Canonical versus grand canonical occupation numbers for simple systems

An ideal gas ofN indistinguishable particles is described by a canonical ensemble (c.e.) and also by a grand canonical ensemble (g.c.e.) which hasN as themean total number of particles, the temperature and volume being the same in both cases. Exact mean occupation numbersn _j(N) are found if the system has only two states 1 and 2 of energiesE ₂E ₁. This should apply to quantum wells and similar simple systems. For systems which have captured one particle, the theory gives the simplest answers, and one find a maximum discrepancy of 17% between the two ensembles for the fermion case. It occurs whenE ₂–E ₁53 meV at room temperature. ForN=1 the mean occupation number for the c.e. is identical for fermions and for bosons, being in both cases given byn ₂(1)={exp[(E ₂-E ₁)/kT]+1}^-1,n ₁(1)=1-n ₂(1) For largeN one reverts to the usual situation and the discrepancy between the ensembles becomes small.     

Positive muon Knight shift in graphite and Grafoil

Positive muon Knight shifts and relaxation rates were measured at room temperature in a graphite crystal and in a stack of Grafoil sheets. The Knight shift was 500 ppm in the single crystal and reduced by 0.702 in Grafoil. Both have the same (large) fractional anisotropy relative to the axis or to the normal to the Grafoil sheets, respectively. The (isotropic) relaxation rates were 0.024(4)s^–1 in the crystal and 0.194(6) ^–1 in the Grafoil. Apparently the ⁺ in Grafoil sees highly aligned bulk crystalline graphite, and does not reach the surfaces of the sheets.We are grateful to Greg Dash, the owner of the graphite crystal, for lending it to Tony Arrott; and to Tony for lending it in turn to us.     

Field sampling demonstration of portable thermal desorption collection and analysis instrumentation

The HAPSITE® (Hazardous Air Pollutants on Site) is a portable gas chromatography-mass spectrometry (GC–MS) unit designed to aid air sampling technicians by identifying and quantifying volatile organic compounds from occupational and environmental sampling. The main goal of the present study was to extend prior laboratory-based work with the portable HAPSITE® ER (extended range model) thermal desorption (TD) capability to real-world field samples from both indoor and outdoor environments using different types of active and passive sampling mechanisms. Understanding the performance of the HAPSITE® ER in a realistic field setting will allow air quality sampling technicians to make improved decisions related to sampling and analysis methods in the field. An important finding was that certain charcoal-based TD sorbents were contraindicated for the HAPSITE® ER because of a substantial hydrocarbon bleed which degraded system performance. A novel time series TD sampler (Logistically Enabled Sampling System-Portable [LESS-P]) was validated using Tenax TA TD tubes against standard active sampling across multiple field sampling sites, and the qualitative analytical trends and compound identities were similar between LESS-P replicates analysed via benchtop GC–MS and HAPSITE® ER. Once validated, the LESS-P was used to determine the reference concentrations for passive sampling calculations. The results confirmed the passive sampling methodology within the benchtop system, but highlighted some systemic sensitivity limitations that must be addressed in order for the HAPSITE® to be accurately applied to passive sampling. We propose that the LESS-P time-series sampler may help to alleviate the requirement for sampling technicians to be on-site during active sampling, allowing for automated sampling throughout the duration of a sampling event.     

One-Dimensional Traps,Two-Body Interactions,Few-Body Symmetries: I. One,Two, and Three Particles

This is the first in a pair of articles that classify the configuration space and kinematic symmetry groups for N identical particles in one-dimensional traps experiencing Galilean-invariant two-body interactions. These symmetries explain degeneracies in the few-body spectrum and demonstrate how tuning the trap shape and the particle interactions can manipulate these degeneracies. The additional symmetries that emerge in the non-interacting limit and in the unitary limit of an infinitely strong contact interaction are sufficient to algebraically solve for the spectrum and degeneracy in terms of the one-particle observables. Symmetry also determines the degree to which the algebraic expressions for energy level shifts by weak interactions or nearly-unitary interactions are universal, i.e. independent of trap shape and details of the interaction. Identical fermions and bosons with and without spin are considered. This article sequentially analyzes the symmetries of one, two and three particles in asymmetric, symmetric, and harmonic traps; the sequel article treats the N particle case.     

Inertial and gravitational mass in quantum mechanics

We show that in complete agreement with classical mechanics, the dynamics of any quantum mechanical wave packet in a linear gravitational potential involves the gravitational and the inertial mass only as their ratio. In contrast, the spatial modulation of the corresponding energy wave function is determined by the third root of the product of the two masses. Moreover, the discrete energy spectrum of a particle constrained in its motion by a linear gravitational potential and an infinitely steep wall depends on the inertial as well as the gravitational mass with different fractional powers. This feature might open a new avenue in quantum tests of the universality of free fall.     

μ+ SR studies on pure MnF2 and site-diluted (Mn0.5Zn0.5)F2SR studies on pure MnF2 and site-diluted (Mn0.5Zn0.5)F2

Positive muon spin rotation and relaxation measurements have been carried out on the antiferromagnets, pure MnF₂ and site-diluted (Mn_0.5Zn_0.5)F₂, above and below the Néel temperature T_N using single-crystal specimens. Two different muon signals have been found in the pure MnF₂; with the precession frequency υ_A for the site A and υ_B for the site B measured in zero external magnetic field at T=5 K. We propose a picture that the signal from the A site represents the “muonium” state, and discuss the characteristic features of muonium in magnetic materials. The spin relaxation rate 1/T₁, measured in zero external field, decreases rapidly with decreasing temperature below T_N. The mechanism of the spin relaxation above T_N is explained by the exchange fluctuations of the Mn moments, while below T_N by the Raman scattering of spin waves. At the same normalized temperature T/T_N, 1/T₁ observed in the diluted (Mn_0.5Zn_0.5)F₂ is significantly larger than that in the pure MnF₂ below T_N. The difference between the pure and diluted systems is related to the large spectral weight of low-energy magnons in (Mn_0.5Zn_0.5)F₂ found by neutron scattering.     

μ +SR study of some magnetic superconductors

The relaxation of the ⁺SR signal has been investigated for the three compounds YRh₄B₄, ErRh₄B₄ and SmRh₄B₄. In the non-magnetic superconducting (T _c 11 K) YRh₄B₄, the data display a Kubo-Toyabe (gaussian) shape for zero (transverse) magnetic fields. ErRh₄B₄ (superconducting below 8.7 K and ferromagnetic below 1 K) shows a dominant signal with very slow relaxation. In contrast SmRh₄B₄ (superconducting below 2.7 K and antiferromagnetic-superconducting below 0.87 K) shows a change in relaxation from gaussian above 60 K to exponential between 1 K and 4 K to two exponential signals (fast and slow) belowT _N=0.9 K. In the region 0.9 K <T < 4.5 K, the relaxation time and the asymmetry both increase withT.supported by the U.S. Department of Energy.Supported by NSERC of Canada.supported by the U.S. Department of Energy.We are grateful to Drs. H. Umezawa and H. Matsumoto for interesting discussions regarding the persistent current screening and the results of self-consistent calculations.