Atomic characteristic functions

Shmueli, U.; Wilson, A. J. C.

doi:10.1107/97809553602060000554

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 2.1, pp. 206-208 | 1 | 2 |

Section 2.1.8.5. Atomic characteristic functions

U. Shmueli^a ^* and A. J. C. Wilson^b ^‡

^a School of Chemistry, Tel Aviv University, Tel Aviv 69 978, Israel, and ^bSt John's College, Cambridge, England
Correspondence e-mail: ushmueli@post.tau.ac.il

2.1.8.5. Atomic characteristic functions

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Expressions for the atomic contributions to the characteristic functions were obtained by Rabinovich et al. (1991a) for a wide range of space groups, by methods similar to those described above. These expressions are collected in Table 2.1.8.1 in terms of symbols which are defined below. The following abbreviations are used in the subsequent definitions of the symbols: $[\eqalign{s_{\pm} &= 2an_{j}\sin(\tau\pm\rho), \cr c_{\pm} &= 2an_{j}\cos(\tau\pm \rho)\quad {\rm and}\cr \sigma_{\pm} &= 2an_{j}\sin(\tau\pm 2\pi/3 + \rho),} ]$ and the symbols appearing in Table 2.1.8.1 are given below: $[\eqalign{ {}^{(a)}L_j(a,\rho) & = \langle J_0(s_+)J_0(s_-) \rangle_\tau \cr & = \textstyle\sum\limits_{k = -\infty}^\infty\cos(4k\rho)J_k^4 \cr & = J_0^4(an_j) + 2\textstyle\sum\limits_{k = 1}^\infty\cos(4k\rho)J_k^4(an_j),\cr {}^{(b)}Q_j^{(1)} (a,\rho) & = \langle J_0^2(s_+)J_0^2(s_-) \rangle_\tau,\cr {}^{(c)}Q_j^{(2)} (a,\rho) & = \langle J_0(s_+)J_0(s_-)J_0(c_+)J_0 (c_-) \rangle_\tau,\cr {}^{(d)}T_j(a,\rho) & = \textstyle\sum\limits_{k = -\infty}^\infty\exp(6ik\rho)J_k^6(an_j) \cr & = J_0^6(an_j)+2\textstyle\sum\limits_{k = 1}^\infty\cos(6k\rho)J_k^6(an_j),\cr {}^{(e)}H_j^{(1)} (a,\mu) & = \left\langle {\cal R} \left [S_j^{(1)} (\tau\hbox{; } a,\mu,0) \right] \right\rangle _\tau, \cr {}^{(f)}H_j^{(2)} (a,\mu) & = \left\langle {\cal R} \left [S_j^{(2)} (\tau\hbox{; } a,\mu,0) \right] \right\rangle _\tau,\cr {}^{(g)}\tilde{H}_j^{(1)} (a,\mu_1,\mu_2,\rho) & = \biggl\langle {{\cal R}} \biggl[S_j^{(1)} (\tau\hbox{; }a,\mu_1,\rho) \cr &\quad\times{}S_j^{(1)} (\tau\hbox{; }a,\mu_2,-\rho) \biggr] \biggr\rangle _\tau, \cr {}^{(h)}\tilde{H}_j^{(2)} (a,\mu_1,\mu_2,\rho) & = \biggl\langle {\cal R} \biggl[S_j^{(2)} (\tau\hbox{; }a,\mu_1,\rho) \cr &\quad\times{}S_j^{(2)} (\tau\hbox{; }a,\mu_2,-\rho) \biggr] \biggr\rangle _\tau, } ]$ where $[S_{j}^{(1)}(\tau\hbox{; }a,\mu,\rho) = \textstyle\sum\limits_{k = -\infty}^{\infty}e^{3ik\mu}J_{k}^{3} (s_{+})]$ and $[S_{j}^{(2)}(\tau\hbox{; }a,\mu,\rho) = \textstyle\sum\limits_{k = -\infty}^{\infty}e^{3ik\mu}J_{k} (s_{+})J_{k}(\sigma_{+})J_{k}(\sigma_{-}).]$

Table 2.1.8.1 | top | pdf |
Atomic contributions to characteristic functions for [p(|E|)]

The table lists symbolic expressions for the atomic contributions to exact characteristic functions (abbreviated as c.f.) for [p(|E|)] , to be computed as single Fourier series (centric), double Fourier series (acentric) and single Fourier–Bessel series (acentric), as defined in Sections 2.1.8.1 and 2.1.8.2. The symbolic expressions are defined in Section 2.1.8.5. The table is arranged by point groups, space groups and parities of the reflection indices analogously to the table of moments, Table 2.1.7.1, and covers all the space groups and statistically different parities of hkl up to and including space group Fd $[\bar{3}]$ . The expressions are valid for atoms in general positions, for general reflections and presume the absence of noncrystallographic symmetry and of dispersive scatterers.

Space group(s)	g	Atomic c.f.	Remarks
Point group: 1
P 1	1	$[ J_{0}(t n_{j})]$
Point group: $[\bar{\bf 1}]$
P $[\bar{1}]$	2	$[J_{0}(2t_{1}n_{j})]$
Point groups: 2, m
All P	2	$[J_{0}^{2}(t n_{j})]$
All C	4	$[J_{0}^{2}(2t n_{j})]$
Point group: $[{\bf 2/}{\bi m}]$
All P	4	$[J_{0}^{2}(2t_{1} n_{j})]$
All C	8	$[J_{0}^{2}(4t_{1} n_{j})]$
Point group: 222
All P	4	$[ L_{j}(t,\Delta)^{(a)}]$
All C and I	8	$[ L_{j}(2t,\Delta)]$
F 222	16	$[ L_{j}(4t,\Delta)]$
Point group: mm2
All P	4	$[ L_{j}(t,0)]$
All C and I	8	$[ L_{j}(2t,0)]$
Fmm 2	16	$[ L_{j}(4t,0)]$
Fdd 2	16	$[L_{j}(4t,0)]$
	16	$[ L_{j}(4t,\pi/4)]$
Point group: mmm
All P	8	$[ L_{j}(2t_{1},0)]$
All C and I	16	$[ L_{j}(4t_{1},0)]$
Fmmm	32	$[ L_{j}(8t_{1},0)]$
Fddd	32	$[ L_{j}(8t_{1},0)]$
	32	$[ L_{j}(8t_{1},\pi/4)]$
Point group: 4
P 4, P4₂	4	$[ L_{j}(t,0)]$
^†	4	$[ L_{j}(t,0)]$
	4	$[ L_{j}(t,\pi/4)]$
I 4	8	$[ L_{j}(2t,0)]$
$[ I4_{1}]$	8	$[ L_{j}(2t,0)]$
	8	$[ L_{j}(2t,\pi/4)]$
Point group: $[\bar{\bf 4}]$
P $[\bar{4}]$	4	$[L_{j}(t,\Delta)]$
I $[\bar{4}]$	8	$[ L_{j}(2t,\Delta)]$
Point group: $[{\bf 4/}{\bi m}]$
All P	8	$[ L_{j}(2t_{1},0)]$
	16	$[ L_{j}(4t_{1},0)]$
	16	$[ L_{j}(4t_{1},0)]$
	16	$[L_{j}(4t_{1},\pi/4)]$
Point group: 422
P 422, P42₁2, P4₂22, P4₂2₁2	8	$[ Q_{j}^{(1)}(t,\Delta)^{(b)}]$
P 4₁22,^† P4₁2₁2^†	8	$[ Q_{j}^{(1)}(t,\Delta)]$
	8	$[ Q_{j}^{(2)}(t,\Delta)^{(c)}]$
I 422	16	$[ Q_{j}^{(1)}(2t,\Delta)]$
I 4₁22	16	$[ Q_{j}^{(1)}(2t,\Delta)]$
	16	$[ Q_{j}^{(2)}(2t,\Delta)]$
Point group: 4mm
All P	8	$[ Q_{j}^{(1)}(t,0)]$
I 4mm, I4cm	16	$[ Q_{j}^{(1)}(2t,0)]$
I 4₁ md, I4₁ cd	16	$[Q_{j}^{(1)}(2t,0)]$
	16	$[ Q_{j}^{(1)}(2t,\pi/4)]$
Point groups: $[\bar{\bf 4}]$ 2m, $[\bar{\bf 4}]$ m2
All P	8	$[ Q_{j}^{(1)}(t,\Delta)]$
I $[\bar{4}]$ 2m, I $[\bar{4}]$ m2, I $[\bar{4}]$ c2	16	$[Q_{j}^{(1)}(2t,\Delta)]$
I $[\bar{4}]$ 2d	16	$[Q_{j}^{(1)}(2t,\Delta)]$
	16	$[Q_{j}^{(2)}(2t,\Delta)]$
Point group: $[{\bf 4/}{\bi mmm}]$
All P	16	$[ Q_{j}^{(1)}(2t_{1},0)]$
,	32	$[Q_{j}^{(1)}(4t_{1},0)]$
$[ I4_{1}/amd]$ , $[I4_{1}/acd]$	32	$[Q_{j}^{(1)}(4t_{1},0)]$
	32	$[ Q_{j}^{(1)}(4t_{1},\pi/4)]$
Point group: 3
All P and R	3	$[ J_{0}^{3}(t n_{j})]$
Point group: $[\bar{\bf 3}]$
All P and R	6	$[ J_{0}^{3}(2t_{1} n_{j})]$
Point group: 32
All P and R	6	$[T_{j}(t,\Delta)^{(d)}]$
Point group: 3m
P 3m1, P31m, R3m	6	$[ T_{j}(t,\pi/2)]$
P 3c1, P31c, R3c	6	$[ T_{j}(t,\pi/2)]$	$[ l=2n\;(P)]$ , $[h+k+l=2n\;(R)]$
	6	$[ T_{j}(t,0)]$	$[ l=2n+1\;(P)]$ , $[h+k+l =2n+1\;(R)]$
Point group: $[\bar{\bf 3}]$ m
P $[\bar{3}]$ m1, P $[\bar{3}]$ 1m, R $[\bar{3}]$ m	12	$[ T_{j}(2t_{1},\pi/2)]$
P $[\bar{3}]$ c1, P $[\bar{3}]$ 1c, R $[\bar{3}]$ c	12	$[ T_{j}(2t_{1},\pi/2)]$	$[ l=2n\;(P)]$ , $[h+k+l=2n\;(R)]$
	12	$[ T_{j}(2t_{1},0)]$	$[ l=2n+1\;(P)]$ , $[h+k+l=2n+1\;(R)]$
Point group: 6
P 6	6	$[H_{j}^{(1)}(t, \pi/2)^{(e)}]$
$[ P6_{1}]$ ^†	6	$[H_{j}^{(1)}(t, \pi/2)]$
	6	$[ H_{j}^{(2)}(t, 0)^{(f)}]$	,
	6	$[H_{j}^{(2)}(t, \pi/2)]$	,
	6	$[ H_{j}^{(1)}(t, 0)]$
$[ P6_{2}]$ ^†	6	$[H_{j}^{(1)}(t, \pi/2)]$
	6	$[H_{j}^{(2)}(t, \pi/2)]$	$[l=3n \pm 1]$
$[ P6_{3}]$	6	$[ H_{j}^{(1)}(t, \pi/2)]$
	6	$[ H_{j}^{(1)}(t, 0)]$
Point group: $[\bar{\bf 6}]$
$[P\bar{6}]$	6	$[H_{j}^{(1)}(t, \Delta)]$
Point group: $[{\bf 6/}{\bi m}]$
	12	$[H_{j}^{(1)}(2t_{1}, \pi/2)]$
$[ P6_{3}/m]$	12	$[ H_{j}^{(1)}(2t_{1}, \pi/2)]$
	12	$[ H_{j}^{(1)}(2t_{1}, 0)]$
Point group: 622
P 622	12	$[\tilde{H}_{j}^{(1)} (t,\pi/2]$ , $[-\pi/2,\Delta)^{(g)}]$
$[ P6_{1}22]$ ^†	12	$[ \tilde{H}_{j}^{(1)}(t,\pi/2]$ , $[-\pi/2,\Delta)]$
	12	$[ \tilde{H}_{j}^{(2)}(t,0,0,\Delta)^{(h)}]$	,
	12	$[ \tilde{H}_{j}^{(2)}(t,\pi/2,\pi/2,\Delta)]$	,
	12	$[ \tilde{H}_{j}^{(1)}(t,0,0,\Delta)]$
$[ P6_{2}22]$ ^†	12	$[ \tilde{H}_{j}^{(1)}(t,\pi/2]$ , $[-\pi/2,\Delta)]$
	12	$[ \tilde{H}_{j}^{(2)}(t,\pi/2,\pi/2,\Delta)]$	$[ l=3n \pm 1]$
$[ P6_{3}22]$	12	$[ \tilde{H}_{j}^{(1)}(t,\pi/2]$ , $[-\pi/2,\Delta)]$
	12	$[ \tilde{H}_{j}^{(1)}(t,0,0,\Delta)]$
Point group: 6mm
P 6mm	12	$[ \tilde{H}_{j}^{(1)}(t,\pi/2,\pi/2,0)]$
P 6cc	12	$[ \tilde{H}_{j}^{(1)}(t,\pi/2,\pi/2,0)]$
	12	$[ \tilde{H}_{j}^{(1)}(t,\pi/2]$ , $[-\pi/2,0)]$
$[P6_{3}cm]$ , $[P6_{3}mc]$	12	$[ \tilde{H}_{j}^{(1)}(t,\pi/2,\pi/2,0)]$
	12	$[ \tilde{H}_{j}^{(1)}(t,0,0,0)]$
Point groups: $[\bar{\bf 6}]$ 2m, $[\bar{\bf 6}]$ m2
P $[\bar{6}]$ 2m, P $[\bar{6}]$ m2	12	$[ \tilde{H}_{j}^{(1)}(t,\Delta,\Delta,0)]$
P $[\bar{6}]$ 2c, P $[\bar{6}]$ c2	12	$[ \tilde{H}_{j}^{(1)}(t,\Delta,\Delta,0)]$
	12	$[ \tilde{H}_{j}^{(1)}(t,\Delta+\pi/2]$ , $[-\Delta-\pi/2,0)]$
Point group: $[{\bf 6/}{\bi mmm}]$
	24	$[\tilde{H}_{j}^{(1)}(2t_{1},\pi/2,\pi/2,0)]$
	24	$[\tilde{H}_{j}^{(1)}(2t_{1},\pi/2,\pi/2,0)]$
	24	$[ \tilde{H}_{j}^{(1)}(2t_{1}]$ , $[\pi/2]$ , $[-\pi/2,0)]$
$[P6_{3}/mcm]$ , $[P6_{3}/mmc]$	24	$[ \tilde{H}_{j}^{(1)}(2t_{1},\pi/2,\pi/2,0)]$
	24	$[ \tilde{H}_{j}^{(1)}(2t_{1},0,0,0)]$
Point group: 23
P 23, P2₁3	12	$[ L_{j}^{3}(t,\Delta)]$
I 23, $[I2_{1}3]$	24	$[ L_{j}^{3}(2t,\Delta)]$
F 23	48	$[ L_{j}^{3}(4t,\Delta)]$
Point group: m $[\bar{\bf 3}]$
Pm $[\bar{3}]$ , Pn $[\bar{3}]$ , Pa $[\bar{3}]$	24	$[ L_{j}^{3}(2t_{1},0)]$
Im $[\bar{3}]$ , Ia $[\bar{3}]$	48	$[ L_{j}^{3}(4t_{1},0)]$
Fm $[\bar{3}]$	96	$[ L_{j}^{3}(8t_{1},0)]$
Fd $[\bar{3}]$	96	$[ L_{j}^{3}(8t_{1},0)]$
	96	$[ L_{j}^{3}(8t_{1},\pi/4)]$

^† And the enantiomorphous space group.

The averages appearing in the above summary are, in general, computed as $[\left\langle f(\tau)\right\rangle = ({{2}/{\pi}})\textstyle\int\limits_{0}^{\pi/2}f(\tau)\;{\rm d}\tau, \eqno(2.1.8.39)]$ except $[H_{j}^{(2)}]$ and $[\tilde{H}_{j}^{(2)}]$ which are computed as $[\left\langle f(\tau)\right\rangle = ({{3}/{\pi}})\textstyle\int\limits_{0}^{\pi/3}f(\tau)\;{\rm d}\tau, \eqno(2.1.8.40)]$ where $[f(\tau)]$ is any of the atomic characteristic functions indicated above. The superscripts preceding the symbols in the above summary are appended to the corresponding symbols in Table 2.1.8.1 on their first occurrence.

References

Rabinovich, S., Shmueli, U., Stein, Z., Shashua, R. & Weiss, G. H. (1991a). Exact random-walk models in crystallographic statistics. VI. P.d.f.'s of for all plane groups and most space groups. Acta Cryst. A47, 328–335.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 2.1, pp. 206-208