Scattering-factor interpolation

Maslen, E. N.; Fox, A. G.; O'Keefe, M. A.

doi:10.1107/97809553602060000600

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. 6.1, p. 565

Section 6.1.1.3.1. Scattering-factor interpolation

E. N. Maslen,^e ^‡ A. G. Fox^b and M. A. O'Keefe^c

6.1.1.3.1. Scattering-factor interpolation

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A general treatment of interpolation is complicated by possible difficulties resulting from singularities in tabulated functions. The interpolation of scattering factors does not involve such problems, however, and a more restricted treatment suffices.

An iterative method, applicable to a function f(x) tabulated at arbitrary values $[x_0,x_1,\ldots,x_n]$ is due to Aitken. $[f(x|x_0,x_1,\ldots,x_k)]$ is the polynominal that coincides with the tabulated values at $[x_0,x_1,\ldots,x_k]$ . $[\eqalignno{f(x|x_0,x_1)&={1\over x_1-x_0}\left\vert\matrix{f_0\,x_0-x\cr f_1\,x_1-x\cr} \right\vert\cr f(x|x_0,x_1,x_2)&={1\over x_2-x_1}\left\vert\matrix{f(x|x_0,x_1)&x_1-x\cr f(x|x_0,x_2)&x_2-x\cr} \right\vert\cr f(x|x_0,x_1,x_2,x_3)&={1\over x_3-x_2}\left\vert\matrix{f(x|x_0,x_1,x_2)&x_2-x\cr f(x|x_0,x_1,x_3)&x_3-x\cr} \right\vert.\cr & &(6.1.1.13)}]$ Iteration is continued until increasing k does not change the interpolated value significantly.

Another interpolation formula, due to Lagrange, is $[f(x)=\textstyle\sum\limits^n_{i=0}l_i(x)\,f_i+R_n(x),]$ where $[l_i(x)={\pi_n(x)\over(x-x_i)\pi_n'(x_i)}]$ and $[R_n(x)=\pi_n(x)[x_0,x_1,\ldots,x_n,x].\eqno (6.1.1.14)]$ $[\pi_n(x)]$ is $[(x-x_0)(x-x_1)\ldots(x-x_n)]$ and $[\pi_n'(x)]$ is its derivative, so that $[\eqalignno{\pi_n'(x_k)&=(x_k-x_0)(x_k-x_1)\ldots(x_k-x_{k-1})\cr &\quad \times(x_k-x_{k+1})\ldots(x_k-x_n)}]$ while $[\eqalign{[x_0,x_1]&={f_0-f_1\over x_0-x_1}\cr [x_0,x_1,x_2]&={[x_0,x_1]-[x_1,x_2]\over x_0-x_2}\cr [x_0,x_1\ldots x_n]&=\sum^n_{k=0} {f_k\over\pi_n'(x_k)}.}]$

For the scattering factors of Tables 6.1.1.1 and 6.1.1.3, the expansion $[f(\sin\theta/\lambda)=\textstyle\sum\limits^4_{i=1}a_i\exp(-b_i\sin^2\theta/\lambda^2)+c\eqno (6.1.1.15)]$ has been found to be particularly effective. The coefficients listed in Table 6.1.1.4 give a close fit to the atomic scattering curves over the range $[0\lt(\sin\theta)/\lambda\lt2.0\,{\rm \AA}^{-1}]$ . Table 6.1.1.4 also contains the maximum and minimum deviations from the true curve, and the mean of the magnitude of the deviation. For $[2.0\,{\rm \AA}^{-1}\lt(\sin\theta)/\lambda\lt6.0\,{\rm \AA}^{-1}]$ , Fox, Keefe & Tabbernor (1989) have shown that (6.1.1.15) is highly inaccurate, and they produced a `logarithmic polynomial' curve-fitting routine based on the equation $[\ln\{\,f[(\sin\theta)/\lambda]\}=\textstyle\sum\limits^3_{i=0}a_is^i\eqno (6.1.1.16)]$ for these high angles. The [a_i] values listed in Table 6.1.1.5 give a close fit to the atomic scattering factor curves over the range $[2.0\lt(\sin\theta)/\lambda\lt6.0\,{\rm\AA}^{-1}]$ . Because f varies slowly with $[(\sin\theta)/\lambda]$ at these high angles, four parameters are all that is necessary for accurate fitting. Confirmation of this is given in Table 6.1.1.5 where the correlation coefficients, C, associated with each fit are also shown, and it can be seen that these are close to 1.0 in every case.

Table 6.1.1.4| top | pdf |
Coefficients for analytical approximation to the scattering factors of Tables 6.1.1.1 and 6.1.1.3

		a₁	b₁	a₂	b₂	a₃	b₃	a₄	b₄	c	Maximum error	$[\sin\theta/\lambda]$ (Å⁻¹)	Mean error
H	SDS	0.493002	10.5109	0.322912	26.1257	0.140191	3.14236	0.040810	57.7997	0.003038	0.000	0.00	0.000
H	HF	0.489918	20.6593	0.262003	7.74039	0.196767	49.5519	0.049879	2.20159	0.001305	0.000	0.17	0.000
H¹⁻	HF	0.897661	53.1368	0.565616	15.1870	0.415815	186.576	0.116973	3.56709	0.002389	0.002	0.09	0.001
He	RHF	0.873400	9.10370	0.630900	3.35680	0.311200	22.9276	0.178000	0.982100	0.006400	0.001	1.01	0.000
Li	RHF	1.12820	3.95460	0.750800	1.05240	0.617500	85.3905	0.465300	168.261	0.037700	0.005	2.00	0.001
Li¹⁺	RHF	0.696800	4.62370	0.788800	1.95570	0.341400	0.631600	0.156300	10.0953	0.016700	0.001	1.78	0.000
Be	RHF	1.59190	43.6427	1.12780	1.86230	0.539100	103.483	0.702900	0.542000	0.038500	0.003	0.56	0.001
Be²⁺	RHF	6.26030	0.002700	0.884900	0.831300	0.799300	2.27580	0.164700	5.11460	−6.1092	0.001	1.97	0.000
B	RHF	2.05450	23.2185	1.33260	1.02100	1.09790	60.3498	0.706800	0.140300	−0.19320	0.002	0.75	0.001
C	RHF	2.31000	20.8439	1.02000	10.2075	1.58860	0.568700	0.865000	51.6512	0.215600	0.006	2.00	0.001
C_val	HF	2.26069	22.6907	1.56165	0.656665	1.05075	9.75618	0.839259	55.5949	0.286977	0.001	0.16	0.000
N	RHF	12.2126	0.005700	3.13220	9.89330	2.01250	28.9975	1.16630	0.582600	−11.529	0.007	0.11	0.002
O	RHF	3.04850	13.2771	2.28680	5.70110	1.54630	0.323900	0.867000	32.9089	0.250800	0.001	0.22	0.000
O¹⁻	HF	4.19160	12.8573	1.63969	4.17236	1.52673	47.0179	−20.307	−0.01404	21.9412	0.011	1.50	0.004
F	RHF	3.53920	10.2825	2.64120	4.29440	1.51700	0.261500	1.02430	26.1476	0.277600	0.001	0.01	0.000
F¹⁻	HF	3.63220	5.27756	3.51057	14.7353	1.26064	0.442258	0.940706	47.3437	0.653396	0.003	0.09	0.001
Ne	RHF	3.95530	8.40420	3.11250	3.42620	1.45460	0.230600	1.12510	21.7184	0.351500	0.002	0.25	0.001
Na	RHF	4.76260	3.28500	3.17360	8.84220	1.26740	0.313600	1.11280	129.424	0.676000	0.009	0.13	0.002
Na¹⁺	RHF	3.25650	2.66710	3.93620	6.11530	1.39980	0.200100	1.00320	14.0390	0.404000	0.001	0.70	0.000
Mg	RHF	5.42040	2.82750	2.17350	79.2611	1.22690	0.380800	2.30730	7.19370	0.858400	0.015	0.08	0.003
Mg²⁺	RHF	3.49880	2.16760	3.83780	4.75420	1.32840	0.185000	0.849700	10.1411	0.485300	0.001	1.34	0.000
Al	RHF	6.42020	3.03870	1.90020	0.742600	1.59360	31.5472	1.96460	85.0886	1.11510	0.018	2.00	0.005
Al³⁺	HF	4.17448	1.93816	3.38760	4.14553	1.20296	0.228753	0.528137	8.28524	0.706786	0.000	1.50	0.000
Si_v	RHF	6.29150	2.43860	3.03530	32.3337	1.98910	0.678500	1.54100	81.6937	1.14070	0.009	2.00	0.002
Si_val	HF	5.66269	2.66520	3.07164	38.6634	2.62446	0.916946	1.39320	93.5458	1.24707	0.001	0.53	0.001
Si⁴⁺	HF	4.43918	1.64167	3.20345	3.43757	1.19453	0.214900	0.416530	6.65365	0.746297	0.000	1.50	0.000
P	RHF	6.43450	1.90670	4.17910	27.1570	1.78000	0.526000	1.49080	68.1645	1.11490	0.003	0.65	0.001
S	RHF	6.90530	1.46790	5.20340	22.2151	1.43790	0.253600	1.58630	56.1720	0.866900	0.005	0.67	0.002
Cl	RHF	11.4604	0.010400	7.19640	1.16620	6.25560	18.5194	1.64550	47.7784	−9.5574	0.007	0.78	0.003
Cl¹⁻	RHF	18.2915	0.006600	7.20840	1.17170	6.53370	19.5424	2.33860	60.4486	−16.378	0.007	0.76	0.003
Ar	RHF	7.48450	0.907200	6.77230	14.8407	0.653900	43.8983	1.64420	33.3929	1.44450	0.029	2.00	0.006
K	RHF	8.21860	12.7949	7.43980	0.774800	1.05190	213.187	0.865900	41.6841	1.42280	0.011	0.90	0.005
K¹⁺	RHF	7.95780	12.6331	7.49170	0.767400	6.35900	−0.00200	1.19150	31.9128	−4.9978	0.011	0.91	0.005
Ca	RHF	8.62660	10.4421	7.38730	0.659900	1.58990	85.7484	1.02110	178.437	1.37510	0.016	0.99	0.006
Ca²⁺	RHF	15.6348	−0.00740	7.95180	0.608900	8.43720	10.3116	0.853700	25.9905	−14.875	0.017	2.00	0.004
Sc	RHF	9.18900	9.02130	7.36790	0.572900	1.64090	136.108	1.46800	51.3531	1.33290	0.014	1.07	0.006
Sc³⁺	HF	13.4008	0.298540	8.02730	7.96290	1.65943	−0.28604	1.57936	16.0662	−6.6667	0.002	1.50	0.000
Ti	RHF	9.75950	7.85080	7.35580	0.500000	1.69910	35.6338	1.90210	116.105	1.28070	0.014	2.00	0.006
Ti²⁺	HF	9.11423	7.52430	7.62174	0.457585	2.27930	19.5361	0.087899	61.6558	0.897155	0.006	1.50	0.001
Ti³⁺	HF	17.7344	0.220610	8.73816	7.04716	5.25691	−0.15762	1.92134	15.9768	−14.652	0.001	0.00	0.000
Ti⁴⁺	HF	19.5114	0.178847	8.23473	6.67018	2.01341	−0.29263	1.52080	12.9464	−13.280	0.002	1.50	0.000
V	RHF	10.2971	6.86570	7.35110	0.438500	2.07030	26.8938	2.05710	102.478	1.21990	0.014	2.00	0.005
V²⁺	RHF	10.1060	6.88180	7.35410	0.440900	2.28840	20.3004	0.022300	115.122	1.22980	0.015	2.00	0.004
V³⁺	HF	9.43141	6.39535	7.74190	0.383349	2.15343	15.1908	0.016865	63.9690	0.656565	0.004	1.50	0.001
V⁵⁺	HF	15.6887	0.679003	8.14208	5.40135	2.03081	9.97278	−9.5760	0.940464	1.71430	0.000	0.34	0.000
Cr	RHF	10.6406	6.10380	7.35370	0.392000	3.32400	20.2626	1.49220	98.7399	1.18320	0.011	2.00	0.004
Cr²⁺	HF	9.54034	5.66078	7.75090	0.344261	3.58274	13.3075	0.509107	32.4224	0.616898	0.002	1.50	0.000
Cr³⁺	HF	9.68090	5.59463	7.81136	0.334393	2.87603	12.8288	0.113575	32.8761	0.518275	0.002	1.50	0.000
Mn	RHF	11.2819	5.34090	7.35730	0.343200	3.01930	17.8674	2.24410	83.7543	1.08960	0.009	2.00	0.004
Mn²⁺	RHF	10.8061	5.27960	7.36200	0.343500	3.52680	14.3430	0.218400	41.3235	1.08740	0.009	2.00	0.002
Mn³⁺	HF	9.84521	4.91797	7.87194	0.294393	3.56531	10.8171	0.323613	24.1281	0.393974	0.001	1.50	0.000
Mn⁴⁺	HF	9.96253	4.84850	7.97057	0.283303	2.76067	10.4852	0.054447	27.5730	0.251877	0.001	1.50	0.000
Fe	RHF	11.7695	4.76110	7.35730	0.307200	3.52220	15.3535	2.30450	76.8805	1.03690	0.011	0.08	0.004
Fe²⁺	RHF	11.0424	4.65380	7.37400	0.305300	4.13460	12.0546	0.439900	31.2809	1.00970	0.008	2.00	0.002
Fe³⁺	RHF	11.1764	4.61470	7.38630	0.300500	3.39480	11.6729	0.072400	38.5566	0.970700	0.008	2.00	0.002
Co	RHF	12.2841	4.27910	7.34090	0.278400	4.00340	13.5359	2.34880	71.1692	1.01180	0.013	0.08	0.004
Co²⁺	RHF	11.2296	4.12310	7.38830	0.272600	4.73930	10.2443	0.710800	25.6466	0.932400	0.006	2.00	0.001
Co³⁺	HF	10.3380	3.90969	7.88173	0.238668	4.76795	8.35583	0.725591	18.3491	0.286667	0.000	1.50	0.000
Ni	RHF	12.8376	3.87850	7.29200	0.256500	4.44380	12.1763	2.38000	66.3421	1.0341	0.014	0.08	0.004
Ni²⁺	RHF	11.4166	3.67660	7.40050	0.244900	5.34420	8.87300	0.977300	22.1626	0.861400	0.003	2.00	0.001
Ni³⁺	HF	10.7806	3.54770	7.75868	0.223140	5.22746	7.64468	0.847114	16.9673	0.386044	0.000	0.57	0.000
Cu	RHF	13.3380	3.58280	7.16760	0.247000	5.61580	11.3966	1.67350	64.8126	1.19100	0.015	0.08	0.005
Cu¹⁺	RHF	11.9475	3.36690	7.35730	0.227400	6.24550	8.66250	1.55780	25.8487	0.89000	0.003	0.24	0.001
Cu²⁺	HF	11.8168	3.37484	7.11181	0.244078	5.78135	7.98760	1.14523	19.8970	1.14431	0.001	0.26	0.000
Zn	RHF	14.0743	3.26550	7.03180	0.233300	5.16520	10.3163	2.41000	58.7097	1.30410	0.016	0.08	0.005
Zn²⁺	RHF	11.9719	2.99460	7.38620	0.203100	6.46680	7.08260	1.39400	18.0995	0.780700	0.001	0.62	0.000
Ga	RHF	15.2354	3.06690	6.70060	0.241200	4.35910	10.7805	2.96230	61.4135	1.71890	0.025	0.08	0.008
Ga³⁺	HF	12.6920	2.81262	6.69883	0.227890	6.06692	6.36441	1.00660	14.4122	1.53545	0.008	1.45	0.000
Ge	RHF	16.0816	2.85090	6.37470	0.251600	3.70680	11.4468	3.68300	54.7625	2.13130	0.024	0.08	0.008
Ge⁴⁺	HF	12.9172	2.53718	6.70003	0.205855	6.06791	5.47913	0.859041	11.6030	1.45572	0.000	0.32	0.000
As	RHF	16.6723	2.63450	6.07010	0.264700	3.43130	12.9479	4.27790	47.7972	2.53100	0.019	0.09	0.008
Se	RHF	17.0006	2.40980	5.81960	0.272600	3.97310	15.2372	4.35430	43.8163	2.84090	0.016	2.00	0.006
Br	RHF	17.1789	2.17230	5.23580	16.5796	5.63770	0.260900	3.98510	41.4328	2.95570	0.012	2.00	0.004
Br¹⁻	RHF	17.1718	2.20590	6.33380	19.3345	5.57540	0.287100	3.72720	58.1535	3.17760	0.016	2.00	0.006
Kr	RHF	17.3555	1.93840	6.72860	16.5623	5.54930	0.226100	3.53750	39.3972	2.82500	0.008	2.00	0.002
Rb	RHF	17.1784	1.78880	9.64350	17.3151	5.13990	0.274800	1.52920	164.934	3.48730	0.028	0.12	0.008
Rb¹⁺	RHF	17.5816	1.71390	7.65980	14.7957	5.89810	0.160300	2.78170	31.2087	2.07820	0.002	1.99	0.001
Sr	RHF	17.5663	1.55640	9.81840	14.0988	5.42200	0.166400	2.66940	132.376	2.50640	0.021	0.13	0.005
Sr²⁺	RHF	18.0874	1.49070	8.13730	12.6963	2.56540	24.5651	−34.193	−0.01380	41.4025	0.008	2.00	0.002
Y	*RHF	17.7760	1.40290	10.2946	12.8006	5.72629	0.125599	3.26588	104.354	1.91213	0.028	0.07	0.006
Y³⁺	*DS	17.9268	1.35417	9.15310	11.2145	1.76795	22.6599	−33.108	−0.01319	40.2602	0.005	2.00	0.001
Zr	*RHF	17.8765	1.27618	10.9480	11.9160	5.41732	0.117622	3.65721	87.6627	2.06929	0.035	0.07	0.008
Zr⁴⁺	*DS	18.1668	1.21480	10.0562	10.1483	1.01118	21.6054	−2.6479	−0.10276	9.41454	0.004	2.00	0.001
Nb	*RHF	17.6142	1.18865	12.0144	11.7660	4.04183	0.204785	3.53346	69.7957	3.75591	0.042	0.08	0.011
Nb³⁺	*DS	19.8812	0.019175	18.0653	1.13305	11.0177	10.1621	1.94715	28.3389	−12.912	0.006	2.00	0.002
Nb⁵⁺	*DS	17.9163	1.12446	13.3417	0.028781	10.7990	9.28206	0.337905	25.7228	−6.3934	0.007	2.00	0.003
Mo	RHF	3.70250	0.277200	17.2356	1.09580	12.8876	11.0040	3.74290	61.6584	4.38750	0.046	0.08	0.012
Mo³⁺	*DS	21.1664	0.014734	18.2017	1.03031	11.7423	9.53659	2.30951	26.6307	−14.421	0.009	2.00	0.003
Mo⁵⁺	*DS	21.0149	0.014345	18.0992	1.02238	11.4632	8.78809	0.740625	23.3452	−14.316	0.010	2.00	0.003
Mo⁶⁺	*DS	17.8871	1.03649	11.1750	8.48061	6.57891	0.058881	0.000000	0.000000	0.344941	0.014	0.00	0.006
Tc	*RHF	19.1301	0.864132	11.0948	8.14487	4.64901	21.5707	2.71263	86.8472	5.40428	0.061	2.00	0.011
Ru	*RHF	19.2674	0.808520	12.9182	8.43467	4.86337	24.7997	1.56756	94.2928	5.37874	0.041	2.00	0.006
Ru³⁺	*DS	18.5638	0.847329	13.2885	8.37164	9.32602	0.017662	3.00964	22.8870	−3.1892	0.013	2.00	0.004
Ru⁴⁺	*DS	18.5003	0.844582	13.1787	8.12534	4.71304	0.36495	2.18535	20.8504	1.42357	0.014	2.00	0.004
Rh	*RHF	19.2957	0.751536	14.3501	8.21758	4.73425	25.8749	1.28918	98.6062	5.32800	0.021	2.00	0.004
Rh³⁺	*DS	18.8785	0.764252	14.1259	7.84438	3.32515	21.2487	−6.1989	−0.01036	11.8678	0.014	2.00	0.004
Rh⁴⁺	*DS	18.8545	0.760825	13.9806	7.62436	2.53464	19.3317	−5.6526	−0.01020	11.2835	0.014	2.00	0.003
Pd	*RHF	19.3319	0.698655	15.5017	7.98929	5.29537	25.2052	0.605844	76.8986	5.26593	0.012	1.08	0.005
Pd²⁺	*DS	19.1701	0.696219	15.2096	7.55573	4.32234	22.5057	0.000000	0.000000	5.29160	0.011	2.00	0.004
Pd⁴⁺	*DS	19.2493	0.683839	14.7900	7.14833	2.89289	17.9144	−7.9492	0.005127	13.0174	0.014	2.00	0.003
Ag	RHF	19.2808	0.644600	16.6885	7.47260	4.80450	24.6605	1.04630	99.8156	5.17900	0.016	1.14	0.007
Ag¹⁺	*DS	19.1812	0.646179	15.9719	7.19123	5.27475	21.7326	0.357534	66.1147	5.21572	0.012	1.13	0.005
Ag²⁺	*DS	19.1643	0.645643	16.2456	7.18544	4.37090	21.4072	0.000000	0.000000	5.21404	0.011	1.14	0.005
Cd	RHF	19.2214	0.594600	17.6444	6.90890	4.46100	24.7008	1.60290	87.4825	5.06940	0.020	2.00	0.008
Cd²⁺	*DS	19.1514	0.597922	17.2535	6.80639	4.47128	20.2521	0.000000	0.000000	5.11937	0.014	1.17	0.007
In	RHF	19.1624	0.547600	18.5596	6.37760	4.29480	25.8499	2.03960	92.8029	4.93910	0.027	2.00	0.009
In³⁺	*DS	19.1045	0.551522	18.1108	6.32470	3.78897	17.3595	0.000000	0.000000	4.99635	0.022	2.00	0.007
Sn	RHF	19.1889	5.83030	19.1005	0.503100	4.45850	26.8909	2.46630	83.9571	4.78210	0.032	2.00	0.009
Sn²⁺	RHF	19.1094	0.503600	19.0548	5.83780	4.56480	23.3752	0.487000	62.2061	4.78610	0.032	2.00	0.009
Sn⁴⁺	RHF	18.9333	5.76400	19.7131	0.465500	3.41820	14.0049	0.019300	−0.75830	3.91820	0.016	2.00	0.004
Sb	RHF	19.6418	5.30340	19.0455	0.460700	5.03710	27.9074	2.68270	75.2825	4.59090	0.035	2.00	0.009
Sb³⁺	*DS	18.9755	0.467196	18.9330	5.22126	5.10789	19.5902	0.288753	55.5113	4.69626	0.028	2.00	0.007
Sb⁵⁺	*DS	19.8685	5.44853	19.0302	0.467973	2.41253	14.1259	0.000000	0.000000	4.69263	0.030	2.00	0.008
Te	*RHF	19.9644	4.81742	19.0138	0.420885	6.14487	28.5284	2.52390	70.8403	4.35200	0.038	2.00	0.009
I	RHF	20.1472	4.34700	18.9949	0.381400	7.51380	27.7660	2.27350	66.8776	4.07120	0.037	2.00	0.009
I¹⁻	RHF	20.2332	4.35790	18.9970	0.381500	7.80690	29.5259	2.88680	84.9304	4.07140	0.038	2.00	0.009
Xe	RHF	20.2933	3.92820	19.0298	0.344000	8.97670	26.4659	1.99000	64.2658	3.71180	0.038	2.00	0.009
Cs	RHF	20.3892	3.56900	19.1062	0.310700	10.6620	24.3879	1.49530	213.904	3.33520	0.032	2.00	0.010
Cs¹⁺	RHF	20.3524	3.55200	19.1278	0.308600	10.2821	23.7128	0.961500	59.4565	3.27910	0.037	2.00	0.009
Ba	RHF	20.3361	3.21600	19.2970	0.275600	10.8880	20.2073	2.69590	167.202	2.77310	0.032	2.00	0.009
Ba²⁺	*DS	20.1807	3.21367	19.1136	0.283310	10.9054	20.0558	0.77634	51.7460	3.02902	0.029	2.00	0.007
La	*RHF	20.5780	2.94817	19.5990	0.244475	11.3727	18.7726	3.28719	133.124	2.14678	0.032	2.00	0.009
La³⁺	*DS	20.2489	2.92070	19.3763	0.250698	11.6323	17.8211	0.336048	54.9453	2.40860	0.028	2.00	0.007
Ce	*RHF	21.1671	2.81219	19.7695	0.226836	11.8513	17.6083	3.33049	127.113	1.86264	0.026	2.00	0.008
Ce³⁺	*DS	20.8036	2.77691	19.5590	0.231540	11.9369	16.5408	0.612376	43.1692	2.09013	0.023	2.00	0.005
Ce⁴⁺	*DS	20.3235	2.65941	19.8186	0.218850	12.1233	15.7992	0.144583	62.2355	1.59180	0.026	2.00	0.007
Pr	*RHF	22.0440	2.77393	19.6697	0.222087	12.3856	16.7669	2.82428	143.644	2.05830	0.021	0.12	0.007
Pr³⁺	*DS	21.3727	2.64520	19.7491	0.214299	12.1329	15.3230	0.975180	36.4065	1.77132	0.019	2.00	0.004
Pr⁴⁺	*DS	20.9413	2.54467	20.0539	0.202481	12.4668	14.8137	0.296689	45.4643	1.24285	0.021	2.00	0.005
Nd	*RHF	22.6845	2.66248	19.6847	0.210628	12.7740	15.8850	2.85137	137.903	1.98486	0.024	0.13	0.007
Nd³⁺	*DS	21.9610	2.52722	19.9339	0.199237	12.1200	14.1783	1.51031	30.8717	1.47588	0.015	2.00	0.003
Pm	*RHF	23.3405	2.56270	19.6095	0.202088	13.1235	15.1009	2.87516	132.721	2.02876	0.026	0.13	0.008
Pm³⁺	*DS	22.5527	2.41740	20.1108	0.185769	12.0671	13.1275	2.07492	27.4491	1.19499	0.012	2.00	0.002
Sm	*RHF	24.0042	2.47274	19.4258	0.196451	13.4396	14.3996	2.89604	128.007	2.20963	0.029	0.13	0.009
Sm³⁺	*DS	23.1504	2.31641	20.2599	0.174081	11.9202	12.1571	2.71488	24.8242	0.954586	0.009	2.00	0.002
Eu	RHF	24.6274	2.38790	19.0886	0.194200	13.7603	13.7546	2.92270	123.174	2.57450	0.031	0.14	0.010
Eu²⁺	*DS	24.0063	2.27783	19.9504	0.173530	11.8034	11.6096	3.87243	26.5156	1.36389	0.004	2.00	0.002
Eu³⁺	*DS	23.7497	2.22258	20.3745	0.163940	11.8509	11.3110	3.26503	22.9966	0.759344	0.006	2.00	0.001
Gd	*RHF	25.0709	2.25341	19.0798	0.181951	13.8518	12.9331	3.54545	101.398	2.41960	0.036	0.15	0.011
Gd³⁺	*DS	24.3466	2.13553	20.4208	0.155525	11.8708	10.5782	3.71490	21.7029	0.645089	0.004	2.00	0.001
Tb	*RHF	25.8976	2.24256	18.2185	0.196143	14.3167	12.6648	2.95354	115.362	3.58324	0.035	0.14	0.012
Tb³⁺	*DS	24.9559	2.05601	20.3271	0.149525	12.2471	10.0499	3.77300	21.2773	0.691967	0.005	0.00	0.001
Dy	*RHF	26.5070	2.18020	17.6383	0.202172	14.5596	12.1899	2.96577	111.874	4.29728	0.037	0.15	0.013
Dy³⁺	*DS	25.5395	1.98040	20.2861	0.143384	11.9812	9.34972	4.50073	19.5810	0.689690	0.003	0.00	0.001
Ho	*RHF	26.9049	2.07051	17.2940	0.197940	14.5583	11.4407	3.63837	92.6566	4.56796	0.040	0.15	0.013
Ho³⁺	*DS	26.1296	1.91072	20.0994	0.139358	11.9788	8.80018	4.93676	18.5908	0.852795	0.003	0.00	0.001
Er	*RHF	27.6563	2.07356	16.4285	0.223545	14.9779	11.3604	2.98233	105.703	5.92046	0.040	0.15	0.015
Er³⁺	*DS	26.7220	1.84659	19.7748	0.137290	12.1506	8.36225	5.17379	17.8974	1.17613	0.003	0.00	0.001
Tm	*RHF	28.1819	2.02859	15.8851	0.238849	15.1542	10.9975	2.98706	102.961	6.75621	0.041	0.15	0.016
Tm³⁺	*DS	27.3083	1.78711	19.3320	0.136974	12.3339	7.96778	5.38348	17.2922	1.63929	0.003	0.00	0.001
Yb	*RHF	28.6641	1.98890	15.4345	0.257119	15.3087	10.6647	2.98963	100.417	7.56672	0.042	0.15	0.016
Yb²⁺	*DS	28.1209	1.78503	17.6817	0.159970	13.3335	8.18304	5.14657	20.3900	3.70983	0.008	0.00	0.003
Yb³⁺	*DS	27.8917	1.73272	18.7614	0.138790	12.6072	7.64412	5.47647	16.8153	2.26001	0.003	0.00	0.002
Lu	*RHF	28.9476	1.90182	15.2208	9.98519	15.1000	0.261033	3.71601	84.3298	7.97628	0.043	0.16	0.016
Lu³⁺	*DS	28.4628	1.68216	18.1210	0.142292	12.8429	7.33727	5.59415	16.3535	2.97573	0.004	0.14	0.002
Hf	*RHF	29.1440	1.83262	15.1726	9.59990	14.7586	0.275116	4.30013	72.0290	8.58154	0.047	0.08	0.016
Hf⁴⁺	*DS	28.8131	1.59136	18.4601	0.128903	12.7285	6.76232	5.59927	14.0366	2.39699	0.002	0.00	0.001
Ta	*RHF	29.2024	1.77333	15.2293	9.37046	14.5135	0.295977	4.76492	63.3644	9.24354	0.049	0.08	0.017
Ta⁵⁺	*DS	29.1587	1.50711	18.8407	0.116741	12.8268	6.31524	5.38695	12.4244	1.78555	0.002	2.00	0.001
W	*RHF	29.0818	1.72029	15.4300	9.22590	14.4327	0.321703	5.11982	57.0560	9.88750	0.051	0.09	0.017
W⁶⁺	*DS	29.4936	1.42755	19.3763	0.104621	13.0544	5.93667	5.06412	11.1972	1.01074	0.001	0.00	0.000
Re	*RHF	28.7621	1.67191	15.7189	9.09227	14.5564	0.350500	5.44174	52.0861	10.4720	0.052	0.09	0.017
Os	*RHF	28.1894	1.62903	16.1550	8.97948	14.9305	0.382661	5.67589	48.1647	11.0005	0.051	0.09	0.017
Os⁴⁺	*DS	30.4190	1.37113	15.2637	6.84706	14.7458	0.165191	5.06795	18.0030	6.49804	0.006	0.29	0.003
Ir	*RHF	27.3049	1.59279	16.7296	8.86553	15.6115	0.417916	5.83377	45.0011	11.4722	0.050	0.09	0.017
Ir³⁺	*DS	30.4156	1.34323	15.8620	7.10909	13.6145	0.204633	5.82008	20.3254	8.27903	0.009	0.28	0.004
Ir⁴⁺	*DS	30.7058	1.30923	15.5512	6.71983	14.2326	0.167252	5.53672	17.4911	6.96824	0.006	0.29	0.003
Pt	*RHF	27.0059	1.51293	17.7639	8.81174	15.7131	0.424593	5.78370	38.6103	11.6883	0.046	0.10	0.016
Pt²⁺	*DS	29.8429	1.32927	16.7224	7.38979	13.2153	0.263297	6.35234	22.9426	9.85329	0.014	0.00	0.006
Pt⁴⁺	*DS	30.9612	1.24813	15.9829	6.60834	13.7348	0.168640	5.92034	16.9392	7.39534	0.006	0.14	0.003
Au	RHF	16.8819	0.461100	18.5913	8.62160	25.5582	1.48260	5.86000	36.3956	12.0658	0.045	0.10	0.015
Au¹⁺	*DS	28.0109	1.35321	17.8204	7.73950	14.3359	0.356752	6.58077	26.4043	11.2299	0.023	0.12	0.009
Au³⁺	*DS	30.6886	1.21990	16.9029	6.82872	12.7801	0.212867	6.52354	18.6590	9.09680	0.009	0.14	0.004
Hg	RHF	20.6809	0.545000	19.0417	8.44840	21.6575	1.57290	5.96760	38.3246	12.6089	0.046	0.10	0.017
Hg¹⁺	*DS	25.0853	1.39507	18.4973	7.65105	16.8883	0.443378	6.48216	28.2262	12.0205	0.027	0.12	0.011
Hg²⁺	*DS	29.5641	1.21152	18.0600	7.05639	12.8374	0.284738	6.89912	20.7482	10.6268	0.013	0.00	0.006
Tl	*RHF	27.5446	0.655150	19.1584	8.70751	15.5380	1.96347	5.52593	45.8149	13.1746	0.059	0.09	0.021
Tl¹⁺	*DS	21.3985	1.47110	20.4723	0.517394	18.7478	7.43463	6.82847	28.8482	12.5258	0.028	0.12	0.011
Tl³⁺	*DS	30.8695	1.10080	18.3481	6.53852	11.9328	0.219074	7.00574	17.2114	9.80270	0.008	0.01	0.004
Pb	RHF	31.0617	0.690200	13.0637	2.35760	18.4420	8.61800	5.96960	47.2579	13.4118	0.060	2.00	0.021
Pb²⁺	*DS	21.7886	1.33660	19.5682	0.488383	19.1406	6.77270	7.01107	23.8132	12.4734	0.020	2.00	0.008
Pb⁴⁺	*DS	32.1244	1.00566	18.8003	6.10926	12.0175	0.147041	6.96886	14.7140	8.08428	0.005	0.31	0.002
Bi	RHF	33.3689	0.704000	12.9510	2.92380	16.5877	8.79370	6.46920	48.0093	13.5782	0.065	2.00	0.020
Bi³⁺	*DS	21.8053	1.23560	19.5026	6.24149	19.1053	0.469999	7.10295	20.3185	12.4711	0.015	2.00	0.006
Bi⁵⁺	*DS	33.5364	0.916540	25.0946	0.39042	19.2497	5.71414	6.91555	12.8285	−6.7994	0.003	0.00	0.001
Po	*RHF	34.6726	0.700999	15.4733	3.55078	13.1138	9.55642	7.02588	47.0045	13.6770	0.066	2.00	0.018
At	*RHF	35.3163	0.685870	19.0211	3.97458	9.49887	11.3824	7.42518	45.4715	13.7108	0.062	2.00	0.015
Rn	RHF	35.5631	0.663100	21.2816	4.06910	8.00370	14.0422	7.44330	44.2473	13.6905	0.054	2.00	0.012
Fr	*RHF	35.9299	0.646453	23.0547	4.17619	12.1439	23.1052	2.11253	150.645	13.7247	0.055	2.00	0.017
Ra	*RHF	35.7630	0.616341	22.9064	3.87135	12.4739	19.9887	3.21097	142.325	13.6211	0.037	2.00	0.012
Ra²⁺	*DS	35.2150	0.604909	21.6700	3.57670	7.91342	12.6010	7.65078	29.8436	13.5431	0.029	2.00	0.006
Ac	*RHF	35.6597	0.589092	23.1032	3.65155	12.5977	18.5990	4.08655	117.020	13.5266	0.030	0.06	0.009
Ac³⁺	*DS	35.1736	0.579689	22.1112	3.41437	8.19216	12.9187	7.05545	25.9443	13.4637	0.021	2.00	0.004
Th	*RHF	35.5645	0.563359	23.4219	3.46204	12.7473	17.8309	4.80703	99.1722	13.4314	0.031	0.07	0.008
Th⁴⁺	*DS	35.1007	0.555054	22.4418	3.24498	9.78554	13.4661	5.29444	23.9533	13.3760	0.014	2.00	0.002
Pa	*RHF	35.8847	0.547751	23.2948	3.41519	14.1891	16.9235	4.17287	105.251	13.4287	0.033	0.06	0.010
U	RHF	36.0228	0.529300	23.4128	3.32530	14.9491	16.0927	4.18800	100.613	13.3966	0.035	0.07	0.010
U³⁺	*DS	35.5747	0.520480	22.5259	3.12293	12.2165	12.7148	5.37073	26.3394	13.3092	0.009	2.00	0.002
U⁴⁺	*DS	35.3715	0.516598	22.5326	3.05053	12.0291	12.5723	4.79840	23.4582	13.2671	0.007	2.00	0.001
U⁶⁺	*DS	34.8509	0.507079	22.7584	2.89030	14.0099	13.1767	1.21457	25.2017	13.1665	0.003	2.00	0.001
Np	*RHF	36.1874	0.511929	23.5964	3.25396	15.6402	15.3622	4.18550	97.4908	13.3573	0.037	0.07	0.011
Np³⁺	*DS	35.7074	0.502322	22.6130	3.03807	12.9898	12.1449	5.43227	25.4928	13.2544	0.006	2.00	0.002
Np⁴⁺	*DS	35.5103	0.498626	22.5787	2.96627	12.7766	11.9484	4.92159	22.7502	13.2116	0.005	2.00	0.001
Np⁶⁺	*DS	35.0136	0.489810	22.7286	2.81099	14.3884	12.3300	1.75669	22.6581	13.1130	0.002	2.00	0.001
Pu	*RHF	36.5254	0.499384	23.8083	3.26371	16.7707	14.9455	3.47947	105.980	13.3812	0.038	0.14	0.013
Pu³⁺	*DS	35.8400	0.484938	22.7169	2.96118	13.5807	11.5331	5.66016	24.3992	13.1991	0.005	2.00	0.001
Pu⁴⁺	*DS	35.6493	0.481422	22.6460	2.89020	13.3595	11.3160	5.18831	21.8301	13.1555	0.003	2.00	0.001
Pu⁶⁺	*DS	35.1736	0.473204	22.7181	2.73848	14.7635	11.5530	2.28678	20.9303	13.0582	0.001	1.36	0.001
Am	*RHF	36.6706	0.483629	24.0992	3.20647	17.3415	14.3136	3.49331	102.273	13.3592	0.040	0.07	0.013
Cm	*RHF	36.6488	0.465154	24.4096	3.08997	17.3990	13.4346	4.21665	88.4834	13.2887	0.041	0.07	0.013
Bk	*RHF	36.7881	0.451018	24.7736	3.04619	17.8919	12.8946	4.23284	86.0030	13.2754	0.042	0.07	0.014
Cf	*RHF	36.9185	0.437533	25.1995	3.00775	18.3317	12.4044	4.24391	83.7881	13.2674	0.043	0.07	0.014

Table 6.1.1.5| top | pdf |
Coefficients for analytical approximation to the scattering factors of Table 6.1.1.1 for the range 2.0 < (sin $[\theta]$ )/λ < 6.0 Å⁻¹ [equation (6.1.1.16)]

	Symbol			(×10)	(×100)
2	He	0.52543	−3.43300	4.80070	−2.54760	1.0000
3	Li	0.89463	−2.43660	2.32500	−0.71949	1.0000
4	Be	1.25840	−1.94590	1.30460	−0.04297	1.0000
5	B	1.66720	−1.85560	1.60440	−0.65981	1.0000
6	C	1.70560	−1.56760	1.18930	−0.42715	1.0000
7	N	1.54940	−1.20190	0.51064	0.02472	1.0000
8	O	1.30530	−0.83742	−0.16738	0.47500	1.0000
9	F	1.16710	−0.63203	−0.40207	0.54352	1.0000
10	Ne	1.09310	−0.50221	−0.53648	0.60957	0.9995
11	Na	0.84558	−0.26294	−0.87884	0.76974	1.0000
12	Mg	0.71877	−0.13144	−1.20900	0.82738	1.0000
13	Al	0.67975	−0.08756	−0.95431	0.72294	1.0000
14	Si	0.70683	−0.09888	−0.98356	0.55631	1.0000
15	P	0.85532	−0.21262	−0.37390	0.20731	1.0000
16	S	1.10400	−0.40325	0.20094	−0.26058	1.0000
17	Cl	1.42320	−0.63936	0.84722	−0.76135	0.9995
18	Ar	1.82020	−0.92776	1.59220	−1.32510	0.9995
19	K	2.26550	−1.24530	2.38330	−1.91290	0.9990
20	Ca	2.71740	−1.55670	3.13170	−2.45670	0.9990
21	Sc	3.11730	−1.81380	3.71390	−2.85330	0.9990
22	Ti	3.45360	−2.01150	4.13170	−3.11710	0.9995
23	V	3.71270	−2.13920	4.34610	−3.22040	0.9995
24	Cr	3.87870	−2.19000	4.38670	−3.17520	1.0000
25	Mn	3.98550	−2.18850	4.27960	−3.02150	1.0000
26	Fe	3.99790	−2.11080	3.98170	−2.71990	1.0000
27	Co	3.95900	−1.99650	3.60630	−2.37050	1.0000
28	Ni	3.86070	−1.88690	3.12390	−1.94290	1.0000
29	Cu	3.72510	−1.65500	2.60290	−1.49760	0.9995
30	Zn	3.55950	−1.45100	2.03390	−1.02160	0.9995
31	Ga	3.37560	−1.23910	1.46160	−0.55471	0.9995
32	Ge	3.17800	−1.02230	0.89119	−0.09884	0.9995
33	As	2.97740	−0.81038	0.34861	0.32231	0.9995
34	Se	2.78340	−0.61110	−0.14731	0.69837	0.9995
35	Br	2.60610	−0.43308	−0.57381	1.00950	0.9995
36	Kr	2.44280	−0.27244	−0.95570	1.27070	0.9995
37	Rb	2.30990	−0.14328	−1.22600	1.45320	1.0000
38	Sr	2.21070	−0.04770	−1.41100	1.55410	1.0000
39	Y	2.14220	0.01935	−1.52240	1.59630	1.0000
40	Zr	2.12690	0.08618	−1.49190	1.51820	1.0000
41	Nb	2.12120	0.05381	−1.50070	1.50150	1.0000
42	Mo	2.18870	−0.00655	−1.25340	1.24010	1.0000
43	Tc	2.25730	−0.05737	−1.07450	1.06630	1.0000
44	Ru	2.37300	−0.15040	−0.77694	0.79060	0.9995
45	Rh	2.50990	−0.25906	−0.44719	0.49443	0.9995
46	Pd	2.67520	−0.39137	−0.05894	0.15404	0.9995
47	Ag	2.88690	−0.56119	0.42189	−0.25659	0.9990
48	Cd	3.08430	−0.71450	0.84482	−0.60990	0.9990
49	In	3.31400	−0.89697	1.35030	−1.03910	0.9990
50	Sn	3.49840	−1.02990	1.68990	−1.29860	0.9990
51	Sb	3.70410	−1.18270	2.08920	−1.61640	0.9990
52	Te	3.88240	−1.30980	2.41170	−1.86420	0.9990
53	I	4.08010	−1.45080	2.76730	−2.13920	0.9990
54	Xe	4.24610	−1.56330	3.04200	−2.34290	0.9990
55	Cs	4.38910	−1.65420	3.25450	−2.49220	0.9995
56	Ba	4.51070	−1.72570	3.41320	−2.59590	0.9995
57	La	4.60250	−1.77070	3.49970	−2.64050	0.9995
58	Ce	4.69060	−1.81790	3.60280	−2.70670	0.9995
59	Pr	4.72150	−1.81390	3.56480	−2.65180	0.9995
60	Nd	4.75090	−1.80800	3.51970	−2.59010	1.0000
61	Pm	4.74070	−1.76600	3.37430	−2.44210	1.0000
62	Sm	4.71700	−1.71410	3.20800	−2.28170	1.0000
63	Eu	4.66940	−1.64140	2.98580	−2.07460	1.0000
64	Gd	4.61010	−1.55750	2.73190	−1.84040	0.9995
65	Tb	4.52550	−1.45520	2.43770	−1.57950	0.9995
66	Dy	4.45230	−1.36440	2.17540	−1.34550	0.9990
67	Ho	4.37660	−1.27460	1.92540	−1.13090	0.9990
68	Er	4.29460	−1.18170	1.67060	−0.91467	0.9990
69	Tm	4.21330	−1.09060	1.42390	−0.70804	0.9990
70	Yb	4.13430	−1.00310	1.18810	−0.51120	0.9990
71	Lu	4.04230	−0.90518	0.92889	−0.29820	0.9990
72	Hf	3.95160	−0.80978	0.68951	−0.09620	0.9990
73	Ta	3.85000	−0.70599	0.41103	0.11842	0.9990
74	W	3.76510	−0.61807	0.18568	0.29787	0.9990
75	Re	3.67600	−0.52688	−0.04706	0.48180	0.9995
76	Os	3.60530	−0.45420	−0.22529	0.61700	0.9995
77	Ir	3.53130	−0.37856	−0.41174	0.75967	0.9995
78	Pt	3.47070	−0.31534	−0.56487	0.87492	0.9995
79	Au	3.41630	−0.25987	−0.69030	0.96224	0.9995
80	Hg	3.37350	−0.21428	−0.79013	1.02850	1.0000
81	Tl	3.34590	−0.18322	−0.84911	1.05970	1.0000
82	Pb	3.32330	−0.15596	−0.89878	1.08380	1.0000
83	Bi	3.31880	−0.14554	−0.90198	1.06850	1.0000
84	Po	3.32030	−0.13999	−0.89333	1.04380	1.0000
85	At	3.34250	−0.15317	−0.83350	0.97641	1.0000
86	Rn	3.37780	−0.17800	−0.74320	0.88510	1.0000
87	Fr	3.41990	−0.20823	−0.64000	0.78354	0.9995
88	Ra	3.47530	−0.25005	−0.50660	0.65836	0.9995
89	Ac	3.49020	−0.25109	−0.49651	0.64340	0.9995
90	Th	3.61060	−0.35409	−0.18926	0.36849	0.9995
91	Pa	3.68630	−0.41329	−0.01192	0.20878	0.9995
92	U	3.76650	−0.47542	0.16850	0.05060	0.9990
93	Np	3.82870	−0.51955	0.29804	−0.06566	0.9990
94	Pu	3.88970	−0.56296	0.42597	−0.18080	0.9990
95	Am	3.95060	−0.60554	0.54967	−0.29112	0.9985
96	Cm	4.01470	−0.65062	0.67922	−0.40588	0.9985
97	Bk	4.07780	−0.69476	0.80547	−0.51729	0.9985
98	Cf	4.14210	−0.73977	0.93342	−0.62981	0.9980

References

Fox, A. G., O'Keefe, M. A. & Tabbernor, M. A. (1989). Relativistic Hartree–Fock X-ray and electron atomic scattering factors at high angles. Acta Cryst. A45, 786–793.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 6.1, p. 565