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.
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.
On considère un gaz d'atomes d'hydrogène supposé stable (pas de recombinaison moléculaire), dont les variables internes de spins électronique et nucléaire sont décrites par un opérateur densité fixé ρSI. Les atomes étant chacun constitué d'une paire de fermions (un proton et un électron), non indissociable du fait de l'existence des collisions d'échange de spin, leur comportement de bosons n'est pas évident. Le but de cet article est de discuter dans quelle mesure la pression du gaz est effectivement celle d'un gaz de bosons indissociables, dans le cadre d'un calcul limité à celui du second coefficient du viriel (correction du second ordre en densité pour un gaz dilué). On trouve que ce coefficient comprend en fait plusieurs termes, soit de statistique pure (effets d'indiscernabilité en l'absence d'interaction entre atomes), soit d'interaction et de statistique combinées (effets des potentiels Vg et Vu). Tous ces termes dépendent explicitement des valeurs moyennes d'un certain nombre d'opérateurs de spin, calculables à partir de ρSI. Le terme de statistique pure, dominant à très basse température, correspond à l'échange d'atomes complets (échange simultané de leurs protons et de leurs électrons), ce qui explique son caractère bosonique. En revanche, dans les termes liés aux interactions apparaissent, non seulement des contributions provenant de l'échange d'atomes complets, mais également des contributions liées à l'échange d'un type de particule seulement (protons ou électrons) qu'on peut relier aux effets des collisions d'échange de spin.
Abstract The quantum-mechanical formulations for the second virial, and the acoustic virial, coefficients are derived for spin-polarized atomic hydrogen (H↓). The dependence of these coefficients on the temperature T as well as the nuclear polarization ζ is analyzed. This is done for the first time. The main inputs in these computations are the scattering phase shifts, which are obtained using the Lippmann-Schwinger equation, with the Silvera triplet-state potential. Starting with these phase shifts, comprehensive calculations of the thermophysical properties for this system are performed for T ranging from 1µK to 100 K, and ζ varying from 0 to 1. These properties include, in addition to the quantum second virial coefficient and the acoustic virial coefficient: the pressure-polarization-temperature (P-ζ-T) behavior, the entropy (per atom), the speed of sound, the second virial correction (to the total internal energy per atom and unit density), and the specific heat capacity (per atom and unit density). The Boyle temperature and the Joule inversion temperature are determined. The T-dependence of these thermophysical properties, and related quantities, is explored. As expected, this dependence is most evident in the low-T limit, where quantum effects predominate. The corresponding ζ-dependence becomes noticeable at 80 K and below. It is observed that the virial coefficients tend to decrease with increasing ζ. Comparison of the present results to previous results are included whenever possible. The overall agreement is very good. As T is lowered, the disruptive effects of thermal energy are weakened relative to the attractive interactions between the atoms. Consequently, the H↓ gas makes a transition to a state of higher order and lower entropy – Bose-Einstein condensation (BEC). This is explored carefully.
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 generalize our previous work on the compatibilty of kinetic equations with second virial corrections to the inclusion of spin and particle indistinguishability; the system is supposed to be sufficiently dilute for higher order virial density corrections (interactions and statistics) to be negligible. We show that the general idea of the “free Winger transform” can be extended to this situation; the function which appears in the kinetic equation becomes here a matrix which acts in the space of spin states of the particles. Assuming that the collisions are described by a hamiltonian which does not act on the spins (a very good approximation for nuclear spins), we write explicitly a kinetic equation which is valid for this case. The right hand side of the equation is an 18 dimension integral, as for spinless distinguishable particles, but here it contains an additional term due to statistics, which introduces commutators and anticommutators. We discuss the local conservation laws in this formalism and find, as expected, a total number of 8 conserved quantities for spin 1/2 particles (including three components of the magnetization). When the gas is at equilibrium, we obtain a pressure dependence which is in agreement with known calculations on spin polarized gases. We finally study the gradient expansion of the collision integral, and show that the zero-order (local) part is identical with the 4 terms (including identical spin rotation terms) obtained previously by Lhuillier et al. The first order (non-local) part contains many terms, wich we compare with those obtained by Silin in a context more closely related to the Landau theory.
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.
Bose–Einstein and Fermi–Dirac statistics versus Boltzmann statistics It was remarked in Chapter 2, §2.1, that there are circumstances in which the “Boltzmann statistics” for an ideal gas – the approximation in Eq. (2.6), in which the total partition function is decomposed into a product of single-molecule partition functions, with division by N 1 ! N 2 ! … to correct for particle identity – is inaccurate even if the gas is still ideal (non-interacting particles). That happens when the number of single-particle states within the energy kT of the ground state is no longer very much greater than the number of particles. (Recall Fig. 2.1 and the accompanying discussion.) We noted that when the Boltzmann statistics is no longer an adequate approximation one must distinguish the particles as fermions, no two of which can be in the same single-particle state (counting states of different spin as different), or as bosons, for which there is no restriction on multiple occupancy of the singleparticle states. Fermions are particles of half-odd-integer spin quantum number; they could be composite particles consisting of an odd number of spin-½ particles. Protons, neutrons, and electrons, all of which are of spin ½, are fermions. Bosons are particles of integer spin; they could be spinless, or of spin 1, etc., or could be composites consisting of an even number of fermions. The nuclei of helium atoms of mass number 4 consist of two protons and two neutrons, and the neutral atom has two electrons outside the nucleus; 4 He atoms are bosons.