Bonding, Interatomic Potentials & Crystal Structure

Bonding and Interatomic Potentials

Cohesive Energy of a Crystal

Cohesive Energy: The energy required to separate completely all constituent atoms or molecules from each other to an infinite distance, starting from their equilibrium positions in the crystal lattice.

The cohesive energy is (minus) the total potential energy of all atoms in the crystal at equilibrium, taking into account all pairwise interactions. It is a key factor in determining the melting and boiling points of the material, and is related directly to the strength of the interatomic bonds.

For a simple diatomic molecule, the cohesive energy is just the negative of the minimum of the interatomic potential. For an extended crystal, however, we must sum over all pairs of atoms which is a much richer problem that introduces the concept of lattice sums.

We can calculate the cohesive energy if we assume that every atom interacts with every other atom in the crystal via an interatomic pair potential such as Lennard-Jones or Morse.

Cohesive Energy per Atom

Consider atom in the crystal. We can write the interaction energy between atom and some other atom as , where is the interatomic potential and is the equilibrium distance between atoms and .

Atom interacts with atoms in the crystal, and the total potential energy of these interactions is minus the cohesive energy. The cohesive energy per atom is:

We do not include in the summation because that would mean atom interacting with itself. The factor of is included because we want the potential energy per atom: the pair potential is the energy of interaction between two atoms, and summing over all counts each pair twice.

Lennard-Jones Cohesive Energy and Lattice Sums

Weak van der Waals interactions between molecules include London dispersion forces arising from charge fluctuations. A suitable description is the Lennard-Jones potential:

Here is the depth of the potential well and is the distance at which the potential crosses zero. The term captures the attractive van der Waals interaction, while models the short-range Pauli repulsion.

Substituting the LJ potential into the cohesive energy equation:

Now write , where is the equilibrium nearest-neighbour separation and is a dimensionless factor depending only on the crystal geometry. For nearest neighbours ; for next nearest neighbours in a simple cubic lattice ; and so on.

where the lattice sums are defined as:

The lattice sums depend only on the crystal lattice, not on the atoms. This is a profound result: the geometric structure of the crystal is fully separated from the chemical identity of the atoms. Different noble gases crystallising in the same FCC structure will have identical lattice sums but different cohesive energies due to different and .

Ionic Bonding and the Madelung Constant

In most covalent bonds, electrons are shared unevenly; ionic bonding is the extreme limit where electrons are transferred completely from one atom to another. The resulting ions are held together by electrostatic forces plus a short-range repulsive term:

We assume the short-range repulsive term is the same for all ion pairs, but the Coulomb term depends on the relative sign of the charges and .

Write and with , where is the sign of the charge and is an integer. Then:

where is the Madelung constant:

Here is the sign of the charge on site relative to site . For the nearest neighbours, which give the largest contributions, the charges are generally opposite so ; overall, is positive. One must be careful in evaluating these sums because they are conditionally convergent – the result depends on the order in which the terms are combined.

For the NaCl structure, .

Machine-Learned Interatomic Potentials

A recent example of a machine-learned interatomic potential (MLIP) is the MACE model, developed by Venkat Kapil and collaborators at UCL. MACE is a foundation model for materials chemistry with wide applicability across many materials classes – from metals and ceramics to molecular systems and water.

Rather than specifying a fixed functional form (like ), MLIPs learn the potential energy surface directly from quantum mechanical training data. They achieve near-quantum accuracy at a fraction of the computational cost, enabling simulations of millions of atoms over nanosecond timescales.

Summary of Bonding Types

Different forms of bonding in solids (molecular, covalent, ionic, or metallic) are manifestations of the same quantum mechanical laws under different conditions. Each can be characterised by its interatomic potential:

1. Molecular (van der Waals): Weak interactions from London dispersion forces (charge fluctuations).

2. Covalent: Electrons occupy new orbitals distributed around at least one other atom. Described by the Morse potential:

3. Ionic: Electrons transferred completely; resulting ions held by electrostatic forces:

4. Metallic: Sharing of electrons between atoms forms a “sea of electrons” surrounding positively charged metallic ions. Not well described by simple pair potentials.

Thermal expansion arises because spacing between atoms increases as they vibrate more at higher temperatures, thanks to the anharmonicity of the interatomic potential – the repulsive wall is steeper than the attractive well.

Crystal Structure

Crystal Lattice and the Unit Cell

A crystal structure is the repeating pattern of atoms, ions, or molecules in a solid material. Long-range order means the arrangement forms a pattern that extends over large distances and gives the material its unique physical and chemical properties.

There are two components to a crystal structure:

• Lattice: A repeating network of identical points in space, defined by three translation vectors.
• Basis: At each lattice point, there are one or more atoms, ions, or molecules, as defined by the basis vectors.

The translation vectors describe the repeating pattern of the lattice in space. Any lattice point can be represented as:

where are integers, and the set of all such points defines the lattice.

Types of Unit Cell

• Unit cell: The parallelepiped defined by the vectors . It can be used to fill all space – with no gaps – by repeated application of the translation vectors. Its volume is:

• Primitive unit cell: The minimum volume cell containing exactly one lattice point.
• Conventional unit cell: A convenient building block chosen to reflect the underlying symmetry of the lattice. It can contain more than one lattice point.
• Wigner-Seitz cell: A primitive unit cell constructed by drawing lines from a given lattice point to all nearby lattice points, then bisecting each line with a perpendicular plane. The smallest volume enclosed is the Wigner-Seitz cell. It possesses the full symmetry of the lattice.

Translational Symmetry

Lattices are symmetric under translation: a lattice is mapped onto itself by translation through any integer multiple of the translation vectors .

If we apply any such translation to the crystal, every atom moves to the position previously occupied by an identical atom, and the crystal looks exactly the same. This is the defining property of a crystal.

Rotational Symmetry

Lattices also have rotational symmetry. In 2D, lattices can be found that are symmetric under rotations of:

These are referred to as one-, two-, three-, four-, and six-fold rotations. No other rotational symmetries are compatible with translational periodicity. In particular, five-fold symmetry is forbidden because pentagons cannot tile a plane without gaps.

The Bravais Lattices

There are an infinite number of possible lattices since all vectors and are allowed. However, we can categorise lattices by considering their symmetry – translational, rotational, and mirror symmetry. From this, we find exactly five types of lattice in 2D, the Bravais lattices:

1. Square: (where is the angle between two vectors)
2. Rectangular:
3. Oblique: (most general)
4. Hexagonal: (six-fold rotational symmetry)
5. Centred rectangular: (with a centred point in the non-primitive cell)

In three dimensions there are 14 Bravais lattices, grouped into 7 crystal systems. The most general is triclinic. The others are monoclinic, orthorhombic, tetragonal, hexagonal, and cubic, each with subcategories: P (primitive), C (base-centred), I (body-centred), F (face-centred), and R (rhombohedral).

Conventional Unit Cells of Cubic Lattices

The three cubic Bravais lattices – simple cubic (SC), body-centred cubic (BCC), and face-centred cubic (FCC) – are the most commonly encountered crystal structures in metals and simple compounds.

The conventional unit cells are convenient to work with and reflect the symmetry of the lattice. Only the SC is a primitive unit cell (one lattice point per cell). The conventional unit cells for BCC and FCC contain 2 and 4 lattice points, respectively.

How to determine the number of lattice points:

• Corner Points: Each contributes because one point is shared by eight cubics.
• Edge Centre: Each contributes because one point is shared by four cubics.
• Face Centre: Each contributes because one point is shared by two cubics.
• Body Centre: Each contributes , not shared.

Simple Cubic (SC)

Lattice points at the eight corners of the cube, each shared by 8 cells: point per cell. The conventional and primitive cells coincide.

Body-Centred Cubic (BCC)

Lattice points at corners () plus one at the body centre: 2 points per conventional cell. The primitive cell is a rhombohedron with edge length and angle between edges. The primitive translation vectors are:

where is the length of the cubic cell edge.

Face-Centred Cubic (FCC)

Lattice points at corners () plus face centres (): 4 points per conventional cell. The primitive cell is a rhombohedron with primitive translation vectors:

Miller Indices

Often, when working with crystals, it is necessary to specify a particular crystal plane. We do this using a system called Miller indices.

The procedure is:

1. Find the intercepts on the axes defined by in terms of multiples of their lengths . If the plane does not intercept a particular axis, its intercept is at infinity.
2. Take the reciprocals of these numbers (noting ) and reduce to three integers having the same ratio, usually the smallest three integers.
3. The result, enclosed in parentheses, is called the index of the plane.

Example: A plane intercepts the axes at . The reciprocals are . The smallest three integers with the same ratio are 2, 3, 3. Thus the Miller indices of the plane are .

Why Miller indices use reciprocals:

• In mathematics, since a non-intercepting plane has intercept at infinity, we express this as a finite quantity: 0.
• In physics, the reciprocal indices are actually the vector coordinates in reciprocal space, which is perpendicular to the real space plane. This makes them natural for describing the normal vector to the plane.

Important planes in cubic crystals include . Families of equivalent Miller planes are expressed as : for example, includes . Directions within the lattice are expressed as . Negative Miller indices are denoted with a bar, e.g. .

Crystal Basis

To form a crystal structure from a lattice, we add a basis – which can be one or more atoms. Bases with multiple atoms lead to a more diverse arrangement of possible structures.

Example: The honeycomb crystal structure can be formed from a hexagonal lattice with a two-atom basis. The hexagonal translation vectors are and , and the two atom basis points (relative to ) are and .

Graphene is an example of a lattice formed with a two-atom basis. Graphene sheets were first isolated in 2004, for which the Nobel Prize in Physics was awarded in 2010 to Novoselov and Geim.

Research Example – Twistronics: Two-dimensional materials are a very active area of modern research. One particularly exciting development is twisted bilayer graphene (TBLG): two graphene layers stacked with a slight twist relative to each other. The twist creates a new repeating pattern (Moire pattern) that alters how electrons behave. At a special “magic angle” of , TBLG can exhibit superconductivity. This discovery (2018) has opened a new field called twistronics.

The Diamond Crystal Structure

The diamond structure has an FCC Bravais lattice, but with two atoms on every FCC lattice site. In terms of the FCC primitive unit cell, there is a two-atom basis with one atom at the origin and another at . Another way to think of the diamond lattice is as two interpenetrating FCC lattices, with one shifted by of the unit cell. This is the crystal structure of carbon (diamond phase), silicon, and germanium – materials at the heart of semiconductor technology.

Coordination Number and Packing Fractions

Coordination Number

The number of nearest neighbours of a given atom in the crystal.

Packing Fraction

The fraction of space filled by hard spheres centred on lattice points and touching their nearest neighbours.

SC packing: Atoms touch along the cube edge, so . One atom per cell:

BCC packing: Atoms touch along the body diagonal , so . Two atoms per cell:

FCC packing: Atoms touch along the face diagonal , so . Four atoms per cell:

FCC is the most efficient packing of identical spheres, equal to the hexagonal close-packed (HCP) structure. This maximum packing fraction of was conjectured by Kepler in 1611 and proven by Thomas Hales in 1998.

Further Explorations

The science of crystallography has a rich history stretching back centuries. Rene-Just Hauy (1784) first proposed that crystals are built from repeating units. Auguste Bravais (1811–1863) classified all possible 3D lattices into 14 types in 1848. The field was transformed by the discovery of X-ray diffraction by William Henry Bragg and William Lawrence Bragg around 1913 (Nobel Prize 1915). Their condition remains the cornerstone of structure determination.

Quasicrystals: In 1982, Dan Shechtman discovered materials with five-fold symmetry – forbidden in periodic crystals. These quasicrystals are ordered but not periodic, exhibiting long-range orientational order without translational periodicity. Shechtman received the Nobel Prize in Chemistry in 2011. Quasicrystals demonstrate that crystalline order is richer than the 14 Bravais lattices alone would suggest.

Connection to Thermal Physics: Crystal structure fundamentally determines a material’s thermal properties. The phonon modes (quantised lattice vibrations) depend directly on the atomic arrangement and interatomic forces. The phonon dispersion relation governs heat capacity (via the Debye model), thermal conductivity, and thermal expansion. Understanding crystal structure is therefore essential for predicting thermal transport – a key concern in thermoelectric devices and thermal management of electronics.