International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 1.6, pp. 112-114

Section 1.6.3. Theoretical background of reflection conditions

U. Shmuelia

1.6.3. Theoretical background of reflection conditions

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We shall now examine the effect of the space-group symmetry on the structure-factor function. These effects are of importance in the determination of crystal symmetry. If [({\bi{W}},{\bi w})] is the matrix–column pair of a representative symmetry operation of the space group of the crystal, then, by definition[\rho({\bi x})=\rho({\bi{W}}{\bi x}+{\bi w}),\eqno(1.6.3.1)]where [\rho({\bi x})] is the value of the electron-density function at the point with coordinates [{\bi x}], [{\bi{W}}] is a matrix of proper or improper rotation and [{\bi w}] is a translation part (cf. Section 1.2.2.1[link] ). It is known that the electron-density function at the point [{\bi x}] is given by[\rho({\bi x})={{1}\over{V}}\sum_{{\bf h}}F({\bf h})\exp(-2\pi i{\bf h}{\bi x}),\eqno(1.6.3.2)]where, in this and the following equations, [{\bf h}] is the row matrix [(h\, k\, l)] and [{\bi x}] is a column matrix containing x, y and z in the first, second and third rows, respectively. Of course, [{\bf h}{\bi x}] is simply equivalent to [hx+ky+lz]. If we substitute (1.6.3.2)[link], with [{\bi x}] replaced by [({\bi{W}}{\bi x}+{\bi w})] in (1.6.3.1)[link] we obtain, after some calculation,[F({\bf h}{\bi{W}})=F({\bf h})\exp(-2\pi i{\bf h}{\bi w}).\eqno(1.6.3.3)]Equation (1.6.3.3)[link] is the fundamental relation between symmetry-related reflections (e.g. Waser, 1955[link]; Wells, 1965[link]; and Chapter 1.4[link] in Volume B). If we write [F({\bf h})=|F({\bf h})|\exp[i\varphi({\bf h})]], equation (1.6.3.3)[link] leads to the following relationships:[|F({\bf h}{\bi{W}})|=|F({\bf h})|\eqno(1.6.3.4)]and[\varphi({\bf h}{\bi{W}})=\varphi({\bf h})-2\pi{\bf h}{\bi w}.\eqno(1.6.3.5)]Equation (1.6.3.4)[link] indicates the equality of the intensities of truly symmetry-related reflections, while equation (1.6.3.5)[link] relates the phases of the corresponding structure factors. The latter equation is of major importance in direct methods of phase determination [e.g. Chapter 2.2[link] in Volume B (Giacovazzo, 2008[link])].

We can now approach the problem of systematically absent reflections, which are alternatively called the conditions for possible reflections.

The reflection h is general if its indices remain unchanged only under the identity operation of the point group of the diffraction pattern. I.e., if [{\bi{W}}] is the matrix of the identity operation of the point group, the relation [{\bf h}{\bi{W}}={\bf h}] holds true. So, if the reflection h is general, we must have [{\bi{W}}\equiv{\bi{I}}], where [{\bi{I}}] is the identity matrix and, obviously, [{\bf h}{\bi{I}}={\bf h}]. The operation [({\bi{I}},{\bi w})] can be a space-group symmetry operation only if w is a lattice vector. Let us denote it by [{\bi w}_{L}]. Equation (1.6.3.3)[link] then reduces to[F({\bf h})=F({\bf h})\exp(-2\pi i{\bf h}{\bi w}_{L})\eqno(1.6.3.6)]and [F({\bf h})] can be nonzero only if [\exp(-2\pi i{\bf h}{\bi w}_{L})=1]. This, in turn, is possible only if [{\bf h}{\bi w}_{L}] is an integer and leads to conditions depending on the lattice type. For example, if the components of [{\bi w}_{L}] are all integers, which is the case for a P-type lattice, the above condition is fulfilled for all h – the lattice type does not impose any restrictions. If the lattice is of type I, there are two lattice points in the unit cell, at say 0, 0, 0 and 1/2, 1/2, 1/2. The first of these does not lead to any restrictions on possible reflections. The second, however, requires that [\exp[-\pi i(h + k + l)]] be equal to unity. Since [\exp(\pi in)=(-1)^{n}], where n is an integer, the possible reflections from a crystal with an I-type lattice must have indices such that their sum is an even integer; if the sum of the indices is an odd integer, the reflection is systematically absent. In this way, we examine all lattice types for conditions of possible reflections (or systematic absences) and present the results in Table 1.6.3.1[link].

Table 1.6.3.1| top | pdf |
Effect of lattice type on conditions for possible reflections

Lattice type[{\bi w}^{T}_{L}][{\bf h}{\bi w}_{L}]Conditions for possible reflections
P (0, 0, 0) Integer None
A [(0, \textstyle{{1}\over{2}} ,\textstyle{{1}\over{2}})] [(k+l)/2] hkl: [k+l=2n]
B [(\textstyle{{1}\over{2}}, 0 ,\textstyle{{1}\over{2}})] [(h+l)/2] hkl: [h+l=2n]
C [(\textstyle{{1}\over{2}}, \textstyle{{1}\over{2}}, 0)] [(h+k)/2] hkl: [h+k=2n]
I [(\textstyle{{1}\over{2}}, \textstyle{{1}\over{2}}, \textstyle{{1}\over{2}})] [(h+k+l)/2] hkl: [h+k+l=2n]
F [(0, \textstyle{{1}\over{2}}, \textstyle{{1}\over{2}})] [(k+l)/2] h, k and l are all even or all odd (simultaneous fulfillment of the conditions for types A, B and C).
  [(\textstyle{{1}\over{2}}, 0, \textstyle{{1}\over{2}})] [(h+l)/2]
  [(\textstyle{{1}\over{2}}, \textstyle{{1}\over{2}}, 0)] [(h+k)/2]
Robv [(\textstyle{{2}\over{3}}, \textstyle{{1}\over{3}}, \textstyle{{1}\over{3}})] [(2h+k+l)/3] hkl: [-h+k+l=3n]
  [(\textstyle{{1}\over{3}}, \textstyle{{2}\over{3}}, \textstyle{{2}\over{3}})] [(h+2k+2l)/3] (triple hexagonal cell in obverse orientation)
Rrev [(\textstyle{{1}\over{3}}, \textstyle{{2}\over{3}}, \textstyle{{1}\over{3}})] [(h+2k+l)/3] hkl: [h-k+l=3n]
  [(\textstyle{{2}\over{3}}, \textstyle{{1}\over{3}}, \textstyle{{2}\over{3}})] [(2h+k+2l)/3] (triple hexagonal cell in reverse orientation)

The reflection h is special if it remains unchanged under at least one operation of the point group of the diffraction pattern in addition to its identity operation. I.e., the relation [{\bf h}{\bi{W}}={\bf h}] holds true for more than one operation of the point group. We shall now assume that the reflection h is special. By definition, this reflection remains invariant under more than one operation of the point group of the diffraction pattern. These operations form a subgroup of the point group of the diffraction pattern, known as the stabilizer (formerly called the isotropy subgroup) of the reflection h, and we denote it by the symbol [\cal{S}_{{\bf h}}]. For each space-group symmetry operation ([{\bi{W}},{\bi w})] where [{\bi{W}}] is the matrix of an element of [\cal{S}_{{\bf h}}] we must therefore have [{\bf h}{\bi{W}}={\bf h}]. Equation (1.6.3.3)[link] now reduces to[F({\bf h})=F({\bf h})\exp(-2\pi i{\bf h}{\bi w}).\eqno(1.6.3.7)]Of course, if [{\bi{W}}] represents the identity operation, w must be a lattice vector and the discussion summarized in Table 1.6.3.1[link] applies. We therefore require that [{\bi{W}}] is the matrix of an element of [\cal{S}_{{\bf h}}] other than the identity. [F({\bf h})] can be nonzero only if the exponential factor in (1.6.3.7)[link] equals unity. This, in turn, is possible only if hw is an integer.

Let us consider a monoclinic crystal with P-type lattice (i.e. with an mP-type Bravais lattice) and a c-glide reflection as an example. Assuming b perpendicular to the ac plane, the [({\bi{W}},{\bi w}]) representation of c is given by[c{:}\ \left[\pmatrix{1 & 0 & 0\cr 0 & \overline{1} & 0\cr 0 & 0 & 1}, \pmatrix{ 0\cr y\cr 1/2}\right].]

The indices of reflections that remain unchanged under the application of the mirror component of the glide-reflection operation must be h0l. The translation part of the c-glide-reflection operation has the form (0, y, 1/2), where y = 0 corresponds to the plane passing through the origin. Hence, for any value of y, the scalar product hw is l/2 and the necessary condition for a nonzero value of an h0l reflection is l = 2n, where n is an integer. Intensities of h0l reflections with odd l will be systematically absent.

Table 1.6.3.2[link] shows the effect of some glide reflections on reflection conditions.4

Table 1.6.3.2| top | pdf |
Effect of some glide reflections on conditions for possible reflections

Glide reflection[{\bi w}^{T}]hConditions for possible reflections
[a\perp[001]] (1/2, 0, z) (hk0) hk0: [h=2n]
[b\perp[001]] (0, 1/2, z) (hk0) hk0: [k=2n]
[n\perp[001]] (1/2, 1/2, z) (hk0) hk0: [h+k=2n]
[d\perp[001]] (1/4, ±1/4, z) (hk0) hk0: [h+k=4n] [(h,k=2n)]

Let us now assume a crystal with an mP-type Bravais lattice and a twofold screw axis taken as being parallel to b. The ([{\bi{W}},{\bi w})] representation of the corresponding screw rotation is given by[2_{1}{:}\ \left[\pmatrix{\overline{1} & 0 & 0\cr 0 & 1 & 0\cr 0 & 0 & \overline{1}}, \pmatrix{ x\cr 1/2\cr z }\right].]The diffraction indices that remain unchanged upon the application of the rotation part of [2_{1}] must be of the form (0k0). The translation part of the screw operation is of the form (x, 1/2, z), where the values of x and z depend on the location of the origin. Hence, for any values of x and z the scalar product hw is k/2 and the necessary condition for a nonzero value of a 0k0 reflection is k = 2n. 0k0 reflections with odd k will be systematically absent. A brief summary of the effects of various screw rotations on the conditions for possible reflections from the corresponding special subsets of hkl is given in Table 1.6.3.3[link]. Note, however, that while the presence of a twofold screw axis parallel to b ensures the condition 0k0: k = 2n, the actual observation of such a condition can be taken as an indication but not as absolute proof of the presence of a screw axis in the crystal.

Table 1.6.3.3| top | pdf |
Effect of some screw rotations on conditions for possible reflections

Screw rotation[{\bi w}^{T}]hConditions for possible reflections
[2_{1}\parallel [100]] (1/2, y, z) (h00) h00: [h=2n]
[2_{1}\parallel [010]] (x, 1/2, z) (0k0) 0k0: [k=2n]
[2_{1}\parallel [001]] (x, y, 1/2) (00l) 00l: [l=2n]
[2_{1}\parallel [110]] (1/2, 1/2, z) (hh0) None
[3_{1}\parallel [001]] (x, y, 1/3) (00l) 00l: [l=3n]
[3_{1}\parallel [111]] (1/3, 1/3, 1/3) (hhh) None
[4_{1}\parallel [001]] (x, y, 1/4) (00l) 00l: [l=4n]
[6_{1}\parallel [001]] (x, y, 1/6) (00l) 00l: [l=6n]

It is interesting to note that some diagonal screw axes do not give rise to conditions for possible reflections. For example, let [{\bi{W}}] be the matrix of a threefold rotation operation parallel to [111] and [{\bi w}^{T}] be given by (1/3, 1/3, 1/3). It is easy to show that the diffraction vector that remains unchanged when postmultiplied by [{\bi{W}}] has the form [{\bf h}=(h h h)] and, obviously, for such h and w, [{\bf h}{\bi w}=h]. Since this scalar product is an integer there are, according to equation (1.6.3.7)[link], no values of the index h for which the structure factor [F(hhh)] must be absent.

A short discussion of special reflection conditions

The conditions for possible reflections arising from lattice types, glide reflections and screw rotations are related to general equivalent positions and are known as general reflection conditions. There are also special or `extra' reflection conditions that arise from the presence of atoms in special positions. These conditions are observable if the atoms located in special positions are much heavier than the rest. The minimal special conditions are listed in the space-group tables in Chapter 2.3[link] . They can sometimes be understood if the geometry of a given specific site is examined. For example, Wyckoff position 4i in space group [P4_{2}22] (93) can host four atoms, at coordinates[4i{:}\  0,\textstyle{{1}\over{2}},z;\, \ \textstyle{{1}\over{2}},0,z+\textstyle{{1}\over{2}};\,\ 0,\textstyle{{1}\over{2}},\overline{z};\, \ \textstyle{{1}\over{2}},0,\overline{z}+\textstyle{{1}\over{2}}.]It is seen that the second and fourth coordinates are obtained from the first and third coordinates, respectively, upon the addition of the vector [t(\textstyle{{1}\over{2}}, \textstyle{{1}\over{2}}, \textstyle{{1}\over{2}})]. An additional I-centring is therefore present in this set of special positions. Hence, the special reflection condition for this set is hkl: [h+k+l=2n].

It should be pointed out, however, that only the general reflection conditions are used for a complete or partial determination of the space group and that the special reflection conditions only apply to spherical atoms. By the latter assumption we understand not only the assumption of spherical distribution of the atomic electron density but also isotropic displacement parameters of the equivalent atoms that belong to the set of corresponding special positions.

One method of finding the minimal special reflection conditions for a given set of special positions is the evaluation of the trigonometric structure factor for the set in question. For example, consider the Wyckoff position 4c of the space group Pbcm (57). The coordinates of the special equivalent positions are[4c{:}\ \ x,\textstyle{{1}\over{4}},0; \quad\overline{x},\textstyle{{3}\over{4}},\textstyle{{1}\over{2}}; \quad\overline{x},\textstyle{{3}\over{4}},0;\quad x,\textstyle{{1}\over{4}},\textstyle{{1}\over{2}}]and the corresponding trigonometric structure factor is[\eqalign{S({\bf h})&=\exp\left[2\pi i\left(hx+{{k}\over{4}}\right)\right]+\exp\left[2\pi i\left(-hx+{{3k}\over{4}}+{{l}\over{2}}\right)\right] \cr &\quad+\exp\left[2\pi i\left(-hx+{{3k}\over{4}}\right)\right]+\exp\left[2\pi i\left(hx+{{k}\over{4}}+{{l}\over{2}}\right)\right]. }]It can be easily shown that[S({\bf h})=2\cos\left[2\pi\left(hx+{{k}\over{4}}\right)\right][1+\exp(\pi il)]]and the last factor equals 2 for l even and equals zero for l odd. The special reflection condition is therefore: [hkl{:}\, l=2n].

Another approach is provided by considerations of the eigensymmetry group and the extraordinary orbits of the space group (see Section 1.4.4.4[link] ). We recall that the eigensymmetry group is a group of all the operations that leave the orbit of a point under the space group considered invariant, and the extraordinary orbit is associated with the eigensymmetry group that contains translations not present in the space group (see Chapter 1.4[link] ). In the above example the orbit is extraordinary, since its eigensymmetry group contains a translation corresponding to [\textstyle{{1}\over{2}}{\bf c}]. If this is taken as a basis vector, we have the Laue equation [\textstyle{{1}\over{2}} {\bf c}\cdot {\bf h}=l^{\prime}], where h is represented as a reciprocal-lattice vector and [l^{\prime}] is an integer which also equals l/2. But for l/2 to be an integer we must have even l. We again obtain the condition [hkl{:}\, l=2n].

These reflection conditions that are not related to space-group operations are given in Chapter 2.3[link] only for special positions. They may arise, however, also for different reasons. For example, a heavy atom at the origin of the space group [P2_{1}2_{1}2_{1}] would generate F-centring with corresponding apparent absences (cf. the special position 4a of the space group [Pbca] and the absences it generates).

We wish to point out that the most common `special-position absence' in molecular structures is due to a heavy atom at the origin of the space group [P2_{1}/c].

References

Giacovazzo, C. (2008). Direct methods. In International Tables for Crystallography, Volume B, Reciprocal Space, edited by U. Shmueli, ch. 2.2, pp. 215–243. Dordrecht: Springer.
Waser, J. (1955). Symmetry relations between structure factors. Acta Cryst. 8, 595.
Wells, M. (1965). Computational aspects of space-group symmetry. Acta Cryst. 19, 173–179.








































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