Application to centric and acentric distributions

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. 200-203 | 1 | 2 |

Section 2.1.7.3. Application to centric and acentric distributions

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

2.1.7.3. Application to centric and acentric distributions

| top | pdf |

We shall summarize here the non-ideal centric and acentric distributions of the magnitude of the normalized structure factor E (e.g. Shmueli & Wilson, 1981; Shmueli, 1982). We assume that (i) all the atoms are located in general positions and have rationally independent coordinates, (ii) all the scatterers are dispersionless, and (iii) there is no noncrystallographic symmetry. Arbitrary atomic composition and space-group symmetry are admitted. The appropriate weight functions and the corresonding orthogonal polynomials are $[\matrix {& & \hbox{Non-ideal} \cr p^{(0)}(|E|) & f_k(x) & \hbox{distribution} \cr\noalign{\hrule}\cr & & \cr (2/\pi)^{1/2}\exp(-|E|^2/2) & He_{2k}(|E|)/[(2k)!]^{1/2} & \hbox{Centric} \cr & & \cr 2|E|\exp(-|E|^2) & L_k(|E|^2) & \hbox{Acentric} \cr\noalign{\hrule} \cr} \eqno(2.1.7.4)]$ where $[He_{k}]$ and $[L_{k}]$ are Hermite and Laguerre polynomials, respectively, as defined, for example, by Abramowitz & Stegun (1972). Equations (2.1.7.2), (2.1.7.3) and (2.1.7.4) suffice for the general formulation of the above non-ideal p.d.f.'s of [|E|] . Their full derivation entails (i) the expression of a sufficient number of moments of [|E|] in terms of absolute moments of the trigonometric structure factor (e.g. Shmueli & Wilson, 1981; Shmueli, 1982) and (ii) calculation of the latter moments for the various symmetries (Wilson, 1978b; Shmueli & Kaldor, 1981, 1983). The notation below is similar to that employed by Shmueli (1982).

These non-ideal p.d.f.'s of [|E|] , for which the first five expansion terms are available, are given by $[p_{c}(|E|) = p_{c}^{(0)}(|E|)\left[1+\sum_{k = 2}^{\infty}{{A_{2k}}\over{(2k)!}} He_{2k}(|E|)\right] \eqno(2.1.7.5)]$ and $[p_{a}(|E|) = p_{a}^{(0)}(|E|)\left[1+\sum_{k = 2}^{\infty}{{(-1)^{k}B_{2k}}\over{k!}} L_{k}(|E|^{2})\right] \eqno(2.1.7.6)]$ for centrosymmetric and noncentrosymmetric space groups, respectively, where $[p_{c}^{(0)}(|E|)]$ and $[p_{a}^{(0)}(|E|)]$ are the ideal centric and acentric p.d.f.'s [see (2.1.7.4)] and the unified form of the coefficients $[A_{2k}]$ and $[B_{2k}]$ , for [k =] 2, 3, 4 and 5, is $[\eqalign{&A_{4}\cr &A_{6}\cr &A_{8}\cr &A_{10}\cr &\cr}\eqalign{&\hbox{ or } \cr &\hbox{ or } \cr &\hbox{ or } \cr &\hbox{ or }\cr &\cr}\eqalign{B_{4} & = a_{4}Q_{4} \cr B_{6} & = a_{6}Q_{6} \cr B_{8} & = a_{8}Q_{8} + U(a_{4}^{2}Q_{4}^{2}-\gamma_{4}^{2})\cr B_{10} & = a_{10}Q_{10} + V(a_{4}a_{6}Q_{4}Q_{6}-\gamma_{4}\gamma_{6}Q_{10})\cr &\quad{}+ W\gamma_{4}^{2}Q_{10} }\eqno(2.1.7.7)]$ (Shmueli, 1982), where U = 35 or 18, V = 210 or 100 and W = 3150 or 900 according as $[A_{2k}]$ or $[B_{2k}]$ is required, respectively, and the other quantities in equation (2.1.7.7) are given below. The composition-dependent terms in equations (2.1.7.7) are $[Q_{2k} = {{\sum_{j = 1}^{m}f_{j}^{2k}}\over{\left(\sum_{n = 1}^{m}f_{n}^{2}\right)^{k}}}, \eqno(2.1.7.8)]$ where m is the number of atoms in the asymmetric unit, $[f_{j},\; j = 1,\ldots,m]$ are their scattering factors, and the symmetry dependence is expressed by the coefficients $[a_{2k}]$ in equation (2.1.7.7), as follows: $[{a_{2k} = (-1)^{k-1}(k-1)!\alpha_{k0} +\textstyle\sum\limits_{p = 2}^{k}(-1)^{k-p}(k-p)!\alpha_{kp} \gamma_{2p},} \eqno(2.1.7.9)]$ where $[\alpha_{kp} = \left(\matrix{ k \cr p } \right) {{(2k-1)!!}\over{(2p-1)!!}} \quad{\rm or}\quad\left(\matrix{ k \cr p } \right) {{k!}\over{p!}} \eqno(2.1.7.10)]$ according as the space group is centrosymmetric or noncentrosymmetric, respectively, and $[\gamma_{2p}]$ in equation (2.1.7.9) is given by $[\gamma_{2p} = {{\langle |T|^{2p} \rangle}\over{\langle |T|^{2} \rangle^{p}}}, \eqno(2.1.7.11)]$ where $[\langle |T|^{k} \rangle]$ is the kth absolute moment of the trigonometric structure factor $[T({\bf h}) = \textstyle\sum\limits_{s = 1}^{g}\exp[2\pi i{\bf h}^{T}({\bf P}_{s}{\bf r}+{\bf t}_{s})] \equiv \xi({\bf h})+i\eta({\bf h}). \eqno(2.1.7.12)]$ In equation (2.1.7.12), g is the number of general equivalent positions listed in IT A (2005) for the space group in question, times the multiplicity of the Bravais lattice, $[({\bf P}_{s},{\bf t}_{s})]$ is the sth space-group operator and $[{\bf r}]$ is an atomic position vector.

The cumulative distribution functions, obtained by integrating equations (2.1.7.5) and (2.1.7.6), are given by $[\eqalignno{N_{c}(|E|)& = {\rm erf}\left({{|E|}\over{\sqrt{2}}}\right)-{{2}\over{\sqrt{\pi}}} \exp\left(-{{|E|^{2}}\over{2}}\right) \cr &\quad\times \left[\sum_{k = 2}^{\infty}{{A_{2k}}\over{(2k)!}} He_{2k-1}(|E|)\right] &(2.1.7.13)}]$ and $[\eqalignno{N_{a}(|E|) & = 1 - \exp(-|E|^{2}) + \exp(-|E|^{2}) \cr &\quad\times \left\{\sum_{k = 2}^{\infty}{{(-1)^{k}B_{2k}}\over{k!}}[L_{k-1}(|E|^{2}) - L_{k}(|E|^2)]\right\}\cr &&(2.1.7.14)}]$ for centrosymmetric and noncentrosymmetric space groups, respectively, where the coefficients are defined in equations (2.1.7.7)–(2.1.7.12) . Note that the first term on the right-hand side of equation (2.1.7.13) and the first two terms on the right-hand side of equation (2.1.7.14) are just the cumulative distributions derived from the ideal centric and acentric p.d.f.'s in Section 2.1.5.6.

The moments $[\langle |T|^{2k} \rangle]$ were compiled for all the space groups by Wilson (1978b) for [k =] 1 and 2, and by Shmueli & Kaldor (1981, 1983) for [k =] 1, 2, 3 and 4. These results are presented in Table 2.1.7.1. Closed expressions for the normalized moments $[\gamma_{2p}]$ were obtained by Shmueli (1982) for the triclinic, monoclinic and orthorhombic space groups except [Fdd2] and [Fddd] (see Table 2.1.7.2). The composition-dependent terms, $[Q_{2k}]$ , are most conveniently computed as weighted averages over the ranges of $[(\sin\theta)/\lambda]$ which were used in the construction of the Wilson plot for the computation of the [|E|] values.

Table 2.1.7.1| top | pdf |
Some even absolute moments of the trigonometric structure factor

The symbols p, q, r and s denote the second, fourth, sixth and eighth absolute moments of the trigonometric structure factor T [equation (2.1.7.12)], respectively, and the columns of the table contain (for some conciseness) [p, q, r/p] and $[s/p^{2}]$ . The numbers in parentheses, appearing beside some space-group entries, refer to hkl subsets which are defined in the note at the end of the table. These subset references are identical with those given by Shmueli & Kaldor (1981, 1983). The symbols q, r and s are also equivalent to $[\gamma_{4}P^{2}]$ , $[\gamma_{6}P^{3}]$ and $[\gamma_{8}P^{4}]$ , respectively, where $[\gamma_{2n}]$ are the normalized absolute moments given by equation (2.1.7.11).

Space groups(s)	p	q		$[s/p^{2}]$
Point group: 1
P1	1	1	1	1
Point group: $[\bar{\bf 1}]$
$[P\bar{1}]$	2	6	10	17½
Point groups: 2, m
All P	2	6	10	17½
All C	4	48	160	560
Point group: $[{\bf 2/}{\bi m}]$
All P	4	36	100	306¼
All C	8	288	1600	9800
Point group: 222
All P	4	28	64	169¾
All C and I	8	224	1024	5432
F222	16	1792	16384	173824
Point group: mm2
All P	4	36	100	306¼
All A, C and I	8	288	1600	9800
Fmm2	16	2304	25600	313600
Fdd2 (1)	16	2304	25600	313600
Fdd2 (2)	16	1280	7168	43264
Point group: *mmm*
All P	8	216	1000	5359 $[3\over8]$
All C and I	16	1728	16000	171500
Fmmm	32	13824	256000	5488000
Fddd (1)	32	13824	256000	5488000
Fddd (2)	32	7680	71680	757120
Point group: 4
$[P4, P4_{2}]$	4	36	100	306¼
$[P4_{1}]$ ^† (3)	4	36	100	306¼
$[P4_{1}]$ ^† (4)	4	20	28	42¼
	8	288	1600	9800
$[I4_{1}]$ (5)	8	288	1600	9800
$[I4_{1}]$ (6)	8	160	448	1352
Point group: $[\bar{\bf 4}]$
$[P\bar{4}]$	4	28	64	169¾
$[I\bar{4}]$	8	224	1024	5432
Point group: $[{\bf 4}/{\bi m}]$
All P	8	216	1000	5359 $[3\over8]$
	16	1728	16000	171500
$[I4_{1}/a]$ (7)	16	1728	16000	171500
$[I4_{1}/a]$ (8)	16	960	4480	23660
Point group: 422
, $[P42_{1}2]$ , $[P4_{2}22]$ , $[P4_{2}2_{1}2]$	8	136	424	1682 $[1\over8]$ ;
$[P4_{1}22]$ ,^† $[P4_{1}2_{1}2]$ ^† (3)	8	136	424	1682 $[1\over8]$
$[P4_{1}22]$ ,^† $[P4_{1}2_{1}2]$ ^† (4)	8	104	208	470 $[1\over8]$
I422	16	1088	6784	53828
$[I4_{1}22]$ (7)	16	1088	6784	53828
$[I4_{1}22]$ (8)	16	832	3328	15044
Point group: 4mm
All P	8	168	640	2970 $[5\over8]$
I4mm, I4cm	16	1344	10240	95060
$[I4_{1}md, I4_{1}cd]$ (7)	16	1344	10240	95060
$[I4_{1}md, I4_{1}cd]$ (8)	16	832	3328	15188
Point groups: $[\bar{\bf 4}{\bf 2}{\bi m},\bar{\bf 4}{\bi m}{\bf 2}]$
All P	8	136	424	1682 $[1\over8]$
$[I\bar{4}m2, I\bar{4}2m, I\bar{4}c2]$	16	1088	6784	53828
$[I\bar{4}2d]$ (5)	16	1088	6784	53828
$[I\bar{4}2d]$ (6)	16	832	3328	15044
*Point group: 4/mmm***
All P	16	1008	6400	51985 $[15\over16]$
,	32	8064	102400	1663550
$[I4_{1}/amd, I4_{1}/acd]$ (5)	32	8064	102400	1663550
$[I4_{1}/amd, I4_{1}/acd]$ (6)	32	4992	33280	265790
Point group: 3
All P and R	3	15	31	71
Point group: $[\bar{\bf 3}]$
All P and R	6	90	310	1242½
Point group: 32
All P and R	6	66	166	508½
Point group: 3m
P3m1, P31m, R3m	6	66	178	604½
P3c1, P31c, (3); R3c (1)	6	66	178	604½
P3c1, P31c, (4); R3c (2)	6	66	154	412½
Point group: $[\bar{\bf 3}{\bi m}]$
$[P\bar{3}1m, P\bar{3}m1, R\bar{3}m]$	12	396	1780	10578¾
$[P\bar{3}1c, P\bar{3}c1]$ (3);	12	396	1780	10578¾
$[R\bar{3}c]$ (1)
$[P\bar{3}1c, P\bar{3}c1]$ (4);	12	396	1540	7218¾
$[R\bar{3}c]$ (2)
Point group: 6
P6	6	90	340	1522½
$[P6_{1}]$ ^† (9)	6	90	340	1522½
$[P6_{1}]$ ^† (10)	6	54	91	161½
$[P6_{1}]$ ^† (11)	6	54	97	193½
$[P6_{1}]$ ^† (12)	6	90	280	962½
$[P6_{2}]$ ^† (13)	6	90	340	1522½
$[P6_{2}]$ ^† (14)	6	54	97	193½
$[P6_{3}]$ (3)	6	90	340	1522½
$[P6_{3}]$ (4)	6	90	280	962½
Point group: $[\bar{\bf 6}]$
$[P\bar{6}]$	6	90	310	1242½
Point group: $[{\bf 6/{\bi m}}]$
	12	540	3400	26643¾
$[P6_{3}/m]$ (3)	12	540	3400	26643¾
$[P6_{3}/m]$ (4)	12	540	2800	16843¾
Point group: 622
P622	12	324	1150	5506¼
$[P6_{1}22]$ ^† (9)	12	324	1150	5506¼
$[P6_{1}22]$ ^† (10)	12	252	577	1537¾
$[P6_{1}22]$ ^† (11)	12	252	583	1601¾
$[P6_{1}22]$ ^† (12)	12	324	1090	4746¼
$[P6_{2}22]$ ^† (13)	12	324	1150	5506¼
$[P6_{2}22]$ ^† (14)	12	252	583	1601¾
$[P6_{3}22]$ (3)	12	324	1150	5506¼
$[P6_{3}22]$ (4)	12	324	1090	4746¼
Point group: 6mm
P6mm	12	396	1930	12818¾
P6cc (3)	12	396	1930	12818¾
P6cc (4)	12	396	1450	6098¾
$[P6_{3}cm, P6_{3}mc]$ (3)	12	396	1930	12818¾
$[P6_{3}cm, P6_{3}mc]$ (4)	12	396	1630	8338¾
Point groups: $[\bar{\bf 6}{\bi m}{\bf 2}, \bar{\bf 6}{\bf 2}{\bi m}]$
$[P\bar{6}m2, P\bar{6}2m]$	12	396	1780	10578¾
$[P\bar{6}c2, P\bar{6}2c]$ (3)	12	396	1780	10578¾
$[P\bar{6}c2, P\bar{6}2c]$ (4)	12	396	1540	7218¾
Point group: 6/*mmm*
P6/mmm	24	2376	19300	224328 $[1\over8]$
P6/mcc (3)	24	2376	19300	224328 $[1\over8]$
P6/mcc (4)	24	2376	14500	106728 $[1\over8]$
P6/mcm, P6/mmc (3)	24	2376	19300	224328 $[1\over8]$
, (4)	24	2376	16300	145928 $[1\over8]$
Point group: 23
P23, $[P2_{1}3]$	12	276	760	2695¼
I23, $[I2_{1}3]$	24	2208	12160	86248
F23	48	17664	194560	2759936
Point group: $[{\bi m}\bar{\bf 3}]$
$[Pm\bar{3}, Pn\bar{3}, Pa3]$	24	1800	9400	67703 $[1\over8]$
$[Im\bar{3}, Ia\bar{3}]$	48	14400	150400	2166500
$[Fm\bar{3}]$	96	115200	2406400	69328000
$[Fd\bar{3}]$ (1)	96	115200	2406400	69328000
$[Fd\bar{3}]$ (2)	96	96768	1484800	28183680
Point group: 432
$[P432, P4_{2}32]$	24	1272	4648	25216 $[7\over8]$
$[P4_{1}32]$ ^† (15)	24	1272	4648	25216 $[7\over8]$
$[P4_{1}32]$ ^† (16)	24	1176	3568	13916 $[7\over8]$
$[P4_{1}32]$ ^† (17)	24	1080	2776	8664 $[7\over8]$
$[P4_{1}32]$ ^† (18)	24	984	2272	6580 $[7\over8]$
I432	48	10176	74368	806940
$[I4_{1}32]$ (15)	48	10176	74368	806940
$[I4_{1}32]$ (17)	48	8640	44416	277276
F432	96	81408	1189888	25822080
$[F4_{1}32]$ (15)	96	81408	1189888	25822080
$[F4_{1}32]$ (18)	96	62976	581632	6738816
Point group: $[\bar{\bf 4}{\bf 3}{\bi m}]$
$[P\bar{4}3m]$	24	1272	5128	32896 $[7\over8]$
$[P\bar{4}3n]$ (1)	24	1272	5128	32896 $[7\over8]$
$[P\bar{4}3n]$ (2)	24	1272	4168	17536 $[7\over8]$
$[I\bar{4}3m]$	48	10176	82048	1052700
$[I\bar{4}3d]$ (15); (20)	48	10176	82048	1052700
$[I\bar{4}3d]$ (15); (21)	48	10176	66688	561180
$[I\bar{4}3d]$ (17)	48	8640	44416	277276
$[F\bar{4}3m]$	96	81408	1312768	33686400
$[F\bar{4}3c]$ (15)	96	81408	1312768	33686400
$[F\bar{4}3c]$ (18)	96	81408	1067008	17957760
Point group: $[{\bi m}\bar{\bf 3}{\bi m}]$
$[Pm\bar{3}m, Pn\bar{3}m]$	48	8784	72160	972717 $[13\over16]$
$[Pn\bar{3}n, Pm\bar{3}n]$ (1)	48	8784	72160	972717 $[13\over16]$
$[Pn\bar{3}n, Pm\bar{3}n]$ (2)	48	8784	56800	488877 $[13\over16]$
$[Im\bar{3}m]$	96	70272	1154560	31126970
$[Ia\bar{3}d]$ (15); (20)	96	70272	1154560	31126970
$[Ia\bar{3}d]$ (15); (21)	96	51840	432640	4497850
$[Ia\bar{3}d]$ (17)	96	70272	908800	15644090
$[Fm\bar{3}m]$	192	562176	18472960	996063040
$[Fm\bar{3}c]$ (1)	192	562176	18472960	996063040
$[Fm\bar{3}c]$ (2)	192	562176	14540800	500610880
$[Fd\bar{3}m]$ (1)	192	562176	18472960	996063040
$[Fd\bar{3}m]$ (2)	192	414720	7782400	205432640
$[Fd\bar{3}c]$ (1)	192	562176	18472960	996063040
$[Fd\bar{3}c]$ (2)	192	414720	6799360	136619840

Note. hkl subsets: (1) [h + k + l = 2n]

; (2)

; (3)

; (4)

; (5)

; (6)

; (7)

; (8)

; (9)

; (10)

; (11)

; (12)

; (13)

; (14)

; (15) hkl all even; (16) only one index odd; (17) only one index even; (18) hkl all odd; (19) two indices odd; (20) [h + k + l = 4n]

; (21)

^†And the enantiomorphous space group.

Table 2.1.7.2| top | pdf |
Closed expressions for $[\gamma_{2k}]$ [equation (2.1.7.11)] for space groups of low symmetry

The normalized moments $[\gamma_{2k}]$ are expressed in terms of $[M_{k}]$ , where $[M_{k} = {(2k)! \over 2^{k}(k!)^{2}} = {(2k - 1)!! \over k!},]$ and [l'] , which takes on the values 1, 2 or 4 according as the Bravais lattice is of type P, one of the types A, B, C or I, or type F, respectively. The expressions for $[\gamma_{2k}]$ are identical for all the space groups based on a given point group, except Fdd2 and Fddd. The expressions are valid for general reflections and under the restrictions given in the text.

Point group(s)	Expression for $[\gamma_{2k}]$
1	1
$[\bar{1}, 2, m]$	$[l'^{k - 1} M_{k}]$
	$[l'^{k - 1} M_{k}^{2}]$
mmm	$[l'^{k - 1} M_{k}^{3}]$
222	$[{l'^{k - 1} \over 2^{k} (k!)^{2}} \sum\limits_{p=0}^{k} (M_{p}M_{k - p})^{3}[p! (k - p)!]^{2}]$

References

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