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 it already helps explain some basic properties of solids. For instance, quartz is an optically active material and can rotate polarization, due to the different propagation speeds of left- and right-circularly polarized light. This behavior is rooted in its atomic structure, which can be viewed as a collection of intertwined “atomic spirals”, as highlighted in the figure.

import { renderQuartz } from "./_interactive/ch01-quartz3d.js"
{
  const W = width;
  const H = Math.round(W * 2 / 3);

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  /* Left panel: cartoon image */
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  leftPanel.style.cssText = "flex:0 0 38%; display:flex; align-items:center; justify-content:center;";
  const img = document.createElement("img");
  img.src = "QuartzCartoon.png";
  img.alt = "Natural quartz crystal";
  img.style.cssText = "max-width:100%; max-height:" + H + "px; border-radius:6px; object-fit:contain;";
  leftPanel.appendChild(img);
  wrapper.appendChild(leftPanel);

  /* Right panel: 3Dmol viewer */
  const rightPanel = document.createElement("div");
  rightPanel.style.cssText = "flex:1 1 60%; min-width:0;";
  const viewer3d = document.createElement("div");
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  rightPanel.appendChild(viewer3d);
  wrapper.appendChild(rightPanel);

  yield wrapper;
  renderQuartz(viewer3d);
}

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. Drag to rotate, scroll to zoom.

1.1 How to define a crystal¹

A crystal like the one depicted in Figure 1.1 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 clarify their geometric meaning and mathematical formulation. This provides a rigorous description of periodic structures and introduces the basic language of solid-state physics. Crystal families and symmetry properties are also introduced building on these concepts.

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 \(\{\mathbf{t}_1, \mathbf{t}_2, \mathbf{t}_3\}\)

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

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 elementary 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” primitive cell, 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, primitive vectors 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 primitive vectors

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

CautionNote: Bravais lattice \(\neq\) crystal…

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 can be 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 primitive vectors often has a fundamental limitation: it cannot always be explicitly symmetric, as illustrated in Figure 1.2. Given the importance of symmetry in physics, this deserves a further discussion.

import { createTriangularCells } from "./_interactive/ch01-triangular-cells.js"
createTriangularCells(width)

Figure 1.2: 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 1.2, the green hexagonal cell contains three lattice points (one at the centre plus six vertices shared with three neighbouring 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 neighbours, and taking the innermost enclosed volume (see violet cell in Figure 1.2). 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 \(2a\cos\theta\).

import { createRestrictionFig } from "./_interactive/ch01-restriction.js"
createRestrictionFig(Math.round(width * 0.75))

Figure 1.3: 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 1.4: there are five Bravais lattices, grouped into four families, as also listed in the table below.

import { createBravais2D } from "./_interactive/ch01-bravais2d.js"
createBravais2D(width)

Figure 1.4: Interactive 2D Bravais lattices. Drag the blue point (\(\mathbf{t}_2\)) to explore the five lattice classes; the red point is \(\mathbf{t}_1 = (1,0)\). Colours: 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 \(\mathcal{BL}\).

Table 1.1: The 5 Bravais lattices in 2D. Families are set by the two unit cell lengths and their internal angle. Note the naming degeneracies: hexagonal = triangular; oblique = monoclinic; centered rectangular = rhombic

Family	Parameters	Bravais lattices
Hexagonal	\(a=b,\,\varphi=60°\)	Simple
Square	\(a=b,\,\varphi=90°\)	Simple
Rectangular	\(a\neq b,\,\varphi = 90°\)	Simple, Centered
Oblique	\(a\neq b,\,\varphi\neq90°\)	Simple

CautionTo be or not to be… centered?!

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 is surely compatible with rotational symmetry, which is surely an option… but this 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 simple and smaller 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 neighbouring hexagonal corners (\(\mathbf{t}\) in Figure 1.5) is not a legitimate primitive translation vector since \(2\mathbf{t}\) is not a good translation.

import { renderCenteredVsHex } from "./_interactive/ch01-centered-vs-hex.js"
{
  const container = document.createElement("div");
  container.style.cssText = "width:100%;";
  yield container;
  renderCenteredVsHex(container);
}

Figure 1.5: 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-neighbour vector t and translating by 2t, one lands on the centre of a hexagon, where there is no point, so this collection of points is not a Bravais lattice.

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:

Table 1.2: The 14 Bravais lattices in 3D. Centering types: P = primitive (also called simple), I = body-centered (Innenzentriert), F = face-centered, C = base-centered.

Family	Parameters	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	\(a \neq b \neq c\), \(\alpha \neq \beta \neq \gamma\)	✓

Each crystal family has distinct characteristics (see Figure 1.6):

• Cubic — the highest symmetry family, it includes most metals (Fe, Cu, Al), ionic crystals (NaCl), and semiconductors (Si, GaAs). The three Bravais variants dominate solid-state physics.
• Tetragonal — a cubic lattice stretched (or compressed) along one axis. Found in high-\(T_c\) superconductors (YBa\(_2\)Cu\(_3\)O\(_7\)), and in several oxides such as TiO\(_2\) (rutile).
• Orthorhombic — three mutually perpendicular axes of unequal length. Realised in sulphur, olivine (Mg\(_2\)SiO\(_4\)), and many layered compounds.
• Hexagonal — characterised by a 6-fold axis; a single Bravais lattice. Describes graphite and many transition-metal dichalcogenides (MoS\(_2\), WS\(_2\)).
• Trigonal (rhombohedral) — can be viewed as a continuous deformation of the FCC lattice obtained by stretching along a body diagonal. Realised in calcite (CaCO\(_3\)), bismuth, and arsenic.
• Monoclinic — one axis not perpendicular to the other two. Common in minerals (gypsum, feldspar).
• Triclinic — the lowest symmetry, found in some minerals such as plagioclase feldspars.

Important naming conventions exist for cubic crystals: cubic lattices are referred to as SC (Simple Cubic, or cP), BCC (Body-Centered Cubic, or cI), and FCC (Face-Centered Cubic, 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). These names are a non-exhaustive subset of the Schoenflies notation, see Section 1.3 for further details. Students are not expected (but neither forbidden!) to remember every single variant by heart, but they are strongly advised to familiarize with the qualitatively different symmetries allowed by these lattices.

import { createBravais3D } from "./_interactive/ch01-bravais3d.js"
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  const H = Math.round(W * 2 / 3);  /* 3:2 aspect ratio */
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  container.style.cssText = "width:100%; max-width:" + W + "px; height:" + H + "px; position:relative; margin:0 auto; display:flex; flex-direction:column;";
  yield container;
  createBravais3D(container);
}

Figure 1.6: 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). Drag to rotate, scroll to zoom.

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.

CautionOn equivalent crystal definitions

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.

CautionWrap-up: the graphene example

Graphene is a 2D crystal containing only C atoms, nevertheless its honeycomb arrangement is not a \(\mathcal{BL}\), as argued before (see Figure 1.5). 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-neighbour 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.

import { renderGrapheneCells } from "./_interactive/ch01-graphene-cells.js"
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  container.style.cssText = "width:90%; max-width:700px; position:relative; border:1px solid #ccc; border-radius:6px; overflow:hidden;";
  yield container;
  /* set height based on actual rendered width (works inside collapsed callouts) */
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    if (W > 0) container.style.height = Math.round(W * 0.72) + "px";
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  const ro = new ResizeObserver(sizeContainer);
  ro.observe(container);
  sizeContainer();
  renderGrapheneCells(container);
}

Figure 1.7: Graphene honeycomb lattice with interactive cell overlay. Use the dropdown to explore: two primitive cell choices, a rectangular conventional cell, and Wigner–Seitz cells centred on each sublattice. Drag to rotate, scroll to zoom.

1.2 Common crystals²

In the following, we recall the most common crystalline structures, which should be learnt by heart.

CautionOn rote memorization in physics

Rote memorization is rarely a goal in physics: we prefer to devote our attention to understanding concepts and mastering derivations. However, since physics describes the real world and not an abstract one, a few facts and figures must be remembered. This includes, at least, basic crystalline forms.

Later on, specific values such as the band gap of Silicon or the orders of magnitude of key parameters will become essential. This is not empty mnemonics: these figures will be the foundation of key models and approximations; if ignored, much of condensed matter physics will lose its context and make little sense.

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), aluminium, 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 neighbours 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 neighbours, 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, crystallize as simple BCC, with a lower coordination number of 8. The interactive Figure 1.8 below shows both the FCC and BCC structures side by side, with the cited primitive cells.

import { renderCubic } from "./_interactive/ch01-xtal-structures.js"
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  const H = Math.round(W * 0.52);
  const container = document.createElement("div");
  container.style.cssText = "width:100%; max-width:" + W + "px; height:" + H + "px; position:relative; overflow:hidden;";
  yield container;
  renderCubic(container);
}

Figure 1.8: Left: BCC structure — Bravais lattice points (blue) with body centres from adjacent cubes (grey). Right: FCC structure — Bravais lattice points (blue) with face centres from adjacent cubes (grey). In both panels the red parallelepiped with semi-transparent faces shows the primitive cell. Drag to rotate, scroll to zoom.

1.2.1 Diamond structure

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 1.9. 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 neighbours at distance \(a\sqrt{3}/4\) and the coordination number is 4. The zincblende (ZB) structure has basically the same crystal as diamond — FCC lattice with a two-atom basis at \((0,0,0)\) and \(a(1,1,1)/4\) — but with a key difference: the two basis atoms belong to different chemical species. This breaks the inversion symmetry in these crystals. See Figure 1.9 for details.

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.

import { renderDiamondZB } from "./_interactive/ch01-xtal-structures.js"
{
  const W = Math.min(width, 760);
  const H = Math.round(W * 0.52);
  const container = document.createElement("div");
  container.style.cssText = "width:100%; max-width:" + W + "px; height:" + H + "px; position:relative; overflow:hidden;";
  yield container;
  renderDiamondZB(container);
}

Figure 1.9: 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. Drag to rotate, scroll to zoom.

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 issue: how can we pack a set of objects (spheres, to fix ideas) as closely as possible?

Figure 1.10 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.

import { renderClosePacking } from "./_interactive/ch01-close-packing.js"
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  yield container;
  renderClosePacking(container);
}

Figure 1.10: 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-centred cube for FCC. Drag to rotate, scroll to zoom.

CautionBCC from square-packed layers

The close-packed construction above starts from triangular layers (the densest 2D packing). A conceptually simpler — though less dense — alternative uses square-packed layers. If a second layer of spheres is placed in the square hollows of the first (AB stacking of square planes), with the in-plane spacing slightly expanded so that spheres touch only along the cube diagonal, the resulting 3D structure is a body-centred cubic (BCC) lattice: each atom sits at the centre of a cube formed by eight neighbours.

The BCC packing fraction (\(\pi\sqrt{3}/8 \approx 68\%\)) is lower than the close-packed value (\(\pi/(3\sqrt{2}) \approx 74\%\)), which is why BCC is never called “close-packed”. Nevertheless, BCC is the equilibrium structure of many metals (Fe at room temperature, W, Cr, Mo, Na, K) because the cohesive energy depends not only on packing density but also on the details of electronic bonding: in several transition metals the \(d\)-band filling favours the higher coordination shell structure of BCC over the denser FCC arrangement.

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 1.11, pay close attention to the content of the primitive cell highlighted in red: both crystals contain four atoms in their basis.

import { renderHexStruct } from "./_interactive/ch01-xtal-structures.js"
{
  const W = Math.min(width, 760);
  const H = Math.round(W * 0.52);
  const container = document.createElement("div");
  container.style.cssText = "width:100%; max-width:" + W + "px; height:" + H + "px; position:relative; overflow:hidden;";
  yield container;
  renderHexStruct(container);
}

Figure 1.11: 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. Drag to rotate, scroll to zoom.

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 1.8 with the trigonal Bravais lattice in Figure 1.6.

1.2.3 NaCl and CsCl structures

Sodium chloride, the common “salt”, crystallises 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 neighbour Na atoms, thus its coordination number is 6. The second-nearest neighbour Cl atoms (second coordination shell) are instead 12.

Another common cubic compound is Caesium 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-centre 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 1.12 for details.

import { renderNaClStruct } from "./_interactive/ch01-xtal-structures.js"
{
  const W = Math.min(width, 760);
  const H = Math.round(W * 0.52);
  const container = document.createElement("div");
  container.style.cssText = "width:100%; max-width:" + W + "px; height:" + H + "px; position:relative; overflow:hidden;";
  yield container;
  renderNaClStruct(container);
}

Figure 1.12: 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 centre; each ion is eightfold coordinated. Note: CsCl is not BCC, since the two sites carry different species. Drag to rotate, scroll to zoom.

1.2.4 Perovskite structure

The perovskite structure (general formula ABO\(_3\), 3D visualization in Figure 1.13) is a cubic structure with: type A atoms at the cube corners, B atoms at the body centre, O atoms at the face centres. 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.

import { renderPerovskite } from "./_interactive/ch01-xtal-structures.js"
{
  const W = Math.min(width, 760);
  const H = Math.round(W * 0.52);
  const container = document.createElement("div");
  container.style.cssText = "width:100%; max-width:" + W + "px; height:" + H + "px; position:relative; border:1px solid #ccc; border-radius:6px; overflow:hidden;";
  yield container;
  renderPerovskite(container);
}

Figure 1.13: Perovskite ABO\(_3\) structure. A atoms (green) at cube corners, B atom (blue) at the body centre, O atoms (red) at face centres. The unit cell contains one formula unit. Drag to rotate, scroll to zoom.

As a final wrap-up, Figure 1.14 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).

import { renderPeriodicTable } from "./_interactive/ch01-periodic-table.js"
{
  const W = width;
  const container = document.createElement("div");
  container.style.cssText = "width:100%; max-width:" + W + "px; position:relative; overflow:visible;";
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      el = el.parentElement;
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    if (el) el.style.minHeight = "0";
  });
  renderPeriodicTable(container);
}

Figure 1.14: Interactive periodic table of the elements colour-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 \(6\).

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).

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).

CautionMore on crystal symmetry 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 centres at the cell corners and cell centre, plus 2-fold centres at the edge midpoints. There are two (!) distinct ways to add mirror lines to this rotational skeleton:

1. p4m — mirrors run through the 4-fold rotation centres, along both cell edges and cell diagonals (four mirror families in total). Every rotation centre sits on a mirror line.
2. p4g — mirrors run at 45° through the 2-fold centres (edge midpoints) only, between adjacent 4-fold centres 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 1.15, where the symmetry elements can be toggled on and off.

import { renderWallpaperP4 } from "./_interactive/ch01-wallpaper-p4.js"
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  container.style.cssText = "width:100%; max-width:" + W + "px; position:relative;";
  wrapper.appendChild(container);
  yield wrapper;
  renderWallpaperP4(container);
}

Figure 1.15: 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 centres; p4g: mirrors between centres, 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 1.16, 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.

import { renderPointGroups } from "./_interactive/ch01-point-groups.js"
{
  const W = Math.min(width, 860);
  const container = document.createElement("div");
  /* width:100% lets the inner layout shrink with the page; max-width caps it
     on wide screens so the figure doesn't blow up. */
  container.style.cssText = "width:100%; max-width:" + W + "px; position:relative; margin:0 auto;";
  yield container;
  renderPointGroups(container);
}

Figure 1.16: Representative molecules for selected 3D point groups. Drag to rotate, scroll to zoom.

More on point groups (Dresselhaus et al. 2008). The elements of the symmetry group \(G\) are - in completely general terms - affine transformations of three-dimensional space, which we write in the Seitz notation \(\{\mathcal{M}|\boldsymbol{\tau}\}\), where \(\mathcal{M}\) is a point-group operation and \(\boldsymbol{\tau}\) is a translation vector. The action on a position vector is

\[ \mathbf{r}\to \{\mathcal{M}|\boldsymbol{\tau}\}\,\mathbf{r}=\mathcal{M}\,\mathbf{r}+\boldsymbol{\tau}, \tag{1.17}\]

from which the composition rule and the inverse follow at once,

\[ \begin{align} \{\mathcal{M}_2|\boldsymbol{\tau}_2\}\{\mathcal{M}_1|\boldsymbol{\tau}_1\} &=\{\mathcal{M}_2\mathcal{M}_1\,|\,\mathcal{M}_2\boldsymbol{\tau}_1+\boldsymbol{\tau}_2\},\\ \{\mathcal{M}|\boldsymbol{\tau}\}^{-1} &=\{\mathcal{M}^{-1}|-\mathcal{M}^{-1}\boldsymbol{\tau}\}. \end{align} \]

Pure translations \(\{\varepsilon|\boldsymbol{\tau}\}\), where \(\varepsilon\) denotes the identity matrix and \(\boldsymbol{\tau}=\mathbf{R}\in\mathcal{BL}\), form an infinite normal subgroup \(T\triangleleft G\). The quotient \(G/T\) is isomorphic to one of the 32 point groups \(P\).

The space group is called symmorphic if it can be written as the semi-direct product \(G = T\rtimes P\), i.e. if there exists a choice of origin such that every generator can be written as \(\{\mathcal{M}|\boldsymbol{0}\}\) or as a pure lattice translation. In a symmorphic group, point-group operations and lattice translations are cleanly separable. This is possible for only 73 such groups out of 230. The remaining 157 groups are non-symmorphic and contain at least one essential element \(\{\mathcal{M}|\boldsymbol{\tau}\}\) in which \(\boldsymbol{\tau}\) is a fractional translation (a fraction of a lattice vector) that cannot be removed by any choice of origin. Two prototypical operations of this kind are:

• Screw axes \(n_m\) — a rotation by \(2\pi/n\) followed by a translation of \(m/n\) along the axis (e.g. HCP crystals contain a fairly obvious \(6_3\) axis, \(60°\) rotation and translation by half the cell size along the stacking direction).
• Glide planes — a mirror reflection followed by a half-lattice translation parallel to the mirror (already encountered in the planar group p4g above).

Why 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”.

About symmetry notations. Parallel notations exist to label point groups, space groups and the individual symmetry elements. The Schoenflies notation is dominant in the spectroscopy communities and it labels PGs by their generating axis: \(C_n\) is a cyclic \(n\)-fold rotation, \(D_n\) adds \(n\) perpendicular two-fold axes, \(T\), \(O\) and \(I\) are the tetrahedral, octahedral and icosahedral groups. The Hermann–Mauguin (or international) notation is the standard of crystallography: rotations are denoted by their order \(1,2,3,4,6\), an overbar denotes inversion-rotations (\(\bar 1\), \(\bar 3\), \(\bar 4\), \(\bar 6\)), \(m\) is a mirror, etc. Space-group symbols additionally carry a leading capital letter: \(P\) for primitive, \(I\) for body-centred, etc. Examples: \(Fd\bar 3m\) (diamond), \(P6_3/mmc\) (HCP).

CautionPlanar groups breakdown

The seventeen planar groups quoted in Section 1.3 can be built up systematically, lattice by lattice, starting from the least symmetric case. Two general facts guide the whole construction: every Bravais lattice is invariant under a 2-fold rotation (inversion in the plane), and each group is conventionally listed under the least symmetric lattice able to host it — a crystal sitting on a rectangular lattice but carrying no mirror at all is still classified under the oblique family.

Oblique lattice. A generic cell supports no mirrors and no rotations beyond order 2:

• p1 — translations only: the “empty” wallpaper, compatible with a fully asymmetric motif;
• p2 — add the 2-fold rotation; four inequivalent classes of rotation centres appear (cell corner, the two edge midpoints, cell centre).

Rectangular lattice. The 90° cell makes mirror lines possible and, with them, their subtle cousin already met in Section 1.3: the glide reflection (mirror followed by a half-period translation along it):

• pm — mirror lines along one direction;
• pg — the mirror is replaced by a glide: no pure reflection survives, yet the pattern is still “mirror-symmetric up to a shift”;
• pmm — mirrors along both directions (two perpendicular mirrors compose into a 2-fold rotation, so p2 is contained automatically);
• pmg — mirrors along one direction, glides along the perpendicular one;
• pgg — glides along both directions, no pure mirror at all.

Centred rectangular (rhombic) lattice. Centering intertwines the previous options: a mirror line now automatically generates a parallel glide line half a cell away, so the “mirror” and “glide” variants merge into a single group per case:

• cm — mirrors, with the inherent parallel glides;
• cmm — mirrors along both directions, plus the corresponding glides.

Square lattice. The new ingredient is the 4-fold rotation:

• p4 — rotations only;
• p4m — mirrors running through the 4-fold centres;
• p4g — mirrors running between the 4-fold centres, which forces glides.

These three are precisely the groups dissected with Escher-style tessellations in Figure 1.15.

Hexagonal lattice. Rotations of order 3 and 6 close the list:

• p3 — 3-fold rotations only;
• p3m1 and p31m — the two inequivalent orientations of the mirror set, rotated by 30° from one another: in p3m1 every 3-fold centre lies on a mirror line, while in p31m one class of centres sits off the mirrors;
• p6 — 6-fold rotations only;
• p6m — 6-fold rotations plus mirrors (the glides come for free: there is no separate “p6g”).

Table 1.3: The 17 planar groups, lattice by lattice.

Lattice family	New ingredient	Groups	#
Oblique	2-fold rotation	p1, p2	2
Rectangular	mirrors or glides	pm, pg, pmm, pmg, pgg	5
Centred rectangular	mirror + glide together	cm, cmm	2
Square	4-fold rotation	p4, p4m, p4g	3
Hexagonal	3- and 6-fold rotations	p3, p3m1, p31m, p6, p6m	5

The same programme in 3D contains nothing conceptually new — there are simply more options at every step: 14 Bravais lattices instead of 5, 32 point groups instead of 10, and screw axes joining glide planes among the composite operations. Carried out in full, the bookkeeping produces the 230 space groups.

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 1.17.

import { renderDefects } from "./_interactive/ch01-defect-0D.js"
{
  const container = document.createElement("div");
  container.style.cssText = "width:100%; position:relative;";
  yield container;
  renderDefects(container);
}

Figure 1.17: 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 1.18.

import { renderLineDefects } from "./_interactive/ch01-defect-1D.js"
{
  const W = width;
  const H = Math.round(W * 0.5);
  const container = document.createElement("div");
  container.style.cssText = "width:100%; max-width:" + W + "px; height:" + H + "px; position:relative; display:flex; flex-direction:column;";
  yield container;
  renderLineDefects(container);
}

Figure 1.18: 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. Drag to rotate, scroll to zoom.

• 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 1.19.

import { renderGrapheneGB } from "./_interactive/ch01-defect-2D.js"
{
  const container = document.createElement("div");
  container.style.cssText = "width:100%; position:relative;";
  yield container;
  renderGrapheneGB(container);
}

Figure 1.19: 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. Drag to rotate, scroll to zoom.

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, for instance the fact that we have simple and base-centered monoclinic but not face-centered monoclinic (but it is interesting to understand why a centered square lattice is useless); even less to know each and every one of the individual space and planar groups (but 2D Bravais lattices are so few that you will probably recall them); know all the crystals linked in Figure 1.20 below…

As with many of the following note chapters, you are rather suggested to take Figure 1.20 as an occasion to 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 1.6? 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 1.20 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.

import { renderCodExplorer } from "./_interactive/ch01-cod-explorer.js"
{
  const W = Math.min(width, 780);
  const container = document.createElement("div");
  container.style.cssText = "width:100%; max-width:" + W + "px; position:relative; margin:0 auto;";
  yield container;
  renderCodExplorer(container);
}

Figure 1.20: Crystal structure explorer based on the Crystallography Open Database (COD). Select a preset or enter any CIF number to fetch and visualise the structure. Use the N₁×N₂×N₃ controls to build a supercell. Drag to rotate, scroll to zoom. Experimental: press Stereo 3D for a parallel or crossed-eye visualization in full screen.

Ashcroft, Neil W., and N. David Mermin. 1976. Solid State Physics. Harcourt College Publishers.

Dresselhaus, Mildred S., Gene Dresselhaus, and Ado Jorio. 2008. Group Theory: Application to the Physics of Condensed Matter. Springer.

Grosso, Giuseppe, and Giuseppe Pastori Parravicini. 2014. Solid State Physics. 2nd ed. Academic Press.

Yu, Peter Y., and Manuel Cardona. 2010. Fundamentals of Semiconductors: Physics and Materials Properties. 4th ed. Springer.

---

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)