1  Crystals and periodicity

The word crystal derives from the Greek κρύσταλλος (krýstallos), meaning “ice.” More than two thousand years ago, the ancient Greeks believed that rock crystal (clear quartz) was water frozen so deeply that it could never melt: Pliny the Elder wrote that it formed “where the winter snow freezes most intensely” (Naturalis Historia, XXXVII, 9). The term was later extended to any solid exhibiting regular faces, long before the atomic basis of this regularity was understood.

Today we know that this regularity reflects the underlying periodic arrangement of atoms, whose full significance became clear only with the advent of quantum mechanics in the 20th century. This concept - periodicity and its consequences - lies at the heart of this course.

In this chapter, we will focus on how to describe periodicity and, for instance, how to specify the structure of quartz shown in Figure 1.1. This will be essential for what follows, but bond geometry already helps explain some basic properties of a solid. For instance, quartz is an optically active material and can rotate polarization, since it supports light propagation at different speeds depending on the left or right circular polarization of electromagnetic waves. This behavior is rooted in the atomic structure of quartz, which can be viewed as a collection of intertwined “atomic spirals”, all with the same chirality, as highlighted in the figure.

Figure 1.1: Quartz crystals exhibit macroscopic regularity stemming from the regular arrangement of Si (ivory) and O atoms (red) in the crystal structure. Atomic spirals highlighted in green.

1.1 How to define a crystal1

A crystal is a solid composed of atoms arranged in a periodic pattern; two fundamental ingredients are needed to describe it:

  1. A Bravais lattice, which encodes the translational symmetry of the crystal;
  2. A basis, which defines the repeated unit.

Here, we want to mathematically clarify these ingredients and the connection between a crystal, its periodicity scheme, the structure of the repeated element and the consequent symmetries, as sketched in Figure 1.2.

Figure 1.2: A crystal consists of a repeated cell, containing one or more atoms (described by the basis), replicated multiple times according to a discrete translational symmetry group (described by the lattice). Note crystals can have additional symmetries (rotation axes, mirror lines/planes), which can be different even in crystals with identical lattices. In the sketch: graphene versus hexagonal boron nitride.

1.1.1 Bravais lattice

Definition 1.1: Bravais Lattice

A Bravais lattice \(\mathcal{BL}\) is an infinite set of discrete translation vectors, often represented as a periodic set of points, generated by integer linear combinations of a set of linearly independent primitive vectors (PVs) \(\{\mathbf{t}_1, \mathbf{t}_2, \mathbf{t}_3\}\)

\[\mathcal{BL} = \left\{\mathbf{t}_\mathbf{n} = n_1\mathbf{t}_1 + n_2\mathbf{t}_2 + n_3\mathbf{t}_3 \mid n_i \in \mathbb{Z}\right\}. \tag{1.1}\]

As for every linearly independent vector set, the number of primitive vectors needed to generate \(\mathcal{BL}\) depends on its dimensionality. The case above corresponds to a 3D lattice while in 2D two primitive vectors will suffice.

Note: many alternative primitive vectors can produce the same lattice and, for instance, \(\{\mathbf{t}_1+n\mathbf{t}_2, \mathbf{t}_2, \mathbf{t}_3\}\) obviously generates the same \(\mathcal{BL}\) for any \(n\in\mathbb{Z}\).

The set \(\mathcal{BL}\) is basically a representation of the translational symmetry group of the crystal. In fact, it is closed under vector addition (if \(\mathbf{a}, \mathbf{b} \in \mathcal{BL}\), then \(\mathbf{a} + \mathbf{b} \in \mathcal{BL}\)) and includes the inverse element (if \(\mathbf{a} \in \mathcal{BL}\), then \(-\mathbf{a} \in \mathcal{BL}\)). Note the lattice only tells how the crystal repeats itself throughout the space, see the basis section for a description of what is exactly repeated.

Definition 1.2: Primitive cell

A primitive cell (PC) is any region that, when translated by every \(\mathbf{t} \in \mathcal{BL}\), tiles the full space (this is why we call it “cell”) exactly once without gaps or overlaps (this is why we qualify it as “primitive”). When talking about “the” PC, one specifically means

\[\left\{\mathbf{r} = x_1\mathbf{t}_1 + x_2\mathbf{t}_2 + x_3\mathbf{t}_3 \mid 0\le x_i < 1\right\}. \tag{1.2}\]

i.e. the parallelogram (in 2D) or parallelepiped (in 3D) spanned by the primitive vectors \(\mathbf{t}_i\). As already mentioned, PVs are not unique, so the PC is not unique either. Nevertheless, its volume is fixed and equals the scalar triple product (in 3D) or, more generally, the absolute value of the determinant built with the PVs

\[\Omega = |\mathbf{t}_1 \cdot (\mathbf{t}_2 \times \mathbf{t}_3)| = \left| \begin{array}{ccc} t_{1x} & t_{1y} & t_{1z} \\ t_{2x} & t_{2y} & t_{2z} \\ t_{3x} & t_{3y} & t_{3z} \end{array} \right|. \tag{1.3}\]

It is useful to reiterate. Please do not carelessly mix the two terms crystal (describing the full periodic arrangement of atoms) and lattice (describing the underlying abstract translational symmetry). In Solid State courses, the two often coincide: replace every “point” in the Bravais lattice with an atom and there is the crystal.

This is a pure accident and common misconception: many crystals, except a few simplest ones, do not work like that. This is only true when the PC, used to tile the crystal space according to the repetition scheme described by the Bravais lattice, contains a single atom. Crystals are most of the times much more complex and, for instance, in protein crystallography the repeated cell can even contain thousands of atoms.

A Bravais lattice is also characterized by rotational or mirror symmetries. The PC built from the PVs often has a fundamental limitation: it is not, or even can not be, always explicitly symmetric, as illustrated in Figure 1.3. Given the importance of symmetry in physics, this deserves a further discussion.

Figure 1.3: Cells on a triangular lattice. Blue and yellow: two equally valid primitive cells. Green: a hexagonal conventional cell (symmetric but non-primitive). Violet: a Wigner–Seitz cell (primitive and symmetric). Hover on the cells to highlight the tiling.

One possible way around the non-unique definition of PC is to introduce the conventional unit cell (also u.c.), which is often non-primitive (i.e. it can be redundant and contain more than one lattice point), but it is explicitly symmetric: in Figure 1.3, the green hexagonal cell contains three lattice points (one at the center plus six vertices shared with three neighboring cells) but its shape immediately reveals the six-fold symmetry of the lattice.

As a further alternative, the Wigner–Seitz (WS) cell is primitive but also it is built directly from the \(\mathcal{BL}\) points and it thus inherits the lattice symmetry by construction.

Definition 1.3: Wigner–Seitz Cell

The Wigner–Seitz cell of \(\mathbf{t}\in\mathcal{BL}\) is the set of all points closer to \(\mathbf{t}\) than to any other point in \(\mathcal{BL}\):

\[\text{WS}(\mathbf{t}) = \bigl\{\mathbf{r} : |\mathbf{r}-\mathbf{t}| < |\mathbf{r}-\mathbf{t}'| \;\;\forall\; \mathbf{t}' \in \mathcal{BL},\; \mathbf{t}' \neq \mathbf{t}\bigr\}. \tag{1.4}\]

Geometrically, the WS cell can be obtained by drawing the perpendicular bisector planes of the segments connecting a lattice point to its nearest neighbors, and taking the innermost enclosed volume (see violet cell in Figure 1.3). Notably, this construction is a special case of a Voronoi tessellation.

The WS cell is automatically primitive and inherits the full symmetry of the lattice. Its shape, however, can be geometrically more complicated, in particular in low-symmetry lattices. The WS becomes especially important in the reciprocal space, where it defines the first Brillouin zone (cf. Section 2.1).

1.1.2 Crystal families

Lattices can be grouped based on symmetry, but not every symmetry is compatible with periodicity.

Theorem 1.1: Restriction Theorem

The only rotations compatible with the translation scheme of a Bravais lattice are those of order \(n = 1\), \(2\), \(3\), \(4\) and \(6\), i.e. rotations by angle

\[\theta = \frac{2\pi}{n} = \{0°, 180°, 120°, 90°, 60°\}. \tag{1.5}\]

Show proof (2D case)

Consider a 2D lattice with a periodicity defined by a set of primitive vectors, including a given \(\mathbf{t}\) of length \(a\). The translation vectors along this direction are

\[\mathcal{T} = \bigl\{m\,\mathbf{t}: m\in \mathbb{Z}\bigr\}. \tag{1.6}\]

Suppose now the lattice is also invariant under a rotation by angle \(\theta\): rotations \(\mathcal{R}(+\theta)\) and \(\mathcal{R}(-\theta)\) create two additional sets of symmetric translation vectors, including the first two rotated vectors \(+\mathcal{R}(+\theta)\mathbf{t}\) and \(-\mathcal{R}(-\theta)\mathbf{t}\). The vector connecting them is also a legitimate translation vector, parallel to \(\mathbf{t}\) and of length \(2a\cos\theta\).

Figure 1.4: Rotating translation vectors \(\pm a\) by \(\pm\theta\) produces a new horizontal translation vector of length \(2a\cos\theta\). For this distance to be a good lattice translation, \(\cos\theta = m/2\) with \(m\) integer, restricting \(\theta\) to only a few options.

The new vector clearly needs to be a multiple of the primitive \(\mathbf{t}\) and thus

\[2a\cos\theta = ma, \qquad m \in \mathbb{Z}. \tag{1.7}\]

and, dividing by \(a\), we obtain \(\cos\theta = m/2\). Since \(-1 \le \cos\theta \le 1\), the only allowed integer values are \(m = \{-2, -1, 0, 1, 2\}\), yielding

\[\theta = 0, \; \pm\frac{\pi}{3}, \; \pm\frac{\pi}{2}, \; \pm\frac{2\pi}{3}, \; \pm\pi. \tag{1.8}\]

These are the angles associated with rotations of order \(n = 1, 2, 3, 4, 6\). The same argument can be easily extended to the 3D case. \(\blacksquare\)

Bravais lattices are classified into lattice families sharing the same rotational/mirror symmetries. In 2D, this leads to finite possibilities that can be explored manually in interactive Figure 1.5: there are five Bravais lattices, grouped into four families:

  • Oblique: this is the least symmetric option and we only have a set of (unavoidable) inversion centers.
  • Rectangular (coming in two flavors, primitive or centered): here we add mirror lines.
  • Square: here symmetry increases and we have 4-fold rotations, plus further diagonal mirror lines.
  • Hexagonal: a different way to increase the symmetry, with 6-fold rotations and many mirror lines.

Note all these symmetries represent the maximal symmetry compatible with a given lattice. The actual symmetry of a crystal will depend on what is the exact repeated unit.

Figure 1.5: Interactive 2D Bravais lattices. Drag the blue point (\(\mathbf{t}_2\)) to explore the five lattice classes; the red point is \(\mathbf{t}_1 = (1,0)\). Colors: blue primitive cell; green conventional (unit) cell; violet Wigner–Seitz cells. Configurations are fully generic and only assume that the vector marked by the red dot is the smallest one in the \(\mathcal{BL}\).

The illustrated families correspond to the rotational symmetries allowed by the restriction theorem, plus we have variants obtained by adding different compatible centering schemes. Adding a Bravais point at the center of the cell is compatible with the cell symmetry since the cell center is one of the inversion centers of the Bravais lattice… but this possibility might give rise to a few legitimate doubts:

  • why do we have a centered rectangular lattice but not a centered square lattice?
  • why don’t we have a hexagonal lattice not including the Bravais lattice point at the center?

The same questions can be posed in the 3D case, where the answer is harder to visualize. In 2D, it is easy to see that a centered square lattice is just a smaller primitive square lattice rotated by 45°, so the centering adds no new lattice type. Conversely, the honeycomb arrangement - in practice, this is the atomic pattern of graphene - is not a simple Bravais lattice and has a basis. Indeed, the vector connecting two neighboring hexagonal corners (\(\mathbf{t}\) in Figure 1.6) is not a legitimate primitive translation vector since \(2\mathbf{t}\) is not a good translation.

Figure 1.6: Left: A centered square lattice in green is just a simple square lattice rotated by 45°, see the blue cell. Right: The honeycomb arrangement: taking the nearest-neighbor vector t and translating by 2t, one lands on the center of a hexagon, where there is no point, so this collection of points is not a Bravais lattice.

A further remark: those mid-points are inversion centers of the lattice, the only places where a further Bravais point can be added without breaking its symmetry.

1.1.3 Lattices in 3D

The extension to 3D is not conceptually complicated, but it is surely less obvious to visualize. There are fourteen Bravais lattices grouped into seven families based on their fundamental symmetries (see Figure 1.7):

  • Cubic: the highest symmetry family, it includes most elemental metals (Fe, Cu, Al), ionic crystals (NaCl), and semiconductors (Si, GaAs). Most crystals studied in these notes belong here.
  • Tetragonal: a cubic lattice stretched along one axis, many symmetries are obviously lost. Found for instance in high-\(T_c\) superconductors (YBa\(_2\)Cu\(_3\)O\(_7\)).
  • Orthorhombic: one more stretching and all high-order rotations are lost, only 2-fold axes remain.
  • Hexagonal: another top symmetry lattice, characterized by a 6-fold axis. This is the lattice of graphite and of many metals touched in these notes.
  • Trigonal: this can be seen as a less symmetric version of cubic lattice (cell deformed along the diagonal), but also a less symmetric version of an hexagonal lattice.
  • Monoclinic: a tilted orthorhombic lattice, only a 2-fold rotation axis and mirror plane survive.
  • Triclinic: one more tilt and we get the least symmetric option, only inversion survives.

Important naming conventions exist for cubic crystals: cubic lattices are referred to as Simple Cubic (SC or cP), Body-Centered Cubic (BCC or cI), and Face-Centered Cubic (FCC or cF). In the figure, key symmetries of the cells are highlighted including in particular: \(n\)-fold rotations \(C_n\) and mirror planes \(\sigma\), with subscript \(h\) (horizontal), \(v\) (vertical) or \(d\) (diagonal).

Figure 1.7: Interactive 3D viewer of the 14 Bravais lattices. Use the right button to show the symmetry operations compatible with the shape of the unit cell (further details in Section 1.3).
Table 1.1: Summary table of the 14 Bravais lattices in 3D. Centering types: P = primitive (also called simple), I = body-centered (Innenzentriert), F = face-centered, C = base-centered.
Family Lengths Angles P C I F
Cubic \(a = b = c\) \(\alpha = \beta = \gamma = 90°\) ✓ ✓ ✓
Tetragonal \(a = b \neq c\) \(\alpha = \beta = \gamma = 90°\) ✓ ✓
Orthorhombic \(a \neq b \neq c\) \(\alpha = \beta = \gamma = 90°\) ✓ ✓ ✓ ✓
Hexagonal \(a = b \neq c\) \(\alpha = \beta = 90°\), \(\gamma = 120°\) ✓
Trigonal \(a = b = c\) \(\alpha = \beta = \gamma \neq 90°\) ✓
Monoclinic \(a \neq b \neq c\) \(\alpha = \gamma = 90° \neq \beta\) ✓ ✓
Triclinic \(a \neq b \neq c\) \(\alpha \neq \beta \neq \gamma\) ✓

The pattern of centering schemes, C, I, F… follows an apparently obscure logic, but it is not so obscure at close inspection. First of all, any collection of Bravais points has an infinite number of inversion centers: for any good translation vector \(\mathbf{t}\), the vector \(-\mathbf{t}\) is also a good translation vector, thus the collection of points is equal to itself if you invert all the axes around any given point. We can extend this further: for any given translation \(\mathbf{t}_0\) the mapping \(\mathbf{t}\to\mathbf{t}_0-\mathbf{t}\), which is an inversion around the fixed point \(\mathbf{t}_0/2\), sends \(\mathcal{BL}\) in itself. This can be recast in the following terms: for any pair of points in the Bravais, the mid-point is an inversion center of the lattice.

Clearly, these are the only locations where it is “possible” to add a further Bravais point (plus all its translated replicas) without disrupting the lattice and its symmetry. If we start from a cubic cell, any point located at (i) the middle of any side of the cube, (ii) the center of any face, (iii) the center of the cube, satisfy this rule. However, if we want to keep the cubic symmetry, we have to rule out the middle of the sides, and if we choose one face center then by symmetry we need to add also all the other ones. Thus the cubic lattice only allows body centering and face centering. Note that base centering used in the orthorhombic case does not make sense for a cubic cell, since the resulting Bravais has tetragonal and not cubic symmetry.

Important note. While any lattice is always symmetric under inversion, this is not the case for any crystal. This depends on whether the repeated cell, with all its atoms, is symmetric or not under inversion. For instance diamond has inversion points, zinc-blende has not. Inversion symmetry is very important in crystals, for instance its presence forbids piezoelectric effects.

1.1.4 Basis

We now move to the second key ingredient required to define a crystal structure: the basis.

Definition 1.4: Basis

Given a crystal, its basis specifies the set of \(N_b\) atoms that are repeated according to the translational scheme defined by \(\mathcal{BL}\). Formally, this is defined by a set of pairs \(\{(\mathbf{d}_\nu,\alpha_\nu)\,|\,\nu=1\dots N_b\}\) where \(\mathbf{d}_\nu\) denotes the position of each atomic species \(\alpha_\nu\) within the cell. The position of the atoms \(\alpha_\nu\) in the crystal is

\[\mathbf{r}_\nu = \mathbf{R} + \mathbf{d}_\nu, \tag{1.9}\]

where \(\mathbf{R}\in\mathcal{BL}\) and \(\nu\) labels the basis element. This is also called a sublattice.

The crystal structure is identified by both the Bravais lattice and the basis. When the basis trivially consists of a single atom (\(\mathbf{d}_1 = \mathbf{0}\)), the crystal is said to be a simple crystal. In such a special case, the lattice points coincide with the crystal atomic positions, but - as already mentioned - this is the exception rather than the rule. Even most “elemental” crystals (meaning they contain a single atomic species) are not simple crystals: for example both silicon and graphite are elemental, but have a non-trivial basis; as discussed in the section devoted to closely-packed stacking, also hexagonal metals are not simple.

1.1.5 Wrap-up: graphene

Graphene is a 2D crystal containing only C atoms, nevertheless its honeycomb arrangement is not a \(\mathcal{BL}\), as argued before (see Figure 1.6). Graphene can be defined by the following primitive vectors

\[\mathbf{t}_{1/2} = a\left(\pm 1,\, \sqrt{3}\right)/2, \tag{1.10}\]

where \(a \approx 2.46\) Å is the lattice constant (the distance between atoms of the same sublattice), plus a basis

\[\begin{cases} \begin{aligned} \mathbf{d}_A &= (0, 0) &\quad \text{C} \\ \mathbf{d}_B &= a(0, 1)/\sqrt{3} &\quad \text{C} \end{aligned} \end{cases} \tag{1.11}\]

The nearest-neighbor C–C distance is \(a/\sqrt{3} \approx 1.42\) Å. The interactive figure below shows the graphene crystal with various cell overlays. The first “Primitive cell” choice corresponds to the description above, but equally legitimate cells can be defined, in particular:

  • a shifted primitive cell, requiring a translation of \(\mathbf{d}_{A/B}\) by \(a/\sqrt{3}\) in the \(y\) direction;
  • a custom rectangular cell, very unusual but still a legitimate primitive cell;
  • a Wigner-Seitz cell built using the site B as the lattice point and cell center;
  • another Wigner-Seitz cell built using a different reference point.

The lattice sites are highlighted by a blue cylinder, which is on purpose visually different from the atoms: the lattice and the crystal are two different concepts, and the lattice site generally does not indicate the position of an atom.

Figure 1.8: Graphene honeycomb lattice with interactive cell overlay. Use the dropdown to explore: two primitive cell choices, a rectangular conventional cell, and Wigner–Seitz cells centered on each sublattice.

Primitive vectors and basis vectors completely define the crystal, but many equivalent choices are possible. It is worth recalling here exactly how this description is not unique:

  • primitive vectors are not uniquely defined as discussed in Section 1.1.1;
  • the basis vectors are even less uniquely defined:
    • origin change, the basis vectors can be collectively shifted by a given constant vector;
    • a basis vector can be translated by a primitive vector: this can lead to a non-unique definition in particular in the case of basis vectors describing atoms located at the edge of the unit cell;
  • orientation change, the overall orientation of the crystal is arbitrary: we don’t get a different crystal if we rotate all primitive and basis vectors; this often leads to alternative descriptions of the graphene crystal, since there are two equally obvious ways to orient the crystal with respect to the Cartesian axes.

1.2 Common crystals2

In the following, we recall the most common crystalline structures. Many technologically and scientifically relevant crystalline structures belong to the cubic family.

Face centered cubic (FCC). The conventional choice (note the cyclic symmetry of the coordinates) for the primitive vectors of this Bravais lattice is

\[\begin{cases}\mathbf{t}_1 = a(0,1,1)/2,\\ \mathbf{t}_2 = a(1,0,1)/2,\\ \mathbf{t}_3 = a(1,1,0)/2.\end{cases} \tag{1.12}\]

and a primitive cell volume \(\Omega = |\det(\mathbf{t}_1, \mathbf{t}_2, \mathbf{t}_3)| = a^3/4\), i.e. one quarter of the conventional cubic cell, consistent with the \(\mathcal{BL}\) points inside the cube: 8 corner points shared between 8 nearby cubes, plus 6 face points shared between 2 nearby cubes, thus \(8/8+6/2=4\). Many metals crystallize as a simple FCC (one atom per primitive cell, i.e. lattice and crystal coincide): typical examples include noble metals (Cu, Ag, Au), aluminum, and several molecular solids. The lattice parameter falls in the \(a \sim 3\)–\(5\) Å range and the coordination number in a simple FCC is 12.

DefinitionCoordination number

The coordination number is defined as the count of atoms that are in direct contact with a central given crystalline site. In the case of FCC, it is easy to see the nearest neighbors are located at

\[a(\pm 1,\pm 1, 0)/2\]

and cyclic permutations, thus we have \(4\times3=12\) positions. The concept can be extended to second nearest neighbors, in this less common case one can talk about “second coordination shell”.

Body centered cubic (BCC). The conventional choice for the primitive vectors is

\[\begin{cases}\mathbf{t}_1 = a(-1,+1,+1)/2,\\ \mathbf{t}_2 = a(+1,-1,+1)/2,\\ \mathbf{t}_3 = a(+1,+1,-1)/2.\end{cases} \tag{1.13}\]

and a primitive cell volume \(\Omega = a^3/2\), as easily confirmed by looking at the \(\mathcal{BL}\) lattice points inside the cube. Again various metals - typically alkali metals, but many others such as Fe - crystallize as simple BCC, with a lower coordination number of 8. The interactive Figure 1.9 below shows both the FCC and BCC structures side by side, with the cited primitive cells.

Figure 1.9: Left: BCC structure — Bravais lattice points (blue) with body centers from adjacent cubes (gray). Right: FCC structure — Bravais lattice points (blue) with face centers from adjacent cubes (gray). In both panels the red parallelepiped with semi-transparent faces shows the primitive cell.

1.2.1 Diamond and zincblende

The crystalline structure of diamond belongs again to the FCC lattice group, but with a non-trivial basis

\[\begin{cases} \begin{aligned} \mathbf{d}_1 &= (0,0,0) &\quad \text{C} \\ \mathbf{d}_2 &= a(1,1,1)/4 &\quad \text{C} \end{aligned} \end{cases} \tag{1.14}\]

describing the location of two C atoms in the primitive cell of this crystal, which is visible in Figure 1.10. In this case one C atom resides at the Bravais lattice positions, while the second one is at \(1/4\) of the cube diagonal: this is the most common basis choice but consider that \(\mathbf{d}_{1/2}=\pm a(1,1,1)/8\) is also a legitimate choice, which is used in different contexts. The name “diamond crystalline structure” is not only used for proper diamond, but also for the structure of Si and Ge. In all these alternative structures, each atom is tetrahedrally coordinated with four nearest neighbors at distance \(a\sqrt{3}/4\) and the coordination number is 4.

The zincblende (ZB) structure is almost the same as diamond one — FCC lattice with a two-atom basis at \((0,0,0)\) and \(a(1,1,1)/4\) — but for a key difference: the two basis atoms belong to different chemical species. This breaks the inversion symmetry in these crystals, see Figure 1.10. Examples include GaAs (\(a = 5.65\) Å), InP (\(a = 5.87\) Å), ZnS (\(a = 5.41\) Å). Each atom of species A is tetrahedrally surrounded by four atoms of species B, and vice versa.

Figure 1.10: Left: Diamond structure (Si, C, Ge) — FCC with two identical atoms. Right: Zincblende structure (SiC, GaAs, ZnS) — same geometry but the two sublattices carry different species. Use the checkboxes to highlight the position of the basis elements along the cube diagonal.

1.2.2 Closely-packed structures

While many metals - including in particular noble metals such as Au or Cu - crystallize in the form of a simple FCC, many other metals have their atoms organized inside a further pattern with coordination number 12, also known as hexagonal close-packed (HCP). The two structures are much more similar than they seem and emerge from the same basic problem: how can we pack a set of objects (spheres, to fix ideas) as compact as possible?

Figure 1.11 illustrates how close-packed structures are built step by step. Starting from a single triangular layer of touching spheres (the densest possible 2D packing), a second layer is placed in the hollows of the first. Notably, there are two inequivalent options when we need to choose the hollows where to place the spheres in the next layer. This ambiguity becomes crucial when we stack a third layer, and we have two qualitatively different possibilities: placing it directly above the first layer (ABA stacking) yields the hexagonal close-packed (HCP) structure, while placing it above the other set of hollows (ABC stacking) yields the cubic close-packed (corresponding to the already discussed FCC) structure. Despite being built from the same layers, the two stackings produce fundamentally different symmetries.

Figure 1.11: Close-packed structures built from triangular layers. Use the dropdown to follow the construction: a single triangular layer, then an AB stack, then the two distinct ways of adding a third layer — ABA (hexagonal close-packed, left) vs ABC (cubic close-packed, right). The last view highlights the conventional unit cells: a hexagonal prism for HCP and a face-centered cube for FCC.

Notably, the HCP structure is not a simple Bravais lattice: it is a hexagonal lattice where \(c=a\sqrt{8/3}\approx 1.633\,a\), with a basis (assuming \(\mathbf{t}_1\) and \(\mathbf{t}_2\) are such that they form a 60° angle)

\[\begin{cases} \mathbf{d}_1 = (0,0,0) \\ \mathbf{d}_2 = (\mathbf{t}_1+\mathbf{t}_2)/3 + \mathbf{t}_3/2 \end{cases} \tag{1.15}\]

which describes the crystalline form of many metals such as Zn, Ti, Mg, Co. As done previously in the case of simple FCC crystals, many further structures can be derived from an HCP arrangement. First of all, we have the wurtzite (WZ) structure, which is the hexagonal analogue of zincblende: a HCP lattice with two different species, each tetrahedrally coordinated (thus, every primitive cell again contains 4 atoms). Examples include ZnO, GaN, AlN. Graphite can also be considered as a related structure: it is hexagonal and implements an ABA stacking scheme of graphene layers, even if the ratio \(c/a\) is not really consistent with a closely-packed scheme (graphene atomic planes are further apart than what is expected for an HCP arrangement). Both these ABA stacking examples are visible in Figure 1.12, pay close attention to the content of the primitive cell highlighted in red: both crystals contain four atoms in their basis.

Figure 1.12: Left: Graphite — ABA stacking of honeycomb carbon layers (gray/violet for alternating layers), stick-only rendering. Right: Wurtzite GaN is the HCP analogue of FCC zincblende, with Ga (pink) and N (violet) tetrahedrally coordinated. For each panel you can visualize the full crystal, the single primitive cell or the hexagonal cell with the highlighted ABA stacking sequence.

Since the ABA stacking of graphene was highlighted to be almost an HCP stacking scheme, except for the “wrong” \(c/a\) ratio, it is also worth noting that an ABC stacking with a “wrong” \(c/a\) ratio is nothing but a structure belonging to the trigonal lattice system as it can be easily verified by comparing the primitive cell of the FCC in Figure 1.9 with the trigonal Bravais lattice in Figure 1.7.

Cannonballs. The question of how to pile identical spheres vastly predates crystallography and it was born in a shipyard. In the 1590s Sir Walter Raleigh asked his mathematician T. Harriot for a rule to count the cannonballs in a pile of given base, a basic bookkeeping problem for a fleet. Harriot solved the counting, then wrote to Kepler about the deeper question hiding behind it: which pile is the tightest?

Kepler’s conjecture. In 1611 Kepler stated without proof that the cubic close packing is “the tightest possible, so that in no other arrangement could more pellets be stuffed into the same container”. The density is

\[\frac{\pi}{\sqrt{18}} = \frac{\pi}{3\sqrt{2}} \approx 0.74048,\]

and both ABC and ABA stacking reach it, as do the uncountably many random stackings ABCACB… in between: close packing does not single out a crystal, only a family.

Gauss, and what was left open. In 1831, reviewing a book by Seeber on ternary quadratic forms, Gauss proved that no lattice packing can beat \(\pi/\sqrt{18}\). That settles the question among regular arrangements, but a pile of cannonballs need not be regular and the possibility of some clever irregular arrangement doing better remained open. Hilbert put the general problem on his 1900 list, as part of the eighteenth problem. The most general problem was only settled very recently (T. Hales et al., A formal proof of the Kepler conjecture, Forum of Mathematics, Pi 5, e2): it took four centuries, and the last step was taken by a machine. In even more recent years, the same problem extended to higher dimensions has hit the news in connection with the role of artificial intelligence.

Metals and BCC packing. Note here that high-density packing is not always the lowest energy choice, even in the case of fairly simple crystal metals. In many cases (notably, in iron!) atoms prefer to organize themselves in square layers, leading to the already discussed BCC lattice. The packing fraction, \(\pi\sqrt{3}/8 \approx 68\%\), is definitely worse and in fact this is never called close-packed structure. Yet BCC is the equilibrium configuration of iron at room temperature, and of W, Cr, Mo, Na, K: cohesion is set by the details of electronic bonding and not by density alone.

Figure 1.13: Left: Examples of stacking built from triangular or square planes of touching spheres. Can you tell if the triangular one is an ABA or ABC sequence? Right: Geometry of a tetrahedron.

Silicon crystal structure and the geometry of a tetrahedron. The silicon structure is built on Si tetrahedra emerging from \(sp^3\) bond coordination, see Figure 1.13. Here make a few deductions from the claimed crystal structure of silicon. The primitive vector \(\mathbf{t}_1\) of the FCC lattice is one of the edges of such Si tetrahedra, of length \(\ell = a\sqrt{2}/2\). From the diamond basis vectors we learn that the Si–Si bond is at \(1/4\) along the cube diagonal (see Figure 1.10), thus \(a_0 = a\sqrt{3}/4 \approx 0.43\,a\), which also equates the distance between a tetrahedron vertex and its barycenter. Hence the silicon crystal structure implies that we should have \(a_0=\ell\sqrt{6}/4\). Since FCC is also an ABC stacking, three atomic planes fit along the cube diagonal so the diagonal is thrice the tetrahedron height, \(d=a\sqrt{3}/3=\ell\sqrt{2/3}\). Combining the two results gives \(a_0=3d/4\): the bond length is \(1/4\) of the diagonal, the interplanar distance \(d\) is \(1/3\) of the diagonal.

Do these relations derived from the silicon crystal structure really match the geometry of a tetrahedron? Start from an equilateral base of side \(\ell\) centered at the origin. Each vertex lies at \(\ell/2/\sin 60° = \ell/\sqrt{3}\) from the center, so the height \(d\) follows from

\[\ell^2/3+d^2 = \ell^2 \implies d = \ell\sqrt{2/3},\]

which matches the previous derivation. For the barycenter, at height \(B\) above the base, the vertex–barycenter distance must satisfy

\[a_0 = d-B = \sqrt{\ell^2/3 + B^2},\]

which gives \(B=d/4\) and therefore \(a_0=3d/4\), again a perfect match.

1.2.3 NaCl and CsCl structures

Sodium chloride, the common “salt”, crystallizes in an FCC lattice with yet another two-atom basis:

\[\begin{cases} \begin{aligned} \mathbf{d}_1 &= (0,0,0) &\quad \text{Na} \\ \mathbf{d}_2 &= a(1,0,0)/2 &\quad \text{Cl} \end{aligned} \end{cases}\]

i.e. NaCl can be described as two interpenetrating FCC sublattices offset by \(a(1,0,0)/2\). Each Na\(^+\) is octahedrally surrounded by six Cl\(^-\) ions. Many ionic compounds share this structure: KCl, MgO, FeO, etc. The NaCl structure is perfect to make a few further considerations on the FCC structure:

  • Violet Na atomic sites reproduce the standard FCC pattern, green Cl sites look different but they are also on a FCC pattern: this is what an FCC looks like if we place a Bravais point at the center of the cubic cell; in this case, the nearest sites are located at the middle of each edge of the cubic cell and the coordination number of FCC is more evident.

  • Each Cl atom is surrounded by 6 nearest neighbor Na atoms, thus its coordination number is 6. The second-nearest neighbor Cl atoms (second coordination shell) are instead 12.

Another common cubic compound is Cesium chloride, having a simple cubic Bravais lattice with a two-atom basis:

\[\begin{cases} \begin{aligned} \mathbf{d}_1 &= (0,0,0) &\quad \text{Cs} \\ \mathbf{d}_2 &= a(1,1,1)/2 &\quad \text{Cl} \end{aligned} \end{cases}\]

Although it looks like a BCC lattice, it is not: Cs and Cl are different species, so the body-center translation is not a symmetry of the crystal. Each Cs\(^+\) is surrounded by eight Cl\(^-\) at the cube corners, and vice versa. Other examples: CsBr, CsI, TlCl. See Figure 1.14 for details.

Figure 1.14: Left: NaCl crystal structure — Na\(^+\) (violet) and Cl\(^-\) (green) on two interpenetrating FCC sublattices, each ion octahedrally coordinated. Right: CsCl crystal structure — simple cubic lattice with Cs\(^+\) (orange) at the corners and Cl\(^-\) (green) at the body center; each ion is eightfold coordinated. Note: CsCl is not BCC, since the two sites carry different species.

1.2.4 Perovskite structure

The perovskite structure (general formula ABO\(_3\), 3D visualization in Figure 1.15) is a cubic structure with: type A atoms at the cube corners, B atoms at the body center, O atoms at the face centers. This translates to the following basis vectors (alternative definitions are possible)

\[\begin{cases} \begin{aligned} \mathbf{d}_1 &= (0,0,0) &\quad \text{A} \\ \mathbf{d}_2 &= a(1,1,1)/2 &\quad \text{B} \\ \mathbf{d}_3 &= a(0,1,1)/2 &\quad \text{O1} \\ \mathbf{d}_4 &= a(1,0,1)/2 &\quad \text{O2} \\ \mathbf{d}_5 &= a(1,1,0)/2 &\quad \text{O3} \end{aligned}\end{cases} \tag{1.16}\]

The perovskite family includes many technologically important materials: BaTiO\(_3\) (ferroelectric), SrTiO\(_3\) (substrate for thin-film growth), and the hybrid organic–inorganic perovskites used in solar cells. Note that many equivalent basis might be used here, for instance \(\mathbf{d}_4\) and \(\mathbf{d}_4-\mathbf{t}_1\) are equivalent sites in the crystal so we could well say that \(O2\) is located in \(a(-1,0,1)/2\). The basis above is simply convenient because the oxygen positions are related by coordinate permutation, thus easier to remember.

Figure 1.15: Perovskite ABO\(_3\) structure. A atoms (green) at cube corners, B atom (blue) at the body center, O atoms (red) at face centers. The unit cell contains one formula unit.

As a final wrap-up, Figure 1.16 summarizes the main crystalline structures of elemental solids (where elemental means that the crystal contains a single atomic species: note that this does not imply the crystal has a trivial monoatomic basis, an obvious counterexample being diamond).

Figure 1.16: Interactive periodic table of the elements color-coded by crystal structure at room temperature. Select a structure type from the dropdown to highlight the corresponding elements and display a generic 3D unit cell. Hover over an element to see lattice parameters. Drag to rotate the unit cell, scroll to zoom.

1.3 Planar and space groups

The symmetry of a crystal depends both on the Bravais lattice and on the symmetry of the repeated cell, which is in turn described in terms of the so-called point group (PG).

Definition 1.5: Point group

Point groups describe norm-preserving symmetry operations that leave the origin fixed, thus they are linear isometries belonging to the orthogonal group \(O(n)\). Symmetry elements include (Schoenflies notation)

  • Identity element indicated as \(\varepsilon\);
  • Mirror reflections indicated as \(\sigma\), with subscript \(h\) (horizontal), \(v\) (vertical) or \(d\) (diagonal);
  • Proper rotations of order \(n\) indicated as \(C_n\);
  • Improper rotations (rotation plus axial mirror) of order \(n\) indicated as \(S_n\);
  • Inversion element indicated as \(i\).

In the context of crystals, the only relevant rotations are those compatible with Theorem 1.1, thus only those restricted to \(n=1\), \(2\), \(3\), \(4\) and \(6\).

2D — 5 Bravais, 10 PGs ⇒ 17 planar groups

In 2D, there are 10 distinct PGs describing the crystal symmetry:

  • Cyclic groups: they only contain rotations and are named \(C_n\), with order \(n=1, 2, 3, 4, 6\);
  • Dihedral groups: they also contain mirror symmetries and are named \(D_n\).

The combination of these PGs with 2D Bravais lattices yields 17 possible planar groups (also named wallpaper groups). This is much less than \(5\times10=50\) since not every point group is relevant for any Bravais lattice. For instance, if we combine a square Bravais lattice with a cell with symmetry \(C_2\), the crystal will not be invariant under \(90°\) rotations nor under mirror symmetry. In addition, combining rotations and mirror lines with translations is less obvious than this and can yield peculiar symmetry operations such as glide reflections (see examples in the optional section below).

3D — 14 Bravais, 32 PGs ⇒ 230 space groups

In 3D, there are 32 distinct PGs and even richer configurations emerge: beyond glide reflections one has to introduce also screw axes, which are rotations combined with fractional Bravais translations along the rotation axis. In Figure 1.1, the crystal is not simply invariant under 120° rotation, and something more subtle occurs: its atomic spirals are invariant under 120° rotation combined with a translation by \(1/3\) of the screw pitch. By applying this operation thrice, a simple translation by a full screw pitch is obtained, which is also one of the primitive vectors of the crystal. Combining the 14 Bravais lattices and the 32 crystallographic point groups yields 230 space groups.

It is worth noting that, when describing electron states, these groups should be further extended to include spin: this leads to double groups, which go beyond the scope of these notes (Dresselhaus et al. 2008).

Two dimensions are the ideal playground to see how rich the classification of crystal symmetries can get: the possible groups are few enough to be listed one by one and drawn on a single page, yet they already display every mechanism at work in the far larger 3D case (230 space groups), where such an exhaustive tour would be hopeless.

The seventeen planar groups quoted in Section 1.3 can be organized according to the five lattices of Figure 1.5. For each family we consider the allowed rotations and then add possible mirror or glide operations.

Rotation patterns. Combining a non-trivial rotation \(\mathcal{Q}\) with a translation \(\boldsymbol{\tau}\) shifts its center. The fixed point of \(\mathbf{r}\to\mathcal{Q}\mathbf{r}+\boldsymbol{\tau}\) is \(\mathbf{r}_0 = (\mathbb{1}-\mathcal{Q})^{-1}\boldsymbol{\tau}\) and the operation is simply a rotation around \(\mathbf{r}_0\), since after easy algebra

\[\mathcal{Q}(\mathbf{r}-\mathbf{r}_0) =\mathcal{Q}\mathbf{r}+\boldsymbol{\tau}-\mathbf{r}_0.\]

The first rotation pattern is visible in the case of an oblique lattice. Discarding cells with no symmetry, leading to group p1 with translations only, this lattice is compatible with 2-fold rotations, which are part of the group p2. In the latter case \(\mathcal{Q}=-\mathbb{1}\), so \(\mathbf{r}_0=\boldsymbol{\tau}/2\): the combination of 2-fold rotations and discrete translation generates a set of rotation centers separated by half the lattice spacing. Note in Figure 1.17 that there are four inequivalent rotation centers.

In the rectangular lattice, mirror and glide symmetries become possible. A pattern without them is still p1 or p2, so we will only consider groups with at least a mirror or glide:

  • pm — one family of parallel mirror lines, no rotation centers;
  • pg — here we turn the mirror into a glide by shifting the triangle;
  • pmm — two perpendicular mirror families, which also generate 2-fold rotations;
  • pmg — here we turn again one mirror into a glide by shifting the symmetric left and right sides of the rhombus;
  • pgg — again we shift the symmetric top and down parts of the pattern.
Figure 1.17: Examples of patterns realizing the 17 planar groups.

The centered rectangular (rhombic) lattice is similar, but composing a mirror with the centering translation \((\mathbf{t}_1+\mathbf{t}_2)/2\) automatically produces a parallel glide between mirror lines. Mirror and glide are therefore no longer independent alternatives, leaving only cm and cmm.

For the square lattice, the new element is the 4-fold rotation:

  • p4 — rotations only (point group \(4\));
  • p4m — mirrors running through the 4-fold centers;
  • p4g — mirrors running through the 2-fold centers instead, together with glide reflections.

These three are precisely the groups dissected with Escher-style tessellations in Figure 1.18.

Finally, the hexagonal lattice gives the purely rotational groups p3 and p6. Adding mirrors to p3 yields two inequivalent arrangements, p3m1 and p31m, while adding them to p6 yields p6m. Glide operations, where present, are already generated by these symmetries and produce no additional groups.

Counting the five families gives \(2+5+2+3+5=17\) groups.

Examples in 2D. To see how PGs can be combined with the translation symmetry, let us consider all the possible crystals with a square lattice and 4-fold rotational symmetry. In the absence of mirror lines, this yields the wallpaper group p4 — the simplest planar group with order-4 rotations, having 4-fold centers at the cell corners and cell center, plus 2-fold centers at the edge midpoints. There are two (!) distinct ways to add mirror lines to this rotational skeleton (the complete classification, with every symmetry element drawn, is the catalog above and Figure 1.17):

  1. p4m — mirrors run through the 4-fold rotation centers, along both cell edges and cell diagonals (four mirror families in total). Every rotation center sits on a mirror line.
  2. p4g — mirrors run at 45° through the 2-fold centers (edge midpoints) only, between adjacent 4-fold centers rather than through them. This forces the appearance of glide reflections, i.e. composite symmetry operations consisting of a reflection followed by a translation of half a lattice period.

These three groups are illustrated in Figure 1.18, where the symmetry elements can be toggled on and off.

Figure 1.18: The three wallpaper groups with 4-fold rotational symmetry, illustrated with tessellation patterns inspired by the work of M.C. Escher (lizards, angels and devils). p4: rotations only; p4m: mirrors through rotation centers; p4g: mirrors between centers, generating glide reflections (dashed lines).

Moving to 3D. The 32 point groups can be distinguished first of all based on rotational symmetry:

  • \(C_n\): cyclic symmetry, with a single rotation axis of order \(n\);
  • \(D_n\): dihedral symmetry, with a principal axis of order \(n\) with \(n\) secondary axes;
  • \(T\): tetrahedral symmetry;
  • \(O\): octahedral (cubic) symmetry.

As visible in the examples of Figure 1.19, groups can be further specified by the subscripts \(h, v, d\), signaling the existence of mirror planes; in the figure labels, \(i\) indicates spatial inversion and \(S_n\) an improper rotational axis of order \(n\). The 32 groups can be organized in a subgroup hierarchy (not shown) with two apexes given by the highest-symmetry groups: \(O_h\) (full cubic symmetry) and \(D_{6h}\) (full hexagonal symmetry). Notable point groups such as icosahedral \(I\) and \(I_h\) are excluded since they have five-fold rotations which are incompatible with translational periodicity.

Figure 1.19: Representative molecules for selected 3D point groups.

Why should one care about symmetry? Group theory controls various important physical features:

  1. Degeneracies. Irreducible representations can force exact degeneracies, or their lifting.
  2. Selection rules. Matrix elements and coupling can be forbidden due to incompatible symmetries.
  3. Response tensors and macroscopic effects. Few examples:
    • birefringence is possible in uniaxial crystals (e.g. calcite), not in cubic ones;
    • chiral crystals (e.g. \(\alpha\)-quartz) can be optically active and rotate light polarization;
    • piezoelectricity is possible in non-centrosymmetric crystals (e.g. quartz again, GaAs, etc.);
    • spin phenomena such as Dresselhaus effect can emerge in non-centrosymmetric crystals.

A full development of these ideas is beyond the scope of these notes, which will only sketch a few relevant observations in optional sections like this one: further details can be found in more specialized sources (Dresselhaus et al. 2008; Yu and Cardona 2010). In practice, the most useful selection rules emerge in the highly symmetric cubic systems, while in most of the other lower-symmetry crystals every irreducible representation is one-dimensional, so every state is non-degenerate, and almost every transition is “allowed”.

1.4 Beyond ideal crystals3

Real crystals are never perfect. Defects can be classified by their dimensionality:

  • Point defects (0D) include vacant lattice sites (vacancies), atoms occupying interstitial positions (self-interstitials), and foreign atoms replacing host atoms (substitutional impurities) or sitting between lattice sites (interstitial impurities). When a vacancy and a self-interstitial form simultaneously — for instance by thermal activation — and are close enough to interact, the result is a Frenkel defect. See Figure 1.20.
Figure 1.20: Point defects in a 2D square lattice. A vacancy is an empty site; a Frenkel defect is a vacancy–self-interstitial pair created when a host atom jumps to a nearby interstitial position; a substitutional impurity replaces a host atom (shown larger); an interstitial impurity sits between lattice sites (shown smaller).
  • Line defects (1D), known as dislocations, are the primary carriers of plastic deformation in metals. An edge dislocation is an extra half-plane of atoms inserted into the lattice, while a screw dislocation arises from a helical distortion of the lattice planes around a line. See Figure 1.21.
Figure 1.21: Line defects (dislocations) in a simple cubic lattice. Left: an edge dislocation highlighted by an orange axis. Right: a screw dislocation: atoms are helically displaced along the dislocation line, creating a spiral ramp around the dislocation axis.
  • Planar defects (2D) include grain boundaries (interfaces between crystallites with different orientations), stacking faults (local errors in the layer stacking sequence, e.g. an ABAB…C…ABAB fault in HCP), and twin boundaries (mirror-related crystal domains). See Figure 1.22.
Figure 1.22: Grain boundaries and twinning planes. Left: A grain boundary is an \(n-1\) dimensional defect connecting two crystals with inconsistent orientations. Right: A twinning plane separates a region with an ABC stacking sequence from another one with a CBA one.

In conclusion…

ImportantTake home messages…

At the end of this chapter you should know…

  • How to define a crystal. Bravais lattice plus a basis define a crystal: how are they defined? how is a crystal different from its lattice? What kind of cells are used to describe the repeated unit?
  • Symmetry and crystals. Restriction theorem, basic symmetry operations, existence of qualitatively different Bravais lattice families. What are planar and space groups, why do they exist?
  • Common crystalline forms. FCC (Al, Au, Ag, Cu…), BCC, diamond (C, Si, Ge…), zincblende (GaAs), wurtzite, etc… common crystalline forms. Connection with stacking schemes. Typical values for the lattice constant.
  • Defects. Classification of deviations from perfect periodicity.

You are not expected: to remember by heart every single 3D Bravais, and even less to know all the subtle aspects behind space and planar groups cited in the optional sections; to know all the crystals linked in Figure 1.23 below…

As with many of the following note chapters, you are rather suggested to take Figure 1.23 as an occasion to think and test your understanding and/or follow your curiosity: try and open some of the suggested crystals… or look for new ones on the cited crystallographic database. Do you see the cell and its symmetries? and the basis? is the cell primitive? can you tell to what Bravais lattice the crystal belongs, among the ones in Figure 1.7? Can you tell if the stoichiometry of Nb\(_3\)Sn actually matches the visible atomic network? can you find the Bravais in the lattice table, do you see the symmetries, rotations, mirror planes, etc.? look at the numbers, \(a\), \(b\), \(c\), how many angstroms?

Test your understanding

As a final exploration tool, Figure 1.23 allows you to upload any crystalline structure from the Crystallographic Open Database (COD) and explore it in a 3D visualization. The only requested information is the number of the correct CIF (Crystallographic Information File) in the database, which you can either already know or search in the website here.

Figure 1.23: Crystal structure explorer based on the Crystallography Open Database (COD). Select a preset or enter any CIF number to fetch and visualize the structure. Use the N₁×N₂×N₃ controls to build a supercell. * Experimental: press Stereo 3D for a parallel or crossed-eye visualization in full screen.*

  1. Grosso and Pastori Parravicini (2014, secs. 2.1–2.3).↩︎

  2. Grosso and Pastori Parravicini (2014, sec. 2.2).↩︎

  3. Ashcroft and Mermin (1976, chap. 30)↩︎