A.1 Indexed Powder Diffraction...
79
Appendix A.1 Indexed Powder Diffraction Patterns Fig. A.1 Indices of peaks in powder diffraction patterns from simple cubic, face-centered cubic, body-centered cubic, diamond cubic, and hexagonal close-packed crystals A.2 Mass Attenuation Coefficients for Characteristic K α X-Rays Table A.1 Mass attenuation coefficients for characteristic K α x-rays [cm 2 /g] Z Cr Co Cu Mo 1H 0.412 0.397 0.391 0.373 2 He 0.498 0.343 0.292 0.202 3 Li 1.30 0.693 0.500 0.198 4 Be 3.44 1.67 1.11 0.256 5B 7.59 3.59 2.31 0.368 Z Cr Co Cu Mo 6C 15.0 7.07 4.51 0.576 7N 24.7 11.7 7.44 0.845 8O 37.8 18.0 11.5 1.22 9F 51.5 24.7 15.8 1.63 10 Ne 74.1 35.8 22.9 2.35 B. Fultz, J. Howe, Transmission Electron Microscopy and Diffractometry of Materials, Graduate Texts in Physics, DOI 10.1007/978-3-642-29761-8, © Springer-Verlag Berlin Heidelberg 2013 681
Transcript of A.1 Indexed Powder Diffraction...
Appendix
A.1 Indexed Powder Diffraction Patterns
Fig. A.1 Indices of peaks in powder diffraction patterns from simple cubic, face-centered cubic,body-centered cubic, diamond cubic, and hexagonal close-packed crystals
A.2 Mass Attenuation Coefficients for Characteristic Kα X-Rays
Table A.1 Mass attenuation coefficients for characteristic Kα x-rays [cm2/g]
Z Cr Co Cu Mo
1 H 0.412 0.397 0.391 0.373
2 He 0.498 0.343 0.292 0.202
3 Li 1.30 0.693 0.500 0.198
4 Be 3.44 1.67 1.11 0.256
5 B 7.59 3.59 2.31 0.368
Z Cr Co Cu Mo
6 C 15.0 7.07 4.51 0.576
7 N 24.7 11.7 7.44 0.845
8 O 37.8 18.0 11.5 1.22
9 F 51.5 24.7 15.8 1.63
10 Ne 74.1 35.8 22.9 2.35
B. Fultz, J. Howe, Transmission Electron Microscopy and Diffractometry of Materials,Graduate Texts in Physics,DOI 10.1007/978-3-642-29761-8, © Springer-Verlag Berlin Heidelberg 2013
681
682 Appendix
Table A.1 (Continued)
Z Cr Co Cu Mo
11 Na 94.9 46.2 29.7 3.03
12 Mg 126 61.9 40.0 4.09
13 Al 155 76.4 49.6 5.11
14 Si 196 97.8 63.7 6.64
15 P 230 115 75.5 7.97
16 S 281 142 93.3 9.99
17 Cl 316 161 106 11.5
18 Ar 342 176 116 12.8
19 K 421 218 145 16.2
20 Ca 490 255 170 19.3
21 Sc 516 269 180 20.8
22 Ti 590 291 200 23.4
23 V 74.7 325 219 26.0
24 Cr 86.8 408 247 29.9
25 Mn 97.5 393 270 33.1
26 Fe 113 57.2 302 37.6
27 Co 124 63.2 321 41.0
28 Ni 144 73.5 48.8 46.9
29 Cu 153 78.0 51.8 49.1
30 Zn 171 87.1 57.9 54.0
31 Ga 183 93.4 62.1 57.0
32 Ge 199 102 67.9 61.2
33 As 219 112 74.7 66.1
34 Se 234 120 80.0 69.5
35 Br 260 133 89.0 75.6
36 Kr 277 142 95.2 79.3
37 Rb 303 156 104 85.1
38 Sr 328 170 113 90.6
39 Y 358 185 124 97.0
40 Zr 386 200 139 16.3
41 Nb 416 216 145 17.7
42 Mo 442 230 154 18.8
43 Tc 474 247 166 20.4
44 Ru 501 262 176 21.7
45 Rh 536 280 189 23.3
46 Pd 563 295 199 24.7
47 Ag 602 316 213 26.5
48 Cd 626 329 222 27.8
49 In 663 349 236 29.5
50 Sn 691 364 247 31.0
51 Sb 723 383 259 32.7
52 Te 740 394 267 33.8
Z Cr Co Cu Mo
53 I 796 425 288 36.7
54 Xe 721 440 299 38.2
55 Cs 760 465 317 40.7
56 Ba 570 480 325 42.3
57 La 225 507 348 44.9
58 Ce 238 535 368 47.7
59 Pr 238 565 390 50.7
60 Nd 251 505 404 53.0
61 Pm 294 400 426 56.3
62 Sm 279 440 434 57.8
63 Eu 309 153 434 60.9
64 Gd 298 161 403 62.6
65 Tb 332 180 321 65.8
66 Dy 325 176 362 68.3
67 Ho 347 187 129 71.3
68 Er 352 191 132 74.4
69 Tm 386 206 140 77.9
70 Yb 387 206 142 80.4
71 Lu 431 229 156 84.0
72 Hf 425 227 155 86.9
73 Ta 432 231 158 90.4
74 W 457 246 168 93.8
75 Re 501 268 187 97.4
76 Os 499 268 184 100
77 Ir 520 278 191 104
78 Pt 541 276 188 107
79 Au 551 295 201 112
80 Hg 541 273 188 115
81 Tl 597 331 226 118
82 Pb 643 343 235 122
83 Bi 666 355 244 126
84 Po 691 370 254 132
85 At 680 363 248 117
86 Rn 734 392 267 108
87 Fr 758 403 277 87.0
88 Ra 743 398 273 88.0
89 Ac 739 461 317 90.8
90 Th 768 406 306 96.5
91 Pa 738 394 271 101
92 U 766 420 288 102
93 Np 800 430 314 42.2
94 Pu 760 408 280 39.9
A.3 Atomic Form Factors for X-Rays 683
Example Calculate the fraction, I/I0, of Mo Kα x-rays transmitted through0.01 cm of metallic Ag (having density 10.5 g cm−3):
I/I0 = exp(−26.5 cm2 g−1 10.5 g cm−3 0.01 cm
) = e−2.78 = 0.062.
A.3 Atomic Form Factors for X-Rays
Table A.2 of x-ray atomic form factors, fx(s), for elements and some ions was ob-tained from calculations with a Dirac–Fock method by D. Rez, P. Rez, I. Grant, ActaCrystallogr. A50, 481 (1994). The column headings are s ≡ sin θ/λ, in units of Å−1.This diffraction vector, s, is converted to the Δk used in the text by multiplicationby 4π .
The tabulated values of fx(s) are in electron units. Conversion to units of cmis performed by multiplying them by the “classical electron radius,” e2m−1c−2 =2.81794 × 10−13 cm.
A.4 X-Ray Dispersion Corrections for Anomalous Scattering
684 Appendix
Tabl
eA
.2A
tom
icfo
rmfa
ctor
sfo
rhi
gh-e
nerg
yx-
rays
s0.
00.
050.
10.
150.
20.
250.
30.
350.
40.
50.
60.
70.
80.
91.
01.
21.
41.
61.
82.
02.
53.
04.
05.
06.
0
He
2.00
1.96
1.84
1.66
1.46
1.26
1.06
0.89
0.74
0.51
0.35
0.25
0.18
0.13
0.10
0.05
0.03
0.02
0.01
0.01
0.00
0.00
0.00
0.00
0.00
Li+
12.
001.
981.
941.
861.
761.
651.
521.
401.
271.
030.
820.
640.
510.
400.
320.
200.
130.
090.
060.
040.
020.
010.
000.
000.
00
Li
3.00
2.71
2.22
1.90
1.74
1.63
1.51
1.39
1.27
1.03
0.83
0.65
0.51
0.41
0.32
0.21
0.14
0.09
0.06
0.05
0.02
0.01
0.00
0.00
0.00
Be+
22.
001.
991.
971.
931.
871.
801.
731.
641.
551.
371.
181.
010.
850.
720.
600.
430.
300.
220.
160.
120.
060.
030.
010.
010.
00
Be
4.00
3.71
3.07
2.47
2.06
1.83
1.69
1.60
1.52
1.36
1.20
1.03
0.88
0.74
0.62
0.44
0.31
0.22
0.16
0.12
0.06
0.03
0.01
0.01
0.00
B5.
004.
734.
063.
322.
702.
271.
981.
801.
681.
531.
401.
281.
151.
020.
900.
690.
530.
400.
300.
230.
130.
070.
030.
010.
01
C6.
005.
755.
124.
333.
572.
962.
502.
181.
951.
691.
541.
431.
321.
221.
120.
920.
740.
590.
470.
370.
220.
130.
050.
020.
01
N7.
006.
786.
185.
394.
573.
833.
222.
752.
401.
941.
701.
551.
451.
351.
271.
090.
920.
770.
640.
530.
330.
210.
090.
040.
02
O8.
007.
807.
256.
475.
634.
814.
093.
493.
012.
341.
951.
721.
571.
461.
381.
221.
070.
930.
790.
680.
440.
290.
140.
070.
04
O−1
9.00
8.71
7.92
6.89
5.84
4.89
4.10
3.47
2.98
2.32
1.94
1.71
1.57
1.46
1.38
1.22
1.07
0.92
0.79
0.67
0.44
0.29
0.13
0.07
0.04
O−2
10.0
09.
598.
547.
225.
964.
904.
063.
422.
942.
301.
931.
711.
571.
471.
381.
221.
070.
920.
790.
670.
440.
290.
130.
070.
03
F9.
008.
828.
307.
566.
715.
865.
064.
363.
762.
882.
311.
961.
741.
591.
481.
331.
191.
060.
930.
810.
570.
390.
190.
100.
06
F−1
10.0
09.
739.
028.
046.
985.
985.
094.
353.
742.
852.
291.
951.
731.
591.
481.
321.
191.
050.
930.
810.
560.
390.
190.
100.
06
Ne
10.0
09.
839.
358.
657.
816.
936.
095.
314.
633.
542.
802.
301.
971.
761.
611.
421.
281.
161.
040.
930.
680.
490.
250.
140.
08
Na+
110
.00
9.88
9.55
9.03
8.38
7.65
6.90
6.17
5.48
4.30
3.40
2.76
2.31
2.00
1.79
1.53
1.37
1.25
1.14
1.03
0.79
0.59
0.33
0.18
0.11
Na
11.0
010
.57
9.76
9.03
8.34
7.62
6.89
6.16
5.48
4.30
3.40
2.76
2.31
2.00
1.79
1.53
1.37
1.25
1.14
1.03
0.79
0.59
0.32
0.19
0.11
Mg+
210
.00
9.91
9.66
9.27
8.76
8.16
7.52
6.86
6.22
5.03
4.05
3.29
2.73
2.32
2.03
1.66
1.46
1.33
1.22
1.12
0.89
0.69
0.40
0.23
0.14
Mg
12.0
011
.51
10.4
89.
518.
748.
087.
456.
826.
205.
044.
073.
302.
732.
322.
031.
661.
461.
331.
221.
120.
890.
690.
400.
240.
14
Al+
310
.00
9.93
9.74
9.43
9.02
8.53
7.98
7.41
6.83
5.70
4.70
3.87
3.21
2.71
2.33
1.85
1.58
1.41
1.29
1.20
0.98
0.78
0.48
0.29
0.18
Al
13.0
012
.44
11.2
310
.06
9.16
8.47
7.88
7.32
6.77
5.70
4.72
3.89
3.23
2.72
2.34
1.84
1.57
1.41
1.29
1.20
0.98
0.78
0.48
0.29
0.18
Si14
.00
13.4
412
.15
10.7
89.
688.
868.
247.
707.
216.
255.
324.
483.
763.
172.
712.
081.
721.
511.
371.
271.
060.
870.
560.
350.
22
P15
.00
14.4
613
.14
11.6
310
.33
9.34
8.60
8.03
7.55
6.68
5.84
5.03
4.29
3.66
3.13
2.37
1.91
1.63
1.45
1.34
1.12
0.94
0.63
0.42
0.27
S16
.00
15.4
814
.18
12.5
811
.11
9.93
9.04
8.38
7.86
7.02
6.26
5.51
4.80
4.15
3.58
2.71
2.14
1.78
1.56
1.41
1.18
1.01
0.71
0.48
0.32
Cl
17.0
016
.51
15.2
413
.60
12.0
010
.64
9.58
8.79
8.19
7.31
6.60
5.92
5.25
4.62
4.03
3.08
2.41
1.97
1.69
1.50
1.24
1.07
0.77
0.54
0.37
Cl−
118
.00
17.3
615
.76
13.8
112
.02
10.5
99.
538.
758.
167.
316.
615.
935.
264.
624.
033.
082.
411.
971.
691.
501.
241.
070.
770.
540.
37
Ar
18.0
017
.54
16.3
014
.66
12.9
611
.45
10.2
39.
288.
577.
586.
886.
265.
655.
044.
473.
472.
722.
201.
851.
621.
301.
120.
840.
600.
43
A.4 X-Ray Dispersion Corrections for Anomalous Scattering 685
Tabl
eA
.2(C
ontin
ued)
s0.
00.
050.
10.
150.
20.
250.
30.
350.
40.
50.
60.
70.
80.
91.
01.
21.
41.
61.
82.
02.
53.
04.
05.
06.
0
K+1
18.0
017
.65
16.6
815
.30
13.7
712
.29
10.9
89.
919.
067.
897.
136.
535.
975.
414.
873.
863.
052.
462.
041.
751.
371.
180.
900.
660.
48
K19
.00
18.2
116
.74
15.2
513
.74
12.2
810
.99
9.92
9.07
7.90
7.13
6.53
5.97
5.41
4.87
3.86
3.05
2.46
2.04
1.75
1.37
1.18
0.89
0.66
0.48
Ca+
218
.00
17.7
216
.94
15.7
814
.42
13.0
311
.72
10.5
99.
648.
277.
406.
776.
245.
735.
224.
243.
402.
742.
261.
911.
451.
230.
950.
720.
53
Ca
20.0
019
.09
17.3
415
.73
14.3
112
.97
11.7
210
.60
9.66
8.28
7.40
6.77
6.23
5.72
5.22
4.24
3.40
2.74
2.25
1.91
1.45
1.23
0.95
0.72
0.53
Sc21
.00
20.1
318
.36
16.6
515
.14
13.7
412
.43
11.2
510
.24
8.70
7.69
7.00
6.47
5.98
5.51
4.58
3.73
3.03
2.49
2.10
1.54
1.28
1.00
0.78
0.58
Ti+
418
.00
17.8
117
.26
16.4
215
.36
14.2
013
.01
11.8
810
.85
9.20
8.05
7.26
6.69
6.22
5.78
4.91
4.08
3.35
2.76
2.31
1.64
1.34
1.04
0.82
0.64
Ti
22.0
021
.17
19.4
117
.64
16.0
514
.58
13.2
111
.96
10.8
69.
168.
027.
256.
686.
215.
764.
884.
053.
322.
742.
301.
641.
341.
040.
830.
63
V+5
18.0
017
.84
17.3
716
.63
15.7
014
.65
13.5
412
.46
11.4
39.
708.
447.
556.
926.
436.
005.
194.
393.
653.
032.
541.
771.
411.
090.
870.
69
V23
.00
22.2
120
.48
18.6
617
.00
15.4
714
.03
12.7
111
.54
9.67
8.38
7.51
6.90
6.41
5.98
5.15
4.34
3.61
3.00
2.51
1.76
1.41
1.09
0.87
0.68
Cr+
420
.00
19.8
019
.23
18.3
417
.23
15.9
814
.68
13.4
212
.25
10.2
78.
837.
847.
146.
626.
195.
404.
633.
913.
282.
751.
901.
481.
130.
910.
73
Cr
24.0
023
.33
21.7
920
.02
18.2
516
.56
14.9
713
.52
12.2
410
.19
8.77
7.80
7.12
6.61
6.18
5.38
4.61
3.88
3.25
2.73
1.89
1.48
1.13
0.92
0.73
Mn+
223
.00
22.7
121
.87
20.6
319
.13
17.5
215
.92
14.4
213
.06
10.8
49.
248.
147.
376.
816.
375.
594.
864.
153.
512.
972.
041.
571.
170.
960.
78
Mn
25.0
024
.28
22.6
120
.76
19.0
117
.36
15.8
114
.36
13.0
410
.85
9.25
8.15
7.38
6.81
6.37
5.59
4.86
4.15
3.51
2.97
2.04
1.57
1.17
0.96
0.78
Fe+2
24.0
023
.71
22.8
921
.65
20.1
418
.51
16.8
715
.32
13.8
911
.50
9.75
8.51
7.65
7.03
6.55
5.78
5.08
4.39
3.76
3.20
2.20
1.66
1.22
1.00
0.82
Fe26
.00
25.3
023
.68
21.8
320
.05
18.3
516
.75
15.2
413
.85
11.5
19.
768.
527.
657.
036.
555.
785.
084.
403.
763.
202.
201.
661.
211.
000.
82
Co+
225
.00
24.7
223
.90
22.6
721
.17
19.5
217
.85
16.2
414
.75
12.2
210
.30
8.93
7.96
7.26
6.75
5.96
5.28
4.62
3.99
3.43
2.37
1.77
1.26
1.03
0.86
Co
27.0
026
.33
24.7
522
.90
21.1
019
.37
17.7
116
.15
14.7
012
.22
10.3
28.
947.
967.
276.
755.
965.
284.
624.
003.
432.
371.
771.
261.
040.
86
Ni+
226
.00
25.7
224
.92
23.7
022
.20
20.5
418
.85
17.2
015
.65
12.9
710
.91
9.39
8.30
7.52
6.95
6.13
5.46
4.82
4.22
3.65
2.55
1.88
1.31
1.07
0.90
Ni
28.0
027
.36
25.8
123
.98
22.1
620
.40
18.7
017
.09
15.5
912
.97
10.9
29.
408.
317.
536.
956.
125.
464.
834.
223.
662.
551.
881.
311.
070.
90
Cu+
227
.00
26.7
325
.94
24.7
423
.24
21.5
819
.86
18.1
716
.57
13.7
711
.56
9.90
8.69
7.81
7.18
6.29
5.62
5.01
4.42
3.86
2.73
2.01
1.36
1.11
0.94
Cu
29.0
028
.38
26.8
725
.05
23.2
221
.44
19.7
118
.06
16.5
013
.76
11.5
79.
918.
707.
827.
186.
295.
635.
024.
433.
872.
732.
011.
361.
110.
94
Zn+
228
.00
27.7
326
.96
25.7
724
.29
22.6
220
.89
19.1
717
.52
14.6
012
.25
10.4
59.
118.
147.
426.
465.
785.
194.
614.
072.
912.
141.
421.
140.
97
Zn
30.0
029
.40
27.9
326
.13
24.3
022
.49
20.7
419
.05
17.4
414
.58
12.2
510
.46
9.12
8.14
7.43
6.46
5.78
5.19
4.62
4.07
2.92
2.14
1.42
1.14
0.97
Ga
31.0
030
.30
28.6
726
.79
24.9
423
.19
21.5
019
.87
18.3
015
.43
13.0
211
.09
9.62
8.52
7.71
6.64
5.93
5.35
4.80
4.27
3.11
2.28
1.48
1.18
1.00
Ge
32.0
031
.28
29.5
327
.50
25.5
723
.80
22.1
520
.58
19.0
716
.25
13.7
911
.76
10.1
78.
958.
046.
846.
085.
504.
974.
453.
292.
431.
551.
221.
03
As
33.0
032
.27
30.4
628
.30
26.2
324
.39
22.7
421
.20
19.7
517
.01
14.5
612
.46
10.7
69.
438.
417.
066.
245.
645.
124.
633.
482.
591.
621.
261.
07
686 Appendix
Tabl
eA
.2(C
ontin
ued)
s0.
00.
050.
10.
150.
20.
250.
30.
350.
40.
50.
60.
70.
80.
91.
01.
21.
41.
61.
82.
02.
53.
04.
05.
06.
0
Se34
.00
33.2
731
.44
29.1
626
.95
25.0
023
.30
21.7
720
.35
17.7
115
.29
13.1
711
.38
9.94
8.82
7.31
6.40
5.78
5.27
4.79
3.67
2.75
1.71
1.30
1.10
Br
35.0
034
.29
32.4
530
.09
27.7
525
.66
23.8
722
.30
20.8
918
.33
15.9
813
.86
12.0
210
.50
9.28
7.59
6.58
5.92
5.41
4.94
3.84
2.92
1.80
1.34
1.13
Br−
136
.00
35.1
232
.91
30.2
427
.73
25.6
123
.82
22.2
720
.88
18.3
315
.99
13.8
612
.02
10.5
09.
287.
596.
585.
925.
414.
943.
842.
921.
801.
341.
13
Kr
36.0
035
.31
33.4
731
.07
28.6
126
.38
24.4
722
.84
21.4
118
.89
16.6
114
.53
12.6
711
.08
9.77
7.91
6.78
6.06
5.54
5.08
4.01
3.08
1.89
1.39
1.16
Rb+
136
.00
35.4
433
.90
31.7
529
.41
27.1
625
.17
23.4
421
.95
19.4
117
.19
15.1
513
.29
11.6
710
.29
8.27
7.01
6.22
5.66
5.21
4.18
3.25
2.00
1.44
1.19
Rb
37.0
035
.95
33.9
131
.69
29.3
827
.16
25.1
823
.45
21.9
519
.41
17.1
915
.15
13.2
911
.67
10.2
98.
277.
016.
225.
665.
214.
183.
252.
001.
441.
19
Sr+2
36.0
035
.53
34.2
032
.29
30.1
027
.91
25.8
824
.08
22.5
219
.92
17.7
215
.73
13.9
012
.25
10.8
28.
657.
266.
385.
795.
334.
333.
412.
111.
501.
23
Sr38
.00
36.8
034
.47
32.1
830
.00
27.8
825
.89
24.1
122
.54
19.9
217
.72
15.7
213
.89
12.2
510
.83
8.66
7.26
6.38
5.79
5.33
4.33
3.41
2.10
1.50
1.23
Y+3
36.0
035
.59
34.4
432
.72
30.7
028
.59
26.5
724
.74
23.1
220
.43
18.2
216
.27
14.4
712
.83
11.3
79.
077.
546.
575.
935.
454.
473.
572.
221.
561.
27
Y39
.00
37.8
235
.37
32.9
130
.65
28.5
026
.51
24.7
023
.10
20.4
318
.23
16.2
714
.47
12.8
311
.37
9.07
7.54
6.57
5.93
5.45
4.47
3.57
2.22
1.56
1.26
Zr+
436
.00
35.6
434
.62
33.0
831
.21
29.2
127
.23
25.3
923
.72
20.9
518
.71
16.7
715
.01
13.3
811
.91
9.51
7.85
6.78
6.07
5.57
4.60
3.72
2.34
1.62
1.31
Zr
40.0
038
.85
36.3
633
.76
31.3
729
.16
27.1
125
.27
23.6
320
.92
18.7
216
.80
15.0
313
.39
11.9
19.
517.
856.
786.
075.
574.
603.
722.
341.
631.
30
Nb+
536
.00
35.6
834
.77
33.3
731
.65
29.7
627
.84
26.0
124
.33
21.4
819
.19
17.2
515
.51
13.9
012
.43
9.97
8.19
7.01
6.23
5.69
4.73
3.87
2.47
1.70
1.35
Nb
41.0
039
.97
37.5
934
.90
32.3
029
.89
27.7
125
.78
24.1
121
.37
19.1
817
.30
15.5
613
.95
12.4
69.
978.
197.
016.
235.
694.
723.
862.
461.
701.
34
Mo+
636
.00
35.7
234
.90
33.6
232
.03
30.2
528
.41
26.6
124
.93
22.0
219
.67
17.7
115
.99
14.4
112
.94
10.4
38.
557.
266.
405.
824.
844.
012.
591.
771.
39
Mo
42.0
041
.00
38.6
435
.89
33.1
830
.66
28.3
926
.38
24.6
421
.82
19.6
217
.75
16.0
614
.47
12.9
910
.45
8.56
7.26
6.41
5.82
4.83
4.00
2.59
1.77
1.38
Ru
44.0
043
.06
40.7
737
.96
35.0
932
.36
29.8
827
.68
25.7
722
.73
20.4
318
.58
16.9
515
.44
14.0
011
.42
9.35
7.85
6.81
6.11
5.05
4.25
2.84
1.94
1.48
Rh
45.0
044
.09
41.8
439
.02
36.0
933
.28
30.6
928
.39
26.3
923
.20
20.8
318
.96
17.3
615
.88
14.4
711
.90
9.77
8.17
7.05
6.28
5.15
4.36
2.96
2.03
1.54
Pd+2
44.0
043
.46
41.9
339
.67
37.0
234
.27
31.6
229
.22
27.1
023
.71
21.2
219
.32
17.7
316
.28
14.9
212
.37
10.2
18.
527.
316.
465.
254.
473.
092.
121.
60
Pd46
.00
45.2
343
.18
40.3
737
.31
34.3
131
.55
29.1
126
.99
23.6
521
.21
19.3
217
.74
16.3
014
.93
12.3
810
.21
8.52
7.31
6.46
5.25
4.47
3.09
2.12
1.59
Ag+
245
.00
44.4
642
.95
40.7
038
.03
35.2
332
.51
30.0
127
.79
24.2
321
.63
19.6
818
.08
16.6
715
.33
12.8
310
.64
8.89
7.58
6.66
5.36
4.57
3.22
2.21
1.66
Ag
47.0
046
.14
43.9
741
.17
38.1
735
.22
32.4
429
.94
27.7
324
.20
21.6
319
.68
18.0
916
.67
15.3
412
.84
10.6
48.
897.
586.
665.
364.
573.
212.
211.
66
Cd+
246
.00
45.4
743
.98
41.7
439
.06
36.2
133
.43
30.8
428
.53
24.7
822
.05
20.0
318
.42
17.0
215
.73
13.2
811
.08
9.27
7.88
6.88
5.47
4.67
3.34
2.31
1.72
Cd
48.0
047
.09
44.8
141
.94
38.9
536
.03
33.2
830
.75
28.4
924
.81
22.0
920
.05
18.4
217
.02
15.7
213
.27
11.0
89.
277.
886.
885.
474.
673.
342.
311.
72
In49
.00
47.9
745
.52
42.6
039
.66
36.8
034
.09
31.5
729
.28
25.4
522
.57
20.4
318
.76
17.3
516
.07
13.6
911
.51
9.66
8.20
7.12
5.59
4.77
3.45
2.42
1.79
Sn50
.00
48.9
246
.33
43.2
940
.31
37.4
934
.83
32.3
430
.05
26.1
223
.10
20.8
319
.09
17.6
616
.40
14.0
811
.94
10.0
58.
537.
385.
714.
863.
572.
521.
86
A.4 X-Ray Dispersion Corrections for Anomalous Scattering 687
Tabl
eA
.2(C
ontin
ued)
s0.
00.
050.
10.
150.
20.
250.
30.
350.
40.
50.
60.
70.
80.
91.
01.
21.
41.
61.
82.
02.
53.
04.
05.
06.
0
Sb51
.00
49.9
047
.21
44.0
240
.95
38.1
235
.50
33.0
530
.78
26.8
123
.67
21.2
719
.44
17.9
816
.72
14.4
512
.34
10.4
58.
887.
665.
854.
953.
682.
631.
93
Te52
.00
50.8
948
.14
44.8
141
.61
38.7
236
.11
33.7
131
.47
27.5
024
.26
21.7
419
.81
18.2
917
.02
14.7
912
.73
10.8
59.
237.
955.
995.
053.
792.
732.
01
I53
.00
51.9
049
.13
45.6
942
.34
39.3
436
.70
34.3
132
.11
28.1
724
.88
22.2
520
.21
18.6
217
.31
15.1
113
.10
11.2
49.
608.
266.
155.
143.
902.
842.
09
I−1
54.0
052
.69
49.5
045
.75
42.2
839
.29
36.6
634
.30
32.1
128
.18
24.8
822
.25
20.2
118
.62
17.3
115
.11
13.1
011
.24
9.60
8.26
6.15
5.14
3.90
2.84
2.09
Xe
54.0
052
.92
50.1
446
.61
43.1
139
.99
37.2
834
.88
32.7
028
.81
25.5
022
.78
20.6
418
.96
17.6
115
.41
13.4
511
.61
9.96
8.57
6.33
5.24
4.00
2.95
2.18
Cs+
154
.00
53.0
950
.64
47.3
643
.92
40.7
337
.92
35.4
633
.27
29.4
126
.10
23.3
321
.10
19.3
317
.92
15.7
013
.78
11.9
810
.32
8.90
6.51
5.34
4.09
3.05
2.27
Cs
55.0
053
.53
50.6
147
.31
43.9
140
.74
37.9
335
.47
33.2
729
.41
26.1
023
.33
21.1
019
.33
17.9
215
.70
13.7
811
.98
10.3
28.
906.
525.
344.
093.
052.
27
Ba+
254
.00
53.2
151
.03
48.0
044
.67
41.4
738
.59
36.0
733
.83
29.9
826
.69
23.8
821
.57
19.7
218
.24
15.9
714
.09
12.3
310
.68
9.23
6.72
5.45
4.19
3.15
2.36
Ba
56.0
054
.35
51.1
347
.85
44.6
141
.48
38.6
236
.09
33.8
429
.98
26.6
823
.88
21.5
719
.72
18.2
515
.97
14.0
912
.33
10.6
89.
236.
725.
454.
183.
162.
36
La+
354
.00
53.3
051
.34
48.5
445
.36
42.1
939
.28
36.6
934
.41
30.5
327
.25
24.4
322
.06
20.1
318
.58
16.2
514
.38
12.6
611
.03
9.57
6.93
5.57
4.27
3.26
2.45
La
57.0
055
.35
51.9
848
.53
45.2
342
.10
39.2
436
.69
34.4
330
.55
27.2
624
.43
22.0
620
.13
18.5
816
.25
14.3
812
.66
11.0
39.
576.
935.
574.
273.
262.
45
Ce+
454
.00
53.3
751
.60
49.0
045
.97
42.8
839
.96
37.3
335
.00
31.0
727
.79
24.9
722
.56
20.5
618
.94
16.5
214
.66
12.9
711
.37
9.90
7.16
5.69
4.36
3.36
2.55
Ce
58.0
056
.39
53.0
549
.58
46.2
443
.06
40.1
337
.50
35.1
731
.18
27.8
224
.93
22.4
920
.51
18.9
016
.51
14.6
512
.96
11.3
69.
897.
165.
694.
353.
362.
54
Pr+3
56.0
055
.31
53.4
050
.62
47.4
144
.16
41.1
138
.37
35.9
431
.82
28.3
825
.44
22.9
420
.89
19.2
316
.77
14.9
113
.25
11.6
710
.20
7.39
5.83
4.44
3.46
2.64
Pr59
.00
57.4
454
.29
50.9
647
.62
44.3
341
.26
38.5
136
.05
31.8
828
.38
25.4
022
.89
20.8
519
.21
16.7
714
.91
13.2
411
.65
10.1
97.
385.
834.
443.
462.
63
Nd+
357
.00
56.3
254
.43
51.6
748
.45
45.1
742
.07
39.2
536
.75
32.5
128
.98
25.9
623
.40
21.2
819
.56
17.0
315
.16
13.5
211
.97
10.5
17.
635.
974.
523.
552.
73
Nd
60.0
058
.47
55.3
452
.02
48.6
645
.34
42.2
139
.38
36.8
632
.56
28.9
825
.92
23.3
521
.24
19.5
417
.02
15.1
613
.52
11.9
610
.49
7.62
5.97
4.52
3.55
2.73
Sm+3
59.0
058
.34
56.5
053
.77
50.5
647
.23
44.0
341
.09
38.4
533
.95
30.2
227
.05
24.3
522
.10
20.2
617
.55
15.6
314
.03
12.5
311
.10
8.12
6.28
4.68
3.73
2.92
Sm62
.00
60.5
357
.46
54.1
550
.77
47.3
944
.16
41.2
038
.55
34.0
030
.23
27.0
224
.31
22.0
620
.23
17.5
515
.63
14.0
312
.52
11.0
98.
116.
284.
673.
742.
91
Eu+
360
.00
59.3
557
.53
54.8
351
.63
48.2
845
.03
42.0
439
.33
34.7
030
.87
27.6
224
.84
22.5
220
.62
17.8
215
.86
14.2
712
.79
11.3
88.
386.
454.
763.
823.
01
Eu
63.0
061
.55
58.5
255
.22
51.8
348
.43
45.1
642
.14
39.4
234
.76
30.8
827
.59
24.8
122
.49
20.5
917
.81
15.8
614
.27
12.7
911
.37
8.37
6.45
4.75
3.82
3.01
Gd+
361
.00
60.3
658
.56
55.8
852
.69
49.3
346
.05
42.9
940
.22
35.4
831
.54
28.2
025
.35
22.9
620
.99
18.0
916
.09
14.5
013
.04
11.6
58.
636.
624.
843.
903.
10
Gd
64.0
062
.55
59.4
155
.98
52.5
749
.20
45.9
642
.95
40.2
135
.48
31.5
528
.21
25.3
622
.96
21.0
018
.09
16.0
914
.50
13.0
411
.65
8.63
6.63
4.83
3.91
3.10
Tb
65.0
063
.60
60.6
457
.37
53.9
850
.54
47.2
044
.08
41.2
436
.33
32.2
528
.79
25.8
423
.38
21.3
518
.36
16.3
214
.72
13.2
811
.91
8.88
6.81
4.92
3.99
3.19
Dy
66.0
064
.63
61.6
958
.44
55.0
651
.61
48.2
445
.08
42.1
737
.15
32.9
729
.41
26.3
923
.85
21.7
518
.65
16.5
414
.93
13.5
112
.16
9.14
7.00
5.00
4.07
3.28
Ho
67.0
065
.65
62.7
559
.51
56.1
452
.69
49.2
946
.08
43.1
337
.99
33.7
030
.06
26.9
524
.33
22.1
618
.94
16.7
715
.14
13.7
412
.41
9.39
7.19
5.09
4.15
3.37
688 Appendix
Tabl
eA
.2(C
ontin
ued)
s0.
00.
050.
10.
150.
20.
250.
30.
350.
40.
50.
60.
70.
80.
91.
01.
21.
41.
61.
82.
02.
53.
04.
05.
06.
0
Er
68.0
066
.68
63.8
160
.59
57.2
353
.77
50.3
547
.10
44.0
938
.84
34.4
530
.72
27.5
324
.83
22.5
819
.25
17.0
015
.35
13.9
512
.64
9.65
7.39
5.18
4.22
3.45
Tm
69.0
067
.70
64.8
661
.66
58.3
154
.85
51.4
148
.13
45.0
739
.72
35.2
231
.39
28.1
225
.34
23.0
319
.56
17.2
415
.55
14.1
612
.87
9.90
7.60
5.27
4.30
3.54
Yb
70.0
068
.72
65.9
162
.73
59.3
955
.93
52.4
849
.16
46.0
740
.60
36.0
032
.08
28.7
325
.87
23.4
819
.89
17.4
815
.75
14.3
613
.08
10.1
47.
815.
374.
373.
62
Lu
71.0
069
.70
66.7
863
.47
60.1
156
.70
53.3
150
.04
46.9
541
.44
36.7
732
.79
29.3
726
.45
23.9
820
.25
17.7
315
.96
14.5
513
.29
10.3
98.
035.
474.
443.
70
Hf
72.0
070
.72
67.7
464
.30
60.8
657
.44
54.0
850
.83
47.7
642
.24
37.5
433
.51
30.0
327
.05
24.5
120
.63
18.0
016
.17
14.7
513
.50
10.6
48.
255.
584.
513.
78
Ta73
.00
71.7
468
.73
65.1
961
.64
58.1
754
.81
51.5
848
.52
43.0
138
.29
34.2
230
.70
27.6
525
.05
21.0
318
.28
16.3
814
.94
13.7
010
.88
8.47
5.69
4.58
3.85
W74
.00
72.7
669
.74
66.1
162
.46
58.9
255
.53
52.2
949
.24
43.7
539
.02
34.9
331
.37
28.2
825
.61
21.4
518
.58
16.6
015
.12
13.8
911
.11
8.70
5.81
4.65
3.93
Re
75.0
073
.78
70.7
667
.06
63.3
159
.70
56.2
552
.99
49.9
444
.45
39.7
235
.62
32.0
428
.90
26.1
821
.89
18.8
916
.82
15.3
114
.08
11.3
48.
935.
944.
734.
00
Os
76.0
074
.80
71.7
968
.04
64.2
060
.49
56.9
853
.69
50.6
245
.12
40.4
036
.30
32.7
029
.54
26.7
722
.34
19.2
217
.06
15.5
014
.26
11.5
69.
166.
074.
804.
07
Ir77
.00
75.8
372
.85
69.0
765
.14
61.3
457
.75
54.4
051
.29
45.7
741
.05
36.9
533
.35
30.1
727
.36
22.8
219
.57
17.3
115
.70
14.4
411
.78
9.39
6.21
4.87
4.14
Pt78
.00
76.9
174
.08
70.3
466
.32
62.3
558
.60
55.1
251
.93
46.3
541
.65
37.5
834
.00
30.8
127
.97
23.3
219
.93
17.5
715
.90
14.6
111
.99
9.62
6.35
4.94
4.21
Au
79.0
077
.94
75.1
571
.40
67.3
263
.27
59.4
355
.87
52.6
246
.97
42.2
538
.19
34.6
231
.43
28.5
723
.83
20.3
217
.85
16.1
014
.79
12.1
99.
856.
505.
024.
28
Hg
80.0
078
.90
76.0
372
.22
68.1
164
.06
60.2
156
.64
53.3
647
.64
42.8
738
.79
35.2
232
.02
29.1
524
.34
20.7
218
.14
16.3
214
.97
12.3
810
.07
6.66
5.10
4.34
Tl
81.0
079
.75
76.6
972
.88
68.8
664
.86
61.0
257
.43
54.1
248
.31
43.4
839
.38
35.8
032
.60
29.7
324
.86
21.1
418
.45
16.5
515
.15
12.5
710
.29
6.82
5.19
4.41
Pb82
.00
80.6
777
.46
73.5
469
.52
65.5
761
.79
58.2
154
.88
49.0
144
.10
39.9
536
.36
33.1
730
.30
25.3
921
.58
18.7
816
.79
15.3
412
.74
10.5
06.
985.
274.
47
Bi
83.0
081
.63
78.2
874
.23
70.1
666
.23
62.4
958
.95
55.6
349
.71
44.7
340
.53
36.9
233
.72
30.8
525
.92
22.0
319
.12
17.0
415
.53
12.9
210
.71
7.16
5.37
4.54
Po84
.00
82.6
179
.16
74.9
670
.79
66.8
563
.15
59.6
556
.36
50.4
245
.37
41.1
137
.46
34.2
631
.39
26.4
422
.48
19.4
817
.31
15.7
313
.08
10.9
17.
335.
464.
60
At
85.0
083
.63
80.1
675
.84
71.5
167
.49
63.7
660
.29
57.0
351
.12
46.0
441
.71
38.0
234
.79
31.9
226
.96
22.9
519
.86
17.5
915
.94
13.2
511
.11
7.52
5.56
4.67
Rn
86.0
084
.65
81.1
876
.76
72.2
968
.14
64.3
760
.90
57.6
751
.81
46.7
042
.32
38.5
835
.32
32.4
327
.47
23.4
220
.25
17.8
916
.17
13.4
011
.30
7.70
5.66
4.73
Fr87
.00
85.2
981
.68
77.4
573
.06
68.8
765
.03
61.5
358
.29
52.4
647
.36
42.9
439
.14
35.8
532
.94
27.9
723
.89
20.6
518
.20
16.4
013
.56
11.4
97.
895.
774.
80
Ra
88.0
086
.11
82.2
278
.01
73.7
669
.61
65.7
362
.18
58.9
253
.09
47.9
943
.55
39.7
136
.38
33.4
528
.46
24.3
521
.05
18.5
216
.65
13.7
211
.66
8.08
5.89
4.87
Ac
89.0
087
.07
82.9
878
.61
74.3
570
.24
66.3
962
.83
59.5
653
.73
48.6
244
.16
40.2
736
.91
33.9
628
.95
24.8
221
.47
18.8
616
.91
13.8
811
.84
8.27
6.01
4.95
Th
90.0
088
.07
83.8
479
.27
74.9
370
.83
67.0
063
.46
60.1
954
.36
49.2
544
.76
40.8
437
.44
34.4
629
.43
25.2
921
.88
19.2
017
.18
14.0
412
.00
8.46
6.14
5.02
Pa91
.00
89.1
385
.04
80.5
476
.11
71.8
467
.84
64.1
560
.79
54.8
749
.76
45.2
941
.38
37.9
834
.99
29.9
425
.76
22.3
119
.55
17.4
514
.20
12.1
68.
656.
265.
10
U92
.00
90.1
686
.08
81.5
477
.04
72.7
068
.61
64.8
561
.43
55.4
450
.31
45.8
341
.92
38.5
035
.50
30.4
326
.23
22.7
319
.92
17.7
414
.36
12.3
18.
846.
405.
18
A.5 Atomic Form Factors 689
A.5 Atomic Form Factors for 200 keV Electrons and Procedurefor Conversion to Other Voltages
Electron form factors can be obtained from the x-ray atomic form factors, fx(s),with the Mott formula (4.113) as:
fel0(s) = 1
s2
(Z − fx(s)
),
where the fx(s) are the values listed in Table A.2. Conversion of fel0(s) to units ofÅ requires multiplication by the factor given in (4.113):
2me2
(4π�)2= 2.3933 × 10−2,
where the extra factor of (4π)−2 originates with the definition s ≡ sin θ/λ (s isconverted to the Δk used in the text by multiplication by 4π ).
For an incident electron with velocity, v, it is necessary to multiply fel0(s) by therelativistic mass correction factor, γ :
γ ≡ 1√
(1 − (v/c)2,
so that:
fel(s) = (2.3933 × 10−2)γfel0(s).
For high-energy electrons of known energy E, the following expression is usuallymore convenient:
γ = 1 + E
mec2� 1 + E [keV]
511.
Form factors for 200 keV electrons are given in the following table. They werederived from the previous table of x-ray atomic form factors, fx(s), calculated witha Dirac–Fock method by D. Rez, P. Rez, I. Grant, Acta Crystallogr. A50, 481 (1994).Form factors at other electron energies can be obtained from x-ray form factors bythe procedure above.
More conveniently, electron form factors for other accelerating voltages can beobtained from the values in the following table for 200 keV electrons by multiplyingby the ratio of relativistic factors. For example, for 100 keV electrons the values inthe table should be multiplied by the constant factor:
γ100
γ200= 1 + 100/511
1 + 200/511= 0.859,
so the values for 100 keV electrons are smaller than those in the table.The column headings in Table A.3 are s ≡ sin θ/λ, in units of Å−1, Δk ≡ 4πs.Table A.3 entries are for 200 keV electrons. The units for all entries are Å. The
column headings are s ≡ sin θ/λ, in units of Å−1. This diffraction vector, s, is con-verted to the Δk used in the text by multiplication by 4π .
690 Appendix
Tabl
eA
.3A
tom
icfo
rmfa
ctor
sfo
r20
0ke
Vel
ectr
ons
s0.
00.
050.
10.
150.
20.
250.
30.
350.
40.
50.
60.
70.
80.
91.
01.
21.
41.
61.
82.
02.
53.
04.
06.
0
He
0.58
10.
569
0.54
00.
498
0.44
80.
397
0.34
70.
302
0.26
20.
198
0.15
20.
119
0.09
50.
077
0.06
30.
045
0.03
30.
026
0.02
00.
017
0.01
10.
007
0.00
40.
002
Li+
1–
13.5
43.
542
1.68
61.
030
0.72
00.
546
0.43
60.
361
0.26
30.
202
0.16
00.
130
0.10
70.
089
0.06
50.
049
0.03
80.
030
0.02
50.
016
0.01
10.
006
0.00
3
Li
4.53
03.
885
2.60
91.
621
1.04
70.
732
0.55
00.
436
0.36
00.
262
0.20
10.
160
0.12
90.
107
0.08
90.
065
0.04
90.
038
0.03
00.
025
0.01
60.
011
0.00
60.
003
Be+
2–
26.7
56.
772
3.07
01.
773
1.17
00.
841
0.64
10.
509
0.35
10.
261
0.20
30.
164
0.13
50.
113
0.08
30.
063
0.04
90.
039
0.03
20.
021
0.01
50.
008
0.00
4
Be
4.22
73.
895
3.10
62.
272
1.61
41.
157
0.85
30.
652
0.51
60.
351
0.25
90.
202
0.16
20.
134
0.11
20.
082
0.06
30.
049
0.03
90.
032
0.02
10.
015
0.00
80.
004
B3.
875
3.66
03.
123
2.48
81.
913
1.45
71.
117
0.87
00.
691
0.46
30.
333
0.25
30.
200
0.16
40.
136
0.10
00.
076
0.06
00.
048
0.04
00.
026
0.01
80.
010
0.00
5
C3.
438
3.29
82.
940
2.47
92.
020
1.62
01.
295
1.04
00.
843
0.57
50.
413
0.31
10.
243
0.19
70.
163
0.11
80.
089
0.07
00.
057
0.04
70.
031
0.02
20.
012
0.00
6
N3.
066
2.97
02.
721
2.38
32.
024
1.68
81.
397
1.15
50.
958
0.67
30.
490
0.37
00.
289
0.23
20.
191
0.13
70.
103
0.08
10.
065
0.05
40.
036
0.02
50.
014
0.00
6
O2.
760
2.69
22.
512
2.25
91.
977
1.69
91.
446
1.22
51.
039
0.75
40.
560
0.42
70.
335
0.26
90.
221
0.15
70.
118
0.09
20.
074
0.06
10.
040
0.02
90.
016
0.00
7
O−1
–−9
.391
0.25
01.
636
1.80
01.
659
1.44
41.
233
1.04
60.
757
0.56
10.
427
0.33
50.
269
0.22
10.
157
0.11
80.
092
0.07
40.
061
0.04
00.
029
0.01
60.
007
O−2
–−2
1.17
−1.7
901.
149
1.69
71.
652
1.45
71.
244
1.05
30.
759
0.56
20.
427
0.33
50.
269
0.22
10.
157
0.11
80.
092
0.07
40.
061
0.04
00.
029
0.01
60.
007
F2.
507
2.45
52.
322
2.12
81.
905
1.67
61.
458
1.26
21.
090
0.81
50.
619
0.47
90.
378
0.30
50.
250
0.17
70.
133
0.10
30.
083
0.06
80.
045
0.03
20.
018
0.00
8
F−1
–−9
.784
−0.0
601.
426
1.68
21.
611
1.44
51.
264
1.09
50.
819
0.62
00.
479
0.37
80.
305
0.25
00.
177
0.13
30.
103
0.08
30.
068
0.04
50.
032
0.01
80.
008
Ne
2.29
52.
255
2.15
32.
002
1.82
31.
633
1.44
81.
275
1.11
90.
860
0.66
60.
523
0.41
80.
339
0.27
90.
198
0.14
80.
115
0.09
20.
076
0.05
00.
035
0.02
00.
009
Na+
1–
14.8
74.
837
2.91
82.
183
1.78
41.
516
1.31
41.
149
0.89
30.
703
0.56
00.
452
0.37
00.
307
0.21
90.
164
0.12
70.
101
0.08
30.
054
0.03
90.
022
0.01
0
Na
6.59
35.
742
4.12
02.
916
2.21
51.
799
1.52
11.
315
1.14
90.
892
0.70
30.
560
0.45
20.
370
0.30
70.
219
0.16
40.
127
0.10
10.
083
0.05
40.
039
0.02
20.
010
Mg+
2–
27.7
87.
781
4.04
42.
701
2.04
51.
658
1.39
61.
203
0.92
80.
735
0.59
20.
482
0.39
80.
332
0.23
90.
179
0.13
90.
111
0.09
10.
059
0.04
20.
024
0.01
1
Mg
7.20
46.
544
5.07
63.
691
2.71
42.
087
1.68
31.
407
1.20
70.
927
0.73
40.
591
0.48
20.
398
0.33
20.
239
0.17
90.
139
0.11
10.
091
0.05
90.
042
0.02
40.
011
Al+
3–
40.8
410
.86
5.28
83.
317
2.38
31.
856
1.51
91.
285
0.97
30.
768
0.62
10.
509
0.42
30.
355
0.25
80.
194
0.15
10.
120
0.09
80.
064
0.04
50.
026
0.01
2
Al
8.16
27.
461
5.88
74.
347
3.19
52.
414
1.89
51.
543
1.29
60.
972
0.76
60.
619
0.50
80.
423
0.35
50.
258
0.19
40.
151
0.12
00.
098
0.06
40.
045
0.02
60.
012
Si8.
005
7.46
76.
177
4.76
73.
597
2.73
72.
133
1.71
21.
413
1.03
30.
803
0.64
70.
533
0.44
50.
376
0.27
60.
209
0.16
30.
130
0.10
60.
069
0.04
90.
028
0.01
3
P7.
616
7.20
96.
191
4.98
33.
887
3.01
62.
366
1.89
31.
550
1.10
80.
847
0.67
80.
557
0.46
60.
395
0.29
20.
222
0.17
40.
139
0.11
40.
074
0.05
20.
030
0.01
4
S7.
185
6.87
26.
070
5.05
64.
070
3.23
42.
574
2.07
11.
694
1.19
60.
901
0.71
30.
583
0.48
70.
414
0.30
70.
236
0.18
50.
148
0.12
10.
079
0.05
50.
032
0.01
5
Cl
6.75
76.
512
5.87
55.
032
4.16
63.
389
2.74
42.
232
1.83
41.
291
0.96
20.
753
0.61
10.
509
0.43
20.
322
0.24
80.
195
0.15
70.
129
0.08
40.
059
0.03
40.
015
Cl−
1–
−4.8
334.
142
4.72
14.
145
3.41
42.
765
2.24
31.
839
1.29
10.
961
0.75
20.
611
0.50
90.
432
0.32
20.
248
0.19
50.
157
0.12
90.
084
0.05
90.
034
0.01
5
Ar
6.36
06.
165
5.65
24.
950
4.19
63.
489
2.87
62.
370
1.96
41.
388
1.02
90.
798
0.64
30.
533
0.45
10.
336
0.26
00.
206
0.16
60.
136
0.08
90.
062
0.03
60.
016
K+1
–17
.99
7.72
25.
470
4.35
43.
577
2.96
62.
471
2.06
81.
479
1.09
80.
847
0.67
80.
559
0.47
10.
350
0.27
10.
215
0.17
40.
144
0.09
40.
066
0.03
80.
017
A.5 Atomic Form Factors 691
Tabl
eA
.3(C
ontin
ued)
s0.
00.
050.
10.
150.
20.
250.
30.
350.
40.
50.
60.
70.
80.
91.
01.
21.
41.
61.
82.
02.
53.
04.
06.
0
K12
.38
10.5
77.
533
5.55
04.
381
3.58
12.
965
2.46
92.
067
1.47
91.
098
0.84
80.
678
0.55
90.
471
0.35
00.
271
0.21
50.
174
0.14
40.
094
0.06
60.
038
0.01
7
Ca+
2–
30.3
410
.20
6.24
54.
644
3.71
53.
062
2.55
92.
156
1.56
21.
166
0.89
90.
716
0.58
70.
492
0.36
40.
282
0.22
50.
182
0.15
10.
099
0.06
90.
040
0.01
8
Ca
13.6
912
.08
8.87
06.
319
4.73
43.
745
3.06
52.
555
2.15
21.
561
1.16
60.
899
0.71
60.
587
0.49
20.
364
0.28
20.
225
0.18
20.
151
0.09
90.
069
0.04
00.
018
Sc12
.87
11.5
58.
795
6.44
24.
880
3.86
93.
170
2.64
92.
240
1.63
91.
231
0.95
10.
756
0.61
70.
516
0.38
00.
293
0.23
40.
190
0.15
70.
104
0.07
30.
042
0.01
9
Ti+
4–
55.8
115
.78
8.26
45.
526
4.15
83.
326
2.75
12.
320
1.70
61.
291
1.00
10.
796
0.64
90.
540
0.39
50.
305
0.24
30.
198
0.16
40.
108
0.07
60.
044
0.02
0
Ti
12.1
411
.02
8.61
76.
459
4.95
73.
956
3.25
42.
730
2.31
81.
711
1.29
41.
003
0.79
70.
649
0.54
10.
396
0.30
50.
243
0.19
80.
164
0.10
80.
076
0.04
40.
020
V+5
–68
.76
18.7
59.
421
6.07
54.
451
3.49
92.
867
2.40
71.
771
1.34
71.
050
0.83
70.
681
0.56
60.
412
0.31
60.
252
0.20
50.
170
0.11
30.
080
0.04
60.
021
V11
.50
10.5
38.
404
6.42
34.
993
4.01
43.
319
2.79
72.
385
1.77
61.
352
1.05
20.
838
0.68
20.
567
0.41
30.
317
0.25
20.
206
0.17
10.
113
0.08
00.
046
0.02
1
Cr+
4–
55.9
215
.88
8.37
35.
639
4.27
53.
447
2.87
52.
446
1.82
81.
403
1.09
90.
877
0.71
50.
593
0.43
00.
329
0.26
10.
213
0.17
70.
118
0.08
30.
048
0.02
2
Cr
9.67
68.
946
7.37
35.
896
4.78
33.
965
3.34
22.
849
2.44
91.
839
1.40
91.
101
0.87
80.
715
0.59
30.
431
0.32
90.
262
0.21
30.
177
0.11
80.
083
0.04
80.
022
Mn+
2–
30.5
510
.41
6.47
14.
891
3.98
63.
358
2.87
52.
485
1.88
71.
458
1.14
60.
917
0.74
80.
620
0.44
90.
342
0.27
10.
221
0.18
30.
122
0.08
70.
050
0.02
2
Mn
10.4
09.
649
7.95
06.
270
4.98
64.
069
3.40
12.
893
2.49
01.
885
1.45
71.
145
0.91
70.
748
0.62
10.
449
0.34
20.
271
0.22
10.
183
0.12
20.
087
0.05
00.
022
Fe+2
–30
.49
10.3
66.
442
4.87
93.
991
3.37
72.
904
2.52
11.
931
1.50
41.
189
0.95
50.
780
0.64
80.
468
0.35
50.
281
0.22
90.
190
0.12
70.
090
0.05
20.
023
Fe9.
934
9.26
17.
726
6.17
24.
958
4.07
43.
424
2.92
62.
529
1.93
01.
502
1.18
80.
955
0.78
00.
648
0.46
80.
355
0.28
10.
229
0.19
00.
127
0.09
00.
052
0.02
3
Co+
2–
30.4
210
.31
6.40
34.
857
3.98
63.
385
2.92
42.
549
1.96
91.
544
1.22
80.
991
0.81
10.
674
0.48
70.
369
0.29
10.
236
0.19
60.
131
0.09
30.
054
0.02
4
Co
9.50
38.
899
7.50
56.
064
4.91
64.
067
3.43
62.
949
2.55
91.
969
1.54
31.
227
0.99
10.
811
0.67
40.
487
0.36
90.
291
0.23
60.
196
0.13
10.
093
0.05
40.
024
Ni+
2–
30.3
510
.25
6.35
74.
829
3.97
33.
387
2.93
72.
571
2.00
11.
581
1.26
51.
025
0.84
20.
701
0.50
60.
383
0.30
10.
244
0.20
30.
136
0.09
70.
056
0.02
5
Ni
9.10
88.
562
7.29
05.
953
4.86
64.
051
3.44
02.
965
2.58
32.
002
1.58
01.
264
1.02
40.
842
0.70
10.
506
0.38
30.
301
0.24
40.
203
0.13
60.
097
0.05
60.
025
Cu+
2–
30.2
810
.18
6.30
84.
795
3.95
53.
382
2.94
32.
586
2.02
91.
614
1.29
81.
057
0.87
10.
727
0.52
50.
397
0.31
20.
253
0.20
90.
140
0.10
00.
058
0.02
6
Cu
8.74
48.
248
7.08
45.
839
4.81
04.
029
3.43
62.
974
2.60
12.
030
1.61
31.
297
1.05
60.
871
0.72
70.
525
0.39
70.
312
0.25
30.
209
0.14
00.
100
0.05
80.
026
Zn+
2–
30.2
110
.12
6.25
64.
758
3.93
23.
372
2.94
52.
597
2.05
11.
642
1.32
91.
087
0.89
90.
752
0.54
40.
411
0.32
30.
261
0.21
60.
144
0.10
30.
059
0.02
7
Zn
8.40
87.
955
6.88
65.
724
4.74
94.
000
3.42
72.
978
2.61
42.
054
1.64
21.
328
1.08
60.
899
0.75
20.
544
0.41
10.
323
0.26
10.
216
0.14
40.
103
0.05
90.
027
Ga
9.93
69.
263
7.75
46.
238
5.04
24.
162
3.51
63.
027
2.64
32.
074
1.66
31.
353
1.11
20.
924
0.77
50.
563
0.42
60.
334
0.26
90.
223
0.14
90.
106
0.06
10.
028
Ge
10.2
69.
654
8.21
76.
658
5.35
44.
369
3.64
43.
104
2.69
12.
098
1.68
41.
375
1.13
60.
948
0.79
80.
582
0.44
00.
345
0.27
80.
229
0.15
30.
109
0.06
30.
029
As
10.2
59.
732
8.45
06.
961
5.63
44.
587
3.79
73.
207
2.75
82.
130
1.70
61.
396
1.15
70.
969
0.81
90.
600
0.45
50.
356
0.28
60.
236
0.15
70.
113
0.06
50.
030
Se10
.11
9.66
48.
541
7.16
35.
867
4.79
43.
960
3.32
52.
841
2.17
01.
730
1.41
61.
177
0.98
90.
838
0.61
70.
469
0.36
70.
295
0.24
30.
162
0.11
60.
067
0.03
0
Br
9.85
19.
473
8.50
57.
264
6.03
64.
975
4.11
93.
451
2.93
62.
221
1.75
91.
437
1.19
61.
007
0.85
70.
634
0.48
30.
378
0.30
40.
250
0.16
60.
119
0.06
90.
031
692 Appendix
Tabl
eA
.3(C
ontin
ued)
s0.
00.
050.
10.
150.
20.
250.
30.
350.
40.
50.
60.
70.
80.
91.
01.
21.
41.
61.
82.
02.
53.
04.
06.
0
Br−
1–
−1.5
546.
972
7.04
96.
049
5.00
54.
137
3.45
92.
939
2.22
01.
759
1.43
61.
195
1.00
70.
857
0.63
40.
483
0.37
80.
304
0.25
00.
166
0.11
90.
069
0.03
1
Kr
9.57
49.
251
8.41
37.
301
6.15
65.
126
4.26
63.
578
3.03
82.
279
1.79
31.
459
1.21
41.
025
0.87
30.
650
0.49
60.
389
0.31
30.
257
0.17
00.
122
0.07
10.
032
Rb+
1–
20.8
310
.33
7.76
86.
321
5.24
14.
378
3.68
63.
133
2.34
31.
832
1.48
51.
233
1.04
20.
889
0.66
40.
510
0.40
00.
322
0.26
50.
175
0.12
50.
073
0.03
3
Rb
16.2
413
.98
10.2
87.
856
6.34
15.
240
4.37
53.
683
3.13
22.
343
1.83
31.
485
1.23
31.
042
0.88
90.
664
0.51
00.
400
0.32
20.
265
0.17
50.
125
0.07
30.
033
Sr+2
–32
.96
12.6
48.
450
6.57
45.
377
4.48
53.
783
3.22
22.
409
1.87
61.
514
1.25
41.
059
0.90
50.
679
0.52
20.
411
0.33
10.
272
0.17
90.
128
0.07
50.
034
Sr18
.09
15.9
211
.77
8.61
16.
659
5.39
24.
480
3.77
63.
217
2.40
81.
876
1.51
41.
254
1.05
90.
905
0.67
90.
522
0.41
10.
331
0.27
20.
179
0.12
80.
075
0.03
4
Y+3
–45
.40
15.2
09.
289
6.91
15.
546
4.59
83.
877
3.30
62.
474
1.92
21.
545
1.27
61.
076
0.92
00.
692
0.53
40.
422
0.34
00.
279
0.18
40.
131
0.07
70.
035
Y17
.52
15.7
412
.09
9.00
96.
955
5.59
24.
623
3.88
83.
310
2.47
41.
921
1.54
41.
276
1.07
60.
920
0.69
20.
534
0.42
20.
340
0.27
90.
184
0.13
10.
077
0.03
5
Zr+
4–
58.0
517
.91
10.2
47.
319
5.75
14.
725
3.97
33.
387
2.53
81.
970
1.57
91.
301
1.09
40.
936
0.70
50.
546
0.43
20.
349
0.28
70.
189
0.13
40.
078
0.03
6
Zr
16.8
515
.34
12.1
39.
233
7.18
45.
778
4.76
84.
005
3.40
62.
542
1.96
81.
577
1.29
91.
094
0.93
50.
705
0.54
60.
432
0.34
90.
287
0.18
90.
134
0.07
80.
036
Nb+
5–
70.8
320
.74
11.2
87.
785
5.99
14.
868
4.07
43.
469
2.60
12.
018
1.61
41.
326
1.11
40.
951
0.71
80.
557
0.44
20.
357
0.29
40.
193
0.13
70.
080
0.03
7
Nb
14.8
913
.77
11.3
49.
026
7.24
45.
921
4.91
74.
136
3.51
62.
615
2.01
81.
611
1.32
41.
112
0.95
00.
717
0.55
70.
442
0.35
70.
294
0.19
30.
137
0.08
00.
037
Mo+
6–
83.7
023
.66
12.4
08.
301
6.26
25.
029
4.18
33.
553
2.66
22.
066
1.65
11.
353
1.13
40.
968
0.73
00.
568
0.45
20.
366
0.30
10.
198
0.14
10.
082
0.03
8
Mo
14.3
113
.33
11.1
89.
044
7.34
26.
041
5.03
74.
246
3.61
32.
688
2.07
01.
648
1.35
01.
132
0.96
60.
730
0.56
80.
452
0.36
60.
301
0.19
80.
141
0.08
20.
038
Ru
13.2
912
.52
10.7
68.
947
7.42
16.
201
5.22
54.
436
3.79
42.
834
2.18
01.
727
1.40
71.
174
0.99
90.
753
0.58
90.
470
0.38
20.
315
0.20
80.
147
0.08
60.
039
Rh
12.8
312
.13
10.5
48.
854
7.41
66.
245
5.29
44.
515
3.87
32.
904
2.23
61.
770
1.43
81.
197
1.01
70.
765
0.59
80.
479
0.39
00.
322
0.21
20.
150
0.08
70.
040
Pd+2
–33
.89
13.5
79.
366
7.47
56.
251
5.31
94.
562
3.93
42.
970
2.29
21.
813
1.47
11.
222
1.03
50.
778
0.60
80.
488
0.39
80.
329
0.21
70.
154
0.08
90.
041
Pd10
.52
10.2
09.
388
8.32
77.
235
6.22
75.
345
4.59
13.
956
2.97
72.
293
1.81
31.
470
1.22
11.
035
0.77
80.
608
0.48
80.
398
0.32
90.
217
0.15
40.
089
0.04
1
Ag+
2–
33.7
813
.49
9.32
07.
465
6.27
25.
361
4.61
93.
998
3.03
32.
347
1.85
71.
505
1.24
71.
054
0.79
00.
618
0.49
60.
405
0.33
60.
222
0.15
70.
091
0.04
2
Ag
12.0
211
.43
10.0
88.
626
7.34
86.
279
5.38
74.
639
4.01
03.
036
2.34
71.
857
1.50
41.
247
1.05
40.
790
0.61
80.
496
0.40
50.
336
0.22
20.
157
0.09
10.
042
Cd+
2–
33.6
713
.40
9.26
37.
443
6.27
95.
392
4.66
54.
053
3.09
32.
400
1.90
11.
539
1.27
31.
075
0.80
30.
627
0.50
40.
412
0.34
20.
227
0.16
00.
093
0.04
3
Cd
12.8
012
.16
10.6
48.
977.
535
6.37
85.
448
4.68
94.
060
3.08
92.
397
1.90
01.
539
1.27
41.
075
0.80
30.
627
0.50
40.
412
0.34
20.
227
0.16
00.
093
0.04
3
In14
.74
13.7
611
.60
9.47
7.77
96.
500
5.51
74.
739
4.10
53.
137
2.44
41.
942
1.57
41.
301
1.09
60.
817
0.63
70.
512
0.41
90.
349
0.23
10.
164
0.09
50.
044
Sn15
.36
14.4
112
.22
9.93
8.07
16.
668
5.61
44.
801
4.15
33.
180
2.48
81.
982
1.60
81.
329
1.11
90.
831
0.64
70.
520
0.42
60.
355
0.23
60.
167
0.09
70.
045
Sb15
.55
14.6
912
.62
10.3
38.
369
6.86
45.
736
4.87
94.
208
3.22
22.
528
2.02
01.
642
1.35
81.
142
0.84
50.
657
0.52
70.
433
0.36
10.
241
0.17
00.
098
0.04
5
Te15
.55
14.7
712
.86
10.6
48.
649
7.07
35.
878
4.97
24.
272
3.26
32.
566
2.05
61.
675
1.38
61.
165
0.86
00.
667
0.53
50.
440
0.36
70.
245
0.17
40.
100
0.04
6
I15
.28
14.6
012
.90
10.8
28.
878
7.27
66.
032
5.08
14.
348
3.30
72.
601
2.09
01.
706
1.41
31.
188
0.87
60.
678
0.54
30.
446
0.37
20.
250
0.17
70.
102
0.04
7
A.5 Atomic Form Factors 693
Tabl
eA
.3(C
ontin
ued)
s0.
00.
050.
10.
150.
20.
250.
30.
350.
40.
50.
60.
70.
80.
91.
01.
21.
41.
61.
82.
02.
53.
04.
06.
0
I−1
–4.
0611
.66
10.7
38.
921
7.30
76.
044
5.08
44.
348
3.30
62.
601
2.09
01.
706
1.41
31.
188
0.87
60.
678
0.54
30.
446
0.37
20.
250
0.17
70.
102
0.04
7
Xe
14.9
814
.38
12.8
710
.94
9.06
77.
465
6.18
85.
199
4.43
43.
355
2.63
62.
121
1.73
61.
440
1.21
20.
892
0.68
90.
551
0.45
30.
378
0.25
40.
180
0.10
40.
048
Cs+
1–
25.4
814
.50
11.3
09.
226
7.60
56.
320
5.31
24.
523
3.40
82.
673
2.15
21.
764
1.46
61.
235
0.90
90.
700
0.56
00.
459
0.38
40.
258
0.18
40.
106
0.04
9
Cs
22.7
519
.57
14.6
111
.39
9.23
37.
600
6.31
75.
310
4.52
33.
409
2.67
32.
152
1.76
41.
466
1.23
50.
909
0.70
00.
560
0.45
90.
384
0.25
80.
184
0.10
60.
049
Ba+
2–
37.2
216
.54
11.8
49.
431
7.74
26.
441
5.41
94.
613
3.46
52.
711
2.18
31.
791
1.49
11.
257
0.92
60.
712
0.56
80.
466
0.38
90.
263
0.18
70.
108
0.05
0
Ba
25.2
022
.00
16.2
112
.06
9.48
57.
737
6.43
05.
412
4.61
13.
466
2.71
22.
183
1.79
11.
491
1.25
70.
926
0.71
20.
568
0.46
60.
389
0.26
30.
187
0.10
80.
050
La+
3–
49.3
418
.84
12.5
39.
694
7.88
96.
557
5.52
14.
701
3.52
52.
752
2.21
41.
818
1.51
61.
279
0.94
20.
724
0.57
70.
472
0.39
50.
267
0.19
00.
110
0.05
0
La
24.6
321
.94
16.7
012
.54
9.80
27.
939
6.57
25.
522
4.69
83.
523
2.75
12.
214
1.81
81.
516
1.27
90.
942
0.72
40.
577
0.47
20.
395
0.26
70.
190
0.11
00.
050
Ce+
4–
61.6
921
.33
13.3
210
.01
8.05
56.
674
5.61
94.
787
3.58
72.
794
2.24
51.
844
1.53
91.
301
0.95
90.
736
0.58
60.
479
0.40
00.
271
0.19
40.
112
0.05
1
Ce
24.0
621
.51
16.5
012
.46
9.78
77.
958
6.61
25.
572
4.75
23.
572
2.79
22.
248
1.84
71.
541
1.30
20.
959
0.73
60.
586
0.47
90.
401
0.27
10.
194
0.11
20.
051
Pr+3
–49
.09
18.6
512
.41
9.64
97.
907
6.61
85.
608
4.79
93.
620
2.83
22.
281
1.87
61.
567
1.32
40.
977
0.74
90.
595
0.48
60.
406
0.27
50.
197
0.11
40.
052
Pr23
.46
20.7
615
.70
11.9
09.
478
7.81
66.
562
5.57
14.
775
3.61
22.
832
2.28
41.
879
1.56
81.
325
0.97
70.
749
0.59
50.
487
0.40
60.
275
0.19
70.
114
0.05
2
Nd+
3–
48.9
618
.54
12.3
39.
611
7.90
06.
635
5.63
94.
839
3.66
22.
870
2.31
31.
904
1.59
21.
347
0.99
40.
762
0.60
50.
494
0.41
20.
279
0.20
00.
115
0.05
3
Nd
22.9
420
.36
15.5
011
.81
9.44
17.
813
6.58
25.
605
4.81
63.
654
2.86
92.
316
1.90
71.
593
1.34
70.
994
0.76
20.
605
0.49
40.
412
0.27
90.
200
0.11
50.
053
Sm+3
–48
.69
18.3
212
.18
9.52
07.
868
6.64
95.
684
4.90
23.
736
2.93
92.
375
1.95
91.
641
1.39
01.
028
0.78
80.
624
0.50
80.
424
0.28
70.
206
0.11
90.
055
Sm21
.98
19.6
215
.11
11.6
19.
347
7.78
56.
600
5.65
34.
882
3.72
92.
939
2.37
71.
961
1.64
21.
391
1.02
80.
788
0.62
40.
509
0.42
40.
287
0.20
60.
119
0.05
5
Eu+
3–
48.5
618
.21
12.0
99.
469
7.84
56.
648
5.69
94.
927
3.76
92.
972
2.40
51.
985
1.66
41.
411
1.04
50.
801
0.63
40.
516
0.43
00.
291
0.20
90.
121
0.05
5
Eu
21.5
219
.27
14.9
211
.52
9.29
77.
765
6.60
25.
670
4.90
83.
762
2.97
12.
406
1.98
71.
666
1.41
21.
045
0.80
10.
634
0.51
60.
430
0.29
10.
209
0.12
10.
055
Gd+
3–
48.4
418
.11
12.0
19.
417
7.81
96.
643
5.71
04.
948
3.79
93.
002
2.43
32.
011
1.68
71.
432
1.06
20.
814
0.64
40.
524
0.43
60.
295
0.21
20.
123
0.05
6
Gd
21.2
319
.28
15.2
911
.87
9.51
77.
886
6.67
45.
722
4.95
23.
799
3.00
22.
432
2.01
11.
687
1.43
21.
062
0.81
40.
644
0.52
40.
436
0.29
50.
212
0.12
30.
056
Tb
20.6
618
.59
14.5
311
.30
9.17
27.
702
6.58
55.
686
4.94
63.
819
3.02
92.
461
2.03
71.
711
1.45
41.
079
0.82
70.
654
0.53
20.
442
0.29
90.
215
0.12
50.
057
Dy
20.2
518
.26
14.3
411
.19
9.10
67.
665
6.57
05.
688
4.95
93.
843
3.05
62.
486
2.06
11.
733
1.47
41.
095
0.84
00.
664
0.53
90.
448
0.30
30.
218
0.12
70.
058
Ho
19.8
617
.94
14.1
511
.08
9.03
87.
626
6.55
25.
686
4.96
83.
864
3.08
02.
511
2.08
41.
754
1.49
31.
111
0.85
30.
675
0.54
70.
455
0.30
70.
221
0.12
90.
059
Er
19.4
817
.63
13.9
710
.97
8.97
07.
584
6.53
25.
682
4.97
53.
884
3.10
32.
534
2.10
61.
775
1.51
21.
127
0.86
60.
685
0.55
60.
461
0.31
10.
224
0.13
10.
060
Tm
19.1
017
.33
13.7
910
.86
8.90
17.
541
6.50
95.
674
4.97
93.
901
3.12
52.
556
2.12
71.
795
1.53
11.
143
0.87
90.
695
0.56
40.
467
0.31
50.
227
0.13
30.
061
Yb
18.7
517
.04
13.6
110
.75
8.83
17.
496
6.48
45.
665
4.98
13.
916
3.14
52.
577
2.14
71.
814
1.54
91.
159
0.89
20.
706
0.57
20.
474
0.31
90.
230
0.13
50.
061
Lu
18.7
617
.25
14.0
411
.14
9.06
67.
618
6.54
55.
699
5.00
63.
937
3.16
62.
597
2.16
61.
831
1.56
61.
174
0.90
50.
716
0.58
00.
480
0.32
30.
233
0.13
60.
062
694 Appendix
Tabl
eA
.3(C
ontin
ued)
s0.
00.
050.
10.
150.
20.
250.
30.
350.
40.
50.
60.
70.
80.
91.
01.
21.
41.
61.
82.
02.
53.
04.
06.
0
Hf
18.3
917
.09
14.1
911
.39
9.27
77.
758
6.63
25.
754
5.04
53.
963
3.18
82.
616
2.18
41.
848
1.58
11.
188
0.91
70.
726
0.58
80.
487
0.32
70.
236
0.13
80.
063
Ta17
.99
16.8
414
.23
11.5
69.
458
7.89
96.
731
5.82
35.
095
3.99
43.
211
2.63
52.
201
1.86
41.
597
1.20
20.
930
0.73
70.
597
0.49
40.
331
0.23
90.
140
0.06
4
W17
.59
16.5
714
.19
11.6
89.
608
8.03
26.
834
5.90
05.
152
4.03
03.
236
2.65
52.
218
1.88
01.
611
1.21
50.
942
0.74
70.
605
0.50
00.
335
0.24
20.
142
0.06
5
Re
17.2
016
.28
14.1
111
.74
9.72
98.
154
6.93
75.
982
5.21
64.
069
3.26
42.
676
2.23
51.
895
1.62
61.
228
0.95
30.
757
0.61
30.
507
0.33
90.
244
0.14
40.
066
Os
16.8
215
.99
14.0
011
.77
9.82
48.
263
7.03
66.
065
5.28
34.
113
3.29
32.
698
2.25
31.
910
1.63
91.
241
0.96
50.
767
0.62
20.
514
0.34
30.
247
0.14
60.
067
Ir16
.39
15.6
413
.82
11.7
49.
871
8.34
27.
121
6.14
35.
350
4.16
03.
326
2.72
22.
271
1.92
51.
653
1.25
30.
976
0.77
60.
630
0.52
10.
348
0.25
00.
147
0.06
7
Pt15
.06
14.4
713
.04
11.3
49.
727
8.33
97.
179
6.22
05.
426
4.21
63.
363
2.74
72.
289
1.94
01.
666
1.26
50.
987
0.78
60.
638
0.52
80.
352
0.25
30.
149
0.06
8
Au
14.6
714
.14
12.8
311
.25
9.72
28.
380
7.24
16.
287
5.49
14.
267
3.40
02.
773
2.30
91.
956
1.67
91.
276
0.99
70.
795
0.64
60.
535
0.35
60.
256
0.15
10.
069
Hg
15.2
114
.64
13.2
311
.52
9.89
58.
493
7.32
26.
351
5.54
54.
310
3.43
52.
800
2.33
01.
972
1.69
31.
287
1.00
70.
805
0.65
40.
541
0.36
00.
259
0.15
30.
070
Tl
17.8
116
.71
14.3
412
.01
10.1
18.
601
7.39
16.
407
5.59
54.
354
3.47
12.
829
2.35
21.
990
1.70
71.
298
1.01
70.
814
0.66
20.
548
0.36
50.
262
0.15
40.
071
Pb18
.83
17.6
915
.13
12.5
210
.39
8.75
27.
479
6.46
75.
644
4.39
53.
506
2.85
82.
375
2.00
71.
722
1.30
91.
027
0.82
20.
670
0.55
50.
369
0.26
50.
156
0.07
2
Bi
19.3
318
.24
15.7
012
.97
10.6
98.
934
7.58
86.
537
5.69
64.
434
3.54
02.
886
2.39
82.
026
1.73
71.
320
1.03
60.
831
0.67
80.
562
0.37
30.
267
0.15
80.
073
Po19
.57
18.5
516
.12
13.3
711
.00
9.13
67.
715
6.61
95.
753
4.47
33.
573
2.91
52.
421
2.04
51.
752
1.33
11.
045
0.83
90.
685
0.56
80.
378
0.27
00.
160
0.07
3
At
19.1
318
.26
16.1
113
.56
11.2
39.
331
7.85
76.
717
5.82
14.
512
3.60
42.
942
2.44
52.
064
1.76
81.
342
1.05
40.
847
0.69
30.
575
0.38
20.
273
0.16
10.
074
Rn
18.7
217
.96
16.0
513
.68
11.4
29.
516
8.00
56.
824
5.89
64.
554
3.63
52.
968
2.46
82.
084
1.78
41.
354
1.06
30.
855
0.70
00.
581
0.38
70.
276
0.16
30.
075
Fr25
.81
22.7
617
.72
14.1
311
.60
9.65
88.
128
6.92
55.
974
4.60
03.
667
2.99
52.
490
2.10
31.
800
1.36
51.
072
0.86
30.
707
0.58
80.
391
0.27
90.
165
0.07
6
Ra
28.3
725
.18
19.2
614
.79
11.8
69.
799
8.23
97.
020
6.05
34.
650
3.70
13.
021
2.51
32.
122
1.81
61.
377
1.08
10.
871
0.71
40.
594
0.39
60.
282
0.16
60.
077
Ac
28.4
825
.67
20.0
615
.38
12.2
09.
994
8.36
77.
114
6.12
74.
699
3.73
53.
048
2.53
52.
142
1.83
31.
389
1.09
00.
878
0.72
10.
600
0.40
00.
286
0.16
80.
078
Th
28.1
125
.68
20.5
315
.88
12.5
510
.21
8.51
07.
216
6.20
44.
747
3.76
93.
075
2.55
82.
161
1.85
01.
401
1.09
90.
886
0.72
80.
606
0.40
50.
289
0.17
00.
079
Pa27
.33
24.8
819
.86
15.4
912
.40
10.2
18.
571
7.29
86.
287
4.81
23.
814
3.10
62.
582
2.18
01.
865
1.41
21.
108
0.89
40.
734
0.61
20.
409
0.29
20.
171
0.07
9
U26
.84
24.5
219
.72
15.4
812
.45
10.2
88.
654
7.37
96.
362
4.86
93.
856
3.13
72.
606
2.19
91.
881
1.42
41.
117
0.90
10.
741
0.61
80.
414
0.29
50.
173
0.08
0
A.6 Indexed Single Crystal Diffraction Patterns: fcc, bcc, dc, hcp 695
A.6 Indexed Single Crystal Diffraction Patterns: fcc, bcc, dc, hcp
696 Appendix
A.6 Indexed Single Crystal Diffraction Patterns: fcc, bcc, dc, hcp 697
698 Appendix
A.6 Indexed Single Crystal Diffraction Patterns: fcc, bcc, dc, hcp 699
700 Appendix
A.6 Indexed Single Crystal Diffraction Patterns: fcc, bcc, dc, hcp 701
702 Appendix
A.6 Indexed Single Crystal Diffraction Patterns: fcc, bcc, dc, hcp 703
704 Appendix
A.7 Stereographic Projections 705
A.7 Stereographic Projections
706 Appendix
A.7 Stereographic Projections 707
708 Appendix
A.8 Examples of Fourier Transforms 709
A.8 Examples of Fourier Transforms
710 Appendix
A.9 Debye–Waller Factor from Wave Amplitude 711
A.9 Debye–Waller Factor from Wave Amplitude
Another approach to calculating the Debye–Waller factor, perhaps simpler than thatof Chap. 10, makes use of the phase relationships in the diffracted wave. The in-stantaneous positions of the atom centers are {r i + δi}, and the intensity, I (Δk), iswritten as ψ∗ψ :
I (Δk) =∑
i
f ∗i e+iΔk·(r i+δi )
∑
j
fj e−iΔk·(rj +δj ), (A.1)
I (Δk) =∑
i
∑
j
f ∗i fj e+iΔk·(r i−rj )e+iΔk·(δi−δj ). (A.2)
We confine our attention to Bragg peaks where Δk · (r i − rj ) = 2π integer, so thefirst exponential in (A.2) is 1:
I (Δk) =∑
i
∑
j
f ∗i fj eiΔk·(δi−δj ). (A.3)
We assume the displacements are small, and expand the exponential in (A.3):
I (Δk) =∑
i
∑
j
f ∗i fj
(1 + iΔk · (δi − δj ) − 1
2
[Δk · (δi − δj )
]2)
. (A.4)
712 Appendix
We simplify further by assuming that the differences, δi − δj , average to zero whensummed over all pairs separated by r i − rj :
I (Δk) = |f |2∑
i
∑
j
(1 − 1
2
[Δk · (δi − δj )
]2)
. (A.5)
From (10.170) the isotropic average of [Δk · (δi − δj )]2 is 1/3Δk2(δi − δj )2 so:
I (Δk) = N2|f |2(
1 − 1
6Δk2(δi − δj )
2)
. (A.6)
Following Sect. 10.2.2, we assume that the displacements of the atom centers, δi andδj , are isotropic random variables with a Gaussian distribution and a characteristicrange, δ. The difference, δi − δj , will therefore have an average range of
√2δ,
allowing us to simplify (A.6) as:
I (Δk) = N2|f |2(
1 − 1
3Δk2δ2
). (A.7)
Approximately, the third factor in (A.7) is the exponential function:
I (Δk) = N2|f |2e−1/3Δk2δ2. (A.8)
The exponential factor in (A.8) is the Debye–Waller factor. It is essentially the sameas (10.59), but with an additional factor of 1/3 in the exponent. The derivation of(10.59) was performed in one dimension, so the 〈x2〉 in (10.59) corresponds to theaverage value of x2 along the direction Δk. Equation (A.8) refers to the average ofthe mean-squared displacement over all directions in space, δ2. It can be importantto specify which average is being reported.
A.10 Time-Varying Potentials and Inelastic Neutron Scattering
Time-Varying Potentials Coherent inelastic neutron scattering is a powerful toolfor studying the wavelengths and energies of elementary excitations in solids, suchas phonons (vibrational waves) and magnons (spin waves). Neutron elastic scatter-ing and neutron diffraction can be understood readily with analogies to x-ray andelectron scattering and diffraction, but inelastic neutron scattering, especially coher-ent inelastic neutron scattering, requires additional concepts. A brief introduction isgiven here.
Equations (4.82) and (4.83) were presented in the context of electron scattering,but nothing specific to electrons was used in obtaining them from the integral formof the Schrödinger equation. They are repeated here (including the time-dependenceof the outgoing wave):
Ψscatt(Δk, r, t) = − m
2π�2
ei(kf·r−ωt)
|r|∫
V(r ′)e−iΔk·r ′
d3r ′. (A.9)
A.10 Time-Varying Potentials and Inelastic Neutron Scattering 713
To use (A.9) for neutron scattering, m denotes the mass of the neutron of course,and we need a potential, V (r), appropriate for neutron scattering. For nuclear scat-tering, we use the “Fermi pseudopotential,” which places all the potential at a pointnucleus:
Vnuc(r) = 4π�
2
2mbδ(r). (A.10)
Here b is a simple constant length (although sometimes it is a complex number).For thermal neutrons, the δ-function is an appropriate description of the shape of anucleus.1
The next step is to place independent Fermi pseudopotentials at the positions{Rj }, of all atomic nuclei in the crystal. We also add one feature essential to inelas-tic scattering by atom vibrations—we allow the centers of the δ-functions to movewith time. Our time-varying potential is:
V (r, t) = 4π�
2
2m
∑
j
bj δ(r − Rj (t)
). (A.11)
Substituting (A.11) into (A.9), we note the elegant cancellation of prefactors:
Ψsc(Q, r, t) = −ei(kf·r−ω0t)
|r|∫ ∑
j
bj δ(r ′ − Rj (t)
)eiQ·r ′
d3r ′. (A.12)
In writing (A.12) we made the substitution Δk → −Q because this new symboland sign are in widespread use for neutron scattering. The integration over the δ-functions of (A.12) fixes the exponentials at the nuclear positions {Rj (t)}:
Ψsc(Q, r, t) = −ei(kf·r−ω0t)
|r|∑
j
bj eiQ·Rj (t). (A.13)
It is convenient to separate the static and dynamic parts of the nuclear positions:
Rj (t) = xj + uj (t), (A.14)
so by substitution:
Ψsc(Q, r, t) = −ei(kf·r−ω0t)
|r|∑
j
bj eiQ·(xj +uj (t)). (A.15)
When Q · u is small, we can expand the exponential in (A.15) to obtain:
1For magnetic scattering, however, an electron spin density is used, and this reflects the shape ofthe atom. Also, the potential for magnetic scattering has vector character.
714 Appendix
Ψsc(Q, r, t) = −ei(kf·r−ω0t)
|r|
×∑
j
bj eiQ·xj
(1 + iQ · uj (t) − 1
2
(Q · uj (t)
)2 + · · ·)
. (A.16)
Elastic Neutron Scattering Neglecting the time-dependence of the scatteringpotential, i.e., setting uj (t) = 0 in (A.16), we recover the case of elastic scattering.The first term in parentheses in (A.16), the 1, involves only the static part of thestructure. We isolate this static term, Ψ el
sc (Q, r), as the elastic part of the scatteredwave:
Ψ elsc (Q, r) = −ei(kf·r−ω0t)
|r|∑
j
bj eiQ·xj . (A.17)
Because b is the neutron equivalent to f (Δk) for the coherent elastic scattering ofelectrons or x-rays, the further development of neutron diffraction takes the samepath that follows (6.18) in Chap. 6.
Phonon Scattering The next term in (A.16), involving Q · uj (t), gives inelasticscattering. To calculate the inelastically-scattered neutron wavefunction, we firstneed the motions of all nuclei. For this we use the phonon expression for collectiveatom motions, uj (ω,q, t):
uj (ω,q, t) = U j (ω,q)√
2Mjωei(q·xl−ωt). (A.18)
Equation (A.18) has a typical form for an elementary excitation in a solid. In par-ticular, this phonon excitation is specified by its combination of wavevector q andfrequency ω. The phase factor, eiq·xj (t), provides the long-range spatial modulationof uj at all atom positions. This spatial modulation has a “polarization,” U j thatidentifies the amplitude and direction of atom motions. The nuclear mass, Mj , isessentially the entire mass of the atom centered at Rj . After substitution of (A.18),the second term in (A.16) gives an inelastically-scattered wave, Ψ inel
sc (Q, r):
Ψ inelsc (Q, r, t) = − iei(kf·r−ω0t)
|r|×
∑
j
bj√2Mjω
(Q · U j (ω,q)
)eiQ·xj ei(q·xj −ωt), (A.19)
Ψ inelsc (Q, r, t) = − ieikf·r
|r| e−i(ω0+ω)t
×∑
j
bj√2Mjω
(Q · U j (ω,q)
)ei(Q+q)·xj . (A.20)
A.11 Review of Dislocations 715
Equation (A.20) identifies two important features about the phase of the neutronwavefunction after coherent inelastic scattering.2 First, the neutron wavefunctionchanges frequency from ω0 to ω0 + ω because the neutron gains an energy �ω byannihilating the phonon of frequency ω. The wavevector in the phase factor for theneutron wavelet scattered from Rj is not the same Q as for elastic scattering, butis Q + q , equivalent to the momentum difference �q . Energy and momentum areconserved, but are transferred between the crystal and the neutron. A spectrometerfor inelastic neutron scattering measures the momentum and energy of scatteredneutrons, and this may be enough information for the experimenter to deduce thefrequencies and wavevectors of the elementary excitations in the sample. There aremany additional considerations in such work, of course.
A.11 Review of Dislocations
Structure of a Dislocation
A dislocation is the only line defect in a solid. A large body of knowledge hasformed around dislocations because their movement is the elementary mechanismof plastic deformation of many crystalline materials. In addition, dislocations insemiconducting crystals are sinks for charge carriers. More than any other experi-mental technique, TEM has revealed the structures and interactions of dislocations.
There are two types of “pure” dislocations. An edge dislocation is the easiest toillustrate. In Fig. A.2, notice how an extra half-plane of atoms has been inserted inthe upper half of the simple cubic crystal. This extra half-plane terminates at the“core” of the edge dislocation line. On the figure is drawn a circuit of 5 × 5 × 5 × 5atoms. This circuit, known as a “Burgers circuit,” does not close perfectly whenit encloses the dislocation line. (It does close in a perfect simple cubic crystal, ofcourse, and it also closes perfectly when it is drawn in a dislocated crystal arounda region that does not contain the dislocation core.) The vector from the end to thestart of the circuit is defined as the “Burgers vector” of the dislocation, b. Dislo-cations are characterized by their Burgers vector and the direction of their dislo-cation line. The magnitude of the Burgers vector parameterizes the strength of thedislocation—dislocations with larger Burgers vectors cause larger crystalline dis-tortions. The “character” of the dislocation is determined by the direction of theBurgers vector with respect to the direction of the dislocation line. In Fig. A.2 theBurgers vector is perpendicular to the dislocation line. This is an “edge dislocation.”
The other type of “pure” dislocation has its Burgers vector parallel to the dislo-cation line. It is a “screw dislocation,” and is illustrated in Fig. A.3. Around the coreof a screw dislocation, the crystal planes form a helix. When we complete a Burgers
2Other features that can be identified are the scaling of the phonon scattering intensity, Ψ inel∗sc Ψ inel
sc ,with Q2 and with the factor b2/Mj . Especially for single crystals, the orientation informationQ · U j is also useful.
716 Appendix
Fig. A.2 Edge dislocation ina cubic crystal. Dislocationline is parallel to y ,b = a〈100〉, and b isperpendicular to thedislocation line
Fig. A.3 Screw dislocationin a cylinder of cubic crystal.Dislocation line is parallel toz, b = a〈001〉, and b isparallel to the dislocation line
circuit in the x–y plane in Fig. A.3, the vector from finish to start lies along z. For ascrew dislocation, b is parallel to the line of the dislocation.
In general, dislocations are neither pure edge dislocations nor pure screw disloca-tions, but rather have their Burgers vectors at some intermediate angle to the line oftheir cores. These are “mixed dislocations.” Whenever a dislocation line is curved,part of the dislocation must have mixed character. An example of a curved disloca-tion line is shown in Fig. A.4, with labels indicating the pure edge and screw parts.All other parts of the dislocation are of mixed character. Notice how the dislocationwas made. The crystal was cut in the lower right corner, and the top (gray) atomswere pushed to the left with respect to the lower (black) atoms. The edge of thecut is the dislocation line. A Burgers circuit around any part of this dislocation linealways gives the same Burgers vector. Since the dislocation line changes direction,however, the character of the dislocation changes along its line.
A dislocation loop, which is mostly of mixed character, is illustrated in Fig. A.5.A planar circular cut is made inside a block of material. The atoms across this cutare sheared as shown in the figure. The edge of the cut is the dislocation line. Onthe left and right edges of this dislocation loop we have edge dislocations (with b
A.11 Review of Dislocations 717
Fig. A.4 Mixed dislocationin a cubic crystal.Quarter-circle of cut plane isin the lower right. All atomsacross the cut are displaced tothe left by b
Fig. A.5 Left: dislocationloop in a cube of crystal. Allatoms across the cut aredisplaced to the left by b.Right: top view of loop
of opposite signs). On the front and back, the dislocation loop has pure screw char-acter (again with b of opposite signs). Everywhere else the dislocation has mixedcharacter.
Strain Energy of a Dislocation (Self Energy)
A dislocation generates large elastic strains in the surrounding crystal, as is evidentfrom Figs. A.2–A.4. The strain in the material in the dislocation core (usually con-sidered to be cylinder of radius 5b) is so large that its excess energy cannot be accu-rately regarded as elastic energy. Sometimes this “core energy” is estimated from theheat of fusion of the crystal. Outside the core region, however, it is reasonable to cal-culate the energy by linear elasticity theory. It turns out that this total elastic energyin the surrounding crystal is typically an order-of-magnitude larger than the energyof the core region. Approximately, therefore, the energy cost of making a unit lengthof dislocation line is equal to the elastic energy per unit length of the dislocation.
We have seen how dislocations can be created by a cut-and-shear process. Thedislocation line is located at the edge of the cut, and the Burgers vector is the vectorof the shear displacement. We seek the energy needed to make the dislocation thisway. First note that the cut itself requires no energy, since the atoms across thecut are properly reconnected after the dislocation is made. The energy needed tomake the dislocation is the energy required to make the shear across the cut surface.Think of the cut crystal as a spring. An elastic restoring force opposes the shear,
718 Appendix
Fig. A.6 Accommodation ofthe same slip by twodislocations or by onedislocation
and this restoring force is proportional to the shear times the shear modulus, G.The distance of displacement across the cut is b. The elastic energy stored in thecrystal is obtained by integrating the force over the distance, x, of shear for 1 cm ofdislocation line:
Eelastic ∝∫ 1 cm
0
∫ b
0Gx dx dz, (A.21)
Eelastic = Gb2K [J/cm]. (A.22)
Here K is a geometrical constant that depends on the size and shape of the crystal(and somewhat on the dislocation character). Neglecting the smaller core energy,the energy cost of creating a unit length of edge dislocation is the Eelastic of (A.22).
Dislocation Reactions
Because the self-energy of a dislocation increases as b2, dislocations prefer Burgersvectors that are as small as possible. Figure A.6 shows how to accommodate twoextra half-planes with either one dislocation of b = 2a, or two dislocations, each ofb = a. The total elastic energy of a crystal with the two separate, smaller disloca-tions is half as large, however. Big dislocations therefore break into smaller ones, sosingle dislocations have the smallest possible Burgers vector. The lower limit to theBurgers vector is set by the requirement that the atoms must match positions acrossthe cut in the crystal. This lower limit is typically the distance between nearest-neighbor atoms. Smaller Burgers vectors are usually not possible, but an exceptionoccurs for fcc and hcp crystals.
Stacking Faults in fcc Crystals
A special dislocation reaction occurs for dislocations on {111} planes in fcc crystals.Figure A.7 shows how the stacking of close-packed planes determines whether thecrystal is fcc or hcp.
The “perfect dislocation” in the fcc crystal has a Burgers vector of the nearest-neighbor separation, b = 1/2[110]. The shifts between the adjacent layers of thefcc structure are smaller than this, however, and we can obtain these shifts by cre-ating a “stacking fault” in the fcc crystal. Specifically, assume that we interrupt the
A.11 Review of Dislocations 719
Fig. A.7 (a) fcc stacking of close-packed (111) planes; perspective view of three layers, with thecubic face marked with the square. (b) Stacking of the three types of (111) planes seen from above.The next layer will be an A-layer, and will locate directly above the dark A-layer at the bottom.(c) hcp stacking of close-packed (0001) basal planes; perspective view of three layers. (d) Stackingof the two types of close-packed planes seen from above. The next layer will be an A-layer, andwill locate directly above the dark A-layer at the bottom
ABCABCABC stacking of the fcc crystal and make a small shift of a {111} planeas: ABCAB|ABCABC. Here we have erred in the stacking by placing an A-layerto the immediate right of a B-layer. The structure is still close packed, but there isa narrow region of hcp crystal (. . .AB|AB. . .). This region of hcp crystal need notextend to the edge of the crystal, however. At the boundary of the hcp region we caninsert a “Shockley partial” dislocation, which has a Burgers vector equal to the shiftbetween an A and a B-layer. This shift is a vector of the type: a/6〈112〉.
Consider a specific dislocation reaction for which the total Burgers vectors acrossthe arrow are equal3:
a/2[110] → a/6[121] + a/6[211]. (A.23)
The energy, proportional to the square of the Burgers vector, is smaller for the twoShockley partials on the right than the single perfect dislocation on the left, as weverify by calculating the energies (A.22):
Eperfect
E2 partials= KGa2/4(12 + 12 + 0)
2KGa2/36(12 + 22 + 12)= 3
2. (A.24)
Equation (A.24) shows that it is energetically favorable for a perfect dislocationin an fcc crystal to split into two Shockley partial dislocations, which then repeleach other elastically (as discussed in the next subsection). There is, however, athin region of hcp crystal between these two Shockley partials (the stacking fault),and the stacking fault energy tends to keep the partials from getting too far apart.Equilibrium separations of Shockley partial dislocations, measured by TEM, are
3The conservation of Burgers vector is equivalent to the fact that a dislocation line cannot terminatein the middle of a crystal, but must extend to the surface or form a loop.
720 Appendix
a means of determining the stacking fault energy of fcc crystals. This stacking faultenergy is qualitatively related to the free energy difference between the fcc and hcpcrystal structures.
Stable Arrays of Dislocations
Look again at the atom positions around the dislocation core in Fig. A.2. Insertingan extra half-plane of atoms in the top half of the crystal causes compressive stressesabove, and tensile stresses below the dislocation line. An edge dislocation line, seenon end in Fig. A.8, is marked with a “⊥” symbol. The circles are lines of constantstrain.
Dislocations interact with each other through their elastic fields, so groups ofdislocations are frequently found in special arrangements. For example, two edgedislocations with the same Burgers vector repel each other when they are situatedon the same glide plane. When they are close together, their compression and tensilestrains add. The elastic energy increases quadratically with the strain field. It istherefore favorable for the dislocations to move apart as in Fig. A.9 (cf., Fig. A.6),so there is an elastic repulsion between these two edge dislocations.
The six dislocations on the left of Fig. A.10 are in a stable configuration, how-ever, since the compressive stress above each dislocation cancels partially the tensilestress below its neighboring dislocation. Perturbing the dislocations out of this lineararray increases the elastic energy. The right side of Fig. A.10 shows in more detailthe extra half-planes of the six edge dislocations in a simple cubic crystal. This dis-location array creates a low-angle tilt boundary between two perfect crystals. Thisparticular example is a symmetric tilt boundary. Other types of tilt boundaries arepossible, as are twist boundaries comprising arrays of screw dislocations. Arraysof (1-dimensional) dislocations are common at 2-dimensional interfaces betweendifferent phases in a material.
Fig. A.8 Compression andtension fields around an edgedislocation
Fig. A.9 Elastic repulsion oftwo edge dislocations on thesame glide plane
A.12 TEM Laboratory Exercises 721
Fig. A.10 Stable dislocationstructure constituting a smallangle tilt boundary
A.12 TEM Laboratory Exercises
This appendix presents the content of a university laboratory course designed tofamiliarize the new user with the practice of microscope calibration, conventionaldiffraction and imaging techniques, and energy-dispersive x-ray spectroscopy. Insuch a course, students have access to the instrument in three hour sessions. Mostlaboratory exercises require 3–4 sessions to complete. Additional time is needed forinstrument startup, data analysis, report writing, and perhaps specimen preparation.
An introduction to the instrument, and the simpler Au and MoO3 exercises inLaboratory 1, require 2–3 sessions. Sample tilting is needed in Laboratory 2 on DFimaging of θ ′ precipitates, and tilting requires practice. Laboratory 3 on EDS ofθ ′ precipitates is straightforward, and could perhaps be performed before Labora-tory 2. Laboratory 4 on dislocations and stacking faults in stainless steel is typical ofphysical metallurgy research with conventional TEM. The instructor may considersubstituting another laboratory on a material more relevant to the research interestsof the student.
The authors often modify the laboratories—some variations are given in theSpecimen or Procedures sections. Of course the instrument alignment proceduresare for a particular microscope, and are found in the microscope manufacturers’manuals. Condensed alignment instructions can be handy references in the labo-ratory. Please read the manufacturer’s manuals, however—they are generally wellwritten and rich in information.
A.12.1 Laboratory 1—Microscope Procedures and Calibrationwith Au and MoO3
The principles of operation and alignment of the transmission electron microscopeshould be covered in the first laboratory session. The Au and MoO3 exercises areoften the first rewarding experiences with a TEM.
722 Appendix
A. Camera Constant Determination
Specimen Polycrystalline Au film evaporated onto a holey carbon film supportedon a 200 mesh copper grid. (Such Au samples are available from vendors of micro-scope supplies.)
Measurements (a) With the microscope at 200 kV and the specimen in the eu-centric position, obtain two focused bright-field (BF) images of the same specimenarea at a medium magnification (∼60 kX) using the largest and smallest objectivelens apertures. Photograph the corresponding electron diffraction patterns (with acamera length ∼100 cm) using the double-exposure technique with the objectiveaperture in to record also the sizes and positions of the two different objective aper-tures.
Explain why the size and position of the objective aperture affects the contrast inthe image.
(b) Photograph two selected-area diffraction (SAD) patterns (with a cameralength of ∼100 cm) from the same specimen area using the largest and smallest in-termediate apertures. Photograph the corresponding BF images, again using a dou-ble exposure with the intermediate aperture in and the objective aperture out (andan appropriate magnification) to record the sizes and positions of the intermediateapertures. Also record the objective, intermediate and projector lens currents.
Calculate the microscope camera constant (λL) from these results.Explain why the size of the intermediate aperture affects the appearance of the
diffraction pattern.
Procedures for Taking Images and SAD Patterns (written for the JEOL2000FX)
Starting with a properly aligned TEM in the magnification (Mag) mode, and thespecimen in the eucentric position:
• Focus the image using the objective lens (focus) controls.• Insert the desired SAD aperture and center it.• Go to the SAD diffraction mode (Diff).• Remove the objective aperture (if it was in).• Center the illumination and spread the beam to obtain sharp diffraction spots.• Focus the spot pattern using the diffraction focus knob. (You can insert the ob-
jective aperture and focus the aperture edge to confirm that the spot pattern is infocus.)
• Center the diffraction pattern on the screen using the projector alignment knobsin the right drawer.
• Set the exposure to approximately 1/3–1/4 of the full-screen meter reading, andphotograph the diffraction pattern. (Alternatively, you may use about 3/4 of thesmall screen reading as an exposure estimate).
• Insert and center the desired objective lens aperture.• Return to the magnification (Mag) mode.
A.12 TEM Laboratory Exercises 723
• Focus and stigmate the image using the objective lens stigmator controls (stigma-tion is required only on the first image). Using the meter reading, photograph theimage.
• (Repeat for all magnifications and diffraction patterns.)
Taking Double Exposures For double exposures, press the photo button to startthe exposure process, and then press it a second time while the screen is raising. Thisprevents the film from advancing after the first exposure. When the first exposureis complete, the photo button light comes back on. Press the button again for thesecond exposure (after setting the desired exposure time).
B. Astigmatism Correction
Specimen Same evaporated Au as above.
Procedures Find a small hole in the holey carbon film that is not covered withgold, i.e., the carbon is exposed around the edge of the hole. Go to a high magnifi-cation (∼500 kX) so the granular features in the carbon film are visible. (You maywant to insert a medium-size objective aperture to increase the contrast from theamorphous carbon. Make sure the aperture is centered!)
View the image on the TV rate camera and correct the astigmatism using thestigmator knobs on the microscope. Remove the TV-rate camera, and use the CCDcamera with a simultaneous live FFT display to perform a final correction of theastigmatism. When the astigmatism is corrected, record three images on the CCDin overfocused, minimum contrast, and underfocused conditions. Print these imagesand their corresponding FFTs, and discuss their features.
C. Rotation Calibration
(written for the Philips EM400T)
Specimen Molybdenum trioxide on carbon substrates. (MoO3 is formed by heat-ing a Mo wire with an oxygen-acetylene torch in air. Carbon substrates supported on200 mesh copper grids are passed through the smoke to collect the MoO3 crystals.)
Experimental Measurements (a) Find a small crystallite of MoO3 with well-defined facets. With the magnification (M) and diffraction (D) modes, use the doubleexposure method to record superimposed BF images of the specimen and its corre-sponding SAD diffraction pattern. Repeat this procedure on the same crystallite foreach magnification (intermediate lens current) in the M mode—magnifications of10, 13, 17, 22, 28, 36, 46, 60, 80 and 100 kX (10 total). (Note: The most commoncamera lengths are typically 575 and 800 mm.)
(b) Record the currents of the objective, diffraction, intermediate and projectorlenses (P1 and P2) for each magnification in the M mode, and for the diffractionpatterns in the D mode, using the display selector knob in the back panel.
724 Appendix
Fig. A.11 Image rotationcalibration of JEOL 120CXmicroscope operated at120 kV. Note abrupt changein image rotation at 40 kX
Data Analysis (a) Using the superimposed BF/SAD images, graph the magnitudeand direction of the image rotation as a function of magnification. Comment on theimportant features of this plot. The crystallography of the MoO3 crystal and itsrelationship to the diffraction pattern are illustrated in Fig. A.11 for a JEOL 100CXmicroscope. There are errors in these features in all four references below, so becareful!
(b) Measure the width of the MoO3 crystal and plot the crystal width as a functionof the dial magnification. (A small crystal is required if its edges are to remain inthe field of view at high magnification.)
(c) On two separate graphs, plot the objective, diffraction, intermediate and pro-jector lens currents for the magnification (M) and diffraction (D) modes as a func-tion of the dial magnification. Discuss the significance of these graphs for imagemagnification and accuracy in SAD.
References for Laboratory 1
1. J.W. Edington, Practical Transmission Electron Microscopy in Materials Science—1. Opera-tion and Calibration of the TEM (Philips Technical Library, Eindhoven, Netherlands, 1974)
A.12 TEM Laboratory Exercises 725
2. J.W. Edington, Practical Transmission Electron Microscopy in Materials Science—2. ElectronDiffraction in the Electron Microscope (Philips Technical Library, Eindhoven, Netherlands,1974), pp. 11–16
3. G. Thomas, M.J. Goringe, Transmission Electron Microscopy of Materials (John Wiley andSons, NY, 1979), pp. 28–33
4. D.B. Williams, Practical Analytical Electron Microscopy in Materials Science (Philips ElectronInstruments, Inc. Mahwah, NJ, 1984), pp. 26–30
A.12.2 Laboratory 2—Diffraction Analysis of θ ′ Precipitates
This experiment introduces the important methods of electron diffraction and dark-field imaging to determine the identity and orientation relationship of precipitates ina matrix. For an introductory laboratory, θ ′ precipitates have proved convenient insize and contrast against the Al matrix. This exercise also provides experience withsample tilt, which may require a prior session of practice. Laboratory 2 coupleswell with the energy-dispersive x-ray analysis in Laboratory 3, but the two can beperformed independently.
Background The θ ′ phase is a metastable precipitate that often forms duringaging of Al–Cu base alloys. It has a tetragonal crystal structure with space groupsymmetry I4/mmm and a = 0.404 nm and c = 0.58 nm. A perspective drawingof the unit cell of the θ ′ phase is shown in Fig. A.12. The unit cell contains fouratoms of Al and two atoms of Cu. The θ ′ precipitates form as thin plates on the100 planes in the Al matrix with the orientation relationship (001)θ ′ ‖ (001)Al and[100]θ ′ ‖ [100]Al.
The θ ′ phase forms as thin plates on all three {001}Al matrix planes. When a thinfoil is viewed along a 〈001〉Al orientation, one variant of θ ′ phase is face-on, whilethe other two variants are edge-on and perpendicular to each another (see Figs. A.12and 13.15). The Al matrix and each variant of θ ′ phase produce different diffractionpatterns. When all three variants are present within the selected area aperture, all ofthese diffraction patterns are superimposed. If a small selected area aperture is used,however, it may be possible to obtain diffraction patterns from only one or twovariants of precipitate. Figure A.13 shows diffraction patterns for the Al matrix in a
Fig. A.12 Left: Labeledcrystal structure of θ ′precipitate. Right:Orientations of three variantsof θ ′ plates in the fcc Almatrix
726 Appendix
Fig. A.13 Indexed 〈001〉 diffraction patterns from fcc Al matrix (left), and two variants of θ ′precipitates within the Al matrix (right)
〈001〉 orientation, and two variants of the θ ′ phase, one face-on along [001]θ ′ and theother edge-on along [100]θ ′ . (The diffraction pattern for the third variant of θ ′ canbe obtained by rotating the [100]θ ′ pattern on the lower right by 90◦.) All three ofthese patterns can then be superimposed to obtain the composite diffraction patternin Fig. A.14. An experimental 〈001〉Al SAD pattern containing all three precipitatevariants (and also double-diffraction spots) is also shown in Fig. A.14.
The different variants of precipitate can be identified by bringing each of theprecipitate diffractions labeled 1, 2 and 3 in the composite pattern onto the opticaxis within a small objective aperture, and making a dark-field (DF) image.
Specimen Electropolished thin foils of Al-4.0 wt% Cu alloy. A sheet of polycrys-talline alloy about 150 µm thick was solution treated for 1 h at 550 °C, quenchedinto water and aged for 12 h at 300 °C to produce well-developed θ ′ precipitateplates. Disks 3 mm in diameter were punched from the sheet and electropolishedin a twin-jet Fischione apparatus using a 25 %HNO3-methanol solution at about−40 °C and 15 V.
(Alternative samples: carbon extraction replicas from a medium carbon steel, orpieces of aluminum beverage cans.)
Procedures (a) Before going to the microscope, photocopy and enlarge the lowindex fcc diffraction patterns in the Appendix of this book. On a second set ofdiffraction patterns you should prepare a set of Kikuchi line patterns. To do so,
A.12 TEM Laboratory Exercises 727
Fig. A.14 Composite diffraction pattern from all three variants of θ ′ precipitate in Al matrix in[100] zone axis. Left: schematic. Right: experimental SAD
draw straight lines through the low index spots. The line through the spot g shouldbe oriented perpendicularly to the direction g (the direction of the spot from theorigin). You may want to plot other low-index diffraction patterns for the θ ′ phaseusing a computer program, if available. Please read some of the four references be-low. They contain information about the crystal structure, morphology, interfacialstructure, and growth kinetics of the θ ′ phase.
(b) Obtain SAD patterns of the matrix and precipitates by tilting the specimento low-index orientations such as 〈001〉Al, 〈011〉Al or 〈112〉Al. Use Kikuchi linepatterns and indexed diffraction patterns to help you. The 〈001〉Al zone axis is theeasiest to interpret, so you should try to obtain this orientation. Orient the specimenso that the pattern is exactly on the zone axis. Spread the illumination and take longexposures when photographing diffraction patterns so the faint precipitate spots willbe sharp and visible. You might test several different exposures to find the optimalexposure (typically about 1/4 of the automatic exposure reading). Don’t forget tofocus the diffraction pattern!
(c) To identify the precipitates in the intermediate aperture that contributed to theSAD pattern, photograph the corresponding BF images using the double-exposuretechnique. You may want to experiment with different size apertures, using a largeaperture to obtain a pattern from all three θ ′ variants, and using a smaller apertureto obtain diffraction patterns from only one or two variants.
(d) Photograph DF images of each of the θ ′ variants on the three {100}Al planes.Do this by tilting the incident beam into the position of the precipitate diffractionspot, so the −g diffraction appears on the optic axis. (Avoid the “amateur mis-take.”) Also photograph the corresponding diffraction patterns. Record the precip-itate diffraction that was used to form the DF image. This can be done by eitherphotographing the beam-stop, or using the double-exposure technique with the ob-jective aperture superimposed on the diffraction pattern for one of the exposures.This record is needed to positively identify each precipitate variant.
728 Appendix
(e) Identify the θ ′ precipitates by fully indexing the diffraction patterns and corre-lating them to the particle morphologies and orientations in the BF and DF images.Your rotation calibration from the previous lab will be useful here. Also determinethe lattice spacings for the θ ′ phase by using the Al diffraction pattern as a standard,with crystallographic data for this phase provided in the references.
(f) On a 〈001〉 stereographic projection, show the orientation relationship be-tween the θ ′ precipitate and matrix. Mark most of the low-index poles for the pre-cipitate and matrix phases. Diffraction programs that also plot stereographic projec-tions are very useful for this.
References for Laboratory 2
1. J.M. Silcock, T.J. Heal, Acta Crystallogr. 9, 680 (1956)2. G.C. Weatherly, R.B. Nicholson, Philos. Mag. A 17, 801 (1968)3. U. Dahmen, K.H. Westmacott, Phys. Stat. Sol. (a) 80, 248 (1983)4. G.W. Lorimer, in Precipitation Processes in Solids (TMS-AIME, Warrendale, PA, 1978), p. 87
A.12.3 Laboratory 3—Chemical Analysis of θ ′ Precipitates
This laboratory could be performed simultaneously with laboratory 2, since it usesthe same specimens of θ ′ precipitates in Al–Cu. The present laboratory demon-strates microbeam chemical analysis with EDS spectroscopy.
Specimen Same electropolished thin foils of Al-4.0 wt% Cu alloy used in Labo-ratory 2.
Procedures (a) Using the same basic probe conditions as in b below, but with thebeam spread over a large area near the edge of the foil, acquire an EDS spectrumwith at least 100,000 counts in the Al Kα peak. Assuming this spectrum representsthe average alloy composition, use this spectrum to determine the k-factor for Aland Cu.
(b) Obtain EDS spectra from about 6 different edge-on θ ′ plates using the sameprobe and counting conditions. Try a small spot size (say 8) for 60 s and work nearthe edge of the foil, i.e., thin-film conditions. If you need more counts, switch toa larger spot size (maybe 6) or a longer counting time. Use the second or thirdcondenser aperture to obtain a well-defined probe.
(c) Take bright-field images of each θ ′ plate. Use the double exposure techniqueto show the size and position of the probe on the plate. Use a magnification of around100 kX.
(d) Find three edge-on θ ′ plates in about the same area (same specimen thickness)but with different plate thicknesses. How do their EDS spectra compare?
(e) Choose three plates, one very near the edge of the foil, one slightly further in,and the third even further in. How do their spectra compare and why?
A.12 TEM Laboratory Exercises 729
(f) If you have time, obtain three more spectra on the same precipitate in a rel-atively thin area with spot sizes of 2, 4, 6 and 8. How does the spot size affect thespectra and why?
(g) If you have time, obtain three spectra along the length of the same precipitateusing the same spot size as in b above. What causes the variation among the spectra?
(h) If you still have time, use a spot size of 8 and take a composition profile acrossthe precipitate/matrix interface. You will need a high magnification to do this.
References for Laboratory 3
Same as for Laboratory 2.
A.12.4 Laboratory 4—Contrast Analysis of Defects
This experiment gives experience in defect identification using contrast analysis.The defect type, plane and displacement vector as well as the Burgers vectors ofisolated perfect dislocations partial dislocations bounding stacking faults will bedetermined. It is more challenging to attempt a full stacking fault analysis as inSect. 8.12.5.
Specimen Electropolished thin foils of AISI Type 302 (or 309) fcc stainless steel,annealed and lightly cold-rolled. Disks 3 mm in diameter were punched from therolled sheet and electropolished in a twin-jet Fischione apparatus using a 10 % per-chloric acid-ethanol solution at about −15 °C and 30 V.
(Alternative samples: Cu-7 % Al sample deformed approximately 5 % in tension,interfacial dislocations on the θ ′ plates used in Laboratory 3, misfit dislocations inSi–Ge heterostructures, dislocations in NiAl deformed a few percent in tension.)
Procedures (a) Before going to the microscope, prepare contrast analysis (g · b)tables for defect visibility, paying particular attention to low-index orientations suchas 〈110〉, 〈100〉, 〈112〉, and 〈111〉. Examples of such contrast tables are Tables 8.2and 8.3. The 〈110〉 orientation is particularly good for analysis because many differ-ent g vectors are available in this orientation. Other microscopists like to start witha 〈100〉 orientation, since it is also a convenient starting place for tilting into otherzone axes. To identify uniquely the dislocation line direction or Burgers vector, youwill need at least two zone axes.
(b) Locate isolated planar defects in the foil (either singly or in groups) and imagethe same area in a strong two-beam, bright-field (BF) condition, and an axial dark-field (DF) condition with s = 0. Try to ensure that the deviation parameter s isidentical for the BF and DF images by tilting the foil so that the relevant extinctioncontour passes through the defect(s) to be analyzed. Record the corresponding SADpatterns. Check the crystallographic orientation on either side of the planar defect.If it is different, record both patterns.
730 Appendix
(c) Continue to image the same defect region under other two-beam BF con-ditions indicated by the contrast tables prepared in a above. Again, pay particu-lar attention to the deviation parameter to ensure that s ≥ 0. Look for evidence ofbounding partial dislocations. Record the corresponding SAD patterns.
(d) Using additional diffraction conditions (as identified in your contrast table),image isolated slip dislocations or dislocation pile-ups present in the foil. Recordthe corresponding SAD pattern.
(e) By trace analysis on an appropriate stereographic projection, identify the de-fect planes and slip planes. Arrange the data to show the nature of the defects anddetermine the Burgers vectors of all dislocations.
References for Laboratory 4
1. J.W. Edington, Practical Electron Microscopy in Materials Science Volume 3—Interpretationof Transmission Electron Micrographs (Philips Technical Library, Eindhoven, 1975), pp. 10–55
2. G. Thomas, M.J. Goringe, Transmission Electron Microscopy of Materials (John Wiley andSons, New York, 1979), pp. 142–169
3. P.B. Hirsch et al., Electron Microscopy of Thin Crystals (R.E. Krieger Pub. Co., Malabar, 1977),pp. 141–147, 162–193, 222–275, 295–316
4. P.H. Humphrey, K.M. Bowkett, Philos. Mag. 24, 225 (1971)5. J.M. Silcock, W.J. Tunstall, Philos. Mag. A 10, 361 (1965)
A.13 Fundamental and Derived Constants
Fundamental Constants
�= 1.0546 × 10−27 erg s = 6.5821 × 10−16 eV s
kB = 1.3807 × 10−23 J/(atom K) = 8.6174 × 10−5 eV/(atom K)
R = 0.00198 kcal/(mole K) = 8.3145 J/(mole K) (gas constant)
c = 2.998 × 1010 cm/s (speed of light in vacuum)
me = 0.91094 × 10−27 g = 0.5110 MeV c−2 (electron mass)
mn = 1.6749 × 10−24 g = 939.55 MeV c−2 (neutron mass)
NA = 6.02214 × 1023 atoms/mole (Avogadro constant)
e = 4.80 × 10−10 esu = 1.6022 × 10−19 coulomb
μ0 = 1.26 × 10−6 henry/m
ε0 = 8.85 × 10−12 farad/m
A.13 Fundamental and Derived Constants 731
a0 = �2/(mee
2) = 5.292 × 10−9 cm (Bohr radius)
e2/(mec2) = 2.81794 × 10−13 cm (classical electron radius)
e2/(2a0) = R (Rydberg) = 13.606 eV (K-shell energy of hydrogen)
e�/(2mec) = 0.9274 × 10−20 erg/oersted (Bohr magneton)
�2/(2me) = 3.813 × 10−16 eV cm2 = 3.813 eV Å
2
Definitions
1 becquerel (B) = 1 disintegration/second
1 Curie = 3.7 × 1010 disintegrations/second
Radiation dose:
1 roentgen (R) = 0.000258 coulomb/kilogram
Gray (Gy) = 1 J/kG
Sievert (Sv) is a unit of “radiation dose equivalent” (meaning that doses of radiationwith equal numbers of Sieverts have similar biological effects, even when the typesof radiation are different). It includes a dimensionless quality factor, Q (Q∼1 forx-rays, 10 for neutrons, and 20 for α-particles), and energy distribution factor, N.The dose in Sv for an energy deposition of D in Grays [J/kG] is:
Sv = Q × N × D [J/kG]Rad equivalent man (rem) is a unit of radiation dose equivalent approximately equalto 0.01 Sv for hard x-rays.
1 joule = 1 J = 1 W s = 1 N m = 1 kg m2 s−2
1 joule = 107 erg
1 newton = 1 N = 1 kg m s−2
1 dyne = 1 g cm s−2 = 10−5 N
1 erg = 1 dyne cm = 1 g cm2 s−2
1 Pascal = 1 Pa = 1 N m−2
1 coulomb = 1 C = 1 A s
1 ampere = 1 A = 1 C/ s
1 volt = 1 V = 1 W A−1 = 1 m2 kg A−1 s−3
732 Appendix
1 ohm = 1 Ω = 1 V A−1 = 1 m2 kg A−2 s−3
1 farad = 1 F = 1 C V−1 = 1 m−2 kg−1 A2 s4
1 henry = 1 H = 1 Wb A−1 = 1 m2 kg A−2 s−2
1 tesla = 1 T = 10,000 gauss = 1 Wb m−2 = 1 V s m−2 = 1 kg s−2 A−1
Conversion Factors
1 Å = 0.1 nm = 10−4 µm = 10−10 m
1 b (barn) = 10−24 cm2
1 eV = 1.6045 × 10−12 erg
1 eV/atom = 23.0605 kcal/mole = 96.4853 kJ/mole
1 cal = 4.1840 J
1 bar = 105 Pa
1 torr = 1 T = 133 Pa
1 C (coulomb) = 6.241 × 1018 electrons
1 kG = 5.6096 × 1029 MeV c−2
Useful Facts
energy of 1 Å photon = 12.3984 keV
hν for 1012 Hz = 4.13567 meV
1 meV = 8.0655 cm−1
temperature associated with 1 eV = 11,600 K
lattice parameter of Si (in vacuum at 22.5 °C) = 5.431021 Å
Neutron Wavelengths, Energies, Velocities
En = 81.81λ−2 (energy-wavelength relation for neutrons [meV, Å])
λn = 3955.4/vn (wavelength-velocity relation for neutrons [Å, m/s])
En = 5.2276 × 10−6v2n (energy-velocity relation for neutrons [meV, m/s])
A.13 Fundamental and Derived Constants 733
Some X-Ray Wavelengths [Å]
Element Kα Kα1 Kα2 Kβ1
Cr 2.29092 2.28962 2.29351 2.08480
Co 1.79021 1.78896 1.79278 1.62075
Cu 1.54178 1.54052 1.54433 1.39217
Mo 0.71069 0.70926 0.71354 0.632253
Ag 0.56083 0.55936 0.56377 0.49701
Relativistic Electron Wavelengths
For an electron of energy E [keV] and wavelength λ [Å]:
λ = h
[2meE
(1 + E
2mec2
)]−1/2
= 0.3877
E1/2(1 + 0.9788 × 10−3E)1/2
kinetic energy ≡ T = 1
2mev
2 = 1
2E
1 + γ
γ 2.
Table A.4 Parameters ofhigh-energy electrons E [keV] λ [Å] γ v [c] T [keV]
100 0.03700 1.1957 0.5482 76.79
120 0.03348 1.2348 0.5867 87.94
150 0.02956 1.2935 0.6343 102.8
200 0.02507 1.3914 0.6953 123.6
300 0.01968 1.587 0.7765 154.1
400 0.01643 1.7827 0.8279 175.1
500 0.01421 1.9785 0.8628 190.2
1000 0.008715 2.957 0.9411 226.3
Bibliography
Further Reading
C.C. Ahn (ed.), Transmission Electron Energy Loss Spectroscopy in Materials Science and theEELS Atlas (Wiley-VCH, Weinheim, 2004). An updated 2nd edition of the Disko, Ahn and Fultzbook by the same name. A practical reference covering EELS instrumentation, quantification, finestructure, and applications to the different classes of materials. Includes a CD ROM with the EELSAtlas.
C.C. Ahn, O.L. Krivanek, EELS Atlas (Gatan, Inc., Pleasanton, CA, 1983). The standard referencepresenting EELS spectra of nearly all the elements in the periodic table and some compounds.
S. Amelinckx, R. Gevers, J. Van Landuyt, Diffraction and Imaging Techniques in Materials Science(North-Holland, Amsterdam, 1978). Excellent chapters on kinematical and dynamical electrondiffraction, the WBDF technique, computed electron micrographs, Kikuchi diffraction and defectsin materials.
L.V. Azároff, Elements of X-Ray Crystallography (McGraw-Hill, New York, 1968), reprintedby TechBooks, Fairfax, VA. Emphasizes crystal structure and symmetry determination by x-raydiffractometry.
B.W. Batterman, H. Cole, Rev. Mod. Phys. 36, 681–717 (1964). A systematic presentation of thedynamical theory of x-ray diffraction based on Maxwell’s equations.
J.M. Cowley, Diffraction Physics, 2nd edn. (North-Holland Publishing, Amsterdam, 1975). Thor-ough but concise presentation of the physical optics approach to diffraction and imaging, scatteringof radiation by atoms and crystals, kinematical and dynamical diffraction, and applications to se-lected topics.
B.D. Cullity, S.R. Stock, Elements of X-Ray Diffraction (Prentice-Hall, Upper Saddle River, NJ,2001). A popular introductory text on x-ray diffraction—provides physical explanations of manytopics.
M. De Graef, Introduction to Conventional Transmission Electron Microscopy (Cambridge Uni-versity Press, Cambridge, 2003). Much more than an introduction, this textbook provides excellentand thorough coverage of electron optics, crystallography, and defect contrast in dynamical theory.Computational methods are presented with an accompanying website.
M.M. Disko, C.C. Ahn, B. Fultz (eds.), Transmission Electron Energy Loss Spectroscopy in Ma-terials Science (Minerals, Metals & Materials Society, Warrendale, PA, 1992). A practical refer-
B. Fultz, J. Howe, Transmission Electron Microscopy and Diffractometry of Materials,Graduate Texts in Physics,DOI 10.1007/978-3-642-29761-8, © Springer-Verlag Berlin Heidelberg 2013
735
736 Bibliography
ence covering EELS instrumentation, quantification, fine structure, and applications to the differentclasses of materials.
J.A. Eades, Convergent-beam diffraction, in Electron Diffraction Techniques, Volume 1, ed. byJ.M. Cowley (International Union of Crystallography, Oxford University Press, Oxford, 1992).Good overall review of the subject.
J.W. Edington, Practical Electron Microscopy in Materials Science, 1. The Operation and Calibra-tion of the Electron Microscope (Philips Technical Library, Eindhoven, 1974). Easy to understanddiscussion of the optics, alignment and calibration of the TEM.
J.W. Edington, Practical Electron Microscopy in Materials Science, 2. Electron Diffraction in theElectron Microscope (Philips Technical Library, Eindhoven, 1975). Thorough discussion of elec-tron diffraction patterns, Kikuchi lines, and their use in the TEM. Has a good appendix on stereo-graphic projections.
J.W. Edington, Practical Electron Microscopy in Materials Science, 3. Interpretation of Transmis-sion Electron Micrographs (Philips Technical Library, Eindhoven, 1975). Excellent discussion ofdiffraction contrast and quantitative defect analysis in the TEM with many useful examples.
J.W. Edington, Practical Electron Microscopy in Materials Science, 4. Typical Electron Micro-scope Investigations (Philips Technical Library, Eindhoven, 1976). A number of illustrative exam-ples of diffraction and imaging analyses in the TEM.
T. Egami, S.J.L. Billinge, Underneath the Bragg Peaks: Structural Analysis of Complex Materials(Pergamon Materials Series, Elsevier, Oxford, 2003). A book on modern powder diffraction ex-periments, with emphasis on total scattering measurements and pair distribution function analysis.Clear and thorough coverage of theory and practice of experiments with synchrotron radiation andneutron scattering for identifying nanoscale structures and disorder in hard condensed matter.
R.F. Egerton, Electron Energy-Loss Spectroscopy in the Electron Microscope, 2nd edn. (PlenumPress, New York, 1996). Thorough, scholarly and rigorous coverage of EELS instrumentation,electron scattering theory, quantitative EELS analysis, and examples in materials research.
C.T. Forwood, L.M. Clarebrough, Electron Microscopy of Interfaces in Metals and Alloys (AdamHilger IOP Publishing Ltd., Bristol, 1991). Excellent reference on computed electron micrographsof interfaces in materials.
P.J. Goodhew, F.J. Humphreys, Electron Microscopy and Microanalysis (Taylor & Francis Ltd.,London, 1988). Easy-to-follow discussions of electron optics in the TEM, electron beam-specimeninteractions, electron diffraction and imaging, and microanalysis.
P. Grivet, Electron Optics, revised by A. Septier, translated by P.W. Hawkes (Pergamon, Oxford,1965). The electromagnetics of electron optics, with emphasis on electron lenses and the TEM.
C. Hammond, The Basics of Crystallography and Diffraction (International Union of Crystallog-raphy, Oxford University Press, Oxford, 1977). Simple and understandable introduction to crys-tallography and diffraction techniques, with worked examples of structure factor calculations anddiffraction analyses.
A.K. Head, P. Humble, L.M. Clarebrough, A.J. Morton, C.T. Forwood, Computed Electron Micro-graphs and Defect Identification (North-Holland Publishing Company, Amsterdam, 1973). Excel-lent reference on computed electron micrographs based on the Howie–Whelan two-beam theoryof diffraction, including applications and limitations of the technique.
P.B. Hirsch, A. Howie, R.B. Nicholson, D.W. Pashley, M.J. Whelan, Electron Microscopy of ThinCrystals (R.E. Krieger, Malabar, Florida, 1977). A reprinted early book on conventional TEM.Excellent discussions of kinematical and dynamical electron diffraction theory and application to
Further Reading 737
defect analysis in materials. It offers a broad coverage of experimental technique, and for manyyears was the essential text on the subject. Includes worked problems.
J.J. Hren, J.I. Goldstein, D.C. Joy (eds.), Introduction to Analytical Electron Microscopy (PlenumPress, New York, 1979). Good overall book on TEM, providing treatment of electron optics, EDS,EELS, CBED, STEM.
The International Union of Crystallography publishes the International Tables for X-ray Crys-tallography (Kynock Press, Birmingham, England, 1952-), which contain the standard tables ofcrystal symmetry plus a wealth of tabulated data on scattering factors, dispersion corrections, andother details and principles of x-ray data analysis.
O. Johari, G. Thomas, The Stereographic Projection and Its Applications (Interscience Publish-ers, John Wiley & Sons, New York, 1969). Provides stereographic projections and presents theirapplications to problems in materials science.
D.C. Joy, A.D. Romig Jr., J.I. Goldstein (eds.), Principles of Analytical Electron Microscopy(Plenum Press, New York, 1986). Provides a good introduction to electron scattering and elec-tron optics, with emphasis on EDS and EELS spectroscopy. Contains worked examples.
R.J. Keyse, A.J. Garratt-Reed, P.J. Goodhew, G.W. Lorimer, Introduction to Scanning Transmis-sion Electron Microscopy (Springer BIOS Scientific Publishers Ltd., New York, 1998). Practicalexplanation of optics, diffraction, imaging and microanalysis—specifically for the STEM.
H.P. Klug, L.E. Alexander, X-Ray Diffraction Procedures (Wiley-Interscience, New York, 1974).Provides an encyclopedic coverage of experimental methods and many principles of x-ray diffrac-tion.
M.A. Krivoglaz, Theory of X-Ray and Thermal Neutron Scattering by Real Crystals (Plenum, NewYork, 1969). An elegant and formal treatment of scattering from fluctuations with analysis of theircorrelation functions.
M.H. Lorretto, Electron Beam Analysis of Materials (Chapman and Hall, London, 1984). Concisediscussion of most TEM topics, including electron diffraction and imaging, CBED, and microanal-ysis.
S.W. Lovesey, Theory of Neutron Scattering from Condensed Matter Vols. 1 and 2 (ClarendonPress, Oxford, 1984). A high-level theoretical treatment of the fundamentals of neutron scatter-ing by nuclei (Vol. 1) and magnetization (Vol. 2). Concise, rigorous, mathematical, and based onadvanced quantum mechanics and correlation functions.
A.J.F. Metherell, Diffraction of electrons by perfect crystals, in Electron Microscopy in MaterialsScience II, ed. by U. Valdre, E. Ruedl (CEC, Brussels, 1975), p. 387. This is probably the mostdetailed and comprehensive article written on materials analysis using the Bloch wave approach todynamical electron diffraction.
I.C. Noyan, J.B. Cohen, Residual Stress (Springer-Verlag, New York, 1987). A thorough devel-opment of the experiment and theory connecting continuum mechanics to x-ray diffractometry.Includes x-ray lineshape analysis.
S.J. Pennycook, D.E. Jesson, M.F. Chisholm, N.D. Browning, A.J. McGibbon, M.M. McGibbon:Z-contrast imaging in the scanning transmission electron microscope. J. Microsc. Soc. Am. 1,234 (1995). An overview of the principles and practice of Z-contrast imaging in the STEM, withemphasis on chemical and structural information on the atomic scale.
R. Pynn, Neutron Scattering—A Primer (Los Alamos Science, Summer 1990). Available athttp://neutrons.ornl.gov/science/ns_primer.pdf. A short overview of neutron scattering, written fora general scientific audience. Efficient as a quick introduction to the subject. Charming illustra-tions.
738 Bibliography
H. Raether, Excitations of Plasmons and Interband Transitions by Electrons (Springer-Verlag,Berlin and New York, 1980). An in-depth treatment of the low-loss part of EELS spectra.
L. Reimer (ed.), Energy-Filtering Transmission Electron Microscopy (Springer-Verlag, Berlin,1995). Contains detailed theoretical discussions of electron-specimen interactions, EELS instru-mentation, spectroscopic diffraction and imaging techniques.
L. Reimer, Transmission Electron Microscopy: Physics of Image Formation and Microanalysis, 4thedn. (Springer-Verlag, New York, 1997). Comprehensive, scholarly, and rigorous coverage of TEMinstrumentation, imaging and diffraction techniques. Strong emphasis on the underlying physics.Extensive references to recent research.
P. Schattschneider, Fundamentals of Inelastic Electron Scattering (Springer-Verlag, Vienna, NewYork, 1986). The theoretical physics of low-loss EELS spectra, with emphasis on the quantummechanics of scattering using many-body theory.
L.H. Schwartz, J.B. Cohen, Diffraction from Materials (Springer-Verlag, Berlin, 1987). Provides athorough treatment of x-ray theory and experiment, including crystallography.
V.F. Sears, Neutron Optics (Oxford University Press, New York and Oxford, 1989). An advancedtheoretical treatment of neutron optics, with a few experimental examples. Includes dynamicaltheory and neutron interferometry.
G. Shirane, S.M. Shapiro, J.M. Tranquada, Neutron Scattering with a Triple-Axis Spectrometer(Cambridge University Press, Cambridge, 2002). Advanced coverage of the experimental tech-niques and methods for measuring coherent inelastic neutron scattering with triple axis spectrom-eters. Covers details of optimizing the instrument optics, spurious features in measurements, andexamples from magnetic and nuclear scattering.
F.G. Smith, J.H. Thomson, Optics, 2nd edn. (John Wiley & Sons, New York, 1988). Although thisbook is concerned with light optics, it provides excellent coverage on the subjects of wave propa-gation, geometrical optics, interference and diffraction, resolution, and phase-amplitude diagrams.
J.C.H. Spence, Experimental High-Resolution Electron Microscopy, 2nd edn. (Oxford Univ. Press,New York, 1988). Wide-ranging coverage of the theory and practice of TEM, emphasizingHRTEM.
J.C.H. Spence, J.M. Zuo, Electron Microdiffraction (Plenum Press, New York, 1992). Excellentdiscussion of dynamical electron diffraction and convergent-beam electron diffraction.
G.L. Squires, Introduction to the Theory of Thermal Neutron Scattering (Dover, Mineola, NewYork, 1996). An elegant and tightly-organized development of the theory of elastic and inelasticneutron scattering. Concise, but rigorous when appropriate. A classic text for the field.
J.W. Steeds, Convergent beam electron diffraction, in Introduction to Analytical Electron Mi-croscopy, ed. by J.J. Hren, J.I. Goldstein, D.C. Joy (Plenum Press, New York, 1979), p. 401. Goodoverall discussion of CBED technique and application to materials.
J.W. Steeds, R. Vincent, Use of high-symmetry zone axes in electron diffraction in determiningcrystal point and space groups. J. Appl. Crystallogr. 16, 317 (1983). Provides a useful sequence ofsteps for determining crystal point and space groups from high-symmetry zone axes.
M. Tanaka, M. Terauchi, Convergent-Beam Electron Diffraction (JEOL Ltd., Nakagami, Tokyo,1985). M. Tanaka, M. Terauchi, T. Kaneyama, Convergent-Beam Electron Diffraction II (JEOLLtd., Musashino 3-chome, Tokyo, 1988). These compilations provide a thorough summary ofCBED procedures such as point and space group determination, lattice parameter measurement,etc.
References and Figures 739
G. Thomas, M.J. Goringe, Transmission Electron Microscopy of Materials (Wiley-Interscience,New York, 1979). Good general discussion of TEM techniques, including kinematical and dynam-ical electron diffraction and imaging. Many TEM images of defects in materials with discussion ofpractice. Includes worked problems.
B.E. Warren, X-Ray Diffraction (Addison-Wesley, Reading, MA, 1969), is now a best buy as aDover reprint (Dover, Mineola, NY, 1990). It provides a rigorous coverage of concepts in x-raypowder diffractometry of imperfect crystals.
D.B. Williams, Practical Analytical Electron Microscopy in Materials Science (Philips ElectronInstruments, Inc., Mahwah, NJ, 1984). In-depth discussion of alignment and calibration of theTEM, quantitative x-ray microanalysis and EELS spectrometry with many useful examples.
D.B. Williams, C.B. Carter, Transmission Electron Microscopy: A Textbook for Materials Science(Plenum Press, New York, 1996). Probably the most comprehensive current book on TEM, cover-ing almost all aspects of the technique. Includes both theory and practical examples.
C.G. Windsor, Pulsed Neutron Scattering, (Taylor and Francis, London, 1981). Detailed but prac-tical coverage of the design and performance of neutron scattering instruments at a pulsed neutronsource. Many insights into instrument design, although missing some modern developments.
References and Figures
Chapter 1 title photograph of Inel Corp. CPS-120 x-ray diffractometer with large-angle position-sensitive detector. Radiation shielding not shown.
1.1 International Centre for Diffraction Data, 12 Campus Boulevard Newtown Square, PA19073-3273, USA. http://www.icdd.com
1.2 H.G.J. Moseley, Philos. Mag. 27, 713 (1914)1.3 F. Richtmyer, E. Kennard, Introduction to Modern Physics (McGraw-Hill, New York, 1947)1.4 A partial list of web sites for synchrotron sources includes (prefixed with http://):
aps.anl.gov/, www.esrf.eu/, www.spring8.or.jp/, www-hasylab.desy.de/, slac.stanford.edu/,www.srs.ac.uk/srs/, www.bessy.de/, www.nsls.bnl.gov/, www.als.lbl.gov/, ssrc.inp.nsk.su/
1.5 L.V. Azároff, Elements of X-Ray Crystallography (McGraw-Hill, New York, 1968). Figurereprinted with the courtesy of TechBooks, Fairfax, VA
1.6 National Institute of Standards and Technology, Standard Reference Materials Program,Bldg. 202, Rm 204, Gaithersburg, MD 20899. http://ts.nist.gov/srm
1.7 J. Nelson, D. Riley, Proc. Phys. Soc. (London) 57, 160 (1945)1.8 H.P. Klug, L.E. Alexander, X-Ray Diffraction Procedures (Wiley-Interscience, New York,
1974). Figure reprinted with the courtesy of John Wiley-Interscience
Chapter 2 title drawing of JEOL JEM-2010F. Figure reprinted with the courtesyof JEOL Ltd., Tokyo.
2.1 B. Demczyk, Ultramicroscopy 47, 433 (1993). Figure reprinted with the courtesy of ElsevierScience Publishing B.V.
2.2 J.M. Howe, W.E. Benson, A. Garg, Y.C. Chang, Mat. Sci. Forum 189–190, 255 (1995).Figure reprinted with the courtesy of Trans Tech Publications Ltd.
2.3 Near the year 2007, manufacturers of TEM instruments include JEOL, FEI, Hitachiand Zeiss. A partial list of web sites for manufacturers of TEM instruments includes:www.jeol.com/, www.fei.com/, www.hitachi-hta.com/, www.smt.zeiss.com/
2.4 Figure reprinted with the courtesy of FEI Company
740 Bibliography
2.5 P.J. Goodhew, F.J. Humphreys, Electron Microscopy and Analysis, 2nd edn. (Taylor & Fran-cis, Ltd., London, 1975). Figure reprinted with the courtesy of Taylor & Francis, Ltd.
2.6 Figure reprinted with the courtesy of Prof. M.K. Hatalis2.7 M. Bilaniuk, J.M. Howe, Interface Sci. 6, 328 (1998). Figure reprinted with the courtesy of
Kluwer Academic Publishers2.8 D.B. Williams, Practical Analytical Electron Microscopy in Materials Science (Philips
Electron Optics Publishing Group, Mahwah, NJ, 1984). Figure reprinted with the courtesyof FEI Company
2.9 D. Alloyeau et al., Nat. Mater. 8, 940 (2009)2.10 A. Hirata et al., Nat. Mater. 10, 28 (2011)2.11 Figure courtesy of Dr. Simon Nieh2.12 L. Reimer, Transmission Electron Microscopy: Physics of Image Formation and Micro-
analysis, 4th edn. (Springer-Verlag, New York, 1997). Figure reprinted with the courtesy ofSpringer-Verlag
2.13 F.W. Sears, M.W. Zemansky, University Physics, 4th edn. (Addison-Wesley–LongmanPublishing, Reading, MA, 1973). Figure reprinted with the courtesy of Addison-Wesley–Longman Publishing
2.14 J.W. Edington, Practical Electron Microscopy in Materials Science, 1. The Operation andCalibration of the Electron Microscope (Philips Technical Library, Eindhoven, 1975). Fig-ure reprinted with the courtesy of FEI Company
Chapter 3 title image shows the Spallation Neutron Source at Oak Ridge NationalLaboratory, Tennessee. The lab and office module is in the foreground, the acceler-ator in the back. The target building with neutron instruments is in the middle of theimage. Figure reprinted with the courtesy of the Oak Ridge National Laboratory.
3.1 E. Svab, G. Meszaros, F. Deak, Mat. Sci. Forum 228, 247 (1996). See also: www.bnc.hu/.Figure reprinted with the courtesy of E. Svab
3.2 Figure reprinted with the courtesy of the Lujan Neutron Scattering Center3.3 Figure reprinted with the courtesy of the Oak Ridge National Laboratory3.4 Unpublished data courtesy of H.L. Smith3.5 Data originally from Neutron News 3, 29 (1992), compiled at www.ncnr.nist.gov/
resources/n-lengths/
Chapter 4 title image conveys the important concept of Fig. 4.7.
4.1 of the Huntington Library, Art Collections, and Botanical Gardens, San Marino, CA4.2 Acta Crystallogr. A49, 231 (1993)4.3 Nature 401, 49 (1999). Figure reproduced with the courtesy of Nature and J.C.H. Spence
Chapter 5 title drawing of Gatan 666 EELS spectrometer. Figure reprinted withthe courtesy of Dr. C.C. Ahn.
5.1 D.H. Pearson, Measurements of white lines in transition metals and alloys using electronenergy loss spectrometry, Ph.D. Thesis, California Institute of Technology, California, 1991.Figure reprinted with the courtesy of Dr. D.H. Pearson
5.2 M.M. Disko, Transmission electron energy-loss spectrometry in materials science, in Trans-mission Electron Energy Loss Spectroscopy in Materials Science, ed. by M.M. Disko,C.C. Ahn, B. Fultz (Minerals, Metals & Materials Society, Warrendale, PA, 1992).Reprinted with courtesy of The Minerals, Metals & Materials Society
5.3 J.K. Okamoto, Temperature-dependent extended electron energy loss fine structure mea-surements from K , L23, and M45 edges in metals, Intermetallic alloys, and nanocrys-talline materials, Ph.D. Thesis, California Institute of Technology, California, 1993. Figurereprinted with the courtesy of Dr. J.K. Okamoto
References and Figures 741
5.4 A. Hightower, Lithium electronic environments in rechargeable battery electrodes, Ph.D.Thesis, California Institute of Technology, California, 2000
5.5 R.F. Egerton, Electron Energy-Loss Spectroscopy in the Electron Microscope, 2nd edn.(Plenum Press, New York, 1996). Figures reprinted with the courtesy of Plenum Press
5.6 D.B. Williams, C.B. Carter, Transmission Electron Microscopy: A Textbook for MaterialsScience (Plenum Press, New York, 1996). Figure reprinted with the courtesy of PlenumPress
5.7 R.D. Leapman, EELS quantitative analysis, in Transmission Electron Energy Loss Spec-troscopy in Materials Science, ed. by M.M. Disko, C.C. Ahn, B. Fultz (Minerals, Metals &Materials Society, Warrendale, PA, 1992). Reprinted with courtesy of The Minerals, Metals& Materials Society
5.8 Figure reprinted with the courtesy of K.T. Moore5.9 D.B. Williams, Practical Analytical Electron Microscopy in Materials Science (Philips
Electron Optics Publishing Group, Mahwah, NJ, 1984). Figure reprinted with the courtesyof FEI Company
5.10 E.H.S. Burhop, The Auger Effect and Other Radiationless Transitions (Cambridge Univer-sity Press, 1952). Figure reprinted with the permission of Cambridge University Press
5.11 Figure reprinted with the courtesy of Dr. K.M. Krishnan5.12 Figure reprinted with the courtesy of C.M. Garland5.13 C. Nockolds, M.J. Nasir, G. Cliff, G.W. Lorimer, in Electron Microscopy and Analysis—
1979, ed. by T. Mulvey (The Institute of Physics, Bristol and London, 1980), p. 4175.14 J.M. Howe, R. Gronsky, Scripta Metall. 20, 1168 (1986). Figure reprinted with the courtesy
of Elsevier Science Ltd.
Chapter 6 title image of electron diffraction pattern from precipitates in an Al–Cu–Li alloy.
6.1 The International Union of Crystallography, International Tables for X-ray Crystallography(Kynock Press, Birmingham, England, 1952-)
6.2 Figure reprinted with the courtesy of Dr. S.R. Singh6.3 Y.C. Chang, Crystal structure and nucleation behavior of {111} Precipitates in an Al-3.9Cu-
0.5Mg-0.5Ag alloy, Ph.D. Thesis, Carnegie Mellon University, Pittsburgh, PA, 1993. Figurereprinted with the courtesy of Dr. Y.C. Chang
6.4 R.J. Rioja, D.E. Laughlin, Metall. Trans., 8A, 1259 (1977). Figure reprinted with the cour-tesy of The Minerals, Metals and Materials Society
6.5 G. Thomas, M.J. Goringe, Transmission Electron Microscopy of Materials (Wiley-Interscience, New York, 1979). Figure reprinted with the courtesy of Wiley-Interscience
6.6 Figure and problem reprinted with the courtesy of Prof. D.E. Laughlin6.7 F.K. LeGoues, H.I. Aaronson, Y.W. Lee, G.J. Fix, in Proceedings of the International Con-
ference on Solid-Solid Phase Transformations ed. by H.I. Aaronson, D.E. Laughlin, R.F.Sekerka, C.M. Wayman (TMS-AIME, Warrendale, PA, 1982), p. 427. Figure reprinted withthe courtesy of The Minerals, Metals and Materials Society
Chapter 7 title image of Kikuchi map of bcc crystal. G. Thomas, M.J. Goringe,Transmission Electron Microscopy of Materials (Wiley-Interscience, New York,1979). Figure reprinted with the courtesy of Wiley-Interscience.
7.1 G. Thomas, M.J. Goringe, Transmission Electron Microscopy of Materials (Wiley-Interscience, New York, 1979). Figure reprinted with the courtesy of Wiley-Interscience
7.2 J.W. Edington, Practical Electron Microscopy in Materials Science, 2. Electron Diffractionin the Electron Microscope (Philips Technical Library, Eindhoven, 1975). Figure reprintedwith the courtesy of FEI Company
7.3 Dr. J.-S. Chen, unpublished results
742 Bibliography
7.4 M. Tanaka, M. Terauchi, Convergent-Beam Electron Diffraction (JEOL Ltd., Nakagami,Tokyo, 1985). Figures reprinted with the courtesy of JEOL, Ltd. Worked thickness exampleon pp. 38–39
7.5 R. Ayer, J. Electron Micros. Tech. 13, 16 (1989). Figure reprinted with the courtesy of AlanR. Liss, Inc.
7.6 S.J. Rozeveld, Measurement of residual stress in an Al–SiCw composite by convergent-beam electron diffraction, Ph.D. Thesis, Carnegie-Mellon University, Pittsburgh, PA, 1991.Figure reprinted with the courtesy of Dr. S.J. Rozeveld
7.7 B.F. Buxton et al., Proc. R. Soc. Lond. A281, 188 (1976). B.F. Buxton et al., Philos. Trans.R. Soc. Lond. A281, 171 (1976). Tables reprinted with the courtesy of The Royal Society,London
7.8 M. Tanaka, H. Sekii, T. Nagasawa, Acta Crystallogr. A39, 825 (1983). Figure reprinted withthe courtesy of the International Union of Crystallography
7.9 M. Tanaka, R. Saito, H. Sekii, Acta Crystallogr. A39, 359 (1983). Figure reprinted with thecourtesy of International Union of Crystallography
7.10 J.M. Howe, M. Sarikaya, R. Gronsky, Acta Crystallogr. A42, 371 (1986). Figure reprintedwith the courtesy of International Union of Crystallography
7.11 The International Union of Crystallography, International Tables for X-ray Crystallography(Kynock Press, Birmingham, England, 1952-)
7.12 J.W. Steeds, R. Vincent, Use of high-symmetry zone axes in electron diffraction in deter-mining crystal point and space groups. J. Appl. Crystallogr. 16 317 (1983)
7.13 J. Gjønnes, A.F. Moodie, Acta Crystallogr. 19, 65 (1965)7.14 M.J. Kaufman, H.L. Fraser, Acta Metall. 33, 194 (1985). Figure reprinted with the courtesy
of Elsevier Science Ltd.
Chapter 8 title image of dislocations in aluminum.
8.1 P.B. Hirsch, A. Howie, R.B. Nicholson, D.W. Pashley, M.J. Whelan, Electron Microscopyof Thin Crystals (R.E. Krieger, Malabar, Florida, 1977). Figure reprinted with the courtesyof R.E. Krieger
8.2 J.W. Edington, Practical Electron Microscopy in Materials Science, 3. Interpretation ofTransmission Electron Micrographs (Philips Technical Library, Eindhoven, 1975). Figurereprinted with the courtesy of FEI Company
8.3 Figure reprinted with the courtesy of Dr. Y.C. Chang8.4 G. Thomas, M.J. Goringe, Transmission Electron Microscopy of Materials (Wiley-
Interscience, New York, 1979). Figure reprinted with the courtesy of Wiley-Interscience8.5 Figure reprinted with the courtesy of Dr. S.R. Singh8.6 J.M. Howe, H.I. Aaronson, R. Gronsky, Acta Metall. 33, 641 (1985). Figure reprinted with
the courtesy of Elsevier Science Ltd.8.7 P.B. Hirsch, A. Howie, M.J. Whelan, Philos. Trans. R. Soc. (London) 252A, 499 (1960)8.8 D.J.H. Cockayne, I.L.F. Ray, M.J. Whelan, Philos. Mag. 20, 1265 (1969). D.J.H. Cockayne,
M.L. Jenkins, I.L.F. Ray, Philos. Mag. 24, 1383 (1971)8.9 L. Reimer, Transmission Electron Microscopy: Physics of Image Formation and Micro-
analysis, 4th edn. (Springer-Verlag, New York, 1997). Figure reprinted with the courtesy ofSpringer-Verlag
8.10 A. Garg, J.M. Howe, Acta Metall. Mater. 39, 1934 (1991). A. Garg, Y.C. Chang, J.M. Howe,Acta Metall. Mater. 41, 240 (1993). Figures reprinted with the courtesy of Elsevier ScienceLtd.
8.11 J.W. Edington, Practical Electron Microscopy in Materials Science, 3. Interpretation ofTransmission Electron Micrographs (Philips Technical Library, Eindhoven, 1975), p. 40.R. Gevers, A. Art, S. Amelinckx, Phys. Stat. Sol. 3, 1563 (1963)
References and Figures 743
8.12 N. Prabhu, J.M. Howe, Philos. Mag. A 63, 650 (1991). Figure reprinted with the courtesyof Taylor & Francis, Ltd.
8.13 M.F. Ashby, Brown, Philos. Mag. 8, 1083 (1963)8.14 H.P. Degischer, Philos. Mag. 26, 1147 (1972). Figure reprinted with the courtesy of Taylor
& Francis, Ltd.8.15 M. Hwang, D.E. Laughlin, I.M. Bernstein, Acta Metall. 28, 629 (1980). Figure reprinted
with the courtesy of Elsevier Science Ltd.8.16 Figure reprinted with the courtesy of Dr. A. Garg
Chapter 9 title figure of (400)fcc diffraction from a nanocrystalline iron alloy(Mo Kα radiation).
9.1 H.P. Klug, L.E. Alexander, X-Ray Diffraction Procedures (Wiley-Interscience, New York,1974), pp. 687–692
9.2 H.P. Klug, L.E. Alexander, X-Ray Diffraction Procedures (Wiley-Interscience, New York,1974), pp. 655–665
9.3 B.E. Warren, X-Ray Diffraction (Dover, Mineola, NY, 1990), pp. 251–2759.4 H. Frase, Vibrational and magnetic properties of mechanically attrited Ni3Fe nanocrystals,
Ph.D. Thesis, California Institute of Technology, California, 1998
Chapter 10 title image conveys the important concept of Fig. 10.3.
10.1 B.E. Warren, X-Ray Diffraction (Dover, Mineola, NY, 1990), pp. 178–19310.2 F. Ducastelle, Order and Phase Stability in Alloys (North-Holland, Amsterdam, 1991), pp.
439–442. This “relaxation energy” is important for the thermodynamics of many alloys10.3 J.A. Rodriguez, S.C. Moss, J.L. Robertson, J.R.D. Copley, D.A. Neumann, J. Major, Phys.
Rev. B 74, 104115 (2006).10.4 B.E. Warren, X-Ray Diffraction (Dover, Mineola, NY, 1990), pp. 206–25010.5 L.H. Schwartz, J.B. Cohen, Diffraction from Materials (Springer-Verlag, Berlin, 1987), pp.
407–40910.6 J.M. Cowley, Diffraction Physics, 2nd edn. (North-Holland Publishing, Amsterdam, 1975),
pp. 152–15410.7 A. Williams, Atomic structure of transition metal based metallic glasses, Ph.D. Thesis, Cal-
ifornia Institute of Technology, California, 198110.8 H.P. Klug, L.E. Alexander, X-Ray Diffraction Procedures (Wiley-Interscience, New York,
1974), pp. 791–85910.9 T. Egami, PDF analysis applied to crystalline materials, in Local Structure from Diffraction,
ed. by S.J.L. Billinge, M.F. Thorpe (Plenum, New York, 1998), pp. 1–2110.10 A. Guinier, X-Ray Diffraction in Crystals, Imperfect Crystals, and Amorphous Bodies
(Dover, Mineola, NY, 1994), pp. 344–349
Chapter 11 title image of Pb precipitate in Al. Figure reprinted with the courtesyof U. Dahmen.
11.1 C. Weissbacker, H. Rose, J. Electron Microsc. (Tokyo) 50, 383 (2001)11.2 J.M. Cowley, A.F. Moodie, Acta Crystallogr. 10, 609 (1957). Ibid. 12, 353, 360, 367 (1959)11.3 M.A. O’Keefe, Electron image simulation: A complementary processing technique, in Pro-
ceedings of the 3rd Pfeffercorn Conference on Electron Optical Systems, Ocean City, MDed. by J.J. Hren, F.A. Lenz, E. Munro, P.B. Sewell, S.A. Bhatt (Scanning Electron Mi-croscopy, Inc., Illinois, 1984), pp. 209–220
11.4 R.R. Meyer, J. Sloan, R.E. Dunin-Borkowski, A.I. Kirkland, M.C. Novotny, S.R. Bailey, J.L.Hutchison, M.L.H. Green, Science 289, 1324 (2000). Figure reproduced with the courtesyof J.L. Hutchison and the American Association for the Advancement of Science
744 Bibliography
11.5 S.D. Hudson, H.T. Jung, V. Percec, W.D. Cho, G. Johansson, G. Ungar, V.S.K. Balagu-rusamy, Science 278, 449 (1997). Figure reproduced with the courtesy of S.D. Hudson andthe American Association for the Advancement of Science
11.6 S.R. Singh, J.M. Howe, Philos. Mag. A 66, 746 (1992). Figure reprinted with the courtesyof Taylor & Francis, Ltd.
11.7 S. Das, J.M. Howe, J.H. Perepezko, Metall. Mater. Trans. 27A, 1627 (1996). Figurereprinted with the courtesy of The Minerals, Metals & Materials Society
11.8 G. Rao, J.M. Howe, P. Wynblatt, Unpublished research11.9 U. Dahmen, Micros. Soc. Am. Bull. 24, 341 (1994). Figure reprinted with the courtesy of
Microscopy Society of America11.10 Figure reprinted with the courtesy of R. Gronsky and D. Acklund11.11 J.M. Howe, S.J. Rozeveld, J. Micros. Res. Tech. 23, 233 (1992). Reprinted with the courtesy
of Wiley-Liss, Inc.11.12 M.M. Tsai, Determination of the growth mechanisms of TiH in Ti using high-resolution
and energy-filtering transmission electron microscopy, Ph.D. Thesis, University of Virginia,Charlottesville, VA, 1997. Figure reprinted with the courtesy of Dr. M.M. Tsai
11.13 Such as Gatan Digital MicrographTM or NIH Image11.14 B. Laird, J.M. Howe, Unpublished research11.15 R. Kilaas, R. Gronsky, Ultramicroscopy 16, 193 (1985). Figure reprinted with the courtesy
of Elsevier Science Publishing B.V.11.16 J.O. Malm, M.A. O’Keefe, Ultramicroscopy 68, 13 (1997)11.17 S.-C.Y. Tsen, P.A. Crozier, J. Liu, Ultramicroscopy 98, 63 (2003)11.18 M.J. Hytch, J.-L. Putaux, J.-M. Penisson, Nature 423, 270 (2003)
Chapter 12 title figure shows HAADF images acquired with a Cs-corrected in-strument. The images were acquired at different values of defocus as labeled. To-gether with other measurements and computational support, the images show howthat La atoms segregate to sites on the surfaces of an Al2O3 crystal, which corre-spond to defocus values of 0 and −8 nm. Bar length is 1 nm. After S. Wang, A.Y.Borisevich, S.N. Rashkeev, M.V. Glazoff, K. Sohlberg, S.J. Pennycook, S.T. Pan-telides, Nat. Mat. 3, 143 (2004).
12.1 After N.D. Browning, D.J. Wallis, P.D. Nellist, S.J. Pennycook, Micron 28, 334 (1997).Reprinted with the courtesy of Elsevier Science Ltd.
12.2 S.J. Pennycook, D.E. Jesson, M.F. Chisholm, N.D. Browning, A.J. McGibbon, M.M.McGibbon, J. Micros. Soc. Am. 1, 234 (1995). Reprinted with the courtesy of MicroscopySociety of America
12.3 A. Amali, P. Rez, Microsc. Microanal. 3, 28 (1997)12.4 A.R. Lupini, S.J. Pennycook, Ultramicroscopy 96, 313 (2003)12.5 P.M. Voyles, D.A. Muller, private communication. See also P.M. Voyles, D.A. Muller, J.L.
Grazul, P.H. Citrin, H.-J.L. Gossmann, Nature 416, 826 (2002)12.6 O.L. Krivanek, N. Dellby, A.R. Lupini, Ultramicroscopy 78, 1 (1999)12.7 S. Uhlemann, M. Haider, Ultramicroscopy 72, 109 (1998)12.8 Q.M. Ramasse, A.L. Bleloch, Ultramicroscopy 106, 37 (2005)12.9 H. Müller, S. Uhlemann, P. Hartel, M. Haider, Microsc. Microanal. 12, 442 (2006)
12.10 M. Lentzen, Microsc. Microanal. 12, 191 (2006)12.11 A.Y. Borisevich, A.R. Lupini, S.J. Pennycook, Proc. Natl. Acad. Sci. 103, 3044 (2006)12.12 K. van Benthem, A.R. Lupini, M. Kim, K.-S. Baik, S. Doh, J.-H. Lee, M.P. Oxley, S.D. Find-
lay, L.J. Allen, J.T. Luck, S.J. Pennycook, Appl. Phys. Lett. 87, 034104 (2005). Reprintedwith the courtesy of the American Institute of Physics
12.13 N.D. Browning, R.P. Erni, J.C. Idrobo, A. Ziegler, C.F. Kisielowski, R.O. Ritchie, Microsc.Microanal. 11 (Suppl 2), 1434 (2005)
References and Figures 745
12.14 J. Frank (ed.), Electron Tomography: Methods for Three-Dimensional Visualization ofStructures in the Cell, 2nd edn. (Springer Science and Business Media, LLC, New York,2006)
12.15 S. Van Aert, K.J. Batenburg, M.D. Rossell, R. Erni, G. Van Tendeloo, Nature 470, 374(2011)
12.16 M. Tanaka, S. Sadamatsu, G.S. Liu, H. Nakamura, K. Higashida, I.M. Robertson. J. Mater.Res. 26, 508 (2011)
Chapter 13 title figure is an enlargement of Fig. 13.15.
13.1 Figure reprinted with the courtesy of Dr. Y.C. Chang13.2 P.B. Hirsch, A. Howie, R.B. Nicholson, D.W. Pashley, M.J. Whelan, Electron Microscopy
of Thin Crystals (R.E. Krieger, Malabar, Florida, 1977), pp. 222–24213.3 N. Prabhu, J.M. Howe, Philos. Mag. A 63, 650 (1991). Figure reprinted with the courtesy
of Taylor & Francis, Ltd.13.4 A.W. Wilson, Microstructural examination of NiAl alloys, Ph.D. Thesis, University of Vir-
ginia, Charlottesville, VA, 1999. Figure reprinted with the courtesy of Dr. A.W. Wilson13.5 P. Rez, Private communication of academic course notes
Index
Aα boundary, 412Aberrations, 98, 211, 213, 605, 607
allowed by symmetry, 603higher order, 605
Absorption and thin-film approximation(table), 229
Absorption correction, 41, 228flat specimen, 41granularity, 45validity, 45
Absorption edge, 185Absorption (electron incoherence), 619, 633,
665Accidental degeneracy, 16Ag precipitate, 613Ag–Cu interface, 568Aharonov–Bohm effect, 63Al–4wt.% Cu alloy, 271Al–Cu, 725Al–Cu θ ′ phase, 284Al–Ge interface, 569Al–Li alloy, 71Al12Mn, 74ALCHEMI, 672Alignment, 721Amateur mistake, 278, 282, 387, 727
small-angle scattering, 520Amorphous material, 496
one-dimensional, 496Analytical TEM, 60, 182Analyzer, 133Anisotropy, elastic, 386, 418, 435Anomalous scattering, 158, 468
partial pair correlations, 505Antibonding orbitals, 186Antiphase boundary, 411
superlattice diffraction, 412Antisite, 261Aperture angle, 66, 86, 111, 608
optimum, 107Apertureless image, 70ARCS, 135Artificial rays, 67Ashby–Brown contrast, 415Astigmatism, 101, 534, 571, 607
correction procedure, 103, 535salt and pepper contrast, 104
Atom, 1, 161, 601, 609as point, 464
Atomic displacement disorder, 475Atomic form factor, 239
dependence on Δk, 176destructive interference at angles, 169effective Bohr radius, 172electron, table of, 689electrons and x-rays, 175model potentials, 170Mott formula, 175physical picture, 167Rutherford, 172screened Coulomb potential, 170sensitivity to bonding electrons, 176shape of atom, 167, 176shapes of V (r), 176Thomas–Fermi, 172x-ray, table of, 684
Atomic periodicities, resolution of, 81Atomic size effect, 483Auger effect, 12, 236Autocorrelation function, 2, 466Average potential of solid, 619Avogadro constant, 730
B. Fultz, J. Howe, Transmission Electron Microscopy and Diffractometry of Materials,Graduate Texts in Physics,DOI 10.1007/978-3-642-29761-8, © Springer-Verlag Berlin Heidelberg 2013
747
748 Index
Axial dark-field imaging, 71, 277, 278, 388Axial divergence, 25, 436
BB2 structure, 258, 261, 569Back focal plane, 65, 69Background, 55
subtraction and integration, 47Backscattered electron image (BEI), 216Backscattered electrons, 216Backscattering spectrometer, 136Bar, 732Barn, 732Barrier penetration, 593Basis vectors, 249Beam propagation, 629Beam representation, 618Beam tilt
coils, 573dislocation position, 585HRTEM, 555, 571
Beams and Bloch waves, 635normalization, 636
Beamtime proposal, 121Beats
acoustic, 633mathematical analysis, 638pattern, 638physical picture, 633
Becquerel, 731Beer’s law, 221Bend contour, 364, 421
Cu–Co, 367diffraction patterns, 366
Bending magnets, 20Bessel function, 268Bethe asymptotic cross-section, 207Bethe ridge, 196Bethe surface, 203, 204Biology, 71, 612Biprism, 63Black cross, 339Bloch waves, 633
amplitudes and dispersion surface, 659change across defect, 662channeled, 595characteristics, 643energies, 642orthogonality, 641propagator, 665representation, 618weighting coefficients, 636
Block diagram of a TEM, 60Blue Boy, 161
Blue sky, 155Boersch effect, 100Bohr magneton, 730Bohr radius
dependence on Z, 172effective, 172
Born approximation, 165, 238, 625first, 165higher order, 165
Bose–Einstein statistics, 482Boundary conditions, 650Bragg–Brentano geometry, 26Bragg’s law, 3, 123Bremsstrahlung
coherent, 57intensity, 15, 23
Bright-field (BF) imaging, 65, 68, 69, 349,364, 546, 550, 654
Brightness, 21, 104compromises, 185conservation of, 105electron gun, 109, 608
Brilliance, 21Broadening of x-ray peaks, 2, 6
complement of TEM, 458dislocation, 458meaning of size and strain, 458stacking faults, 450
Brockhouse, 133Buckled specimen, 364, 426Bullet, 119Burgers circuit, 715
in HRTEM image, 81Burgers vector, 373, 567, 715
conservation of, 719fcc, 376
CCs corrector, 79, 557, 602Calorie, 732Camera constant, 75
calibration, 76determination of, 722
Camera equation, 74Camera-length, 74, 86Carrier, 81Catalyst, 565Cauchy function, 438CCD cameras, 387, 578CdSe, 610Center of gravity, 509Center of the goniometer, 48Channeling, 590Characteristic x-ray, 12, 15, 23
Index 749
Chemical bonding, 187Chemical disorder, 475, 486Chemical map, 60Chemical ordering, 258Chemical short-range order, 486, 490Children’s jacks, 280Chromatic aberration, 99, 212, 213, 607
importance of thin specimens, 100Classical electron radius, 139, 155, 730Cliff–Lorimer factor, 225
calculation, 228experimental determination, 228
Coherence, 148Coherent bremsstrahlung, 57Coherent elastic scattering, 149Coherent imaging, 587Coherent scattering, 145
forward direction, 168, 507inelastic, 149, 712phases, 154
Coherent scattering length, 140Cold neutrons, 131Column approximation, 452Column lengths
distribution, 454neighbor pairs in column, 456random termination, 455
Coma, 607Complementarity of BF and DF, 71Complementary variables, 118Compton scattering, 158
incoherence, 159Computer control, 605Concentration variables, 262Condenser lens
aperture, 85convergence (C2), 85spot size (C1), 85
Constant Q scan, 133Constants, 730Constructive interference, 3, 145Contrast transfer function, 553
damping of, 556incoherent vs. coherent, 598
Conventional modes, 68Conventional TEM, 76Convergence angle control, 223Convergent-beam electron diffraction (CBED),
77, 318α-Ti, 334, 341, 342BF disk symmetry, 329black cross, 339DF disk symmetry, 329diffraction group, 329, 330
disk and crystal symmetry, 330disk intensity nonuniformity, 77Ewald sphere, 318FeS2, 339Friedel’s law, 328G disk symmetry, 330Gjønnes–Moodie lines, 339glide plane, 339HOLZ lines and lattice parameter, 325HOLZ radius Gn, 323illumination, 77intensity oscillations in disk, 320point group, 328positions of disks, 324
orthorhombic examples, 325projection diffraction group, 329sample thickness determination, 320screw axis, 339semi-angle of convergence, 319space group, 335, 339, 341special positions, 337symmetric many-beam, 337unit cell, 322whole pattern symmetry, 329
Conversion factors, 732Convolution
commutative property, 469defined, 438delta function, 465, 470example, 437Gaussians, 438Lorentzians, 438of potential and beams, 624theorem, 441Voigt function, 439
Core excitations, table of energies, 195Core hole, 188
decay, 182Cornu spiral, 533, 584Corrector
Cc/Cs, 557Cs, 603
Correlations, 490short-range, 490
Costs, 607Coulombic interaction, 198, 597Coupled harmonic oscillators, 631, 678Cowley–Moodie method, 676Critical angle, 130, 131, 592Crystal potential
inversion symmetry, 625real, 625
Crystal symmetry elements, 330Crystal system notation, 255
750 Index
Crystallite sizesdistribution and TEM, 459Patterson function, 473TEM and x-ray, 458
CsCl, 261Cu–Co, 418Cu2O, 178Curie, 731
Dδ boundary, 413d-orbitals (shapes), 178Damping function, 563Dark-field (DF) imaging, 65, 68, 69, 349, 546,
550, 654De Broglie wavelength, 119Dead time, 29Debye model, 481Debye–Scherrer, 10Debye–Waller factor, 188, 478, 589, 711
calculation of, 481concept, 480conventions, 481
Deconvolution, 439Fourier transform procedure, 439frequency filter, 445procedure with noise, 444
Defects, 350Delta function, 465Density, 43Density heterogeneity, 502Density of unoccupied states, 187Density-density correlations, 514Depth of field, 86, 111, 608Depth of focus, 86, 111Detector
analytic TEM, 33annular dark-field, 588beryllium window, 32calorimetric, 30charge sensitive preamplifier, 34count rates, 29dead layer, 31energy resolution, 29escape peak, 31gas-filled proportional counter, 30intrinsic semiconductor, 31position-sensitive, 33quantum efficiency, 29scintillation counter, 30SDD, 33Si[Li], 33silicon drift, 32solid state, 31
table of characteristics, 30x-ray, 29
Deviation parameter, s, 272, 351, 355effective, 651Kikuchi lines, 313
Deviation vector, s , 272, 351in dynamical theory, 623
Differential scattering cross-section, 152inelastic, 200
DiffractionΔk and θ , 242beams across defect, 664coherence, 243effect of apertures, 100electron, 238fine structure, 280forbidden diffractions, 253Fourier transform of potential, 242frequency and time, 241incident wave, 241line broadening, 429rel-rods, 269shape factor, 249structure factor, 249structure factor rules, 251translational invariance in plane, 8vectors and coordinates, 240wave, 240wavevectors, 242
Diffraction contrast, 60, 69, 349dynamical
dislocation, 661interface, 661stacking fault, 661
dynamical without absorption, 660null contrast, 369strain fields, 368
Diffraction coupling, 184Diffraction lens, 86Diffraction mode, 67Diffraction pattern
background, 55bcc, 697, 698chemical composition, 6crystallite sizes, 8dc, 699, 700fcc, 695, 696hcp, 701–704indexed powder, 681internal strains, 6, 433internal stresses, 434peak broadening, 6silicon, 4size effect broadening, 6, 431
Index 751
Diffraction vector, 80Diffuse scattering, 463
chemical disorder, 489, 595displacement disorder, 483Huang, 484short-range order (SRO), 493thermal, 480, 596
DigitalMicrographTM software, 581Dilatation, 433Dipole approximation, 207Dipole oscillator, 154Dirac δ-function, 234, 465Dirac equation, 16Dirac notation, 197Dirty dark-field technique, 71Disk of least confusion, 99
resolution, 107Dislocation, 349, 370, 583, 715
g · b analysis, 729Burgers vector, 715, 730charge sinks, 715contrast tables, 729core, 717dipole, 377double image, 383dynamical contrast, 386, 662edge, 372, 715fcc and hcp, 718groups of, 720image width, 385interactions, 720loop, 716mixed, 716partial, 719phase-amplitude diagram, 371plastic deformation, 715position of image, 371, 372, 383reactions, 718screw, 374, 715self energy, 717strain field, 720superdislocation, 377tilt boundary, 720tomogram, 614weak-beam dark-field method, 387
Dispersion corrections, 683Dispersion surface, 639, 658Dispersions of excitations, 133Displacement disorder
dynamic, 475static, 475
Displacement field, 582Divergence
thick hexapole, 603
DO19 structure, 568Domains of order, 264, 287Dopant, 601Double diffraction
forbidden diffractions, 316tilting experiment, 316
Double exposures, 723Double-differential cross-section, 201Double-tilt holder, 290DPPC, 132Drift of sample, 387Duane–Hunt rule, 13Dynamical absences
space group, 341Dynamical theory
boundary conditions, 650concepts, 617eigenvalue problem, 675extinction distances, 675multibeam, 624, 673multibeam and HRTEM, 673multislice method, 676phase grating, 675propagator, 675vs. kinematical theory, 625, 629
EEffective deviation parameter, 355Effective extinction distance, 355Eigenfunctions for electrons, 618Elastic anisotropy, 450, 486Elastic cross-section
Rutherford, 173Elastic scattering, 149Electric dipole radiation, 154Electric dipole selection rule, 18, 57Electron
holography, 63Electron coherence length, 85Electron energy-loss near-edge structure
(ELNES), 187Electron energy-loss spectrometry (EELS), 60,
598, 610background in spectrum, 185chemical analysis, 208energy filter, 577experimental intensities, 202fine structure, 187, 188M4,5 edge, 214magnetic prism, 210Ni spectrum, 185nomenclature for edges, 186partial cross-section, 206plasmon peak, 185
752 Index
Electron energy-loss spectrometry (EELS)(cont.)
spectrometer, 183, 590aperture, 205diffraction-coupled, 184entrance aperture, 183, 203image-coupled, 184monochromator, 610parallel or serial, 183
spectrumbackground, 208edge jump, 235multiple scattering, 209
thickness gradients, 577typical spectrum, 185white lines, 185, 189zero-loss peak, 185, 610
Electron form factors, table of, 689Electron gun
brightness, 104filament saturation, 83self-bias design, 83thermionic triode, 83
Electron interaction parameter, 560Electron mass, 730Electron microprobe, 217, 221Electron probe size, 223Electron scattering
Born approximation, 162coherent elastic, 162Green’s functions, 164
Electron tomography, 610Electron wave probability, 162Electron wavelengths, table of, 733Electron-atom interactions, 12Electronic transition nomenclature, 186Electropolishing, 726Elegant collar, 161, 178Elemental mapping, 36Energy, 149Energy transfer, 9Energy-dispersive x-ray spectrometry (EDS),
29, 60, 216, 598artifacts, 224background, 225compositional accuracy, 231confidence level, 232detector take-off angle, 221electron trajectories in materials, 216escape path, 221hole count, 224k-factor determination, 728microchemical analysis, 220
minimum detectable mass (MDM), 231minimum mass fraction (MMF), 231practice, 728quantification, 222, 225sensitivity versus Z, 182spectrometer, 221spurious x-rays, 224statistical analyses, 231Student-t distribution, 231typical spectrum, 221, 234
Energy-filtered TEM (EFTEM), 577chemical mapping, 212diffraction contrast, 213energy-filtered TEM imaging, 209filters, 211instrumentation, 210spatial resolution, 214
Epithermal neutrons, 122Equatorial divergence, 25Eucentric tilt, 98Everhart–Thornley detector, 219Ewald sphere
and Bragg’s Law, 275axial dark-field imaging, 278construction, 274curvature, 274dynamical theory, 655Laue condition, 274manipulations, 276
Excitation error, sg , 622in dynamical theory, 623
Extended electron energy-loss fine structure(EXELFS), 188
Extended x-ray absorption fine structure(EXAFS), 190
Extinction distance, 355, 618, 622and structure factor, 626effective, 355table of, 356
Extracted particle, 73
FFactors of 2π , 247, 464, 652Faraday cage, 222Fast Fourier transform, 563
deconvolution, 460Fe–Cu (grain boundaries), 511Fe–Ni, 137Fe3Al, 412FeCo, 258, 261, 287Fermi, 134
chopper, 134Field effect transistor, 34
Index 753
Field emission gun, 83cold, 83Schottky, 84
Filament lifetime, 83Fingerprinting, 4First-order Laue zone (FOLZ), 278, 324Fission, 120Fluorescence correction, 229Fluorescence yield, 219Flux (in scattering), 151Focused ion-beam milling, 224Focusing circle, 26Focusing strength, 64Forbidden diffractions, 253, 256
double diffraction, 316Forbidden transitions, 17Form factor
electron, 167table of, 689
physical picture, 167x-ray, 157
table of, 684Forward scattering (coherence), 507Fourier transform
bare Coulomb, 172complex, 442cutoff oscillations, 445decaying exponential, 171deconvolution, 439Gaussian, 444Lorentzian, 171, 444low-pass filter, 444scattered wave, 167table of pairs, 709
Frank interstitial loopHRTEM image of, 81
Fraunhofer region, 524Fresnel fringes, 585
astigmatism, 104at edge, 533focus, 103, 535, 585spacing, 534
Fresnel integrals, 532Fresnel propagator, 536Fresnel region, 524Fresnel zones, 528Friedel’s law, 453, 468
CBED, 328
Gg · b rule, 372
Burgers vector, 373GaAs, 80Gas gain, 30
Gaussian damping function, 562Gaussian focus, 550, 571Gaussian function, 457Gaussian image plane, 99Gaussian thermal displacements, 712Geiger, 173Generalized oscillator strength (GOS), 202,
203Geometric phase analysis, 580Geometrical optics, 63Gjønnes–Moodie (GM) lines, 339Glass lens, 88
concave, 91Fermat’s principle, 93phase shifts, 92shape of surface, 91spherical surface, 92
Goniometer, 24, 290Bragg–Brentano, 26circle, 26TEM sample, 64
Grain boundary, 413, 567Graphene, 605Gray, 731Green’s function, 164
spherical wavelet, 523wave equation, 536
Growth ledges, 421Guinier approximation, 508, 510Guinier radius, 510Guinier–Preston zones, 271
HHAADF imaging, 60, 587
defocus, 599electron channeling, 590, 600electron scattering, 595electron tunneling, 593resolution, 598sample drift, 590source of incoherence, 589vs. HRTEM images, 598
Half-width-at-half-maximum (HWHM), 430Heterogeneous ordering, 287Hexagonal close packed
interplanar spacings, 53structure factor rule, 53
Hexapole lens, 602Hf, 609HFIR, 130High-resolution TEM (HRTEM), 79
as interference patterns, 81compensate aberration with defocus, 544effect of defocus, 542
754 Index
High-resolution TEM (HRTEM) (cont.)effect of spherical aberration, 543experimental, 542image matching, 558lens characteristics, 550microscope parameters, 561simple interpretations, 569specimen parameters, 559total error in phase, 545
High-resolution TEM practiceanomalous spot intensities, 579beam tilt effects, 573defocus, 571doubling of spot periodicities, 575FFTs from local regions, 577minimum contrast condition, 571sample thickness, 576surface layers, 579use of EELS, 576
High-resolution TEM simulationsbeam convergence, 562diffuse scattering, 564measurement of parameters, 564microscope instabilities, 563other helpful programs, 577procedure, 558quantifying parameters, 563size of array and unit cell, 564specimen and microscope, 570
Higher-order Laue zone (HOLZ), 278, 324dynamical absences, 341excess and deficit lines, 326, 327lines and lattice parameter, 325
Hole count, 224Holography, 63Homogeneous medium, plane wave in, 522Homogeneous ordering, 287Hönl dispersion corrections, 157Howie–Whelan–Darwin equations, 624Huang scattering, 484Huygens principle, 526
spherical wave analysis, 526Hydrogenic atom, 204
IIdeal gas, 508Illumination angle, 85Illumination system
convergence (C2), 85lenses, 82, 85point source, 85spot size (C1), 85
Image coupling, 184
Image shift, 607Imaging lens system, 86
cross-overs, 97image inversions, 86
Imaging mode, 67Imaging plates, 34, 387In-situ studies, 62Incident plane wave, 162Incoherence, 145, 148Incoherent elastic scattering, 149Incoherent imaging, 587Incoherent inelastic scattering, 149Incoherent scattering, 140, 148, 587Index of refraction, 89Indexing diffraction patterns
concept, 4, 290easy way, 292indexed patterns, 681row and column checks, 295start with diffraction spots, 296start with zone axis, 292
Inelastic, 149Inelastic electron scattering, 597Inelastic form factor, 199, 597Inelastic scattering, 9, 128, 149Information limit, 554, 607
HAADF imaging, 598Insertion device, 20Instrument function, 441Instrumental broadening, 436Integral cross-section, 207Integral inelastic cross-section, 235Interband transition, 610Interface
coherent, 568crystal-liquid, 577incoherent, 569semicoherent, 568
Intermediate aperture, 73Intermediate lens, 68, 86Internal interfaces
displacement vector, 393phase shifts, 393phase-amplitude diagram, 396
Internal stress, 434International Centre for Diffraction Data, 4Interphase boundaries, 567Interstitial loop, 414Ionization, 12
cross-section, 220Isomorphous substitutions, 468Isotopic substitutions, 468Isotropic averages, 494
Index 755
JJEOL 200CX, 571JEOL 2010F, 59JEOL 4000EX, 555, 571Johansson crystals, 26Jump-ratio image, 212
KK–B mirror, 28Kikuchi lines
deviation parameter, 313indexing, 309Kikuchi maps, 314Kossel cones, 308measure of s, 351, 388origin, 306sign of s, 313specimen orientation, 311visibility, 308
Kinematical theorydisorder, 463validity, 238, 351, 359vs. dynamical theory, 625, 629, 677
Kinematics of inelastic scattering, 196Knock-on damage, 233Kossel cones, 308
LL10 structure, 568LaB6 thermionic electron source, 83Laboratory exercises, 721Lattice fringe imaging, 546Lattice parameter measurement, 48Lattice translation vectors
primitive, 244Laue condition, 247
and Bragg’s law, 247Ewald sphere, 274
Laue method, 10backscatter Laue of Si, 10
Laue monotonic scattering, 489, 493Laue zones, 278, 324
symmetry and specimen tilt, 278Ledges, 567Lens, 211, 537
aberrations, 98, 605as phase shifter, 537curvature of glass, 90double convex, 90electrostatic, 557glass, 89ideal phase function, 537magnetic, 93performance criteria, 98, 605
phase transfer function, 546transfer, 604
Lens and propagator rules, 537Lens design
phase shifts, 92ray tracing, 90
Lens formula, 65, 112, 539L’Hôspital’s rule, 266Light in transparent medium, 525Line of no contrast, 415, 417Lipid bilayer, 132Liquid crystal, 566Lobe aberration, 607Long-range order, 261Lorentz factor, 38, 41Lorentz force, 603Lorentz microscopy, 61Lorentzian function, 454, 457
second moment divergence, 462
MMagnetic field
applied, 139Magnetic lens
electron trajectory, 95focusing action, 95image rotation, 96, 724Lorentz forces of solenoid, 94pole pieces, 94post-field, 96rotation calibration with MoO3, 96, 723
Magnification, 66Main amplifier, 35Manufacturers (TEM), 63, 185, 224Marsden, 173Mass attenuation coefficients, 160
x-ray, table of, 681Mass-thickness contrast, 70, 350Materials, 1
chemical compositions, 1crystal structure, 1diffraction pattern, 2microstructures, 1
Matrix C or C−1, 637Maxwellian distribution, 121Mean inner potential, 178Measured intensities, 43Metallic glass, 5, 79, 500Metals, cold-worked, 458Microchemical analysis, 182Microstructure, 1, 59, 349Miller index, 3Minimum contrast condition, 571Missing wedge, 611
756 Index
Mixed methods, 612Moderation, 121Moderator
poisoned, 125Modulation, 81Moiré fringes, 397, 422
parallel, 398rotational, 398
Momentum transfer, 9Monochromatic radiation, 10Monochromator, 26, 122, 133
asymmetrically-cut crystal, 27diffracted beam, 28electron, 184incident beam, 28
Monte Carlo, 216Moseley’s laws, 17, 233Mott formula, 175Multi-body spatial correlations, 504Multi-lens systems, 67Multichannel analyzer, 36Multiphonon scattering, 589, 596Multiplicity, 42Multislice method
accuracy, 585defocus, 562deviation parameter, 585in k-space, 561microscope parameters, 561phase shifts in, 541projected potential, 560slice thickness, 560
NNanobeam diffraction, 78Nanocrystal
CeO2 and Pd, 565Fe–Cu, 451KI, 565, 566Ni3Fe, 450
Nanodiffraction, 73Nanostructure, 125, 564, 601Nanotube
single-wall carbon, 565, 566Nearest-neighbor shells, 495Nelson–Riley lattice parameter determination,
49Neutron
chopper, 122free, 120gas, 120guide, 131magnetic scattering, 139mass, 730
moderation, 121moderator, 121polarized, 139reactor source, 120scattering, 117, 118spallation source, 120time-of-flight monochromator, 122velocity, 119wavelength, 730
Neutron scattering, 117Ni film, 132Ni–Fe, 139NIST SRM, 44Nobel prizes, 2Nomenclature
EELS edges, 186electronic transitions, 186x-ray, 17, 19
Non-dipole transitions, 208Normal stress, 434Normalization of vectors, 293Nuclear scattering, 140Null contrast condition, 369Null water, 141, 143
OObjective aperture, 65Objective lens, 63
construction, 86pole pieces, 86
Octupole lens, 604Optical fiber principle, 590Order parameter, L, 261Ordered structures, 258Ordering, 287, 491Orientation for diffraction, 37Orientation relationship
crystallographic, 305image and diffraction pattern, 86
Orthogonality condition, 440Orthogonality relationships, 621Osmium staining, 71
Pπ boundary, 412Pair distribution function, 501
synchrotron source, 504Pair probability (conditional), 490Partial cross-section, 206Partial dislocation, 400, 719
Frank, 400Shockley, 400
Partial pair correlations, 504
Index 757
Patterson function, 452, 463atomic displacement disorder, 475average crystal, 474chemical disorder, 489definition of, 465deviation crystal, 474example, 472graphical construction, 467homogeneous disorder, 475infinite δ series, 470perfect crystal, 469random displacements, 476SRO, 493thermal spread, 479
Pauli principle, 200Peak width vs. Δk method, 446Pearson VII function, 51Peltier cooler, 33Pendellösung, 631Penetration of radiation, 119Periodic boundary conditions, 563Perturbation theory, 594, 642Phase
and materials, 539of electron wavefront, 521velocity, 146
Phase contrast, 60, 350Phase errors, 81
constructive interference, 551lens accuracy, 92
Phase fraction determination, 44integrated areas, 47internal standard method, 46retained austenite, 46
Phase grating, 560, 627approximation, 679
Phase image, 582Phase problem, 468
anomalous scattering, 468Phase relationships, 81, 145, 453Phase transfer function, 539Phase wave, 126Phase-amplitude diagram, 350, 356, 357,
678bend contour, 364dislocation, 371Fresnel zones, 530in dynamical theory, 629interfaces, 393moiré fringes, 397of white noise, 443screw dislocation, 381, 382stacking fault, 402thickness fringes, 360
Phase-space transform chopper, 288Philips EM400T, 223Philips EM430, 550Phonon, 481, 595, 714
multiphonon scattering, 589, 596scattering, 149, 714
Photoelectric scattering, 157Planck’s constant, 9, 112, 730Plasmon, 185, 191
data, table of, 194lifetime, 192mean free path, 192, 233specimen thickness, 192, 233
Point resolution, 551Poisson ratio, 435Polar net, 301Polarization correction, 42Polarized incident radiation, 45Pole-zero cancellation, 35Poly-DCH polymer, 76Polychromatic radiation, 10Polycrystalline Au, 722Polymer (liquid crystal), 566Porod law, 512, 519Porod plot, 514
fractal particles, 514surface area, 514
Position-sensitive detector, 26area detector, 33charge-coupled-device, 33, 578delay line, 33imaging plates, 34, 578measured intensities, 43pixelated diodes, 34resistive wire, 33
Powder average for x-ray diffractometry,44
Powder diffractometer, 122Powder method, 11Precipitate
coherency, 415fringe contrast, 410image of coherent, 418incoherent, 419orientation relationship, 728semi-coherent, 419variants, 725
Principal quantum number, 16Principal strains, 434Program advisory committee, 121Projected potential, 560Projector lens, 68, 86
distortion, 292Propagator, 536, 561
758 Index
Protium, 122Pseudo-Voigt function, 51, 439
QQuadrupole lens, 102, 604Quantum dot, 610Quantum efficiency, 29Quantum electrodynamics, 12Quantum mechanics, 9, 16Quantum numbers, 16Quasi-elastic, 435Quasielastic scattering, 137
RRadial distribution function, 190, 501, 515
small-angle scattering, 516Radio analogy for HRTEM, 81Radius of gyration, 510Ray diagram, 63
for TEM, 110Ray tracing, 69, 90Real image, 64Receiving slit, 25Reciprocal lattice, 245
dimensionality, 289primitive translation vectors, 246
Reciprocal lattice vectorsfcc, bcc, sc, 248uniqueness, 246
Reciprocityin optics, 599
Reduced diffraction intensity, 503Reduced x-ray interference function, 505Reference lattice, 581Refinement methods, 49
constraints, 52parameters, 50peak shape, 51
Reflected waves, 524Reflectometry, 131Refractive index, 89Rel-disk, 281Rel-rods, 269Relativistic correction, 112, 733Relaxation energy, 483Representations in quantum mechanics, 618,
639Residual contrast, 374, 381Resolution, 106, 557
energy, 184limit in HRTEM, 108limit in STEM, 588optimal, 551point, 551
point-to-point, 553state-of-the-art in 2007, 81vertical, 608
Resonance scattering, 140Richardson’s constant, 109Rietveld refinement, 49Right-hand rule, 291
zone-axis convention, 295Roentgen, 731Ronchigram, 605Rotating anode source, 24Rutherford cross-section, 173Rutherford scattering, 216
in HAADF imaging, 589, 596Rydberg, 16, 730
SSample shape for x-ray diffractometry, 44Sample thickness
example, 144SANS, 129, 130, 139Sb in Si, 601Scanning electron microscopy (SEM), 216,
218, 221Scanning transmission electron microscopy
(STEM), 60, 587, 588Scattered wave, 523Scattering
complementarity of different methods, 117differential cross-section, 152phase lag, 526total cross-section, 153
Scattering factorelectron, 560
Scattering potential, 238time-varying, 712
Scherrer equation, 432Scherzer defocus, 553, 571, 586Scherzer resolution, 551, 553
in HAADF imaging, 588Schrödinger equation, 16, 522, 620
Green’s function, 164, 523Secondary electron imaging (SEI), 218Secondary electrons, 218Seemann–Bohlin diffractometer, 26Selected-area diffraction (SAD), 73
spherical aberration, 113Selection rule, 57Semiconductor device, 609Shape factor, 250, 353, 452, 472, 507
and s, 273column of atoms, 452definition, 250envelope function, 267
Index 759
Shape factor (cont.)intensity, 353, 472rectangular prism, 264rel-rods, 269sphere, disk, rod, 269
Shear strain, 434Shielding by core electrons, 17Shockley partial dislocation, 719Short-range order (SRO), 486, 490
single crystal, 495Warren–Cowley parameters, 491
Shubnikov groups, 140Si, 4, 80, 582, 601Si crack and dislocations, 614Si dumbbells, 600Si–Ge superlattice, 601Side-centered orthorhombic lattice, 286Side-entry stage, 98Sideband, 81Sievert, 731SIGMAK, SIGMAL, 207Sign of s, 313Signal-to-noise ratio, 29Simultaneous strain and size broadening, 446Single channel analyzer, 36Single crystal methods, 10Single-wall carbon nanotube, 565SiO2, 609Size broadening, 430, 452, 462, 473Skilled microscopist, 60, 103, 225, 542, 573,
590Slit width, 38, 436Small-angle scattering, 129, 506
concept, 506from continuum, 507Guinier radius, 510neutron (SANS), 516Porod plot, 514x-ray (SAXS), 516
Solid mechanics, 436Solid-solid interfaces by HRTEM, 567Soller slits, 25, 436Space group (CBED), 341Space groups, 140Spectral brilliance, 21Spectrum image, 61, 210Spherical aberration, 98, 602, 605
and defocus in HRTEM, 99, 542and underfocus for SAD, 73correction, 579, 602effect on SAD, 113negative, 608phase distortion, 81
Spin, 16Spin wave scattering, 149Spin-echo spectrometer, 137Spin-orbit splitting, 18Spot size control (C1), 223Stacking fault, 400, 450
analysis example, 407asymmetry of images, 670bounding partials, 404, 407diffraction peak broadening, 450diffraction peak shifts, 451dynamical theory, 665, 667dynamical treatment, 404energy, 720extrinsic/intrinsic rule, 406graphite, 420HRTEM image of, 82kinematical treatment, 400, 403tetrahedra, 414top of specimen, 406, 407visibility, 404widths in images, 409
Staining, 71Star aberration, 607Statistical scatter, 29, 50, 55, 442Stereographic projection, 705–708
construction, 298electron diffraction patterns, 299examples, 302Kurdjumov-Sachs relationship, 305polar net, 301poles, 298rules for manipulation, 300, 301twinning, 304Wulff net, 300, 705, 708
Stigmation, 102procedure, 534, 550, 723stigmator, 103
Stokes correction, 439Storage ring, 20Strain, 583Strain broadening
distribution of strains, 433, 447, 482heterogeneity of strains, 482origin, 432
Strain fields, 352, 368Stray fields, 607Strip chart recorder, 461Structural image, 548Structure factor, 250
and extinction distance, 626and s, 273bcc, 254dc, 4, 255
760 Index
Structure factor (cont.)definition, 250fcc, 254hcp, 53lattice, 255phase factor, 244sc, 251simple lattice, 244
Sublattice, 261Sum peak, 35Supercell, 560Superlattice diffractions, 258, 260
B1 structure, 259B2 table of, 259L10-ordered structure, 260L12-ordered structure, 261
Symmetry elements and diffraction groups,331
Synchrotron radiation, 20, 208beamlines, 21pair distribution function, 504user and safety programs, 22
Systematic absencesglide planes, 257screw axes, 257
Tθ ′ precipitate, 725Take-off angle, 24TEM laboratory practice
apertures, 727laboratory exercises, 721preparation, 727procedures, 721
Thermal diffuse scattering, 475, 478Thermal field emission gun, 84Thermal vibrations, 517Thermionic electron gun, 83Thickness contours, 360
effect of absorption, 363wedge-shaped specimen, 362
Thin-film approximation, 225Thomas Gainsborough, 161Thompson scattering, 155Three dimensional imaging, 608Three-dimensional analysis, 611Three-window image, 212Through-focus series, 564, 571, 575Ti–Al, 376, 567Ti–Al–Mo alloy, 569Tilt of beam or crystal, 561, 611Time-of-flight, 123Timing, 135
Tomography, 610Torr, 732Total internal reflection, 591Total scattering cross-section, 153Transfer lens, 604Transmutation, 140Transparency broadening, 436Triple-axis spectrometer, 133Tritium, 122Tungsten filament, 82Tunneling, 593Turbulence of air, 607Twin, 422
boundary, 413Two-beam BF images, 354
antiphase boundary, 413contrast of dislocation, 385dislocation, 373, 379, 380moiré fringes, 400stacking fault, 404, 409twin boundary, 413
Two-beam dynamical theory, 630, 635,645
Two-lens system, 68
UUndulator, 21Uniform strain, 482Unmixing, 491
VVacancy, 414
loop, 414Valence electrons, 177, 187Vector ψ or φ, 637Vegard’s law, 484Vertical resolution, 608Vibrations, 607Videorecording for kinetics, 62Visualization of tomograms, 611Void, 415
Fresnel effect, 415Voigt function, 439
second moment divergence, 462Voltage center alignment, 573
WWarren–Cowley SRO parameters, 491Wave amplitudes, 148Wave crests, 146
match at interface, 89Wave equation
Green’s function, 536Wavefront modulations, 627
Index 761
Wavelengthselectron, table of, 733x-ray, table of, 733
Wavelet (defined), 145, 237Wavevector of electron in solid, 620Weak phase object, 548Weak-beam dark-field method, 387
g–3g, 387analysis of, 389deviation parameter, s, 389dislocations in Si, 393Kikuchi lines, 388stationary phase, 390
Wehnelt electrode, 83White lines, 185, 189White noise, 444Wien filter, 184Wiggler, 21Window discriminator, 36Wobbling, 573Wulff net, 300, 705, 708
XX-ray
absorption, 41absorption coefficients, table of, 681anomalous scattering, 158, 179
chart, 683bremsstrahlung, 13characteristic, 12characteristic depth, 160classical electrodynamics of scattering, 154coherent bremsstrahlung, 57Compton scattering, 158detector, 29dispersion corrections, 157electric dipole radiation, 154
energy spectrum, 36energy-wavelength relation, 14form factors, table of, 684generation, 12line broadening, 429mapping, 36mass attenuation (absorption), 160mirror, 28near-resonance scattering, 156notation, 19photoelectric scattering, 157scattering, 154scattering dependence on atomic number,
157spectrometer, 33spectroscopy system, 35spurious, 224synchrotron radiation, 20tube, 22wavelength distribution, 14wavelengths, table of, 733
YYoung’s modulus, 435, 450
ZZ-contrast imaging, see HAADF imaging,
350, 587ZAF correction, 228Zemlin tableau, 605Zero-loss peak, 185, 610Zero-order Laue zone, ZOLZ, 278Zero-point vibrations
diffuse scattering from, 482Zone axis, 291Zr–Ni, 79
