Condensed matter physics begins with a deceptively simple question: why do
\(10^{23}\) atoms, left to themselves, arrange into the beautifully regular
patterns we call crystals?
The answer starts with the force between a pair of atoms. At large separations
the atoms attract one another — through covalent, ionic, metallic or van der
Waals interactions, depending on the species involved. At short separations the
closed electron shells overlap and the Pauli principle forces a steep repulsion.
The net interaction energy \(U(r)\) therefore has a minimum at some separation
\(r_{e}\), and at that separation the force vanishes:
The equilibrium separation \(r_{e}\) is, for hard-sphere-like atoms of radius \(r\),
just \(2r\): the atoms touch. This single number sets the scale of everything that
follows, because it fixes the lattice constant, and through the lattice constant
it fixes the density, the elastic moduli, the phonon frequencies and the band
structure.
A solid forms when \(k_{B}T\) falls well below the depth of that potential well,
so that atoms can no longer wander past one another. What kind of solid forms
depends on how much time and freedom the atoms had while they were settling.
Classification of matter. This book is
concerned almost entirely with the left-hand branch, and for the first several
chapters with single crystals in particular.
Definition 1.1. (Crystalline solid)
A solid in which the atoms, ions or molecules are arranged in an orderly,
periodic array that repeats over distances large compared with the atomic
spacing. Pure copper, silicon and rock salt are crystalline.
Definition 1.2. (Amorphous solid)
A solid in which the constituents are packed with well-defined local
coordination but no long-range periodicity. Window glass, most polymers and
rubber are amorphous.
The distinction is not that amorphous solids are disordered in every sense. In
vitreous silica each silicon still sits at the centre of a nearly perfect oxygen
tetrahedron; what is lost is the correlation between one tetrahedron and another
a hundred bond lengths away. The technical statement is that amorphous solids
possess short-range order but not long-range order.
Two ways of packing identical
atoms into a plane. On the left every atom sees an identical environment,
identically oriented; on the right the local coordination is similar
everywhere but the pattern never repeats.
Within the crystalline branch we distinguish further:
A single crystal (or monocrystalline solid) maintains its
periodicity, without interruption, across the whole specimen. Gem-quality
diamond and the silicon boules used by the semiconductor industry are single
crystals.
A polycrystal is an aggregate of many small single-crystal
grains, each with its own orientation, joined at grain boundaries.
Ordinary copper wire, structural steel and most metals as we encounter them
are polycrystalline.
Remark. The classification above is not exhaustive. Quasicrystals, discovered by
Shechtman in 1982, are ordered but not periodic: their diffraction patterns show
sharp spots with five-fold symmetry, which the section on symmetry and Symmetry Operations will show is
impossible for a periodic lattice. They are the exception that proves how
strong the periodicity assumption is.
Why periodicity matters
Nearly every result in this book — Bloch's theorem, energy bands, phonon
dispersion, Bragg diffraction — follows from translational periodicity rather
than from the details of the interatomic potential. Establishing the language of
periodicity carefully now pays for itself many times over later.
Lattice, Basis and Crystal Structure
The single most important idea in crystallography is the separation of a crystal
structure into two logically independent pieces: an abstract periodic array of
points (the lattice), and the group of atoms attached to each point (the
basis).
The lattice is a mathematical abstraction — a set of points, not of atoms.
The basis is real matter. The crystal structure is what a diffraction experiment
actually sees.
The Bravais lattice
Definition 1.3. (Bravais lattice, constructive definition)
A three-dimensional Bravais lattice is the infinite set of points with position
vectors
where \(\avec{1},\avec{2},\avec{3}\) are any three vectors not all lying in one
plane. These are called primitive vectors of the lattice, and the point
\(\sum_{i} n_{i}\avec{i}\) is reached by taking \(n_{i}\) steps of length \(a_{i}\)
along \(\avec{i}\) for \(i=1,2,3\).
Definition 1.4. (Bravais lattice, symmetry definition)
A Bravais lattice is an array of discrete points that looks exactly the same —
in arrangement and in orientation — when viewed from any point of the
array.
The two definitions describe the same object, and it is worth keeping both in
mind: the first is what you compute with, the second is what you test against.
The phrase “and in orientation” in the second definition is not decorative. It
is precisely what the honeycomb net of the figure above fails.
Lattice plus a one-atom basis.
(a) Every point of the oblique
net is an integer combination of \(\avec{1}\) and \(\avec{2}\). (b) Decorating each
lattice point with the same basis, in the same orientation, produces the
crystal structure.
A general (oblique) plane lattice.
The basis
Definition 1.5. (Basis)
The group of atoms attached to every lattice point, specified by the positions
\(\vb{r}_{j}\) of its members relative to that point:
Every basis in a given crystal is identical in composition, in internal
arrangement, and in orientation. The number of atoms in the basis may be one or
many:
Monatomic basis — one atom per lattice point. Copper, sodium,
the noble-gas solids.
Diatomic basis — two atoms. NaCl, CsCl, diamond, silicon,
GaAs. (In diamond the two atoms of the basis are identical, which is
a useful reminder that a multi-atom basis is not the same thing as a
multi-element compound.)
Polyatomic bases — CaF\(_{2}\) has three atoms per lattice
point; protein crystals may have tens of thousands.
Lattice points are not atoms
A lattice point need not coincide with an atom, and an atom need not sit at a
lattice point. In the honeycomb structure of graphene neither of the two carbon
atoms in the basis sits at a lattice point in the conventional choice of origin.
The examiner's favourite question — “what is the Bravais lattice of
NaCl/CsCl/diamond?” — is testing exactly this distinction.
A structure that is not a Bravais lattice
Consider the vertices of a two-dimensional honeycomb, the figure above. The
array looks identical when viewed from \(P\) or from \(Q\). But the view from \(R\) is
rotated by \(180^{\circ}\) relative to the view from \(P\): at \(P\) the three bonds
point up-left, up-right and down, while at \(R\) they point down-left, down-right
and up. Arrangement is preserved; orientation is not. The honeycomb is therefore
not a Bravais lattice.
The vertices of a
honeycomb net. The environments of \(P\) and \(Q\) are identical; the environment
of \(R\) is that of \(P\) rotated through \(180^{\circ}\). Not every vertex is
equivalent, so the vertex set is not a Bravais lattice.
The cure is the decomposition of the section on lattice, Basis and Crystal Structure: the honeycomb is a
triangular (hexagonal) Bravais lattice with a two-atom basis, one atom of
the basis sitting on each of the two inequivalent sublattices. This is exactly
the structure of a single sheet of graphene, and we shall return to it in
the section on important Crystal Structures.
Test for a Bravais lattice
Stand at each point in turn and look around. If any two points give views that
differ by so much as a rotation, the set of points is not a Bravais lattice —
but it can always be described as some Bravais lattice with a multi-point basis.
Unit Cells
Definition 1.6. (Unit cell)
A region of space which, when translated through some subset of the lattice
vectors, tiles all of space without gaps and without overlaps, thereby
generating the complete crystal.
Primitive and non-primitive cells
Definition 1.7. (Primitive unit cell)
A volume of space that, when translated through all the vectors of the
Bravais lattice, fills space exactly once. Equivalently: a cell containing
precisely one lattice point.
The primitive cell built on primitive vectors \(\avec{1},\avec{2},\avec{3}\) is
the parallelepiped they span, whose volume is
Two facts about the relation above deserve emphasis. First, the shape of a
primitive cell is enormously non-unique: infinitely many different
parallelograms, and indeed infinitely many non-parallelogram regions, contain
exactly one lattice point. Second, the volume is completely unique. Every
primitive cell of a given lattice has the same volume, namely \(1/n\) where \(n\) is
the number density of lattice points. This is obvious once stated — one point
per cell, so the cell volume is the volume per point — but it is the workhorse
behind half the numerical problems in this chapter.
Definition 1.8. (Non-primitive (conventional) unit cell)
A cell containing more than one lattice point, chosen because it displays the
symmetry of the lattice more clearly than any primitive cell does. The
conventional cubic cell of the fcc lattice, with four lattice points, is the
standard example.
Three cells drawn on the same
square lattice. The teal and blue cells have different shapes but the same
area and each contains one lattice point; the amber cell contains two.
Counting lattice points in a cell
Points on the boundary of a cell are shared with neighbouring cells, so they must
be counted fractionally. In two dimensions the cleanest bookkeeping uses interior
angles: a corner of interior angle \(\theta\) claims a fraction
\(\theta/360^{\circ}\) of the point sitting there. For the primitive parallelogram
of the figure above with angles \(60^{\circ},120^{\circ},60^{\circ},
120^{\circ}\),
so that cell is non-primitive. In three dimensions the corresponding rules for a
parallelepipedal cell are the familiar ones: a corner site counts \(1/8\), a site
on a face counts \(1/2\), a site on an edge counts \(1/4\), and an interior site
counts \(1\).
Worked example 1.1. How many distinct primitive cells?
A common examination question asks in how many distinct ways a primitive
cell can be constructed for a given two-dimensional lattice, where cells related
by a lattice translation are not counted as distinct. The recipe: count the
inequivalent pairs of primitive vectors \((\avec{1},\avec{2})\) that can be drawn
from one lattice point to its neighbours such that the parallelogram they span
has the minimum area.
For a square lattice of spacing \(a\), the shortest lattice vectors are the four
\(\pm a\uv{x}, \pm a\uv{y}\); pairing them subject to the area condition
\(|\avec{1}\times\avec{2}| = a^{2}\) and discarding pairs related by symmetry
leaves the single familiar square cell. For a centred rectangular lattice the
same construction yields several distinct parallelograms of equal area. The
practical point is that the number of choices is finite and countable,
while the area is fixed at \(a^{2}\) in every case.
The Wigner–Seitz cell
Among the infinitely many primitive cells, one is canonical: it is built without
making any arbitrary choice of primitive vectors, and it inherits the full point
symmetry of the lattice.
Definition 1.9. (Wigner–Seitz cell)
The region of space about a lattice point that is closer to that point than to
any other lattice point.
Construction
Pick a lattice point as origin.
Draw vectors from it to all nearby lattice points.
Bisect each vector with a perpendicular plane (a perpendicular line in
two dimensions).
The smallest volume enclosed by these bisectors is the Wigner–Seitz
cell.
The Wigner–Seitz cell is primitive: it contains exactly one lattice point, and
translating it through all lattice vectors tiles space without overlap. It is
also as symmetric as the Bravais lattice itself. We will meet its counterpart in
reciprocal space — the first Brillouin zone — in the section on the Reciprocal Lattice, and
essentially all of band theory is written on that object.
Wigner–Seitz construction on an oblique
plane lattice. Dotted lines join the central point to its near neighbours; the
shaded hexagon is bounded by the perpendicular bisectors of those lines.
Worked example 1.2. Area of the Wigner–Seitz cell of graphene
The carbon atoms of graphene sit at the vertices of a honeycomb of side \(a\)
(the carbon–carbon bond length), but the Bravais lattice is the
triangular lattice whose primitive vectors have length \(\sqrt{3}\,a\). Taking
Equivalently: the Wigner–Seitz cell of the triangular lattice is a regular
hexagon made of six equilateral triangles of side \(a\) each of area
\(\sqrt{3}a^{2}/4\), giving \(6 \times \sqrt{3}a^{2}/4 = 3\sqrt{3}a^{2}/2\). Note
that this hexagon contains two carbon atoms, consistent with the two-atom basis.
Symmetry and Symmetry Operations
Definition 1.10. (Symmetry operation)
A transformation that maps the crystal onto itself: after the operation, the
crystal is indistinguishable from what it was before.
Symmetry operations divide into two families according to whether or not they
leave at least one point fixed.
Point-group operations are performed about a point, a line or a
plane, and leave at least one point of the crystal unmoved: rotations,
reflections, inversion and their combinations.
Space-group operations include the lattice translations, and the
compound operations (screw axes and glide planes) that combine a rotation or
reflection with a fractional translation.
The four elementary operations
1. Translation.
Displacing the crystal by a lattice vector
\(\Rvec = n_{1}\avec{1}+n_{2}\avec{2}+n_{3}\avec{3}\) carries every lattice point
onto another lattice point. Translational symmetry is the defining property of a
crystal, and it is the symmetry from which Bloch's theorem follows.
2. Rotation.
If rotating the crystal by \(\theta = 360^{\circ}/n\)
about some axis reproduces it, that axis is an \(n\)-fold rotation axis,
written \(n\) in Hermann–Mauguin notation (\(C_{n}\) in Schönflies notation). As
the section on symmetry and Symmetry Operations proves, only \(n = 1,2,3,4,6\) can occur in a periodic
crystal.
3. Reflection.
If a plane divides the crystal into two halves that
are mirror images of one another, that plane is a mirror plane, denoted
\(m\).
4. Inversion.
If replacing every position vector \(\vb{r}\) measured
from some point by \(-\vb{r}\) leaves the crystal unchanged, that point is a
centre of inversion, denoted \(\bar{1}\) (read “one bar”). Inversion is
intrinsically a three-dimensional operation: in two dimensions it coincides with
a two-fold rotation.
Three of the four elementary symmetry
operations. Translation, the fourth, cannot be drawn within a finite figure.
Compound operations
Combining the elementary operations generates a few more:
Rotoinversion \(\bar{n}\): rotate by \(360^{\circ}/n\), then invert.
The allowed values are again \(\bar{1},\bar{2},\bar{3},\bar{4},\bar{6}\), and
\(\bar{2}\) is the same operation as a mirror \(m\).
Screw axis \(n_{m}\): rotate by \(360^{\circ}/n\) and translate by
\(m/n\) of the lattice repeat along the axis.
Glide plane: reflect, then translate by half a lattice vector
parallel to the plane.
Combining the \(32\) crystallographic point groups with the \(14\) Bravais lattices,
and allowing screw axes and glide planes, produces exactly \(230\) distinct space
groups in three dimensions. Cataloguing them is the business of crystallography
proper; for our purposes the important message is that the list is finite, and
finite because of the theorem we prove next.
The crystallographic restriction theorem
Textbooks often state, without proof, that only \(1\)-, \(2\)-, \(3\)-, \(4\)- and
\(6\)-fold rotation axes are compatible with a lattice, offering the hand-waving
argument that “you cannot tile a floor with pentagons”. That statement is true,
but the theorem deserves better, because the proof is three lines long and is
occasionally examined directly.
Theorem 1.1. (Crystallographic restriction)
A Bravais lattice can possess an \(n\)-fold rotation axis only for
\(n = 1, 2, 3, 4, 6\).
Proof. Let \(A\) and \(B\) be two lattice points separated by a primitive vector of length
\(a\), chosen to be the shortest lattice vector perpendicular to the rotation axis.
Suppose the lattice has an \(n\)-fold axis through every lattice point, and let
\(\theta = 2\pi/n\).
Rotating \(B\) about \(A\) by \(+\theta\) carries it to a lattice point \(B'\); rotating
\(A\) about \(B\) by \(-\theta\) carries it to a lattice point \(A'\). Both \(B'\) and
\(A'\) must be lattice points, because the rotation is a symmetry of the lattice.
By construction \(A'B'\) is parallel to \(AB\), so the vector \(\overrightarrow{A'B'}\)
is a lattice vector and its length must be an integer multiple of \(a\):
\begin{equation*}
\left|A'B'\right| = a + 2a\cos\theta = m\,a, \qquad m \in \mathbb{Z}.
\end{equation*}
Therefore
\begin{equation}
\cos\theta = \frac{m-1}{2}, \qquad m \in \mathbb{Z}.
\end{equation}
Since \(-1 \le \cos\theta \le 1\), the only admissible integers are
\(m = -1,0,1,2,3\), giving
that is \(n = 2,3,4,6,1\). No other rotation is possible.
Geometry of the
crystallographic restriction theorem. Rotating \(B\) about \(A\) and \(A\) about \(B\)
through \(\theta\) produces two further lattice points whose separation must
itself be an integer multiple of \(a\).
Remark. Equation the relation above forbids five-fold lattice symmetry, not
five-fold symmetry as such. Individual molecules, viruses and quasicrystals are
free to be five-fold symmetric; what they cannot be is periodic while doing so.
Crystal Systems and the Bravais Lattices
Classifying lattices by their symmetry gives \(5\) distinct Bravais lattices in two
dimensions (belonging to \(4\) crystal systems) and \(14\) in three dimensions
(belonging to \(7\) crystal systems).
Two dimensions: five plane lattices
The four plane crystal systems and five plane Bravais lattices.
System
Bravais lattice(s)
Axes
Angle
Oblique
oblique (primitive)
\(a \neq b\)
\(\gamma \neq 90^{\circ}\)
Rectangular
primitive rectangular
\(a \neq b\)
\(\gamma = 90^{\circ}\)
centred rectangular
\(a \neq b\)
\(\gamma = 90^{\circ}\)
Square
square
\(a = b\)
\(\gamma = 90^{\circ}\)
Hexagonal
hexagonal (triangular)
\(a = b\)
\(\gamma = 120^{\circ}\)
Four systems, five lattices
Examinations ask both “how many plane crystal systems?” (four) and “how many
plane Bravais lattices?” (five). The rectangular system supports two distinct
lattices — primitive and centred — because the centred rectangular lattice
cannot be described by a primitive rectangular cell without losing the mirror
symmetry. Read the question carefully.
Three dimensions: seven systems, fourteen lattices
The seven crystal systems and fourteen Bravais lattices in three
dimensions. P = primitive, I = body-centred, F = face-centred, C = base-centred
(side-centred).
Students meeting the table above for the first time invariably ask why
certain obvious-looking entries are missing. Why is there no base-centred cubic
lattice? Why no face-centred tetragonal? These are excellent questions, and the
answers are instructive, because in each case the missing lattice is not
forbidden — it simply is not new.
Base-centred cubic does not exist as a separate lattice.
Add lattice
points at the centres of the top and bottom faces of a cube. The resulting array
is certainly a Bravais lattice, but it no longer has the four three-fold axes
along the body diagonals that define cubic symmetry: the \(z\) direction is now
inequivalent to \(x\) and \(y\). What we have built is in fact a
body-centred tetragonal lattice with \(a_{t} = b_{t} = a/\sqrt{2}\) and
\(c_{t}=a\), viewed in an inconvenient orientation. Since it is already on the
list, it is not counted twice.
Face-centred tetragonal is body-centred tetragonal in disguise.
By
the same argument, choosing a new cell rotated by \(45^{\circ}\) about the
\(c\) axis, with \(a' = a/\sqrt{2}\), converts a face-centred tetragonal array into a
body-centred tetragonal one. The rule for the whole table is:
The counting rule
Two centred lattices are counted as one whenever a change of cell converts one
into the other without loss of point symmetry. The fourteen surviving
lattices are those which cannot be reduced to one another in this way.
Worked example 1.3. Which Bravais lattice is this?
A cubic array of points at coordinates \((n_{x},n_{y},n_{z})\) has points removed
so that only those with \(n_{x}+n_{y}+n_{z}\) even survive. Which lattice
results? And what if only the odd sums survive?
Solution. Take the even case. The surviving points include
\((0,0,0)\) and \((1,1,0)\), \((1,0,1)\), \((0,1,1)\) — that is, the corners of a cube
of side \(2\) together with the centres of its faces. Rescaling by a factor of two,
this is precisely the fcc lattice with conventional cube edge \(2\).
Now the odd case. The surviving points are \((1,0,0)\), \((0,1,0)\), \((0,0,1)\),
\((1,1,1)\), and so on. Translating the whole set by \((1,0,0)\) — which is
allowed, since a lattice is defined only up to choice of origin — maps “odd”
onto “even”. So this is the same lattice: also fcc.
Both lattices are fcc. The trap in this question is to assume that if one parity
gives fcc, the other must give bcc; a moment's thought about translating the
origin shows the two parities are equivalent.
The Cubic Lattices
The cubic system is where most of solid-state physics is done, partly because it
is simple and partly because a large fraction of elemental solids adopt one of
its three lattices.
Simple cubic
Lattice points sit only at the eight corners of the cube. Each corner is shared
among eight adjacent cells, so each contributes \(1/8\):
\begin{equation}
N_{\text{sc}} = 8 \times \tfrac{1}{8} = 1 \quad\text{lattice point per cell.}
\end{equation}
The conventional cell is therefore already primitive, with primitive vectors
\(\avec{1}=a\uv{x}\), \(\avec{2}=a\uv{y}\), \(\avec{3}=a\uv{z}\). Nearest neighbours
touch along a cube edge, so \(a = 2R\). Polonium is the only element that
crystallises in this structure under ordinary conditions — a strong hint that
simple cubic is not a good way to pack spheres.
Body-centred cubic
Add one lattice point at the body centre. It lies wholly inside the cell:
whose parallelepiped has volume \(a^{3}/2\) — half the conventional cell, as it
must be for two lattice points per conventional cell. Nearest neighbours lie
along the body diagonal, of length \(\sqrt{3}\,a\), so the nearest-neighbour
distance is \(\sqrt{3}a/2\) and \(4R = \sqrt{3}a\). The alkali metals (Na, K, Li,
Cs), chromium and \(\alpha\)-iron are bcc.
Face-centred cubic
Add lattice points at the centres of all six faces. Each face point is shared
between two cells:
spanning a volume \(a^{3}/4\). Nearest neighbours lie along a face diagonal of
length \(\sqrt{2}\,a\), giving a nearest-neighbour distance \(a/\sqrt{2}\) and
\(4R = \sqrt{2}a\). Copper, silver, gold, aluminium, nickel and lead are fcc.
Base-centred cubic — a cautionary entry
For completeness: placing extra points at the centres of two opposite faces gives
\(8\times\frac{1}{8} + 2\times\frac{1}{2} = 2\) points per cell. As
the section on crystal Systems and the Bravais Lattices explained, this array is a body-centred
tetragonal lattice in disguise and is not one of the fourteen.
bcc: 2 points/cell
fcc: 4 points/cell
The three cubic Bravais lattices,
drawn with their conventional cubic cells. Only in the simple cubic case is
the conventional cell primitive.
sc: 1 point/cell
Coordination shells
Definition 1.11. (Coordination number)
The number of nearest neighbours of a lattice point. Because every point of a
Bravais lattice has an identical environment, the coordination number is a
property of the lattice, not of the point.
Examination questions frequently reach beyond the nearest neighbours to the
second and third shells, so it is worth tabulating them once and for all. The
method is always the same: list the lattice vectors in order of increasing
length, and count how many there are of each length.
Coordination shells of the cubic lattices, in units of the
conventional cube edge \(a\). The entry “\(12\) at \(\sqrt{2}a\)” means twelve
lattice points at distance \(\sqrt{2}\,a\).
Simple cubic
bcc
fcc
1st shell
\(6\) at \(a\)
\(8\) at \(\tfrac{\sqrt{3}}{2}a \approx 0.866a\)
\(12\) at \(\tfrac{a}{\sqrt{2}} \approx 0.707a\)
2nd shell
\(12\) at \(\sqrt{2}a\)
\(6\) at \(a\)
\(6\) at \(a\)
3rd shell
\(8\) at \(\sqrt{3}a\)
\(12\) at \(\sqrt{2}a\)
\(24\) at \(\tfrac{\sqrt{6}}{2}a \approx 1.225a\)
4th shell
\(6\) at \(2a\)
\(24\) at \(\tfrac{\sqrt{11}}{2}a\)
\(12\) at \(\sqrt{2}a\)
Continue reading — it's free
Register with your mobile number to unlock the full chapter and all 100+ chapters of the Physics Library.
