3 Fundamentals of Crystal Structure – Fundamentals of Solid-State Physics: Thermal Properties

Learning Objectives

Define a Bravais lattice, a basis, and a crystal structure in a precise real-space language.
Distinguish primitive, conventional, and Wigner–Seitz cells and explain why they serve different purposes.
Identify the basic symmetry operations used in crystallography and relate them to the seven crystal systems and the 14 Bravais lattices.
Describe the real-space geometry of the common structure types SC, BCC, FCC, HCP, NaCl, CsCl, diamond, zincblende, and graphite.
Explain why the basis matters not only geometrically but also for symmetry and later for lattice dynamics and diffraction.

3.1 Roadmap

Translational order: Bravais lattice, basis, and crystal structure.
Primitive, conventional, and Wigner–Seitz cells.
Symmetry operations, crystal systems, and the 14 Bravais lattices.
Common elemental and binary structure types in real space.
Why the basis matters, and how today’s real-space picture prepares the next lecture.

Symbol	Meaning
$\mathbf a_i$	primitive translation vectors of the direct lattice
$\mathbf R_n$	Bravais-lattice vector
$\mathbf T$	lattice translation vector
$\mathbf r_\mu$	position of atom $\mu$ within the basis
$\mathbf r_{n\mu}$	position of atom $\mu$ in cell $n$
$p$	number of atoms in the basis, i.e. atoms per primitive cell
$a,b,c$	edge lengths of the conventional cell
$\alpha,\beta,\gamma$	conventional-cell angles
$P,I,F,C,R$	primitive, body-centered, face-centered, base-centered, rhombohedral centering symbols
$1,2,3,4,6$	allowed rotation orders of a periodic crystal lattice
$m,\bar{1}$	mirror plane, inversion center
SC, BCC, FCC, HCP	simple cubic, body-centered cubic, face-centered cubic, hexagonal close packed

3.2 From Translational Order to Crystal Structure

A crystalline solid is characterized by discrete translational symmetry in three linearly independent directions. Choosing primitive vectors $\mathbf a_1$, $\mathbf a_2$, and $\mathbf a_3$, every lattice translation can be written as

\[ \mathbf R_n = n_1 \mathbf a_1 + n_2 \mathbf a_2 + n_3 \mathbf a_3, \qquad n_1,n_2,n_3 \in \mathbb Z . \]

The set of all points $\mathbf R_n$ is the Bravais lattice. It contains only the translational skeleton of the crystal and does not yet specify which atoms sit at those positions.

3.2.1 Basis and Atomic Positions

If the basis contains $p$ atoms at internal positions $\mathbf r_\mu$ with $\mu=1,\dots,p$, then all atomic positions are

\[ \mathbf r_{n\mu} = \mathbf R_n + \mathbf r_\mu, \qquad \mu = 1,\dots,p . \]

Thus,

Crystal structure $=$ Bravais lattice $+$ basis.

A monatomic Bravais lattice has $p=1$, but a crystal made of only one chemical element may still require $p>1$. Diamond and graphite are important examples.

3.2.2 Interpretation and Sanity Checks

Same Bravais lattice, different basis $\rightarrow$ different crystal structure. Translational symmetry alone does not determine the full structure.
Not every visually regular arrangement is a Bravais lattice. The honeycomb pattern is not a Bravais lattice; it is a triangular Bravais lattice with a two-atom basis.
The basis may contain identical atoms. A multi-atom basis is not restricted to compounds; it may also occur in elemental solids.

Figure 3.1: Figure placeholder. Example showing that a honeycomb arrangement is not itself a Bravais lattice and must be described as a Bravais lattice plus a two-atom basis.

3.3 Primitive, Conventional, and Wigner–Seitz Cells

3.3.1 Primitive Cell

A primitive cell is any cell that contains exactly one lattice point and tiles all space by translation through the Bravais-lattice vectors. For the primitive vectors $\mathbf a_1$, $\mathbf a_2$, and $\mathbf a_3$, the primitive-cell volume is

\[ V_{\mathrm c} = (\mathbf a_1 \times \mathbf a_2)\cdot \mathbf a_3 . \]

The primitive cell is not unique. The simplest algebraic choice is often the parallelepiped spanned by the primitive vectors.

3.3.2 Conventional Cell

A conventional cell is chosen to display the symmetry of the lattice as clearly as possible. It may therefore contain more than one lattice point and need not be primitive. This is why BCC and FCC are usually drawn as cubes although their primitive cells are not cubic.

3.3.3 Wigner–Seitz Cell

A particularly symmetric primitive cell is the Wigner–Seitz cell: the region of space closer to a chosen lattice point than to any other. It is constructed by drawing perpendicular bisecting planes between a lattice point and its neighbors.

3.4 Symmetry Operations and Crystal Classification

So far we have used only translations. A crystal, however, may possess additional symmetries that map it onto itself.

3.4.1 Basic Symmetry Operations

For periodic crystals the most important point symmetries are:

rotation about an axis,
reflection in a mirror plane,
inversion through a center,
and combinations such as rotoinversion.

Translations form the translation group. Operations that leave at least one point fixed form the point group. Combining point operations with translations leads to the space group.

A compact way to write the simplest examples is:

\[ \mathbf r \mapsto \mathbf r + \mathbf T \qquad \text{(translation)}, \]

\[ (x,y,z) \mapsto (-x,y,z) \qquad \text{(reflection in the $yz$ plane)}, \]

\[ (x,y,z) \mapsto (-x,-y,-z) \qquad \text{(inversion)}. \]

3.4.2 Why Only 1-, 2-, 3-, 4-, and 6-Fold Rotations?

A periodic lattice can be invariant only under rotation orders compatible with translational tiling. In three-dimensional crystals this leaves

\[ n = 1,2,3,4,6 . \]

Fivefold rotational symmetry is incompatible with periodic translational order in an ordinary crystal lattice.

3.4.3 Point Groups, Crystal Systems, and Space Groups

For actual crystal structures one encounters:

7 crystal systems,
32 crystallographic point groups,
230 space groups.

For the Bravais-lattice skeleton alone, however, only 14 Bravais lattices remain. This is because the basis need not preserve the full symmetry of the underlying lattice.

This distinction is central:

the Bravais lattice classifies the translational framework,
the basis may lower the symmetry of the crystal structure built on that framework.

Figure 3.2: Figure placeholder. The standard crystallographic symmetry operations used in point-group classification: rotations, inversion, mirror reflection, and rotoinversion.

Figure 3.3: Figure placeholder. Two crystals built on the same cubic cell but with different bases, illustrating that the basis may preserve or reduce the visible symmetry of the structure.

3.5 Crystal Systems and the 14 Bravais-Lattice Types

The seven crystal systems are defined by metric relations among the conventional-cell edges and angles, together with characteristic rotational symmetry.

Crystal System	Metric Relations of the Conventional Cell	Characteristic Rotation Symmetry	Bravais Types
triclinic	$a \neq b \neq c$, $\alpha \neq \beta \neq \gamma$	none beyond inversion	$P$
monoclinic	$a \neq b \neq c$, $\alpha = \gamma = 90^\circ \neq \beta$	one twofold axis	$P,\ C$
orthorhombic	$a \neq b \neq c$, $\alpha = \beta = \gamma = 90^\circ$	three mutually perpendicular twofold axes	$P,\ C,\ I,\ F$
tetragonal	$a=b\neq c$, $\alpha = \beta = \gamma = 90^\circ$	one fourfold axis	$P,\ I$
trigonal / rhombohedral	$a=b=c$, $\alpha = \beta = \gamma \neq 90^\circ$	one threefold axis	$R$
hexagonal	$a=b\neq c$, $\alpha = \beta = 90^\circ$, $\gamma = 120^\circ$	one sixfold axis	$P$
cubic	$a=b=c$, $\alpha = \beta = \gamma = 90^\circ$	four threefold axes along space diagonals	$P,\ I,\ F$

Adding the numbers of Bravais types by crystal system gives

\[ 1+2+4+2+1+1+3 = 14. \]

This count is not accidental. Some centerings do not generate new lattice types. For example, a base-centered tetragonal description can be re-expressed as a primitive tetragonal lattice with a different choice of axes.

3.5.1 Interpretation

The seven crystal systems classify metrics and characteristic rotational symmetries.
The 14 Bravais lattices classify distinct translational skeletons.
The actual crystal structure requires an additional basis.

Figure 3.4: Figure placeholder. The 14 Bravais lattices grouped by crystal system and centering type.

3.6 Common Elemental Structure Types in Real Space

3.6.1 Simple Cubic, Body-Centered Cubic, and Face-Centered Cubic

Structure	Pearson Symbol	Underlying Bravais Lattice	Coordination Number	Packing Fraction	Key Real-Space Feature
SC	$cP1$	simple cubic	$6$	$0.524$	open cubic arrangement
BCC	$cI2$	body-centered cubic	$8$	$0.680$	extra site at body center of conventional cube
FCC	$cF4$	face-centered cubic	$12$	$0.740$	close-packed cubic arrangement

A few geometric interpretations are especially useful:

SC is the simplest cubic structure but comparatively open.
BCC is not close packed, yet the next-neighbor shells are still fairly dense.
FCC can be viewed as close-packed triangular layers stacked as $ABCABC\ldots$.

3.6.2 Counting Lattice Points in Cubic Conventional Cells

For the cubic conventional cells:

\[ N_{\mathrm{lp}}^{\mathrm{SC}} = 8\times \frac18 = 1 , \]

\[ N_{\mathrm{lp}}^{\mathrm{BCC}} = 8\times \frac18 + 1 = 2 , \]

\[ N_{\mathrm{lp}}^{\mathrm{FCC}} = 8\times \frac18 + 6\times \frac12 = 4 . \]

Here $N_{\mathrm{lp}}$ is the number of lattice points in the conventional cell.

This immediately explains why the common monatomic cubic structures are labeled $cP1$, $cI2$, and $cF4$.

Figure 3.5: Figure placeholder. Primitive, conventional, and Wigner–Seitz cells, including the nonprimitive conventional cubic descriptions of BCC and FCC.

3.6.3 Primitive Lattice Vectors and Primitive-Cell Volume of the Cubic Bravais Lattices

For the cubic Bravais lattices, it is useful to distinguish clearly between the conventional cubic cell of edge length $a$ and the primitive cell, which contains exactly one lattice point. The primitive-cell volume is given by

\[ V_{\mathrm{prim}} = \left| \mathbf a_1 \cdot (\mathbf a_2 \times \mathbf a_3) \right| . \]

For SC, the conventional cubic cell is already primitive. For BCC and FCC, by contrast, the conventional cubic cell is chosen for symmetry reasons, while the primitive cell is smaller.

3.6.3.1 Simple Cubic

For the simple cubic lattice, a natural choice of primitive vectors is

\[ \mathbf a_1 = a(1,0,0), \qquad \mathbf a_2 = a(0,1,0), \qquad \mathbf a_3 = a(0,0,1). \]

The primitive-cell volume is therefore

\[ V_{\mathrm{prim}}^{\mathrm{sc}} = \left| \mathbf a_1 \cdot (\mathbf a_2 \times \mathbf a_3) \right| = a^3. \]

So for SC, the primitive cell and the conventional cubic cell coincide.

3.6.3.2 Body-Centered Cubic

For the body-centered cubic lattice, a convenient choice of primitive vectors is

\[ \mathbf a_1 = \frac{a}{2}(-1,1,1), \qquad \mathbf a_2 = \frac{a}{2}(1,-1,1), \qquad \mathbf a_3 = \frac{a}{2}(1,1,-1). \]

Their triple product gives

\[ V_{\mathrm{prim}}^{\mathrm{bcc}} = \left| \mathbf a_1 \cdot (\mathbf a_2 \times \mathbf a_3) \right| = \frac{a^3}{2}. \]

This is consistent with the fact that the conventional cubic BCC cell contains two lattice points, so the primitive-cell volume must be half of the conventional-cell volume.

3.6.3.3 Face-Centered Cubic

For the face-centered cubic lattice, a standard choice of primitive vectors is

\[ \mathbf a_1 = \frac{a}{2}(0,1,1), \qquad \mathbf a_2 = \frac{a}{2}(1,0,1), \qquad \mathbf a_3 = \frac{a}{2}(1,1,0). \]

The primitive-cell volume is

\[ V_{\mathrm{prim}}^{\mathrm{fcc}} = \left| \mathbf a_1 \cdot (\mathbf a_2 \times \mathbf a_3) \right| = \frac{a^3}{4}. \]

This agrees with the fact that the conventional cubic FCC cell contains four lattice points.

3.6.4 Hexagonal Close Packed

The HCP structure has the same close-packing efficiency as FCC, but a different translational symmetry. It is built from close-packed triangular layers stacked as

\[ ABAB\ldots \]

A convenient description uses a hexagonal Bravais lattice with a two-atom basis. In one common fractional-coordinate convention the basis is

\[ \left(\frac23,\frac13,0\right), \qquad \left(\frac13,\frac23,\frac12\right). \]

and the ideal close-packing ratio is

\[ \frac{c}{a}=\sqrt{\frac{8}{3}} \approx 1.633 . \]

HCP and FCC therefore share:

coordination number $12$,
the same maximum packing fraction $0.740$,

but they differ in stacking sequence and therefore in their Bravais-lattice description.

Note

Important Distinction. FCC is itself a Bravais lattice. HCP requires a two-atom basis on a hexagonal Bravais lattice.

Figure 3.6: Figure placeholder. Comparison of close-packed stacking in HCP and FCC, showing the $ABAB\ldots$ and $ABCABC\ldots$ sequences.

Optional / Appendix: Ideal Geometry of the HCP Ratio

Treat the atoms as touching hard spheres of radius $r$. In the close-packed basal plane the nearest-neighbor spacing is

\[ a = 2r. \]

The atom in the next layer sits above the center of an equilateral triangle of side $2r$. The four sphere centers form a regular tetrahedron of edge $2r$, whose height is

\[ h=\sqrt{\frac{2}{3}}(2r). \]

The hexagonal repeat contains two such layer spacings, so

\[ c=2h = 2\sqrt{\frac{2}{3}}(2r)/2 = 2\sqrt{\frac{8}{3}}r \]

and therefore

\[ \frac{c}{a} =\sqrt{\frac{8}{3}}. \]

3.6.5 Diamond

Diamond is an elemental structure with directional covalent bonding. Its underlying Bravais lattice is FCC, but the crystal requires a two-atom basis:

\[ (0,0,0), \qquad \left(\frac14,\frac14,\frac14\right). \]

Each atom is tetrahedrally coordinated and has coordination number $4$.

A useful mental picture is that diamond consists of two interpenetrating FCC sublattices displaced by one quarter of the body diagonal of the conventional cube. Because the bonding is strongly directional, the structure is much more open than close-packed structures.

3.6.6 Graphite

Graphite is the second important crystalline allotrope of carbon. Its structure consists of parallel graphene-like layers. Within each layer, carbon atoms form a honeycomb arrangement with strong covalent $sp^2$ bonding; neighboring layers are coupled only weakly by van der Waals forces.

The key real-space facts are:

within a layer, each carbon atom has three nearest neighbors,
the in-plane C–C bond length is about $1.42,\text{\AA}$,
the interlayer spacing is much larger, about $3.35,\text{\AA}$.

This large separation of energy scales explains the pronounced anisotropy of graphite: it behaves like a strongly bound two-dimensional solid within the planes and a weakly bound layered solid perpendicular to them.

A single graphene sheet is not a Bravais lattice; it is a triangular Bravais lattice with a two-atom basis. Graphite then results from stacking such sheets, most commonly in an $ABAB\ldots$ sequence for hexagonal graphite.

Figure 3.7: Figure placeholder. Comparison of diamond and graphite as two crystalline allotropes of carbon: tetrahedral three-dimensional bonding in diamond versus layered honeycomb sheets in graphite.

3.7 Common Binary Structure Types in Real Space

3.7.1 A Comparison Table

Prototype	Pearson Symbol	Underlying Bravais Lattice	Basis / Site Occupancy	Coordination	Structural Lesson
NaCl (rock salt)	$cF8$	FCC	two interpenetrating FCC sublattices, one shifted relative to the other	$6$ and $6$	octahedral coordination on an FCC framework
CsCl	$cP2$	simple cubic	one species at corners, the other at body center	$8$ and $8$	same picture as “BCC-like” occupancy, but not a BCC Bravais lattice
zincblende (ZnS)	$cF8$	FCC	two different atoms on the two diamond-like sublattices	$4$ and $4$	diamond geometry with two atomic species

3.7.2 Sodium Chloride

In the NaCl structure both ion species occupy interpenetrating FCC sublattices displaced with respect to one another. Each ion is surrounded by six oppositely charged neighbors, so the coordination polyhedron is octahedral.

A useful interpretation is that one ion species occupies the FCC lattice sites while the other occupies the octahedral sites of that same framework.

3.7.3 Cesium Chloride

In the CsCl structure the Bravais lattice is simple cubic and the basis is

\[ (0,0,0), \qquad \left(\frac12,\frac12,\frac12\right). \]

Each ion has eight nearest neighbors of the opposite kind.

The important conceptual point is again that the body-center site is not an independent Bravais-lattice point of the decorated structure, because the two sites carry different atoms.

3.7.4 Zincblende

Zincblende is the binary analogue of the diamond structure. The geometry is the same as in diamond, but the two sites of the basis are occupied by different atoms, for example Zn and S.

Every atom is tetrahedrally coordinated by four atoms of the other species.

Compared with diamond, the most important structural consequence is that inversion symmetry is lost once the two identical sublattices are occupied by different atomic species.

Figure 3.8: Figure placeholder. Comparison of NaCl, CsCl, and zincblende structure types, emphasizing how different bases on simple cubic or FCC frameworks generate different coordinations and symmetries.

3.8 Why the Basis Matters

At this stage the basis may still look like a bookkeeping device. It is more than that.

3.8.1 The Basis Can Change the Symmetry

The Bravais lattice may have a certain point symmetry, but the crystal structure built from it can have less symmetry if the basis is not compatible with all symmetry operations of the lattice. This is why one must clearly distinguish:

symmetry of the lattice,
symmetry of the decorated crystal.

3.8.2 The Basis Determines Local Coordination

NaCl and CsCl already show that changing the basis on a cubic framework changes the coordination number from $6$ to $8$. Diamond and graphite show even more strongly that the local geometry is controlled by the basis and bonding, not by translational periodicity alone.

3.8.3 The Basis Matters for Later Physics

If there are $p$ atoms in the primitive cell, then later in lattice dynamics the crystal will have $3p$ displacement degrees of freedom per wave vector, not just three. A multi-atom basis is therefore the structural prerequisite for optical phonon branches.

Likewise, when diffraction is introduced, the relative positions of atoms within the basis will control interference and reflection intensities.

These are future topics. The important point for today is simpler: the Bravais lattice tells us where repetition occurs, the basis tells us what is repeating.

3.9 Bridge to the Next Lecture

Today we stayed entirely in real space. We defined the direct lattice, distinguished primitive and conventional cells, introduced the basic symmetry operations, and classified the 14 Bravais lattices and the most important prototype structures.

In the next lecture we will keep the same crystals but change viewpoint:

from real-space periodicity to reciprocal-space periodicity,
from lattice planes and symmetry to diffraction conditions,
from atomic basis positions to interference and structure factors.

That transition will work only because the real-space description is now in place.

Take-Home Messages

A crystal structure is a Bravais lattice plus a basis.
Primitive, conventional, and Wigner–Seitz cells describe the same lattice from different viewpoints and are not interchangeable.
The basic point symmetries of periodic crystals are built from rotations, reflections, inversion, and their combinations.
Ordinary periodic crystals allow only 1-, 2-, 3-, 4-, and 6-fold rotational symmetry.
The seven crystal systems organize lattice metrics and characteristic rotations, while the 14 Bravais lattices classify distinct translational frameworks.
FCC is a Bravais lattice, whereas HCP requires a two-atom basis on a hexagonal lattice.
NaCl, CsCl, diamond, zincblende, and graphite show that changing the basis changes coordination, symmetry, and physical character even when the translational scaffold is similar.
The basis is the structural ingredient that later gives rise to optical phonons and nontrivial diffraction intensities.

3.10 Problem Set

Bravais Lattice or Not? Decide whether each of the following is a Bravais lattice or a Bravais lattice plus basis: honeycomb net, monatomic FCC crystal, HCP crystal, diamond crystal, single graphene sheet.
Counting Lattice Points. Count the number of lattice points in the conventional SC, BCC, and FCC cubic cells. State in each case whether the conventional cell is primitive.
Crystal Systems and Bravais Types. List the seven crystal systems and the Bravais-lattice types that belong to each of them. Check explicitly that the total number is 14.
NaCl Versus CsCl. Compare NaCl and CsCl in terms of underlying Bravais lattice, basis, and coordination number. Explain why CsCl is not a BCC Bravais lattice.
Diamond, Zincblende, and Graphite. Compare these three structures in terms of underlying Bravais lattice or layered framework, basis, coordination number, and qualitative bonding geometry.