Michael J. Stephen

Spectrum of Light Scattered from Charged Macromolecules in Solution

Many macromolecules of biological interest, when dissolved in water, acquire charges and these charges are shielded by an atmosphere of counterions. The effect of the ion atmosphere on the diffusion constants of the molecules and hence on the spectrum of scattered light from such a solution is investigated here using a simple model. It is found that the ionic atmosphere can have an important effect on the diffusion of the macromolecules and that measurement of the spectrum of scattered light may provide a way to measure the residual charge on the macromolecules.

Collapse of a polymer chain

Abstract The behavior of a polymer chain near the Flory temperature is determined. The discussion is based on the analogy proposed by de Gennes between a polymer chain near the Flory temperature and a magnetic system near the tricritical point.

Feynman graph expansion for tricritical exponents

Abstract The determination of the fixed points of the Wilson classical spin model by the ϵ-expansion is discussed. Expansions in ϵ for d = 3−ϵ are given for the tricritical exponents γ, η and α.

Resistance fluctuations in disordered one-dimensional conductors

Exact calculations of the low moments of the resistance for models of random one-dimensional conductors are given. The fluctuations in resistance grow with length more rapidly than the average resistance, so that the latter is not representative of the probability distribution. The probability distribution is calculated in the limit of large disorder.

Percolation with long-range interactions

The concentration dependence of the ferromagnetic ordering temperature, Tc(p) in a dilute bond Ising model,, is studied for several long-range interactions near the limit p to 0, Tc to 0. The critical region of the random Ising model transition, occurring at all finite Tc, is shown to shrink to zero as p approaches zero. For dipolar interactions, spin glass ordering may occur at sufficiently small p.

Magnetic susceptibility of percolating clusters

Abstract The diamagnetic susceptibility of a superconducting percolating cluster is related to the fraction of superconducting links in connected loops and the effective area of these loops in the cluster. The susceptibility is evaluated for a simple two-dimensional fractal model of a cluster. The average susceptibility is predicted not to diverge at the percolation point.

Conformal transformations and the application of complex variables in mechanics and quantum mechanics

The use of complex variables and conformal mapping have long been useful tools in many branches of physics. However, these methods have not been widely applied in mechanics and elementary quantum mechanics. In this paper, it is shown that there are many useful applications of complex variables and conformal mapping in these subjects. It will be proven that central force problems in the plane have conformal duals. The inverse fifth power law force is self‐dual in all dimensions. These results are extended to noncentral forces. Many of these results also apply to the quantum mechanics of a particle. For central force problems and certain noncentral force problems in the plane the Schrodinger equation preserves its form under conformal mapping and the inverse fifth power law force is again self‐dual in all dimensions. The methods may also be applied in the presence of magnetic fields.

Yang-Lee zeros of the Potts model

The Yang-Lee zeros of the three-component ferromagnetic Potts model in one dimension in the complex plane of an applied field are determined. The phase diagram consists of a triple point where three phases coexist. Emerging from the triple point are three lines on which two phases coexist and which terminate at critical points (Yang-Lee edge singularity). The zeros do not all lie on the imaginary axis but along the three two-phase lines. The model can be generalized to give rise to a tricritical point which is a new type of Yang-Lee edge singularity. Gibbs phase rule is generalized to apply to coexisting phases in the complex plane.

Universality and tricritical points in three dimensions

Amplitude ratios near tricritical points in three dimensions are discussed and shown to be universal and in agreement with the classical values when logarithmic corrections are included. The spherical model of a tricritical point is shown to be a special case exhibiting non-universality because of the absence of logarithmic corrections.

Pseudo-Hamiltonians for the Potts model at the critical point

Abstract The diagonal transfer matrix of the 2-dimensional, q -component Potts model on a square lattice is shown to commute with a linear operator at the critical point. In the 4-component model the linear operator is equivalent to the linear Heisenberg chain.