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Transport properties in a spin polarized gas (II)
Claire C. Lhuillier, Franck Laloë
To cite this version:
Claire C. Lhuillier, Franck Laloë. Transport properties in a spin polarized gas (II). Journal de
Physique, 1982, 43, pp.225-241. �hal-00003387�
J. Physique 43 (1982) 225-241
Classification
Physics Abstracts
51.10 - 67.20
FÉVRIER 1982, PAGE 225
v-
Transport properties in a spin polarized gas, II
C. Lhuillier and F. Laloë
(Reçu le 7juillet-1981, accepté le 1er octobre 1981)
. Laboratoire de Spectroscopie Hertzienne de l'EN.S., 24, rue Lhomond, F 75231 Paris Cedex 05, France
Résumé. - Partant des résultats généraux obtenus dans l'article précédent, on étudie les conséquences des effets
quantiques d'indiscernabilité sur les propriétés de transport d'un gaz polarisé à basse température. La théorie
suppose que l'orientation nucléaire M est quelconque (pas de développement en puissances de M).
On commence par un calcul simple du courant de spin en présence d'un gradient d'orientation, compte tenu des
cohérences de spin et de leur évolution due à l'effet de rotation des spins identiques; ce dernier joue en fait un
rôle essentiel à la limite des faibles températures. Les équations hydrodynamiques d'évolution de M sont non
linéaires et anisotropes. Les effets d'indiscernabilité introduisent également un caractère oscillatoire dans l'évo-
lution des composantes transverses de l'orientation.
Un calcul plus élaboré permet ensuite d'étudier les phénomènes quantiques de couplage entre diffusion de spin
et conduction de la chaleur. De tels phénomènes pourraient être à la base de méthodes de surpolarisation ther-
miques d'un échantillon gazeux.
Enfin, on s'intéresse aux mélanges des deux isotopes 3He et 4He où, en sus du couplage classique entre concen-
tration isotopique et conduction thermique, apparaît un couplage supplémentaire quantique avec la diffusion
de spin.
Abstract
- The general results obtained in the preceding article are applied to the study of transport phenomena
in a spin polarized gas at low temperatures, with particular emphasis on the particle indistinguishability effects
in collisions. The effects of the nuclear orientation Mare treated exactly (no M expansion).
First, a simple theory of spin diffusion is presented and the response of the gas to a spin orientation gradient is
calculated. The spin coherences and their evolution due to the « identical-spin rotation effect » are taken into
account; it is found that they can play an important role, especially in the low temperature limit. The hydrodynamic
equations of evolution ofM are non-linear and anisotropic. Particle indistinguishability effects give an oscillatory
character to the evolution of the transverse components of M (spin oscillations).
Then, a more elaborate variational method is used to study the coupling between spin diffusion and heat conduction.
This effect is, again, a sheer consequence of quantum interference phenomena introduced by particle indis-
tinguishability.
Finally, isotopic mixtures of 3He and 4He are considered. ln addition to coupling between isotopic diffusion and
heat conduction, which exists for classical systems, quantum collision effects introduce a coupling between these
two modes and spin diffusion.
Introduction.- ln the preceding article, we have
studied the consequences of particle indistinguishabi-
litY effects on the collision term of the Boltzmann
equation for a dilute (non-degenerate) spin polarized
gag. We have seen that these effects introduce several
terms which have the form of commutators or anti-
commutators between spin density operators; the
former correspond to what we have called the « iden-
tical spin rotation effect ». We have also applied our
results to the sindy of the heat conduction and visco-
sity coefficients and found that both may depend
strongly on the nuclear polarization M of the gas;
the correlations between spin variables and velocities
of the atolls have also been discussed. Nevertheless,
it so happens that only the anticommutators give a
contribution to the viscosity and heat conduction
coefficients, the identical spin rotation eITectplaying
no foie in both cases. This is because the commutators
disappear from the calculations whenever a spin
rotation invariance argument ensures that the average
spin density operator p~ ahd the solution Ops of the
linearized Boltzmann equation can be diagonalized
'
,
'
I
i 1
! 1.
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ii' .
Ji"
t
Il. '
P'
I!.
226
JOURNAL DE PHYSIQUE
N"2
in the saille basis. ln other words, no spin coherence
effects e) are involved in the applications presented
in the preceding article, and this is why an results
where physically interpretable in terms of a semi-
classical mixture model (the atolls with up or clown
spins are considered as two different atomic species,
with cross sections obtained froID simple considera-
tions concerning quantum effects in collisions).
ln the present article, we wish to study situations
where spin coherences and identical spin rotation
effects do play a foie and where the semi-classical
model of a gag mixture is not sufficient ln spin diffu-
sion, the average direction of the atomic spins changes
o'Ver the sample, so that aIl spin operators do DOt
necessarily commute with each other. Conceptually,
the problem of spin diffusion is more complex than,
for example, heat conduction, one reason being that
the distinguishable or indistinguishable character
ofthe atolls is governed by the spin orientation, which
changes in space and lime. We have already mention-
ed these difficulties in the introduction of the preced-
ing article, as weIl as the contribution of Emery [1]
who clarified the situation by pointing out that, for
an unpolarized gag, the particle indistinguishability
effects in collisions give a negligible contribution to the
spin diffusion coefficient ln what follows, we shan
confirm this point of view, but we shall also find that
the presence of a significant nuclear polarization can
radically change the situation: in fact, the spin diffu-
sion phenomenon is then dominated by identical
spin rotation effects, and the corresponding hydrody-
namic equations become highly non-linear and ani-
sotropic. Instead of remaining a purely dissipative
process, the spin diffusion phenomenon also acquÎTes
an oscillatory character; ibis effect may even become
domiriant at very low temperatures. Such properties
are rather unusual in a gag,and their origin is quantum
interference effects during collisions between identi-
cal atolls having a high nuclear polarization (they
1'1 .
III.
Il .
Ii
J,
l,
l
,
l'
1.:
1.
l
,i' '
1:
!
i' ,
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r ..
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i
Il
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i
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i
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:
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disappear if M = 0). They are reminiscent of the
effects predicted by Leggett and Rice [2]and Leggett [3]
in normalliquid 3He at low temperatures, where they
are induced by the strong degeneracy of the quantum
liquid.
Another aspect of the transport properties in a
spin polarized gag is the coupling between modes:
since particle indistinguishability effects create corre-
lations between atomic spin orientations and velocities,
one can expect that the response of the gag to a heat
gradient may include the production of a macroscopic
spin enTrent ln the theory of the preceding article,
the possibiIity of such a spin enTrent was ignored froID
the beginning by the very choice of the trial density
operator but, here, we shan develop more elaborate
calculations and obtain qualitative predictions for
these effects.
The sindy of gaseous mixtures of 3He and 4He offers
additional possibilities for studying quantum effects
in transport properties. ln ibis case, three modes can
be coupled : diffusion of the isotopic concentration,
spin diffusion, and heat conduction. This triple coupl-
ing is studied in the last section of the present article.
1. Spin diffusion in spin polarized gases : simple
theory. - ln tbis section, we sindy the spin diffusion
in a spin polarized gas with non negligible polariza-
lion M, in the simple situation where there is no tem-
perature or pressure gradient To derive the spin
diffusion equation, we shall solve by an approximate
method the linearized Boltzmann equation obtained
in the preceding article, which we recall here for
completeness :
1t [ips(r, p) + ~ . Vr { p~(r, p)} = IcoII[o,osCr,p)]
(la)
where the collision integral Icollis given by :
IcoII[bPs(r,p)] = fd3q' ~ fd2q fO(P2) { (1k(f})(bPS(P'I) - bPS(PI) + p~ Tr { bps(p~) - bPS(P2)}) +
+ 1 (1~X'(f})[~, bps(p~)+ bPsCP'I) - bPS(PI) ~bPS(P2)]+ + ~ L;;X'(f})[p~,o,os(p~)~ Ops(pm}
+
~ fd3q ~ Lf~d.(k)fo(P - q) [p~, bps(p) - o,os(P- q)] (lb)
and:
Pl = P
P2 = P - q
e) Following a common usage in optical pumping,
« spin populations» or « spin coherences» refer to dia-
gonal or off diagonal elements of the spin density matrix
respectively (longitudinal or transverse coinponent of the
magnetization).
P'I = p + t q' - t q
p~ = p - t q' - -! q (le)
(f) is the angle between vectors q and Q', which both
have the saille modulus; k is defined by k = qj2/i).
IIi ibis expression, the first terms are proportional to
(1k(f})and do not depend on particle indistinguishability
effects «<classical terms »), but aIl other terms are
N°2
TRANSPORT PROPERTIES lN A SPIN POLARIZED GAS II
227
direct consequences of these effects and have an oppo-
site sign for bosons (e = + 1)or fermions(e = - 1).
The anticommutators are proportional to (JZx,(O)
and they account for the variations of the scattering
cross sections between atoms either in the saille, or
orthogona~ spin states. The commutator in ..r.(O)
corresponds to the «identica1 spin rotation effect»
and the commutator in "~~d.to the Saille effect in the
forward direction «<nuclear spin Faraday effect »).
It should be emphasized thaï the collision operator
has been linearized with respect to the deviations form
a local equilibrium situation, but thaï the nuclear
polarization effectsare treated exactly. ln other words,
we do .not restrict ourselves to situations where the
nuclear polarization M is near ilS value at thermal
equilibrium, which is ordinarily very 10w, except
in extreme situations [no M expansion has been
necessary to obtain equation (1)]. Since the longitu-
dinal relaxation lime TI is usually very long in a
dilute gag [4, 5], a highly polarized sample can be
considered to be in a metastable state.
1.1 A FIRST APPROXIMATION FOR THE SPIN CURRENT.
- The mathematical approximation methods we shaH
use are the saille as in t4e preceding article (first order
Chapman-Enskog expansion, truncated basis method).
Here, we give only the main steps of the calculation,
without dwelling on details.
For the problem of spin diffusion, the local equili-
brium density operator (equation (40) of the preceding
article) is given by :
p~(r, p) = fo(p) p~(r) .
with, for spin i- atoms (we simplify the notation Mo
toM):
p~(r) = Hl + G.M(r)J
(in contrast to the case of heat conduction and visco-
sity, the function fo is independent of r). The corres-
pouding drift term for Dps is then :
Il'
- p.Vr[~(r)J =
2
- L p. VMi(r) (Ji
m m i=I,2,3
where i = x, y, z and the (Ji'Sare the Pauli matrices.
Since this expression has no component on the unit
2 x 2 matrix, the drift term is orthogonal to the atomic
number density nef), linear momentum and energy
densities :T'.(r)and 'ill(r); since it is an odd function ofp,
it is also orthogonal to the spin orientation density
.At(r). The angular dependence of the drift term is
given by spherical harmonics Yt' (fi), where fi is the
unit vector pfp. As a consequence, we can write the
first order Chapman-Enskog term of the density ope-
Tatar in the form :
{)Ps = i-[CO(P).P + ~Ci(P).P (J].
(3b)
As in the preceding article, we shan use a simpleTvaria-
tional form. ln this section, we choose the following
trial density operator :
(jps =! L c;.p (Ji
;=1,2,3
(3e)
(2a)
which depends only on 9 real parameters (the compo-
nents of CI' C2and C3)'Two simplifications have been
made to write (3e) : first, the P dependence of the vec-
tors c's has been ignored; second, since the drift
term (3a) is traceless (no component on the unit
2 x 2 matrix), the saille property is assumed to be
valid for (jps and the term in co.p has been sup-
pressed e) from (3b). Since the collision operator
Tc given by (tb) is not in general diagonal in p and
can change the trace of operators, expression (3e)
is only an approximation, which will be useful to
discuss in simple terms the main characteristics
of the spin diffusion phenomenon. ln the next sec-
tion, we shaH use more elaborate calculations and
introduce corrections to the simple approximation
(3e).
As in the preceding article, we shaH choose a (local)
reference frame Oxyz with axis Oz parallel to the
nuclear polarization M. An important difference is
thaï this choice does not in general imply that the drift
operator [equation (3a)J becomes diagonal; thug,
we cannot restrict the summation over i in (3e) to
only the term i = 3. As a consequence, the commuta-
tors, which simply vanished in the case of viscosity
and heat conduction, now play a Tale in the calcula-
tions. ln fact, with the trial density operator (3e), the
cancellation actually oœurs for the anticommutators,
not the commutators; this is because Dps is a linear
function ofp and
(2b)
Pl + P2 = p~+ p~
(3a)
(momentum conservation in a collision). Despite
these differences, the calculations remain similar to
those of the preceding article and give :
1
ïJVMx = nXI Cl- nX4 eMc2
1
ïJ VMy = nX4 BMcl + nXl C2
1
ïJVMz = nX 1C3
(4a)
e) If the vector Co were not zero, it could not be inde-
pendent of p; this is because the trial density operator has
to be orthogonal to the linear momentum density :l'.
N°2
TRANSPORT PROPERTIES lN A SPIN POLARIZED GAS II
227
direct consequences of these effects and have an oppo-
site sign for bosons (B = + 1) or fermions (B = - 1).
The anticommutators are proportional to O"~x'(e)
and they account for the variations of the scattering
cross sections between atoms either in the saille, or
orthogonal, spin states. The commutator in !~x'(e)
corresponds to the «identical spin rotation effect»
and the commutator in !~~d.to the Saille effect in the
forward direction «<nuclear spin Faraday effect »).
It should be emphasized that the collision operator
has been linearized with respect to the deviations form
a local equilibrium situation, but that the nuclear
polarization effectsare treated exactly. ln other words,
we do. not restrict olirselves to situations where the
nuclear polarization M is near its value at thermal
equilibrium, which is ordinarily very low, except
in extreme situations [no M expansion has been
necessary to obtain equation (1)]. SinGe the longitu-
dinal relaxation time TI is usually very long in a
dilute gas [4, 5], a highly polarized sample cao be
considered to be in a metastable state.
1.1 A FIRST APPROXIMATION FOR THE SPIN CURRENT.
- The mathematical approximation methods we shaH
use are the saille as in the preceding article (first order
Chapman-Enskog expansion, truncated basis method).
Here, we give only the main steps of the ca1culation,
without dwelling on details.
For the problem of spin diffusion, the local equili-
brium density operator (equation (40) of the preceding
article) is given by :
p~(r, p) = fo(p) p~(r) .
with, for spin t atoms (we simplify the notation Mo
toM) :
p~(r)= Hl + a.M(r)]
(in contrast to the case of heat conduction and visco-
sity, the function fo is independent of r). The corres-
pouding drift term for Ops is then :
l p.Vr[~(r)J =
2
~ L p.VMi(r) ~j
m m j=1,2,3
where i = x, y, z and the O"i'Sare the Pauli matrices.
SinGe this expression has no component on the unit
2 x 2 matrix, the drift term is orthogonal to the atomic
number density n(r), linear momentum and energy
densities Ir(r) and W(r); sinGeit is an odd function ofp,
it is also orthogonal to the spin orientation density
.At(r). The angular dependence of the drift term is
given by spherical harmonies Yt' (fi), where fi is the
unit vector p/p. As a consequence, we cao write the
first order Chapman-Enskog term of the density ope-
rator in the form :
Ops =t[co(P).P + ;;: Cj(P)'p O"i}
(3b)
As in the preceding article, we shall use a simpleTvaria-
tional form. ln this section, we choose the following
trial density operator :
/jps = t L Ci'P O"i
i= 1,2,3
(3e)
(2a)
which depends only on 9 real parameters (the compo-
nents of CI' C2and C3)'Two simplifications have been
made to write (3e) : first, the P dependence of the vec-
tors c's has been ignored; second, sinGe the drift
term (3a) is traceless (no component on the unit
2 x 2 matrix), the same property is assumed to be
valid for /jps and the term in co'P has been sup-
pressed e) from (3b). Since the collision operator
Fc given by (1b) is flot in general diagonal in P and
Gan change the trace of operators, expression (3e)
is only an approximation, which will be useful to
discuss in simple terms the main characteristics
of the spin diffusion phenomenon. ln the next sec-
tion, we shall use more elaborate calculations and
introduce corrections to the simple approximation
(3e).
As in the preceding article, we shaH choose a (local)
reference frame Oxyz with axis Oz paraiIel to the
nuclear polarization M. An important difference is
that this choice does not in general imply that the drift
operator [equation (3a)] becomes diagonal; thug,
we cannot restrict the summation over i in (3e) to
only the term i = 3. As a consequence, the commuta-
tors, which simply vanished in the case of viscosity
and heat conduction, now play a role in the calcula-
tions. ln fact, with the trial density operator (3e), the
cancellation actually occurs for the anticommutators,
not the commutators; this is because Ops is a linear
function ofp and
(2b)
PI + P2 = P'l + p~
(3a)
(momentum conservation in a collision). Despite
these differences,the calculations remain siinilar to
those of the preceding article and give :
1
7JVMx = nXl c1- nX4 BMc2
1
7JVMy = nX4 BMcI + nXl C2
1
7J VMz = nX1 C3
(4a)
e) If the vector Cowere not zero, it couId flot be inde-
pendent of p; this is because the trial density operator has
to be orthogonal to the linear momentum density 3'.