On the Order Parameter of the Continuous Phase Transition in the Classical and Quantum Mechanical limits
C. A. M. dos Santos, F. S. Oliveira, M. S. da Luz, J. J. Neumeier
OOn the Order Parameter of the Continuous Phase Transition in the Classical andQuantum Mechanical limits
C. A. M. dos Santos , F. S. Oliveira , M. S. da Luz , and J. J. Neumeier Universidade de S˜ao Paulo , Escola de Engenharia de Lorena , Lorena - SP , , Brazil Universidade Federal do Triˆangulo Mineiro , Instituto de Ciˆencias Tecnol´ogicas e Exatas , Uberaba - MG , , Brazil and Montana State University , Physics Department , Bozeman - MT , , USA
The mean field theory is revisited in the classical and quantum mechanical limits. Taking intoaccount the boundary conditions at the phase transition and the third law of the thermodynamicsthe physical properties of the ordered and disordered phases were reported. The equation for theorder parameter predicts the occurrence of a saturation of Ψ = 1 near Θ S , the temperature belowthe quantum mechanical ground state is reached. The theoretical predictions are also comparedwith high resolution thermal expansion data of SrTiO monocrystalline samples and other someprevious results. An excellent agreement has been found suggesting a universal behavior of thetheoretical model to describe continuous structural phase transitions. a r X i v : . [ c ond - m a t . o t h e r] F e b I. INTRODUCTION
Mean field theory, first developed by Landau [1–3], has successfully described most of the continuous phase tran-sitions, such as structural distortions [4] , magnetic [5], and superconducting transitions [6], by introducing an orderparameter (Ψ) which describes many physical properties based upon the fraction of both order and disordered phasescoexisting in a given temperature below the critical temperature of the phase transition [3].This theory is better applied near the phase transition ( T ∼ T C ), where the density of the ordered phase, givenby Ψ is small, because the free energy can be computed by a power series of Ψ. The solution to minimize the freeenergy near T C provides Ψ ∝ ( T C − T ) β , with β between 0.25 and 0.50 [7, 8]. Some authors have recognized this asthe classical limit of the mean field theory [9, 10].On the other hand, describing the physical properties at low temperature limit ( T (cid:28) T C ) is a challenge since thedensity of the ordered phase is high (Ψ ∼
1) and free energy cannot be expressed by a mathematical series [11]. Thisis the quantum limit in which physical properties must reach saturations due to a quantum mechanical ground state[12–14].One of the most successful theoretical description which takes into account the saturation of order parameter hasto do with Thomas, Salje and collaborators [11, 15–18], who have included a harmonic oscillation term in the freeenergy due to the soft phonon modes related to the continuous displacive phase transitions [19–22]. The model wassuccessfully applied to described several physical properties of the many materials [4, 11, 15, 17–19, 21–25] and seemsto hold an universal behavior for this type of structural phase transition (see for instance figure 1 in reference [16]).Despite the successful of this model, our recent results on high resolution thermal expansion measurements (HRTE)[26], which has relative resolution 100 to 1000 times better than diffractometric techniques [27] as well as thousandsof data points in each measurement, performed in SrTiO single crystals [28], have brought some important insightsregarding to the order parameter saturation, especially due to the saturation of the volumetric thermal expansionat low temperatures, which must respect the third law of thermodynamics, i. e. the thermal expansion coefficient(Ω = 1 /V C d (∆ V ) /dT ) must be zero as the temperature approaches absolute zero.Thus, we have revisited the theoretical model by Salje et al. [16] in order to carefully take into account the boundaryconditions at the phase transition ( T = T C ), which should respect the continuity of the free energy ( G ), volume ( V ),entropy ( S ), and energy ( U ) of the ordered and disordered phases, and, at zero temperature, in which S and Ω mustbe equal to zero in order to attend the third law of the thermodynamics [29, 30]. In addition, the equations forthe physical properties in the classical limit should be recovered when the characteristic temperature that holds theground state in quantum mechanical limit vanishes.HRTE measurements performed in SrTiO single crystal shown unambiguously quadratic temperature dependencesin a large temperature interval below the phase transition. The cubic to tetragonal structural transition in thiscompound has been well described by the model reported here. We found a direct experimental evidence that thethermal expansion coefficient is the best physical property to describe the order parameter of this transition. II. CLASSICAL MODEL
Taking the classical mean field theory by Landau [1–3] for a continuous phase transition, the Gibbs free energy isgenerically given by G = G D + a ( T C − T ) Ψ + b Ψ + ... (1)where a and b are constants, and G D is the Gibbs free energy of the disordered phase, when Ψ = 0. Keeping only thefirst three terms of the series, the equilibrium order parameter can be obtained by dGd (Ψ ) = 0 , (2)which implies Ψ = 0 , (3)for disordered phase (D) at T > T C and, Ψ = − a ( T C − T )2 b , (4)which describes the order parameter and the density of the ordered phase (O) at T ≤ T C . Inserting equation 4 into1 provides G = G D − a b ( T C − T ) . (5)Taking into account only the effects of entropy and volume in a structural phase transition, the Gibbs free energy isa function of the temperature and pressure, G = G ( T, P ), that implies dG = − SdT + V dP (6)in which − S = (cid:18) ∂G∂T (cid:19) P , (7)and V = (cid:18) ∂G∂P (cid:19) T . (8)Comparing equations 5 and 6, and remembering that the entropy of the disordered phase is assumed to be temper-ature independent, which is given by S D = − (cid:18) ∂G D ∂T (cid:19) P , (9)one can find G D = G − S D T, (10)where G is a constant which defines a reference for the free energy.Furthermore, taking into account the boundary conditions G = G D = G O , S = S D = S O = S C , and Ψ = 0 at thephase transition ( T = T C ), and S = 0 and Ψ = 1 at T = 0, due to third law of thermodynamics, it is possible toshow that a = S C , (11) b = 12 S C T C , (12)Ψ = ( T C − T ) T C = 1 − TT C , (13)and G = G − S C T − S C ( T C − T ) Ψ + 12 S C T C Ψ , (14)or G = G − S C T − S C T C ( T C − T ) . (15)Taking the derivative of equation 14 with regard to temperature, it is possible to find that S = S C (cid:0) − Ψ (cid:1) = S C TT C . (16)Furthermore, such as V and S are independent variables, one can use the relation − (cid:18) ∂S∂P (cid:19) T = (cid:18) ∂V∂T (cid:19) P = Ω , (17)to demonstrate that − (cid:18) ∂S∂P (cid:19) T = − (cid:18) ∂S C ∂P (cid:19) T (cid:0) − Ψ (cid:1) − S C (cid:32) ∂ (cid:0) Ψ (cid:1) ∂P (cid:33) T , (18)or Ω = Ω D (cid:0) − Ψ (cid:1) + S C TT (cid:18) dT C dP (cid:19) , (19)where T C holds all the pressure dependence of Ψ in the equation 18 and dT C /dP measures the pressure dependenceof the critical temperature.But at T = T C , Ψ = 0, which implies that the thermal expansion coefficient of the ordered phase (Ω O ) is differentthan that of the disordered phase (Ω D ) due to the lambda-type jump at the transition temperature. Thus, fromequation 19 one can write Ω D = Ω O − S C T C (cid:18) dT C dP (cid:19) , (20)which put back into equation 19, remembering that (cid:0) − Ψ (cid:1) = T /T C from equation 13, leads toΩ = Ω O (cid:0) − Ψ (cid:1) . (21)Thus, Ψ can be described as a function of the fundamental thermodynamic properties T , S , or Ω asΨ = 1 − TT C , (22)Ψ = 1 − SS C , (23)and, Ψ = 1 − ΩΩ O , (24)which predict linear dependencies of S and Ω as a function of the temperature. III. QUANTUM MECHANICAL MODEL
Regarding to the saturation of order parameter at low temperature Thomas, Salje, and other coworkers [11, 15–18]have proposed a modification of the free energy to take into account quantum mechanical aspects, especially theharmonic oscillations due to soft modes, which are developed below the critical temperature of the structural phasetransition. The free energy given by equation 1 from classical limit can be rewritten in the following form related tothe quantum mechanical limit (25) G = G D + a S [coth (Θ S /T ) − coth (Θ S /T C )] Ψ + b . As far as we know, this equation has appeared for the first time in the report by Salje et al. in 1991 (see equation 37in reference [16]). They have applied it to describe the behavior of many displacive transitions in several compounds.The Θ S measures a temperature in which ground state in quantum mechanical limit becomes relevant.After an extensive mathematical work using similar procedure and the same boundary conditions at T C and at zerotemperature to find the equations for the classical limit, we were able to find the physical parameters a , b , and G D of Table I. Some thermodynamic properties in the classical and quantum mechanical limits for the mean field model described inthis work. Last column displays the equations for the quantum mechanical limit for Θ S much smaller than T and T C . Parameter Classical limit Quantum mechanical limit T ≈ T C (cid:29) Θ S G D G − S C T G − S C Θ S coth (Θ S /T ) G − S C TS D S C S C (Θ S /T ) csch (Θ S /T ) S C G G D − S C2 ( T C − T ) T C G D − S CΘS2 [ coth ( ΘS /T C ) − coth ( ΘS /T )] [ coth ( ΘS /T C ) − ] G D − S C2 ( T C − T ) ( T C − ΘS )Ψ ( T C − T ) /T C [ coth ( ΘS /T C ) − coth ( ΘS /T )][ coth ( ΘS /T C ) − ] ( T C − T ) / ( T C − Θ S ) S S D (cid:0) − Ψ (cid:1) Ω Ω O (cid:0) − Ψ (cid:1) equation 25, and thermodynamic properties of the continuous phase transition in the quantum mechanical limit (seeAppendix). The Gibbs free energy can be rewritten as (26) G = G − S C Θ S coth (Θ S /T ) − S C Θ S [coth (Θ S /T C ) − coth (Θ S /T )] Ψ + 12 S C Θ S [coth (Θ S /T C ) −
1] Ψ . The first important difference from the previous reports [11, 15–17, 22, 23] has to do with the first two termsof equation 25, which are related to free energy ( G D ) of the disordered phase, that has a temperature dependencemore complicated than in the classical limit ( G D ∝ T ). The discussion afterwards will demonstrate that this termplays an important role in the quantum mechanical description of the total entropy in the low temperature regime.Furthermore, the second relevant observation has to do with the third law of thermodynamics, which requires S =zero at zero temperature, implying a saturation in the order parameter at Ψ = 1 as T approaches zero ( T < Θ S ).This is an important difference since previous reports [11, 16] in which it predicts a saturation of Ψ at a fraction ofone. This has to do with the pre-factor b in equation 25 which normalizes Ψ ( T ) between zero at T = T C and 1 at T →
0. Equation 26 allowed us to find the order parameter in the low temperature phase asΨ = coth (Θ S /T C ) − coth (Θ S /T )coth (Θ S /T C ) − . (27)Futhermore, equation 26 yields the analytical determination of the thermodynamic properties in the quantum me-chanical limit, as shown in table 1 (see details in the Appendix). They are compared with those from the classical limit.Equations for the classical limit are naturally recovered when the quantum mechanical characteristic temperature Θ S is vanished (compare first and last columns).In order to better understand the equations in this model, in figure 1 are plotted the behavior of the main propertiesfor the quantum mechanical (black and red lines) and classical limits (blue and green lines) using Θ S = 20 K and T C = 100 K. The results compare the behavior of the properties below T C , which are composed by the contribution ofboth order and disordered phase densities balanced by the order parameter, with those related only by the disorderedphase, indicated with subindex D.In figure 1(a) are shown the free energy behavior taking G = zero for simplicity. Both curves of each limit reachthe same G value at T = T C , since at this point the phase has the same free energy. In addition, the free energy ofthe ordered phase is lower than in the disordered phase, in both classical and quantum mechanical limits, as expecteddue to the earlier phase be energetically favourable.Figure 1(b) displays the expected behavior for the heat capacity at constant volume as a function of the temperature.In the classical limit, C v ∝ T in the ordered phase and zero at disordered phase ( S D is supposed to be constant).In figure 1(c) are shown the behaviors for the total entropy in the ordered phase ( S ) and the entropy related tothe disordered phase ( S D ). In the classical model, the entropy due to disordered phase is considered temperatureindependent (green line), and total entropy decreases linearly proportional to the temperature based upon the ratio T /T C , from T C down to the ground state at absolute zero (blue line). On the other hand, the results for the quantummechanical regime show temperature dependencies which must be carefully discussed. First of all, S (red line) tendsto zero at finite temperature of the order of Θ S , which is in agreement with the expected by the quantum groundstate and the third law of the thermodynamic. Interesting is the behavior of the total entropy, which is almost linearas a function of the temperature in the interval T /T C = 0.3 to 1, for the Θ S and T C values used in the figure 1. Thisobservation has to do with the weak dependence of S D of approximately 10 % in this temperature interval. Thisseems to explain why the classical model, Ψ ∝ ( T C − T ), has been frequently used to describe quantum mechanicalphase transitions (compare equations for order parameter in table 1). In order to clarify that, we also plotted thetotal entropy in the linear regime (see blue dash line in fig. 1(c)), which is given by S = S C ( T − Θ S ) / ( T C − Θ S ).Besides of the expected the agreement near T C , interesting is to notice that the extrapolation to S = zero yields T = Θ S directly.Figure 1(d) shows the behavior of the Ψ as a function of the temperature. Ψ in the classical limit is linear fromzero up to T C , while in the quantum mechanical regime shows a clear saturation at Ψ = 1 in temperatures aboveabsolute zero, which is clearly related to the ground state with S = 0. The expected behavior of Ψ for T (cid:29) Θ S isalso shown in figure 1(d) (see the dashed line). Its extrapolation to Ψ = 1 yields directly Θ S = 20 K, which agreeswith the fact that of a quantum mechanical ground state is reached close to this temperature.Furthermore, the temperature dependence of S D is very important for the reduction of the total entropy of theordered state which reaches the ground state ( S = 0) in a thermal energy of the order of k B Θ S . Interesting is to notethat not only S goes to zero near Θ S but also S D . Thus, we can understand Θ S as the temperature below which leadsthe compound to a quantum mechanical ground state making S to vanish faster than in the classical limit ( S = 0only at T = 0).The effect of Θ S on S D is displayed in figure 2(a) for several different Θ S values, remembering that T C does notplay any role on S D . It is possible to observe that the higher Θ S the easier the ground state is reached. Furthermore,if one makes T = Θ S , the equation for S D given in table 1 leads to S D = S C csch (1) = 0 . S C , which is shown bythe dashed line in figure 2(a). Interesting is to observe that making Θ S = 0, the classical limit is recovered in which S D = S C = constant (black line).In order to evaluate how Θ S chances the behavior of the order parameter, in figure 2(b) are shown some Ψ curvesas a function of the temperature for different Θ S values using a constant T C = 100 K. It is possible to observe that Ψ reaches the ground state at finites temperatures, if Θ S (cid:54) = 0. Furthermore, the classical behavior is recovered makingΘ S = 0, which represents the linear temperature dependence, Ψ ∝ ( T C − T ).Another important aspect is the shape of the Ψ curves, which are extremely dependent of the Θ S /T C ratio. Thehigher is Θ S /T C , the higher is the saturation due to the ground state (Ψ = 1 and S = 0). Additionally, one cannote that if Θ S /T C ratio tends to infinite, Ψ becomes a step-like function (see ref. [31] and references therein). Thisbehavior reminds a discontinuous (or first order) phase transition, in which the transition from high (disordered)to low (ordered) temperature phase happens abruptly at T = T C (the origin of this observation will be addressedelsewhere).Inset of the figure 2(b) displays the behavior of Ψ near T C for the different Θ S values. All the curves show lineartemperature dependence given by Ψ = ( T C − T ) / ( T C − Θ S ) . (28)Due to this linear behavior near T C , probably many authors have used the classical mean field theory to describephase transitions instead taking into account the quantum mechanical effects.Although there are many similarities between the model reported here with that reported previously by Salje andcoworkers [16], but some important differences can be noticed. The most important is, in their results Ψ neverreaches 1, even at T = 0. We understand this difference because Ψ = 1 must happen at T = 0, since a ground statewith S = 0 is required due to the third law of the thermodynamics. We have a direct experimental evidence for thatusing Ω determined by HTRE experiments performed in SrTiO single crystals as shown in the next section. IV. COMPARISON WITH EXPERIMENTS
Recent HRTE measurements have been performed by us in SrTiO single crystals [28]. Figure 3 shows the tem-perature behavior of the volumetric thermal expansion (∆ V /V C ) measurement performed using a capacitance quartzcell, which shows a clear quadratic behavior in a large temperature interval below the phase transition temperature(105.65 K). This result was also observed in other two oxygen vacancy doped SrTiO single crystals. Thanks toHRTE [26] which has resolution 100 to 1000 times better than diffraction methods [27], has better precision thanmetallic cells [32, 33], and provides thousands of data points in the temperature measurement interval.Figure 3 also shows the volumetric thermal expansion coefficient (Ω). A clear linear behavior is observed from ∼
30 K up to near the critical transition temperature. Furthermore, a saturation at low temperature is also clearlynoticed, in which Ω approaches zero at low temperature. Based upon these results, we see a direct connection with themodel for the quantum mechanical limit described in section 2. Additionally, as pointed out in several previous works[11, 24], the rotation angle ϕ , which measures the antidistortive angle from the cubic to tetragonal in the transition Figure 1. Predicted behavior in the classical and quantum mechanical limits for the (a) free energy, (b) heat capacity atconstant volume, (c) entropy, and (d) order parameter. The curves were plotted using Θ S = 20 K and T C = 100 K. Figure 2. (a) Temperature dependence of the entropy for the disordered phase for different Θ S values. The dashed horizontalline indicates the points where T = Θ S . When Θ S = zero the classical limit of the mean field thery is recovered. In (b) areshown Ψ as a function of the temperature for different Θ S /T C values. Inset shows the linear temperature dependence near T C .Figure 3. (Right scale) Volumetric thermal expansion (∆ V /V C ) measurement of SrTiO single crystal (for details see reference[28]). A clear quadratic temperature dependence can be observed and is indicated by the fitting displayed by the blue line. Insetdisplays a magnification at low temperature in which the expected deviation of the quadratic behavior due to the saturationof ∆ V /V C is observed. (Left scale) Thermal expansion coefficient determined from Ω = d (∆ V /V C ) /dT . The linear behaviorand the saturation at low temperature can be noticed. T C , Ω O , Ω D , and ∆Ω are directly determined from linear extrapolationnear the phase transition. Only 25% of the data points are shown in both curves [34]. Figure 4. Comparison of the order parameter determined from HRTE data for SrTiO single crystal with theoretical prediction. T C , Θ S , and Ω O were determined in Fig. 3. Blue dashed line describes the expected behavior near T ≈ T C . In (b) is shownthe distortive angle ϕ for the cubic to tetragonal phase transition in SrTiO determined from HRTE measurements [28] andby numerical integration using the equation for Ω. Other theoretical predictions reported previously are plotted along in orderto compare the fittings [11, 35, 36]. Insert compares the experimental data with the theoretical predictions near T C . of the SrTiO compound has been directly related as the order parameter. However, our recent results on HRTE [28]suggest that is better to use Ω as the order parameter instead ϕ , since the last one is proportional to (cid:82) Ω dT , whichis not necessarily zero at T = 0. Thus, in agreement to the model developed in this work Ω, should be related tothe Ψ , such as shown in table 1 and in Appendix. Hereafter, we discuss the implication of these observations on theHRTE results obtained in SrTiO .First of all, the equation for thermal expansion coefficient in the quantum mechanical limit near the critical tem-perature can be written as Ω = Ω O (cid:18) T − Θ S T C − Θ S (cid:19) . (29)Thus, taking the temperature at the peaks as the critical temperature T C = 105.65 K and making a linear fit (shownby the dashed blue line) yields directly Ω O = 2 . × − K − and Θ S = 19.5 K, without much efforts to find thefitting parameters as in previous reports [11]. Additionally, one must keep in mind that Ω must be taken subtractingoff the background in order to make Ψ zero right above T C , as required by the mean field theory.Now, the temperature dependence of the Ψ and Ω, given by the correspondent equations in the table 1, can becompared. An excellent agreement between experimental data and theoretical curves in full temperature intervalbelow T C can be noticed. Saturation near Ψ = 1 can be clearly observed, as expected. Additionally, the blue dashedline fits well the behavior near T C and extrapolates to T = Θ S at Ψ = 1. This demonstrate that the Ω instead ϕ isthe best order parameter to describe the antidistortive phase transition in the SrTiO , in agreement with our recentwork [28].0 Figure 5. In (a) is displayed the collapse of the order parameter data previously reported in the references [16, 24, 25] usingequation 30, plotted along with the HRTE data for the SrTiO single crystal. Insert shows the way T C and Θ S are determined.They agree with the data reported for LaAlO [24]. (b) It is shown the collapse in linear form for the data shown in (a). Insetdisplays the collapse around the saturation of Ψ . In order to show the quality of the agreement between the experimental data and the theoretical description inthis work, we compare in figure 4(b) the results for the angle ϕ , obtained from HRTE measurements in SrTiO samples, with the theoretical prediction for Ω( T ), using T C , Ω O , and Θ S directly obtained from figure 3(a) (for moredetails, see reference [28]) with some other theoretical curves reported previously [11, 18, 35, 36]. Although all thetheoretical models show fits close to the experimental data, the model proposed here shows the best fit. Furthermore,the previous reports [11] are based upon fittings which need 4 to 6 parameters. In the present work, the directdetermination of parameters from Ω( T ) near the phase transition allows us to find the temperature dependence of ϕ in the full temperature range, which suggest that the model is correct.Finally, we compare the theoretical model for the quantum mechanical limit with some data available on literature,especially reported by Salje and coworkers [16]. Figure 5 displays the scaling of the data for our SrTiO data [28],along with SiO [23], LaAlO [24], and Pb (PO ) [25], all related to the n = 2 in equation 37 reported by Salje [17](the data available for n = 4 will be addressed elsewhere) based upon the following equation. (cid:0) − Ψ (cid:1) [coth (Θ S /T C ) −
1] = f (Θ S /T ) , (30)where f (Θ S /T ) = coth (Θ S /T ) − T / Θ S is the reduced temperature which measures the ration between thermalenergy and quantum mechanical energy that leads the ordered phase to the ground state.An excellent collapse, shown in figure 5(a) for the data of all samples and the theoretical prediction displayed bythe black lines, are clearly observed, suggesting a universal behavior, despite the definition of the order parameterchosen in the previous reports [11, 16, 24]. The linear behavior displayed in the figure 5(b) confirms the agreementbetween the experimental and the quantum mechanical model with the corrections introduced in this works.1 V. CONCLUSION
Quantum mechanical model for the order parameter has been revisited. Taking into account the boundary con-ditions, at the continuous phase transition and at the absolute zero temperature, which must obey the third law ofthermodynamics, the pre-factor terms of the free energy equation were naturally found.Based upon free energy equation, the temperature dependencies of the physical properties related to the order-disordered phase transition were derived. The theoretical model showed that the entropy of the disordered phaseplays a very important role in the ordered state, since it has a strong temperature dependence, which reaches aground state near the characteristic temperature, Θ S , defined previously by Salje et al. [16]. Furthermore, it alsocarries on the total entropy to zero near the same temperature.Interesting is to note that the model predicts that the order parameter is related to one of the three fundamentalproperties, temperature, entropy, or thermal expansion coefficient. The experimental results on HRTE performed inSrTiO single crystals [28] provide direct evidence that the volumetric thermal expansion coefficient is the appropriatedfundamental physical property to describe the order parameter of the cubic to tetragonal distortive phase transitionin this compound, instead the antidistortive angle ϕ [11, 16, 24]. Another evidence that the model works well isthe universal collapse of the previous results for the order parameter, both in linear limit (Ψ →
0) and also at thesaturation regime (Ψ → T C and Θ S , in comparison with previous reports, which have to find 4 to6 fitting parameters [11].Finally, preliminary analyses of other experimental data suggest that the theoretical model reported here can alsobe applied to other types of continuous phase transitions, such as magnetic and superconducting transitions. In suchcases, the energy related to each transition must be added to the entropy term in the free energy equation. ACKNOWLEDGMENTS
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