Exact values of some two-dimensional lattice sums

Lattice sum for a hexagonal close-packed structure and its dependence on the c/a ratio of the hexagonal cell parameters

We continue the work by Lennard-Jones and Ingham, and later by Kane and Goeppert-Mayer, and present a general lattice sum formula for the hexagonal close packed (hcp) structure with different c/a ratios for the two lattice parameters a and c of the hexagonal unit cell. The lattice sum is expressed in terms of fast converging series of Bessel functions. This allows us to analytically examine the behavior of a Lennard-Jones potential as a function of the c/a ratio. In contrast to the hard-sphere model, where we have the ideal ratio of c/a=sqrt[8/3] with 12 kissing spheres around a central atom, we observe the occurrence of a slight symmetry-breaking effect and the appearance of a second metastable minimum for the (12,6) Lennard-Jones potential around the ratio c/a=2/3. We also show that the analytical continuation of the (n,m) Lennard-Jones potential to the domain n,m<3 such as the Kratzer potential (n=2,m=1) gives unphysical results.

Analytic form of a two-dimensional critical distribution

This paper explores the possibility of establishing an analytic form of the distribution of the order parameter fluctuations in a two-dimensional critical spin-wave model, or width fluctuations of a two-dimensional Edwards-Wilkinson interface. It is shown that the characteristic function of the distribution can be expressed exactly as a gamma function quotient, while a Charlier series, using the convolution of two Gumbel distributions as the kernel, converges to the exact result over a restricted domain. These results can also be extended to calculate the temperature dependence of the distribution and give an insight into the origin of Gumbel-like distributions in steady-state and equilibrium quantities that are not extreme values.

The Lennard-Jones Potential Revisited: Analytical Expressions for Vibrational Effects in Cubic and Hexagonal Close-Packed Lattices

Analytical formulas are derived for the zero-point vibrational energy and anharmonicity corrections of the cohesive energy and the mode Grüneisen parameter within the Einstein model for the cubic lattices (sc, bcc, and fcc) and for the hexagonal close-packed structure. This extends the work done by Lennard-Jones and Ingham in 1924, Corner in 1939, and Wallace in 1965. The formulas are based on the description of two-body energy contributions by an inverse power expansion (extended Lennard-Jones potential). These make use of three-dimensional lattice sums, which can be transformed to fast converging series and accurately determined by various expansion techniques. We apply these new lattice sum expressions to the rare gas solids and discuss associated critical points. The derived formulas give qualitative but nevertheless deep insight into vibrational effects in solids from the lightest (helium) to the heaviest rare gas element (oganesson), both presenting special cases because of strong quantum effects for the former and strong relativistic effects for the latter.

The asymmetric Wigner bilayer

We present a comprehensive discussion of the so-called asymmetric Wigner bilayer system, where mobile point charges, all of the same sign, are immersed into the space left between two parallel, homogeneously charged plates (with possibly different charge densities). At vanishing temperatures, the particles are expelled from the slab interior; they necessarily stick to one of the two plates and form there ordered sublattices. Using complementary tools (analytic and numerical), we study systematically the self-assembly of the point charges into ordered ground state configurations as the inter-layer separation and the asymmetry in the charge densities are varied. The overwhelming plethora of emerging Wigner bilayer ground states can be understood in terms of the competition of two strategies of the system: net charge neutrality on each of the plates on the one hand and particles' self-organization into commensurate sublattices on the other hand. The emerging structures range from simple, highly commensurate (and thus very stable) lattices (such as staggered structures, built up by simple motives) to structures with a complicated internal structure. The combined application of our two approaches (whose results agree within remarkable accuracy) allows us to study on a quantitative level phenomena such as over- and underpopulation of the plates by the mobile particles, the nature of phase transitions between the emerging phases (which pertain to two different universality classes), and the physical laws that govern the long-range behaviour of the forces acting between the plates. Extensive, complementary Monte Carlo simulations in the canonical ensemble, which have been carried out at small, but finite temperatures along selected, well-defined pathways in parameter space confirm the analytical and numerical predictions within high accuracy. The simple setup of the Wigner bilayer system offers an attractive possibility to study and to control complex scenarios and strategies of colloidal self-assembly, via the variation of two system parameters.

Rich Polymorphic Behavior of Wigner Bilayers

Self-assembly into target structures is an efficient material design strategy. Combining analytical calculations and computational techniques of evolutionary and Monte Carlo types, we report about a remarkable structural variability of Wigner bilayer ground states, when charges are confined between parallel charged plates. Changing the interlayer separation, or the plate charge asymmetry, a cascade of ordered patterns emerges. At variance with the symmetric case phenomenology, the competition between commensurability features and charge neutralization leads to long range attraction, appearance of macroscopic charges, exotic phases, and nonconventional phase transitions with distinct critical indices, offering the possibility of a subtle, but precise and convenient control over patterns.

Wigner-crystal formulation of strong-coupling theory for counterions near planar charged interfaces

We present a new analytical approach to the strong electrostatic coupling regime (SC) that can be achieved equivalently at low temperatures, high charges, low dielectric permittivity, etc. Two geometries are analyzed in detail: one charged wall first, and then two parallel walls at small distances that can be likely or oppositely charged. In all cases, only one type of mobile counterions is present, and ensures electroneutrality (salt-free case). The method is based on a systematic expansion around the ground state formed by the two-dimensional Wigner crystal(s) of counterions at the plate(s). The leading SC order stems from a single-particle theory, and coincides with the virial SC approach that has been much studied in the last 10 years. The first correction has the functional form of the virial SC prediction, but the prefactor is different. The present theory is free of divergences and the obtained results, both for symmetrically and asymmetrically charged plates, are in excellent agreement with available data of Monte Carlo simulations under strong and intermediate Coulombic couplings. All results obtained represent relevant improvements over the virial SC estimates. The present SC theory starting from the Wigner crystal and therefore coined Wigner SC, sheds light on anomalous phenomena like the counterion mediated like-charge attraction, and the opposite-charge repulsion.

Counterions at highly charged interfaces: from one plate to like-charge attraction

We present an analytical approach for similarly and highly charged planar interfaces in the presence of counterions. The procedure is physically transparent and based on an exact low temperature expansion around the ground state formed by the two-dimensional Wigner crystal of counterions. The one plate problem is worked out, together with the two plates situation. Unlike previous approaches, the expansion is free of divergences, and is shown to be in excellent agreement with available data of Monte Carlo simulations under strong Coulombic couplings. In the two plates case, the present results shed light on the like-charge attraction regime.

Symmetrization of the Hurwitz zeta function and Dirichlet L functions

We consider the Hurwitz zeta function ζ ( s , a ), and form two parts ζ + and ζ − by symmetric and antisymmetric combinations of ζ ( s , a ) and ζ ( s ,1− a ). We consider the properties of ζ + and ζ − , and then show that each may be decomposed into parts denoted by and , each of which obeys a functional equation of the Dirichlet L type, with a multiplicative factor of −1 for the functions . We show the results of this procedure for rational a = p / q , with q =1, 2, 3, 4, 5, 6, 7, 8, 10, and demonstrate that the functions and have some of the key properties of Dirichlet L functions.

Search for universal roughness distributions in a critical interface model

We study the probability distributions of interface roughness, sampled among successive equilibrium configurations of a single-interface model used for the description of Barkhausen noise in disordered magnets, in space dimensionalities d = 2 and 3. The influence of a self-regulating (demagnetization) mechanism is investigated, and evidence is given to show that it is irrelevant, which implies that the model belongs to the Edwards-Wilkinson universality class. We attempt to fit our data to the class of roughness distributions associated to 1/falpha noise. Periodic, free, "window," and mixed boundary conditions are examined, with rather distinct results as regards quality of fits to 1/falpha distributions.