Elasticity and antiferromagnetism of metallic antiferromagnetics

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Elasticity and antiferromagnetism of metallic antiferromagnetics

R. Street, J.H. Smith

To cite this version:

R. Street, J.H. Smith. Elasticity and antiferromagnetism of metallic antiferromagnetics. J. Phys.

Radium, 1959, 20 (2-3), pp.82-87. �10.1051/jphysrad:01959002002-308200�. �jpa-00236072�

ELASTICITY AND ANTIFERROMAGNETISM OF METALLIC ANTIFERROMAGNETICS

By

^{R. STREET} and J. H.

SMITH,

Department ^of Physics, ^The University, Sheffield, England.

Résumé. 2014 On peut prévoir ^{une variation du module} d’Young ^{avec la} température ^au point

de Néel sur les substances antiferromagnétiques. Nous donnons ici les résultats des mesures rela- tives aux alliages CuMn(03B3) ^{et aux} alliages ^{CuMn à} plusieurs phases (03B1 + y) ^{On trouve sur ces}

derniers la variation prévue ^{due à Mn 03B1 et} également ^une anomalie, ^{vers 130} °K, attribuée à la présence ^{de MnCu y} précipité. Cette dernière phase ^est ferromagnétique au-dessous de 130 °K.

Une variation régulière ^{avec la} température ^{du module} d’Young ^{de Pd} confirme le fait qu’aux

basses températures ^{Pd ne serait} pas antiferromagnétique.

Abstract. 2014 If an antiferromagnetic ^is spontaneously ^{deformed on} cooling through ^{the Néel} temperature, ^{then the} application ^{of an external stress} results in a redistribution of domain vectors,

e.g. they may ^{rotate or} antiferromagnetic domain walls may ^move. This causes an additional strain component which will be apparent ^{as an} anomalous variation of the Young’s modulus with the temperature. ^The results of measurements of the temperature dependence ^of Young’s ^modulus

for antiferromagnetic 03B3-CuMn alloys ^{and mixed} phase (03B1 ⁺ 03B3) ^CuMn alloys ^are reported.

The (03B1 + y) alloys ^show (a) ^a Young’s modulus variation of the expected form which is due to the contained 03B1-Mn, (b) ^a Young’s ^modulus anomaly ^{at about 130 °K} associated with the preci- pitated 03B3-CuMn (containing ^{40 atomic} percent Mn). ^{It is} shown that the latter phase ^below

130 °K exhibits ferromagnetic characteristics.

A smooth temperature variation of Young’s modulus has been obtained for Pd which is consistent with the assumption ^{that Pd is not} antiferromagnetic ^{at low} temperatures.

PHYSIQUE 20, FÉVRIER 1959,

Introduction. - The

dependence

^of

Young’s

modulus ^on the ^state of

magnetization

^{of ferro-}

magnetic

materials has been known for many years, the

phenomenon

^{is known as the AE-effect.}

For a

ferromagnetic

^{of non-zero}

magnetostriction

an external stress Z in

general

^affects the distri- bution of the

magnetization

^vectors,

by rotating

them _{away from}

preferred

^{axes or}

by

^domain

boundary

^{movement, and the resultant rate of}

change

^of

intensity

^of

magnetization

^{with stress}

(è)l/è)Z)H

^is

equal

^{to the rate of}

change

^{of magneto-}

strictive deformation with field

(è)À/H)z.

^When

an external stress is

applied,

^{in addition to the}

normal elastic strain of the

lattice,

e, there is another

component

^of

strain, cm, produced by the magnetostrictive

deformation

accompanying

^the

redistribution of the

magnetization

^{vectors. At}

magnetic saturation,

^the

magnetization

^{vectors are}

all

aligned along

the field

direction,

^hence

è)À/è)H

⁼

)7/t)Z

^{= 0 and E is the}

only

component

of strain

produced by

^{the stress Z. Thus in the}

unsaturated state the value of

Young’s

^modulus

ZI(F-

+ Sjn) is less than its value at

magnetic

^satu-

ration.

It follows that in zero

applied

^{field the}

Young’s

modulus of a

ferromagnetic

should decrease when cooled

through

^{the Curie}

temperature.

^{Above the}

Curie

temperature

^no spontaneous

ferromagnetic

order exists and _e;m ⁼

0 ;

below the Curie tempe- rature e;m =1= ^0.

X-ray techniques

^{have been}

employed

^{to show}

that

antiferromagnetic

^{materials are}

spontaneously

deformed when

they

^{are heated or cooled}

through

their Néel

temperatures (Tombs

^and

Rooksby,

1950)

^{and thus exhibit}

antiferromagnetostriction.

Neutron diffraction studies show that the direc- tions of

antiferromagnetism generally

^{coincide with}

crystallographic

directions of low order. It is therefore

possible

^for

boundary

^walls

separating

domains

having

^different directions of antiferro-

magnetism

^{to move and the} directions ^of antiferro-

magnetism

^{to rotate} away from

preferred

^direc-

tions under the influence of an external stress.

Measurements of the temperature variation of

Young’s

môdulus of

pressed

^{bars of cobalt and}

nickel oxides show a very

pronounced

decrease in modulus near the Néel

temperatures

^{of the two}

materials

(Street

^and

Lewis, 1951 ;

g

Fine, 1953).

The

object

of this communication is to report ^some observations made

recently

^{on the} temperature

variations of

Young’s

^modulus of metallic antifer-

romagnetic

^materials.

Expérimental.

^{- The method of} measurement

was similar to that described

by

^Zacharias

(1933).

The

specimens,

^of

rectangular

^{cross section}

2.00 mm X 3.00 _mm, were cemented to quartz

crystals,

of identical cross

section,

^{which were cut}

so that

longitudinal

vibrations

along

^the

length

^of

the

composite

oscillator were excited

by

^{an alter-}

nating p.d. applied

^to electrodes

deposited

^on

opposite

^{side faces of the}

crystal.

^{The resonant}

frequencies

of oscillation of the

composite

^oscil- lator,

f o,

^are

given by

the relation

where

fa, fq

^are

respectively

^the fundamental fre-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphysrad:01959002002-308200

83

quencies

^{of resonant}

longitudinal

oscillation of the

spécimen

^{and the} quartz

crystal

alone ; ms, mq ^are the masses of the

specimen

^{and the}

crystal. Experi- mentally f o

^and

f q

^{are measured as functions of} temperature from which ’

/8

may be calculated.

Hence

Young’s

modulus is determined

by

^{the rela-}

tion ^’

ts

^and

f,

^were

always arranged

^{to be} very

nearly equal by adjusting

^the

length

^{of the}

speci-

men and

by using

quartz

crystals having

^various

values of

f q.

In this way, the

perturbing

^{effect of}

the adhesive at the

junction

^{of the}

crystal

^and

specimen

^{was minimized:}

The

composite

oscillator was mounted in ^an

-enclosure

which could be evacuated or filled with low

pressure

^{helium as a heat}

exchanger

^{and means}

of

heating

^or

cooling

^were

provided.

^{It was} pos- sible to determine relative values of

Young’s

modulus to within 1 part ^{in 104.}

Results and discussion

Cu-Mn

Alloys.

^{- The}

antiferromagnetic

pro-

perties

^of

single phase

y CuMn

alloys containing

more than about 70 atomic percent of manganese have been

investigated by

^{Bacon et al.}

(1957).

Below a transition temperature,

Tt,

^which

depends

on

composition,

^the

alloys

^are

antiferromagnetic

and have a face-centred

tetragonal

structure. At the transition temperature ^the

alloys undergo

^a

Fie. 1. - Temperature ^variation qf ^Young’s modulus for y-CuMn alloys containing ^{80 and 85 atomic} percentage

Mn. ^,

martensitic

transformation,

^above

Tt they

^{are no}

longer antiferromagnetic

^{and have a} face-centred cubic structure. The

extrapolated

^Néel tempe- rature,

TN,

determined from the temperature

dependence

^{of the}

intensity

^{of the}

(110) (magnetic)

neutron diffraction

peak

^is

^always

^{greater than}

Tt.

The values of

Young’s

^{modulus as functions of}

temperature

^{for two}

typical alloys

of this series are

shown in

Figure

^{1. As}

expected,

^{these curves}

show

pronounced

minima which occur at the tran- sition temperature ^{of the}

alloys.

The

interpretation

^of the results obtained is uncertain in view of the fact that Tt

TN.

However, investigations

^{of the stress}

dependence

^of

the

intensity

^{of the}

(110) (magnetic)

^{neutron dif-}

fraction

peak (Bacon

^et

al.,

^loc.

cit.)

suggests ^that the additional ^strain component Em ^{in the antiferro-}

magnetic

^{state arises from}

boundary

^{wall move-}

ment rather than from rotation of domain magne- tization vectors.

Only boundary

^walls

separating

domains

having digerent

directions ^of antiferro-

magnetism

^{will move on the}

application

^{of an}

external stress. Other domain types may exist e.g. if

along

^a line within a domain the

spin

^direc-

tions are

t 1 t 1

t

t,

^a

boundary

wall may ^occur between "

changed-step

^" domains such that on

moving through

^{the wall the}

spin

directions are

formally represented by t t t t t t t t t.

^The

two domains

separated by

^a

boundary

^{of this} type

have the same

antiferromagnetostrictive

^strain

along

any

direction,

^the

application

^{of stress will}

not result

directly

^{in wall movement and the}

contribution _{to z.} will be zero. From the expres- sions for the

Young’s

^modulus

change

^summarized

in the

Appendix

^{it will be seen that when domain}

boundary

^{movements are involved}

comparison

^of

experimental

results with theoretical

predictions

^is

difficult as

knowledge

^{of the area of domain walls}

and the forces

impeding

^{wall movement are}

requir-

ed.

(oc

⁺

y)

^CuMn Alloys. ^- It is of interest to inves-

tigate

^the temperature

dependence

^of

Young’s

modulus of iI..-Mn as a useful

preliminary

^{to the}

consideration of the

antiferromagnetic

^domain

structure of the element. The

investigation

^cannot

be undertaken

directly using

pure oc-Mn

samples

since the material is

extremely

^{brittle and cannot}

be machined into the

required

regular

shape.

^The

difficulty

^{has been overcome in the}

following

way.

Starting

^materials

containing

^various

proportions

of copper ^were

prepared by melting

^{in an} argon ^arc furnace and then after

heating

for extended

periods

of time at temperatures ^{near the}

melting point they

^were

rapidly quenched,

^thus

retaining

^the

y solid solution. Materials

prepared

ⁱⁿ this way

containing

^{as much as 95 atomic} percent of manga-

nese were

relatively

easy ^to machine and it was

possible

^to produce

specimens

^{of the}

required

^form

from them. The

specimens

^{were then trans-}

formed to the mixed

(oc

+

y) phase by heating

^for

many hours ^at 600 °C. Thus the

specimens

^con-

taining high proportions

of manganese consisted of

a matrix of oc-Mn with a

precipitated y-phase containing approximately

^{40 atomic} percent ^man-

ganese. From the

phase diagram

^{of this} system

determined

by

^{Dean et al.}

(1945)

^a transformed

specimen having

^{a total of} 95 atomic

percent

manganese contains ^over 90

%

oc-Mn. The elec- trical

resistivity

^vs.

temperature

^{curves of the}

FIG. 2. ^- Temperature variation of resistivity ^of (oc + y)

CuMn alloys containing 80, ^{85 and 92.5 at.} % ^Mn.

Ordinates- pl Po ^where P ⁼ resistivity ^at temperature T,

po ⁼ resistivity at ⁰ OC,

FIG. 3 (a) ^- Temperature variation of Young’s ^{modulus of} (a + y) ^CuMn alloys containing ^{95 at.} % ^Mn (upper curve) ^{and 90 at} % ^Mn (lower curve). ^Ordinates EJE,

where F ⁼ Young’s ^{modulus at} temperature T, .Eo ⁼ Youngs ^{modulus at 0 °C.}

specimens (fig. 2)

^{have the form} characteristic of ce-Mn

(White

^and

Woods, 1957).

In

figures 3(a)

^and

(b),

values of

Young’s

modulus

expressed

^as ratios of the 0 OC value are

plotted

^{as a} function of temperature ^for (oc +

y)

^CuMn

alloys containing

various propor- tions of manganese. All the curves taken with

specimens containing

^more than 70 atomic per- cent Mn show a

stepwise

variation in

Young’s

modulus at about 104 OK which is rather

higher

than the Néel

temperature

^{of oc-IVIn -} i.e. 100 OK deduced from neutron diffraction measurements

by

Shull and Wilkinson

(1953)

and 95 oR. deduced from

spécifie

^heat measurements

by

^{Tauer and}

Weiss

(1957). The decrease

ⁱⁿ

Young’s

^modulus

on

cooling through the

^Néel

temperature implies

that ce-Mn must

undergo antiferromagnetostrictive

"

deformations when

antiferromagnetic ordering

^sets

in. It is difficult to estimate the

magnitude

^{of the}

antiferromagnetostriction

^{since there is no infor-}

mation on the

crystalline anisotropy

energy ^or the number of domain

boundary

^{walls and the}

impe-

dance to their motion.

FIG. 3(b). ^- Variation of Youngs modulus of (a + y)

CuMn alloys containing 50, 69, ^{80 and 90 at.} % ^Mn.

E ⁼⁼ Youngs modulus at temperature T, Eo ⁼ Young’s

modulus at 0 °C. The origin of the ordinate axis is

arbitrary.

An obvious feature of the results shown

parti-

cularly

ⁱⁿ

figure 3(b) is

^the occurrence of maxima at about 130

OK,

^{due to}

magnetic ordering

^{in the}

85

y-CuMn

^{contained in the}

specimens.

^Measu-

rements have been made of the

magnetic

^suscep-

tibility

^{of the mixed}

phase (oc

+

y)

^{materials pre-}

pared

^as described above and also on

specimens

^of

single phase y-CuMn alloys containing

^{40 and}

50 atomic percent ^Mn.

Typical

^{results of measu-}

rements of the temperature

dependence

^{of the}

force

acting

^{on the}

specimens placed

^{in a non-}

uniform magnetic ^{field are shown in}

figure

^4.

When the

specimens

^{are cooled}

through

^{130 OK in}

zero field the

susceptibility

^{reaches a broad maxi-}

mum at 130 °K which suggests

antiferromagnetic ordering. However, preliminary investigations

^of

the neutron diffraction patterns ^{of the 50 atomic} percent

y-CuMn alloy

^{at 80 OK do not indicate}

long-range antiferromagnetic ordering

^{of the} type observed with Mn-rich

y-CuMn alloys.

^{In addi-}

tion the

specimens

^{exhibit an unusual ferro-}

magnetic

^behaviour which appears

by allowing

them to cool

through

^{130 OK in an}

applied

^field.

This

ferromagnetic

^{behaviour is also exhibited}

by

copper rich

y-MnCu alloys (Owen

^et

al., 1957).

The results shown in

figure 4 were

^obtained

by

FiG. 4. ^- Temperature variation of force acting ^on spe- cimen of 40 at. percent ^Mn y-CuMn alloys. ^{The three}

curves were obtained with

measuring

^{fields of} 7,10 ^and

11 kOe. (For explanation of sections ^{a and} p ^see text.)

allowing

^the

specimens

^{to cool to 80 OK in an}

applied

^{field of 10 kOe.} Measurements were then made ^{of the} forces

acting

^on the specimens ^{as their}

temperature slowly

increased : ^- the results

plotted

^{on the curves marked} p ^were obtained with the various

measuring

^fields

acting parallel

^to

the field

applied during cooling ;

^{the curves a show}

the results obtained with the

measuring

^fields

acting antiparallel

^to the field

applied during cooling.

^These results shows that

cooling

^{in a}

magnetic

field results in a " frozen-in "

magnetic

moment,

parallel

^{to the}

field,

which is magne-

tically hard,

^{reverse fields of 11 kOe are not suffi-}

cient to reverse its

direction ;

^fields up ^to 7 kOe have little effect on the moment as may be seen

from

figure

^{5. At the}

higher

^reverse fields the

FIG. 5. ^- Temperature variation of permanent ^moment

for 40 at. percent ^Mn y-CuMn alloy cooled in field of 10 kOe. The values of a are averages calculated ^from

the separations of the a and _p sections of curves

typified ⁱⁿ figure ^{4. Fields} applied ^{were 3} kOe, ^{X 5} kOe,

+ ⁷ kOe, 0 ¹⁰ kOe, Ll11 ^kOe.

ferromagnetic

^{moment is time}

dependent

^{but this}

phenomenon

^{has not been}

investigated

^{in detail}

at the present ^time.

Palladium and chromium. ^- The temperature

variation of

Young’s

modulus for a

specimen

^of

palladium

^{shows no}

anomaly

^{at 80 OK at which} temperature ^{there is a broad}

peak

^{in the} suscep-

tibility

^vs. temperature ^curve

(Hoare

^and

Matthew, ^1952).

^The

elasticity

measurements thus support ^{the view that the}

susceptibility

^maximum

is due to electronic ^{band structure}

changes

^and

not due to the occurrence of

antiferromagnetism.

There ^{is some} doubt as to whether chromium is

antiferromagnetic (Shull

^and

Wollan, 1956).

^The

original investigations (Shull

^and

Wilkinson, 1953)

indicated ^a Néel

temperature 470’DK,

^{but there is}

no

anoinaly

^{in the}

Young’s

^{rnodulus at this}

tempe-

, rature which would be

expected

^{if antiferro-}

’

magnetism

^{were to occur.}

However,

the absence of an

anomaly

ⁱⁿ

Young’s

modulus is not

absolutely

conclusive as ^this could arise in an

antiferromagnetic having very high

^uniaxial

crystalline anisotropy.

This situation

probably

^occurs in rhombo- hedral

Cr203 (Street

^and

Lewis, 1956)

^{but it would}

seem to be

unlikely

^{in cubic materials.}

Appendix.

^{- Let À be the}

component,

^resolved

along

the direction of

applied measuring

^{stress, of}

the

magnetostrictive

^{strain of an}

antiferromagnetic

domain system. À may be

expressed

^{in terms of}

an

antiferromagnetostriction

coefficient Àc ^{and a}

function, T(M)

^of aparameter .NI which is characte- ristic of t he domain process

considered,

^e. g. if ^rota- tion of domain

magnetization

^vectors occurs, M is the

angle

of rotation away from ^a

preferred

^direc-

tion,

^{if domain}

boundary

^{movement occurs M is the}

positional

co-ordinate of the

boundary.

^{Thus we}

write À = Àc

T(M).

For an

applied

^{stress Z the} component ^{of strain}

energy

density

^{due to domain}

changes

^is

1

where the

prime

^denotes differentiation with res-

pect

^{to M.}

Hence f’(M) = - ^z Àc ’Y/(M).

The energy

density

^{of the domain} system ^will

have other contributions

arising

^from

crystalline

and internal strain

anisotropy, boundary

^wall energy etc. These contributions do not involve M

explicitly

^{and in sum are}

represented by fA(M).

Rotations :

Against high internal strain anisotropy.

Against ^uniaxial crystalline anisotropy.

Away ^from preferred directions of antiferroma-

gnetism ^{which are} [100] directions (cubic ^mate-

rials.

Movement of boundaries :

Separating change-step ^domains

Separating domains, ^{in cubic} materials, with ortho-

gonal directions of a. f. m.

This

analysis

is treated in greater detail and

applied particularly

^{to the AE-eflect and}

magnetic susceptibility

^of

ferromagnetics by

Street and Lewis

(1958).

’

Considering

reversible

changes only,

^the

equi-

librium value of lVl for a

given

^{stress will be}

given by f’(M) + f’A(M)

^{= 0 or}

This is an

implicit

^{relation between M and Z}

from which

dM/dZ

may be derived :

hence

at zero

applied

^stress.

Thus

(dÀ/dZ)z=0 ⁼ (dM/dZ) (dX/dM) =

lÀc ’Y’(O)]2/f(O)

d À/dZ

^{has the} dimensions of

compliance (inverse elasticity)

and is the contribution to the total com-

pliance produced by

^{the strain} component ^{em men-}

tioned in the text.

Below the Néel temperature the measured com-

pliance 1 jEb

⁼ ( s +

Em) /Z

^and the value deter- mined

by extrapolation

^from measurements above the Néel temperature

1 jEa

⁼

C JZ.

^Hence

Thus values

of A Ê appropriate.to

1 ^{any domain} process may be evaluated

by appropriate

^substi-

tution in this

general equation (Street

^and

Lewis, 1958).

^{Values of}

A ( 1 JE)for

various’domain _proces-

ses in

randomly

^oriented

polycrystalline

^materials

assuming uniform

^stress distribution are summar-

ized below :

Zi = internal stress.

X ⁼ (3 /2) Xc[cos2 ^0- (1/3)].

Where 0 ⁼ angle between direction of antiferro-

magnetism ^and measuring ^direction.

Anisotropy ^energy density ^is /(7]) ^{z sin2} q + ^...

Àll1 = conventional single crystal magnetostrictive

coefficient.

K, ^{= first} crystalline anisotropy energy coefficient.

S = area of boundaries per unit volume.

f’:t (0) ^{= second} differential w. r. t. position ^of

energy of domain boundary walls per unit volume.

REFERENCES

BACON (G. E.), ^DUNMUR (I. W.), ^SMITH (J. H.) ^{and STREE} (R.), ^Proc. Roy. Soc., 1957, ^A 241, ^223.

DEAN (R. S.), ^LONG (J. R.), ^GRAHAM (T. R.), POTTER

87

(E. V.) ^{and HAYES} (E. T.), ^Trans. Amer. Soc. Metals, 1945, 34, 443.

FINE (M. E.), ^{Rev. Mod.} Physics, 1953, 25, 158.

HOARE (F. E.) and MATTHEWS (J. C.), ^Proc. Roy. Soc., 1952, ^A 212,137.

OWEN (J.), ^BROWNE (M. E.), ^ARP (V.) ^{and KIP} (A. F.),

J. Phys. ^Chem. Solids, 1957, 2, ^85.

SHULL (C. G.) ^{and WOLLAN} (E. O.), Solid State Physics,

, 1956, ^{vol. 2} (New York : Academic Press), p. 181.

SHULL (C. G.) and WILKINSON (M. K.), ^{Rev. Mod.} Physics, 1953, 25, 100.

STREET (R.) ^{and LEWIS} (B.), Nature, London, 1951, 168, 1036 ; ^Phil. Mag., 1956, 1, 663 ; Proc. Phys. Soc., 1958. 72, 604.

TAUER (K. J.) ^{and WEISS} (R. J.), ^J. Phys. ^Chem. Solids, 1957, 2, 237.

TOMBS (N. C.) and ROOKSBY (H. P.), Nature, London, 1950, 165, 442.

WHITE (G. K.) ^{and WOODS} (S. B.), ^Canad. J. Phys., 1957, 35, 346.

ZACHARIAS (J.), Phys. Rev., ^1933, 44, ^116.

DISCUSSION

Mr.

Meiklejohn.--

^{I should like to} suggest ^that the reasôn for the

difficulty

ⁱⁿ

reversing

the magne- tization is due to an interaction between antiferro-

magnetic

^and

ferromagnetic

^or

superparamagnetic

regions.

^{If true the}

hystérésis

^{curve will not be}

symmetrical

^about the M axis as has been found in the Cobalt-cobaltous oxide system.

Mr. Street. ^- 1 would agree that it is

possible

that the

magnetic

^{hardness in Cu-Mn}

alloys

^may

be due ^to

antiferromagnetic-ferromagnetic

^inter-

action. However this case is

obviously

^{not as}

clear cut as that of the Co-CoO system ^{referred to}

by

^Dr.

Meiklejohn.

^At

high

^Mn content, ^Cu-Mn

alloys

have well

developed long-range

^antiferro-

magnetic

structures but this type of structure is not observed with

alloys containing

^{less than about}

70 atomic percent Mn. Neutron diffraction studies of Cu-Mn

alloys containing

less than 70 atomic per cent manganese indicate

short-range magnetic ordering.

^{If there is}

antiferromagnetic coupling

in the

alloys exhibiting ferromagnetic

^behaviour

the

ordering

^of

spins

^is

certainly

^{différent from}

that observed at

higher

manganese ^contents. As

an alternative to the interaction mechanism _{it may} be

suggested that

^there

is short-range ferromagnetic coupling

^of

spins,

characterized

by high

magneto-

crystalline anisotropy.

1 agree that information

on the

hysteresis

^{curves of the}

alloys

^{would be}

very useful in

helping

^to decide the

question.