Hyperfine and electronic spin waves in atomic hydrogen : the role of the spin exchange process

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Hyperfine and electronic spin waves in atomic hydrogen : the role of the spin exchange process

J.-P. Bouchaud, C. Lhuillier

To cite this version:

J.-P. Bouchaud, C. Lhuillier. Hyperfine and electronic spin waves in atomic hydrogen : the role of the spin exchange process. Journal de Physique, 1985, 46 (11), pp.1781-1795.

�10.1051/jphys:0198500460110178100�. �jpa-00210130�

1781

LE JOURNAL ^DE PHYSIQUE

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 Supérieure, 24,

Lhomond,

75231 Paris Cedex 05, ^France

(Reçu ^{le 12} fivrier ^1985, accepté

forme difinitive ^{le 1 er} juillet 1985)

Résumé.

Le mécanisme microscopique qui sous-tend le phénomène d’ondes de spin nucléaire récemment mises

évidence dans des systèmes ^gazeux

dégénérés : ^3He [Paris] ^et H~ [Cornell] ^{est la rotation des} spins

des particules identiques. ^C’est

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 pro-

blème, ^{il est montré}

quelques exemples que les modes prévus ^existent

réalité mais qu’ils ^sont

^maintes

conditions fortement amortis. La première ^cause intrinsèque d’amortissement est liée

phénomène d’échange

de spin. ^Nous étudions, ^tant

^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}

appendice.

Abstract

The identical spin ^rotation effect is the quantum mechanical collisional effect underlying ^{the transverse}

nuclear spin

recently

^{in dilute}

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

^cohe-

rences waves). Starting ^from

quantum ^Boltzmann equation recently derived for the study of 4-component ^atomic hydrogen,

show that the current of any transverse quantity (hyperfine

^Zeeman coherence) does indeed precess around molecular exchange ^fields. Unhappily, ^{in most} cases, the oscillatory modes associated with these transverse quantities

heavily damped. ^The first intrinsic cause of damping ^{of these}

^is spin exchange.

We study the conditions under which this damping ^{is minimized in both}

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.

J. Physique ⁴⁶ (1985) ^{1781-1795 NOVEMBRE} 1985, ¹

Classification Physics ^Abstracts

51.10

67.20

05.30

34.00

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 ^under- lying ^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 ^interfe-

rences : due to the Pauli principle, the interaction

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198500460110178100

between two atoms in the same spin ^{state is diffe-}

rent 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 _sym- metrization. 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 now the collisions of two atoms with spin

not parallel ^to each other. We can choose for sim-

,

plicity ^the quantization ^axis along ^{one of the two} spins say spin 1 ). ^{The two atoms are then described}

in spin space by ^{the kets}

and it is easily ^seen that this situation is not a statio- nary ^one during collision. Due to the above-men- tioned consequences of the symmetry’s requirement,

the two components ^a and fl ^{do not} experience ^the

same dephasing through their interaction with 1 ),

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

resultant of the two encounters, and each of them rotates a certain amount 0 : this is the « identical spin

rotation effect » (1).

(1) ^This presentation ^must be considered

hand-waving reasoning and any extrapolation ^{made with} great care;

a

slightly

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

that it bears

on transition amplitudes (and ^not probabilities), quantum interferences

essential, ^{and the} spin rotation effect is

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 funda-} mentally ^{the same} physical origin

the molecular exchange

field in dense degenerate ^Fermi systems, Levy and Rucken- stein [5] have shown that this alternative approach ^allows

a

qualitative description of nuclear spin

ⁱⁿ spin polarized ^atomic hydrogen.

Such an effect cannot manifest itself in a two-level system ^{with a} spatially homogeneous magnetization,

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 well- defined orientation, ^thus leading ^{to a} cumulative rota- tion 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} polari-

zation 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 _propor- tional 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} tempera- ture, the dominant interference effect is the one bet-

ween the non interacting ^wave (characteristic length

scale A) and the scattered amplitude ^{in the backward-}

forward direction (characteristic length ^scale a);

the total efficiency ^{of the} process that benefits of this

heterodynage scales like Aa. But as the thermal de

Broglie wavelength A ^{varies as} 11ft, ^{identical spin}

rotation effects can in fact supersede (at ^low enough temperatures) randomization effects due to colli-

sions, ^and propagative ^modes (spin waves) ^{can be}

then observed with a Q-factor varying ^like Àala2.

These propagative modes reduce to the usual diffu- sive 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.

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 micros-} copic point ^of view, ^this unexpected effect of indis-

tinguishability ^should appear in the collision of _any quantum system exhibiting coherences (optical ^cohe-

rences as well as spin coherences, ^for example).

But to our knowledge, ^none of the searches for elec- tronic spin ^{waves in atomic} hydrogen (University ^of

Amsterdam and University of British Columbia)

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 coherence-

wave observation are. This is illustrated qualitatively

and quantitatively ^{in the case of} 4-component ^atomic hydrogen (that is, ^{the case where the 4} hyperfine ^suble-

vels 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, hydro- dynamic ^{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

very 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 mF

0 sublevels of hydrogen ^{in zero} magnetic ^field (hypqrfine transition, ^see Fig. 1) showing ^{that in}

Fig. ^1.

^Zeeman diagram ^of hydrogen.

(2) Methods used

standard ones in Chapman- Enskog 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

Di) ^{has been} carefully explained ⁱⁿ [1]. Some intermediate stages ^{in the} calculation

be found in [10] and all the final results

[algebraic ^and numerical] useful for the. spin ^{current calcu-}

lations

given ⁱⁿ Appendices.

most cases they ^are strongly damped by the electro- nic transfer _process. The best intrinsic conditions of observation are discussed and numerically ^deter-

mined 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 ^con-

clusion as to the observability of coherence waves.

2. The microscopic ^collision processes in 4-component hydrogen.

Elastic collisions between hydrogen ^{atoms in their}

electronic ground ^{state can} develop along ^{two chan-}

nels characterized by ^the symmetrical ^or antisym-

metrical 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 anti-} symmetrical 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} eigen-

states (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 _pro-

cesses 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 undisso- ciable 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 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

have attributed the label

for

«

exchange ») ^without ambiguity.

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

the perturbations ^{in the} system ^and govern the ^return to equilibrium (all the relevant cross sections weight- ing 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 inter- nal off-diagonal degrees of freedom of the system and are the generalization ^of the identical spin ^rota-

tion effect described in the introduction. All the terms of this second class are indistinguishability ^manifes-

tations 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

3. Propagative mode associated with the 0-0 cohe-

rence in low magnetic ^field

In this section, ^we shall consider a situation describ- ed in zero or low magnetic ^field by ^the following ^den- sity ^matrix (in coupled basis) :

This is the experimental ^{situation that can be observ-}

ed 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 hyper-

fine coupling ^{can be} neglected during collision, ^so

the present calculation will be valid down to a few tens of millikelvins.

The two important quantities ^{in this} problem ^are

the difference in populations ^{of the two} mF

⁰ sublevels : x(r, p, t) and the coherence itself: z(r, p, t),

that oscillates at the hyperfine frequency ^{(J). The} popu- lation of the b and c levels ç(r, p, t) plays ^a signifi-

cant role, ^as does the total population ^{of the 4 suble-} vels n = ç + t + v.

For the sake of simplicity ^and without loss of

important physical features of the situation, ^{we can}

assume that the population ç ^{of the two} mF

0 levels is constant over the sample ^{as well as the total} population ^n.

Specializing the Boltzmann equation (Eq. ^{(18) of} [8]) ^{to this} specific situation and neglecting ^{all non-}

resonant 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], equa- tions (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 :

d3p ^{z(r, p, t)) and J(x)} (resp. J(z)) ^{is the flux of the}

molecular quantity x (resp. Z)

The bracket notation ) stands for the thermal

averaging ^over the Boltzmann distribution, ^while

0 ^{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

En

1 ).

(4) This secular approximation ^{is valid inasmuch as} COTI, >

1, Ti. being ^the intercollision time and

the precession ^fre-

quency of the coherence

The approximation applies ^to

a

very large ^{domain of densities} up to 1022 atoms/cm3.

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 well- known 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 amplitu-

des products). ^{On the other} hand, ^the hyperfine ^fre-

quency 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 equa- tion. This frequency ^{shift is} entirely ^{due to the « iden-}

tical 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 indistinguisha- bility 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

have no action on itself : A x A

0.

In figure ^{2 we} report the results of a new computa- tion 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.

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 Chapman- Enskog procedure (cf. [ 10]) ^{and we} obtain the follow-

ing :

(’) ^The computation has been done with the

recent Kolos-Wolniewics determinations ^{of the} Vg ^[12]

and V. [13] potentials. Our results reported ⁱⁿ figure ⁶

in general agreement with those of Morrow and Ber- linski [14]; they

markedly different from those of [15]

(the Vu potential used in this last reference is different from that used here and in [14]).

T;’( s-1)

800

n = 1014 cm-3

.600

400

200

0 2 4 6 8 T (K)

a)

A-1

n 1014 cm-3

1000-

T(K)

1 2 3 4 5 9

b)

Fig. ^2.

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 :}

A

8e -( ⁺ ) 8n - ⁾

A 8 ^A-).

As we have predicted from the form of the Boltz-

mann equation, ^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

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} equation (7) ^of [lb],

called hereafter LL2 :

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 interfe-}

rences, accounting for the slower decay of Me ^{as the}

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

fl/n :

(6) ^We discuss here only ^the intrinsic collisional damping

of those waves and shall postpone

brief overlook at

extrinsic causes of damping, ^{which should} be controlled if one is to observe these waves experimentally, ^until

tion 5.

Fig. ^3.

a) p ^{factors of} the 0-0 coherence wave

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

showing the relative importance ^of the different indistin-

guishability processes. The variation in temperature ^{of the} three components

/t,,nl Jle and M.

for the A = 0 case

is given ⁱⁿ figure ^3b.

ii) ^{for the} transverse polarization M+ = F/n :

In general, ^{the two} coupled equations (8) ^{must be}

solved 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

observability ^{of this} hydrodynamic ^{mode is then}

The first inequality ^{states that the rate of the} damping

transfer _process is negligible before the current pre- cession frequency; ^{the last} inequality insures that the system ^{is in the} hydrodynamic regime (and ^{not in the}

ballistic one). These conditions determine an inter- mediate 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 trans-}

verse 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 restor-} ing force and the overwhelming weight of the colli- sions on the walls will probably impede any observa- tion 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 ^{as :}

From this approximation, ^{one sees the} major ^diffe-

rence between 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 :

In figure ^{4 we} report ^{the domain of} density ^{and tem-}

perature ^{in which the} Q factor ^{of these new} spin ^waves

exceeds one. As it can be $een, the domain of such a

hydrodynamic ^{mode is} quite limited in temperature and density.

4. Electronic spin ^{waves in} high magnetic ^field.

In the high magnetic field limit using ^{the usual nota-}

tions reported ⁱⁿ figure 1, ^{we shall now focus on}

Fig. ^4.

Density ^X characteristic length

tempe- rature, showing ^the region in which 0-0 coherence

(described ^{in Sect.} 3)

^be

^{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

^{with an effective} quality ^factor greater ^{than 1}

^{be seen. This} figure has been drawn for A

1; ^the choice of different values of A would have induced only ^minor changes in the full line.

situations involving ^an electronic coherence between the states b and c described by ^a density matrix of the

following form in the abcd basis :

Such a situation is easily obtained in an E.P.R. expe- riment 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 distribu- tion function leads to the following continuity equa- tion for the _{averages x} and z (7) :

(’) The collision process is described in the rotating frame,

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]).

where

whe

I and S are the longitudinal components of the nuclear and electronic spins :

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 * (8). What is nevertheless new

and possibly ^of great 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 two-}

level 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 Boltz-

mann equation for X ^{and z} by ^{the usual} Chapman- Enskog 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

Xln, ^S being ^{defined as in} equation (13f).

(1) ^{It can 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.

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.

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 recombi- nation 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

and the system ^{is in} equilibrium ^with regard ^{to the}

transfer _process. The set of equations (13) ^and (14)

then reduces, ^{to the} following :

This equation ^describes transverse electronic spin

waves. 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. ^Never- theless, this electronic transport process is funda-

mentally 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}

Fig. ^5. - ,u factor for the b-c coherence

(in high field)

for 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).

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= 0 ; xo

n j2 ; ^S

^{1 and an} inhomoge-

neous coherence ^z will evolve according ^to equations (13) ^and (14) ^{in the} following way :

This situation will, ⁱⁿ general, ^{be a little more favou-}

rable 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 (Ti-1) ^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

length ^on the order of 1 mm and can be equally

well described by the S.W. domain delimited in

figure ^4.

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 cohe-

rence 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

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} prima- rily 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 trans-}

verse component ^{of a} conservative molecular _quan-

tity, ^and

-

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 experi-

mental observation of hyperfine ^{or Zeeman electronic} spin ^{waves should} take into account difficulties associated with external causes of damping : ^{a radio-} frequency 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-to-

noise 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 inhomo- geneous 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 ^a high enough t4

strong gradients ^of longitudinal polarization ^become

instable against transverse fluctuations of magneti- zation, ^the Reynolds number of this instability being

Appendix ^1.

In this 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} quan- tities.

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 :

A) DIFFERENTIAL CROSS SECTIONS.

The different

cross sections relevant to the discussion of spin

waves arise from two kinds of scattering.

i) ^For forward-backward scattering, ^the generic

definition is

The _upper label is related to forward or backward

scattering, while the subscript ^a labels the direct

or transfer _process.

ii) ^{For lateral} 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 :}

In order to obtain the phase ^shift expansions ^{of the} quantities ^{related to direct or transfer} processes, ^one should use relations (A .1 ) ^and (A. 2).

B) ^{ANGULAR AVERAGED CROSS SECTIONS.}

^In describing spin equilibrium ^or transport properties, ^{one is} 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 Q 0 ^and Q 1.

One then readily obtains, ^for example,

after a few trigonometric manipulations, ^{one has}

The analogous quantities ^{for aex(O) and} Tex (0) ^{are obtained} by substituting

For example, ^{one has}

The notations introduced in [ 11,14,15] ^{are related to ours} by

As far 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 not} possess the same property. ^The corresponding formulae, echoing (A. 17), ^are

The following quantity ^is interesting ^because it appears in the expression giving ^the frequency shift of the 0-0 coherence due to transfer collisions :

are the notations used in [14] ^and [15]).

C) ^THERMALLY AVERAGED CROSS SECTIONS : COLLISION INTEGRALS.

In the expression ^{of several} transport coefficients, ^the following integral usually appears :

where f ^{can be} any function of the wave vector k. In particular, f ^{can be one of the} angular ^averages Q or 6.

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 mo- menta 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 OF THE QUALITY FACTORS AND DIFFUSION COEFFICIENTS OF SPIN WAVES AS FUNCTIONS OF COLLISIONAL INTEGRALS.

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,

11,

b) ^The quality ^factor p.

p is defined in equation (6) by : p

Dz 3tz ^{with the} following

2. The b-c coherence ( a) The diffusion coefficient (

This coefficient is now

b) ^The quality ^factor.

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.

T 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.

References

[1] a) LHUILLIER, ^{C. and} LALOË, F., ^J. Physique ⁴³ (1982) 197 ;

b) LHUILLIER, ^{C. and} LALOË, F., J. Physique ⁴³ (1982)

225.

[2] a) NACHER, ^P. J., TASTEVIN, G., LEDUC, M., CRAMP- TON, S. B. AND LALOË, F., J. Physique ^{Lett. 45} (1984) 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).