Exponents, magnetic dependence

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. 

Rb and 1H SLR rate as a function of temperature is a very important parameter which shows the suppression of phase transition and reveals the frustration in the mixed system. Temperature dependence of Ti in any ordered system can be described by the well known Bloembergen-Purcell-Pound (BPP) type expression. However, disordered systems show deviations from BPP behaviour, showing a broad distribution of relaxation times. The magnetization recovery shows a stretched exponential recovery of magnetization following M(t)=Mo(1 — 2 exp (— r/Ti) ) where a is the stretched exponent. 

Sobol et al.8 have measured proton SLR time (7 ) in Rb1 x(NH4)xH2As04 systems in the range 100-4.2 K. Magnetization recovery was found to be non-exponential in the entire range of temperature. The MR data fit to a stretched exponential recovery and the exponent a was found to be temperature dependent implying broadening of the distribution of microscopic correlation times p(r ) with decreasing temperature. 

Applying superposition approximations to the Ising model, one finds an evidence for the phase transition existence but the critical parameter to is systematically underestimated (To is overestimated respectively). Errors in calculation of to are greater for low dimensions d. Therefore, the superposition approximation is effective, first of all, for the qualitative description of the phase transition in a spin system. In the vicinity of phase transition a number of critical exponents a, /3,7,..., could be introduced, which characterize the critical point, like oc f-for . M oc (i-io), or xt oc i—io for the magnetic permeability. Superposition approximations give only classical values of the critical exponents a = ao, 0 = f o, j — jo, ., obtained earlier in the classical molecular field theory [13, 14], say fio = 1/2, 7o = 1, whereas exact magnitudes of the critical exponents depend on the space dimension d. To describe the intermediate order in a spin system in terms of the superposition approximation, an additional correlation length is introduced, 0 = which does not coincide with the true In the phase... 

But rather than considering spatially-dependent spectral properties, which often are essential for image contrast, time-dependent gradients shall be admitted to illustrate the basic concepts of space encoding. Then the space-dependent magnetization phase - yGrf in the exponent of the integrand has to be replaced by... 

Infrared singular behaviour is characterized by power law singularities in thermodynamic quantities with singular (or critical) exponents which usually vary continuously with system parameters. Depending on the physical interpretation of these parameters in the different situations, this implies 1) and 2) singular ground-state properties, a special particle spectrum and instabilities for correlation functions 3) the phase transition of continuous order for 2-d magnetic systems and other systems with continuous... 

In HF doped ice, Kopp (personal communication) found exponents of 0.4 and 0.6 for the concentration dependence of the dielectric relaxation and of the nuclear magnetic relaxation, respectively. He suggests that the mass-action law does not hold for the relaxations. 

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