---
title: The Structure of Solids
module: Crystal Structure
moduleNumber: 3
lessonNumber: 1
order: 301
summary: >
  A crystal is a unit cell repeated in three dimensions. We classify the common
  cubic lattices, compute the Coulomb energy of an ionic crystal through the
  Madelung constant, and show how the divergent naive lattice sum is tamed by
  cubic shells. The cohesive energy that results predicts melting points and
  connects the diatomic bond of an earlier lesson to the bulk solid.
topics: [Crystal Structure]
draft: false
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 10 — Solid State Physics; §10-1 The Structure of Solids"
  - book: Kittel
    ref: "Ch. 1 — Crystal Structure"
---

The same bonding mechanisms that hold a
[diatomic molecule](/condensed-matter/molecules-and-bonding/bonding-mechanisms)
together — ionic, covalent, metallic, and van der Waals — bind atoms into solids,
now on a scale of Avogadro's number of atoms. When a liquid is cooled slowly its
molecules arrange into a regular array that maximizes the number of bonds and
minimizes the potential energy: a **crystal**. Cooled too fast, they freeze in a
disordered snapshot: an **amorphous** solid such as glass.

> **Definition (Crystalline and amorphous solids).** A **crystalline** solid has
> **long-range order** — a single unit structure repeated over many atomic
> diameters — and a sharp melting point. An **amorphous** solid has only
> short-range order and merely softens as it warms. Most common solids are
> **polycrystalline**: aggregates of small single crystals.

The smallest repeating structure is the **unit cell**. Its geometry depends on
the bonding and, when more than one kind of atom is present, on their relative
sizes.

## Cubic lattices

The three cubic unit cells differ only in where atoms sit relative to the corners
of a cube.

$$
% caption: The three cubic unit cells: simple cubic (atoms at corners only),
% body-centered cubic with one atom at the cube center, and face-centered cubic
% with an atom at the center of each face.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % --- simple cubic ---
  \begin{scope}
    \draw[black] (0,0) rectangle (1.4,1.4);
    \draw[black] (0.5,0.5) rectangle (1.9,1.9);
    \draw[black] (0,0) -- (0.5,0.5);
    \draw[black] (1.4,0) -- (1.9,0.5);
    \draw[black] (0,1.4) -- (0.5,1.9);
    \draw[black] (1.4,1.4) -- (1.9,1.9);
    \foreach \p in {(0,0),(1.4,0),(0,1.4),(1.4,1.4),(0.5,0.5),(1.9,0.5),(0.5,1.9),(1.9,1.9)}
      \fill[acc] \p circle (2.4pt);
    \node[anchor=north] at (0.95,-0.25) {simple cubic};
  \end{scope}
  % --- body-centered ---
  \begin{scope}[xshift=4.2cm]
    \draw[black] (0,0) rectangle (1.4,1.4);
    \draw[black] (0.5,0.5) rectangle (1.9,1.9);
    \draw[black] (0,0) -- (0.5,0.5);
    \draw[black] (1.4,0) -- (1.9,0.5);
    \draw[black] (0,1.4) -- (0.5,1.9);
    \draw[black] (1.4,1.4) -- (1.9,1.9);
    \foreach \p in {(0,0),(1.4,0),(0,1.4),(1.4,1.4),(0.5,0.5),(1.9,0.5),(0.5,1.9),(1.9,1.9)}
      \fill[acc] \p circle (2.4pt);
    \fill[acc] (0.95,0.95) circle (2.8pt);
    \node[anchor=north] at (0.95,-0.25) {body-centered};
  \end{scope}
  % --- face-centered ---
  \begin{scope}[xshift=8.4cm]
    \draw[black] (0,0) rectangle (1.4,1.4);
    \draw[black] (0.5,0.5) rectangle (1.9,1.9);
    \draw[black] (0,0) -- (0.5,0.5);
    \draw[black] (1.4,0) -- (1.9,0.5);
    \draw[black] (0,1.4) -- (0.5,1.9);
    \draw[black] (1.4,1.4) -- (1.9,1.9);
    \foreach \p in {(0,0),(1.4,0),(0,1.4),(1.4,1.4),(0.5,0.5),(1.9,0.5),(0.5,1.9),(1.9,1.9)}
      \fill[acc] \p circle (2.4pt);
    \fill[acc] (0.7,0.7) circle (2.6pt);
    \fill[acc] (1.2,1.2) circle (2.6pt);
    \fill[acc] (0.95,0.25) circle (2.6pt);
    \node[anchor=north] at (0.95,-0.25) {face-centered};
  \end{scope}
\end{tikzpicture}
$$

Sodium chloride crystallizes in the **face-centered-cubic (fcc)** structure: each
$\text{Na}^+$ has six $\text{Cl}^-$ nearest neighbors and vice versa, and the ions
are **not** paired into molecules. Cesium chloride adopts a different cubic
arrangement with eight nearest neighbors. The structure is whatever minimizes the
total potential energy given the ion sizes.

$$
% caption: A face of the NaCl crystal: sodium and chlorine ions alternate on a
% cubic grid, each ion surrounded by nearest neighbors of the opposite charge.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % sodium (small, acc) on even i+j
  \foreach \p in {(0,0),(1.1,1.1),(2.2,0),(0,2.2),(2.2,2.2),(3.3,1.1),(1.1,3.3),(3.3,3.3),(0,0)}
    \fill[acc] \p circle (4pt);
  \foreach \p in {(2.2,0),(0,2.2),(2.2,2.2),(3.3,1.1),(1.1,3.3),(3.3,3.3)}
    \fill[acc] \p circle (4pt);
  % chlorine (large, neutral) on odd i+j
  \foreach \p in {(1.1,0),(0,1.1),(2.2,1.1),(1.1,2.2),(3.3,0),(0,3.3),(3.3,2.2),(2.2,3.3)}
    \fill[black] \p circle (6pt);
  \node[acc, anchor=west] at (3.9,3.3) {sodium ion (small)};
  \node[black, anchor=west] at (3.9,2.6) {chlorine ion (large)};
\end{tikzpicture}
$$

## The Madelung constant

The attractive Coulomb energy of one ion in the crystal is not simply $-ke^2/r$,
because that ion feels every other ion. Writing the net attractive energy as

$$
U_{\text{att}} = -\alpha\,\frac{ke^2}{r},
$$

the dimensionless **Madelung constant** $\alpha$ collects the geometry. Each ion
has 6 opposite-charge nearest neighbors at distance $r$, then 12 like-charge ions
at $\sqrt{2}\,r$, then 8 opposite at $\sqrt{3}\,r$, and so on, so the naive sum is

$$
\alpha \overset{?}{=} 6 - \frac{12}{\sqrt{2}} + \frac{8}{\sqrt{3}} - \frac{6}{\sqrt{4}} + \cdots
$$

$$
% caption: An ion in the NaCl lattice sees successive shells of neighbors: 6
% opposite charges at r, 12 like charges at √2 r, 8 opposite at √3 r; the
% alternating pull and push is what the Madelung constant sums.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \fill[black] (0,0) circle (2.6pt);
  \node[anchor=north east, font=\scriptsize] at (-0.05,-0.05) {ion};
  \draw[acc, thick] (0,0) circle (1.5);
  \draw[black, thick, dashed] (0,0) circle (2.4);
  \draw[acc, thick] (0,0) circle (3.1);
  \node[acc, anchor=west, font=\scriptsize] at (3.5,1.1) {6 opposite, shell 1};
  \node[black, anchor=west, font=\scriptsize] at (3.5,0.2) {12 like, shell 2};
  \node[acc, anchor=west, font=\scriptsize] at (3.5,-0.7) {8 opposite, shell 3};
  \draw[acc, ->] (3.45,1.1) -- (0.9,1.15);
  \draw[black, ->] (3.45,0.2) -- (1.55,1.8);
  \draw[acc, ->] (3.45,-0.7) -- (2.05,2.3);
\end{tikzpicture}
$$

This alternating series **does not converge**. The resolution is physical: real
crystals are electrically neutral in compact regions, so the sum must be
organized into neutral cubic shells rather than spherical ones. Done that way it
converges to $\alpha = 1.7476$ for NaCl (and the fcc lattices LiBr, KCl, RbF, all
share it), while CsCl gives $\alpha = 1.7627$.

Adding the exclusion-principle repulsion $A/r^n$ and minimizing, the total
potential energy per ion pair at equilibrium is

$$
U(r_0) = -\alpha\,\frac{ke^2}{r_0}\left(1 - \frac{1}{n}\right).
$$

The exponent $n$ is found from the measured **dissociation energy**. For NaCl the
lattice dissociation energy is $770\ \text{kJ/mol} = 7.98\ \text{eV}$ per ion
pair; with $r_0 = 0.282\ \text{nm}$ and $\alpha = 1.75$, solving the equation
gives $n \approx 9$.

**Worked example — spacing of NaCl from its density.** Treating each ion as
occupying a cube of side $r_0$, the mass of one mole ($58.4\ \text{g}$) fills a
volume $2 N_A r_0^3$. From the density $\rho = 2.16\ \text{g/cm}^3$,

$$
r_0^3 = \frac{m}{2 N_A \rho} = \frac{58.4\ \text{g}}{2(6.02\times10^{23})(2.16\ \text{g/cm}^3)} = 2.24\times10^{-23}\ \text{cm}^3,
$$

so $r_0 = 2.82\times10^{-8}\ \text{cm} = 0.282\ \text{nm}$, matching X-ray
diffraction.

## Cohesive energy

The dissociation energy is measured per ion pair; the **cohesive energy** is the
same well depth expressed per atom pair, which is the quantity comparable across
all bonding types. For NaCl, removing an $\text{Na}^+ + \text{Cl}^-$ pair costs
$7.98\ \text{eV}$, but forming $\text{Cl}^-$ from Cl requires $+3.62\ \text{eV}$
while forming $\text{Na}^+$ from Na releases $5.14\ \text{eV}$, so removing the
neutral pair costs

$$
7.98 - 3.62 + 5.14 = 6.46\ \text{eV per pair}.
$$

$$
% caption: Cohesive-energy well of an ionic crystal: the Madelung Coulomb
% attraction and the exclusion-principle repulsion give a deep minimum at r_0,
% whose depth is the crystal's cohesive energy and sets the melting point.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,-2.6) -- (0,2.4) node[left] {$U$};
  \draw[->, black] (0,0) -- (7.0,0) node[below] {$r$};
  \draw[acc, very thick, domain=0.95:6.6, samples=120, variable=\x]
    plot ({\x},{4.0/(\x*\x*\x*\x*\x) - 3.4/\x});
  \fill[acc] (1.56,-1.75) circle (2pt);
  \draw[black, dashed] (1.56,-1.75) -- (1.56,0) node[above] {$r_0$};
  \draw[black, dashed] (1.56,-1.75) -- (0,-1.75) node[left] {cohesive};
\end{tikzpicture}
$$

A large cohesive energy means strongly bound atoms, a high melting point, and a
hard crystal — ionic and covalent crystals (NaCl, diamond) are hard with high
melting points, while van der Waals crystals of noble gases are soft and melt at
very low temperatures. Metallic bonding, the fourth mechanism, holds the atoms
together by valence electrons that detach entirely and roam the whole lattice.
That electron sea is the subject of a
[later module](/condensed-matter/free-electron-fermi-gas/free-electron-gas-and-conduction),
where it also carries the electric current. The lattice geometry itself is the
next concern: the [Bravais lattices](/condensed-matter/crystal-structure/bravais-lattices-and-crystal-systems)
that classify every crystalline arrangement, and the reciprocal space in which
their diffraction is read.