acoustic signatures of the phases and phase transitions in the … · 2019. 7. 30. · reseaarticle...

Research ArticleAcoustic Signatures of the Phases and Phase Transitions in theBlume Capel Model with Random Crystal Field

Gul Gulpinar

Department of Physics Dokuz Eylul University 35160 Izmir Turkey

Correspondence should be addressed to Gul Gulpinar gulgulpinardeuedutr

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

Academic Editor Oleg Derzhko

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

Sound propagation in the Blume Capel model with quenched diluted single-ion anisotropy is investigated The sound dispersionrelation and an expression for the ultrasonic attenuation are derived with the aid of the method of thermodynamics of irreversibleprocesses A frequency-dependent dispersion minimum that is shifted to lower temperatures with rising frequency is observed inthe ordered region The thermal and sound frequency (120596) dependencies of the sound attenuation and effect of the Onsager ratecoefficient are studied in low- and high-frequency regimes The results showed that 120596120591 ≪ 1 and 120596120591 ≫ 1 are the conditions thatdescribe low- and high-frequency regimes where 120591 is the single relaxation time diverging in the vicinity of the critical temperatureIn addition assuming a linear coupling of sound wave with the order parameter fluctuations in the system and 120576 as the temperaturedistance from the critical point we found that the sound attenuation follows the power laws 120572(120596 120576) sim 1205962120576minus1 and 120572(120596 120576) sim 12059601205761 inthe low- and high-frequency regions while 120576 rarr 0 Finally a comparison of the findings of this study with previous theoretical andexperimental 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 ofa magnetic ordering transition points provide insight intothe critical dynamics of spin systems Resonant ultrasoundspectroscopy study of CoF2 has shown the existence of apeak in sound attenuation as the Neel point is approached[1] Magnetic phase diagram of multiferroic MnWO4 hasbeen obtained by sound velocity and attenuation measure-ments [2] Investigation of the critical dynamics of soundpropagation near continuous phase transition points notonly provides valuable information about phase changemechanisms but also enables the determination of the criticalindices that characterize these transitions [3] Investigationof nonequilibrium processes probed by ultrasound wavesin the spin-ice materials such as Yb2Ti2O7 and Dy2Ti2O7

[4 5] and the frequency-dependent anisotropy of soundvelocity and attenuation of the acoustic wave propagationthrough the nematic liquid crystals [6ndash8] represent examplesto studies that are related to ultrasonic propagation insystems that undergo phase transitions Further considerable

attention has been focused on the investigation of soundattenuation in disordered conductors [9] the behavior ofsound propagation near the 120582 point of the confined liquid4He [10ndash13] and absorption of ultrasound near the criticalmixing point of a binary liquid [14ndash16] Finally one shouldnote that a great variety of magnetic materials includinghalides metals oxides intermetallics and sulphides exhibitanomalies in their elastic properties due to the fact that order-disorder transitions are typically accompanied by small latticedistortions [17]

The Blume Capel (BC) model formulated independentlyby Blume [18] and Capel [19] plays a fundamental role inthe multicritical phenomena associated with various physicalsystems such as liquid crystals [20 21] metallic alloys[22] proteins [23] and polymeric systems [24 25] Onthe other hand the introduction of randomness changesthe critical behaviors of a spin model considerably that israndom fields can altogether eliminate the phase transitionsin low dimensions and affect the numerical values of thecritical exponents in higher dimensions [26ndash28] Since theninvestigation of the effect of crystal field disorder on the

HindawiAdvances in Condensed Matter PhysicsVolume 2018 Article ID 3175068 13 pageshttpsdoiorg10115520183175068

2 Advances in Condensed Matter Physics

equilibrium phase diagram of spin-1 Ising models has beena research interest for many authors The BC model withrandom single-ion anisotropy provides a microscopic modelfor phase transitions of 3He-4He mixtures in silica aerogel[29 30] and relevant to the study of the phase separationin porous media in the vicinity of the superfluid transition[31 32] The interplay between quenched disorder providedby a random field and network connectivity in the BC modelis investigated by using the replica method [33] Moreoverthe BC model with infinite-range ferromagnetic interactionsand under the influence of a quenched disorder has beeninvestigated and a classification of the phase diagrams interms of their topology is presented in [34]

In this study we investigate the critical dynamics of soundwave propagation in the BCmodel with bimodal crystal fieldby combining the statistical equilibrium theory and the ther-modynamics of linear irreversible processes This approachhas been utilized to investigate the sound propagation in agreat variety of model systems Making use of the lowestapproximation of the cluster variation method and linearresponse theory of irreversible processes Erdem and Keskinperformed the calculations of the sound attenuation nearthe critical point in the Blume-Emery-Griffiths (BEG) modelwith zero crystal field [35ndash37] Gulpinar investigated thecritical behavior of ultrasound wave absorption coefficientin the metamagnetic Ising model within the mean-fieldapproximation [38] Later the absorption of sound in thespin-32 Ising model on the Bethe lattice is obtained and itstemperature variance is analyzed near the phase transitionpoints [39] Recently Cengiz and Albayrak studied the soundattenuation phenomena for a finite crystal field BEG modelon the Bethe lattice in terms of the recursion relationsby using the Onsager theory [40] Due to mathematicalcomplexity the properties of critical sound propagation havenot been studied in any spin system with random bondrandom magnetic field or random crystal field terms in theHamiltonian expression To the best of our knowledge thecritical dynamics of the sound propagation of the BC modelwith quenched diluted single-ion anisotropy have not beenstudied by the methods of irreversible thermodynamics It isassumed in this manuscript that the sound wave is coupled tothe order parameter fluctuations that decay mainly via orderparameter relaxation process and the steady-state dynamicsof the BCmodel with random diluted crystal field are formu-lated while the system is under the effect of a propagatingsound wave of frequency 120596 which let us obtain the sounddispersion relation and sound attenuation coefficient for alltemperatures and frequencies that contain effectively onlyone phenomenological rate coefficient Temperature varianceof sound dispersion and absorption has been investigatedin the vicinity of the critical point Finally the frequencybehavior of the attenuation coefficient for temperatures ispresented in the vicinity of second-order transition fromordered to disordered phase

The paper is organized as follows the model and itsstatic properties are presented briefly in Section 2 Nextthe free energy production near the equilibrium is statedand the order parameter relaxation time is obtained in

Section 3 The steady solution of the kinetic equation of theorder parameter and expressions for the sound dispersionrelation and ultrasonic attenuation coefficient are obtained inSection 4 Finally frequency and temperature behaviors of theultrasonic attenuation are analyzed and the discussion of theresults is given in Section 5

2 The Model and Its Equilibrium Properties

The BC model with random single-ion anisotropy in thepresence of an external magnetic field 119867 is described by theHamiltonian

= minus119869sum⟨119894119895⟩

120590119894120590119895 minus 119867sum119894

120590119894 + sum119894

Δ 1198941205902119894 (1)

with 120590119894 = minus1 0 1 Here 119869 gt 0 is the exchange interaction dueto ferromagnetic coupling and Δ 119894 is the crystal field acting onsite i with a distribution function

119865 (Δ 119894) = 119901120575 (Δ 119894 minus 119863) + (1 minus 119901) 120575 (Δ 119894) (2)

where 119901 is concentration of the spins on the lattice whichare influenced by a crystal field The mean-field free energyfor a nonvanishing external field is given by the followingexpression [41 42]

Ψ = Ψ0 (119881 (119886) 119879) + 1198731199111198692 1198982

+ 119873119879int ln(1 minus 2 cosh ((119869119911119898 + 119867) 119879)2 cosh ((119869119911119898 + 119867) 119879) + 119890Δ 119894119879)119865 (Δ 119894) 119889Δ 119894(3)

where Ψ0(119881(119886) 119879) is the lattice-free energy that is indepen-dent of spin configuration Here 119886 119881 and 119898 =lt 120590119894 gt are thelattice constant volume and order parameter of the systemrespectively In addition lt gt denotes thermal expectationvalue If one makes use of (2) the mean-field free energybecomes

Ψ = Ψ0 (119881 (119886) 119879) + 1198731199111198691198982

2 + 119873119901Δminus 119901119873119896119861119879 ln(2 cosh 119869119911119898 + 119867119879 + 119890Δ119879)+ (1 minus 119901)119873119896119861119879 ln(2 cosh 119869119911119898 + 119867119879 + 1)

(4)

For the sake of simplicity we will take Boltzmann constantas unity (119896119861 = 1) from now on The equilibrium conditions(120597Ψ120597119898)|119890119902 = 0 and (120597Ψ120597119886)|119890119902 = 0 give the followingself-consistent equations for themagnetization and the latticeconstant

Advances in Condensed Matter Physics 3

119898 = 2119901 sinh ((119869119911119898 + 119867) 119879)2 cosh ((119869119911119898 + 119867) 119879) + 119890Δ119879 + 2 (1 minus 119901)sdot sinh ((119869119911119898 + 119867) 119879)2 cosh ((119869119911119898 + 119867) 119879) + 1

(5)

120597Ψ012059711988610038161003816100381610038161003816100381610038161003816119890119902 = 119873119911120597119869120597119886 [minus1198982

2+ 2119901119911119898 sinh ((119869119911119898 + 119867) 119879)2 cosh ((119869119911119898 + 119867) 119879) + 119890Δ119879+ 2 (1 minus 119901) 119911119898 sinh ((119869119911119898 + 119867) 119879)2 cosh ((119869119911119898 + 119867) 119879) + 1]

(6)

The numerical solutions of the equation state given by (5)for vanishing external field have been performed and it hasbeen reported that the BC model with quenched dilutedsingle-ion anisotropy exhibits three distinct phase diagramtopologies in the (119879Δ) plane depending on 119901 [43 44] For1 ge 119901 gt 0945 the form of the phase diagram is identicalto that of the pure BC model with homogenous crystalfield where the ferromagnetic phase is separated from theparamagnetic phase by a phase boundary that is of secondorder up to a tricritical point (TCP) at which the transitionbecomes first order For 0945 gt 119901 ge 0926 TCP stillexists but 120582minus and first-order lines have reentrant parts Thephase diagram of the system changes dramatically in natureif one increases the dilutence further For 0926 gt 119901 ge89 there exists discontinuous phase transitions betweenthe ferromagnetic and paramagnetic phases at strong crystalfields and low temperatures The transition becomes con-tinuous at higher temperatures Further the 120582-line displaysreentrance A portion of the second-order transition lineis masked by a first-order transition line This situationcauses two distinct multicritical points to appear criticalendpoint (CEP) and double critical endpoint (DCP) Undera threshold value of the crystal field concentration (119901 le 89)the quenched disorder completely eliminates the first-orderphase transitions and the whole phase boundary is of secondorder

3 Relaxation Dynamics of the BC Model withQuenched Diluted Single-Ion Anisotropy

One may assume the existence of a small uniform externalfield for a short while in order to be able to formulate therelaxation dynamics of the BC model with random single-ion anisotropy In addition the amplitude of the external fieldshould be sufficiently small to allow the spin system to bein the neighborhood of equilibrium where linear responsetheory can be utilized In other words we investigate thefinal stage of the approach to equilibrium In the case of theexistence of a small deviation of the magnetic field from itsequilibrium value 120575119867 = 119867 minus 1198670 the system will be removedslightly from equilibrium and a finite free energy productionΔΨ will arise

ΔΨ = Ψ (119898119867 119886) minus Ψ0 (11989801198670 1198860) (7)

where ΔΨ corresponds to an increase in the correspondingthermodynamic potential related to the deviation of 119898 119886119867from their equilibrium values andΨ0 is the equilibrium valueof the free energy while 119898 = 1198980 119886 = 1198860 and 119867 =1198670 In the neighborhood of equilibrium the free energyproduction may be written as a Taylor-series expansionin which deviations in the thermodynamic quantities areretained to second order

ΔΨ = 1198602 (119898 minus 1198980)2 minus 119861 (119898 minus 1198980) (119867 minus 1198670)+ 1198622 (119867 minus 1198670)2 + 119863 (119898 minus 1198980) (119886 minus 1198860)+ 119864 (119886 minus 1198860) (119867 minus 1198670) + 1198652 (119886 minus 1198860)2+ 119866 (119867 minus 1198670)

(8)

Here the coefficients 119860 to119866 are the so-called free energy pro-duction coefficients and they are calculated by the followingsecond-order derivatives

119860 = 1205972Ψ1205971198982

100381610038161003816100381610038161003816100381610038161003816119890119902 119861 = minus 1205972Ψ120597119898120597119867

100381610038161003816100381610038161003816100381610038161003816119890119902 119862 = 1205972Ψ1205971198672

100381610038161003816100381610038161003816100381610038161003816119890119902 119863 = 1205972Ψ120597119898120597119886

100381610038161003816100381610038161003816100381610038161003816119890119902 = 1205972Ψ120597119898120597119869100381610038161003816100381610038161003816100381610038161003816119890119902 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902 119864 = 1205972Ψ120597119867120597119886

100381610038161003816100381610038161003816100381610038161003816119890119902 = 1205972Ψ120597119867120597119869100381610038161003816100381610038161003816100381610038161003816119890119902 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902 119865 = 1205972Ψ1205971198862

100381610038161003816100381610038161003816100381610038161003816119890119902= 1205972Ψ01205971198862

100381610038161003816100381610038161003816100381610038161003816119890119902 + 1205972Ψ1205971198692100381610038161003816100381610038161003816100381610038161003816119890119902 ( 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902)2 + 120597Ψ120597119869

10038161003816100381610038161003816100381610038161003816119890119902 12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902

119866 = 120597Ψ12059711986710038161003816100381610038161003816100381610038161003816119890119902

(9)

where the subscript ldquo119890119902rdquo denotes the thermal equilibriumthus the derivatives are evaluated for 119898 = 1198980 119886 = 1198860 and119867 = 1198670 The explicit expressions for the above-mentionedfree energy production coefficients are given in theAppendix

If the system is shifted from its equilibrium by thedeviation of the external field from its equilibrium value(120575119867 = 0) andor by the volume change of the crystal which isproportional to 119886minus1198860 the generalized force119883119898 arises whichcan be regarded as the force that brings magnetization backto its equilibrium value The generalized force conjugate to


generalized current may be obtained by differentiating thefree energy production with respect to (119898 minus 1198980)

119883119898 = minus 120597ΔΨ120597 (119898 minus 1198980)= minus [119860 (119898 minus 1198980) minus 119861 (119867 minus 1198670) + 119863 (119886 minus 1198860)]

(10)

In the realm of theory of irreversible thermodynamicsthe time derivative of magnetization () is treated as thegeneralized current conjugate to119883119898

= 119889 (119898 minus 1198980)119889119905 (11)

If the deviation from the equilibrium condition is small onecan write a linear relation between the current and the force = 119871119883119898 Thus the dynamics of the order parameter for theBCmodel with quenched diluted crystal field are ruled by thefollowing rate equation

= minus119871 [119860 (119898 minus 1198980) minus 119861 (119867 minus 1198670) + 119863 (119886 minus 1198860)] (12)

where 119871 is the order parameter Onsager coefficient Weshould note that in this study the most simple temperaturedependence is assumed for 119871 which must be found either inprinciple by a more powerful theory such as path probabilitymethod [45 46] or in practice by fit with the experimentalfindings

For the case of vanishing external stimulation (119867 = 1198670119886 = 1198860) one obtains the relation that describes the rate ofchange in the order parameter relaxing to its equilibrium stateas follows

= minus119871119860 (119898 minus 1198980) (13)

The solution of the kinetic equation given by (13) is of theform (119898 minus 1198980 ≃ 119890minus119905120591) where 120591 is the relaxation time of theorder parameter Thus the relaxation time corresponds to

120591 = 1119871119860 (14)

The temperature behavior of the order parameter relax-ation time near the phase transition points of the BC modelwith bimodal crystal field has been investigated in detail in[44] and a rapid increase in the single relaxation time isobserved when the temperature approaches the critical andmulticritical phase transition temperatures In addition 120591presents a scaling relation 120591 sim |119879 minus 119879119862|minus1 which correspondsto well-known phenomena of critical slowing down Finallyas a signature of the first-order phase transition a jump-discontinuity has been observed in the relaxation time neara first-order phase transition

4 Derivation of the SoundAttenuation Coefficient

In this section we will study the transport properties for theBC model with bimodal crystal field near its magnetic phasetransition points With this aim we will consider the case in

which the lattice is stimulated by the soundwave of frequency120596 Due to nature of the linear response theory if you perturbthe system at a frequency 120596 the response will take place atsame frequency Thus one can find the steady-state solutionof (12) with an oscillating external force 119886 minus 1198860 = 1198861119890119894120596119905 asfollows

119898 minus 1198980 = 1198981119890119894120596119905 (15)

Introduce this expression into (12)And assuming that 119867 = 1198670 one obtains the following

nonhomogenous equation for1198981

1198941205961198981119890119894120596119905 = minus1198711198601198981119890119894120596119905 minus 1198711198631198861119890119894120596119905 (16)

Solving (16) for 11989811198861 gives11989811198861 = minus 119871119863119894120596 + 119871119860 (17)

In addition if onemakes use of (14) for the relaxation timeof the BC model with a random crystal field (17) becomes

11989811198861 = minus 1198711198631205911 + 119894120596120591 (18)

The response in the pressure 119901 minus 1199010 is obtained bydifferentiating the minimum work with respect to 119881 minus 1198810

119901 minus 1199010 = 120597ΔΨ120597 (119881 minus 1198810) = minus 11988631198810 120597ΔΨ120597 (119886 minus 1198860) (19)

Then using (8) one obtains

119901 minus 1199010 = minus 119886031198810 [119863 (119898 minus 1198980) + 119865 (119886 minus 1198860)] (20)

On the other hand the derivative of pressure with respect tovolume gives

( 120597119901120597119881)119904119900119906119899119889

= minus 11988620911988120

[11986311989811198861 + 119865] (21)

where119863 and119865 are the free energy production coefficients thatare given in (9) If one introduces (18) and the density 120588 =1198721198810 into (21)

(120597119901120597120588)119904119900119906119899119889

= 119886209119872 [119865 minus 11987111986321205911 + 119894120596120591] (22)

Finally using the definition 119888119904119900119906119899119889 = (120597119901120597120588)12119904119900119906119899119889

and thefact that the order parameter relaxation time tends to verylarge values in the vicinity of the continuous phase transitionpoints one obtains the complex effective elastic constant(complex velocity of sound) of the BC model with quencheddiluted crystal field

119888119904119900119906119899119889 cong 11988603 radic 119865119872 [1 minus 11987111986321205912119865 (1 + 119894120596120591)] (23)


It is a well-known fact that the lag between the oscillationsof the pressure and the excitation of a given mode leads to thedissipation of energy and dispersion of the sound wave

119888 (120596) = Re 119888119904119900119906119899119889 = 11988603radic119872radic119865 2119865 minus 11987111986321205912119865 (1 + 12059621205912) (24)

Since we assume a linear coupling of sound wave with theorder parameter fluctuations in the system the dispersionwhich is the relative sound velocity change with frequencydepends not on the sound wave amplitude but on frequency120596 Finally if one makes use of the following definition for thesound attenuation constant

120572 (120596) = Im minus120596119888119904119900119906119899119889= Im(11988603 radic 119865119872)minus1 [ minus2120596119865 (1 + 119894120596120591)2119865 minus 1198711198632120591 + 1198942119865120596120591] = 3radic11987221198860 1

(radic119865)3119871119863212059621205912

(1 minus 11987111986321205912119865 ) + 12059621205912 (25)

A finite attenuation above the critical temperature cannotbe obtained due to the fact that the attenuation constantis proportional to free energy production constant 119863 thatvanishes above the critical temperature where the orderparameter is equal to zero (see (A1)) This is entirely dueto the mean-field approximation and one may expect afinite attenuation constant above the critical temperature forhigher-order approximations (ie Bethe approximation)

Now if one rewrites the free-energy production coeffi-cient 119865 given in (9) the following expressions are obtainedforradic119865 and (radic119865)minus3

radic119865 ≃ radic 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902

1 + 12 ( 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902)

minus1

sdot [ 1205972Ψ1205971198692100381610038161003816100381610038161003816100381610038161003816119890119902 ( 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902)2 + 120597Ψ120597119869

10038161003816100381610038161003816100381610038161003816119890119902 12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902]

(26)

(radic119865)minus3 ≃ ( 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902)

minus32 1 minus 32 ( 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902)

minus1

sdot [ 1205972Ψ1205971198692100381610038161003816100381610038161003816100381610038161003816119890119902 ( 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902)2 + 120597Ψ120597119869

10038161003816100381610038161003816100381610038161003816119890119902 12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902]

(27)

In addition if one uses the definition 1198882infin = (119886209119872)(1205972Ψ01205971198862)|119890119902 (24) and (25) become

119888 (120596) = 119888infin 1 + 12 ( 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902)

minus1

sdot [ 1205972Ψ1205971198692100381610038161003816100381610038161003816100381610038161003816119890119902 ( 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902)2 + 120597Ψ120597119869

10038161003816100381610038161003816100381610038161003816119890119902 12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902]

sdot 2119865 minus 11987111986321205912119865 (1 + 12059621205912)

(28)

120572 (120596) = 1198862018119872 11198883infin1 minus 32 ( 1205972Ψ01205971198862

100381610038161003816100381610038161003816100381610038161003816119890119902)minus1

sdot [ 1205972Ψ1205971198692100381610038161003816100381610038161003816100381610038161003816119890119902 ( 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902)2 + 120597Ψ120597119869

10038161003816100381610038161003816100381610038161003816119890119902 12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902]

sdot 119871119863212059621205912(1 minus 11987111986321205912119865) + 12059621205912

(29)

where 119888infin is the velocity of sound for very high frequencies atwhich order parameter can no longer follow the sound wavemotion

5 Results And Discussion

The calculated data that provides information about thethermal variation of the sound velocity and attenuationcoefficient in the ordered and disordered phases and thefrequency dependence of isothermal attenuation coefficientin the ordered phase of the BC model with quenched dilutedcrystal field will be discussed in the following In additionthe behavior of sound absorption in the neighborhood ofcritical temperature is analyzed according to various values ofphenomenological rate coefficient Finally by making use ofdouble logarithmic plots of the sound attenuation coefficientversus the distance from the critical temperature the dynami-cal critical exponents of the sound absorption are obtained forlow- and high-frequency regimes Before starting to discussthe above-mentioned results it is convenient to introduce thefollowing reduced quantities

120579 = 119879119911119869 119889 = 119863119911119869 ℎ = 119867119911119869

(30)

here 120579 119889 and ℎ are the reduced values of the temperaturecrystal field and magnetic field respectively One shouldstress that we focus on zero field case (ℎ = 00) and 1198883infin is takenas unity for the sake of simplicity throughout this section

Figure 1 displays the temperature variation of thefrequency-dependent sound velocity (dispersion) of the BCmodel with quenched diluted crystal field for several valuesof sound wave frequency while 119871 = 002 119901 = 098


015 030 045

=20

=05

025

050

c

(a)

075

100

Figure 1 Sound dispersion at different frequencies of sound for 119871 =002 119901 = 098 and 119889 = 119889119862 = 045 The number accompanying eachcurve represents the value of the sound frequency 120596

and 119889 = 045 The number accompanying each curvedenotes the value of 120596 and one observes a characteristicsound velocityminimum that shifts to the lower temperatureswith increasing frequency The minima become deeper withdecreasing 120596 and ultrasonic dispersion reaches the finitevalue of 10 at 120579 = 120579119862 and remains temperature-independentafter then Similar temperature dependence of the sounddispersion has been reported near the order-disorder phasetransition point of the BEG model [47] Comparably theexistence of the minima of the sound velocity in the orderedphase and its shift to lower temperatures with increasingfrequency have been observed in the classical liquid crystals[8] liquid 4He confined in a microfluidic cavity [10] andthe magnetic compound RbMnF3 [48] In addition it can beseen from Figure 1 that the change in sound velocity becomesfrequency-independent as one approaches to critical value ofthe reduced temperature and this behavior is in parallel withthe findings of the molecular field theory [17 49] dynamicalrenormalization group theory for the propagation of soundnear the continuous structural phase transitions [50] andBrillouin scattering studies of A2MX6-crystals [49]

Figure 2 illustrates the double logarithmic plot of theisothermal 120572 versus 120596 for two distinct values of the reducedtemperature (120579 = 010 0390 lt 120579119862) in the ferromagneticphase while 119901 = 098 119871 = 002 120579119862 = 039747 and 119889119862 = 045One can observe the usual 1205962 dependence of the attenuationin the low-frequency region where 120596120591 ltlt 1 This result isin accordance with the theoretical and experimental findingsfor CoF2 [1] and MnF2 [51 52] The ultrasonic attenuationhas different frequency dependence in the high-frequencyregime at which order parameter can no longer follow soundwave motion this part of the plot has zero slope thusultrasonic attenuation behaves as 120572 prop 1205960 while 120596120591 gtgt 1 Inaddition there is a crossover behavior in attenuation betweenlow- and high-frequency regimes The theoretical treatmentof the sound propagation by mode-mode coupling method

0

minus6

minus12

minus18

minus8 minus4 0 4

ＦＩＡ10

ＦＩＡ10

=010

=0397

slope=

20

slope=00

(a)

Figure 2 Double logarithmic plot of attenuation versus frequencyfor 119871 = 002 119901 = 098 119889 = 119889119862 = 045 and 120579119862 = 039747The number accompanying each curve represents the value of thereduced temperature

predicts that the sound attenuation coefficient scales as 120572 sim120596119910 for the sound frequencies that are quite larger than thecharacteristic frequency of the order parameter fluctuations120596119888 and an extensive study of the frequency dispersion ofthe critical attenuation of KMnF3 above 1862 K measured119910 = 013 plusmn 005 [53ndash55] In addition similar frequencydependence of the sound attenuation coefficient has beenreported for various model systems such as spin-1 and spin-32 Isingmodels on theBethe lattice [39 40] zero crystal fieldBEG model [37] and metamagnetic Ising model [38] withinmean-field theory

Figures 3(a)-3(c) show the calculated sound attenuationcoefficient as a function of the reduced temperature 120579 near thecritical point for various values of the sound frequency whosevalue is denoted by the number accompanying each curveand the vertical dashed lines refer to critical temperature 120579119888while 119901 = 098 119871 = 002 and 119889 = 045 One can observefrom these figures that the attenuation rises to a maximumin the ordered phase for all values of 120596 But the frequencydependencies of the location form and amplitude of theabsorption peak is different for low- and high-frequencyregions Figure 3(a) presents the temperature variance of 120572for two values of the sound frequency 120596 = 12 times 10minus5 120596 =18 times 10minus5 (blue and red curves) which satisfy the condition120596120591 ≪ 1 for the low-frequency regime In the low-frequencyregion the sound attenuation peak is rather sharp and locatedvery near to the critical temperature 120579119888 and its amplitudeincreases with rising sound frequency

Figure 3(b) presents the temperature dependence of120572 for higher values of the sound frequency (120596 =0002 0005 02 05 10) It is observed from this figure thatthe absorption peak is rather broad and its amplitude growswith rising frequency for these ranges of the frequencyFurther the temperature at which attenuation maxima take


80times10minus5

40times10minus5

00

=18times10minus5

12times10minus5

＝

0392 0396 0400

(a)

＝

10times10minus1

50times10minus2

00

01 02 03 04

=10times101

20times10minus1

20times10minus3

(b)

＝

01 02 03 04

008

004

000

=10times101

=10times102

=10times103

(c)

Figure 3 Sound attenuation120572 as a function of the reduced temperature at several sound frequency values120596 in the neighborhood of a criticalpoint for 119901 = 098 119889 = 119889119862 = 045 and 119871 = 002 The number accompanying each curve denotes the value of the frequency120596 and the verticaldotted line represents the critical temperature in the ordered phase

place moves to lower temperatures with rising frequencyThefrequency shift of the attenuation maxima is also observedin the experimental and theoretical studies of the ultrasonicinvestigation of liquid helium [56 57] and Rb2ZnCl4 andK2SeO4 [58] as well as other Ising models [37ndash40] andelastically isotropic Ising system above the critical point onthe basis of a complete stochastic model [59]

In Figure 3(c) we aim to illustrate the temperaturedependence of the sound attenuation coefficient for threedistinct values of the sound frequency (120596 = 1 10 1000) forwhich120596120591 gtgt 1 and the order parameter can no longer followsound wave motion Both the location and the amplitude ofthe absorption peak do not change with the sound frequencyand this is in accordance with the power law 120572 sim 1205960 observed


＝

= 60times10minus3

L=05

1040

0026

0013

0000

038 039 040

(a)

＝

()

08

04

00

01 02 03 04

01

02

04

L = 08 = 20times10minus1

(b)

Figure 4 Ultrasonic attenuation as a function of reduced temperature for (a) 120596 = 0006 and (b) 120596 = 02 while 119901 = 098 and 119889 = 119889119862 = 045The number accompanying each curve denotes the value of the order parameter Onsager coefficient (119871) and the arrow denotes the criticaltemperature

in Figure 2 while 120596120591 gtgt 1 Similar results are observed formetamagnetic Ising and BEG models in the high-frequencyregion But since these models have two order parametersand their relaxation dynamics are governed by two relaxationtimes their attenuation coefficient have two peaks for somefrequency values in the low-frequency region as expected[37 38]

It is clear from (14) and (28) that both the singlerelaxation time and sound attenuation coefficient depend onthe phenomenological Onsager rate coefficient (119871) Detailedinvestigation of the relaxation dynamics of the BC modelwith bimodal random crystal field revealed the fact that therelaxation time increases and the order parameter relaxationslows down with rising 119871 [44] Due to fact that thermalfluctuations become very large and the correlation distanceextends over large number of spins the coupling betweensound waves and order parameter relaxation gives rise toan increase in the absorption of the sound waves as thetemperature approaches its critical value As the transitiontemperature of a solid is closely approached the orderedparameter relaxes slow enough the internal irreversible pro-cesses that tend to restore local equilibrium are switched onand they make the entropy increase which results in energydissipation in the system [17 57] Due to these facts wefound it interesting to study the effect of 119871 on the temperaturevariance of the sound attenuation near the second-orderphase transition point Figures 4(a) and 4(b) illustrate theattenuation coefficient versus reduced temperature at severalvalues of 119871 for 119901 = 098 and 119889 = 045 at two distinctvalues of the sound wave frequency 120596 and the arrows referto critical point temperature It is apparent from Figure 4(a)which displays the calculated 120572 for 120596 = 0006 that the

maxima of 120572 broaden and move to lower temperatures withdecreasing 119871 Meanwhile the value of the sound attenuationmaxima slightly increases with decreasing 119871 The attenuationpeak broadens and its amplitude decreases and shifts to lowertemperatures with decreasing 119871 while 120596 = 02 We shouldnote that for systems that have twoormore order parametersthe relation between the generalized forces and currentsmay be written in terms of a matrix of phenomenologicalrate coefficients and the effect of the nondiagonal Onsagercoefficient has been investigated for BEG and metamagneticIsing models in the mean-field approximation [37 38] andspin-32 and spin-1 Ising models on the Bethe lattice [39 40]

The critical behavior of the sound attenuation coefficientin the neighborhood of the second-order phase transitionpoints is characterized by the dynamical sound attenuationcritical exponent 120588 As discussed above isothermal soundattenuation coefficient behaves as 120572 prop 1205962 for frequenciesthat satisfy the condition 120596120591 ltlt 1 and 120572 prop 1205960 for120596120591 gtgt 1 In addition the comparison of Figures 3(a)ndash3(c)reveals the fact that the constant frequency temperaturevariance of 120572 near the critical temperature is different in low-and high-frequency regions Consequently one may expecttwo distinct power law relations and two distinct exponents(120588119897 120588ℎ) for these two regions

120572 (120596 120576) asymp 1205962120576minus120588119897 for 120596120591 ≪ 11205960120576minus120588ℎ for 120596120591 ≫ 1 (31)

where 120576 = 120579 minus 120579119862 is the expansion parameter which isa measure of the distance from the critical temperature


ＦＩＡ10

ＦＩＡ10|1minus＝|

minus12

minus16

minus20

minus1040 minus975 minus910 minus845

slope=minus10

=110minus5

=110minus6

=110minus7

(a)

ＦＩＡ10


minus600

minus675

minus750

minus825


slope = 10

=110minus2=1102

(b)

Figure 5 Double logarithmic plots of attenuationversus |1minus120579120579119862| for (a) low-frequency and (b) high-frequency regimes in the neighborhoodof a critical point for 119901 = 098 119889 = 119889119862 = 045 and 119871 = 002

The critical exponents for the function 120572(120596 120576) are definedrespectively as

120588119897 = minuslim120576rarr0

ln |120572 (120596 120576)|ln |120576| for 120596120591 ≪ 1

120588ℎ = minuslim120576rarr0

ln |120572 (120596 120576)|ln |120576| for 120596120591 ≫ 1 (32)

In order to calculate the values of these exponents onemay sketch logminus log plots of the sound attenuation coefficientversus |1minus120579120579119862| and find the slopes of the linear parts of thesecurves whichwill be equal tominus120588119897 andminus120588ℎ for frequencies thatsatisfy the conditions 120596120591 ≪ 1 and 120596120591 ≫ 1 respectively Withthis aim Figures 5(a) and 5(b) illustrate the log10120572minus log10|1 minus120579120579119862| plots for the constant values of sound frequencies inlow- and high-frequency regions respectively Figure 5(a)displays three distinct curves for 120596 = 10 times 10minus7 (blackcurve) 120596 = 10 times 10minus6 (blue curve) and 120596 = 10 times 10minus5(green curve) which obey the 120596120591 ≪ 1 condition As onecan see from this figure for all three frequencies in the low-frequency region the slopes of the linear part of log10120572 minuslog10|1 minus 120579120579119862| curves are equal to minus1 namely 120588119897 = 1Utilizing (31) and (30) one obtains the power law 120572(120596 120576) sim1205962120576minus1 of the sound attenuation coefficient for the BC modelwith bimodal random crystal field in the low-frequencyregimeThis result is in accordance with the scaling behaviorof the sound attenuation coefficient 120572119871119870 obtained withinthe Landau-Khalatnikov theory which applies to bilinearquadratic coupling of strain-order parameter coupling [1757] 120572119871119870 sim 1205962120591 and the order parameter relaxation timebehaves as 120591 sim 120576minus1 as 120576 rarr 0 within the mean-field

approximation [44] A strong anomalous increase of thesound attenuation as the temperature approaches its criticalvalue has been also observed for systems that undergo (elasticand distortive) structural phase transitions The dynamicalrenormalization group calculations for distortive structuralphase transitions report 120588120590 = (119911] + 120572120590) where 120572120590 =120572 + 2(120601 minus 1)(1 minus 1205751205901) where 119911 ] 120601 and are 120572 are thedynamical correlation length heat capacity and crossovercritical exponents respectively Utilizing the same methodSchwabl showed that the sound attenuation coefficient scalesas 120572 sim 120576minus321205962 in the vicinity of the critical point of the elasticstructural phase transitions [50] Ultrasonic measurementsreported small but positive attenuation exponents in isotropicgarnet structured magnets such as Y3Fe5O12 and Gd3Fe5O12

[60] Recently resonant ultrasound spectroscopy studies ofCoF2 have shown that 120588 = 09 to 11 for the resonance peaksnear 487 860 1300 and 1882 kHz which depend strongly onthe shear modulus [1]

Finally Figure 5(b) shows the linear part of the log120572versus log |1 minus 120579120579119862| curves for frequencies 120596 = 01 (bluesymbols) and 120596 = 100 (black symbols) while 120596120591 ≫ 1 Onecan see from this figure that the slope of the curves is +1thus 120588ℎ = minus1 If one again makes use of (30) and (31) thefollowing power law can be written for the sound absorption120572(120596 120576) sim 12059601205761 while 120596120591 ≫ 1 and verifies the convergenceof the sound absorption to zero as the reduced temperatureapproaches its critical value (see Figure 3(b)) We shouldnote that similar behavior is observed in the high-frequencyregion of the sound attenuation for the BEG model [37] Ithas been shown by the phenomenological theory that soundattenuation reaches the saturation value 120572119901ℎ119890 rarr 119861120591minus1 for very


high frequencies where 119861 is independent of temperature andfrequency for a system with a single relaxation process [3] Ifone keeps in mind that 120591 sim 120576minus1 as 120576 rarr 0 this corresponds to120572119901ℎ119890(120596 120576) asymp 12059601205761 which is in parallel with our result

6 Concluding Remarks

The present study is devoted to critical dynamics of soundpropagation in BC model with a random single anisotropyby using a method that combines the statistical equilibriumtheory of phase transitions with the linear response theory ofirreversible thermodynamics With this aim it is supposedthat the system is stimulated by a sound wave oscillatingat an angular frequency 120596 and the complex effective elasticconstant and sound velocity and sound attenuation coef-ficient are obtained for all temperatures and frequenciesby studying the steady-state solutions We have studiedthe temperature behavior of the sound velocity for variousvalues of the sound frequency and observed a characteristicminimum that shifts to the lower temperatures with risingfrequency and a frequency-independent sound velocity in thevicinity of critical temperatureThenwe have investigated thetemperature frequency and order parameter Onsager ratecoefficient dependencies of the sound attenuation near thesecond transition temperatures Frequency-dependent shiftsin attenuation maxima to lower temperatures with increasingfrequency for temperatures less than the second-order phasetransition temperatures are obtained and possible mecha-nisms for these phenomena are discussed Our investigationof the frequency variance of the isothermal sound attenuationrevealed the fact that sound attenuation follows the powerlaw 120572 sim 1205962 for frequencies that satisfy the condition 120596120591 ≪1 and 120572 sim 1205960 while 120596120591 ≫ 1 Consequently 120596120591 is thequantity that determines the separation of the low- and high-frequency regimes By making use of this information andobtaining the slope of the log 120572 versus log |1 minus 120579120579119888| plotswe have obtained that sound absorption follows the scalingrelation 120572(120596 120576) sim 1205962120576minus1 in the low-frequency regime andthe power law 120572(120596 120576) sim 12059601205761 for frequencies that obey thecondition 120596120591 ≫ 1 Finally we should note that the high-frequency behavior provides information about the dampingmechanism

In this study the sound attenuation related to the relax-ation process of the order parameter for BC model witha random single anisotropy is studied We would like toemphasize that the above-mentioned theory is limited neitherto the systems with only scalar spin nor to the magneticphase transitions It could also apply to a crystalline solid andother systems whenever the order parameter dynamics arerelaxation dynamics We should note that there are additionalpossible reasons for attenuation peaks in magnetic systemswhich are not taken into account in the current analysis suchas coupling of the sound waves to modes domain effectsand nuclear acoustic resonance [17] In addition in the caseof real crystals with many structural defects the presenceof disorder will affect the critical behavior of the soundattenuation in solid media The field-theoretic description ofdynamic critical effects of the disorder on acoustic anomalies

near the temperature of second-order phase transition forthree-dimensional Ising-like systems has shown that thepresence of impurities brings additional couplings changingthe value of the critical exponent of the sound attenuationand strongly affects the order parameter shape function at thelow-frequency region in the vicinity of the critical point [61]

Appendix

The free energy production coefficients A to G that aredefined in (9) are given as follows

119860 = 119873119869119911 minus 11987311986921199112119879 [ 2119901 cosh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 119890Δ119879minus ( 2radic119901 sinh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 119890Δ119879)2

+ 2 (1 minus 119901) cosh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 1minus (2radic1 minus 119901 sinh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 1 )2]

119861 = 119873119869119911119879 [ 2119901 cosh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 119890Δ119879minus ( 2radic119901 sinh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 119890Δ119879)2


119862 = minus119873119879 [ 2119901 cosh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 119890Δ119879minus ( 2radic119901 sinh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 119890Δ119879)2

+ 2 (1 minus 119901) cosh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 1minus (2radic1 minus 119901 sinh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 119890Δ119879)2]

119863 = 119873119911 12059711986912059711988610038161003816100381610038161003816100381610038161003816119890119902 [1198980 minus 21198691199111198980119901 cosh (1198691199111198980119879)119879 [2 cosh (1198691199111198980119879) + 119890Δ119879]

minus 2119901 sinh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 119890Δ119879


+ 41198691199111198980119901 sinh2 (1198691199111198980119879)119879 [2 cosh (1198691199111198980119879) + 119890Δ119879]2minus 21198691199111198980 (1 minus 119901) cosh (1198691199111198980119879)119879 [2 cosh (1198691199111198980119879) + 1]minus 2 (1 minus 119901) sinh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 1+ 41198691199111198980 (1 minus 119901) sinh2 (1198691199111198980119879)119879 [2 cosh (1198691199111198980119879) + 1] ]



119865 = 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902 + 1198731199111198980

12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902 [11989802

minus 2 (1 minus 119901) cosh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 1minus 2119901 cosh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 119890Δ119879] + 11987311991121198982

0119879 ( 12059711986912059711988610038161003816100381610038161003816100381610038161003816119890119902)

2

times [ 2119901 cosh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 119890Δ119879minus ( 2radic119901 sinh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 119890Δ119879)2


119866 = 2119873 (1 minus 119901) cosh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 1minus 2119873119901 sinh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 119890Δ119879

(A1)

Data Availability

The data that support the findings of this study are availablefrom the corresponding author upon reasonable request

Disclosure

The research and publication of this article was performedas a part of the employment of the author Departmentof Physics Dokuz Eylul University Department of PhysicsIzmir Turkey

Conflicts of Interest

The author declares that there are no conflicts of interest

References

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[2] V Felea P Lemmens S Yasin et al ldquoMagnetic phase diagram ofmultiferroic MnWO4 probed by ultrasoundrdquo Journal of PhysicsCondensed Matter vol 23 no 21 Article ID 216001 2011

[3] A Pawlak ldquoCritical soundpropagation inmagnetsrdquo inHorizonsin World Physics L Pedroza and M Everett Eds vol 268 pp1ndash69 Nova Science Publishers Hauppauge NY USA 2009

[4] S Bhattacharjee S Erfanifam E L Green et al ldquo Acousticsignatures of the phases and phase transitions in rdquo PhysicalReview B Condensed Matter and Materials Physics vol 93 no14 2016

[5] S Erfanifam S Zherlitsyn J Wosnitza et al ldquoIntrinsic andextrinsic nonstationary field-driven processes in the spin-icecompound Dy2Ti2O7rdquo Physical Review B Condensed Matterand Materials Physics vol 84 no 22 Article ID 220404 2011

[6] S S Turzi ldquoViscoelastic nematodynamicsrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 94 no 6Article ID 062705 2016

[7] P Biscari A DiCarlo and S S Turzi ldquoAnisotropic wavepropagation in nematic liquid crystalsrdquo Soft Matter vol 10 no41 pp 8296ndash8307 2014

[8] C Grammes J K Kruger K-P Bohn et al ldquoUniversalrelaxation behavior of classical liquid crystals at hypersonicfrequenciesrdquo Physical Review E Statistical Nonlinear and SoftMatter Physics vol 51 no 1 pp 430ndash440 1995

[9] A Shtyk andM FeigelrsquoMan ldquoUltrasonic attenuation via energydiffusion channel in disordered conductorsrdquoPhysical Review BCondensed Matter and Materials Physics vol 92 no 19 ArticleID 195101 2015

[10] X Rojas B D Hauer A J MacDonald P Saberi Y Yang andJ P Davis ldquo Ultrasonic interferometer for first-soundmeasure-ments of confined liquid rdquo Physical Review B CondensedMatterand Materials Physics vol 89 no 17 2014

[11] E Krotscheck and M D Miller ldquoThird sound and stability ofthin 3He-4He filmsrdquo Physical Review B vol 73 article 134514pp 1ndash14 2006

[12] J M Valles R M Heinrichs and R B Hallock ldquo 3He-4HeMixture FilmsTheHe coverage dependence of the 3He bindingenergyrdquo Physical Review Letters vol 56 pp 1704ndash1707 1986

[13] P A Sheldon and R B Hallock ldquoThird sound and energeticsinrdquo Physical Review B Condensed Matter andMaterials Physicsvol 50 no 21 pp 16082ndash16085 1994

[14] D B Fenner ldquoNonsingular absorption of ultrasound near thecritical mixing point of a binary liquidrdquo Physical Review A


Atomic Molecular and Optical Physics vol 23 no 4 pp 1931ndash1940 1981

[15] P K Khabibullaev S Z Mirzaev and U Kaatze ldquoCriticalfluctuations and noncritical relaxations of the nitrobenzene-isooctane system near its consolute pointrdquo Chemical PhysicsLetters vol 458 no 1-3 pp 76ndash80 2008

[16] S Z Mirzaev and U Kaatze ldquoScaling function of criticalbinarymixtures Nitrobenzene-n-hexanedata revisitedrdquoChem-ical Physics vol 393 no 1 pp 129ndash134 2012

[17] B Luthi Physical Acoustics in the Solid State Springer BerlinGermany 2005

[18] M Blume ldquoTheory of the first-order magnetic phase change inUO2rdquoPhysical ReviewA AtomicMolecularandOptical Physicsvol 141 no 2 pp 517ndash524 1966

[19] H W Capel ldquoOn the possibility of first-order phase transitionsin Ising systems of triplet ions with zero-field splittingrdquo PhysicaA Statistical Mechanics and its Applications vol 32 no 5 pp966ndash988 1966

[20] P E Cladis R K Bogardus W B Daniels and G N TaylorldquoHigh-pressure investigation of the reentrant nematic - Bilayer-smectic-A transitionrdquo Physical Review Letters vol 39 no 11 pp720ndash723 1977

[21] G P Johari ldquoThe Tammann phase boundary exothermicdisordering and the entropy contribution change on phasetransformationrdquo Physical Chemistry Chemical Physics vol 3 no12 pp 2483ndash2487 2001

[22] W Sinkler C Michaelsen R Bormann D Spilsbury andN Cowlam ldquoNeutron-diffraction investigation of structuralchanges during inverse melting ofsrdquo Physical Review B Con-densed Matter and Materials Physics vol 55 no 5 pp 2874ndash2881 1997

[23] M I Marques J M Borreguero H E Stanley and N VDokholyan ldquoPossible mechanism for cold denaturation ofproteins at high pressurerdquo Physical Review Letters vol 91 no13 2003

[24] S Rastogi G W Hohne and A Keller ldquoUnusual Pressure-Induced Phase Behavior in Crystalline Poly(4-methylpentene-1) Calorimetric and Spectroscopic Results and Further Impli-cationsrdquoMacromolecules vol 32 no 26 pp 8897ndash8909 1999

[25] A L Greer ldquoToo hot to meltrdquo Nature vol 404 no 6774 pp134-135 2000

[26] A N Berker ldquoOrdering under randomfields Renormalization-group argumentsrdquo Physical Review B Condensed Matter andMaterials Physics vol 29 no 9 pp 5243ndash5245 1984

[27] Y Imry and SMa ldquoRandom-field instability of the ordered stateof continuous symmetryrdquo Physical Review Letters vol 35 no 21pp 1399ndash1401 1975

[28] G Grinstein and S-K Ma ldquoSurface tension roughening andlower critical dimension in the random-field Ising modelrdquoPhysical Review B Condensed Matter andMaterials Physics vol28 no 5 pp 2588ndash2601 1983

[29] A Falicov and A N Berker ldquoCorrelated Random-Chemical-Potential Model for the Phase Transitions of Helium Mixturesin PorousMediardquoPhysical Review Letters vol 74 no 3 pp 426ndash429 1995

[30] M Paetkau and J R Beamish ldquoTricritical point and superfluidtransition in 3He-4He mixtures in silica aerogelrdquo PhysicalReview Letters vol 80 no 25 pp 5591ndash5594 1998

[31] L B Lurio N Mulders M Paetkau M H W Chan and S GJ Mochrie ldquoSmall-angle x-ray scattering measurements of themicrostructure of liquid helium mixtures adsorbed in aerogelrdquo

Physical Review E Statistical Nonlinear and SoftMatter Physicsvol 76 no 1 Article ID 011506 2007

[32] TR PriskC PantaleiHKaiser andP E Sokol ldquoConfinement-driven phase separation of quantum liquid mixturesrdquo PhysicalReview Letters vol 109 no 7 Article ID 075301 2012

[33] R Erichsen A A Lopes and S G Magalhaes ldquoMulticriticalpoints and topology-induced inverse transition in the random-fieldBlume-Capelmodel in a randomnetworkrdquoPhysical ReviewE Statistical Nonlinear and Soft Matter Physics vol 95 no 6Article ID 062113 2017

[34] P V Santos F A da Costa and J M de Araujo ldquoMean-fieldsolution of the Blume-Capel model under a random crystalfieldrdquo Physics Letters A vol 379 no 22-23 pp 1397ndash1401 2015

[35] M Keskin and R Erdem ldquoCritical behaviors of the soundattenuation in a spin-1 Ising modelrdquo The Journal of ChemicalPhysics vol 118 no 13 pp 5947ndash5954 2003

[36] R Erdem and M Keskin ldquoOn the temperature dependence ofthe sound attenuationmaximum as a function of frequency andmagnetic field in a spin-1 Ising model near the critical regionrdquoPhysics Letters A vol 326 no 1-2 pp 27ndash31 2004

[37] R Erdem and M Keskin ldquoSound attenuation in a spin-1 isingsystem near the critical temperaturerdquoPhysics Letters A vol 291no 2-3 pp 159ndash164 2001

[38] G Gulpinar ldquoCritical behavior of sound attenuation in ametamagnetic Ising systemrdquo Physics Letters A vol 372 no 2pp 98ndash105 2008

[39] T Cengiz and E Albayrak ldquoThe Bethe lattice treatment ofsound attenuation for a spin- 32 Ising modelrdquo Physica AStatistical Mechanics and its Applications vol 391 no 10 pp2948ndash2956 2012

[40] T Cengiz and E Albayrak ldquoThe crystal field effects on soundattenuation for a spin-1 Isingmodel on the Bethe latticerdquo Journalof Statistical MechanicsTheory and Experiment vol 2012 no 7Article ID P07004 2012

[41] G Gulpınar R Erdem andM Agartıoglu ldquoCritical and multi-critical behaviors of static and complexmagnetic susceptibilitiesfor the mean-field Blume-Capel model with a random crystalfieldrdquo Journal of Magnetism and Magnetic Materials vol 439pp 44ndash52 2017

[42] G Gulpinar and R Erdem ldquoHigh-frequency magnetic fieldon crystal field diluted S=1 Ising system Magnetic relaxationnear continuous phase transition pointsrdquo Canadian Journal ofPhysics 2018

[43] A Benyoussef M Saber T Biaz and M Touzani ldquoThe spin-1 Ising model with a random crystal field The mean-fieldsolutionrdquo Journal of Physics C Solid State Physics vol 20 no32 pp 5349ndash5354 1987

[44] G Gulpinar and F Iyikanat ldquoDynamics of the Blume-Capelmodel with quenched diluted single-ion anisotropy in theneighborhoodof equilibrium statesrdquo Physical Review E Statisti-cal Nonlinear and Soft Matter Physics vol 83 no 4 Article ID041101 2011

[45] R Kikuchi ldquoThe path probability methodrdquo Progress of Theoret-ical Physics Supplement vol 35 pp 1ndash64 1966

[46] A Erdinc and M Keskin ldquoEquilibrium and nonequilibriumbehavior of the BlumendashEmeryndashGriffiths model using the pairapproximation and path probability method with the pairdistributionrdquo International Journal of Modern Physics B vol 18no 10n11 pp 1603ndash1626 2004

[47] R Erdem and M Keskin ldquoSound dispersion in a spin-1 Isingsystem near the second-order phase transition pointrdquo PhysicsLetters A vol 310 no 1 pp 74ndash79 2003


[48] T J Moran and B Luthi ldquoHigh-frequency sound propagationnear magnetic phase transitionsrdquo Physical Review B CondensedMatter and Materials Physics vol 4 no 1 pp 122ndash132 1971

[49] W Henkel J Pelzl K H Hock and H Thomas ldquoElasticconstants and softening of acoustic modes in A2MX6-crystalsobserved by Brillouin scatteringrdquo Zeitschrift fur Physik B Con-densed Matter and Quanta vol 37 no 4 pp 321ndash332 1980

[50] F Schwabl ldquoPropagation of sound at continuous structuralphase transitionsrdquo Journal of Statistical Physics vol 39 no 5-6pp 719ndash737 1985

[51] A Ikushima ldquoAnisotropy of the ultrasonic attenuation inMnF2near the Neel temperaturerdquo Journal of Physics and Chemistry ofSolids vol 31 no 5 pp 939ndash946 1970

[52] A Pawlak ldquoCritical soundpropagation inMnF2rdquoPhysica StatusSolidi (c) vol 3 pp 204ndash207 2006

[53] K Kawasaki ldquoSound Attenuation and Dispersion near theLiquid-Gas Critical Pointrdquo Physical Review A vol 1 pp 1750ndash1757 1970

[54] Y Shiwa and K Kawasaki ldquoThe Mode-Coupling Approach toSound Propagation in a Critical Fluid I ndash Correction Dueto Hydrodynamic Interactions ndashrdquo Progress of Theoretical andExperimental Physics vol 66 no 1 pp 118ndash128 1981

[55] M Suzuki ldquoDynamical scaling and ultrasonic attenuation inKMnF3 at the structural phase transitionrdquo Journal of Physics CSolid State Physics vol 13 no 4 article no 013 pp 549ndash5601980

[56] D T Sprague N Alikacem P A Sheldon and R B Hallockldquo 3He binding energy in thin helium-mixture filmsrdquo PhysicalReview Letters vol 72 no 3 pp 384ndash387 1994

[57] L D Landau I M KhalatnikovldquoOn the anomalous absorptionof sound near a second-order phase transition pointrdquo DoklAkad Nauk SSSR 96 469 (1954) reprinted in ldquoCollected Papersof L D Landaurdquo Ed D TerHaar Pergamon London UK 1965

[58] AM Schorgg andF Schwabl ldquoTheory of ultrasonic attenuationat incommensurate phase transitionsrdquo Physical Review B Con-densed Matter and Materials Physics vol 49 no 17 pp 11682ndash11703 1994

[59] A Pawlak ldquoTheory of critical sound attenuation in Ising-typemagnetsrdquo Acta Physica Polonica A vol 98 no 1-2 pp 23ndash392000

[60] I K Kamilov and K K Aliev ldquoUltrasonic studies of the criticaldynamics of magnetically ordered crystalsrdquo Physics-Uspekhivol 41 no 9 pp 865ndash884 1998

[61] P V Prudnikov andVV Prudnikov ldquoCritical sound attenuationof three-dimensional lsing systemsrdquo Condensed Matter Physicsvol 9 no 2 pp 403ndash410 2006

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equilibrium phase diagram of spin-1 Ising models has beena research interest for many authors The BC model withrandom single-ion anisotropy provides a microscopic modelfor phase transitions of 3He-4He mixtures in silica aerogel[29 30] and relevant to the study of the phase separationin porous media in the vicinity of the superfluid transition[31 32] The interplay between quenched disorder providedby a random field and network connectivity in the BC modelis investigated by using the replica method [33] Moreoverthe BC model with infinite-range ferromagnetic interactionsand under the influence of a quenched disorder has beeninvestigated and a classification of the phase diagrams interms of their topology is presented in [34]

In this study we investigate the critical dynamics of soundwave propagation in the BCmodel with bimodal crystal fieldby combining the statistical equilibrium theory and the ther-modynamics of linear irreversible processes This approachhas been utilized to investigate the sound propagation in agreat variety of model systems Making use of the lowestapproximation of the cluster variation method and linearresponse theory of irreversible processes Erdem and Keskinperformed the calculations of the sound attenuation nearthe critical point in the Blume-Emery-Griffiths (BEG) modelwith zero crystal field [35ndash37] Gulpinar investigated thecritical behavior of ultrasound wave absorption coefficientin the metamagnetic Ising model within the mean-fieldapproximation [38] Later the absorption of sound in thespin-32 Ising model on the Bethe lattice is obtained and itstemperature variance is analyzed near the phase transitionpoints [39] Recently Cengiz and Albayrak studied the soundattenuation phenomena for a finite crystal field BEG modelon the Bethe lattice in terms of the recursion relationsby using the Onsager theory [40] Due to mathematicalcomplexity the properties of critical sound propagation havenot been studied in any spin system with random bondrandom magnetic field or random crystal field terms in theHamiltonian expression To the best of our knowledge thecritical dynamics of the sound propagation of the BC modelwith quenched diluted single-ion anisotropy have not beenstudied by the methods of irreversible thermodynamics It isassumed in this manuscript that the sound wave is coupled tothe order parameter fluctuations that decay mainly via orderparameter relaxation process and the steady-state dynamicsof the BCmodel with random diluted crystal field are formu-lated while the system is under the effect of a propagatingsound wave of frequency 120596 which let us obtain the sounddispersion relation and sound attenuation coefficient for alltemperatures and frequencies that contain effectively onlyone phenomenological rate coefficient Temperature varianceof sound dispersion and absorption has been investigatedin the vicinity of the critical point Finally the frequencybehavior of the attenuation coefficient for temperatures ispresented in the vicinity of second-order transition fromordered to disordered phase

The paper is organized as follows the model and itsstatic properties are presented briefly in Section 2 Nextthe free energy production near the equilibrium is statedand the order parameter relaxation time is obtained in

Section 3 The steady solution of the kinetic equation of theorder parameter and expressions for the sound dispersionrelation and ultrasonic attenuation coefficient are obtained inSection 4 Finally frequency and temperature behaviors of theultrasonic attenuation are analyzed and the discussion of theresults is given in Section 5

2 The Model and Its Equilibrium Properties

The BC model with random single-ion anisotropy in thepresence of an external magnetic field 119867 is described by theHamiltonian

= minus119869sum⟨119894119895⟩

120590119894120590119895 minus 119867sum119894

120590119894 + sum119894

Δ 1198941205902119894 (1)

with 120590119894 = minus1 0 1 Here 119869 gt 0 is the exchange interaction dueto ferromagnetic coupling and Δ 119894 is the crystal field acting onsite i with a distribution function

119865 (Δ 119894) = 119901120575 (Δ 119894 minus 119863) + (1 minus 119901) 120575 (Δ 119894) (2)

where 119901 is concentration of the spins on the lattice whichare influenced by a crystal field The mean-field free energyfor a nonvanishing external field is given by the followingexpression [41 42]

Ψ = Ψ0 (119881 (119886) 119879) + 1198731199111198692 1198982

+ 119873119879int ln(1 minus 2 cosh ((119869119911119898 + 119867) 119879)2 cosh ((119869119911119898 + 119867) 119879) + 119890Δ 119894119879)119865 (Δ 119894) 119889Δ 119894(3)

where Ψ0(119881(119886) 119879) is the lattice-free energy that is indepen-dent of spin configuration Here 119886 119881 and 119898 =lt 120590119894 gt are thelattice constant volume and order parameter of the systemrespectively In addition lt gt denotes thermal expectationvalue If one makes use of (2) the mean-field free energybecomes

Ψ = Ψ0 (119881 (119886) 119879) + 1198731199111198691198982

2 + 119873119901Δminus 119901119873119896119861119879 ln(2 cosh 119869119911119898 + 119867119879 + 119890Δ119879)+ (1 minus 119901)119873119896119861119879 ln(2 cosh 119869119911119898 + 119867119879 + 1)

(4)

For the sake of simplicity we will take Boltzmann constantas unity (119896119861 = 1) from now on The equilibrium conditions(120597Ψ120597119898)|119890119902 = 0 and (120597Ψ120597119886)|119890119902 = 0 give the followingself-consistent equations for themagnetization and the latticeconstant



(5)

120597Ψ012059711988610038161003816100381610038161003816100381610038161003816119890119902 = 119873119911120597119869120597119886 [minus1198982


(6)




ΔΨ = Ψ (119898119867 119886) minus Ψ0 (11989801198670 1198860) (7)



(8)


119860 = 1205972Ψ1205971198982

100381610038161003816100381610038161003816100381610038161003816119890119902 119861 = minus 1205972Ψ120597119898120597119867

100381610038161003816100381610038161003816100381610038161003816119890119902 119862 = 1205972Ψ1205971198672

100381610038161003816100381610038161003816100381610038161003816119890119902 119863 = 1205972Ψ120597119898120597119886

100381610038161003816100381610038161003816100381610038161003816119890119902 = 1205972Ψ120597119898120597119869100381610038161003816100381610038161003816100381610038161003816119890119902 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902 119864 = 1205972Ψ120597119867120597119886

100381610038161003816100381610038161003816100381610038161003816119890119902 = 1205972Ψ120597119867120597119869100381610038161003816100381610038161003816100381610038161003816119890119902 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902 119865 = 1205972Ψ1205971198862

100381610038161003816100381610038161003816100381610038161003816119890119902= 1205972Ψ01205971198862

100381610038161003816100381610038161003816100381610038161003816119890119902 + 1205972Ψ1205971198692100381610038161003816100381610038161003816100381610038161003816119890119902 ( 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902)2 + 120597Ψ120597119869

10038161003816100381610038161003816100381610038161003816119890119902 12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902

119866 = 120597Ψ12059711986710038161003816100381610038161003816100381610038161003816119890119902

(9)






(10)


= 119889 (119898 minus 1198980)119889119905 (11)





= minus119871119860 (119898 minus 1198980) (13)


120591 = 1119871119860 (14)





119898 minus 1198980 = 1198981119890119894120596119905 (15)



1198941205961198981119890119894120596119905 = minus1198711198601198981119890119894120596119905 minus 1198711198631198861119890119894120596119905 (16)



11989811198861 = minus 1198711198631205911 + 119894120596120591 (18)






( 120597119901120597119881)119904119900119906119899119889

= minus 11988620911988120

[11986311989811198861 + 119865] (21)


(120597119901120597120588)119904119900119906119899119889

= 119886209119872 [119865 minus 11987111986321205911 + 119894120596120591] (22)



119888119904119900119906119899119889 cong 11988603 radic 119865119872 [1 minus 11987111986321205912119865 (1 + 119894120596120591)] (23)






(radic119865)3119871119863212059621205912

(1 minus 11987111986321205912119865 ) + 12059621205912 (25)



radic119865 ≃ radic 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902

1 + 12 ( 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902)

minus1

sdot [ 1205972Ψ1205971198692100381610038161003816100381610038161003816100381610038161003816119890119902 ( 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902)2 + 120597Ψ120597119869

10038161003816100381610038161003816100381610038161003816119890119902 12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902]

(26)

(radic119865)minus3 ≃ ( 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902)

minus32 1 minus 32 ( 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902)

minus1

sdot [ 1205972Ψ1205971198692100381610038161003816100381610038161003816100381610038161003816119890119902 ( 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902)2 + 120597Ψ120597119869

10038161003816100381610038161003816100381610038161003816119890119902 12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902]

(27)


119888 (120596) = 119888infin 1 + 12 ( 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902)

minus1

sdot [ 1205972Ψ1205971198692100381610038161003816100381610038161003816100381610038161003816119890119902 ( 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902)2 + 120597Ψ120597119869

10038161003816100381610038161003816100381610038161003816119890119902 12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902]

sdot 2119865 minus 11987111986321205912119865 (1 + 12059621205912)

(28)

120572 (120596) = 1198862018119872 11198883infin1 minus 32 ( 1205972Ψ01205971198862

100381610038161003816100381610038161003816100381610038161003816119890119902)minus1

sdot [ 1205972Ψ1205971198692100381610038161003816100381610038161003816100381610038161003816119890119902 ( 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902)2 + 120597Ψ120597119869

10038161003816100381610038161003816100381610038161003816119890119902 12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902]

sdot 119871119863212059621205912(1 minus 11987111986321205912119865) + 12059621205912

(29)




120579 = 119879119911119869 119889 = 119863119911119869 ℎ = 119867119911119869

(30)




015 030 045

=20

=05

025

050

c

(a)

075

100




0

minus6

minus12

minus18

minus8 minus4 0 4

ＦＩＡ10

ＦＩＡ10

=010

=0397

slope=

20

slope=00

(a)






80times10minus5

40times10minus5

00

=18times10minus5

12times10minus5

＝

0392 0396 0400

(a)

＝

10times10minus1

50times10minus2

00

01 02 03 04

=10times101

20times10minus1

20times10minus3

(b)

＝

01 02 03 04

008

004

000

=10times101

=10times102

=10times103

(c)





＝

= 60times10minus3

L=05

1040

0026

0013

0000

038 039 040

(a)

＝

()

08

04

00

01 02 03 04

01

02

04


(b)









ＦＩＡ10


minus12

minus16

minus20


slope=minus10

=110minus5

=110minus6

=110minus7

(a)

ＦＩＡ10


minus600

minus675

minus750

minus825


slope = 10

=110minus2=1102

(b)



120588119897 = minuslim120576rarr0

ln |120572 (120596 120576)|ln |120576| for 120596120591 ≪ 1


ln |120572 (120596 120576)|ln |120576| for 120596120591 ≫ 1 (32)











Appendix








119863 = 119873119911 12059711986912059711988610038161003816100381610038161003816100381610038161003816119890119902 [1198980 minus 21198691199111198980119901 cosh (1198691199111198980119879)119879 [2 cosh (1198691199111198980119879) + 119890Δ119879]

minus 2119901 sinh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 119890Δ119879





119865 = 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902 + 1198731199111198980

12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902 [11989802


0119879 ( 12059711986912059711988610038161003816100381610038161003816100381610038161003816119890119902)

2




(A1)

Data Availability


Disclosure




References





































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery







(5)

120597Ψ012059711988610038161003816100381610038161003816100381610038161003816119890119902 = 119873119911120597119869120597119886 [minus1198982


(6)




ΔΨ = Ψ (119898119867 119886) minus Ψ0 (11989801198670 1198860) (7)



(8)


119860 = 1205972Ψ1205971198982

100381610038161003816100381610038161003816100381610038161003816119890119902 119861 = minus 1205972Ψ120597119898120597119867

100381610038161003816100381610038161003816100381610038161003816119890119902 119862 = 1205972Ψ1205971198672

100381610038161003816100381610038161003816100381610038161003816119890119902 119863 = 1205972Ψ120597119898120597119886

100381610038161003816100381610038161003816100381610038161003816119890119902 = 1205972Ψ120597119898120597119869100381610038161003816100381610038161003816100381610038161003816119890119902 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902 119864 = 1205972Ψ120597119867120597119886

100381610038161003816100381610038161003816100381610038161003816119890119902 = 1205972Ψ120597119867120597119869100381610038161003816100381610038161003816100381610038161003816119890119902 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902 119865 = 1205972Ψ1205971198862

100381610038161003816100381610038161003816100381610038161003816119890119902= 1205972Ψ01205971198862

100381610038161003816100381610038161003816100381610038161003816119890119902 + 1205972Ψ1205971198692100381610038161003816100381610038161003816100381610038161003816119890119902 ( 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902)2 + 120597Ψ120597119869

10038161003816100381610038161003816100381610038161003816119890119902 12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902

119866 = 120597Ψ12059711986710038161003816100381610038161003816100381610038161003816119890119902

(9)






(10)


= 119889 (119898 minus 1198980)119889119905 (11)





= minus119871119860 (119898 minus 1198980) (13)


120591 = 1119871119860 (14)





119898 minus 1198980 = 1198981119890119894120596119905 (15)



1198941205961198981119890119894120596119905 = minus1198711198601198981119890119894120596119905 minus 1198711198631198861119890119894120596119905 (16)



11989811198861 = minus 1198711198631205911 + 119894120596120591 (18)






( 120597119901120597119881)119904119900119906119899119889

= minus 11988620911988120

[11986311989811198861 + 119865] (21)


(120597119901120597120588)119904119900119906119899119889

= 119886209119872 [119865 minus 11987111986321205911 + 119894120596120591] (22)



119888119904119900119906119899119889 cong 11988603 radic 119865119872 [1 minus 11987111986321205912119865 (1 + 119894120596120591)] (23)






(radic119865)3119871119863212059621205912

(1 minus 11987111986321205912119865 ) + 12059621205912 (25)



radic119865 ≃ radic 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902

1 + 12 ( 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902)

minus1

sdot [ 1205972Ψ1205971198692100381610038161003816100381610038161003816100381610038161003816119890119902 ( 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902)2 + 120597Ψ120597119869

10038161003816100381610038161003816100381610038161003816119890119902 12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902]

(26)

(radic119865)minus3 ≃ ( 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902)

minus32 1 minus 32 ( 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902)

minus1

sdot [ 1205972Ψ1205971198692100381610038161003816100381610038161003816100381610038161003816119890119902 ( 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902)2 + 120597Ψ120597119869

10038161003816100381610038161003816100381610038161003816119890119902 12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902]

(27)


119888 (120596) = 119888infin 1 + 12 ( 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902)

minus1

sdot [ 1205972Ψ1205971198692100381610038161003816100381610038161003816100381610038161003816119890119902 ( 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902)2 + 120597Ψ120597119869

10038161003816100381610038161003816100381610038161003816119890119902 12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902]

sdot 2119865 minus 11987111986321205912119865 (1 + 12059621205912)

(28)

120572 (120596) = 1198862018119872 11198883infin1 minus 32 ( 1205972Ψ01205971198862

100381610038161003816100381610038161003816100381610038161003816119890119902)minus1

sdot [ 1205972Ψ1205971198692100381610038161003816100381610038161003816100381610038161003816119890119902 ( 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902)2 + 120597Ψ120597119869

10038161003816100381610038161003816100381610038161003816119890119902 12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902]

sdot 119871119863212059621205912(1 minus 11987111986321205912119865) + 12059621205912

(29)




120579 = 119879119911119869 119889 = 119863119911119869 ℎ = 119867119911119869

(30)




015 030 045

=20

=05

025

050

c

(a)

075

100




0

minus6

minus12

minus18

minus8 minus4 0 4

ＦＩＡ10

ＦＩＡ10

=010

=0397

slope=

20

slope=00

(a)






80times10minus5

40times10minus5

00

=18times10minus5

12times10minus5

＝

0392 0396 0400

(a)

＝

10times10minus1

50times10minus2

00

01 02 03 04

=10times101

20times10minus1

20times10minus3

(b)

＝

01 02 03 04

008

004

000

=10times101

=10times102

=10times103

(c)





＝

= 60times10minus3

L=05

1040

0026

0013

0000

038 039 040

(a)

＝

()

08

04

00

01 02 03 04

01

02

04


(b)









ＦＩＡ10


minus12

minus16

minus20


slope=minus10

=110minus5

=110minus6

=110minus7

(a)

ＦＩＡ10


minus600

minus675

minus750

minus825


slope = 10

=110minus2=1102

(b)



120588119897 = minuslim120576rarr0

ln |120572 (120596 120576)|ln |120576| for 120596120591 ≪ 1


ln |120572 (120596 120576)|ln |120576| for 120596120591 ≫ 1 (32)











Appendix








119863 = 119873119911 12059711986912059711988610038161003816100381610038161003816100381610038161003816119890119902 [1198980 minus 21198691199111198980119901 cosh (1198691199111198980119879)119879 [2 cosh (1198691199111198980119879) + 119890Δ119879]

minus 2119901 sinh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 119890Δ119879





119865 = 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902 + 1198731199111198980

12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902 [11989802


0119879 ( 12059711986912059711988610038161003816100381610038161003816100381610038161003816119890119902)

2




(A1)

Data Availability


Disclosure




References





































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery








(10)


= 119889 (119898 minus 1198980)119889119905 (11)





= minus119871119860 (119898 minus 1198980) (13)


120591 = 1119871119860 (14)





119898 minus 1198980 = 1198981119890119894120596119905 (15)



1198941205961198981119890119894120596119905 = minus1198711198601198981119890119894120596119905 minus 1198711198631198861119890119894120596119905 (16)



11989811198861 = minus 1198711198631205911 + 119894120596120591 (18)






( 120597119901120597119881)119904119900119906119899119889

= minus 11988620911988120

[11986311989811198861 + 119865] (21)


(120597119901120597120588)119904119900119906119899119889

= 119886209119872 [119865 minus 11987111986321205911 + 119894120596120591] (22)



119888119904119900119906119899119889 cong 11988603 radic 119865119872 [1 minus 11987111986321205912119865 (1 + 119894120596120591)] (23)






(radic119865)3119871119863212059621205912

(1 minus 11987111986321205912119865 ) + 12059621205912 (25)



radic119865 ≃ radic 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902

1 + 12 ( 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902)

minus1

sdot [ 1205972Ψ1205971198692100381610038161003816100381610038161003816100381610038161003816119890119902 ( 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902)2 + 120597Ψ120597119869

10038161003816100381610038161003816100381610038161003816119890119902 12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902]

(26)

(radic119865)minus3 ≃ ( 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902)

minus32 1 minus 32 ( 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902)

minus1

sdot [ 1205972Ψ1205971198692100381610038161003816100381610038161003816100381610038161003816119890119902 ( 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902)2 + 120597Ψ120597119869

10038161003816100381610038161003816100381610038161003816119890119902 12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902]

(27)


119888 (120596) = 119888infin 1 + 12 ( 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902)

minus1

sdot [ 1205972Ψ1205971198692100381610038161003816100381610038161003816100381610038161003816119890119902 ( 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902)2 + 120597Ψ120597119869

10038161003816100381610038161003816100381610038161003816119890119902 12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902]

sdot 2119865 minus 11987111986321205912119865 (1 + 12059621205912)

(28)

120572 (120596) = 1198862018119872 11198883infin1 minus 32 ( 1205972Ψ01205971198862

100381610038161003816100381610038161003816100381610038161003816119890119902)minus1

sdot [ 1205972Ψ1205971198692100381610038161003816100381610038161003816100381610038161003816119890119902 ( 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902)2 + 120597Ψ120597119869

10038161003816100381610038161003816100381610038161003816119890119902 12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902]

sdot 119871119863212059621205912(1 minus 11987111986321205912119865) + 12059621205912

(29)




120579 = 119879119911119869 119889 = 119863119911119869 ℎ = 119867119911119869

(30)




015 030 045

=20

=05

025

050

c

(a)

075

100




0

minus6

minus12

minus18

minus8 minus4 0 4

ＦＩＡ10

ＦＩＡ10

=010

=0397

slope=

20

slope=00

(a)






80times10minus5

40times10minus5

00

=18times10minus5

12times10minus5

＝

0392 0396 0400

(a)

＝

10times10minus1

50times10minus2

00

01 02 03 04

=10times101

20times10minus1

20times10minus3

(b)

＝

01 02 03 04

008

004

000

=10times101

=10times102

=10times103

(c)





＝

= 60times10minus3

L=05

1040

0026

0013

0000

038 039 040

(a)

＝

()

08

04

00

01 02 03 04

01

02

04


(b)









ＦＩＡ10


minus12

minus16

minus20


slope=minus10

=110minus5

=110minus6

=110minus7

(a)

ＦＩＡ10


minus600

minus675

minus750

minus825


slope = 10

=110minus2=1102

(b)



120588119897 = minuslim120576rarr0

ln |120572 (120596 120576)|ln |120576| for 120596120591 ≪ 1


ln |120572 (120596 120576)|ln |120576| for 120596120591 ≫ 1 (32)











Appendix








119863 = 119873119911 12059711986912059711988610038161003816100381610038161003816100381610038161003816119890119902 [1198980 minus 21198691199111198980119901 cosh (1198691199111198980119879)119879 [2 cosh (1198691199111198980119879) + 119890Δ119879]

minus 2119901 sinh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 119890Δ119879





119865 = 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902 + 1198731199111198980

12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902 [11989802


0119879 ( 12059711986912059711988610038161003816100381610038161003816100381610038161003816119890119902)

2




(A1)

Data Availability


Disclosure




References





































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery










(radic119865)3119871119863212059621205912

(1 minus 11987111986321205912119865 ) + 12059621205912 (25)



radic119865 ≃ radic 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902

1 + 12 ( 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902)

minus1

sdot [ 1205972Ψ1205971198692100381610038161003816100381610038161003816100381610038161003816119890119902 ( 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902)2 + 120597Ψ120597119869

10038161003816100381610038161003816100381610038161003816119890119902 12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902]

(26)

(radic119865)minus3 ≃ ( 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902)

minus32 1 minus 32 ( 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902)

minus1

sdot [ 1205972Ψ1205971198692100381610038161003816100381610038161003816100381610038161003816119890119902 ( 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902)2 + 120597Ψ120597119869

10038161003816100381610038161003816100381610038161003816119890119902 12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902]

(27)


119888 (120596) = 119888infin 1 + 12 ( 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902)

minus1

sdot [ 1205972Ψ1205971198692100381610038161003816100381610038161003816100381610038161003816119890119902 ( 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902)2 + 120597Ψ120597119869

10038161003816100381610038161003816100381610038161003816119890119902 12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902]

sdot 2119865 minus 11987111986321205912119865 (1 + 12059621205912)

(28)

120572 (120596) = 1198862018119872 11198883infin1 minus 32 ( 1205972Ψ01205971198862

100381610038161003816100381610038161003816100381610038161003816119890119902)minus1

sdot [ 1205972Ψ1205971198692100381610038161003816100381610038161003816100381610038161003816119890119902 ( 120597119869120597119886

10038161003816100381610038161003816100381610038161003816119890119902)2 + 120597Ψ120597119869

10038161003816100381610038161003816100381610038161003816119890119902 12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902]

sdot 119871119863212059621205912(1 minus 11987111986321205912119865) + 12059621205912

(29)




120579 = 119879119911119869 119889 = 119863119911119869 ℎ = 119867119911119869

(30)




015 030 045

=20

=05

025

050

c

(a)

075

100




0

minus6

minus12

minus18

minus8 minus4 0 4

ＦＩＡ10

ＦＩＡ10

=010

=0397

slope=

20

slope=00

(a)






80times10minus5

40times10minus5

00

=18times10minus5

12times10minus5

＝

0392 0396 0400

(a)

＝

10times10minus1

50times10minus2

00

01 02 03 04

=10times101

20times10minus1

20times10minus3

(b)

＝

01 02 03 04

008

004

000

=10times101

=10times102

=10times103

(c)





＝

= 60times10minus3

L=05

1040

0026

0013

0000

038 039 040

(a)

＝

()

08

04

00

01 02 03 04

01

02

04


(b)









ＦＩＡ10


minus12

minus16

minus20


slope=minus10

=110minus5

=110minus6

=110minus7

(a)

ＦＩＡ10


minus600

minus675

minus750

minus825


slope = 10

=110minus2=1102

(b)



120588119897 = minuslim120576rarr0

ln |120572 (120596 120576)|ln |120576| for 120596120591 ≪ 1


ln |120572 (120596 120576)|ln |120576| for 120596120591 ≫ 1 (32)











Appendix








119863 = 119873119911 12059711986912059711988610038161003816100381610038161003816100381610038161003816119890119902 [1198980 minus 21198691199111198980119901 cosh (1198691199111198980119879)119879 [2 cosh (1198691199111198980119879) + 119890Δ119879]

minus 2119901 sinh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 119890Δ119879





119865 = 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902 + 1198731199111198980

12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902 [11989802


0119879 ( 12059711986912059711988610038161003816100381610038161003816100381610038161003816119890119902)

2




(A1)

Data Availability


Disclosure




References





































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






015 030 045

=20

=05

025

050

c

(a)

075

100




0

minus6

minus12

minus18

minus8 minus4 0 4

ＦＩＡ10

ＦＩＡ10

=010

=0397

slope=

20

slope=00

(a)






80times10minus5

40times10minus5

00

=18times10minus5

12times10minus5

＝

0392 0396 0400

(a)

＝

10times10minus1

50times10minus2

00

01 02 03 04

=10times101

20times10minus1

20times10minus3

(b)

＝

01 02 03 04

008

004

000

=10times101

=10times102

=10times103

(c)





＝

= 60times10minus3

L=05

1040

0026

0013

0000

038 039 040

(a)

＝

()

08

04

00

01 02 03 04

01

02

04


(b)









ＦＩＡ10


minus12

minus16

minus20


slope=minus10

=110minus5

=110minus6

=110minus7

(a)

ＦＩＡ10


minus600

minus675

minus750

minus825


slope = 10

=110minus2=1102

(b)



120588119897 = minuslim120576rarr0

ln |120572 (120596 120576)|ln |120576| for 120596120591 ≪ 1


ln |120572 (120596 120576)|ln |120576| for 120596120591 ≫ 1 (32)











Appendix








119863 = 119873119911 12059711986912059711988610038161003816100381610038161003816100381610038161003816119890119902 [1198980 minus 21198691199111198980119901 cosh (1198691199111198980119879)119879 [2 cosh (1198691199111198980119879) + 119890Δ119879]

minus 2119901 sinh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 119890Δ119879





119865 = 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902 + 1198731199111198980

12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902 [11989802


0119879 ( 12059711986912059711988610038161003816100381610038161003816100381610038161003816119890119902)

2




(A1)

Data Availability


Disclosure




References





































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






80times10minus5

40times10minus5

00

=18times10minus5

12times10minus5

＝

0392 0396 0400

(a)

＝

10times10minus1

50times10minus2

00

01 02 03 04

=10times101

20times10minus1

20times10minus3

(b)

＝

01 02 03 04

008

004

000

=10times101

=10times102

=10times103

(c)





＝

= 60times10minus3

L=05

1040

0026

0013

0000

038 039 040

(a)

＝

()

08

04

00

01 02 03 04

01

02

04


(b)









ＦＩＡ10


minus12

minus16

minus20


slope=minus10

=110minus5

=110minus6

=110minus7

(a)

ＦＩＡ10


minus600

minus675

minus750

minus825


slope = 10

=110minus2=1102

(b)



120588119897 = minuslim120576rarr0

ln |120572 (120596 120576)|ln |120576| for 120596120591 ≪ 1


ln |120572 (120596 120576)|ln |120576| for 120596120591 ≫ 1 (32)











Appendix








119863 = 119873119911 12059711986912059711988610038161003816100381610038161003816100381610038161003816119890119902 [1198980 minus 21198691199111198980119901 cosh (1198691199111198980119879)119879 [2 cosh (1198691199111198980119879) + 119890Δ119879]

minus 2119901 sinh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 119890Δ119879





119865 = 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902 + 1198731199111198980

12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902 [11989802


0119879 ( 12059711986912059711988610038161003816100381610038161003816100381610038161003816119890119902)

2




(A1)

Data Availability


Disclosure




References





































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






＝

= 60times10minus3

L=05

1040

0026

0013

0000

038 039 040

(a)

＝

()

08

04

00

01 02 03 04

01

02

04


(b)









ＦＩＡ10


minus12

minus16

minus20


slope=minus10

=110minus5

=110minus6

=110minus7

(a)

ＦＩＡ10


minus600

minus675

minus750

minus825


slope = 10

=110minus2=1102

(b)



120588119897 = minuslim120576rarr0

ln |120572 (120596 120576)|ln |120576| for 120596120591 ≪ 1


ln |120572 (120596 120576)|ln |120576| for 120596120591 ≫ 1 (32)











Appendix








119863 = 119873119911 12059711986912059711988610038161003816100381610038161003816100381610038161003816119890119902 [1198980 minus 21198691199111198980119901 cosh (1198691199111198980119879)119879 [2 cosh (1198691199111198980119879) + 119890Δ119879]

minus 2119901 sinh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 119890Δ119879





119865 = 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902 + 1198731199111198980

12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902 [11989802


0119879 ( 12059711986912059711988610038161003816100381610038161003816100381610038161003816119890119902)

2




(A1)

Data Availability


Disclosure




References





































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






ＦＩＡ10


minus12

minus16

minus20


slope=minus10

=110minus5

=110minus6

=110minus7

(a)

ＦＩＡ10


minus600

minus675

minus750

minus825


slope = 10

=110minus2=1102

(b)



120588119897 = minuslim120576rarr0

ln |120572 (120596 120576)|ln |120576| for 120596120591 ≪ 1


ln |120572 (120596 120576)|ln |120576| for 120596120591 ≫ 1 (32)











Appendix








119863 = 119873119911 12059711986912059711988610038161003816100381610038161003816100381610038161003816119890119902 [1198980 minus 21198691199111198980119901 cosh (1198691199111198980119879)119879 [2 cosh (1198691199111198980119879) + 119890Δ119879]

minus 2119901 sinh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 119890Δ119879





119865 = 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902 + 1198731199111198980

12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902 [11989802


0119879 ( 12059711986912059711988610038161003816100381610038161003816100381610038161003816119890119902)

2




(A1)

Data Availability


Disclosure




References





































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery











Appendix








119863 = 119873119911 12059711986912059711988610038161003816100381610038161003816100381610038161003816119890119902 [1198980 minus 21198691199111198980119901 cosh (1198691199111198980119879)119879 [2 cosh (1198691199111198980119879) + 119890Δ119879]

minus 2119901 sinh (1198691199111198980119879)2 cosh (1198691199111198980119879) + 119890Δ119879





119865 = 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902 + 1198731199111198980

12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902 [11989802


0119879 ( 12059711986912059711988610038161003816100381610038161003816100381610038161003816119890119902)

2




(A1)

Data Availability


Disclosure




References





































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery









119865 = 1205972Ψ01205971198862100381610038161003816100381610038161003816100381610038161003816119890119902 + 1198731199111198980

12059721198691205971198862100381610038161003816100381610038161003816100381610038161003816119890119902 [11989802


0119879 ( 12059711986912059711988610038161003816100381610038161003816100381610038161003816119890119902)

2




(A1)

Data Availability


Disclosure




References





































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery



























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery





acoustic signatures of the phases and phase transitions in the … · 2019. 7. 30. · reseaarticle...

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