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Physical Review

Mobility of Ions in a System of Interacting Bose Particles

Ryuzo Abe, Ko Aizu

Published Jul 1, 1961• 68 citations
Publisher

Abstract

The transport property of an ion in a dilute Bose-Einstein gas subject to an external electric field is investigated by means of the Boltzmann equation. The interaction Hamiltonian which describes the ion-phonon scattering processes is obtained by the use of the Bogoliubov transformation and the cross section for the scattering of the ion by phonons is calculated. The solution of the Boltzmann equation is obtained by applying a variation principle and the temperature dependence of the ion mobility is shown to be ${T}^{\ensuremath{-}4}$ at very low temperatures. A comparison of the results with the experimental data in liquid helium and the Khalatnikov and Zharkov theory is given and also the ion mobility in a Fermi system is briefly discussed.

DOI
10.1103/physrev.123.10

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Physical Review•Aug 15, 1962•7 citations
The transport property of an ion in a dilute Bose-Einstein gas subject to an external electric field is investigated by means of the Boltzmann equation. The interaction Hamiltonian which describes the ion-phonon scattering processes is obtained by the use of the Bogoliubov transformation and the cross section for the scattering of the ion by phonons is calculated. The solution of the Boltzmann equation is obtained by applying a variation principle and the temperature dependence of the ion mobility is shown to be T\ensuremath−4{T}^{\ensuremath{-}4}T\ensuremath−4 at very low temperatures. A comparison of the results with the experimental data in liquid helium and the Khalatnikov and Zharkov theory is given and also the ion mobility in a Fermi system is briefly discussed.
The transport property of an ion in a dilute Bose-Einstein gas subject to an external electric field is investigated through Boltzmann equation approach. By means of the Bogolyubov transformation, the interaction hamiltonian between ion and elementary excitation is obtained, and the cross section for the scattering of the ion from phonons or rotons is calculated. The solution to the Boltzmann equation is obtained by the use of a variation principle, and the temperature dependence of ion mobility is shown to be UPSILON /sup -4/ at very low temperature regions where phonon excitations are more important than rotons. At higher temperatures where roton excitations are important the temperature dependence is shown to be UPSILON /sup -1/ exp( DELTA /k UPSILON ), where DELTA is the excitation energy of roton. The agreement in absolute magnitude with the experimental data in liquid helium is obtained by choosing reasonable value of parameter both in higher and lower temperature regions. Comparison with the Khalatnikov and Zharkov theory is given, and also the ion mobility in Fermi system is briefly discussed. (auth)
We investigate transport dynamics of a single low-energy ionic impurity in a Bose-Einstein condensate. The impurity is implanted into the condensate starting from a single Rydberg excitation, which is ionized by a sequence of fast electric field pulses aiming to minimize the ion's initial kinetic energy. Using a small electric bias field, we study the subsequent collisional dynamics of the impurity subject to an external force. The fast ion-atom collision rate, stemming from the dense degenerate host gas and the large ion-atom scattering cross section, allow us to study a regime of frequent collisions of the impurity within only tens of microseconds. Comparison of our measurements with stochastic trajectory simulations based on sequential Langevin collisions indicate diffusive transport properties of the impurity and allows us to measure its mobility. Our results open a novel path to study dynamics of charged quantum impurities in ultracold matter.
Transport coefficients of superfluid dipolar Fermi gas are calculated by using the Boltzmann equation approach. The interaction between Bogoliubov-Bogoliubov quasiparticles in the collision integral is considered in binary process. The shear viscosities nxz↑ = nyy↑,nxy↑; nxz↑ ≈ nyz↑; and nu↑ are proportional to T-6; T-8; and T-10 , respectively. Also, we have found the elements of the diffusive thermal conductivities Kxx↑= Kyy↑ with temperature dependency T-5 and Ku↑ which is proportional to T-7 and other components which are zero.
This article reviews recent experimental and theoretical work on the mobility of negative ions in the superfluid A and B phases of liquid 3He. In the normal Fermi liquid at temperatures below ∼ 50 mK and also in the superfluid close to the superfluid transition temperature, Tc , the mobility of a negative ion may simply be considered as limited by the elastic scattering of 3He quasiparticles. This explains the constancy of the ion mobility in the normal phase. However, underlying the rapid increase of the measured mobility in the superfluid phases there is a subtle quantum-mechanical scattering effect. Detailed solutions of the 3He quasiparticle—negative ion scattering process in the pair-correlated state provide a simple physical picture of an energy-dependent forward-peaking phenomenon. This yields quantitative theoretical results for the ion mobility in the quasi-isotropic B phase and for the ion mobility tensor in the anisotropic A phase which agree with the experimental data.
The viscosity of a dilute Bose–Einstein condensate is obtained for a range of temperatures and densities. The kinetic equation used to derive the viscosity is based on Bogoliubov mean field theory and is a Boltzmann-like equation for the Bogoliubov excitations in the condensate. The viscosity can be assigned a numerical value for any gas for which the mass and scattering length are known. The viscosity decreases slowly as the temperature is decreased, but at low temperature (around 0.01 T c ), begins to show a much more rapid decrease with decreasing temperature. This interesting behavior of the viscosity is related to the behavior of the eigenvalues of the bogolon collision operator at these low temperatures.
The stability of phonons in Bose systems is considered. It is noted that the phonons in helium II are stable against decay into two or more phonons, in contrast to the low-density Bose gas where the lifetime for decay of one phonon into two is finite and has been calculated by several authors. The consequences of phonon stability for allowable mechanisms of phonon-phonon interaction are considered. It is concluded that virtual three-phonon processes may still occur although energy-conserving three-phonon processes cannot take place. The inclusion of such virtual processes makes it plausible that the interactions between excitations can be weak, even though the linewidths measured in neutron scattering are larger than the line shifts. The quantum hydrodynamics of Landau and Khalatnikov including nonlinear terms is considered as a calculational model. In this theory the stability of the phonons is indeterminate and must be legislated. If the phonons are assumed unstable against decay, the phonon lifetime in the low-density and long-wavelength limit can be calculated, and is in agreement with the result for the atomic theory of the low-density Bose gas. This lends considerable plausibility to the quantitative validity of the Landau-Khalatnikov calculation of phonon-phonon scattering in helium II via an almost energy-conserving intermediate one-phonon state. The paper concludes by comparing the results of an idealized neutron scattering experiment as calculated from the Feynman-Cohen point of view, and from the point of view of the theory of the low-density Bose gas.