Symmetry in Crystallography Notes (2023)

Index

• Symmetry in Crystals
• Rotationssimetria
• stereographic projections
• Crystallographic point groups
• micro translations
• Symmetry of the reciprocal lattice
• systematic absences
• space groups
• Space group determination
• special positions
• references

OInternet siteit was translated into Romanian by Alexander Ovsov.

introduction

In crystallography, symmetry is used to characterize crystals, identify repeating parts of molecules, and simplify data collection and almost all calculations. In addition, the symmetry of the physical properties of a crystal, such as thermal conductivity and optical activity, must include the symmetry of the crystal.¹then athoroughKnowledge of symmetry is essential for a crystallographer. A clear and concise description of crystallographic symmetry has been prepared byRobert Von Drele.

An object is described assymmetricalabout aTransformationwhen, after transformation, the object appears to be in a state identical to its initial state. In crystallography, most types of symmetry can be described by an apparentMovementof the object as a kind of rotation or translation. Apparent motion is called symmetry.Operation. The locations where symmetry operations occur, such as an axis of rotation, a mirror plane, a center of inversion, or a translation vector are described aselements of symmetry.

There are two different methods for describing rotational symmetry operations. These two sets of descriptors are theHerman MauguinNomenclature¹¹it's atnice fliesNomenclature.¹²OKarl Hermann-Karl MauguinSystem is normally used to describe crystals and crystallographic symmetry. OArthur SchoenfliesThe convention is mainly used to describe symmetry in discrete molecules, in optical spectroscopy and in quantum mechanics. In these notes, the Hermann-Mauguin notation is listed first, followed by the corresponding Schönflies notation in parentheses.

Symmetry in Crystals

Our discussion of symmetry in crystallography must begin with a description of crystals. Crystals are defined as solids that have an atomic structure with long-range three-dimensional order. Unfortunately, this long-range order cannot be absolutely confirmed by any method other than some diffraction technique. However, there are several observations that can be made that strongly suggest that it is samplingcrystallinebefore performing a diffraction experiment.

Typically, crystals have flat faces and sharp edges. In addition, many crystals have one or more directions that can be cleaved cleanly. Specimens with a naturally rounded shape or specimens with a conchoidal fracture pattern are almost always designated asglassno significant long-range 3-D order. Likewise, materials that can easily be pierced with a probe and retain the deformed shape are gels or plastic materials and therefore do not have long-range 3-D order.

If you look at several crystals of the same material, you will quickly realize that although the crystals may be different sizes, all crystals have the same shape orhabit. In particular, the angles between certain pairs of faces on different crystals are the same. This observation was first made byNicolau Steno1669.²This observation became known asLaw of constancy of interface angles.

Steno and others in the 17th century became interested in the specific composition of crystals that would allow them to maintain the same angles between pairs of faces.⁴These scientists believed that crystals must be composed of a few regularly repeating components. Through these early studiesRené-Just Haüycould postulate that if calcite crystals and cubic garnets were constructed from many small blocks, those blocks could easily be used to describe the faces of these crystals in terms of rational indices.⁵OLaw of Rational Indexesforms the basis of optical crystallography.

unit cells

These regularly repeating blocks are now known asunit cells. The dimensions of a unit cell are described by the lengths of the three axes,A,B, EC, and the three interaxial angles α, β and γ. In most published articles, axial lengths are expressed in Å (Ångstroms) and interaxial angles in ° (degrees).

There are many ways to repeat blocks in any grid. The main principles that define the network are that each point in the network should be in an identical neighborhood to every other point in the network and that the individual blocks in the network should have the smallest possible volume. There are often many ways to choose the vectors between the grid points and even the positions of the grid points themselves. These unique lattice vectors are calledBasevectors orBasePhrase. Some two-dimensional examples of this grid selection are shown below.

When researchers discuss a given material, they should start from a standard or conventional description of the unit cell of that material. Therefore, crystallographers have chosen the following criteria for selecting unit cells. By convention, the unit cell boundaries are chosen to beright(A×Bis the direction ofC) to have thisgreater symmetry, and have thissmaller zellvolume. If other symmetry considerations do not take precedence, the cell is chosen as suchA≤B≤C, and α, β and γ all < 90° or all ≥ 90°. This type of cell is calledreducedCell. There are several other rules to preserve the conventionalreduced cellfor a specific material.⁷

Crystalline materials are separated into 7 different crystalline systems. These crystalline systems are more easily identified by the limitations of cellular parameters. Note, however, that cell parameter constraints are only necessary conditions. Thus, a given sample may have cellular parameters that appear to fall within a category within experimental error, but are actually of lesser symmetry.

The 7 crystal systems are listed in Table 1 below. In the system of lesser symmetry, triclinic, there are no restrictions regarding the values ​​of cellular parameters. In the other crystalline systems, symmetry reduces the number of unique lattice parameters, as shown in the table. Certain conventions were 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. By convention, this axis is selected asBorCAxis such that β or γ ≥ 90°. note thatC-single monoclinic cells are common in the French literature andB-single cells are common in most other languages. In tetragonal, trigonal, and hexagonal systems, one axis contains greater symmetry. By convention, this axis is selected asCAxle.

Each of the seven crystalline systems describes different ways in which simple three-dimensional networks can be constructed. As with all lattice systems, crystalline lattices are considered to be availablegrid pointsat the corners of the unit cell. The grid points are chosen so that the local neighborhood around any particular grid point is identical to the neighborhood around any other grid point.

Some trigonal lattices can be expressed in terms of a hexagonal or rhombohedral lattice. These grids are shown in the drawings below. Note that the rhombohedral lattice vectors below can be expressed in an inverted orientation a) or an inverted orientation b).

Bravais Gitter

It is sometimes possible to create a grid withgreater symmetryif the lattice vectors are chosen such that one or more lattice points also lie in the middle of a lattice face or within the unit cell. Those networks with additional network points are described ascenteredgrid. Grids with grid points only at the corners are calledPrimitiveand are marked with the symbolP. Observe thatreducedthe cells described above are always primitive. 1849,August Bravaisfound that all regular crystals can be described with only 14 lattice types for the 7 crystal systems.⁸Remember that the neighborhood around any grid point is exactly the same as the neighborhood around any other grid point.

Non-primitive networks may have one, two or three additional network points per unit cell. Grid with an additional grid point such asA-,B-, orCCentered cells have an additional lattice point located in the center of a crystal face. The additional grid points inA-Centered cells appear in theb-cfaces. Likewise, additional grid points point to it.B-Centered cells appear in thea-cFaces and grid points appear on thea-bfaces ofCcentered cells. body centered grid,EU-centered, cells have an extra grid point in the center of the cell. Rhombohedral cells based on a hexagonal lattice conventionally have lattice points at (2/3, 1/3, 1/3) and (1/3, 2/3, 2/3). centered face,FCentered cells have grid points on all faces.

In general, the greatest metric symmetry is identified by computer programs. However, these centered grids can sometimes be identified simply by looking at cell parameters. If two cell lengths of a reduced lattice are approximately equal, and the two corresponding cell angles are also approximately equal, then the cell is likely to be centered.

The table below lists the 14 Bravais grille types. Bravais symbols are a combination of the crystal system and the lattice designation. Triclinic types begin with the letterAthis meansanortiscoof the mineral anorthite, a mineral with triclinic symmetry. The other types of lattices generally start with the initial of the crystalline system.

† OSThe monoclinic network icon represents a network withA,C, orEUcentralization (B-exclusive) orA,B, orEUcentralization (C-exclusive).
* OSThe orthorhombic lattice symbol represents one of the three types of side-centered lattice,A,B, orC.
# FromPtrigonal network and aPhexagonal lattices are identical in appearance, these two systems are considered to form only one type of Bravais lattice.

Rotationssimetria

There are two basic types of rotational symmetry operations.CorrectRotations move an object, but do not change the object's handedness.improperRotations include a proper rotation plus a component that reverses the object's handedness.

correct curves

AN-to bend (C_N) The proper rotation process is a counterclockwise movement of (360/n)° around an axis passing through the object.NThe double rotation process is repeatedNtimes, then the object returns to its original position. Crystals with a periodic lattice can only have axes with 1, 2, 3, 4, and 6 times axes of symmetry. The following description of rotational symmetry operations is similar to Prof. Stephen Nelson.³In the drawings below, the axis of symmetry extends perpendicularly from the page.

• Rotation of 1 Facha.A 1 (EThe double rotation operation implies either a 0° rotation or a 360° rotation and is denoted as an arrayidentityOperation.

  

• 2 face rotation.A double (C2) the rotate operation moves the object by (360/2)°=180°. The symbol used to denote a double shaft is a filled oval.

  

• giro 3x.A triple (C3) the rotate operation moves the object by (360/3)°=120°. The symbol used to designate a triple axis is a solid equilateral triangle.

  

• 4x rotation.4 times (C4) rotation operation moves the object by (360/4)°=90°. The symbol used to designate a 4x axis is a filled square.

  

• 6x rotation.A 6x (C6) rotation operation moves the object by (360/6)°=60°. The symbol used to designate a 6-fold axis is a filled hexagon.

  

Improper Curves

Improper rotation can be seen as occurring in two parts, first a proper rotation is performed, followed by an inversion through a specific point on the axis of rotation. In H-M nomenclature, false rotations are sometimes called roto-inversions. In Schönflies' scheme, improper rotations are rotational reflection axes, as they are a rotation followed by a reflection perpendicular to the axis of rotation. Wrong turns are marked with the symbolN, WoNrepresents the correct rotation component type of the feature. Just like real rotation operations1(EU=S₂),M=2(σ =S₁),3(S₆),4(S₄), E6(S₃) wrong rotations are often observed in crystals. These axes are pronounced as3 barin the United States andmeasure 3in many European countries. Therefore3in H-M correspondsS₆em Schönflies.

Note that it is not necessary for the rotate feature or flip center to exist as a group feature for the wrong rotation axis to exist, e.g. O4(S₄) the feature does not contain a 4x rotation axis (C₄) another center of inversion.

• 3Roto-Inversion.This operation involves a (360/3)° rotation followed by a flip at the center of the object. The symbol is a filled triangle with an open circle in the middle. This is the only improper rotation that also contains the actual rotation axis and an inversion center.

• 4Roto-Inversion.This operation involves a (360/4)° rotation followed by a rotation at the center of the object. The symbol is a 4-sided open diamond with a filled oval in the middle.

• 6Roto-Inversion.This operation involves a (360/6)° rotation followed by a flip at the center of the object. The symbol is an open hexagon with a filled triangle inside.

A website that illustrates clusters of points based on molecular species is available at:http://symmetry.otterbein.edu/gallery/. Please note that this website uses JMOL software, therefore Java must be enabled in your browser.

stereographic projections

Drawing the three-dimensional symmetry operations on a two-dimensional surface like this page was a difficult problem. One way to solve this problem is to use astereographic projection. These figures are also effective for describing the angular relationships between the faces of a crystal.

To create a stereographic projection, imagine that the object with a certain symmetry or surface area is at the center of a sphere. Imagine that the sphere has a polar axis divided by an equatorial plane. Project features of interest onto the object from the center to the surface of the sphere. Then project the points on the surface of the sphere across the equatorial plane to the point where the polar axis intersects the sphere on 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, its projection on the equatorial plane is represented as aMore.Points emanating from the southern hemisphere are marked with aKreis.Points created by incorrect symmetry operations are also sometimes marked with a comma to indicate opposite handedness.

The unit cell axis with the greatest symmetry is generally chosen as the polar axis. Rotation axes that are not on 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 the projection and end in arrows. Mirrored planes are drawn as thick lines. Inversion centers are drawn as open circles in the middle of the polar axis.

E.J.W. Whittaker prepared a fuller discussion of thisstereographic projections.

Crystallographic point groups

Symmetry operations can be combined to create other symmetry operations. When writing these operations mathematically, the operations are applied from right to left. Therefore, in the following expression, object x is processed first by C and then by B. The surgery icon*it just indicates that a transformation is being applied to the object.

A * x = B * C * x

These operations can therefore be combined to form agroupof symmetry operations. These groups of operations are calledpoint groupsbecause all symmetry elements of these operations pass through a single point of the object. When studying groups, mathematicians have discovered that such groups always have the following properties.

1. IdentityThere must be a group operation that, when performed before or after another group operation, produces the same transformation as the other group operation. The identity operation is sometimes called the "do nothing" operation. (A*E=E*A=A, E is the identity operation)

2. Vice-versaFor each operation in the group there must be a second operation which, combined with the first, produces the identity operation. The reverse 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. AssociativityThe order in which the operations are combined does not matter. [A*(B*C) = (A*B)*C]

In addition to these properties, all crystallographic symmetry groups have the next property.

4. ClosingIf any two operations in the group are combined, the resulting operation must be a member of the group. (A = B * C, A, B, C are group operations)

Finally, some groups also have the property ofcommutativity. The order of operations on commutative groups does not matter. (A*B=B*A)

When combined according to the group rules, the correct and incorrect rotation operations described above produce a total of 32 unique crystallographic point groups. These groups are listed in the table below. Centrosymmetrical point groups are shown in bold.

* The symbolmm2 also represents the 2 point groupsmmEM2M.
†These groups of points represent sets of groups, for example, 32 means 321 and 312

By convention, the following rules have been adopted for describing point groups. When an axis of rotation is followed by a bar and aM,so this mirror is perpendicular to the axis of rotation. In orthorhombic systems, the three signs describe symmetry along the three axes,A,B, EC, respectively For cells of the tetragonal, trigonal and hexagonal type, theCThe axis is single and the first symbol in the point group shows symmetry along the single axis. In tetragonal systems, the second symbol shows symmetry along the [100] and [010] directions, and the third symbol shows symmetry along the [110] and [1] directions.10] directions. In trigonal and hexagonal cells, the second symbol shows symmetry along [100], [010] and [110], and the third symbol shows symmetry along [210], [120] and [120]. In rhombohedral systems on rhombohedral axes, the first symbol shows symmetry along [111] and the second symbol shows symmetry along [110], [011], E [101]. Cubic symbols show [100], [010], [001] in the first symbol, [111], [111], [111], [111] on the second symbol and [110], [110], [011], [011], [101] and [101] on the third symbol.

The following examples illustrate how stereographic projections can help you understand point groups.

In addition to the points in the previous figure, this projection shows a circle in the lower left corner with coordinates (-x, -y, -z), which is a point created 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 perpendicular to the 2-fold. The relative coordinates of a point with respect to the mirror inBare (x, -y, z). Creating a mirror by adding a flip center to a dual axis is an example of the group closure property. The mirror is indicated by the strong outer circle of the projection.

The other groups of points and their symmetry-related coordinates can be derived in a similar way as shown above. All 32 crystallographic point groups are shown in the stereographic projections below.

micro translations

Rotational symmetry operations can be combined with translations of part of the unit cell, giving rise to entirely new symmetry operations. Proper rotations in combination with translations result in operations described asscrewAxes. Mirror levels that are combined with translations are createdSlideaircraft operations.

The symbol for a screw shaft is n_Mwhere n indicates the type of rotation and translation (m/n) of the unit cell. to 2₁The operation is a rotation followed by a translation of 1/2 of the cell parallel to the axis of rotation. There are a total of 11 types of screw shafts, 2₁, 2₁, 3₁, 3₂, 4₁, 4₂, 4₃, 6₁, 6₂, 6₃, 6₄, e 6₅. Some pairs of screw axes rotate objects in opposite directions. The opposite pairs are the 3₁e 3₂, die 4₁e 4₃, die 6₁e 6₅, and the 6₂e 6₄.

Floating operations occur when a translation follows a mirroring operation. Translation directions are parallel to a unit cell direction or parallel to a combination of cell directions. Slip planes are described by the direction of translation. The direction of the mirror plane is given explicitly or is implied, for example, in the space group symbol. Slip plane operations exist in all three directions and in pairs of directions. Gliders that move in the middle of the cell in two different directions are summonedNslide. An object undergoes aNThe fluctuation reflected in the plane (001) would be translated by (A+B)/2.

There is an additional type of glider, the diamond glide,D. Occurs only in groups of cell spaces centered on the face or body and is characterized by a translation of (±A±B)/4, (±B±C)/4, (±C±A)/4 or similar translations. As the denominator says, 4 in a rowDSlips are required to return an object to a gridded version of itself.

In a more recent version ofInternational Tables of Crystallography, Vol. A, to diegoodcalled a landslidee-Slip is described.¹⁴OeSlip occurs only in centered cells and is defined as two separate slips reflecting in parallel planes and shifting one of the other two cell directions. This type of landslide was suggested by the thirdnaming reportthe IUCr.¹⁵

Symmetry of the reciprocal lattice

The actual cell parameters are determined by the relative positions of the reciprocal grid points. In fact, the reciprocal parameters of the cell are determined during a process known asindexingthe diffraction pattern. The real cell parameters are then calculated from the reciprocal cell parameters according to the relationships below.

A=B*×C*= (B* C*sen α*) / V*

B=C*×A*= (C* A*sen b*) / V*

C=A*×B*= (A* B*sen c*) / V*

V* = 1/V =A*B*C*(1-cos²a*-cos²β*-cos²γ* + 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* sen α*)

cos γ = (cos α* cos β* - cos γ) / (sin α* sin β*)

Laue's fleet

The relative intensities in a diffraction pattern depend on the electron density distribution of the sample. This electron density distribution must follow the symmetry of the crystal itself. This symmetry is calledhall of Laue. The Laue class for a sample is described as one of 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.

lei by Friedel

The additional center of symmetry is at least approximately due to the fact that diffraction or interference effects are inherently centrosymmetric. The intensity of (h k l) point in the reciprocal lattice arises from the dispersion of electron density parallel to (h k l) planes in the crystal. Likewise, the intensity of (h k l)-point results from the density of electrons in planes parallel to (h k l) planes in the crystal. But since (h k l) planes and the (h k l) planes are simply directed in opposite directions, so the intensities of (h k l) E (h k l) The points in the reciprocal lattice must be at least approximately equal. This relationship of equality between the intensities of (h k l) E (h k l) is calledlei by Friedel.

For intensity data of a chiral compound, Friedel's law can be broken by anomalous scattering of heavy atoms. In these data sets, the reciprocal lattice has the same symmetry as the symmetry of the crystal point group. Thus, if the symmetry of the crystal's point group is shown as 222, then the intensities would have symmetry 222. The anomalous scattering of heavy atoms is not a strong effect, so the intensities will still show approximately Friedel's law.

Equivalent Intensities of Symmetry

A simple way to determine which data should have equivalent intensities due to symmetry is to look at the stereographic projection of the sample point group. From a simple study of the stereographic projection, the coordinates related to the symmetry (x,y,z) can be found. Just convert the x coordinates toH, the y coordinates took, and the z coordinates foreuValues. Then, the intensities equivalent to the symmetry are determined.

A monoclinic crystal has Laue symmetry of 2/M. The equivalent coordinates assuming aB-Unique line, specified as (x, y, z), (-x, y, -z), (-x, -y, -z) and (x, -y, z). Assim, as intensities do (h k l), (Hkeu), (H k eu), E (Hkeu) The grid points must have equivalent values. Note that this also means that the intensities of (h keu), (Hk l), (H keu), E (Hk eu) must also be equivalent, but are not necessarily equivalent to (h k l), etc

If a monoclinic compound is chiral, one would expect the intensities to have only 2 points of group symmetry. So, the intensities of (h k l) the data are equivalent to the intensities of (Hkeu) Data. Similar,EU(h keu) =EU(Hk l);EU(Hkeu) =EU(H k eu); EEU(Hk eu) =EU(H keu).

If a crystal has all three cell angles = 90.0° within experimental error, most researchers would suspect that the sample has orthorhombic symmetry. In most cases, this assumption would be correct, but not in all cases. ifEU(h k l) =EU(Hkeu) EEU(Hk eu) =EU(H keu), ButEU(h k l) ≠EU(Hk eu), then the sample has monoclinic rather than orthorhombic Laue symmetry. The symmetry of the Laue class is determined by the symmetry of thereciprocal grid intensitiesnot the apparent symmetry of the cell parameters. The cell parameters only determine whichpossiblegreater symmetry of the sample.

systematic absences

Some symmetry operations can be easily identified by specific information on the diffraction pattern intensities. In particular, cell centering operations, screw axes and sliding planes can be recognized by the fact that they result in systematically missing certain groups of inflection points. All these symmetry operations involve amicro translation.

Consider a data set withCFloating reflected in the plane normal toBAxle. Symmetry operations can 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 following summaries are above thisJAtoms and range from 1 to N/2.

F(hkl) = ∑F_Jexp 2πEU(hx_J+ky_J+lz_J) +
∑F_Jexp 2πEU[hx_J-ky_J+eu(1/2 +z_J)]

Take a look at the datak= 0. For this data, the structure factors become:

F(H0eu) = ∑F_Jexp 2πEU(hx_J+lz_J) +
∑F_Jexp 2πEU(hx_J+lz_J) exp2πEU(eu/2)

F(H0eu) = ∑F_Jexp 2πEU(hx_J+lz_J) [1 + exp peu]

Seeuis an odd integer, so exp πeu= -1 eF(H0eu) = 0. Seeuis then an even integerF(H0eu) is probably not 0 (but it could be accidental). The systematic absence conditions for other symmetry operations can be derived similarly to the previous one.

A table of these absence conditions is shown below. The reflection condition indicates data that may be present, other data with the same conditions would be systematically absent. The characterNcan be any integer. So the condition forhkl,k+eu= 2n+1 indicates this for the general class of peakshklthat aboutk+eumust be positive for these peaks to have measurable intensity, and for these peaks to havek+euNegative must not have any measurable intensity when theACentering symmetry is present in the symmetry of the data set.

It is best to always check for systematic absences and therefore translational symmetry operations in the following order: cell centering operations, then slip planes, and finally screw axes. This order is important because higher symmetry operations such as cell centering can mask lower symmetry operations such as slip planes and screw axes. Also, this sequence of conditions with the highest possible number of absences works for the conditions with the fewest possible absences for a given sample.

Note that if a crystal satisfies the systematic absence condition forCcell centering, then the fact that for 0Kl,k= 2n does not necessarily mean oneBglide reflectedA. This condition does not provide new information as it is part of the cell centering condition.

space groups

When the 7 crystalline systems are combined with the 14 Bravais lattices, the 32 point groups, screw axes and glide planes, Arthur Schönflies¹², Evgraph S. Federow¹⁶e H. Hilton¹⁷could describe the 230 unique space groups. A space group is a set of symmetry operations that combine to describe the symmetry of a region of three-dimensional space, the unit cell. In point groups, all symmetry elements pass through a point on the object. In space groups, symmetry elements need not intersect at a single point, although some operations can intersect at different points in the cell.

Chiral compounds produced as a single enantiomer can only crystallize in a subset of the 65 space groups that do sonohave improper rotations, such as centers of symmetry, mirror planes, slips, or3,4, or6Axes.

A space group is denoted by a capital letter identifying the type of network (P,A,F, etc.) followed by the point group symbol, in which the rotation and reflection elements are extended by screw axes and glide planes. Note that the symmetry of the point group for a given space group can be determined by removing the centering symbol from the space group cell and replacing all screw axes with similar rotation axes and replacing all glide planes with planes of rotation. mirror. The point group symmetry for a space group describes the true symmetry of its reciprocal lattice.

OBilbao Crystallographic Serverprovides 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 thisspace group patternwhich illustrates the various combinations of point group symmetry elements in space groups. The table below lists the unique space groups separated by crystal system and Laue class.

The space groups inclearlyare centrosymmetric.

The previous table lists the mathematically distinct groups of spaces. In addition, there are many groups of non-standard rooms, some of which are listed in theInternational Tables of Crystallography, Volume A.¹⁸For example, the space groupsP2₁/AEP2₁/Nare variants of the space groupP2₁/Cthat can often be found in the literature. Some non-standard space groups are described in an article to facilitate the comparison of cellular parameters of several related compounds. For example, when the main compound of a work is in the monoclinic space groupC2/C, but a related compound is triclinic, some authors would determine the structure of this second compound in the non-standard space groupC1if the cell parameters inC1were similar to the cellular parameters of the lead compound. Finally, some authors would publish structures in non-standard space groups because the basis vectors of related cells would produce cell angles closer to 90°. For example, structures are often published inPninstead ofStkor inEU2/Ainstead ofC2/Cbecause angle β would be much closer to 90°. When the β angles are closer to 90°, the refinement of the x and z coordinates of the atoms show much less correlation and they converge to form chemically reasonable structures more quickly.

The following tables list graphical and typed symbols used to describe symmetry operations inInternational Tables of Crystallography, Volume A.¹⁹Note that the ⊥ symbol is used to indicate that the feature is perpendicular (normal) to the page.

Table 6. Symmetry planes perpendicular to the projection plane

Symmetrieebene	graphic icon	translation	Symbol
Reflexionsebene		none	M
slip plane		1/2 along the line	A,B, orC
slip plane		1/2 normal to the plane	A,B, orC
double glider		1/2 along the line & 1/2 normal to the plane	e
diagonal slide plane		1/2 along the line & 1/2 normal to the plane	N
diamond glider		1/4 along the line & 1/4 normal to the plane	D

Table 7. Symmetry planes parallel to the projection plane

Symmetrieebene	graphic icon	translation	Symbol
Reflexionsebene		none	M
slip plane		1/2 along the arrow	A,B, orC
double glider		1/2 along one of the arrows	e
diagonal slide plane		1/2 along the arrow	N
diamond glider		1/8 or 3/8 along the arrows	D

Table 8. Axes of symmetry

element of symmetry	graphic icon	translation	Symbol
identity	none	none	1
2 times ⊥ side		none	2
2 folds on the side		none	2
2 sub 1 ⊥ side		1/2	21
2 sub 1 side		1/2	21
3-small		none	3
3 part 1		1/3	31
3 below 2		2/3	32
4-small		none	4
4 under 1		1/4	41
4 under 2		1/2	42
4 under 3		3/4	43
6-small		none	6
6 part 1		1/6	61
6 part 2		1/3	62
6 under 3		1/2	63
6 under 4		2/3	64
6 under 5		5/6	65
reversal		none	1
3 bar		none	3
4bar		none	4
6 bar		none	6= 3/M
2 times and reversal		none	2/M
2 sub 1 and reversal		none	21/M
4 times and inversion		none	4/M
4 sub 2 and inversion		none	42/M
6 times and inversion		none	6/M
6 sub 3 and reversal		none	63/M

Determining the space group of a material

Identifying the correct space group for a given sample and its diffraction pattern is usually straightforward. First, the Laue class is determined. Then the possible systematic absences are determined in order to identify the appropriate cell centering conditions, slip planes and screw axes, if any. Based on the Laue class and the symmetry operations identified by systematic absences, the choice of space group(s) is usually restricted to one or a few.

If the space group is not uniquely determined, the solution and structure refinement steps are attempted with the various possible space groups until the structure determination is complete. First try the space groups that start with the greatest symmetry. If the space group chosen is not the space group with the highest symmetry, check the structure for additional symmetry elements using the PLATON program or the checkcif program.^20,21

Additional connection information is often used to narrow down the space group choice to a single space group. 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 computing the fusionRunof the equivalent symmetry data for the given Laue class. a reasonableRunto identify the Laue class, a value would be &0.06. Larger values ​​generally indicate that the sample has crystallized into a less symmetric Laue group.

Data of equivalent symmetry strength are merged using the following relationship

F²= ∑ω_J F_J²/ ∑ω_J

where the sums are over the equivalent symmetry data set. SometimesTo escapeare removed from the merged set or given lower weights to reduce their impact on subsequent refinement.

Run= ∑ [ ∑ |F_J²- <F²>| ] / ∑ [ (∑F_J²)/N]

where the inner sums are over reflections equivalent to the symmetry and the outer sums are over the only reflections.hklData. The term n is the number of equivalent dates for a given valuehklbe merged.

Tests for a center of symmetry

If the compound is not a single enantiomorph, statistical tests for a center of symmetry can be performed.

Probability distributions for centric and acentric unit cells were derived²²and 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/ (piS)^1/2] exp(-|F|²/ 2S) (centered)

P₁(|F|) = (2 |F| /S) exp(-|F|²/S) (acentric)

Wo

S= ∑F_J²

Sis a function of the scattering factors for the atoms and therefore decreases with increasing sin (EU/eu). To further simplify this functional dependency on distributionsEU, most statistical tests use normalized structure factors,E_hkl.

E_hkl²=F_hkl²/ (e∑F_J²)

WoF_J=F_J^Öexp(-B sen²EU/eu₂) is the scattering factor for the jth atom andeis an integer, 1 or greater, which corrects for some classes of reflections having mean values ​​less than ∑F_J²by a whole amount.

From <|E_H|²> = 1 =S, the distributions become

P_-1(|E|) = [(2 /pi)^1/2] exp(-|E|²/ 2) (centered)

P₁(|E|) = 2 |E| exp(-|E|²) (acentric)

These two distributions are completely independent of the variation of θ and differ significantly from each other. Differences in the two distributions indicate that centric unit cells should have more reflections with very strong and very weak values ​​relative to acentric unit cells. Acentric unit cells typically have a more uniform data distribution with significantly less faint data than centric cells. Comparing the distribution of a set of measured data with the two theoretical distributions can determine the presence or absence of a center of symmetry in the crystal in question.

Means for a variety of functions |E| can be estimated from the two theoretical distributions. These values ​​are shown in the table below. Most computer programs that test centric unit cells average these functions fromEand compare these experimental values ​​with the theoretical values.

There are three caveats to note when using these tests. First, the intensity data must be relatively strong. Weak records, withF²/P< 6.0, has too weak data to provide 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 an unusual distribution of atoms causes the experimental values ​​to adopt a centric distribution. If the tests indicate an acentric unit cell, then the tests are probably correct. If the tests suggest a centric unit cell, the tests may be correct or simply indicate a skewed electron density distribution in the unit cell. Finally, if the tests indicate a hypercentric distribution, e.g. <|E²-1|> < 0.6, then the sample is likely to beswells.

The definitive discussion of space groups and space group symmetry is available atInternational Tables of Crystallography, Volume A.¹⁸A good introduction to symmetry andInternational Tableswas prepared by L. S. Dent Glasser on the IUCr Teaching Brochures website.²³

polar space groups

Some space groups do not contain symmetry operators that fix the unit cell origin in one or more directions. These space groups can be identified by examining the equivalent symmetry positions for the space group. There are no symmetry operations for any of the three coordinate directions that invert this coordinate (x -> -x), (y -> -y), or (z -> -z). For thisPolar-space groups, the origin must be determined by the atoms. This can be achieved by fixing the coordinate(s) in the polar direction(s) during refinement of an atom, or by applying a constraint to find a function, usually the mean, of all coordinates in the atom. given direction(s) to maintain. Firmly.²⁴When the polar axis is fixed by specifying the coordinate or coordinates of an atom, choosing the heaviest atom in the structure for this purpose results in a more robust refinement.

Inversion of a crystal structure

Very few methods of solving a structure are able to correctly choose the right path.absolute settinga crystalline structure. For these samples, the structure is approximately determined and refined. The configuration is then verified by anomalous scattering or by knowing and verifying the configuration of the compound itself. For example, naturally occurring peptides should have most helices with a right-handed twist. There may be a few short left-twisting spirals, but almost all of them should be right-twisting.

If the structure is found to have the wrong hand, it is simply inverted by a point of symmetry such as (0, 0, 0) or (0.5, 0.5, 0.5) and the refinement is complete. The space group must also be switched if the structure crystallizes into one of the 11 pairs of enantiomorphic space groups. If the sample crystallizes into one of the 7 special space groups, the inversion must take place through a special point.

The 230 space groups include 11 space groups that occur as enantiomorphic pairs:P3₁,P3₂;P3₁12,P3₂12;P3₁21,P3₂21;P4₁,P4₃;P4₁22,P4₃22;P4₁2₁2,P4₃2₁2;P6₁,P6₅;P6₂,P6₄;P6₁22,P6₅22;P6₂22,P6₄22; EP4₁32,P4₃32. When a sample crystallizes into one of these space groups, one of the two is arbitrarily chosen for the solution of the structure and at least for partial refinement. If it is later discovered that the wrong space group of the pair was originally chosen, the atoms are reversed and the alternate space group of the pair is used in further refinements.

For the other 7 groups of spaces, the inversion must be performed by some point of the structure other than (0, 0, 0) or (0.5, 0.5, 0.5). This problem was first described in print by Parthe and Gelato²⁵and Bernardinelli and Flack²⁶. A table listing these seven space groups and their corresponding inversion points is provided below.

* The command in the SHELX refiner to invert the atoms.

special positions

Many space groups contain simple symmetry operations such as inversion centers, rotation axes, and mirror planes. Typically, the positions 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 (symmetry centers), straight lines (rotation axes) or planes (mirror planes). There are also combinations of symmetry operations, such as 2/m points or mm lines.

An atom in one of these positions has fewer symmetry-related positions in the cell than an atom in a general position. OInternational Tables, Vol. A¹⁴lists the special positions, often called Wyckoff positions, and the relative number of positions related to symmetry.

Because the atoms in some of these special positions are at a constant distance from their symmetry-related atoms, these special positions can result in systematically missing intensity data. These systematic absences can be derived similarly to the systematic absence conditions for slip planes and screw axes derived above.

Note that these systematic absence conditions apply only to atoms situated in the respective special positions. This can be a problem when the atoms in a given position are very heavy and the remaining atoms in general positions are much lighter and generally evenly distributed in the cell. Under these circumstances, the systematic absence due to the atom in the special position can be confused with a general systematic absence that obscures the symmetry of the space group. So when one structure cannot be resolved, and othertricksattempted, it may be necessary to decrease the symmetry to try to resolve the structure.

space group names

Although Hermann-Mauguin space group names are always used in publications and in the cifs that accompany publications, a different set of space group names are also included in cif files. This alternative description of space groups was developed by Sid Hall²⁷to give each room group and room group configuration a unique name. However, all ambiguities in H-M space group names are resolved by considering cell parameters or symmetry operators.

references

1. This is a brief descriptionNeumann's principle. Verhttp://reference.iucr.org/dictionary/Neumann's_principle.
2. N. Steno,1669,About the Solid Within the Naturally Solid Dissertation Content Pródromo.
3. By Stephen A. Nelson emhttp://www.tulane.edu/~sanelson/eens211/introsymmetry.htm.
4. J.P. Glusker, M. Lewis e M. Rossi1994,Crystal structure analysis for chemists and biologists.VCH Publishers: New York, 5-6 (and references).
5. René-Just Haüy,1801,Treaty of Mineralogy.
6. From the IUCr inhttp://reference.iucr.org/dictionary/Law_of_rational_indices.
7. Hans Wondratschek, "Matrices, Mappings, and Crystallography" em International Union of Crystallographers Teaching Panfletos, disponível em:http://www.iucr.org/education/pamphlets.
8. P.M. deWolff emInternational Tables of Crystallography, Vol. A, Section 9.3, Kluwer: Boston (1996) pp. 741-748.
9. A. Bravais,1849,J. de Math.,14, 137-180.
10. Outside ofhttp://www.physics.ucla.edu/demoweb/demomanual/matter_and_thermodynamics/matter/fourteen_bravais_lattices.html.
11. C. Hermann,1928,Z. Krist.,68, 257-287, and ch. Mauguin,1931,Z. Krist.,76, 542-558.
12. A. Schoenflies,Crystalline systems and crystalline structure.B. G. Teabuer: Leipzig (1891), e A. Schönflies,Crystal structure theoryBoratrager Brothers: Berlim, (1923).
13. E. J. W. Whittaker,1984,The stereographic projectionin the teaching brochures of the International Union of Crystallography, available at:http://www.iucr.org/education/pamphlets/11.
14. Th.Hahn aInternational Tables of Crystallography, Vol. A, Section 1.3.2, Kluwer:Boston (1996) p. 6.
15. 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,Crystal Act., A48, 727-732.
16. E.S. Federov,1885 The elements of the study of configurations. Trans. do St. Petersburg Min. Soc., part 21.
17. H. Hilton,Mathematical crystallography and the theory of motion groups.(1903) Oxford: Clarendon Press.
18. Th.Hahn aInternational Tables of Crystallography, Vol. A, Klüwer: Boston (1996).
19. Th.Hahn aInternational Tables of Crystallography, Vol. A, Kluwer: Boston (1996) Section 1.4, pp. 7-10.
20. AL Spek,2007, PLATON, a multipurpose crystallographic tool, Utrecht University, Utrecht, The Netherlands.
21. The checkcif program is available at:http://checkcif.iucr.org/.
22. A. J. C. Wilson,1949,Crystal Act.,2, 318-321, e A.J.C. Wilson,1942,Nature,150, 151-152.
23. L. S. Dent Glasser,1997,symmetryin the teaching brochures of the International Union of Crystallography, available at:http://www.iucr.org/education/pamphlets/11.
24. H. D. Flack e D. Schwarzenbach,1988,Crystal Act., A44, 499-506.
25. E. Parthe and L.M. Ice-cream,1984,Crystal Act., A40, 169-183.
26. G. Bernardinelli e H. D. Flack,1985,Crystal Act., A41, 500-511.
27. Salon S.R.,1981,Crystal Act., A37, 517-525 with errata1981,Crystal Act., A37, 921.