Nuclear magnetic ordering in Ca(OH)2. II. Restricted trace approximation

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Nuclear magnetic ordering in Ca(OH)2. II. Restricted

trace approximation

G.D.F. van Velzen, W. Th. Wenckebach

To cite this version:

Nuclear

_{magnetic ordering}

in

_Ca(OH)2.

II. Restricted

trace

approximation

G. D. F. van Velzen and W. Th. Wenckebach

Kamerlingh

Onnes

Laboratory

P.O. Box 9504, 2300 RA Leiden The Netherlands

(Reçu

le 1 er décembre 1989, révisé le 5 mars 1990,

accepté

le 21 mars

₁₉₉₀₎

Résumé. 2014 Les

_phases

_magnétiques

du

_{système quasi}

bi-dimensionnel des

_spins

des _protons dans

(OH)2Ca

sont étudiées en utilisant la méthode de la trace restreinte. Les résultats sont utilisés

pour la

description

de l’évolution du

_système

des

_spins

nucléaires

_pendant

une désaimantation

adiabatique

dans le référentiel tournant

_(DART)

utilisée pour refroidir ce

_système

de

_spins.

Le cas où la

_température

des

spins

nucléaires est

_négative

et le

champ

magnétique

extérieur est orienté

_{parallèlement}

à l’axe c du cristal est étudié en détail. Pour des

_champs

effectifs nuls on

prévoit

un état

_{antiferromagnétique. Néanmoins,}

on trouve

_qu’il

est

_{possible qu’on}

_{passe par} un

état

_{ferromagnétique pendant}

la _DART, ce résultat

_expliquant

les observations

_{expérimentales}

antérieures.

Abstract. 2014 The

magnetic phases

of the

_quasi

two-dimensional

_{proton spin}

_system in

_Ca(OH)2

are studied

_using

the restricted trace

_{approximation.}

The results are used to describe the evolution of the nuclear

_spin

_system

_during

the adiabatic

demagnetization

in the

rotating

frame

(ADRF),

which is used to cool this

spin

system. The case that the nuclear

_spin

_temperature is

negative

and the external

_magnetic

field is oriented

_parallel

to the

_crystalline

c-axis is studied in detail. If the effective field is

_equal

to zero, the

ground

state is found to be

antiferromagnetic.

However,

during

the ADRF one _{may pass}

_through

a

_{ferromagnetic phase, explaining}

earlier

experimental

results. Classification

Physics

Abstracts 75.30K - 76.60

1. Introduction.

This _paper concerns the

_{ferromagnetic ordering}

of the

_quasi

two-dimensional

_proton

_spin

system

in a

_{Ca(OH)2 (Fig. 1),}

observed

_by

J. Marks

[1]

]

and J. C. M.

_Sprenkels

[2].

Their studies were

_performed

_using

a three

_step

_{cooling procedure,}

introduced

_{by Abragam}

and coworkers

_[3].

_First,

the

_sample

is cooled to 0.5 K

_using

a

’He

_evaporation

_cryostat

and a

magnetic

field

_B°

of about 3 T is

_applied.

_Then,

the

_polarization

_p,, =

1 z)

II

of the

_proton

spins

I

= 1/2 ,

is increased to

typically p ° =

_0.7,

_{using dynamic}

nuclear

_{polarization (DNP) [4],}

corresponding

to a reduction of the

_entropy

of nuclear

_spin

_system

to

where N is the total number of

_spins.

We have rendered the

_entropy

dimensionless

_by

_dividing

it

_by

Boltzman’s constant k.

Fig.

1. -

(a) Hexagonal

unit cell of

Ca(OH)2 showing

the

_position

of the ions and the cell _parameters.

(b)

The structure of a

_layer

of protons shown in a

0,

0, 1

_cross

section of the unit cell.

_{Open spheres}

2

are

0.355 Â

above and solid

_spheres

are

0.355 Â

below

_{0, 0, 1}

].

In the final _step of the

cooling

process, the Zeeman order which has been

produced by

the

DNP,

is transformed into

_dipolar

order. This is achieved

_by

an adiabatic

_{demagnetization}

in the

_rotating

frame

(ADRF) [4].

In this

_step

the

_{externally applied magnetic}

field

Bo

is

_kept

constant

_contrary

to the

_practice

in a real adiabatic

demagnetization.

Instead an rf field

_{2 B1}

cos w t is

_applied

in a direction

_{perpendicular}

to the

_applied

static

_magnetic

field

Bo.

Its

_{frequency w}

is chosen in the

_{neighbourhood}

of the nuclear Larmor

_frequency

wo =

y

1 Bo 1 ,

where _{y = 2 7r .} 42.5759 _x

106 s-1

T-1

1 is the

gyromagnetic

ratio of the

proton

spins. During

the

ADRF,

one first varies this

_frequency

until it is

_equal

to _-wo.

_{Subsequently,}

the

_amplitude

_{21 B11 1}

of the rf-field is reduced to zero.

In order to describe the nuclear

_spin

system

during

an

_ADRF,

one uses a frame of reference

(x,

y,

z )

rotating

at the rf

_frequency

around its

_z-axis,

which is chosen

_parallel

to the

externally applied magnetic

field

_Bo.

_Furthermore,

the x-axis of this

_rotating

frame is chosen

parallel

to the

_component

of the rf

_field,

which rotates in the same direction as the

_rotating

frame. In this

_rotating

frame of

_reference,

the Hamiltonian is

_given

in

_frequency

units

_{by :}

where d _{= w0 - w} is the so-called

_longitudinal

effective field in the

_rotating

frame and

w ₁ the transverse effective field.

_Furthermore,

_{JC’ D}

is the truncated

_dipolar

Hamiltonian :

Here h is Plancks constant,

_r;j

is the vector

_connecting

the

_spins

I’

and

Ii

and

Oi. .1

is the

_angle

between

_ri

_and

the

_{externally applied magnetic}

field

_Bo.

As in our

previous

paper

[5],

we assume

the shape

of the

sample

such that

Using

this Hamiltonian in the

_rotating

_frame,

an ADRF

simply

consists of

reducing

the effective fields L1 and w 1

adiabatically

to 0.

Thus,

this ADRF leads to a reduction of the

temperature

in the

_rotating

frame. In order to find the nature of the various

_{magnetic phases}

during

and after such an

_ADRF,

we need to determine the

_{phase diagram}

of the nuclear

spin

system

at constant

_entropy

as a function of the effective fields 4 and _{w 1. In} a

previous

_paper

[5],

we

_performed

such

_{calculations,}

_using

the molecular field

approximation,

and

applying

to the circumstances of our

_experiments,

i.e. the case that the

_longitudinal

effective field L1 is

negative

before the

ADRF,

so the

_spin

_temperature

is also

negative

and while the

externally

applied magnetic

field is

_parallel

to the

_crystalline

c-axis.

In these molecular field calculations we considered the

paramagnetic phase

shown in

figure

2a and two

_types

of ordered

_phases,

a

ferromagnet exhibiting

a domain _structure, where the domains have the

_shape

of

_pancakes

i.e. consist of _many

_planes

of

_proton

_spins

as shown in

figure

2b and furthermore an

antiferromagnetic

state which is shown in

_figure

2c. In both ordered

_phases,

the nuclear

spins

within a

_{crystal plane}

are

_parallel.

The two cases differ however in the direction of the

_{magnetization}

in successive

_planes.

While in the

_{ferromagnetic}

state the

_spins

are

_parallel

in _many successive

crystal planes,

the orientation of the

spins

alternates from

_plane

to

_plane

in the

_{antiferromagnetic}

state.

The

_{phase diagram resulting}

from the molecular field

_{approximation}

is shown in

_figure

3. It

shows that the

_{experimentally}

observed

ferromagnetic phase

is

_expected

to occur for all values of the effective fields 2l and w 1 such that

where

_{A F}

is a

_dipole

sum :

taken over all

_spins

Ii

situated in the same

_{crystal plane}

as

I i.

However,

if 4 =

0,

the

ferromagnetic

and

antiferromagnetic phases

_are

degenerate,

and one cannot decide

_{theoretically,}

which one will

_actually

occur. This

_degeneracy

arises from the

_highly

two-dimensional nature of the

_proton

_spin

_system

in

_Ca(OH)2,

which is caused

_by

the fact that the

_proton

_spins

within one

_{crystal plane}

are much nearer to each other than those in different

_planes.

Then the molecular field that the

_spins

in a

given crystal plane

experience,

is

_completely

due to the

_dipolar

interaction between the

_spins

in that

_crystal

plane,

while the contribution of other

_crystal

_planes

is

_negligible.

The present paper is devoted to a more elaborate theoretical treatment of the ordered state

of the _proton

_spin

_system

in

_Ca(OH)2.

We will

_incorporate

short _range correlations

_using

the

restricted trace

_{approximation}

_{[6, 7].}

We will show that short _range order lifts the

_degeneracy

between the two

_possible

ordered states, but that

_contrary

to the

_experimental

results the

Fig.

2. -

The three

_{magnetic phases}

of the _proton

_spin

_system in

_Ca(OH)2

that are considered in this

paper. The directions x, y and z are defined in

figure

1.

_(a)

The

_{paramagnetic phase. (b)}

The

Fig.

3. _{- w 1 -} 4

_{phase diagram}

of the nuclear

spin

system in

_Ca(OH)2

for T 0 and

_{Bo parallel}

to the

crystalline

c-axis and

_according

to the molecular field

approximation.

The initial

_polarization

is

Po

= 0.4.

assume that a

_{ferromagnetic}

state is created

_during

the ADRF and that a transition to

antiferromagnetic

order does not take

place

afterwards because the time constant for such a transition is too

_long.

2.

_Principles

of the calculation of the

_{phase diagram.}

Basic to the method of the restricted trace

_{approximation}

is that each of the

_{possible phases}

of a

_system

of

interacting spins

can be characterized

_by

the

_polarization

of the individual

_spins.

_E.g.,

in the

_{paramagnetic phase}

these

_{polarizations}

are

_equal.

On the other

_hand,

in a

_{ferromagnetic phase}

_showing

domains of

type

A,

B,

etc. these

_{polarizations}

have a constant value

_{p’ =}

_pA,

_{pi =}

_{p ,}

etc. in each

_type

of

_domain,

while in an

antiferromagnetic phase having

two sub-lattices A and B of

_equal

size,

these

_{polarizations}

are

pA

and

_pB

_{respectively.}

The determination of the

_magnetic

_{phase diagram}

at constant

_entropy

now consists of two

steps.

For each of the

_{possible phases,}

we first calculate the values of the

polarizations

p’

of the individual

_spins

and the value of the

_{temperature T,}

for

_given

values of the

_{entropy s}

and the

_{externally applied magnetic}

field

_Bo.

For this purpose, we use the

_{thermodynamic}

principle

that in

_{thermodynamic equilibrium}

the

_{polarizations}

_p’

should be

such,

that

s is the dimensionless

_{entropy, /3 =}

_h/kT

is the so-called inverse

_temperature,

is the energy of the

spin

system

and J6 is the Hamiltonian of the nuclear

_spin

_system

as

_given

by equation (1.2).

Subsequently,

we determine the

_{phase diagram by investigating}

for each value of the

externally applied magnetic

field

_Bo,

which

_phase

is most favourable at a

_given

value of the

entropy.

Thus,

if the

_temperature

is

_{positive, equation}

_(2.2)

_implies

that we have to choose the

_phase

with the lowest energy

E,

while at

_negative

_temperature,

we have to choose the

phase

with the

_highest

_energy E.

Both

_steps

in the determination of the

_phase

_{diagram require}

that we obtain

_expressions

of the energy and the

entropy

as a function of the

_{polarizations}

p’

for each of the

_possible

_phases.

As a

_{starting point}

for

_obtaining

these

_expressions,

the restricted trace

_{approximation}

considers a

_{magnetic phase}

at a certain

_strength

of the

_{externally applied magnetic}

field and a

given

value of the

_entropy.

One then considers all

_quantum

mechanical

_{states 03C8>}

for which the

_polarization

of each

_spin

I

has

the value

_pi

which characterizes this

_phase.

So for each of these

_quantum

mechanical states :

Subsequently,

one defines the restricted trace Tr’ as the trace over the

_subspace

of Hilbert

space

containing

all these

_{states |03C8>.}

The value of such a restricted trace is

_evidently

determined

_by

the

_{polarizations}

_p’

of the individual

_spins,

so one may consider the set of all

possible

restricted traces of an

_operator

as a function of these

polarizations.

_Hence,

if one succeeds in

_expressing

the

_entropy

and the energy of the

spin

system

in restricted traces, one has succeeded in

_expressing

these

_{thermodynamic quantities}

in the

_{polarizations}

_pi.

The restricted trace

_{approximation}

makes two

_important

_assumptions.

_First,

it assumes that one _may

replace equation (2.2), stating

that

thermodynamic equilibrium corresponds

to a maximum of ln

L,

by

the

_requirement

that the values of the

_{polarizations}

should be such that

where

Thus one

_{replaces equation (2.2) by equation (2.6)}

to calculate the values of the

_{polarizations}

for each of the

_{possible phases}

of the

_spin

_system.

The second

_assumption

of the restricted trace

_{approximation}

is that the maximum of In

Li’

is so

_sharp

that one _may

_approximate

Using

these two

_assumptions,

the

_{phase diagram}

at constant _entropy is now obtained as

follows.

_First,

one needs to evaluate the restricted trace

_given

in

_equation

_(2.7)

for each of the

possible phases.

For each of these

_phases

one

_subsequently

calculates the

_{polarizations}

p’

of the

_spins

and the inverse

_temperature

_/3

_{by solving}

_equation

_(2.6)

under the condition

that the

_entropy

as

_{given by equation (2.10)}

has a constant value

s°.

_Finally

one determines

the _energy for each

_{phase using equation (2.9)}

and decides which

_phase

is most favourable.

3. Séries

_expansion

in the restricted trace

_{approximation.}

The evaluation of the restricted trace in

_{equation (2.7)}

is still too difficult to be

_performed

exactly.

Therefore it is

_{approximated by}

a series

_expansion.

For this purpose the Hamiltonian is

_split

into two

_parts

_Eo

and 9Y. We choose the first

_part,

such that it is constant in the

subspace

of Hilbert space over which the restricted trace is taken.

Hence,

like a restricted

trace, it can be considered as a function of the

polarizations

p i .

Then

For _{this purpose, Goldman}

_[7]

_{simply splits}

the Hamiltonian in the Zeeman term and the

dipolar

interaction term. In the case of

_Ca(OH)2

_however,

we found such a

_splitting

to lead to

divergences.

We avoid these

_divergences

_by

first

_splitting

each

_spin

_operator

in the Hamiltonian in its

_expectation

value in the restricted

_part

of Hilbert space and a so-called

_spin

deviation

_operator

_{Iiu, :}

Then the Hamiltonian

_{given by equation (1.2)}

is

_split

in a

_part

which is constant in the

_part

of Hilbert _space over which the restricted trace is taken and two terms

_{containing spin}

deviation

operators :

The first constant term in jc is

given by

Thus,

Eo (pi )

corresponds

to the

_expression

for the energy of the nuclear

spin

system

in the molecular field

_{approximation. Using}

the results of our

_previous

_paper

_[5],

we _may therefore

equate piy = pjy = 0.

The second term in JC contains the terms linear in the

_spin

deviation _{operators :}

Inserting

the Hamiltonian

_(3.3)

in

_{expression (2.7) yields}

As a

_following

_step

in the restricted trace

_{approximation,}

one

_expands

the second term in this

expression.

In first order restricted trace

_{approximation,}

this is done to second order in

_{3.

However,

the occurence of the term

_{{3 ’v}

_{1 in}

the

_exponent

_{slows down the convergence} of this

expansion considerably.

This

_problem

is

_easily

solved

_using

the fact that

_according

to

equations (2.5)

and

_(3.2)

the

_expectation

value of the second term

_Vi

₁ of the

Hamiltonian,

for all

_{states 11/1),}

over which the restricted traces are taken.

Hence,

‘v’ does not contribute

to the energy of the nuclear

spin

system

and

Using

this

fact,

we add a

term 1 B (V1),>

to the

_right

hand side of

_{equation (3.7)}

without

modifying

the result :

Now,

we insert

_{equation (3.9)}

in

_{equation (3.10)}

and

_expand

up to second order in

/3.

We note

that

_{using equation}

_(3.8)

and Goldman’s method for the calculation of restricted traces

_[7],

one finds that several lower order terms in the

_expansion

vanish :

As a result :

As remarked

above,

the restricted traces are taken over the

_subspace

of Hilbert space where

the

_{polarizations}

are

_{given by}

p’.

Hence the term

_{ln Tr’ {1}}

and the

_quantity

are functions of these

_{polarizations.}

Thus we have

_expressed

In L’ in the

polarizations

pi.

Using

equation

(2.9),

we can now _{also express the energy of the}

_spin

_system

in the

Furthermore,

using

equation (2.10)

we find for the

_entropy

where

The determination of the

_{phase diagram}

now consists of the

following procedure.

We first calculable the

_{polarizations}

_p’

and the inverse

temperature

/3

for each

_type

of

_magnetic

ordering, by maximizing

In

Li’ or

_{by solving}

at a constant

value s°

of the

_entropy.

This means we

_{require simultaneously}

that

Then we calculate _{the energy} of that

_particular

state,

using equation (3.14). Finally,

we obtain the

_{phase diagram by choosing}

the state with the lowest or

_highest

_energy,

_depending

on whether the

temperature

is

_positive

or

_negative.

The molecular field

_{approximation corresponds}

to

_taking

the zero order terms in the

expansion only.

This means that we

_neglect

the terms with

_T(pi).

_Thus,

in the molecular field

approximation

the

_{polarizations}

are

pô

and the inverse

_temperature

is

₁₃₀

and

_they

are solutions of

where we

_require

that the

entropy

has the constant value

s°,

, so

Furthermore,

the energy is

given by

Thus,

the zero order terms in the

_{expressions (3.14)}

and

_(3.15)

_{for the energy} and the

entropy,

simply correspond

to the

_expressions

for these

_quantities

as obtained in the molecular field

_{approximation.}

_However,

one should be aware that the actual value of these

terms

_might

differ from the value obtained in the molecular field

_{approximation,}

as the values of the

polarizations

p’

might change

upon

including

the

_higher

order terms.

4. Calculation of the restricted traces.

As outlined in the

_previous

_section,

we can calculate which of the three

_{possible phases}

will

actually

occur,

by solving

the

equations (3.17)

and

(3.18).

In order to do

this,

for each of the

phases,

first all contributions to these

equations

must be determined. For the

_expressions

for the energy

Eo

and the

_entropy

so

following

from the molecular field

approximation,

we refer

approximation [7].

A

_{straightforward application}

of this method to our Hamiltonian

_(1.2)

yields

the

following

results for each of the three

_phases.

4.1 THE PARAMAGNETIC PHASE. - As is shown in

figure

2a,

a

_{paramagnetic phase}

is

characterized

_by

a nuclear

_polarization

which is

_equal

for all

_proton

_spins

in the

_sample.

Furthermore,

in our case the

_{externally applied}

effective

_magnetic

field 4 and w _{1 are} oriented in the z- and x-directions

_respectively

so the

_component

of the

polarization parallel

to the

y-axis is

equal

to zero.

_Then,

_according

to our

previous

paper

[5] :

The contribution of the

_dipolar

interactions vanishes because we have chosen the

_shape

of the

sample

such that

_{equation (1.5)}

holds. Furthermore

were

The restricted trace

Tp(p1)

is

_{given by}

where

is a sum which converges

rapidly

with the 6th power of the distance between the

spins

I

and Ij.

We determined

D 2

for the case of

Ca(OH)2

in the same _way as

_{Sprenkels et}

al.

[2]

and found

It should be noted that

_{T(pi )}

has been calculated

_{previously by}

Goldman for the case of a

paramagnetic phase

in the

_special

situation that p, = 0. For this purpose, he used a

_high

temperature

expansion [10],

but as is to be

_expected

his and our results are the same.

4.2 THE FERROMAGNETIC PHASE. - As shown in

_figure

2b,

the

_{ferromagnetic phase}

in

Ca(OH)2

is characterized

_by

two

_types

of domains A and B. The

_{polarizations}

in the z and x

directions of all

_spins

in the domains of

_type

A are

given by Pz and px respectively,

while for those in the domains of

_type

B _{these are - pZ and px. Because in} our case the

_externally

applied

effective

_magnetic

fields 4 and 1 are oriented in the z- and x-directions

respectively,

the

_component

of the

_{polarization parallel}

to the

_y-axis

is

_equal

to zero in both

_types

of domains. The total number of

_spins

in domains

_{of type}

A is

_equal

to xN and of those

_{of type}

B is

_{( 1 - x)}

N.

_Then,

_according

to our

previous

_paper

The molecular field value of the

_entropy,

_so, is

_given

_by

where 1

For the restricted trace

T F(pi)

we find

where

is a sum which converges

_rapidly

with the 6th power of the distance between the

spins

I

and Ii.

Because of this

_rapid

_{convergence and because in the}

_{ferromagnetic phase}

the domains are

_{large compared}

to the interatomic

distances,

the sums

D 2

and

D 2

can be considered to be

_equal

up to a _very

_high

_{accuracy :}

4.3 THE ANTIFERROMAGNETIC PHASE. As shown in

_figure

2c,

the

_{antiferromagnetic phase}

in

Ca(OH)2

is characterized

_by

two

_types

of

sublattices,

A and

B,

each

_containing

an

_equal

number of

_spins.

AU

_spins

in sublattice A have a

_polarization

pA

with

_components

pzA

and

_px,

while those in sublattice B have a

_polarization

pB

with

_components

pz B

and

_pxB.

Then,

in the molecular field

approximation,

the energy is

given by

In this

_{equation, A AF}

is a

dipole

sum, which for

_Ca(OH)2

is

_{given by [2]}

The molecular field value of the

_entropy

of the

_{antiferromagnetic phase}

is

_equal

to

where

D 2 was

defined above and

The latter sum converges

rapidly

with the 6th power of the distance between the i-th

and j-th

spin.

We have calculated it in the same manner as the sum

D 2 and

found it to be

_{given by}

5. Calculation of the

_{phase diagram.}

Now we have obtained

_expressions

for the restricted traces

_occurring

in the

_{equations (3.17)}

and

_(3.18),

we can solve these

equations

for each value of the effective fields d and

w

1 and

for each of the three

possible phases. Subsequently,

we can determine which

_phase

is

most

_favourable,

i.e. has the

_highest

energy,

by inserting

the solutions of

equations (3.17)

and

(3.18)

in

_{expression (3.14)}

for _{the energy.}

In our

_previous

_{paper, where} we used the molecular field

_{approximation,}

we were able to

perform

this task

_analytically

for all cases,

except

for the

_{antiferromagnetic}

_phase

when

A = o.

In the

_present

treatment,

using

the restricted trace

_{approximation,}

the more

_complex

equations

to be solved render such

_analytical

solutions

_impossible,

_except

for some _very

special

cases.

_Therefore,

we solved the

_problem

_{numerically, using}

the Newton-Revson method for non-linear

_{equations [8].}

Figure

4 shows the

_{phase diagram resulting}

from this

_calculation,

as a function of the

effective fields 4 and w _l’ for the case

that p ° =

0.4.

Upon

comparison

with the results of the

molecular field

_{approximation presented}

in

_figure

3,

one sees an

_important

difference. The

Fig.

4. -

w 1 - A

phase diagram

of the nuclear

spin

system in

Ca(OH)2

for T « 0 and

_{Bo parallel}

to the

crystalline

c-axis and

_according

to the restricted trace

_{approximation.}

The initial

_polarization

is

Po =

0.4. ADRF-1 is the

_path

of the ADRF as used

_by

Sprenkels.

ADRF-2 is a

_{proposed path avoiding}

a

latter

_{approximation yields}

an

_{antiferromagnetic phase only}

if 4 =

0,

while under those circumstances it is still

_degenerate

with the

_{ferromagnetic phase.}

On the other

hand,

in the restricted trace

_{approximation}

this

_degeneracy

is lifted.

Thus,

as

_long

as d is

_small,

the

antiferromagnetic phase

is favourable for all values of w 1.

It is

_interesting

to note, that our observation that short range correlations favour the

antiferromagnetic phase

with

_respect

to the

_{ferromagnetic phase,}

also follows from the

following simple

argument

originating

from J.-F.

_{Jacquinot [9].}

Using

a classical

_description

like the molecular field

approximation,

the _energy of the

ferromagnetic

as well as the

_{antiferromagnetic phases}

is determined

by

terms of the

_type

A ii

I’i z Ij z

only.

Contributions to the _energy

_originating

in terms of the

_type

_{Aij (I x i}

_IjX

+

I

Ijy )

are

neglected.

In a

_quantum

mechanical treatment, one may not

_{always neglect}

these latter

terms. Consider e.g. an

_{antiferromagnetic}

state, where half the

_spins

has while

>

the other half has

₍

- - 2 1 .

If we

neglect

terms of the

_type

_{A i (I X I X + I .v ‘}

_{I y ),}

this state is

degenerate

with a second state where all

_spins

are inverted. If

_however,

we take this latter

term into account it mixes both states,

leading

to one with a

_higher

_energy an’d one with a lower energy, thus

rendering

one of these mixed states more favourable than each of the two

original

states. On the other

_hand,

_{ferromagnetic}

states do not show this

_type

of

_degeneracy.

So in that case, the extra terms do not lead to such a

_{qualitatively}

different result between a .

quantum

mechanical and a classical

_description.

_Then,

if a classical

_{description yields}

the same energy for an

_{antiferromagnetic}

and a

_{ferromagnetic phase,}

one may

expect

that in a

quantum

mechanical

_description,

the extra contribution to _{the energy}

_originating

in the term

(7 7+7 7),

should favour the

_{antiferromagnetic phase.}

Our

_result,

that for

_larger

values of

à,

the

_{ferromagnetic phase}

still becomes more favourable than the

_{antiferromagnetic phase,}

can be understood

completely by using

the molecular field

_{approximation only.}

The

_{parallel susceptibility}

is much

_larger

for a

ferromagnet

than for an

_{antiferromagnet.}

_Therefore,

the former

_gains

more _{energy than the} latter when a

_parallel

effective field is

_{applied, eventually rendering}

it more favourable.

Equivalently,

for small values of L1 but

_increasing

values of _{w i,} the

_spin

_system

remains

antiferromagnetic,

because the transverse

_{susceptibilities}

of both ordered

_phases

are the same.

6.

_Comparison

with

_experiments

and conclusion.

We now _compare our calculations with

experimental

results obtained

by Sprenkels et

al.

[2].

His

_experiments

were

_performed

under the conditions described in section 1.

However,

the ADRFs were not

_{always complete.}

I.e. first the

_parallel

effective field d was reduced to a value

₄₀

which was not

_{necessarily equal}

to zero.

_{Subsequently,}

the transverse effective field

1 was turned off. The

development

of the effective fields

during

such an ADRF is indicated

in

_figure

4 as ADRF-1. After each of these

_ADRFs,

the NMR

_signal

was recorded. From this

signal

the average

polarization

in the z-direction

and _{the energy of the nuclear}

_spin

_system was

_determined,

_using

the methods described in reference

_[4].

The results for

_ff,

as a function of

_4°

are shown in

_figure

5 for the case that

Fig.

5. -

Experimental

results for pz versus

_Ao.

The solid curve

_gives

the theoretical result

_using

the

restricted trace

_{approximation.}

F :

_{Ferromagnetic phase.}

AF :

_{Antiferromagnetic phase.}

In order to _compare these

_experimental

results with the theoretical treatment

_presented

above,

we calculate the

_phase

_diagram

for

_{po =}

0.7. We take into account that in the

experiment

of

_Sprenkels,

the

_shape

of the

_sample

is such that

_{demagnetizing}

fields can not be

neglected.

For his

_{sample, A 0 =}

2 7T . 32.12 x

103 s-1 [2].

As a

_result,

the

_phase

transitions are shifted to lower values of d than shown in

_figure

4,

where

A 0 =

0. We find that a

_phase

transition between the

_{ferromagnetic}

and

_{antiferromagnetic phase}

should occur at

4 = 2 7T . 0.4 x

10 3 s-1.

_Hence,

the two ADRFs to

₄₀

= 0 shown in

_figure

5,

should have lead to an

_{antiferromagnetic phase.}

_However,

this theoretical

_prediction

is in

_strong

contradiction with the

_{experimentally}

observed

decay

of the energy of the nuclear

spin

system,

from which

_Sprenkels

deduced that the nuclear

_spin

_system

was

_{really ferromagnetic.}

In order to

_get

more information about this

_disagreement

between

_experiment

and

_theory,

we use the restricted trace

_{approximation}

in _{the way described}

above,

to calculate

Pz

for both the

_{antiferromagnetic}

and

_{ferromagnetic phase.}

The results are also

plotted

in

figure

5.

_Clearly,

the

_experimental

results

_at

_{4° 2 r 0.4 x 10 3 s-1}

1 _{agree with the}

predictions

for a

ferromagnetic phase

and hence

_they

are consistent with the

_analysis

of the

decay

of the energy after the ADRF.

_{Unfortunately,}

there are insufficient data at

4°

1

2 7T 0.4

x 10 3 s-1

to obtain similar extra

_support

for the existence of a

_{ferromagnetic}

phase

at these lower values of the

_longitudinal

effective field.

However,

the

_consistency

between the observed

_decay

of the energy

and pZ

at

_higher

values

of

l.do 1,

does confirm that it is correct to use the

_decay

of the energy to conclude whether the nuclear

_spin

_system

is

_{ferromagnetic}

or not.

_Hence,

_Sprenkels

conclusion that the nuclear

_spin

system

is

_{ferromagnetic}

at

₄₀

= 0 _must be _correct. In this _{way, these} determinations of

Pz

still

_yield

extra

_support

for our observation that at

₄₀

= 0

experimentally

_a

ferromagnetic

phase

is

_observed,

while

_{theoretically}

an

_{antiferromagnetic phase}

is

_predicted.

of this order of

_magnitude,

the

_temperature

rises to above the critical

temperature.

On the other

_hand,

the

_phase

transition from a

_{ferromagnetic}

to an

_{antiferromagnetic phase}

is first order and it takes a finite time to

_change

from a

_macroscopic

domain structure to a

microscopic

sublattice structure. In

_Sprenkels

_case, the domains are _very

_large

- he estimates a thickness of the order of

103

atomic

_layers.

_Then,

this

_phase

transition

_might

take too

_long

compared

to the

heating

of the nuclear

_spin

_{system. Hence,}

the

_{ferromagnetic phase}

is

observed,

though

in fact it is metastable.

If this

_explanation

is correct, one _may avoid this

_problem

and

really

observe the

antiferromagnetic phase

as follows.

_Then,

the ADRF is

_{performed using}

a _{very strong} rf

_field,

as indicated in

_figure

4

_by

ADRF-2.

_So,

one does not _pass

_through

an intermediate

ferromagnetic phase

that would

_persist

as a metastable stable after

finishing

the ADRF.

Acknowledgments.

This work is

_part

of the research program of the «

_Stichting

Fundamenteel Onderzoek der Materie

(FOM) »

and has been made

_{possible by}

financial

_support

from the « Nederlandse

Organisatie

voor

_{Wetenschappelijk}

Onderzoek

(NWO)

».

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MARKS J., WENCKEBACH W.

_Th.,

POULIS N. J.,

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SPRENKELS J. C. M., WENCKEBACH W. Th., POULIS N. J., J.

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