Marder M.P. Condensed Matter Physics

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Symmetries of Lattices with Bases

2.5 Symmetries of Lattices with Bases

2.5.1 Thirty-Two Crystallographic Point Groups

The general task of finding all possible symmetries of lattices is far from finished

because one must continue to discuss the symmetries of lattices with bases.

Figure 2.14. A stereogram is a compact two-dimensional representation of the symmetries

of a three-dimensional object, shown in (A). Identify all the symmetry planes and axes of

the object and inscribe them upon a sphere, as shown in (B). Next, find the shadow of the

symmetry planes and axes as shown in (C). This projection is the stereogram, shown in

Table 2.9 as T

First, one has to construct all the relevant point groups. There is a grand total

of 32 crystallographic point groups, which are illustrated in Table 2.9. The conven-

tional way to describe these groups among crystallographers is through the use of a

stereogram, whose construction is illustrated in Figure 2.14. Conventionally, stere-

ograms are decorated with symbols, shown in Table 2.10, that describe all possible

symmetry axes. A dashed line in a stereogram denotes a plane that is not a mirror

plane of the object.

There are two sets of notation for the crystallographic point groups. The

Schönüies notation appears in the upper part of each cell in Table 2.9. In some

cases there are two versions of the Schönflies notation, and both are listed.

C = Cyclic; allows successive rotation about main axis.

D - Dihedral; contains two-fold axes perpendicular to main axis.

S = Spiegel; unchanged after combination of reflection and rotation.

T = Tetragonal.

O = Octahedral.

34 Chapter 2. Three-Dimensional Lattices

Table 2.9. The 32 crystallographic point groups

Triclinic Monoclinic Ortho- Trigonal Tetragonal Hexagonal Cubic

rhombic

Point groups

are

illustrated

stereograms that display

the

symmetry axes

each group.

The symmetry axes

are

described

by the

symbols defined

Table 2.10.

solid line always

describes

mirror plane, while

dashed line refers

a plane about which mirror symmetry

is not present.

Symmetries of Lattices with Bases 35

Table 2.10. Symmetry axes, together with their conventional symbols

Schönflies

International Operation

Axis type Notation Notation

The proper rotations are identical in the Schönflies and International systems. However, the

improper rotations are conceived differently. For example, S3 directs one to rotate coun-

terclockwise by 120°, and then reflect across the mirror plane perpendicular to the rotation

axis.

This operation is the inverse of 6, which rotates by 60° and then applies an inversion.

The two- to sixfold axes do not necessarily have perpendicular mirror planes. However, a

twofold rotoinversion axis is identical to a mirror plane perpendicular to the axis, a threefold

rotoinversion axis always includes an inversion center, and sixfold rotoinversion axes always

have mirror planes perpendicular to the axis.

Chapter 2. Three-Dimensional Lattices

A subscript n =

. . . 6 denotes the order of a rotation axis, and subscripts h,

v, and d denote the three types of mirror plane described in Table 2.10.

International notation appears in the lower part of each cell in Table 2.9, with

complete designations on the left, and conventional abbreviations to the right. The

notation associates each group with a list of its symmetry axes, in accord with Table

2.10. Notation such as 6m refers to a mirror plane containing a sixfold axis, while

^ refers to a mirror plane perpendicular to a sixfold axis.

Figure 2.15. A solid may have pentagonal symmetry, but one cannot tile the plane with it,

so one cannot build a lattice with this symmetry.

Table 2.9 is not a complete list of all possible point groups. Rather, it is a list

of point group symmetries that can be exhibited by a lattice. Of course, one can

build an object with a fivefold axis, as shown in Figure 2.15. But there is no way to

build any sort of completely regular lattice that shares this symmetry, as shown in

Problem 1.4. A lattice with a fivefold axis is crystallographically forbidden. See,

however, Section 5.10.

2.5.2 Two Hundred Thirty Distinct Lattices

The grand total of distinct lattices with bases is 230, exhaustively discussed by

Hahn and Cochran (1992) and Bradley and Cracknell (1972) or in a more peda-

gogical manner by Borchardt-Ott (1993). A first group of lattices, the symmorphic

lattices, can be constructed by taking objects of the symmetries described in Ta-

ble 2.9 and placing them with various orientations at the lattice sites of the 14

Bravais lattices. There are 73 lattices of this type. The remaining lattices are

nonsymmorphic, which means that they contain symmetry operations in which a

translation and rotation or reflection applied simultaneously leave a lattice invari-

ant, although neither the translation nor the rotation applied independently would

do so. A structure containing a symmetry of this type was pictured in Figure 1.9.

When a translation followed by rotation leaves the lattice invariant, one has a screw

axis,

while when a translation followed by reflection leaves it unchanged, one has

a glide plane. Problem 5 shows that the hep lattice contains both glide planes and

screw axes.

Decorating lattice points with quantum-mechanical spins creates even more

elaborate possibilities for symmetry groups, because a spin 1/2 particle acquires a

phase of

— 1

after rotation through 2π radians. The symmetry groups obtained in

this way are called the Shubnikov, color, or magnetic groups; there are 1651 of

them, and they are discussed by Vainshtein (1994) and Lifshitz (1997).

Some Macroscopic Implications of Microscopic Symmetries 37

Large though the number of crystal space groups may be, it is small com-

pared to the number of distinct crystals realizing these symmetries, which is with-

out limit. Compilations of crystal structures may be found in Wyckoff

(1963—

1971) , Landolt and Börnstein (New Series), vols. 5-7, 10, and 14, and Pearson

(1985).

Computer automated hardware and software are making it progressively

easier to analyze new and complicated crystal structures. The Cambridge Struc-

tural Database contains a listing of over 400,000 organic and metal organic com-

pounds. The NIST Crystal Data Base contains data for over 230,000 materials.

These sources require a paid subscription. For the purposes of most physicists, the

3500 minerals catalogued in the freely available American Mineralogist Crystal

Structure Database will be sufficient.

2.6 Some Macroscopic Implications of Microscopic Symmetries

Sometimes symmetries of an object that are evident on a macroscopic scale can be

used to restrict the possible lattices from which it could possibly be constructed.

Here are a few examples.

2.6.1 Pyroelectricity

Pyroelectnc crystals, such as tourmaline, have a bulk dipole moment as a result of

having a dipole moment in each unit cell. The rather problematic nature of these

dipole moments will be discussed in Section 22.2. Under normal conditions, stray

electrons in the environment shield this dipole, making it invisible, but upon heat-

ing or cooling, the ends of the crystal develop momentarily a measurable charge.

Taking the dipole moment to point along the axis of maximum symmetry, one must

rule out all point groups that contain a perpendicular mirror plane, or any sort of

rotoinversion axis, because the dipole changes sign under any of these symmetry

operations, or any rotation axis perpendicular to the main one. The cubic groups

can also be ruled out because they all contain threefold axes that are incompatible

with a dipole moment. Consulting Table 2.9, one sees that the only point groups

remaining are C„ (n =

. . . 6), C

(n = 2 . . . 6), and Ci/,.

2.6.2 Piezoelectricity

The piezoelectric crystals have no dipole moment in their ground state, but acquire

one upon application of a mechanical deformation in some direction. Quartz is the

most common example displaying such behavior. Suppose that one deforms the

crystal by moving each lattice point a small distance u(r), which varies smoothly

as one travels through the crystal. The

strain

tensor

a matrix of deformations of

the crystal defined by

1 / du

dug\

g a = — 1 — · If

βχχ

is nonzero, the crystal has been stretched in (2.10)

2 \ Örß dr

I the x direction, while if e^ is nonzero, the crystal

^ ' has been sheared in the x

—

y plane.

38 Chapter 2. Three-Dimensional Lattices

it will be discussed at length in section 12.3. Assuming that the polarization P

arises as a linear function of the applied strains, it must take the form

= Σ'Β

αβΊ

αβ

(2.11)

aß

where 23 is some tensor describing a general linear relationship between dipole

moment and the strain. Now if one performs an inversion on the crystal, sending

—>

— r, the strain tensor is invariant, but the polarization must change sign, and

so therefore must

Œ5.

This means that if the crystal is invariant under inversion, B

must vanish. All piezoelectric crystals must therefore be noncentrosymmetric, and

their point groups exclude

D„h

and S2·

2.6.3 Optical Activity

Certain crystals, such as S1O2 (low quartz) are able to rotate the plane of incoming

polarized light. Such behavior is only possible if the unit cells are

chiral,

meaning

that there is a mirror image of the cell which cannot be related to it by any trans-

lation or rotation. Finding the point groups that have chiral unit cells is the subject

of Problem 8.

Problems

Face-centered and body-centered cubic lattices: Consider the face-centered

and body-centered cubic lattices to be simple cubic lattices decorated with a

basis.

(a) Determine the number of basis vectors that is necessary, and write down the

basis vectors for each of these lattices.

(b) Determine in each case the ratio of the volume of the conventional unit cell

to the primitive unit cell.

Face-centered cubic lattice as stacked triangles:

(a) Let a\ and

be primitive vectors for a triangular lattice. Referring to Figure

2.2 (C), find a third primitive vector that stacks two-dimensional triangular

lattices so as to produce an fee lattice.

(b) The cube shown in Figure 2.2(D) can be brought into the orientation depicted

there by a first rotation around the y axis, followed by a rotation around the i

axis.

Find the two rotation angles.

Fourfold coordination: What is the angle between neighboring bonds in the

fourfold coordinated diamond and wurtzite lattices?

Hexagonal close-packed lattice:

(a) Verify that the basis vectors given by Eq. (2.6) result in placing atoms di-

rectly over the centers of triangles along the c axis.

Problems 39

(b) Verify that the hep lattice forms a close-packed structure when c/a =

γ/8/3.

Glide and screw axes: Let the lower left point in Figure 2.16 be the origin.

Figure 2.16. Top view of

hep lattice.

(a) The hep lattice contains a screw axis in the c direction. Referring to Figure

2.16, describe the Cartesian coordinates of a point through which this axis

can pass, so that the lattice remains invariant under translation along c by c/2

followed by rotation through 60°.

(b) The hep also has a glide plane, which is parallel to a plane containing both

the a and c axes. Describe where this plane may be located, so that translation

along c/2 followed by reflection about the plane leaves the lattice invariant.

6. Packing fractions: It is difficult to avoid asking why elements choose the

crystal structure they do. Without a quantitative theory of cohesion, developed

in Chapter 11, this question cannot be answered in detail. However, simple

arguments can explain a great deal. Many atoms are held together by isotropie

attraction that favors packing them as closely as possible. The concept of

packing can be addressed by setting the distance between nearest neighbors

in a given lattice to 2, putting unit spheres on each lattice site, and asking

what portion of space is occupied by the spheres. This number is the packing

fraction and is displayed in Table 2.11.

Verify the packing fractions listed in Table 2.11. The packing argument ex-

plains the great frequency of the hep and fee structures, although it cannot

explain how crystals would choose between them.

Three-dimensional ground states: Consider a collection of particles in

three dimensions whose energy is

= \ Σ

Φ(

ν)

with

(2·

)

Chapter 2. Three-Dimensional Lattices

Table 2.11. Packing fractions for various crys-

tal structures

Structure

hep

fee

bec

diamond

Packing Fraction

0.74

0.68

0.52

0.34

)

J0oexp(-r)(p3-l) ifr<1.5

(2 13)

I 0 else,

where r

;;

- is the distance (measured, say, in Â) between particles / and

Com-

pare the energies of the bec, fee, and hep lattices. Find the lattice spacing

leading to the minimum energy for each of these three lattices, and find the

state of lowest overall energy.

8. Optical activity: Find all the crystallographic point groups allowed for crys-

tals which rotate the plane of incoming polarized light. The group C\ is one

of them; it is called asymmetric, and all the others are called chiral.

9. Second law of crystal habit:

Figure 2.17. Setting for the second law of crystal habit in two dimensions.

The second law of crystal habit, or Haiiy's law of rational parameters, relates

the possible faces of a crystal to a set of integers. The prescription, found

empirically for real crystals follows:

Pick any three linearly independent edges of a crystal (they will not generally

be orthogonal) and use them as reference axes, a\, «2, and «3 (see Figure

2.17). Take any face of the crystal and find the three points where it cuts

the reference axes; call these points (p\, p2, PÏ). Next, take any other face

of the crystal, and find again the three points (q\, q2,

q-})

where this second

plane cuts the axes. Then p\jq\ = n, p2/<?2 =

>"2,

and

p^jq-i

= r? are rational

numbers, and the ratios between r\, ri, and rj can be written as the ratios of

integers.

References 41

(a) Prove that the law of rational parameters holds for any two-dimensional crys-

tal that is created by drawing straight lines between points of a Bravais lattice.

(b) Generalize the result to a three-dimensional lattice.

References

W. Borchardt-Ott (1993), Crystallography, Springer-Verlag, Berlin.

C. J. Bradley and A. P. Cracknell (1972), The Mathematical Theory of Symmetry in Solids; Repre-

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A. Bravais

(

1850),

Mémoire on systems formed by points distributed regularly on a plane or in space

Journal de

l'Ecole

Polytechnique, XIX(33),

1-128.

In French.

P. Ewald, ed. (1962), Fifty

Years

ofX-Ray Diffraction, International Union of Crystallography.

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209-252. In German.

T. Hahn and A. J. Cochran, eds. (1992), International

Tables

for Crystallography, vol. A: Spac

Group Symmetry, 3rd ed., Kluwer Academic Publishers, Dordrecht.

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the

Crystal Maze, Oxford

University Press, Oxford.

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and

Technology,

New Series, Group III, Springer-Verlag, Berlin.

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W. B. Pearson (1985), Handbook ofCry stallo graphic Data for Intermetallic Phases, American So-

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Readex Microprint, 1972 (Landmarks of Science).

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Crystals:

Symmetry and Methods of Structural

Crystal-

lography, vol.

of Modern Crystallography, 2nd ed., Springer-Verlag, Berlin.

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