Symmetry in Crystallography
Table of Contents
- Symmetry in Crystals
- Rotational Symmetry
- Stereographic Projections
- Crystallographic Point Groups
- Micro-translations
- Symmetry of the Reciprocal Lattice
- Systematic Absences
- Space Groups
- Determining the Space Group
- Special Positions
- References
This web page has been translated into Romanian by Alexander Ovsov.
Introduction
In crystallography, symmetry is used to characterize crystals, identify repeating parts of molecules, and simplify both data collection and nearly all calculations. Also, the symmetry of physical properties of a crystal such as thermal conductivity and optical activity must include the symmetry of the crystal.^{1} Thus, a thorough knowledge of symmetry is essential to a crystallographer. A clear, brief description of crystallographic symmetry was prepared by Robert Von Dreele.
An object is described as symmetric with respect to a transformation if the object appears to be in a state that is identical to its initial state, after the transformation. In crystallography, most types of symmetry can be described in terms of an apparent movement of the object such as some type of rotation or translation. The apparent movement is called the symmetry operation. The locations where the symmetry operations occur such as a rotation axis, a mirror plane, an inversion center, or a translation vector are described as symmetry elements.
Two distinct methods of describing rotational symmetry operations exist. These two sets of descriptors are the Hermann-Mauguin nomenclature^{11} and the Schönflies nomenclature.^{12} The Carl Hermann- Charles Mauguin system is typically used to describe crystals and crystallographic symmetry. The Arthur Schönflies convention is primarily used to describe symmetry in discrete molecules, in optical spectroscopy, and in quantum mechanics. In these notes, Hermann-Mauguin notation will be listed first followed by the corresponding Schönflies notation in parentheses.
Symmetry in Crystals
Our discussion of symmetry in crystallography should begin with a description of crystals. Crystals are defined as solids that have an atomic structure with long-range, 3-dimensional order. Unfortunately, this long-range order cannot be absolutely confirmed by any other method than some diffraction technique. However, there are several observations that can be made that will strongly suggest that a sample is crystalline before a diffraction experiment is undertaken.
Typically, crystals have flat faces and sharp edges. Also, many crystals
will have one or more directions that can be cleaved cleanly.
Samples with a naturally round shape, or samples that have a concoidal
fracture pattern are nearly always described as a glass
having
no significant, long-range, 3-D order. Similarly, materials that can be
gently poked with a probe, and retain the deformed shape are gels or plastic
materials and hence have not long-range, 3-D order.
When you look at several crystals from one material, you will soon notice that, although the crystals may have different sizes, all crystals have the same shape or habit. In particular, the angles between certain pairs of faces of the different crystals will be the same. This observation was first made by Nicholas Steno in 1669.^{2} This observation became known as the law of constancy of interfacial angles.
Figure 1. Models of crystals showing constant angles.^{3}
Steno and others in the 17th century were interested in the specific make up of crystals that would allow them to maintain the same angles between pairs of faces.^{4} These scientists believed that crystals must be made of some regularly-repeating components. Through these early studies, René-Just Haüy was able to postulate that if crystals of calcite and cubic garnets were built from many small blocks, then these blocks could easily be used to describe the faces of these crystals in terms of rational indices.^{5} This law of rational indices forms the basis of optical crystallography.
Figure 2. Models from Haüy's Traité de minéralogie (1801) - the crystal form have been redrawn in red.^{5}
Unit Cells
These regularly-repeating blocks are now known as unit cells. The dimensions of a unit cell are described by the lengths of the three axes, a, b, and c, and the three interaxial angles, α, β, and γ. In most published papers the axial lengths are expressed in terms of Å (Ångströms), and the interaxial angles are expressed in terms of ° (degrees).
Figure 3. Unit cell showing cell parameters.
There are many choices of repeating blocks in any given lattice. The main principles defining the lattice is that each lattice point must be in an identical environment as any other lattice point, and that the individual blocks in the lattice must have the smallest volume possible. Often there are many ways to select the vectors between lattice points and even the locations of the lattice points themselves. These unique lattice vectors are called basis vectors or basis set. Some 2-dimensional examples of these lattice choices are shown below.
Figure 4. Different choices of lattice vectors and lattice points.^{6}
When researchers discuss a particular material, they need to work from one standard or conventional description of the unit cell for that material. Thus crystallographers have chosen the following criteria for selecting unit cells. By convention the unit cell edges are chosen to be right-handed (a × b is the direction of c), to have the highest symmetry, and to have the smallest cell volume. If other symmetry considerations do not override, then the cell is chosen so that a ≤ b ≤ c, and α, β, and γ all < 90 ° or all ≥ 90 °. This type of cell is called the reduced cell. There are several other rules for obtaining the conventional reduced cell for a given material.^{7}
Crystalline materials are separated into 7 crystal different systems. These crystal systems are most easily identified by the constraints on the cell parameters. Note, however, that the cell parameter constraints are only necessary conditions. Thus a particular sample could have cell parameters that appear to fall into one category within experimental error, but are actually of lower symmetry.
The 7 crystal systems are listed in Table 1 below. In the lowest symmetry system, triclinic, there are no restrictions on the values of the cell parameters. In the other crystal systems, symmetry reduces the number of unique lattice parameters as shown in the Table. Certain conventions have been followed in tabulating the parameters. In the monoclinic system, one of the axes is unique in the sense that it is perpendicular to the other two axes. This axis is selected by convention as either the b or c axis so that either β or γ are ≥ 90°, respectively. Note that c-unique monoclinic cells are common in French literature and b-unique cells are common in most other languages. In the tetragonal, trigonal, and hexagonal systems, one axis contains higher symmetry. By convention this axis is selected as the c axis.
Crystal System | # | Cell Parameters | Symmetry |
---|---|---|---|
Triclinic | 6 | a ≠ b ≠ c; α ≠ β ≠ γ | 1 |
Monoclinic | 4 | a ≠ b ≠ c; α = γ = 90°, β ≥ 90° | 2/m |
Orthorhombic | 3 | a ≠ b ≠ c; α = β = γ = 90° | mmm |
Tetragonal | 2 | a = b ≠ c; α = β = γ = 90° | 4/m, 4/mmm |
Trigonal | |||
Hexagonal | 2 | a = b ≠ c; α = β = 90°, γ = 120° | 3, 3m |
Rhombohedral | 2 | a = b = c; α = β = γ ≠ 90° | 3, 3m |
Hexagonal | 2 | a = b ≠ c; α = β = 90°, γ = 120° | 6/m, 6/mmm |
Cubic | 1 | a = b = c; α = β = γ = 90° | 2/m3, m3m |
The seven crystal systems each describe separate ways that simple
3-dimensional lattices may be constructed. As with all lattice systems,
crystalline lattices are considered to have lattice points
on the
corners of the unit cell. Lattice points are selected so that the local environment
around any particular lattice point is identical to the environment around any other
lattice point.
Some trigonal lattices can be expressed on the basis of either a hexagonal or rhombohedral lattice. These lattices are shown in the drawings below. Note that rhombohedral lattice vectors can be expressed in an obverse orientation a), or in a reverse orientation b) below.
Figure 5. Rhombohedral lattices in the obverse and reverse orientations.
Bravais Lattices
It is sometimes possible to generate a lattice with higher symmetry if the lattice vectors are chosen so that one or more lattice points are also on the center of a face of the lattice or inside of the unit cell. These lattices with additional lattice points are described as centered lattices. Lattices with lattice points only on the corners are called primitive and are designated with the symbol P. Note that reduced cells described above are always primitive. In 1849, Auguste Bravais found that all regular crystals could be described in terms of only 14 lattice types for the 7 crystal systems.^{8} Remember that the environment around any one lattice point is exactly the same as the environment around any other lattice point.
Figure 6. A c-centered lattice showing the a and b lattice vectors and the a' and b' vectors of the primitive cell.^{9}
Non-primitive lattices can have one, two, or three additional lattice points per unit cell. Lattices with one additional lattice point such as an A-, B-, or C-centered cells have an additional lattice point located on the center of some face of the crystal. The additional lattice points in A-centered cells appear on the b-c faces. Similarly, additional lattice points in B-centered cells appear on the a-c faces, and lattice points will appear on the a-b faces of C-centered cells. Lattices with body-centered, I-centered, cells have an additional lattice point in the center of the cell. Rhombohedral cells that are based on a hexagonal lattice conventionally have lattice points at (2/3, 1/3, 1/3) and (1/3, 2/3, 2/3). Face-centered, F-centered cells have lattice points on all faces.
Generally higher metric symmetry is identified by computer programs. However, these centered lattices can sometimes be identified by a simple examination of the cell parameters. When two cell lengths of a reduced lattice are approximately equal and the two corresponging cell angles are also approximately equal, then the cell is probably centered.
The following table lists the 14 Bravais lattice types. The Bravais
symbols are a combination of the crystal system and the lattice designation.
Triclinic types begin with the letter a
that stands for
anorthic
from the mineral anorthite a mineral found to have triclinic
symmetry. The other lattice types generally begin with the first letter of the
crystal system.
Crystal System |
Bravais Lattice |
Laue Symmetry |
---|---|---|
Triclinic | aP | 1(C_{i}) |
Monoclinic | mP, mS† | 2/m(C_{2h}) |
Orthorhombic | oP, oS*, oF, oI | mmm(D_{2h}) |
Tetragonal | tP, tI | 4/m(C_{4h}), 4/mmm(D_{4h}) |
Trigonal# | ||
hexagonal | hP | 3(C_{3i}), 3m(D_{3d}) |
rhombohedral | rR | 3(C_{3i}), 3m(D_{3d}) |
Hexagonal | hP | 6/m(C_{6h}), 6/mmm(D_{6h}) |
Cubic | cP, cF, cI | m3(T_{h}), m3m(O_{h}) |
† The S symbol for monoclinic lattices represents a
lattice with A, C, or I centering (b-unique)
or A, B, or I centering (c-unique).
* The S symbol for orthorhombic lattices stands for any of
the three side-centered lattice types, A, B, or C.
# Since P trigonal lattice and a P hexagonal lattice are
identical in appearance, these two systems are considered to make up only one
Bravais lattice type.
Figure 7. Bravais Lattices^{10}
Rotational Symmetry
There are two basic types of rotational symmetry operations. Proper rotations move an object, but do not change the handedness of the object. Improper rotations include a proper rotation as well as a component that inverts the handedness of the object.
Proper Rotations
An n-fold (C_{n}) proper rotation operation represents a counter-clockwise movement of (360/n)° around an axis through the object. If an n-fold rotation operation is repeated n times, then the object returns to its original position. Crystals with a periodic lattice can only have axes with 1-, 2-, 3-, 4-, and 6-fold symmetry axes. The following description of rotational symmetry operations is similar to that given by Prof. Stephen Nelson.^{3} In the following drawings, the symmetry axis extends perpendicular from the page.
1-Fold Rotation. A 1(E)-fold rotation operation implies either a 0° rotation or a 360° rotation, and is referred to as the identity operation.
2-Fold Rotation. A 2-fold(C2) rotation operation moves the object by (360/2) ° = 180 °. The symbol used to designate a 2-fold axis is a solid oval.
3-Fold Rotation. A 3-fold(C3) rotation operation moves the object by (360/3) ° = 120 °. The symbol used to designate a 3-fold axis is a solid equilateral triangle.
4-Fold Rotation. A 4-fold(C4) rotation operation moves the object by (360/4) ° = 90 °. The symbol used to designate a 4-fold axis is a solid square.
6-Fold Rotation. A 6-fold(C6) rotation operation moves the object by (360/6) ° = 60 °. The symbol used to designate a 6-fold axis is a solid hexagon.
Improper Rotations
An improper rotation may be thought of as occurring in two parts,
first a proper rotation is performed, followed by an inversion through a particular
point on the rotation axis. In the H-M nomenclature, improper rotations are
sometimes called roto-inversions. Improper rotations are designated by the symbol
n, where n
represents the type of proper rotation component of the operation. As in
the proper rotation operations, only 1 (i),
2 (σ),
3 (S_{6}),
4 (S_{4}), and
6 (S_{3}) improper rotations are commonly
observed in crystals. These axes are pronounced as 3 bar
in the United States
and bar 3
in many European countries. Note that in the Schöflies notation,
improper rotation axes are considered as roto-reflection axes--a proper rotation followed
by a reflection through the plane perpendicular to the rotation axis. Thus
3 in H-M is
equivalent to S_{6} in Schönflies.
Certain types of improper rotation axes occur frequently
and are given special designations. These include an inversion center
(or center of symmetry) and a mirror plane. A 1 operation
(i) is simply an inversion center. A 2 operation (σ)
represents a mirror operation that is perpendicular to the corresponding proper
rotation axis. In the H-M notation, mirrors are labeled as m
.
Note that it is not necessary for either the rotation operation or the inversion center to exist as an operation of the group for the improper rotation axis to exist, e.g. the 4 (S_{4}) operation contains neither a 4-fold rotation axis (C_{4}) nor an inversion center.
3 Roto-inversion. This operation involves a rotation by (360/3) ° followed by an inversion through the center of the object.
4 Roto-inversion. This operation involves a rotation by (360/4) ° followed by an inversion through the center of the object.
6 Roto-inversion. This operation involves a rotation by (360/6) ° followed by an inversion through the center of the object.
A web site that illustrates the point groups based on molecular species is available at: http://symmetry.otterbein.edu/gallery/. Note that this site uses JMOL software, so Java must be activated on your web browser.
Stereographic Projections
Drawing the three-dimensional symmetry operations on a two-dimensional surface such as this page has been a difficult problem. One way to overcome this problem is through the use of a stereographic projection. Such figures are also effective in describing the angular relations among the faces of a crystal.
To construct a stereographic projection, imagine that the object with
a given symmetry or surface is at the center of a sphere. Consider the sphere to
have a polar axis that is bisected by an equatorial plane. Project features of
interest on the object from the center, out to the surface of the sphere. Then
project the points on the surface of the sphere through the equatorial plane to
the point where the polar axis intersects the
sphere in the opposite hemisphere. The stereographic projection is then given by
the equatorial plane and all intersections of the plane by the projected
points. If the projection point started in the northern hemisphere then its
projection onto the equatorial plane is represented as a plus.
Points
originating in the southern hemisphere are denoted with a
circle.
Sometimes points generated by improper symmetry operations
are also denoted with a comma to indicate opposite handedness.
The unit cell axis with highest symmetry is usually selected as the polar axis. Rotation axes not in the equatorial plane are drawn with the symbol representing the type of axis at the projection point on the equatorial plane. Rotation axes in the equatorial plane are drawn outside of the projection terminating in arrows. Mirror planes are drawn as thickened lines. Inversion centers are drawn as open circles in the center of the polar axis.
Figure 8. Stereographic projection.^{13}
E. J. W. Whittaker has prepared a more thorough discussion of Stereographic Projections.
Crystallographic Point Groups
Symmetry operations can be combined to generate other symmetry operations. When
these operations are written mathematically, the operations are applied in a
right-to-left order. Thus in the expression below, the object x is operated
on first by C and then by B. The operation symbol *
only indicates
that some transformation is being applied to the object.
A * x = B * C * x
These operations can be thus combined to form a group of symmetry operations. These groups of operations are called point groups because the symmetry elements of these operations all pass through a single point of the object. In the study of groups, mathematicians have found that such groups always have the following properties.
1. Identity One operation of the group must exist that when operated either before or after any other group operation produces the same transformation as that of the other group operation. Sometimes the identity operation is called the "do nothing" operation. (A * E = E * A = A, E is the identity operation)
2. Inverse For each operation of the group there must exist a second operation that when combined with the first operation produces the identity operation. The inverse operation cancels the effect of any operation. The identity operation is also its own inverse. (A * B = B * A = E, E is the identity, B is the inverse of A)
3. Associativity The order of combining the operations does not matter. [A * (B * C) = (A * B) * C]
In addition to these properties, all crystallographic symmetry groups possess the next property.
4. Closure When any two operations of the group are combined, then the resultant operation must be a member of the group. (A = B * C, A, B, C are operations of the group)
Finally, some groups also exhibit the property of commutativity. The order of operations in commutative groups does not matter. (A * B = B * A)
When the proper and improper rotation operations described above are combined following the rules of groups, they yield a total of 32 unique crystallographic point groups. These groups are listed in the following table. The centrosymmetric point groups are shown in bold.
System | Essential | Point |
---|---|---|
Symmetry | Groups | |
Triclinic | none | 1, 1 |
Monoclinic | 2 or m | 2, m, 2/m |
Orthorhombic | 222 or mm2 # | 222, mm2, mmm |
Tetragonal | 4 or 4 | 4, 422, 4, 4/m, 4mm, 42m, 4/mmm |
Trigonal | 3 or 3 | 3, 3, 32 †, 3m †, 3m^{2} |
Hexagonal | 6 or 6 | 6, 622, 6, 6/m, 6mm, 62m, 6/mmm |
Cubic | 23 | 23, 2/m3, 432, 43m, 4/m32/m = m3m |
* The symbol mm2 also represents the point groups 2mm and
m2m.
† These point groups represent sets of groups, e.g., 32 represents 321 and
312
By convention the following rules have been adopted to describe point
groups. When a rotation axis is followed by a slash and an m,
then this mirror is perpendicular to the rotation axis. For orthorhombic systems
the three characters describe the symmetry along the three axes, a,
b, and c, respectively. For
tetragonal, trigonal, and hexagonal type cells, the c axis is
unique, and the first symbol in the point group shows the symmetry along the unique
axis. In tetragonal systems, the second symbol shows the symmetry along the {[100]
and [010]} directions and the third symbol shows the symmetry along the {[110] and
[110]} directions. In trigonal and hexagonal cells,
the second symbol shows the symmetry along {[100], [010] and
[110]}, and the third symbol shows
symmetry along {[110], [120], and
[210]}. In rhombohedral systems on rhombohedral
axes, the first symbol shows symmetry along [111], and the second symbol shows
symmetry
along {[110], [011], and
[101]}. Cubic symbols show {[100], [010], [001]} in the
first symbol, {[111],
[111],
[111],
[111]} in the second symbol and
{[110], [110], [011], [011],
[101], and [101]} in the third symbol.
The following examples demonstrate how stereographic projections can help understand point groups.
Figure 9. Stereographic projection of 2-fold rotation axis. The cross in the upper, right side of the drawing is assigned the coodinates (x, y, z), the identity operation. If the 2-fold symmetry element is parallel with the b axis then the cross in the lower, left side of the drawing is assigned the coordinates (-x, y, -z).
Figure 10. Stereographic projection of 2/m rotation. In addition to the points in the previous figure, this projection shows a circle in the lower left region with the coordinates (-x, -y, -z) that is a spot generated by the inversion center. Note that the combination of the 2-fold rotation axis and the inversion center leads to another operation--a mirror plane normal to the 2-fold. The relative coordinates of a point related by the mirror in b are (x, -y, z). The generation of a mirror by adding an inversion center to a 2-fold axis is an example of the closure property of the group. The mirror is shown by the strong outer circle of the projection.
The other point groups and their symmetry-related coordinates can be derived in a similar manner to that shown above. All 32 crystallographic point groups are shown in the stereographic projections below.
Figure 11. Stereographic projections of the 32 crystallographic point groups
Micro-translations
Rotational symmetry operations can be combined with translations of part of the unit cell creating entirely new symmetry operations. Proper rotations combined with translations give rise to operations described as screw axes. Mirror planes that are combined with translations give rise to glide plane operations.
The symbol for a screw axis is n_{m} where n indicates the type of rotation and the translation is (m/n) of the unit cell. Thus a 3_{1} screw axis is a 3-fold counter-clockwise rotation followed by a translation of 1/3 of the unit cell. Performing this operation three times is equivalent to a full unit cell translation. Note that a 3_{2} screw axis rotates in the opposite direction as the 3_{1}.
Glide operations occur when a mirror operation, the glide plane, is followed by a translation, the glide vector. The translation directions either are parallel with a unit cell direction or are parallel with a combination of cell directions. When glides are described separately, they are given the symbols f_{g} in which the letter g indicates the direction of the mirror component and f indicates the direction of translation. Thus an a_{b} glide is an a glide plane in the b direction, which means that the object is reflected in a plane parallel with the (010) planes and then translated by a/2. Glide plane operations exist in all three directions and in pairs of directions. Glides that translate by half of the cell in two different directions are called n glides. An object undergoes an n_{c} operation when it is reflected in the (001) plane, and translated by (a + b)/2. Applying two identical glide operations to an object, is equivalent to applying a unit cell translation to that object.
An example of an a, b, or c glide plane.
An example of an n glide plane.
There is one additional type of glide plane, the diamond glide, d. It occurs only in space groups with face- or body-centered cells, and is characterized by a translation of (±a ± b)/4, (±b ± c)/4, (±c ± a)/4, or similar translations. As the denominator implies, 4 consecutive d glides are required to return an object to a lattice-translated version of itself.
In a recent version of the International Tables for Crystallography, Vol. A, the double glide called the e-glide is described.^{14} The e glide occurs only in centered cells and is defined by one plane with two perpendicular glide vectors related by a centering operation. This type of glide was proposed by the third Nomenclature Report of the IUCr.^{15}
Symmetry of the Reciprocal Lattice
The real cell parameters are determined by the relative positions of the reciprocal lattice points. Actually the reciprocal cell parameters are determined during a process known as indexing the diffraction pattern. From the reciprocal cell parameters the real cell parameters are then calculated according to the relations below.
a = b* × c* = (b* c* sin α*) / V*
b = c* × a* = (c* a* sin β*) / V*
c = a* × b* = (a* b* sin γ*) / V*
V* = 1/V = a*b*c* (1-cos^{2} α*-cos^{2} β*-cos^{2} γ* + 2 cos α* cos β* cos γ*)
cos α = (cos β* cos γ* - cos α) / (sin β* sin γ*)
cos β = (cos γ* cos α* - cos β) / (sin γ* sin α*)
cos γ = (cos α* cos β* - cos γ) / (sin α* sin β*)
Laue Class
The relative intensities in a diffraction pattern are dependent on the electron density distribution of the sample. This electron density distribution must follow the symmetry of the crystal itself. This symmetry is called the Laue class. The Laue class for a sample is described as one of the 11 centrosymmetric point groups. Note that the appropriate centrosymmetric point group or Laue class for a sample can be identified by adding a center of symmetry to the point group operations of the particular crystallographic point group of the sample.
Friedel's Law
The additional center of symmetry is due, at least approximately, to the fact that diffraction or interference effects are inherently centrosymmetric. The intensity of the (h k l) point in the reciprocal lattice comes from scattering of the electron density that is parallel to the (h k l) planes in the crystal. Similarly, the intensity of the (h k l) point comes from the electron density in planes parallel with the (h k l) planes in the crystal. But since the (h k l) planes and the (h k l) planes are simply opposite in direction, then the intensities of the (h k l) and (h k l) points in the reciprocal lattice should be, at least approximately the same. This equality relationship between the intensites of (h k l) and (h k l) is called Friedel's law.
For the intensity data from a chiral compound, Friedel's law can be broken by the anomalous scattering of heavy atoms. In these data sets the reciprocal lattice has the same symmetry as the symmetry of the point group of the crystal. Thus if the point group symmetry of the crystal is shown to be 222 then the intensities would exhibit 222 symmetry. The anomalous scattering of heavy atoms is not a strong effect, so the intensities will still approximately exhibit Friedel's law.
Symmetry-Equivalent Intensities
One simple way to determine which data should have equivalent intensities based on symmetry is to consider the stereographic projection of the point group of the sample. From a simple examination of the stereographic projection, you can determine the symmetry-related (x, y, z) coordinates. Simply convert the x coordinates to h, the y coordinates to k, and the z coordinates to l values. The symmetry-equivalent intensities are then determined.
A monoclinic crystal, has the Laue symmetry of 2/m. The equivalent coordinates, assuming a b-unique axis, are given as (x, y, z), (-x, y, -z), (-x, -y, -z), and (x, -y, z). Thus the intensities of the (h k l), (h k l), (h k l), and (h k l) lattice points should have equivalent values. Note that this also means that the intensities of the (h k l), (h k l), (h k l), and (h k l) should also be equivalent to each other but are not necessarily equivalent to (h k l), etc.
If a monoclinic compound is chiral then the intensities would be expected to have only 2 point group symmetry. Thus the intensities of the (h k l) data are equivalent to the intensities of the (h k l) data. Similarly, I(h k l) = I(h k l); I(h k l) = I(h k l); and I(h k l) = I(h k l).
If a crystal happens to have all three cell angles = 90.0° within experimental error then most workers would guess that the sample had orthorhombic symmetry. In most cases this guess would be correct, but not in all cases. If I(h k l) = I(h k l) and I(h k l) = I(h k l), but I(h k l) ≠ I(h k l), then the sample has monoclinic not orthorhombic Laue symmetry. The symmetry of the Laue class is dictated by the symmetry of the reciprocal lattice intensities not the apparent symmetry of the cell parameters. The cell parameters only dictate the possible highest symmetry of the sample.
Systematic Absences
Some symmetry operations can be readily identified by specific information in the intensities of the diffraction pattern. In particular, cell centering, screw axes, and glide plane operations can be identified by the fact that they cause certain groups of diffraction points to be systematically absent. These symmetry operations all include a micro-translation.
Consider a data set that has a c glide operation reflecting in the plane normal to the b axis. The symmetry operations could be (x, y, z) and (x, -y, z+1/2). If there are N atoms in the unit cell, then there are N/2 unique atoms. The summations below are over the j atoms and run from 1 to N/2.
F(hkl) = ∑
f_{j}
exp 2πi(hx_{j} + ky_{j} +
lz_{j}) +
∑ f_{j} exp 2πi[hx_{j} -
ky_{j} + l(1/2 + z_{j})]
Consider the data with k = 0. For these data the structure factors become:
F(h0l) = ∑
f_{j} exp 2πi(hx_{j} +
lz_{j}) +
∑ f_{j} exp 2πi(hx_{j} +
lz_{j}) exp2πi(l/2)
F(h0l) = ∑ f_{j} exp 2πi(hx_{j} + lz_{j}) [1 + exp πil]
If l is an odd integer, then exp πil = -1 and F(h0l) = 0. If l is an even integer, then F(h0l) is probably not (but could accidentally be) 0. The systematic absence conditions for other symmetry operations can be derived in a similar manner as was done above.
A table of these absence conditions is shown below. The reflection condition
indicates the data that may be present, other data with the same conditions would be
systematically absent. The
character n
can be any integer. Thus the condition that
for hkl, k + l = 2n+1, indicates that for the general
class of peaks hkl that the sum of k + l should be
positive for these peaks to have a measurable intensity, and that those peaks with
k + l negative must not have a measurable intensity if the
A centering symmetry is present in the symmetry of the data set.
It is always best to check for systematic absences, and hence for translation-containing symmetry operations, in the following order cell centering operations, then glide planes, and finally screw axes. This order is important because higher symmetry operations such as cell centering can mask lower symmetry operations such as glide planes and screw axes. Also, this order works from conditions that have the largest number of possible absences to the conditions with the least possible absences for a given sample.
Note that if a crystal exhibits the systematic absence condition for C cell centering then the fact that for 0kl, k = 2n does not necessarily indicate a b glide reflecting in a. This condition gives no new information, because it is a part of the cell centering condition.
Symmetry Element | Types | Reflection Condition |
---|---|---|
A centered | hkl | k + l = 2n |
B centered | h + l = 2n | |
C centered | h + k = 2n | |
F centered | k + l = 2n, h + l = 2n, h + k = 2n | |
I centered | h + k + l = 2n | |
R (obverse) | -h + k + l = 3n | |
R (reverse) | h - k + l = 3n | |
Glide reflecting in a | 0kl | |
b glide | k = 2n | |
c glide | l = 2n | |
n glide | k + l = 2n | |
d glide | k + l = 4n | |
Glide reflecting in b | h0l | |
a glide | h = 2n | |
c glide | l = 2n | |
n glide | h + l = 2n | |
d glide | h + l = 4n | |
Glide reflecting in c | hk0 | |
b glide | k = 2n | |
a glide | h = 2n | |
n glide | k + h = 2n | |
d glide | k + h = 4n | |
Glide reflecting in (110) | hhl | |
b glide | h = 2n | |
n glide | h + l = 2n | |
d glide | h + k + l = 4n | |
Screw || [100] | h00 | |
2_{1}, 4_{2} | h = 2n | |
4_{1}, 4_{3} | h = 4n | |
Screw || [010] | 0k0 | |
2_{1}, 4_{2} | k = 2n | |
4_{1}, 4_{3} | k = 4n | |
Screw || [001] | 00l | |
2_{1}, 4_{2}, 6_{3} | l = 2n | |
3_{1}, 3_{2}, 6_{2}, 6_{4} | l = 3n | |
4_{1}, 4_{3} | l = 4n | |
6_{1}, 6_{5} | l = 6n | |
Screw || [110] | hh0 | |
2_{1} | h = 2n |
Space Groups
When the 7 crystal systems are combined with the 14 Bravais lattices, the 32 point groups, screw axes, and glide planes, Arthur Schönflies^{12}, Evgraph S. Federov^{16}, and H. Hilton^{17} were able to describe the 230 unique space groups. A space group is a group of symmetry operations that are combined to describe the symmetry of a region of 3-dimensional space, the unit cell. In point groups, the symmetry elements all pass through one point in the object. In space groups, symmetry elements need not intersect at a single point, although some operations may coincidentally intersect at various points in the cell.
Chiral compounds that are prepared as a single enantiomer can crystallize in only a subset of 65 space groups that do not have improper rotations such as centers of symmetry, mirror planes, glides, or 3, 4, or 6 axes.
A space group is designated by a capital letter identifying the lattice type (P, A, F, etc.) followed by the point group symbol in which the rotation and reflection elements are extended to include screw axes and glide planes. Note that the point group symmetry for a given space group can be determined by removing the cell centering symbol of the space group and replacing all screw axes by similar rotation axes and replacing all glide planes with mirror planes. The point group symmetry for a space group describes the true symmetry of its reciprocal lattice.
The Bilbao Crystallographic Server gives a description of all space groups and their symmetry generators, general positions, Wyckoff positions, related subgroups and supergroups, and normalizers. M. W. Meier has prepared an interesting set of notes on space group patterns that illustrates the various combinations of point group symmetry elements into space groups. The following table lists the unique space groups separated by crystal system and Laue class.
Crystal System |
Laue Class |
Space Group |
---|---|---|
triclinic | 1 | P1, P1 |
monoclinic | 2/m | P2, P2_{1}, C2, Pm, Pc, Cm, Cc, P2/m, P2_{1}/m, C2/m, P2/c, P2_{1}/c, C2/c |
orthorhombic | mmm | P222, P222_{1}, P2_{1}2_{1}2, P2_{1}2_{1}2_{1}, C222_{1}, C222, F222, I222, I2_{1}2_{1}2_{1}, Pmm2, Pmc2_{1}, Pcc2, Pma2, Pca2_{1}, Pnc2, Pmn2_{1}, Pba2, Pna2_{1}, Pnn2, Cmm2, Cmc2_{1}, Ccc2, Amm2, Aem2, Ama2, Aea2, Fmm2, Fdd2, Imm2, Iba2, Ima2, Pmmm, Pnnn, Pccm, Pban, Pmma, Pnna, Pmna, Pcca, Pbam, Pccn, Pbcm, Pnnm, Pmmn, Pbcn, Pbca, Pnma, Cmcm, Cmce, Cmmm, Cccm, Cmme, Ccce, Fmmm, Fddd, Immm, Ibam, Ibca, Imma |
tetragonal | 4/m | P4, P4_{1}, P4_{2}, P4_{3}, I4, I4_{1}, P4, I4, P4/m, P4_{2}/m, P4/n, P4_{2}/n, I4/m, I4_{1}/a |
tetragonal | 4/mmm | P422, P42_{1}2, P4_{1}22, P4_{1}2_{1}2, P4_{2}22, P4_{2}2_{1}2, P4_{3}22, P4_{3}2_{1}2, I422, I4_{1}22, P4mm, P4bm, P4_{2}cm, P4_{2}nm, P4cc, P4nc, P4_{2}mc, P4_{2}bc, I4mm, I4cm, I4_{1}md, I4_{1}cd, P42m, P42c, P42_{1}m, P42_{1}c, P4m2, P4c2, P4b2, P4n2, I4m2, I4c2, I42m, I42d, P4/mmm, P4/mcc, P4/nbm, P4/nnc, P4/mbm, P4/mnc, P4/nmm, P4/mcc, P4_{2}/mmc, P4_{2}/mcm, P4_{2}/nbc, P4_{2}/nnm, P4_{2}/mbc, P4_{2}/mnm, P4_{2}/nmc, P4_{2}/ncm, I4/mmm, I4/mcm, I4/amd, I4/_{1}acd |
trigonal | 3 | P3, P3_{1}, P3_{2}, R3, P3, R3, |
trigonal | 3m | P312, P321, P3_{1}12, P3_{1}21, P3_{2}12, P3_{2}21, R32, P3m1, P31m, P3c1, P31c, R3m, R3c, P31m, P31c, P3m1, P3c1, R3m, R3c |
hexgonal | 6/m | P6, P6_{1}, P6_{5}, P6_{2}, P6_{4}, P6_{3}, P6, P6/m, P6_{3}/m |
hexagonal | 6/mmm | P622, P6_{1}22, P6_{5}22, P6_{2}22, P6_{4}22, P6_{3}22, P6mm, P6cc, P6_{3}cm, P6_{3}mc, P6m2, P6c2, P62m, P62c, P6/mmm, P6/mcc, P6_{3}/mcm, P6_{3}/mmc |
cubic | m3 | P23, F23, I23, P2_{1}3, I2_{1}3, Pm3, Pn3, Fm3, Fd3, Im3, Pa3, Ia3 |
cubic | m3m | P432, P4_{2}32, F432, F4_{1}32, I432, P4_{3}32, P4_{1}32, I4_{1}32, P43m, F43m, I43m, P43n, F43c, I43d, Pm3m, Pn3n, Pm3n, Pn3m, Fm3m, Fm3c, Fd3m, Fd3c, Im3m, Ia3d |
The space groups in bold are centrosymmetric.
The previous table lists the mathematically-unique space groups. In addition to these there
are many non-standard space groups, some of which are listed in the International
Tables for Crystallography, Vol A.
^{18} For example, the space groups
P2_{1}/a and P2_{1}/n are variants of the
space group P2_{1}/c that are often seen in the literature. Some
non-standard space groups are reported in a publication so that the cell parameters of
several related compounds can be more easily compared. For example, if the main compound in a paper
is found to be in the monoclinic space group C2/c, but a related compound is found
to be triclinic, some authors would determine the structure of this second compound in the non-standard
space group C1 if the cell parameters in
C1 were similar to the cell parameters of the main compound. Finally
some authors would publish structures in non-standard space groups because the basis vectors of the
related cells would produce cell angles that were closer to 90 °. For example, structures are often
published in Pn rather than Pc or in I2/a rather than
C2/c because the β angle would be much
closer to 90 °. When β angles are closer to 90 °, the refinement of the x and z coordinates
of the atoms show much less correlation, and converge to form chemically-reasonable structures more
quickly.
The following tables list the graphical and typed symbols used to describe symmetry operations in
the International Tables for Crystallography, Vol A.
^{19} Please note that the
symbol ⊥ is used to indicate that the operation is perpendicular (normal) to the page.
Symmetry plane | Graphical symbol | Translation | Symbol |
---|---|---|---|
Reflection plane | None | m | |
Glide plane | 1/2 along line | a, b, or c | |
Glide plane | 1/2 normal to plane | a, b, or c | |
Double glide plane | 1/2 along line & 1/2 normal to plane |
e | |
Diagonal glide plane | 1/2 along line & 1/2 normal to plane |
n | |
Diamond glide plane | 1/4 along line & 1/4 normal to plane |
d |
Symmetry plane | Graphical symbol | Translation | Symbol |
---|---|---|---|
Reflection plane | None | m | |
Glide plane | 1/2 along arrow | a, b, or c | |
Double glide plane | 1/2 along either arrow | e | |
Diagonal glide plane | 1/2 along the arrow | n | |
Diamond glide plane | 1/8 or 3/8 along the arrows | d |
Symmetry Element | Graphical Symbol | Translation | Symbol |
---|---|---|---|
Identity | None | None | 1 |
2-fold ⊥ page | None | 2 | |
2-fold in page | None | 2 | |
2 sub 1 ⊥ page | 1/2 | 2_{1} | |
2 sub 1 in page | 1/2 | 2_{1} | |
3-fold | None | 3 | |
3 sub 1 | 1/3 | 3_{1} | |
3 sub 2 | 2/3 | 3_{2} | |
4-fold | None | 4 | |
4 sub 1 | 1/4 | 4_{1} | |
4 sub 2 | 1/2 | 4_{2} | |
4 sub 3 | 3/4 | 4_{3} | |
6-fold | None | 6 | |
6 sub 1 | 1/6 | 6_{1} | |
6 sub 2 | 1/3 | 6_{2} | |
6 sub 3 | 1/2 | 6_{3} | |
6 sub 4 | 2/3 | 6_{4} | |
6 sub 5 | 5/6 | 6_{5} | |
Inversion | None | 1 | |
3 bar | None | 3 | |
4 bar | None | 4 | |
6 bar | None | 6 = 3/m | |
2-fold and inversion | None | 2/m | |
2 sub 1 and inversion | None | 2_{1}/m | |
4-fold and inversion | None | 4/m | |
4 sub 2 and inversion | None | 4_{2}/m | |
6-fold and inversion | None | 6/m | |
6 sub 3 and inversion | None | 6_{3}/m |
Determining the Space Group of a Material
Identifying the proper space group for a given sample and its diffraction pattern is often straight forward. The Laue class is determined first. Then the possible systematic absences are determined to identify the appropriate cell centering conditions, glide planes, and screw axes, if any. From the Laue class and the symmetry operations identified by the systematic absences, the choice of space group(s) is usually limited one unique or to a very few.
If the space group is not uniquely determined then the structure solution and refinement steps are tried with the different possible space groups until the structure determination is completed. Try the space groups beginning with the highest symmetry first. If the chosen space group is not the highest symmetry space group, then be sure to check for additional symmetry elements in the structure using either the PLATON program or the checkcif program.^{20,21}
Additional information about the compound is often used to reduce the choice of space group to a single one. If the compound is chiral and is expressed as a single enantiomorph, for example, then the compound can only crystallize in one of the 65 chiral space groups.
Merging Data to Determine Laue Class
The Laue class is usually found by calculating the merging Rint of the symmetry-equivalent data for the given Laue class. An appropriate Rint for identifying the Laue class would be a value & 0.06. Values larger than this usually suggests that the sample crystallized in a lower symmetry Laue group.
Symmetry-equivalent intensity data are merged using the following relationship
F^{2} = ∑ ω_{j} F_{j}^{2} / ∑ ω_{j}
where the summations are over the set of symmetry-equivalent data. Sometimes outliers are either removed from the merged set or are given lower weights to reduce their effects on the subsequent refinement.
Rint = ∑ [ ∑ |F_{j}^{2} - <F^{2}>| ] / ∑ [ (∑ F_{j}^{2})/n ]
where the inner sums are over the symmetry-equivalent reflections and the outer sums are over the unique hkl data. The term n is the number of equivalent data for a given hkl being merged.
Tests for a Center of Symmetry
If the compound is not a single enantiomorph, then statistical tests for a center of symmetry can be performed.
Probability distributions for centric and acentric unit cells have been derived^{22} and are given below. These derivations assume that the electron density is randomly and uniformly distributed throughout the unit cell.
P_{-1}(|F|) = [(2)^{1/2} / (πS)^{1/2}] exp(-|F|^{2} / 2 S) (centric)
P_{1}(|F|) = (2 |F| / S) exp(-|F|^{2} / S) (acentric)
where
S = ∑ f_{j}^{2}
S is a function of the scattering factors for the atoms and thus will decrease for increasing sin (θ / λ). To simplify this functional dependence of the distributions on θ, most statistical tests utilize normalized structure factors, E_{hkl}.
E_{hkl}^{2} = F_{hkl}^{2} / (ε ∑ f_{j}^{2})
where f_{j} = f_{j}^{o} exp(-B sin^{2}θ / λ_{2}) is the scattering factor for the jth atom and ε is an integer, 1 or greater, that corrects for the fact that some classes of reflections have expectation values that are less than ∑ f_{j}^{2} by an integer amount.
Since <|E_{h}|^{2}> = 1 = S, the distributions become
P_{-1}(|E|) = [(2 / π)^{1/2}] exp(-|E|^{2} / 2) (centric)
P_{1}(|E|) = 2 |E| exp(-|E|^{2}) (acentric)
These two distributions are completely independent of the variation of θ and are significantly different from each other. The differences in the two distributions indicate that centric unit cells should have more reflections with very strong and very weak values compared to acentric unit cells. Acentric unit cells typically tend to have more even distributions of data with significantly fewer weak data than centric cells. Comparing the distribution of a measured data set with the two theoretical distributions could establish the presence or absence of a center of symmetry in the crystal under consideration.
Mean values for a variety of functions of |E| may be estimated from the two theoretical distributions. These values are presented in the following Table. Most computer programs that test for centric unit cells calculate the mean values for these functions of E and compare these experimental values with the theoretical values.
Centric | Acentric | |
---|---|---|
<|E|^{2}> | 1.000 | 1.000 |
<|E^{2}-1|> | 0.968 | 0.736 |
<|E|> | 0.798 | 0.886 |
Three caveats must be remembered when using these tests. First, the intensity data must be relatively strong. Weak data sets, having F^{2}/σ < 6.0, have too many weak data to give reliable results for this test. Second, these theoretical values were derived assuming a uniform distribution of electron density in the unit cell. The presence of heavy atoms or of an unusual distribution of atoms tends to make the experimental values assume a centric distribution. When the tests indicate an acentric unit cell, the tests are likely to be correct. When the tests suggest a centric unit cell, the tests may be correct or may simply indicate a skewed distribution of electron density in the unit cell. Finally, if the tests indicate a hyper acentric distribution, e.g. <|E^{2}-1|> < 0.6, then it is likely that the sample is twinned.
The definitive discussion of space groups and space group symmetry is
available from the International Tables for Crystallography, Volume
A.
^{18} A good introduction to symmetry and to the International
Tables
has been prepared by L. S. Dent Glasser at the IUCr teaching pamphlets
web site.^{23}
Polar Space Groups
Some space groups do not contain symmetry operators that fix the origin of the unit cell in one or more directions. These space groups can be identified by the examining the symmetry-equivalent positions for the space group. For any of the three coordinate directions there are no symmetry operations that invert that coordinate (x -> -x), (y -> -y), or (z -> -z). For these polar space groups, the origin must be fixed by the atoms. This may be accomplished either by holding the coordinate(s) in the polar direction(s) fixed for one atom during refinement or by applying a constraint to hold some function, usually the mean value, of all coordinates in the given direction(s) fixed.^{24} If the polar axis is established by fixing the coordinate or coordinates of one atom, then selecting the heaviest atom in the structure for this purpose will lead to a more stable refinement.
Inversion of a Crystal Structure
Very few methods of solving a structure are able to properly pick the correct absolute configuration of a crystal structure. For these samples, the structure is roughly determined and refined. The configuration is then checked either by means of anomalous scattering or by knowing and checking the configuration of the compound itself. For example, naturally occuring peptides should have helices with a left-handed twist.
If the structure is determined to have the incorrect handedness, then the structure is simply inverted through some symmetry point such as (0, 0, 0) or (0.5, 0.5, 0.5) and the refinement is completed. The space group must also be changed if the structure crystallizes in one of the 11 enantiomorphous pairs of space groups. If the sample crystallizes in one of 7 special space group, then inversion must be through a special point.
The 230 space groups include 11 space groups that occur as enantiomorphous pairs: P3_{1}, P3_{2}; P3_{1}12, P3_{2}12; P3_{1}21, P3_{2}21; P4_{1}, P4_{3}; P4_{1}22, P4_{3}22; P4_{1}2_{1}2, P4_{3}2_{1}2; P6_{1}, P6_{5}; P6_{2}, P6_{4}; P6_{1}22, P6_{5}22; P6_{2}22, P6_{4}22; and P4_{1}32, P4_{3}32. When a sample crystallizes in one of these space groups, one of the pair is arbitrarily chosen for the structure solution and at least partial refinement. If it is later determined that the wrong space group of the pair was chosen initially, then the atoms are inverted and the alternate space group of the pair is used in further refinements.
For 7 other space groups inversion must be carried out through some point in the structure other than (0, 0, 0) or (0.5, 0.5, 0.5). This problem was first described in print by Parthe and Gelato^{25} and Bernardinelli and Flack^{26}. A table listing these seven space groups and appropriate inversion points is listed below.
Space Group | Inversion Point | SHELXTL MOVE* |
---|---|---|
Fdd2 | 0.125 0.125 0.500 | MOVE 0.25 0.25 1 -1 |
I4_{1} | 0.500 0.250 0.500 | MOVE 1 0.5 1 -1 |
I4_{1}22 | 0.500 0.250 0.125 | MOVE 1 0.5 0.25 -1 |
I4_{1}md | 0.500 0.250 0.500 | MOVE 1 0.5 1 -1 |
I4_{1}cd | 0.500 0.250 0.500 | MOVE 1 0.5 1 -1 |
I42d | 0.500 0.250 0.125 | MOVE 1 0.5 0.25 -1 |
F4_{1}32 | 0.125 0.125 0.125 | MOVE 0.25 0.25 0.25 -1 |
* The command in the SHELX refinement program to invert the atoms.
Special Positions
Many space groups contain simple symmetry operations, such as inversion centers, rotation axes, and mirror planes. Usually the locations of these symmetry elements are either fixed by convention, or fixed relative to other symmetry operations in the cell. These special positions are located on points (centers of symmetry), lines (rotations axes) or planes (mirror planes). Also, there are combinations of symmetry operations such as 2/m points or mm lines.
An atom at one of these locations will have fewer symmetry-related positions in the cell than would an atom in a general position. The International Tables, Vol. A^{14} lists special positions often called Wyckoff positions as well as the relative number of symmetry-related positions.
Because atoms on some of these special positions are a constant distance from their symmetry-related atoms, these special positions can give rise to systematically absent intensity data. These systematic absences can be derived in a similar manner as the systematic absence conditions for glide planes and screw axes derived above.
Note that these systematic absence conditions apply only to atoms that
sit on the respective special positions. This can be a problem when
the atom(s) on a special position are very heavy and the remaining atoms
are in general positions are much lighter and are, in general, evenly distributed
in the cell. Under these circumstances, the systematic absence due to the
atom on the special position could be confused as a general systematic
absence, obscuring the space group symmetry. Thus when a structure cannot
be solved, and other tricks
have been tried, it may be necessary
to lower the symmetry to try to solve the structure.
Space Group Names
Although the Hermann-Mauguin names for space groups will always be used in publications and in the cifs accompanying the publications, another set of names for space groups is also included in cif files. This alternate description of space groups was developed by Sid Hall^{27} to give a unique name to each space group and space group setting. However, all of the ambiguities in H-M space group names are resolved by looking at either the cell parameters or the symmetry operators.
References
- This is a brief description of Neumann's Principle. See http://reference.iucr.org/dictionary/Neumann's_principle.
- N. Steno, 1669, De Solido Intra Solidum Naturaliter Contento Dissertationis Prodromus.
- From Stephen A. Nelson in http://www.tulane.edu/~sanelson/eens211/introsymmetry.htm.
- J. P. Glusker, M. Lewis, and M. Rossi 1994,
Crystal Structure Analysis for Chemists and Biologists.
VCH Publishers:New York, 5-6 (and references therein). - René-Just Haüy, 1801, Traité de minéralogie.
- From the IUCr at http://reference.iucr.org/dictionary/Law_of_rational_indices.
- Hans Wondratschek, "Matrices, Mappings, and Crystallography" in International Union of Crystallographers Teaching Pamphlets available at: http://www.iucr.org/education/pamphlets.
- P. M. deWolff in
International Tables for Crystallography, Vol. A
, Section 9.3, Kluwer:Boston (1996) pp.741-748. - A. Bravais, 1849, J. de Math., 14, 137-180.
- From http://www.physics.ucla.edu/demoweb/demomanual/matter_and_thermodynamics/matter/fourteen_bravais_lattices.html.
- C. Hermann, 1928, Z. Krist., 68, 257-287, and Ch. Mauguin, 1931, Z. Krist., 76, 542-558.
- A. Schönflies,
Krystallsysteme und Krystalstruktur.
B. G. Teabuer: Leipzig (1891), and A. Schönflies,Theorie der Kristalstruktur
Gebrüder Boratrager: Berlin, (1923). - E. J. W. Whittaker, 1984,
The Stereographic Projection
in the International Union of Crystallography Teaching Pamphlets available at: http://www.iucr.org/education/pamphlets/11. - Th. Hahn in
International Tables for Crystallography, Vol. A.
, Section 1.3.2, Kluwer:Boston (1996) p. 6. - P. M. de Wolff, Y. Billiet, J. D. H. Donnay, W. Fischer, R. B. Galiulin, A. M. Glazer, Th. Hahn, M. Senechal, D. P. Shoemaker, H. Wondratschek, A. J. C. Wilson, & S. C. Abrahams, 1992, Acta Cryst., A48, 727-732.
- E. S. Federov, 1885
The Elements of the Study of Configurations.
Trans. of the St. Petersburg Min. Soc., Part 21. - H. Hilton,
Mathematical Crystallography and the Theory of Groups of Movements.
(1903) Oxford: Clarendon Press. - Th. Hahn in
International Tables for Crystallography, Vol. A.
, Kluwer: Boston (1996). - Th. Hahn in
International Tables for Crystallography, Vol. A.
, Kluwer: Boston (1996) Section 1.4, pp 7-10. - A. L. Spek, 2007, PLATON, A Multipurpose Crystallographic Tool, Utrecht University, Utrecht, The Netherlands.
- The checkcif program available at: http://checkcif.iucr.org/.
- A. J. C. Wilson, 1949, Acta Cryst., 2, 318-321, and A. J. C. Wilson, 1942, Nature, 150, 151-152.
- L. S. Dent Glasser, 1997,
Symmetry
in the International Union of Crystallography Teaching Pamphlets available at: http://www.iucr.org/education/pamphlets/11. - H. D. Flack and D. Schwarzenbach, 1988, Acta Cryst., A44, 499-506.
- E. Parthe and L. M. Gelato, 1984, Acta Cryst., A40, 169-183.
- G. Bernardinelli and H. D. Flack, 1985, Acta Cryst., A41, 500-511.
- S. R. Hall, 1981, Acta Cryst., A37, 517-525 with erratum 1981, Acta Cryst., A37, 921.