The quantum Boltzmann equation governing the evolution of the one-body internal density matrix of an atomic hydrogen (or alkali) gas is established. Indistinguishability of both nuclei and electrons explains the presence in the collision term of a large number (eleven) of « exchange cross sections ». The different collision terms fall into two categories : these which contribute to the damping of perturbations (similar to the usual collision term in the classical regime) and those which are Hamiltonian like and originate from the « identical spin rotation effect ». Due to those numerous exchange terms, all transport properties in atomic hydrogen are very sensitive to the internal state of the gas.
The quantum Boltzmann equation governing the evolution of the one-body internal density matrix of an atomic hydrogen (or alkali) gas is established. Indistinguishability of both nuclei and electrons explains the presence in the collision term of a large number (eleven) of « exchange cross sections ». The different collision terms fall into two categories : these which contribute to the damping of perturbations (similar to the usual collision term in the classical regime) and those which are Hamiltonian like and originate from the « identical spin rotation effect ». Due to those numerous exchange terms, all transport properties in atomic hydrogen are very sensitive to the internal state of the gas.
The identical spin rotation effect is the quantum mechanical collisional effect underlying the transverse nuclear spin waves recently seen in dilute non degenerate gases in Paris [3He] and at Cornell University [H↓]. This mechanism could in principle give rise, at low enough temperatures, to hydrodynamic oscillatory modes associated with any non diagonal elements of the atomic internal density matrix (generalized spin waves or coherences waves). Starting from a quantum Boltzmann equation recently derived for the study of 4-component atomic hydrogen, we show that the current of any transverse quantity (hyperfine or Zeeman coherence) does indeed precess around molecular exchange fields. Unhappily, in most cases, the oscillatory modes associated with these transverse quantities are heavily damped. The first intrinsic cause of damping of these waves is spin exchange. We study the conditions under which this damping is minimized in both zero field and in high magnetic field. Numerical values of all relevant parameters for the description of spin transport properties in atomic hydrogen are given in the appendices.
The general results recently obtained for the Boltzmann collision term in a gas of atomic hydrogen are applied to the study of longitudinal spin diffusion in high magnetic fields. This phenomenon can be described as a diffusion of concentration in a four component reactive mixture (the reaction being here the electronic transfer between nuclei). A numerical discussion of all the coefficients of the diffusion matrix between 100 mK to 100 K is given in the text, together with a discussion of the nature of the hydrodynamic modes in such a situation. The variations of the diffusion coefficients with the internal state of the gas are important for temperatures lower than 500 mK or higher than 10 K, when the spin exchange process is not negligible compared to the direct process. Finally, the relative importance of these two processes determines the nature of the hydrodynamic modes in the system.
Polarization phenomena in collisions between one-electron (alkali-like) atoms are calculated with density-matrix techniques. The kinetic energies for which this treatment is valid are limited from above by the use of the Born-Oppenheimer approximation, and from below by the assumed degeneracy of the hyperfine states. The basic mechanism for polarization changes is the electron-exchange effect, as suggested by Purcell and Field and by Wittke and Dicke. The special situation of unpolarized targets is treated in some detail. In this case, there are two numbers which characterize the scattering through any angle: the absolute values of certain direct and exchange scattering amplitudes, Fd and Fx. All cross sections must be expressible in terms of ∣Fd∣ and ∣Fx∣, and this is explicitly carried out in two cases of special interest. One refers to the case in which only the electron-spin polarization is measured, and the other to measurements of the polarization of the hyperfine states of the scattered atom. Effects of the identity of the atoms are also studied in detail.
The effect of spin exchange collisions on the internal variables (orientation, alignment, etc.) of colliding atoms is discussed, with particular emphasis on the role played by the Pauli principle. The calculation presented is valid for distinguishable or identical particles, and the initial introduction of an effective exchange hamiltonian for the spins is avoided. The one-atom density matrix of the internal variables is calculated after collision, in an operatorial form which enables the discussion of various physical effects. The first term in the density matrix evolution is a transfer term which does not originate from particle indistinguishability. The following terms are indeed introduced by the Pauli antisymmetrization principle applied to electrons and nuclei, and they can be described as electronic or nuclear apparent magnetic field (or electric gradient). In addition, terms arising from the combined exchange of electrons and nuclei are obtained, in particular from the exchange of identical whole atoms (bosons in the case of hydrogen). The evolution equations for the density operators are applied to a few particular cases in order to evaluate spin exchange effects in various situations : weak or strong hyperfine decoupling, Zeeman or hyperfine resonances, etc...
We consider a gas of atomic hydrogen which is supposed to be stable against molecular recombination and whose internal variables (electronic and nuclear spins) are described by a given density matrix ρSI. Since each atom consists of a pair of fermions (one proton and one electron) which is not indissociable due to the possibility of spin exchange collisions, their bosonic behaviour is not obvious a priori. Our aim is to discuss to what extent the pressure of such a gas is actually the pressure of a gas of bosons, our calculation being restricted to the quantum-mechanical second virial coefficient (second order correction in density to the pressure of a dilute gas). We find that this coefficient is the sum of several terms, due either to pure statistics (particle-density effects in the absence of interactions) or to the combined effects of interaction and quantum statistics (role of Vg and Vu potentials). All these terms depend explicitly on the average over ρSI of some spin operators. The pure quantum statistical contribution, which is dominant at very low temperature, is related to the exchange of two atoms as a whole (protons and electrons exchanged at the same time), which accounts for its bosonic behaviour. On the other hand, the interaction effects give rise to contributions associated with the exchange of one kind of particle only (protons or electrons) related to the spin exchange collisions, in addition to contributions arising from the exchange of whole atoms.
This is the first of a series of articles where we develop a theory for the description of the dynamics of dilute quantum gases, at or out of equilibrium, with an accurate treatment of the correlation effects introduced by binary collisions (these correlations are not taken into account when the Boltzmann equation is used). For this purpose, we propose the introduction of a variant of the Wigner transform, the « free Wigner transform », which wipes out the effects of the potential at short distances. If initially factorized, the entering part of the free transform remains exactly so during all collision, so that it is better adapted to a molecular chaos assumption than the ordinary transform. Once the free transform is known, the effects of the short distance correlations can be reconstructed from it. Therefore, the problem is to find an appropriate equation of evolution which gives its time variations. With this aim in mind, we study in the present article the evolution of the free Wigner transform in a binary collision. We show that its evolution equation includes coefficients which can be exactly obtained from the « on shell » terms of the T matrix (S matrix, phase shifts), even while the particles are interacting. In addition to the usual terms which involve the scattering cross sections, as usual with the Boltzmann equation, the calculation provides other terms which correspond to collision retardation effects and « refraction » effects (refraction of the wave function of the test atom by density gradients of the collision partners). These terms will play an important role in the following article, and introduce the second virial corrections to the equilibrium and transport properties of the gas.