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.     

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.     

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.     

Time asymmetric quantum theory and the ambiguity of the Z-boson mass and width

Relativistic Gamow vectors emerge naturally in a time asymmetric quantum theory as the covariant kets associated to the resonance pole in the second sheet of the analytically continued S-matrix. They are eigenkets of the self-adjoint mass operator with complex eigenvalue and have exponential time evolution with lifetime . If one requires that the resonance width (defined by the Breit-Wigner lineshape) and the resonance lifetime always and exactly fulfill the relation , then one is lead to the following parameterization of in terms of resonance mass and width : . Applying this result to the -boson implies that and $\Gamma_R \approx \Gamma_Z-1.2\mbox{MeV}$ are the mass and width of the {\it Z}-boson and not the particle data values or any other parameterization of the Z-boson lineshape. Furthermore, the transformation properties of these Gamow kets show that they furnish an irreducible representation of the causal Poincaré semigroup, defined as a semi-direct product of the homogeneous Lorentz group with the semigroup of space-time translations into the forward light cone. Much like Wigner's unitary irreducible representations of the Poincaré group which describe stable particles, these irreducible semigroup representations can be characterized by the spin-mass values . Received 8 June 2000 / Published online: 27 November 2000     

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.