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296 Handbook on the Physics and Chemistry of Rare Earths with n the correlation length critical exponent. Similarly, the susceptibility associated with the order (eg, the magnetic susceptibility w1⁄4@M for a ferro- @H magnet) also diverges at the critical point, which defines the susceptibility exponent w$jtjg: (4) A list of most commonly used exponents is shown in Table 1. All of those exponents are accessible in experiments, for example, table III in Kornblit and Ahlers (1975) includes all the critical exponents for the ferromagnetic transition in EuO. Because fluctuations at all length scales become equally important, many of the microscopic details of the system become irrelevant. As a result, impor- tant physical quantities can be expressed as scaling functions of the diverging correlation length and other macroscopic parameters. For example, for many CPTs the free energy density is just a function of the correlation length itself, f$xd, where d is the dimensionality of the system. If this is the case, it follows that the specific heat exponent is related to the correlation length exponent by the hyperscaling relation 2 a 1⁄4 nd. Other such scaling relations, that can be directly derived from the notion of physics-at-all-length-scales, are 2a1⁄42b+g, 2a1⁄4b(d+1) and Fishers relation g1⁄4(2)n. TABLE 1 The Definition of Several Critical Exponents for a Magnetic Material Exhibiting a CPT Finite Quantum Phase Quantity Exponent Temperature Transition Order parameter b m$(t)b m$(ggc)b Specific heat a C$jtja – Susceptibility g w$jtjg w$jggcjg Critical isotherm d B$jmjd B$jmjd (at t1⁄40 or g1⁄4gc) Correlation length n x$jtjn x$jggcjn Correlation function G(r)$jrjd+2 G(r)$jrjd+2 (at t1⁄40 or g1⁄4gc) Dynamics z – xt$xz The first column indicates the physical observable, the second the exponent. The third column indicates how the observable scales near a finite temperature phase transition as a function of the reduced temperature t 1⁄4 T Tc . The fourth column shows how the observable scales near a QPT, Tc as a function of the control parameter ggc.PDF Image | HANDBOOK ON THE PHYSICS AND CHEMISTRY OF RARE EARTHS
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