Magnetic properties of an antiferromagnetic two-singlet system. II. Results on terbium aluminum garnet

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Magnetic properties of an antiferromagnetic two-singlet system. II. Results on terbium aluminum garnet

A. Gavignet-Tillard, J. Hammann, L. de Seze

To cite this version:

A. Gavignet-Tillard, J. Hammann, L. de Seze. Magnetic properties of an antiferromagnetic two- singlet system. II. Results on terbium aluminum garnet. Journal de Physique, 1973, 34 (1), pp.27-34.

�10.1051/jphys:0197300340102700�. �jpa-00207353�

MAGNETIC PROPERTIES

OF AN ANTIFERROMAGNETIC TWO-SINGLET SYSTEM.

II. RESULTS ON TERBIUM ALUMINUM GARNET (*)

A.

GAVIGNET-TILLARD,

^J. HAMMANN and L. DE SEZE Service de

Physique

^{du Solide et} de Résonance

Magnétique

Centre d’Etudes Nucléaires de

Saclay,

^{BP n°}

2, 91,

Gif-sur-Yvette

(Reçu

^{le 7}

juin 1972)

Résumé. ²⁰¹⁴ Les

propriétés métamagnétiques

^du grenat de terbium et d’aluminium

(TbAlG), qui

est un

antiferromagnétique

^{à six} sous-réseaux

(TN

⁼

1,35 °K),

^sont étudiées ici : on donne les courbes d’aimantation et de

susceptibilite

différentielle en fonction du

champ magnétique appliqué

pour différentes directions

cristallographiques

^{et des} températures variant de 0,36 ^°K à 1 °K. Les résultats sont

interprétés

dans le cadre du

champ

moléculaire et du modèle à deux

singulets qui

est

approprié

pour TbAlG, ^en prenant ^en considération

l’échange

^à

ajouter

^aux interactions

dipolaires qui

^restent

prépondérantes

^{dans les} grenats ^{de terre rare et} d’aluminium. Les valeurs des constantes d’interaction ainsi déterminées sont confirmées _par

l’énergie magnétique

^{à 0 °K}

déduite de mesures de chaleur

spécifique

^entre 0,35 et

4,2

^°K.

Abstract. ²⁰¹⁴ The

metamagnetic properties

of terbium aluminum garnet, ^{which is a} six-sublattice

antiferromagnet (TN

^{= 1.35}

°K)

^are studied : the

magnetization

^and differential

susceptibility

curves as functions of the external

magnetic

^{field are}

reported

^{for different}

crystallographic

^directions

and temperatures from 0.36 °K to 1 °K. These results are

analysed

within the frame of the molecular- field

approximation

^{in terms of the}

two-singlet

^model

appropriate

^for

TbAlG, taking

^{into account}

some

exchange along

^{with the}

dipolar

interactions

preponderant

in rare-earth aluminum garnets.

The interaction constants found are confirmed

by

the value of the

magnetic

energy ^at 0 °K deduced from

specific

^heat measurements between 0.35 °K and 4.2 °K.

Classification

Physics abstracts : 17.68

1. Introduction. ^- In a

preceding

paper

[ 1 ],

^which

we shall henceforth refer to as

I,

^we discussed from a

theoretical

point

of view the

magnetic

^{behaviour of}

two-singlet systems,

and outlined the main

properties

of their

phase diagrams

in presence of ^a

magnetic

field.

We shall now concentrate on the

practical

^{case of}

terbium aluminum

garnet (TbAIG).

^The

susceptibility

and

specific

^heat measurements of Cooke et al.

[2]

showed that TbAIG

undergoes

^a transition to an

antiferromagnetic

^{state at}

TN

⁼ 1.35 OK. Its

magnetic

structure was determined

by

^neutron diffraction

experiments performed

^{at 0.31 OK}

by

^Hammann

[3] ;

it is

analogous

^{to the structure}

of dysprosium

^aluminum

garnet (DAG) :

^six sublattices

a-a’, fi-fi’

^and

y-y’

^with

magnetic

^moments

lying along

^{the z, x}

^{and y} crystallo- graphic

^axes of the cubic cell of

garnets (see

^Table

I).

TbAIG is known to fit the

two-singlet

^{model : the}

spectroscopy

^{data of}

Koningstein et

^al.

[4]

^{show that}

Tb3 +

in TbAIG has two fundamental

singlets

^well

separated

^{from the upper} levels. This is confirmed

by

highly anisotropic properties [5]

and with group

theory

considerations

[6].

When an external

magnetic

^{field is}

applied along

direction

[111 ],

TbAIG behaves like a two-sublattice

metamagnet ; magnetization

measurements

along [111 ]

at 0.36 OK in the

paramagnetic phase (i.

^{e. above the}

threshold

field)

^{in fields} up ^to 60 kG were

analysed

ⁱⁿ

terms of the

two-singlet

^{model and}

dipolar

interactions

(the preponderance

^of

dipolar

interactions was

deduced from the value of the

spontaneous

^moment measured

by

^neutron diffraction

[7]).

The energy

splitting

^{Li and the}

only non-vanishing component

^of the

magnetic

^moment between the two states were

found to be :

In this

analysis,

^the presence of upper levels of

Tb3 +

was taken into account as a linear contribution to the

magnetization [5]. Crystal-field

calculations

[6] proved

the

consistency

^{of such an}

approximation.

^{In the}

present

^{paper, we intend to}

study

^the

magnetic phase diagram

^{of TbAIG and thus restrict the} discussion to low

magnetic

^fields

(less

^{than 5}

kG) ;

^{this Van Vleck}

term is then

unimportant,

^{and we shall not include it}

(*) Cet article fait partie de la thèse de Doctorat d’Etat pré- sentée à l’Université Paris VI par l’un des auteurs (A. G.-T.).

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197300340102700

28

when

calculating magnetization

curves, for the sake

of clarity.

TbAIG allies the

simplicity

^{of the}

two-singlet

^model

to the fact that

dipolar

interactions can be

computed easily ;

^{therefore its}

magnetic properties

^{can be}

predicted

^and confronted to

experiment.

^Moreover

TbAIG appears ^as the two

singlet analog

^{of DAG}

(Dy3 +

^{in DAG exhibits a} fundamental doublet with

highly anisotropic properties)

^{where the} interactions

are

mainly dipolar,

^{and which has been}

extensively

studied

[8]-[11].

^A

comparison

^{of the}

magnetic

pro-

perties

^{of the two}

garnets

^will

emphasize

the role of the energy

splitting

^{of the two}

singlets.

A

preliminary report

^{on the}

experimental phase diagram

^{of TbAIG at} 0.36 OK in presence of a

magnetic

field has

already

^been

published [12].

^We

present

^here

more extensive

experimental

^{results :}

magnetization,

differential

susceptibility

^and

specific

^{heat measure-}

ments between 0.35 OK and 4.2 OK. Their

interpre-

tation will lead to the

analysis

of the interactions and to the determination of the amount of

exchange

^{that must}

be added to the

dipolar

interactions : it will

provide

^a

good

illustration of the theoretical results of paper I.

2.

Expérimental.

^{- 2. 1 THE CRYOSTAT. - The}

^He3 cryostat

^we used has been described

by

^Testard

[13] :

the

sample

is maintained

through

^an

exchange

gas

(He3)

^{in contact with a}

pot

^{in which}

liquid ^He3

^boils

under reduced pressure. The

exchange

gas ^{can even-}

tually

be removed _very fast

by

^an active-carbon _pump.

The

liquid ^He3

^{can be}

brought

^{to a low} pressure with either a diffusion _pump or an active-carbon pump

placed

^{above the}

^He3

^{bath. This last}

technique

^was

used to obtain the lowest

temperature (N

^0.31

OK).

The

temperature

^could be stabilized

anywhere

^between

0.31 OK and 1 OK with

good

accuracy

(-

^0.001

OK) by regulating

the pressure above the

He3

bath. In order to obtain

temperatures ranging

from 1 OK to 4.2

OK,

the

He3 pot

^was

emptied

^{and the}

temperature

^stabilized

through

^the pressure above the

Re4

bath.

A

magnetic

^field

parallel

^to the axis of the

cryostat

was created

by

^a

superconducting

^solenoid

equiped

with ^a

persistence

^{switch and}

performing

up ^to 20 kG for a current of 35 A. This coil had an homo-

genity

^{at its center of}

^10-3

^{in a volume of 1}

^cm3.

2.2 THERMOMETRY. - Our thermometer was a

standard

1/2

^{W-470 S2}

Speer resistor,

^without any

special

^{treatment like the}

grinding

^out of the surface.

It was coated with 7031 GE

low-temperature

^vanish

and

wrapped

^{in a} thin copper foil in order to ensure a

good

^{thermal contact with} its environment.

Its resistance was measured with an AC Wheatstone

bridge operating

^at 1 000 Hz and

using phase

^detec-

tion

techniques.

The lead resistances were eliminated

by

^the three-lead

technique

^{and the} power

input

was

approximately ^10-9

^W

allowing

^{a relative} pre- cision of

10-4.

The thermometer was calibrated

against

the vapor

pressures of

Re4

and

He3

in the

respective tempe-

rature ranges

(with

corrections for the thermomo- lecular pressure ^near 0.3

OK).

For the

magnetization

measurements, ^{we were} after an absolute

precision

of 0.01 OK. Therefore the

Speer

^{resistor was} calibrated once for diffèrent values of the

magnetic

^field

(up

^{to 8}

kG) ;

^{we found that}

such

relatively

^low

magnetic

^{fields did not alter the}

resistance measurement

significantly (i.

^{e. within}

a

precision

^{of 0.01}

OK)

^{and we checked that from}

one run to another our calibration was still exact within 0.01 OK.

For the

specific

^heat measurements, ^the thermo-

meter was

carefully

recalibrated every time it ^was warmed _up and its resistance R was fitted in the

He3

and

He4 temperature

ranges to ^an

expression

with the aid of the linear

regression

program of Tournarie

[14].

The program

rejected

^the

obviously

erroneous

points

and the accuracy of the calibration

was then around 0.001 op.

2. 3 MAGNETIZATION MEASUREMENTS. ^-

They

^were

performed

^{on a}

spherical monocrystalline sample (diameter

^{= 3.95 mm,}

weight

^{= 0.1892}

g)

grown from ^a flux

by

^{J. Marechal}

(LETI,

³⁸

Grenoble, France).

^The

garnet

^{structure was checked}

by X-ray

diffraction and the

sample

^{was oriented so that it}

could be rotated about a

[110]

^axis

perpendicular

to the external vertical

magnetic field, using

^the

device described

by

Bidaux et al.

[9].

The

sphere

^was

placed

^{at the center of one of two}

coils

(with

^vertical

axes) connected astatistically,

so that the

sample

^{alone was}

responsible

^{for flux}

variations

through

^{the two} coils. The

magnetic

field

being

^{set at a definite}

value,

^the

sample

^was

extracted from the upper coil and the

resulting

^flux

variation

(proportional

^{to the}

magnetization

^{of the}

sample)

^{was read with a} fluxmeter. With this

method,

the

heating

up of the

sample

^{when it is moved does}

not affect the measure itself.

Nevertheless,

^{it takes}

approximately

five minutes for a return to thermal

equilibrium,

and this makes each measurement

quite long.

2. 4 DIFFERENTIAL SUSCEPTIBILITY. ^- It is

interesting

to measure the differential

susceptibility dM jdH (M

⁼

magnetization,

^{H = external}

field)

^{versus the}

magnetic ^field,

for it allows an easy determination of the order of a

metamagnetic transition, provided

that the

sample

^{used has a}

precisely ellipsoidal shape :

^{a first order} transition is then characterized

by

^{a constant value of}

dM/dH.

In order to measure

dM/dH,

^we

placed

^the

sample

in the

top

^{coil and let H} vary

linearly

^{with time.}

The

output signal

^of the astatic coils is propor-

tional to

dM/dt (t referring

^to

time).

^Two

types

^of

problems

^arise

during

this process :

a) dH/dt

^must be small in order to

keep

^{the tem-}

perature

^{of the}

sample constant ;

b) dM/dt

is very sensitive ^to fluctuation of

dH/dt,

which must be

regulated.

We solved the difhculties as follows : the coils

were connected to a

galvanometric amplifier (whose gain ranged

^from

¹⁰⁴

^to

106) capable

^of

giving

^{a full}

range derivation for

dM/dt

^{as low as 5}

pV.

^Its

output signal

^{was connected to the Y}

input

^{of an X-Y}

recorder.

As for

dH/dt,

^{the current}

through

^the

superconduct- ing

^{solenoid was}

produced by

^a power

supply

^regu-

lating

the tension across the

solenoid ;

^this

gives

far better results than a current

regulation.

^The

X-input

of the recorder was fed with the tension

across a 0.001 Q standard-resistor connected in series with the

superconducting

coil. The

sweeping

rate

dH/dt

^was

typically

^0.5

kG/mn.

2.5 SPECIFIC ^{HEAT. -} We took six

monocrystals (total weight :

^1.0725

g) having

^{the same}

origin

as the

monocrystalline sphere

used in the magne- tization measurements and

glued

^them around the

Speer-resistor

thermometer

(see

^the

thermometry section)

^with

silicon-grease charged

^with

tiny pieces

of copper, in order to

improve

^{the thermal contact.}

The whole

thing

^was

wrapped

^{in a} thin copper foil and a heater made of

manganin

^{wire was wound}

around it and held with 7031 GE vanish. The

manganin

wire had a diameter of 0.05 mm and the resistance of the heater was

approximately

^{200 Q.}

The

sample

^was

suspended

^{in the}

experimental

chamber

by

the thermometer and heater leads made of

manganin

wire 0.05 mm in

diameter ; they

^were

coated with tin solder in order to avoid heat dissi-

pation

^{in the leads. The}

sample

^{was cooled}

by

^the

He3 exchange

^{gas which could be}

pumped by

^active-

carbon. The measurements were made

by

^{the usual}

heat

pulse

^{method :} electrical power is

supplied

^to

the heater

during

^{a short}

period

^{and the}

change

^of

temperature

^is determined from drift rates before and after the

heating.

The current

through

^{and the}

voltage

^{across the}

heater were measured with

five-digit

multimeters

(the potential

^leads

being

^connected inside the expe- rimental

chamber),

^{and the}

heating period

^{was timed}

with a

frequencymeter.

The power used was

typically 10-5 _W,

for

lengths

^{of time}

ranging

^{from 10 to}

30 s.

After the heat

pulse

^{30 s were}

generally

necessary to go back ^{to a} stable drift rate of the

temperature.

Measurements of the drift were continued for

typi- cally

five minutes. The overall _accuracy ^on the

tempe-

rature rise was

approximately

²

% ;

^{this is}

mainly

due to the fact that _{the power}

supplied

^{to the ther-}

mometer warmed _up our small

samples

^{and the}

fairly important

^{drift rate} limited the width of the

temperature jump

^on the chart recorder. This was

particularly

^{true at low}

temperatures,

^the

specific

heat

being

then small.

3.

Dipolar

interactions in TbAIG. - As TbAIG has the same

anisotropy

directions as DAG

[6],

the

problem

^{of the}

dipolar

interaction is identical in both cases. Bidaux et al.

[8]

^{showed that in}

order to understand the

magnetic

^{behaviour of such}

garnets, twenty-four

sublattices can be considered.

They

^{are defined as follows}

(using

^{the same rotations}

as ref.

[8]).

The sublattices

oci, a2, a3

^and

a4

^{are deduced}

by

^a

(0, 0, 1/2)

translation. Sublattices

B1,

^...,

B4,

^and

yi, ^...,

Y4, along

with their

anisotropy direction,

^are

deduced from _al, ...,

a4 by

^{rotations of ± 2}

Tr/3

^about

the

[111] ]

^direction.

The

dipolar

^tensor

T ij

^is

where u is the unit tensor and

rij

⁼

rj -

ri

(ri being

the location of the ith ion in the

crystal).

The molecular field

acting

^{upon the ith} ion is thus

mi

>

being

the thermal average of the

magnetic

moment

mi of the jth

^ion.

In fact the

symmetry operations

^of the space group reduce to six the number of non-zero

dipolar

^sums

that are, when calculated in a

spherical

^{volume :}

(a

^{= lattice}

parameter

= 12.01

A

for

TbAIG).

The

problem

^{now is to find out what}

magnetic

structure is stable at 0 OK in presence of a

magnetic

field and

purely dipolar

interactions. Bidaux et al.

[8] investigated

this in the case of DAG

(assuming g_, 0 0,

gx ⁼

gy

⁼

0) ;

^the transition

temperature TN

of a

given

^structure

(leading

^{to a} molecular-field constant

fl)

^{is :}

30

TABLE 1

AF, ^AF’ and F

magnetic

structures

of

garnets. ^The sublattices are

defined

^{in the text}

The stable structure is the one

leading

^{to the}

highest TN.

In the two

singlet

^{case the}

quantity

^on the left side of the

equality

^is

replaced by

(The

^two

expressions

become identical when 4 -

0.)

We see that the

problem

of the stable structure is the ^{same as} in

DAG, except

that the transition condi- tion 2

M2

> L1 must be fulfilled.

Therefore in the absence of an external

magnetic

field the stable structure, ^{if it}

exists,

^{is the one labelled}

AF

(see

^Table

I).

^{When a}

large magnetic

^{field H is}

applied along [001],

the sublattices a-a’ are

polarized

and

B-B’, y-y’

^are

uncoupled

from H and _may reorder in the AF’ structure

(see

^Table

I).

^{If it is}

parallel

^to

[110],

^x-(x’ may take a structure

of type

^F

(see

^Table

I).

The transition conditions for the

AF,

^{AF’ and F}

structures are written

explicitly

^{in Table I.}

When these conditions are

applied

^{to the case of}

TbAIG

(with

^{L1 = 2.5 oK}

and m,, =

^7.6

/lB)’

^{it is}

easy ^to check that neither the AF’ not the F structure

are allowed

(*). Therefore,

^{we are}

always dealing with

a transition of the AF --+ P

type (P referring

^{to the}

saturated

paramagnetic phase).

^{Due to this}

simplifi- cation,

^{when we} write the molecular-field

equations

with the

applied

^field

along [111], [110]

^and

[001],

the

dipolar

^sums appears in three groups that ^are

and the sets of

equations

^{to be solved are :}

Direction

[111] :

TABLE Il

Interaction constants and the

corresponding

calculated values

of

^the

threshold fields

^{in the}

purely dipolar

case, and with the parameters determined

from

^the

experiments.

^The

experimental

^threshold

fields

^are

given

^{in the}

first

^column.

with

Let us note that the interaction

parameters

ⁱⁿ direction

[111] ] (TbAIG

^{is then}

analogous

^{to a two-}

sublattice

antiferromagnet),

^when

reported

ⁱⁿ

figure

⁸

of paper

I,

indicate that the

metamagnetic

^transition

at 0 OK is a first order one.

When the three sets of

equations

^{are solved at}

T ⁼ 0.36 OK

(with

^{L1 = 2.5 oK}

and m, =

^7.6

,uB),

a first order transition is found in the three directions

(see Fig. 1,

^{2 and}

3) ;

the values of the threshold fields can be found in table II.

Let us examine now the

experimental

^{data and}

compare them in detail to the results of this section.

FIG. 1. ^- Magnetization ^versus internal field in direction [1111 ]

at 0.36 °K. Here m ⁼ (ma + ma-)/2 (see text). ^{For the} experi- mental _{curve, m} is taken as the magnetic moment per Tb3+ ion measured in direction [111] multiplied by

/3.

^{The other two}

curves correspond ^to the interaction constants given in table II.

FIG. 2. ^- Magnetization ^{versus internal} field in direction [110]

at 0.36 °K. Here m ⁼

(mB

⁺ mB’)/2 (see text). ^{For the} experi- mental curve, m is taken as the magnetic moment per Tb3+ ion measured in direction [110] multiplied by

/2.

^{The other two}

curves correspond ^to the interaction constants of table II.

FIG. 3. ^- Magnetization ^versus internal field in direction [001] ]

at 0.36 OK. Here m = (m« ⁺ m«-)/2 (see text). ^{For the} experi- mental curve m is the magnetic moment per Tb3+ ion measured in direction [001]. ^{The other two curves} correspond ^{to the}

interaction constants of table II.

32

4. Results and discussion. ^- 4.1 EXPERIMENTAL

PHASE DIAGRAM AT 0.36 °K AS A FUNCTION OF THE DIRECTION OF THE MAGNETIC FIELD. ^- The

experi-

mental

magnetization

^curves in directions

[111 ], [110]

^and

[001] ]

^{at T = 0.36 oK are}

reported

^on

figures 1,

² and 3. The

metamagnetic

^transition

is a first order one in all three

directions,

ⁱⁿ

agreement

with the

prediction

^of

dipolar interactions ;

^the

experimental

values of the threshold fields are listed in table II. The differences with the values of

dipolar

interactions

suggest

^{that some}

exchange

^{must be}

taken into account, ^{as was}

already pointed

^out

[12] ;

this ^can be done

by replacing

^the

dipolar

^sums

A,

B and C

by phenomenological parameters A’, B’

and C’ that must be determined. The

difficulty

^{in this}

fitting

lies in the fact that ^not

only

^{the threshold}

fields,

^{but the}

shapes

^{of the}

magnetization

^curves

must be

adjusted.

^The

procedure

is rather tedious since no

analytical

^{form of the net}

magnetization

versus

magnetic

^{field is available.}

In direction

[111 ],

TbAIG behaves like ^a two- sublattice

antiferromagnet.

^The

temperature

T ⁼ 0.36 OK is low

enough compared

^to

TN

^{to make}

a

zero-temperature approximation,

and the formula

(6)

of _paper 1 can be used to determine the threshold field for a

given

^set

(A’, B’ -

²

C’).

As A’ appears alone in the three sets of

equations concerning [111 ], [110]

^and

[001] (see

^the

preceding section)

^{two magne-}

tization curves

only

^{are needed} for the determination of

(A’, B’, C’).

^The

procedure

^{is as follows : direction}

[111] ]

^must be considered

first ;

among the

couples

of values

(A’,

^{B’ - 2}

C’)

consistent with

H,x’ [ 111 ],

we

pick

up the ^one

leading

^{to a}

good

fit of the _magne- tization curve

(especially

^{in the AF}

phase

^{which is}

more sensitive to the interactions than the P

phase).

Then A’

being given,

^we look for values

(B’, C’)

such that

fitting

^{well the}

magnetization

^curve in either the

[110]

or the

[001] ]

direction. We remarked that

[001] ]

^is

more sensitive to variations of B’ and C’ than

[110].

Therefore we chose

[001].

We were able to

get

^an excellent fit in direction

[111] ] (see Fig. 1) ;

^{in direction}

[001],

^{we could}

adjust

^the

rounding

^{of the}

magnetization

^{in the}

4F ^phase.

But the value of the threshold field was

always

^a

little

larger

^{than the}

experimental

^one

(*). Finally

the best set

(A’, B’, C’) gives

^{a value}

H;alc [001] ]

5

%

^above

Hsexp [001] ] (see

Table II and

Fig. 3) ;

this may be due ^{to a} small misorientation of the

crystal,

^to which direction

[001] ]

^{is more sensitive}

than

[110]

^or

[111 ] (as

^{can be seen on the}

experimental phase diagram

^{of TbAIG}

given

^{in ref.}

[12]) ;

^this

assumption

^is

supported by

^the

good

fit obtained in direction

[110] (see

Table II and

Fig. 2).

4.2 EXPERIMENTAL PHASE DIAGRAM IN DIRECTION

[111 ]

AS A FUNCTION OF THE TEMPERATURE. ^- When H lies

along [111] ]

^{it is}

interesting

^to

investigate

^the

effect of the

temperature

upon the

metamagnetic

transition AF --> P and compare the results to the theoretical

predictions

^of paper I.

The

magnetization

^{curves at different}

temperatures

are

reported

^on

figure

^{4 and the} differential suscep-

tibility

^{results on}

figure

^{5. It can be seen that above}

a critical

temperature

FIG. 4. ^- Magnetization ^versus internal field in direction 111] ] for different temperatures. ^{Here m =} (ma. + m«,)/2 (see text)

it is taken as the magnetic moment per Tb3+ in

direction"[111]

^] multiplied by

/3.

FIG. 5. ^- dM/dt ^{versus internal} magnetic ^{field at dînèrent} temperatures. dM/dt ^is proportional ^{to the} differential _suscep-

tibility, dH/dt being ^constant.

(*) ^{Let us} mention that slightly ^{different values of} (A’, B’, C’) which give ^a first order transition in [111] at ^{0 °K lead to a} second order one in direction [001].

the

metamagnetic

transition becomes a second order one, ^as

expected.

The differential

susceptibility

^at

0.36 °K was measured with

increasing

^and

decreasing magnetic fields,

^{and did not} exhibit _any

hysteresis.

The

phase diagram

^is

reported

^on

figure 6,

^where

H$ [111 ]

^is

plotted

^{as a function of the}

temperature

(above Tc,, HS [l l l] ]

^{is taken as} the inflexion

point

of the

magnetization curve).

FIG. 6. ^- Magnetic phase diagram ^{of TbAIG in direction} [111] ]

as a function of temperature. The threshold ^field is plotted against the temperature.

Using

^the

phenomenological parameters

^{of table II} the critical

temperature TcalcCR

^{as well as the Néel}

temperature TN lc

^can be calculated and

compared

to the

experimental

results. As was

pointed

^{out in}

paper

I,

the relevant

quantity

is in fact tanh

(d/2 kT).

We

get

^{then :}

This is the usual result when the molecular-field

approximation

is used with the

Ising

^{model: a calculated}

temperature (here

^{it is}

^{replaced by (tanh} (d /2 kT j) -1 )

is above the

experimental

^one

by approximately

25

% [17].

4.3 ENERGY AND ENTROPY. ^- The

specific

^heat

(per ^{Tb3 +} ion)

^{of TbAIG} between 0.35 OK and 4.2 OK is shown on

figure

^{7. This curve} is obtained from the

experimental

results when the contributions of the lattice of

TbAIG,

^of the copper

sample

^{holder and}

of the

silicon-grease

^are substracted. The first two contributions were calculated : for the lattice

part,

we used the value

OD

⁼ 500 OK for the

Debye

^tem-

perature

^of

TbAIG, following

Landau et al.

[10] ;

for the

sample

^{holder we took for} the electronic

specific

^{heat and}

Debye temperature

of copper the values

given by

Sato et al.

[16].

^Both contributions

were found to be

quite

^small

(less

^0.1

%

^{of the total}

specific heat).

As for the

silicon-grease,

^{we fitted the}

remaining specific

^heat between 2 OK and 4 OK to

a

Schottky anomaly plus

^a

^T3

^lattice

term, represent- ing

^the

silicon-grease part ;

^at 4.2 OK it amounted to

30 %

^{of the total}

specific

^heat.

FIG. 7. ^- Specific ^heat per Tb3+ ion versus temperature.

The

Â-type anomaly

^{in the}

specific

^heat

(see Fig. 6) peaks

^at

temperature TN -

1.31 OK. We noticed that this value differs

slightly

^from

susceptibility

and

specific

^{heat results of Cooke et al.}

[1]

^and

from the observation of Hammann

[3]

^{in his neutron}

diffraction

experiments.

^The

fitting

^{of the}

specific

heat ^« tail »

(above

²

OK) gives

^{for the}

splitting dSchottky

of the levels :

This must be

compared

^{to d =} 2.5 OK. The difi’e-

rence between the two values is due to the fact that the

T-2 magnetic

contribution above the transition

was included in the fit.

On the low

temperature

^{side of the Â}

anomaly,

the rise in the

specific

heat is due to the nuclear contri- bution that becomes

important

^{below 0.5 °K : the}

constant A in the

hyperfine coupling ( - AI.J)

^can

be estimated :

4.3.1

Magnetic

energy. ^- In order to determine the total _energy ^at T ⁼ 0

OK,

^{we must evaluate} the

integral

^{of the}

specific

heat from 0 OK to

infinity, leaving

^out the nuclear contribution. The

specific

heat between 0 °K and 0.5 OK can be

extrapolated

and the

integration

between 0 OK and 4.2 OK can

34

be made

graphically.

Above 4.2

OK,

^the

specific

heat ^can be taken ^{as a}

Schottky anomaly (with dSchottky -

^2.87

°K), allowing

the calculation of its

integral.

In total we

get

^{for the}

energy E

per

Tb3 +

ion :

This must be

compared

^to

^Ecalc given by (see

paper

1) : E calc -

where _mo is the

spontaneous magnetic

^{moment at}

T ⁼ 0 oK. We

get

^{then :}

The

agreement

between the two values is excellent

considering

^the

approximation

used in the deter- mination of E.

4.3.2

Entropy.

^{- If we use the same}

techniques

as in the former section to determine the

integral

of

C/T,

^we find for the

entropy S

per

Tb3 +

ion

The theoretical value for a

two-singlet system

^{is :}

5. Conclusion. ^- The

experimental

results show that TbAIG is indeed ^a

good example

^{of a two-}

singlet system

which fits well the results of paper I.

We were able to determine from the

magnetization

curves at 0.36 °K in different

crystallographic

^direc-

tions the ^amount of

exchange

that exists

along

^with

the

preponderant dipolar interactions,

^{and we could}

interpret

^the

phase diagram

^{in direction}

[1111 ]

^{as a}

function of the

temperature.

The Néel and critical

temperatures

calculated with the molecular-field

approximation

^are

slightly

above the

experimental values, indicating

^{that more elaborate}

approximations

should be introduced in order to account for the

properties

^at

high temperatures.

^The

magnetic

energy at 0 °K determined from

specific

measurements confirms’the

validity

^{of the}

phenomenological

^inter-

action

parameters

of table II.

As for AF’ and F structures, it should be worth- while to check whether or not

they

^{appear at tem-}

peratures

lower than 0.36 OK under the influence of mechanisms that we did not

consider,

like the nucleus- induced

ordering [15].

Acknowledgments.

^{- We wish to} thank M. R.

Gérard-Deneuville for his most

competent

^technical assistance and Dr. G. Jehanno who took care of the X-rav orientation of the crystal.

References

[1] ^BIDAUX R., GAVIGNET-TILLARD A. and HAMMANN J.,

to be

published

^{in J.}

Physique.

[2]

^{COOKE A.} H., THORP T. L. and WELLS M. R., ^Proc.

Phys.

^{Soc. 92}

(1967)

^400.

[3]

^HAMMANN J., ^Acta Cryst. ^{B 25}

(1969)

^1853.

[4] KONINSTEIN J. A. and SCHAACK G.,

Phys.

^{Rev. B2}

(1970)

^1242.

[5]

GAVIGNET-TILLARD A. and HAMMANN J., Proc. 12th Conf. on Low Temperature

Physics (Academic

Press of

Japan) (1971)

^697.

[6]

GAVIGNET-TILLARD

A.,

HAMMANN J. and DE SEZE

L.,

to be

published

^{in J.}

Phys.

Chem. Solids.

[7]

^HAMMANN

J.,

Thèse,

Rapport

^{CEA n° 3866}

(1969).

[8]

BIDAUX R. and VIVET B., ^J.

Physique

²⁹

(1968)

^57.

[9]

^BIDAUX R., CARRARA P. and VIVET B., ^J.

Physique

29

(1968)

^357.

10] ^{LANDAU D.} P., ^{KEEN B.} E., SCHNEIDER B. and WOLF W. P.,

Phys.

^{Rev. B 3}

(1971)

^2310.

[11]

^{WOLF W.} P., ^SCHNEIDER B., ^{LANDAU D. P. and KEEN} B. E., ^{to be}

published.

References

[10]

^and

[11]

are the first two of a series of papers ^on DAG, where references to the

previous

work of their authors can be found.

[12] GAVIGNET-TILLARD A. and HAMMANN J., Proc. 17th Conference in

Magnetism

^and

Magnetic

^Materials

(American

Institute of

Physics) (1972)

^675.

[13]

^TESTARD

O.,

^Thèse

d’Université, Orsay (1966), Rap-

port ^{CEA R 3073.}

[14]

^TOURNARIE M., ^J.

Physique

³⁰

(1969)

^737.

[15] ^MURAO T., ^J.

Phys.

^Soc. Japan ³¹

(1971)

^683.

[16]

^SATO Y., SIVERSTEN J. M. and TOTH L. E.,

Phys.

Lett. A 28

(1968)

^118.

[17]

^{DOMB C. and MIEDEMA A.} R.,

Progress

^{in Low} Temperature

Physics,

^{Vol. IV}

(North-Holland

Pub.

Company) (1964),

p. 296.