Origin specification

Giacovazzo, C.

doi:10.1107/97809553602060000555

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

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International Tables for Crystallography (2006). Vol. B. ch. 2.2, pp. 210-215 | 1 | 2 |

Section 2.2.3. Origin specification

C. Giacovazzo^a ^*

^aDipartimento Geomineralogico, Campus Universitario, I-70125 Bari, Italy
Correspondence e-mail: c.giacovazzo@area.ba.cnr.it

2.2.3. Origin specification

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(a) Once the origin has been chosen, the symmetry operators $[{\bf C}_{s} \equiv ({\bf R}_{s}, {\bf T}_{s})]$ and, through them, the algebraic form of the s.f. remain fixed.

A shift of the origin through a vector with coordinates $[{\bf X}_{0}]$ transforms $[\varphi_{\bf h}]$ into $[\varphi'_{\bf h} = \varphi_{\bf h} - 2\pi {\bf h} \cdot {\bf X}_{0} \eqno(2.2.3.1)]$ and the symmetry operators $[{\bf C}_{s}]$ into $[{\bf C}'_{s} = ({\bf R}'_{s}, {\bf T}'_{s})]$ , where $[{\bf R}'_{s} = {\bf R}_{s}\hbox{;} \;{\bf T}'_{s} = {\bf T}_{s} + ({\bf R}_{s} - {\bf I}) {\bf X}_{0} \quad s = 1, 2, \ldots, m. \eqno(2.2.3.2)]$
(b) Allowed or permissible origins (Hauptman & Karle, 1953, 1959) for a given algebraic form of the s.f. are all those points in direct space which, when taken as origin, maintain the same symmetry operators $[{\bf C}_{s}]$ . The allowed origins will therefore correspond to those points having the same symmetry environment in the sense that they are related to the symmetry elements in the same way. For instance, if $[{\bf T}_{s} = 0]$ for $[s = 1, \ldots, 8]$ , then the allowed origins in Pmmm are the eight inversion centres.

To each functional form of the s.f. a set of permissible origins will correspond.

(c) A translation between permissible origins will be called a permissible or allowed translation. Trivial allowed translations correspond to the lattice periods or to their multiples. A change of origin by an allowed translation does not change the algebraic form of the s.f. Thus, according to (2.2.3.2), all origins allowed by a fixed functional form of the s.f. will be connected by translational vectors $[{\bf X}_{p}]$ such that $[({\bf R}_{s} - {\bf I}) {\bf X}_{p} = {\bf V}, \quad s = 1, 2, \ldots, m, \eqno(2.2.3.3)]$ where V is a vector with zero or integer components.

In centred space groups, an origin translation corresponding to a centring vector $[{\bf B}_{v}]$ does not change the functional form of the s.f. Therefore all vectors $[{\bf B}_{v}]$ represent permissible translations. $[{\bf X}_{p}]$ will then be an allowed translation (Giacovazzo, 1974) not only when, as imposed by (2.2.3.3), the difference $[{\bf T}'_{s} - {\bf T}_{s}]$ is equal to one or more lattice units, but also when, for any s, the condition $[({\bf R}_{s} - {\bf I}) {\bf X}_{p} = {\bf V} + \alpha {\bf B}_{v}, \quad s = 1, 2, \ldots, m\hbox{;} \quad \alpha = 0, 1 \eqno(2.2.3.4)]$ is satisfied.

We will call any set of cs. or ncs. space groups having the same allowed origin translations a Hauptman–Karle group (H–K group). The 94 ncs. primitive space groups, the 62 primitive cs. groups, the 44 ncs. centred space groups and the 30 cs. centred space groups can be collected into 13, 4, 14 and 5 H–K groups, respectively (Hauptman & Karle, 1953, 1956; Karle & Hauptman, 1961; Lessinger & Wondratschek, 1975). In Tables 2.2.3.1 –2.2.3.4 the H–K groups are given together with the allowed origin translations.

Table 2.2.3.1 | top | pdf |
Allowed origin translations, seminvariant moduli and phases for centrosymmetric primitive space groups

	H–K group
	$[(h, k, l){\underline P}(2, 2, 2)]$		$[(h + k, l){\underline P}(2, 2)]$		$[(l){\underline P}(2)]$	$[(h + k + l){\underline P}(2)]$
Space group	$[P\bar{1}]$	Pmna	$[\displaystyle P {4\over m}]$	$[\displaystyle P {4\over n} mm]$	$[\displaystyle P\bar{3}]$	$[R\bar{3}]$
	$[\displaystyle P {2\over m}]$	Pcca	$[\displaystyle P {4_{2}\over m}]$	$[\displaystyle P {4\over n}cc]$	$[\displaystyle P \bar{3}1m]$	$[R \bar{3}m]$
	$[\displaystyle P {2_{1}\over m}]$	Pbam	$[\displaystyle P {4\over n}]$	$[\displaystyle P {4_{2}\over m}mc]$	$[\displaystyle P \bar{3}1c]$	$[R \bar{3}c]$
	$[\displaystyle P {2\over c}]$	Pccn	$[\displaystyle P {4_{2}\over n}]$	$[\displaystyle P {4_{2}\over m}cm]$	$[\displaystyle P \bar{3}m1]$	$[\displaystyle Pm \bar{3}]$
	$[\displaystyle P {2_{1}\over c}]$	Pbcm	$[\displaystyle P {4\over m}mm]$	$[\displaystyle P {4_{2}\over n}bc]$	$[\displaystyle P \bar{3}c1]$	$[\displaystyle Pn \bar{3}]$
	Pmmm	Pnnm	$[\displaystyle P {4\over m}cc]$	$[\displaystyle P {4_{2}\over n}nm]$	$[\displaystyle P {6\over m}]$	$[\displaystyle Pa \bar{3}]$
	Pnnn	Pmmn	$[\displaystyle P {4\over n}bm]$	$[\displaystyle P {4_{2}\over m}bc]$	$[\displaystyle P {6_{3}\over m}]$	$[\displaystyle Pm \bar{3}m]$
	Pccm	Pbcn	$[\displaystyle P {4\over n}nc]$	$[\displaystyle P {4_{2}\over m}nm]$	$[\displaystyle P {6\over m}mm]$	$[\displaystyle Pn \bar{3}n]$
	Pban	Pbca	$[\displaystyle P {4\over m}bm]$	$[\displaystyle P {4_{2}\over n}mc]$	$[\displaystyle P {6\over m}cc]$	$[\displaystyle Pm \bar{3}n]$
	Pmma	Pnma	$[\displaystyle P {4\over m}nc]$	$[\displaystyle P {4_{2}\over n}cm]$	$[\displaystyle P {6_{3}\over m}cm]$	$[\displaystyle Pn \bar{3}m]$
	Pnna				$[\displaystyle P {6_{3}\over m}mc]$
Allowed origin translations	(0, 0, 0);	$[(0, {1\over 2}, {1\over 2})]$	(0, 0, 0)		(0, 0, 0)	(0, 0, 0)
	$[({1\over 2}, 0, 0)]$ ;	$[({1\over 2}, 0, {1\over 2})]$	$[(0, 0, {1\over 2})]$		$[(0, 0, {1\over 2})]$	$[({1\over 2}, {1\over 2}, {1\over 2})]$
	$[(0, {1\over 2}, 0)]$ ;	$[({1\over 2}, {1\over 2}, 0)]$	$[({1\over 2}, {1\over 2}, 0)]$
	$[(0, 0, {1\over 2})]$ ;	$[({1\over 2}, {1\over 2}, {1\over 2})]$	$[({1\over 2}, {1\over 2}, {1\over 2})]$
Vector $[{\bf h}_{s}]$ seminvariantly associated with $[{\bf h} = (h, k, l)]$					(l)
Seminvariant modulus $[\boldomega _{s}]$	(2, 2, 2)		(2, 2)		(2)	(2)
Seminvariant phases	$[\varphi_{eee}]$		$[\varphi_{eee}\hbox{; } \varphi_{ooe}]$		$[\varphi_{eee}\hbox{; } \varphi_{eoe}]$	$[\varphi_{eee}\hbox{; } \varphi_{ooe}]$
Seminvariant phases					$[\varphi_{oee}\hbox{; } \varphi_{ooe}]$	$[\varphi_{oeo}\hbox{; } \varphi_{eoo}]$
Number of semindependent phases to be specified	3		2		1	1

Table 2.2.3.2 | top | pdf |
Allowed origin translations, seminvariant moduli and phases for noncentrosymmetric primitive space groups

	H–K group

Space group	P1	P2	Pm	P222	Pmm2	P4	$[P\bar{4}]$	P3	P312	P31m	P321	R3	R32
		$[P2_{1}]$	Pc	$[P222_{1}]$	$[Pmc2_{1}]$	$[P4_{1}]$	P422	$[P3_{1}]$	$[P3_{1}12]$	P31c	$[P3_{1}21]$	R3m	P23
				$[P2_{1}2_{1}2]$	Pcc2	$[P4_{2}]$	$[P42_{1}2]$	$[P3_{2}]$	$[P3_{2}12]$	P6	$[P3_{2}21]$	R3c	$[P2_{1}3]$
				$[P2_{1}2_{1}2_{1}]$	Pma2	$[P4_{3}]$	$[P4_{1}22]$	P3m1	P6	$[P6_{1}]$	P622		P432
					$[Pca2_{1}]$	P4mm	$[P4_{1}2_{1}2]$	P3c1	$[P\bar{6}m2]$	$[P6_{5}]$	$[P6_{1}22]$		$[P4_{2}32]$
					Pnc2	P4bm	$[P4_{2}22]$		$[P\bar{6}c2]$	$[P6_{4}]$	$[P6_{5}22]$		$[P4_{3}32]$
					$[Pmn2_{1}]$	$[P4_{2}cm]$	$[P4_{2}2_{1}2]$			$[P6_{3}]$	$[P6_{2}22]$		$[P4_{1}32]$
					Pba2	$[P4_{2}nm]$	$[P4_{3}22]$			$[P6_{2}]$	$[P6_{4}22]$		$[P\bar{4}3m]$
					$[Pna2_{1}]$	P4cc	$[P4_{3}2_{1}2]$			P6mm	$[P6_{3}22]$		$[P\bar{4}3n]$
					Pnn2	P4nc	$[P\bar{4}2m]$			P6cc	$[P\bar{6}2m]$
						$[P4_{2}mc]$	$[P\bar{4}2c]$			$[P6_{3}cm]$	$[P\bar{6}2c]$
						$[P4_{2}bc]$	$[P\bar{4}2_{1}m]$			$[P6_{3}mc]$
							$[P\bar{4}2_{1}c]$
							$[P\bar{4}m2]$
							$[P\bar{4}c2]$
							$[P\bar{4}b2]$
							$[P\bar{4}n2]$
Allowed origin translations	(x, y, z)	(0, y, 0)	(x, 0, z)	(0, 0, 0)	(0, 0, z)	(0, 0, z)	(0, 0, 0)	(0, 0, z)	(0, 0, 0)	(0, 0, z)	(0, 0, 0)	(x, x, x)	(0, 0, 0)
		$[(0, y, {1\over 2})]$	$[(x, {1\over 2}, z)]$	$[({1\over 2}, 0, 0)]$	$[(0, {1\over 2}, z)]$	$[({1\over 2}, {1\over 2}, z)]$	$[(0, 0, {1\over 2})]$	$[({1\over 3}, {2\over 3}, z)]$	$[(0, 0, {1\over 2})]$		$[(0, 0, {1\over 2})]$		$[({1\over 2}, {1\over 2}, {1\over 2})]$
		$[({1\over 2}, y, 0)]$		$[(0, {1\over 2}, 0)]$	$[({1\over 2}, 0, z)]$		$[({1\over 2}, {1\over 2}, 0)]$	$[({2\over 3}, {1\over 3}, z)]$	$[({1\over 3}, {2\over 3}, 0)]$
		$[({1\over 2}, y, {1\over 2})]$		$[(0, 0, {1\over 2})]$	$[({1\over 2}, {1\over 2}, z)]$		$[({1\over 2}, {1\over 2}, {1\over 2})]$		$[({1\over 3}, {2\over 3}, {1\over 2})]$
				$[(0, {1\over 2}, {1\over 2})]$					$[({2\over 3}, {1\over 3}, 0)]$
				$[({1\over 2}, 0, {1\over 2})]$					$[({2\over 3}, {1\over 3}, {1\over 2})]$
				$[({1\over 2}, {1\over 2}, 0)]$
				$[({1\over 2}, {1\over 2}, {1\over 2})]$
Vector $[{\bf h}_{s}]$ seminvariantly associated with $[{\bf h} = (h, k, l)]$	(h, k, l)	(h, k, l)	(h, k, l)	(h, k, l)	(h, k, l)					(l)	(l)
Seminvariant modulus $[\boldomega_{s}]$	(0, 0, 0)	(2, 0, 2)	(0, 2, 0)	(2, 2, 2)	(2, 2, 0)	(2, 0)	(2, 2)	(3, 0)	(6)	(0)	(2)	(0)	(2)
Seminvariant phases	$[\varphi_{000}]$	$[\varphi_{e0e}]$	$[\varphi_{0e0}]$	$[\varphi_{eee}]$	$[\varphi_{ee0}]$	$[\varphi_{ee0}]$	$[\varphi_{eee}]$	$[\varphi_{hk0}]$ if	$[\varphi_{hkl}]$ if	$[\varphi_{hk0}]$	$[\varphi_{hke}]$	$[\varphi_{h, \, k, \, \bar{h} + \bar{k}}]$	$[\varphi_{eee}]$ ; $[\varphi_{ooe}]$
Seminvariant phases						$[\varphi_{oo0}]$	$[\varphi_{ooe}]$	(mod 3)	(mod 6)				$[\varphi_{oeo}]$ ; $[\varphi_{ooe}]$
Allowed variations for the semindependent phases	$[\\|\infty\\|]$	$[\\|\infty\\|]$ , $[\\|2\\|]$ if	$[\\|\infty\\|]$ , $[\\|2\\|]$ if	$[\\|2\\|]$	$[\\|\infty\\|]$ , $[\\|2\\|]$ if	$[\\|\infty\\|]$ , $[\\|2\\|]$ if	$[\\|2\\|]$	$[\\|\infty\\|]$ , $[\\|3\\|]$ if	$[\\|2\\|]$ if $[h \equiv k]$ (mod 3) $[\\|3\\|]$ if $[l \equiv 0]$ (mod 2)	$[\\|\infty\\|]$	$[\\|2\\|]$	$[\\|\infty\\|]$	$[\\|2\\|]$
Number of semindependent phases to be specified	3	3	3	3	3	2	2	2	1	1	1	1	1

Table 2.2.3.3 | top | pdf |
Allowed origin translations, seminvariant moduli and phases for centrosymmetric non-primitive space groups

	H–K group
	$[(h, l) {\underline C} (2, 2)]$	$[(k, l) {\underline I}(2, 2)]$	$[(h + k + l) {\underline F}(2)]$	$[(l) {\underline I} (2)]$	$[{\underline I}]$
Space groups	$[\displaystyle C {2\over m}]$	Immm	Fmmm	$[\displaystyle I {4\over m}]$	$[Im \bar{3}]$
	$[\displaystyle C {2\over c}]$	Ibam	Fddd	$[\displaystyle I {4_{1}\over a}]$	$[Ia \bar{3}]$
	Cmcm	Ibca	$[Fm \bar{3}]$	$[\displaystyle I {4\over m} mm]$	$[Im \bar{3} m]$
	Cmca	Imma	$[Fd \bar{3}]$	$[\displaystyle I {4\over m} cm]$	$[Ia \bar{3} d]$
	Cmmm		$[Fm \bar{3} m]$	$[\displaystyle I {4_{1}\over a} md]$
	Cccm		$[Fm \bar{3} c]$	$[\displaystyle I {4_{1}\over a} cd]$
	Cmma		$[Fd \bar{3} m]$
	Ccca		$[Fd \bar{3} c]$
Allowed origin translations	(0, 0, 0)	(0, 0, 0)	(0, 0, 0)	(0, 0, 0)	(0, 0, 0)
	$[(0, 0, {1\over 2})]$	$[(0, 0, {1\over 2})]$	$[({1\over 2}, {1\over 2}, {1\over 2})]$	$[(0, 0, {1\over 2})]$
	$[({1\over 2}, 0, 0)]$	$[(0, {1\over 2}, 0)]$
	$[({1\over 2}, 0, {1\over 2})]$	$[({1\over 2}, 0, 0)]$
Vector $[{\bf h}_{s}]$ seminvariantly associated with $[{\bf h} = (h, k, l)]$				(l)
Seminvariant modulus $[\boldomega _{s}]$	(2, 2)	(2, 2)	(2)	(2)	(1, 1, 1)
Seminvariant phases	$[\varphi_{eee}]$	$[\varphi_{eee}]$	$[\varphi_{eee}]$	$[\varphi_{eoe}]$ ; $[\varphi_{eee}]$ $[\varphi_{ooe}]$ ; $[\varphi_{oee}]$	All
Number of semindependent phases to be specified	2	2	1	1	0

Table 2.2.3.4 | top | pdf |
Allowed origin translations, seminvariant moduli and phases for noncentrosymmetric non-primitive space groups

	H–K group

Space group	C2	Cm	Cmm2	C222	Amm2	Imm2	I222	F432	F222	I4	I422	$[I\bar{4}]$	Fmm2	I23
		Cc	$[Cmc2_{1}]$	$[C222_{1}]$	Abm2	Iba2	$[I2_{1}2_{1}2_{1}]$	$[F4_{1}32]$	F23	$[I4_{1}]$	$[I4_{1}22]$	$[I\bar{4}m2]$	Fdd2	$[I2_{1}3]$
			Ccc2		Ama2	Ima2			$[F\bar{4}3m]$	I4mm	$[I\bar{4}2m]$	$[I\bar{4}c2]$		I432
					Aba2				$[F\bar{4}3c]$	I4cm	$[I\bar{4}2d]$			$[I4_{1}32]$
										$[I4_{1}md]$				$[I\bar{4}3m]$
										$[I4_{1}cd]$				$[I\bar{4}3d]$
Allowed origin translations	(0, y, 0)	(x, 0, z)	(0, 0, z)	(0, 0, 0)	(0, 0, z)	(0, 0, z)	(0, 0, 0)	(0, 0, 0)	(0, 0, 0)	(0, 0, z)	(0, 0, 0)	(0, 0, 0)	(0, 0, z)	(0, 0, 0)
	$[(0, y, {1\over 2})]$		$[({1\over 2}, 0, z)]$	$[(0, 0, {1\over 2})]$	$[({1\over 2}, 0, z)]$	$[({1\over 2}, 0, z)]$	$[(0, 0, {1\over 2})]$	$[({1\over 2}, {1\over 2}, {1\over 2})]$	$[({1\over 4}, {1\over 4}, {1\over 4})]$		$[(0, 0, {1\over 2})]$	$[(0, 0, {1\over 2})]$
				$[({1\over 2}, 0, 0)]$			$[(0, {1\over 2}, 0)]$		$[({1\over 2}, {1\over 2}, {1\over 2})]$			$[({1\over 2}, 0, {3\over 4})]$
				$[({1\over 2}, 0, {1\over 2})]$			$[({1\over 2}, 0, 0)]$		$[({3\over 4}, {3\over 4}, {3\over 4})]$			$[({1\over 2}, 0, {1\over 4})]$
Vector $[{\bf h}_{s}]$ seminvariantly associated with $[{\bf h} = (h, k, l)]$	(k, l)	(h, l)	(h, l)	(h, l)	(h, l)	(h, l)	(h, l)			(l)	(l)		(l)
Seminvariant modulus $[\boldomega _{s}]$	(0, 2)	(0, 0)	(2, 0)	(2, 2)	(2, 0)	(2, 0)	(2, 2)	(2)	(4)	(0)	(2)	(4)	(0)	(1, 1, 1)
Seminvariant phases	$[\varphi_{e0e}]$	$[\varphi_{0e0}]$	$[\varphi_{ee0}]$	$[\varphi_{eee}]$	$[\varphi_{ee0}]$	$[\varphi_{ee0}]$	$[\varphi_{eee}]$	$[\varphi_{eee}]$	$[\varphi_{hkl}]$ with $[h + k + l \equiv 0]$ (mod 4)	$[\varphi_{hk0}]$	$[\varphi_{hke}]$	$[\varphi_{hkl}]$ with $[(2k - l) \equiv 0]$ (mod 4)	$[\varphi_{hk0}]$	All
Allowed variations for the semindependent phases	$[\matrix{\\|\infty\\|,\cr \\|2\\|\cr \hbox{ if } k = 0\cr}]$	$[\\|\infty\\|]$	$[\matrix{\\|\infty\\|,\cr \\|2\\|\cr \hbox{ if } l = 0\cr}]$	$[\\|2\\|]$	$[\matrix{\\|\infty\\|,\cr \\|2\\|\cr \hbox{ if } l = 0\cr}]$	$[\matrix{\\|\infty\\|,\cr \\|2\\|\cr \hbox{ if } l = 0\cr}]$	$[\\|2\\|]$	$[\\|2\\|]$	$[\\|2\\|]$ if $[h + k + l \equiv 0]$ (mod 2) $[\\|4\\|]$ if $[\equiv 1]$ (mod 2)	$[\\|\infty\\|]$	$[\\|2\\|]$	$[\\|2\\|]$ if $[h + k + l \equiv 0]$ (mod 2) $[\\|4\\|]$ if $[2k-l\equiv]$ 1 (mod 2)	$[\\|\infty\\|]$	All
Number of semindependent phases to be specified	2	2	2	2	2	2	2	1	1	1	1	1	1	0

(d) Let us consider a product of structure factors $[\eqalignno{F_{{\bf h}_{1}}^{A_{1}} \times F_{{\bf h}_{2}}^{A_{2}} \times \ldots \times F_{{\bf h}_{n}}^{A_{n}} &= \textstyle\prod\limits_{j = 1}^{n} F_{{\bf h}_{j}}^{A_{j}}\cr &= \exp \left(i \textstyle\sum\limits_{j = 1}^{n} A_{j}\varphi_{{\bf h}_{j}}\right) \textstyle\prod\limits_{j = 1}^{n} |F_{{\bf h}_{j}}|^{A_{j}}, &\cr&&(2.2.3.5)}]$ $[A_{j}]$ being integer numbers.

The factor $[\textstyle\sum_{j = 1}^{n} A_{j}\varphi_{{\bf h}_{j}}]$ is the phase of the product (2.2.3.5). A structure invariant (s.i.) is a product (2.2.3.5) such that $[\textstyle\sum\limits_{j = 1}^{n} A_{j} {\bf h}_{j} = 0. \eqno(2.2.3.6)]$ Since $[|F_{{\bf h}_{j}}|]$ are usually known from experiment, it is often said that s.i.'s are combinations of phases $[\textstyle\sum\limits_{j = 1}^{n} A_{j}\varphi_{{\bf h}_{j}}, \eqno(2.2.3.7)]$ for which (2.2.3.6) holds.

$[F_{0}]$ , $[F_{\bf h} F_{-{\bf h}}]$ , $[F_{\bf h} F_{\bf k} F_{\overline{{\bf h} + {\bf k}}}]$ , $[F_{\bf h} F_{\bf k} F_{\bf l} F_{\overline{{\bf h} + {\bf k} + {\bf l}}}]$ , $[F_{\bf h} F_{\bf k} F_{\bf l} F_{\bf p} F_{\overline{{\bf h} + {\bf k} + {\bf l} + {\bf p}}}]$ are examples of s.i.'s for .

The value of any s.i. does not change with an arbitrary shift of the space-group origin and thus it will depend on the crystal structure only.
(e) A structure seminvariant (s.s.) is a product of structure factors [or a combination of phases (2.2.3.7)] whose value is unchanged when the origin is moved by an allowed translation.

Let $[{\bf X}_{p}]$ 's be the permissible origin translations of the space group. Then the product (2.2.3.5) [or the sum (2.2.3.7)] is an s.s., if, in accordance with (2.2.3.1), $[\textstyle\sum\limits_{j = 1}^{n} A_{j} ({\bf h}_{j} \cdot {\bf X}_{p}) = r, \quad p = 1, 2, \ldots \eqno(2.2.3.8)]$ where r is a positive integer, null or a negative integer.

Conditions (2.2.3.8) can be written in the following more useful form (Hauptman & Karle, 1953): $[\textstyle\sum\limits_{j = 1}^{n} A_{j} {\bf h}_{s_{j}} \equiv 0 \quad (\hbox{mod}\ \boldomega _{s}), \eqno(2.2.3.9)]$ where $[{\bf h}_{s_{j}}]$ is the vector seminvariantly associated with the vector $[{\bf h}_{j}]$ and $[\boldomega _{s}]$ is the seminvariant modulus. In Tables 2.2.3.1 –2.2.3.4, the reflection $[{\bf h}_{s}]$ seminvariantly associated with $[{\bf h} = (h, k, l)]$ , the seminvariant modulus $[\boldomega _{s}]$ and seminvariant phases are given for every H–K group.

The symbol of any group (cf. Giacovazzo, 1974) has the structure $[{\bf h}_{s} L \boldomega _{s}]$ , where L stands for the lattice symbol. This symbol is underlined if the space group is cs.

By definition, if the class of permissible origin has been chosen, that is to say, if the algebraic form of the symmetry operators has been fixed, then the value of an s.s. does not depend on the origin but on the crystal structure only.
(f) Suppose that we have chosen the symmetry operators $[{\bf C}_{s}]$ and thus fixed the functional form of the s.f.'s and the set of allowed origins. In order to describe the structure in direct space a unique reference origin must be fixed. Thus the phase-determining process must also require a unique permissible origin congruent to the values assigned to the phases. More specifically, at the beginning of the structure-determining process by direct methods we shall assign as many phases as necessary to define a unique origin among those allowed (and, as we shall see, possibly to fix the enantiomorph). From the theory developed so far it is obvious that arbitrary phases can be assigned to one or more s.f.'s if there is at least one allowed origin which, fixed as the origin of the unit cell, will give those phase values to the chosen reflections. The concept of linear dependence will help us to fix the origin.
(g) n phases $[\varphi_{{\bf h}_{j}}]$ are linearly semidependent (Hauptman & Karle, 1956) when the n vectors $[{\bf h}_{s_{j}}]$ seminvariantly associated with the $[{\bf h}_{j}]$ are linearly dependent modulo $[\boldomega _{s}]$ , $[\boldomega _{s}]$ being the seminvariant modulus of the space group. In other words, when $[\textstyle\sum\limits_{j = 1}^{n} A_{j} {\bf h}_{s_{j}} \equiv 0 \quad (\hbox{mod}\ \boldomega _{s}), \qquad A_{q} \not\equiv 0 \quad (\hbox{mod}\ \boldomega _{s}) \eqno(2.2.3.10)]$ is satisfied. The second condition means that at least one exists that is not congruent to zero modulo each of the components of $[\boldomega _{s}]$ . If (2.2.3.10) is not satisfied for any n-set of integers $[A_{j}]$ , the phases $[\varphi_{{\bf h}_{j}}]$ are linearly semindependent. If (2.2.3.10) is valid for and , then $[{\bf h}_{1}]$ is said to be linearly semidependent and $[\varphi_{{\bf h}_{1}}]$ is an s.s. It may be concluded that a seminvariant phase is linearly semidependent, and, vice versa, that a phase linearly semidependent is an s.s. In Tables 2.2.3.1 –2.2.3.4 the allowed variations (which are those due to the allowed origin translations) for the semindependent phases are given for every H–K group. If $[\varphi_{{\bf h}_{1}}]$ is linearly semindependent its value can be fixed arbitrarily because at least one origin compatible with the given value exists. Once $[\varphi_{{\bf h}_{1}}]$ is assigned, the necessary condition to be able to fix a second phase $[\varphi_{{\bf h}_{2}}]$ is that it should be linearly semindependent of $[\varphi_{{\bf h}_{1}}]$ .

Similarly, the necessary condition to be able arbitrarily to assign a third phase $[\varphi_{{\bf h}_{3}}]$ is that it should be linearly semindependent from $[\varphi_{{\bf h}_{1}}]$ and $[\varphi_{{\bf h}_{2}}]$ .

In general, the number of linearly semindependent phases is equal to the dimension of the seminvariant vector $[\boldomega _{s}]$ (see Tables 2.2.3.1 –2.2.3.4). The reader will easily verify in (h, k, l) P (2, 2, 2) that the three phases $[\varphi_{oee}]$ , $[\varphi_{eoe}]$ , $[\varphi_{eoo}]$ define the origin (o indicates odd, e even).
(h) From the theory summarized so far it is clear that a number of semindependent phases $[\varphi_{{\bf h}_{j}}]$ , equal to the dimension of the seminvariant vector $[\boldomega _{s}]$ , may be arbitrarily assigned in order to fix the origin. However, it is not always true that only one allowed origin compatible with the given phases exists. An additional condition is required such that only one permissible origin should lie at the intersection of the lattice planes corresponding to the origin-fixing reflections (or on the lattice plane h if one reflection is sufficient to define the origin). It may be shown that the condition is verified if the determinant formed with the vectors seminvariantly associated with the origin reflections, reduced modulo $[\boldomega _{s}]$ , has the value ±1. In other words, such a determinant should be primitive modulo $[\boldomega _{s}]$ .

For example, in $[P\bar{1}]$ the three reflections $[{\bf h}_{1} = (345), {\bf h}_{2} = (139), {\bf h}_{3} = (784)]$ define the origin uniquely because $[\left|\matrix{3 &4 &5\cr 1 &3 &9\cr 7 &8 &4\cr}\right| {_{\rm reduced\; mod\; (2, 2, 2)} \atop \hbox{\rightarrowfill}} \left|\matrix{1 &0 &1\cr 1 &1 &1\cr 1 &0 &0\cr}\right| = -1.]$ Furthermore, in $[[{\bf h}_{s} = (h + k, l), \boldomega _{s} = (2, 0)]]$ $[{\bf h}_{1} = (5, 2, 0), \quad{\bf h}_{2} = (6, 2, 1)]$ define the origin uniquely since $[\left|\matrix{7 &0\cr 8 &1\cr}\right| {_{\rm reduced\; mod\; (2, 0)} \atop \hbox{\rightarrowfill}} \left|\matrix{1 &0\cr 0 &1\cr}\right| = 1.]$
(i) If an s.s. or an s.i. has a general value φ for a given structure, it will have a value −φ for the enantiomorph structure. If $[\varphi = 0]$ , π the s.s. has the same value for both enantiomorphs. Once the origin has been assigned, in ncs. space groups the sign of a given s.s. $[\varphi \neq 0]$ , π can be assigned to fix the enantiomorph. In practice it is often advisable to use an s.s. or an s.i. whose value is as near as possible to $[\pm \pi/2]$ .

References

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