Specific heat critical exponent

Cv Pc 8 a an P Y c f specific heat reduced density critical exponent for the critical isotherm critical exponent for the specific heat critical exponent for the specific heat along isot r critical exponent for the order parameter critical exponent for the susceptibility reduced temperature friction coefficient... 

For magnetic systems, the effect of weak quenched uncorrelated point-like disorder on the critical behavior is usually predicted by the Harris criterion [40] disorder changes the critical exponents only if the specific heat critical exponent ap re of the pure (undiluted) system is positive ... 

Fig. 60. Concentration dependence of various properties of polycrystalline Y(Ni xPt )2B2C obtained by specific heat measurements transition temperature Tc exponent a and parameter Hc2 from eq. (6) upper critical field Hc2(0) at T =0, where the dotted line schematically describes the dirty limit corresponding to the isotropic single band case (in reality there is a finite intersection with the field-axis for the dotted asymptotic line, see Shulga and Drechsler 2002) exponent fi of eq. (8) for the curvature of the electronic specific heat in the mixed state and Sommerfeld constant xn (after Lipp et al. 2001).

Additional comment deserve magnetostriction measurements near the ordering temperature 7c reflecting critical phenomena. Few data for critical expansion is available, such as have been reported by Dolejsi and Swenson (1981) for the case of Gd metal. The thermal expansion coefficient in the critical region should assume the form 1(7 — Tc)/Tc °-The critical exponent or should be the same as for the specific heat and depend only on the universality class (dimensionality, No. of degrees of freedom) of the system. For Gd metal this universality class has been determined by Frey et al. (1997). 

Maybe the major achievement provided by the discovery of the fe rfim is the possibility to study the complete set of critical exponents on a ferroic system for the first time after their prediction [9,10]. Table 15.1 shows the results as compared with predictions from theory and simulations. Most remarkably, the order parameter exponent ft (Figure 15.10) clearly deviates from the prediction ft 0 and achieves a value which comes close to that observed recently on the standard rfim system, the dilute uniaxial antiferromagnet Fci ,Zn.,F2, x = 0.15, in an external magnetic field [50], Further, the most disputed value, namely the specific heat exponent a [48] (Figure 15.12) clearly describes the same logarithmic divergence as that found on Fci. Zn. I 2, a 0 [10], which still lacks theoretical confirmation. 

Figure 15.12 Specific heat of SBN as measured with pulsed heating techniques displaying criticality with an exponent a 0 [48].

Salje EKH, Wrack B (1983) Specific-heat measurements and critical exponents of the ferroelastic phase transiton in Pb3(P04)2 and Pb3(Pi xAsx04)2. Phys Rev B 28 6510-6518 Salje EKH, Devarajan V, Bismayer, U, Guimaraes DMC (1983) Phase transitions in Pb3(Pi.xAsx04)2 influence of the central peak and flip mode on the Raman scattering of hard modes. J Phys C 16 5233-52343... 

Small exponents. Evidence for Ising criticality can be provided by some properties showing weak divergences, which are absent in the mean-field case. One such case is the specific heat, which diverges with the exponent a. Kaatze and coworkers [112] have indeed shown the presence of such an a anomaly in EtNH3N03 + //-octanol, but as already mentioned, this system shows an anomalous location of the critical point, indicating that non-Coulomb interactions play a considerable role in driving the phase separation. 

According to the Harris criterion, disorder is irrelevant at a critical point if the pure specific heat exponent a < 0. We show that a more general criterion is required for the... 

For a pure type critical point irrelevant disorder) a < 0 where a is the specific heat exponent (i.e. c T — Tc ° ) of the pure system. In this case the fluctuation in Tc is set by the size, i.e. 

Table 8.1 Definitions and values of the major critical exponents. The quantities Kt and Cy are the isothermal compressibility and constant volume specific heat capacity respectively, piiq and Pvap are the densities of the coexisting liquid (liq) and vapour (vap) phases, T the temperature, and and the critical temperature and pressure respectively.

Critical exponents obtained for various spin-glass systems. Some of the values are deduced from scaling laws (e.g. specific heat with a = 2- y - 2j3). 

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