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Crystallography In a NutshellBy Charles E. Strouse
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PrefaceThis brief description of the fundamental aspects of x-ray crystallography is designed for the novice investigator who intends to carry out a crystallographic structure determination. It is intended to remove some of the mystery from the highly automated procedures used for the routine determination of x-ray crystal structures. Careful study of this document should allow one to avoid most of the common pitfalls of these procedures. In particular, an aspect of structural analysis that most commonly leads to grief, for the novice and the experienced crystallographer alike, is the correct assignment of crystallographic symmetry. This assignment is of necessity a subjective judgment based primarily on negative evidence. Only if one is familiar with the implications of crystallographic symmetry can one hope to make this judgment wisely. The Lattice and the Unit CellA crystal is made up of molecules or ions arranged in a regular and periodic fashion. A repeating unit in this three-dimensional structure is called a "unit cell". The only restriction on the choice of the unit cell is that all the corners of the cell must have identical environments. This means that the origin, the size, and the shape of the cell are highly arbitrary. Three intersecting edges of the unit cell are conventionally labeled "a", "b", and "c" and the angles between these edges are labeled α, β, and γ. In the following figure several possible unit cells have been chosen in a periodic array of letters.
All these cells except the one in the lower right are "primitive"; they contain only one repeating unit. All the primitive cells have the same volume (area in this two-dimensional example); no cell with smaller volume can be chosen. The unit cell in the lower right is "non-primitive"; it has a volume twice that of the primitive cells. This particular non-primitive cell is said to be "centered" since a point in the center of the cell has exactly the same environment as a point at the corner. Much larger non-primitive cells could be chosen. It is always possible to choose a primitive unit cell, but in materials that possess some additional crystallographic symmetry it is sometimes computationally convenient to choose a non-primitive unit cell whose edges correspond to symmetry axes. The five types of centered cells shown in the following figure are used in the conventional descriptions of crystallographic symmetry.
In the "end-centered" cells (A,B,and C centered) the center of one end of the unit cell is equivalent to the corner. The "face centered" cell is simultaneously A, B, and C centered. In the "body centered" cell, the center of the unit cell is equivalent to the corner. One usually discusses the crystal symmetry in terms of a mathematical "lattice", an INFINITE array of points, each with the same environment. Implicit in the concept of a lattice is the definition of a translational symmetry operation. Any translation of the lattice that superimposes two lattice points leaves the lattice unchanged. We will make use of this translational symmetry in later sections. First, however, we will explore some of the basic features of x-ray scattering. X-Ray ScatteringAn electron, acting under the influence of an oscillating electric field, acts as a source of "scattered" radiation. This is to say that x-rays are scattered strongly by electrons. Neutrons, on the other hand are scattered by nuclei and by unpaired electrons. This difference accounts for the different capabilities of x-ray and neutron diffraction. In what follows we will be concerned with the scattering caused by the electronic charge contained in a differential volume located at coordinates x,y,z in the unit cell; x, y, and z are conventionally used to represent the fractional unit cell coordinates in the "a", "b", and "c" directions respectively. For the sake of brevity this scattering will be referred to as the scattering from the "point" x,y,z. DIFFRACTION OCCURS, FOR RADIATION OF A GIVEN WAVELENGTH, ONLY IF THE CRYSTAL IS ORIENTATED WITH RESPECT TO A PARALLEL BEAM OF RADIATION SUCH THAT ALL TRANSLATIONALLY EQUIVALENT "POINTS" IN THE CRYSTAL SCATTER IN PHASE. This condition requires that the phase of the radiation scattered from a "point" one unit cell in the "a" direction from an arbitrary origin "point" is given by the expression: fa = fo + 2p h where h is an integer. Similarly for the b and c directions fb = fo + 2p k fc = fo + 2p l where k and l are also integers. It is easy to see that, for example, if h was slightly greater than an integer, the net scattering, summed over all the equivalent "points" in the "a" direction would approach zero. The integers h, k, and l are referred to as "Miller indices" and can be used to enumerate all the conditions that give rise to diffraction. Since the phase is a linear function of the position of the scatterer with respect to the origin, the phase of the radiation scattered from a "point" at fractional coordinates x,y,z must be given by fx,y,z = fo + 2 π (hx + ky + lz) In complex notation, the scattered wave is represented by e 2π i (hx + ky + lz) The relative amplitude of the scattered radiation can then be expressed by integration over the volume of the unit cell F(hkl) = ∫ρxyz e 2π i (hx + ky + lz) dV where ρxyz is the electron density at point x,y,z , ∫ dV represents integration over the volume of the unit cell, and F(hkl) is called the "structure factor". (The absolute amplitude, of course, depends on the intensity of the incident radiation and the number of unit cells present, i.e. the size of the crystal.) This is a very useful result because it shows that the amplitudes of the diffracted waves are given by the Fourier transform of the electron density distribution in the unit cell. The inverse transform thus gives the electron density in terms of the scattered amplitudes ρxyz = ∑hkl F(hkl) e -2π i (hx + ky + lz) where ∑hkl represents the sum over all h, k, and l. Before we consider the implications of this expression, it is useful to study some of the geometric consequences of this derivation. Let us consider the locus of points that scatter in phase (mod 2π) with an arbitrary origin "point". For the case h=1, k=0, l=0, this locus is a family of planes defined by the b and c directions.
+-----+-----+-----+-----+-----+-----+
+-----+-----+-----o-----+-----+-----+---- b
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+-----+-----+-----+-----+-----+-----+
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| a All points in the plane containing the origin scatter exactly in-phase with the origin; all points in the adjacent planes scatter 2π out of phase with the origin, etc. This places two constraints on the relative orientations of the incident beam, the crystal, and the diffracted beam. In the following diagram it can be seen that all points in a plane will scatter in-phase only if θ' = θ.
This leads directly to the crystallographic version of Bragg's law λ = 2 d sin θ where d is the interplanar spacing. These "lattice planes" are useful for visualizing the geometry of a diffraction experiment, but it is important to notice that the derivation preceding this discussion made no use of them. The Reciprocal Lattice and the Ewald ConstructionA simplification of the graphical representation of a diffraction experiment can be achieved by transformation of each family of planes (i.e. each orientation of the crystal with respect to the incident beam that obeys Bragg's law) to a single point. If one constructs vectors from an arbitrary origin perpendicular to each family of planes with lengths equal to the reciprocal of the d-spacing of each family, one generates a new lattice in a space with linear dimensions of the reciprocal of length. The coordinates in this discrete reciprocal space are h, k, and l, and the coordinate axes are designated a*, b*, and c*. This definition requires that a* is perpendicular to b and c etc. In the following figure the family of planes indicated in real space transforms to the point x in reciprocal space. It is useful to notice that a family of planes in reciprocal space transforms into a lattice point in real space.
The real and reciprocal lattices are in fact related to each other by a Fourier transform. The utility of the reciprocal lattice is demonstrated by the graphical expression of Bragg's law developed by Ewald. In this construction the incident beam enters from the left and passes through the crystal, the origin of real space. A sphere whose radius is the reciprocal of lambda is constructed about this origin, and the origin of reciprocal space is taken as the intersection of the sphere and the beam that has passed through the crystal. It is left as an exercise for the reader to show that diffraction will occur when a reciprocal lattice point touches the sphere, and that the direction of the diffracted beam is determined by construction of a vector from the real origin through the reciprocal lattice point on the sphere.
The Rotation PhotographAlthough the Ewald construction has many uses, in our discussion we will consider only its application to the interpretation of a simple type of x-ray photograph. In a rotation photograph a real crystallographic axis of the crystal (e.g. the "a" axis) is oriented perpendicular to a monochromatic x-ray beam. The crystal is then rotated about the real axis and the resulting diffraction pattern is recorded on a piece of photographic film. The following figure shows the geometry of the experiment. The x-ray beam enters from the left and strikes the crystal, "C". The rotation axis is the "a" axis. Consider the family of planes in reciprocal space perpendicular to the real "a" axis; this family of planes has an interplanar spacing d* = 1/a. These planes intersect the Ewald sphere to define a set of circles. As the crystal rotates, reciprocal lattice points pass through the surface of the sphere, and diffraction occurs. All x-rays diffracted from the crystal must pass through the sphere on one of these circles. If one considers only the diffracted beams in the plane of the paper, the film will be exposed at the point indicated by "s". The direct beam will strike the film, at the point indicated by "c".
It is possible to determine the length of a real axis (in this case "a") from the following equation: Arctan ( (s-c)/(C-c) ) = arcsin ( (1/a)/ (1/λ) ). Crystallographic SymmetryIt is the translational symmetry of crystals that gives rise to diffraction phenomena. Crystals, however, often exhibit additional elements of rotational symmetry. In this section we will be concerned with the consequences of combining these two types of symmetry. We will begin with a brief review of point symmetry, and then move on to the basic aspects of lattice symmetry. Point SymmetryThe point symmetry of an object can be described in terms of two fundamental types of rotational symmetry operations; a "proper rotation" is a simple rotation, while an "improper rotation" is an operation that involves a change in handedness. In the Schoenflies notation commonly used by spectroscopists, these operations are denoted by the symbols Cn and Sn where Cn corresponds to a rotation about some axis by an angle of 360/n degrees and Sn corresponds to a rotation of 360/n degrees followed by reflection through a plane perpendicular to the rotation axis. ( Sn is called a rotoreflection operation). In the Hermann-Mauguin notation used by crystallographers the proper rotations are denoted by numerals Proper Rotations
Improper rotations in the Hermann-Mauguin notation are expressed in terms of rotoinversion operations instead of rotoreflection operations. A rotoinversion operation is denoted by a numeral with a bar over it, and it consists of a proper rotation followed by an inversion. The correspondence between the two notations is shown below. Improper Rotations
One often needs to consider the coordinate transformations that correspond to these operations. For example, a two-fold rotation (2 or C2) about the x axis in an orthogonal coordinate system transforms the point x,y,z to the point x,-y,-z, while a mirror operation perpendicular to the x axis transforms a point x,y,z to a point -x,y,z. Transformations of this kind can be used to demonstrate the equalities expressed above.
Exercise: Write down the coordinate transformations that correspond to each of the improper rotations indicated above. Lattice SymmetryTranslational Operations A lattice is an infinite array of points each of which has an identical environment. This definition gives rise to the concept of translational symmetry, since if one translates the lattice in such a way as to move one lattice point to the former position of another, the lattice is unchanged. The edges of the unit cell discussed above correspond to three vectors that can serve as representations of three independent translational symmetry operations. (In non-primitive unit cells there are additional translational symmetry operations that correspond to vectors between symmetry equivalent points.) All other translational operations can be constructed as linear combinations of these. Rotational Operations Of the infinite number of possible rotational operations in point group symmetry, only two-fold, three-fold, four-fold and six-fold operations are compatible with a lattice. Construction of a lattice containing five-fold axes, for example, would be equivalent to construction of a tile floor with regular pentagons. Translational-Rotational Operations The combination of translational symmetry and rotational symmetry leads to the creation of an additional class of symmetry operations that are made up of combinations of rotational and translational components. These rotation-translation operations are again of two types, proper (screw axes) and improper (glide planes). Proper Rotation + Translation = Screw Axis Improper Rotation + Translation = Glide Plane The following is a tabulation of all the translational-rotational operations. In the examples discussed it will be assumed that the symmetry element passes through the origin. Screw axes 21 31 32 41 42 43 61 62 63 64 65 where Nm is N followed by translation of m/N parallel to rotation axis Examples: A 21 axis parallel to "b" is the combination of a two-fold rotation that takes the point x,y,z to the point -x,y,-z, and a translation that adds 1/2 to the y coordinate. Therefore a 21 takes x,y,z to -x,1/2+y,-z. A 31 axis parallel to "c" (in a coordinate system where α=β=90° and γ=120°) takes the point x,y,z to the point -y,x-y,1/3+z. Two consecutive 31 operations take x,y,z to y-x,-x,2/3+z. Notice that a 32 operation takes x,y,z to -y,x-y,2/3+z. The 31 and 32 axes are thus enantiomers of each other. Glide planes a = m + a/2 b = m + b/2 c = m + c/2 n = m + ( (a+b)/2 or (b+c)/2 or (a+c)/2 ) d = m + ( (a+b)/4 or (b+c)/4 or (a+c)/4 ) except in cubic or tetragonal space groups where d = m + (a+b+c)/4 Examples: An a-glide perpendicular to "b" takes x,y,z to 1/2+x,-y,z. An n-glide perpendicular to "a" takes x,y,z to -x,1/2+y,1/2+z. A Practical ObservationSeveral examples have been given in which the equivalent positions were derived from the symmetry operation. It is useful to be able to quickly go the other way. Notice that for symmetry in the "c" direction, a four-fold axis will exchange x and y, three-fold and six-fold operations will mix x and y, and two-fold operations and inversions will involve only sign changes on x and y. Examples:
Notice that an inversion involves three sign changes, a proper two-fold rotation two sign changes, and and improper two-fold (a mirror) one sign change. The position of the negative signs indicates the orientation of the symmetry element. The Space GroupAs in the case of point symmetry, all the symmetry operations of a lattice form a mathematical group. In the case of lattice symmetry this group is called the space group. It should be noticed first that for any choice of the unit cell there is only a finite number of unique translational symmetry operations (three for a primitive cell). Since there is only a finite number of rotational and translational-rotational operations, and since these can be oriented in only a finite number of ways consistent with the translational symmetry, there is a finite number of unique space groups. This number is 230. These space groups are commonly divided into the following seven classes.
Our problem in a structural investigation is to determine to which of these classes and ultimately to which of the 230 space groups a particular structure belongs. To see how this is done we must examine the effect of the crystallographic symmetry on the structure factor expression derived earlier. Laue Symmetry and Systematic AbsencesThe observed intensity of a diffracted beam is proportional to the magnitude squared of the structure factor. I(hkl) = F(hkl)F*(hkl) If each point, hkl, in reciprocal space is assigned a weight I(hkl), the resulting intensity weighted reciprocal space represents a three dimensional diffraction pattern. The symmetry of this diffraction pattern is called the "Laue symmetry". We shall see that the Laue symmetry is related to (but not identical to) the symmetry of the crystal structure. From the definition of the structure factor F(-h-k-l) = F*(hkl) and thus I(-h-k-l) = I(hkl), i.e. the diffraction pattern is always centrosymmetric. (This ignores the effect of anomalous scattering which we will take up later.) In general, if a function is real, its Fourier Transform is Hermitian. Another consequence of this is that if rho(xyz) is centrosymmetric (Hermitian and real), F(hkl) is real. This greatly simplifies the solution of the phase problem for centrosymmetric structures. In the case where there is a rotational symmetry element, we can rewrite the structure factor expression as an integral over the independent portion of the unit cell and the sum over the symmetry operations. F(hkl) = ∫ ρxyz ∑ e 2πi (hx+ky+lz) dV If the only symmetry operation is a two-fold axis parallel to the "b" direction, which takes a point x,y,z to -x,y,-z, the structure factor expression is F(hkl) = ∫ ρxyz [e 2πi (hx+ky+lz) + e 2πi (-hx+ky-lz) ]dV. In this case F(hkl) = F(-hk-l) and I(hkl) = I(-hk-l) Thus the two-fold axis in the structure gives rise to a two-fold axis in the diffraction pattern. Common sense, of course, requires the same result. Since the diffraction pattern is always centrosymmetric, the presence of a two-fold axis requires the presence of a mirror perpendicular to it 2/m symmetry). Similarly mirror symmetry in the crystal requires 2/m symmetry in the diffraction pattern. A single 21 axis parallel to "b" give a somewhat more complex structure factor expression. Since the point x,y,z is transformed by the 21 axis to the point -x,y+1/2,-z, F(hkl) = ∫ ρxyz [e 2πi (hx+ky+lz) + e 2πi (-hx+ky-lz) e πik ]dV This again requires that I(hkl) = I(-hk-l). Notice that e 2πi is 1 if n is an even integer and -1 if n is an odd integer. A 21 axis in the structure produces a two-fold axis in the diffraction pattern. There is, however, a manifestation of the translational component of the 21 in the diffraction pattern. Notice that for reflections with h=0 and l=0, F(hkl) = ∫ ρxyz [e 2πi ky ( 1 +e πik ) ]dV . Those reflections with k odd must have I(0k0)=0. These reflections are said to be systematically absent. Only symmetry operations with translational components give rise to systematic absences. There are thus three classes of absences. Three dimensional absences (true for all hkl) arise from pure translations (cell centering). Unit cell Absent A hkl, k+l = 2n+1 (odd) B hkl, h+l = 2n+1 C hkl, h+k = 2n+1 I hkl, h+k+l = 2n+1 F hkl, mixed (not all even or all odd) Glide planes cause two-dimensional absences. The following examples should be sufficient to illustrate the pattern Absent a-glide perpendicular to b h0l, h = 2n+1 c-glide perpendicular to b h0l, l = 2n+1 n-glide perpendicular to b h0l, h+l = 2n+1 a-glide perpendicular to c hk0, h = 2n+1 As in the example above, one-dimensional absences arise from screw axes. For example Absent 21 parallel to a h00, h = 2n+1 31 parallel to c 00l, l = 3n+1, 3n+2 41 parallel to c 00l, l = 4n+1, 4n+2, 4n+3 42 parallel to c 00l, l = 2n+1 All of these conditions can be derived from a simple expansion of the type illustrated for a 21 axis. Experimental Determination of the Space GroupIn a structure determination an initial goal is the assignment of the lattice symmetry. Knowledge of the Laue symmetry and the systematic absences in some cases leads to an unambiguous assignment of the space group, and in others it only narrows the possibilities. In all cases, however, there is a subjective judgment to be made. In the determination of the Laue symmetry one must decide whether two (or more) sets of observed intensities are "identical". In the detection of the absences one is working with negative evidence; are the reflections in question absent or just too weak to be detected in the background. These problems are not, however, sources of many errors. Rather, most problems are caused when legitimate symmetry elements go undetected. The first step in the space group determination is the assignment of the Laue symmetry. The Laue symmetry corresponds to the point group of intensity weighted reciprocal space. Determination of the Laue symmetry is usually accomplished before the I(hkl)s are measured. We shall discuss the standard procedures for determination of the Laue symmetry in more detail when we describe the data collection procedure, but here we can at least mention the Laue technique. A Laue photograph is taken with the same geometry as a rotation photograph, but the crystal is stationary and the x-ray source used is polychromatic. From what we have said above, if there is a two-fold axis perpendicular to the beam, or if there is a mirror plane that contains the beam, the resulting photograph will exhibit a mirror plane. After the I(hkl) have been measured, a search is made for three, two and one dimensional absences. The three dimensional absences reveal any centering (if centering is present, it is most efficient to detect it at the beginning of the data collection to avoid measuring the intensities of all the absent reflections). Any two-dimensional absence that is not a special case of a three-dimensional absence indicates the presence of a glide plane, and any one-dimensional absence that is not a special case of three- or two-dimensional absences indicates the presence of a screw axis. The presence of two identifiable symmetry elements sometimes requires the presence of a third operation that would be otherwise undetectable. The notation used for the designation of space group symmetry will be presented along with examples of space group determinations for each of the seven crystal classes.
Triclinic - Laue symmetry ī The only unique space groups are designated P1 and Pī. The P indicates a primitive unit cell. One could, if one wanted to fly in the face of convention, choose a non-primitive cell. For example, the space group C1 is not incorrect, just unconventional. There is no qualitative information in the diffraction pattern that indicates the presence or absence of the inversion center, but a statistical analysis of the intensities can often provide this information. The "General" (equivalent) positions in Pī are x,y,z and -x,-y,-z. For computational convenience the origin of the unit cell is always located at an inversion center if there is one. It is important to notice that inversion centers at the corners of the unit cell require the presence of additional inversion centers at the center of each cell face, the center of each cell edge, and at the center of the cell. There are eight crystallographically distinct inversion centers in Pī . These centers constitute "Special" positions in that, for example, if there was only one atom of a particular kind per unit cell in a Pī structure, it would have to be located at one of these inversion centers.
Monoclinic - Laue symmetry 2/m Example 1 -- no systematic absences. Possible space groups P2, Pm, P2/m. P2 has only a two-fold, Pm has only a mirror, and P2/m has a two-fold perpendicular to a mirror. The identification of the correct space group is usually indicated by successful solution and refinement of the structure (i.e. trial and error).
The "Special Positions" indicated above are loci of points related to themselves by symmetry. In the case of P2/m the special positions are the eight inversion centers, the four two-fold axes, and the two mirrors. Example 2 -- h0l, l=2n+1 absent and 0k0, k=2n+1 absent. The first absence indicates the presence of a c-glide plane perpendicular to "b", and the second indicates the presence of a 21 axis parallel to "b". The space group is P21/c. This is far and away the most common space group for non-optically-active substances. The presence of the 21 and the c requires the presence of an inversion center. If one, as a first guess, chooses the origin of the unit cell at the intersection of the 21 and the c-glide, one generates positions -x,1/2+y,-z and x,-y,1/2+z equivalent to the point x,y,z. The combination of these two operations gives rise to an additional equivalent position, -x,1/2-y,1/2-z. This corresponds to an inversion center at x=0, y=1/4, z=1/4. Since one wishes to have the origin located at the inversion center, it is necessary to relocate the 21 at 1/4 along "c" and to move the c-glide to 1/4 along "b". This results in new equivalent positions x,y,z; -x,1/2+y,1/2-z; x,1/2-y,1/2+z; -x,-y,-z. It is important that you completely understand and can duplicate this transformation.
Orthorhombic - Laue symmetry 2/m 2/m 2/m Example 1 -- no systematic absences. Possible space groups are P222, Pmm2, and Pmmm. The three symbols following the P indicate the symmetry in the a, b, and c directions. The fact that there are no absences indicates that the unit cell is primitive and that there are only two-folds or mirror planes or both in all three directions. There are only three unique possibilities since the product of two orthogonal two-fold operations is a two-fold in the third direction (i.e. P22 requires P222), and the product of two orthogonal mirrors also requires a two-fold in the third direction (i.e. Pmm requires Pmm2). Three orthogonal mirrors thus require three two-folds (i.e. Pmmm implies P 2/m 2/m 2/m). This result carries over to all orthorhombic space groups in that there will always be three proper rotation axes, two improper and one proper, or three of each. Example 2 -- hkl, h+k+l=2n+1; 0kl, k=2n+1; h0l, h=2n+1. Possible space groups Iba2 and Ibam. Iba2 has only the operations indicated in the symbol. Ibam must also have two-fold axes in all three directions and also an inversion center. Equivalent positions: The b-glide and a-glide require the equivalent positions x,y,z ; -x,1/2+y,z; 1/2+x,-y,z; and 1/2-x,1/2-y,z. This last position corresponds to a two-fold axis at x=1/4,y=1/4. It would be more conventional to locate the two-fold at x=0,y=0. This would require that the glide planes be located at x=1/4 and y=1/4 respectively, and give general positions x,y,z 1/2-x,1/2+y,z 1/2+x,1/2-y,z -x,-y,z . Inclusion of the I centering gives the additional positions 1/2+x,1/2+y,1/2+z -x,y,1/2+z x,-y,1/2+z 1/2-x,1/2-y,1/2+z The second and third of these correspond to c-glides in the a and b directions, and the last to a 21 axis at x=1/4,y=1/4. These positions correspond to space group Iba2. In Ibam, addition of a mirror perpendicular to the two-fold axis (the c direction) generates an inversion center at their intersection. This gives eight more general positions related to the eight above by inversion.
Tetragonal - Laue symmetry 4/m or 4/m 2/m 2/m In tetragonal space groups the three positions in the space group symbol (or Laue symmetry symbol) have a different meaning than they have in orthorhombic space groups. The first symbol after the lattice type indicates the symmetry along the principal symmetry axis, which is conventionally taken to be the "c" direction. This will be some kind of four-fold axis. The second symbol, if there is one, indicates the symmetry in the "a" and "b" directions (these axes are, of course, identical). The third symbol indicates any symmetry on the diagonal between "a" and "b". The fact that there are tetragonal space groups with two Laue symmetries has one important practical consequence. If the Laue symmetry is 4/m, one must, as in the case of orthorhombic symmetry, measure I(hkl) for all positive values of h,k and l. If the Laue symmetry is 4/m 2/m 2/m, one need only measure half of these (e.g. those with k greater than or equal to h), since the other half is related by the diagonal mirror. Example 1 -- Laue symmetry 4/m 2/m 2/m, no systematic absences. Enumeration of all the possible space groups gives P422, P4mm, P-42m, P-4m2, and P4/mmm.
Trigonal - Laue symmetry -3 or -3 2/m 2/m and Hexagonal - Laue symmetry 6/m or 6/m 2/m 2/m Conventionally, in cases of trigonal or hexagonal symmetry the unit cell is chosen so that "c" is the principal symmetry axis α =β= 90o and γ = 120o. The first position in the Laue symmetry (or the first position after the lattice type in the space group symbol) indicates the symmetry in the "c" direction, the second position the symmetry in the "a" and "b" directions, and the third position the symmetry on the diagonal between "a" and "b". Example 1 -- Laue symmetry -3, no systematic absences. Possible space groups P3 and P-3. There is a pure translational operation that is compatible with trigonal symmetry and which makes the cell chosen above non-primitive (but not centered in the sense described earlier). This operation makes points 1/3,2/3,2/3 and 2/3,1/3,1/3 equivalent to the origin. This operation results in the systematic absences hkl, -h+k+l=3n+1 and 3n+2. The presence of this translational symmetry is indicated by the symbol R in the lattice type. This R stands for rhombohedral and indicates that there exists a primitive unit cell with a three-fold axis on the body diagonal. It is important to recognize, however, that the remaining positions in the space group symbol refer to the directions indicated above in the non-primitive cell. Computationally, it is more convenient to choose the non-primitive (hexagonal) axes in the definition of the unit cell. Example 2 -- Laue symmetry 3m, absences hkl, -h+k+l=3n+1 and 3n+2. Possible space groups R32, R3m, R`3m. Example 3 -- Laue symmetry 3m, absences hkl, -h+k+l=3n+1 and 3n+2; & hhl (i.e. -h,h,l), l=2n+1. Possible space groups R3c and R`3c. Notice that hhl defines a plane in reciprocal space parallel to a c-glide plane.
Cubic - Laue symmetry 2/m -3 or 4/m -3 2/m The cubic unit cell is chosen so that the four three-fold axes define the body diagonals of the cell. In the symmetry symbols, the first position indicates the symmetry in the "a" (or "b" or "c") direction, the second position indicates the symmetry along the body diagonal, and the third position indicates the symmetry along a face diagonal. Example: Laue symmetry 2/m 3, no absences. Possible space groups P23 and Pm3. SummaryThe following table contains the 230 unique space groups separated according to Laue symmetry. It is important to recognize that there are many alternate designations for these space groups that result from alternate choices of the unit cell. For example, Pban could just as well be Pcna or Pncb. In practice, where an arbitrary choice of axes is required before the symmetry has been completely established, such "non-standard" space groups are common. There is seldom a compelling reason to convert all parameters to the standard "setting" ( and there is an obvious practical reason not to attempt such a transformation). The fact that there are many more than 230 settings makes it impractical to, given the Laue symmetry and absences, determine the space group by "exhaustion". To have any confidence in a space group assignment, one must be able to derive the space group symbol and the equivalent positions in a systematic way. A useful compilation of space groups is found in the International Tables for X-ray Crystallography. This presentation gives drawings and tabulations of the general and special positions and systematic absences.
Laue Symmetry Space Groups
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