Virial Coefficients from Unified Statistical Thermodynamics of Quantum Gases Trapped under Generic Power Law Potential in d Dimension and Equivalence of Quantum Gases

From the unified statistical thermodynamics of quantum gases, the virial coefficients of ideal Bose and Fermi gases, trapped under generic power law potential are derived systematically. From the general result of virial coefficients, one can produce the known results in d=3 and d=2. But more importantly we found that, the virial coefficients of Bose and Fermi gases become identical (except the second virial coefficient, where the sign is different) when the gases are trapped under harmonic potential in d=1. This result suggests the equivalence between Bose and Fermi gases established in d=1 (J. Stat. Phys. DOI 10.1007/s10955-015-1344-4). Also, it is found that the virial coefficients of two-dimensional free Bose (Fermi) gas are equal to the virial coefficients of one-dimensional harmonically trapped Bose (Fermi) gas.     

Magnetic anisotropies of some aromatic molecules

A common method for the estimation of uncertainties introduced by surface and impurity effects into experimental measurements of virial coefficients is described. The sign and the amplitude of the second virial coefficient response to perturbation caused by adsorption of molecules on the internal surface of the vessel have been determined. It has been shown that the magnitude of the second virial coefficient distortion depends on such competing factors as adsorption-impurity perturbation parameter, mixture composition which has been corrected taking into account this perturbation, and the nature of the impurity expressed in terms of its second virial coefficient and of the solvent-impurity cross second virial coefficient. The character of the Lennard-Jones 12–6 potential parameters perturbation, caused by the adsorption-impurity effects, is determined using second virial coefficient data inversion technique. Numerical estimates are made for nitrogen, helium, argon, xenon, their binary mixtures, and also for krypton-sulphur hexafluoride gaseous mixtures.     

Virial expansion for a strongly correlated Fermi system and its application to ultracold atomic Fermi gases

A strongly correlated Fermi system plays a fundamental role in very different areas of physics, from neutron stars, quark–gluon plasmas, to high temperature superconductors. Despite the broad applicability, it is notoriously difficult to be understood theoretically because of the absence of a small interaction parameter. Recent achievements of ultracold trapped Fermi atoms near a Feshbach resonance have ushered in enormous changes. The unprecedented control of interaction, geometry and purity in these novel systems has led to many exciting experimental results, which are to be urgently understood at both low and finite temperatures. Here we review the latest developments of virial expansion for a strongly correlated Fermi gas and their applications on ultracold trapped Fermi atoms. We show remarkable, quantitative agreements between virial predictions and various recent experimental measurements at about the Fermi degenerate temperature. For equations of state, we discuss a practical way of determining high-order virial coefficients and use it to calculate accurately the long-sought third-order virial coefficient, which is now verified firmly in experiments at ENS and MIT. We discuss also virial expansion of a new many-body parameter—Tan’s contact. We then turn to less widely discussed issues of dynamical properties. For dynamic structure factors, the virial prediction agrees well with the measurement at the Swinburne University of Technology. For single-particle spectral functions, we show that the expansion up to the second order accounts for the main feature of momentum-resolved rf-spectroscopy for a resonantly interacting Fermi gas, as recently reported by JILA. In the near future, more practical applications with virial expansion are possible, owing to the ever-growing power in computation.     

Effective pair potentials in fluids in the presence of three-body forces. II

Earlier calculations are extended to the second order in density thus providing the non-additive contributions to the fourth virial coefficient in the presence of a weak three-body force given by the triple-dipole dispersion potential of Axilrod and Teller. We compare the Percus-Yevick approximations of Rushbrooke and Silbert and of Rowlinson at the level of the fourth virial coefficient and find that the former is more accurate. We also compare our calculations of the effective pair potential of liquid argon with the results of Mikolaj and Pings obtained from X-ray diffraction measurements.     

Universal properties of a trapped two-component fermi gas at unitarity

We treat the trapped two-component Fermi system, in which unlike fermions interact through a two-body short-range potential having no bound state but an infinite scattering length. By accurately solving the Schr?dinger equation for up to N=6 fermions, we show that no many-body bound states exist other than those bound by the trapping potential, and we demonstrate unique universal properties of the system: Certain excitation frequencies are separated by 2variant Planck's over 2piomega, the wave functions agree with analytical predictions and a virial theorem is fulfilled. Further calculations up to N=30 determine the excitation gap, an experimentally accessible universal quantity, and it agrees with recent predictions based on a density functional approach.     

Second Virial Coefficient of a Two-Component Quantum Plasma

The screened second virial coefficient of a plasma which was given in a previous paper is discussed. Numerical values differ at high temperatures from those of the second virial coefficient with DEBYE potential (i.e. BETH -UHLENBECK formula with DEBYE potential).     

Controlling the motion of solitons in BEC by a weakly periodic potential

By developing multiple-scale method combined with Wentzel--Kramer--Brillouin expansion, this paper analytically studies the modulating effect of weakly periodic potential on the dynamical properties of the Bose--Einstein condensates (BEC) trapped in harmonic magnetic traps. A black--grey soliton transition is observed in the BEC trapped in harmonic magnetic potential, due to the weakly periodic potential modulating effect. Meanwhile, it finds that with the slight increase of the weakly periodic potential strength, the velocity of the soliton decreases, while its width firstly decreases then increases, a minimum exists there. These results show that the amplitude, velocity, and width of matter solitons can be effectively managed by means of a weakly periodic potential.     

Critical temperature of infinitely long chains from Wertheim's perturbation theory

Wertheim's theory is used to determine the critical properties of chains formed by m tangent spheres interacting through the pair potential u(r). It is shown that within Wertheim's theory the critical temperature and compressibility factor reach a finite non-zero value for infinitely long chains, whereas the critical density and pressure vanish as m ^-1.5. Analysing the zero density limit of Wertheim's equation or state for chains it is found that the critical temperature of the infinitely long chain can be obtained by solving a simple equation which involves the second virial coefficient of the reference monomer fluid and the second virial coefficient between a monomer and a dimer. According to Wertheim's theory, the critical temperature of an infinitely long chain (i.e. the Θ temperature) corresponds to the temperature where the second virial coefficient of the monomer is equal to 2/3 of the second virial coefficient between a monomer and dimer. This is a simple and useful result. By computing the second virial coefficient of the monomer and that between a monomer and a dimer, we have determined the Θ temperature that follows from Wertheim's theory for several kinds of chains. In particular, we have evaluated Θ for chains made up of monomer units interacting through the Lennard-Jones potential, the square well potential and the Yukawa potential. For the square well potential, the Θ temperature that follows from Wertheim's theory is given by a simple analytical expression. It is found that the ratio of Θ to the Boyle and critical temperatures of the monomer decreases with the range of the potential.     

Non-Abelian Chern-Simons particles in an external magnetic field

The quantum mechanics and thermodynamics of SU(2) non-Abelian Chern-Simons particles (non-Abelian anyons) in an external magnetic field are addressed. We derive the N-body Hamiltonian in the (anti-) holomorphic gauge when the Hilbert space is projected onto the lowest Landau level of the magnetic field. In the presence of an additional harmonic potential, the N-body spectrum depends linearly on the coupling (statistics) parameter. We calculate the second virial coefficient and find that in the strong magnetic field limit it develops a step-wise behavior as a function of the statistics parameter, in contrast to the linear dependence in the case of Abelian anyons. For small enough values of the statistics parameter we relate the N-body partition functions in the lowest Landau level to these of SU(2) bosons and find that the cluster (and virial) coefficients dependence on the statistics parameter cancels.     

Virial theorem and universality in a unitary fermi gas

Unitary Fermi gases, where the scattering length is large compared to the interparticle spacing, can have universal properties, which are independent of the details of the interparticle interactions when the range of the scattering potential is negligible. We prepare an optically trapped, unitary Fermi gas of 6Li, tuned just above the center of a broad Feshbach resonance. In agreement with the universal hypothesis, we observe that this strongly interacting many-body system obeys the virial theorem for an ideal gas over a wide range of temperatures. Based on this result, we suggest a simple volume thermometry method for unitary gases. We also show that the observed breathing mode frequency, which is close to the unitary hydrodynamic value over a wide range of temperature, is consistent with a universal hydrodynamic gas with nearly isentropic dynamics.     

Radial distribution function and second virial coefficient for interacting bosons

We have studied the radial distribution function and the second virial coefficient of interacting bosons. The second virial coefficient has been deduced theoretically and is in good agreement with experimental values. The third virial coefficient has been calculated from the experimental values of the pressure.     

Second virial coefficient and mechanical moduli of metallic glasses

The relationship between the bulk, shear moduli and second virial coefficient of amorphous materials is derived according to their dependences with the radial distribution function. Lennard-Jones–Gaussian potential is used to investigate the relationship between second virial coefficient and temperature, where Lennard-Jones potential represents interactions with the nearest neighbor atoms, and Gaussian potential is responsible for the multi-atom interactions including the next nearest neighbor atoms and heterogeneous structures for a metallic glass. The results show that deep potential well formed by Gaussian potential causes a large second virial coefficient at low temperatures, which is very obvious for the larger fragility glasses. The quadratic form relationship of shear modulus and compositions is proposed, and confirmed by the experimental results of Pd_xNi_100−x−20P₂₀ alloy.     

First and second sound in a two-dimensional harmonically trapped Bose gas across the Berezinskii–Kosterlitz–Thouless transition

We theoretically investigate first and second sound of a two-dimensional (2D) atomic Bose gas in harmonic traps by solving Landau’s two-fluid hydrodynamic equations. For an isotropic trap, we find that first and second sound modes become degenerate at certain temperatures and exhibit typical avoided crossings in mode frequencies. At these temperatures, second sound has significant density fluctuation due to its hybridization with first sound and has a divergent mode frequency towards the Berezinskii–Kosterlitz–Thouless (BKT) transition. For a highly anisotropic trap, we derive the simplified one-dimensional hydrodynamic equations and discuss the sound-wave propagation along the weakly confined direction. Due to the universal jump of the superfluid density inherent to the BKT transition, we show that the first sound velocity exhibits a kink across the transition. These predictions might be readily examined in current experimental setups for 2D dilute Bose gases with a sufficiently large number of atoms, where the finite-size effect due to harmonic traps is relatively weak.     

Unitary quantum three-body problem in a harmonic trap

We consider either 3 spinless bosons or 3 equal mass spin-1/2 fermions, interacting via a short-range potential of infinite scattering length and trapped in an isotropic harmonic potential. For a zero-range model, we obtain analytically the exact spectrum and eigenfunctions: for fermions all the states are universal; for bosons there is a coexistence of decoupled universal and efimovian states. All the universal states, even the bosonic ones, have a tiny 3-body loss rate. For a finite range model, we numerically find for bosons a coupling between zero angular momentum universal and efimovian states; the coupling is so weak that, for realistic values of the interaction range, these bosonic universal states remain long-lived and observable.     

An approximate multi-property empirical isotropic interatomic pair potential for the krypton dimer

An approximate empirical isotropic interatomic potential for krypton interaction is developed by simultaneously fitting the Morse-Morse-Morse-Spline-van der Waals potential form to the pressure second virial coefficient, viscosity, thermal conductivity and depolarized interaction-induced light scattering data. Absolute zeroth and second moments of the two-and three-body spectra, the pressure third virial coefficient and isotopic thermal diffusion factor have been measured and compared with theoretical calculations using various models for the interatomic potential. The results show that it is the most accurate potential yet reported for this system. The use of the new potential in lattice sum calculations yields good results for several properties of solid krypton.     

H-H and C-H coupling constants in neutral and protonated azines

New measurements are reported of the density dependent depolarization ratio for argon, krypton, xenon, methane and sulphur hexafluoride, and the results are analysed to provide values for the second and third depolarization virial coefficients. The relationships between the second depolarization virial coefficient, the zeroth moment of the two-body Rayleigh spectrum and the second Kerr virial coefficient are considered, and it is shown that they now provide consistent results for the collision-induced pair polarizability anisotropy. Former inconsistencies are attributed to insufficient allowance for the effects of three-body interactions. Calculations of the second and third depolarization virial coefficients based on the DID model and using the Maitland-Smith potential are in excellent agreement with the experimental results for argon, krypton and xenon.     

On the formal equivalence of the Frisch-Berne and Watts perturbation theories to evaluate transport coefficients

A system of hard spheres in contact with a structureless wall is considered. The potential between the wall and the particles is of Lennard-Jones type. The second, the third and the fourth three-dimensional virial coefficient of the adsorption isotherm are calculated. The result being that over an extremely large temperature range the two-dimensional virial treatment is a rather good approximation. Conclusions are drawn about the deviations from Henry's law and the coverage, which can be described by virial expansions. It is doubted whether, from experimentally determined virial coefficients, information about the potential between the adatoms can be extracted.     

Bose–Einstein condensation transition studies for atoms confined in Laguerre–Gaussian laser modes

Multiply-connected traps for cold, neutral atoms fix vortex cores of quantum gases. Laguerre–Gaussian laser modes are ideal for such traps due to their phase stability. We report theoretical calculations of the Bose–Einstein condensation transition properties and thermal characteristics of neutral atoms trapped in multiply connected geometries formed by Laguerre–Gaussian (LG_p^l) beams. Specifically, we consider atoms confined to the anti-node of a LG₀¹ laser mode detuned to the red of an atomic resonance frequency, and those confined in the node of a blue-detuned LG₁¹ beam. We compare the results of using the full potential to those approximating the potential minimum with a simple harmonic oscillator potential. We find that deviations between calculations of the full potential and the simple harmonic oscillator can be up to 3%–8% for trap parameters consistent with typical experiments.     

Contribution of quantum molecular flexibility to the second virial coefficient of water vapor

The contribution of essentially quantum internal molecular motions to the second virial coefficient B2 of water vapor is analyzed in the framework of the path integral approach. A general purpose ab initio polarizable force field QMPFF2 or a nonpolarizable three-site water model are used with oscillator and Morse valence potentials. It is demonstrated that the contribution may be significant but depends strongly on the form of the intramolecular potential. In the case of the more realistic stretching Morse potential, inclusion of quantum molecular flexibility into the simulation reduces the virial coefficient by 20%-40%. Also, the internal modes make a contribution to the difference in the virial coefficient for light and heavy water, which is opposite to that of the intermolecular motions, so that the net effect can even change the sign at higher temperatures.     

Ab initio calculation of the refractivity and hyperpolarizability second virial coefficients of neon gas

Second virial coefficients for the density dependence of a number of electric properties are calculated for neon gas. Employing an accurate CCSD(T) potential for the Ne² van der Waals dimer and interaction-induced electric dipole polarizabilities and hyperpolarizabilities obtained from CCSD response theory, we evaluated the dielectric, refractivity, Kerr and ESHG second virial coefficients using both a semiclassical and a quantum statistical approach. The results cover a wide range of temperatures and are expected to be more reliable than the available experimental and empirical data. Quantum effects are found to be important only for temperatures below 100 K. The frequency-dependence of the refractivity virial coefficient is found to be small, but not negligible. For frequencies in the visible region it accounts for a few percent of the final results. For the ESHG virial coefficient of neon, frequency dependence is found to be very important, accounting for 20–25% of the second virial coefficient at the typical frequencies employed in experiments.