Longitudinal electronic and nuclear spin diffusion in atomic hydrogen J.P. Bouchaud, C. Lhuillier To cite this version: J.P. Bouchaud, C. Lhuillier. Longitudinal electronic and nuclear spin diffusion in atomic hydrogen. Journal de Physique, 1985, 46 (8), pp.13351344. <10.1051/jphys:019850046080133500>. <jpa00210077> HAL Id: jpa00210077 https://hal.archivesouvertes.fr/jpa00210077 Submitted on 1 Jan 1985 HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
67.20 Partant The Zeeman J. Physique 46 (1985) 13351344 AOÛT 1985, 1335 Classification Physics Abstracts 51.10 05.30 Longitudinal electronic and nuclear spin diffusion in atomic hydrogen J. P. Bouchaud and C. Lhuillier Laboratoire de Spectroscopie Hertzienne de l E.N.S., 24 rue Lhomond, 75231 Paris Cedex 05, France (Reçu le 25 janvier 1985, accepté le 16 avril 1985) 2014 Résumé. de l équation de Boltzmann quantique régissant la dynamique de l hydrogène atomique, on étudie la diffusion de la composante longitudinale des spins électronique ou nucléaire en haut champ magnétique. Ce problème est équivalent à un problème de diffusion de concentration dans un mélange réactif à quatre composantes (la réaction étant ici le transfert électronique). L importance des divers coefficients de la matrice de diffusion est présentée numériquement dans un très large domaine de température (100 mk100 K). Il est montré comment ces coefficients varient avec l état interne du système. Ces variations sont considérables pour toutes les températures inférieures à 500 mk ou supérieures à 10 K, beaucoup plus faibles dans le domaine des températures intermédiaires à cause de la faiblesse des phénomènes d échange de spin dans ce domaine de température. L importance relative des phénomènes d échange de spin visàvis des processus directs de collision détermine la nature des modes hydrodynamiques décrivant la diffusion des variables internes d un tel système. 2014 Abstract. 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. 1. Introduction. Theoretical interest in electronic spin polarized hydrogen (H!), a twocomponent bosonic system expected to undergo Bose condensation in gaseous phase [13], has renewed the interest in atomic hydrogen. Many experiments have been designed (Amsterdam, M.I.T., Vancouver, Comell, Harvard and Grenoble) to study low temperature properties of atomic hydrogen with a view to realizing either Bosecondensed Hi, or spin polarized beams and targets for nuclear physics, or else ultra stable low temperature masers [4]. All these experiments have in common the fact that at some stage they deal, not with the idealized Hi (i.e. a system of atoms with complete electronic spin polarization and only one internal degree of freedom, namely the nuclear spin), but with a statistical mixture of the four first internal levels of atomic hydrogen schematized in figure 1. The kinetic behaviour of such mixtures is important in many experimental situations : electronic and nuclear spin diffusion (the Fig. 1. diagram of hydrogen. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019850046080133500
independently respectively in by in 1336 subject of this paper) is the main phenomenon that governs the shape of NMR and ESR spectra as well as the efficiency of the StemGerlach sorting of the 4component mixture in polarized beam achievement. In order to be able to study the transport properties of such gases, we recently established the Boltzmann equation governing the evolution of the internal density matrix of hydrogenlike systems [5]. Without going into details (which can be found in reference [5]), one can try to explain why the dynamics of fourcomponent hydrogen is substantially more complex than that of Hi : H! atoms can be considered in this range of thermal energy as undissociable bosons interacting through one molecular potential Vu. At variance in the 4component hydrogen, two molecular channels are open for elastic interactions : the even one (molecular potential Vg) and the odd one (potential Vu). Going from the molecular to the atomic point of of any view, this property explains indistinguishability effect ([5] and references therein) the existence of two elastic collision processes : the direct, and the transfer, usually named spin exchange process, in which the two nuclei exchange their electrons. As a result, H atoms can no longer be considered as undissociable entities and statistical effects due to the indistinguishability of individual protons and electrons become important : four component hydrogen does not behave as a pure boson. This very brief account of the various effects involved in the collision process explains the great complexity of the general collision term of the Boltzmann equation of reference [5]. We shall not comment further here on this general equation but apply it to solve the simplest transport problem in such a system, that is electronic and nuclear longitudinal spin diffusion. In other words, we shall try to answer the following question : how do the different components of hydrogen (levels a, b, c, d of figure 1) diffuse relative to each other? The general transport equation of reference [5] allows the description of the species diffusion in any external field from zero magnetic field to very high fields : we shall in the following focus on the results in the high magnetic field limit, but the general qualitative results we shall obtain for such mixtures large variability of the diffusion coefficients with internal would be state of the mixtures and with temperature equally valid whatever the external field. The paper is organized as follows : in section 2, we derive from the general theory of reference [5] the kinetic equations governing the evolution of the four populations of the a, b, c, d levels of hydrogen in high magnetic field. We discuss very briefly on physical grounds, the form of the collision term of these equations and the nature of the equilibrium distribution of both external and internal degrees of freedom. In section 3, we derive the general expressions for the spin currents and comment briefly on the very large numerical variations of these coefficients versus temperature and internal composition. Section 4 is devoted to the study of the hydrodynamical regimes : the general form of the hydrodynamical equations is established as well as their specialization in the two limiting cases where transfer processes are either negligible or as important as the direct processes. We summarize and conclude by discussing the nature of the spin diffusion regime versus density and temperature. All formulae necessary for computing spin diffusion properties in hydrodynamical regimes are given in Appendix together with a table of numerical values of the important collisions integrals. 2. Kinetic equation and nature of the equilibrium. A dilute, non degenerate mixture of the four first levels of atomic hydrogen can be specified the simplest case, where no coherences exist the four distributions functions f.(r, p), fb(r, p), f,,(r, p), fd(r, p) describing the populations of the four a, b, c, d species (see Fig. 1) in phase space. The expressions of the a, b, c, d states in the decoupled I mi, M s) basis allows then a direct use of equation (18) of reference [5] to calculate the rate of change of the f (r, p) through collisions. For the sake of simplicity we shall write these equations in the high magnetic field limit where the a, b, c, d levels can be described the decoupled I mi, Ms > basis as I tj >, I ii ), Iii X. I TT > ( ) In this case, equation (18) of reference [5] is diagonal, all commutators disappear and we obtain for the collisional rate of change of fa : (1) This is equivalent, as a first approximation, to ignoring the hyperfine coupling A. Taking into account this coupling would introduce minor modifications of the order of Alp, H to the matrix of diffusion coefficients : an effect generally negligible with regard to the range of variation of these various coefficients.
We of Equation 1337 and three similar equations for fb, fe, fa. (The passage from equation (18) of reference [5] to this equation requires a manipulation of differential cross sections easily done with the help of formulas (Al12) of reference [5].) The notations used in (1) are the usual ones f.,( 1 ) stands forfi(r, p 1, t) (local collisions), the momentum of ingoing and outgoing particles are related by the usual relations for elastic collisions : It is important to note that in the Boltzmann equation, the label is primarily associated with the momentum and thus with the nucleus. The reaction schematized in equation (2) is a pure electronic transfer phenomenon with no exchange degeneracy, which justifies the weight 6t. The second transfer term is associated with the following scheme : and v, is the velocity of the relative motion ( qilm). The numerous cross sections appearing in this equation are related to the various mechanisms described in the introduction. Their exact definitions are given in reference [5] and are recalled in appendix 1. Be (resp En) are the signatures of an exchange process involving electrons (resp nuclei). In our case electrons and protons are fermions, Ee 6n 1. 2.1 PHYSICAL DISCUSSION. do not want here to duplicate the discussion of the general Boltzmann equation of reference [5] but wish to underline that, in the present simple situation, the form of the Boltzmann collision term is in fact quite natural : the first four contributions are those expected for a classical non «reactive mixture», the last two being those associated with the «reaction» taking place in the mixture due to the spin exchange process. i) The first two terms of equation (la) describe collisions aa or ab that is, collisions between atoms in the same electronic spin state : due to this constraint, the interaction potential is necessarily Vu and the two terms are readily obtained by considering the nuclear spin of the encounters : either they are in the same nuclear spin state and thus indistinguishable and the process must be described by a symmetrized cross section ( au + Be Bn 6u ) or they are in different nuclear spin states and thus distinguishable which justifies the use of the unsymmetrized Qu cross section. It can be remarked that the problem of Hi reducing to the consideration of only these two kinds of collisions, the general approach leads then to the results obtained directly in reference [6] where the particles were taken to be undissociable. ii) The third and fourth terms of (1) (collisions ac and ad) describe collisional processes of atoms in different electronic spin states without electronic transfer : the use of the direct cross section ad appears as normal. The third term involves encounters which can be distinguished both by their nuclear and electronic spin, whereas in the fourth term the nuclear spins are indistinguishable. This explains the s. a factor related to the nuclei exchange degeneracy. iii) The transfer terms are more involved : the first one is associated with the following spin flip : In this reaction, there is a nuclear exchange degeneracy which explains the general form of the cross section (a, + 8n dt The collision term thus appears to be relatively simple, even though the exact form of the exchange cross sections could hardly have been predicted a priori. 2.2 THE LOCAL EQUILIBRIUM. (1) (with the three similar equations for fb, fe, fa) implies that the local equilibrium has the following two properties : First, it is Boltzmannian : that is such that Second, it obeys the internal equilibrium law The first condition is a well known consequence of the collisional invariance of mass, momentum and kinetic energy and of the Boltzmann H theorem. Condition (5) is its internal counterpart, a consequence of the collisional invariance of both electronic and nuclear spins; it implies that the populations of the internal levels necessarily scale like : exp (ams + ymi). Insofar as hyperfine coupling as well as all the magnetic couplings are negligible (which is assumed throughout this paper), the system may exhibit nuclear and electronic temperatures different from that defined by the equilibrium distribution of velocities. 2.3 RELAXATION TOWARDS EQUILIBRIUM. The relaxation towards local equilibrium is governed by processes with characteristic rates dominated by the thermal averages ( nvud), nvuu) and nva"> while the rate of the transfer «reaction» Ft is measured by an average of the transfer cross section (Tt N ( nv at>) [7]. It is a peculiar feature of hydrogen [8a, b] great interest for the realization of Hi that, in the low temperature range (1 K10 K), the cross section of the transfer process is at least an order of magnitude smaller than that of the direct
Let that 1338 process. As a consequence, it is possible to observe, during limited lapses of time, local equilibrium situations which obey equation (4) but violate equation (5) : we shall denote such situations as metastable mixtures. The return to the homogeneous equilibrium is indeed different depending on if it concerns situations where local equilibrium obeys both conditions (4), (5) (o equilibrium» mixtures) or «metastable» mixtures in which only equation (4) is obeyed. We shall, in a first step ( 3), give the expressions of spin currents in the general case where the local equilibrium only obeys equation (4); with convenient constraints, these results easily apply to the case where the local equilibrium obeys both equations (4) and (5). In a second step, we shall discuss the various hydrodynamical regimes that can occur in each situation and end with a numerical discussion of the domain of validity of each regime versus temperature and density of the gas. 3. Expressions of the spin currents in an homogeneous 4components mixture. 3.1 ANALYTICAL FORM OF THE SPIN DIFFUSION COEF FICIENTS. us then consider a situation where the total density n(r), the mean velocity of the gas u(r) and the mean temperature T(r) are homogeneous over the sample : the inhomogeneity of the problem is in the internal composition of the mixture and is described to first order by the gradient of each species, four quantities related by the constraint : This defines a 4 x 4 matrix of diffusion coefficients. However the concentration gradients are subjected to the constraint (6) insuring that the flux under consideration is a purely diffusive effect (not mixed with sound) : this implies that the separate study of the 16 coefficients of the above mentioned matrix has no physical meaning in this situation. As the problem depends now on three independent internal variables, it is much more meaningful to discuss it in such a basis. Nevertheless, many choices of independent internal variables are possible and in the following we shall in fact use two of them. In order to discuss the analytical and numerical form of the spin currents in high magnetic field, it is particularly convenient to consider the three following variables : the electronic polarization of the system : (S 1 for H 1) and what we call nuclear polarization of the up and down electronic components, IT and Ii : With these variables, the expression of the fluxes J becomes : Using a standard ChapmanEnskog approximation scheme, it is easy to obtain the fluxes of the four species as linear combination of their concentration gradients. where 1 is the unit 3 x 3 matrix and 6 the following 3 x 3 matrix : the (a, P, y, q, 6, s, 0) are linear combinations of collisions integrals involving the transfer cross sections (at, adt, dt while Wd is only related to the direct cross section. The precise expression of these coefficients is given in Appendix 2, together with the numerical results for the relevant collisions integrals (Table I). Their numerical values are reported on figures 2a and 2b. 3.2 PHYSICAL DISCUSSION AND NUMERICAL RESULTS. i) For any internal situation, the matrix (1 + E) can then be inverted, giving the matrix of diffusion coefficients of interest. In the general case, these dofiusion coefficients will strongly depend (through the matrix E) on the internal state of the system. This means as can easily be checked the diffusion of any species a, b, c, d, relative to each other is equally strongly dependent on the composition of the mixture (a result at variance with that of a twocomponent mixture). ii) However, due to the weak efficiency of the transfer process compared to the direct process in the 110 K temperature range, the coefficients of the 8 matrix are then small or very small compared to 1
Numerical In Table I. values of the a collision integrals defined in Appendix 2. Those integrals enter in the definition of the diffusion matrix and more generally in all the diffusive phenomena. T is in Kelvin and the various W in A2. (Detailed tables are available from the publisher on request.) 1339 (see Fig. 2) and the diffusion matrix reduces then to : We can thus conclude that in this temperature range, the electronic and nuclear spins diffuse approximately in the same way (within a 10 % approximation) : the nondiagonal elements being equally of the order of 10 % of the diagonal coefficients. With the same accuracy, one can say that the diagonal diffusion constants are then roughly independent of the internal state of the mixture. At temperatures lower than 1 K or much higher than 10 K, this approximation fails and the matrix (1 + 8) must be inverted. iii) A simple case (IT I I 0). In order to illustrate the limits of validity of approximation (12), we have calculated exactly all the diffusion coefficients in the specific case where : I T 1! 0 (i.e. na nd and n, nd) and S is arbitrary. Such a situation can occur either in presence of a very efficient wall relaxation for the nuclear spin or after a n/2 NMR pulse. The matrix can then be readily inverted giving the following results : a) The flux of the electronic component is only related to the gradient of the electronic spin and decoupled from any gradient of nuclear polarization; Fig. 2. these figures are shown all the coefficients of interest in the discussion of longitudinal spin diffusion (Sect. 3.1). (a) This figure shows the temperature dependence of Wd (introduced in formula (10); its precise definition and numerical value are given in Appendix 2). This thermally averaged cross section is the main contribution to the diagonal coefficients of the diffusion matrix (Eq. 10)). (b) Those curves show the different ratios a a/ Wa, P fl/ W,., appearing in equation (11), as functions of the temperature.
In 1340 the relevant diffusion coefficient Ds is then independent of the polarization. This coefficient is shown over a very large range of temperature in figure 3a. b) The fluxes of the two nuclear polarizations are coupled and depend on the electronic polarization : For an electronically unpolarized system S 0 (Fig. 3b), the two diagonal coefficients are equal as well as the two nondiagonal coefficients The order of magnitude of the diagonal coefficient for nuclear spin diffusion is the same as that for the electronic spin diffusion, but it can be seen by comparing figures 3a and 3b that they nevertheless deviate strongly in both the low and high temperature ranges. In the electronically polarized system (S 1), the symmetries of the coefficients of (13) are lost. As it should be, the Df; 1 Ds + 1 coefficient is identical to the nuclear spin diffusion coefficient ofh! obtained in reference [9]. For almost complete electronic polarization (S n, 1) the diffusion in the weak components Dr; 1 is quite different from that in the strongly populated components D and the flux of I T depends extremely weakly on a possible gradient of I I (see Fig. 3c). The flux of 7J, does not depend at all on VI T : All those results are also valid for the S 1 situation if Ii and 1 T are interchanged. Fig. 3. these three figures all the nonzero coefficients of the spin diffusion matrix (Eq. (10)) are represented as functions of temperature. S is arbitrary, IT Ii 0 and the problem has been treated exactly. (a) This curve gives the electronic spin diffusion coefficient D, in the situation where IT and Ii are both zero. In this case, the calculation is easy and exact. Ds does not depend on the value of S. Comparison with the approximation Do of equation (12) shows the limits of validity of this approximation. (b) On the other hand, the diffusion matrix linking the currents of IT and Ii to their gradients is not diagonal and does depend on the value of S. We show on figure 3b the coefficients when S 0. In this case, the matrix is symmetrical and with equal diagonal coefficients. (c) For S 1, the symmetry of the matrix is lost : the diagonal elements are not equal and one of the nondiagonal element is zero.
1341 Intermediate electronic polarization gives intermediate results between those illustrated in figures 3b and 3c for the two extreme cases. This specific example well illustrates the general comments made before. It emphazises the fact that in the intermediate temperature range (110 K), a perturbation expansion of (10) is allowed and that for lower or higher temperature, the matrix must be inverted exactly. In the low temperature range T 500 mk, the situation is very involved, each flux being strongly coupled to all the internal gradients : all coefficients of the 3 x 3 matrix are of the same order of magnitude (cf. Fig. 2) and each experimental situation must be studied with great care, the spin diffusion coefficients then being extremely sensitive to the internal state of the mixture. The existence of a local Htheorem nevertheless rules out the possibility of occurrence of an instability (i.e. the determinant of the characteristic matrix is always positive and finite for any set of values of S, I I and I T). As a last comment, it must be underlined that the 7(a 0) coefficients appearing in (11) are linear combinations of collision integrals of the various transfer cross sections (see Appendix 2). As such they depend on differences in scattering effects in the Vg and Yu potentials and may be much more sensitive to the exact forms of the potentials than the Wd coefficient which is related to the direct process. a measurement of some of these diffusion Conversely coefficients would be a crucial test of the quality of the potentials (the details on the potentials and on the numerical computation are given in Appendix 2). 4. Hydrodynamical equations. By integrating equation (1), one obtains the hydrodynamical equation for each species of the fourcomponent mixture; as an example, the hydrodynamical equation for the aspecies reads : where T(t ), the usual longitudinal relaxation time related to the spin exchange process [7], is defined by : where n is the total density, x the momentum of the relative motion, p the reduced mass and Q[P,t] the angular averaged transfer cross section : In the notations of reference [8], this quantity is equal to 1/2(6+ + a). Combination of the four equations similar to (16) gives the conservation equations of atomic density (here taken to be a constant) and of nuclear and electronic longitudinal spin density : with with that are direct consequences of conservation of the molecular quantities during collisions (cf. Ref. [5]). The last information contained in the four equations (16) can then be written under the form : where A is defined as : This last equation has a different structure containing both a diffusive conservative term (div J(A)) and a relaxation term which describes the effect of the «spin exchange reaction» taking place in the mixture. This set of three equations (17,18,19) supplemented by the expressions (10) of the spin currents gives a complete description of the diffusion in the 4component mixture. It is then easy to account for the transport processes in any situation ranging from the «metastable mixture» case to the case where the electron transfer effect imposes at each time the internal equilibrium (na n, nb nd or equivalently A IS). As we saw in the previous paragraph (3.2), the leading contribution to the fluxes of I, S and A can be described by a unique diffusion coefficient Do that determines a characteristic time Th for the diffusive mode of spatial extension L of the order L 2 /29)0. The set of hydrodynamical equations (17)419) can then easily be solved in the two extreme situations : i) Th TP); the transport phenomenon is a diffusion controlled process in a metastable chemical mixture; one can neglect the relaxation term of equation (19) and the problem becomes a usual diffusion process in a fourcomponent nonreactive mixture that can be described either with the three independent variables S, I, A or with the previous set of variables introduced in 3. 1 (1 (1 T + I )/2; A (Ii I )/2). ii) T,( ) Th; the spin exchange process is then sufficiently efficient to restore at each time a local
The 1342 internal equilibrium (that is A I S or equivalently na n, nb nd). The hydrodynamical evolution of A is at each time governed by the evolution of I and S that are the only two independent variables left in the problem. The use of the equilibrium constraint (A 1S) allows us to derive from the general equations (10) the expression of the fluxes of electronic and nuclear spin densities versus their local gradients : In the temperature range (110 K) where the transfer collision integrals are small compared to the direct collision integrals, the diffusion matrix is then diagonal (up to ;2) and takes the form : where Wl, W2, W4, W5 are collisions integrals of transfer cross sections small compared to Wd. Their algebraic expressions and numerical values are given in full in Appendix 2 and table I. iii) For characteristic times Th of the order of Tt(l), it must be remembered that the equations (1719) are coupled nonlinearly and must be solved together. To summarize and conclude on the various results obtained in this section, it is important to note that for a given experiment the frontiers between the different regimes studied here are temperature and density dependent. The characteristic times for the return towards local Boltzmann equilibrium Td nvud >1 or to the internal equilibrium T, nv at > are inversely dependent on the density, whereas the characteristic hydrodynamical time (Th L 2 /2 00) is linearly dependent on density. The first criterion of validity of the calculations done in this work is that Td should be short compared to the hydrodynamical characteristic time Th (in order to insure the validity of the ChapmanEnskog solution of the Boltzmann equation), that is, using expression (10) of the diffusion coefficient : In the (n, T) diagram, for a given experiment (that is for a given macroscopic length L), this condition allows us to determine a first critical line n(l)(t) separating the molecular Knudsen regime (region I of figure 4) from the region where the hydrodynamical approach is valid. In this last region, one can further discriminate between a region where diffusion takes place in a «metastable» mixture (i.e. a mixture where Fig. 4. two curves drawn in figure 4 divide the plane (density n x characteristic length L versus temperature) into three regions : the low density region (I) corresponds to a rarefied gas regime (Knudsen regime) where the mean free path is of the same order as the macroscopic length L. At higher density (regions II and III) hydrodynamical approach becomes valid. In the high density region (region III), transfer collisions are important, thus imposing an internal equilibrium, (A IS) to the system. The inbetween region (region II) corresponds to a situation in which the gas is both metastable (any concentration of species is stationary) and can be treated by hydrodynamical equation. This (small) region of interest is situated around 1016 at./cm for a sample of size 1 mm. any concentration of the four species is metastable) when :
For For We 1343 that is : The exchange cross sections u:p are defined by the generic formula : and the region of higher density where the mixture is everywhere in internal equilibrium (na nc nb nd) : region III of figure 4. In the Knudsen regime (region I of Fig. 4), the present ChapmanEnskog approximation fails and the Boltzmann equation must be solved directly. In the metastable domain (region II of Fig. 4), all formulae obtained in 3 are directly useful, the set of hydrodynamical equations (1719) reducing then to a set of three conservative equations. It must be emphasized that in this domain the approximation (12) is nearly everywhere a very good approximation. Finally, in the domain III, one can distinguish between the low and high temperature regions : T 10 K, the two independent fluxes of electronic and nuclear spins described by equation (20) are very well approximated by expression (21). The hydrodynamical equations of interest are equations (17:19) supplemented by the condition A IS. temperatures higher than 10 K, or lower than 5 K, one has to use the hydrodynamical equations (1719) supplemented by the general expression (10) of the fluxes : expression that must be constrained by the condition A 1S (and the related condition VA I VS + S VI). Acknowledgments. The authors are grateful to D. Kleppner for pointing out the experimental interest of the subject. They thank H. Guignes and G. Berthaud for helpful assistance in computational work. Appendix 2. THE DIFFUSION MATRIX. present here the definitions of the useful coefficients for the diffusion matrix in terms of «collisional integrals». The notations introduced in 3 are specified by the following formulae : where W are integrals defined by : Appendix 1. We recall very briefly in this appendix the definitions of the different cross sections relevant for the present discussion. The reader is referred to the appendix of references [5,12J for a more complete discussion of the properties and phase shift expansions of these quantities. To the two molecular potentials Vg and Vu are associated the usual transition matrices Tg and Tu and the related transition matrices for the direct and transfer processes : The different cross sections involved in equation (1) a., Qd, at are obtained from the respective transition matrices through formulas : We recall that the bracket notation [...] stands for p, v,, d3qi d2qf f b(pi) f b(p2) with notations defined in section 2, formulas (lb). Curves showing a/w d, fl/wd,..., OjWd are given in figure 2. Precise values of the integrals W,, W29 W3, W4, W5 and Wd are given in table I. The numerical
1344 method used for this calculation has been detailed in reference [9]. In order to achieve a good numerical accuracy within the double precision calculation, a very small mesh of discretization (102 to 103 A) was used for the small and intermediate interatomic distances in the G channel of collisions. The more recent determinations of Kolos and Wolniewicz were used for the Yg [10] and Yu potentials [11]. Hyperfine interactions and magnetic couplings have been ignored which is probably a rather poor approximation in the low temperature limit (T 100 mk). References [1] Proceedings of the Aussois meeting on Spin Polarized Quantum Gases in J. Physique Colloq. 41 (1980) C71. [2] SILVERA, I. F. and WALRAVEN, J., Scientific American 246 (1982) 66. [3] GREYTAK, T. J. and KLEPPNER, D., Lectures on spin polarized hydrogen at 1982 Summer School Les Houches. Session XXXVIII. New trends in atomic Physics. Vol. 2, G. Grymberg et R. Stora ed. (North Holland Physics Publishing) 1974. [4] ANDERSON, K. E., CRAMPTON, S. B., JONES, K. M., NUNES, G. Jr. and SOUZA, S. P. : contributed paper to LT17, Karlsruhe, FRG, Aug. 1522, 1984. [5] BOUCHAUD, J. P. and LHUILLIER, C., J. Physique 46 (1985) 1101. [6] LHUILLIER, C. and LALÖE, F., J. Physique 43 (1982) 197 and 43 (1982) 225. [7] BALLING, L. C., HANSON, R. J. and PIPKIN, F. M., Phys. Rev. 133A (1964) 607. [8] a) BERLINSKY, A. J. and SHIZGAL, B., Can. J. Phys. 58 (1980) 881. b) MORROW, M. and BERLINSKY, A. J., Can. J. Phys. 61 (1983) 1042. [9] LHUILLIER, C., J. Physique 44 (1983) 1. [10] KOLOS, N., WOLNIEWICZ, L., J. Mol. Spectrosc. 54 (1975) 303. [11] KOLOS, N., WOLNIEWICZ, L., Chem. Phys. Lett. 24 (1974) 457. [12] BOUCHAUD, J. P., PhD. Thesis, Paris (1985).