Blake A.J.(ed.) Crystal Structure Analysis

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12 Introduction to symmetry and diffraction

the basic building block of the whole structure, which can be regarded

as being assembled by placing identical copies of the unit cell together

to ﬁll space. Each unit cell contains the equivalent of one lattice point

(there are lattice points at all eight corners, but each is shared by the

eight unit cells that meet there). The three basic vectors are the three

different edges of the parallelepiped, and are also called the unit cell

edges. Their three lengths and the three angles are often referred to as

either unit cell parameters or lattice parameters; these two terms are

interchangeable and equivalent.

For any given lattice, many different choices of unit cell are possi-

ble, but there is always at least one for which the cell edges are the

three shortest non-coplanar vectors of the lattice, and this is preferred

by convention; it is called the reduced cell. (Actually, there are differ-

ent deﬁnitions of the term ‘reduced cell’, of which this is a particular

one that is probably most widely used and most clearly deﬁned; its full

name is the Niggli reduced cell. For completeness of the deﬁnition, the

base vector directions are chosen to make all three cell angles <90

◦

all three ≥90

◦

, and the axes are ordered in length, a ≤ b ≤ c.)

In the absence of any rotation or reﬂection symmetry in the crystal

structure, the three unit cell axes are normally of different lengths and

the three angles differ from each other and from special values such as

◦

, although close approximation to equality or to such values may

fortuitously be found sometimes.

It is the translation symmetry of crystalline materials that gives rise to

X-ray diffraction. Any regularly spaced arrangement of objects can act

as a diffraction grating for waves having a wavelength comparable to

the repeat distance(s) between the identical objects. Thus, regularly and

closely spaced parallel lines give a one-dimensional diffraction grating

for infra-red and visible light in spectrometers, enabling the separa-

tion of different wavelengths, and a diffraction effect can be observed

whenmonochromaticlight fromasodium streetlamp is viewed through

a ﬁnely woven material such as an umbrella. Unit cell dimensions in

crystals are comparable to the wavelengths of X-rays (and of electrons

and neutrons moving at appropriate velocities), so crystals act as three-

dimensional diffraction gratings. The basic mathematical relationships

are given in the introductory chapter (Chapter 1) and in the chapter on

data-collection theory (Chapter 5).

2.3 Symmetry of individual molecules, with

relevance to crystalline solids

In applying point group symmetry to molecules, chemists learn about

two basic types of symmetry operations and symmetry elements, and

these include some particularly common examples that are treated spe-

cially. Crystallographers use the same symmetry operations (they are

fundamentalpropertiesof nature!),but detailed deﬁnitions and notation

are different; this is unfortunate but is for good reasons.

2.3 Symmetry of individual molecules, with relevance to crystalline solids 13

A symmetry element is a physically identiﬁable point, line, or plane

in a molecule (or any other individual object) about which symmetry

operations are applied; a symmetry operation is an inversion through

such a point, a rotation about a line, or a reﬂection in a plane, that leaves

the molecule afterwards with an identical appearance. Each symmetry

element thus provides a number of possible symmetry operations (for

example, a rotation axis gives rise to rotations by any multiple of its

basic minimum angle: rotations of 120

◦

, 240

◦

and 360

◦

are all valid sym-

metry operations associated with a three-fold rotation axis; the last of

these is equivalent to the identity operation, i.e. not doing anything at

all, and we have just seen a similar situation for translation symmetry,

where any lattice translation can be regarded as a combination of multi-

ples of the three fundamental lattice translations for a particular crystal

structure).

For individual molecules, all symmetry operations can be classiﬁed

as one of two types: proper rotations (rotations by a certain fraction of

360

◦

about a rotation axis), and improper rotations (the combination

of a rotation and a simultaneous reﬂection in a plane perpendicular

to the axis and passing through the centre of the molecule); this is the

deﬁnition (known as the Schoenﬂies convention) used by chemists for

spectroscopyand bonding applications. Because of the particular impor-

tance of the inversion centre in crystallographic symmetry (see later), we

use a different convention, called the Hermann–Maugin or international

convention, where an improper (or inversion) axis is the combination of

a rotation and a simultaneous inversion through a point at the cen-

tre of the molecule. For the operations that can occur in crystalline

solids, the correspondence between the two conventions is shown in

Table 2.1, together with the conventional symbols. Note that inversion

and reﬂection operations are just special cases of improper rotations.All

improper operations involve a change of hand (a left hand is reﬂected

Table 2.1. Symbols for symmetry operations and elements.

Crystallography Spectroscopy Notes

(Hermann–Maugin) (Schoenﬂies)

Proper rotations

1 C

(or E) identity operation

2 C

3 C

4 C

6 C

Improper rotations

1 i (= S

) inversion

2 (or m) σ(= S

) reﬂection

3 S

4 S

6 S

(= C

+ σ

)

14 Introduction to symmetry and diffraction

or inverted into a right hand), while proper rotations retain the same

handedness; this has important implications for the crystal structures of

chiral molecules as well as the molecules themselves.

Symmetry elements can be combined only in certain ways that are

consistent with each other. For a single molecule, all symmetry ele-

ments present must pass through a common point at the centre of the

molecule; if there is an inversion centre, there can be only one and it

is at this point. For this reason, the total collection of all the symmetry

operations for a molecule is called its point group, and each point group

has its own characteristic properties and a conventional symbol (again,

different symbols are used by spectroscopists and by crystallographers).

In principle, any order of rotation axis (the number of individual min-

imum rotation operations that must be repeated in order to achieve a

total of 360

◦

rotation) is possible within a molecule, although high-order

rotations are rare, and two-fold rotation (C

) is the most common. By

contrast, the combination of othersymmetry with translation inthe crys-

talline state puts restrictions on the types of symmetry operation that

are possible, because some orders of rotation are incompatible with the

repeat nature of a lattice. Thus, the only orders possible in crystalline

solids are 1, 2, 3, 4, and 6, for both proper and improper rotations. This

does not mean that molecules with other symmetry elements can not

crystallise! However, such symmetry elements can not apply to the sur-

roundings of the molecules in the crystal, i.e. to the crystal structure as

a whole; atoms that are symmetry-equivalent in an isolated molecule,

such as all 10 carbon atoms of ferrocene, have different environments

in a crystal and are no longer fully equivalent (they would give differ-

ent solid-state

C NMR signals, for example). Because of this restriction,

there are only 32 point groups that are relevant to crystallography; these

are discussed later.

All three-dimensional lattices have inversion symmetry, whether or

not the individual unit cell contents are centrosymmetric, and so the

presence of inversion symmetry in a crystal structure does not put

any restrictions on unit cell parameters; they can still adopt any arbi-

trary values that give a sensible overall packing of the molecules. Any

rotation or reﬂection symmetry in the solid state, however, imposes

restrictions and special values on the unit cell parameters. For exam-

ple, four-fold rotation symmetry means that the unit cell must have two

square faces exactly opposite each other with an axis perpendicular to

them both (parallel to the rotation axis), so two of the cell axes are equal

in length and all three angles are 90

◦

. On the basis of these restrictions,

crystal symmetry is broadly divided into seven types, called the seven

crystal systems. Table 2.2 shows their names, the minimum symmetry

characteristicof each one, and therestrictionsontheunit cell parameters.

For some crystal structures with rotation and/or reﬂection symmetry,

it is convenient and conventional to choose a unit cell containing more

than one lattice point. For example, take an orthorhombic structure, for

which each of the three unit cell axes is associated with either a two-

fold rotation along the axis, a reﬂection perpendicular to it, or both of

2.3 Symmetry of individual molecules, with relevance to crystalline solids 15

Table 2.2. The seven crystal systems. For the essential symmetry, each type of rotation axis is generic; it could be a proper or improper

rotation or a screw axis, and mirrors can also be glide planes. The centred cell types shown in parentheses can be converted into standard

types not in parentheses by a different choice of axes, but are used in some cases in order to satisfy other conventions or conveniences

regarding symmetry and geometry.

Crystal system Essential symmetry Unit cell restrictions Cell types

triclinic none none P

monoclinic 2 and/or m for one axis α = γ = 90

◦

P, C (I)

orthorhombic 2 and/or m for three axes α = β = γ = 90

◦

P, C (A), I, F

tetragonal 4 for one axis a = b; α = β = γ = 90

◦

P, I

trigonal 3 for one axis a = b; α = β = 90

◦

, γ = 120

◦

P (R)

hexagonal 6 for one axis a = b; α = β = 90

◦

, γ = 120

◦

cubic 3 for four directions a = b = c; α = β = γ = 90

◦

P, I, F

these; the unit cell is a rectangular parallelepiped with all axes mutually

perpendicular, like a standard building brick; the 90

◦

cell angles are a

necessary consequence of the rotation/reﬂection symmetry and so are

an indicator that such symmetry is probably present in the structure.

Now consider a similar structure in which there is a lattice point added

at the centre of the unit cell. Since lattice points are all equivalent by

deﬁnition, this means the point at the centre of each unit cell is entirely

equivalent to the points at the cell corners. Remember that the unit cell

is just a convenient way of joining up lattice points to indicate the geom-

etry; the lattice is a fundamental property of the structure, but the unit

cell is an arbitrary deﬁnition. It would be possible to choose a smaller

unit cell, since the distance from one corner to the centre of the cell is

shorter than the longest of the three original unit cell axes. This cell

would have half the volume of the original cell and there would be one

lattice point per unit cell overall. It would, however, have some strange

cell angles and the convenience of the 90

◦

angles and rectangular cell

shape are lost. In such cases, the larger unit cell with more than one

lattice point is usually chosen by convention, so that the geometrical

properties in Table 2.2 still apply. Unit cells with one lattice point are

referred to as primitive (P), and those with more than one lattice point

are called centred. Different kinds of centring are possible in the various

crystal systems, and these may involve lattice points at the centres of

opposite pairs of faces (A, B,orC depending on which faces arecentred),

at the centres of all faces (F), or at the body centre of the cell (I). We omit

hereconsideration of alternative cellchoices for sometrigonal structures

(R for rhombohedral); the treatment needed is more than completeness

is worth. Some forms of centring are not relevant for particular crystal

systems (e.g. there is no advantage in using any form of centred cell for

a triclinic structure), and the essentially different possible combinations

of lattice symmetry with primitive and centred cell choices leads to 14

distinct results, known as the 14 Bravais lattices (strictly speaking, this

term is incorrect, as it is the unit cell and not the lattice that is centred).

The appropriate combinations are included in Table 2.2 as ‘cell types’.

16 Introduction to symmetry and diffraction

2.4 Symmetry in the solid state

The impossibility of having some kinds of symmetry element, such

as a ﬁve-fold rotation axis, in the crystalline solid state might seem

to reduce the types of symmetry available in crystals compared with

individual molecules. This is, however, not the case, as the presence

of translation makes other kinds of symmetry possible. In a single

molecule, the repeated use of one particular symmetry operation even-

tually reproduces the original orientation of the molecule (for example,

two successive reﬂections in the same plane, orfour 90

◦

rotationsabout a

four-foldaxis), and thisis anecessary property of any symmetryelement

in any point group. Imagine now a mirror plane in a crystal structure,

containing two of the unit cell axes (say a and c) and perpendicular to

the third axis (b), but with an operation that combines reﬂection with

translation equal to half a unit cell repeat along one of the axes (a) instead

of just reﬂection. Two successive operations brings the structure back,

not to the same position, but exactly one unit cell removed along the

a-axis, which is entirelyequivalent because of the pure lattice translation

symmetry. Such a symmetry operation is possible in an inﬁnite lattice

in the solid state, but not for an individual non-polymeric molecule.

This is called a glide plane. The glide direction in this example could

equally well be along the c-axis, or along a and c simultaneously, i.e.

along a unit cell face diagonal, in each case with a distance equal to half

the corresponding lattice repeat, so there are different possible ways of

combining reﬂection with translation. These are given symbols show-

ing the glide direction: a, b, c for a glide along one of the axes, n for

any diagonal glide, just as m stands for a pure reﬂection plane. (For

centred unit cells in some crystal systems, it is possible to have glide

planes with a translation component of one-quarter rather than half a

unit cell repeat unit, because two successive operations correspond to

translation from a cell corner to a centringlattice point insteadof another

corner; such glide planes are called d, and they are not common.) In a

similar way, translation can be combined with rotation axes, the trans-

lation always being along the direction of the rotation axis and by an

amount equal to a multiple of 1/n of the lattice repeat in that direction,

where n is the order of the rotation axis. This gives a screw axis, for

which the conventional symbol is a number for the order of the axis

together with a subscript for the multiple of the smallest possible trans-

lation: thus, 4

and 4

are possible four-fold screw axes. Some screw

axes, like screw threads, have handedness, so it is important to have a

convention about the combined directions of rotation and translation;

this occurs when the translation is not one half of the lattice repeat. A

axis is taken as a positive rotation of 90

◦

when viewed along the

axis with one-quarter translation away from the viewer, equivalent to

a right-handed screw thread. Examination of the combined effects of a

screw axis with pure lattice translation properties shows that the cor-

responding left-handed four-fold screw axis is 4

, and there are similar

pairs of left-handed and right-handed screw axes for other orders of

2.4 Symmetry in the solid state 17

Table 2.3. Symmetry elements with translation components.

Rotations (screw axes)

Two-fold 2

Three-fold 3

Four-fold 4

Six-fold 6

Reﬂections (glide planes)

Translation parallel to cell axes abc

Translation parallel to diagonals n

Translation half-way to centring lattice point d

rotation. The full set of possible glide planes and screw axes is shown

in Table 2.3.

In contrast to the situation in single molecules, symmetry elements

in the solid state do not all pass through one point; instead they are

regularly arranged parallel to each other, the symmetry elements of

each unit cell being repeated identically in all other unit cells. Just as

in molecules, however, there are only certain ways in which symmetry

elements can be put together consistently. The total number of possi-

ble arrangements of symmetry elements in the crystalline solid state is

exactly 230, and these arrangements are called space groups, by analogy

with point groups. Their symmetry properties are well established and

are available in standard reference books and tables, the most important

being the International Tables for Crystallography, Volume A, the contents

of which we will look at in Chapter 4.

The notation used for space groups is an extension of that for point

groups in crystallography, and this is one reason for using different

notation from that followed by spectroscopists. Each space group sym-

bol consists of a single capital letter to denote the cell centring, followed

by a combination of numbers (in some cases with a horizontal bar over

the top) and lower-case letters to show the presence of rotation, reﬂec-

tion and inversion symmetry in the structure. Because the combination

of some symmetry operations necessarily implies the presence of others

as well, not all the symmetry needs to be indicated, and there are con-

ventions about which take precedence in the symbols chosen. The rules

are different for each of the crystal systems, as follows.

Triclinic: no rotations or reﬂections, no need for centred cells, the only

question is whether inversion symmetry is absent or present, giving two

possible space groups P1 and P

1, respectively.

Monoclinic: there is symmetry about the unique axis, normally taken

as the b-axis; this may be a two-fold rotation (2 or 2

) along the axis, a

reﬂection plane (mirror m, or glide a, c or n) perpendicular to this axis, or

both of these together, e.g. C2, Pn, P2

/c. There are 13 monoclinic space

groups.

Orthorhombic: the possibilities for monoclinic symmetry apply to all

three axes. Where both rotation and reﬂection are present for an axis,

only the reﬂection is given. Possible combinations are rotation only for

18 Introduction to symmetry and diffraction

all three axes, rotation and reﬂection for all three axes, and reﬂection for

two axes with rotation for the third; in the last case, it is conventional to

take the rotation axis along c. Examples are P2

, Aba2, Cmcm, Fdd2.

There are 59 orthorhombic space groups.

Tetragonal, trigonal and hexagonal: symmetry (including some kind

of 4-, 3- or 6-fold rotation, possibly with perpendicular reﬂection) for

the unique c-axis is given ﬁrst, then symmetry along the a- and b-axes

(equivalent to each other), then any symmetry lying between a and b,

e.g. P4

2, R3c, P6

/mmc. There are 68 tetragonal, 25 trigonal, and 27

hexagonal space groups.

Cubic: symmetry is given ﬁrst along the cell axes, then along the four

body diagonals (always 3 or

3), then along the face diagonals, e.g. P2

3c. There are 36 cubic space groups.

The occurrence of the different space groups in real structures is

far from equally distributed. Some space groups are extremely rare,

while others are very common. Most molecular materials crystallize in

triclinic, monoclinic or orthorhombic space groups, while higher sym-

metries are more commonly found for inorganic ionic and network

structures and minerals. Around one third of all crystal structures of

molecular compounds have space group P2

/c, though the symbol for

thisspace groupmay be P2

/a or P2

/n if the unit cell axes arechosen dif-

ferently. Generally, screw axes and glide planes are more common than

pure rotations and reﬂections in crystal structures, except where indi-

vidual molecules themselves have these symmetry elements, because

the presence of translation components means that the molecules,

with irregular shapes and charge distributions, pack together more

effectively.

Returning to a point raised earlier, in the construction of a lattice,

the choice of the equivalent points in the crystal structure is arbitrary.

Another way of expressing this is that the choice of origin of the unit cell

(the point within one repeat unit that is assigned three zero co-ordinates)

is arbitrary. In high-symmetry inorganic crystal structures such as sim-

ple ionic salts, drawings and models often show atoms or ions at the

corners of unit cells, but this is not necessary and, in fact, it is unusual for

the structures of molecules that have no internal symmetry elements.

By convention in most space groups, the origin is chosen to lie on a

symmetry element; in particular, for centrosymmetric space groups, the

origin is normally placed on an inversion centre, because this simpliﬁes

the mathematics of Fourier calculations. In most cases symmetry ele-

ments lie between molecules, relating them together, rather than inside

molecules.

2.5 Diffraction and symmetry

The symmetry of a diffraction pattern is closely related to the symmetry

of the structure producing it, allowing us to deduce something about the

space group from the observed pattern. The symmetry of a diffraction

2.5 Diffraction and symmetry 19

pattern is seen in the equivalence of different regions of it, which may be

related to each other by certain symmetry elements; this refers to both

the positions (due to the directions of individual diffracted beams) and

the intensities of the various reﬂections on a recorded pattern. A diffrac-

tion pattern has a central point (corresponding to the reﬂection with all

zero indices) and so its symmetry is expressed in terms of a point group,

whereas the crystal structure has a space group; how are these related?

One important aspect of X-ray diffraction is that, in the absence of an

effect called anomalous scattering or anomalous dispersion, which is

rarely a large effect and is signiﬁcant usually only when heavier ele-

ments are present in a structure, every diffraction pattern has inversion

symmetry, whether or not the crystal structure is centrosymmetric; this

is known as Friedel’s Law. Therefore, of the 32 crystallographic point

groups, only 11 are possible as the symmetry of a diffraction pattern,

and these are known as the 11 Laue classes. Each space group has a

corresponding point group to which it is related (and is the point group

symmetry that a crystal would have if grown under ideal conditions),

and a corresponding Laue class. Clearly, with 230 space groups, 32 point

groups,and11 Laue classes,mostof the Laueclasseshave a largenumber

of related space groups.

To derive the point group froma space group, the initial capital letter is

ignored(this refers to the cell centring, which is irrelevantto point group

symmetry), then all screw axis symbols are replaced by the correspond-

ing pure rotation and all glide planes are replaced by pure reﬂections.

Thus, for example, 2

becomes 2, 6

becomes 6, and all of a, b, c, n and

d become m: P2

/c → 2/m, Aba2 → mm2, I4

/acd → 4/mmm. To derive

the Laue class from the point group, add an inversion centre if there is

not already one; in many cases, this will automatically generate further

symmetry elements, and the result is just one of the 11 point groups

that have inversion symmetry. For each of the three most common (and

lowest-symmetry)crystalsystems, thereisjust one Laue class; for each of

theotherstherearetwo. The correspondenceofdifferentcrystal systems,

point groups and Laue classes is shown in Table 2.4.

We thus have the following forms of symmetry that are important in

crystallography: the space group (choice of 230) is the complete symme-

try of the crystal structure and is one property that has to be established

as part of determining the structure by diffraction methods; the Laue

class (choice of 11) is the point-group symmetry of the diffraction pat-

tern if Friedel’s Law applies, and is directly observed in the diffraction

experiment; a point group is the collection of all symmetry operations

(excluding any with translation components) about a particular central

point in a single object. Point group symmetry can be used to refer to

the shape of a crystal, the shape of a single unit cell, or any chosen

point within a crystal structure, e.g. the position of an atom or the cen-

tre of a molecule. The point group symmetry of a molecule in a crystal

structure, taking account not only of the molecule itself but also of its

environment, may be equal to or lower than the point group symmetry

of the same molecule in isolation; it can not be higher without changing

20 Introduction to symmetry and diffraction

Table 2.4. Crystal systems, point groups and Laue classes. In each case, the Laue class is also one of the possible point groups. The

corresponding Schoenﬂies point group symbols are given in the same order. In some cases a different choice of axes can lead to a different

order of the symbols for a particular point group, e.g.

62m instead of 6m2, but these are the same point group.

System Laue class Other point groups Corresponding Schoenﬂies

triclinic 11 C

monoclinic 2/m 2, mC

, C

orthorhombic mmm mm2, 222 D

, D

tetragonal 4/m 4, 4 C

, S

4/mmm 4mm, 422, 4m2 D

, D

trigonal 33 S

3m 32, 3mD

, D

hexagonal 6/m 6, 6 C

, C

6/mmm 6mm, 622, 6m2 D

, D

cubic m323 T

3m 432, 43mO

O, T

the shape of the molecule or by invoking structural disorder, a topic

not covered here. If Friedel’s Law does not apply in the presence of sig-

niﬁcant anomalous dispersion for a non-centrosymmetric structure, the

symmetry of the diffraction pattern will be one of the point groups not

included in the list of 11 Laue classes, the point group associated with

the space group of this structure.

2.6 Further points

In pure lattice translation terms, the repeat unit of a crystal structure is

either one complete unit cell (primitive cells) or a well-deﬁned fraction

of one unit cell (one half for A, B, C or I centring, one third for rhombo-

hedral R structures described on a conventional trigonal unit cell, one

quarter for F centring). If any inversion, rotation or reﬂection symmetry

is present in the structure (including screw axes and glide planes), then

these additional symmetry elements relate atoms and molecules to each

other within each unit cell as well as between unit cells, so the unique,

symmetry-independent part of the structure is only a fraction of the

lattice repeat unit, the fraction depending on the amount of symmetry

present. This unique structural portion is called the asymmetric unit of

the structure, and it may consist of a single molecule, a group of more

than one molecule, a fraction of a molecule possessing symmetry within

itself, or a combination of different molecules and/or ions, for example

including co-crystallized solvent molecules. Operation of all the space

group symmetry (translation, inversion, rotation and reﬂection) gener-

ates the complete crystal structure from the asymmetric unit. The details

of the asymmetric unit, together with the unit cell parameters and the

space group, are what need to be determined in order to describe the

crystal structure.

2.6 Further points 21

Any point in a crystal structure that does not lie on a rotation axis,

a mirror plane or an inversion centre has point group symmetry 1 (C

)

and will be related by symmetry to a number of other equivalent points

in the same unit cell. This is called a general position, and each space

group has a ﬁxed value for the number of equivalent general positions,

ranging from 1 for space group P1 (no symmetry other than pure lat-

tice translation, so every point in the unit cell is unique) to 192 for the

highest-symmetry cubic space groups with F unit cells. For a point

on a symmetry element without translation component, operation of

that element leaves the point unchanged, so the number of equivalent

points in the unit cell is lower by a fraction depending on the nature

of the symmetry element (2, 3, 4 or 6 for a rotation axis, 2 for reﬂec-

tion or inversion). Such positions are called special positions, and can

be occupied only by molecules that themselves possess the appropriate

symmetry. In some space groups there are no special positions. In oth-

ers there are special positions where two or more symmetry elements

intersect; the point group symmetry of such special positions is corre-

spondingly higher, and the number of equivalent points in the unit cell

correspondingly lower. Screw axes and glide planes do not give rise

to special positions, because of their translation components. Atoms on

special positions may require symmetry-imposed constraints on their

co-ordinates and/or displacement parameters during reﬁnement; most

of these are generated automatically by modern programs.

For a crystal structure containing just one kind of molecule (simi-

lar arguments apply to structures containing more than one kind of

molecule,suchassolvatesandco-crystals,andtoioniccrystalstructures,

but there are some additional complications), the number of molecules

in each unit cell is conventionally known as Z. A related parameter of

interest is the number of molecules in the asymmetric unit, which is

given the symbol Z’. The most common value for this is 1, but it is

greater than one in a signiﬁcant number of structures, where there is

more than one molecule (chemically identical but not related by crys-

tallographic symmetry) in the asymmetric unit, and it is less than one if

there is one independent molecule that itself lies on a crystallographic

symmetry element (rotation axis, mirror plane or inversion centre).

One other symmetry term needs to be recognized before these prin-

ciples are applied in trying to work out the possible space groups for a

material from its observed diffraction and other properties. The metric

symmetry is the observed symmetry of the unit cell of a structure with-

out reference to its contents. It takes no account of information from

the Laue symmetry and looks only at the shape of the unit cell. It can

thus lead to false conclusions. For example, if a monoclinic structure

fortuitously has a β angle close to 90

◦

, the unit cell has the same shape

as for genuine orthorhombic symmetry. The metric symmetry is always

at least as high as the Laue symmetry, and may be higher; for the four

high-symmetry crystal systems, the metric symmetry is the same as the

higher of the two possible Laue classes in each case. Trigonal primitive

and hexagonal unit cells are indistinguishable in terms of their basic