Exponents, magnetic

An essential feature of mean-field theories is that the free energy is an analytical fiinction at the critical point. Landau [100] used this assumption, and the up-down symmetry of magnetic systems at zero field, to analyse their phase behaviour and detennine the mean-field critical exponents. It also suggests a way in which mean-field theory might be modified to confonn with experiment near the critical point, leading to a scaling law, first proposed by Widom [101], which has been experimentally verified. 

To detennine the critical exponents y and S, a magnetic interaction temi -hm is added to the free energy and... 

Figure A2.3.29 Calculation of the critical temperature and the critical exponent y for the magnetic susceptibility of Ising lattices in different dimensions from high-temperature expansions.

A third exponent y, usually called the susceptibility exponent from its application to the magnetic susceptibility x in magnetic systems, governs what m pure-fluid systems is the isothennal compressibility k, and what in mixtures is the osmotic compressibility, and detennines how fast these quantities diverge as the critical point is approached (i.e. as > 1). 

Exponent values derived from experiments on fluids, binary alloys, and certain magnets differ substantially from all those derived from analytic (mean-field) theories. Flowever it is surprising that the experimental values appear to be the same from all these experiments, not only for different fluids and fluid mixtures, but indeed the same for the magnets and alloys as well (see section A2.5.5). 

The exponents apply not only to solid systems (e.g. order-disorder phenomena and simple magnetic systems), but also to fluid systems, regardless of the number of components. (As we have seen in section A2.5.6.4 it is necessary in multicomponent systems to choose carefully the variable to which the exponent is appropriate.)... 

To eliminate the ambiguities in the subject of electricity and magnetism, it is convenient to add charge q to the traditional I, m and t dimensions of mechanics to form the reference dimensions. In many situations permittivity S or permeabiUty ]1 is used in Heu of charge. For thermal problems temperature Tis considered as a reference dimension. Tables 2 and 3 Hst the exponents of dimensions of some common variables in the fields of electromagnetism and heat. 

A is a constant and p is the critical exponent which adopts values from 0.3 to 0.5. Values around p = 0.5 are observed for long-range interactions between the particles for short-range interactions (e.g. magnetic interactions) the critical exponent is closer to p 0.33. As shown in the typical curve diagram in Fig. 4.2, the order parameter experiences its most relevant changes close to the critical temperature the curve runs vertical at Tc. 

For ordinary temperatures and ordinary magnetic fields, the exponent is very small and the exponential can be accurately approximated by the expansion, e x 1 — jc. Thus... 

The largest Lyapunov exponents for the relativistic hydrogenlike atom in a uniform magnetic field... 

Magnetic finishing drums, 15 446 Magnetic flocculation, 16 639 Magnetic flow meters, 20 681 Magnetic flux, exponents of dimensions, 8 585t... 

Magnetic flux density, 15 434 exponents of dimensions, 8 585t Magnetic flux leakage tests, 17 419-420 Magnetic flux leakage technique, in... 

Fig. 1. Magnetic field dependences of the proton spin-lattice relaxation time of water in Bioran B30 and Vycor glasses at temperatures above 27°C and below the temperature where the non-surface water freezes ( —25°C and —35°C). The solid lines represent the power law in the Larmor frequency with an exponent of 0.67 (34).

Three characteristics of the MRD profile change when the protein is hydrated with either H2O or D2O. Both terms of Eq. (6) are required to provide an accurate fit to the data. The second or perpendicular term dominates once the transverse modes become important. The power law for the MRD profile is retained, but the exponent takes values between 0.78 and 0.5 depending on the degree of hydration. A low frequency plateau is apparent for samples containing H2O which derives from two sources the field limitation of the local proton dipolar field as mentioned above, and from limitations in the magnetization transfer rates that may be a bottleneck in bringing the liquid spins into equilibrium with the solid spins. 

It is those functions with an inverse sixth-power dependence on the separation that are our main concern in this chapter. Those power laws with exponents greater or less than 6 are included in Table 10.1 mainly to emphasize the point that many types of interactions exist and that these are governed by different relationships. The interactions listed are by no means complete Interactions of quadrupoles, octapoles, and so on might also be included, as well as those due to magnetic moments however, all of these are less important than the interactions listed. Let us now examine Table 10.1 in greater detail. 

For state separations of the order of kT the first- and second-order Zeeman effect coefficients for equation (65) are both important and the exponents vary with temperature there results complicated expressions for the magnetic behaviour for which a general expression has been developed.2-28... 

The main difficulty in l3C NMR is the low natural abundance of the carbon-13 nucleus (1.108%) and its low gyromagnetic ratio y, which yields a much smaller Boltzmann exponent 2yp0B0ikT than that of protons. Low natural abundance and small gyro-magnetic ratio are the reasons why 13C NMR is much less sensitive (1.59%) than H NMR (100%). A common measure of sensitivity in NMR is the signal to noise of a reference sample, e.g. 1% ethyl benzene in deuteriochloroform. Several methods are available for improving the signal noise in 13C NMR. 

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