Field-entropy phase diagram of a nuclear dipolar antiferromagnet

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Field-entropy phase diagram of a nuclear dipolar antiferromagnet

C. Urbina, J.F. Jacquinot, M. Goldman

To cite this version:

C. Urbina, J.F. Jacquinot, M. Goldman. Field-entropy phase diagram of a nuclear dipolar antifer- romagnet. Journal de Physique, 1982, 43 (10), pp.1461-1467. �10.1051/jphys:0198200430100146100�.

�jpa-00209527�

Field-entropy phase diagram ^{of a nuclear dipolar} antiferromagnet

C. Urbina, ^{J. F.} Jacquinot ^{and M. Goldman}

SPSRM, ^{Orme des} Merisiers, 91191 Gif-sur-Yvette Cedex, ^France (Reçu ^{le 16 mars} 1982, accepté ^le 2 juin 1982)

Résumé. 2014 Ce travail est une étude théorique ^et expérimentale ^du diagramme ^de phase champ-entropie ^d’un antiferromagnétique dipolaire nucléaire dans CaF2. ^{Dans une} grande région ^du plan champ-entropie ^{l’état du} système ^{est une structure «} mixte » dans laquelle ^{des domaines} paramagnétiques ^et antiferromagnétiques ^coexistent

en des proportions bien définies. Les prédictions théoriques, ^{basées sur} le modèle de la « trace restreinte », sont en très bon accord avec les résultats expérimentaux obtenus par des ^mesures de R.M.N.

Abstract.

This work is a theoretical and experimental study ^{of the} field-entropy phase diagram ^{of a nuclear} dipolar antiferromagnet ⁱⁿ CaF2. ^{There is a} large region ^{in this} plane ^{where the state of the system is a « mixed »}

structure in which paramagnetic ^and antiferromagnetic domains coexist in well defined proportions. The theore- tical predictions, ^{based on the «} restricted trace » model, ^{are in} very good agreement ^{with the} experimental ^results

obtained by ^N.M.R. measurements.

Classification

Physics ^Abstracts

75.25

76.50

75.60C

1. Introduction.

The production of nuclear

dipolar magnetic ordering ⁱⁿ diamagnetic crystals

and its study by ^nuclear magnetic resonance or neu-

tron diffraction techniques, have been extensively

described in a number of publications [1-4]. ^Their

essential aspects ^are briefly reviewed below.

1) ^Because of the weakness of the nuclear magnetic

moments, the critical temperature ^{of the} ordering, Tc,

is in the microkelvin _range.

2) ^{It is} only the nuclear spins ^{that are cooled}

whereas the other degrees of freedom in the crystal ( the lattice ») ^{remain at a} relatively high temperature.

This is achieved by ^a two-step process : dynamic

nuclear polarization ^{in a} high ^field Ho, ^followed by

an adiabatic demagnetization ^{in the same} d.c. field

by ^{means of an} appropriate radiofrequency ^field (A.D.R.F.).

In high field, the Hamiltonian of a system ^{of N}

spins ^I subjected ^to dipolar interactions is :

where _coo

_y, Ho ^{is the Larmor} frequency ^and

Ki is the secular part ^{of the} dipole-dipole interaction :

with

where rij is the distance between spins ⁱ and j, ^and °ij

is the angle between the applied ^field Ho ^and rij- After the dynamic polarization ^the spin tempera-

ture is in the mK range. To perform ^the second cool-

ing step ^{we introduce an r.f.} field, H1, rotating ^around Ho ^{at the} frequency ^{cv. The} Hamiltonian in the rotat-

ing ^{frame is :}

where A

(wo - w) ^{is the} longitudinal ^effective

field in frequency units, ^and w, = - yl H 1. ^This

irradiation has two effects; it cools down the Zeeman term by ^{a factor} 4 /cvo, and it establishes a thermal

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198200430100146100

1462

contact between Zeeman and dipolar energies ^when ,J H’ L (where HL is the local field in the rotating frame). The A.D.R.F. is performed by ^{a familiar}

fast _passage stopped at resonance, i.e. by ^adiaba- tically sweeping ^the frequency ^cv up ^to a)o.

If the nuclear polarization reached in the first step ^is high enough, this process will lead ^{to a} dipolar

ordered state. The ordered structure so obtained, depends ^on the orientation of Ho ^with respect ^{to the} crystalline ^{axes and on the} sign of the nuclear spin temperature.

Nearly all of the studies of nuclear A.F. published

up ^{to now} have been performed ^{in zero effective}

field. They ^{were devoted to an} understanding ^{of the} properties ^{of the} dipolar ^{ordered states without}

attention to the various steps ^{in the} ordering process

as the effective field is decreased. This article is a

theoretical and experimental study ^of the field- entropy phase diagram ^{in the} rotating ^{frame, of the} 19F spin system ⁱⁿ CaF2, ^{at T 0 and with} HO,11[100].

2. Theory.

^The main features of nuclear dipolar magnetic ordering ^are correctly ^described by ^a

model based on a ^mean field approach. ^{The «local}

Weiss field » approximation, ^{as well as its} simplest improvement, namely ^{the «} restricted trace » approxi- mation, ^{have been} extensively described elsewhere [1-4]

and we will use here their results without further

explanation.

According ^to the local Weiss field approximation,

which neglects ^the short-range correlations, ^the pola- lization li >/2 ^{of a} spin is related to the total field it experiences through

where roT ^{is the Larmor} frequency corresponding

to the total field. The frequency roT ^{is the sum of}

contributions from the external effective field and the Weiss field :

Fig. ^1.

Antiferromagnetic ^{structure in zero} effective field in CaF2 ^{for T 0 and} Ho/ 100].

In this approximation the total _energy is

where the factor 1/2 ^before on ^accounts for the fact that it is a self _energy.

The particular antiferromagnetic ^{structure consi-}

dered here is defined by ^{a vector} Ko

[0, 0, 7r/a]

(where a is the lattice parameter ^{of the s.c. fluorine} lattice) ^and consists of two sublattices A and B with spin polarizations pA and _pB respectively (Fig. 1).

In this _case, the Weiss frequencies turn out to be,

for a spherical sample

where A(Ko) ^is the Fourier transform of the dipolar coupling ^constants Aij ^over the lattice :

The entropy ^{is fixed} by ^the polarization ^{p before}

A.D.R.F. by ^the expression

where N is the number of spins.

2.1 THE MIXED PHASE. QUALITATIVE DISCUSSION.

The behaviour of a nuclear antiferromagnet ⁱⁿ

non-zero effective field is best understood by compa- rison with that of an electronic antiferromagnet.

We begin ^therefore by recalling briefly ^{the well-}

known properties of the latter and in particular ^the

appearance in ^a finite field of the so called « spin- flop » phase.

One should remember that by contrast to nuclear

spins, the electronic spins ^are always ^{at a} positive

temperature and the stable structures are those of lowest _energy.

The interactions responsible for electronic anti-

ferromagnetism ^are usually Heisenberg interactions,

which are isotropic ^and short-range. ^{In addition}

there is usually ^a much weaker anisotropy energy which lifts the degeneracy associated with the iso- tropy ^{of the} Heisenberg interactions, by selecting

a preferential orientation for the sublattice pola-

rizations.

The appearance of the spin-flop phase ^{can be}

understood as follows; ^at _very low temperature,

the parallel susceptibility ^{of the} antiferromagnet

is vanishingly small, ^{so that a} moderate field applied parallel ^{to the} anisotropy axis would not affect the

antiferromagnetic structure. Its bulk magnetization

remains zero and there is no Zeeman contribution to the _energy of the system. On the other hand, ^{let us}

consider an antiferromagnetic ^structure differing ^from

the previous ^one by just ^{a 900} rotation of spin pola- rizations, ^{so as to} bring ^{them into the} plane perpen- dicular to the anisotropy axis; ^{there is no} change

in the isotropic Heisenberg interaction _energy, but the anisotropy energy vanishes, ^{and the new structure} is, ^{in zero} field, less stable than the initial longitudinal antiferromagnet. However, ^{this structure has a} large

transverse susceptibility ^with respect ^{to a field} parallel

to the anisotropy axis : the spin polarizations ^can

rotate so as to become parallel ^to the total field

they experience. This is the spin-flop phase; ^the

increase in anisotropy energy is compensated by

a decrease of Zeeman energy. Above a threshold external field H(’) ^the energy of the spin-flop phase (including ^{the Zeeman} contribution) is lower than that of the longitudinal phase, ^{and the} system ^under-

goes ^a first-order transition to the spin-flop phase.

At exactly the critical field H(’), ^which depends only ^{on the} temperature, ^both phases ^{can coexist}

in _any proportion. ^{This is a} characteristic property of first-order transitions in systems ^with short-range interactions, ^{where the two} phases ^{do not influence}

each other. Slightly ^above H(’) the whole system is in the spin-flop phase. ^{When H} increases, ^the angle

between the sublattice polarizations decreases until it vanishes at a critical field H(’), ^{where the} system undergoes ^a second-order transition to the _para-

magnetic ^{state in} which the polarizations ^{of all} spins ^are equal ^and parallel ^to the external field.

Keeping in mind the fact that in the dipolar ^nuclear

case the interactions are long-ranged ^and anisotropic

we can apply ^a procedure ^{similar to the one used}

above for the electronic case : in an applied ^external

field we stabilize the system by allowing ^{it to} include,

besides the low susceptibility phase which is stable in zero field, regions corresponding ^{to a different}

structure, ^{with a} comparable spin-spin energy but

a larger susceptibility. ^{From now on, we consider}

nuclear spin systems ^{at a} negative temperature ^so that the stablest structures are those of highest energy.

First, ^{we should} point ^{out that we} cannot use,

as in the electronic _case, the same antiferromagnetic

structure rotated by 900, because for the anisotropic dipolar interactions this would cause such an enor- mous change ^{in the} spin-spin energy that no concei- vable change ⁱⁿ Zeeman energy would be sufficient to balance it. In fact, ^a simple inspection ^{of the trun-}

cated dipolar Hamiltonian (2) ^shows that such a

rotation changes ^the dipolar energy by ^{a factor} - 1/2.

We must then look for another ordered structure with large transverse susceptibility ^{but whose} spin-

spin energy is close to that of the antiferromagnetic

structure. In CaF2, ^{when T 0 there is} just ^one

such structure : it is a ferromagnetic ^{state with thin}

domains perpendicular ^to Ho [2] (Fig. 2). Its energy in the Weiss field approximation, EF, ^is independent

of the orientation of the external field

with

This should be compared ^{with the} antiferromagnetic

energy ^as given by equation (7) ^{for L1 = 0}

We construct then a « mixed state » as a compro- mise between the original antiferromagnetic ^struc-

ture and this ferromagnetic ^{state. It} consists in an

alternation of paramagnetic ^and antiferromagnetic domains, spread uniformly through ^the sample,

whose shapes and orientations are the same as in the above mentioned ferromagnet (Fig. 3). ^{As a conse-}

quence of their large longitudinal susceptibility,

the paramagnetic slices stabilize the whole structure in an applied ^{field. In fact, it will} appear below that

they ^{hinder to a certain extent the} penetration ^{of the}

field into the A.F. domains. This is analogous ^{to the}

case of type ^II superconductors ^{in the} region ^between

the two critical fields, where there is partial pene- tration of flux and the microscopic ^{structure is a}

« mixed state » consisting ^of both normal and _super-

conducting regions [5].

2 . Z THE MIXED PHASE. QUANTITATIVE DISCUSSION.

The « field-entropy » region where this mixed state

can exist during ^the remagnetization (or ^{the dema-} gnetization) ^{of the} antiferromagnetic structure, ^is determined by ^a variational calculation outlined below.

Fig. ^2.

Ferromagnetic with domains structure at T 0 in CaF2. ^When HO,11[100] its energy is slightly ^{below that}

of the fundamental antiferromagnetic structure, but it has

the largest longitudinal susceptibility.

1464

Fig. ^{3. - o Mixed state »} arising ^when a non-zero effec- tive field is applied ^{on the A.F. structure of} figure ^1.

Let ^x be the relative volume of the paramagnetic

domains and _pc their spin polarization. The relative volume of the antiferromagnetic domains is (1 - x)

and their sublattice polarizations ^{are PA and PB-}

The calculation of the total frequencies ^WT expe- rienced by the various spins ^{is based on a standard} magnetostatic method : the dipolar frequency ^for spin Ii ^is computed by considering first the contri- bution from the other spins ^{in the same domain} (the so-called «internal frequency >> Win,), ^{and then} treating ^{the rest of the} sample (which ^{we recall is} spherical) ^as if it has a homogeneous polarization ^p

The contribution of the spins ^out of the domain is not zero because of the long-range character of the dipolar interactions. It combines with the applied

field A to form the so-called « external frequency ^»

Oext

The results are

where q is the Weiss field factor (12) associated with the bulk polarization ^{of a} thin domain perpendicular

to the external field.

The total frequencies ^are

In zero effective field, ^{when the state is} purely antiferromagnetic, ^{one has x}

⁰ and the antiferro-

magnetic frequencies ^are

The expressions ^{for the} energy and the entropy per spin in the Weiss field approximation ^are

where s( p) ^{is defined} by equation (10).

We now consider fixed values of the applied ^{held d}

and the inverse temperature ^and determine the

equilibrium ^{values of} pA, pB, pc and x by imposing

the condition that they maximize - #F ^{= S -} BE.

The four equilibrium equations obtained in this way ^are

and

and must be solved numerically ^to yield ^{the values} P;. ({3, L1) ^and x({3, L1). The first three conditions yield

the Weiss equations :

The value of (- PF) ^{in the} mixed state, calculated with the equilibrium values of P;.(P, .1) ^and x(P, .1)

obtained above, ^is larger ^{than in a} purely parama-

gnetic ^or purely antiferromagnetic ^{state in a certain} region ^{of A and} j8. ^{In this} region, the mixed state is then the stable one.

A remarkable result of this theory ^{is that} at constant

temperature ^{pA, PB and pc are} independent ^of d, and so are the total frequencies according ^to (21)

and the external and internal frequencies according

to (14). ^This means, after (15), ^{that the} proportion

of paramagnetic ^{domains x} depends linearly ^{on the} applied ^{field .1}

where the critical fields A c (’) ^and A c (2) correspond ^to

x

0 and x

1 respectively. According ^to (15) they ^{are related} through

In the _range of coexistence of the two phases,

,j c (1) j j c (2), the total frequencies ^{in the anti-}

ferromagnetic ^domains are, after carrying (22) ^and

(23) ^into (15)

The second term in the right hand side of _equa- tion (24) is just ^the antiferromagnetic contribution (16).

The first term is the resultant of all other contribu- tions arising ^{from the} applied ^{field d} and the others domains. As J(’) J, ^it appears that the _para-

magnetic regions ^{screen to a certain extent the A.F.}

domains from the applied ^field.

As the experimental quantities ^{x, pa are measured} during ^{an adiabatic} demagnetization, i.e. while varying

d at constant entropy rather than at constant tempe- rature, it is ^more convenient to express them ^as functions of S and J rather than f3 ^{and J. This is}

achieved directly by calculating S (f3, J), using ⁱⁿ (18) ^the equilibrium ^values x(fl, J) ^and p;.,(f3, J) ^and extracting ^{from it} f3 (8, J ).

The theoretical transition fields J(l) and j (2)

are plotted against the initial polarization (which ^is

a well-defined function of entropy) ⁱⁿ figure ^{4. The}

first remark about the phase diagram ^obtained by

the Weiss field approximation ^is that it is obviously

a poor approximation ^at low initial polarization.

The theory predicts incorrectly that, however small the initial polarization ^{of the} spins, adiabatic dema-

gnetization ^will produce antiferromagnetism. ^{This is}

related to the basic simplifying idea of the Weiss field approximation ^{which is to} neglect completely

the short-range correlations. The simplest way ^to take into account, ^at least partially, ^the short-range

correlations between spins ^{is to use the « restricted}

trace » approximation [6], ^which predicts ^the phase diagram ^also plotted ⁱⁿ figure ^4. The calculation

procedure ^is very similar to the previous one, the

only difference being that the restricted-trace _expres- sions are used for the _energy and the entropy ^instead of the Weiss field expressions (17) ^and (18).

A second remark concerns an important ^diffe-

rence between a dipolar ^nuclear antiferromagnet

and usual electronic antiferromagnets with short- range interactions. In the nuclear case the external

frequency Wext in a given domain, given by ^{the last} equation (14), ^{is not} simply ^the applied field J, ^because

of the long-range dipolar contribution from the spins

outside this domain. This dipolar contribution to the external field depends ^on the relative volume ^x of the two phases. ^{When the} applied ^{field is varied}

this relative volume varies so as to keep Wext constant;

at constant temperature, ^{what was a} single ^transition

field in an electronic Heisenberg antiferromagnet

is replaced ^{in a nuclear} dipolar antiferromagnet by

a transition field interval, ^between J c (’) and j c (2).

3. Experimental methods and results [7].

^{All the}

measurements have been performed ^{on a} spherical sample of calcium fluoride, ^{1.3 mm in} diameter, doped ^{with TM21 ions at} concentration Tm2 + /F -

of 1.6 ^x 10- 5. The fluorine spins ^are dynamically polarized through ^the Solid Effect with the Tm2 + in a field of 27 kG, ^{with a} microwave irradiation

frequency of 130 GHz. The maximum fluorine pola-

Fig. ^{4. - Field} polarization phase diagram of the A.F.

structure of figure ^{1. Broken curve : Weiss field} approxima-

tion predictions. ^{Solid curve : lst-order} restricted trace

approximation predictions. Solid circles : experimental

results from fast-passage measurements. Open ^{circles :} experimental results from 43Ca signal. ^{The « initial »} polarization used in the abscissa is the average of the polari-

zations measured before and after a « demagnetization- remagnetization » cycle ^{in order} to account for the non

adiabaticity ^{of the} complete sequence.

rization obtained in this sample ^{was of the order}

of 80 %. ^The sample ^is cooled down to 0.270 K

by pumping ^{over the} liquid ^’He bath in which it is

immersed, and the adiabatic demagnetization ^{in the} rotating ^{frame is} performed by applying ^{an r.f. field} H, t

30 mG, ^and sweeping ^the frequency ^{at a rate}

of 4 kHz/s. ^{The 19F N.M.R.} absorption signal ^is

detected with a Q-meter (107 MHz) ^{and the 43Ca} signal by ^{means of an} hybrid junction. ^{The two} signals

are recorded in a multichannel analyser during ^linear

sweeps of the magnetic field and then stored in a cal- culator where their areas and moments can be com-

puted.

The first reported experimental ^{results were obtained}

many years ago [1], ^when the existence of a mixed state was not suspected. They ^were derived from observations of the fast-passage signal which suffers

a qualitative change ⁱⁿ shape ^and amplitude ^{at a}

well-defined transition effective field L1 o. ^{Later on}

it appeared, by considering the existence of the mixed state, that this transition field 4 0 corresponds exactly

to the _upper critical field A(2) introduced above.

This method gives ^{then a measure of} ’A c (2) but reveals

nothing about the structure or even the existence of the mixed state. The results of this method are

plotted ^as solid circles in figure ^4.

1466

More recent experimental ^{results have} been obtain- ed by ^a completely ^{different method,} which consists in observing ^the absorption signal of nuclear probe spins (43Ca) during ^the demagnetization ^{of the fluo-}

rine spins.

Calcium fluorine contains, ^besides fluorine, ^calcium

atoms which form an f.c.c. lattice and are located at the centres of cubes of the fluorine lattice. A small fraction of the calcium nuclei (0.13 % ) ^{occurs as}

the magnetic isotope 43Ca, ^of spin 7/2, ^{whereas most}

nuclei are the spinless isotope ^{4°Ca. Because of their}

low concentration, the influence of the 43Ca spins

on the bulk properties ^{of the 19F} spin system ^is

negligibly ^{small, but} they ^{can be used as local} magne- tic probes ^since they ^sense the different dipolar ^fields

in the sample. However, they ^{need to be} highly polarized for their resonance signal ^to be observed with a sufficient signal-to-noise ratio. The solid effect used for polarizing ^{the 19F} spin ^is ineffective for polarizing ^{the 43Ca.} The latter are polarized by

thermal contact in the rotating ^frame [8] ^{with the}

cold dipolar reservoir of 19F spins. ^43Ca polariza-

tions as high ^as 90 % ^can be achieved in this _{way in}

a few seconds. A small r.f. field is then used to observe the 43Ca resonance signal during slightly saturating (9 %) ^linear sweeps of the external field (325 G/s).

The strongest experimental evidence for the actual existence of a mixed state is given by ^the following experiment. ^The system ^of highly polarized ^19F spins ^is demagnetized, ^{the A.D.R.F.} being stopped

at about 3 G from resonance; the polarized ^43Ca signal ^{is then observed to consist of two lines of} unequal intensities (Fig. 5). This observation is incon-

Fig. ^5.

^{43Ca N.M.R.} absorption spectra ^{as a function} of 19F effective field A : a) ^{for 4 »} y HL, ^{i.e. for a} purely paramagnetic ^{19F state;} b) ^{for an} intermediate fluorine effective field I AIT, I -- ³ G, ^{i.e. for} 19F in the mixed state.

43Ca signal ^is splitted ^{into two} components showing ^the

existence in the system ^{of two kind of} macroscopic domains;

c) ^{for A}

^{0, i.e. in a} purely antiferromagnetic ^{19F state.}

sistent with a structure of the 19F spins, ^either purely paramagnetic ^or purely antiferromagnetic. ^The average

dipolar ^{field at} the calcium sites is zero in both cases.

One observes indeed only ^one line when the fluorine system is either in a high ^field (paramagnetic) ^{or in}

zero effective field (A.F.).

By contrast, if the fluorine system ^{is in the mixed} state, ^the 43Ca spins experience ^non-zero dipolar

fields which are different in the paramagnetic ^and antiferromagnetic domains, ^thus giving ^{rise to the}

appearance of two resonance lines. These dipolar fields, hpara ^{and hA.F., can be} easily computed ^{from the} 19F dipolar frequencies (15), by noting ^{that the}

external effective field L1 concerns only ^{the 19F} spins,

and that the antiferromagnetic contributions (16)

vanish at the 43Ca sites, because of their location at the centre of the 19F cubes.

The line intensities are proportional ^to the relative volumes of the _{para and} antiferromagnetic domains,

that is x and (1 - x) respectively.

According ^to equations (25) ^{the centre of} gravity

of the spectrum ^{is not shifted}

Experimentally, ^{the 43Ca and 19F} absorption

N.M.R. signals ^are recorded for different values of the 19F effective field L1 during ^a fluorine o dema-

gnetization-remagnetization » cycle performed by

steps. ^{For each value of L1} we measure the relative

intensity ^{of the two 43Ca lines as well as} their shifts with respect ^{to the} single ^{unshifted 43Ca} line before 19F demagnetization. ^{The two 19F} critical fields

,A c (’) ^and A c (2) ^{are obtained} by extrapolation, ^{as the}

fields at which the intensity ^{of one} of the line vanishes.

The results of a typical ^{run are shown in} figure ^6.

The _open circles are the relative intensities of the

paramagnetic ^{line as a} function of the ’9F effective field. The crosses represent the shift of the two lines with respect ^to the unshifted paramagnetic ^line.

This set of results was obtained with an initial 19F polarization ^{of 77} %. After the remagnetization ^the polarization ^was 60%. The losses are due to the fact that the A.D.R.F. can not be perfectly ^adia-

batic [9] ^{and to} spin-lattice relaxation. In our experi-

mental conditions (H - ^{27 kG and T - 0.27} K) 7BD ¹⁵ min. This short relaxation time turns out to be (in ^{addition to the} poor signal-to-noise ratio) ^{one of the most} important difficulties in these

experiments.

By changing the initial polarization ^{one can in}

principle ^construct the entire field polarization phase

diagram.

Fig. ^6.

^43Ca lineshifts and relative intensities as a func- tion of the effective field on 19F spins. The initial fluorine

polarization ^{was 77} %. Open circles : relative intensity ^of

the Ca line corresponding ^{to the 19F} paramagnetic ^domains.

Crosses : 43Ca (A.F. ^and paramagnetic) lineshifts with respect ^{to the} paramagnetic line before fluorine demagne-

tization. The critical fields A c (’) ^and A c (2) ^are determined by extrapolation.

In practice, ^{in order to} determine the relative intensities of the two lines, ^their splitting hA,F. - hpara ^I

must be larger ^{than their} width, ^which limits the mea-

surements to low entropies ( pi > 60 % ).

The results are plotted ^as open circles in figure ^4.

Together with the results previously ^obtained by

fast _passage measurements they provide convincing

evidence for the actual existence of the mixed state

predicted ^on theoretical grounds. ^Within experi-

mental uncertainty the critical fields are in reasonable agreement ^{with the} predictions of the first-order restricted trace approximation.

4. Conclusion.

The occurrence of a «mixed»

state during ^the demagnetization ^{of a nuclear} dipolar antiferromagnet ^is clearly established by ^the experi-

ments described in this work. We have determined

a large portion ^{of the} field-entropy phase diagram :

in a large region ^{there is a} coexistence of _parama-

gnetic ^and antiferromagnetic ^{domains in the} sample.

It must be stressed that the theoretical curves as

well as the experimental results involve no adjustable parameters.

Although ^this phase diagram ^{is in some} way similar to that of a spin-flop phase ^{for a} Heisenberg

electronic antiferromagnet, ^{there are} important ^diffe-

rences arising ^{from the} long-range ^{nature of the} dipo-

lar interactions. For example, the existence of a large

domain of coexistence of two phases ^{in a} proportion

which is a well-defined function of the external parameters ^{is a} consequence of the long-range ^inter-

actions between different regions ^{of the} sample.

Acknowledgments.

^{We are} very much indebted to R. Bidaux who helped ^{us to correctly} identify

the nature of the mixed phase. This work has greatly

benefited from the stimulus and interest of Professor A. Abragam.

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