Interatomic Forces & Crystal Bonding

1. Cohesive Energy of Crystalline Solids

The cohesive energy (or binding energy) $E_{\text{coh}}$ of a crystal is defined as the net energy required to disassemble the solid into its constituent neutral free atoms (or ions in ionic physics) at infinite mutual separation at zero temperature ($T = 0\text{ K}$):

Cohesion originates entirely from electromagnetic interactions governed by quantum mechanics. Solids are classified into five primary bonding categories based on the distribution of their valence electrons:

1. Ionic Crystals (e.g., $\text{NaCl}, \text{CsCl}, \text{LiF}$): Electrostatic attraction between positive cations and negative anions formed by complete valence electron transfer. Highly localized electron density, strong binding ($E_{\text{coh}} \approx 5 - 10\text{ eV/atom}$), high melting points, electrical insulators in the solid state.
2. Covalent Crystals (e.g., Diamond, $\text{Si}, \text{Ge}, \text{GaAs}$): Cohesion via quantum mechanical sharing of electron pairs localized along directional bonds formed by overlapping hybridized atomic orbitals ($sp^3$). Extreme hardness, high melting points, semiconductor or insulator band structure.
3. Metallic Crystals (e.g., $\text{Na}, \text{Cu}, \text{Fe}, \text{Al}$): Valence electrons delocalize completely into an itinerant electron gas ("Fermi sea") permeating an array of positively charged ion cores. Non-directional bonding, high electrical and thermal conductivities, ductility.
4. Van der Waals (Molecular) Crystals (e.g., Solid $\text{Ar}, \text{Kr}, \text{CH}_4$): Weak cohesion ($E_{\text{coh}} \approx 0.05 - 0.2\text{ eV/atom}$) arising from fluctuating quantum dipole-induced dipole interactions. Low melting and boiling points, soft mechanical properties.
5. Hydrogen-Bonded Crystals (e.g., Ice $\text{H}_2\text{O}, \text{HF}$, nucleic acids): Directional electrostatic dipole attractions mediated by an electropositive bare proton sandwiched between electronegative lone pairs (O, N, F). Intermediate strength ($0.1 - 0.5\text{ eV/bond}$).

1. Electrostatic Madelung Energy Formalism

Consider an ionic crystal consisting of $2N$ ions ($N$ positive cations and $N$ negative anions with charges $\pm q$). Let $R$ be the nearest-neighbor interionic separation distance. The total electrostatic Coulomb potential energy of ion $i$ interacting with all other ions $j \neq i$ in the crystal is:

where $r_{ij} = p_{ij} R$ denotes the distance between ions $i$ and $j$ measured in units of nearest-neighbor distance $R$, and the sign is positive for unlike charges (attraction) and negative for like charges (repulsion).

The dimensionless sum is defined as the Madelung Constant $\alpha$:

Multiplying by $N$ ion pairs (to avoid double-counting the pair interactions), the total electrostatic Madelung energy of the crystal is:

2. Calculation of the Madelung Constant for a One-Dimensional Ionic Chain

Consider an infinite 1D chain of alternating cations and anions with nearest-neighbor distance $R$. Choosing an arbitrary positive ion as the origin, its neighbors are situated at distances $R, 2R, 3R, \dots$ to both the left and right:

• Two opposite ions at distance $1R$: contribution $+2 \times (1/1)$
• Two identical ions at distance $2R$: contribution $-2 \times (1/2)$
• Two opposite ions at distance $3R$: contribution $+2 \times (1/3)$

The 1D Madelung constant $\alpha_{\text{1D}}$ is therefore given by the alternating harmonic series:

Recalling the Taylor series expansion of $\ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + \dots$ evaluated at $x = 1$:

3. Madelung Constants for 3D Ionic Lattices

In three dimensions, the summation is conditionally convergent and requires specialized summing techniques (e.g., the Evjen method of neutral concentric polyhedra or the Ewald summation method in reciprocal space):

• Rock-Salt ($\text{NaCl}$): $\alpha = 1.747565$
• Cesium Chloride ($\text{CsCl}$): $\alpha = 1.762675$
• Zincblende ($\text{ZnS}$): $\alpha = 1.63805$
• Wurtzite ($\text{ZnS}$): $\alpha = 1.64132$
• Fluorite ($\text{CaF}_2$): $\alpha = 5.03878$

1. Total Cohesive Energy with Born-Mayer Repulsion

At short interionic separations, the electron clouds of adjacent ions overlap. By the Pauli Exclusion Principle, overlapping closed electron shells must occupy higher unpopulated quantum states, generating a steep quantum mechanical repulsive potential. The total potential energy per ion pair $U_{\text{tot}}(R)$ is modeled by the Born-Mayer equation:

where $z$ is the coordination number (number of nearest neighbors), $B$ is a repulsive strength constant, and $\rho \approx 0.33\text{ \AA}$ is the characteristic range parameter of core repulsion.

2. Equilibrium Interionic Separation $R_0$

At equilibrium ($R = R_0$), the net mechanical force vanishes: $\left.\frac{dU_{\text{tot}}}{dR}\right|_{R=R_0} = 0$:

Substituting this repulsive term back into $U_{\text{tot}}(R_0)$ yields the celebrated Born-Mayer cohesive energy formula per ion pair:

Since $\rho / R_0 \approx 0.33\text{ \AA} / 2.82\text{ \AA} \approx 0.12$, core repulsion reduces the pure electrostatic Coulomb binding energy by approximately $10 - 15\%$.

3. Derivation of Bulk Modulus $B$

The isothermal bulk modulus $B$ measures the resistance of the crystal to uniform hydrostatic compression: $B = - V \frac{dP}{dV}$. At $T = 0\text{ K}$, the hydrostatic pressure is $P = - \frac{dU_{\text{cryst}}}{dV}$, which implies:

For a rock-salt cubic crystal with volume per ion pair $V = 2 R^3$ (where $V_0 = 2 R_0^3$), changing variables via $\frac{dU}{dV} = \frac{dU/dR}{dV/dR} = \frac{1}{6R^2} \frac{dU}{dR}$ leads directly to:

1. Origin of London Dispersion Forces

In inert gas crystals (solid $\text{Ne}, \text{Ar}, \text{Kr}, \text{Xe}$), atoms possess completely closed electronic shells with spherical symmetry and zero permanent dipole moment ($\langle \vec{p} \rangle = 0$). However, quantum fluctuations in electron cloud positions produce an instantaneous fluctuating electric dipole $\vec{p}_1(t) \sim e \vec{x}(t)$. This instantaneous dipole generates an electric field at distance $R$:

This electric field induces a dipole moment in the adjacent neutral atom proportional to its electronic polarizability $\alpha_{\text{pol}}$: $\vec{p}_2 = \alpha_{\text{pol}} \vec{E}_1 \propto \frac{\alpha_{\text{pol}} p_1}{R^3}$. The resulting dipole-dipole electrostatic interaction energy is attractive and scales inversely with the sixth power of distance:

2. The Lennard-Jones (6-12) Potential

When two inert gas atoms approach within the distance of their closed electron shells, Pauli core exclusion generates a steep repulsive potential, empirically parameterized as an inverse-twelfth power $1/R^{12}$. The total interaction potential between a pair of atoms separated by distance $R$ is the Lennard-Jones (6-12) potential:

where:

• $\varepsilon$: Depth of the potential well (binding energy minimum).
• $\sigma$: Finite distance at which the interatomic potential is zero ($U_{LJ}(\sigma) = 0$).

The minimum of the two-body potential occurs where $\frac{dU_{LJ}}{dR} = 0$:

At this equilibrium separation, the potential energy minimum is exactly $U_{LJ}(R_{\text{min}}) = -\varepsilon$.

3. Cohesive Energy of Inert Gas FCC Crystals

In an FCC crystal of $N$ inert gas atoms, summing over all pairs with $R_{ij} = p_{ij} R$ yields the total crystal potential energy:

For an FCC lattice, the lattice sums over all neighbors are geometric constants:

Minimizing $U_{\text{tot}}(R)$ with respect to $R$ yields the equilibrium crystal nearest-neighbor separation $R_0$ and the total cohesive energy per atom: