Displacement-displacement Correlation Function Research Articles

Ultrafast entropy production in pump-probe experiments

The ultrafast control of materials has opened the possibility to investigate non-equilibrium states of matter with striking properties, such as transient superconductivity and ferroelectricity, ultrafast magnetization and demagnetization, as well as Floquet engineering. The characterization of the ultrafast thermodynamic properties within the material is key for their control and design. Here, we develop the ultrafast stochastic thermodynamics for laser-excited phonons. We calculate the entropy production and heat absorbed from experimental data for single phonon modes of driven materials from time-resolved X-ray scattering experiments where the crystal is excited by a laser pulse. The spectral entropy production is calculated for SrTiO3 and KTaO3 for different temperatures and reveals a striking relation with the power spectrum of the displacement-displacement correlation function by inducing a broad peak beside the eigenmode-resonance.

Theory of anharmonic phonons in two-dimensional crystals

Anharmonic effects in an atomic monolayer thin crystal with honeycomb lattice structure are investigated by analytical and numerical lattice dynamical methods. Starting from a semi-empirical model for anharmonic couplings of third and fourth order, we study the in-plane and out-of-plane (flexural) mode components of the generalized wave vector dependent Gr\"uneisen parameters, the thermal tension and the thermal expansion coefficients as function of temperature and crystal size. From the resonances of the displacement-displacement correlation functions we obtain the renormalization and decay rate of in-plane and flexural phonons as function of temperature, wave vector and crystal size in the classical and in the quantum regime. Quantitative results are presented for graphene. There we find that the transition temperature from negative to positive thermal expansion is lowered with smaller system size and by renormalization of the flexural mode frequency. Extensive numerical analysis throughout the Brillouin zone explores various decay and scattering channels. The relative importance of Normal and Umklapp processes is investigated. The work is complementary to crystalline membrane theory and computational studies of anharmonic effects in two-dimensional crystals.

X-ray absorption Debye-Waller factors fromab initiomolecular dynamics

An ab initio equation of motion method is introduced to calculate the temperature-dependent mean-square vibrational amplitudes ${\ensuremath{\sigma}}^{2}$ which appear in the Debye-Waller factors in x-ray absorption, x-ray scattering, and related spectra. The approach avoids explicit calculations of phonon modes, and is based instead on calculations of the displacement-displacement time correlation function from ab initio density functional theory molecular dynamics simulations. The method also yields the vibrational density of states and thermal quantities such as the lattice free energy. Illustrations of the method are presented for a number of systems and compared with other methods and experiment.

Smectic layer displacement fluctuations in solid substrate supported smectic-A films.

In this paper we present results of calculations of static and dynamic characteristics of smectic layer displacement fluctuations in solid substrate supported smectic-A films with due regard for asymmetric profiles of the bending elastic constant K and the smectic layer compressibility B. We also take into account difference in properties of boundary surfaces of the film, namely, the surface tension of the free surface is taken to be finite whereas that of a film-substrate interface is assumed to be infinitely large. Profiles of the smectic layer displacement fluctuations and correlations between them are calculated for films formed of liquid crystalline compounds with the bulk smectic-A to nematic phase transition. The calculations are performed at temperatures much lower than that of the bulk phase transition and at maximum temperatures of existence of films of given thickness. The time dependent displacement-displacement correlation functions for thermal smectic layer displacement fluctuations and time dependent intensity-intensity correlation functions for diffuse x-ray scattering from the films are also calculated. It is shown that unlike free-standing smectic-A films, an effect of the temperature on dynamics of smectic layer displacement fluctuations in solid substrate supported smectic-A films can be observed in experiments on coherent x-ray dynamic scattering from not very thick films (N approximately 20) and at significantly smaller wave vector transfer component in the film plane.

Quantum stabilization in anharmonic crystals.

For a model of interacting quantum particles of mass m oscillating in a double-well crystalline field, a mechanism of its stabilization by quantum effects is described. In particular, a stability condition involving m, the interaction intensity, and the parameters of the crystalline field is given. It is independent of the temperature and is satisfied if m is small enough and/or the tunneling frequency is big enough. It is shown that under this condition the infinite-volume free energy density is an analytic function of the external field and the displacement-displacement correlation function decays exponentially; hence, no phase transitions can arise at all temperatures. This gives a complete and rigorous answer to the question about the influence of quantum effects on structural phase transitions, the discussion of which was initiated in [T. Schneider, H. Beck, and E. Stoll, Phys. Rev. B 13, 1123 (1976)]].

Structure and fluctuations of smectic membranes

Smectic membranes are perfect model systems for studying low-dimensional phase transitions and the associated fluctuations. During the last two decades we have seen important progress in the understanding of the structure and fluctuation behavior of these systems, driven by both new experimental techniques and theoretical developments. Phase transitions are reviewed involving liquid, hexatic, and crystalline layers, which provide several types of model system for low-dimensional melting. The authors discuss the influence of the surfaces on the physical properties of the membranes as well as the crossover from threeto two-dimensional behavior. The layer-displacement fluctuations in smectic membranes have been investigated by specular and diffuse x-ray reflectivity. Theoretical and experimental aspects of the displacement-displacement correlation function are discussed. Of special interest is the quenching or enhancement of fluctuations at surfaces, which is directly related to the phenomenon of surface ordering. The authors consider the conditions under which fluctuations are conformal throughout a membrane, and then the dynamic aspects of the layer-displacement correlation function, which include the effects of finite size, surface tension, and viscous dissipation. This leads in smectic membranes to a discrete spectrum of elastic and viscous relaxation modes, which have been studied experimentally with coherent x rays at third-generation synchrotron sources. The fluctuating character of crystalline-B membranes is also considered. Finally, the article looks briefly at thinning transitions, smectic membranes of chiral molecules, smectic films on substrates, and applications to biologically relevant systems. Open questions and future trends in the field are discussed.

Dynamics of free-standing smectic-A films.

The eigenvibrations and time-dependent layer displacement-layer displacement correlation functions are analyzed in a free-standing thin smectic-A films with the help of a discrete layer model. The film motions are described using the Chebyshev polynomials of second kind, U(n)(x). The eigenfrequencies problem is essentially simplified within the framework of this approach since the numerical solution of the high degree algebraic characteristic equation is replaced by the analytical solution of a rather simple trigonometrical equation. The dependences of eigenmodes on wave number q(perpendicular) were analyzed. For small q(perpendicular) one mode is a low frequency acoustic wave and other modes are high frequency optical oscillations. As the wave number q(perpendicular) increases all modes successively turn into relaxation when starting with the acoustic mode. The rather simple expression for susceptibility matrix and for spectral densities of layer displacement correlation functions were obtained using the Chebyshev polynomials. It was shown that the frequency dependences of spectral densities are sensitive to wave number q(perpendicular). For small q(perpendicular) the spectral densities of displacement-displacement correlation functions have a sharp peak and for large q(perpendicular) they turn into a contours of Lorentzian type.

Multiple-scattering x-ray-absorption fine-structure Debye-Waller factor calculations

An efficient local equation-of-motion method is introduced for calculations of the mean-square half-path-length fluctuations ${\ensuremath{\sigma}}_{j}^{2}$ in multiple-scattering x-ray-absorption fine-structure Debye-Waller factors in aperiodic systems. Given a few local force constants, the method yields ${\ensuremath{\sigma}}_{j}^{2}$ via projected densities of modes or via the displacement-displacement correlation function in real time, over a few vibration cycles. The calculation scales linearly with the system size and does not rely on any symmetry considerations. Sample applications are presented for crystalline Cu and Ge, and zinc tetraimidazole.

Anharmonic correlated Einstein-model Debye-Waller factors

An anharmonic correlated Einstein model is derived for local vibrational amplitudes in x-ray-absorption fine structure ~XAFS! that takes into account all near neighbors of absorber and backscattering atoms. The model is based on quantum thermodynamic perturbation theory and includes anharmonic effects based on empirical potentials. Calculations are presented for the second and third cumulants in XAFS as well as the net thermal expansion and thermal expansion coefficient. This model avoids full lattice dynamical calculations yet provides reasonable agreement with experiment. The generalization to displacement-displacement correlation functions and multiple-scattering Debye-Waller factors is also discussed. @S0163-1829~97!03125-1#

Ab Initio XAFS Debye-Waller Factors

An efficient, finite temperature, equation of motion method is introduced for calculations of the mean-square relative displacements σ 2 in XAFS Debye-Waller factors. Instead of calculating projected densities of modes, the calculations are based on the displacement-displacement correlation function and done in real time, over a few vibration cycles. Results for the Debye-Waller factors are found to be accurate to within about 15% percent at all temperatures. We also discuss how force constants used in the calculations can be obtained from ab initio Car-Parrinello/molecular dynamics calculations. Tests of the method are presented based on the Wiggs-Jonsson plane-wave pseudopotential code [Comp. Phys. Comm. 87, 319 (1995)]. Application to XAFS Debye-Waller factors in biological molecules is discussed.

Evaluation of the displacement-displacement correlation function of a one-dimensional crystal: A comparison of the results of perturbation theory with the Mori-Dupuis method.

We calculate the spectral shape of the displacement-displacement correlation function of a one-dimensional chain of atoms interacting through a nearest-neighbor potential. We make use of the perturbation treatment developed by Maradudin and Fein. Our calculations are parallel to the corresponding work done using the frequency moments method. As expected the two methods agree best at low temperatures.

Fluctuations and light scattering in thin smectic films.

An analysis of the layer-displacement fluctuations in thin smectic-A liquid-crystal films is presented. Explicit formulas for the displacement-displacement correlation function, taking into account the surface tension and the anchoring strength, are derived. It is shown that the boundary conditions essentially affect the fluctuations not only on the surface but in the interior of the film as well. The light scattering on the displacement-associated director fluctuations is considered. This consideration is carried out on the basis of the Green's function of the electromagnetic field for an optically anisotropic film. The calculations include multiple-reflection effects for incident and scattered light beams. It is shown that the scattered light intensity differs essentially from the intensity for the same volume in an infinite smectic medium. There is no forward scattering in the case of normal incidence and an undulation behavior of the intensity as a function of scattering angle takes place. There are maxima in the angular distribution. The magnitudes of the maxima are sensitive to the surface-tension coefficient. The possibility of measuring the surface-tension coefficient by means of an optical experiment is discussed.

Dynamic correlations of the classical and quantum Toda lattices.

The dynamic correlations of classical and quantum Toda lattices are approached by moment expansion. For the classical model, the moments of the spectral shape of the displacement-displacement correlation function are exactly calculated up to the eighth one, while, for the quantum system, their evaluation is limited to the sixth one, using the effective-potential method in low-coupling approximation. The spectral shape is calculated using the continued-fraction expansion. The relevance of quantum effects is clearly shown, in dependence on temperature and quantum coupling. At all wave vectors, the spectral shape presents a single-peak structure, both in the classical and in the quantum regime.

Long-time correlations in a model of structural phase transitions with infinite range interaction

The long-time value of the displacement-displacement correlation function is calculated within theφ4-lattice model used to simulate structural phase transitions. Its temperature dependence and dispersion is discussed for infinite range interaction. Special attent is devoted to the natural explanation of a central peak in the scattering function using nonvanishing long-time correlations.

Landau-Peierls instability, x-ray-diffraction patterns, and surface freezing in thin smectic films

A model of x-ray diffraction for thin smectic-A liquid-crystal films is presented. The effect of the smectic-layer displacement fluctuations and correlations and the molecular form factor on the interlayer structure and the x-ray-diffraction pattern is discussed. In thin films the influence of the displacement-displacement correlation function on the x-ray-diffraction pattern is very small and can be neglected in the analysis of experimental data. On the other hand, both the displacement-fluctuation term (Debye-Waller factor) and molecular form factor produce strong measurable effects and so can be determined. We discuss the dependence of displacement fluctuations, calculated in the framework of the Landau--de Gennes model, on the smectic elastic constants and the smectic-vapor surface tension and show that these constants can be determined from the x-ray-diffraction pattern. The analysis of the hydrodynamic (collective) layer fluctuations and the individual molecular-motion fluctuations shows that the latter can be neglected in comparison to the former. The fluctuation amplitudes predicted by the model agree within 5% with the recent experimental measurements performed on smectic-I on -C films.In thin smectic films the fluctuation amplitudes \ensuremath{\sigma}(r)=[〈${\mathit{u}}^{2}$(r)〉${]}^{1/2}$ are only \ensuremath{\sim}4 \AA{} compared with \ensuremath{\sim}8 \AA{} in a macroscopic sample. The fluctuations are suppressed at the two free surfaces by the large surface tension, grow away from each surface, and have a parabolic profile near the center of the film. We argue that one of the reasons for surface freeezing in smectic liquid crystals is the quench of the layer fluctuations by the large surface tension. However, we also show that in systems with small surface tension the fluctuations at the surface are in fact larger than the ones in the interior of the system. The growth of the diffuse scattering, due to the displacement-displacement correlations, with the thickness of the smectic film is discussed and shown to evolve towards the structure predicted for large samples by Gunther, Imry, and Lajzerowicz [Phys. Rev. A 20, 1733 (1980)]. The model for the displacement layer fluctuations including the director as an independent variable is presented. Furthermore, the coupling between the layer fluctuations and the nematic order parameter in smectic liquid crystals is qualitatively discussed. It is argued that the compressional modes induce the nematic order-parameter fluctuations and that a large fluctuation profile may induce the smectic-A--smectic-C phase transition in thin films. Eventually in tilted smectic liquid crystals the layer fluctuation profile may induce a tilt profile. Finally, it is shown that the presented model can be applied to smectic systems other than smectic-A; we give explicit formulas for the x-ray-scattering intensity from the smectic-${\mathit{A}}_{\mathit{d}}$ films and also calculate the fluctuations amplitudes for the stratified smectic-I on -C system.

Mode coupling approximation in a model of structural phase transitions with infinite range interaction

The long-time value of the displacement-displacement correlation function and the mean square displacements are calculated within the Φ4-lattice model with infinite range interaction by use of a mode coupling approximation. Their temperature dependence is compared with exact results. While the mean square displacements (zero decoupling times) agree with exact results except the low temperature range for weak coupling, the long time correlations from MCA (infinite decoupling times) are wrong. Even a separation of the displacement function into two parts representing the two different time scales is of no use since the distribution function doesn't separate.

“Frozen” phonon lmto-calculations for ferroelectric perovskites

Abstract “Frozen” phonon calculations based on Density-Functional-Theory were performed for BaTiO3 and SrTiO3 with a full potential version of the linear-muffin-tin-orbital method. It was demonstrated that the ferroelectric instability in perovskites is the result of a delicate balance between forces produced by bonding and nonbonding states. Calculations of the electric field gradient at finite temperatures and its comparison to recent NMR-measurements in the tetragonal phase of BaTiO3 allow conclusions about the displacement-displacement correlation function.

Thermal amplitudes of surface atoms on Si(111) 2 x 1 and Si(001) 2 x 1.

Atomic displacement-displacement correlation functions for the surface atoms of the Si(111) 2\ifmmode\times\else\texttimes\fi{}1 and Si(001) 2\ifmmode\times\else\texttimes\fi{}1 surfaces are calculated using phonon energies and eigenvectors obtained from a tight-binding theory of the lattice dynamics of Si. The temperature dependence and the possible anisotropies of the amplitudes of vibration of the surface atoms are investigated. For the \ensuremath{\pi}-bonded chain model of Si(111) 2\ifmmode\times\else\texttimes\fi{}1 it is found that the surface atoms have larger vibrational amplitudes perpendicular to the surface than parallel to it, and this anisotropy (1.8 in the mean-squared amplitudes at 270 K) increases rapidly with increasing temperature. This is attributed to the existence of a small-wavelength acoustic phonon with an unusually low energy. For the surface atoms on Si(001) 2\ifmmode\times\else\texttimes\fi{}1 the anisotropy between perpendicular and parallel vibrations is not as marked. The interatomic vibrational correlations between surface atoms reveal the strength of the new bonds that are formed in the reconstruction of these surfaces.

Hydrodynamic Fluctuations in Heavy Fermion Systems

A theory of hydrodynamic fluctuations is presented. We find that the displacement-displacement correlation function for the lattice contains an overdamped hydrodynamic mode which is due to electro density fluctuations. Since the Landau-Placzek ratio is 5–6 orders of magnitude larger than in ordinary metals the electron-diffusion mode should be experimentally observable. In fact it seems that they have been observed already. We also discuss the density mode at small wave vectors outside the hydrodynamic regime. The electron-diffusion mode converts into a propagating mode which has features of an acoustic plasmon or zero sound. In addition we find high energy plasmons and a low energy optical plasmon which is expected when more than one heavy quasiparticle band intersects the Fermi level.

Collective excitations in metallic glasses: A model of dynamic structure.

Glasses lack the diffusive nature of dense liquids and the long-range order of crystals, two essential features which, by an elementary argument on the form of the density of collective modes, may be incorporated into a simple model for the displacement-displacement correlation function of a glassy metal. This, in turn, is used to evaluate the dynamic structure factor, and from an application of the theory to a typical system, a dispersion relation for collective excitations is obtained which is found to exhibit a rotonlike minimum.