How can one find the minimum total energy of an infinite number of particles? A proof showing that, for certain interactions, periodic ‘ground states’ exist provides a new perspective on this, one of the oldest questions in physics.
When deciding how particles move and interact, nature has an affinity for minimum values. In newtonian mechanics, particles choose trajectories that minimize action, a quantity with the dimensions energy × time. In thermodynamics, a collection of particles at a fixed volume and temperature will settle into a state that minimizes the so-called free energy of the system, a quantity that, at absolute zero, reduces to the total energy — that is, the sum of all the interactions between pairs of the particles.
Simple as such statements may seem, the task of finding ‘ground-state’ configurations, and proving that they do indeed minimize the energy of a system, is notoriously difficult. The standard approach is more or less trial and error: one chooses a number of configurations, calculates their energies, and picks the structure with the smallest energy as the ground state.
Writing in Physical Review Letters1, András Sütő short-circuits this process. He demonstrates with a strikingly powerful mathematical argument that, in certain cases, the ground-state configuration sought when a material freezes into a regular crystal structure is a periodic one. Certain formal criteria must be met: first, the pair- interaction potential ϕ(r), which describes the potential energy of a two-particle system in terms of their separation, r, must possess a mathematical analogue known as a Fourier transform. (This transform expresses the potential as a series of oscillating sine and cosine terms that depend on a parameter k, the wavenumber, which represents an inverse wavelength.) Additionally, the Fourier-transformed potential, which is written (k), must be positive for all values of k below some threshold value, K0, and zero for all values above it.
The Fourier description of the potential is convenient for potentials such as those considered by Sütő, because it allows the total lattice energy — the energy required to bring all particles infinitely far apart from each other — to be expressed as a simple mathematical formula. The advantage of the Fourier representation is clearly seen when the potential not of particle pairs, but of a periodic lattice of particles, is considered. Such a spatial lattice — a direct, or Bravais, lattice — also has a Fourier representation, known as the reciprocal lattice.
In the reciprocal lattice, the total energy of the system (the quantity we wish to minimize) is simply the sum, up to K0, of the lattice potentials (K) for all reciprocal-lattice vectors K that build up the lattice. (For a three-dimensional reciprocal lattice, each K will have three wavenumber components, and so uniquely define a point in the lattice.) Whereas the term at K=(0, 0, 0) represents the origin of the coordinate system and is therefore the same for all types of lattices, all other allowed energy contributions, and thus the total energy of the system, depend on the exact spatial configuration of the particles in the direct lattice.
Imagine now a direct lattice (call it X) for whose reciprocal lattice all non-zero reciprocal-lattice vectors define points outside a sphere of radius K0. In this case, the total energy of the system will include only the term at K=(0,0,0). But for all other lattices with at least one value of K smaller than K0, the total energy can only be larger, because the additionally contributing potentials (K) are, by our definition, positive. So X must have the minimum possible energy.
Sütő1 proved his point by showing that, for a three-dimensional lattice and at the minimum particle density for which his results apply, given by ρ*=K03/(8√2π3), the body-centred cubic (bcc) lattice that many metals assume indeed fulfils the above requirements. With increasing density, other lattice types join the group, producing an infinite number of ground-state configurations. Moreover, Sütő demonstrated that periodic ground-state configurations are stable against arbitrary deviations from periodicity, and that aperiodic unions of periodic configurations minimize energy.
These are remarkable achievements. But why did such a clear and powerful proof take so long? The answer probably lies in the peculiar form of ϕ(r), which implicitly demands a finite potential at zero separation — the equivalent of permitting two particles to overlap each other fully. In the case of interacting atoms, however, a strong repulsive force is created between bound electron shells and their quantum-mechanical wavefunctions because of the Pauli exclusion principle (which holds that no two particles such as electrons may occupy the same quantum state), and the atomic-pair potential diverges steeply towards infinity as r approaches zero. And at the mesoscopic scale — that between the microscopic and the macroscopic — the rigid core of colloidal particles ensures that they just ‘bounce off’ each other at low separations, and also do not overlap. Neither the atomic nor the mesoscopic potentials even possess a Fourier transform; for a long time, finite interactions of the type considered by Sütő1 were simply considered ‘unphysical’.
But effective potentials between the centres of mass of entities such as block copolymers2, polymer chains or dendrimers can indeed remain finite at zero separation (Fig. 1). Theory3 and simulation4 have shown that the effective potential between two chains has a gaussian form, and calculations have shown that its ground-state configuration is face-centred cubic (fcc) at low densities and bcc at high densities5,6. Fourier transformation does not change a gaussian form, so (k) is positive in this case. There is, however, no value of K0 above which (k) is zero, so Sütő's result is not valid here. Nevertheless, his work joins a rapidly growing body of research7, spurred on by developments in soft-matter physics, on the properties of bounded interaction potentials in one-component systems, in which the properties of (k) are used to obtain information on the ground-state configuration or the topology of the phase diagram.
A potential (k) that vanishes above a threshold seems difficult to realize physically. Sütő shows that the requirement for this to occur leads to an interaction potential that shows oscillatory behaviour combined with a power-law decay at large values of r. This is strikingly similar to the RKKY (Ruderman-Kittel-Kasuya-Yosida), or Friedel, potential8,9,10 that arises as an effective interaction potential between ions when the effect of free electrons is incorporated as a statistical average. For metals of valency Z, the ion density ρ is related to Sütő's minimum density by ρ=ρ*π√2/(3Z), so that ρ exceeds ρ* by 48% for Z=1 and trails it by 26% for Z=2. So unfortunately there is no happy agreement between the real and the Sütő densities that would hint at why certain metals crystallize into the bcc structure. The power laws governing the behaviour of the RKKY and Sütő potentials at large values of r are slightly different, and effective ion potentials also include a steeply repulsive region at small values of r (ref. 10) that is absent in Sütő's interaction. So the long- and short-range parts of the potential must be considered independently. Sütő provides us with a suitable reference frame to deal with the former.
In the late 1970s, a celebrated and controversial article11 was published entitled ‘Should all crystals be bcc?’. It contained a theory of freezing that involved density waves — modulated density profiles — at the set of reciprocal lattice vectors K as parameters in a mathematical expression for the free energy. On the basis that the bcc crystal is the only spatial configuration that combines the shortest reciprocal-lattice vectors lying on the surface of a sphere, it was concluded that the stablest crystals should all be bcc. This argument, although brilliant, was flawed in making certain simplifying assumptions. Sütő's work comes from a very different direction to again put the bcc lattice in a distinctive position among all Bravais lattices. In this way, it recasts the question of why crystals form as they do in a fresh and thought-provoking way.
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Communications in Mathematical Physics (2011)