The form factors used in the calculations of the cross sections for magnetic scattering of neutrons are defined in Subsection 6.1.2.3
as in which U(r) is the radial wavefunction for the unpaired electrons in the atom, k is the length of the scattering vector, and is the lth-order spherical Bessel function.
Tables 4.4.5.1–4.4.5.8 give the coefficients in an analytical approximation to the 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) but allowing three instead of two exponential terms: where s is the value of in Å−1.
Atom or ion | A | a | B | b | C | c | D | e |
---|
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 |
|
Atom or ion | A | a | B | b | C | c | D | e |
---|
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 |
|
Ion | A | a | B | b | C | c | D | e |
---|
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 |
|
Ion | A | a | B | b | C | c | D | e |
---|
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 |
|
Atom or ion | A | a | B | b | C | c | D | e |
---|
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 |
|
Atom or ion | A | a | B | b | C | c | D | e |
---|
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 |
|
Ion | A | a | B | b | C | c | D | e |
---|
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 |
|
Ion | A | a | B | b | C | c | D | e |
---|
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 4.4.5.9–4.4.5.14 give coefficients in the approximation used by Lisher & Forsyth (1971) to the , , and form factors for the same series of atoms and ions, again using three rather than two exponential terms, viz for :
Atom or ion | A | a | B | b | C | c | D | e |
---|
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 |
|
Atom or ion | A | a | B | b | C | c | D | e |
---|
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 |
|
Ion | A | a | B | b | C | c | D | e |
---|
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 |
|
Ion | A | a | B | b | C | c | D | e |
---|
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 |
|
Ion | A | a | B | b | C | c | D | e |
---|
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 |
|
Ion | A | a | B | b | C | c | D | e |
---|
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) in terms of Slater-type functions in the form by using the identity:
The form factors have been calculated from these relationships in the range 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) and Desclaux & Freeman (1978) using Dirac–Fock theory. The constants given in Tables 4.4.5.3, 4.4.5.4, 4.4.5.7, 4.4.5.8, and 4.4.5.11–4.4.5.14 have been fitted to the results of these calculations. For the rare-earth ions, the form factors are given in the range 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: where is the difference between the ith fitted point and its theoretical value. The sum is over the N points included in the fitting procedure.