Advances in Condensed Matter Physics

Volume 2018, Article ID 3175068, 13 pages

https://doi.org/10.1155/2018/3175068

## Acoustic Signatures of the Phases and Phase Transitions in the Blume Capel Model with Random Crystal Field

Department of Physics, Dokuz Eylül University, 35160 Izmir, Turkey

Correspondence should be addressed to Gul Gulpinar; rt.ude.ued@raniplug.lug

Received 5 April 2018; Accepted 8 May 2018; Published 11 June 2018

Academic Editor: Oleg Derzhko

Copyright © 2018 Gul Gulpinar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Sound propagation in the Blume Capel model with quenched diluted single-ion anisotropy is investigated. The sound dispersion relation and an expression for the ultrasonic attenuation are derived with the aid of the method of thermodynamics of irreversible processes. A frequency-dependent dispersion minimum that is shifted to lower temperatures with rising frequency is observed in the ordered region. The thermal and sound frequency () dependencies of the sound attenuation and effect of the Onsager rate coefficient are studied in low- and high-frequency regimes. The results showed that and are the conditions that describe low- and high-frequency regimes, where is the single relaxation time diverging in the vicinity of the critical temperature. In addition, assuming a linear coupling of sound wave with the order parameter fluctuations in the system and as the temperature distance from the critical point, we found that the sound attenuation follows the power laws and in the low- and high-frequency regions, while . Finally, a comparison of the findings of this study with previous theoretical and experimental studies is presented and it is shown that a good agreement is found with our results.

#### 1. Introduction

The attenuation studies of acoustic waves in the vicinity of a magnetic ordering transition points provide insight into the critical dynamics of spin systems. Resonant ultrasound spectroscopy study of has shown the existence of a peak in sound attenuation as the Néel point is approached [1]. Magnetic phase diagram of multiferroic has been obtained by sound velocity and attenuation measurements [2]. Investigation of the critical dynamics of sound propagation near continuous phase transition points not only provides valuable information about phase change mechanisms but also enables the determination of the critical indices that characterize these transitions [3]. Investigation of nonequilibrium processes probed by ultrasound waves in the spin-ice materials such as Yb_{2}Ti_{2}O_{7} and Dy_{2}Ti_{2}O_{7} [4, 5] and the frequency-dependent anisotropy of sound velocity and attenuation of the acoustic wave propagation through the nematic liquid crystals [6–8] represent examples to studies that are related to ultrasonic propagation in systems that undergo phase transitions. Further, considerable attention has been focused on the investigation of sound attenuation in disordered conductors [9]: the behavior of sound propagation near the point of the confined liquid ^{4}He [10–13] and absorption of ultrasound near the critical mixing point of a binary liquid [14–16]. Finally, one should note that a great variety of magnetic materials including halides, metals, oxides, intermetallics, and sulphides exhibit anomalies in their elastic properties due to the fact that order-disorder transitions are typically accompanied by small lattice distortions [17].

The Blume Capel (BC) model formulated independently by Blume [18] and Capel [19] plays a fundamental role in the multicritical phenomena associated with various physical systems, such as liquid crystals [20, 21], metallic alloys [22], proteins [23], and polymeric systems [24, 25]. On the other hand, the introduction of randomness changes the critical behaviors of a spin model considerably; that is, random fields can altogether eliminate the phase transitions in low dimensions and affect the numerical values of the critical exponents in higher dimensions [26–28]. Since then investigation of the effect of crystal field disorder on the equilibrium phase diagram of spin-1 Ising models has been a research interest for many authors. The BC model with random single-ion anisotropy provides a microscopic model for phase transitions of ^{3}He-^{4}He mixtures in silica aerogel [29, 30] and relevant to the study of the phase separation in porous media in the vicinity of the superfluid transition [31, 32]. The interplay between quenched disorder provided by a random field and network connectivity in the BC model is investigated by using the replica method [33]. Moreover, the BC model with infinite-range ferromagnetic interactions and under the influence of a quenched disorder has been investigated and a classification of the phase diagrams in terms of their topology is presented in [34].

In this study, we investigate the critical dynamics of sound wave propagation in the BC model with bimodal crystal field by combining the statistical equilibrium theory and the thermodynamics of linear irreversible processes. This approach has been utilized to investigate the sound propagation in a great variety of model systems. Making use of the lowest approximation of the cluster variation method and linear response theory of irreversible processes, Erdem and Keskin performed the calculations of the sound attenuation near the critical point in the Blume-Emery-Griffiths (BEG) model with zero crystal field [35–37]. Gulpinar investigated the critical behavior of ultrasound wave absorption coefficient in the metamagnetic Ising model within the mean-field approximation [38]. Later, the absorption of sound in the spin- Ising model on the Bethe lattice is obtained and its temperature variance is analyzed near the phase transition points [39]. Recently, Cengiz and Albayrak studied the sound attenuation phenomena for a finite crystal field BEG model on the Bethe lattice in terms of the recursion relations by using the Onsager theory [40]. Due to mathematical complexity, the properties of critical sound propagation have not been studied in any spin system with random bond, random magnetic field, or random crystal field terms in the Hamiltonian expression. To the best of our knowledge, the critical dynamics of the sound propagation of the BC model with quenched diluted single-ion anisotropy have not been studied by the methods of irreversible thermodynamics. It is assumed in this manuscript that the sound wave is coupled to the order parameter fluctuations that decay mainly via order parameter relaxation process and the steady-state dynamics of the BC model with random diluted crystal field are formulated, while the system is under the effect of a propagating sound wave of frequency , which let us obtain the sound dispersion relation and sound attenuation coefficient for all temperatures and frequencies that contain effectively only one phenomenological rate coefficient. Temperature variance of sound dispersion and absorption has been investigated in the vicinity of the critical point. Finally, the frequency behavior of the attenuation coefficient for temperatures is presented in the vicinity of second-order transition from ordered to disordered phase.

The paper is organized as follows: the model and its static properties are presented briefly in Section 2. Next, the free energy production near the equilibrium is stated and the order parameter relaxation time is obtained in Section 3. The steady solution of the kinetic equation of the order parameter and expressions for the sound dispersion relation and ultrasonic attenuation coefficient are obtained in Section 4. Finally, frequency and temperature behaviors of the ultrasonic attenuation are analyzed and the discussion of the results is given in Section 5.

#### 2. The Model and Its Equilibrium Properties

The BC model with random single-ion anisotropy in the presence of an external magnetic field is described by the Hamiltonianwith . Here is the exchange interaction due to ferromagnetic coupling and is the crystal field acting on site* i* with a distribution function:where is concentration of the spins on the lattice which are influenced by a crystal field. The mean-field free energy for a nonvanishing external field is given by the following expression [41, 42]:where is the lattice-free energy that is independent of spin configuration. Here , , and are the lattice constant, volume, and order parameter of the system, respectively. In addition, denotes thermal expectation value. If one makes use of (2), the mean-field free energy becomesFor the sake of simplicity, we will take Boltzmann constant as unity () from now on. The equilibrium conditions and give the following self-consistent equations for the magnetization and the lattice constant:The numerical solutions of the equation state given by (5) for vanishing external field have been performed and it has been reported that the BC model with quenched diluted single-ion anisotropy exhibits three distinct phase diagram topologies in the () plane depending on [43, 44]. For , the form of the phase diagram is identical to that of the pure BC model with homogenous crystal field, where the ferromagnetic phase is separated from the paramagnetic phase by a phase boundary that is of second order up to a tricritical point (TCP) at which the transition becomes first order. For , TCP still exists but and first-order lines have reentrant parts. The phase diagram of the system changes dramatically in nature if one increases the dilutence further: For , there exists discontinuous phase transitions between the ferromagnetic and paramagnetic phases at strong crystal fields and low temperatures. The transition becomes continuous at higher temperatures. Further, the -line displays reentrance. A portion of the second-order transition line is masked by a first-order transition line. This situation causes two distinct multicritical points to appear: critical endpoint (CEP) and double critical endpoint (DCP). Under a threshold value of the crystal field concentration (), the quenched disorder completely eliminates the first-order phase transitions, and the whole phase boundary is of second order.

#### 3. Relaxation Dynamics of the BC Model with Quenched Diluted Single-Ion Anisotropy

One may assume the existence of a small uniform external field for a short while in order to be able to formulate the relaxation dynamics of the BC model with random single-ion anisotropy. In addition, the amplitude of the external field should be sufficiently small to allow the spin system to be in the neighborhood of equilibrium, where linear response theory can be utilized. In other words, we investigate the final stage of the approach to equilibrium. In the case of the existence of a small deviation of the magnetic field from its equilibrium value , the system will be removed slightly from equilibrium and a finite free energy production will arise:where corresponds to an increase in the corresponding thermodynamic potential related to the deviation of from their equilibrium values and is the equilibrium value of the free energy, while , , and . In the neighborhood of equilibrium, the free energy production may be written as a Taylor-series expansion, in which deviations in the thermodynamic quantities are retained to second order:Here, the coefficients to are the so-called free energy production coefficients and they are calculated by the following second-order derivatives:where the subscript “” denotes the thermal equilibrium; thus the derivatives are evaluated for , , and . The explicit expressions for the above-mentioned free energy production coefficients are given in the Appendix.

If the system is shifted from its equilibrium by the deviation of the external field from its equilibrium value () and/or by the volume change of the crystal which is proportional to , the generalized force arises, which can be regarded as the force that brings magnetization back to its equilibrium value. The generalized force conjugate to generalized current may be obtained by differentiating the free energy production with respect to :

In the realm of theory of irreversible thermodynamics, the time derivative of magnetization () is treated as the generalized current conjugate to :If the deviation from the equilibrium condition is small, one can write a linear relation between the current and the force: . Thus, the dynamics of the order parameter for the BC model with quenched diluted crystal field are ruled by the following rate equation:where is the order parameter Onsager coefficient. We should note that, in this study, the most simple temperature dependence is assumed for , which must be found either in principle by a more powerful theory such as path probability method [45, 46] or in practice by fit with the experimental findings.

For the case of vanishing external stimulation (, ), one obtains the relation that describes the rate of change in the order parameter relaxing to its equilibrium state as follows:The solution of the kinetic equation given by (13) is of the form (), where is the relaxation time of the order parameter. Thus, the relaxation time corresponds to

The temperature behavior of the order parameter relaxation time near the phase transition points of the BC model with bimodal crystal field has been investigated in detail in [44] and a rapid increase in the single relaxation time is observed when the temperature approaches the critical and multicritical phase transition temperatures. In addition, presents a scaling relation , which corresponds to well-known phenomena of critical slowing down. Finally, as a signature of the first-order phase transition, a jump-discontinuity has been observed in the relaxation time near a first-order phase transition.

#### 4. Derivation of the Sound Attenuation Coefficient

In this section, we will study the transport properties for the BC model with bimodal crystal field near its magnetic phase transition points. With this aim, we will consider the case in which the lattice is stimulated by the sound wave of frequency . Due to nature of the linear response theory, if you perturb the system at a frequency , the response will take place at same frequency. Thus, one can find the steady-state solution of (12) with an oscillating external force as follows:Introduce this expression into (12).

And assuming that , one obtains the following nonhomogenous equation for :Solving (16) for gives

In addition, if one makes use of (14) for the relaxation time of the BC model with a random crystal field, (17) becomes

The response in the pressure is obtained by differentiating the minimum work with respect to :Then, using (8), one obtainsOn the other hand, the derivative of pressure with respect to volume giveswhere and are the free energy production coefficients that are given in (9). If one introduces (18) and the density into (21),Finally, using the definition and the fact that the order parameter relaxation time tends to very large values in the vicinity of the continuous phase transition points, one obtains the complex effective elastic constant (complex velocity of sound) of the BC model with quenched diluted crystal field:

It is a well-known fact that the lag between the oscillations of the pressure and the excitation of a given mode leads to the dissipation of energy and dispersion of the sound wave.Since we assume a linear coupling of sound wave with the order parameter fluctuations in the system, the dispersion which is the relative sound velocity change with frequency depends not on the sound wave amplitude but on frequency . Finally, if one makes use of the following definition for the sound attenuation constant,

A finite attenuation above the critical temperature cannot be obtained due to the fact that the attenuation constant is proportional to free energy production constant that vanishes above the critical temperature, where the order parameter is equal to zero (see (A.1)). This is entirely due to the mean-field approximation and one may expect a finite attenuation constant above the critical temperature for higher-order approximations (i.e., Bethe approximation).

Now, if one rewrites the free-energy production coefficient given in (9), the following expressions are obtained for and :

In addition, if one uses the definition , (24) and (25) becomewhere is the velocity of sound for very high frequencies at which order parameter can no longer follow the sound wave motion.

#### 5. Results And Discussion

The calculated data that provides information about the thermal variation of the sound velocity and attenuation coefficient in the ordered and disordered phases and the frequency dependence of isothermal attenuation coefficient in the ordered phase of the BC model with quenched diluted crystal field will be discussed in the following. In addition, the behavior of sound absorption in the neighborhood of critical temperature is analyzed according to various values of phenomenological rate coefficient. Finally, by making use of double logarithmic plots of the sound attenuation coefficient versus the distance from the critical temperature, the dynamical critical exponents of the sound absorption are obtained for low- and high-frequency regimes. Before starting to discuss the above-mentioned results, it is convenient to introduce the following reduced quantities:here, , , and are the reduced values of the temperature, crystal field, and magnetic field, respectively. One should stress that we focus on zero field case () and is taken as unity for the sake of simplicity throughout this section.

Figure 1 displays the temperature variation of the frequency-dependent sound velocity (dispersion) of the BC model with quenched diluted crystal field for several values of sound wave frequency, while , , and . The number accompanying each curve denotes the value of and one observes a characteristic sound velocity minimum that shifts to the lower temperatures with increasing frequency. The minima become deeper with decreasing and ultrasonic dispersion reaches the finite value of at and remains temperature-independent after then. Similar temperature dependence of the sound dispersion has been reported near the order-disorder phase transition point of the BEG model [47]. Comparably, the existence of the minima of the sound velocity in the ordered phase and its shift to lower temperatures with increasing frequency have been observed in the classical liquid crystals [8], liquid He confined in a microfluidic cavity [10], and the magnetic compound [48]. In addition, it can be seen from Figure 1 that the change in sound velocity becomes frequency-independent as one approaches to critical value of the reduced temperature and this behavior is in parallel with the findings of the molecular field theory [17, 49], dynamical renormalization group theory for the propagation of sound near the continuous structural phase transitions [50], and Brillouin scattering studies of -crystals [49].