The dissipative modulated structures in a distribution of interacting vacancies in the elastically anisotropic body-centred cubic crystals under an isothermal irradiation

ABSTRACT

Dissipative mechanism of formation of a modulated structure of vacancies in a body-centred cubic (b.c.c.) crystal is considered as its intrinsic property under an isothermal irradiation irrespectively of elastic-anisotropy factor sign. Conditions of self-organisation of a precursor of formation of a ‘superlattice’ of nanovoids, namely, the modulated structure in a spatial distribution of the diffusing vacancies generated by irradiation, because of the instability of their homogeneous distribution as a result of interaction between them in a b.c.c. host crystal under irradiation, are analysed. As shown for the first time, the ‘cohesive’ and ‘elastic’ couplings between interacting (and diffusing) vacancies have a crucial role in a pattern-selection process accordingly with different (positive or negative) elastic-anisotropy factors.     

Hamiltonian and Brownian systems with long-range interactions: V. Stochastic kinetic equations and theory of fluctuations

We developed a theory of fluctuations for Brownian systems with weak long-range interactions. For these systems, there exists a critical point separating a homogeneous phase from an inhomogeneous phase. Starting from the stochastic Smoluchowski equation governing the evolution of the fluctuating density field of Brownian particles, we determine the expression of the correlation function of the density fluctuations around a spatially homogeneous equilibrium distribution. In the stable regime, we find that the temporal correlation function of the Fourier components of density fluctuations decays exponentially rapidly, with the same rate as the one characterizing the damping of a perturbation governed by the deterministic mean field Smoluchowski equation (without noise). On the other hand, the amplitude of the spatial correlation function in Fourier space diverges at the critical point T=T_c (or at the instability threshold k=k_m) implying that the mean field approximation breaks down close to the critical point, and that the phase transition from the homogeneous phase to the inhomogeneous phase occurs sooner. By contrast, the correlations of the velocity fluctuations remain finite at the critical point (or at the instability threshold). We give explicit examples for the Brownian Mean Field (BMF) model and for Brownian particles interacting via the gravitational potential and via the attractive Yukawa potential. We also introduce a stochastic model of chemotaxis for bacterial populations generalizing the deterministic mean field Keller-Segel model by taking into account fluctuations and memory effects.     

The changes in electronic structure and optical properties of magnetic semiconductors at magnetic phase transitions

The influence of magnetic phase transitions on electronic structure and optical properties of magnetic semiconductors is discussed. Europium chalcogenides and chromium chalcogenide spinels are the main subjects of the investigation. It is shown, that many-body effects are responsible for the changes of optical properties and non-rigid band behavior of electronic structure. Magnetic phase transition leads to energy shift of wide bands and change in density of states of “magnetic” d(f)-electrons without any significant shift of their energies. The influence of fluctuations at T ～ T _c and antiferromagnetic semiconductors are also considered.     

On fluctuations in liquids and gases

The Lagrangian formalism is used to derive a system of nonlinear inhomogeneous dissipative differential equations describing the nonlinear dynamics of interrelated fluctuations of density, δρ, and temperature, δT, in a medium. With these equations, the unstable (with respect to initial conditions) phase trajectory describing parameter fluctuations in the ρ-T plane was obtained. By numerically solving the equations, we show that δρ and δT oscillate in time almost periodically, which is typical of fluctuations.     

Specific heat and thermal expansion of the Bi and Tl high temperature superconductors nearT c

We present measurements of the specific heat and of the thermal expansion of the most prominent phases of the Bi and Tl high temperature superconductors (Bi-2212, Bi-2223, Tl-2212, Tl-2223) in the neighbourhood ofT _c . In all these systems we observe small but sharp anomalies which have very little similarity with the mean field jump found in the conventional superconductors. The analysis of these anomalies shows clear evidence for the presence of strong fluctuations of the order parameter. For temperatures more than 5 K away fromT _c ,2D Gaussian fluctuations are found, while within ±5 K ofT _c the fluctuation contribution is best fitted by critical fluctuations. The shape of the thermal expansion anomalies is similar to that of the specific heat. Combination of both measurements predicts dT _c /dp+0.2 K/kbar for Bi- and Tl-2223 systems.     

Magnetic field dependence ofT c and temperature dependence ofH c2 in layered superconductors with open normal state Fermi surface

A theoretical framework for treating the effects of magnetic fieldH on the pairing theory of superconductivity is considered, where the field is taken in an arbitrary direction with respect to crystal axes. This is applicable to closed, as well as open normal state Fermi surface (FS), including simple layered metals. The orbital effects of the magnetic field are treated semiclassically while retaining the full anisotropic paramagnetic contribution. Explicit calculations are presented in the limits |H| → |H _c2(T)|,T ∼ 0 andT →T _c(|H|), |H| ∼ 0. Effects of weak nonmagnetic impurity scattering, without vertex corrections, have also been taken into account in a phenomenological way. The final results for the case of open FS and layered materials are found to differ considerably from those of the closed FS. For example, an important parameter,h(T=0)=|H_c2(0)|/[-Tδ|H _c2 T|δT]_T{s0} for the case of a FS open ink _z-direction with thek _z-bandwidth, 4t ₃, very small compared to the Fermi energy,E _F, is close to 0.5906, compared to 0.7273 for the closed FS, in the clean limit. Analytical results are given for the magnetic field dependence ofT _c and the temperature dependence of H _c2 for a model of layered superconductors with widely open FS. For a set of band structure parameters for YBa₂Cu₃O₇ used elsewhere, we find reasonable values for the upper critical fieldH _c2(0), the slope (dH _c2/dT)_{T c0}, anisotropic coherence lengths ζ_i(T=0),i=x, y, z, and (dT _c/d|H|)_{|H| → 0}.     

Dynamic phase transition in the kinetic Blume--Emery--Griffiths model: Phase diagrams in the temperature and interaction parameters planes

As a continuation of our previously published work, the dynamic phase transitions are studied further, within a mean-field approach, in the kinetic Blume--Emery--Griffiths model in the presence of a time varying (sinusoidal) magnetic field by using the Glauber-type stochastic dynamics. The dynamic phase transitions are obtained and the phase diagrams are constructed in two different planes, namely in the reduced temperature (T) and biquadratic interaction (k) plane and found eight fundamental types of phase diagrams for various values of reduced crystal-field interaction (d) and magnetic field amplitude (h), and in the (T,?d) plane and obtained six distinct topologies for different values of k and h. Phase diagrams exhibit one or two dynamic tricritical points and a dynamic double critical end point, dynamic triple and quadruple points, and besides disordered and ordered phases, three coexistence phase regions exist in which occurring of these strongly depend on the values of d, k and h.     

Influence of the ordering of carbon vacancies on the electronic properties of TiC0.625

Annealing of f.c.c. TiC_0.625 for 10 days between 780 and 750°C leads to a trigonal superstructure by ordering of carbon vacancies. Long-range ordering decreases slightly the residual electrical resistivity ϱ(1.6 K) and the electronic specific heat and increases the temperature coefficient of the resistivity dϱ/dT between 100 and 300 K. A minimum in ϱ(T), more apparent in the disordered samples (around 65 K) is attributed to the scattering of electrons by the strong atomic disorder.     

Scaling properties of hadron production in the Ising model for quark-hadron phase transition

Earlier study of quark-hadron phase transition in the Ginzberg-Landau theory is reexamined in the Ising model, so that spatial fluctuations during the transition can be taken into account. Although the dimension of the physical system is 2, as will be argued, bothd=2 andd=4 Ising systems are studied, the latter being theoretically closer to the Ginzberg-Landau theory. The normalized factorial momentsF _q are used to quantify multiplicity fluctuations, and the scaling exponentν is used to characterize the scaling properties. It is found by simulation on the Ising lattice thatν becomes a function of the temperatureT nearT _c . The average value ofν over a range ofT<T _c agrees with the value of 1.3 derived analytically from the Ginzberg-Landau theory. Thus the implications of the mean-field theory are not invalidated by either the introduction of spatial fluctuations or the restriction to a 2D system.     

Scaling behaviour in the Bose gas

The scaling behaviour of fluctuations of the Bose field (f) in the ergodic infinite volume equilibrium states of ad-dimensional Bose gas at temperatureT and density , can be classified in terms of the testfunctionsf. In the low density regime, the space of testfunctions splits up in two subspaces, leading to two different types of non-commuting macroscopic field fluctuation observables. Testfunctionsf with Fourier transform yield normal fluctuation observables. The local fluctuations of the field operators (f) must be scaled subnormally (i.e. with a negative scaling index) if the testfunctionf has . The macroscopic fluctuations of these fields can then again be described by a Bose field. The situation changes when the density of the gas exceeds the critical density. The field operators which have normal fluctuations in the low density regime need to be scaled abnormally in the high density regime, yielding classical macroscopic fluctuation observables. Another difference with the low density regime is that the space of testfunctions with splits up in two subspaces when the critical density is reached: for a first subspace the algebraic character of the macroscopic field fluctuation observables in also classical because it is necessary to scale the fluctuations of the field operators normally, while for the remaining subclass, the same negative scaling index is required as in the low density regime and hence also the algebraic character of these macroscopic fluctuations is again CCR.     

A re-examination of the induction (crystal field) energy contribution to the relative stabilities of the f.c.c. and h.c.p. rare-gas crystal structures

A new calculation of the induction (crystal field) energy of the f.c.c. and h.c.p. structures of the rare-gas crystals of neon and argon gives an energy in favour of f.c.c. which is more than sufficient to counter the summed pair-potential which favours h.c.p. Predicted values of (E _f.c.c. - E _h.c.p.)/E _f.c.c. for neon and argon are 0·9 × 10^-3 and 1·3 × 10^-3 respectively. These values are similar to those that have been obtained for a modified dispersion energy and it is concluded that they should be added to these dispersion energies. Calculations on solid helium show a negligible difference in the induction energy for the two close-packed structures.     

Role of vacancies of a strongly strained crystal in the melting process

It has been shown that, apart from classical vacancies formed as a result of the thermal fluctuations, a crystal can contain so-called nonclassical vacancies of nonfluctuation nature. The latter vacancies appear when the temperature exceeds a critical value T _C . The factor responsible for their formation is a mechanical instability of an ideal crystal. The temperature T _C is a second-order phase transition point. The vacancies formed as a result of this phase transition are joined together into small clusters with sizes of the order of several atoms. The above transition makes a substantial contribution to the premelting process observed experimentally.     

Approach to field-induced stationary state in a gas of hard rods

The Boltzmann equation describing one-dimensional motion of a charged hard rod in a neutral hard rod gas at temperatureT = 0 is solved. Under the action of a constant and uniform field the charged particle attains a stationary state. In the long time limit the velocity autocorrelation function decays via damped oscillations. In the reference system moving with the mean particle velocity the decay of fluctuations in the position space is governed (in the hydrodynamic limit) by the diffusion equation. Both the stationary current and the diffusion coefficient are proportional to the square root of the field. It is conjectured that this result also holds forT > 0 in a strong field limit.On leave from the Institute of Theoretical Physics, University of Warsaw, Hoza 69, 00-081 Warsaw, Poland.     

Dynamics and kinetic roughening of interfaces in two-dimensional forced wetting

We consider the dynamics and kinetic roughening of wetting fronts in the case of forced wetting driven by a constant mass flux into a 2D disordered medium. We employ a coarse-grained phase field model with local conservation of density, which has been developed earlier for spontaneous imbibition driven by capillary forces. The forced flow creates interfaces that propagate at a constant average velocity. We first derive a linearized equation of motion for the interface fluctuations using projection methods. From this we extract a time-independent crossover length ξ_×, which separates two regimes of dissipative behavior and governs the kinetic roughening of the interfaces by giving an upper cutoff for the extent of the fluctuations. By numerically integrating the phase field model, we find that the interfaces are superrough with a roughness exponent of χ= 1.35 ±0.05, a growth exponent of β= 0.50 ± 0.02, and ξ_×∼v^-1/2 as a function of the velocity. These results are in good agreement with recent experiments on Hele-Shaw cells. We also make a direct numerical comparison between the solutions of the full phase field model and the corresponding linearized interface equation. Good agreement is found in spatial correlations, while the temporal correlations in the two models are somewhat different.     

Equilibrium Fluctuations for Interacting Ornstein-Uhlenbeck Particles

 We study the hydrodynamic density fluctuations of an infinite system of interacting particles on ℝ ^d . The particles interact between them through a two body superstable potential, and with a surrounding fluid in equilibrium through a random viscous force of Ornstein-Uhlenbeck type. The stationary initial distribution is the Gibbs measure associated with the potential and with a given temperature and fugacity. We prove that the time-dependent density fluctuation field converges in law, under diffusive scaling of space and time, to the solution of a linear stochastic partial differential equation driven by white noise. Received: 10 July 2001 / Accepted: 9 September 2002 Published online: 8 January 2003 RID="*" ID="*" We thank J. Fritz for fruitful discussions, in particular about the existence of the infinite dynamics. A special thanks to L. Bertini for help in the proof of the spectral gap estimate (cf. Appendix B). Communicated by H. Spohn     

Free Molecular Flow with Boundary Effect

We study the dissipative effect of the boundary condition in the kinetic theory. We focus our study on the simplest situation of the free molecular flow with diffuse reflection boundary condition and constant boundary temperature, T _*. The geometry is also chosen to be the simplest ones, a bounded symmetric domain in ${\mathbb{R}^d}$ : an interval for d = 1, a disk for d = 2, and a ball for d = 3. It is shown that the solution converges to the global Maxwellian with the given boundary temperature T _*. We obtain the optimal convergence rates of (t + 1)^?d . The stochastic formulation of Shih-Hsien Yu is refined and generalized for our analysis.     

Interaction of a Hopf bifurcation and a symmetry-breaking bifurcation: Stochastic potential and spatial correlations

The multivariate master equation for a general reaction-diffusion system is solved perturbatively in the stationary state, in a range of parameters in which a symmetry-breaking bifurcation and a Hopf bifurcation occur simultaneously. Thestochastic potential U is, in general, not analytic. However, in the vicinity of the bifurcation point and under precise conditions on the kinetic constants, it is possible to define a fourth-order expansion ofU around the bifurcating fixed point. Under these conditions, the domains of existence of different attractors, including spatiotemporal structures as well as the spatial correlations of the fluctuations around these attractors, are determined analytically. The role of fluctuations in the existence and stability of the various patterns is pointed out.     

Properties of ordered,continuously degenerate systems

Ordered continuously degenerate systems are shown to have some common properties associated with strong transverse fluctuations of the order parameter. Longitudinal fluctuations are related to the transverse ones in three-dimensional systems. The longitudinal susceptibility is infinitely large at zero external field and the longitudinal scattering is rather strong at low wavevectors.

Properties of two-dimensional degenerate systems depend on the dimensionality n of the order parameter. There is a phase transition at finite temperature T ₀ for n=2. At temperatures below T ₀ a ‘super-fluid density’ or ‘transverse rigidity’ arises. Probably no phase transition takes place for n≥3 since the effective temperature increases with the scale of fluctuations.

The role of singularities in phase transitions and a general topological classification of singularities is considered. Applications of the theory to magnets, liquid crystals, superfluids, superconductors and plasmas are demonstrated.     

Field theory of critical behavior in a driven diffusive system

We use a field theoretic renormalization group method to study the critical properties of a diffusive system with a single conserved density subject to a constant uniform external field. A fixed point stable belowd _c=5 is found to govern the critical behavior. Scaling forms of density correlation functions are derived and critical exponents are obtained to all orders in =5–d. Spatial correlations are found to be very anisotropic with elongated correlations along the external field. Long wavelength transverse fluctuations are suppressed completely to yield mean field transverse exponents.