Relativistic effects

Cowley, J. M.

doi:10.1107/97809553602060000593

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 4.3, pp. 260-261

Section 4.3.1.4. Relativistic effects

J. M. Cowley^b ^‡

4.3.1.4. Relativistic effects

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It has been shown by Fujiwara (1961) that, at least for electron energies up to 1 MeV or so, the relativistic effects on diffraction amplitudes and geometry are adequately described by the use of relativisitically corrected values for the mass and wavelength of the electrons; $[m=m_0(1-\beta^2)^{-1/2} \eqno (4.3.1.26)]$ $[\eqalignno{ \lambda&= h\bigg/\left[2em_0E\left(1+{eE\over2m_0c^2}\right)\right]^{1/2} =\lambda_c{(1-\beta^2)^{1/2}\over\beta} \cr &=12.2639/(E+0.97845\times10^{-6}\,E^2)^{1/2}, &(4.3.1.27)}]$ where [m_0] is the rest mass, $[\lambda_c]$ is the Compton wavelength $[\beta=\nu/c]$ , and λ is given in Å if E is in volts. Consequently, σ varies with the incident electron energy as $[[1+h^2/m^2_0c^2\lambda^2)]^{1/2}]$ , or $[\sigma=2\pi/\{\lambda E[1+(1-\beta^2)^{1/2}]\}.]$ Values of λ, $[\lambda^{-1},m/m_0,]$ β = ν/c, and σ are listed for various values of the accelerating voltage, E, in Table 4.3.2.1 with λ in Å and E in volts.

Table 4.3.2.1| top | pdf |
Parameters useful in electron diffraction as a function of accelerating voltage, E

E (keV)	λ	1/λ		v/c	σ
1	0.387629	2.57979	1.00196	0.06247	0.0081126
2	0.273961	3.65016	1.00391	0.08821	0.0057448
3	0.223579	4.47270	1.00587	0.10788	0.0046975
4	0.193530	5.16715	1.00783	0.12439	0.0040741
5	0.173015	5.77986	1.00978	0.13887	0.0036493

6	0.157863	6.33460	1.01174	0.15191	0.0033361
7	0.146082	6.84548	1.01370	0.16384	0.0030931
8	0.136581	7.32168	1.01566	0.17490	0.0028975
9	0.128707	7.76958	1.01761	0.18524	0.0027358
10	0.122043	8.19383	1.01957	0.19498	0.0025991

15	0.099407	10.05963	1.02935	0.23711	0.0021374
20	0.085882	11.64383	1.03914	0.27186	0.0018641
25	0.076632	13.04940	1.04892	0.30184	0.0016790
30	0.069789	14.32899	1.05871	0.32837	0.0015433
35	0.064459	15.51381	1.06849	0.35227	0.0014386

40	0.060153	16.62414	1.07828	0.37406	0.0013548
45	0.056580	17.67403	1.08806	0.39410	0.0012859
50	0.053551	18.67366	1.09784	0.41268	0.0012280
55	0.050941	19.63072	1.10763	0.43000	0.0011786
60	0.048659	20.55115	1.11741	0.44622	0.0011357

65	0.046642	21.43968	1.12720	0.46147	0.0010982
70	0.044843	22.30012	1.13698	0.47586	0.0010650
75	0.043223	23.13560	1.14677	0.48948	0.0010354
80	0.041756	23.94874	1.15655	0.50239	0.0010087
85	0.040418	24.74173	1.16634	0.51467	0.0009847

90	0.039190	25.51646	1.17612	0.52637	0.0009628
95	0.038060	26.27454	1.18591	0.53754	0.0009428
100	0.037013	27.01738	1.19569	0.54822	0.0009244
120	0.033491	29.85866	1.23483	0.58667	0.0008638
140	0.030739	32.53222	1.27397	0.61956	0.0008180

160	0.028509	35.07642	1.31310	0.64810	0.0007820
180	0.026654	37.51759	1.35224	0.67314	0.0007529
200	0.025079	39.87466	1.39138	0.69531	0.0007289
250	0.021986	45.48412	1.48922	0.74101	0.0006839
300	0.019687	50.79517	1.58707	0.77652	0.0006526

350	0.017891	55.89295	1.68491	0.80483	0.0006297
400	0.016439	60.83109	1.78276	0.82786	0.0006122
450	0.015233	65.64563	1.88060	0.84691	0.0005984
500	0.014212	70.36195	1.97845	0.86286	0.0005873
550	0.013334	74.99858	2.07629	0.87638	0.0005783

600	0.012568	79.56945	2.17414	0.88794	0.0005707
650	0.011893	84.08529	2.27198	0.89793	0.0005644
700	0.011292	88.55452	2.36983	0.90661	0.0005590
750	0.010755	92.98385	2.46767	0.91421	0.0005543
800	0.010269	97.37874	2.56552	0.92091	0.0005503

850	0.009829	101.74364	2.66336	0.92684	0.0005468
900	0.009427	106.08226	2.76121	0.93212	0.0005437
950	0.009058	110.39769	2.85905	0.93684	0.0005410
1000	0.008719	114.69256	2.95690	0.94108	0.0005385
1100	0.008115	123.22919	3.15259	0.94836	0.0005344

1200	0.007593	131.70646	3.34828	0.95436	0.0005310
1300	0.007136	140.13516	3.54397	0.95936	0.0005282
1400	0.006733	148.52355	3.73966	0.96358	0.0005259
1500	0.006374	156.87810	3.93535	0.96718	0.0005240

References

Fujiwara, K. (1961). Relativistic dynamical theory of electron diffraction. J. Phys. Soc. Jpn, 16, 2226–2238.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 4.3, pp. 260-261