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

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

Section 2.1.8.5. Atomic characteristic functions

U. Shmuelia* and A. J. C. Wilsonb

aSchool 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[link]) for a wide range of space groups, by methods similar to those described above. These expressions are collected in Table 2.1.8.1[link] 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[link] 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[link] and 2.1.8.2[link]. The symbolic expressions are defined in Section 2.1.8.5[link]. 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[link], 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)gAtomic c.f.Remarks
Point group: 1      
P1 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)]  
F222 16 [ L_{j}(4t,\Delta)]  
Point group: mm2      
All P 4 [ L_{j}(t,0)]  
All C and I 8 [ L_{j}(2t,0)]  
Fmm2 16 [ L_{j}(4t,0)]  
Fdd2 16 [L_{j}(4t,0)] [ h+k+l=2n]
  16 [ L_{j}(4t,\pi/4)] [ h+k+l=2n+1]
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)] [ h+k+l=2n]
  32 [ L_{j}(8t_{1},\pi/4)] [ h+k+l=2n+1]
Point group: 4      
P4, P42 4 [ L_{j}(t,0)]  
[P4_1] 4 [ L_{j}(t,0)] [ l=2n]
  4 [ L_{j}(t,\pi/4)] [ l=2n+1]
I4 8 [ L_{j}(2t,0)]  
[ I4_{1}] 8 [ L_{j}(2t,0)] [2h+l=2n]
  8 [ L_{j}(2t,\pi/4)] [ 2h+l=2n+1]
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)]  
[ I4/m] 16 [ L_{j}(4t_{1},0)]  
[ I4_1/a] 16 [ L_{j}(4t_{1},0)] [ l=2n]
  16 [L_{j}(4t_{1},\pi/4)] [ l=2n+1]
Point group: 422      
P422, P4212, P4222, P42212 8 [ Q_{j}^{(1)}(t,\Delta)^{(b)}]  
P4122, P41212 8 [ Q_{j}^{(1)}(t,\Delta)] [ l=2n]
  8 [ Q_{j}^{(2)}(t,\Delta)^{(c)}] [ l=2n+1]
I422 16 [ Q_{j}^{(1)}(2t,\Delta)]  
I4122 16 [ Q_{j}^{(1)}(2t,\Delta)] [ 2k+l=2n]
  16 [ Q_{j}^{(2)}(2t,\Delta)] [ 2k+l=2n+1]
Point group: 4mm      
All P 8 [ Q_{j}^{(1)}(t,0)]  
I4mm, I4cm 16 [ Q_{j}^{(1)}(2t,0)]  
I41 md, I41 cd 16 [Q_{j}^{(1)}(2t,0)] [2k+l=2n]
  16 [ Q_{j}^{(1)}(2t,\pi/4)] [2k+l=2n+1]
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)] [2h+l=2n]
  16 [Q_{j}^{(2)}(2t,\Delta)] [2h+l=2n+1]
Point group: [{\bf 4/}{\bi mmm}]      
All P 16 [ Q_{j}^{(1)}(2t_{1},0)]  
[I4/mmm], [I4/mcm] 32 [Q_{j}^{(1)}(4t_{1},0)]  
[ I4_{1}/amd], [I4_{1}/acd] 32 [Q_{j}^{(1)}(4t_{1},0)] [ l=2n]
  32 [ Q_{j}^{(1)}(4t_{1},\pi/4)] [ l=2n+1]
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      
P3m1, P31m, R3m 6 [ T_{j}(t,\pi/2)]  
P3c1, 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      
P6 6 [H_{j}^{(1)}(t, \pi/2)^{(e)}]  
[ P6_{1}] 6 [H_{j}^{(1)}(t, \pi/2)] [ l=6n]
  6 [ H_{j}^{(2)}(t, 0)^{(f)}] [ l=6n+1], [6n+5]
  6 [H_{j}^{(2)}(t, \pi/2)] [l=6n+2], [6n+4]
  6 [ H_{j}^{(1)}(t, 0)] [ l=6n+3]
[ P6_{2}] 6 [H_{j}^{(1)}(t, \pi/2)] [ l=3n]
  6 [H_{j}^{(2)}(t, \pi/2)] [l=3n \pm 1]
[ P6_{3}] 6 [ H_{j}^{(1)}(t, \pi/2)] [ l=2n]
  6 [ H_{j}^{(1)}(t, 0)] [ l=2n+1]
Point group: [\bar{\bf 6}]      
[P\bar{6}] 6 [H_{j}^{(1)}(t, \Delta)]  
Point group: [{\bf 6/}{\bi m}]      
[ P6/m] 12 [H_{j}^{(1)}(2t_{1}, \pi/2)]  
[ P6_{3}/m] 12 [ H_{j}^{(1)}(2t_{1}, \pi/2)] [ l=2n]
  12 [ H_{j}^{(1)}(2t_{1}, 0)] [l=2n+1]
Point group: 622      
P622 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)] [l=6n]
  12 [ \tilde{H}_{j}^{(2)}(t,0,0,\Delta)^{(h)}] [ l=6n+1], [6n+5]
  12 [ \tilde{H}_{j}^{(2)}(t,\pi/2,\pi/2,\Delta)] [ l=6n+2], [6n+4]
  12 [ \tilde{H}_{j}^{(1)}(t,0,0,\Delta)] [ l=6n+3]
[ P6_{2}22] 12 [ \tilde{H}_{j}^{(1)}(t,\pi/2], [-\pi/2,\Delta)] [ l=3n]
  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)] [ l=2n]
  12 [ \tilde{H}_{j}^{(1)}(t,0,0,\Delta)] [ l=2n+1]
Point group: 6mm      
P6mm 12 [ \tilde{H}_{j}^{(1)}(t,\pi/2,\pi/2,0)]  
P6cc 12 [ \tilde{H}_{j}^{(1)}(t,\pi/2,\pi/2,0)] [ l=2n]
  12 [ \tilde{H}_{j}^{(1)}(t,\pi/2], [-\pi/2,0)] [ l=2n+1]
[P6_{3}cm], [P6_{3}mc] 12 [ \tilde{H}_{j}^{(1)}(t,\pi/2,\pi/2,0)] [l=2n]
  12 [ \tilde{H}_{j}^{(1)}(t,0,0,0)] [l=2n+1]
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)] [ l=2n]
  12 [ \tilde{H}_{j}^{(1)}(t,\Delta+\pi/2], [-\Delta-\pi/2,0)] [ l=2n+1]
Point group: [{\bf 6/}{\bi mmm}]      
[P6/mmm] 24 [\tilde{H}_{j}^{(1)}(2t_{1},\pi/2,\pi/2,0)]  
[ P6/mcc] 24 [\tilde{H}_{j}^{(1)}(2t_{1},\pi/2,\pi/2,0)] [ l=2n]
  24 [ \tilde{H}_{j}^{(1)}(2t_{1}], [\pi/2], [-\pi/2,0)] [ l=2n+1]
[P6_{3}/mcm], [P6_{3}/mmc] 24 [ \tilde{H}_{j}^{(1)}(2t_{1},\pi/2,\pi/2,0)] [l=2n]
  24 [ \tilde{H}_{j}^{(1)}(2t_{1},0,0,0)] [ l=2n+1]
Point group: 23      
P23, P213 12 [ L_{j}^{3}(t,\Delta)]  
I23, [I2_{1}3] 24 [ L_{j}^{3}(2t,\Delta)]  
F23 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)] [h+k+l=2n]
  96 [ L_{j}^{3}(8t_{1},\pi/4)] [h+k+l=2n+1]
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[link] on their first occurrence.

References

First citation 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 [|E|] for all plane groups and most space groups. Acta Cryst. A47, 328–335.Google Scholar








































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