Marder M.P. Condensed Matter Physics

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Symmetries

Point group symmetries: Choose any lattice point as the origin. Both rectangular

and centered rectangular lattices can be reflected about the x or y axes, and

each is invariant under a 180° rotation.

Because the rectangular and centered rectangular lattices share the same point

group symmetries, they are said to belong to the same crystal system, but they

are not the same lattice.

One might protest that the rectangular and centered rectangular lattices are ob-

viously different because their primitive vectors are different. However, making

this argument would lead one to conclude that two square lattices of different size

are different as well. The correct question to ask in determining if two lattices

are the same is whether one structure can be deformed continuously into the other

without losing symmetries along the way. For example, if one lattice is twice the

size of another, but otherwise the same, one would want to call them the same. Al-

though centered rectangular and rectangular lattices share point group symmetries,

they are different lattices and have different space groups. They are different be-

cause there exists no way to deform the first continuously into the second without

temporarily destroying some symmetries, as indicated in Figure 1.10.

Mirror plane.

..broken..

...and restored

Rectangular

Centered Rectangular

Figure 1.10. In deforming the rectangular lattice into the centered rectangular lattice,

reflection symmetry about the y axis is destroyed.

A more formal expression of this same idea uses the idea of a change of co-

ordinates. Suppose one has a first group of symmetry operations, G =

and

a second group G'

—

Jl' + a'. Then the two groups are equivalent if there exists a

single matrix S (change of coordinates) such that

3lS + S~

ä =

3i'

+ a.

(1.8)

In other words, G and G' must be the same up to linear changes of coordinate

systems. This definition of equivalent lattices is the same as the previous one,

because once one has the matrix S, then there is a family of matrices

= (!-/)+»,

(1.9)

14 Chapter 1. The Idea of

Crystals

which varies smoothly between the unit matrix and S as t varies between 0 and

It generates explicitly a smooth deformation of one lattice into the other while

preserving the group operations.

1.3.3 Role of Symmetry

Perhaps symmetry is more important for physicists to understand the world than it

is for the world

itself.

Most of

the

exact statements in physics result from symmetry

arguments, and often symmetry provides the only path to making any substantive

statement about complicated assemblies of matter. Its importance persists in con-

densed matter physics, although the discipline's domain includes disordered and

noisy systems. Helpful books on the formal theory of symmetry in physics include

Heine (1960) and Tinkham (1964).

Problems

Honeycomb lattice:

(a) Verify that the honeycomb lattice described by Eq. (1.5) has properly been

constructed so that the distance between all neighboring points is identical.

(b) From Table 2.1, the lattice spacing of graphene is 2.46Â. Find the distance

between nearest neighbors (Figure 1.3).

Hexagonal lattice:

(a) The hexagonal lattice may be viewed as a special case of

the

centered rectan-

gular lattice. Referring to Figure 1.4 and to the conventional unit cell depicted

there, find the ratio c/a for which the centered rectangular lattice would be-

come hexagonal.

(b) Enumerate the symmetries of the hexagonal lattice, and compare them with

the symmetries of the centered rectangular lattice.

Nanotube structures: To form a single-walled nanotube, one rolls up a sin-

gle atomic layer of graphite, graphene, as shown in Figure 1.11. Referring

to Figure 1.11 (A) and (B), all nanotubes can be indexed with two integers m

and n where c = ma\ + nai and a\ and

a.2

are primitive vectors; one created

in this way is an (m, n)-nanotube.

(a) Are all structures labeled by distinct pairs of integers m, n

(—oo, oo)

dif-

ferent from each other?

(b) What are the indices for the nanotube appearing in (C)?

that the crystal is rolled into a cylinder by pulling together two atoms sepa-

rated by vector c as shown in Figure 1.11 (A). Write down an explicit expres-

sion for locations of atoms after the sheet has been rolled into a tube.

Problems 15

Figure 1.11. How to roll a sheet of graphene into a nanotube.

Allowed symmetry axes:

(a) Consider a two-dimensional Bravais lattice that is left invariant after rotation

by angle

around the origin. Suppose the lattice to have points at (0, 0) and

(a, 0). By requiring the image of

(a,

0) under rotations through ±θ to be in the

Bravais lattice, find

simple expression that implicitly specifies all possible

rotation axes.

(b) Prove that the only allowed axes are twofold, threefold, fourfold, and sixfold.

In particular,

is impossible for a Bravais lattice to have

fivefold rotation

axis.

Two-dimensional ground states: Portions of this problem are most easily

carried out with the aid

of a

computer algebra program or brief compiled

programs.

(a) Consider a collection of particles in two dimensions whose energy is

«w

where

^(r)

= <

Uoexp(-r)(^-l) if r

1.5

(U1)

else,

and r,j is the distance (measured, say, in Â) between particles

and

Find

the crystal structure in Figure 1.4 which provides a minimum energy state for

this potential, and the equilibrium lattice spacing, assuming no particles are

at a distance less than 1. The potential has been chosen so that only nearest

neighbors interact in the ground state. Do not check all crystal structures

explicitly, only the square and hexagonal lattices.

16 Chapter 1. The Idea of

Crystals

(b) Suppose that φ is replaced by

4>(r)

= \

&>exp(-r)(—-1) ifr< 1.5

(L12

)

else.

Show that particles would collapse into

state of high density. It is not nec-

essary to perform sums numerically: Consider what happens when particles

are so closely spaced that they can be thought of as constituting a continuous

distribution.

φ(ή

φο^χ

(-ή(^-ή

(1.13)

and assuming that the ground state is a lattice of the same symmetry as in part

(a),

find the equilibrium lattice spacing and energy per particle within 10%.

This sum does need to be performed numerically.

References

J. Emsley (1998), The Elements, 3rd ed., Clarendon Press, Oxford.

H. Goldstein (1980), Classical Mechanics, 2nd ed., Addison-Wesley, Reading, MA.

Groth (1906-1919), Chemische Krystallographie, W. Engelmann, Leipzig.

R. J. Haiiy (1801), Treatise on Minerology, Conseil des Mines, Paris.In French.

V. Heine (1960), Group Theory in Quantum Mechanics, Pergamon, New York.

J. C. Heyraud and J.

Métois (1980), Establishment of the equilibrium shape of metal crystallites

foreign substrate: Gold on graphite, Journal of Crystal Growth, 50, 571-574.

S. G. Lipson (1987), Helium crystals, Contemporary Physics, 28, 117-142.

C. H. MacGillavry (1976), Fantasy & Symmetry: The Periodic Drawings of

C. Escher, H.

Abrams, New York.

J. B. Marion and S. T. Thornton (1988), Classical Dynamics of Particles and Systems, 3rd ed., Har-

court Brace Jovanovich, San Diego.

J. C. Meyer, A. K. Geim, M.

Katsnelson, K. S. Novoselov, T. J. Booth, and S. Roth (2007), The

structure of suspended graphene sheets, Nature, 446,

60-63.

Steno (1671), The Prodromus to

Dissertation Concerning Solids Naturally Contained Within

Solids: Laying

Foundation

for the Rendering

Rational Accompt Both of

the

Frame and the

Several Changes

the Masse

the Earth,

also

the Various Productions

the Same,

London, F. Winter. Translation of: De solido intra solidum naturaliter. Readex Microprint, 1973

(Landmarks of Science)

J. J. Thomson (1907), The Corpuscular Theory of Matter, Charles Scribner's Sons, New York.

M. Tinkham (1964), Group Theory and Quantum Mechanics, McGraw-Hill, New York.

Three-Dimensional Lattices

2.1 Introduction

In order to classify the crystalline solids found in nature, one must study three-

dimensional lattices. While the subject involves few concepts not already present in

the two-dimensional case, the number of possible structures is bewilderingly large.

Fortunately, a large number of the elements adopts particularly simple structures.

On the other hand, alloys and compounds explore countless different forms, of

which over 400,000 have now been cataloged; these cannot be summarized in any

neat way despite the aid provided by the theory of symmetry.

There are 32 distinct point groups consistent with crystalline symmetry in

three dimensions, which were first enumerated by Hessel (1830), spurred on by

the desire to classify the shapes of naturally occurring rock crystals. The three-

dimensional Bravais lattices were first correctly enumerated by Bravais (1850);

solution of this problem may seem easier than that of finding all point groups, but

it is more abstract, since crystal surfaces are visible, while lattices are deduced as

an economical explanation for their appearance. Listing of lattices with bases was

begun by Sohncke (1879), who found 65 lattices. The full number is 230, and

these were enumerated by Fedorov (1895) and Schonflies (1891). Fedorov had

priority in most respects, but only following correspondence between the two sci-

entists were the final errors corrected; in one of Schönflies' early papers, 227 space

groups appear. Many other historical details are related by Ewald (1962), Phillips

(1971),

and Hoddeson et al. (1992).

This chapter is designed more for reference than for recreational reading. At

first the listing of crystal structures may appear more like a dull catalog of animals

in some distant land than basic physics. Yet knowledge of crystal structures is

the foundation on which much of the rest of condensed matter physics rests. The

detailed calculation of electronic and mechanical properties of solids depends on

knowing where the atoms lie.

Distribution Among Elements.

A comprehensive account of everything known about crystal structures cannot

be confined to one volume, or ten, let alone a chapter. Still, it is worth giving a

sense of the types of information available, first the elements and then some of

the more common compound structures. The low-temperature crystal structures of

the elements are shown in Table 2.1. Room-temperature crystal structures of the

elements are shown in the periodic table inside the front cover. Not all structures

are known with certainty; boron has a huge unit cell that continues to resist precise

determination. The crystal structures mentioned in the chart and in the periodic

table are defined in the following sections.

Condensed Matter

Physics,

Second Edition

by Michael P. Marder

Chapter 2. Three-Dimensional Lattices

Table 2.1. Low temperature crystal systems and crystal structures of the ele-

ments

Element

Ac Actinium

Ag Silver

Al Aluminum

Am Americium

Ar Argon

As Arsenic

Au Gold

Ba Barium

Be Beryllium

Bi Bismuth

B Boron

Br2 Bromine

C Carbon

Ca Calcium

Cd Cadmium

Ce Cerium

Chlorine

Co Cobalt

Cr Chromium

Cs Cesium

Cu Copper

Dy Dysprosium

Er Erbium

Eu Europium

F Fluorine

Fe Iron

Ga Gallium

Ge Germanium

Gd Gadolinium

H2 Hydrogen

He Helium

Hf Hafnium

Hg Mercury

Ho Holmium

I2 Iodine

In Indium

Ir Iridium

K Potassium

Kr Krypton

La Lanthanum

Li Lithium

Lu Lutetium

Mg Magnesium

Mn Manganese

# Lattice a c or a

89 CUB

47 CUB

13 CUB

95 CUB

18 CUB

33 RHO

79 CUB

56 CUB

4 HEX

83 RHO

5 TET

35 ORC

6 CUB

6 HEX

20 CUB

48 HEX

58 CUB

HEX

17 ORC

27 CUB

HEX

24 CUB

CUB

HEX

55 CUB

29 CUB

66 HEX

68 HEX

63 CUB

26 CUB

CUB

31 ORC

32 CUB

64 HEX

1 HEX

2 HEX

72 HEX

80 RHO

67 HEX

53 ORC

49 TET

77 CUB

19 CUB

36 CUB

57 CUB

HEX

3 CUB

71 HEX

12 HEX

25 CUB

CUB

TET

fee 5.31

fee 4.09

fee 4.05

fee 4.89

fee 5.26

4.13 54°

fee 4.08

bec 5.02

hep 2.29 3.58

4.75 57°

8.74 5.03

6.67 4.48

dia 3.57

2.46 6.70

fee 5.58

hep 2.98 5.62

fee 5.16

hep 3.65 5.96

6.24 4.48

fee 3.55

hep 2.51 4.07

bee 2.88

fee 3.68

hep 2.72 4.43

bec 6.05

fee 3.61

hep 3.59 5.65

hep 3.55 5.59

bec 4.61

bec 2.87

fee 3.59

4.51 7.86

dia 5.66

hep 3.56 5.80

hep 3.75 6.49

hep 3.57 5.83

hep 3.20 5.06

2.99 71°

hep 3.58 5.62

4.79 9.78

bet 3.24 4.94

fee 3.84

bec 5.23

fee 5.72

fee 5.30

hep 3.75 6.07

bec 3.50

hep 3.50 5.55

hep 3.21 5.21

8.89

6.30

fct 3.77 3.53

At 4.2 K.

Unit cell has 2 atoms.

Unit cell has at least 50 atoms.

6=8.72.

At 123 K. 8 atoms in unit cell.

Diamond.

Graphite.planar structure.

£=8.26.

At 113 K. 8 atoms in unit cell.

At5K.

At 49 K.

b axis 1.001 times a axis.

hep lattice formed by H2 molecules.

At 2 K and 26 atm pressure.

At5K.

b=l.25. Unit cell has 8 atoms, set in pairs.

At5K.

At 58 K.

Unit cell has 58 atoms.

Unit cell has 20 atoms.

(Continued)

Introduction

Element

Mo Molybdenum

N2 Nitrogen

Na Sodium

Nb Niobium

Nd Neodymium

Ne Neon

Ni Nickel

Np Neptunium

O2 Oxygen

Os Osmium

P Phosphorus

Pa Protactinium

Pb Lead

Pd Palladium

Po Polonium

Pr Praseodymiurr

Pt Platinum

Pu Plutonium

Rb Rubidium

Re Rhenium

Rh Rhodium

Ru Ruthenium

S Sulfur

Sb Antimony

Sc Scandium

Se Selenium

Si Silicon

Sm Samarium

Tin

Sr Strontium

Ta Tantalum

Tb Terbium

Te Tellurium

Th Thorium

Ti Titanium

TI Thallium

Tm Thulium

U Uranium

V Vanadium

W Tungsten

Xe Xenon

Y Yttrium

Yb Ytterbium

Zn Zinc

Zr Zirconium

# Lattice

CUB

HEX

CUB

HEX

ORC

HEX

ORC

CUB

TET

CUB

59 CUB

HEX

CUB

MON

CUB

HEX

CUB

HEX

ORC

RHO

CUB

HEX

CUB

RHO

CUB

TET

CUB

HEX

CUB

HEX

CUB

HEX

CUB

HEX

CUB

HEX

bcc

fee

bcc

hep

fee

hep

bet

fee

hep

fee

bcc

hep

fee

hep

fee

hep

dia

fee

bcc

hep

fee

hep

fee

bcc

hep

bcc

fee

hep

fee

hep

3.15

4.16

5.64

4.29

3.30

3.66

4.43

3.52

2.65

4.72

5.50

2.74

3.31

7.17

3.93

4.95

3.89

5.16

3.67

3.92

6.18

5.59

2.76

3.80

2.70

c or

5.90

4.33

6.66

3.82

4.32

10.5

3.24

5.92

10.97

4.46

4.28

10.47

24.5

4.50

4.54

3.31

4.36

5.43

9.00

6.49

5.82

6.08

3.31

3.60

4.45

5.08

2.95

3.46

4.84

3.88

3.54

3.47

3.02

3.16

6.20

3.65

5.49

2.66

3.23

58°

5.27

4.93

23°

3.17

5.69

5.91

4.69

5.53

5.55

5.73

4.95

5.15

K. Unit cell

has 8

atoms.

At4K.

6=5.87.

Unit cell has

atoms.

At3K.

Black Phosphorus, 6=4.38, unit cell

has 8

atoms.

White phosphorus.

6=4.82,

a=102°, 16 atoms

unit cell.

At5K.

6=12.9.

Unit cell

has

128 atoms.

Unit cell has

atoms.

Unit cell has

atoms forming spiral chains.

Unit cell

has 3

atoms.

Gray

tin.

White tin. Unit cell has

atoms.

Unit cell

has 3

atoms forming spiral chains.

At 58

CUB=cubic, TET=tetragonal, ORC=orthorhombic, MON=Monoclinic, TRI=Triclinic,

HEX=hexagonal, RHO=Rhombohedral. Source: Wyckoff (1963-1971), vol. 1.

Chapter 2. Three-Dimensional Lattices

2.2 Monatomic Lattices

Monatomic lattices are those formed from entirely from atoms of a single element.

In the simplest of these, the atoms form a Bravais lattice. This means that atomic

positions are given by sums over three primitive vectors, which must be linearly

independent:

R = n\a\ + «2^2 +

«3«3·

"i- "2·

and

are

inte

ers

The

subscripts on the (2.1 )

vectors a, do not refer to components of the vec-

tors,

but to three separate three-vectors.

2.2.1 The Simple Cubic Lattice

Figure 2.1. The simple cubic lattice, showing the lattice spacing a. To view the stereo pair,

hold the image at a distance of around half a meter, cross eyes until left and right images

overlap, and focus on the combined image.

The simple cubic (sc) lattice is described by primitive vectors

a\=a{\

0 0), a

= a(0 1 0), α

= α(0 0 I) (2.2)

and pictured in Figure 2.1. The only element ever to choose this structure as its

ground state is polonium, partly because the simple cubic lattice has large holes

in it, and most elements prefer to take advantage of the fact that other configura-

tions pack space more efficiently. However, the simple cubic lattice does provide a

starting point for constructing more common structures.

2.2.2 The Face-Centered Cubic Lattice

The conventional unit cell of the face-centered cubic lattice (fee) is constructed by

putting atoms on the corners of a cube and then putting an atom on each face, as

shown in Figure 2.2. A set of primitive vectors describing an fee lattice of spacing

a is

2i = |(l 1 0), 3

= |(1 0 1), 3

= |(0 1 1). (2.3)

Notice that the lattice spacing, or lattice constant a describes the distance between

adjacent corners of the cube. It does not give the distance from an atom to its

nearest neighbor. The Wigner-Seitz cell is shown in Figure 2.2 (B).

Monatomic Lattices

Figure 2.2. (A) The conventional unit cell of the fee lattice, showing primitive vectors

and lattice spacing a (stereo pair). (B) Wigner-Seitz cell of the fee lattice. (C) If spheres

in an fee lattice are given radii a/(2\/2), they pack together. (D) The fee lattice can be

constructed by stacking two-dimensional triangular lattices in a regular sequence (stereo

pair).

The fee lattice is sometimes referred to as cubic close-packed, because if the

fee lattice is composed of balls with radius a/(2\/2), they stack neatly together, as

shown in Figure 2.2(C). Within the planes normal to the vector

[1,1,1],

the atoms

of an fee lattice lie in a two-dimensional triangular lattice, and one can view the

lattice as being built from layers of triangular lattices stacked one upon the other,

as shown in Figure 2.2(D). Primitive vectors to construct the lattice from this point

of view are the subject of Problem 2.

Chapter 2. Three-Dimensional Lattices

Figure 2.3. (A) The conventional unit cell of the body-centered cubic lattice (stereo pair).

(B) Wigner-Seitz cell of the bcc lattice.

The conventional unit cell of the fee lattice is four times as large as a primitive

unit cell, as one may determine by considering the fee lattice as a cubic lattice with

a basis (Problem 1).

More than twenty of the elements adopt the fee lattice, including copper, silver,

gold, and the noble gases at low temperatures.

2.2.3 The Body-Centered Cubic Lattice

The conventional unit cell of the body-centered cubic (bcc) lattice is constructed

by putting atoms on the corners of a cube and then putting an atom in the middle

to fill up the big hole in the center, as shown in Figure 2.3(A). A symmetrical set

of primitive vectors for a lattice of spacing a is

«i = |(1 1 -1), «2 = |(-1 1 1), «3 = |(1 "I 1), (2-4)

and the Wigner-Seitz cell appears in Figure 2.3 (B). As in the case of the fee lattice,

the lattice constant a refers to the distance between corners of the cube, not to the

distance between nearest neighbors.

The conventional unit cell of the bcc lattice is twice as large as a primitive

unit cell (Problem 1). Sixteen elements adopt the bcc structure at low temperature,

including iron and uranium.