Nuclear dipolar magnetic ordering

NUCLEAR DIPOLAR MAGNETIC ORDERING

M. GOLDMAN Commissariat a l’Energie Atomique, Division de la Physique, Service de Physique du Solide et de Resonance Magnétique, CEN-Saclay, Boite Postale N°2,91190 Gif-s/Yvette, France

~1~C

NORTH-HOLLAND PUBLISHING COMPANY

—

AMSTERDAM

PHYSICS REPORTS (Section C of Physics Letters) 32, No. 1(1977) 1—67. NORTH-HOLLAND PUBLISHING COMPANY

NUCLEAR DIPOLAR MAGNETIC ORDERING M. GOLDMAN Commissariat

a I’Energie Atomique, Division de Ia Physique, Service de Physique du So/ide et de Resonance Magnétique, CEN-Saclay, Boite Postale N°2,91190 Gif-s/ Yvette, France Received September 1976

Contents: 1. Introduction 1 .1. Principle of production of nuclear magnetic ordering 2. Prediction of the ordered structures 2.1. The method of Villain 2.2. The method of Luttinger and Tisza 2.3. The stable Structures in calcium fluoride 2.4. The stable structures in lithium fluoride 3. Approximate theories of ordering 3.1. Weiss-field and high temperature theories 3.2. Spin-wave and random phase approximations 3,3. Restricted-trace approximation 4. Experimental investigation of antiferromagnetism 4.1. Production of ordering in calcium fluoride 4.2. Transverse susceptibility in zero field 4.3. Longitudinal susceptibility in zero field

3 4 9 9 12 13 16 17 17 19 20 22 22 25 28

4.4. Non-uniform longitudinal susceptibility of the fluorine spins 4.5. Spin-lattice relaxation 4.6. NMR investigation of lithium fluoride 5. Experimental investigation of ferromagnetism 5.1. Characteristic properties of the ferromagnet 43Ca resonance, evidence for ferromagnetism 5.2. The 5.3. Dipolar field as a function of energy 5.4. Reproducibility of the domains 5.5. Thickness of the domains 5.6. Transverse susceptibility in zero field 5.7. Ferromagnetic—anriferromagnetic transition 6. Experimental studies under way 6.1. Field-entropy phase diagram 6.2. Neutron diffraction 7. Conclusion References

Single orders for this issue PHYSICS REPORTS (SectionCof PHYSICS LETFERS) 32,No. 1(1977)1—67. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check. Single issue price Dfl. 26.—, postage included.

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M Goldman, Nuclear dipolar magnetic ordering

3

Abstract: This article reviews the subject of magnetic ordering in nuclear spin systems subjected to dipole—dipole interactions. This study has been developed so far in only one laboratory, the laboratory of Nuclear Magnetism at Saclay. The impetus was given by Professor A. Abragam, who invented a method for producing the very low temperatures necessary for achieving nuclear magnetic ordering [1,2]. The first experimental investigation, by M. Chapellier, Vu Hoang Chau and the author, was directed at producing antiferromagnetism in calcium fluoride. Later were involved J.F. Jacquinot, working on calcium fluoride, S.F.J. Cox — a threeyear visitor — and V. Bouffard, who looked for antiferromagnetism in lithium fluoride, W.Th. Wenckebach who during a one-year visit contributed to the study of ferromagnetisin in CaF 2 and to that of antiferromagnetism in LiF, and Y. Roinel who works with V. Bouffard on lithium hydride. Two technicians, J. Vaissidre and C. Pasquette, contribute to the experimental work. The study of nuclear magnetic ordering is still under progress, and many of the expected phenomena still await experimental verification. The decision as to what is the proper time to write a review article is then by necessity largely arbitrary. This article will essentially consist of two parts: a description of the principles and of the theoretical approximations used to study nuclear magnetic ordering, followed by a description of the experimental investigation already completed or under way. No attempt has been made to give detailed credit to every one for his particular contribution, since everybody has been involved in the building and clarification of the physical concepts and experimental ideas through innumerable discussions.

1. Introduction When a system of nuclear spins, subjected to dipole—dipole interactions, is cooled to a sufficiently low temperature, one may expect that it will undergo a transition to a magnetic ordered structure, in complete analogy with electronic spin systems subjected to Heisenberg interactions, whose magnetic phase transitions have been extensively studied for a long time. The production and study of magnetic ordering in nuclear spin systems raise however problems that differ widely from those encountered when studying the magnetic ordering of electronic spin systems. The first, and most obvious problem is that of producing the ordering, which requires exceedingly low temperatures, because of the smallness of nuclear dipole—dipole interactions. As a rough order of magnitude, the critical temperature T~is such that kB T~is comparable with the interaction energy between nuclei nearest neighbours. The latter is typically a few kHz and, since: kB —2

X

iO~MHz K-’

6K. The cooling technique for achieving such low we expect thatwill T~is the range to different 10— temperatures by in necessity be l0~ entirely from those currently used in cryogenics. Secondly, the magnetizations and energies involved in nuclear magnetic ordering being smaller, by many orders of magnitude, than those of electronic ordering, the methods of observation of the system will also be very different. As shown below, a great many of the observation methods use the technique of nuclear magnetic resonance. The great variety of measurement methods afforded by nuclear magnetic resonance have been developed initially for nuclear spin systems at high temperature. For the investigation of nuclear magnetic ordering, each one of these methods has to be reexamined critically and adapted to the low-temperature domain, and at times new measurement methods have to be devised. An important part of the theoretical effort must therefore be aimed at establishing a rigorous connection between the observations being made and the often indirect information they yield on the properties of the system. Finally, the approximate theoretical methods which are successful for describing nuclear magnetic ordering need not be the same as those used for electronic ordering. This results from the facts that dipolar interactions, which are long-ranged but do not have a constant sign, are very

4

M Goldman, Nuclear dipolar magnetic ordering

different from the usual short-ranged Heisenberg or Ising interactions; that the quantities which are measured are not the same for instance it is much easier to measure the entropy or the energy of the system than its temperature ; and that the accuracy of the measurements is in general much less than for electronic spin systems. In particular it would be of little interest in the present state of the study to concentrate on spin-wave excitations or on critical exponents. It is worth noting that the dipolar Hamiltonian is known with certainty, without any adjustable parameter, so that the study of nuclear magnetic ordering yields in principle a very direct test of the validity of the statistical theories used to describe it. This advantage is partly mitigated by the fact that the samples always contain a small percentage of paramagnetic impurities whose perturbing influence cannot be neglected. In the rest of this introduction we present the method of production of nuclear magnetic ordering and we discuss some of its features. —

—

1.1. Principle of production of nuclear magnetic ordering

In order that magnetic ordering takes place in a nuclear spin system it is necessary both that the temperature be low, as seen above, and that the external field be small. If the field is too high, the state of the nuclear system is determined by the Zeeman interactions of the spins rather than by their dipolar interactions, and the system is paramagnetic. The critical field is of the order of a few gauss. The problem of cooling the system to a low temperature in a low field is solved by a two-step process, which is more easily discussed in terms of entropy rather than temperature. The first step is achieved in a high external field. It consists in decreasing the entropy of the nuclear system to a very low value, through the increase of its polarization. In high field, when the state of the system is determined essentially by its Zeeman interaction, the entropy depends only on the polarization. For spins for instance, the entropy is equal to: ~,

S

kBN{in 2— ~[(1 +p) in (1 +p) +(l —p)ln (1 —p)]},

(1)

where N is the number of spins. It varies from kBN ln 2 to 0 when the polarization varies from 0 to 1, so that increasing the polarization corresponds indeed to decreasing the entropy. The practical method for increasing the polarization, known as the “solid effect”, is described below. The second step for producing nuclear ordering consists in removing the unwanted high field while preserving the low value of the entropy achieved in the first step. That is, it is an adiabatic demagnetization of the nuclear spins. The low entropy of the system at the end of the process corresponds to a low temperature. This low temperature is imparted only to the nuclear spins, and not to the other degrees of freedom of the sample (the “lattice”), which remain at relatively high temperature. Two conditions are then required for the production of nuclear ordering: 1) The nuclear spin system must be very loosely coupled to the lattice: its spin-lattice relaxation must be long in order that, once cooled, enough time is available for observing its properties before it warms up above the transition. This requirement can be met by a proper choice of the experimental conditions. 2) When isolated from the lattice the properties of the nuclear spin system must be describable in terms of temperature. The assumption that such a description is valid is the basis of the spin temperature theory 1131. Introduced many years ago in nuclear magnetism as a hypothesis, the concept of spin temperature has been firmly established by a great number of experiments and is

M. Goldman, Nuclear dipolar magnetic ordering

5

now a standard tool for the practice of nuclear magnetic resonance in solids. Its failure to describe physical phenomena is limited to very special circumstances [4] and does not disprove its validity under most experimental conditions. Since however most if not all investigations of the spin ternperature concept had been performed in the high temperature domain, one had to assume that it remained valid at very low temperature for the present study. The two steps of the nuclear cooling: dynamic polarization and adiabatic demagnetization are now described in turn. 1.1.1. Dynamic nuclear polarization The dynamic polarization of nuclear spins by the method of the “solid effect” is extensively used for producing polarized targets for nuclear or high-energy physics experiments. Its description can be found in a number of places [5—71.We will content ourselves with a brief sketch of its principle. We consider for simplicity a solid containing nuclear spins I = ~ plus a small proportion of electronic spins S of Larmor frequencies w1 and w5, respectively. In the absence of interactions ~,

between spins I and S, the only transitions that could be induced by an rf field would either be purely electronic (~S~ = ±1, ~ = 0), at the frequency w~,or purely nuclear (~S~ = 0, ~ = ±1), at the frequency w1. However, the dipolar interaction between spins I and S mixes their states, and makes it possible to induce transitions at the frequencies w~±w~,which correspond to the mutual flips of an electronic and a nuclear spin. One of these transitions flips the two spins in the same direction (flip—flip), and the other in opposite directions (flip—flop). The mutual-flip 6. The polarizationtransischeme tion rates depend strongly on the distance between spins: they vary as r is based on two properties: i) It is possible to choose field and temperature conditions such that at thermal equilibrium the electronic polarization is very close to unity, whereas the nuclear polarization is very small. Consider for instance Tm2~ions (g = 4.83) in CaF 2, in a field of 25 kOe and at a temperature TL = 0.3K. We have then MHz, 130 GHz, and the thermal equilibrium polarizations of 2~spins and w1 the 19F100 spins are respectively: the Tm ~ = tanh(hws/2k TL) (1 _7) X 10’°, peq. = tanh(hw 1/2k TL) 0.008. .‘-,~

—

ii) The electronic spins are much more strongl3i coupled to the lattice than the nuclear spins, and the electronic spin-lattice relaxation time T15 is much shorter than the nuclear one, T,1. For an idealized description, we assume that p~q = 1, pq = 0, and T,1 We suppose that initially all spins S are pointing up, and that equal proportions of spins I are pointing up and down. We apply an r.f. field inducing flip—flop transitions. The polarization process goes as follows. 1) The system absorbs an r.f. photon. One spin S flips down while a nearby spin I which was pointing down flips up. 2) The spin S flips back up by spin-lattice relaxation, whereas the orientation of the spin I which has justed flipped is “frozen”. 3) Through absorption of another photon, the spin S flips down again while another spin I which was pointing down flips up. Etc... The continuation of such processes results in a polarization of the spins I in the vicinity of the spins 5, where the flip—flop transition rate is fast. This polarization diffuses toward the distant ~.

6

M Goldman, Nuclear dipolar magnetic ordering

nuclear spins through nuclear flip—flop transitions mediated by the nuclear dipole—dipole interactions. Thanks to the nuclear spin diffusion, the dynamic polarization works effectively with a concentration of electronic spins much less than that of the nuclear spins. In an ideal case, the steady-state nuclear polarization is equal to ~ or to _p~, depending on whether the irradiation frequency is that which corresponds to flip—flop, or to flip—flip transitions, respectively. The discussion of the various limiting factors to the nuclear polarization, as well as of a more sophisticated model, are outside the scope of the present article. As a practical example, in calcium fluoride doped with Tm2~ions at a concentration of 1.2 X l0~ relative to ‘9F spins, and with the following conditions: field H 27 kOe irradiation frequency w 130 GHz sample temperature TL 0.7 K, the fluorine polarization P 1 90% is obtained after approximately 3 hours of irradiation. — — —

1.1.2. Nuclear adiabatic demagnetization Once the nuclear spins are polarized, we cut off the microwave power. We assume that we can forget the paramagnetic impurities, which are at low concentration, and the nuclear spin-lattice relaxation, which is supposed to be very slow. We are then left with an isolated nuclear spin system at low entropy in a high magnetic field, which we want to subject to an adiabatic demagnetization. There are two methods for performing nuclear adiabatic demagnetization: in the laboratory frame and in the rotating frame, which we describe in turn. The adiabatic demagnetization in the laboratory frame corresponds simply to a slow decrease of the applied magnetic field to zero. The demagnetization is adiabatic if the system is at all times very close to equilibrium, that is if its density matrix is always of the form: o

=

exp(—~3~lC)/Tr{exp(---j3lC)} ,

(2)

where j3 is the inverse spin temperature of the system and 1C its Hamiltonian: ~C=—yHIz+~KD=woIZ+~KD (expressed in frequency units). w~I~ is the Zeeman interaction and ~CD the dipole—dipole interactions: 2h ~ 3J~ J~ D ~ (~ r. l~y 1)(1 r11) ‘~i r11 where r.1 is the distance between spins I~and I~,and is the unit vector along the direction joining them. The variation of temperature during the demagnetization is obtained by writing that the entropy S is constant, where: -~

5—kBTr(olnu),

~—

(4)

and using eq. (2) for a. It is particularly simple in the high temperature domain to show that adiabatic demagnetization corresponds to a cooling. To the lowest order in 13 we have, according to eqs. (2) and (4):

M Goldman, Nuclear dipolar magnetic ordering

7

k~f3~ Tr(~C2)+ Ct. kB132 {w~Tr(I~)+ Tr(~C~)} + Ct.

S=

—~

=

—~

=

—~k~132 Tr(I~)X (w~+D~)+ Ct.

(5)

where we define the local frequency DL in the laboratory frame through: =

Tr(lC~)/Tr(I~).

(6)

The constancy of S yields: /3

cx

(w~+D~~”2.

(7)

Whence, starting from a high field (w 0 Pfinal/I~initiaI

=

wo/DL

~‘

~‘

DL)

and demagnetizing to zero field we obtain:

1

It is only when w0 is not too large compared with DL that the achievement of equilibrium is fast. The rate of thermal mixing betweenwZeeman and dipolar interactions, which is of the order 1 (typically 10 to 100 ~isec)when of Dj 0 ~ DL decreases very steeply when w0 is increased. In high field, that is when w0 DL’ it is legitimate to truncate ~ so as to retain its secular part ~ that is that part which commutes with I~: ~‘

A [2I~’II

=

—

I~I~ I,~I~I —

,

with

2 011) rj~3, 3 cos is the angle between r 11 and the external field H. We use then instead of the full Hamiltonian the following truncated one:

A11 = where

!y2~(1

(8) (9)

—

(10) which is a sum,of two quasi-constants of the motion, w0I~and ~ It is assumed, and well verified in the high-temperature domain, that both acquire separately a spin temperature in a time T2 Dj’, i.e. that the density matrix becomes of the form: aexp(—aw0I~ —j3lC~)/Tr{exp(...)}.

(11)

The coefficients a and /3 are the Zeeman and dipolar inverse temperatures, respectively. One can establish a thermal contact between these terms by irradiating the sample with an r.f. field H1 perpendicular to H and rotating with a frequency w close to the Larmor frequency w~. In a frame rotating with frequency w, the rate of change of the density matrix:

o

=

exp(iwl5t) a exp(—iwI~t),

is given by: dO/dt=_i[~C*,O]

,

where ~1C~ is the effective Hamiltonian:

(12)

8

M. Goldman, Nuclear dipolar magnetic ordering

(13) with = w. When ~ is not too large, ~C’~is a low-field Hamiltonian. Since I~commutes neither with I~nor with ~ there is no obvious breakdown of ~IC~into a sum of constants of the motion. We extend to that case the spin temperature hypothesis, i.e. we assume that the density matrix in the rotating frame evolves toward the equilibrium form: ~

0

=

—

exp(__/3~C*)/Tr{exp(_/3~JC*)} ,

(14)

where ~3is the inverse spin temperature in the rotating frame. The adiabatic demagnetization in the rotating frame is performed by a familiar fast passage. In complete analogy with the adiabatic demagnetization in the laboratory frame, it works as follows. One first applies an r.f. field at a large distance from resonance, and then sweeps slowly the field toward resonance. If the field sweep is sufficiently slow, eq. (14) is nearly satisfied at all times and the entropy remains constant. By analogy with eq. (7) we have, in the high-temperature limit: 2)”2, (15) /3cr(~2 +w~+D where w 1 = —-yH,, and the local frequency D in the rotating frame is defined through: Tr(~JC~2)/Tr(I).

(16) D The longitudinal demagnetization (z~= 0) can be followed by a transverse demagnetization, by decreasing w, to zero. When o., = 0, it is unnecessary to use a rotating frame. As viewed from the laboratory frame, the effect of this demagnetization is to make a = 0 in eq. (11). If the demagnetization is started with z~~D, w 1, we have, according to eq. (15): 2

z.~/D 1 . (17) In practice, the demagnetization is not completely adiabatic, but the increase of entropy can be kept small. As an example, a complete fast passage (i.e. adiabatic demagnetization followed by remagnetization) in calcium fluoride with the external field parallel to a four-fold axis, performed with an r.f. field amplitude H, = 30 mG and with a sweep rate dIJ/dt = 1 G/sec, results in a relative decrease of the polarization amplitude of about 13%. At the time of writing, the experimental study has been limited to adiabatic demagnetizations in the rotating frame. The study of nuclear magnetic ordering, produced as described above, has the following features. 1) The most directly accessible parameter is not the temperature of the system, but its entropy. As shown below, it is also easy to measure the energy in zero effective field. Both quantities are very sensitive to short-range correlations, which are difficult to calculate accurately by any theory. 2) The nuclear spin system being isolated from the lattice, and its energy spectrum being bounded upwards as well as downwards, it is possible to choose the sign of its temperature to be either positive or negative by starting the fast passage with the effective field parallel or antiparallel to the magnetization, respectively. At zero positive temperature, the state of the system is that of lowest energy, whereas at zero negative temperature it is that of highest energy. These states correspond to different ordered structures. 3) After adiabatic demagnetization, the Hamiltonian of the system is ~fCDif the demagnetization !3final//3inht

=

~“

M Goldman, Nuclear dipolar magnetic ordering

9

is performed in the laboratory frame and it is JC~if it is performed in the rotating frame. The form of the truncated Hamiltonian 3Cj, depends through the coefficients A11 (eq. (9)) on the orientation of the external field with respect to the crystalline axes. By changing this orientation it is possible to produce different kinds of ordering. Even at those orientations of the field for which the ordered structure in the rotating frame is the same as in the laboratory frame, the different Hamiltonians in both cases yield different properties. In particular it will be shown that the transition entropy is higher in the rotating frame than in the laboratory frame, i.e. the initial polarization required to produce ordering is lower. 4) When studying magnetic ordering in the rotating frame, since a large external magnetic field is still present, it is possible to use nuclear magnetic resonance methods to observe the properties of the system, and to benefit from the sensitivity and versatility of these methods. Furthermore it is in general easier to have a long nuclear spin-lattice relaxation time in high field than in zero field. 5) Finally the approximate theories used to describe nuclear magnetic ordering turn out to be somewhat simpler with the truncated Hamiltonian JC, than with the full dipolar Hamiltonian ~CD.

2. Prediction of the ordered structures There exists no rigorous theory for the prediction of the ordered structures corresponding to a given Hamiltonian, and approximations are used, whose validity must be checked by experiment. We use two different approaches, the method of Villain and the method of Luttinger and Tisza. Their application to nuclear dipolar magnetic ordering is described in refs. [81 and [9], respectively. We give here only a simplified description of these 9F methods. consider first aThen system spins in We calcium fluoride. weof identical spins with the eventual specialization to the ‘ consider briefly how the theory is modified when there are several spin species, with the case of lithium fluoride as an example. We limit ourselves throughout to the truncated Hamiltonian ~ ~,

2.1. The method of Villain [10] This method is based on the local Weiss-field approximation, i.e. on the neglect of short-range correlations. The energy corresponding to the Hamiltonian (8) is: (~C~)~ ~

(18)

,

where (...) are thermal equilibrium averages at a given spin temperature. We replace the rigorous eq. (18) by the approximation: ~

.1A 11[2(I~)(I~1)_(I~Up_c12Up]

,

(19)

which can be written: ~

w~(I~),

(20)

10

M Goldman, Nuclear dipolar magnetic ordering

where w1 is the Larmor frequency corresponding to the local Weiss-field experienced by the spin i. It is equal to: (21) that is, according to eq. (19): =

—

=

—

~

A11(I~)

~

A,~1(I~)

w~ ‘2 ~I~A11I~>.

(22)

The next step is to assume that (Ii) is the thermal equilibrium value in the field H, spins this yields: (I,> = —-i- ~ tanh(~f3Iw1).

=

—w~/y.For

~,

(23)

2 Iw,I (23), together with (22) represents a system of 3N equations, for the components of the N spins of the sample. For every value of inverse temperature j3, there are a number of solutions. The only acceptable solutions are those for which (Ii), (J~>and KI~)are real numbers. The stable solution must be selected among these according to a thermodynamical stability criterion. This crjterion is slightly more elaborate than usually, because of the possibility of negative temperatures. The procedure is as follows: 1) We choose the zero of energy to be that of the system at infinite temperature. Systems at positive temperature then have negative energies, and systems at negative temperature have positive energies. 2) The stable state has to be determined independently for positive temperatures among the states with negative energies, and for negative temperatures among the states with positive energies. 3) Within each set, the stable structure is that for which at constant temperature, the free energy has maximum absolute value (F < 0 at T> 0 and F> 0 at T < 0. Both cases correspond to —/3F = S f3E maximum), at constant energy, the entropy S is maximum, at constant entropy, the energy E has maximum absolute value. The energy E is given by eq. (19) and the entropy is to within a factor kB given by: —

—

— —

S

~ {ln 2— ~[(1 +p~)ln(1 +p~)+(l —p~)ln(1 —p~)]}

,

(24)

where p, is the modulus of the polarization p~= 2(1,). The only trouble is that solving eqs. (23) is so formidably complicated as to be well-nigh impossible. The “second-best” procedure adopted by Villain is to solve these equations in the immediate vicinity of the transition. The polarizations being there very small, one can replace the hyperbolic tangents by their arguments and obtain linear equations:

M Goldman, Nuclear dipolar magnetic ordering

11

(25) or else X(I~),

(26)

X=—4I/3~.

(27)

(01 =

with

In this limit we have: SNln2—~~p~,

(28)

and: E~ ~w1•(I1)~X~p~

(29)

=~X(S—Nln2).

(30)

According to the third form of the stability criterion, the stable tructures are: at T> 0 the one for which A is minimum at T < 0 the one for which A is maximum. Equations (26) are solved by using space Fourier transforms: 2 ~RI~> exp(ik r,), 1(k) = N”

(31)

A(k) = ~A 11exp[ik

(i-,

—

r1)]

(32)

,

where r, is the vector joining the origin (taken at a lattice site) to the spin i, and k is a vector belonging to the first Brillouin zone of the nuclear lattice. Eq. (32) is meaningful only if the sum is independent of subscript j. As discussed in ref. [8] this is not true when k~ is comparable with the sample size. These “pathological” k values correspond to a very small volume of the Brillouin zone and we assume that they can be neglected. We choose the sample shape to be an ellipsoid, in order~that eq. (32) be meaningful for k = 0. Equations (26) become, according to eqs. (22), (31) and (32): [—A(k) A]I~(k)= 0, —

[—A(k) A]I~(k)= 0, —

[2A(k)

—

X]I~(k)= 0.

(33)

The 3N solutions of these equations are discussed in ref. [8]. Here, we limit ourselves to the longitudinal structures, for which ~~ip= (17 = 0. For each vector k = k0 we can build a longitudinal structure by taking: I~(k0)= I;(—k0) ~ 0, and I~ (k)

=

0 for k

A=2A(k0).

±k0.The corresponding coefficient A is equal to: (34)

12

M. Goldman, Nuclear dipolar magnetic ordering

The stable structures at positive and negative temperature correspond to A(k0) minimum and maximum, respectively. The next problem is to find out how the structure which is stable near ~ is modified the 3~is when such that: temperature is lowered. The only tractable case is that when the structure stable at / 2l(1~)I p. p is independent of subscript i. Such a structure satisfies eqs. (23) at all temperatures, to within a scaling of the value of p. As shown below, the stable structures derived for CaF 2 are of this “permanent” type. We assume that such a structure, which was shown to be stable at the transition, remains stable at lower temperatures. That such a guess is reasonable is substantiated by an alternative approximation method described below. 2.2. The method of Luttinger and Tisza We look for the stable structure when the temperature is zero. Disregarding the case when the dipolar field at the sites of some nuclei would be zero, the polarization of each spin is equal to 1. The entropy being zero, the stable structure (at positive temperature, say) is that which minimizes the energy: (35) where o~is given by eqs. (22), while satisfying the N conditions: (J) (Ii) = .

(36)

,

which we call the “strong constraint”. This problem is formally solved by the Lagrange-multiplier method, which yields: (37)

(O.A1(1,).

Solving these equations with multipliers A, different for each spin is an impossible task. To by-pass this difficulty, we use the ansatz of Luttinger and Tisza: we minimize the energy (35) under the condition, much less stringent than eqs. (36), that:

~II~ (I,)

(Ii)

=

~N,

(38)

which we call the “weak constraint”. Since we have only one condition, we use only one Lagrange multiplier and we obtain: =

A(1,)

,

(39)

which is identical with eqs. (26). The stable structure corresponding to the weak constraint is therefore the same as in the preceding section. Now, since the weak constraint (38) satisfies the conditions (36), we are sure that the stable solution of eqs. (39) has an energy lower than, or at most equal to that of the stable solution of eqs. (37). As a consequence, if the stable solution of eqs. (39) (weak constraint) happens to satisfy eqs. (36), we know that it is also the stable solution corresponding to the strong constraint. The

M Goldman, Nuclear dipolar magnetic ordering

13

conclusion is the same for the stable structure at negative temperature. To summarize, what we have proved within the two approximation methods used here is that if the structure which is stable at the transition is of the “permanent” type (as defined above), it is also stable at zero temperature. There is then a good probability that it is also the stable structure at intermediate temperatures. We recall that the proof holds only when the system contains one spin species. Even then, we will see later that in some cases the short-range correlations can cause a first-order transition between two different ordered structures. 2.3. The stable structures in calcium fluoride The Fourier transforms of the dipolar interaction coefficients in various cubic lattices have been tabulated for a number of points k regularly spaced in the Brillouin zone [11]. This enables one to calculate the A(k) for all orientations of the external field. In a simple cubic lattice of spins such as the 19F spins in CaF2, the preceding theory predicts, ~,

depending on the field orientation and of the sign of the temperature, the occurrence of five different ordered structures. Three are antiferromagnetic, one is ferromagnetic with domains, and the fifth one is also probably ferromagnetic, but its exact nature is not yet firmly established. They are listed below for typical orientations of the external field H0, and shown on fig. 1.

Structure I 1>0

Structure ~ 1>0

Structure ~E 1<0

Fig. 1. Predicted antiferromagnetic structures in a simple cubic lattice of spins ~ subjected to truncated dipole—dipole interactions.

a) Positive temperature Structure I, antiferromagnetic. H0 ii [001]. —

=

p exp(ik, r,)

=

(ir/a, ir/a, 0),

with k,

where a is the lattice parameter, 2h/(2a3) = ~A A(k1) = —5.352 y 1 7KinCaF = —hA,/(4k~)=3.4X lO 2 It is a two-sublattice antiferromagnetic structure where successive planes perpendicular to [110] carry magnetizations alternatively parallel and antiparallel to [001], the direction of the d.c. field. Structure II, antiferromagnetic. H0 111110]. —

14

M Goldman, Nuclear dipolar magnetic ordering

k11

(0, 0, ir/a)

=

2h/(2a3), A(k,,) = —4.843 y T~ 3.06X lO7KinCaF 2 Successive planes perpendicular to 110011 carry magnetizations alternatively parallel and antiparallel to [110]. b) Negative temperature —

Structure III, antiferromagnetic. H0 1110011.

k,11

(0, 0, ~r/a), 2h/(2a3), A(k111) = 9.687 y = —6.13 X 107K in CaF =

k,,

=

2 Successive planes perpendicular to [0011 carry magnetizations alternatively parallel and antiparallel to [0011. A(k111) differs from A(k11) despite the fact that k111 = k,1, because the orientation of H0 is not the same in both cases. Structure IV, ferromagnetic. H0 II [111]. The way by which we reach the conclusion that this structure is ferromagnetic deserves some comments. For this orientation of H0, the maximum and minimum values of A (k) correspond to small values of k, that is k a ~ 1, where a is the lattice parameter. We still limit ourselves to k~ much smaller than the sample dimensions in order that A(k) be well defined. It can be shown that when k is small, A(k) depends only on the angle °k between H0 and k, and neither on the magnitude of k nor on the orientation of the crystalline axes. It is given by: 2Ok —1). (40) A(k) 1’[_1!~ ~ (3 cos 2a3 3 Its extremum values are: —

A(k)max

=

87r ‘y2h ~

for k II H

-~

0

2a A(k)

2~

A .

mm.

=

—

~

3

~

2a

~

for k

I

H

(41)

0

We obtain a solution of eqs. (26) with A maximum, i.e. acceptable at T < 0 near the transition, by taking (I~(r))cx exp(ik0 r) .

,

(42)

where k0 is any small vector parallel to H0. This solution however is not “permanent”, insofar as it does not satisfy eqs. (23) well below the transition. This difficulty is by-passed by noting that the solution (42) is degenerate with all those corresponding to vectors k II H0 and small. In particular we can use a linear combination of solutions corresponding to multiples of k0, of the form:

M Goldman, Nuclear dipolar magnetic ordering

(J~(r)>cx

sin[(2n

~ n0

+

2n+l

15

l)k0 r}.

If the sum were extended to n oo, this would correspond to a square-wave modulation of (Jr), that is a permanent structure consisting of ferromagnetic domains in the form of slices perpendicular to H0, of width d = ink0. In fact, the sum must be restricted to values of n such that (2ii + 1) k0a ~ 1. This corresponds to a rounding-off of the variation of at the edges of the domains, on a distance comparable with a, the lattice parameter. Since d a, we have then constructed a quasi-permanent structure satisfying the stability criterion, which is a ferromagnetic structure wi~thdomains, whose domain walls are thin and may depend on temperature. In order to decide whether we can have a single-domain ferromagnet, it is necessary to consider the values of A(0). They are the following, for typical ellipsoidal sample shapes: Sphere: -+

~~Z> ~‘-

—

A(0)=0. —

—

(43)

Infinitely long needle parallel to H0: 4 2~ 3 3 2a Infinitely flat disk perpendicular to H

(43’) 0:

3 3 2a We conclude from this that at T < 0, an infinitely flat disk perpendicular to H

(43”)

0 might give rise to single-domain ferromagnetism, but that for oiher shapes, the ferromagnetic structure will necessarily be composed of domains. The structure V concerns the case of positive temperature when H0 is around [111]. By analogy with structure IV, we can construct an acceptable ferromagnetic structure whose domains consist of thin slices parallel to H0. This structure is degenerate with respect to the orientation of the slices, and also with other structures where the transverse magnetization would be non-zero. This case has not yet been studied experimentally and we will leave it aside. The domains of stability of these structures, derived from the variation of the A (k)’s with the orientation of H0, are shown on fig. 2.

a)

1>0

b)

1<0

Fig. 2. Stable structures in a cubic system of spins with truncated dipole—dipole interactions as a function of orientation of external field with respect to crystalline axes. Structures I—Ill are those of fig. 1. Structure IV is a ferromagnet with domains. Structure V is not analyzed.

16

M Goldman, Nuclear dipolar magnetic ordering

2.4. The stable structures in lithium fluoride We give only a very brief discussion of how the theory is modified when the system contains two spin species of different gyromagnetic ratios and different spins (~for ‘9F and ~ for 7Li). We call I the 19F spins and S the 7Li spins, and we forget the existence of ‘-~7%of the isotope 6Li, for simplicity. The foregoing discussion, specialized to the case of LiF, is also valid for compounds such as NaF or LiH. The local Weiss-fields experienced by individual spins I, and S,~,w~and w~,respectively, depend on the polarizations of both spins I and S. In the vicinity of the transition we have: (Ii)

=

—/3~w~I(I+ 1)73

—~J3~w~

(1= ~) (S~).

If we write: (0~ =

A~(1~);

(0

=

As(S)

(44)

,

we have then A 1 = 5A5. Eqs. (44) are solved as before through the use of space Fourier transforms. If N is the number of spins I (or S) the energy and the entropy are respectively: E/N=~~I~(w~-(J,)+w~ .(Sv))

2+~l(S~,)I2)

=~A1~I~ (I(I~)l

S/N=Ct. —2~ (](J)]2 +~I(S~)l2). The stable structures at positive and negative temperatures correspond then respectively to A 1 minimum and maximum. On the other hand, at zero temperature, we use two different conditions for the weak constraint: ~ (J). (1,~)= N/4,

~ KS) KS)

=

9N/4.

We must use two Lagrange multipliers when looking for the maximum and the minimum of the energy and we obtain, in place of eq. (39): 1,) ; ~ = A = AJ( 5(S). The ratio AJ/A5 has no reason to be the same as near /3~ and it depends on the structure. The energy is equal to: E/N(A1

+

9X5)/8 -

In contradistinction with the case when there is only one spin species, the problems are not the same near /3c and at 13 = oo~The reason is that for a given structure the ratio V1,)I/I(S)I is different in both cases. A consequence of this fact is that for a given orientation of H0 the structure which

M. Goldman, Nuclear dipolar magnetic ordering

17

is stable near /3~is not necessarily the same as that which is stable at zero temperature. The calculation predicts for LiF the same ordered structures as for CaF 2 in roughly the same domains of orientation of H0. This is a reasonable result since LiF consists of a simple cubic array of magnetic moments of comparable magnitude: Iy1]Sy~ 0.8. However, in the antiferromagnetic structures, each sublattice contains both spins I and spins 5, whose polarizations are not the same and vary differently with temperature. Such structures might be called “antiferrimagnetic”. Similarly, the domains of the ferromagnetic structures contain both spins I and S. Another difference with the preceding case is that the field orientations which correspond to the transition between different structures depend on the entropy of the system. That is, for given orientations one predicts the occurrence of a first-order transition between two ordered structures, besides the transition between paramagnetism and ordering. The discussion of this point will not be pushed further.

3. Approximate theories of ordering We sketch the principle of several approximations used to describe the properties of the ordered structures in their range of existence. These approximations are standard and their description will be very brief. Their predictions for specific properties will not be given in this section, but rather when we describe the experimental studies of these properties. These methods are: the Weiss-field approximation, supplemented with the high-temperature approximation for spin-temperature theory, the spin-wave and random phase approximations, and the restricted-trace approximation. As an example, we will treat the case of an antiferromagnetic structure in CaF2. 3.1. Weiss-field and high temperature theories [81 The Weiss-field approximation is the simplest approximation for magnetic ordering. It is a very useful guide for the qualitative properties of ordered structures. When it is notably insufficient, we supplement it in the present section with a high-temperature expansion for spin-temperature theory. We consider an antiferromagnetic structure with two sublattices .cz( and ~3. This structure is defined by a vector K0: when the origin is taken at a lattice site of sublattice, exp(iK0. r,) equals +1 or —l depending on whether the spin i belongs to sublattice .ss( or ~3respectively. All spins in .9! have the same vector polarization PA (that and allis spins in ~3 the same polarization toA and ~l3 the precession frequencies in thePB~ Weiss fields) are, The Weiss frequencies according to eqs. (22), (31) and (32): X

WA ~A

~

A(V ~

=

4 =

~

A

B\

o.”.P~ PxJ —

I Arn\( —

—

WB(PA,PB)’

B

4~’~’)~PxPx

—~A(K 0)(p~ p~) ~A(0)(p~

w~

A +

—

+

p~)

A(K0)(p~—p~’)+~A(0)(p~ +p~) WA(PB’PAL

(45)

18

M. Goldman, Nuclear dipolar magnetic ordering

If the precession frequency corresponding to the external field is ~°e’ the spins frequencies are: 0~0e+0)A

We have, according to eq. (23): tanh (~/3I~I), PB = 1~Al The energy and the entropy are respectively: PA

———~—

—

~

S

~ =

~ N{s(pA) + S(PB)}

B

1

tanh (~PIw~D.

(46)

‘

(47)

‘~B

PB)}

(48)

,

where s(p) is the entropy per spin of polarization p in the Weiss-field approximation (eq. (24)). Eqs. (45) to (48) describe all properties of the ordered structure. An obviously incorrect prediction of the Weiss-field approximation is that of the transition entropy. To see this let us recall the process of production of ordering. We start in high field; all spins are parallel, they have the same polarization p 1, and the entropy is: S1

Ns(p~).

After adiabatic demagnetization to zero field, we have PB function of polarization, the final entropy is

=

~PA

and, since the entropy is an even

Sf =Ns(pA). Since the demagnetization takes place at constant entropy, we must have PA = p~.The theory predicts incorrectly that, however small the initial polarization of the system, adiabatic demagnetization will produce antiferromagnetism. This can be corrected only by taking into account the entropy associated with short-range order. The simplest (and rather crude) procedure is the following. i) We take for granted the critical inverse temperature predicted by the Weiss-field theory: =

—2/A (K0) -

ii) We use the high-temperature approximation from spin temperature theory (eqs. (1 5) and (17)) to calculate the initial polarization ~ = ~~/3init ~init that will yield the value ~ for /3final after adiabatic demagnetization to zero field. We obtain thus a “critical initial polarization” p0, related to the critical entropy through eq. (1). The result is: p0D/IA(K0)l.

(49)

However approximate, this result tells us grossly how much initial polarization is required to produce ordering by adiabatic demagnetization. The predicted values of p0 for structures Ito IV are listed below. Structure I. H0 [001], T> 0. 2h/(2a3), A(K0) = —5.3 52 y D3.l6~y2h/(2a3), —

M Goldman, Nuclear dipolar magnetic ordering

19

p0=0.59.

:1.

Structure II. H0 [1101, T> 0. 2h/(2a3), A(K0) = —4.843 y D l.96’y2h/(2a3), —

p 0

=

0.405.

Structure III. H0 [001], 2h/(2a3) A(K0) = 9.687 y D=3.16y2h/(2a3), —

T< 0.

p 0

=

0.326.

Structure IV. H0 [ill], 21~/(2a3) A(K0) = 8.378 y D= 1.31 y2h/(2a3), —

T< 0.

p 0

=

0.156.

According to these figures, structures III and IV are the easiest to produce. It is to these structures that the experimental studies in CaF2 have been almost entirely devoted so far. Note: The preceding results correspond to ordered structures produced by adiabatic demagnetization in the rotating frame. When the adiabatic demagnetization is performed in the laboratory frame, the theory of section 2 predicts that the ordered structures at positive and negative temperatures are identical with structures I and III, respectively, and correspond to the same values of A(K0). The local frequency D1 in the laboratory frame is however larger than D, because ~ D consists of~C~ plus extra terms. The calculation yields:

2h/(2a3),

DL = 5.02 7 whence, by analogy with eq. (49) p 0

DL/IA(Ko)I

=

0.518 0.938

for T< 0, for

T>0.

To produce the same ordered structure, it is necessary to start from a much higher polarization when performing the demagnetization in the laboratory frame than in the rotating frame. 3.2. Spin-wave and random phase approximations [9] These well-known approximations are much better than the Weiss-field approximation at very low temperature. The domain of temperatures where they are useful is however not attainable by experiment in the present state of the art. We have used the spin-wave approximation to check the consistency of the structures predicted at zero temperature, and the random-phase approximation to see how well do other approximations extrapolate to low temperature.

20

M. Goldman, Nuclear dipolar magnetic ordering

In zero effective field, the Hamiltonian ~ tor I~in the Heisenberg representation is:

is given by eq. (8). The rate of change of the opera-

(50)

~JI~ ~

The gist of the spin wave approximations for antiferromagnets is to replace I~and I~by +~ or depending on the sublattice to which these spins belong. The linearization of the equations of evolution of the operators 1~makes it possible to define elementary excitations, approximately boson-like and called spin waves, and to derive the sublattice polarizations from the thermal equilibrium populations of the elementary excitations. It is a peculiarity of antiferromagnetism that even at zero temperature, the sublattice polarizations deviate from complete alignment of the spins in each sublattice. In order for an antiferromagnetic structure, derived in the preceding section to be acceptable in the spin wave theory, it is necessary that all elementary excitation energies be of the proper sign: positive for the stable structure at positive temperature and negative for the stable structure at negative temperature, and also that the spin deviation at zero temperature be small. The calculations have been performed for the antiferromagnetic structures I and III with H0 ii [0011, and II with H0 [1101. The elementary —~

excitation energies have the correct sign, and the zero-point spin deviations are equal to 5.4%, 4.2%, and 1 .2% for structures I, II and III respectively. The structures derived by the approximations of Villain, and of Luttinger and Tisza are then quite acceptable for spin-wave theory. In the random phase approximation, one replaces in eq. (50) all J~1by their expectation value ±R/2,where R is the reduced sublattice magnetization. The elementary excitations are the same as the spin-wave excitations, but their energy is R times that of the spin waves. The reduced magnetization R is equal to the thermal average of an appropriate linear combination of transverse spin correlations. These transverse correlation thermal averages can in turn be related to the Rdependent elementary excitation energies through the fluctuation-dissipation theorem. This allows a self-consistent calculation of R as a function of temperature [121. It is also possible to calculate the energy and the entropy of the system. For practical purposes, it is simpler to use the Green-function formalism, rather than that of the equation-of-motion. The reader is referred to ref. [9] for more details. The calculations have been limited to structures I and III. The critical initial polarizations predicted by R.P.A. are p0 = 0.55 for structure land p0 = 0.12 for structure III. Especially in this last case, the RPA value for p0 is much less than the Weiss-field value and also, as will appear in section 4, than the experimental one. The conclusion which can be drawn is that R.P.A. ceases to be good when extrapolated too close to the transition, and that therefore it is of little use in the present state of the experimental investigation. 3.3. Restricted-trace approximation [13] The restricted-trace approximation represents an attempt to go beyond the Weiss-field approximation by taking at least partially into account the short-range correlations between spins. Its principle can be illustrated as follows. Let us consider a transition to an antiferromagnetic state of known structure, with sublattices s~’i and 13 . I. and S~are the spins belonging to .9! and ‘13 respectively. What we need is to calculate the partition function: ,

ZTr{exp(—31C)} .

(51)

M. Goldman, Nuclear dipolar magnetic ordering

21

We choose as a basis the eigenkets of both I~= ~ J~and S~= ~ S, and we split the Hilbert space of the system into a sum of subspaces corresponding to well-defined eigenvalues of I~and S~: (IZ)~NpA

and

(S~)~Np~

The trace (51) can then be written as a sum of partial traces restricted to the various subspaces: =

~

Tr’{exp(—/3~C)}=

(pA,PB)

~ ~‘(P~, PB)~ (PA~PB)

(52)

At this point, the assumption is made that the restricted traces ~‘(PA’ PB) exhibit a sharp maximum for those values of PA and PB which correspond to the sublattice polarizations in the ordered phase at the inverse temperature j3, and that we can replace eq. (52) by: ~

Tr’{exp(—13~C)}

(53)

,

where Tr’ is the maximum restricted trace. In practice we write: ZTr’l X [Tr’~exp(—13~IC)}/Tr’l]

(54)

,

where 1 is the unit operator, or else: ln 2~ ln(Tr’l) +ln[Tr’{exp(—j3~C)}/Tr’l]

-

(54’)

The first term on the right-hand side is to within the factor kB equal to the entropy in the Weissfield approximation. As for the second term, it is very difficult to calculate exactly, and it is approximated by an expansion in powers of ~3.The values of PA and PB are finally obtained by the conditions: —~—ln~Z~——ln~0. (55) öPA

aPB

When the power expansion, in eq. (54’), is limited to the term linear in /3, one recovers exactly the Weiss-field approximation. From the term in j32 and up, one obtains corrections to the Weiss-field approximation originating from short-range correlations between spins. We have used expansions up to j32 and to /33, corresponding to the so-called 1st-order and 2nd-order restricted trace approximation, respectively. As will be seen by comparison with experiment, the predictions yielded by these approximations, although qualitatively correct, are still far from satisfactory. The advantage of the restricted-trace approximation, however, is that it is the simplest approximation beyong that of the Weiss-field which yields correct qualitative predictions for the properties of the ordered states and also for the transition between paramagnetism and ordering. As an example, fig. 3 reproduces the predicted variation of sublattice polarization in zero effective field as a function of initial polarization, for the antiferromagnetic structure III. The predictions of the 1st and 2nd-order restricted trace approximations are compared to those of Weiss-field and RPA. They are physically much more reasonable than the latter. The predicted values for the transition initial polarization, 38% and 43% respectively for 1st and 2nd-order approximations, are reasonably close to the value obtained above (3 2.6%), and in semi-quantitative agreement with experiments described below.

22

M Goldman, Nuclear dipolar magnetic ordering

,~,/,

/ ‘7/ ,/1/

o~d //

C

,;V//

Weiss

\

2

a

‘///

~O5 Restricted Trace 1~0rder

RPA

0 ~

05 2~Order Initial polarization Pi

Fig. 3. Sublattice polarizations in zero field as a function of initial polarization for structure III, according to Weiss-field, RPA and 1st and 2nd-order restricted-trace approximations.

4. Experimental investigation of antiferromagnetism It is not the purpose of this review to discuss at length the technical problems involved in the experimental work, but rather the physical principles used in devising the experimental procedures. The technical part will be limited to the forthcoming subsection, where we give as an example a short account of the experimental set up and procedure used in the work with calcium fluoride. This will be followed by the description of several properties of the antiferromagnetic structure III in CaF2 (produced by adiabatic demagnetization in the rotating frame at negative temperature with the field parallel to a fourfold axis). Finally, we will describe some preliminary results on antiferromagnetism in lithium fluoride. 4.1. Production of ordering in calcium fluoride All studies on calcium fluoride have been performed with spherical samples of diameter between 1.3 and 1.5 mm. The sample, glued to a piece of copper, is located in a helium 3 cryostat made of quartz. It fits into the narrow tail of this cryostat (of inner and outer diameters 1.5 and 3.5 mm, respectively) where it can rotate and move vertically in a controlled way. The rotation axis is as a rule aligned with a [110] crystalline direction, as adjusted to within 10 by X-ray diffraction. The cryostat is immersed in a pumped helium 4 bath. Its tail is surrounded with the microwave irradiation system (a piece of 4 mm-waveguide with a 4 mm diameter hole drilled in it) and several NMR coils. 2~ions as the paramagnetic impurity, at a conMost studies have used samples containing Tm centration Tm2~/F = 1.2 X l0~. Their g-factor is g = 3.45. The dynamic polarization is performed in a field of 27 kG, produced by an electromagnet, with a microwave irradiation of frequency 130 GHz. The helium 3 is pumped and liquefied, and flows back into the cryostat. The

M. Goldman, Nuclear dipolar magnetic ordering

23

sample temperature is then around 0.7 K. The nuclear polarization reaches a steady-state limit after 3 to 4 hours. The micro-wave is then turned off and the system is cooled to about 0.26 K in one shot with a charcoal pump. Finally, the adiabatic demagnetization of ‘9F is performed with an r.f. field of 30mG at 107 MHz. The r.f. field is applied at 30 G from resonance, and the d.c. field is swept toward resonance at a rate of I G/sec. Various calibrations are necessary. They are made as follows. I) Orientation of the sample. The rotation axis, perpendicular to the field, being parallel with a [011] axis, it is possible to make the field parallel to a fourfold, a threefold and a twofold axis. The calibration is made by measuring as a function of orientation the second moment of the nuclear absorption signal. According to theory [141, the second moment in CaF 2 is proportional to: M2

cx

[X~ + A~+ A~ 0.1952] —

where A1, A2, A3 are the cosines of the d.c. field direction with respect to the crystalline axes. The absorption signal is recorded on a multi-channel analyzer during a linear sweep of the field at a rate of 700 G/sec. The r.f. amplitude is small, so as not to saturate the spins: typically, the magnetization loss caused by one field sweep is 5 X l0~. II) Amplitude of the r.f field. The calibration of the r.f. field amplitude H1 produced in the NMR coils is performed by measuring the loss of magnetization caused by a succession of slightly saturating linear field sweeps through the resonance line. The relative loss caused by each sweep is according to theory [1 5] equal to: =

—inyH~/(dH/dt)

This loss is monitored through the change of the area of the absorption signal. III) Calibration of the nuclear polarization and of the receiver gain. The absorption signal v, in a NMR experiment, is proportional to the magnetization in quadrature with the r.f. field. If we call Ox the direction of the latter in the rotating frame, we have then: 1)

=

where ~is the gain of the receiver. It can be proved [16] that, whatever the spin temperature, the polarization p = 2 (I~)/N is related to the area of the absorption signal through:

fv(~)dz~

=

~ f(I~)(~) d~= —~rw,(Ii).

(56)

This allows a calibration of the received gain if one has an independent knowledge of the polarization. In spherical samples of CaF2 the most practical method of calibration of the polarization is through the variation of the second moment of the absorption line [1 5]. In an impurity-free sample, this second moment varies according to: 2). (57) M2(p)M2(0)X(l —p

24

M Goldman, Nuclear dipolar magnetic ordering

When paramagnetic impurities are present, the dipolar fields they create at the nuclear sites alter the shape of the nuclear absorption signal and yield an extra contribution to its second moment. The latter is determined by comparing the observed second moment, extrapolated to zero polarization, to its theoretical value. For instance, in CaF2 spheres doped with Tm2~ions at a concentration of 1.2 X I O~,one has, with H 0 [100]: 2, M 2. M2(O)theor = 12.95 G 2(O)exp. = 13.35 G When proper care is taken of the influence of paramagnetic impurities, this method yields an absolute calibration of the polarization with an accuracy of 3%. IV) Impurity concentration. The most reliable calibration, of the Tm2~concentration is obtained by comparing the areas of the resonance signal of the bulk fluorine spins and that of fluorine spins close to the impurities, whose resonance frequency is shifted by the electronic superhyperfine field. At each field orientation, the number of fluorine spins experiencing a given superhyperfine field is a known small multiple of the number of electronic spins. The ratio of fluorine spins with a given frequency shift to unshifted fluorine spins then yields directly the ratio Tm2~/F.When comparing the areas of the resonance signals, one has to assume equal polarizations for both the normal and the shifted spins. Several experimental observations make this assumption plausible. The figure obtained in the sample used: Tm2~/F 1.2 X lO~ is probably accurate to better than 20%. —

V) Contribution of impurities to the local frequency. The paramagnetic impurities present in the sample cause a perturbation of the system through the dipolar field they create a the nuclear sites. Fluorine spins close to the impurities experience such large frequency shifts that they are no longer in thermal contact with the bulk spins. However the thermal contact is not suppressed among the spins whose frequency shifts differ by less than the nuclear linewidth. One has to use, in place of the bare dipolar Hamiltonian ~ and “effective non-Zeeman Hamiltonian”:

;Jc~~ =R~,+~I~ ~ where the ~ 1’sare the superhyperfine frequency shifts. The effective local frequency D’, characterizing the heat capacity of this thermal bath, is given by: 2 = Tr(~IC~~)/Tr(I~) = {Tr(W~2)+ Tr(~II~ t5~I~)2}/Tr(I~) D’ (58) where D is defined by eq. (16). An example of a method for measuring D’ is given in ref. [17]. The experimental result with the samples used is, for H 0 111100]: 2/~y2 5.23 G2 D’ whereas the theoretical value of D2 (eq. (16)) in an impurity-free sample is: ,

M. Goldman, Nuclear dipolar magnetic ordering

25

D2/y2 4.33 G2. The perturbation caused by the impurities, although non-negligible, is reasonably small for this orientation of the field. —

4.2. Transverse susceptibility in zero field One of the most characteristic properties of antiferromagnets is that below the transition their transverse susceptibility is independent of temperature. The simplest way to see this is through the Weiss-field approximation, which states that the spins are aligned with the total field they experience. When one irradiates the system at exact resonance the transverse r.f. field H 1 looks static when viewed in the rotating frame, whereas the longitudinal component of the effective field vanishes. The total effective field, which reduces to H1, is then perpendicular to the normal orientation of the sublattice polarizations, and one has by symmetry: p~—p~

and

pp~.

Eqs. (45) and (46) then yield: w1 —~A(O)p~~,

c4~, 0,

c4~A(K0)p~

and /

PAX/PAz

— —

T/

T

WAx/W~

whence:

xi

=p~’/wi= l/[A(K0)+~A(0)] ,

(59)

which is independent of sublattice polarization, that is of temperature. The orientations of the various fields and polarizations are shown on fig. 4. Both the spin-wave and the random-phase approximations yield the same expression (59) for x±.It should be noted that the constancy of x1 is predicted only when the system contains one spin species. As seen below a different result is predicted in lithium fluoride. Equation (59) is valid only below the transition but, according to the incorrect prediction of the Weiss-field approximation it should hold whatever the initial polarization prior to the adiabatic

~-

Ay

~1x1I I

Fig. 4. Sublattice polarizations and fields for an antiferromagnet in the presence of a small transverse field. The figure corresponds to the case of a positive temperature.

26

M Goldman, Nuclear dipolar magnetic ordering

I

I

I

I

I

I

Restricted Trace

I

-

Weiss

E

0.1

0.2 0.3 0.1.

Initial

0.5 0.6 0.7 0.8 0.9 1

polarization

P~

Fig. 5. Transverse susceptibility in zero field as a function of initial polarization for structure Ill, according to high-temperature and Weiss-field approximations. (a) infinitely flat disc perpendicular to H0 (b) spherical sample;(c) infinitely long needle parallel to H0.

0

05 Initial polarization p

1 1

Fig. 6. Transverse susceptibility as a function of initial polarization for CaF2 in structure Ill (H0 [100j, T < 0). The curves are the predictions of 3d order high-temperature cxpansion, Weiss-field and 1st order restricted-trace approximations. The value of the plateau is adjusted to the Weiss-field prediction: xi(Weiss) = 0.15 7 G’.

demagnetization. We can compare this behaviour with that predicted by the high-temperature approximation to spin-temperature theory for the paramagnetic state. The result, which is detailed in ref. [8] is: Xipara

=

p1/[D

+

~A(O)p1] ,

(60)

where p1 is the initial polarization. The values (59) and (60) coincide when the initial polarization is equal to p1 = D/IA(K0)I, which is identical with the critical initial polarization p0 calculated above (eq. (49)). One can expect that Xi will undergo a smooth transition from the behaviour predicted by eq. (60) to that predicted by eq. (59) when the initial polarization is increased. The variation of x1 with ~ according to eqs. (59) and (60) is shown for various sample shapes in fig. 5. More elaborate calculations have been made for spherical samples (for which A(O) = 0): a thirdorder expansion in inverse temperature /3 for the paramagnetic domain, and the 1st-order restricted trace approximation. Their predictions will be described together with the experimental results. Three different kinds of measurements of the transverse susceptibility have been performed with spherical samples. The first measurement is performed by observing on the run the dispersion signal u during a fast passage, which is nothing but an adiabatic demagnetization followed by an adiabatic remagnetization. The dispersion signal is proportional to the magnetization in phase with the r.f. field. At the centre of the fast passage, the longitudinal effective field is zero, and u is proportional to w1 Xi~ The initial and final values of the entropy are obtained by measuring the nuclear polarization before and after the fast passage. The magnetization loss being relatively small (about 1 3%), the entropy at the centre of the passage is taken to be the average of its initial and final values. We

M. Goldman, Nuclear dipolar magnetic ordering

27

take as “initial polarization” the polarization which would have yielded this entropy had the passage been rigorously adiabatic. The variation of Xi with initial polarization is shown in fig. 6, together with theoretical curves. One observes, as expected, that Xi is nearly constant at high initial polarization. The value of x1 in the plateau has been adjusted to the Weiss-field theoretical value. With this fit, one obtains a very good agreement at low initial polarizations with the nonlinear predictions of the 3d-order expansion in j3. The agreement is not so good with the restrictedtrace predictions. This last approximation does not predict the non-linearities observed in the paramagnetic domain (as expected from a 1st order approximation) and, in the antiferromagnetic domain, an almost constant value of Xi is predicted, which is however approximately 20% higher than the Weiss-field value. This has long been considered as a weakness of the restricted trace approximation on the following basis: Xi is constant within experimental uncertainty and is reasonably expected to keep the same value down to very low temperature. In this domain one is tempted to trust the spin wave approximation which predicts a temperature-independent value of Xi identical with eq. (59). The question has been reopened by recent absolute measurements discussed below. A second type of measurements yields in principle the variation of Xi with dipolar energy without any adjustable parameter. Experimentally, after having performed the adiabatic demagnetization in the rotating frame to zero effective field, one observes at regular time intervals the nuclear absorption signal during linear field sweeps, with a small non-saturing r.f. field. The gain of the receiver being calibrated as explained above, one obtains calibrated values of energy and transverse susceptibility as follows. i) The transverse susceptibility in zero effective field, which is the (real) susceptibility to a steady transverse field, is related to the imaginary part of the frequency-dependent transverse susceptibility through the Kramers—Kronig relation: Xi

=

x’(O)

=

—~

~f

~ x”(~)d~.

(61)

The susceptibility Xi is defined by: Xi

=

p~/w1= 2(I~)/(Nw1),

and the absorption signal v is equal to: v = ~(I~>(i.~) = ~~Nw1 x”(~) whence: ~ v(~)dz~—~Nw1xi.

(62)

ii) When the system contains only one nuclear species, it can be shown [16] that the dipolar energy is related to the first moment of the absorption line through:

f ~v(~)d~

=

3~w1(~C~).

(63)

In practice the absorption signal is recorded in a multichannel analyzer and the integrals (62)

28

M Goldman, Nuclear dipolar magnetic ordering

I

I

I

Restricted Trace

I

0i~

I

—

1st Order

I I

-

2ti2/2&) Dipolar Energy per spin

~,

2/203)

Fig. 7. Calibrated transverse susceptibility as a function of dipolar energy in Ca F 2, as deduced from the Kramers—Kronig transform of the absorption signal. Same approximations as in 2l1/2a3 = 2.72 kHz. fig. 6. In Ca F2, ~

Dipolar Energy per spin (T

Fig. 8. Transverse susceptibility as a function of dipolar energy in Ca F2. xi measured as a function of time and variation of energy with time tocomputed afterwards plateau adjusted the Weiss-field value.(see text). xj in the

and (63) are computed on-line by a small calculator. The accuracy on the calibration of Xi and (3~~) is of the order of 5%. The results obtained by this measurement are shown in fig. 7. The value of Xi in the plateau is definitely larger than the prediction common to Weiss-field, spin wave and random phase approximations, but very close to the prediction of the 1st order restrictedtrace approximation. It is not clear at present whether this is due to the perturbing effect of the paramagnetic impurities or whether it is an intrinsic effect. It will be seen in section 5 that the same excess of ~ with respect to Weiss-field is observed in the ferromagnetic state. Finally, the variation of x 1 as a function of dipolar energy was obtained as a by-product of a measurement of the longitudinal susceptibility, described in the next section. The variation of Xi with energy is shown in fig. 8, with Xi in the plateau adjusted to the Weiss-field value. To within a scaling of the ordinate, these results coincide with those of fig. 7. The net result of this study is that one observes a plateau of Xi as expected for an antiferromagnet. The value of Xi in the plateau is qualitatively as expected, as well as the values of entropy and energy at which the plateau begins. 4.3. Longitudinal susceptibility in zero field In zero effective field, the sublattice polarizations are opposite, and the bulk magnetization of the sample is zero. In the presence of a small longitudinal field, the sublattice polarization amplitudes cease to be equal, and there appears a bulk magnetization proportional to the field. We define the longitudinal susceptibility through:

x11

(p~+p~)/~,

(64)

where ~ is the Larmor frequency corresponding to the effective field. The property which, together with the constancy of x1~is characteristic of antiferromagnets is that their longitudinal

M. Goldman, Nuclear dipolar magnetic ordering

29

susceptibility decreases below the transition and vanishes at zero temperature. The easiest way to see this is through the Weiss-field approximation. At zero temperature, the sublattice polarizations are unity and their orientation is determined by that of the field they experience (at negative temperature, the sublattice magnetization is antiparallel to the field). A small longitudinal field will change slightly the amplitudes but not the directions of the sublattice fields. There will therefore be no change in the sublattice polarizations and the bulk magnetization will remain zero, whence = 0. When the temperature is not zero, let us write: p~p+e and p~=—p+e’. We have, according to eqs. (45) and (46): p

tanh{—~j3[A(K0)p + ~A(O)(

+

+ ‘) +

~]}

—p+e’—tanh~—~j3[—A(K0)p+~A(0)(+e’)+z~]}. To the first order in z~, and e’, this yields ptanh{—~/3A(K0)p},

2)[A(0)e

= ‘

whence: =

=

+

—~j3(1 p

(65) (66)

~] ,

—

e/~= —~/3(l

—

p2)/[

1

+

~j3(l

—

p2)A(O)]

-

It is easily verified through eq. (65) that the predicted value of 1x

(67)

11 I is maximum at the transition. The variation of the longitudinal susceptibility has been subjected to two experimental investigations in spherical samples of CaF2. The first study [18] was simply aimed at proving that x is indeed decreasing below the transition. Its principle is to detect the bulk magnetization of the CaF2 sphere through the dipolar field it creates in its neighbourhood, by observing a shift in the NMR resonance frequency of liquid helium 3 surrounding the sample. The experimental procedure is schematically as follows. After adiabatic demagnetization of the fluorine spins, the r.f. field at 1 07 MHz is frequency-modulated with a frequency of 0.1 Hz and an amplitude of 2 kHz. This without is equivalent to an adiabatic 9F spins actually changing themodulaexternal tion of the longitudinal effective field seen by the ‘ field experienced by the helium 3. This produces a modulation of the sample magnetization proportional to XI. It is detected by recording the modulation of the lock-in signal of 3He in a coil located just below the CaF 2 sphere. The modulation of the dipolar field is very small (a few tens 3He of mG) and its observation severe stability conditions for the to d.c.the field the spectrometer frequency. In requires practice very the d.c. field is stabilized with respect 3Heand frequency through the lock-in signal of 3He in an auxiliary coil sufficiently removed from the CaF 2 sample for the dipolar field of the latter to be negligible. In the course of time, the fluorine system warms up under the effect of spin—lattice relaxation, and one expects that the amplitude of modulation of the helium signal will first increase and then decrease, when the system becomes paramagnetic. The results for a typical run are shown in fig. 9. There is a small initial increase of the modulation amplitude. It is not much larger than the scatter of experimental points, but it has been repeatedly observed in all runs.

30

M. Goldman, Nuclear dipolar magnetic ordering

Time (minI

Fig. with of the longitudinal the system warms up, for Ca F bility9.isVariation monitored by time the resonance frequencysusceptibility shift of a ~Heasprobe. 2 in structure Ill. The

19F suscepti-

The second study [19] yields the variation of X 11 with dipolar energy. It is based on the following facts. i) when a change occurs in the longitudinal polarization, its value can be extracted from the NMR absorption signal; ii) one can devise a sequence of variations of the d.c. field which induces changes of the longitudinal polarization, simply and unambiguously related to X11 iii) a clean theoretical analysis can be made of the decrease of energy caused by the non-adiabatic variations of the field; iv) the decrease of energy caused by spin-lattice relaxation is known from an independent study described in a section 4.5. By comparing the variation of x11 as a function of time, with that of the energy, one obtains finally the longitudinal susceptibility as a function of dipolar energy. These points are now analyzed in turn. The variation of (Ii) is, according to eq. (12), given by (‘Z)

~Tr~I~ ~ d}=Tr{—iI~[~C~~ ~]} =

Tr{—i[I~, ~*]~}

=

(—i[I~, ~f*])

The effective Hamiltonian ~1C” being given by eq. (13), we obtain:
=

w~(I~).

(68)

This result, which is very general, is used in the following experimental sequence. After adiabatic demagnetization in the rotating frame of the polarized nuclei and with the r.f. field still present, the d.c. field is subjected to a square-wave modulation, with a period of a few seconds, between and +~ from resonance, where ~ is small compared with local and Weiss frequencies (2~/y= 1 G). Each field jump, which corresponds to a change 2~of the effective frequency in the rotating frame, is followed by an evolution of the system toward a new equilibrium with respect to its new —~

M Goldman, Nuclear dipolar magnetic ordering

31

effective Hamiltonian. The modulation period is long enough to allow equilibrium to be reached between successive field jumps. The variation of longitudinal polarization after each field jump is then equal to: =

~

It is measured by integrating the absorption signal following the field jump. This sequence is schematically shown in fig. 10. One observes not only the absorption signal, but also the dispersion signal. The latter is not different at ±Z~ from resonance than at exact resonance and it yields relative values of Xi~

..-

0

U

Fig. 10. Field-jump sequence and

v—

times

19F absorption signal for the measurement of the longitudinal susceptibility in Ca F 2.

In the course of the sequence, the system warms up both because of relaxation and of the sudden field jumps, and the susceptibilities eventually decrease. The values of the susceptibilities and of the energy in the course of the experiment are determined afterwards as follows. i) The value of Xi in the plateau is adjusted to the Weiss-field theoretical value. ii) For the points at the end of the sequence, when the system is warm and x1 is much less than in the plateau, the values of x11 and (~C~) are adjusted to those of Xi according to the 3d order expansion in inverse temperature for the paramagnetic state. This fitting yields the values of x11 for all points of the sequence. iii) The variation with time of the dipolar energy is determined by two factors: the spin-lattice relaxation, which has been studied independently and is described in section 4.5. the non-adiabatic evolution following each field jump. Let the effective Hamiltonian before a given field jump be: —

—

+

=

+ wl’x

It is transformed by the field jump into: =

~Iz

+

~c;:~ + W1I~.

32

M Goldman, Nuclear dipolar magnetic ordering

The energy corresponding to this effective time-independent Hamiltonian remains constant during the subsequent evolution of the system. w1 being small, we can neglect the heat capacity of the term ~

and we obtain: 2X = ~

=

—N~

11 ,

(69)

which we can compute using the value of x11 measured as explained above. The two contributions to the variation of the energy being known, the value of (~JCJ~.,)at the time of each field jump is determined step by step, backwards in time, starting from its calibrated low value at the end of the sequence. As a final check the initial value of (~C~) calculated by this method, can be compared to that obtained independently by measuring in a separate experiment the first moment of the fluorine signal immediately after an adiabatic demagnetization started with the same initial polarization. Both values coincide within experimental uncertainty. The variation of x~with dipolar energy has been described above. The variation of X11 is shown in fig. 11, together with several theoretical curves: high-temperature expansion, Weiss-field, RPA, and 1st and 2nd order restricted trace approximations. The combination of high-temperature plus Weiss-field approximations, although it lacks thermodynamic consistency, yields predictions very close to those of the restricted-trace approximations, except at high energy, that is close to zero temperature where the latter are known to be invalid. The experimental points are in semi-quantitative agreement with these predictions. The decrease of ~11at high energy is small, but as expected. We have ignored the perturbing effect of the impurities, which is small. The results would have been but slightly affected if we had used the calibrated value of Xi in the plateau rather than the Weiss-field value. The combined results on Xi and x provide a convincing evidence for the reality of nuclear antiferromagnetism in the rotating frame as predicted. The variation of these quantities is smooth, however and no marked singularity is observed nor expected at the transition from paramagnetism to antiferromagnetism thus precluding a precise measurement of the critical energy or entropy. —

I

—

II

Restricted trace iSiOrder 2°~Order -

o

2t~2/2a3) 3~Or~r~ R.PA/ ~ Dipolar Energy per spin (y I

Fig. 11. Longitudinal susceptibility as a function of dipolar energy in Ca F 2. The values of x~are calibrated against those of xi (fig. 8). Theoretical curves for 3d order high-temperature expansion,Weiss-field, RPA, 1st and 2nd-order restricted-trace approximations.

M Goldman, Nuclear dipolar magnetic ordering

33

It is another property, whose study will now be described, which has a marked critical behavjour allowing the determination of the critical energy. 4.4. Non-uniform longitudinal susceptibility of the fluorine spins [20] Calcium fluoride 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%) consist of the magnetic isotope 43Ca, of spin ~, whereas most nuclei are the spinless isotope 40Ca. The influence of the 43Ca spins on the bulk properties of the fluorine spin system is very weak, because of their low concentration and of their small gyromagnetic ratio. Their weak influence can however be detected by observing the resonance signal of the 43Ca spins themselves. This influence is the following. In the dipolar field of the 43Ca spins the fluorine spins acquire polarizations departing slightly from their undisturbed values. These departures are non-uniform. They affect the shape of the 43Ca-resonance signal in a way depending on the non-uniform longitudinal susceptibility of the ‘9F spins to the dipolar field of the 43Ca spins. Another consequence of the low concentration and small magnetic moment of the 43Ca spins is that each one can be treated independently. This is because: i) The heat capacity of the 43Ca—43Ca dipolar interactions ~ is much less than that of 1C~ and ~1C~J (S and I refer to spins of 43Ca and 19F, respectively). ii) The mean time between successive flip—flops, for two 43Ca spins at the average distance, is of the order of 1 hour, whereas internal equilibrium among the 19F spins takes place within a small fraction of a msec. One can then assume that each 43Ca spin is in a well-defined and timeindependent state S~= m. iii) The dipolar field of a 43Ca spin falls down to a negligible value at distances much less than the average distance between neighbouring 43Ca spins. There are then negligibly few ‘9F spins influenced by the dipolar field of more than one 43Ca spin. The dipolar field created by a calcium spin in the state S~ m is proportional to m. In the linear approximation, the ‘9F polarization departures are also proportional to m, as well as the dipolar field that they create at the site of the calcium spin, which shifts the resonance frequency of the latter. 43Ca spins with opposite values of m experience opposite frequency shifts. Furthermore since spins with m < 0 give an absorption signal and spins with m> 0 an emission signal, the total resonance signal of 43Ca will look like the derivative of a bell-shaped curve. This feature is standard for spins in the demagnetized state [3]. In the present case it has a particularly simple physical interpretation. These considerations will be put on a more quantitative basis through the local Weiss-field approximation. The dipolar Hamiltonian is equal to: +

=

(70)

.

We neglect the term ~

~fC~is given by eq. (8), and TICS is equal to:

~ck~=~1 ~

(71)

with: C~= y

2OlM)/rj~ . 1’y~h(l 3 c~s —

(72)

34

M. Goldman, Nuclear dipolar magnetic ordering

As for most NMR signals in solids, it is impossible to give a detailed theoretical description of the 43Ca resonance signal. A simple and well-defined information is however yielded by its first moment. When a system contains two spin species landS, eq. (63) can be generalized 1161. The first moments of the two resonance signals are equal to: M~~ 1iw~{3(~C1)+(W5>} ,

(73) (74)

and, since (~tC1~i ~‘

~

M~ 37r~1w~(~1CJ>, S

S

(73’)

‘

M1 i~w1(~C~), (74) what can be done experimentally is then to measure as a function of (~JC~J). We begin by calculating (IC~)as a function of temperature. Let us consider the spins I in the vicinity of a particular spin S with S~= m. The polarization p1 of a spin i is in the local Weiss-field approximation related to its frequency w1 through: p1

=

tanh(—~I3w~) .

(75)

If the spin S were absent, the undisturbed antiferromagnetic structure would correspond to 1 = ±p, pi = p~ and =

o4~

A(K 0)p~.

In the presence of the spin S we have: 0 + pi = p1 w~w?+öw.w9+~DA..e.+mC.

.

(76)

By expanding eq. (75) to first order with respect to Sw~we obtain: ej_~/3(l_p2)(~DAjjei+mCi,~).

(77)

Through a space-Fourier transformation and by using the Weiss-field value of the inverse critical temperature: =

—2/A(K 0),

eq. (77) yields:

(k)

m C(k) (i3/13~X1_— , (78) 112A(K 2)A(k)/A(K N1 0) 1 (f3//3~)(l p 0) where N1 is the number of spins I and k is a vector of the first Brillouin zone of the s.c. fluorine lattice. According to eq. (78), e(k) is proportional to the component of wave-vector k of the calcium dipolar field, m C(k), and we can write: =

—

—

M. Goldman, Nuclear dipolar magnetic ordering

e(k)mC(k)X11(k).

35

(78’)

This is a consequence of the fact that p~° I = p is the same for both sublattices. If the effective field experienced by the fluorine system were not zero, Ip~I would be different in both sublattices, and (k) would depend both on C(k) 2) andis C(K0 k). at This will notand be equal analyzed. maximum thecase transition to I, so that when It is easily shown that (/3//3~)(l p A(k)/A(K 0) is close to 1, X11(k) is sharply peaked at the transition. The spins S being treated independently, we have: —

—

~

~im9m m

~C~~p~(/3,m),

(79)

1

where ~ is the population of spins S with 5~= m. The spins S being located at the centres of cubes of spins I, it can be shown that C(K0) = 0, whence:

~C~p~° ccpC(K0)0. The energy (~C~) is then: =

m m ~m

~

C,,2 ,

2l~m~m~J~C(—k)(k) ~NJ~~2

m

~

k

~C(k)2XII(k), m

where we have used the fact that C(k)

(~C’ IS’\S(S~2\P(l z ‘t-”.

(80)

k

2\J

=

C(—k). We obtain finally, according to eq. (78): C(k)2

k

When the spins S are unpolarized, (Sf2)

=

~S(S

+

.

~8l

1). In the sum, the terms for which A(k)/A(K

0) is close to unity cause the energy to be sharply peaked at the transition. In a rough makeshift approximation (~1C~) is calculated as a function of j3 by the 1st order restricted trace approximation in order to obtain the variation of (~K~) with (1C1). The resulting curve for structure III (H0 II [100] and T< 0) is shown on fig. 12, together with high-temperature and Weiss-field predictions, and compared with the results. 9F,experimental the calcium resonance signal was recorded at Experimentally, after the ‘ regular time intervals anddemagnetization its first momentof computed. The variation with time of (3C~)being known from independent measurements, one obtains the points of fig. 12. They correspond to three different runs. The maximum expected is clearly exhibited and the overall variation is as predicted. None of the approximations however is very good. This is probably because the local longitudinal fluorine susceptibility depends very much on short-range correlations. The combination of Weiss-field plus high-temperature expansion is completely inadequate. The restricted-trace prediction, although qualitatively correct is still very approximate.

36

M. Goldman, Nuclear dipolar magnetic ordering

120

-

-

Weiss

Temp 3~Order

~.

F_F Dipalar energy per F spin (kHz)

Fig. 12.

19F—43Ca energy as a function of 19F—19F energy in Ca F

2 demagnetized withH0 I [100] and T < 0, together with the predictions of high-temperature, Weiss-field and restricted-trace approximations.

The energy per spin at which (~C~) is maximum, which we identify with the transition energy, is about 2.5 kHz, whereas the values predicted by the restricted-trace method are 1.62 kHz and 1.93 kHz for the 1st-order and the 2nd-order approximations, respectively. 4.5. Spin-lattice relaxation [21] Once the nuclear spin system has been cooled by adiabatic demagnetization in the rotating frame, it warms up toward the lattice temperature under the effect of spin-lattice relaxation. In solids, this relaxation is known to originate from the coupling of the nuclear spins with paramagnetic impurities. It depends on their nature, their concentration and their coupling with the lattice. The problem investigated in the present section is this: in the presence of given impurities how will the spin-lattice relaxation of the nuclear spin system depend on its internal state, paramagnetic or antiferromagnetic. In the high-temperature limit, one can define unambiguously a spin-lattice relaxation time T1. Most if not all measurable quantities are proportional to the inverse temperature j3, which varies exponentially toward a steady-state value. At low spin temperatures, and in particular in an ordered state, the various physical quantities are not proportional to each other and none is a priori expected to vary exponentially. We are concerned here with the variation under the effect of spinlattice relaxation of the dipolar energy of the nuclear spin system, which can easily be measured. Experimentally, after the adiabatic demagnetization, one records at regular time intervals the fluorine absorption signal during slightly saturating linear passages, and then computes its first moment. As in standard treatments of spin-lattice relaxation (see e.g. [3, Ch. 3]), the spin orientations of the paramagnetic impurities are considered as stochastic variables which vary randomly around their average value under the effect of their own electronic relaxation. The dipolar field they create at the sites of the nuclei has the same random character. The perturbation responsible for the relaxation of the nuclear spins has then the form of a Zeeman coupling with this random field.

M Goldman, Nuclear dipolar magnetic ordering

37

At low lattice temperature, the correlation time r~of this field is as a rule much longer than the nuclear Larmor period, i.e.: w0r~~ 1 It can be shown that in that case only the longitudinal components of the random field are effective for relaxing the nuclei. The spin-lattice coupling responsible for relaxation is then of the form:

1c1(t) = VF(t),

(82)

where: V=~I~h1I~,

(83)

and F(t) is a random function satisfying: (F(t))av

=

0,

(F(0)F(t))av = exp(—ItI/r~).

(84)

The master equation for the rate of change of the dipolar energy is of the form:

V(t)], V])dt, where: V(t) = exp(i~1C’Dt)Vexp(—i~FC~t).

(85) (86)

We call T2 the decay time of the expectation value under the integral, which we assume to be comparable with the inverse of the nuclear linewidth. We will consider two limiting cases: ~ T2 and r~~ T2. 1st case: ~ T2. In the integrand of eq. (85) the exponential decays much faster than the trace. We can then neglect the decay of the latter, whence: V], V]),

(87)

or else, according to eqs. (8) and (83): ~

(88)

The time-derivative of the energy is proportional to the correlations of the transverse spin components. This enables one to guess the qualitative features of relaxation. At high nuclear spin temperature, that is at low dipolar energy, both the energy and the transverse correlations are proportional to the inverse spin temperature /3. Below the transition, on the other hand, the energy increases very steeply, because of the energy associated with long-range order, whereas the transverse correlations are expected to increase but slowly. Consequently, the relaxation rate, defined as:

38

M Goldman, Nuclear dipolar magnetic ordering

—

~—

(~c115)I(~c~),

is expected to remain approximately constant in the paramagnetic phase and to decrease with temperature in the ordered phase. For a more quantitative although still crude estimate of this rate we have used the 1st order restricted trace approximation. The calculations, for structure III are described in ref. [211 and will not be reproduced here. The results are reproduced in figs. 13, 14 and 1 5 and will be discussed

/Efl~a5(W~ISS) -

Extropol. from low energy t~(

12

r~>)T2 Theory

N

—

aU, U, 0.

w a

a 0

0

Reduced time

t/T1

Fig. 13. Reduced time-variation of dipolar energy in Ca F2 under the effect of spin—lattice relaxation. Theoretical curves for ~ T2 and r~~. T2, and experimental results. The value of T1 is adjusted to the low-energy portion of the decay. 5

I

C

a-

Theory

Dipotor Energy per spin

6

(kHz)

Fig. 14. Variation as a function of dipolar energy ofthe reduced time-derivative of this energy in Ca F2. The points are deduced from those of fig. 13 and the curves correspond to the same approximations in both figures.

M Goldman, Nuclear dipolar magnetic ordering

2-

~

0

I

\~~~)>T, Tc<
0

0

if

O~0

000

00

0

0

1

2

3

4

5

6

Dipolar Energy per spin (kKz) Fig. 15. Variation as a function ofdipolar energy of its reduced spin—lattice relaxation rate. Experimental points and theoretical curves deduced from those of fig. 13.

together with the experimental results. The theoretical formulae are fitted to the value of the spin—lattice relaxation time T1 at high temperature, which is treated as an adjustable parameter. 2nd case: T~. The result yielded by eq. (85) in that case can most simply be obtained by the following physical argument. If the coupling ~JC1(t) = VF(t) were time-independent (i.e. F(t) = F(0) = 1), the terms ~1C1~, and V would merge into a single thermal reservoir, and a time of the order of ~‘2 would be required for it to reach thermal equilibrium. If F(t) varies slowly (i.e. r, ~ T2), both terms are practically at thermal equilibrium at all times, and the relaxation rate of the reservoir is a weighted average of the partial rates of its components. Since only the part V is in contact with the lattice, 1,we obtain: and its partial relaxation rate being r~ ~ ‘~JC~+ V)=_-L(V>, ~‘

or, if we assume that (~1C~) ~ (V): (89) The term V is the magnetic coupling of the nuclear spins with the dipolar field of the relaxing impurities. It does not differ qualitatively from the 43Ca—’9F dipolar coupling discussed in the preceding section, and it has the same behaviour: in the paramagnetic phase (V) increases faster than the energy, it goes through a maximum at the transition and then decreases. One then expects

40

M Goldman, Nuclear dipolar magnetic ordering

that the relaxation rate of the energy will exhibit a maximum at the transition. For a quantitative estimate, we use the same approximations as in the preceding section: as a function of /3, (V) is calculated by the local Weiss-field approximation, linearized with respect to the impurity dipolar field, and (~C~) is calculated by the 1st order restricted-trace approximation. The only difference with the preceding case is that we must discard the nuclei which are too close to the impurities, because as a consequence of their large resonance frequency shift they are not in thermal contact with the bulk nuclei. The “cut-off” distance from the impurities being much larger than the internuclear distance, the dipolar field of the impurity varies slowly between neighbouring nuclei, and one can restrict oneself to the small k components of its Fourier spectrum. The experimental variation of the energy as a function of reduced time is shown in fig. 13, together with the theoretical curves. These experimental points are used to compute —T1 d(~C~)/dt and —T1 d(ln(JC~)/dt,which are plotted as a function of energy in figs. 14 and 15, respectively. The only adjusted parameter is T1, the relaxation time at low energy. The results are clearly consistent with a relaxation corresponding to r~~ T2. Therepresent is experi2~ions in mental evidence, not discussed here, that this relaxation is not due to the Tm the sample, but to some other unknown paramagnetic impurity. The striking quantitative agreement with the theoretical curves for r~~ T 2 is accidental, since as seen in the preceding section the true transition energy is about 50% higher than that predicted by the 1St order restricted trace approximation. What is shown by the experimental results is then that the relaxation rate begins to decrease already in the paramagnetic phase. The dipolar energy depends both on longitudinal and transverse spin correlations whereas, according to eq. (88), its decay rate depends only on transverse correlations. This means that at the approach of the transition the longitudinal correlations increase faster than the transverse ones, which is physically plausible. 4.6. NMR investigation of lithium fluoride [221 Several measurements have been made on LiF samples in the form of flat rectangular platelets. The paramagnetic centres used for the dynamic polarization were F centers created by irradiating the samples with electrons at liquid nitrogen temperature. Their concentration relative to the nuclei was about l0~. The polarization was performed in a field of 55 kG with a microwave irradiation at 140 GHz in a pumped helium 3 cryostat. The [100] axis was in the plane of the platelet. The external field was parallel to this axis, and the demagnetization in the rotating frame was performed at negative temperature. Under these conditions one expects an9F antiferromagnetic-structure identical and 7Li spins (the spins I and 5). with structure III, but where each sublattice contains both ‘ The high impurity concentration, necessary to obtain sufficiently large initial polarizations, causes an important perturbation of the system, which precludes the possibility of a good quantitative agreement with theory. The ambition in these studies has been so far limited to observing qualitative features in agreement with the predictions of Weiss-field plus 1st order high-temperature approximations. In the present article, we limit ourselves to a brief account of the transverse susceptibility measurements as a function of entropy, with emphasis on the difference with calcium fluoride. Experimentally one performs the fast passage simultaneously for both spin species, by applying two r.f. fields whose frequencies correspond to resonance at the same field H 0, for the spins I and the spins 5, and by sweeping slowly the field through H0. One observes the dispersion signals of

M Goldman, Nuclear dipolar magnetic ordering

41

both the spins land the spins S. Their values at resonance yield Xi(J) and Xi(S). We give but the principle of the calculation of these susceptibilities as a function of entropy in the Weiss-field approximation. In the antiferromagnetic structure we must distinguish four types of spins, labelled as follows: A and B for spins I in sublattices .ss{ and ~3, respectively; C and D for spin Sin sublattices .~ and~I3. Four steps are involved in the theory. They consist in calculating the following quantities: i) Weiss frequencies as a function of spin polarizations, ii) Transverse susceptibilities as a function of spin polarizations, iii) Spin polarizations as a function of temperature, iv) Entropy. The Weiss frequency experienced by each type of spins is a linear combination of the four polarizations. In the presence of r.f. fields at exact resonance, one has by symmetry: PBZ

=

PBX

PAz

PDz

PAx

PDX

=

~~PCz PcX

The transverse polarization PAx’ for instance, is obtained by writing: PAx/PAz

=

WAx/WAz

which yields a susceptibility Xi(I) of the form: Xi(l)

PAZ/UPAZ +

where 1 and m are constants depending on dipolar sums. The main difference with the case when there is only one spin species is that, since the ratio PAZ/PCz depends on temperature, the transverse susceptibility x ~(I) is not constant in the antiferromagnetic state. The same is true for X1(S). With I = ~ and S = ~, the polarizations at the inverse temperature /3 are given by: PAZ = =

tanh(xA), ~[4 cotanh(4x~) cotanh(x~)] —

where: XA

=

—~/3wA

and

x~= —~/3w~

As for the entropy of N spins land N spins 5, it is equal to: 1 pA)ln(l PA)] SNkB[ln2 ~[(l +pA)ln(l ~PA)~( +3xc(l —p~)+ln-f[l —exp(—8x~)]/[l —exp(—2xc)]}] The calculation of the susceptibilities in the paramagnetic domain is performed by a standard 1st order expansion in inverse temperature [3] with proper care given to the Weiss-field associated with the transverse polarizations. All calculations assumed that the sample was an infinitely flat ellipsoid. Fig. 16 shows the predicted variations of Xi(l) and X 1(S) as a function of the parameter: 1~’2, R = [(Se S)I2NkB] —

42

M Goldman, Nuclear dipolar magnetic ordering 01

1

_____

1~

Eitrcçy pcri~JT~eterRV((S~S)/2NkB)

Reduced Entropy parameter R=[(S

Fig. 16. Theoretical variation of transverse susceptibilities of 7Li and 19F as a function of entropy parameter R in a platelet of lithium fluoride, after adiabatic demagnetization with H 0 ii [100] and T< 0, according to 1st order high-temperature expansion and Weiss-field approximation. In the definition of R, S is the entropy and S0 its value at infinite temperature.

Fig. 17. Experimental variation of transverse susceptibilities of 7Li and 19F in lithium fluoride, as a function of entropy parameter R, following an adiabatic demagnetization with

0-S)/2 NI~]

H0 II [100] and T< 0.

where S~is the entropy at infinite temperature. In the high-temperature limit, i.e. R small, R is proportional to /3, so that the susceptibilities are linear functions of R. Except at very low entropy, the antiferromagnetic susceptibilities are expected to be nearly constant. The transition occurs at R 0.2, which corresponds to initial polarizations p1

30%

and

Ps

20%.

The experimental results are shown in fig. 17. The susceptibilities seem to reach a plateau, in accordance with expectation. The initial slopes are however about twice larger than predicted by the high temperature approximation, which is due to the dipolar field of the impurities. The plateau is reached at R 0.5 and the values of Xi are 30—40% higher than calculated by the Weiss-field theory, a feature already found in calcium fluoride. Their ratio however: XiO’)/Xi(I~1)exp.

=

1.63

agrees reasonably with the theoretical value 1.46, within the precision of the calibration. The levelling-off of the susceptibilities is not observed at positive temperature, where the transition is expected at a higher value of R. However qualitative, these results point toward the existence of the expected antiferromagnetic phase. A few other results supporting the same conclusion are described in ref. [22].

M Goldman, Nuclear dipolar magnetic ordering

5. Experimental investigation

43

of ferromagnetism

This section describes the investigation of the ferromagnetic structure produced in CaF2 by adiabatic demagnetization at negative temperature. The external field is as a rule oriented along a [Ill] axis. We begin by deriving the main characteristic properties of the ferromagnetic structure according to Weiss-field theory. Then we describe the experiments which establish beyond doubt the existence of this structure. Finally we describe several of its experimental properties. 5.1. Characteristic properties of the ferro magnet We have seen in section 2 that for a spherical sample the ferromagnetic structure in zero field consists of domains in the form of thin slices perpendicular to the external field, carrying opposite magnetizations. The relative volumes of both types of domains are equal, so that the bulk magnetization vanishes. When an external longitudinal field is introduced, it is a characteristic property of ferromagnets with domains that the relative size of the domains will vary through the motion of the domain walls. We will use the Weiss-field approximation to calculate the polarizations and relative sizes of the domains as a function of field and entropy in a spherical sample. The field being longitudinal, all polarizations are longitudinal. We neglect the effect of the domain walls. Let PA and PB be the polarizations in domains 1~qand c13, and x and (1 x) their relative volumes, respectively. The successive steps of the calculation are the following. i) One computes the total frequency experienced by each type of spins (i.e. the Larmor frequency in the field they experience). It is the sum of the external frequency ~ plus the dipolar frequency. Let us consider for instance a particular spin I, in a domain .9q ; its dipolar frequency is a sum of two contributions: a) The contribution from the spins located in the same domain as the spin I,. It is equal to q PA, where, according to eq. (43”): 2h/2a3). q = (8ir/3)(y b) The contribution from the spins outside this domain. In so far as the domains are small we can treat the whole sample outside the domain where the spin sits, as a continuum of average polarization: —

Pay. =XPA+(l —x)pB.

Its contribution to the frequency experienced by the spin I,, proportional tOPav is written r Pay. The coefficient r is found by considering the case when P~. = PA, that is PB = PA~In a spherical sample where all spins have equal polarizations, the total dipolar frequency vanishes (eq. (43)). We have then: 0

(q +r)PA,

that is r

=

—q.

The total frequency experienced by the spins I, of domains .~r( is finally: = Z~ + WA = ~ + q(l X)(PA PB). —

—

(90)

44

M. Goldman, Nuclear dipolar magnetic ordering

One obtains in a similar fashion: (90’) These frequencies are related to the polarizations and to the inverse temperature /3 by the Weissfield equations: uA=tanh(pA)= —~/3w~, 1 (PB) = —~13w~ UB = tanh

(91)

ii) The value of x is that for which —j3F = S entropy per spin are respectively:

E/N = ~ L~{XPA+ (1

—

X)PB}

~ ~{XP~ + (1

—

X)pB} +

=

S/NxsA+(l

j3E is maximum at /3 constant. The energy and

—

+ ~ {XWAPA + (1

~qx(l

—

X)(PA

—

—

x)WBpB} PB)2,

—x)sB,

(92) (93)

where s~stands for s(px), the entropy per spin of polarization p~,(eq. (24)). The condition (SF/ax) SA

—

5B

=

=

0 yields:

~/3(PA— PB)[~ + ~q(1

—

2X)(PA

PB)] .

(94)

One could have treated the three unknown quantities PA’ PB and x on the same footing, and used the conditions: aF/apA

=

aF/apB

=

0.

It is easily verified that they yield the Weiss-field eqs. (91). This approach is that used in the Bragg—Williams theory [23]. It is completely equivalent to the standard Weiss-field approach, where it is eq. (93) (with s~given by eq. (24)) which is derived from eqs. (91) and (92). By combining eqs. (90), (90’), (91) and (94), one obtains: SA

—

SB

=

2(uA + uB)(PA

—

PB).

(95)

It can easily be proved that eq. (95) implies: PA

=

(96)

~PB .

The two possible structures are locally stable with respect to small changes of PA’ PB or x. The stable one is found by comparing either the free energies of the two structures at constant temperature, or else their energies at constant entropy, which will eventually prove simpler. When PA = PB’ the polarization is homogeneous in the whole sample. This paramagnetic phase is obviously the stable one in high field. When PA = ~~PB’ the structure is composed of domains; it is this one that we call ferromagnetic. The domain polarizations are opposite, irrespective of the value of x. We begin by analyzing this last phase, and then compare its stability to that of the paramagnetic phase. Since the entropy is an even function of polarization, eq. (93) yields: S/N=sA

~(PA)-

This means that during an adiabatic variation of the external field the polarizations of the domains

M Goldman, Nuclear dipolar magnetic ordering

(when they exist) remain constant. Their absolute value is equal to the initial polarization p. (94) becomes: 0Øp[~+q(l—2x)p],

45 Eq.

(97)

that is: x~(l+L~qp).

(98)

The relative domain sizes vary linearly with external field. Since x I ~ 1, the ferromagnetic structure with domains can possibly exist only when: I’~I~~qp.

(99)

At constant entropy, the ferromagnet will be stable compared with the paramagnet if Epara > 0. According to eqs. (92) and (98), one has: Eferro Epara = (~ qp)2/4q ~ 0,

Eferro

—

—

so that the ferromagnet is stable for all values of I ~I ~ ~ ~c is then the critical field of transition between paramagnetism and ferromagnetism with domains. The domain frequencies are according to eqs. (90—90’) and (98): and

wT~qp

~4—---qp.

When p is constant, eq. (91) yields j3 = cte: the isentropic magnetization is also isothermal within the range of existence of the domains. The bulk polarization is equal to: Pav.P(X)P(2~~)P, that is, according to eq. (98): Pay.

=

(100)

~/q.

It varies linearly with ~ in the whole range of existence of the domains. The parallel susceptibility X 11P5~./~

l/q,

(101)

is independent of p, that is of temperature. It can be shown, by a calculation similar to that leading to eq. (59) that it is equal to the transverse susceptibility Xi. The constancy of X11 is a characteristic difference of ferromagnets with domains, with respect to antiferromagnets. 43Ca resonance, evidence for ferromagnetism [24,25]

5.2. The

The most direct evidence that, after demagnetization at negative temperature with H 0 II [111], the phase of the fluorine system is a ferromagnet with domains, is provided by the observation of 43Ca resonance. theThis observation is made with highly polarized 43Ca spins. Experimentally, the solid effect used for polarizing the 19F is ineffective for polarizing the 43Ca. An obvious limitation, among others, to the polarization process is the smallness of the spin diffusion constant in the dilute 43Ca spin system, which severely limits the rate at which the polarization of the 43Ca spins close to the paramagnetic impurities can be carried to the distant ones. The polarization of the calcium is obtained

46

M Goldman, Nuclear dipolar magnetic ordering

as follows [26]. After the adiabatic demagnetization of the fluorine spins, one applies a strong r.f. field at a frequency close to the resonance frequency of the 43Ca. The low temperature of the spin—spin reservoir ~C~J+ ~ is imparted through thermal mixing to the effective Zeeman interaction of the 43Ca spins in their rotating frame, which results in an increase of their polarization. Polarizations as high as 80% are readily produced. One then uses a small r.f. field to observe the 43Ca resonance signal during slightly saturating linear sweeps of the external field. At low entropy, the 43Ca signal consists of two well-separated lines, which proves that there are two types of calcium spins experiencing different and well-defined dipolar fields superimposed on the external field. This observation rules out the possibility of a two-sublattice antiferromagnetic structure, in which the dipolar field at the calcium sites would be zero. in the antiferromagnetic state previously investigated, the calcium resonance signal is indeed observed to consist of a single line. In contradistinction, the observation of two lines follows immediately from the existence of macroscopic domains with opposite fluorine polarizations, since 43Ca spins located in different domains will experience different non-zero dipolar fields from the fluorine spins. Figs. 1 8 and 1 9

--—~

Fig. 18. 43Ca and

191.-

_

resonance signals (left and right, respectively) inCa F 2, in the ferromagnetic structure with domains

produced by adiabatic demagnetization with H0 Ii [1111 and T < 0.

M. Goldman, Nuclear dipolar magnetic ordering

Fig. 19.

47

43Ca and 19F resonance signals (left and right, respectively) in Ca F 2, in the antiferromagnetic structure produced by

adiabatic demagnetization with H0 II [100] and T< 0.

43Ca resonance signals observed after fluorine adiabatic demagnetization at negative temperashowwith H ture 9F resonance signals 0 parallel to [Ill] and [100], respectively. Also shown are the ‘ observed in both cases. They look very much alike, and their observation would not discriminate between the two different ordered structures. Another remark is that the relative intensities of the 43Ca lines, which are equal in zero effective fluorine field, vary monotonically with effective field during an adiabatic remagnetization of the fluorine system. The intensity of each line being proportional to the number of calcium spins present in each type of domains, this shows that the relative sizes of the different domains vary continuously with external field, i.e. that these domains are of macroscopic sizes with respect to the interatomic spacing. This completes the proof that the fluorine state is a ferromagnet with domains. Quantitatively, the dipolar fields experienced by the 43Ca spins in each type of domains is equal to that experienced by fluorine spins. To see this, let us consider a calcium spin in a slice of fluorine polarization p, and let us surround it with a sphere within the slice, of radius much larger than the internuclear spacing. The dipolar field created by the fluorine spins inside the sphere is zero. For computing the field created by the 19F spins inside the slice but outside the sphere, we can

48

M Goldman, Nuclear dipolar magnetic ordering

treat their magnetization as continuously distributed, and we obtain a contribution: —4irM~+ ,iTMF = —~,irMF= —~ir(y1h/2a3). The contribution from the other domains is computed by averaging their polarizations, in a fashion similar to that leading to eqs. (90), (90’). The final result is: hA

—2(q/-y

=

1)p(l hB

—

x)

(102)

2(q/y1)px,

=

where q is the same as in eq. (90). According to eq. (102) these dipolar fields are indeed the same 9F spins. as Using those experienced byfor thex ‘one finds: the value (98) hA

=

(—qp

hB

=

(qp

+

~

(103)

+

The difference of dipolar fields: hA

—

hB

=

—2qp/y~,

(104)

is independent of ~, and the centre of gravity of the 43Ca resonance is constant: h 0

xhA +(l —x)hB

=

0,

according to eqs. (102). Experimentally, one performs an adiabatic demagnetization of the fluorine system to zero effective followed series of after partialeach adiabatic remagnetizations toexperiment various effective 43Ca field, resonance signalbyisaobserved of these steps. The whole takes fields. place The in time short compared with the fluorine spin—lattice relaxation time (about 1 hour), so that one can assume that the entropy remains constant. The fluorine polarization is also measured at each field, from the area of the fluorine resonance signal. The observed 43Ca line intensities, 43Ca resonance frequencies and ‘9F polarizations as a function of effective fluorine frequency z~are reported on figs. 20, 21 and 22, respectively. The solid lines are the Weiss-field predictions of eqs. (98), (100) and (103). One used the theoretical value: qfy 1

55 G,

~L5

9F Effective field (B I ‘

Fig. 20. Relative intensities of 43Ca resonance lines as a function of effective field, in Ca F 2 ferromagnetic with domains. The solid lines are the Weiss-field predictions for p

=

0.7.

49

M Goldman, Nuclear dipolar magnetic ordering

19F Effective field (B)

5

Fig. 21. Resonance frequencies of43Ca resonance lines as a function of effective field, in Ca F 2 ferromagnetic with domains. The solid lines are the Weiss-field predictions for p = 0.7.

“:

centre of gravity of each pair of lines.

19F Effective field (B)

Fig. 22. Fluorine polarization as a function of effective field in Ca F 2 ferromagnetic with domains. The theoretical approximation is the same as in figs. 20 and 21.

and, for a best fit with experiment, the polarization p tative fit with these predictions.

=

0.7. The results exhibit a reasonable quali-

5.3. Dipolar field as a function of energy 9F polarizations is equal to difference between the resonance dipolar fields of It domains with opposite ‘ theThe splitting between the 43Ca lines. was measured as a function of dipolar energy, as deduced from the first moment of the fluorine resonance signal. The fluorine system was in zero effective field, and its energy was varied either by successions of slightly saturating passages, or by spin—lattice relaxation, depending on the experiments. According to the Weiss-field theory, the energy per spin is equal to E/N~qp2, whence, according to eq. (104): =

(4/y

1”2. 1)(qE/N)

(105)

50

M Goldman, Nuclear dipolar magnetic ordering

(0

w a’

C

~.~10 O

I

0

~

Wens

0

Restricted Trace 1

Dipolar

Energy per spin

Order

C

6

kHz

43Ca resonance lines as a function of dipolar energy in Ca F Fig. 23. Field separation between 19F spins is zero. The curves correspond to the predictions of the Weiss-field 2 ferromagnetic and 1st-order with restricted-trace domains. The effective field for the approximations.

The experimental results are shown in fig. 23, together with the prediction of Weiss-field and 1storder restricted-trace approximations. In the latter approximation, we did not take into account the impurity dipolar fields, which is very crude in the present case since one computes, for H 0 II [Ill]: 2/y2 0.74 G2, D whereas one measures: D’2/ 2 1.59 G2 7 (D2 and D’2 are defined by eqs. (16) and (58)). The measurement of small 43Ca line splittings is not very accurate, because of the finite width of the individual resonance lines. The results are qualitatively as expected. The transition energy seems to be higher than the restricted-trace prediction, within experimental accuracy. What is more serious is that for the largest splittings, the energy per spin seems to be smaller than the Weiss-field value. Since the actual energy must be larger than this, because of the short-range energy, one is led to think that a sizeable fraction of the fluorine spins is not ferromagnetically ordered and does not experience the normal dipolar field, i.e. that the ferromagnetic structure is not perfectly realized. 5.4. Reproducibility of the domains When one produces a ferromagnet with domains, by adiabatic demagnetization of the 19F in the rotating frame, at negative spin temperature and with H 0 I [111], the positions of the individual domains are nearly reproducible from one experiment to the other. This is demonstrated by the following experiment. i) After production of the ferromagnet by fluorine demagnetization, one can materialize the 43Ca spins in different domains. For instance, domains by imparting different polarizations to the since the two calcium resonance lines are due to 43Ca spins located in different types of domains,

M Goldman, Nuclear dipolar magnetic ordering

51

one can saturate the 43Ca polarization in one type of domains, by irradiating the corresponding calcium resonance line, while leaving unaffected the calcium polarization in the other type of domains. Alternatively, one can irradiate the 43Ca spins at exact resonance, i.e. midway between the two absorption lines. Calcium spins located in domains with opposite fluorine polarizations experiencing opposite dipolar fields, they will acquire opposite polarizations. The differential polarization of the calcium spins in the different fluorine domains constitutes an imprint of these domains. Let us suppose for the sake of definiteness that the calcium is prepared with opposite polarizations in the two types of domains. ii) The fluorine spins are remagnetized. All fluorine polarizations become equal and the domains disappear. Their calcium imprint is however still present as long as spin diffusion among the calcium spins has not homogeneized their polarization. We suppose for the moment that calcium spin diffusion is negligible during the whole experiment. iii) The fluorine system is demagnetized back to zero effective field. The calcium signal is the same as in step i), except for a decrease in intensity of 30% to 40%. This proves that the new domains occupy the same place, to within about 30%, and have the same orientation of their polarization, as the initial ones. If indeed the new domains were randomly located with respect to the old ones, each type of domain would contain on the average as many 43Ca spins polarized up as down, and the calcium signal would vanish. This reproducibility of the domains was quite unexpected and has not received a detailed interpretation. Several of its features were investigated. They are the following. a) After production of the ferromagnet with H 0 [[111] and the materialization of its domains by opposite calcium polarizations, one rotates slowly the sample along an axis perpendicular to the external field H0. In a large range of orientation of the field around the direction [1111 the stable state is still expected to be a ferromagnet whose domains are thin slices perpendicular to the field. During the rotation, the calcium imprint of the initial domains remains fixed with respect to the crystal, i.e. the slices with opposite calcium polarizations become tilted with respect to H0. If the fluorine domains evolved so as to be at equilibrium at all times, they would be tilted with respect to the calcium imprint, and the calcium signal would decrease with rotation angle, and vanish when 0 d/L ~ 1, where d is the domain thickness and L its transverse dimension. Experimentally, the calcium signal remains unaffected when the rotation angle is less than 10°. At 00 10°,it disappears suddenly, which shows that the fluorine domains suffer an abrupt reorientation. If then the crystal is rotated back to its initial orientation, the calcium signal reappears, with an intensity decreased by 30 to 40%: the fluorine domains have come back to their initial positions. It is a tempting hypothesis that the positions of the domains are not determined by surface boundary conditions but rather by their pinning to some well-defined centres inside the sample. If this is the case, the energy of pinning of the domains can be estimated by stating that the tilted fluorine domains reorient when the corresponding gain in dipolar energy equals the pinning energy. For slices whose axis makes an angle 0 with the field H0, the dipolar energy in the Weiss-field approximation is equal to: 2-(3 cos2O 1)/2. -~

—

E(0)~~Nqp

Let 00 be the rotation angle at which the domains reorient. We obtain, for the pinning energy per spin:

52

M. Goldman, Nuclear dipolar magnetic ordering =

[E(0) —E(00)]/Na’jqp2 sin2O0

Forp = 0.7, and 0~= 10°,this yields c/IV~ 0.12 kHz. 43Ca spins. The b) The reproducibility the domains is notbut, affected by thefluorine dipolarremagnetization field of the fluorine—calcium dipolar of energy is very small since after the 43Ca imprint is the only remmant of the initial domains, this energy might conceivably trigger the formation of the domains during the second fluorine demagnetization. This possibility is ruled out by comparing the results of two experiments. They differ by the imprint of the domains produced after the first fluorine demagnetization. In the first experiment the calcium spins are polarized up in the domains where the fluorine polarization is pointing up, and saturated in the domains with fluorine polarized down. It is the opposite in the second experiment: the calcium is saturated in the domains up, and polarized up in the domains down. The average 43Ca dipolar fields at the sites of the fluorine spins are opposite in these two cases. If the formation of the domains after the second demagnetization were determined by the dipolar field of 43Ca, the final fluorine polarizations of the domains would be opposite in experiments I and 2. Experimentally, the final domain polarizations are always parallel to the initial ones. c) The reproducibility of the domains is the same whether the effective field in the second demagnetization is parallel or antiparallel to that in the first demagnetization. The two possibilities are schematically shown in fig. 24. In both cases, during the first demagnetization small domains of polarization down appear in a sea of fluorine spins polarized up. The same happens during the second demagnetization in case A, whereas it is the opposite in case B: small domains polarized up appear in a sea of spins polarized down. It could have happened that the centres of nucleation of domains up and those of nucleation of domains down were independent. If this had been the case, the domains at the end of experiment B wou~dnot recover their initial positions and the final calcium signal would vanish, or at least be much smaller than at the end of experiment A. This is ruled out since the final calcium signal is the same in both cases.

~i;

~xper~ent

ci~

•~i

~0 SI

110 >

time

a

a a

Experiment B

Fig. 24. Schematic variation of the effective field in two experiments on the reproducibility of ferromagnetic domains (see text).

M Goldman, Nuclear dipolar magnetic ordering

53

5.5. Thickness of the domains It was found in section 2 that the ferromagnetic domains were thin slices perpendicular to the field, but nothing was said of their thickness, nor of that of the domain walls. An approximate estimation of these quantities will be presented first, before describing the experimental investigation. The calculation assumes that the slices extend through the whole sample. It is very crude and aimed at giving only an order of magnitude for the domain thickness. An elaborate treatment would be pointless since, as seen below, the actual domain thickness is much smaller than expected. Let us approximate the domain to a disc of radius R and thickness d. The polarization is equal to p throughout the disc, except in the wall and close to the edge of the disc, because of surface effects. We use an idealized description where the polarization is zero in a plane sheet of thickness e (the wall) and in a ring of thickness csd, where ~ is a coefficient of the order of unity. This idealized domain is shown in fig. 25. In the equilibrium configuration the ratio of polarized-tototal volume is maximum. We have: 1”poi. = ir(R ad)2(d —

—

~rR2d thatis, withd/R, e/d~ I:

_

—

2csd/R

—

e/d.

(106)

This ratio is maximum for: d2

=

eR/(2a).

(107)

The domain thickness depends on the wall thickness e, which must now be estimated. We use the local Weiss-field approximation to relate the ndividual spin polarizations to their local frequencies, and we assume that the polarizations are constant in planes perpendicular to the direction Oz of the external field. That is: p(z)

=

tanh[—~/3w(z)] .

(108)

The Fourier transforms of p and w are related through: w(k)A(k)p(k).

(108’)

The only non-vanishing components are in the model those for which k II Oz. A(k) was computed

!‘IIIII’ 11111!~ Fig. 25. Model of a ferromagnetic domain. The polarization is zero in the dashed part and homogeneous in the central part.

54

M. Goldman, Nuclear dipolar magnetic ordering

for several discrete values of k as explained in section 2, according to the tabulated figures of ref. [11]. They fit closely a variation of the form: A(k)

=

q

exp(—~2k2),

(109)

with q = (8ir/3)(’y2h/2a3)

and

0.5 a

where a is the fluorine lattice parameter. The system of eqs. (108) to (109) is difficult to solve exactly. We can guess from eq. (109) that the wall thickness is comparable with ~, that is with a. To make this estimate more precise, although still approximate, we can use a trial function for the variation of p(z) which must both be mathematically simple and exhibit the qualitative behaviour expected for p(z). The simplest form is an error function: p(z)-~-~1/~fexp(_y2/2?72)dy.

(110)

p(z) is equal to ±p 0for z =

±o°,

that is in the bulk of the adjacent domains.

We obtain: 0

122

p(k)—2--j,~exp(—~k~ ),

and: 2(~2+ w(k) = 2-~~exp[—~k whence: pq’.f2 exp[—~y2/(~2+

_~_f

W(Z)

~J~j2

For z

=

±°°,

p2)]

dy.

+112)0

eq. (108) then yields:

p 0

tanh(—~/3qp0),

=

which is as expected in the bulk of the domain. The inverse temperature is equal to: ~3z —(2/q)(u0/p0), with u0 = tanh’(p0). The coefficient ~ is adjusted so that, in the limit when z is small, we have:

whence, according to eqs. (110) and (111): =

~

(ill)

M Goldman, Nuclear dipolar magnetic ordering

55

With these forms of p and w, eq. (108) is not strictly verified between the limits z small and z infinite. This could be amended by using for p(z) an appropriate combination of error functions. Such a complication is not warranted by the approximate character of the theory. The wall thickness is of the order of 2~,that is: c/a~2~/2~(u~/p~l)_h/2. —

(112)

This equation predicts that the wall thickness c is a decreasing function of p0, as well as the domain thickness, according to eq. (107). For p0 = 0.7, a value typically achieved in the experiments, eq. (112) yields: e/a~1.37, whence c 3.7 A. Inserting this value into eq. (107) with R 0.7 mm yields: 2A. (113) d~3600(a)” However crude this estimate, one expects a domain width of a few thousand angströms in the central portion of the spherical sample, and which is larger the smaller the domain polarization. Experimentally, once the ferromagnetic structure produced by adiabatic demagnetization, the positions and therefore the sizes of the domains do not change during the subsequent warming of the system through spin--lattice relaxation. This conclusion is reached by observing the change with time of the 43Ca resonance signal when the calcium spins have opposite polarizations in the initial domains of opposite fluorine polarizations. If the sizes of the domains changed in the course of relaxation, one would soon reach a state where on the average each type of domains would contain as many calcium spins polarized up as down, and the calcium signal would vanish. This iS not what is observed, and the experimental decrease of the calcium signal amplitude is as expected from the sole decrease of the distance between resonance frequencies. The first attempt to measure the domain thickness is based on fluorine spin-diffusion. The phenomenon of spin—diffusion is well known in nuclear magnetism (see for instance [14, ch. VI): it is produced by flip—flop transitions between like spins and tends to homogeneize their polarization throughout the sample. If a ferromagnet with domains, i.e. with a highly inhomogeneous polarization, is to be stable at all, there must be some mechanism inhibiting spin diffusion among the fluorine spins. This mechanism is best understood by considering first the simpler case of spindiffusion in an external inhomogeneous field [27]: a flip—flop between two spins experiencing different fields does not conserve Zeeman energy, and can take place only if the Zeeman energy imbalance is compensated by a change of local dipole—dipole energy. The change of energy of the dipole—dipole interactions is accompanied by a variation of their entropy. Eventually, the decrease of short-range dipolar entropy limits the continuation of flip—flop transitions and the equilibrium nuclear state in an inhomogeneous field does not correspond in general to a homogeneous polarization. The situation is very similar in a ferromagnet with domains, except that the inhomogeneous field is a dipolar Weiss field created by the spins themselves. The flip—flops between fluorine spins in different domains do not conserve the long-range energy. They create therefore short-range energy, and the flip—flops are limited by the decrease of short-range entropy. One is then naturally led to distinguish between long-range order and short-range order. In a ferro-

0

56

M. Goldman, Nuclear dipolar magnetic ordering

magnet with domains at thermal equilibrium, there is a well-defined ratio between long-range and short-range entropies (or energies). A natural idea at this point is that if one were able to saturate continuously the short-range order, nothing would hamper flip—flops and the fluorine polarization would decrease to zero by spin diffusion. Starting from a square-wave distribution of polarizations: p(z,0)~p0 71

~

~

noddfl

(114)

d

which corresponds to domains of thickness d, we would have at time p(z,

t)-~p0

~

t:

1-sin (~~)exp(_~ -

2n

(115)

t) .

The term n = 1 becomes quickly The first moment of thedecay fluorine signal, is 2)av,very should tenddominant. asymptotically to an exponential with time which constant: proportional to (P(1) rd2/(27r2D), (116) whence: dir~/ib~.

(117)

In CaF 2, the diffusion constant D is of the order of [28, 29]: 12cm2s. D— l.6X l0 It is then sufficient to measure the time constant r to obtain the average domain thickness d. For d 3600 A (i.e. a = 1) one expects:

r

40 sec.

The method for saturating the local dipolar order consists in applying a strong r.f. field whose frequency is modulated by a few kHz around the 43Ca resonance value [30,3]. The effective calcium Zeeman interaction in the rotating frame is thermally coupled to ~ which is itself thermally coupled to ~ The non-adiabatic modulation of the former results in a heating of the latter. The heating of ~1CJ is local, because it takes place via ~ which is a local interaction. ideally, the experiment should go as follows: i) One performs an adiabatic demagnetization with an initial polarization so low that the final state is paramagnetic. One applies the strong modulated r.f. field on the 43Ca spins and observes the decay of the first moment of the ‘9F signal. Its decay time is equal to the dipolar saturation time Tsat~ ii) One repeats the same experiment, but with a high initial polarization, so that the final state is ferromagnetic. The fluorine first moment will decay with the diffusion time r, provided it is longer than Tsat~ Experimentally, the shortest saturation time obtained was Tsat 1 sec, and no difference was observed whether the initial polarization was low or high. We must then have: -~

M Goldman, Nuclear dipolar magnetic ordering

57

r~lsec, or else, according to eq. (117) (118)

d~<560A,

which is substantially smaller than expected. An independent estimate of d was obtained by observing the spin-diffusion among the 43Ca spins. In the first experiment described in section 5.4: fluorine demagnetization, materialization of the domains with differential 43Ca polarizations, fluorine remagnetization, second fluorine demagnetization, the calcium signal “memory” of 0.6 to 0.7 is observed when the second demagnetization is performed shortly after the remagnetization. As a function of the time spent in the remagnetized state, the memory of the calcium signal after demagnetizing back decreases with a time constant of the order of 20h. This variation is shown in fig. 26. It cannot be due to an evolution of the fluorine system since, as seen above, it takes less than 1 sec for the fluorine polarization to diffuse through a domain. This time is also much longer than the fluorine dipolar spin—lattice relaxation time (at most a few hours). The decay of the calcium signal memory is attributed to the spin diffusion among the 43Ca spins, which tends to homogeneize their polarization in space. The calcium memory being proportional to the difference of average calcium polarizations in the two types of domains, the thickness of the latter is given by — —

— —

(119)

d=7r\/T~~.

The spin-diffusion coefficient can be approximately estimated by a theory which will not be detailed here. The result is: 8cm2s,

D

(120)

5 1.8X l0~ which yields: ‘—‘

d—’ 100 A. 1 .8

I

I

-

0

-

500

1000

1500

Time in remagnetized state (mm

Fig. 26. Variation of the “memory” of the 43Ca resonance signal as a function of the time spent in the remagnetized state. The solid curve corresponds to eq. (115), with r = d2/(2ir2D) = 1200 mm.

58

M. Goldman, Nuclear dipolar magnetic ordering

This estimate is very rough, because of the theoretical uncertainty on D5, the experimental uncertainty on the decay time, 43Ca and also obtained for d iswhether not much larger than the spinsbecause (about the 30 value A), and it is doubtful a diffusion equation average distance between is valid over such small distances. All one can say is that the domain thickness is of the order of a few hundred A, that is considerably less than expected. This result, together with the studies of domain reproducibility, points strongly towards the existence of well-defined and fixed centres in the crystal which determine the positions of the domains. Their exact nature is not known. Possible candidates are the Tm2~ions, whose mean distance R 43Ca 5 the 130experiment A is comparable withrotation, the domain thickness. only However, the transverse observed displitting and on sample are consistent with both domain mensions L much larger than d, i.e. much larger than the mean impurity spacing R~.In that case, it is far from evident that randomly distributed impurities should lead to so precisely reproducible domain positions. ‘--‘

5. 6. Transverse susceptibility in zero field The transverse susceptibility has been measured as a function of energy by the method described in section 4.2: one computes the Kramers—Kroning transform and the first moment of the fluorine absorption signal observed with a non-saturating r.f. field. The theory of section 5.1 predicts that in the ferromagnetic state, x 1 must be constant and equal to: Xi

=

l/q.

Measurements have been made for several orientations of the magnetic field. The rotation axis, perpendicular to H0, was aligned with a binary crystalline axis. We call 0 the angle between H0 and the fourfold crystalline axis perpendicular to the rotation axis. As an example, fig. 27 shows the experimental variation of Xi as a function of(~fC~) for 0 = 65°, in the domain of stability of the ferromagnet. Xi reaches a quasi-plateau, but with a slight monotonic increase. This increase is observed for all orientations where the system is ferromagnetic. In contradistinction, at all orientations where the system is antiferromagnetic, Xi always decreases

b

/

-

~ I ~

00

~

.

‘~“Weiss

~

//00

!/ 0

~“~High Temp. 1~ Order

2

1 Dipolor Energy

per

spin

3 (

kHz

1.

)

Fig. 27. Transverse susceptibility as a function of dipolar energy in CaF2 ferromagnetic with domains. The angle between H0 and 1100] is 0 = 650. The angles between H0, and 1010] and [0011are equal.

M Goldman, Nuclear dipolar magnetic ordering

59

slightly as a function of energy. The absolute value of Xi in the ferromagnet is larger than the Weiss-field value, as in the antiferromagnetic state. The 1st order restricted trace approximation, predicts a value of Xi that is indeed larger than the Weiss-field value, but which decreases as a function of energy, contrary to experiment. There is some arbitrariness in comparing the values of x1 at different field orientations, since its variation with energy is not always the same. Fig. 28 reproduces the values of Xi as a function of angle 0 at constant energy: 2.6 kHz per spin. Except for an excess of 10—20%, the variation is consistent with that predicted by the Weiss-field approximation. The slight variation, if real, of Xi in the ferromagnetic domain has received no interpretation.

3~

[ba] Fig. 28. Transverse susceptibility in CaF2 as a function of angle The solid curve is the Weiss-field prediction.

[iii] 0

Lou]

betweenH0 and 11001, at constant energy: E

=

2.6 kHz/spin.

It is not possible from these experiments to determine with any accuracy the angle of transition between ferromagnetism and antiferromagnetism. This is accomplished by a different experiment, to be described now. 5. 7. Ferromagnetic—an tiferromagnetic transition The cleanest testsignal, for discriminating ferromagnetism and antiferromagnetism is the 43Ca which consistsbetween of two lines and one line, respectively. The energy range shape where of thethe system is ordered is known from the variation of Xi~ Starting from a ferromagnetic state, with H 0 1 [111], the sample was rotated by small steps so as to decrease the angle 0 between H0 and the axis [100], and the calcium signal was observed at each orientation. This signal undergoes an abrupt change at an angle 0 comprised between 19°and 22°: instead of two lines symmetrically shifted with respect to the Larmor frequency, one observes a strong central line with two symmetrical satellites, which corresponds to the partial transforma-

60

M. Goldman, Nuclear dipolar magnetic ordering

tion of the ferromagnet into an antiferromagnet. The satellites disappear at smaller values of 0. The transformation is reversible, when rotating back the crystal. According to the Weiss-field approximation, the transition angle is equal to 17.5°.It is independent of entropy. This results from the fact that the variation of entropy S with reduced energy E/q, where q is the Weiss-field coefficient, is a universal curve. Curves S = f(E) for different structures will therefore either be identical or will not cross. Physically, it is well known that for a first-order transition, there is a temperature at which both phases can coexist in various proportions. This implies that the curves S = f(E) for the two phases must cross. This is shown in fig. 29, where the curves S = f(E) are schematically drawn for two structures between which takes place a first-order transition. The inverse temperature at the transition is equal to the slope of the straight line AB. Both phases coexist between A and B.

E~ TtureiB~truct~e

0

Energy

E

I

D

Fig. 29. Schematic entropy-energy diagram for two different structures. The stable states are: along CB: Structure I; along AB: Structure II; along AB: Both structures I and II in various proportions.

The correction to the Weiss-field curve S = f(E) arises entirely from short-range correlation effects. In an attempt to take these into account, we have used the 1st order restricted trace approximation both for the antiferro and the ferromagnetic phases. In the last case we have neglected wall effects. The predicted phase diagram, as a function of initial polarization and field orientation, is shown in fig. 30, for a rotation around a binary axis. The dashed portion corresponds to the coexistence of both phases. At constant orientation, one predicts indeed an entropy interval of coexistence (along PQ, for instance). A less usual prediction is that at constant entropy, there must be an angular interval of coexistence. The average transition angle, 0 21°,is consistent with the experimental result. We have looked for a qualitative test of the restricted trace prediction, namely that at appropriate field orientations the antiferromagnetic phase is stable at lower entropy than the ferromagnetic phase. The experiment consists in demagnetizing the system with the highest possible initial polarization, so as to end up at point P of fig. 30. One must then observe a single calcium resonance line. As the system warms up by spin—lattice relaxation, one expects to see two side lines, corresponding to the appearance of the ferromagnet. These side lines should grow at the expense of the central line and then coalesce, when the system finally becomes paramagnetic. Experimentally, one observes the converse: there are at first two calcium lines and then appears ‘—-

M Goldman, Nuclear dipolar magnetic ordering

~21I

20

61

Para’

I

I

Initial

Antiferro Polarization

Fig. 30. Domains of stability of the paramagnetic, antiferromagnetic and ferromagnetic phases in CaF

2 at negative temperature, as a function of initial polarization and angle 0 between H0 and [1001, according to the 1st order restricted-trace approximation. Both ferro and antiferromagnetic phases coexist in the dashed part.

a small central line which grows at the expense of the initial lines. This preliminary result shows clearly that, contrary to the restricted trace theory, it is the ferromagnetic phase which is stable at the lowest entropy.

6. Experimental studies under way We have postponed until this section the description of two experimental studies in progress at the time of writing. Their physical principle is clearly defined, but the results are either still inexistant or too scarce to warrant the presentation of these experiments on the same footing as those described above. Both studies concern antiferromagnetism. The first one deals with the fieldentropy phase diagram of an antiferromagnet, and the second one is aimed at performing neutrondiffraction experiments.

6.1. Field—entropy phase diagram The purpose of this section is to investigate the dependence on external longitudinal effective field and on entropy of the transition between paramagnetism and antiferromagnetism. Although it is necessary to use an r.f. field to ensure thermal equilibrium in the rotating frame, we neglect the Zeeman coupling with this r.f. field and the transverse components of the spin polarizations. We use the Weiss-field approximation to discuss the qualitative features of the phase diagram. The first and most naive approach is to compare at constant entropy the energies of the paramagnetic and antiferromagnetic structures as a function of external field. The entropy is fixed by the polarization p in high field. For a paramagnetic structure in a field —.~/y,the polarization of each spin is still equal to p and

62

M. Goldman, Nuclear dipolar magnetic ordering

the energy is equal to: Epara

=

(N/2) ~p -

(121)

The calculations for the antiferromagnetic structure are schematically as follows. i) Let PA and PB the sublattice polarizations. Each sublattice contains N/2 spins. The entropy being the same as in high field, we have: 2s(p). (122) At constant p we can, for every possible value of PA’ calculate PB so as to satisfy eq. (122). At this stage, both values PB = ± PB I are acceptable. ii) Given suitable values of PA and PB’ we calculate the value of L~consistent with the Weissfield eqs. (45) and (46). With the notations: ~(PA)

uA

~~(PB)

=

=

tanh’(pA)

we have: = —j3w~ wi uB

—~I3w~ w~

and

uB

=

tanh’(pB),

Z~.+~A(KO)(PA PB)

(123)

—

~—~A(Ko)(pA—pB)

whence: L~=~A(Ko)(pA —pB)(uA+uB)/(uA —uB).

(124)

iii) We compare the energy (121) of the paramagnetic phase with that of the antiferromagnetic phase: EAF

=

~N~(pA

+ PB) +

~N(wApA

+ WBPB)

(125)

=~Nl~(pA +PB)+~NA(Ko)(PA PB)2-

The stable structure is that whose energy is higher. The result of this approach is that the transition should be of second order for initial polarizations such that: 2ptanh’(p)~ 1 i.e. p

‘~

0.647,

and of first order at higher initial polarizations. The prediction of first-order transitions leads us to suspect that the above approach is oversimplified, and one must look for the possibility of coexistence of both phases in a suitable range of field and entropy. This calculation is sketched below. We consider a structure consisting of paramagnetic and antiferromagnetic domains, of relative volume x and (1 x). The dipolar fields, and therefore the thermodynamic properties, depend on the shape of the domains. It can be proved a posteriori that the most favourable configuration is that when the domains are thin slices perpendicular to the external field. The theory follows the same steps as for the study of ferromagnetism described in section 5.1. Let PA’ PB and Pc be the polarizations in sublattices and ~i3 of the antiferromagnetic domains, and in the paramagnetic domains, respectively. —

-~

M Goldman, Nuclear dipolar magnetic ordering

63

i) The calculation of the frequencies experienced by the various spins is similar to that leading to eqs. (90), (90’). The results are:

W~

~

L~+W~

z~+ = +(l x)q[p~ ~(PA ~PB)], where q is the same as in eqs. (90), (90’). ii) The expressions for the energy and the entropy are: =

F/N

=

x{~~Pc

~

+ ~

--

Wcpc}

+

(126)

—

~(l

—

x){~~(PA + PB) + ~(WAPA + WBPB)},

S/N=xs(p~)+~(l—X){5(PA)+5(PB)}.

(127) (128)

iii) We set equal to zero the partial derivatives of —j3F = S 13E with respect to PA’ PB’ Pc and x, which yields these four quantities as a function of j3. The first three equations yield as they should the Weiss-field equations: —

p~=tanh[—~I3wfl

(XA,B,C).

A result of the theory is that at constant temperature PA’ PB and Pc remain constant and that x depends linearly on z~in the range of coexistence of the two phases. The theoretical field of transition is plotted as a function of initial polarization in fig. 31. As before, the Weiss-field approximation is inadequate at low initial polarization. For using the restricted-trace approximation, the procedure is very similar to the previous one, the only difference being that one uses the restricted-trace expressions for the energy and the entropy in place of eqs. (127) and (128). The results obtained by the 1st-order restricted-trace approximation are also shown in fig. 31.

//

5

//

Paramagnetic

/

-4. /

/

//

.5

~ 3

Restricted

/‘

WeissN,,~,/

~ 2 UI

Mixed phase para-antiferro

,//

1

~

~--

Trace

.]~I___ ~

,/

Antiferromagnetic

0 Initiat

.5 Polarization

Fig. 31. Phase-diagram of the para—antiferromagnetic transition in CaF

1

2, after adiabatic demagnetization at T < 0 with H0 1111001. The experimental points are the “turn-over” field values of the fast-passage signals, and the curves are the predictions of the Weissfield and 1st-order restricted-trace approximations.

:

64

M. Goldman, Nuclear dipolar magnetic ordering

J~~~o28

—20 —15 —10 —5 0 5 10 15 DIstance from resonance(G) Fig. 32. Fast-passage dispersion signal of with initial polarizations Pi

=

20

19F in CaF

2, at negative spin temperature with H0 Ii [1001. Right: experimental signals 0.28 and 0.59. Left: Weiss-field prediction for the fast-passage with Pm = 0.59.

Experimentally, the only hint obtained so far at the value of the critical field was from the observation of the fast-passage signal. We have discussed in section 4.2 the amplitude of the fastpassage signal at exact resonance. As for the shape of the full signal, it suffers a qualitative change below the transition: whereas it has the usual Lorentzian-like shape in the paramagnetic phase, it exhibits a flat top in the antiferromagnetic phase. As an example, fig. 32 shows the fast passage signals observed with initial polarizations p~= 28% and 59%, respectively, that is at entropies above and below the transition. Also shown is the fast-passage signal predicted by the Weiss-field approximation for p, = 59%. The restricted-trace approximation yields a very similar curve. The qualitative agreement with the observed signals is satisfactory. (In ref. [81, where the mixed para— antiferromagnetic phase was ignored, the Weiss-field signal shape predicted for a second-order transition between paramagnetism and antiferromagnetism contrasted much more with experiment than that of fig. 32.) The field values where the flat tops begin are reported as a function of initial polarization in fig. 31. There is an overall reasonable agreement with the predictions of the 1 st-order restricted-trace approximation for the upper critical field. An experimental indication for the actual existence of a mixed phase has been yielded by the following preliminary experiment. The system of polarized fluorine spins was subjected to an adiabatic demagnetization in the rotating frame was stopped at about 3.5 Ghas from resonance. 43Ca which resonance signal. The 43Ca spins been polarThis was followed by an observation of the ized in a previous experiment as explained in section 5.2. The calcium signal consisted of two lines of unequal intensities. This observation is inconsistent with a structure of the fluorine system either purely paramagnetic or purely antiferromagnetic. The average dipolar field at the calcium sites is indeed zero in both cases, and one observes only one line when the fluorine system is either in high field or in zero effective field. On the contrary, if the fluorine system is in a mixed phase, the calcium spins experience non-zero dipolar fields which are different in the paramagnetic and antiferromagnetic domains, whence the appearance of two resonance lines. The frequency shifts are calculated as in section 5.2. They are equal to: Wpara

WAF

where q’(y 5/y1)q,

=

=

(1 —x)q’[p~ —xq’[p~

—

—

~(PA

~(PA +

~PB)]

PB)]

‘

‘

(129)

M. Goldman, Nuclear dipolar magnetic ordering

65

whereas the line intensities are proportional to the respective volumes of the para- and antiferromagnetic domains, that is x and (1 x). The observation which has been made is roughly consistent with these expectations. One can hardly observe the calcium resonance by the same method as that described in section 5.2 for the ferromagnetic phase, where one performs successive remagnetizations to various distances from fluorine resonance and records at each time the calcium signal. The reason is that in the available samples of CaF2, the dipolar spin-lattice relaxation is very short when H0 Ii [100], —

less than 3 mm, whereas it is about 1 h for H0 II [111]. The reason for this anisotropy is not yet understood. In the experiment currently being planned, the calcium signal would be recorded on the run at regular time intervals in the course of the fluorine demagnetization. A prerequisite to this experiment is an important increase of the signal-to-noise ratio of the calcium resonance detector.

6.2. Neutron diffraction We mention an experiment still in a preliminary stage, of neutron diffraction by nuclear antiferromagnets. Neutron diffraction is a unique method for giving a direct confirmation of the existence and the structure of an antiferromagnetic phase [31], based on the dependence of the neutron scattering amplitude on the spin of the scattering centres. In the case of paramagnetic ions, this spin-dependence is due to the magnetic interaction between the neutron and the electrons. Nuclear magnetic moments being smaller by three orders of magnitude than electronic magnetic moments, the sole magnetic interaction would not yield an observable spin-dependence of the neutron—nucleus scattering. There is however another source of spin-dependent scattering, the strong interaction between the neutron and the nuclei, which can be much larger than the magnetic one. The spin-dependent part of the neutron scattering amplitude has been measured for a number of nuclei by the so-called “pseudomagnetic precession” method [32]. One defines a nuclear pseudomagnetic moment ~ which is the moment whose magnetic interaction with the 9F, 7Li neutron would yield the same spin-dependent scattering amplitude as that observed. For ‘ and 1H the values of p~are, in Bohr magnetons: -

‘Li :

=

—0.017

=

—0.62

±0.002

±0.03

1H :Ii*=+5.4. The value of j~ is so low for 19F that it is hopeless to observe antiferromagnetism in Ca F 2 by 7Li nuclei, but the best candidate is the proton. neutron diffraction. This might be possible with The neutron diffraction experiments in preparation will be done with antiferromagnetic lithium fluoride and lithium hydride. 7. Conclusion The possibility of producing and studying magnetic ordering in nuclear spin systems, subjected to dipole—dipole interactions arose from the conjunction of two ideas: firstly that the dynamic polarization of the nuclear spins was an effective practical way for reducing their entropy; and

66

M. Goldman, Nuclear dipolar magnetic ordering

secondly that the spin-temperature concept, which had proved so fruitful at high spin temperature, could be extended to very low temperatures. From this starting point on, it was shown how the combination of the theoretical machinery of spin-temperature theory with simple approximations to magnetic ordering (essentially the Weiss-field and the restricted-trace approximations), could be used to predict the nature of the ordered structures as well as their main properties. The experimental results, still limited in number, establish convincingly the physical reality of ordered structures and the overall consistency of the theoretical description of their properties. The exact nature of an antiferromagnetic structure could be unambiguously ascertained by a neutron diffraction experiment. This is ruled out in calcium fluoride by an accidental physical limitation: the smallness of the spin-dependent neutron—fluorine scattering amplitude, but can be attempted with other compounds, such as LiF or Lull. The evidence for the actual production of the expected antiferromagnetic structure in calcium fluoride can only result from the agreement, at least qualitative, between experiment and theory for as many properties as possible. The results obtained for an expected antiferromagnet: variation of parallel and perpendicular susceptibilities as a function of entropy or energy, transition field, non-uniform parallel susceptibility to the 43Ca inhomogeneous dipolar field, spin—lattice relaxation, all of which agree qualitatively with the prediction of approximate theories, constitute a converging set of evidence that we have indeed produced the anticipated structure. The existence of the ferromagnetic structure of calcium fluoride is firmly established by the behaviour of the 43Ca resonance signal. A very analogous use of a nuclear probe was made long ago for the study of ordered phases of electronic moments [33]. Several properties of the ferromagnet in CaF 2 depart significantly from those predicted, as for instance the size of the domains. Other properties, such as the reproducibilities of the domains, were completely unexpected. The departure from theoretical prediction might be due to the perturbing influence of the paramagnetic impurities. Accounting for them, even qualitatively, is a difficult task as yet not undertaken. It is however an asset for NMR experimental techniques to have demonstrated their ability to reveal these “facts of life” with some detail. This is particularly true for the reproducibility of the domains, which could hardly have been detected by other methods. Confronted as we were with a novel physical field, it seemed preferable to begin by concentrating on a few ordered structures and to perform on them a variety of different studies, rather than to perform the same type of measurements (such as for instance that of the transverse susceptibility) on many different substances. Apart from strengthening the plausibility of the predicted structures, it illustrates more fully the versatility of the experimental approach to the properties of nuclear magnetic ordered phases. Another reason for this choice is that the production of high dynamic nuclear polarizations with few paramagnetic impurities present, raises new and difficult problems for every new compound, one might even say for every new sample. Not much was said of the experimental difficulties, which are numerous and severe not only for the production of the dynamic polarization but at all steps of the study for obtaining results with sufficient reliability and precision, and which are the bottleneck limiting the overall yield of significant results per unit time. The study of nuclear magnetic ordering has proved the validity of the spin temperature concept at very low temperature and given a physical meaning to temperature below one microkelvin which, for other systems, would hardly be distinguishable from absolute zero. Its major originality however, is the possibility of performing the adiabatic demagnetization in the rotating frame and of choosing at will the sign of the temperature. The production of ordering at negative temperature

M. Goldman, Nuclear dipolar magnetic ordering

67

with respect to effective interactions as viewed from a rotating frame, and the possibility of changing the nature of the ordered structure by a small rotation of the sample are privileges of nuclear spin systems, unknown in the rest of magnetism.

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