Spin waves in paramagnetic Boltzmann gases

Abstract

As the temperature is lowered we get an interesting temperature regionɛd≪T≪ħ2/mr02(whereɛd is the quantum degeneracy temperature,m the mass of a gas molecule,r0 the radius of interparticle interaction) in which the thermal de Broglie wavelength Λ of a particle is considerably greater than its sizer0 though Λ turns out to be less than the mean interparticle distanceN−1/3≫Λ≫r0. Although the gas molecules obey the classical Boltzmann-Maxwell statistics the system as a whole begins to exhibit a larger number of essentially quantum macroscopic collective features. One of the most interesting and dramatic features is the possibility of propagation of weakly damped spin oscillations in spin-polarized gases (external magnetic field, optical pumping). Such oscillations can propagate both in the low-frequencyθτ≪1 regime and the high frequencyθτ≫1. The last case is highly non-trivial for a Boltzmann gas with a short range interaction between particles. The weakness of relaxation damping of spin modes implies that the degree of polarization is high enough 1>/|α|≫|a|/Λ, whereα=(N+−N−)N,a is the two-particles-wave scattering length. Under these conditions the spectrum of magnons has the form (Bashkin 1981, 1984; Lhuillier and Laloe 1982) $$\omega = \Omega _H + \left( {{{K^2 \nu _{\rm T}^2 } \mathord{\left/ {\vphantom {{K^2 \nu _{\rm T}^2 } {\Omega _{int} }}} \right. \kern-\nulldelimiterspace} {\Omega _{int} }}} \right)\left( {{{1 - i} \mathord{\left/ {\vphantom {{1 - i} {\Omega _{int} }}} \right. \kern-\nulldelimiterspace} {\Omega _{int} }}\tau } \right), \Omega _{int} = {{ - 4\pi ahN\alpha } \mathord{\left/ {\vphantom {{ - 4\pi ahN\alpha } m}} \right. \kern-\nulldelimiterspace} m}, \nu _{\rm T}^2 = {T \mathord{\left/ {\vphantom {T m}} \right. \kern-\nulldelimiterspace} m}$$ (1) where ΩH is the Larmor precession frequency for spins in the magnetic fieldH. Collisionless Landau damping restricts the region of applicability of (1) with not too large wave vectorsKvT≪|Ωint|. The existence of collective spin waves has been experimentally confirmed in NMR-experiments with gaseous atomic hydrogen H↑ (Johnsonet al 1984). The presence of undamped spin oscillations means automatically the existence of long range correlations for transverse magnetization. Such correlations decrease with the distance according to the power law $$\delta _{ik} \left( r \right) = 2\left| a \right|\frac{{\left( {\beta N\alpha } \right)^2 }}{\gamma }\delta _{ik} $$ (2) . Hereβ is the molecule magnetic moment. Spin waves being even damped can nevertheless reveal themselves atT≳ħ2/mr02 or when |α|≲r0/Λ. The first experimental discovery or damped spin waves in gaseous3He↑ has been done in Nacheret al 1984. Oscillations of magnetization can also propagate in some condensed media such as liquid3He-4He solutions, semimagnetic semiconductors etc.