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

International Tables for Crystallography (2006). Vol. C, ch. 4.4, pp. 454-461

Section 4.4.5. Magnetic form factors

P. J. Browna

4.4.5. Magnetic form factors

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The form factors used in the calculations of the cross sections for magnetic scattering of neutrons are defined in Subsection 6.1.2.3[link] as [\langle\, j_l(k)\rangle =\textstyle \int\limits^\infty_0\,U^2(r)\, j_l(kr)4\pi r^2\,{\rm d} r, \eqno (4.4.5.1)]in which U(r) is the radial wavefunction for the unpaired electrons in the atom, k is the length of the scattering vector, and [j_l(kr)] is the lth-order spherical Bessel function.

Tables 4.4.5.1[link][link][link][link][link][link][link]–4.4.5.8[link] give the coefficients in an analytical approximation to the [\langle \,j_0\rangle] magnetic form factors for the 3d and 4d transition series, the 4f electrons of rare-earth ions, and the 5f electrons of some actinide ions. The approximation has the form used by Forsyth & Wells (1959[link]) but allowing three instead of two exponential terms: [\eqalignno{ \langle\, j_0(s)\rangle &=A\exp(-as^2)+B\exp(-bs^2) \cr &\quad +C\exp(-cs^2)+D, &(4.4.5.2)}]where s is the value of [(\sin\theta)/\lambda] in Å−1.

Table 4.4.5.1| top | pdf |
<j0> form factors for 3d transition elements and their ions

Atom or ionAaBbCcDe
Sc 0.2512 90.030 0.3290 39.402 0.4235 14.322 −0.0043 0.2029
Sc+ 0.4889 51.160 0.5203 14.076 −0.0286 0.179 0.0185 0.1217
Sc2+ 0.5048 31.403 0.5186 10.990 −0.0241 1.183 0.0000 0.0578
Ti 0.4657 33.590 0.5490 9.879 −0.0291 0.323 0.0123 0.1088
Ti+ 0.5093 36.703 0.5032 10.371 −0.0263 0.311 0.0116 0.1125
Ti2+ 0.5091 24.976 0.5162 8.757 −0.0281 0.916 0.0015 0.0589
Ti3+ 0.3571 22.841 0.6688 8.931 −0.0354 0.483 0.0099 0.0575
V 0.4086 28.811 0.6077 8.544 −0.0295 0.277 0.0123 0.0970
V+ 0.4444 32.648 0.5683 9.097 −0.2285 0.022 0.2150 0.1111
V2+ 0.4085 23.853 0.6091 8.246 −0.1676 0.041 0.1496 0.0593
V3+ 0.3598 19.336 0.6632 7.617 −0.3064 0.030 0.2835 0.0515
V4+ 0.3106 16.816 0.7198 7.049 −0.0521 0.302 0.0221 0.0433
Cr 0.1135 45.199 0.3481 19.493 0.5477 7.354 −0.0092 0.1975
Cr+ −0.0977 0.047 0.4544 26.005 0.5579 7.489 0.0831 0.1114
Cr2+ 1.2024 −0.005 0.4158 20.548 0.6032 6.956 −1.2218 0.0572
Cr3+ −0.3094 0.027 0.3680 17.035 0.6559 6.524 0.2856 0.0436
Cr4+ −0.2320 0.043 0.3101 14.952 0.7182 6.173 0.2042 0.0419
Mn 0.2438 24.963 0.1472 15.673 0.6189 6.540 −0.0105 0.1748
Mn+ −0.0138 0.421 0.4231 24.668 0.5905 6.655 −0.0010 0.1242
Mn2+ 0.4220 17.684 0.5948 6.0050 0.0043 −0.609 −0.0219 0.0589
Mn3+ 0.4198 14.283 0.6054 5.469 0.9241 −0.009 −0.9498 0.0392
Mn4+ 0.3760 12.566 0.6602 5.133 −0.0372 0.563 0.0011 0.0393
Fe 0.0706 35.008 0.3589 15.358 0.5819 5.561 −0.0114 0.1398
Fe+ 0.1251 34.963 0.3629 15.514 0.5223 5.591 −0.0105 0.1301
Fe2+ 0.0263 34.960 0.3668 15.943 0.6188 5.594 −0.0119 0.1437
Fe3+ 0.3972 13.244 0.6295 4.903 −0.0314 0.350 0.0044 0.0441
Fe4+ 0.3782 11.380 0.6556 4.592 −0.0346 0.483 0.0005 0.0362
Co 0.4139 16.162 0.6013 4.780 −0.1518 0.021 0.1345 0.1033
Co+ 0.0990 33.125 0.3645 15.177 0.5470 5.008 −0.0109 0.0983
Co2+ 0.4332 14.355 0.5857 4.608 −0.0382 0.134 0.0179 0.0711
Co3+ 0.3902 12.508 0.6324 4.457 −0.1500 0.034 0.1272 0.0515
Co4+ 0.3515 10.778 0.6778 4.234 −0.0389 0.241 0.0098 0.0390
Ni −0.0172 35.739 0.3174 14.269 0.7136 4.566 −0.0143 0.1072
Ni+ 0.0705 35.856 0.3984 13.804 0.5427 4.397 −0.0118 0.0738
Ni2+ 0.0163 35.883 0.3916 13.223 0.6052 4.339 −0.0133 0.0817
Ni3+ 0.0012 35.000 0.3468 11.987 0.6667 4.252 −0.0148 0.0883
Ni4+ −0.0090 35.861 0.2776 11.790 0.7474 4.201 −0.0163 0.0966
Cu 0.0909 34.984 0.4088 11.443 0.5128 3.825 −0.0124 0.0513
Cu+ 0.0749 34.966 0.4147 11.764 0.5238 3.850 −0.0127 0.0591
Cu2+ 0.0232 34.969 0.4023 11.564 0.5882 3.843 −0.0137 0.0532
Cu3+ 0.0031 34.907 0.3582 10.914 0.6531 3.828 −0.0147 0.0665
Cu4+ −0.0132 30.682 0.2801 11.163 0.7490 3.817 −0.0165 0.0767

Table 4.4.5.2| top | pdf |
<j0> form factors for 4d atoms and their ions

Atom or ionAaBbCcDe
Y 0.5915 67.608 1.5123 17.900 −1.1130 14.136 0.0080 0.3272
Zr 0.4106 59.996 1.0543 18.648 −0.4751 10.540 0.0106 0.3667
Zr+ 0.4532 59.595 0.7834 21.436 −0.2451 9.036 0.0098 0.3639
Nb 0.3946 49.230 1.3197 14.822 −0.7269 9.616 0.0129 0.3659
Nb+ 0.4572 49.918 1.0274 15.726 −0.4962 9.157 0.0118 0.3403
Mo 0.1806 49.057 1.2306 14.786 −0.4268 6.987 0.0171 0.4135
Mo+ 0.3500 48.035 1.0305 15.060 −0.3929 7.479 0.0139 0.3510
Tc 0.1298 49.661 1.1656 14.131 −0.3134 5.513 0.0195 0.3869
Tc+ 0.2674 48.957 0.9569 15.141 −0.2387 5.458 0.0160 0.3412
Ru 0.1069 49.424 1.1912 12.742 −0.3176 4.912 0.0213 0.3597
Ru+ 0.4410 33.309 1.4775 9.553 −0.9361 6.722 0.0176 0.2608
Rh 0.0976 49.882 1.1601 11.831 −0.2789 4.127 0.0234 0.3263
Rh+ 0.3342 29.756 1.2209 9.438 −0.5755 5.332 0.0210 0.2574
Pd 0.2003 29.363 1.1446 9.599 −0.3689 4.042 0.0251 0.2453
Pd+ 0.5033 24.504 1.9982 6.908 −1.5240 5.513 0.0213 0.1909

Table 4.4.5.3| top | pdf |
<j0> form factors for rare-earth ions

IonAaBbCcDe
Ce2+ 0.2953 17.685 0.2923 6.733 0.4313 5.383 −0.0194 0.0845
Nd2+ 0.1645 25.045 0.2522 11.978 0.6012 4.946 −0.0180 0.0668
Nd3+ 0.0540 25.029 0.3101 12.102 0.6575 4.722 −0.0216 0.0478
Sm2+ 0.0909 25.203 0.3037 11.856 0.6250 4.237 −0.0200 0.0408
Sm3+ 0.0288 25.207 0.2973 11.831 0.6954 4.212 −0.0213 0.0510
Eu2+ 0.0755 25.296 0.3001 11.599 0.6438 4.025 −0.0196 0.0488
Eu3+ 0.0204 25.308 0.3010 11.474 0.7005 3.942 −0.0220 0.0356
Gd2+ 0.0636 25.382 0.3033 11.212 0.6528 3.788 −0.0199 0.0486
Gd3+ 0.0186 25.387 0.2895 11.142 0.7135 3.752 −0.0217 0.0489
Tb2+ 0.0547 25.509 0.3171 10.591 0.6490 3.517 −0.0212 0.0342
Tb3+ 0.0177 25.510 0.2921 10.577 0.7133 3.512 −0.0231 0.0512
Dy2+ 0.1308 18.316 0.3118 7.665 0.5795 3.147 −0.0226 0.0315
Dy3+ 0.1157 15.073 0.3270 6.799 0.5821 3.020 −0.0249 0.0146
Ho2+ 0.0995 18.176 0.3305 7.856 0.5921 2.980 −0.0230 0.1240
Ho3+ 0.0566 18.318 0.3365 7.688 0.6317 2.943 −0.0248 0.0068
Er2+ 0.1122 18.122 0.3462 6.911 0.5649 2.761 −0.0235 0.0207
Er3+ 0.0586 17.980 0.3540 7.096 0.6126 2.748 −0.0251 0.0171
Tm2+ 0.0983 18.324 0.3380 6.918 0.5875 2.662 −0.0241 0.0404
Tm3+ 0.0581 15.092 0.2787 7.801 0.6854 2.793 −0.0224 0.0351
Yb2+ 0.0855 18.512 0.2943 7.373 0.6412 2.678 −0.0213 0.0421
Yb3+ 0.0416 16.095 0.2849 7.834 0.6961 2.672 −0.0229 0.0344

Table 4.4.5.4| top | pdf |
<j0> form factors for actinide ions

IonAaBbCcDe
U3+ 0.5058 23.288 1.3464 7.003 −0.8724 4.868 0.0192 0.1507
U4+ 0.3291 23.548 1.0836 8.454 −0.4340 4.120 0.0214 0.1757
U5+ 0.3650 19.804 3.2199 6.282 −2.6077 5.301 0.0233 0.1750
Np3+ 0.5157 20.865 2.2784 5.893 −1.8163 4.846 0.0211 0.1378
Np4+ 0.4206 19.805 2.8004 5.978 −2.2436 4.985 0.0228 0.1408
Np5+ 0.3692 18.190 3.1510 5.850 −2.5446 4.916 0.0248 0.1515
Np6+ 0.2929 17.561 3.4866 5.785 −2.8066 4.871 0.0267 0.1698
Pu3+ 0.3840 16.679 3.1049 5.421 −2.5148 4.551 0.0263 0.1280
Pu4+ 0.4934 16.836 1.6394 5.638 −1.1581 4.140 0.0248 0.1242
Pu5+ 0.3888 16.559 2.0362 5.657 −1.4515 4.255 0.0267 0.1287
Pu6+ 0.3172 16.051 3.4654 5.351 −2.8102 4.513 0.0281 0.1382
Am2+ 0.4743 21.776 1.5800 5.690 −1.0779 4.145 0.0218 0.1253
Am3+ 0.4239 19.574 1.4573 5.872 −0.9052 3.968 0.0238 0.1054
Am4+ 0.3737 17.862 1.3521 6.043 −0.7514 3.720 0.0258 0.1113
Am5+ 0.2956 17.372 1.4525 6.073 −0.7755 3.662 0.0277 0.1202
Am6+ 0.2302 16.953 1.4864 6.116 −0.7457 3.543 0.0294 0.1323
Am7+ 0.3601 12.730 1.9640 5.120 −1.3560 3.714 0.0316 0.1232

Table 4.4.5.5| top | pdf |
<j2> form factors for 3d transition elements and their ions

Atom or ionAaBbCcDe
Sc 10.8172 54.327 4.7353 14.847 0.6071 4.218 0.0011 0.1212
Sc+ 8.5021 34.285 3.2116 10.994 0.4244 3.605 0.0009 0.1037
Sc2+ 4.3683 28.654 3.7231 10.823 0.6074 3.668 0.0014 0.0681
Ti 4.3583 36.056 3.8230 11.133 0.6855 3.469 0.0020 0.0967
Ti+ 6.1567 27.275 2.6833 8.983 0.4070 3.052 0.0011 0.0902
Ti2+ 4.3107 18.348 2.0960 6.797 0.2984 2.548 0.0007 0.0640
Ti3+ 3.3717 14.444 1.8258 5.713 0.2470 2.265 0.0005 0.0491
V 3.7600 21.831 2.4026 7.546 0.4464 2.663 0.0017 0.0556
V+ 4.7474 23.323 2.3609 7.808 0.4105 2.706 0.0014 0.0800
V2+ 3.4386 16.530 1.9638 6.141 0.2997 2.267 0.0009 0.0565
V3+ 2.3005 14.682 2.0364 6.130 0.4099 2.382 0.0014 0.0252
V4+ 1.8377 12.267 1.8247 5.458 0.3979 2.248 0.0012 0.0399
Cr 3.4085 20.127 2.1006 6.802 0.4266 2.394 0.0019 0.0662
Cr+ 3.7768 20.346 2.1028 6.893 0.4010 2.411 0.0017 0.0686
Cr2+ 2.6422 16.060 1.9198 6.253 0.4446 2.372 0.0020 0.0480
Cr3+ 1.6262 15.066 2.0618 6.284 0.5281 2.368 0.0023 0.0263
Cr4+ 1.0293 13.950 1.9933 6.059 0.5974 2.346 0.0027 0.0366
Mn 2.6681 16.060 1.7561 5.640 0.3675 2.049 0.0017 0.0595
Mn+ 3.2953 18.695 1.8792 6.240 0.3927 2.201 0.0022 0.0659
Mn2+ 2.0515 15.556 1.8841 6.063 0.4787 2.232 0.0027 0.0306
Mn3+ 1.2427 14.997 1.9567 6.118 0.5732 2.258 0.0031 0.0336
Mn4+ 0.7879 13.886 1.8717 5.743 0.5981 2.182 0.0034 0.0434
Fe 1.9405 18.473 1.9566 6.323 0.5166 2.161 0.0036 0.0394
Fe+ 2.6290 18.660 1.8704 6.331 0.4690 2.163 0.0031 0.0491
Fe2+ 1.6490 16.559 1.9064 6.133 0.5206 2.137 0.0035 0.0335
Fe3+ 1.3602 11.998 1.5188 5.003 0.4705 1.991 0.0038 0.0374
Fe4+ 1.5582 8.275 1.1863 3.279 0.1366 1.107 −0.0022 0.0327
Co 1.9678 14.170 1.4911 4.948 0.3844 1.797 0.0027 0.0452
Co+ 2.4097 16.161 1.5780 5.460 0.4095 1.914 0.0031 0.0581
Co2+ 1.9049 11.644 1.3159 4.357 0.3146 1.645 0.0017 0.0459
Co3+ 1.7058 8.859 1.1409 3.309 0.1474 1.090 −0.0025 0.0462
Co4+ 1.3110 8.025 1.1551 3.179 0.1608 1.130 −0.0011 0.0374
Ni 1.0302 12.252 1.4669 4.745 0.4521 1.744 0.0036 0.0338
Ni+ 2.1040 14.866 1.4302 5.071 0.4031 1.778 0.0034 0.0561
Ni2+ 1.7080 11.016 1.2147 4.103 0.3150 1.533 0.0018 0.0446
Ni3+ 1.4683 8.671 0.1794 1.106 1.1068 3.257 −0.0023 0.0373
Ni4+ 1.1612 7.700 1.0027 3.263 0.2719 1.378 0.0025 0.0326
Cu 1.9182 14.490 1.3329 4.730 0.3842 1.639 0.0035 0.0617
Cu+ 1.8814 13.433 1.2809 4.545 0.3646 1.602 0.0033 0.0590
Cu2+ 1.5189 10.478 1.1512 3.813 0.2918 1.398 0.0017 0.0429
Cu3+ 1.2797 8.450 1.0315 3.280 0.2401 1.250 0.0015 0.0389
Cu4+ 0.9568 7.448 0.9099 3.396 0.3729 1.494 0.0049 0.0330

Table 4.4.5.6| top | pdf |
<j2> form factors for 4d atoms and their ions

Atom or ionAaBbCcDe
Y 14.4084 44.658 5.1045 14.904 −0.0535 3.319 0.0028 0.1093
Zr 10.1378 35.337 4.7734 12.545 −0.0489 2.672 0.0036 0.0912
Zr+ 11.8722 34.920 4.0502 12.127 −0.0632 2.828 0.0034 0.0737
Nb 7.4796 33.179 5.0884 11.571 −0.0281 1.564 0.0047 0.0944
Nb+ 8.7735 33.285 4.6556 11.605 −0.0268 1.539 0.0044 0.0855
Mo 5.1180 23.422 4.1809 9.208 −0.0505 1.743 0.0053 0.0655
Mo+ 7.2367 28.128 4.0705 9.923 −0.0317 1.455 0.0049 0.0798
Tc 4.2441 21.397 3.9439 8.375 −0.0371 1.187 0.0066 0.0645
Tc+ 6.4056 24.824 3.5400 8.611 −0.0366 1.485 0.0044 0.0806
Ru 3.7445 18.613 3.4749 7.420 −0.0363 1.007 0.0073 0.0533
Ru+ 5.2826 23.683 3.5813 8.152 −0.0257 0.426 0.0131 0.0830
Rh 3.3651 17.344 3.2121 6.804 −0.0350 0.503 0.0146 0.0545
Rh+ 4.0260 18.950 3.1663 7.000 −0.0296 0.486 0.0127 0.0629
Pd 3.3105 14.726 2.6332 5.862 −0.0437 1.130 0.0053 0.0492
Pd+ 4.2749 17.900 2.7021 6.354 −0.0258 0.700 0.0071 0.0768

Table 4.4.5.7| top | pdf |
<j2> form factors for rare-earth ions

IonAaBbCcDe
Ce2+ 0.9809 18.063 1.8413 7.769 0.9905 2.845 0.0120 0.0448
Nd2+ 1.4530 18.340 1.6196 7.285 0.8752 2.622 0.0126 0.0461
Nd3+ 0.6751 18.342 1.6272 7.260 0.9644 2.602 0.0150 0.0450
Sm2+ 1.0360 18.425 1.4769 7.032 0.8810 2.437 0.0152 0.0345
Sm3+ 0.4707 18.430 1.4261 7.034 0.9574 2.439 0.0182 0.0510
Eu2+ 0.8970 18.443 1.3769 7.005 0.9060 2.421 0.0190 0.0511
Eu3+ 0.3985 18.451 1.3307 6.956 0.9603 2.378 0.0197 0.0447
Gd2+ 0.7756 18.469 1.3124 6.899 0.8956 2.338 0.0199 0.0441
Gd3+ 0.3347 18.476 1.2465 6.877 0.9537 2.318 0.0217 0.0484
Tb2+ 0.6688 18.491 1.2487 6.822 0.8888 2.275 0.0215 0.0439
Tb3+ 0.2892 18.497 1.1678 6.797 0.9437 2.257 0.0232 0.0458
Dy2+ 0.5917 18.511 1.1828 6.747 0.8801 2.214 0.0229 0.0439
Dy3+ 0.2523 18.517 1.0914 6.736 0.9345 2.208 0.0250 0.0476
Ho2+ 0.5094 18.515 1.1234 6.706 0.8727 2.159 0.0242 0.0560
Ho3+ 0.2188 18.516 1.0240 6.707 0.9251 2.161 0.0268 0.0503
Er2+ 0.4693 18.528 1.0545 6.649 0.8679 2.120 0.0261 0.0413
Er3+ 0.1710 18.534 0.9879 6.625 0.9044 2.100 0.0278 0.0489
Tm2+ 0.4198 18.542 0.9959 6.600 0.8593 2.082 0.0284 0.0457
Tm3+ 0.1760 18.542 0.9105 6.579 0.8970 2.062 0.0294 0.0468
Yb2+ 0.3852 18.550 0.9415 6.551 0.8492 2.043 0.0301 0.0478
Yb3+ 0.1570 18.555 0.8484 6.540 0.8880 2.037 0.0318 0.0498

Table 4.4.5.8| top | pdf |
<j2> form factors for actinide ions

IonAaBbCcDe
U3+ 4.1582 16.534 2.4675 5.952 −0.0252 0.765 0.0057 0.0822
U4+ 3.7449 13.894 2.6453 4.863 −0.5218 3.192 0.0009 0.0928
U5+ 3.0724 12.546 2.3076 5.231 −0.0644 1.474 0.0035 0.0477
Np3+ 3.7170 15.133 2.3216 5.503 −0.0275 0.800 0.0052 0.0948
Np4+ 2.9203 14.646 2.5979 5.559 −0.0301 0.367 0.0141 0.0532
Np5+ 2.3308 13.654 2.7219 5.494 −0.1357 0.049 0.1224 0.0553
Np6+ 1.8245 13.180 2.8508 5.407 −0.1579 0.044 0.1438 0.0585
Pu3+ 2.0885 12.871 2.5961 5.190 −0.1465 0.039 0.1343 0.0866
Pu4+ 2.7244 12.926 2.3387 5.163 −0.1300 0.046 0.1177 0.0490
Pu5+ 2.1409 12.832 2.5664 5.152 −0.1338 0.046 0.1210 0.0491
Pu6+ 1.7262 12.324 2.6652 5.066 −0.1695 0.041 0.1550 0.0502
Am2+ 3.5237 15.955 2.2855 5.195 −0.0142 0.585 0.0033 0.1120
Am3+ 2.8622 14.733 2.4099 5.144 −0.1326 0.031 0.1233 0.0727
Am4+ 2.4141 12.948 2.3687 4.945 −0.2490 0.022 0.2371 0.0502
Am5+ 2.0109 12.053 2.4155 4.836 −0.2264 0.027 0.2128 0.0414
Am6+ 1.6778 11.337 2.4531 4.725 −0.2043 0.034 0.1892 0.0387
Am7+ 1.8845 9.161 2.0746 4.042 −0.1318 1.723 0.0020 0.0379

Tables 4.4.5.9[link][link][link][link][link]–4.4.5.14[link] give coefficients in the approximation used by Lisher & Forsyth (1971[link]) to the [\langle \,j_2\rangle], [\langle\,j_4\rangle], and [\langle\,j_6\rangle] form factors for the same series of atoms and ions, again using three rather than two exponential terms, viz for [l\neq0]: [\eqalignno{ \langle \,j_l(s)\rangle &=As^2\exp(-as^2)+B s^2\exp(-bs^2) \cr &\quad+Cs^2\exp(-cs^2)+Ds^2. & (4.4.5.3)}]

Table 4.4.5.9| top | pdf |
<j4> form factors for 3d atoms and their ions

Atom or ionAaBbCcDe
Sc 1.3420 10.200 0.3837 3.079 0.0468 0.118 −0.0328 0.1343
Sc+ 7.1167 15.487 −6.6671 18.269 0.4900 2.992 0.0047 0.1624
Sc2+ −1.6684 15.648 1.7742 9.062 0.4075 2.412 0.0042 0.1105
Ti −2.1515 11.271 2.5149 8.859 0.3555 2.149 0.0045 0.1244
Ti+ −1.0383 16.190 1.4699 8.924 0.3631 2.283 0.0044 0.1270
Ti2+ −1.3242 15.310 1.2042 7.899 0.3976 2.156 0.0051 0.0820
Ti3+ −1.1117 14.635 0.7689 6.927 0.4385 2.089 0.0060 0.0572
V −0.9633 15.273 0.9274 7.732 0.3891 2.053 0.0063 0.0840
V+ −0.9606 15.545 1.1278 8.118 0.3653 2.097 0.0056 0.1027
V2+ −1.1729 14.973 0.9092 7.613 0.4105 2.039 0.0067 0.0719
V3+ −0.9417 14.205 0.5284 6.607 0.4411 1.967 0.0076 0.0569
V4+ −0.7654 13.097 0.3071 5.674 0.4476 1.871 0.0081 0.0518
Cr −0.6670 19.613 0.5342 6.478 0.3641 1.905 0.0073 0.0628
Cr+ −0.8309 18.043 0.7252 7.531 0.3828 2.003 0.0073 0.0781
Cr2+ −0.8930 15.664 0.5590 7.033 0.4093 1.924 0.0081 0.0631
Cr3+ −0.7327 14.073 0.3268 5.674 0.4114 1.810 0.0085 0.0505
Cr4+ −0.6748 12.946 0.1805 6.753 0.4526 1.800 0.0098 0.0644
Mn −0.5452 15.471 0.4406 4.902 0.2884 1.543 0.0059 0.0488
Mn+ −0.7947 17.867 0.6078 7.704 0.3798 1.905 0.0087 0.0737
Mn2+ −0.7416 15.255 0.3831 6.469 0.3935 1.800 0.0093 0.0577
Mn3+ −0.6603 13.607 0.2322 6.218 0.4104 1.740 0.0101 0.0579
Mn4+ −0.5127 13.461 0.0313 7.763 0.4282 1.701 0.0113 0.0693
Fe −0.5029 19.677 0.2999 3.776 0.2576 1.424 0.0071 0.0292
Fe+ −0.5109 19.250 0.3896 4.891 0.2810 1.526 0.0069 0.0375
Fe2+ −0.5401 17.227 0.2865 3.742 0.2658 1.424 0.0076 0.0278
Fe3+ −0.5507 11.493 0.2153 4.906 0.3468 1.523 0.0095 0.0314
Fe4+ −0.5352 9.507 0.1783 5.175 0.3584 1.469 0.0097 0.0360
Co −0.4221 14.195 0.2900 3.979 0.2469 1.286 0.0063 0.0400
Co+ −0.4115 14.561 0.3580 4.717 0.2644 1.418 0.0074 0.0541
Co2+ 0.4759 14.046 0.2747 3.731 0.2458 1.250 0.0057 0.0282
Co3+ −0.4466 13.391 0.1419 3.011 0.2773 1.335 0.0093 0.0341
Co4+ −0.4091 13.194 −0.0194 3.417 0.3534 1.421 0.0112 0.0622
Ni −0.4428 14.485 0.0870 3.234 0.2932 1.331 0.0096 0.0554
Ni+ −0.3836 13.425 0.3116 4.462 0.2471 1.309 0.0079 0.0515
Ni2+ −0.3803 10.403 0.2838 3.378 0.2108 1.104 0.0050 0.0474
Ni3+ −0.4014 9.046 0.2314 3.075 0.2192 1.084 0.0060 0.0323
Ni4+ −0.3509 8.157 0.2220 2.106 0.1567 0.925 0.0065 0.0352
Cu −0.3204 15.132 0.2335 4.021 0.2312 1.196 0.0068 0.0457
Cu+ −0.3572 15.125 0.2336 3.966 0.2315 1.197 0.0070 0.0397
Cu2+ −0.3914 14.740 0.1275 3.384 0.2548 1.255 0.0103 0.0394
Cu3+ −0.3671 14.082 −0.0078 3.315 0.3154 1.377 0.0132 0.0534
Cu4+ −0.2915 14.124 −0.1065 4.201 0.3247 1.352 0.0148 0.0579

Table 4.4.5.10| top | pdf |
<j4> form factors for 4d atoms and their ions

Atom or ionAaBbCcDe
Y −8.0767 32.201 7.9197 25.156 1.4067 6.827 −0.0001 0.1031
Zr −5.2697 32.868 4.1930 24.183 1.5202 6.048 −0.0002 0.0855
Zr+ −5.6384 33.607 4.6729 22.338 1.3258 5.924 −0.0003 0.0674
Nb −3.1377 25.595 2.3411 16.569 1.2304 4.990 −0.0005 0.0615
Nb+ −3.3598 25.820 2.8297 16.427 1.1203 4.982 −0.0005 0.0724
Mo −2.8860 20.572 1.8130 14.628 1.1899 4.264 −0.0008 0.0410
Mo+ −3.2618 25.486 2.3596 16.462 1.1164 4.491 −0.0007 0.0592
Tc −2.7975 20.159 1.6520 16.261 1.1726 3.943 −0.0008 0.0657
Tc+ −2.0470 19.683 1.6306 11.592 0.8698 3.769 −0.0010 0.0723
Ru −1.5042 17.949 0.6027 9.961 0.9700 3.393 −0.0010 0.0338
Ru+ 1.6278 18.506 1.1828 10.189 0.8138 3.418 −0.0009 0.0673
Rh −1.3492 17.577 0.4527 10.507 0.9285 3.155 −0.0009 0.0483
Rh+ −1.4673 17.957 0.7381 9.944 0.8485 3.126 −0.0012 0.0487
Pd −1.1955 17.628 0.3183 11.309 0.8696 2.909 −0.0006 0.0555
Pd+ −1.4098 17.765 0.7927 9.999 0.7710 2.930 −0.0006 0.0530

Table 4.4.5.11| top | pdf |
<j4> form factors for rare-earth ions

IonAaBbCcDe
Ce2+ −0.6468 10.533 0.4052 5.624 0.3412 1.535 0.0080 0.0522
Nd2+ −0.5416 12.204 0.3571 6.169 0.3154 1.485 0.0098 0.0519
Nd3+ −0.4053 14.014 0.0329 7.005 0.3759 1.707 0.0209 0.0372
Sm2+ −0.4150 14.057 0.1368 7.032 0.3272 1.582 0.0192 0.0319
Sm3+ −0.4288 10.052 0.1782 5.019 0.2833 1.236 0.0088 0.0328
Eu2+ −0.4145 10.193 0.2447 5.164 0.2661 1.205 0.0065 0.0516
Eu3+ −0.4095 10.211 0.1485 5.175 0.2720 1.237 0.0131 0.0494
Gd2+ −0.3824 10.344 0.1955 5.306 0.2622 1.203 0.0097 0.0363
Gd3+ −0.3621 10.353 0.1016 5.310 0.2649 1.219 0.0147 0.0494
Tb2+ −0.3443 10.469 0.1481 5.416 0.2575 1.182 0.0104 0.0280
Tb3+ −0.3228 10.476 0.0638 5.419 0.2566 1.196 0.0159 0.0439
Dy2+ −0.3206 12.071 0.0904 8.026 0.2616 1.230 0.0143 0.0767
Dy3+ −0.2829 9.525 0.0565 4.429 0.2437 1.066 0.0092 0.0181
Ho2+ −0.2976 9.719 0.1224 4.635 0.2279 1.005 0.0063 0.0452
Ho3+ −0.2717 9.731 0.0474 4.638 0.2292 1.047 0.0124 0.0310
Er2+ −0.2975 9.829 0.1189 4.741 0.2116 1.004 0.0117 0.0524
Er3+ −0.2568 9.834 0.0356 4.741 0.2172 1.028 0.0148 0.0434
Tm2+ −0.2677 9.888 0.0925 4.784 0.2056 0.990 0.0124 0.0396
Tm3+ −0.2292 9.895 0.0124 4.785 0.2108 1.007 0.0151 0.0334
Yb2+ −0.2393 9.947 0.0663 4.823 0.2009 0.965 0.0122 0.0311
Yb3+ −0.2121 8.197 0.0325 3.153 0.1975 0.884 0.0093 0.0435

Table 4.4.5.12| top | pdf |
<j4> form factors for actinide ions

IonAaBbCcDe
U3+ −0.9859 16.601 0.6116 6.515 0.6020 2.597 −0.0010 0.0599
U4+ −1.0540 16.605 0.4339 6.512 0.6746 2.599 −0.0011 0.0471
U5+ −0.9588 16.485 0.1576 6.440 0.7785 2.640 −0.0010 0.0493
Np3+ 0.9029 16.586 0.4006 6.470 0.6545 2.563 −0.0004 0.0470
Np4+ −0.9887 12.441 0.5918 5.294 0.5306 2.263 −0.0021 0.0583
Np5+ −0.8146 16.581 −0.0055 6.475 0.7956 2.562 −0.0004 0.0600
Np6+ 0.6738 16.553 −0.2297 6.505 0.8513 2.553 −0.0003 0.0623
Pu3+ −0.7014 16.369 −0.1162 6.697 0.7778 2.450 0.0000 0.0546
Pu4+ −0.9160 12.203 0.4891 5.127 0.5290 2.149 −0.0022 0.0520
Pu5+ −0.7035 16.360 −0.0979 6.706 0.7726 2.447 0.0000 0.0610
Pu6+ −0.5560 16.322 −0.3046 6.768 0.8146 2.426 0.0001 0.0596
Am2+ −0.7433 16.416 0.3481 6.788 0.6014 2.346 0.0000 0.0566
Am3+ 0.8092 12.854 0.4161 5.459 0.5476 2.172 −0.0011 0.0530
Am4+ −0.8548 12.226 0.3037 5.909 0.6173 2.188 −0.0016 0.0456
Am5+ −0.6538 15.462 −0.0948 5.997 0.7295 2.297 0.0000 0.0594
Am6+ −0.5390 15.449 −0.2689 6.017 0.7711 2.297 0.0002 0.0729
Am7+ −0.4688 12.019 −0.2692 7.042 0.7297 2.164 −0.0011 0.0262

Table 4.4.5.13| top | pdf |
<j6> form factors for rare-earth ions

IonAaBbCcDe
Ce2+ −0.1212 7.994 −0.0639 4.024 0.1519 1.096 0.0078 0.0388
Nd2+ −0.1600 8.009 0.0272 4.028 0.1104 1.068 0.0139 0.0363
Nd3+ 0.0416 8.014 −0.1261 4.040 0.1400 1.087 0.0102 0.0367
Sm2+ 0.1428 6.041 0.0723 2.033 0.0550 0.513 0.0081 0.0450
Sm3+ −0.0944 6.030 −0.0498 2.074 0.1372 0.645 −0.0132 0.0387
Eu2+ −0.1252 6.049 0.0507 2.085 0.0572 0.646 0.0132 0.0403
Eu3+ −0.0817 6.039 −0.0596 2.120 0.1243 0.764 −0.0001 0.0206
Gd2+ −0.1351 5.030 0.0828 2.025 0.0315 0.503 0.0187 0.0453
Gd3+ −0.0662 6.031 −0.0850 2.154 0.1323 0.891 0.0048 0.0371
Th2+ −0.0758 6.032 −0.0540 2.158 0.1199 0.890 0.0051 0.0488
Tb3+ −0.0559 6.031 −0.1020 2.237 0.1264 1.107 0.0167 0.0170
Dy2+ −0.0568 6.032 −0.1003 2.240 0.1401 1.106 0.0109 0.0463
Dy3+ −0.0423 6.038 −0.1248 2.244 0.1359 1.200 0.0188 0.0350
Ho2+ −0.0725 6.045 −0.0318 2.243 0.0738 1.202 0.0252 0.0634
Ho3+ −0.0289 6.050 −0.1545 2.230 0.1550 1.260 0.0177 0.0351
Er2+ 0.0648 6.056 −0.0515 2.230 0.0825 1.264 0.0250 0.0409
Er3+ −0.0110 6.061 −0.1954 2.224 0.1818 1.296 0.0149 0.0455
Tm2+ 0.0842 4.070 0.0807 0.849 −0.2087 0.039 0.2095 0.0360
Tm3+ 0.0727 4.073 0.0243 0.689 3.9459 0.002 −3.9076 0.0502
Yb2+ −0.0739 5.031 0.0140 2.030 0.0351 0.508 0.0174 0.0434
Yb3+ −0.0345 5.007 −0.0677 2.020 0.0985 0.549 −0.0076 0.0359

Table 4.4.5.14| top | pdf |
<j6> form factors for actinide ions

IonAaBbCcDe
U3+ −0.3797 9.953 0.0459 5.038 0.2748 1.607 0.0016 0.0345
U4+ −0.1793 11.896 −0.2269 5.428 0.3291 1.701 0.0030 0.0472
U5+ −0.0399 11.891 −0.3458 5.580 0.3340 1.645 0.0029 0.0444
Np3+ −0.2427 11.844 −0.1129 5.377 0.2848 1.568 0.0022 0.0368
Np4+ −0.2436 9.599 −0.1317 4.101 0.3029 1.545 0.0019 0.0500
Np5+ −0.1157 9.565 −0.2654 4.260 0.3298 1.549 0.0025 0.0495
Np6+ −0.0128 9.569 −0.3611 4.304 0.3419 1.541 0.0032 0.0520
Pu3+ −0.0364 9.572 −0.3181 4.342 0.3210 1.523 0.0041 0.0496
Pu4+ −0.2394 7.837 −0.0785 4.024 0.2643 1.378 0.0012 0.0414
Pu5+ −0.1090 7.819 −0.2243 4.100 0.2947 1.404 0.0015 0.0477
Pu6+ −0.0001 7.820 −0.3354 4.144 0.3097 1.403 0.0020 0.0513
Am2+ −0.3176 7.864 0.0771 4.161 0.2194 1.339 0.0018 0.0374
Am3+ −0.3159 6.982 0.0682 3.995 0.2141 1.188 −0.0015 0.0281
Am4+ −0.1787 7.880 −0.1274 4.090 0.2565 1.315 0.0017 0.0419
Am5+ −0.0927 6.073 −0.2227 3.784 0.2916 1.372 0.0026 0.0485
Am6+ 0.0152 6.079 −0.3549 3.861 0.3125 1.403 0.0036 0.0732
Am7+ 0.1292 6.082 −0.4689 3.879 0.3234 1.393 0.0042 0.0475

For the transition-metal series, the coefficients of the approximation have been obtained by fitting to form factors calculated from the Hartree–Fock wavefunctions given by Clementi & Roetti (1974[link]) in terms of Slater-type functions in the form [U(r)=\textstyle\sum\limits_{nj}\,N_{nl}\,r^2A_{nlj}\exp(-a_{nlj}\,r) \eqno (4.4.5.4)]by using the identity: [\eqalignno{ &\int^\infty_0\,j_l(kr) r^n\exp(-pr)4\pi r^2\,{\rm d} r \cr &\quad ={\pi^{3/2}\Gamma(n+l+3)k^l \over 2^{l-1}\Gamma(l+3/2)(k^2+p^2)^{(n+l+3)/2}} \cr &\qquad\times {_2}F_1\left({n+l+3\over 2};\, {l-n+3\over 2};\, l+{3\over2};\, {k^2\over k^2+p^2}\right). \cr&& (4.4.5.5)}]

The form factors have been calculated from these relationships in the range [(\sin\theta)/\lambda=0] to 1.5 Å−1 at intervals of 0.05 Å−1, and the coefficients of the exponential expansion fitted by a least-squares procedure at the calculated points.

For the atoms of the rare-earth and actinide series, the wavefunctions and form factors have been calculated by Freeman & Desclaux (1979[link]) and Desclaux & Freeman (1978[link]) using Dirac–Fock theory. The constants given in Tables 4.4.5.3[link], 4.4.5.4[link], 4.4.5.7[link], 4.4.5.8[link], and 4.4.5.11[link][link][link]–4.4.5.14[link] have been fitted to the results of these calculations. For the rare-earth ions, the form factors are given in the range [(\sin\theta)/\lambda=0] to 0.5 Å−1 at intervals of 0.5 Å−1 and in the range 0.5 to 1.2 Å−1 at intervals of 0.1 Å−1. For the actinide ions, the calculations extend to 1.5 Å−1. All the values given in the publications cited were included in the fitting procedure. The accuracy with which the exponential expansions fit the theoretical form factors can be judged from the value of the parameter e given in the tables, and defined by: [e=100\left(\,{\sum_i\delta^2_i \over N}\,\right)^{1/2}, \eqno (4.4.5.6)]where [\delta_i] is the difference between the ith fitted point and its theoretical value. The sum is over the N points included in the fitting procedure.

References

Clementi, E. & Roetti, C. (1974). Roothan–Hartree–Fock atomic wave functions. At. Data Nucl. Data Tables, 14, 177–478.
Desclaux, J. P. & Freeman, A. J. (1978). Dirac–Fock studies of some electronic properties of actinide ions. J. Magn. Magn. Mater. 8, 119–129.
Forsyth, J. B. & Wells, M. (1959). On an analytic approximation to the atomic scattering factor. Acta Cryst. 12, 412–414.
Freeman, A. J. & Desclaux, J. P. (1979). Dirac–Fock studies of some electronic properties of rare-earth ions. J. Magn. Magn. Mater. 12, 11–21.
Lisher, E. J. & Forsyth, J. B. (1971). Analytic approximations to form factors. Acta Cryst. A27, 545–549.








































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