Calculated time (s)		Ratio of Calculated time between two methods
No. of diffraction	BW	DPSM	Ratio of Calculated time between two methods
800	49.11	19.44	2.53
631	25.09	7.31	3.43
523	14.56	4.10	3.55
393	6.37	1.78	3.58
285	2.48	0.72	3.44
225	1.25	0.36	3.47
165	0.51	0.14	3.64
121	0.21	0.06	3.50
93	0.10	0.03	3.33

Table 2 Comparison of calculated time by the DPSM and BW methods for Kx?=?1.645?nm-1, Ky?=?0.3903?nm-1, and Kz?=?398.4030?nm-1

No. of diffraction	Calculated time (s)		Ratio of calculated time between the BW and the DPSM methods
No. of diffraction	BW	DPSM
800	40.08	18.82	2.13
627	20.63	7.79	2.63
524	12.44	4.32	2.88
396	5.70	1.96	2.90
289	2.37	0.79	3.00
233	1.28	0.42	3.05
165	0.48	0.16	3.00
124	0.22	0.06	3.67
95	0.11	0.03	3.67

Fig.1 Plots of exit wave functions of 2?×?2 unit cells of the crystal MgAl2O4 calculated by the DPSM and the BW methods with parameters: zone axis [001], sample thickness z?=?12.8?nm, the wavelength λ?=?0.00251?nm and Kx?=?Ky?=?0.0, Kz?=?398.406?nm-1. (a) Number of electron diffraction?=?800, calculated by the BW method; (b) Number of electron diffraction?=?393, calculated by the BW method; (c) Number of electron diffraction?=?121, calculated by the BW method; (d) Number of electron diffraction?=?800, calculated by the DPSM method; (e) Number of electron diffraction?=?393, calculated by the DPSM method; (f) Number of electron diffraction?=?121, calculated by the DPSM method.

Fig.2 Plots of exit wave functions of 2?×?2 unit cells of the crystal MgAl2O4 calculated by the DPSM and BW methods with parameters: zone axis [001], sample thickness z?=?12.8?nm, the wavelength λ?=?0.00251?nm, Kx?=?1.645?nm-1, Ky?=?0.3903?nm-1, and Kz?=?398.4030?nm-1. (a) Number of electron diffraction?=?800, calculated by the BW method; (b) Number of electron diffraction?=?393, calculated by the BW method; (c) Number of electron diffraction?=?121, calculated by the BW method; (d) Number of electron diffraction?=?800, calculated by the DPSM method; (e) Number of electron diffraction?=?393, calculated by the DPSM method; (f) Number of electron diffraction?=?121, calculated by the DPSM method.

Table 3 Mean square root errors of different numbers of electron diffraction with respect to BW800 for Kx?=?Ky?=?0.0, Kz?=?398.406?nm-1 (800 means that number of electron diffraction?=?800, BW means that the simulation is carried out by the BW method)

Number of electron diffraction	Errors between BW and BW800	Errors between DPSM and BW800	Errors between two methods
93	0.01509448	0.01509249	<10^-5
121	0.01147363	0.01147300	<10^-5
165	0.00724817	0.00724791	<10^-5
225	0.00447864	0.00447913	<10^-5
285	0.00293397	0.00293578	<10^-5
393	0.00145200	0.00145786	<10^-5
523	0.00056101	0.00060484	<10^-4
631	0.00016675	0.00028689	<2?×?10^-4

Table 4 Mean square root errors of different numbers of electron diffraction with respect to BW800 for Kx?=?1.645?nm-1, Ky?=?0.3903?nm-1, and Kz?=?398.4030?nm-1 (800 means that number of electron diffraction?=?800, BW means that the simulation is carried out by BW method)

Number of electron diffraction	Errors between BW and BW800	Errors between DPSM and BW800	Errors between two methods
95	0.01536972	0.01537063	<10^-5
124	0.01062718	0.01062893	<10^-5
165	0.00679219	0.00679083	<10^-5
233	0.00393093	0.00393123	<10^-5
289	0.00268337	0.00268428	<10^-5
396	0.00127854	0.00128232	<10^-5
524	0.00048330	0.00052793	<10^-4
627	0.00018016	0.00028892	<2?×?10^-4

Fig.3 Potential projection of the crystal MgAl2O4 along [001] direction.

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