Hyperfine and electronic spin waves in atomic hydrogen : the role of the spin ..... en are the signatures of the statistics obeyed by the electrons and nuclei (in our case, .... other hand, the main contribution to Pn and Pen arises from backward ...
Tome 46
No 11
NOVEMBRE 1985
LE JOURNAL DE J.
PHYSIQUE
Physique 46 (1985) 1781-1795
NOVEMBRE
1985,1
1781
Classification Physics Abstracts 51.10
-
67.20
-
05.30
-
34.00
Hyperfine and electronic spin waves in atomic hydrogen : the role of the spin exchange process J.-P. Bouchaud and C. Lhuillier Laboratoire de Spectroscopie Hertzienne, Ecole Normale 75231 Paris Cedex 05, France
(Reçu
le 12 fivrier 1985,
Supérieure, 24,
rue
Lhomond,
accepté sous forme difinitive le 1 er juillet 1985)
Résumé. Le mécanisme microscopique qui sous-tend le phénomène d’ondes de spin nucléaire récemment mises en évidence dans des systèmes gazeux non dégénérés : 3He [Paris] et H~ [Cornell] est la rotation des spins des particules identiques. C’est un phénomène de mécanique quantique extrêmement général qui conduit à prévoir l’existence possible de modes propagatifs associés à la dynamique des éléments non diagonaux de la matrice densité à basse température (ondes de spins généralisées). Cette prédiction est étudiée en détail dans le cas de l’hydrogène atomique à 4 composantes. Partant d’une équation de Boltzmann établie récemment pour ce problème, il est montré sur quelques exemples que les modes prévus existent en réalité mais qu’ils sont en maintes conditions fortement amortis. La première cause intrinsèque d’amortissement est liée au phénomène d’échange de spin. Nous étudions, tant en bas champ magnétique qu’en haut champ, les conditions permettant de minimiser cet effet d’amortissement. Les valeurs numériques permettant de calculer toutes les caractéristiques du transport de spin dans l’hydrogène atomique sont données en appendice. 2014
Abstract 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. 2014
1. Introduction. The « identical spin rotation effect >> [1] describing the role of indistinguishability in the collisions of two spin 1/2 atoms is the microscopic phenomenon underlying the macroscopic spin waves recently observed in Paris [2] and at Cornell University [3] on dilute, non degenerate samples of 3He and H 1. This effect can be described as follows : when two
identical atoms carrying a 1/2 spin meet, the result of the transient exchange degeneracy occurring during the overlap of the wave packets is the rotation of the transverse components of the two spins around their sum.
The quantum mechanical origin of such an effect is not related to any spin dependent Hamiltonian but to indistinguishability and quantum interferences : due to the Pauli principle, the interaction
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198500460110178100
1782
between two atoms in the same spin state is different from that between two atoms in orthogonal spin states. In the first case, the total two-body wave function must be correctly symmetrized with respect to the exchange of the two encounters leading to an interacting wave packet modulated by interferences of period A (de Broglie wavelength of the particles). In the second case, on the contrary-insofar as the interaction is spin independent- all the interference terms disappear, as do all consequences of the symmetrization. This is actually a very satisfying result : in this last case, the particles are in fact distinguishable by their spin state, which can be considered as a meaningful label because ’
i) the asymptotic states are well defined, and ii) the interaction cannot flip the spins. As a consequence, inasmuch as the period A of the interferences is not small before the range of the interatomic potential a, the sampling of this potential during the collision will depend on the initial spins
configuration. Consider
the collisions of two atoms with spin each other. We can choose for simto parallel the plicity quantization axis along one of the two spins say spin1 ). The two atoms are then described now
not ,
in
spin
space
by
the kets
and it is
easily seen that this situation is not a statioduring collision. Due to the above-mentioned consequences of the symmetry’s requirement, the two components a and fl do not experience the same dephasing through their interaction with1 ), so during the collision, without any spin-dependent interaction, the spin of atom 2 rotates in spin space. In fact, due to the total spin conservation during the process, the rotation takes place around the spin nary
one
resultant of the two encounters, and each of them rotates a certain amount 0 : this is the « identical spin rotation effect » (1).
an effect cannot manifest itself in a two-level with a spatially homogeneous magnetization, system but it will be present in all situations where there is a change in the polarization direction in the sample. Let us consider, for example, a situation in which the polarization can be described as the sum of a roughly constant longitudinal component X,, and a small transverse inhomogeneous one X,. First, the random motion of the atoms through the cell will tend to wipe out the spatial inhomogeneities of .X_L. But the identical spin rotation effect described above will be superimposed upon this common effect obeying a usual diffusive law (Fick’s law) : if the magnetization X_, is not zero, the axis along which this spin rotation takes place has on the average a welldefined orientation, thus leading to a cumulative rotation of Kj_ at each collision. If the efficiency of a collision for producing this cumulative rotation is greater than that for restoring homogeneity through randomization, then a new collective mode will appear in the system, namely a transverse spin wave. The Q factor of these waves, proportional to the polarization of the sample, is a measure of the ratio of the identical spin rotation efficiency to that of the usual diffusion phenomenon : in the very low temperature limit, the cross section for spin diffusion is proportional to the square of the scattering length a2. As we have said before, the identical spin rotation effect is associated with interference effects; at low temperature, the dominant interference effect is the one between the non interacting wave (characteristic length scale A) and the scattered amplitude in the backwardforward direction (characteristic length scale a); the total efficiency of the process that benefits of this heterodynage scales like Aa. But as the thermal de identical spin Broglie wavelength A varies as rotation effects can in fact supersede (at low enough temperatures) randomization effects due to collisions, and propagative modes (spin waves) can be then observed with a Q-factor varying like Àala2. These propagative modes reduce to the usual diffusive mode as the temperature is increased : higher temperatures mean a decrease in the de Broglie wavelength that is, a decrease in the spatial period of the modulation of the interacting wave packet which describes particles in the same spin state. This explains why macroscopic exchange effects (spin waves, for example) disappear at high temperatures.
Such
11ft,
-
(1)
This presentation must be considered as a hand-waving reasoning and any extrapolation made with great care; a slightly more sophisticated, but still simple, approximate presentation has been given by Johnson [4], and the precise quantum mechanical reasoning can be found in [1]. In fact, the essential features of the reasoning are that it bears on transition amplitudes (and not probabilities), quantum interferences are essential, and the spin rotation effect is a physical phenomenon affecting the coherences of the system that cannot manifest itself in equilibrium statistical situations where the postulate of random phases is generally the rule. Using the fact that this phenomenon has fundamentally the same physical origin as the molecular exchange field in dense degenerate Fermi systems, Levy and Ruckenstein [5] have shown that this alternative approach allows a qualitative description of nuclear spin waves in spin polarized atomic hydrogen.
The understanding of the very general nature of this quantum mechanical effect led some physicists to predict the existence of new spin waves in dilute paramagnetic gases [6, 7]. In fact, from a microscopic point of view, this unexpected effect of indistinguishability should appear in the collision of any quantum system exhibiting coherences (optical coherences as well as spin coherences, for example). But to our knowledge, none of the searches for electronic spin waves in atomic hydrogen (University of Amsterdam and University of British Columbia)
1783
have been presently successful. The aim of this paper is to point out why observation of spin waves should generally be more difficult to observe in any complex system than in ’He or Hi. We also wish to clarify what the first imperative requirements for coherencewave observation are. This is illustrated qualitatively and quantitatively in the case of 4-component atomic hydrogen (that is, the case where the 4 hyperfine sublevels of the electronic ground state of atomic hydrogen can be involved in collisions). The results discussed in this paper are mainly
macroscopic ones (conservation equations, hydrodynamic modes and so on), but they are all extracted from a recently derived Boltzmann equation for the 4-component atomic hydrogen [8]. This Boltzmann equation being rather complex, we have deliberately avoided any exposition of the algebraic derivation of the transport equations (2), and have tried to put the emphasis on the physical meaning of the results and on their relation with the microscopical effects. The paper is divided into four parts. In section 2, we briefly recall the main physical microscopic effects underlying the quantum Boltzmann equation in 4-component hydrogen and their consequences of interest for the problem discussed here. (This is not meant to be a general discussion of the Boltzmann equation, which must be looked for in [8].) In section 3, we analyse the coherence waves between the two 0 sublevels of hydrogen in zero magnetic field mF (hypqrfine transition, see Fig. 1) showing that in very
=
they are strongly damped by the electronic transfer process. The best intrinsic conditions of observation are discussed and numerically determined In section 4, we study the electronic spin wave in high magnetic field (b-c transition). Section 5 contains a summary of the results and a general conclusion as to the observability of coherence waves. most cases
microscopic collision hydrogen. 2. The
processes in
4-component
Elastic collisions between hydrogen atoms in their electronic ground state can develop along two channels characterized by the symmetrical or antisymmetrical form of the total electronic cloud. The Vg molecular potential (which is the binding potential of H2) corresponds to the symmetrical state; the V, potential corresponds to the antisymmetrical state; all the phenomena described in this paper can be calculated with only the knowledge of these two molecular potentials. But, in fact, this molecular point of view is not the most suitable one to use to describe atomic collisions and transport in dilute gases. From the atomic point of view at infinite separation, the electrons are well localized on each nuclei, and their electronic state can only be described as a linear superposition of the two molecular eigenstates (1 Eg and 3 Eu). During a collision, the G and U states, corresponding to different energies, do not develop in the same way in time and the electrons independent of any indistinguishability effects oscillate between the two nuclei. After the collision, there are only two possibilities : either each nucleus has kept its original electron, or the electrons have been swapped We shall call these two collision processes the direct collision and the transfer collision (3 ) (the corresponding transition amplitudes are recalled -
-
in
Appendix I).
Due to the existence of the transfer process, atoms as a whole can no longer be considered as undissociable entities, and the symmetrization principle must be applied separately to electrons on the one hand and to protons on the other hand. The collision can no
longer be considered as
Fig.
1.
-
Zeeman
diagram of hydrogen.
U
(2) Methods used are standard ones in ChapmanEnskog transport theory of gases [9] and their application to the problem of atoms with one internal degree of freedom (namely, the nuclear spin of 3He, Hi or Di) has been carefully explained in [1]. Some intermediate stages in the calculation can be found in [10] and all the final results [algebraic and numerical] useful for the. spin current calculations are given in Appendices.
a problem of two interacting bosons but must be considered as a problem of four interacting fermions. Simple theoretical approaches like those developed in [1] and [5] to deal with 3He on spin polarized hydrogen H J, are then no longer sufficient, and it is necessary to go back to the very first principles and write the Boltzmann collision term relevant to this situation. This is the problem we deal with in [8]. We do not want to reproduce the discussion of this Boltzmann collision term here
(3) This second process is usually called the spin exchange process. We shall avoid this terminology here in order to separate this phenomenon from the indistinguishability effects (to which we have attributed the label ex for «
exchange ») without ambiguity.
1784
but wish to underline a qualitative feature directly relevant to the present problem. The Boltzmann collision term for atomic hydrogen contains not less than thirteen independent cross sections falling in two categories :
i) first, those which contribute to the damping of perturbations in the system and govern the return to equilibrium (all the relevant cross sections weighting these processes have been called Q with various subscripts and superscripts for identification; see Appendix 1), ii) and second, the class of the Hamiltonian-like terms. These terms always appear as commutators of the internal density matrices. They describe the conservative evolution during collisions of the internal off-diagonal degrees of freedom of the system and are the generalization of the identical spin rotathe
tion effect described in the introduction. All the terms of this second class are indistinguishability manifestations of electrons, nuclei or atoms as a whole; the corresponding « cross sections >> have been called r (see Appendix 1). These terms exclusively act on non diagonal internal quantities (coherences) shifting their resonance frequencies and inducing a precession of their fluxes in inhomogeneous conditions. They are at the origin of the new coherence waves that we shall now discuss from a more macroscopical point of view. Focusing on two situations of experimental interest, we shall first consider the mode associated with the coherence between the mF 0 levels in zero or low magnetic field (3.) and, second, study the electronic spin wave (b-c coherence mode) in high magnetic field
The two important quantities in this problem are 0 the difference in populations of the two mF sublevels : x(r, p, t) and the coherence itself: z(r, p, t), that oscillates at the hyperfine frequency (J). The population of the b and c levels ç(r, p, t) plays a significant role, as does the total population of the 4 suble=
vels n = ç + t + v.
simplicity and without loss of of the situation, we can features important physical 0 assume that the population ç of the two mF levels is constant over the sample as well as the total For the sake of
=
population n. Specializing the Boltzmann equation (Eq. (18) of [8]) to this specific situation and neglecting all nonresonant terms (4) leads to a set of coupled equations for the evolution of the two distribution functions z(r, p, t) and x(r, p, t). We will not write out these equations in full (they can be found in [10], equations (54-55)), but will only discuss their main physical consequences. 3.1 CONSERVATION EQUATIONS. - Averaging these equations over the momentum distribution leads to the following continuity equations in the rotating frame :
=
Propagative mode associated with the 0-0 cohein low magnetic field In this section, we shall consider a situation described in zero or low magnetic field by the following density matrix (in coupled basis) :
3.
rence
d3p
z(r, p, t)) and J(x) (resp. J(z)) molecular quantity x (resp. Z)
is the flux of the
) stands for the thermal the Boltzmann distribution, while is the angular average of the (7 cross section (their precise definitions are given in Appendix 1). Be and en are the signatures of the statistics obeyed by the electrons and nuclei (in our case, Be 1, and 1 ). En The bracket
averaging
notation
over
0
This is the experimental situation that can be observed after a R.F. pulse on a diagonal density matrix. To study the dynamics of this system, we apply the Boltzmann equation (Eq. (18) of [8]) to the present case. We recall that this equation has been established for a dilute, non degenerate gas in which the hyperfine coupling can be neglected during collision, so the present calculation will be valid down to a few tens of millikelvins.
=
=
-
-
(4)
This secular approximation is valid inasmuch as COTI, > co the precession frequency of the coherence z. The approximation applies to a very large domain of densities up to 1022 atoms/cm3.
1, Ti. being the intercollision time and
1785
Equations (2) describe both the relaxation (T1, T2) and the hyperfine frequency shift (A) due to transfer collisions. This is a new presentation of a very wellknown phenomenon [11]. Referring to the general discussion of the Boltzmann equation, we underline the point that the relaxation phenomenon only involves at cross sections (defined, as recalled in Appendix 1, from the real part of transition amplitudes products). On the other hand, the hyperfine frequency shift is related to the i cross sections (imaginary part of transition amplitude products; cf Appendix 1) and originates from the commutators appearing in the collision term of the general Boltzmann equation. This frequency shift is entirely due to the « identical spin rotation effect >> as made apparent by the presence of the Be and en coefficients in front of the i cross sections. It involves either the indistinguishability of electrons (term in Be) or of nuclei (term in Bn) and could be described as the rotation of the internal variables around electronic or nuclear molecular exchange fields. Such a term does not appear in the conservation equation in the case of ’He or spin polarized Hi, which is not surprising, as in such cases there is only one internal variable : the nuclear spin. The molecular field should thus be directed along the nuclear spin density A(r) and could consequently 0. have no action on itself : A x A In figure 2 we report the results of a new computation of the relaxation time and frequency shift due to the transfer effects (5). It is amusing to note that a measurement of this shift gives direct information about the statistical character of both constituents of hydrogen atoms, electrons and protons.
T;’( s-1) 800
n =1014
cm-3
.600
400
200
0
6
4
2
T (K)
8
a)
A-1
=
n 1014 cm-3 1000-
T(K) 3.2 HYDRODYNAMICAL REGI1VIES. - In order to solve equations (2a) and (2b), it is necessary to have an expression of the fluxes J(x) and J(z). As is usual in the hydrodynamic regime, we draw these quantities from the Boltzmann equation using the ChapmanEnskog procedure (cf. [ 10]) and we obtain the follow-
2
1
3
4
5
9
b)
ing : a) Inverse relaxation time T11 of the population difference between 0-0 levels. b) Quantities nv(£+ ± /L) entering in the definition of the frequency shift (Eq. (4)). More precisely, this shift can be written with notation originating from [15] as :
Fig.
2.
-
8e -( ) 8n - A-). A 8) A
As
has been done with the more determinations of the Vg [12] and V. [13] potentials. Our results reported in figure 6 are in general agreement with those of Morrow and Berlinski [14]; they are markedly different from those of [15] (the Vu potential used in this last reference is different from that used here and in [14]).
(’)
The
computation
-
have
+
predicted from the form of the Boltzequation, the longitudinal quantity x has a purely diffusive behaviour (Fick’s equation (5a)), whereas the flux of the transverse quantity z acquires a reactive component along both Vz and Vx (Eq. (5b)). These reactive components are directly related to the « identical spin rotation effect », 5tz and 5tx’ being linear combination of r’s collision integrals, whereas the diffusive behaviour measured by Dx and Dz mann
recent Kolos-Wolniewics
=
we
1786
is only related to the J cross sections. (A full derivation of these coefficients is reported in [10]. Their final algebraic expressions are reported in Appendix 2). The use of equations of flux (5) to close equations (2) gives rise to a non-linear wave equation. In order to simplify the physical discussion, we will focus on the following situation in which x is homogeneous over the sample whereas the coherence z is not (but is small compared to X). Such a situation is encountered after the use of a short inhomogeneous pulse of R.F. on the (0, 0) transition (Vx is then of second order in z). Neglecting second order and higher order terms, the equations of flux (5) then reduce to
This last expression similar to called hereafter LL2 :
equation (7) of [lb],
with a p coefficient (= Dz :Rz) shown in figure 3a. It is interesting to note that this p appears naturally as the sum of three terms (Fig: 3b)
each of which is
clearly associated with one type of exchange (electrons alone, nuclei alone and atoms as a whole). As they should be, Pe and Pn are related to the transfer process while Yen is related to the direct
process. This explains the relative weakness of Me and gn compared to Yen at low temperatures. On the other hand, the main contribution to Pn and Pen arises from backward interferences (the nuclei being exchanged) while p, results from forward interferences, accounting for the slower decay of Me as the
Fig. 3. a) p factors of the 0-0 coherence wave versus T for two values of alignment A, A 0 and A l.b) From its very definition (cf. Appendix 2), it can be seen that p can be written under the form -
=
=
-
temperature increases.
Unhappily, in this situation, the p coefficient is not the effective quality factor of the new propagating mode associated with the reactive component of the current (6). In fact, combination of the continuity equations (2) with the expression of flux (6) leads in this linear approximation to the following equations of motion : i)
for the
longitudinal polarization Mz
=
showing the relative importance of the different indistinguishability processes. The variation in temperature of the for the A = 0 case three components /t,,nl Jle and M. is given in figure 3b. -
ii) for the
transverse
-
polarization M+ = F/n :
fl/n :
(6) We discuss here only the intrinsic collisional damping of those waves and shall postpone a brief overlook at some extrinsic causes of damping, which should be controlled if one is to observe these waves experimentally, until section 5.
general, the two coupled equations (8) must be together, and the relaxation of the longitudinal polarization Mz strongly affects the driving field of the spin waves Dz pM,,I I + u2 Mi . The conditions of In
solved
1787
of this
observability
hydrodynamic
mode is then
The first inequality states that the rate of the damping transfer process is negligible before the current precession frequency; the last inequality insures that the system is in the hydrodynamic regime (and not in the ballistic one). These conditions determine an intermediate domain for the product (nL) of the density by the characteristic length of the sample in which spin waves can be observed For higher densities, the relaxation process predominates and destroys the longitudinal polarization (Xln) as well as the transverse polarization (z/n) ; no spin waves can then exist. For lower densities (in the ballistic domain), the effect of « identical spin rotation » does persist at each collision, but the disappearance of the diffusive restoring force and the overwhelming weight of the collisions on the walls will probably impede any observation of a macroscopic effect. In the in-between domain of metastable mixtures spin waves do exist, but their quality factor is nevertheless reduced by the transfer process. Insofar as the damping constant T 11 is small before the characteristic frequencies of the spin waves, Mz is a slow variable, and the dispersion equation of the waves can be approximated alge-
braically
Density X characteristic length versus tempeshowing the region in which 0-0 coherence waves (described in Sect. 3) can be seen. The dashed line separates the low density Knudsen regime from the region in which hydrodynamic modes exist. The full line delimits the region (SW) in which spin waves with an effective quality factor greater than 1 can be seen. This figure has been drawn Fig.
4.
-
rature,
for A 1; the choice of different values of A would have induced only minor changes in the full line. =
-
as :
situations involving an electronic coherence between the states b and c described by a density matrix of the following form in the abcd basis :
From this approximation, one sees the major diffebetween these spin waves and those predicted in LL2, which have been observed in Hi at Comell University [4, 5]. The spin transfer process is now at the origin of a shift of the frequency of the spin waves and, overall, of an extra damping process that will considerably diminish the effective quality factor Q of these waves : rence
figure 4 we report the domain of density and temperature in which the Q factor of these new spin waves
In
exceeds
one.
Such a situation is easily obtained in an E.P.R. experiment on the b-c transition. The essential parameters of the problem are the difference in population X between the c and b states and the z coherence. Averaging the Boltzmann equation over the distribution function leads to the following continuity equation for the averages x and z (7) :
As it can be $een, the domain of such a mode is quite limited in temperature
hydrodynamic and density.
spin waves in high magnetic field. In the high magnetic field limit using the usual notations reported in figure 1, we shall now focus on 4. Electronic
(’)
The collision process is described in the rotating an approximation that should be valid in fields of 10 tesla down to temperatures of the order of 500 mK (see discussion in [8] and [18]).
frame,
1788
where
The nuclear spin-dependent diffusion coefficient D* measures the damping of the spatial perturbation of the z coherence; it is a combination of collision integrals of the a’s cross sections, whereas It! and It! are ratio of i’s collision integrals (measuring the efficiency of the « identical spin rotation effects ») to a’s collision integrals (damping effects). (The analytical expressions for these three coefficients, as well as their numerical values, can be found in Appendix 2.) In this case, as in the previously studied one, the reactive part of the flux (terms containing It! and M*) is at the origin of spin waves (here electronic spin waves) obeying a set of equations ( 13)-( 14) quite similar to those obeyed by the 0-0 coherence in low magnetic field (2)-(6). Nevertheless, because of the specificity of the relaxation due to the transfer process, various physical situations can appear, and we must distinguish between two cases.
whe
I and S are the longitudinal components of the nuclear and electronic spins :
i) n. -- nc L-- nd -0. In high magnetic fields and at low temperatures, the Boltzmann equilibrium leads to S - - 1 and nd - 0. On the other hand, we know that recombination is a very efficient mechanism for destroying the a component of the mixture [16, 17]. Therefore, an EPR experiment on the b-c transition can probably meet the above requirement na - nd --- 0. In such a situation
These equations of motion describe phenomena qualitatively similar to those observed in low magnetic field, i.e. longitudinal and transverse relaxation due to the exchange process as well as a frequency shift of the coherence A * and possibly of great
(8).
What is nevertheless
new
importance is the fact that the damping phenomena necessarily involves the presence of a or d atoms. In the absence of a and d atoms (full nuclear polarization :I = - 1, and x nS), the twolevel system (b-c) behaves like a system with only one internal metastable degree of freedom S (the electronic spin is a conservative variable during collisions and the nuclear spin just a spectator), and we can predict that its dynamics from the collisional point of view =
will reduce to that described in LL2. In order to solve the dynamics of the system in the general case, it is necessary to couple equations (13) with the expressions of flux : in the hydrodynamic regime, we obtain these expressions from the Boltzmann equation for X and z by the usual ChapmanEnskog procedure. In the special case in which S and x are constant over the sample, the flux of x is null
and the flux of z reduces to
with M
(1)
=
Xln, S being defined as in equation (13f).
be noticed that this frequency shift is induced by purely electronic field (see Eq. (13e)), which is not surprising because the coherence arises between two identical nuclear states. a
It
can
"
and the system is in equilibrium with transfer process. The set of equations then reduces, to the following :
regard to the (13) and (14)
equation describes transverse electronic spin As we have already noticed in this two-level system, the nuclear spin is just a spectator in the collisions and in this nuclearly polarized hydrogen, the general features of the theory of LL2 are quite valid, in particular, the metastable character of the only internal degree of freedom, here the electronic spin. The identity of form of the hydrodynamic equations of motion is not, therefore, a real surprise. Nevertheless, this electronic transport process is fundamentally different from the nuclear spin transport in H J, on one particular point : the interaction does not depend on the nuclear spin, while it does depend indirectly on the electronic one (through the Vg and Yu molecular potentials). It was thus necessary to develop the whole analysis of the collision done in [8] in order to calculate ab initio the diffusion coefficient DZ as well as the coefficient Jli + It* measuring the quality factor of these waves. Their numerical values can be checked in figure 5. It is of This
waves.
1789 on the order of 1 mm and can be equally well described by the S.W. domain delimited in figure 4.
length
5.
Summary
and conclusions.
In this paper,
we have studied the equations of motion of two off-diagonal elements of the density matrix of atomic hydrogen : that of the 0-0 coherence in zero or low magnetic field and that of the b-c coherence in high magnetic field If we suppose that the only ihhomogeneity of the problem lays in the spatial repartition of this coherence z (we have shown in each case the experimental plausability of such an hypothesis), then the equation of evolution of the coherence in the rotating frame is (in the linear limit) of the general form
5. - ,u factor for the b-c coherence wave (in high field) two values of the nuclear polarization : I 0, p = */2 + M* and I = - 1, p = It* + Jl! (Eqs. (15) and (16) of the text). One should note that in order to avoid coherence decay due to spin exchange collisions, it is better to work with I = - 1 (only b states).
Fig.
for
=
the utmost importance to underline that insofar as na ’" nd ’" 0, the observation of these electronic spin waves is not limited in density by the spin exchange process, as it was the case for the mode associated with the coherence 0-0 in low magnetic field
ii) On the other hand, if the elimination of the a and d components is incomplete, then damping phenomena associated with the transfer process become non-negligible (right-hand side of equations (13a) and (13b)), and the range of density where the spin waves can be observed becomes strictly limited. For example, consider a situation in which a relaxation process (a wall relaxation process for instance) rapidly destroys the nuclear polarization. A situation with initial homogeneous conditions na = nb n j2 ; 1 and an inhomoge1= 0 ; xo n j2 ; S neous coherence z will evolve according to equations (13) and (14) in the following way : =
=
-
=
-
.
This situation will, in general, be a little more favourable than the low-field case studied in section 3 because the driving fields for the spin waves p* Xln + ,u2 S do not relax to zero. Nevertheless, the damping of the coherence will decrease the quality factor of the waves in a way quite similar to that discussed in the previous section. Here also the range of density where spin waves can be seen is strictly limited to l Ol s-l O16 at/cm’ for a characteristic
(Ti-1)
The first two terms of the right-hand side describe two well-known phenomena [11] : the relaxation process and the energy shift associated with spin exchange collisions. The last two are new and related to the specific form of the Boltzmann equation in systems with internal degrees of freedom. In this last part, the parameter f1 can take various, more or less complicated forms (Eqs. (4) or (14)), but it is always related to the longitudinal « polarization )) of the sample. Equation (17) is an equation of a propagating mode, more or less damped. The term, responsible for the propagation ( - ipA V2z), originates from the. « identical spin rotation » effects via the Hamiltonian-like terms of Boltzmann collision integral. This effect being a quite general quantum-mechanical effect, it is evident that spin waves could in principle be associated with the transport of any non-diagonal atomic quantity in hydrogen and alkalis. Moreover, quantum mechanics tells us that the p factor will always diverge at a low enough temperature (like A-1a). Nevertheless, and the present study is a very good illustration of this fact, the observation of these coherence waves requires that no other phenomenon should introduce extra damping (the p factor is generally not the effective quality factor of the waves). The damping phenomena can be divided into two classes : those which originate from the collisional process itself (we shall call them intrinsic), and those which come from the experimental environment (we shall call them extrinsic). In this paper, we are primarily concerned with the intrinsic collisional processes, and we have shown that minimization of collisional damping requires two conditions : the coherence under study must be the transcomponent of a conservative molecular quanand
-
verse
tity,
1790
this internal degree of freedom must be, within very good approximation, completely decoupled from the other variables of the problem, in order to avoid a relaxation of the longitudinal polarization through statistical homogeneization of the populations. -
Those two criteria, which insure the metastability of the quantity under consideration, are totally satisfied in the case of nuclear spin waves in His or 3He and in the case of electronic spin waves in
doubly polarized hydrogen. Those two criteria are not obeyed in the case of the (0-0) coherence in low field. Nevertheless, due to the relatively weak efficiency of the spin exchange process at low temperature, there exists a limited domain of temperature and density in which the coherence waves are not overdamped by the collisional process (see Fig. 4). To conclude, it must be underlined that experimental observation of hyperfine or Zeeman electronic spin waves should take into account difficulties associated with external causes of damping : a radiofrequency detection would necessitate low Q detection circuitry in order to minimize radiation damping (to give an order of magnitude, the radiation damping rate [19] equals the spin exchange damping rate at 1 K for a Q factor of the order of 5). If the signal-tonoise ratio in such a low Q system is sufficient, then hyperfine spin waves could effectively be observed in the density domain of figure 4. For high field ESR, one must worry not only about radiation damping but also about inhomogeneous field broadening (a very stringent condition with respect to the frequency shift due to « identical spin rotation »). In these conditions, it is perhaps easier to probe equation (17) indirectly, for example by looking at the instability mechanism recently
proposed by Castaing [20]. (For
ii)
For lateral
a
high enough
t4
strong gradients of longitudinal polarization become instable against transverse fluctuations of magnetization, the Reynolds number of this instability being
Appendix
1.
Appendix, we recall very briefly the definitions of the different cross sections relevant to the present discussion. The reader is referred to the appendices of [8] and [10] for a more complete discussion of the properties and phase shift expansions of these quantities. The two molecular potentials Vg and V. are associated with the usual transition matrices Tg and Tu and the related transition matrices for the direct and transfer processes : In this
DIFFERENTIAL CROSS SECTIONS. The different cross sections relevant to the discussion of spin waves arise from two kinds of scattering.
A)
i) For forward-backward definition is
-
scattering,
the
generic
The upper label is related to forward or backward scattering, while the subscript a labels the direct or transfer process.
scattering,
where a and fl labels stand for direct or transfer processes (repeated labels are omitted; equation (A. 5) then coincides with the usual definition of a differential cross section a, the L(%(% being null), and 0 (k;, kf). Introducing the usual phase shift expansion of the transition operator, one can express the different cross sections as phase-shift series : =
1791
In order to obtain the phase shift use relations (A .1 ) and (A. 2).
expansions of the quantities related to direct
or
transfer processes,
one
should
In describing spin equilibrium B) ANGULAR AVERAGED CROSS SECTIONS. primarily interested in the following angular averages : -
for the
description of static equilibrium effects and
:
for the description of fluxes in transport properties. In terms of the phase shifts, one has the following for
One then
after
For
a
few
or
Q 0 and Q 1.
readily obtains, for example,
trigonometric manipulations,
one
has
The analogous quantities for aex(O) and Tex (0) are obtained by substituting1 example, one has
The notations introduced in
[ 11,14,15]
are
related to
ours
by
transport properties,
one
is
1792
As far
the
as T cross
sections
are
concerned,
one
also has
Contrarily to the algebra of the Q’s in which afwd, abwd can be written in terms of (2’°, the T’s do property. The corresponding formulae, echoing (A.17), are
not possess
same
The following quantity is interesting because it appears in the 0-0 coherence due to transfer collisions :
are
the notations used in
[14]
and
shift of the
[15]).
THERMALLY AVERAGED CROSS SECTIONS : COLLISION coefficients, the following integral usually appears :
C)
where
expression giving the frequency
f can be any function of the wave vector k.
In
INTEGRALS.
-
In the
expression
of several transport
particular, f can be one of the angular averages Q or 6.
1793
In this case, the usual notation is
and the analogous definition for 0(s,’). If f is a T function, which does not reduce to a Q or
(2, the temperature average will be written
Having introduced those wave vector averages, the collision integrals appearing in the Chapman-Enskog theory of transport can be expressed as follows :
where the bracket notation [...] stands for P1 vr d3q; d2qf d3p1 X fb(p1) fb (P2) where pl, P2 are the two momenta of the impinging atoms in the laboratory frame, qi, qf the input and output momenta in the c.m. frame, and vr the relative velocity. A complete description of those standard notations can be found in [ 1-9].
Appendix 2. EXPRESSION INTEGRALS.
OF THE -
QUALITY FACTORS AND DIFFUSION COEFFICIENTS OF SPIN WAVES AS FUNCTIONS OF COLLISIONAL
The generic equation describing a coherence wave (of a fictitious
1/2 spin wave) is the following :
Thus, the important quantities are u and D. 1. The 0-0 coherence (low field)
(Sect. 3).-
a) The diffusion coefficients. The
Dz coefficient (introduced in equation (5b)) is A (alignment)-dependent through :
We remember that
We also give the expression of
Dx (Eq. (5a)) describing the diffusion of x :
One has :
JOURNAL DE
PHYSIQUE.
-
T.
46, N" 11,
NOVEMBRE
1985
110
1794
b) The quality factor p. p is defined in
2. The b-c
equation (6) by : p
=
Dz 3tz with the following
coherence(
a) The diffusion coefficient ( This coefficient is
now
The quality factor.
b)
We must
give here the effective quality factor p defined by :
and
Numerical values of these collision integrals versus temperature are given in table I.
Table I.
collision integrals involved in the definition of the various quality factors of the investigated waves (c£ Appendix 2) and, more generally, in the amplitude of the generalized« identical spin rotation » effect. -
T
References C. and LALOË, F., J. Physique 43 (1982) 197 ; b) LHUILLIER, C. and LALOË, F., J. Physique 43 (1982)
[1] a) LHUILLIER, 225.
[2] a) NACHER,
P.
J., TASTEVIN, G., LEDUC, M., CRAMPF., J. Physique Lett. 45 (1984)
TON, S. B. AND LALOË,
L-441 ;
b) TASTEVIN, G., NACHER, P. J., LEDUC, M. and LALOË, F., J. Physique Lett. (1985). [3] JOHNSON, B. R., DENKER, J. S., BIGELOW, N., LEVY, L. P., FREED, J. H. and LEE, D. M., Phys. Rev. Lett. 52 (1984) 1508. [4] JOHNSON, B. R., Ph. D. Thesis, Cornell University (1984).
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[5] LEVY, L. P., RUCKENSTEIN, A. E., Phys. Rev. Lett. 52 (1984) 1512. [6] BASHKIN, E. P., Pis’ma Zh. Eksp. Teor. Fiz. 33 (1981) 11; JETP Lett. 33 (1981) 8. [7] SILVERA, I., Quotation in Physics Today, June 1984, p. 19.
[8] BOUCHAUD, J. P. and LHUILLIER, C., J. Physique 46 (1985) 1101. [9] CHAPMAN, S. and COWLING, T. G., The mathematical theory of non-uniform gases (Cambridge University Press) 1952. [10] BOUCHAUD, J. P., Thesis (1985) Paris. [11] BALLING, L. C., HANSON, R. J. and PIPKIN, F. M., Phys. Rev. A 133 (1964) 607. [12] KOLOS, N., WOLNMWICZ, L., J. Mol. Spectrosc. 54 (1975) 303.
[13] KOLOS, N., WOLNIEWICZ, L., Chem. Phys. Lett. 24 (1974) 457. [14] MORROW, M. and BERLINSKY, A. J., Can. J. Phys. 61 (1983) 1042. A. J. and SHIZGAL, B., Can. J. Phys. 58 BERLINSKY, [15] (1980) 881. [16] CLINE, R. W., GREYTAK, T. J., KLEPPNER, D., Phys. Rev. Lett. 47 (1981) 1195. [17] SPRIK, R. K., WALRAVEN, J. T. M., VAN YPEREN, G. H. and SILVERA, I. F., Phys. Rev. Lett. 49 (1982) 153. [18] UANG, Y. H., FERRANTE, R. F., STWALLEY, W. C., J. Chem. Phys. 74 (1981) 6256. [19] BLOEMBERGEN, N. and POUND, R. V., Phys. Rev. 95 (1954) 8. [20] CASTAING, B., Proceedings of LT 17, Physica B + C 126 (1984) 212.
