Magnetic form factors

Brown, P. J.

doi:10.1107/97809553602060000594

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

pdf | chapter contents | chapter index | related articles

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

Section 4.4.5. Magnetic form factors

P. J. Brown^a

4.4.5. Magnetic form factors

| top | pdf |

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 $[\langle\, j_l(k)\rangle =\textstyle \int\limits^\infty_0\,U^2(r)\, j_l(kr)4\pi r^2\,{\rm d} r, \eqno (4.4.5.1)]$ 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 –4.4.5.8 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) 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 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

Table 4.4.5.2| top | pdf |
〈j₀〉 form factors for 4d atoms and their ions

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

Table 4.4.5.3| top | pdf |
〈j₀〉 form factors for rare-earth ions

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

Table 4.4.5.4| top | pdf |
〈j₀〉 form factors for actinide ions

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

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

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

Table 4.4.5.6| top | pdf |
〈j₂〉 form factors for 4d atoms and their ions

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

Table 4.4.5.7| top | pdf |
〈j₂〉 form factors for rare-earth ions

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

Table 4.4.5.8| top | pdf |
〈j₂〉 form factors for actinide ions

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 $[\langle \,j_2\rangle]$ , $[\langle\,j_4\rangle]$ , 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 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

Table 4.4.5.10| top | pdf |
〈j₄〉 form factors for 4d atoms and their ions

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

Table 4.4.5.11| top | pdf |
〈j₄〉 form factors for rare-earth ions

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

Table 4.4.5.12| top | pdf |
〈j₄〉 form factors for actinide ions

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

Table 4.4.5.13| top | pdf |
〈j₆〉 form factors for rare-earth ions

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

Table 4.4.5.14| top | pdf |
〈j₆〉 form factors for actinide ions

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 $[U(r)=\textstyle\sum\limits_{nj}\,N_{nl}\,r^2A_{nlj}\exp(-a_{nlj}\,r) \eqno (4.4.5.4)]$ 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) 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 $[(\sin\theta)/\lambda=0]$ 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.Google Scholar

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