Nuclear magnetic ordering in Ca(OH)2. III. Experimental determination of the critical temperature

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Submitted on 1 Jan 1990

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C.M.B. van der Zon, G.D.F. van Velzen, W. Th. Wenckebach

To cite this version:

1479

Nuclear

_{magnetic ordering}

in

_Ca(OH)2.

III.

_Experimental

determination of the

critical

_temperature

C. M. B. van der

_Zon,

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 On

_présente

une étude

_{expérimentale}

de l’ordre

_magnétique

des

spins

des protons

dans

_(OH)2Ca.

L’état ordonné est

_produit

_par

_{polarisation dynamique}

nucléaire suivie par désaimantation

_adiabatique

dans le référentiel tournant. La

température

de

_spin

est

_négative,

lorsque

le

_champ

_magnétique

extérieur est

_parallèle

à l’axe c du cristal. On a déterminé l’état

magnétique

en utilisant les résultats

_{expérimentaux}

sur l’aimantation

_parallèle

au

_champ

effectif. On a mesuré

l’énergie

et

_l’entropie

du

_système

de

_spins

et déterminé la

_température

de

_spin

à

partir

de ces résultats. On trouve que la transition de

phase

se

_produit

à - 0.9 _03BCK, en bon accord avec la

_{prédiction théorique}

de

_{l’approximation}

des

_champs

moléculaires.

Abstract. 2014 An

_{experimental study}

of the

_{magnetic ordering}

of the _proton

_spin

_system in

Ca(OH)2

is

_presented.

The ordered

_phase

is reached via

_dynamic

nuclear

polarization

followed

by

an adiabatic

_{demagnetization}

in the

_rotating

frame. The

_spin

temperature is

negative

while the external

magnetic

field is

_parallel

to the

_crystalline

c-axis. Measurements of the

_{magnetization}

parallel

to the effective

_magnetic

field are used to determine the

_magnetic

_phase

of the nuclear

spin

system. From a measurement _{of the energy of the nuclear}

_spin

system as a function of the

entropy, the nuclear

_spin

_temperature is derived. The

_phase

transition is found to occur at

- 0.9

03BCK,

which is in

_good

_agreement with the theoretical

_prediction

of the molecular field

approximation.

J.

_Phys.

France 51

₍₁₉₉₀₎

1479-1488 1 er JUILLET 1990,

Classification

Physics

Abstracts 75.30K - 76.60

1. Introduction.

Nuclear

_{spins interacting}

via

_{dipole-dipole}

interactions show a

_phase

transition to a

magnetically

ordered state at a

_temperature

ouf the order of 1

...,K.

This

_extremely

low

temperature

is reached

_using

a

_{multi-step cooling}

_process introduced

_{by Abragam}

and

co-workers

_{[1, 2]}

and used later

_by

Marks et al.

_[3]

to create a

_{ferromagnetic ordering}

of the

proton

spins

in

_Ca(OH)2.

First,

the

_sample

is cooled

_by

_cryogenic

methods to about 0.5 K and

placed

in a

_strong

_magnetic

field.

Then,

the nuclear

_spin

_system

is cooled further to about 2 mK

_by

means of

_Dynamic

Nuclear Polarization

_{(DNP) [4, 5]. Finally,}

an Adiabatic

Demagnetization

in the

_Rotating

Frame

_{(ADRF) [4] brings}

the

_temperature

of the nuclear

spin

system

below the transition

_temperature.

This _paper concems an

_{experimental study}

of the

_ordering

of the

proton

spins

in

_Ca(OH)2,

which has the

_crystal

structure shown in

_figure

1. This

_hexagonal

structure is of the

_CdI2-type,

with space

symmetry

group

D 3 -

P3m.

The

_positions

of the nuclei and the dimensions of the

Fig.

1.

_{- (a) Hexagonal}

unit cell of

Ca(HO)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, § ] .

unit cell were determined

_by

neutron diffraction at low

_temperatures

_[6].

The

_protons

are

arranged

in almost two

_{dimensional layers.}

The

_arrangement

of the

_protons

in these

_layers

is shown in

_figure

1 b. The

_{proton-proton}

distance in the

_layers

is 2.186

_Á,

while the distance between the

_layers

is 4.880

Á.

In order to describe the

_proton

_spin

_system

of

_{Ca(OH)2 during}

and after the

_ADRF,

we use a frame of reference with its z-axis

_parallel

to the external static

_magnetic

field. The ADRF is

performed using

a

_saturating

rf-field 2

_BI

cos _{w t, which is oriented in the}

_xy-plane.

_Then,

the nuclear

_spin

_system

is in thermal

_equilibrium

if our frame of reference rotates about the z-axis with the

_{frequency w}

of this rf-field. In this

_rotating

_frame,

its Hamiltonian is

_{given by}

In this

_{equation, d}

=

yBo - w

is the so-called effective field in the

rotating

frame of

reference,

while

_Bo

is the

_amplitude

of the static

_magnetic

field and y is the nuclear

gyromagnetic

ratio.

Furthermore,

K§

is the truncated

_dipolar

Hamiltonian,

i.e. the

part

of the

dipolar

interaction that commutes with the Zeeman interaction :

where

1481

vector

_connecting

these

spins

and the external

magnetic

field.

Furthermore,

_I)

and

1:

are the

_components

of the i-th

_{spin. Finally,}

_{cc o = 4 r x 10 -}

and h is Planck’s constant.

At the start of the

ADRF,

we choose the

frequency w

of the rf-field such that 4 is _many

times the NMR line width

_4NMR.

Then,

w is

_swept

until 4 s

4NMR

and the rf-field is switched

off. In the

_special

case that we choose 4 =

0,

we call the ADRF

_complete.

Otherwise,

it is

called

_incomplete.

Directly

after this

ADRF,

the nuclear

_spin

_system

is still in internal thermal

_equilibrium

in the

_rotating

frame of reference. So it is described

_by

a

_density

matrix :

where T =

h/kf3

is the

_temperature

of the nuclear

spin

_system

after the

ADRF,

while k is

Boltzman’s constant.

In our

_experiments

4 « 0 at the

_beginning

of the

ADRF,

so this final

_temperature

is also

negative.

Furthermore,

the

_{externally applied}

static

_magnetic

field is oriented

_parallel

to the

crystalline

c-axis.

According

to the

_experiments

of

_{Sprenkels et}

al.

[7],

the

proton

spins

then order as a

_{longitudinal ferromagnet}

with a domain structure. There are two

_types

of

_domains,

in one the

_polarization

of the

_proton

_spins

is

_parallel

to the

_{externally applied}

static

_magnetic

field,

in the other

_{anti-parallel.}

I.e. in one

_type

it is

_parallel

and in the other

_{anti-parallel}

to

the

_crystalline

c-axis.

_Furthermore,

the domains have a flat

_{pancake-like shape,}

with the

planes

of the domains

_{perpendicular}

to the

_crystalline

c-axis,

so each domain consists of a

large

number of

_{adjacent layers}

of

_proton

_spins

of the

_type

shown in

_figure

lb.

We studied this

_{ferromagnetic ordering}

more

elaborately by measuring

the

properties

of

the ordered

_spin

system

as a function of its

_entropy.

We

_present

measurements

_yielding

the

transverse

_{susceptibility}

_y, and of the

_{parallel magnetization p,}

=

(Iz)

II

as a function of the

longitudinal

effective field A The results are

_{interpreted using}

the restricted trace

approximation

[8]

as elaborated for the case of the

_{magnetic ordering}

of the

_proton

_spins

in

Ca(OH)2 by

van Velzen et al.

_{[9]. Thus,}

we determine the

_{magnetic phase}

of the nuclear

spin

system

as a function of its

_entropy.

Subsequently,

we show measurements of the energy of the nuclear

spin

system

as a function of its

_entropy.

From these

results,

the nuclear

_spin

temperature

is calculated.

Then,

the

magnetic phase being

known from the

_{previous experiment,}

the critical

_temperature,

where the transition to the

_{ferromagnetic phase}

occurs, is determined.

2.

_Experiments.

Our

_experiments

are

_performed

on a

_{cylindrically shaped single crystal}

of

_Ca(OH)2,

with a

diameter of 3.78 mm and a

_length

of

_{0.18 mm,}

while the axis of the

_cylinder

is oriented

parallel

to the

_crystalline

c-axis. For the

_DNP,

the

_sample

is irradiated with a 1.5 MeV electron beam

_{creating 02 -centres,}

at a concentration of 6 x

10- 5

centres _per

_proton,

_[10].

The

_sample

is cooled to 0.5 K

_using

a

3He

_evaporation

_cryostat.

A

_{superconducting}

solenoid created a

_magnetic

field of 2.73 T

_along

the

_crystalline

c-axis. The

_proton

_spins

are

_polarized

by

means of DNP

_using

75 GHz microwave irradiation.

_{Subsequently,}

the external

_magnetic

field is raised to 5.63 T in order to reduce the

_spin-lattice

relaxation rate.

_Finally,

an ADRF is

performed

as described in the

_previous

section and in such a _way that the final

_temperature

is

negative.

2.1 THE ENTROPY. - All the

_properties

of the nuclear

_spin

_system

are measured for four

values of the nuclear

_polarization

_po

before the

_ADRF,

which is determined from the area

using

the method of reference

_[11]

_]

to determine the

_{factor g}

for our

cylindrical sample.

The

_{entropy S}

of the

_proton

_spin

_system

is almost

_completely

conserved

_during

the ADRF.

Furthermore,

this

_entropy

is related to the

_polarization

_{p °}

before the ADRF

_{by [4] :}

Hence,

by performing

all measurements as a function of the initial

polarization

p °,

, we

effectively perform

them as a function of the

_entropy

of the nuclear

spin

_system.

We note that

_during

the

_ADRF,

the

_entropy

increases

by

about 9

%,

as was checked

_by

performing

ADRFs and

_{subsequent remagnetizations}

in the

_rotating

frame. In the

_following

sections,

we will use values

_{of p °}

that are corrected for this

_slight

increase of

_entropy.

The four corrected values

_{of p °}

are

_given

in the first column of table I.

Table I. - Results

_of

the transverse

_{susceptibility}

measurements.

2.2 THE TRANSVERSE SUSCEPTIBILITY. - We have also determined the transverse

suscepti-bility

of the

_proton

_spin

_system

after a

complete

ADRF. It is defined in the

rotating

frame of

reference. In

_{equation (2.3),}

_{Px is the}

_{magnetization}

in this

_rotating

frame due to a transverse

field

_{B1 =}

_{y -1}

1 li)

1

applied

in the x-direction of this

rotating

frame. We have obtained

X 1 from NMR

absorption signals

recorded

directly

after a

_complete

ADRF and

computing

their

_{Kramers-Kronig}

transforms

_{[4] :}

where the calibration

_{factor g}

is the same as above in

equation (2.1 ).

The results are

_given

in

the second column of table 1.

2.3 THE PARALLEL POLARIZATION. - After the

ADRF,

the

_proton

_spin

_system

is described

by

the Hamiltonian JC

_{given by equation ( 1.1 )}

and the

_density

matrix

_{given by equation (1.4).}

If the ADRF is

_incomplete,

the effective

_longitudinal

field A differs from zero.

_Then,

also the

1483

We have measured these

_{polarizations}

as a function of the effective field

d,

by performing

incomplete

ADRFs and

_{by measuring}

the nuclear

_{polarizationpz}

after these ADRFs

_using

the

same method as described above for

p °.

The results for the four values

of p °

given

in table 1 are

_presented

as data

_point

in

_figures

_2a,

_b,

c and d. The curves are the result of theoretical

calculations that will be treated in section 3.

Fig. 2. - pz plotted

as a function of

_{A/2 w,}

for four different values of the initial

polarization

p °.

The curves are discussed in section 3.

2.4 THE INFLUENCE OF THE DEMAGNETIZING FIELD. - The

_crystal

structure of

_Ca(OH)2

is non-cubic and the

_sample

has a

_{non-spherical shape.}

Therefore the

_polarized

_proton

_spins

create a

_{demagnetizing}

field that should be added to the effective fields in the

_rotating

frame.

Thus,

the

_protons

_experience

a total

longitudinal

effective field

For our

cylindrical sample,

A 0

is calculated in reference

_{[ 11 ] .}

Its value is

_given

in the third column of table I.

The influence of the

_{demagnetizing}

field on Pz will be accounted for in the next section. Its influence on the observed

_{perpendicular susceptibility}

_{X 1} is

_easily

calculated

[12] :

where _{X 1, o} is the

_{susceptibility}

in a

_sample

where

A ° - 0. Applying

this relation to the

experimental

values for _{X 1,} we _{obtain X 1, o} as a function of the initial

_polarization

Po.

These results are

_given

in the last column of table

I,

and furthermore

_plotted

in

_figure

3. The solid squares

represent

our results. Another result of

_{Sprenkels [7]}

is shown as a _open

square.

Finally,

small dots

represent

values for _{X 1, 0,} obtained

_by

Marks

[3]

in a

_completely

different way, i.e.

by

recording

the

_imaginary

_component of the

_{frequency dependent}

susceptibility during

the ADRF.

2.5 THE DIPOLAR ENERGY. -

_Finally,

we determined the first

_{moments m}

1 of the NMR

absorption signals directly

after the

_complete

ADRFs. From these first moments, the

_dipolar

energy T ! ri1 1

Fig.

3. - The transverse

_{susceptibility}

_{X 1.. o} versus the initial

polarization

p 0.

₍₀₎

results from this work.

(e)

results of Marks.

_(D)

result of

_Sprenkels.

The curves are determined

_by

the restricted trace

1485

Fig.

4. - The

_dipolar

energy per

spin

Ed/2 ir N

versus the entropy per

spin

SI Nk.

The

slopes

of the dotted lines 1 and 2 _represent the

_dipolar

_temperature at 1 and 2

_kHz/spin.

of the

_proton

_spin

_system

was calculated

_{using [4] :}

where the calibration

_{factor g}

is the same as above.

As was

_explained

_above,

the initial

_polarization

_{p °}

is related to the

entropy S

of the

_proton

spin

system

via

_{equation (2.2).}

Hence

_{using equation (2.2),}

one can

_plot

the

_dipolar

_energy

Ed

as a function of the

_entropy

S. The result is shown in

_figure

4.

As is well

_known,

the

_{thermodynamical}

relation

allows us to calculate the

_temperature

T of the nuclear

_spin

_system

after the ADRF from the results

presented

in

_figure

4. Since the number of

_{experimental points}

is

_small,

we restrict

ourselves to an _{estimate of the average}

_dipolar

_temperature

for two values of

_Ed/2

’TT N

_{only :}

1 and 2

_kHz/spin.

From the

_slope

of the dashed lines 1 and 2

_through

the

_{experimental points}

in

_figure

4,

we find :

3. Theoretical

_{interpretation.}

applied

to the conditions of the

_{experiments given}

above.

In our

previous

_paper, a

_major

effort was concerned with the determination of the

_magnetic

phase diagram

of the nuclear

_spin

_system

in

_Ca(OH)2.

Three

_phases

are

_expected

to occur, a

paramagnetic phase,

where all

_spins

are oriented in the same

direction,

a

_{ferromagnetic phase}

with domain structure as described in section

1,

and an

_{antiferromagnetic phase,}

where the nuclear

_spins

in successive

_{crystal planes}

have

_{opposite polarizations.}

We note that in our

_experiments

all these

_{phases might}

occur. This can be seen most

_cléarly

in

_figure

2,

showing

our

_experimental

results for the

_{parallel polarization}

_p, as a function of

the

_longitudinal

effective field 4 and the initial

_polarization

_Po.

_Using

the calculations of reference

[9],

we have determined for each value of 4 and

p °,

, which of the three

phases

is

most favourable. For the most favourable

_phase,

we have

subsequently

calculated the

_parallel

polarization

Pz as a function of d. The results are

_plotted

in

figure

2

by

solid curves. P

indicates the

_{paramagnetic phase,}

F the

_{ferromagnetic phase}

and AF the

_{antiferromagnetic}

phase.

It can be seen that for the case

that p 0 = 0. 19

_only

the

paramagnetic phase

is

expected

and no

_ordering

should be observable.

_However,

for

higher

values

of po

all

three

phases

are

expected.

’-Then,

for small values of

d,

the

theory predicts

the

_{antiferromagnetic}

_phase

with a small

parallel susceptibility

For

_higher

values of

à,

it

_predicts

the

_{ferromagnetic phase}

with a much

higher

value of

XII

Finally,

at still

higher longitudinal

effective

fields,

the

spin

system

is

predicted

to become

paramagnetic

and p, reaches a

_nearly

constant value

_approaching

_po.

We note that the theoretical results are _very

_detailed,

while the

_experimental

_accuracy is

limited. The accuracy

of pZ

is

mainly

determined

by

our

_previous

determination of the

calibration

factor g

and is about 1 %

_{[ 11 ].}

However,

the _accuracy of à is much

_less,

because the

_{externally applied magnetic}

field drifts

_slowly.

_Therefore,

before each

ADRF,

we

determined the

_frequency

_yB°

of the rf-field for which 4 = 0. This determination is

performed by recording

the NMR line of the

_polarized

_proton

spins

and

_{by fitting}

yB°

in such a _way that the ratio of the moments _m°

_{and ml}

of this NMR line

_corresponds

to

the value

_given

in reference

_[11].

As can be seen from that

reference,

the accuracy of this

procedure

is about 0.5 kHz.

As a

result,

our

experiments

are not

_{precise enough}

to show the

_possible

existence of the

predicted antiferromagnetic phase.

However,

in a

_previous

_paper

_[9],

we

_analysed

an earlier

experiment

of

_{Sprenkels [7]}

which was

_performed

at a

_higher

value

of p ° =

0.70. This

analysis

showed that the

_{antiferromagnetic phase}

does not occur. As an

_explanation,

we

proposed

that

the

_spin

_system,

_{which passes}

_through

the

_{ferromagnetic phase during}

the

ADRF,

remains frozen in that

_phase

when 4 = 0.

Hence,

we conclude that all data

points

in

figure

2a concern the

paramagnetic phase,

while

those in

_figures

2c and 2d concern the

_{ferromagnetic phase.}

The scatter in the

_datapoints

in

figure

2b is too

_large

to _{draw any} conclusions. From

_figures

2c and

2d,

we also see

_that,

for the

ferromagnetic phase,

the

_slope

of our

experimental

curve for Pz versus 4

corresponds

well with the theoretical

_prediction

of the restricted trace

_{approximation.}

But

_figure

2a shows

that,

for the

_{paramagnetic phase,}

the

_{experimentally}

found

_slope

is less than

_predicted

theoretical-ly.

As the

_slope

of these curves

_corresponds

to the

_{parallel susceptibility}

Yll, for the

1487

restricted trace

_{approximation.}

_However,

the

_prediction

for the

_{paramagnetic phase}

seems to

be less

_good.

3.2 THE TRANSVERSE SUSCEPTIBILITY.

_{- Using}

the restricted trace

_{approximation,}

we also

calculated the transverse

_{susceptibility}

of the nuclear

_spin

system

as a function of the initial

polarization

p °.

The results are

_plotted

in

figure

3

_together

with the

_experimental

results discussed above. The curve indicated

_by

F

_represents

the result for the

_{ferromagnetic phase,}

the curve indicated

_by

P the

_{paramagnetic phase.}

For

_{p ° > 0. 3,}

the

_experimental

values of

X 1 are more or less

_independent

of po

. which agrees with the theoretical

prediction

for the

ordered

_phase.

At lower values

_{of p 0}

_the

_experimental

values diminish

_rapidly

as

_{predicted by}

the theoretical curve for the

_{paramagnetic phase.}

We note

_however,

that for these lower values of

_{p °,}

the observed transverse

_{susceptibility}

seems to be

_considerably

lower than

predicted by

the restricted trace

_{approximation.}

This observation is in

_agreement

with earlier

experiments

on

_{CaF2 [8],}

where also

_{systematically}

lower transverse

_{susceptibilities}

were

observed than

_{predicted by}

the restricted trace

_{approximation.}

3.3 THE TRANSITION TEMPERATURE. - As discussed in section

_2.5,

the

_spin

_temperature

is

experimentally

found to be

_equal

to - 1.1

_uK

when the

_dipolar

energy is

equal

to

Ed/2

irN = 1

kHz/spin,

while it is - 0.7

uK

when

Ed/2

irN = 2

kHz/spin.

Furthermore,

our

analysis

of our measurements of the

_{parallel polarization}

shows that the nuclear

_spin

_system

is

paramagnetic

in the first case, while it is

ferromagnetic

in the latter case. This conclusion is confirmed

_by

our measurements of the transverse

_{susceptibility.}

_Hence,

the transition

temperature

should lie between these two values :

This value for the critical

_temperature

agrees very well with the value

predicted by

the molecular field

_{approximation}

_{[7] :}

4. Conclusions.

In this _paper, we

_presented

a

_study

of the

_ordering

of the

_proton

_spins

in

_Ca(OH)2

that occurs

at

_{negative spin}

_temperature.

The

_experimental

values of the

_parallel

nuclear

_polarization

are

interpreted using

the restricted trace

_{approximation.}

The results show that a

_longitudinal

ferromagnetic ordering

with domain structure is created. The

_{antiferromagnetic ordering,}

which is

_predicted

for _very small values of the effective

_longitudinal

field could not be observed. In the _present case, this

_negative

result

might

very well

originate

in

inadequate

experimental

accuracy,

contrary

to the case of

_Sprenkels

_experiments

as

_analyzed

in a

previous

paper

[9].

From the measurements of the

_dipolar

_energy versus the

_entropy

we determined the

_dipolar

temperature

after the ADRF.

_Combining

these results with those for the

_{parallel polarization,}

we have

_determined

_the

_experimental

_{value of the critical}

_temperature.

It

_corresponds

_very well to the value

_{predicted by}

the restricted trace

_{approximation.}

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

References

[1]

ABRAGAM A., C.R. Hebdo. Acad. Sci. Paris 251

₍₁₉₆₀₎

_{225-227 ;} ibid. 254

₍₁₉₆₂₎

1267-1269.

[2]

CHAPELLIER M., GOLDMAN M., VU HOANG CHAU, ABRAGAM A., C.R. Hebdo. Sci. Paris 268

(1969)

1530-1533.

[3]

MARKS _J., WENCKEBACH W. Th., POULIS N. J.,

Physica

B 96B

₍₁₉₇₉₎

337-340.

[4]

ABRAGAM A., GOLDMAN M., Nuclear

_{magnetism ;}

Order and disorder

_(Clarendon

Press,

Oxford)

1982.

[5]

VAN DER ZON C. M. B., MARKS _J., WENCKEBACH W. Th., POULIS N. _J.,

_Physica

B 145B

₍₁₉₈₇₎

153-164.

[6]

BUSING W. _R., LEVI H. _A., J. Chem.

_Phys.

26

₍₁₉₅₇₎

563-568.

[7]

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

_Phys.

C. 15

₍₁₉₈₂₎

_{L941-L947 ;} ibid.

16

₍₁₉₈₃₎

4425-4445.

[8]

GOLDMAN M., SARMA G., J.

_Phys.

France 36

₍₁₉₇₅₎

1353-1362.

[9]

VAN VELZEN G. D. _F., WENCKEBACH W. Th., J.

_Phys.

France 51

₍₁₉₉₀₎

1463.