R. Orbach*,     Ecole Supérieure de Physique et de Chimie Industrielles de la Ville de Paris, 10, rue Vauquelin, 75005 Paris, France

A brief list of current areas of research in high field physics is presented covering most of the presentations at this Symposium. More detailed description is given for three topics for which high magnetic fields are required, and which possess unusual interest. These are: 1) Density of states for vibrational states on fractals “Fractons”, 2) Thermodynamic properties of exchange enhanced systems, and 3) p-state pairing in thin film or layered superconductors.

1 INTRODUCTION

This Symposium follows at least two others in the rapid development of high magnetic field physics.1,2 In addition, a survey, now rather aging, of opportunities for research in high magnetic fields has been prepared.3 The purpose of the present paper is to briefly classify the character of those papers to be presented at this Symposium, and then to describe in outline form three areas which are of particular interest to the author.

The general areas of research in high magnetic fields to be discussed at this Symposium can very roughly be titled as:

No list is complete, but this can serve as a rough outline of topics unique to high magnetic field research.

This paper will explore three of the seven areas listed above, The remaining four will be well covered by others at this Symposium. Only one of these three represents original work by this author. However, the significance of the other two warrants some attention.

Each of the three topics is described below in terms of the physical ideas which have been developed, and the possible experimental probes. Space limitations require that the reader be referred to the original treatments for the complete details.

2 DENSITY OF VIBRATIONAL STATES ON A FRACTAL, “FRACTONS”

Fractals are open, self similar structures, with interesting properties as a function of the length scale.4 A specific example would be a percolation arrangement where the number of sites on the infinite cluster (p > pc, where pc is the percolation threshold concentration) increases not as

where r is the distance and d the Euclidean dimensionality, but rather as

where is an effective dimensionality, equal to d − (β/ν) in terms of the usual percolation exponents.5 This behavior occurs for short length scales in comparison to the coherence length for percolation, ξp. For larger lengths one finds usual Euclideanp properties. If now one examines diffusion along the infinite cluster, the “dead ends” cause a length dependence for the diffusion constant:

where again for percolation , t being the conductivity exponent.

The diffusion problem along a fractal can be solved, leading to the ensemble averaged autocorrelation function6

(1)

where the particle has assumed to have been localized at the origin at time t = 0.

One now notes that the form of the diffusion equation (Master Equation) is the same as, for example, the harmonic vibrational problem, with a simple replacement of the first time derivative by the second. This mapping allows us to regard the inverse Laplace transform of Eq. (1) as the lattice vibrational density of states (with ω2 replacing the Laplace transform spectral parameter ε) for a fractal arrangement of masses and springs. One finds

(2)

For Euclidean systems, p = d-1, so we are led to define, for mode counting purposes, a reciprocal space of effective dimensionality

(3)

We refer to these states, when quantized, as “fractons.” Their properties are most interesting. Before we outline them in more detail, some experimental examples are of interest.

Our attention to this problem was aroused by the work of Stapleton et al.7 who measured the spin-lattice relaxation time for low-spin Fe(3+) in three hemoproteins. These large molecules were shown by x-ray measurements (counting the increase of the number of alpha carbons with distance for myoglobin at 250 K) to yield a value for , certainly not integral. Their data for the spin-lattice relaxation time as a function of temperature for myoglobin azide (MbN3) are copied below:

Their interpretation relied on the use of the usual two-phonon integral for the Raman process relaxation rate, the integrand being proportional to the square of the vibrational density of states. Keeping all other factors the same as for Euclidean space, they extracted the exponent

(4)

They did not report other data which would enable us to obtain an independent estimate for . The use of self avoiding random walks as a model for these proteins is inappropriate. For such systems in d = 3, 8

(5)

leading to , representing one dimensional vibrational behavior. The essential condition for application of these ideas to physical systems is that the length scale be less than the Euclidean correlation length. For lattice vibrations, this implies that the frequency be greater than a crossover frequency, ωc.o., itself related to by the following expression6:

(6)

where L is the size of the fractal object (e.g., the percolation correlation length, or the size of the molecule) in units of the monomer length, and the frequency scale is that of the Debye frequency appropriate to the fractal object. For example, Stapleton et al. state that the temperature range 1–20 K is associated with wavelengths of from 10 to 103 bonds. For a large molecule, this would certainly be consistent with the requirement for fractal behavior.

There are other properties of fractal vibrational states. For example, the vibrational eigenfunctions are local and should not contribute to the thermal conductivity. This behavior (though not with an identification of fractal properties) has recently been reported by Kelham and Rosenberg for epoxy resin, for the energy range of 8–50 K (their measurements spanned the range of 0.1–80 K).9

It is clear that the identification with fractal behavior depends on the condition (6), which then leads to a vibrational density of states (2). The experimental consequences are immediate. The one phonon, or direct relaxation process rate, is directly proportional to the vibrational density of states. If one performs an electron spin lattice relaxation time measurement at sufficiently high magnetic fields, it is possible that one can obtain a direct measurement of the fracton density of states. The field must be sufficiently high that the condition (6) is satisfied. Then the field dependence of the relaxation rate will give the energy dependence of the density of fracton states, and hence a value for p using Eq. (2). The crossover magnetic field will separately give an estimate of using Eq. (6). That is, is not a free parameter, in that it is determined by the crossover behavior. Finally the factor d can also be determined from x-ray measurements, over-determining all the fractal parameters.

There are other interesting consequences of fractal behavior. The eigenstates are supposed to be localized. This could lead to rather interesting magnetic resonance bottleneck effects in that the spatial transfer of excitation will be diffusive rather than wavelike. This might lead to strong bottleneck conditions for both the direct and resonance relaxation processes. Here too, large magnetic fields would be useful to unravel the dynamical properties of a bottleneck. For example, if a bottleneck is found for the resonance relaxation process, the field dependence of the strength of the bottleneck will give a direct estimate of the fracton lifetime (the analysis is similar to that of Gill,10 but using fractons instead of phonons).

3 THERMODYNAMIC PROPERTIES OF EXCHANGE ENHANCED SYSTEMS

The effect of magnetic fields upon the thermodynamic properties of Fermion systems [e.g., the nonlinear magnetic susceptibility and the field dependent specific heat], depends on the relation between field H and Fermi temperature TF. Even for extreme fields, this ratio is small (100 Tesla is the equivalent of 170 K in Zeeman splitting). Exchange...