Mullin, William
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Professor Emeritus, Department of Physics, College of Natural Sciences
Last Name
Mullin
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William
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Physics
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Introduction
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Publication EXACT TRANSPORT-PROPERTIES OF DEGENERATE, WEAKLY INTERACTING, AND SPIN-POLARIZED FERMIONS(1983) Mullin, WJ; MIYAKE, KExact results for the transport coefficients of a highly degenerate spin-polarized Fermi system are presented. The case of weakly interacting particles, such as dilute3He in4He or D darr, is treated. The results are compared to the variational treatment and corrections as large as 8% are found. Detailed behavior of the polarization of an ideal Fermi gas as a function of temperature and magnetic field is discussed as a model for dilute3He in4He. Limits of applicability of the formulas are given. We find that at appropriate fields and temperatures, the two spin species may be considered to be a mixture of a degenerate and a classical gas.Publication ANALYSIS OF CERTAIN BINARY COLLISION APPROXIMATION CLOSURES OF THE BBGKY HIERARCHY(1995) SNIDER, RF; Mullin, WJ; LALOE, FThe closure of the BBGKY hierarchy to obtain the Boltzmann equation requires, in particular, restricting particle interactions to include only isolated binary collisions. Boercker and Dufty accomplish this by approximating the three-particle reduced density operator in a particular manner that favours correlation between two of the particles, while ignoring the correlation with the third. The tradition of most other closures has more closely followed Boltzmann's original thinking to completely neglect any reference to three-particle effects while assuming a generalized form of molecular chaos for the pair density operator. The two closures are compared in two ways: (a) by finding iterated series solutions of the BBGKY hierarchy and of the Boltzmann equation; (b) by computing an exact correction to the quantum Boltzmann equation. A consequence of the comparison of the iterated series shows that an important, but little emphasized, difference between the BBGKY and Boltzmann hierarchies is the effective instantaneousness of binary collisions in the latter. The form for the correction found is shown to vanish for either closure provided the instantaneousness of the binary collisions is imposed. It is shown moreover that the correction is closely related to the three-body collision integral arising in the standard theory of the density corrections to the Boltzmann equation. We also comment on the related work of Klimontovich, who introduces an approximation analogous to that of Boercker and Dufty.Publication LONGITUDINAL RELAXATION-TIME FOR DILUTE QUANTUM GASES(1990) Mullin, WJ; LALOE, F; RICHARDS, MGWe calculate the longitudinal relaxation timeT 1 for a polarized spin-1/2 Fermi gas, in zero magnetic field, for conditions of temperatureT and densityn such that Boltzmann statistics are valid. Our results show generally thatT 1 is independent of polarization of the gas. At highT, where the thermal wavelength lambda is small compared to the scattering lengtha, T 1 is proportionalT 1/2, while at lowT, such that lambda is greater thana, T 1 is proportional toT –1/2.T 1 thus has a minimum at some intermediate temperature confirming the numerical results of Shizgal. Physical arguments show that the existence of the minimum does not depend on the presence of an attractive part of the potential. As an example of the expected temperature dependence we calculateT 1 numerically, via the distorted-wave Born approximation, for the case of a gas interacting via a hard core. We also computeT 1 for a spin-1/2 Bose gas, which also shows a minimum.Publication NMR RELAXATION-TIMES FOR A DILUTE POLARIZED FERMI GAS(1990) Mullin, WJ; LALOE, F; RICHARDS, MGWe calculate the longitudinal relaxation time T1 for a polarized Boltzmann Image Fermi gas. We show that T1 is independent of polarization of the gas. At high T, where the thermal wavelength a is small compared to the scattering length a, T1 is proportional Image , while at low T, such that λ is greater than a, T1 is proportional to Image . T1 thus has a minimum at some intermediate temperature confirming the numerical results of Shizgal. The existence of the minimum does not depend on the presence of an attractive part of the potential. As an example of the expected temperature dependence we calculate T1 numerically for a hard core gas.Publication Bose-Einstein condensation in a harmonic potential(1997) Mullin, WJWe examine several features of Bose-Einstein condensation (BEC) in an external harmonic potential well. In the thermodynamic limit, there is a phase transition to a spatial Bose-Einstein condensed state for dimensionD≥2. The thermodynamic limit requires maintaining constant average density by weakening the potential while increasing the particle numberN to infinity, while of course in real experiments the potential is fixed andN stays finite. For such finite ideal harmonic systems we show that a BEC still occurs, although without a true phase transition, below a certain “pseudo-critical” temperature, even forD=1. We study the momentum-space condensate fraction and find that it vanishes as 1/sqrt(N) in any number of dimensions in the thermodynamic limit. InD≤2 the lack of a momentum condensation is in accord with the Hohenberg theorem, but must be reconciled with the existence of a spatial BEC inD=2. For finite systems we derive theN-dependence of the spatial and momentum condensate fractions and the transition temperatures, features that may be experimentally testable. We show that theN-dependence of the 2D ideal-gas transition temperature for a finite system cannot persist in the interacting case because it violates a theorem due to Chester, Penrose, and Onsager.Publication SOLUTION OF THE KINETIC-EQUATION FOR POLARIZED FERMI SYSTEMS AT ALL TEMPERATURES(1987) JEON, JW; Mullin, WJWe have solved the kinetic equation for a dilute, polarized Fermi system for a range of temperatures from the degenerate limit to the Boltzmann case. The solution is possible because we have been able to reduce the collision integral to two-fold form. We calculate the spin diffusion constant and find the expected results for the degenerate and Boltzmann limits, improved results for the high polarization regime in which one spin species is degenerate and one Boltzmann, and values for all intermediate temperatures as well. Because of a relation between the collision time and the “spin rotation quality parameter”, µ, we give results for that quantity valid for all temperatures. A similar analysis should allow the computation of other transport coefficients.Publication LONGITUDINAL AND TRANSVERSE SPIN DIFFUSION IN POLARIZED HE-3-HE-4 SOLUTIONS(1991) JEON, JW; Mullin, WJSpin dynamics for arbitrarily polarized and very dilute 3He-4He solutions are described. We generalize previous work to include 3He-4He phenomenological interactions, and we calculate longitudinal and transverse spin diffusion coefficients and the identical-particle spin-rotation parameter. Good agreement is found in comparison with recent data. The s-wave approximation is found to be inadequate and mean-field corrections are important.Publication A study of Bose-Einstein condensation in a two-dimensional trapped gas(1998) Mullin, WJWe examine the possibility of Bose-Einstein condensation (BEC) in two-dimensional (2D) system of interacting particles in a trap. We use a self-consistent mean-field theory of Bose particles interacting by a contact interaction in the Popov and WKB approximations. The equations show that the normal state has a phase transition at some critical temperature T c but below T c the Bose-Einstein condensed state is not a consistent solution of the equations in the thermodynamic limit. This result agrees with a theorem recently discussed by the author that shows that a BEC state is impossible for an interacting gas in a 2D trap in the thermodynamic limit.Publication KINETIC-EQUATION FOR DILUTE, SPIN-POLARIZED QUANTUM-SYSTEMS(1988) JEON, JW; Mullin, WJA kinetic equation, which includes the effects of degeneracy, is derived for dilute, polarized systems by the Green's function method of Kadanoff and Baym. When the Born approximation is used for the self-energy, the equation reduces to a result due to Silin. In the Boltzmann limit our result is equivalent to the equation of Lhuillier and Laloë, with the addition of a mean-field drift term analogous to that appearing in the Landau-Silin equation. Our kinetic equation is used to derive an expression for the transverse spin-diffusion relaxation time, τ>, for a Fermi system. In the Boltzmann and low-polarization limits τ> reduces to τ ∥, the longitudinal relaxation time. However, in a highly polarized degenerate system τ> can be very much shorter than τ ∥.Publication BOSE CONDENSATION OF IDEALIZED SPIN-POLARIZED ATOMIC-HYDROGEN IN EQUILIBRIUM(1980) Mullin, WJA model of spin-polarized hydrogen (H↑) is treated in which interactions between atoms are neglected while the single-atom Zeeman and hyperfine interactions are treated exactly. These magnetic terms in the Hamiltonian are found to affect substantially the Bose-Einstein condensation and the various thermodynamic variables. Computations are discussed of the condensation temperature, condensate density, and specific heat in order to indicate how changes in magnetic field strength might be expected to affect future measurements on this quantum system.