Non-ideal distributions: the correction-factor approach

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. 199-203 | 1 | 2 |

Section 2.1.7. Non-ideal distributions: the correction-factor approach

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. Non-ideal distributions: the correction-factor approach

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2.1.7.1. Introduction

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The probability density functions (p.d.f.'s) of the magnitude of the structure factor, presented in Section 2.1.5, are based on the central-limit theorem discussed above. In particular, the centric and acentric p.d.f.'s given by equations (2.1.5.11) and (2.1.5.8), respectively, are expected to account for the statistical properties of diffraction patterns obtained from crystals consisting of nearly equal atoms, which obey the fundamental assumptions of uniformity and independence of the atomic contributions and are not affected by noncrystallographic symmetry and dispersion. It is also assumed there that the number of atoms in the asymmetric unit is large. Distributions of structure-factor magnitudes which are based on the central-limit theorem, and thus obey the above assumptions, have been termed `ideal', and the subjects of the following sections are those distributions for which some of the above assumptions/restrictions are not fulfilled; the latter distributions will be called `non-ideal'.

We recall that the assumption of uniformity consists of the requirement that the fractional part of the scalar product [hx+ky+lz] be uniformly distributed over the [0, 1] interval, which holds well if [x, y, z] are rationally independent (Hauptman & Karle, 1953), and permits one to regard the atomic contribution to the structure factor as a random variable. This is of course a necessary requirement for any statistical treatment. If, however, the atomic composition of the asymmetric unit is widely heterogeneous, the structure factor is then a sum of unequally distributed random variables and the Lindeberg–Lévy version of the central-limit theorem (cf. Section 2.1.4.4) cannot be expected to apply. Other versions of this theorem might still predict a normal p.d.f. of the sum, but at the expense of a correspondingly large number of terms/atoms. It is well known that atomic heterogeneity gives rise to severe deviations from ideal behaviour (e.g. Howells et al., 1950) and one of the aims of crystallographic statistics has been the introduction of a correct dependence on the atomic composition into the non-ideal p.d.f.'s [for a review of the early work on non-ideal distributions see Srinivasan & Parthasarathy (1976)]. A somewhat less well known fact is that the dependence of the p.d.f.'s of [|E|] on space-group symmetry becomes more conspicuous as the composition becomes more heterogeneous (e.g. Shmueli, 1979; Shmueli & Wilson, 1981). Hence both the composition and the symmetry dependence of the intensity statistics are of interest. Other problems, which likewise give rise to non-ideal p.d.f.'s, are the presence of heavy atoms in (variable) special positions, heterogeneous structures with complete or partial noncrystallographic symmetry, and the presence of outstandingly heavy dispersive scatterers.

The need for theoretical representations of non-ideal p.d.f.'s is exemplified in Fig. 2.1.7.1(a), which shows the ideal centric and acentric p.d.f.'s together with a frequency histogram of [|E|] values, recalculated for a centrosymmetric structure containing a platinum atom in the asymmetric unit of $[P\bar{1}]$ (Faggiani et al., 1980). Clearly, the deviation from the Gaussian p.d.f., predicted by the central-limit theorem, is here very large and a comparison with the possible ideal distributions can (in this case) lead to wrong conclusions.

Figure 2.1.7.1 | top | pdf |

Atomic heterogeneity and intensity statistics. The histogram appearing in (a) and (b) was constructed from [|E|] values which were recalculated from atomic parameters published for the centrosymmetric structure of C₆H₁₈Cl₂N₄O₄Pt (Faggiani et al., 1980). The space group of the crystal is $[P\bar{1}]$ , [Z=2] , i.e. all the atoms are located in general positions. (a) A comparison of the recalculated distribution of [|E|] with the ideal centric [equation (2.1.5.11)] and acentric [equation (2.1.5.8)] p.d.f.'s, denoted by $[\bar{1}]$ and 1, respectively. (b) The same recalculated histogram along with the centric correction-factor p.d.f. [equation (2.1.7.5)], truncated after two, three, four and five terms (dashed lines), and with that accurately computed for the correct space-group Fourier p.d.f. [equations (2.1.8.5) and (2.1.8.22)] (solid line).

Two general approaches have so far been employed in derivations of non-ideal p.d.f.'s which account for the above-mentioned problems: the correction-factor approach, to be dealt with in the following sections, and the more recently introduced Fourier method, to which Section 2.1.8 is dedicated. In what follows, we introduce briefly the mathematical background of the correction-factor approach, apply this formalism to centric and acentric non-ideal p.d.f.'s, and present the numerical values of the moments of the trigonometric structure factor which permit an approximate evaluation of such p.d.f.'s for all the three-dimensional space groups.

2.1.7.2. Mathematical background

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Suppose that [p(x)] is a p.d.f. which accurately describes the experimental distribution of the random variable x, where x is related to a sum of random variables and can be assumed to obey (to some approximation) an ideal p.d.f., say $[p^{(0)}(x)]$ , based on the central-limit theorem. In the correction-factor approach we seek to represent [p(x)] as $[p(x) = p^{(0)}(x)\textstyle\sum\limits_{k}d_{k}\; f_{k}(x), \eqno(2.1.7.1)]$ where $[d_{k}]$ are coefficients which depend on the cause of the deviation of [p(x)] from the central-limit theorem approximation and $[f_{k}(x)]$ are suitably chosen functions of x. A choice of the set $[\{f_{k}\}]$ is deemed suitable, if only from a practical point of view, if it allows the convenient introduction of the cause of the above deviation of [p(x)] into the expansion coefficients $[d_{k}]$ . This requirement is satisfied – also from a theoretical point of view – by taking $[f_{k}(x)]$ as a set of polynomials which are orthogonal with respect to the ideal p.d.f., taken as their weight function (e.g. Cramér, 1951). That is, the functions $[f_{k}(x)]$ so chosen have to obey the relationship $[\textstyle\int\limits_{a}^{b}f_{k}(x)f_{m}(x)p^{(0)}(x)\;{\rm d}x = \delta_{km} = \cases{ 1, & if\quad $k = m$\cr 0, & if\quad $k \neq m $}\;, \eqno(2.1.7.2)]$ where [[a,b]] is the range of existence of all the functions involved. It can be readily shown that the coefficients $[d_{k}]$ are given by $[d_{k} = \textstyle\int\limits_{a}^{b}f_{k}(x)p(x)\;{\rm d}x = \langle f_{k}(x) \rangle = \textstyle\sum\limits_{n = 0}^{k}c_{n}^{(k)}\langle x^{n} \rangle, \eqno(2.1.7.3)]$ where the brackets $[\langle \; \rangle]$ in equation (2.1.7.3) denote averaging with respect to the unknown p.d.f. [p(x)] and $[c_{n}^{(k)}]$ is the coefficient of the nth power of x in the polynomial $[f_{k}(x)]$ . The coefficients $[d_{k}]$ are thus directly related to the moments of the non-ideal distribution and the coefficients of the powers of x in the orthogonal polynomials. The latter coefficients can be obtained by the Gram–Schmidt procedure (e.g. Spiegel, 1974), or by direct use of the Szegö determinants (e.g. Cramér, 1951), for any weight function that has finite moments. However, the feasibility of the present approach depends on our ability to obtain the moments $[\langle x^{n} \rangle]$ without the knowledge of the non-ideal p.d.f., [p(x)] .

2.1.7.3. Application to centric and acentric distributions

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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	$[{l'^{k - 1} \over 2^{k} (k!)^{2}} \sum\limits_{p=0}^{k} (M_{p}M_{k - p})^{3}[p! (k - p)!]^{2}]$

2.1.7.4. Fourier versus Hermite approximations

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As noted in Section 2.1.8.7 below, the Fourier representation of the probability distribution of [|F|] is usually much better than the particular orthogonal-function representation discussed in Section 2.1.7.3. Many, perhaps most, non-ideal centric distributions look like slight distortions of the ideal (Gaussian) distribution and have no resemblance to a cosine function. The empirical observation thus seems paradoxical. The probable explanation has been pointed out by Wilson (1986b). A truncated Fourier series is a best approximation, in the least-squares sense, to the function represented. The particular orthogonal-function approach used in equation (2.1.7.5), on the other hand, is not a least-squares approximation to $[p_{c}(|E|)]$ , but is a least-squares approximation to $[p_{c}(|E|)\exp(|E|^{2}/4). \eqno(2.1.7.15)]$ The usual expansions (often known as Gram–Charlier or Edgeworth) thus give great weight to fitting the distribution of the (compararively few) strong reflections, at the expense of a poor fit for the (much more numerous) weak-to-medium ones. Presumably, a similar situation exists for the representation of acentric distributions, but this has not been investigated in detail. Since the centric distributions $[p_{c}(|E|)]$ often look nearly Gaussian, one is led to ask if there is an expansion in orthogonal functions that (i) has the leading term $[p_{c}(|E|)]$ and (ii) is a least-squares (as well as an orthogonal-function)² fit to $[p_{c}(|E|)]$ . One does exist, based on the orthogonal functions $[f_{k} = n(x)He_{k}(2^{1/2}x), \eqno(2.1.7.16)]$ where [n(x)] is the Gaussian distribution (Myller-Lebedeff, 1907). Unfortunately, no reasonably simple relationship between the coefficients $[d_{k}]$ and readily evaluated properties of $[p_{c}(|E|)]$ has been found, and the Myller-Lebedeff expansion has not, as yet, been applied in crystallography. Although Stuart & Ord (1994, p. 112) dismiss it in a three-line footnote, it does have important applications in astronomy (van der Marel & Franx, 1993; Gerhard, 1993).

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