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Spin-lattice relaxation of a nuclear antiferromagnet
M. Goldman, J.F. Jacquinot
To cite this version:
M. Goldman, J.F. Jacquinot. Spin-lattice relaxation of a nuclear antiferromagnet. Journal de
Physique, 1976, 37 (5), pp.617-625. �10.1051/jphys:01976003705061700�. �jpa-00208456�
SPIN-LATTICE RELAXATION OF A NUCLEAR ANTIFERROMAGNET
M. GOLDMAN and J. F.
JACQUINOT
Service de
Physique
du Solide et de RésonanceMagnétique
Centre d’Etudes Nucléaires de
Saclay,
B.P. n°2, 91190 Gif-sur-Yvette,
France(Reçu
le 19 décembre1975, accepté
le 21janvier 1976)
Résumé. 2014 Nous considérons l’influence de l’ordre antiferromagnétique nucléaire sur la relaxation spin-réseau de l’énergie dipolaire nucléaire par des
impuretés paramagnétiques.
Les prédictionsdiffèrent selon que le temps de corrélation 03C4c du couplage
spin-réseau
est plus court ou plus long que T2. Lorsque 03C4c ~ T2, la relaxation est plus lente dans l’état antiferro que dans l’étatparamagnétique,
lorsque 03C4c ~ T2, la vitesse de relaxation présente un maximum à la transition. Les résultats obtenus dans des cristaux de fluorure de calcium sont conformes aux prévisions théoriques correspondantà 03C4c ~ T2.
Abstract. 2014 We consider the influence of antiferromagnetic nuclear ordering on the spin-lattice
relaxation of the nuclear dipolar energy by
paramagnetic impurities.
Thepredictions
are different depending whether the correlation time 03C4c of thespin-lattice coupling
is shorter or longer than T2.When 03C4c ~ T2, relaxation is slower in the antiferromagnetic than in the paramagnetic state. When 03C4c ~ T2, the relaxation rate is maximum at the transition. The
experimental
results in calcium fluoridecrystals agree with the theoretical predictions for the case 03C4c ~ T2.
Classification Physics Abstracts
8.544 - 8.664
1. Introduction. - The purpose of this article is to
study
the influence of themagnetic ordering
of anuclear
spin
system on itsspin-lattice
relaxation.In the
high-temperature
limit(the
system is then in aparamagnetic state),
it is well known thatspin-
lattice relaxation is
generally exponential.
A conditionfor this is that the state of the system be characterized
by
aspin temperature.
This is the case for instance indiamagnetic
solids where relaxation is causedby paramagnetic impurities
at lowconcentration,
except for the few nuclei in the immediatevicinity
of theimpurities.
The inversetemperature P
of the system variesexponentially
with a time constantTl
as a conse-quence of the fact that the
density
matrix is linearin
fi,
in thishigh
temperature limit. Since most measur-able
quantities
areproportional
tofl, they
vary toward their thermalequilibrium
value with the sametime constant
Tl.
Let us consider for instance a system which has beendemagnetized
in therotating
frame
(see
for instance ref.[1]). Examples of quantities
which are
proportional
to the inverse temperature in therotating
frame are : theamplitude
of theresonance
signal,
observed either in theabsorption
or in the
dispersion
mode with a small nonsaturating
r.f.
field ;
or thedispersion signal
observed with asaturating
r.f. fieldapplied
at exact résonance ; or theamplitude
of thefree-decay signal following
an r.f.pulse [2].
As for the entropy, whosedeparture
fromits
infinite-temperature
value isproportional
top2,
it varies with a time
constant 2 Tl.
When on the other hand the nuclear
spin
system is in an ordered state, there is no reasonwhy
its inverse temperature shoulddecay exponentially. Furthermore,
the variousphysical properties
of the system arecomplicated
and different functions offl,
so that eachof them is
expected
to vary in a different non-expo- nential way. It is then necessary tospecify
which ofthem is under
investigation.
In thefollowing,
wefocus our attention on the time evolution under the effect of
spin-lattice relaxation
of thedipolar
energy of the system, aquantity
which caneasily
be mea-sured
[3].
The method of
production
of nuclearmagnetic ordering
has been described elsewhere[4].
We recallbriefly
itsprinciple.
Nuclearmagnetic ordering
isproduced by
a two-step process.First,
the nuclearspins
aredynamically polarized
inhigh
fieldby
theSolid
Effect,
which reduces their entropy to a lowvalue; secondly,
thepolarized
nuclearspins
areadiabatically demagnetized
to zero field. Under the effect of theirdipole-dipole coupling,
andprovided
their entropy is
sufficiently
lowthey undergo
a transi-tion to an ordered state. In all
experiments performed
so
far,
we have used adiabaticdemagnetization
in therotating frame,
thatis,
with the external d.c. fieldalways
present, we haveperformed
adiabatic fastArticle published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01976003705061700
618
passages
stopped
at resonance. The ordered structureproduced
in that waydepends
both on the orientation of the d.c. field with respect to thecrystalline
axes,and on the
sign
of thespin
temperature. Inparticular,
in a
simple
cubic system of nuclearspins 2,
such as19F in calcium
fluoride,
onepredicts
the occurrenceof the
antiferromagnetic
structurepictured
infigure 1,
when the adiabaticdemagnetization
in therotating
frame is
performed
atnegative
temperature with the external d.c. fieldparallel
to a four-foldcrystalline
axis. A number of
experimental
studies have been devoted to that case[4, 5],
and it is to that case thatwe will
eventually specialize
thestudy
ofspin-lattice
relaxation.
FIG. l. - Antiferromagnetic structure of CaF2 after adiabatic demagiictilat ion n the rotating frame, with Ho // [100] and T 0.
We will consider the case when the relaxation is caused
by paramagnetic impurities.
For the compa- rison betweentheory
andexperiment,
the timeconstant Tl
of theexponential decay
in the low- energy limit(that
is athigh temperature)
will be takenas an
adjustable
parameter to be fitted to the expe- rimental results.2.
Theory.
- We consider a nuclearspin
system whose Hamiltonian is of the form :The main Hamiltonian
Jeo
is the sum of a Zeemaninteraction
Wo Iz plus
adipole-dipole
interaction termJCD’
We are inhigh
field and as usual we truncateJeD,
so as to retainonly
the partJCà
which commuteswith
I..
It is of the form :with :
where
Oij
is theangle
between theapplied
fieldHo
and rij-
Jel (t)
is thespin-lattice coupling responsible
for thenuclear
spin-lattice
relaxation.The derivation of the
equation governing
theevolution of the
dipolar
energy is made in a standard fashion(see
for instance ref.[1]
ch.III).
Itsmajor
steps are thefollowing.
The nucleardensity
matrix 6varies
according
to :We use a canonical transformation
through
aunitary
operator :
and we
define,
for everyoperator Q :
We
have,
in thisrepresentation :
We
expand
thisequation
to the second order inRi,
then we go back to the
Schrôdinger representation,
and we obtain :
where
(...)av.
means an ensemble(or time)
averageover the random
coupling Jel (t).
We suppose that(JC 1 (t».,.
=0,
so that the first term on theright-
hand side of eq.
(7)
vanishes. A condition for thevalidity
of eq.(7)
is thatwhich can be assumed to be valid even when the system is in an
antiferromagnetic
state which breaks the symmetry of the Hamiltonian. A short discussion of thispoint
can be found in reference[3].
During
the evolution we are interestedin,
the inversespin-temperature
is so muchhigher
than the lattice inverse temperature that we can take the latter to bezero. In any case, when the inverse
spin-temperature
is
high,
it is notpermissible
toreplace
in eq.(7),
as is usual in the
high-temperature approximation, u(0) by (a(0) - UL),
where UL is thedensity
matrixat thermal
equilibrium
at the lattice temperature.By multiplying
both sides of eq.(7) by JCD
andtaking
the traces we obtain :where we use the notation :
Let us now tum to the form of
Jel (t).
We suppose that the relaxation is causedby
thedipolar coupling
of the nuclear
spins
I with thespins S
ofparamagnetic impurities
at low concentrationrandomly
distributed in thesample.
Thespin
orientation of eachimpurity
varies
randomly
around an average value under the effect either of its ownspin-lattice relaxation,
or offlip-flop
transitions withnearby impurities (we
sup- pose in that case that the electronicspin-spin
reservoiris much more
strongly coupled
to the lattice than to the nuclearspin-spin reservoir,
so that its temperature isalways equal
to that of thelattice).
We call Te the correlation time of this fluctuation and we suppose that oeo Le » 1. In that case thecoupling
terms whichare effective for the
spin-lattice
relaxation are of the form :where the sum is restricted to the nuclear
spins
outsidethe diffusion barrier of the
impurities.
In this treat-ment we will
neglect
thepossibility
of a diffusion-limited relaxation.
Eq. (11)
can be written in the form :where :
and
F(t)
is a random function of time such that :and
Since V commutes with
I,,
we have :Since furthermore
Rg
commute withJeo,
and accord-ing
to the condition(8),
eq.(9)
becomes :We assume that the
decay
time of the trace under theintegral
isroughly
of the order ofT2,
the free-induc-tison decay
time athigh
temperature. We will consider twolimiting
cases,corresponding
tor,,, « T2
andLe »
T2, respectively.
i) rc, « T2.
- Under theintegral
of eq.(16),
theexponential
factordecays
much morerapidly
thanthe trace. We can then
neglect
thedecay
of thelatter,
and we obtain :or
else, according
to eqs.(2)
and(13) :
ii)
tc >T2.
-According
to eq.(15),
the traceon the
right-hand
side of eq.(16)
isequal
to :Eq. (16)
can then beintegrated by parts :
In the limit when te is very
long,
this can be written :Under this
form,
the limit is seen to beproportional
to the linear response of the
quantity
V to the adiabaticintroduction of the
perturbation V, according
tolinear-response theory [6] ;
it is alsoequal
to theexpectation value V )
when the system is at thermalequilibrium
with respect to the HamiltonianJC’ D
+ V. We can then write :This result can be
given
asimple physical interpre- tation, already
discussed in reference[7].
If thecoupling JCi(t)
=VF(t)
weretime-independent (i.e.
F(t)
=F(O)
=1),
then the termskà
and V would reach thermalequilibrium
in a time of the order ofT2.
If
F(t)
variesslowly (i.e.
te »T2),
both terms stillconstitute at all times a
single
thermal reservoir in astate characterized
by
asingle
inversetemperature fl.
More
explicitly, let fl
andfl’
be the inverse tempe-ratures
corresponding
tokà and V, respectively.
We can write the
following
evolutionequations :
where
dldt ... >SL
stands for the evolution due to thespin-lattice coupling.
Sinceonly
the term V iscoupled
to thelattice,
we have :620
and
The term g causes an evolution
of fl
andfi’
towarda common value at a rate of the order of
T21
»ré 1.
We have then at all times
03B2’ ~ fi.
We get rid of the unknownfunction g by summing
eqs.(22)
and(22’),
which
yields :
If now we suppose
that V(fl) ) « « Kà(fl) ),
werecover eq.
(21).
CALCULATION OF THE RELAXATION RATES WITHIN APPROXIMATE THEORIES OF NUCLEAR ORDERING. -
Eq. (18)
and(21)
are exactexpressions
for the relaxa-tion rate of the
dipolar
energy in the twolimiting
casesthat we considered. We will now solve them with the
help
ofspecific approximations
to nuclear magne- ticordering.
Weanalyze
in tum the-limiting
caseswhen r,,, «
T2
and rc, >T2,
and wespecialize
fromnow on the
study
to theantiferromagnetic structure of figure
1.1 st case Tc «
T2.
-According
to eq.( 18),
the relaxa- tion rate of the energy isproportional
to the corre-lation of the transverse
spin
components. Athigh temperature,
that is at low energy in theparamagnetic
state, the
dipolar
energy and the correlation functionsare afl
proportional
to the inverse temperaturep.
We have then :
and the relaxation of the energy is
exponential,
as is well known. When the temperature decreases below the transition temperature, the energy increases very
steeply,
because of the energy associated withlong-range order,
whereas the correlation between transversespin components
varies butslowly.
As aconsequence, -
d (In JC’ »
must decrease. Wedt C D
expect then on
physical grounds
aslowing
down ofthe relaxation in the ordered state, as
compared
to theparamagnetic
state, which is confirmedby
the follow-ing approximate
calculation.We calculate the
expectation
values on theright-
hand side of eq.
(18) by
the restricted-traceapproxi-
mation
[8],
whoseprinciple
is thefollowing.
In theantiferromagnetic
state,thé
systemsplits
into twosublattices A and % in which the
spins
carry differentlongitudinal polarizations pA
and p.. The latter aredetermined
by
the conditionthat they
maximize the restrictedpartition
function :where the trace is restricted to the Hilbert
subspace corresponding
to well-defined values of pA and pB.The lowest non-trivial order of
approximation
consistsin
expanding
this restricted trace up top2.
Withinthe same order of
approximation,
theexpectation
value of an operator
Q,
which is takenequal
to :is
expanded
to the first orderin fl :
When the effective field is zero,
JC = jeD
andWe call
Ko
=(0, 0, nia)
the vectorcorresponding
tothe
antiferromagnetic
structure; thatis,
when theorigin
is taken at a lattice site of sublattice A, thepolarization
of thespin
located at ri is :To the first-order restricted-trace
approximation,
the
dipolar
energy has thefollowing
value[8] :
where N is the number of nuclear
spins
in thesample,
and :
where a is the lattice parameter,
This
approximation yields,
for eq.(18) :
In order to estimate the
importance
of the term inp2,
let us make the
rough assumption
that on the average the random fields are uncorrelated :independently
of thesubscripts i
andj.
We then obtain :
tp2
isdistinctly
smaller thanD,
and within theapproxi-
mation used
here,
it is sufficient to write :where ç
ispositive
constant.In the
high
temperaturelimit, p
= 0 and we have :whence :
If we call
Tl the
characteristic time-constant of theexponential decay
in thislimit,
wefinally
obtain :We have
plotted
the variationswith JeD >
of thequantities - Tl :t Je;,) and - Tl :t (ln Je;, »)
in
figures
2 and 3respectively.
Asanticipated,
therelaxation rate remains constant at low energy and decreases in the ordered state. The
corresponding
variation
of Je’D >
as a function of(t/Tl)
isplotted
on
figure
4. Thefigures correspond
to the case ofFIG. 2. - Variation as a function of dipolar energy of the reduced time-derivative of this same energy under the effect of spin-lattice
relaxation in CaF2. The curves correspond to theory in the limiting
cases Te « T2 and tc » T2. The experimental points are deduced
from those of figure 4. ’
FIG. 3. - Variation as a function of dipolar energy of its reduced
spin-lattice relaxation rate in CaF2. Theoretical curves for rc « T2 and Tc » T2, and expérimental points.
calcium fluoride. The
origin
oftime,
infigure 4, is
of coursearbitrary.
It has to beadjusted
to theinitial value of the energy.
2nd case
r,,, » T2.
- We use forcalculating
thesusceptibility
x of eq.(21)
a makeshift combination of local Weiss-field and restricted-traceapproxima-
tions. The
paramagnetic impurities
create at the sitesof the nuclear
spins
aninhomogeneous dipolar
fieldsuperimposed
on the Weiss-fieldexperienced by
thesenuclear
spins
in the ordered state. This causes aninhomogeneous departure
of the individualspin polarizations
withrespect
to theperfect
orderedstate. This
departure
in turnchanges
the Weiss-fieldseen
by
the nuclearspins,
which contributes self-consistently
to theirinhomogeneous change
ofpolari-
zation. The
susceptibility
XV-+V isequal
to the Zeemanenergy of the nuclear
spins
in the field of theimpurities.
As shown
below,
thissusceptibility
goesthrough
amaximum at the transition temperature. What is
predicted
is thenthat, starting
from theparamagnetic
state,
theslope-d (In ( Rg »)/dut
increases with energyup to the transition energy, and then decreases.
According
to the local Weiss-fieldapproximation,
the
polarization
of aspin 1/2 Ii
is related to the totalfield it
experiences through :
where roi is the Larmor
frequency corresponding
tothat total field. The
frequency wi
is a sum of contribu- tions from the Weiss-field in theperfect
antiferro-magnetic
state, theimpurity field,
and the Weiss-fieldarising
from thedeparture
fromperfect ordering.
Since the
impurity
concentration islow,
we canconsider
separately
the effect of eachimpurity
and622
then sum up their effects. We
write,
in thevicinity
of aparticular impurity :
where the terms on the
right-hand
sitecorrespond
tothe normal
Weiss-field,
theimpurity
field and themodified
Weiss-field, respectively.
As arough approxi- mation,
weexpand
forsimplicity
theright-hand
sideof eq.
(29)
to the first order inhi
andôi :
Since in the
perfect
ordered state, thepolarizations
are :
their
departure
fromperfect ordering
is :We now introduce the space Fourier transforms of
all localized
quantities :
,where N is the number of nuclear
spins
and k is avector of the
reciprocal
latticebelonging
to the firstBrillouin zone. The
origin
is taken at the site of theimpurity.
As thisorigin
is a centre of symmetry andhi
is a
dipolar field,
we haveh(k)
=h(- k).
On the otherhand we
have, according
to the local Weiss-fieldapproximation :
whence :
with :
We also have
A(k)
=A(- k).
By taking
the Fourier transforms of both sides of eq.(31),
we obtain :Or
else, according
to eq.(32) :
which
yields :
1
This describes the
departure
fromperfect
ordercreated
by
oneimpurity.
Its contribution to thesusceptibility
XV~V is :The n
impurities
of thesample yield equal
contribu-tions to x, and we
obtain, according
to eqs.(21)
and
(33) :
-It can be shown
that -1 fl(1 - p2)
is minimum forfl = Pc (we
recallthat fl 0).
This is due to the fast increase of thepolarization
p below the transition.It is
easily proved by
anexpansion
of the Weiss- fieldequation :
around the transition inverse temperature :
For those modes k for which
A(k)
is close toA(Ko),
the
polarization departure e(k)
issharply peaked
atthe transition. As a consequence, the sum on the
right-hand
side of eq.(34)
is maximumat fl
=Pc’
Since on the other
hand Rà )
is a monotonic func- tion of03B2,
we find that bothare maximum
at fl
=Pc,
asanticipated.
The Weiss-field
approximation
is not suitable fordescribing
the variation ofd JCp >/dt
as a functionof energy, because it
predicts
that the latter vanishes in theparamagnetic phase.
It is notpossible
eitherto use eq.
(34)
whilecalculating
the energy as a func- tion offl through
the restricted-trace formula(24).
The reason is that the value
of 1 Pc 1 in
the restricted- traceapproximation
ishigher
than thatpredicted by,
eq.
(36),
which would cause aspurious divergence
of the
right-hand
side of eq.(34)
around the transition.To avoid this we express eq.
(34)
as a function ofreduced temperature, that is we write
according to eq. (36) :
We insert this value into eq.
(34)
and we computeboth Rà )
andA through
the restricted-traceapproximation.
We use as before the relaxation timeTl
in thehigh-temperature limit,
so as to elimi- nate the unknownquantities,
and we obtainfinally :
It is in fact
possible
to avoid theapproximation (37)
and to calculate XV~V within the restricted-trace
approximation
too. This however does notmodify
the
qualitative
features of eq.(38),
and the effort is not warrantedby
theapproximate
character of thetheory.
There remains to estimate the sums in eq.
(38).
The fields
hi
aredipolar
fields createdby
the para-magnetic impurities,
whose Fourier transforms have beencomputed [9].
We make a furtherapproximation,
as follows. The nuclei which are close to the
impurities experience
alarge
electronic field and are not in thermal contact with the average nuclearspins.
The
latter,
which are located outside the diffusionbarrier, experience
electronic fields which varyslowly
over an internuclear distance. It is then reasonable to assume that
only
small k values(i.e.
ka «1)
contribute to the sums on the
right-hand
side ofeq.
(38).
In thislimit,
the Fourier transforms have thefollowing
verysimple
values[9] :
were ek
is theangle
between k and the external d.c.field,
andA
straightforward
but tedious calculationyields finally :
with :
and :
Eq. (40)
is validonly
when a >0,
that is atnegative
temperature
(A(Ko)
>0).
Theexpression
isslightly
different for an ordered structure at
positive
tempe-rature :
In the
present
case thepredicted variations,
as afunction
of je X
ofare
plotted
inusures
2 and3, respectively.
The corres-ponding
variationof JeD >
with(tltl)
isplotted
in
figure
4.FIG. 4. - Reduced time-variation of dipolar energy under the effect of spin-lattice relaxation. Theoretical curves for tc « T2, and rc » T2, and expérimental results. The value of Tl is adjusted
to the low-energy portion of the decay.
The
peak
of relaxation rate at the transition isprobably over-emphasized by
thepresent approxi-
mation. We believe however that it should
exist,
in the
light
of a relatedstudy :
thesusceptibility
ofthe
dipolar coupling
between19F
and43Ca spins
tothe
dipolar
field of the calcium nuclei[10].
Aspredicted by
atheory analogous
to thepresent
one, this suscep-tibility
is observed to gothrough
a maximum at thetransition. This
study
is outside the scope of thepresent
article.To
summarize,
thequalitative
features of thespin-
lattice
relaxation,
aspredicted by
the presenttheory,
are the
following.
For rc « T2. - - d Rà >/dt
isapproximately
proportional
to the energy in theparamagnetic
state,624
and it increases much more
slowly
with energy in theantiferromagnetic
state.Accordingly,
the relaxation rate -d(ln H’D »/dt
is
approximately
constant in theparamagnetic
state, and then decreases.For r,, » T2. - -d JeD >/dt
increases faster than( Rà )
in theparamagnetic
state, it reaches a maxi-mum at the transition and then decreases.
Accordingly, - d(ln Rà »Idt
increases with energy up to the transition and then decreases.3.
Experimental
methods and results. - The expe- riments have beenperformed
onspherical samples
of calcium
fluoride,
of 1.3 mmdiameter, doped
withTm2+
ions at a concentrationTm2+ /F-
of 1.2 x10- 5.
The fluorine
spins
aredynamically polarized through
the Solid Effect with the
Tm2+
in a field of 27kG,
with a microwave irradiationfrequency
of 130 GHz.The fluorine
polarization
obtained after 3 to 4 hours is of the order of 90%.
The microwave power is then cutoff,
thesample
is cooled down to below 0.3 Kby pumping
over théliquid ’He
bath in which it isimmersed,
and the adiabaticdemagnetization
in therotating
frame isperformed by applying
a r.f. fieldHl ~
30 mG andsweeping
the external field at arate of 1
G/s
down to resonance.The nuclear
signal
at the output of aQ-meter
isrecorded on a multichannel
analyzer during
linearsweeps of the
magnetic
field at a rate of 700G/s.
The field sweeps are
grouped
in sequences of ten passages made at 1 sinterval,
and the cumulativesignal corresponding
to each sequence is stored in acalculator. The sequences are
repeated
every 20 to 30 s. Thedipolar
energy for each sequence is deduced from the first moment of thecorresponding
nuclearabsorption signal [3].
Asexplained
in thisreference,
thegain
of the receiver is determinedby measuring
both the area and the second moment of the fluorine
resonance
signal prior
to thedemagnetization.
Thisallows an absolute calibration of the
dipolar
energy with an accuracy of about 5%.
The relative saturation ofdipolar
energy causedby
each passage is 1.5 x10-4.
The
dipolar
relaxation time in thelow-energy
limitvaries from 160 to 180 s
depending
on thesample
and on the
temperature
achieved in each run. This relaxation is not due to theTm2+ ions,
asproved by
its
temperature dependence.
The relaxation rate causedby
these ions is indeedproportional
to(1 2013 p2)/p,
where P is the electronic thermalequi-
librium
polarization.
Inhigh
field and at low tempe- rature, thiscorresponds
to :Experimentally, Tl
variesaccording
to eq.(41)
downto 0.6 K and increases much more
slowly
than pre- dictedby
thisequation
at lower temperatures, which indicates that the relaxation is then causedby
adifferent
impurity.
We have so far not determined itsnature and we have thus no direct
knowledge
ofwhether the correlation time Te for the relaxation of the
dipolar
energy islonger
or shorter than thenuclear
T2.
The
experimental
variationof JeD >
with time isshown in
figure
4. Thepoints correspond
to twodifferent
experiments.
Theonly adjustment
is thatof
Tl
in thelow-energy
limit. These results have been used to compute the variationswith Rà )
ofwhich are shown in
figures
2 and3, respectively.
The
qualitative
ratures of these results are suf-ficiently
clean to substantiate the conclusion that we are in the casewhen r,,, « T2.
Thestriking
agreement with the numericalpredictions
for that case is acci-dental,
as evidencedby
the results obtained whenstudying
the variation of the43Ca-19F
energy as a function of the19F-19F
energy[10].
This laststudy yields
for the transition energy the value of - 2.5 kHz perspin, approximately 50 % higher
than the value1.62 kHz per
spin predicted by
the 1 st order restricted-trace
approximation.
Onphysical grounds
onepredicts
the occurrence in the
paramagnetic
domain of a fastincrease of the energy associated with the
longitudinal
correlations at the
approach
of the transition. This isclosely
related to thédivergence
of thesusceptibility
to a
longitudinal periodic
field of wave-vectorKo (the
so-calledstaggered susceptibility),
as evidencedfor instance
by
anapproximation
of the Ornstein-Zernike
type (see
for instance[11]).
Since the energyassociated with transverse correlations is not
expected
to
experience
such alarge
increase we areled,
in viewof eq.
(18),
to thequalitative
conclusion that the relaxation rate of thedipolar
energy mustalready
decrease in the
paramagnetic
domain at theapproach
of the
transition,
which agrees with the results. This discussion establishes that there is noinconsistency
between the
present
results and those of reference[10].
It also sets a limit to the
quantitative validity
of the presenttheory
of relaxation.4. Conclusion. - We have
developed
thetheory
ofspin-lattice
relaxation of thedipolar
energy in anuclear
spin
system whichundergoes
a transition toan
antiferromagnetic phase.
We have assumed that the relaxation was due to thecoupling
of the nuclei with fixedparamagnetic impurities.
On verygeneral grounds
onepredicts
aqualitatively
differentbeha-
viour when the correlationtime te
of thiscoupling
is
longer
or shorter than the nuclear inverse linewidthT2.
When te ~T2,
the relaxation rate isexpected
todecrease
steadily
as a function of energy below thetransition,
whereas whenr,,, » T2
this rate isexpected
to be
peaked
at the transition and then to decrease.As the
theory
isessentially
concerned with thechange
in relaxation rate whengoing
from the para-magnetic
to the orderedphase,
it is essential to useapproximations
which cover bothphases
with thesame formalism. This is the case for the
approxima-
tions used here : local Weiss-field for the relaxation rate and restricted-trace
approximation
for the energy.The
experimental
results obtained in calcium fluoride exhibit the featuresexpected
for the casewhen
rc « T2.
Thecomparison
with a differentexperiment [10]
reveals however that the relaxation ratebegins
to decreasealready
in theparamagnetic phase,
which can bequalitatively interpreted.
Despite
its obviousshortcomings,
the combination of local Weiss-field and restricted-traceapproxima-
tions
yields
a correct order ofmagnitude
for thevariation of the
spin-lattice
relaxation rate.Acknowledgments.
- Weacknowledge
the contri- bution of Dr. M.Chapellier
to this work. We aremuch indebted to Professor A.
Abragam
for hisconstant interest and advice.
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