Marder M.P. Condensed Matter Physics

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Monatomic Lattices

2.2.4 The Hexagonal Lattice

A hexagonal lattice is pictured in Figure 2.4. This structure does not occur among

the elements, except as the starting point for the hexagonal close-packed structure.

Its primitive vectors are

ai = (αλ/3/2 a/2 0), a

= (aVÌ/2 -a/2 0), a

= (0 0 c), (2.5)

Figure 2.4. Hexagonal lattice described by lattice constants c and a (stereo pair).

2.2.5 The Hexagonal Close-Packed Lattice

The hexagonal close-packed (hep) structure, a common ground state among the el-

ements, is like the honeycomb lattice a lattice with a

basis.

As shown in Figure 2.5,

it is formed by creating a triangular lattice of lattice constant a in two dimensions.

Then a copy of this lattice is stacked vertically over the first one, in such a way that

particles of the second lattice are directly over centers of triangles of the first one,

at height c/2. A new copy of the original triangular lattice is now stacked directly

over the first one, at height c, and the stacking sequence begins over again. One

way to demonstrate that the hep lattice is not a Bravais lattice is to view it from the

top as in Figure 2.16 (Problem 5); from this angle it appears to be a honeycomb

lattice, so it must be described as a lattice with a basis.

To construct the hep lattice, begin with the primitive vectors in Eq. (2.5) and

supplement them with basis vectors

(0 0 0) (a/y/3 0 c/2) In Cartesian coordinates. (2.6a)

(0 0 0) (1/3 1/3 1/2). In coordinates described by multiples of the (2.6b)

primitive vectors, as in Eq. (1.6).

There is no necessary relation between the lattice constants c and a. However,

if c = ^/8/3a, the atoms are

just the right location that if expanded to a radius of

Figure 2.5. Hexagonal close-packed lattice, produced by two interlaced hexagonal lattices

(stereo pair).

24 Chapter 2. Three-Dimensional Lattices

a/2, they form a perfect close-packed structure. The difference between this close-

packed structure and the one formed by the face-centered cubic lattice lies in the

way that successive triangular layers are stacked upon one another. In Figure 2.5,

notice that one has two choices as to where to put the topmost layer of atoms. In

the hep lattice, the layers are stacked so that the structure repeats, after two layers,

while to produce the fee lattice, one stacks them so that they repeat after three, as

in Figure 2.2(D). More than 25 elements adopt the hep lattice, including zinc and

titanium.

2.2.6 The Diamond Lattice

The diamond lattice is constructed by taking the fee lattice described by Eq. (2.3)

and making a copy of it, displaced relative to the original by (a/4 a/4 a/4).

Every atom in the structure has precisely four nearest neighbors, symmetrically

organized around it, as shown in Figure 2.6. Only four elements commonly assume

this structure — carbon, silicon, germanium, and tin — but the cultural status of

diamond and technological importance of silicon make this a structure of great

significance.

Figure 2.6. Diamond lattice. Each atom has four nearest neighbors, a structure produced

by removing four atoms from the bec structure, as shown in the lower left, and then assem-

bling these tetrahedra as shown in the stereo pair to the right.

2.3 Compounds

Compounds are crystals made of more than one element. Since at least two types

of atoms are involved in their construction, they cannot be Bravais lattices, and all

must be described as lattices with a basis. Hundreds of thousands of different crys-

tals have been cataloged, and only a few common arrangements are listed below.

Compounds

2.3.1 Rocksalt—Sodium Chloride

The sodium chloride structure (NaCl) alternates sodium

and

chlorine

points

a simple cubic lattice,

shown

Figure

2.7.

This structure

can be

described

fee

lattice

lattice constant

with

basis

of a

sodium atom

(000),

and a

chlorine atom

at a/2(\ 0 0)

(see Table

2.2).

Figure 2.7. The sodium chloride structure can be viewed

as a

simple cubic lattice

which

two different atoms alternate,

or as two

interpenetrating

fee

lattices (stereo pair). Figure

11.3 provides

alternate view

NaCl, showing the sizes conventionally attributed to

the

two ions.

Table 2.2. Crystals with

the

sodium chloride structure

Crystal

AgBr

AgCl

AgF

BaO

BaS

BaSe

BaTe

CaS

CaSe

CaTe

CdO

5.77

5.55

4.92

5.52

6.39

6.60

6.99

5.69

5.91

6.35

4.70

Crystal

CrN

CsF

FeO

KBr

KC1

LiBr

LiCl

LiF

LiH

4.14

6.01

4.31

6.60

6.30

5.35

7.07

5.50

5.13

4.02

4.09

Crystal

Lil

MgO

MgS

MgSe

MnO

MnS

MnSe

NaBr

NaCl

NaF

Nal

6.00

4.21

5.20

5.45

4.44

5.22

5.49

5.97

5.64

4.62

6.47

Crystal

NiO

PbS

PbSe

PbTe

RbBr

RbCl

RbF

Rbl

SnAs

SnTe

SrO

4.17

5.93

6.12

6.45

6.85

6.58

5.64

7.34

5.68

6.31

5.16

Crystal

SrS

SrSe

SrTe

TiC

TiN

TiO

ZrC

ZrN

6.02

6.23

6.47

4.32

4.24

4.18

4.13

4.68

4.61

Lattice constants

a in

Â. Source: Wyckoff (1963-1971), vol.

Chapter 2. Three-Dimensional Lattices

2.3.2 Cesium Chloride

The cesium chlonde structure (CsCI) alternates cesium and chlorine on the points

of a body-centered cubic lattice, as shown in Figure 2.8. This structure can be

described as a simple cubic lattice of lattice constant a, with a basis of a cesium

atom at (000), and a chlorine atom at a/2(l 1 1) (see Table 2.3).

Figure 2.8. The cesium chloride structure can be viewed as a body-centered cubic lattice

with an atom of

second type inhabiting the interior of

the

cube, (stereo pair).

Table 2.3. Crystals with the cesium chloride structure

Crystal a Crystal a Crystal a

AgCd 3.33 CsCÎ 4TÏ2 NÎÂÎ ÏM

AgMg 3.28 CuPd 2.99 TiCl 3.83

AgZn 3.16 CuZn 2.95 Til 4.20

CsBr 4.29 NH

C1 3.86 TISb 3.84

Lattice constants a in Â. Source: Wyckoff

(1963-1971),

vol. 1.

2.3.3 Fluorite—Calcium Fluoride

The calcium fluoride structure (CaFl2) may be described as a simple cubic lattice

with basis (in units of the lattice spacing a and Cartesian coordinates)

(0 0 0) (0 1/2 1/2) (1/2 0 1/2) (1/2 1/2 0) (2.7)

of a first type of atom (calcium) and

(1/4 1/4 1/4) (1/4 3/4 3/4) (3/4 1/4 3/4) (3/4 3/4 1/4) (2.8a)

(3/4 3/4 3/4) (3/4 1/4 1/4) (1/4 3/4 1/4) (1/4 1/4 3/4) (2.8b)

of a second (fluorine) (see Figure 2.9 and Table 2.4).

Compounds

Figure 2.9. The calcium fluoride structure (stereo pair) can be viewed as a face-centered

cubic lattice made from a first type of atom (calcium), with the center of the cell inhabited

by a cubic arrangement of the second type of atom (fluorine).

Table 2.4. Crystals with the calcium fluoride structure

Crystal a Crystal a Crystal a

BaF?

CaF

CdF

Ce0

Cm0

6.20

5.46

5.39

5.41

5.37

CoSi

5.36

Hf0

5.12

0 4.62

S 5.71

Pb 6.84

SrCl

6.39

6.77

6.53

6.98

5.47

Lattice constants a in Â. Source: Wyckoff (1963-1971), vol. 1.

Figure 2.10. The zincblende structure can be viewed as a diamond lattice, in which alter-

nating lattice points are occupied by two alternating elements, (stereo pair).

2.3.4 Zincblende—Zinc Sulfide

The zincblende structure (ZnS) is identical to diamond, except that two species of

atoms alternate between sites (see Figure 2.10 and Table 2.5).

28 Chapter 2. Three-Dimensional Lattices

Table 2.5. Crystals with the zincblende structure

Crystal

Agi

AlAs

AIP

AlSb

BeS

BeSe

BeTe

CdS

6.47

5.62

5.45

6.13

4.85

5.07

5.54

5.82

Lattice constants a in Â.

Crystal

CdTe

CuBr

CuCl

Cui

GaAs

GaP

GaSb

HgS

6.48

5.69

5.41

6.04

5.63

5.45

6.12

5.85

Source: Wyckoff

(

1963-

-1971)

Crystal

HgSe

HgTe

InAs

InP

InSb

SiC

ZnS

ZnTe

, vol. 1.

6.08

6.43

6.04

5.87

6.48

4.35

5.41

6.09

2.3.5 Wurtzite—Zinc Oxide

The wurtzite structure (ZnO) is an hexagonal lattice with basis

(0 0 0) (α/2α/2ν

3 c/2) First species; Cartesian coordinates. (2.9a)

(0 0 Cu) (a/2 a/2V3 c/2 + eu) Second species; Cartesian coordinates. (2.9b)

(0 0 0) (1/3 1/3 1/2)) First species; primitive vector (2.9c)

' ' ' " coordinates: see Eq. (2.5).

(0 0 u) (1/3 1/3 U+l/2)) Second species; primitive vector (2.9d)

' ' ' " coordinates.

When u = 3/8 and c/a =

^/8/3,

every atom is equidistant from four neighbors;

although these extra symmetries are not inevitable for the definition of the structure,

natural crystals display them (see Figure 2.11 and Table 2.6).

Table 2.6. Crystals with the wurtzite structure

Crystal a c Crystal a

A1N

BeO

CdS

CdSe

CuH

3.11 4.98

2.70 4.38

4.13 6.75

4.30 7.02

2.89 4.61

MgTe

SiC

ZnO

ZnS

4.52

4.39

3.08

3.25

3.81

7.33

7.02

5.05

5.23

6.23

Lattice constants a and c in Â. In all cases, u = 3c/c

Source: Wyckoff (1963-1971), vol. 1.

2.3.6 Perovskite—Calcium Titanate

The perovskite structure (CaTiC>3) is built from three types of atoms. The calcium

fill out a simple cubic lattice, the oxygens lie on the face centers, and titanium

occupy body centers (see Figure 2.12 and Table 2.7).

Compounds 29

Figure 2.11.

The

wurtzite (ZnO) structure

is an

hexagonal structure built with

two

types

of atoms,

and it

allows each atom

have four nearest neighbors

the opposite type.

can

viewed

two interlaced hep lattices, just as

the

diamond

and

zincblende structures

are built

as two

interlaced

fee

lattices (stereo pair).

Figure

2.12. The

perovskite structure

is a

simple cubic lattice

calcium, with oxygens

occupying face centers,

and

titanium occupying

the

body center, (stereo pair).

Table

2.7.

Crystals with

the

perovskite structure

Crystal

BaTi0

CaSn0

CaTi0

CaZr0

CsCdBr

CsHgBr

4.01

3.92

3.84

4.02

5.33

5.77

Crystal

CsHgCl

CsI0

κιο

KMgF

KNiF

KZnF

5.44

4.66

4.41

3.97

4.01

4.05

LaA10

LaGa0

RbI0

SrTi0

SrZr0

YA10

3.78

3.88

4.52

3.91

4.10

3.68

Lattice constants

a in Â.

Source: Wyckoff (1963-1971), vol.

30 Chapter 2. Three-Dimensional Lattices

2.4 Classification of Lattices by Symmetry

There is a complete classification of lattices by symmetry, which includes no less

than 230 distinct types and which will be discussed only briefly.

2.4.1 Fourteen Bravais Lattices and Seven Crystal Systems

There are precisely seven distinct point symmetry groups that arise when one con-

structs Bravais lattices out of identical point particles. They are called the seven

crystal systems and are shown in Table 2.8. A related but separate question is how

many distinct three-dimensional Bravais lattices can be built out of identical point

particles. The answer is 14, and all are listed in Table 2.8.

Cubic system: The first symmetry to strike the eye after inspecting the cube has to

do with the square sides. However, the cubic symmetry class derives its name

from three threefold axes about lines [1,1,1] between opposite corners of the

cube.

The cube is also invariant under rotations of 90° about any of three

perpendicular axes, and under reflections about the three planes perpendicular

to these axes. It has twofold rotation and mirror symmetries about the axis

[1,1,0].

The three Bravais lattices possessing this set of symmetries are the

simple cubic, face-centered cubic, and body-centered cubic lattices.

Tetragonal system: If one stretches the cube to make four of the sides into rect-

angles, then one has an object with the symmetry of the

tetragonal

group:

the

threefold symmetry is lost, as well as the 90° rotation symmetry about two

of the axes, but the solid is still symmetric under reflections about planes that

Figure 2.13. (A) If one stretches a bcc lattice by a factor of y/2 along one of its axes, it

becomes an fee lattice with lattice constant \[2 times larger, as shown by simply reconnect-

ing the lines between atoms. (B) Deforming tetragonal lattices along their axes produce

the simple and centered orthorhombic lattices. (C) Deforming tetragonal lattices along

diagonals produces base-centered and face-centered orthorhombic lattices.

Classification of Lattices by Symmetry

Table 2.8. The seven crystal systems and fourteen Bravais lattices in three

dimensions

Simple Base- Body- Face-

Centered Centered Centered

Chapter 2. Three-Dimensional Lattices

bisect it. Stretching a simple cubic lattice produces a simple

tetragonal

lat-

tice,

and stretching either a face-centered cubic or a body-centered cubic lat-

tice produces a centered

tetragonal

lattice. That one can produce only a single

structure from deformation of

the

fee and bec lattices follows from the slightly

surprising fact that an fee lattice can be obtained by expanding a bec lattice

along one axis by a factor of \/2, as illustrated in Figure 2.13(A). Therefore,

any distortion of the bec lattice could have equally well been produced by a

distortion of the fee lattice.

Orthorhombic system: Next, one can deform the top and bottom squares of the

tetragonal solid into rectangles, eliminating the last of the 90° rotation sym-

metries. This solid has orthorhombic symmetry. Deforming the simple tetrag-

onal lattice along one of its axes produces a simple orthorhombic lattice, and

deforming the centered tetragonal lattice produces a body-centered orthogo-

nal lattice, as in Figure 2.13(B). There are two additional orthorhombic lat-

tices,

however, which can be produced by stretching the two tetragonal solids

along face diagonals, as illustrated in Figure 2.13(C). Deforming the simple

tetragonal lattice in this way produces the base-centered orthorhombic lattice,

while deforming the centered tetragonal in this way produces a face-centered

orthorhombic lattice.

Monoclinic system: One generates a solid with monoclinic symmetry by squeez-

ing a tetragonal solid across a diagonal so as to eliminate the 90° angles on the

top and bottom faces, leaving the sides built out of rectangles. There are only

two distinct monoclinic lattices. The simple monoclinic lattice results from

distortion of the simple orthorhombic lattice, while the centered monoclinic

lattice results from appropriate distortion of the face-centered, base-centered,

and body-centered orthorhombic lattices.

Triclinic system: Final in this progression is the triclinic symmetry which is pro-

duced by pulling the top of a monoclinic solid sideways relative to the bottom

so that all faces become diamonds. The only symmetry now remaining is in-

version symmetry, and there is only one lattice of

this

type, the

triclinic

lattice.

Rhombohedral or trigonal system: There are two crystal classes still missing

from the list. If one starts with a cube and stretches it across a body diag-

onal, one gets a solid with rhombohedral or trigonal symmetry. Stretching

any of the three cubic Bravais lattices in this way produces the same Bravais

lattice, called the rhombohedral lattice or the

trigonal

lattice.

Hexagonal system: Finally, one can form a solid with a hexagon at the base and

perpendicular walls to illustrate the hexagonal symmetry. There is only one

Bravais lattice of this type, the hexagonal lattice.

In this way, one has

point groups for lattices built out of points with spherical

symmetry.