Cohesive Energy, Lattice Energy, and the Shape of \(U(R)\)
Two energies that are not the same
Two quantities are used to describe how tightly a crystal is bound, and
examinations exploit the fact that students often treat them as synonyms.
Definition 3.1. (Cohesive energy)
The cohesive energy is the energy that must be supplied to a crystal to
separate it into neutral free atoms at rest, infinitely far apart, each
in its ground-state electronic configuration.
Definition 3.2. (Lattice energy)
The lattice energy, used for ionic crystals, is the energy that must be
supplied to separate the crystal into free ions at rest, infinitely far
apart.
For a crystal built from neutral atoms — argon, copper, silicon — the two
coincide, because the free particles are neutral atoms either way. For an ionic
crystal they differ, and by a large amount. Pulling NaCl apart into
\(\ce{Na+}\) and \(\ce{Cl-}\) costs about 7.9 eV per formula unit;
pulling it apart into neutral Na and Cl costs about
6.4 eV. The difference is exactly the energy recovered by
handing the electron back from \(\ce{Cl-}\) to \(\ce{Na+}\), namely
\(I_{\text{Na}} - A_{\text{Cl}} = 5.14 - 3.61 = \text{1.53 eV}\).
Which energy is the question asking for?
If a problem quotes a Madelung constant, or a repulsive term \(B/R^{m}\), it is
about the lattice energy — the ions are the objects being separated.
If it quotes an ionisation potential or an electron affinity, it is converting
between the two. The Born–Haber cycle of the section on bulk Modulus, the Born Exponent, and the Born–Haber Cycle is precisely the
bookkeeping that connects them.
Two further conventions matter for signs. The potential energy \(U(R_{0})\) at
equilibrium is negative — the bound state lies below the separated
state. The cohesive or lattice energy is the positive number \(|U(R_{0})|\). A
question that asks for “the binding energy” wants the positive number; a
question that asks for “the potential at equilibrium separation” wants the
negative one. Both appear, and they differ by a sign, not a factor.
The universal shape of the interatomic potential
Whatever the bonding mechanism, the potential energy of a pair of atoms as a
function of their separation \(R\) has the same qualitative form:
At large \(R\) there is an attraction. Its origin differs between
bond types — fluctuating dipoles, Coulomb attraction of ions, sharing of
electrons — but it always falls off as some power \(R^{-n}\) with
\(n \ge 1\).
At small \(R\) there is a steep repulsion. Its origin is the
same in every crystal: when closed shells overlap, the Pauli
exclusion principle forces electrons into higher orbitals, and the energy
rises. It is not electrostatic repulsion between nuclei, which is far too
weak at these distances because of screening.
In between there is a minimum, at the equilibrium separation
\(R_{0}\), of depth \(|U(R_{0})|\).
The force between the atoms is \(F(R) = -\,\mathrm{d}U/\mathrm{d}R\). It vanishes
at \(R_{0}\), is repulsive (positive) inside, attractive (negative) outside, and
— crucially — reaches a most negative value at some separation
\(R_{c} > R_{0}\) before dying away. That extremum is the maximum tension the bond
can carry, and \(R_{c}\) is the separation at which the bond breaks.
the section on the General Power-Law Potential computes both.
The universal shape of the
interatomic interaction, drawn here for the Lennard–Jones form. Top:
the potential \(U(R)\) (solid) is the sum of a steep repulsion \(+B/R^{m}\)
(dashed red) and a longer-ranged attraction \(-A/R^{n}\) (dashed teal). It
crosses zero at \(\sigma\) and has its minimum \(-\epsilon\) at \(R_{0}\).
Bottom: the force \(F = -\mathrm{d}U/\mathrm{d}R\) on the same
horizontal scale. It vanishes at \(R_{0}\), and its most negative value
\(F_{\min}\), reached at \(R_{c} > R_{0}\), is the greatest tension the bond can
sustain.
Four separations, four different questions
Do not confuse them:
\(\sigma\): where \(U = 0\) (the “collision diameter”).
\(R_{0}\): where \(U\) is a minimum, so \(F = 0\). The bond length.
\(R_{c}\): where \(F\) is a minimum, so \(\mathrm{d}^{2}U/\mathrm{d}R^{2} = 0\).
The inflection point of \(U\); the breaking separation.
\(R \to \infty\): free atoms, \(U \to 0\).
Always \(\sigma < R_{0} < R_{c}\).
The five kinds of binding
The attractive mechanism is what classifies the bond. All five are electrostatic
in origin; they differ in what the valence electrons are doing.
The five bonding mechanisms at a glance. Cohesive energies are per
atom, except for ionic crystals where the figure is per ion pair.
Coulomb attraction of oppositely charged ions, \(\propto R^{-1}\)
\(6\)–\(10\)
NaCl, CsCl, LiF, MgO
Covalent
electron pair shared between two atoms, spins antiparallel
\(3\)–\(8\)
C (diamond), Si, Ge, \(\ce{SiC}\)
Metallic
valence electrons delocalised over the whole crystal
\(1\)–\(9\)
Na, Cu, Fe, W
Hydrogen
proton shared between two electronegative atoms
\(0.1\)–\(0.5\)
ice, HF, \(\ce{KH2PO4}\), DNA
Three qualitative correlations run through the table above and are worth
committing to memory, because they answer a great many one-mark questions
without any calculation:
Cohesive energy tracks melting point. Solid neon melts at
24 K, NaCl at 1074 K, tungsten at 3695 K
— in the same order as their cohesive energies.
Directionality tracks brittleness. Covalent bonds have a
definite angular arrangement, so covalent solids cannot deform by sliding
planes past one another: they are hard and brittle. Metallic bonding is
non-directional, which is why metals are malleable and ductile.
Conductivity tracks electron mobility, not bond strength.
Diamond and tungsten have comparable cohesive energies; one is an insulator
and the other a good conductor. What matters is whether the electrons are
localised in bonds or free to move.
Real crystals are rarely pure cases. GaAs is mostly covalent with an ionic
admixture; graphite is covalent within a sheet and van der Waals between sheets;
transition metals combine metallic and covalent character.
the section on covalent, Metallic and Hydrogen Bonding quantifies the mixture.
Crystals of the Inert Gases
The inert gases form the simplest crystals in existence, and are therefore the
right place to start. Their outermost electron shells are full, so the charge
distribution of a free atom is spherically symmetric. Their ionisation energies
are high (from 12.1 eV for xenon to 24.6 eV
for helium) while their cohesive energies are tiny — a hundredth of an
electronvolt or so. Because the energy available on condensation is so much
smaller than the energy needed to disturb an electron shell, the charge
distribution in the crystal is barely different from that of the free atom.
The consequence is that these crystals can be treated as an assembly of rigid,
spherically symmetric, weakly attracting balls. They are transparent insulators,
they melt below 170 K, and — because spheres pack best that way —
all of them except helium crystallise in the face-centred cubic structure.
The van der Waals–London interaction
Here is the puzzle. If the charge distribution of each atom really were rigid and
spherically symmetric, the electrostatic potential outside a neutral atom would
vanish identically: by Gauss's law the electronic charge exactly cancels the
nuclear charge as seen from outside. Two such atoms would not interact at all,
inert gases would show no cohesion, and they would never condense. They do
condense. Something is missing from the rigid picture.
What is missing is that the charge distribution is not static. At any instant the
electrons of atom 1 are somewhere in particular, giving the atom an instantaneous
dipole moment even though its time-averaged moment is zero. That
instantaneous dipole produces a field at atom 2, which polarises atom 2, and the
two dipoles then attract. Because the induced moment is proportional to the
inducing field, the interaction survives the averaging: the average of a product
of correlated fluctuations is not the product of the averages.
The
Drude–London model. Each inert-gas atom is replaced by a one-dimensional
harmonic oscillator: a fixed charge \(+e\) and a mobile charge \(-e\) joined by a
spring of force constant \(c = m\omega_{0}^{2}\). The nuclei are held a distance
\(R\) apart; \(x_{1}\) and \(x_{2}\) are the instantaneous charge separations. The
Coulomb coupling between the two oscillators lowers the zero-point energy, and
that lowering is the van der Waals attraction.
Setting up the model
Model each atom as a one-dimensional harmonic oscillator, as in
the figure above. Charges \(\pm e\) are separated by \(x_{1}\) on atom 1
and \(x_{2}\) on atom 2, and both oscillate along the line joining the atoms with
force constant \(c\). The unperturbed Hamiltonian is
the four terms being, in order, nucleus–nucleus,
electron–electron, nucleus 1 with electron 2, and electron 1 with
nucleus 2.
For \(|x_{1}|,|x_{2}| \ll R\), expand each term with
\((1+u)^{-1} \simeq 1 - u + u^{2}\). The zeroth-order terms cancel in pairs, the
first-order terms cancel as well, and what survives at second order is
with the inverse relations \(x_{1} = (x_{s}+x_{a})/\sqrt{2}\),
\(x_{2} = (x_{s}-x_{a})/\sqrt{2}\), and the corresponding momenta
\(p_{1} = (p_{s}+p_{a})/\sqrt{2}\), \(p_{2} = (p_{s}-p_{a})/\sqrt{2}\). Since
\(x_{1}x_{2} = \tfrac{1}{2}\left(x_{s}^{2}-x_{a}^{2}\right)\) and
\(x_{1}^{2}+x_{2}^{2} = x_{s}^{2}+x_{a}^{2}\), the total Hamiltonian separates:
The ground-state energy of the coupled pair is
\(\tfrac{1}{2}\hbar(\omega_{s}+\omega_{a})\). Adding the two expansions in
the relation above, the linear terms cancel exactly — one mode
softens by as much as the other stiffens — and the leading surviving
correction is second order and negative:
where \(\alpha_{p}\) is the atomic polarisability.
The van der Waals attraction
\begin{equation*}
U_{\text{vdW}}(R) = -\frac{A}{R^{6}}, \qquad A \propto \hbar\omega_{0}\alpha_{p}^{2}.
\end{equation*}
It varies as the inverse sixth power of the separation. It is a
quantum-mechanical effect — \(\hbar\) appears explicitly — arising from
correlated zero-point fluctuations, and it is present between all atoms,
though in ionic, covalent and metallic crystals it is swamped by stronger terms.
It is also called the London interaction, or the induced-dipole–induced-dipole
interaction.
Remark. The one-dimensional model gives the prefactor \(\tfrac{1}{2}\hbar\omega_{0}
\alpha_{p}^{2}\); a proper three-dimensional treatment gives
\(\tfrac{3}{4}\hbar\omega_{0}\alpha_{p}^{2}\). The numerical factor depends on the
model, but the \(R^{-6}\) dependence and the proportionality to
\(\hbar\omega_{0}\alpha_{p}^{2}\) do not. Note that \(A\) grows rapidly with
polarisability, which is why the cohesive energies increase steadily from neon to
xenon.
The other two \(R^{-6}\) terms
The London term is one of three contributions that all fall off as \(R^{-6}\) and
are collectively called van der Waals forces. Examinations occasionally ask
which is which:
Keesom (orientation) term, between two permanent dipoles
that are free to rotate. Thermal averaging over orientations leaves
\(U = -\,\mu^{4}/\left(3k_{B}T R^{6}\right)\) — note the temperature
dependence, unique to this term.
Debye (induction) term, between a permanent dipole and the
dipole it induces in a neighbour: \(U = -2\mu^{2}\alpha_{p}/R^{6}\).
London (dispersion) term, the relation above, between two
instantaneous dipoles. This is the only one that survives between
atoms with no permanent moment — and therefore the only one operating in an
inert-gas crystal.
The repulsive term
As two atoms approach closely enough for their charge clouds to overlap, the
electrostatic energy of the system changes and the interaction becomes strongly
repulsive. The dominant reason is not nuclear Coulomb repulsion, which is
well screened, but the Pauli exclusion principle: two closed shells cannot
occupy the same region of space in their ground configurations, so overlap
forces some electrons into higher-energy orbitals.
Electronic charge
distributions overlap as atoms approach; the solid circles are the nuclei.
In (a) the clouds are separate and only the weak \(R^{-6}\) attraction acts. In
(b) the shaded region is doubly occupied, and the exclusion principle forces
electrons into higher orbitals: the energy rises steeply. Empirically the rise
is well described by \(B/R^{12}\) for the inert gases.
Experimental data on the inert gases are fitted well by an empirical repulsive
potential \(B/R^{12}\) with \(B\) a positive constant. There is no deep theoretical
justification for the exponent 12 — an exponential \(\lambda e^{-R/\rho}\)
represents the physics better, and is what the section on ionic Crystals and the Madelung Energy uses for ionic
crystals — but the twelfth power is analytically convenient, because it is the
square of the sixth power already present.
The Lennard–Jones potential
Adding the two contributions gives the potential energy of a pair of inert-gas
atoms:
It is conventional to re-parametrise using an energy \(\epsilon\) and a length
\(\sigma\), defined by \(A = 4\epsilon\sigma^{6}\) and \(B = 4\epsilon\sigma^{12}\), so
that
This is the Lennard–Jones (or 6–12) potential. In this form
\(U(\sigma) = 0\) and the minimum has depth exactly \(\epsilon\), at
\(R_{0} = 2^{1/6}\sigma = 1.122\,\sigma\); both are read off directly from
the figure above. The parameters \(\epsilon\) and \(\sigma\) are fixed
experimentally, historically from the second virial coefficient and the
viscosity of the dilute gas — not from the crystal, which is why the
crystal predictions below are genuine tests rather than fits.
Lennard–Jones parameters for the inert gases, obtained from
gas-phase data, and the resulting predictions for the fcc crystal compared
with experiment.
{\(\epsilon\) (eV)}
{\(\sigma\) (Å)}
\(R_{0}\) (Å)
cohesive energy (eV/atom)
{calc. \(1.09\sigma\)}
{obs.}
{calc. \(8.6\epsilon\)}
{obs.}
Ne
0.0031
2.74
2.99
3.13
0.027
0.020
Ar
0.0104
3.40
3.71
3.76
0.089
0.080
Kr
0.0140
3.65
3.98
4.01
0.120
0.116
Xe
0.0200
3.98
4.34
4.35
0.172
0.170
From a pair to a crystal: the lattice sums
The Lennard–Jones potential describes one pair. To get the energy of a crystal
of \(N\) atoms, sum over all pairs. Write the distance from a chosen reference atom
to its \(j\)-th neighbour as \(p_{j}R\), where \(R\) is the nearest-neighbour distance
and \(p_{j}\) is a pure number fixed by the geometry (\(p = 1\) for the nearest
shell, \(\sqrt{2}\) for the next in fcc, and so on). Then
Lennard–Jones lattice sums, and the resulting equilibrium spacing
and cohesive energy.
Structure
{\(A_{12}\)}
{\(A_{6}\)}
{\(R_{0}/\sigma\)}
{\(U_{0}/N\epsilon\)}
simple cubic
6.2022
8.4019
1.067
-5.69
bcc
9.1142
12.2533
1.068
-8.24
fcc
12.1319
14.4539
1.090
-8.61
hcp
12.1323
14.4549
1.090
-8.61
Both sums converge quickly — the nearest shell already supplies most of the
total — because \(p^{-12}\) and \(p^{-6}\) die fast. Minimising
the relation above with respect to \(R\) gives the two results that
the table above tabulates.
Equilibrium and cohesive energy of an inert-gas crystal
Writing \(x = (\sigma/R)^{6}\), the relation above reads
\(U = 2N\epsilon\left(A_{12}x^{2} - A_{6}x\right)\), a parabola in \(x\) minimised at
\(x = A_{6}/2A_{12}\). Hence
For the fcc structure, \(R_{0} = 1.09\sigma\) and
\(U_{0} = -8.6\,N\epsilon\) — the same two numbers for every inert
gas, since \(\epsilon\) and \(\sigma\) scale out.
The agreement in the table above is good, and the pattern of the residual
disagreement is itself informative. The predicted \(R_{0}\) is always slightly
too small and the predicted cohesive energy always slightly too
large, and the discrepancy shrinks steadily from neon to xenon: 28%
for Ne, 10% for Ar, 6% for Kr, 1% for
Xe.
The reason is zero-point motion, which the classical calculation ignores. Atoms
in the crystal are not at rest at \(R_{0}\); they oscillate with an irreducible
zero-point energy \(\sim \tfrac{1}{2}\hbar\omega\) which reduces the binding
and expands
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