First-neighbour Interactions Research Articles

Thermodynamics of ordering and mixing in plagioclase feldspars: atomistic modelling in favour of Landau theory

Plagioclase feldspars are key components of terrestrial planets and deriving their phase diagram and activity–composition relations has been the subject of many studies. Three main approaches have been used for thermodynamic modelling of the behaviour of plagioclase feldspars, and all three disagree in terms of the magnitude and relative importance of mixing energies and ordering energies. The present study uses Monte Carlo simulations based on classical atomistic modelling to provide independent constraints over ordering and mixing for anorthite and its solid solution with albite. Unsurprisingly, Al-avoidance plays a key role. The competition between first-neighbour and second-neighbour interactions is highlighted, providing simple explanations to the high degree of short-range order in macroscopically disordered plagioclase. The mixing properties obtained in this study support strong non-ideality, with negative mixing enthalpy and temperature-dependent mixing entropy. Ordering energies are consistent with studies using Landau theory. Unfortunately, the form of the Landau equations hindered parametrisation of a full thermodynamic model from albite to anorthite, even though the thermodynamic formalism remains simple. Reasons, perspectives, and implications for the phase diagram of ternary feldspars are discussed.

One-dimensional q-state Potts model with multi-site interactions

A one-dimensional (1D) q-state Potts model with N sites, m-site interaction K in a field H is studied for arbitrary values of m. Exact results for the partition function and the two-point correlation function are obtained at H = 0. The system in a field is shown to be self-dual. Using a change of Potts variables, it is mapped onto a standard 2D Potts model, with first-neighbour interactions K and H, on a cylinder with helical boundary conditions (BC). The 2D system has a length and a transverse size m. Thus the Potts chain with multi-site interactions is expected to develop a 2D critical singularity along the self-duality line, , when and .

One-dimensional Ising model with multispin interactions

We study the spin-1/2 Ising chain with multispin interactions K involving the product of m successive spins, for general values of m. Using a change of spin variables the zero-field partition function of a finite chain is obtained for free and periodic boundary conditions and we calculate the two-spin correlation function. When placed in an external field H the system is shown to be self-dual. Using another change of spin variables the one-dimensional Ising model with multispin interactions in a field is mapped onto a zero-field rectangular Ising model with first-neighbour interactions K and H. The 2D system, with size m × N/m, has the topology of a cylinder with helical BC. In the thermodynamic limit , , a 2D critical singularity develops on the self-duality line, .

Impact of Substrate Defects on the Equilibrium One-Dimensional Island Size Distribution

As long as only first-neighbour interactions are considered, equilibrium island size distributions of monoatomic islands in one dimension follow an exponential law regardless of the strength and the repulsive or attractive character of the adsorbate–adsorbate interactions. However, one-dimensional atomic wires obtained via nucleation at the step edges have a monomodal island size distribution. In this paper, we present a simple one-dimensional Monte Carlo model that shows how the monomodal distribution observed experimentally can be obtained by including surface defects that only suppress the interaction between two successive adsorbates.

Analytical determination of the Landau-Ginzburg parameters of (100) metal homoepitaxial systems

In the initial stages of homoepitaxial growth on the (100) surface of metals such as Ag, Fe, Cu, Ni, and Pd, where the clean surface does not reconstruct, two-dimensional islands with compact, near-square shapes are formed. In order to determine the phenomenological material parameters of the nonlinear and nonlocal Landau-Ginzburg theory, which describes the metal homoepitaxial systems mentioned above, an atomistic model for these systems is developed. Based on this model, we derive analytical relationships between the Landau-Ginzburg parameters A, B, C, and D, and the parameters of the homoepitaxial system (such as coverage, first-neighbour interaction energy, etc.). We find that the Landau-Ginzburg parameters of the system depend on the specific material as well as on the coverage of the surface. We then apply the method to the Ag/Ag(100) system.

Urotropin azelate: a rather unwilling co-crystal.

Urotropin (U) and azelaic acid (AA) form 1:1 co-crystals (UA) that give rise to a rather complex diffraction pattern, the main features of which are diffuse rods and bands in addition to the Bragg reflections. UA is characterized by solvent inclusions, parasite phases, and high vacancy and dislocation densities. These defects compounded with the pronounced tendency of U to escape from the crystal edifice lead to at least seven exotic phase transitions (many of which barely manifest themselves in a differential scanning calorimetry trace). These involve different incommensurate phases and a peritectoid reaction in the recrystallization regime (T(h) > 0.6). The system may be understood as an OD (order-disorder) structure based on a layer with layer group P(c)c2 and cell a(o) approximately 4.7, b approximately 26.1 and c approximately 14.4 A. At 338 K the layer stacking is random, but with decreasing temperature the build-up of an orthorhombic MDO (maximal degree of order) structure with cell a(1) = 2a(o), b(1) = b, c(1) = c and space group Pcc2 is begun (at approximately 301 K). The superposition structure of the OD system at T = 286 (1) K with space group Bmmb and cell â = 2a(o), b = b and ĉ = c/2 owes its cohesion to van der Waals interactions between the AA chains and to three types of hydrogen bonds of varied strength between U-U and U-AA. Before reaching completion, this MDO structure is transformed, at 282 K, into a monoclinic one with cell a(m) = -a(o) + c/4, b(m) = b, c(m) = -2(a(o) + c/2), space group P2(1)/c, spontaneous deformation approximately 2 degrees, and ferroelastic domains. This transformation is achieved in two steps: first a furtive triggering transition, which is not yet fully understood, and second an improper ferroelastic transition. At approximately 233 K, the system reaches its ground state (cell a(M) = a(m), b(M) = b, c(M) = c(m) and space group P2(1)/c) via an irreversible transition. The phase transitions below 338 K are described by a model based on the interaction of two thermally activated slip systems. The OD structure is described in terms of a three-dimensional Monte Carlo model that involves first- and second-neighbour interactions along the a axis and first-neighbour interactions along the b and c axes. This model includes random shifts of the chains along their axes and satisfactorily accounts for most features that are seen in the observed diffraction pattern.

LOW-TEMPERATURE SERIES EXPANSIONS FOR SQUARE-LATTICE ISING MODEL WITH FIRST AND SECOND NEIGHBOUR INTERACTIONS

We have calculated the low-temperature series expansions of the spontaneous magnetization and the zero-field susceptibility of the square-lattice ferromagnetic Ising model with first-neighbour interaction J1 and second-neighbour interaction J2 to the 30th and 26th order respectively by computer. Our results extend the previous calculations by Lee and Lin to six more orders. We use the Padé approximants to estimate the critical exponents and the critical temperature for different ratios of R = J2/J1. The estimated critical temperature as a function of R agrees with the estimation by Oitmaa from high-temperature series expansions.

Contrasting antiferromagnetic order between FePS 3 and MnPS 3

Transition-metal thiophosphates, MPS3 (M = Fe, Mn, etc.), make up a class of antiferromagnetic materials with quasi-two-dimensional magnetic behaviour. The metal atoms occupy a honeycomb lattice. In MnPS3, the first-neighbour interaction is 400 times stronger than the exchange between the planes and the Mn moments show little anisotropy. The antiferromagnetic order is not well established perpendicular to the planes. Here we present the first powder neutron diffraction patterns of FePS3. The absence of trailing edges on the magnetic Bragg peaks, such as have been observed in MnPS3, indicates that the magnetic order is three-dimensional in FePS3.

The Low-Temperature Three-Dimensional Structures of the Second-Stage Caesium Graphitide

The three-dimensional (3D) structure of the second-stage CSC24 graphitide was investigated by recording hk.l monochromatic rotating-crystal diffractograms. At low temperature, the two-dimensional (2D) structure is formed of an incommensurate (2.53 x 2.53 R 14.5°) and a commensurate (2 x 2 R 0°) lattice. Below 140 K, whereas the Cs incommensurate layers become 3D ordered, the 2 x 2 R 0° commensurate layers are randomly distributed in the structure. The commensurate structure, with composition CsC16 (stage two), probably equilibrates the lack of Cs induced by the incommensurate structure (composition CSC26), in order to reach the global composition CSC24 of the compound. Depending on the temperature, two kinds of stackings appear. Below 140 K, the incommensurate Cs layers are arranged in a 2D-modulated initial stacking, transformed at a lower temperature into a rhombohedral stacking in which Cs atoms are modulated along the c axis. Simulation of the selection rules allowed the determination of models of the incommensurate layers stacking along the c axis, with translations from one Cs layer to another layer commensurate with the graphite lattice. The initial stacking is described by a pseudo-two-layer model with successive random translations and a short-range order. The structure appearing at a lower temperature is described by a pseudorhombohedral stacking with definite translations. The transition from the initial stacking with a first-neighbour interaction to the pseudorhombohedral stacking with a second-neighbour interaction could be a result of the change in interactions between the Cs atoms and the graphitic host, due to the dominating influence of the graphite periodic potential at low temperature.

Hyperfine field and ordering in bcc CoFe bulk alloys studied by 59Co NMR and Monte-Carlo simulation

Bcc CoFe alloys and ordered compounds with Fe concentrations of 50 at.%, 40 at.% and 30 at.% have been studied by 59Co NMR. Spectra with regularly spaced satellites are observed and interpreted as resulting from a step increase of the resonance frequency with each additional Fe impurity in the first coordination shell. The influence of the second-neighbour shell is also demonstrated. Satellite position analysis leads to the relationship between the hyperfine field and the first- and secondneighbour shell composition, which is found valid for the three compositions both in ordered compounds and disordered alloys. This relationship is used to determine the chemical short range order ruling the compound structure. The analysis reveals that the probability for Co to be found in perfectly ordered stoichiometric regions is higher than expected from a random distribution of the excess Co in off-stoichiometric compounds. Monte-Carlo simulations indicate that such short range order would result not only from first-neighbour interactions but also from rather large second- and/or third-neighbour interactions. However, the observations made during isothermal annealing show that the phase transformation takes place through a nucleation and growth process and not by way of homogeneous ordering. Therefore, the final short range order is also discussed in terms of kinetics of the phase transformation.

Tricritical behaviour of the ferromagnetic Blume-Capel model with first- and second-neighbour interactions

A ferromagnetic Blume-Capel model with first- and second-neighbour interactions is investigated by the use of the effective-field theory with correlations. The general expressions for evaluating the second-order phase transition and the tricritical point are obtained. Particular emphasis is given to the square lattice for which the dependence of the position of the tricritical point on the relative strengths of the second- and first-neighbour interactions is investigated.

Phase diagram of the one-dimensional extended Hubbard model with attractive and/or repulsive interactions at quarter filling.

We study the phase diagram of the one dimensional (1D) $U-V$ model at quarter filling in the most general case where the on-site and first-neighbour interactions $U$ and $V$ can be both attractive and repulsive. The results have been obtained using exact diagonalization of small clusters and variational techniques, as well as exact results in various limits. We have analyzed four properties of the groundstate: i)~whether it is insulating or metallic; \hbox{ii)~whether} it is homogenous or phase separated; iii)~whether it has a spin gap; iv)~whether it has dominant superconducting fluctuations. With eight phases, the resulting phase diagram is unexpectedly rich. The four phases not found in the weak coupling limit are: i) an insulating phase when $U$ and $V$ are large enough; ii) a region of phase separation when $V$ is attractive; iii) another region of phase separation when $V$ is large enough and $U$ small; iv) a region with dominant superconducting fluctuations when $V$ is intermediate and $U$ small. The actual nature of this last phase, which has pairs but no spin gap, is not fully clear yet.

The effect of second neighbour repulsion on fcc binary alloy phase diagrams: A Monte Carlo study

We report the results of the Monte Carlo simulation of the phase diagram of fcc binary alloys using a 3-D Ising model with nearest and next-nearest neighbour repulsive interactions. Calculations are carried out for a ratio of second- to first-neighbour interaction energies of 0.4. The resulting phase diagram contains three superstructures A2B2, A2B and A5B, each separated by a disordered fcc phase. There was no evidence for the formation of an A3B phase. These results are in qualitative agreement with CVM results.

Point defects and sink strength in h.c.p. metals

Abstract We present in this paper a comprehensive study of the static and dynamic properties of point defects and of their interaction with dislocations and grain boundaries. First, by using short-range pair potentials fitted to physical parameters of Mg and Zr, we calculate defect configurations and associated energies. Two different sets of potentials are used: one weakly repulsive below first-neighbour interaction range and the second strongly repulsive. Both predict the basal octahedral interstitial to be the most stable, and either the axial dumbbell or the trigonal respectively as the second equilibrium configuration. Based on these results and those for the vacancy, we predict the corresponding anisotropic diffusion coefficients. We find that for all cases the interstitial moves faster along basal plane directions than along the c axis. For the vacancy, the ratio between diffusivities depends on the potential being used. Sink strengths are calculated for grain boundaries and dislocations. Those str...

The interaction between 119Sn and solute atoms dissolved in iron; the case of beryllium alloy

Be is a small atom (V=4.89 cm3 mol-1) compared with Fe and Sn (V=7.09 and 16.30 cm3 mol-1 respectively). As a result, size mismatch theories predict an attractive component of the potential between Be and Sn when dissolved in iron. The author presents spectra of 119Sn in a FeBe alloy, from which he deduces repulsive interactions at first- and second-neighbour distances. The first-neighbour interaction is in fair agreement with the semi-empirical theory of Miedema and Krolas. The repulsive interaction at the second-neighbour distance contradicts the expectation developed in earlier papers. He also makes some comments on the influence of solute-solute interactions on the deduction of probe-solute interaction potentials.

The triangular antiferromagnet with infinite nearest-neighbour coupling

The authors show how the exact interface free energy F for the Ising antiferromagnet on the triangular lattice with infinite first-neighbour and finite second-neighbour interactions can be determined analytically. From considerations of the angular dependence of the interface free energy they conclude that the transition is not second order. An upper bound for the transition temperature is given.

The effect of quantum fluctuations on the ferromagnet-helix transition

The authors study the effect of quantum zero-point motion on the zero-temperature phase diagram of a Heisenberg system with competing interactions on a hexagonal lattice. Within the basal plane, first-neighbour interactions J1 are ferromagnetic while second- and third-neighbour interactions J2 and J3 can have either sign. The inter-plane interaction J' is taken to be small. They find that quantum fluctuations enhance the tendency towards a first-order transition from ferromagnetic to helical order. Even a very small single-ion anisotropy suffices to explain the small first-order jump observed in NiBr2.

Lattice gas model on tetrahedral sites of a BCC lattice

A lattice gas model is suggested for describing the order-disorder transition of mobile particles in BCC cage lattices such as alpha -AgI, Ag2S, NbHx, etc. The effect of repulsive first- and second-neighbour interactions is studied by a six sub-lattice mean-field approximation. With a fixed number of mobile particles determined by the composition of AgI the mean-field approximation predicts that the disordering process strongly depends on the ratio of the second- and first-neighbour interactions. Beside the low-temperature ordered state and the high-temperature random distribution two further intermediate states are found to be stable. This fact results in three different kinds of disordering process. The results suggest a new interpretation of the alpha phase of AgI.

The triangular Ising antiferromagnet with first- and second-neighbour interactions

The transition temperature for the triangular-lattice Ising model with first- and second-neighbour interactions has been calculated using the interface method. All interactions are assumed to be antiferromagnetic with an arbitrary interaction strength ratio. The authors include the possibility that the second-neighbour interactions are anisotropic and present explicit results, shown in figures 3, 4 and 5, for three special model classes. These special cases correspond to next-nearest-neighbour interactions in one, two or three directions, the nonzero couplings having equal strengths. The results agree with the exact critical temperature in limiting cases where it is known. They call attention to an entanglement problem that one may also encounter in other applications of the interface method if the arrangement is not chosen with care. Going beyond the interface method they discuss the exact limiting behaviour when one of the two interactions is very weak. When the first-neighbour interaction approaches zero, in particular, they argue that the transition temperature should vanish exponentially for the one-direction model, should have an algebraic singularity of order 4/7 for the two-direction model and should vanish linearly for the three-direction model.

Phase diagram of the square-lattice Ising model with first- and second-neighbour interactions

The phase diagram of the square-lattice Ising model with first- and second-neighbour interactions in an external magnetic field has been calculated by the interface method. The first-neighbour interactions are assumed to be antiferromagnetic while the interaction ratio, R, assumes arbitrary values. Closed-form second-order phase boundaries are obtained for three ordered phases (here called antiferromagnetic, layered and degenerate phases). This is not in agreement with Monte Carlo (MC) computations which proposed two ordered phases (called antiferro- and superantiferromagnetic phase). A proof is given that the layered and degenerate phase are two separate phases. For R>or=0 all transitions are second-order order-disorder transitions. For R<0 the transition from the ordered (antiferromagnetic) to the disordered phase is first order at low temperatures, and a tricritical-point estimate, which is in good agreement with MC and renormalisation-group calculations, is found. An estimate for the critical chemical potential of the hard-squares lattice gas with first- and second-neighbour exclusion is found. This is about 70% higher than the currently accepted value.