Crystal Structure and DynamicsPaolo G. Radaelli, Michaelmas Term 2014

Part 1: Symmetry in the solid stateLectures 1-5

Web Site:http://www2.physics.ox.ac.uk/students/course-materials/c3-condensed-matter-major-option

Bibliography

◦ C. Hammond The Basics of Crystallography and Diffraction, Oxford University Press (fromBlackwells). A well-established textbook for crystallography and diffraction.

◦ D.S. Sivia, Elementary Scattering Theory for X-ray and Neutron Users, Oxford UniversityPress, 2011. An excellent introduction to elastic and inelastic scattering theory and theFourier transform.

◦ Martin T. Dove, Structure and Dynamics: An Atomic View of Materials, Oxford Master Seriesin Physics, 2003. Covers much the same materials as Lectures 1-11.

◦ T. Hahn, ed., International tables for crystallography, vol. A (Kluver Academic Publisher,Dodrecht: Holland/Boston: USA/ London: UK, 2002), 5th ed. The International Tables forCrystallography are an indispensable text for any condensed-matter physicist. It currentlyconsists of 8 volumes. A selection of pages is provided on the web site. Additional samplepages can be found on http://www.iucr.org/books/international-tables.

◦ C. Giacovazzo, H.L. Monaco, D. Viterbo, F. Scordari, G. Gilli, G. Zanotti and M. Catti, Fun-damentals of crystallography (International Union of Crystallography, Oxford UniversityPress Inc., New York)

◦ Paolo G. Radaelli, Symmetry in Crystallography: Understanding the International Tables, Oxford University Press (2011). Contains a discussion of crystallographic symmetryusing the more abstract group-theoretical approach, as described in lectures 1-3, but inan extended form.

◦ ”Visions of Symmetry: Notebooks, Periodic Drawings, and Related Work of M. C. Escher”,W.H.Freeman and Company, 1990. A collection of symmetry drawings by M.C. Escher,also to be found here:http://www.mcescher.com/Gallery/gallery-symmetry.htm

◦ Neil W. Ashcroft and N. David Mermin, Solid State Physics, HRW International Editions,CBS Publishing Asia Ltd (1976) is now a rather old book, but, sadly, it is probably still thebest solid-state physics book around. It is a graduate-level book, but it is accessible tothe interested undergraduate.

1

Contents

1 Lecture 1 — Translational symmetry in crystals: unit cells and lattices 31.1 Introduction: crystals and their symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Lattices and coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Bravais lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Choice of basis vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.3 Centred lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Distances and angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Dual basis and coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Dual basis in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6 Dot products in reciprocal space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.7 A very useful example: the hexagonal system in 2D . . . . . . . . . . . . . . . . . . . 111.8 Reciprocal lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.8.1 Centring extinctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Lecture 2 — Rotational symmetry 162.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 Generalised rotations expressed in terms of matrices and vectors . . . . . . . . . . . . 16

2.2.1 Rotations about the origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.2 Rotations about points other than the origin . . . . . . . . . . . . . . . . . . . . 172.2.3 Roto-translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.4 Generalisation of symmetry to continuous functions and patterns . . . . . . . . 18

2.3 Some elements of group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3.1 Symmetry operators as elements of crystallographic groups . . . . . . . . . . . 192.3.2 Optional : the concept of group class . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Symmetry of periodic patterns in 2 dimensions: 2D point groups and wallpaper groups 212.4.1 Point groups, wallpaper groups, space groups and crystal classes . . . . . . . 222.4.2 Point groups in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4.3 Bravais lattices in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Lecture 3 — Symmetry in 3 dimensions 263.1 New symmetry operators in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.1.1 Inversion and roto-inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.1.2 Glide planes and screw axes in 3D . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Understanding the International Tables of Crystallography . . . . . . . . . . . . . . . . 293.2.1 ITC information about crystal structures . . . . . . . . . . . . . . . . . . . . . . 293.2.2 ITC information about diffraction patterns . . . . . . . . . . . . . . . . . . . . . 29

4 Lecture 4 — Fourier transform of periodic functions 334.1 Fourier transform of lattice functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Atomic-like functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.3 Centring extinctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.4 Extinctions due to roto-translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.5 The symmetry of |F (q)|2 and the Laue classes . . . . . . . . . . . . . . . . . . . . . . 35

5 Lecture 5 — Brillouin zones and the symmetry of the band structure 375.1 Symmetry of the electronic band structure . . . . . . . . . . . . . . . . . . . . . . . . . 385.2 Symmetry properties of the group velocity . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2.1 The oblique lattice in 2D: a low symmetry case . . . . . . . . . . . . . . . . . . 415.2.2 The square lattice: a high symmetry case . . . . . . . . . . . . . . . . . . . . . 42

5.3 Symmetry in the nearly-free electron model: degenerate wavefunctions . . . . . . . . 442

1 Lecture 1 — Translational symmetry in crystals: unitcells and lattices

1.1 Introduction: crystals and their symmetry

In 1928, Swiss physicist Felix Bloch obtained his PhD at University of Leipzig,under the supervision of Werner Heisenberg. In his doctoral dissertation, hepresented what we now call the quantum theory of crystals, and what a mo-mentous occasion that was! Much of the technology around us — from digitalphotography to information processing and storage, lighting, communication,medical imaging and much more is underpinned by our understanding of thebehavior of electrons in metals (like copper) and semiconductors (like silicon).Today’s contemporary physics also focuses on crystals, in particular on prop-erties that cannot be described in terms of simple one-electron physics, suchas high-temperature superconductivity.

One may ask what is special about a crystal as compared, for example, toa piece of glass. From previous introductory courses, you already know partof the answer: crystals have a lattice, i.e., atoms are found regularly on thecorners of a 3-dimensional grid. You also know that some crystals have abase, comprising several atoms within the unit cell, which is replicated byplacing copies at all nodes of the lattice.

At a more fundamental level, crystals differ from glasses and liquids becausethey have fewer symmetries. In a liquid or glass, properties look identical onaverage if we move from any point A to any point B (translation) and if welook in any direction (rotation). By contrast in a crystal, properties vary ina periodic manner by translation, and are only invariant by certain discreterotations. The theory of symmetries in crystals underpin all of the crystalphysics we have mentioned and much more. Here, we will start by describingtranslational symmetry in a more general way that you might have seen so far,suitable for application to crystals with non-orthogonal lattices. We will thenlook at ways of classifying rotational symmetries systematically, and we willlearn how to use the International Tables of Crystallography (ITC hereafter),an essential resource for condensed matter physics research.

1.2 Lattices and coordinates

Let us start by looking at the cubic, tetragonal and orthorhombic lattices youshould already be familiar with. The nodes (i.e., unit cell origins) of an or-

3

thorhombic lattice are expressed by means of vectors 1 of the form:

rn = a inx + b jny + c knz (1)

where a, b, c are the lattice constants (a = b for the tetragonal lattice, a = b = c

for the cubic lattice) and i, j, k are unit vectors along the x, y and z axes,while nx, ny and nz are arbitrary integers. We can interpret these vectors asinvariant translation operators — if points A and B are separated by one suchvector, the properties of the crystal are identical in A and B.

The position of points in the crystal that are not on lattice nodes can be ex-pressed as

r = a ix + b jy + c k z (2)

One can see the relation between x, y and z (which are dimensionless) andthe usual Cartesian coordinates, which have dimensions. We can define thebasis vectors (not to be confused with the unit cell base) a = a i, b = b jand c = c k. In the cubic, tetragonal and orthorhombic lattices, the basisvectors are orthogonal, but this is not so in other most important cases —for example, that of hexagonal crystal like graphite, where the angle betweena = b 6= c, α = β = 90◦ and γ = 120◦. We can, however, generalise thenotion of basis vectors so that it applies to cases such as this and even themost general (triclinic) lattice. In synthesis:

• In crystallography, we do not usually employ Cartesian coordinates. In-stead, we employ coordinate systems associated with appropriate ba-sis vectors.

• Basis vectors have the dimension of a length, and coordinates (positionvector components) are dimensionless.

• We can denote the basis vectors as ai , where the correspondence withthe usual crystallographic notation is

a1 = a; a2 = b; a3 = c (3)1Here, we will call vector an object of the form shown in eq. 2, containing both components

and basis vectors. In this form, a vector is invariant by any coordinate transformation, whichaffect both components and basis vectors. Both basis vectors and components (coordinates)are written as arrays — see below. For a concise introduction to this kind of notation, see ITCvolume B, chapter 1.

4

• We will sometimes employ explicit array and matrix multiplication for clarity.In this case, the array of basis vectors is written as a row2, as in [a] =

[a1 a2 a3].

• With this notation, a generic position within the crystal will be defined bymeans of a position vector, written as

r =∑i

aixi = ax+ by + cz (4)

Points within the unit cell at the origin of the lattice have 0 ≤ x ≤ 1,0 ≤ y ≤ 1, 0 ≤ z ≤ 1 — the so-called fractional coordinates.

• We can also define other vectors on the lattice, for example, the instanta-neous velocity of a vibrating atom. The components of a generic vectorv will be denoted as vi, where

v1 = vx; v2 = vy; v3 = vz; (5)

• Components will be expressed using column arrays, as in [v] =

v1

v2

v3

• A generic vector is then written as

v =∑i

aivi = a1v1 + a2v

2 + a3v3 (6)

1.2.1 Bravais lattices

From eq. 4, we can deduce the position of the lattice points in the mostgeneral case:

rn =∑i

aini = anx + bny + cnz (7)

As you already know, in 3 dimensions there are 14 Bravais lattices (from 19-century French physicist Auguste Bravais) — see fig. 1. The lattices havedistinct rotational symmetries, belonging to one of the 7 crystal systems (see

2By writing basis vectors as rows and coordinates or components as columns, vectorsbecome the usual product of a row and a column array. Moreover, upon coordinate trans-formations, the transformation matrices for basis vectors and components are inverse of oneanother. The subscript and superscript notation also emphasises this — note that vectorshave fully “contracted” indices.

5

below), and distinct topologies, since some crystal systems admit both primi-tive and centred lattices.

1.2.2 Choice of basis vectors

There can be many choices of basis vectors. In order to describe the samelattice, the two transformation matrices converting one basis set to the otherand vice versa must be integer matrices, which implies that their determinantis exactly 1 or -1 (depending on the handedness of the basis sets). All thesechoices are equivalent from the point of view of the translational symmetry.Whenever possible, the basis vectors are chosen so that the correspondingunit cell has the full rotational symmetry of the lattice.

1.2.3 Centred lattices

Making such a choice is not possible for the so-called centred lattices, so onehas to make a choice of either using a primitive cell or using a non-primitivecell (the conventional cell) with the full rotational symmetry of the lattice. Youhave already encountered some important cases of centred lattices, such asthe BCC and FCC cubic lattices of many metals — the others are shown infig. 1. Body centred cells (e.g., cubic BCC) and cells with one pair of centredfaces (A, B or C centring) have a 2-atom base. Cells with all faces centred(such as cubic FCC) have a 4-atom base. The conventional hexagonal cellof rhombohedral (trigonal) lattices have a 3-atom base. All lattice points de-fine invariant translations as before, regardless of whether points are at thecorners of the unit cell or at ”centring positions”. For these lattices, we canalways choose to use either primitive or conventional basis vectors:

• When primitive basis vectors are used, coordinates of points related bytranslation will differ by integral values of x,y and z.

• When conventional basis vectors are used, coordinates of points relatedby translation will differ by either integral or simple fractional (either n/2or n/3) values of x,y and z.

1.3 Distances and angles

• The dot product between two position vectors is given explicitly by

6

Figure 1: The 14 Bravais lattices in 3D. The R-centred hexagonal cell can be used as aconventional cell for the trigonal lattice.

7

r1 · r2 = a · ax1x2 + b · bx1x2 + c · c z1z2 +

+a · b [x1y2 + y1x2] + a · c [x1z2 + z1x2] + b · c [y1z2 + z1y2] =

= a2 x1x2 + b2 x1x2 + c2 z1z2 + ab cos γ [x1y2 + y1x2]

+ac cosβ [x1z2 + z1x2] + bc cosα [y1z2 + z1y2] (8)

• More generically, the dot product between two vectors is written as

v · u =∑i

aivi ·∑j

ajvj =∑i,j

[ai · aj ]uivj (9)

• The quantities in square bracket represent the elements of a symmetricmatrix, known as the metric tensor. The metric tensor elements havethe dimensions of length square.

Gij = ai · aj (10)

Calculating lengths and angles using the metric tensor

• You are generally given the lattice parameters a, b, c, α, β and γ. In terms of these, themetric tensor can be written as

G =

a2 ab cos γ ac cosβab cos γ b2 bc cosαac cosβ bc cosα c2

(11)

• To measure the length v of a vector v:

v2 = |v|2 = [ v1 v2 v3 ]G

v1

v2

v3

(12)

• To measure the angle θ between two vectors v and u:

cos θ =1

uv[ u1 u2 u3 ]G

v1

v2

v3

(13)

1.4 Dual basis and coordinates

• Let us assume a basis vector set ai for our vector space as before, and letus consider the following set of new vectors.

8

bi = 2π∑k

ak(G−1)ki (14)

From Eq. 10 follows:

ai · bj = ai · 2π∑k

ak(G−1)ki = 2π∑k

Gik(G−1)kj = 2πδji (15)

• Note that the vectors bi have dimensions length−1. Since the bi are linearlyindependent if the ai are, one can use them as new basis vectors, form-ing the so-called dual basis. This being a perfectly legitimate choice,can express any vector on this new basis, as

q =∑i

qibi (16)

• We can write any vector on this new basis, but vectors expressed us-ing dimensionless coordinates on the dual basis have dimensionslength−1, and cannot therefore be summed to the position vectors.

• We can consider these vectors as representing the position vectors of aseparate space, the so-called reciprocal space.

• The dot product between position vectors in real and reciprocal space is adimensionless quantity, and has an extremely simple form (eq. 17):

q · v = 2π∑i

qixi (17)

• In particular, the dot product of integral multiples of the original basis vec-tors (i.e., direct or real lattice vectors), with integral multiples of thedual basis vectors (i.e., reciprocal lattice vectors) are integral multi-ples of 2π. This property will be used extensively to calculate Fouriertransforms of lattice functions.

9

Recap of the key formulas for the dual basis

• From direct to dual bases (eq. 14)

bi = 2π∑k

ak(G−1)ki

• Dot product relation between the two bases (eq. 15)

ai · bj = 2πδji

• Dot product between vectors expressed on the two different bases (eq. 17)

q · v = 2π∑i

qixi

1.5 Dual basis in 3D

• In 3 dimensions, there is a very useful formula to calculate the dual basisvectors, which makes use of the properties of the vector product:

b1 = 2πa2 × a3

a1 · (a2 × a3)(18)

b2 = 2πa3 × a1

a1 · (a2 × a3)

b3 = 2πa1 × a2

a1 · (a2 × a3)

Note that

v = a1·(a2 × a3) = abc(1− cos2 α− cos2 β − cos2 γ + 2 cosα cosβ cos γ

)1/2(19)

is the unit cell volume.

• In crystallographic textbooks, the dual basis vectors are often written as a∗,b∗ and c∗.

1.6 Dot products in reciprocal space

• As we shall see later, it is very useful to calculate the dot product betweentwo vectors in reciprocal space. This can be tricky in non-Cartesiancoordinate systems.

10

• A quick way to do this is to determine the reciprocal-space metric tensor,which is related to the real-space one.

• The reciprocal-space metric tensor is G = (2π)2G−1, so that, for two reciprocal-space vectors q and r:

q · r =∑i,j

Gijqirj (20)

1.7 A very useful example: the hexagonal system in 2D

• The hexagonal system in 2D has a number of important applications in con-temporary solid-state physics problems, particularly for carbon-basedmaterials such as graphene and carbon nanotubes.

• By crystallographic convention, the real-space basis vectors form an anglea ∧ b of 120◦ (this is also true in 3D).

• The real-space metric tensor is therefore:

G = a2[

1 −1/2−1/2 1

](21)

• From eq. 20, we can find the reciprocal-space metric tensor :

G =(2π)2

a24

3

[1 1/2

1/2 1

](22)

• It follows, for example, that the length of a vector in reciprocal space isgiven by:

q =2π

a

√4

3(h2 + hk + k2) (23)

1.8 Reciprocal lattices

Having defined the dual basis, we can use it to construct a lattice in reciprocalspace, known as the reciprocal lattice.

• Reciprocal-space vectors are described as linear combinations of the re-ciprocal or dual basis vectors, (dimensions: length−1) with dimen-sionless coefficients.

11

• Reciprocal-lattice vectors (RLV ) are reciprocal space vectors with inte-gral components. These are known as the Miller indices and are usu-ally notated as hkl.

It can be easily shown that all equivalent primitive cells in real space de-fine equivalent primitive dual bases and reciprocal space lattices 3. However,a problem arises when dealing with conventional bases, since the recipro-cal lattice they generate is patently different from that of any of the primitivebases. This is illustrated with an example in fig. 2. One can see that theconventional reciprocal-space unit cell (associated with the dual basis a∗c andb∗c) is smaller than the primitive reciprocal-space unit cell (a∗p and b∗p), andtherefore generates more reciprocal lattice points. Bragg scattering is neverobserved at these points, for the reasons explained here below — we will saythat they are extinct by centering.

1.8.1 Centring extinctions

• The dot product of real and reciprocal space vectors expressed in theusual coordinates is

q · v = 2π∑i

qivi (24)

• The dot product of real and reciprocal lattice vectors is:

- If a primitive basis is used to construct the dual basis, 2π times aninteger for all q and v in the real and reciprocal lattice, respectively.In fact, as we just said, all the components are integral in this case.

- If a conventional basis is used to construct the dual basis, 2π

times an integer or a simple fraction of 2π. In fact the componentsof the centering vectors are fractional.

• Therefore, if a conventional real-space basis is used to construct the dualbasis, only certain reciprocal-lattice vectors will yield a 2πn dotproduct with all real-lattice vectors — a necessary (bot not suffi-cient, see below) condition to observe Bragg scattering at these points.These reciprocal-lattice vectors are exactly those generated by thecorresponding primitive basis.

3This is due to the fact that the transformation matrix of the dual basis is the inverse matrixof the transformation matrix of the real-space basis. In this course, I will shy away from a fulltreatment of crystallographic coordinate transformations — a huge subject in itself, but onethat at this level does not add much to the comprehension of the subject.

12

ap*  

bp*  

bc*  

ac*  

bc*  

ac*  

ac  

bc  

ap*  

bp*  

ap  bp  

Conven,onal  u.c.   Primi,ve  u.c.  

Real  Space  

Reciprocal  Space  

Figure 2: C-centred cells: construction of reciprocal basis vectors on the real-space lattice(top) and corresponding reciprocal lattice (bottom). One can see that the primitive reciprocalbasis vectors generate fewer points (closed circles) than the conventional reciprocal basisvectors (open an closed points). Only closed points correspond to observed Bragg peaks, theothers are extinct by centring.

13

• Each non-primitive lattice type has centring extinctions, which can be ex-pressed in terms of the Miller indices hkl (see table 1).

Table 1: Centering extinction and scattering conditions for the centered lat-tices. The “Extinction” columns lists the Miller indices of reflections that areextinct by centering, i.e., are “extra” RLV generated as a result of using aconventional basis instead of a primitive one. The complementary “Scatter-ing” column corresponds to the listing in the International Tables vol. A, andlists the Miller indices of “allowed” reflections. “n” is any integer (positive ornegative).

Lattice type Extinction Scattering

P none all

A k + l = 2n+ 1 k + l = 2nB h+ l = 2n+ 1 h+ l = 2nC h+ k = 2n+ 1 h+ k = 2n

F k + l = 2n+ 1 or k + l = 2n andh+ l = 2n+ 1 or h+ l = 2n andh+ k = 2n+ 1 h+ k = 2n

I h+ k + l = 2n+ 1 h+ k + l = 2n

R −h+ k + l = 3n+ 1 or −h+ k + l = 3n−h+ k + l = 3n+ 2

• The non-exctinct reciprocal lattice points also form a lattice, which is natu-rally one of the 14 Bravais lattices. For each real-space Bravais lattice,tab. 2 lists the corresponding RL type.

14

Table 2: Reciprocal-lattice Bravais lattice for any given real-space Bravaislattice (BL).

Crystal system Real-space BL Reciprocal-space BLTriclinic P P

Monoclinic C C

Orthorhombic P PA or B or C A or B or C

I FF I

Tetragonal P PI I

Trigonal P PR R

Hexagonal P P

Cubic P PI FF I

15

2 Lecture 2 — Rotational symmetry

2.1 Introduction

While the first experimental confirmation of translational symmetry in crystalswas provided by the X-ray diffraction experiments performed in 1914 by W.Hand W.L. Bragg (after and son), the existence of rotational symmetry hadbeen known for centuries, simply by the observation of crystal shapes. In thepresence of rotational symmetry, some atoms will have “companions”, relatedto them by rotation around a certain axis or mirroring by certain planes. Otheratoms, positioned on top of certain symmetry elements, will have fewer or nocompanions. The number of symmetry-related companions is known as themultiplicity of that atomic site. It is therefore evident that knowing the fullcrystal symmetry (translation + rotation) allows a simplified description of thecrystal structure and calculations of its properties in terms of fewer uniqueatoms.

2.2 Generalised rotations expressed in terms of matrices andvectors

2.2.1 Rotations about the origin

In Cartesian coordinates, a rotation about the origin is expressed as:

x(2)y(2)z(2)

= R

x(1)y(1)z(1)

(25)

where x(1), y(1), z(1) are the coordinates of the original point, x(2), y(2), z(2)

are the coordinates of the companion point and R in an orthogonal matrix withdet(R) = +1. Matrices with det(R) = −1 represent the so-called improperrotations, which include mirror reflections (we will see other examples later).

In crystallographic (non-Cartesian) coordinates, eq. 25 is modified as:

x(2)y(2)z(2)

= D

x(1)y(1)z(1)

(26)

where in general D is a non-orthogonal matrix, which, however, still hasdet(D) = ±1

16

2.2.2 Rotations about points other than the origin

In general, the internal rotational symmetry of crystals will include rotationsaround points that are not the origin, although, unlike the case of a liquidor glass, only a discrete set of such rotations exists in crystals. In Cartesiancoordinates, the expression for a proper of improper rotation through the pointx0, y0, z0, is

x(2)y(2)z(2)

= R

x(1)− x0y(1)− y0z(1)− z0

+

x0y0z0

=

x(2)y(2)z(2)

= R

x(1)y(1)z(1)

+

txtytz

(27)

where

txtytz

=

x0y0z0

− R

x0y0z0

(28)

for example, a mirror plane perpendicular to the x axis and located at x = 1/4

will produce the following transformation:

x(2)y(2)z(2)

=

−x(1) + 1/2y(1)z(1)

(29)

Symmetry operators written in the form of eq. 27 are said to be in normalform. They are written as the composition (application of two operators insuccession) of a rotation (proper or improper) — the rotational part, followedby a translation — the translational part.

Note that when using non-Cartesian coordinates the normal form of symme-try operators remains the same:

x(2)y(2)z(2)

= D

x(1)y(1)z(1)

+

txtytz

(30)

In general, D is not orthogonal, but its determinant is still ±1.

17

2.2.3 Roto-translations

A special class of symmetries in crystallography is represented by proper orimproper rotations followed by a non-lattice translation. A typical example ofthis is the screw axis, which produces a companion point by rotating (e.g.,by 120◦) and then translating along the rotation axis (e.g., by 1/3 of a latticetranslation — the notation of this operation is 31, see below). Clearly we canstill use the normal form in eq. 30 to express roto-translations, it is sufficientto add the additional translation to the vector t.

2.2.4 Generalisation of symmetry to continuous functions and patterns

Up to now, we have discussed the symmetry of crystals in terms of atomsand their companions. However, the symmetry of crystals is much deeperthan this: the electron density in a crystal (a continuous function) has thesame symmetry as the atoms. Continuous functions in reciprocal space (e.g.,phonon or electron dispersion relations) also have crystallographic symme-tries that are related to the symmetry of the crystal (see below). In the spiritof describing real experiments, this means that smaller regions of the recip-rocal space need to be measured (a quadrant, an octant etc.) to reconstructthe whole dispersion. Symmetry also restricts, for example, certain compo-nents of the group velocity. We will use the word pattern to mean, generically,sets of points (e.g., atoms) or continuous functions, and with reference to the“wallpaper” patterns we will discuss below.

In describing the symmetry of isolated objects or periodic systems, one de-fines operations (or operators) that describe transformations of the pattern.We create the new pattern from the old pattern by associating a point p2 toeach point p1 (as shown in eq. 30 in terms of coordinates) and transferringthe “attributes” (for example, the electron density) of p1 to p2. This trans-formation preserves distances and angles, and is in essence a combinationof translations, reflections and rotations. If the transformation is a symmetryoperator, the old and new patterns are indistinguishable.

2.3 Some elements of group theory

If two symmetry operators are known to exist in a crystal, say O1 and O2,then we can construct a third symmetry operator by applying O1 and O2 oneafter the other. This means that the set of all symmetry operators is closed by“application in succession” (a form of composition — see below). Likewise,

18

we can always create the inverse of an operator, e.g., by reversing the senseof rotation and the direction of translations etc. These considerations indicatethat the set of all symmetry operators of a crystal has the mathematical struc-ture of a group. A group is defined as a set of elements with a defined binaryoperation known as composition, which obeys the following rules.

• A binary operation (usually called composition or multiplication) must bedefined. We indicated this with the symbol “◦”.

• Composition must be associative: for every three elements f , g and h ofthe set

f ◦ (g ◦ h) = (f ◦ g) ◦ h (31)

• The “neutral element” (i.e., the identity, usually indicated with E) must exist,so that for every element g:

g ◦ E = E ◦ g = g (32)

• Each element g has an inverse element g−1 so that

g ◦ g−1 = g−1 ◦ g = E (33)

• A subgroup is a subset of a group that is also a group.

• A set of generators is a subset of the group (not usually a subgroup) thatcan generate the whole group by composition. Infinite groups (e.g., theset of all lattice translations) can have a finite set of generators (theprimitive translations).

2.3.1 Symmetry operators as elements of crystallographic groups

Note that, in the case of crystallographic operators:

• Composition of two symmetry operators is the application of these oneafter another. When denoting the composition O1 ◦ O2, we mean thatthe operator to the right is applied first.

• Composition is not commutative. As shown in the example in fig. 3, theorder of application of rotations and translations does matter. Therefore,crystallographic groups are discrete non-Abelian groups.

19

m10 45º

m11

4+

������

����

m10 4+

������

����

45º

m11

m10◦4+ 4+◦m10

Figure 3: Left: A graphical illustration of the composition of the operators 4+ and m10 togive 4+ ◦m10 = m11. The fragment to be transformed (here a dot) is indicated with ”start”, andthe two operators are applied in order one after the other, until one reaches the ”end” position.Right: 4+ and m10 do not commute: m10 ◦ 4+ = m11 6= m11.

• Graphs or symmetry elements are sets of invariant points of a pattern un-der one or more symmetry operators.

Symmetry  operator:    transforms  one  part  of  the  pa.ern  into  another  (in  this  case  by  reflec5on)  

Symmetry  element  or  graph:  mirror  plane.    A  set  of  invariant  points  under  the  symmetry  operator  

Another  symmetry  element  or  graph,  related  to  the  other  mirror  plane  by  4-­‐fold  rota5on.    Belongs  to  the  same  class.  

Figure 4: Symmetry operators and symmetry elements (graphs). A symmetry operatoris the point by point transformation of one part of the pattern into another (arrow). In thelanguage of coordinates, the attributes of point x(1), y(1), z(1), e.g., colour, texture etc., willbe transferred to point x(2), y(2), z(2). Symmetry elements (shown by lines) are points leftinvariant by a given set of transformations.

2.3.2 Optional: the concept of group class

Group classes are an optional and non-examinable topic. However, they arehugely important for a more advanced understanding of symmetry in crystal-

20

lography, so I include here a few basic notions.

• In group theory, classes are subset of a group (not in general subgroups)related by are conjugation, which is the equivalent of similarity trans-formations in matrices. h′ is conjugated with h if there is a g ∈ G sothat:

h′ = g ◦ h ◦ g−1 (34)

Figure 5: Left. A showflake by by Vermont scientist-artist Wilson Bentley, c. 1902. RightThe symmetry group of the snowflake, 6mm in the ITC notation. The group has 6 classes, 5marked on the drawing plus the identity operator E. Note that there are two classes of mirrorplanes, marked “1” and “2” on the drawing. One can see on the snowflake picture that theirgraphs contain different patterns.

• In crystallography, there is an extremely important relation between conju-gation of symmetry operators and symmetry elements: if two symmetryoperators O1 and O2 are conjugated as O2 = H ◦ O1 ◦H−1, then theirsymmetry elements (graphs) are transformed into one another by thesymmetry of H.

2.4 Symmetry of periodic patterns in 2 dimensions: 2D pointgroups and wallpaper groups

We will start our description of the crystallographic symmetry groups by ex-amining the problem in 2 dimensions. This is much simpler than the full 3Dcase, and it is amenable to be explored both by the more conventional method

21

Graph  representa+on  

Secondary  symmetry  direc+on  Ter+ary  symmetry  direc+on  

Site  mul+plicity  Wychoff  le8er  

Site  symmetry  

Equivalent  points  for  general  posi+on  (here  12  c)  Primary  symmetry  direc+on  

Figure 6: An explanation of the most important symbol in the Point Groups subsection of theInternational Tables for Crystallography. All the 10 2D point groups (11 if one counts 31m and3m1 as tow separate groups) are reproduced in the ITC (see lecture web site). The graphicalsymbols are described in fig. 10. Note that primary, secondary and tertiary symmetries neverbelong to the same class.

of matrices and vectors and with the ore abstract and visual method of thesymmetry elements.

2.4.1 Point groups, wallpaper groups, space groups and crystal classes

• Point groups (in any dimensions) are finite groups of pure rotations/reflectionsaround a fixed point (origin).

• Space groups in 3D describe the full translational and rotational symme-try of crystals. They are infinite discrete groups, and they contain theinfinite group of translations as a subset.

• Plane (or wallpaper) groups are the equivalent to space groups in 2D.They are so called because they describe the symmetry of all possiblewallpapers.

• Crystal classes (in any dimension, not to be confused with conjugation

22

classes — see above) are point groups obtained from a space or wall-paper group by removing the translational part of the operators4. It iseasy to see how this is done by writing the operators in normal form,eq. 30

Table 3: The 17 wallpaper groups. The symbols are obtained by combiningthe 5 Bravais lattices with the 10 2D point groups, and replacing g with msystematically. Strikeout symbols are duplicate of other symbols.crystal system crystal class wallpaper groups

oblique1 p12 p2

rectangularm pm, cm,pg, cg

2mm p2mm, p2mg (=p2gm), p2gg, c2mm, c2mg, c2gg

square4 p4

4mm p4mm, p4gm, p4mg

hexagonal3 p3

3m1-31m p3m1, p3mg, p31m, p31g6 p6

6mm p6mm, p6mg, p6gm, p6gg

2.4.2 Point groups in 2D

There is an infinite number finite point groups in 2D and above. Examplesare groups or rotations about an axis by angles that are rational fractions of2π.

However, only a small number of these groups are relevant for crystallogra-phy, since all others are incompatible with a lattice. There are 10 crystallo-graphic point groups in 2D (corresponding to the 10 crystal classes in tab. 3)and 32 in 3D (see web site for a complete list of crystallographic point groupsand their properties from the ITC)5.

2.4.3 Bravais lattices in 2D

In 2D, there are 5 types of lattices: oblique, p-rectangular, c-rectangular,square and hexagonal (see Fig. 7 — more on this topic in the Supplementary

4In fact, it can be easily shown that the rotational part of g ◦ f is RgRf . Therefore, therotational parts of the operators of a wallpaper group or a space group form themselves agroup, which is clearly one of the 10 point groups in 2D. This is called the crystal class of thewallpaper group.

5A very interesting set of “sub-periodic” groups is represented by the so-called friezegroups, which describe the symmetry of a repeated pattern in 1 dimension. There are only7 frieze groups, which are very simple to understand given the small number of operatorsinvolved. For a description of the frieze groups, with some nice pictorial example, see theSupplementary Material.

23

Material). In the c-rectangular lattice, no primitive cell has the full symmetryof the lattice, as in the 3D centred lattices. It is therefore convenient to adopta non-primitive centred cell having double its volume. As in the 3D case, theorigin of the unit cell is to a large extent arbitrary. It is convenient to choose itto coincide with a symmetry element.

�p�� �c�

Oblique  

Rectangular  

Square   Hexagonal  

Figure 7: The 5 Bravails lattices in 2 dimensions.

• Although there are an infinite number of symmetry operators (and symme-try elements), it is sufficient to consider elements in a single unit cell.

• Glides. This is a composite symmetry (roto-translation — see above),which combines a translation with a parallel reflection, neither of whichon its own is a symmetry operator. In 2D groups, the glide is indicatedwith the symbol g. Twice a glide translation is always a symmetry trans-lation: in fact, if one applies the glide operator twice as in g ◦ g, one ob-tains a pure translation (since the two mirrors cancel out), which there-fore must be a lattice (symmetry) translation.

24

• Walpaper groups. The 17 Plane (or Wallpaper) groups describe the sym-metry of all periodic 2D patterns (see tab. 3). The decision three in fig.8 can be used to identify wallpaper groups. 6

64

Has

mir

rors

?

32

Has

mir

rors

?H

as m

irro

rs?

Has

mir

rors

?

p6p6

mm

Axe

s o

n

mir

rors

?

p4

p4m

mp4

gm

All

axes

on

mir

rors

?

p3

p3m

1

1

p31m

Has

ort

ho

go

nal

mir

rors

?

Has

glid

es?

p2p2

gg p2m

g

Has

ro

tati

on

so

ff m

irro

s?

c2m

mp2

mm

Has

mir

rors

?

Has

glid

es?

pm

cm

Has

glid

es?

pg

p1

Axi

s o

f h

igh

est

ord

er

Figure 8: Decision-making tree to identify wallpaper patterns. The first step (bottom) is toidentify the axis of highest order. Continuous and dotted lines are ”Yes” and ”No” branches,respectively. Diamonds are branching points.

6For a more complete description of wallpaper groups and of the symmetry of the underly-ing 2D lattices, see the Supplementary Material. Examples of Escher drawings with differentwallpaper group symmetries will be shown in the lectures (see lecture slides on the web site).

25

3 Lecture 3 — Symmetry in 3 dimensions

Symmetry in 3 dimensions is not dissimilar from the 2D case, but it is richer,having more Bravais lattices (14 instead of 5) and many more space groups(219 instead of 17 — the ITC lists 230 because it distinguishes left- and right-handed versions of otherwise identical groups). The rotation axes in 3D arethe same as in 2D, but they are combined in different ways, most notably, inthe cubic groups.

3.1 New symmetry operators in 3D

3.1.1 Inversion and roto-inversion

In 3D, we can also distinguish between proper and improper rotations, be-cause the inversion operator I, which has matrix representation −1 (minusthe identity matrix) in all coordinate systems, has a negative determinant. Allimproper rotations can therefore be obtained by composing a proper rota-tion with the inversion — itself an improper rotation — as I ◦ r. The identitycommutes with all other operators. Here below are some more properties ofimproper operators in 3D:

• I ◦ 2 = m⊥, so the mirror plane is the improper operator corresponding toa 2-fold rotation axis perpendicular to it.

• There are three other significant improper operators in 3D, known as roto-inversions. They are obtained by composition of an axis r of order

higher than two with the inversion, as I ◦r. These operators are 3 ( ),

4 ( ) and 6 ( ), and their action is summarized in Fig. 9. The sym-bols are chosen to emphasize the existence of another operator insidethe ”belly” of each new operator. Note that 3◦ 3◦ 3 = 33 = I, and 34 = 3,i.e., symmetries containing 3 also contain the inversion and the 3-foldrotation. Conversely, 4 and 6 do not automatically contain the inversion.In addition, symmetries containing both 4 (or 6) and I also contain 4 (or6).

3.1.2 Glide planes and screw axes in 3D

Glide planes and 2-fold screw axes in 3D are the improper and proper version(respectively) of the same 2D operator, which we have indicated with the

26

-­‐  +  

-­‐   +  

+   -­‐  

+/-­‐  

+/-­‐  

+/-­‐  

+  

-­‐  

+  

-­‐  

Figure 9: Action of the 3 , 4 and 6 operators and their powers. The set of equivalent pointsforms a trigonal antiprism, a tetragonally-distorted tetrahedron and a trigonal prism, respec-tively. Points marked with ”+” and ”-” are above or below the projection plane, respectively.Positions marked with ”+/-” correspond to pairs of equivalent points above and below theplane.

symbol g. Glide planes in 3D consist in the sequential application of a mirrorplane and of a non-lattice translation parallel to the plane (any perpendicularcomponent being equivalent to a translation of the plane itself). Glide planeshave different symbols, depending on the orientation of the glide vector withrespect to the basis vectors and/or to the plane of the projection (fig. 10).

There is also a new type of operator, resulting from the compositions ofproper rotations (including two-fold axes) with translation parallel tothem. These are known as screw axes. For an axis of order n, the nth

power of a screw axis is a pure translation, and must therefore be a lat-tice translation t . Therefore, the translation component must be m

n t, wherem is an integer. We can limit ourselves to m < n, all the other operators be-ing composition with lattice translations. Roto-translation axes are thereforeindicated as nm, as in 21, 63 etc. (fig. 10).

27

m a,b or c n e

31

32

41 43

42 61 65

62 64

63

2 21

d

18

38

2 3 4 6 m g

1 3 4 6

Figure 10: The most important graph symbols employed in the International Tables to de-scribe 3D space groups. Fraction next to the symmetry element indicate the height (z coordi-nate) with respect to the origin.

28

3.2 Understanding the International Tables of Crystallography

The ITC have been designed to enable scientists with minimal understandingof group theory to work with crystal structures and diffraction patterns. Mostof the ITC entries, except for the graphical symbols that we have alreadyintroduced, are self-explanatory (see fig. 11 and 12).

3.2.1 ITC information about crystal structures

By using the ITC, one can (amongst other things):

• Given the coordinates of one atom, determine the number and coordinatesof all companion atoms. This greatly simplifies the description of crystalstructures.

• Determine the local point-group symmetry around each atom, which is thesame as the symmetry of the electron density and is connected to thesymmetry of the atomic wavefunction. This is particularly important forcertain types of spectroscopy.

Atomic positions are listed in order of decreasing multiplicity, starting fromthe most general position (see 12). Positions of lower multiplicity are locatedon symmetry elements (graphs), and therefore have a special site symmetry.Each unique site is aligned a letter, known as the Wyckoff letter, starting witha for the site of lowest multiplicity. It is quite possible for sites to have thesame site symmetry and yet not be companions of each other — in this case,these sites are assigned different Wyckoff letters (see fig. 12, Wyckoff lettersa, b, c, d corresponds to set of sites having the same point symmetry 1). In thelecture, we will discuss the case of Ag2O2, a simple oxide that crystallise inspace group P21/c (no. 14) — see fig. 13 .

3.2.2 ITC information about diffraction patterns

The ITC lists centring extinction conditions as well as extinctions associatedwith glide planes and screw axes — all these extinctions conditions are knownas “General” because they affect the contribution to scattering from all atoms.Also, atoms on certain sites do not contribute to certain Bragg peaks — theseextinctions are known as “Special” and are specific to each Wyckoff site.

29

SG  symbols  (Hermann-­‐Mauguin  nota6on)  

SG  symbol  (Schoenflies  nota6on)  

Crystal  class  (point  group)  

Crystal  system  

b  is  orthogonal  to  a  and  c.    Cell  choce  1  here  means  that  the  origin  is  on  an  inversion  centre.  

SG  diagrams  projected  along  the  3  direc6ons  

A  generic  posi6on  and  its  equivalent.    Dots  with  and  without  commas  are  related  by  an  improper  operator  (molecules  located  here  would  have  opposite  chirality)  

Asymmetric  unit  cell:  the  smallest  part  of  the  paGern  that  generates  the  whole  paGern  by  applying  all  symmetry  operators.  A  symmetry  operator  notated  as  follows:    a  2-­‐fold  axis  with  an  

associated  transla6on  of  ½  along  the  b=axis,  i.e.,  a  screw  axis.    This  is  located  at  (0,y,¼)  (see  SG  diagrams,  especially  top  leO).    Only  a  subset  of  the  symmetry  operators  are  indicated.  All  others  can  be  obtained  by  composi6on  with  transla6on  operators.  

IT  numbering  

Figure 11: Explanation of the most important symbols and notations in theSpace Group entries in the International Tables. The example illustrated hereis SG 14: P21/c .

30

Group  generators  

Refers  to  numbering  of  operators  on  previous  page  

Primi4ve  transla4ons  

4  pairs  of  equivalent  inversion  centres,  not  equivalent  to  centres  in  other  pairs  (not  in  the  same  class).    They  have  their  own  dis4nct  Wickoff  leEer.  

Equivalent  posi4ons,  obtained  by  applying  the  operators  on  the  previous  page  (with  numbering  indicated).    Reflects  normal  form  of  operators.  

Reflec4on  condi4ons  for  each  class  of  reflec4ons.    General:    valid  for  all  atoms  (i.e.,  no  h0l  reflec4on  will  be  observed  unless  l=2n;  this  is  due  to  the  c  glide).    Special:    valid  only  for  atoms  located  at  corresponding  posi4ons  (leI  side),  i.e.,  atoms  on  inversion  centres  do  not  contribute  unless  k+l=2n  

Figure 12: IT entry for SG 14: P21/c (page 2).

31

Figure 13: Crystal structure data for silver (I,III) oxide (Ag2O2), space group P21/c (no. 14).Data are from the Inorganic Crystal Structures Database (ICSD), accessible free of chargefrom UK academic institutions at http://icsd.cds.rsc.org/icsd/.

32

4 Lecture 4 — Fourier transform of periodic functions

4.1 Fourier transform of lattice functions

• It can be shown (see Supplementary Material on the web site) that theFourier transform of a function f(r) (real or complex) with the periodicityof the lattice can be written as:

F (q) =1

(2π)32

∑ni

e−2πi∑

i qini

∫u.c.

d(x)f(x)e−iq·x

=v0

(2π)32

∑ni

e−2πi∑

i qini

∫u.c.

dxif(xi)e−2πi∑

i qixi

(35)

where v0 is the volume of the unit cell

• The triple infinite summation in ni = nx, ny, nz is over all positive and nega-tive integers. The qi = qx, qy, qz are reciprocal space coordinates on thedual basis. The xi = x, y, z are real-space crystallographic coordinates(this is essential to obtain the qini term in the exponent). The integral isover one unit cell.

• F (q) is non-zero only for q belonging to the primitive RL. In fact, if q be-longs to the primitive reciprocal lattice, then by definition its dot productto the symmetry lattice translation is a multiple of 2π, the exponentialfactor is 1 and the finite summation yields N (i.e., the number of unitcells). If q does not belong to the primitive reciprocal lattice, the expo-nential factor will vary over the unit circle in complex number space andwill always average to zero.

• It is the periodic nature of f(r) that is responsible for the discrete nature ofF (q).

4.2 Atomic-like functions

Although the result in eq. 35 is completely general, in the case of scatteringexperiments the function f(x) represents either the density of nuclear scat-tering centres (neutrons) or the electron density (x-rays, electrons). Thesefunctions are sharply pointed around the atomic positions, and it is thereforeconvenient to write:

33

f(x) =∑j

fj(x) (36)

where the summation if over all the atoms in the unit cell, and fj(x) is (some-what arbitrarily) the density of scatterers assigned to a particular atom (seenext lectures for a fully explanation). With this, eq. 35 becomes:

F (q) =v0

(2π)32

∑ni

e−2πi∑

i qini∑j

e−2πi∑

i qixij

∫u.c.

dxifj(xi)e−2πi

∑i qi(x

i−xij) =

=v0

(2π)32

∑ni

e−2πi∑

i qini∑j

e−2πi∑

i qixijfj(q) (37)

When evaluated on reciprocal lattice points, the rightmost term is now thestructure factor Fhkl.

4.3 Centring extinctions

In particular, we can show that those “‘conventional”’ RLV that we called ex-tinct by centring are indeed extinct, that is, F (q) = 0. For example, in thecase of an I-centrel lattice, we have:

Fhkl =∑j

e−2πi(hx+ky+lz)(

1 + e−2πi(h2+ k

2+ l

2

)fj(q) (38)

and this is = 0 for h+ k + l 6= 2m, which precisely identifies the non-primitiveRL nodes.

4.4 Extinctions due to roto-translations

Similarly, the presence of roto-translation operators (glide planes and screwaxes) produces extinction conditions, which, however, do not affect whole RLsub-lattices but only certain sections of it. In the case of glides, there areplanes of hkl’s in which some of the reflections are extinct. In the case ofscrews, there are lines of hkl’s in which some of the reflections are extinct.Here, I will show only one example, that of a glide plane ⊥ to the x axis andwith glide vector along the y direction, which generates the two companionsx, y, z and −x, y + 1

2 , z. We have:

34

Fhkl =∑j

e−2πi(ky+lz)(e−2πihx + e−2πi(−hx+

k2))fj(q) (39)

We can see that, within the plane of reflections having h = 0, all reflectionswith k 6= 2m are extinct.

4.5 The symmetry of |F (q)|2 and the Laue classes

• It is very useful to consider the symmetry of the RL when |F (x)|2 is asso-ciated with the RL nodes. In fact, this corresponds to the symmetry ofthe diffraction experiment, and tells us how many unique reflections weneed to measure.

• Translational invariance is lost once |F (x)|2 is associated with theRL nodes.

• Let R be the rotational part and t the translational part of a generic symme-try operators. One can prove that

F (q) =N

(2π)32

∫u.c.

d(x)f(x)e−i(R−1q)·xe−iq·t = F (R−1q)e−iq·t (40)

Eq. 40 shows that the reciprocal lattice weighed with |F (q)|2 has the fullpoint-group symmetry of the crystal class.

• This is because the phase factor e−iq·t clearly disappears when taking themodulus squared. In fact, there is more to this symmetry when f(x) isreal, i.e., f(x) = f∗(x): in this case

F ∗(q) =N

(2π)32

∫u.c.

dxf∗(x)eiq·x (41)

=N

(2π)32

∫u.c.

dxf(x)eiq·x = F (−q)

• Consequently, |F (q)|2 = F (q)F (−q) = |F (−q)|2 is centrosymmetric. Aswe shall shortly see, the lattice function used to calculate non-resonantscattering cross-sections is real. Consequently, the |F (q)|2-weighed RL(proportional to the Bragg peak intensity) has the symmetry of the crys-tal class augumented by the center of symmetry. This is necessarilyone of the 11 centrosymmetryc point groups, and is known as the Laueclass of the crystal.

35

Fridel’s law

For normal (non-anomalous) scattering, the reciprocal lattice weighed with |F (q)|2 has the fullpoint-group symmetry of the crystal class supplemented by the inversion. This symmetry isknown as the Laue class of the space group.In particular, for normal (non-anomalous) scattering, Fridel’s law holds:

|F (hkl)|2 = |F (hkl)|2 (42)

Fridel’s law is violated for non-centrosymmetric crystals in anomalous conditions.Anomalous scattering enables one, for example, to determine the orientation of a polar crystalor the chirality of a chiral crystal in an absolute way.

36

5 Lecture 5 — Brillouin zones and the symmetry ofthe band structure

• The electronic, vibrational and magnetic phenomena occurring in a crystalhave, overall, the same symmetry of the crystal.

• However, individual excitations (phonons, magnons, electrons, holes) breakmost of the symmetry, which is only restored because symmetry-equivalentexcitations also exist and have the same energy (and therefore popula-tion) at a given temperature.

• The wavevectors of these excitations are generic (non-RL) reciprocal-spacevectors.

• When these excitations are taken into account, one finds that inelastic scat-tering of light, X-rays or neutrons can occur. In general, the inelasticscattering will be outside the RL nodes.

• Various Wigner Seitz constructions 7 are employed to subdivide the recip-rocal space, for the purpose of:

� Classifying the wavevectors of the excitations. The Bloch theo-rem states that “crystal” wavevectors within the first Brillouin zone(first Wigner-Seitz cell) are sufficient for this purpose.

� Perform scattering experiments. In general, an excitation with wavevec-tor k (within the first Brillouin zone) will give rise to scattering at allpoints τ +k, where τ is a RLV . However, the observed scatteringintensity at these points will be different. The repeated Wigner-Seitz construction is particularly useful to map the “geography” ofscattering experiments.

� Construct Bloch wave functions (with crystal momentum withinthe first Brillouin zone) starting from free-electron wave-functions(with real wavevector k anywhere in reciprocal space). In partic-ular, apply degenerate perturbation theory to free-electron wave-functions with the same crystal wavevector. Here, the extendedWigner-Seitz construction is particularly useful, since free-electronwavefunctions within reciprocal space “fragments” belonging to thesame Brillouin zone form a continuous band of excitations.

7A very good description of the Wigner-Seitz and Brillouin constructions can be found inthe book by Ashcroft and Mermin (see bibliography). See also the Supplementary Material fora summary of the procedure.

37

• The different types of BZ constructions were already introduced last year.They are reviewed in the lecture and in the supplementary material onthe web site.

5.1 Symmetry of the electronic band structure

• We will here consider the case of electronic wavefunctions, but it isimportant to state that almost identical considerations can be appliedto other wave-like excitations in crystals, such as phonons and spinwaves (magnons).

• Bloch theorem: in the presence of a periodic potential, electronic wave-functions in a crystal have the Bloch form:

ψk(r) = eik·ruk(r) (43)

where uk(r) has the periodicity of the crystal. We also recall that thecrystal wavevector k can be limited to the first Brillouin zone (BZ). Infact, a function ψk′(r) = eik

′·ruk′(r) with k′ outside the first BZ can berewritten as

ψk′(r) = eik·r[eiτ ·ruk′(r)

](44)

where k′ = τ+k, k is within the 1st BZ and τ is a reciprocal lattice vector(RLV ). Note that the function in square brackets has the periodicity ofthe crystal, so that eq. 44 is in the Bloch form.

• The application of the Bloch theorem to 1-dimensional (1D) electronic wave-functions, using either the nearly-free electron approximation or thetight-binding approximation leads to the typical set of electronic disper-sion curves (E vs. k relations) shown in fig. 14. We draw attention tothree important features of these curves:

• Properties of the electronic dispersions in 1D

� They are symmetrical (i.e., even) around the origin.

� The left and right zone boundary points differ by the RLV 2π/a, andare also related by symmetry.

� The slope of the dispersions is zero both at the zone centre and atthe zone boundary. We recall that the slope (or more generally thegradient of the dispersion is related to the group velocity of thewavefunctions in band n by:

38

k  

Figure 14: A set of typical 1-dimensional electronic dispersion curves in thereduced/repeated zone scheme.

vn(k) =1

~∂En(k)

∂k(45)

• Properties of the electronic dispersions in 2D and 3D

� They have the full Laue (point-group) symmetry of the crystal. Thisapplies to both energies (scalar quantities) and velocities (vectorquantities)

� Zone edge centre (2D) or face centre (3D) points on opposite sidesof the origin differ by a RLV and are also related by inversion sym-metry.

� Otherwise, zone boundary points that are related by symmetry do notnot necessarily differ by a RLV .

� Zone boundary points that differ by aRLV are not necessarily relatedby symmetry.

� Group velocities are zero at zone centre, edge centre (2D) or facecentre (3D) points.

� Some components of the group velocities are (usually) zero at zoneboundary points (very low-symmetry cases are an exception —see below).

39

� Group velocities directions are constrained by symmetry on symme-try elements such as mirror planes and rotation axes.

• The dispersion of the band structure, En(k), has the Laue symmetry of thecrystal8. This is because:

� En(k) is a macroscopic observable (it can be mapped, for example,by Angle Resolved Photoemission Spectroscopy — ARPES) andany macroscopic observable property of the crystal must have atleast the point-group symmetry of the crystal (Neumann’s principle— see later).

� If the Bloch wavefunction ψk = eik·ruk(r) is an eigenstate of theSchroedinger equation

[− ~2

2m∇2 + U(r)

]ψk = Ekψk (46)

then ψ†k = e−ik·ru†k(r) is a solution of the same Schroedinger equa-tion with the same eigenvalue (this is always the case if the poten-tial is a real function). ψ†k has crystal momentum −k. Therefore,the energy dispersion surfaces (and the group velocities) must beinversion-symmetric even if the crystal is not. 9

5.2 Symmetry properties of the group velocity

• The group velocity at two points related by inversion in the BZ must beopposite.

• For points related by a mirror plane or a 2-fold axis, the components of thegroup velocity parallel and perpendicular to the plane or axis must beequal or opposite, respectively.

• Similar constraints apply to points related by higher-order axes — in partic-ular, the components of the group velocity parallel to the axis must beequal.

• Since eik·ruk(r) = ei(k+τ)·r{e−τ ·ruk(r)} and the latter has crystal momen-tum k + τ , Ek must be the same at points on opposite faces across theBrillouin zone (BZ), separated by τ .

8Strictly speaking, this is only true if the crystal point-group symmetry contains the timereversal operator. This is not the case if the crystal is ferromagnetic or ferrimagnetic.

9Note that this derivation does not include the spin — an omission that becomes importantin the presence of spin-orbit interaction. In this case, states with the same energy and oppositemomenta have opposite spins.

40

Figure 15: the first and two additional Wigner-Seitz cells on the oblique lat-tice (construction lines are shown). Significant points are labelled, and somepossible zone-boundary group velocity vectors are plotted with arrows.

• If the band dispersion Ek is smooth through the zone boundary, then thegradient of Ek (and therefore vn(k)) must be the same at points onopposite faces across the Brillouin zone (BZ), separated by τ . 10

5.2.1 The oblique lattice in 2D: a low symmetry case

• The point-group symmetry of this lattice is 2, and so is the Laue class ofany 2D crystal with this symmetry (remember that the inversion and the2-fold rotation coincide in 2D).

• Fig. 15 shows the usual Wigner-Seitz construction on an oblique lattice.

• Points on opposite zone-boundary edge centres (A-A and B-B) are relatedby 2-fold rotation–inversion (so their group velocities must be opposite)and are also related by a RLV (so their group velocities must be thesame). Follows that at points A and B, as well as at the Γ point (whichis on the centre of inversion) the group velocity is zero.

• For the other points, symmetry does not lead to a cancelation of the groupvelocities: For example, points a1 and b1 are related by a RLV but

10This leaves the possibility open for cases in which vn(k) jumps discontinuosly across theBZ boundary and is therefore not well defined exactly at the boundary. This can only happenif bands with different symmetries cross exactly at the BZ boundary.

41

Figure 16: Relation between the Wigner-Seitz cell and the conventionalreciprocal-lattice unit cell on the oblique lattice. The latter is usually chosenas the first Brillouin zone on this lattice, because of its simpler shape.

not by inversion, so their group velocities are the same but non-zero.Similarly, points c1 and c2 are related by inversion but not by a RLV , sotheir group velocities are opposite but not zero.

• Points a1, a2, b1 etc. do not have any special significance. It is thereforecustomary for this type of lattice (and for the related monoclinic andtriclinic lattices in 3D) to use as the first Brillouin zone not the Wigner-Seitz cell, but the conventional reciprocal-lattice unit cell, which has asimpler parallelogram shape. The relation between these two cells isshown in fig. 16.

5.2.2 The square lattice: a high symmetry case

• A more symmetrical situation is show in fig. 17 for the square lattice (Lauesymmetry 4mm). A tight-binding potential has been used to calculateconstant-energy surfaces, and the group velocity field has been plottedusing arrows.

• By applying similar symmetry and RLV relations, one finds that the groupvelocity is zero at the Γ point, and the BZ edge centres and at the BZcorners.

• On the BZ edge the group velocity is parallel to the edge.

• Inside the BZ, the group velocity of points lying on the mirror planes isparallel to those planes.

42

Figure 17: Constant-energy surfaces and group velocity field on a squarelattice, shown in the repeated zone scheme. The energy surfaces have beencalculated using a tight-binding potential.

43

5.3 Symmetry in the nearly-free electron model: degenerate wave-functions

• one important class of problems involves the application of degenerate per-turbation theory to the free-electron Hamiltonian, perturbed by a weakperiodic potential U(r):

H = − ~2

2m∇2 + U(r) (47)

• Since the potential is periodic, only degenerate points related by a RLV

are allowed to “interact” in degenerate perturbation theory and give riseto non-zero matrix elements.

• The 1D case is very simple:

� Points inside the BZ have a degeneracy of one and correspond totravelling waves.

� Points at the zone boundary have a degeneracy of two, since k = π/a

and therefore k − (−k) = 2π/a is a RLV . The perturbed solutionsare standing wave, and have a null group velocity, as we have seen(fig. 14).

• The situation is 2D and 3D is rather different, and this is where symmetrycan help. In a typical problem, one would be asked to calculate theenergy gaps and the level structure at a particular point, usually but notnecessarily at the first Brillouin zone boundary.

• The first step in the solution involves determining which and how many de-generate free-electron wavefunctions with momenta differing by a RLVhave a crystal wavevector at that particular point of the BZ.

• Symmetry can be very helpful in setting up this initial step, particularly if thesymmetry is sufficiently high. For the detailed calculation of the gaps,we will defer to the “band structure” part of the C3 course.

44

Figure 18: Construction of nearly-free electron degenerate wavefunctions forthe square lattice (point group 4mm). Some special symmetry point are la-belled. Relevant RLV s are also indicated.

Nearly-free electron degenerate wavefunctions

• Draw a circle centred at the Γ point and passing through the BZ point you are asked toconsider (either in the first or in higher BZ — see fig. 18). Points on this circle correspondto free-electron wavefunctions having the same energy.

• Mark all the points on the circle that are symmetry-equivalent to your BZ point.

• Among these, group together the points that are related by a RLV . These points representthe degenerate multiplet you need to apply degenerate perturbation theory.

• Write the free-electron wavefunctions of your degenerate multiplet in Bloch form. You will findthat all the wavefunctions in each multiplet have the same crystal momentum. Functionsin different multiplets have symmetry-related crystal momenta.

• This construction is shown in fig. 18 in the case of the square lattice (pointgroup 4mm) for boundary points between different Brillouin zones. Onecan see that:

• The degeneracy of points in the interior of the first BZ is always one,since no two points can differ by a RLV . This is not so for higherzones though (see lecture). For points on the boundary of zonesor at the interior of higher zones, the formula DN/Mn = D1/M1

45

holds, where DN is the degeneracy of the multiplet in the higherzone (the quantity that normally we are asked to find), MN is themultiplicity of that point (to be obtained by symmetry) and D1 andM1 are the values for the corresponding points within or at theboundary of the 1st BZ.

� X-points: there are 4 such points, and are related in pairs by aRLV . Therefore, there are two symmetry-equivalent doublets offree-electron wavefunctions (which will be split by the periodic po-tential in two singlets).

� Y-points: there are 8 such points, and are related in pairs by a RLV .Therefore, there are four symmetry-equivalent doublets of free-electron wavefunctions (which will be split by the periodic potentialin two singlets).

� M-points: there are 4 such points, all related by RLV s. Therefore,there is a single symmetry-equivalent quadruplet of free-electronwavefunctions (which will be split by the periodic potential in twosinglets and a doublet).

� X2-points: these are X-point in a higher Brillouin zone. There are 8such points, related by RLV s in groups of four. Therefore, thereare two symmetry-equivalent quadruplets of free-electron wave-functions (each will be split by the periodic potential in two singletsand a doublet). These two quadruplets will be brought back above(in energy) the previous two doublets in the reduced-zone scheme.

46