Nuclear magnetic ordering in Ca(OH)2. I. Molecular field approximation

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Nuclear magnetic ordering in Ca(OH)2. I. Molecular

field approximation

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

To cite this version:

Nuclear

_{magnetic ordering}

in

_Ca(OH)2.

I. Molecular field

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,

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 des

_champs

moléculaires. Les résultats sont

utilisés _pour la

description

de l’évolution du

système

des

spins

nucléaires

pendant

une

désaimantation

_adiabatique

dans le référentiel tournant

_(DART)

utilisé pour refroidir ce

_système

des

_spins.

_Ainsi, dans le cas où la

_température

des

spins

nucléaires est

_négative

et le

champ

magnétique

extérieur est orienté

_{parallèlement}

à l’axe c du

_cristal,

on

_prévoit

un état

ferromagnétique.

Abstract. 2014 The

magnetic phases

of the

quasi

two-dimensional _proton

_spin

_system in

_Ca(OH)2

are studied

_using

the molecular field

_{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. Thus, for the case that the nuclear

_spin

temperature is

negative

and the external

_magnetic

field is oriented

_parallel

to the

_crystalline

c-axis, a

_{ferromagnetic phase}

is

_expected.

Classification

Physics

Abstracts 75.30K - 76.60

1. Introduction.

The

_study

of

_{magnetic orderings}

of nuclear

_spin

_systems

_{having dipolar}

_{interactions,}

was

initiated

_{by Abragam}

and coworkers

_{[1]. They}

were the first to create

_dipolar

ordered

_phases

of the fluorine

_spins

in the three dimensional

_system

_CaF2,

_{using dynamic}

nuclear

polarization (DNP)

followed

_by

an adiabatic

_{demagnetization}

in the

_rotating

frame

_(ADRF),

to cool the nuclear

_spin

_system.

_Following

their

_pioneering

work,

we

_investigated

experimen-tally

the

_{magnetic ordering}

of the

proton

spins

in

Ca(OH)2

(see

Fig.

1)

and J. Marks

[2, 3]

was the first to observe an ordered

_phase.

He found it to occur at

_{negative spin}

_temperature,

while the

_{externally applied magnetic}

field is oriented

_parallel

to the

_crystalline

c-axis. The

ferromagnetic

nature of this ordered

_phase

was _proven

_{experimentally by}

J. C. M.

_Sprenkels

[4].

He showed that the observed

_ferromagnet

exhibits 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.

In order to understand their

_experimental

results,

Marks and

_Sprenkels

used the molecular field

_{approximation}

in the manner described

_by

Goldman

_[1].

When

_applied

to the case of

Ca(OH)2

at

_{negative spin}

_temperature

and with the

_magnetic

field

_parallel

to the

_crystalline

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

axis,

this

_{approximation yields,}

that after

_completing

the final

_cooling

_step,

the

ADRF,

two

different ordered

_phases

are

_degenerate.

So one cannot decide which of the two

_phases

should

actually

be observed

_[4].

These two

_phases

are the observed

_{ferromagnetic phase}

shown in

figure

2b and furthermore an

_{antiferromagnetic phase}

which is shown in

_figure

2c. In both cases 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 phase}

the

spins

are

_parallel

in _many successive

_{crystal planes,}

the orientation of the

_spins

alternates from

_plane

to

_plane

in the

_{antiferromagnetic phase.}

The

_degeneracy

of these two ordered

_phases

as calculated

_by

the molecular field

approximation,

arises from the

_highly

two-dimensional nature of the

_proton

_spin

_system

in

Ca(OH)2.

As shown in

_figure

_1,

the

_proton

_spins

in this substance are

_arranged

in

_planes

perpendicular

to the

_crystalline

_c-axis,

where the distance between

_spins

within one

_crystal

plane

is much smaller than the distance between the

_planes.

As a

_result,

a

_spin

in a

_given

crystal plane experiences

a molecular field which is almost

_{completely composed}

of

_dipolar

fields of

_spins

in that same

_{crystal plane,}

while the

_dipolar

fields of

_spins

in other

_{crystal planes}

yield

a

_negligible

contribution to this molecular field.

Hence,

this molecular field is the same for the

_{ferromagnetic}

and

_{antiferromagnetic phases}

shown in

_figures

2b and 2c and the molecular field

_{approximation}

will

_{yield equal energies}

for theses two

_phases.

The

_present

paper is devoted to a

study

of the evolution of the

_proton

spin

_system

in

Ca(OH)2 during

the

_ADRF,

which is used as the final

step

to cool the nuclear

_spin

_system.

For _{this purpose} we use the molecular field

_{approximation}

to calculate the

_{magnetic phase}

diagram

of the nuclear

_spin

system

as a function of the effective fields that the

spins

experience during

this ADRF. It will be shown

_that

_during

the

_ADRF,

the nuclear

_spin

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)}

Thé

to the

antiferromagnetic phase, though

at the end of the

ADRF,

when the effective fields are

zero, both

phases

are

degenerate.

Thus the

experiments,

which show

_{ferromagnetic}

order,

can be understood if one takes into account that a

_{ferromagnetic phase}

is created

_during

this

ADRF,

while

afterwards,

apparently

a transition to the

_{antiferromagnetic phase}

does not

take

_place.

However,

the molecular field

_{approximation neglects}

_{short range} correlations which may lift the

_degeneracy

of the

_{antiferromagnetic}

and

_{ferromagnetic}

_phases

at effective fields

_equal

to zero and

change

the nature of the

_phase

_diàgram

_{considerably.}

_Therefore,

in a

following

paper

[5],

we will use the restricted trace

_{approximation}

_[6,

_7]

to extend the

_present

treatment

to include such short _range correlations.

2. The nuclear

_spin

_system

of

Ca(OH)2.

The nuclear

_spin

_system

in

_Ca(OH)2

consists of

_proton

_{spins 1}

= 1/2

_having

a

_gyromagnetic

2

ratio

_{y = 2}

7T 42.5759 x

106 s-1 1 T- 1.

To

_investigate

its ordered

_phases,

it is necessary to

lower the

_temperature

below the critical

_temperature

which is - 0.853

uK,

according

to the molecular field

_{approximation [3].}

Such an ultra low

_temperature

is reached in three

steps.

First,

the

_sample

is cooled to 0.5 K

_using

a

3He

_evaporation

_cryostat.

Then the

_polarization

p, =

(Iz)

II

of the nuclear

spins

is increased

by

means of

_dynamic

nuclear

polarization

(DNP) [8, 9]

to a value

p °

which is

typically

0.7. The

application

of DNP

corresponds

to a

strong

increase of the order of the nuclear

_spin

_system

_or,

_{equivalently,}

to a

_strong

reduction of the

_entropy

to a value

[1]

]

where N is the total number of nuclear

_spins.

_Furthermore,

we render the

_entropy

dimensionless,

by dividing

it

_by

Boltzman’s constant k.

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) [1].}

In such an

_ADRF,

the

extemally applied magnetic

field

Bo

is

_kept

constant,

contrary

to the

_practice

in a real adiabatic

demagnetization.

Instead,

an rf-field is

used,

with a

_frequency

in the

_{neighbourhood}

of the nuclear Larmor

frequency

wo =

yBo.

In order to describe the nuclear

_spin

_system

under influence of this

rf-field,

we consider it in a frame of reference

_(x,

y,

z), rotating

at the

_{rf-frequency w}

around its z-axis. The z-axis is chosen

_parallel

to the static

_magnetic

field

_Bo,

which is

_applied

in the direction of the

crystalline

c-axis. The rf-field

_{2 B1 cos w t,}

which is oriented in the

_{crystalline aa-plane,}

is

decomposed

into two

_rotating

components,

each with an

amplitude

_B1.

Then the x-axis of the

rotating

frame is chosen

_parallel

to the

_component

_rotating

in the same direction. The Hamiltonian in this

_rotating

frame is

_{given by}

where W 1 =

’YB

1 is the so-called transverse effective field in the

rotating

frame,

1;

and

Iix

are the z- and

_x-components

of the i-th

spin

and A _{= Wo - w} is the

longitudinal

effective

field. We have reduced this Hamiltonian to

_frequency

units

_{by dividing}

it

_by

Planck’s constant

HD

is the truncated

_dipolar

Hamiltonian

_representing

the

_dipolar

interactions between the

spins :

where

Here,

u o = 4 ir X 10-7 ,

rij

is the vector

_connecting

the

_spins

I

and Ij

and

_{0 ij}

is the

_angle

between

_rii

and the

_{externally applied magnetic}

field

_Bo.

The

_importance

of

_using

a

_rotating

frame of

reference,

arises from the fact

that,

in the presence of a

_strong

_rf-field,

the nuclear

_spin

_system

is in internal thermal

equilibrium

in this

rotating

frame

_[8],

so we may define a nuclear

spin

_temperature

T in this

rotating

frame. An ADRF now consists of

_reducing

the effective fields 4 and w ₁

adiabatically

to

_0,

_by

first

varying

the

_{frequency co}

of the rf-field and

_{subsequently reducing}

its

_{amplitude 2 B1.}

Just as a normal adiabatic

_{demagnetization}

leads to a reduction of the normal

_temperature,

this ADRF leads to a reduction of the

temperature

in the

_rotating

frame. As a

_result,

we obtain the

extremely

low

_spin

temperature

needed to order

_the

nuclear

_spins.

In order to find the nature of the ordered

_{phase during}

and after the

_ADRF,

we will determine

_{phase diagrams}

of the nuclear

_spin

_system

as a function of the effective fields 4 and _{W 1.} As an ADRF is

_adiabatic,

_entropy

is conserved

_during

its

_application.

To obtain the evolution of the nuclear

_spin

_system

_during

an

_ADRF,

we will therefore need

_{phase diagrams}

for

_given,

constant

values s°

of the

_entropy.

_{Equivalently, using equation (2.1),}

we will need

phase diagrams

for

given

constant values

_{p ° of}

the nuclear

_polarization

before the ADRF. We will

perform

calculations

_yielding

such

_{phase diagrams}

for the circumstances

_applying

to our

experiments,

i.e. for

_{negative spin}

_temperature

and for the

_{externally applied magnetic}

field

parallel

to the

_crystalline

c-axis.

3.

_Principles

of the calculation of the

_{phase diagram.}

The

_{magnetic phase diagram}

is

calculated,

using

the molecular field

_{approximation.}

For our determination of the

_{phase diagram}

of the

_proton

_spin

_system

in

_Ca(OH)2

we consider three

possible phases,

the

_{paramagnetic phase}

and the ferro- and

_{antiferromagnetic phases.}

Each of these

_phases

is characterized

_by

the

_polarization

_{p’ =}

_(Ii>

_/1

of the nuclear

_spins.

As

illustrated in

_figure

2a these

_{polarizations}

are

_equal

in the

_{paramagnetic phase.}

On the

contrary,

the

_{ferromagnetic phase}

shown in

_figure

2b shows two

_types

of domains. In one

_type

the nuclear

_polarization

_{p’ = pA}

while in the other

_type

_{p’ =}

_pB.

The relative sizes of the two

types

of domains are x and 1 - x

_{respectively. Finally,}

the

_{antiferromagnetic phase}

shown in

figure

2c has two sub-lattices of

_equal

size with nuclear

_{polarizations}

_pA

and

_pB

_{respectively.}

The calculation of the

_{magnetic phase diagram}

at constant

_entropy

now consists of two

steps.

For each of the three

_{possible phases,}

we first determine the nuclear

_{polarizations}

P’

and the

_temperature

T for

_given

values of the

_{entropy s}

and the effective fields 4 and _{w 1.}

_According

to

_{thermodynamics}

these

_{polarizations}

can be calculated from

_[10]

temperature,

which has the dimension of inverse

_frequency,

and

is the energy of the

spin

system

in

_frequency

units.

Subsequently,

we determine the

_{phase diagram by investigating}

for each value of the effective fields à and _{w 1,} which

_phase

is most

_favourable,

at a

_given

value of the

_entropy.

As we do this determination at

negative

_temperature,

we have to choose the

_phase

with the

highest

energy.

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.}

The essence of the molecular field

_{approximation}

is the

_assumption

that the nuclear

_spins

are

statistically independent,

i.e. there are no short range correlations between them. With this

approximation

the

_{energy E}

is

_{given by}

It should be noted that the actual value of the _energy calculated in this way,

depends

on the

shape

of the

_sample.

This is due to the so-called

_{demagnetizing}

fields that occur because of the

long

range character of the

dipolar

interaction

_[1].

In order to avoid this

_problem,

we assume that all

_experimental

data are

taken,

_using

a

_sample

with such a

_shape

that these

demagnetizing

fields are

_equal

to zero.

Specifically

we assume the

_shape

of the

_sample

such that

It should be

_noted,

that it is

_{generally possible,}

to reduce data obtained in any

ellipsoidally

shaped sample

to the values

_they

would have had in a

_sample

with such a

_shape

that

equation

(3.4)

holds.

The molecular field value of the

_{entropy s}

is the same as for a

_system

of

_{non-interacting}

spins

and is

_{given by [1] :}

where

_fO(pi)

is defined in

_{equation (2.1 )}

and

We now

_proceed

with the first

_step

of the calculation of the

_{phase diagram.}

We insert this

expression

for the

_entropy

and

_{equation (3.3)}

for the energy in

equation (3.1 ) :

Then,

for each of the

_{possible phases,}

we determine the nuclear

polarizations

p’

and the

In the second

step

we calculate the _{energy E} of each

_{possible phase}

at the

_given

values of the effective fields and the

_entropy.

_Then,

we obtain the

_{magnetic phase diagram}

at constant

entropy

by determining

for each value of the effective fields 4 and w 1, which

phase

has the

highest

energy.

4. 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

fields 4

_{and cv}

_{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.

_So,

for the

paramagnetic phase,

Inserting

these values in

_{equation (3.3) yields :}

The contribution of the

_dipolar

interaction

_vanishes,

because we have chosen the

_shape

of the

sample

such that

_{equation (3.4)}

holds.

_Furthermore,

_{using equation (3.5)}

we find

As

_explained

in section

_3,

we must insert the

_{expressions (4.2)}

and

_(4.3)

_{for the energy and} the

_entropy

in

_{equations (3.7)}

and

(3.8)

in order to obtain the molecular field values of the nuclear

_{polarizations}

and the inverse

_temperature

at the constant

value s°

of the

_entropy.

We first insert

_{equation (4.3)}

for the

_entropy

in

_{equations (3.8)}

and

(2.1 )

to obtain the

_following

very

simple

result. The molecular field value of the

_{modulus p}

of the nuclear

_{polarization,}

in a

paramagnetic

state

_{occurring during}

and after an

_ADRF,

is

_{always equal}

to p °,

i.e. the nuclear

polarization

before this ADRF.

Inserting

this result and

_{equation (4.2)}

in

_{equation (3.7)}

and

_solving

the

_latter,

furthermore

yields

that

Inserting equation

(4.5)

in

_equation

_{(4.2) yields :}

Thus

_providing

the molecular _{field value of the energy of the}

_spin

_system

in the

_paramagnetic

phase,

as a function of the effective

_magnetic

fields and at a

_given

value of the

_entropy,

which is obtained

_{by inserting}

_po

in

_{equation (2.1 ).}

5. The

_{ferromagnetic phase.}

The

_{ferromagnetic}

_phase

in

_Ca(OH)2

shown in

_figure

2b,

is a

_{longitudinal ferromagnet.}

Such a

ferromagnetic

was observed for the first time to occur in

_{CaF2 [11].}

General

_expressions

for the molecular field values of its

_properties

at low values of the transverse effective field w 1 and any value of the

longitudinal

effective field

4,

were obtained

by

Goldman

[1].

This ordered

_phase

is characterized

_by

two

_types

of

domains,

A and B. All

_spins

in the domains of

type

A have a

_polarization

pA

and those in the domains of

_type

B a

_polarization

pB.

In the absence of

_{externally applied}

effective

_magnetic

fields à

_{and w i,}

_PA

is oriented

parallel

to the + z-axis while

pB

is

equal

in

magnitude,

but oriented in the

opposite

direction,

i.e.

_along

the - z-axis. An effective field à in the z-direction will result in a relative

growth

of the size of the domains of

_type

_A,

at the _expense of the domains of

_type

B. An effective field llJ

1 in

the

x-direction,

will lead to

canting

of all

spins

in the x-direction. Because in our case no

effective

_magnetic

field is

_applied

in the

_y-direction,

the

component

of the

_polarization

parallel

to the

_y-axis

is

_equal

to zero in both

_types

of domains.

_Thus,

for domains of

_type

_A,

the

_{polarizations}

are

_{given by :}

while for domains of

_type

B :

Finally,

we define xN as the total number of

_spins

in domains of

_type

A.

Then,

the number of

spins

in domains of

_type

B is

_{(1 - x)}

N.

Inserting

these values in

_{equation (3.3) yields :}

In this

_{equation A F}

is a

_dipole

_sum,

_representing

the molecular field which a

_{spin experiences}

from all other

_{spins :}

to different domains. This

_dipole

sum has been calculated

_by

Sprenkels

et al.

_[4].

It has the

important

feature

that,

up to _very

_high

_accuracy,

_only

those

_spins

Ii

contribute,

that are

situated in the same

_{crystal plane}

as the

_spin

Ii’,

for which the

demagnetizing

field is

calculated. This feature reflects the

_highly

two-dimensional character of the

proton

spin

system

in

_{Ca(OH)2. Sprenkels et}

al. found

_{A F}

to be

_equal

to

Furthermore,

using equation (3.5),

we find for the molecular field value of the

_entropy

in the

_{ferromagnetic phase :}

As in the

_previous

_section,

we must follow the routine of section 3 and insert the

expressions (5.3)

and

_(5.5)

in

_{equations (3.7)}

and

_(3.8),

in order to obtain the molecular field values of the nuclear

_{polarizations}

and the inverse

temperature.

The

_beauty

of the

ferromagnetic phase

is,

that it allows for

_analytical

solutions for these

_quantities

for all values of the effective fields and the

_entropy.

The results for the

_{polarizations}

are :

while the relative domain sizes follow from

and the inverse

_temperature

is

_{given by}

Inserting

these values in

_{equation (5.3) yields}

the molecular _{field value of the energy of the} nuclear

_spin

_system

in the

_{ferromagnetic phase,}

as a function of the effective fields d

_{and cv l,}

for a

_given

value

p °

of the

_polarization

before the ADRF :

It should be

_noted,

that the solutions of the molecular field

_{approximation corresponding}

to

the

_{ferromagnetic}

_phase,

do not exist for all values of the effective fields 2l and

Cù. This follows from the obvious restriction that the relative size of each

_type

of domain should

_always

be

_positive,

so 0 x 1.

_Then,

_according

to

_{equations (5.6)}

and

_(5.7),

6. The

_{antiferromagnetic phase.}

The

_{antiferromagnetic phase}

in

_Ca(OH)2

shown in

_figure

_2c,

is a

_{longitudinal antiferromagnet}

of a

_type

which was first observed to occur in

_{CaF2 [12].}

General

_expressions

for the molecular field values of its

_properties

at low values of the effective fields à and cv 1, were obtained

_by

Goldman

_[1].

This ordered

_phase

is characterized

_by

two

_sublattices,

A and

_B,

each

_containing

an

_equal

number of

_spins.

All

_spins

in sublattice A have a

polarization

pA

while those in sublattice B have a

_polarization

pB.

In the absence of effective

fields,

pA

is oriented

_parallel

to the +

z-axis,

while

_pB

is

_equal

in

_magnitude,

but oriented in the

_opposite

direction,

i.e.

_along

the - z-axis. As in a

_ferromagnet,

an effective field cv ₁ in the x-direction results in

_canting

in the x-direction of the

_{polarizations}

in both sublattices. But

_contrary

to the

_{ferromagnetic}

_phase,

an effective field d in the z-direction cannot

_change

the relative number of

_spins

in the two sublattices and hence it will

_change

the

_magnitude

of the

_{polarizations}

themselves.

_Thus,

the

_{polarizations}

of the

_spins

in sublattice A are

_{given by}

while the

_{polarizations}

in the other sublattice are

equal

to

Inserting

these values in

_{equation (3.3) yields :}

In this

_{equation, A AF}

is a

dipole

sum

_representing

the molecular field which a

spin experiences

from all other

_{spins :}

where gij =

+ 1 if the

_spins

I

and Ij

_belong

to the same sublattice and

_{giy = -}

1 if

_{they belong}

to different sublattices. The

_dipole

sum

_AAF

has been calculated

_by

_{Sprenkels et}

al.

_[4].

As in the

_{ferromagnetic}

_case, it has the

_important

feature,

that

_only

those

_spins

Ii contribute,

that are situated in the same

_{crystal plane}

as

Ii’.

As can be seen from

_figures

2b and

_2c,

all

_spins

situated in the same

_{crystal plane}

are oriented in the same

_direction,

in the

_{ferromagnetic}

as well as in the

_{antiferromagnetic}

_phase.

_Hence,

for

_spins

in the same

_{crystal plane,}

the factor

The molecular field value of the

_entropy

of the

_{antiferromagnetic phase}

follows

_directly

from

_{equation (3.5) :}

As in the case of the

_{ferromagnetic phase,}

we must follow the routine of section 3 and insert the

_{expressions (6.3)}

and

(6.5)

in

_{equations (3.7)}

and

_(3.8),

in order to obtain the

_equilibrium

values of the nuclear

_{polarizations}

and the inverse

_temperature.

Unfortunately,

unlike the

ferromagnetic phase, analytic

solutions cannot be

obtained

for

_general

values of the effective

fields.

However,

such

_analytic

solutions can still be found for the

special

case that the

_longitudinal

effective field d is

_equal

to zero. In the

following

subsection we therefore first treat this

special

case.

Then,

in the next

_subsection,

we consider the

general

case

_{where A #}

0. 6.1 THE ANTIFERROMAGNETIC

PHASE, à

= 0. In the _case that d =

0,

the

analytic

results

for the molecular field values of the

_{polarizations}

are :

while the inverse

_temperature

is

_{given by}

Inserting

these values in

_{equation (6.3), yields}

the energy of the nuclear

spin

system

in the

antiferromagnetic phase,

as a function of the effective

_{field W}

_{1 on} the

x-direction,

for the

special

case that à =

0,

and for a

_given

value

_{p °}

of the

_polarization

before the ADRF :

It should be

_noted,

that for this

_special

case, where 4 =

0,

the molecular field values of all

quantities

are the same in the

_{antiferromagnetic phase}

as well as in the

_{ferromagnetic phase.}

6.2 THE ANTIFERROMAGNETIC PHASE, GENERAL CASE. - In the

general

case, where ,J:o

0,

we cannot find exact solutions for the molecular field values of the nuclear

The zero order terms are the solutions for 4 = 0 as obtained

_by

in the

_previous

section.

Thus,

03B2

(0) is

_{given by equation (6.7)}

and

_PA(O),

etc.

_{by equation (6.6).}

We note that the

_symmetry

of the

_{antiferromagnetic phase}

is such that upon

inverting

the effective

_longitudinal

field

2l,

the values of

_/3

_{AF, Pz}

_{and px}

should remain the same, while those

of

_6p,

and

_dpx

should invert as well.

_Thus,

the former

_quantities

are even functions of the 4 while the latter are uneven functions.

_Up

to second

_order,

the power series

expansions (6.9)

therefore read as :

Inserting

these

_expansions

in

_{equations (6.3)}

and

(6.5),

allows us to write _{the energy and} the

_entropy

also as a _{power series in} A In the first case, the calculation is

straightforward.

Using equation (6.6)

for

_{p z (0)}

and

_px(o),

one finds

This

_expression

is seen to be

_symmetric

in

d,

as is to be

_expected

from the

symmetry

of the

antiferromagnetic phase

upon inversion of that

quantity.

In order to obtain a power series

expansion

for the

_entropy,

we first

_expand

the functions

f o (p A)

and

_{f o (p B),}

_occurring

in

_{equation (6.5) giving}

the

_entropy

of an

_{antiferromagnet,}

around the value

Next,

we

_expand

pA

and

_{p B,}

_{using equations (6.10).}

_Inserting

the results

_(6.6)

for

PZ (0)

and

_px°,

we find :

where one chooses the +

_-sign

for

p A

and the -

-sign

for

pB.

where P and

_Q

are

_given

in

equation (6.13).

As in the case _{of the energy, this}

_expansion

is a

symmetric

function of

d,

which was to be

expected

from the

_symmetry

_arguments

discussed

above.

The

_higher

order contributions

_/3

_{pZ2, 8pZ 1,}

etc. to the inverse

_temperature

and the nuclear

_{polarizations}

can now be found

_{by inserting}

these

_expansions

_{for the energy and the}

entropy

in

_{equations (3.7)}

and

_(3.8)

and

_solving

them. We first consider

_{equation (3.8).}

In the

present

case, it

yields

So,

up to second order

which can be rewritten as

This result allows us to eliminate the second order contributions

_{pz (2)}

and

_pX2

in the

_expansion

(6.11)

for the energy

EAF :

Thus,

in order calculate

EAF

up to second

order,

we

only

need to know the

_first

order

contributions

_{8 p Z 1}

and

_{8 p Z 1 .}

To find these first order terms, we need to solve

_{equations (3.7)}

up to first order for the

special

case of an

_{antiferromagnetic phase.}

_{For e.g.}

p

= p A,

we need to solve

where we

_expand

the various terms _up to first order in 4. The results are

and

We now have obtained an

_{analytical expression}

for the energy

EAF

of the

antiferromagnetic

phase

as a function of the effective fields d and _w

1 and

the

polarization

p °

before the ADRF. This

_expression

is obtained as an

expansion

_up to second order in the effective

_longitudinal

field 2l. It is

_{given by}

_equation

_(6.18),

wherein

_{p z (0)}

and

_{p x (0)}

are

_{given by equation (6.6)}

and

8pZ 1

and

_8pxl

_{by equation (6.20).}

In the next

_section,

we will use this result

_together

with

those for the

_paramagnetic

and

_{ferromagnetic phases}

to determine the

_{phase diagram}

of the nuclear

_spin

_system.

7. The

_{phase diagram.}

As described in section

3,

the final

_step

in the calculation of the

_{phase diagram}

consists of

determining

for each value of the effective fields 4

_{and cv}

_{1 the phase}

with the

_highest

energy. In the

_present

section we will

perform

this calculation

_using

the molecular field

approxi-mation,

i.e. we will

neglect

short _range correlations. For this _purpose, we will use the molecular field values

jE,

EF and EAF

of the

_energies

of the

_{paramagnetic,}

the

_{ferromagnetic}

and the

_{antiferromagnetic phases}

as

_given

in

_{equations (4.6), (5.9), (6.8)}

and

(6.18).

Figure

3 shows these values as a function of the effective field Li in the z-direction for the

_special

case that _{w 1 =} 0

_{and po =}

0.4.

Two different scales are used for the vertical axis. One scale

_applies

for the case of

Ca(OH)2 only

and reads

_E/2

wN,

so it

_yields

_{the energy in units of kHz} per

spin.

The other scale

_applies

for any

spin

system

that can be described

by

the Hamiltonian

_(2.3)

and which has a

_highly

two-dimensional character. This scale reads

El 1

_NAF(P

°)2,

so it

_yields

the energy in

4

dimensionless units.

_Similarly,

of two horizontal

_scales,

one reads

_d/2

_7T,

_yielding

a

_reading

of the effective field in

kHz,

for the

_special

case of

_Ca(OH)2.

Again,

the other scale

applies

more

_generally

and reads

_{d /A F p °,}

_yielding

this effective field in dimensionless units.

As in

_large

intervals of the effective field

d,

the difference between the molecular field values of the

_energies

of the various

_phases

is very

small,

also these

differences,

EAF - E F

and

EP -

_{E F}

are

_plotted

in

_figure

3b.

Figure

3

_clearly

shows two different

_regions

of interest.

_First,

for d

_{A F p °,}

one finds the

ferromagnetic phase

to have the

_highest

molecular _{field value for the energy.}

So,

one

_expects

the

_{ferromagnetic phase}

to occur.

_Secondly,

for 4 >

_{A F p °,}

_only

the

_{paramagnetic phase}

can be

_expected,

because

_according

to

_{equations (5.10)}

no solutions for the

_{ferromagnetic phase}

exist.

Still,

one

_point

in

_figure

3 has to be considered in more detail. When 4 =

0,

the molecular

field values of the

_energies

of the

_{ferromagnetic phase}

and the

_{antiferromagnetic phase}

appear to be

_equal.

_Thus,

a more detailed calculation

_might

still

_yield

that the

antiferromagne-tic

_phase

is more favourable in a small

_region

near this

_point.

In order to find out whether this is the case, one should

_incorporate

short range

correlations,

e.g.

by using

the restricted trace

approximation.

Such an extension of the

present

treatment will be considered in a

_following

paper

[5].

We

_investigated

the influence of an extension of the treatment of section 6 to

_higher

order

Fig.

3. -

(a)

The energy of the three

magnetic phases

as a function à for W 1 = 0 and

Po

= 0.4. The units _are

explained

in the text.

_(b)

The energy differences

E AF - E F and E P - E F.

For the

former case, both the series

_{expansion (6.18)}

and the exact numerical solution are shown.

equations (6.10)

and

_{(6.11), using}

the

_{Newton-Raphson}

method for non-linear

_{equations [113].}

The

_resulting

values of the nuclear

_{polarizations}

and the inverse

_temperature

were inserted in

equation (6.3)

in order to obtain the molecular field value of _{the energy of the nuclear}

_spin

system

as a function of the effective fields and the

_polarization

_{p °}

before the ADRF. For the case that _w 1 - 0 and

Po =

0.4,

the result is

plotted

in

figure

3b.

As one sees _very

_clearly

from

_figure

3b,

the numerical calculations and the series

region 4 «

0.3

_{A Fp °,}

the series

_expansion

seems to _{bc very}

_good,

the relative difference of the energy obtained

by

the two methods

being

about 0.001. Hence the molecular field

approximation

does

_{really predict}

a

_degeneracy

of the

_{ferromagnetic}

and the

antiferromagne-tic

_phases

at the

_{point 4}

= 0.

We conclude this section

_{by considering}

the

_{complete phase diagram}

as a function of L1 and w ₁ shown in

figure

4. From

equation (5.11)

we find that for

_large

effective

fields,

where

the molecular field

_{theory predicts}

the nuclear

_spin

_system to be

_{paramagnetic.}

From

comparing

the molecular field values of the _energy of the various

_phases

as

_given

_by

_equations

(4.6), (5.9), (6.8)

and

_(6.18),

we furthermore find that the

_spin

_system

is

_expected

to be

ferromagnetic

at smaller effective fields.

_However,

at 4 =

0,

the

ferromagnetic

and the

antiferromagnetic phase

are

_degenerate

so the molecular field

_theory

cannot decide which of the two will

_actually

occur.

Fig.

4. _{- 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

p ° -

0.4.

8. Conclusion.

In

_experiments

on the nuclear

_{dipolar ordering}

of the

_proton

spins

in

_Ca(OH)2

the ordered

phases

are reached via an

_ADRF,

i.e. an adiabatic reduction of the effective fields d and _{w i.} It is clear from

_figure

_4,

that one

_always

_passes

through

a

ferromagnetic phase

during

such an ADRF.

_Thus,

the

_present

treatment

_explains

the

_experiments

of

_Sprenkels

However,

for a

longitudinal

effective field 4

equal

to zero, the molecular field

approxi-mation

_predicts

that the observed

_{ferromagnetic}

_ordering

is

_degenerate

with an

antifer-romagnetic phase.

Therefore one needs a more

_{sophisticated}

treatment,

taking

into account

short range

correlations,

in order determine the real

ground

state under these conditions.

Such a _treatment,

_using

the restricted trace

_{approximation}

will be

_given

in a

_subsequent

_paper

[5].

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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