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Submitted on 1 Jan 1986

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The paramagnetic response of nickel at high temperature

P.J. Brown, H. Capellman, J. Déportes, D. Givord, S.M. Johnson, K.R.A.

Ziebeck

To cite this version:

P.J. Brown, H. Capellman, J. Déportes, D. Givord, S.M. Johnson, et al.. The paramag- netic response of nickel at high temperature. Journal de Physique, 1986, 47 (3), pp.491-496.

�10.1051/jphys:01986004703049100�. �jpa-00210229�

(2)

The paramagnetic response of nickel

at

high temperature

P. J. Brown

(+),

H.

Capellman (°),

J.

Déportes (*),

D. Givord

(*),

S. M. Johnson

(+)

and K. R. A. Ziebeck

(+)

(+) Institut Laue-Langevin, 156X, 38042 Grenoble Cedex, France

(°) lnstitut fur Theoretische Physik, Technische Hochschule, D-5100 Aachen, F.R.G.

(*) Laboratoire Louis Néel, C.N.R.S., 166X, 38042 Grenoble Cedex, France (Reçu le 8 aout 1985, révisé le 5 novembre, accepté le 6 novembre 1985)

Résumé. 2014 La réponse paramagnétique du nickel a été mesurée entre 1,25 et 2 Tc par diffusion de neutrons polarisés

avec analyse de polarisation, jusqu’à des transferts en énergie de 180 meV (~3 Tc). Les résultats sont en accord

avec ceux obtenus par Steinsvoll et al. dans un domaine plus restreint de températures et moments. Ni les calculs

CPA ni l’analyse en terme de Lorentzienne, proposée par Steinsvoll et al., ne sont capables de rendre compte des résultats obtenus dans aucun des domaines en moments. Ces deux analyses sont adaptées à des systèmes de moments locaux, et ne considèrent donc pas la physique microscopique appropriée à des systèmes d’électrons itinérants.

Abstract. 2014 The paramagnetic response of nickel has been measured using polarized neutron scattering with polarization analysis from 1.25 to 2 Tc and for energy transfers up to 180 meV (~3 Tc). In the restricted tempe-

rature and momentum range covered by Steinsvoll et al., the present results were found to be in satisfactory agre-

ement. However, CPA calculations are unable to account for wavevector and frequency dependence, as well as

absolute magnitude of the observed scattering.

Classification

Physics Abstracts

75.20

1. Introduction.

Recent

paramagnetic

neutron

scattering

measure-

ments have shown that the

paramagnetic phase

of

iron is characterized

by

strong

spatial

correlations and a

suppression

of the thermal fluctuations with short

wavelengths [1,

2]. The enhanced

long

wave-

length

fluctuations have low

energies

and contribute

significantly

to the free energy,

characterizing

the

thermal

properties

of iron above

Tc.

The response

integrated

over all thermal

energies (2 KTc) yields

an

amplitude

per iron atom of

1.55 PB

much smaller than the ground state value of 2.2 MB. Thus the

magni-

tude, wavevector and energy

dependence

of the

scattering

is

incompatible

with the concept of ther-

mally

disordered atomic moments

(close

to the

ground

state

value)

and with the

predictions

of CPA calcu- lations [3-7].

A local moment model based on a

single

site CPA

has been

proposed

for nickel

[5]

and

S(Q, cv)

is pre- dicted to be similar to that of iron except that the

amplitude

of the moment may increase with tempe-

rature. A disordered atomic moment

description

for

the

paramagnetic phase

of nickel is

surprising

since

the

ground

state moment 0.61 pB involves less than

one electron

(or

hole). In order to test the

validity

of such

hypothesis preliminary

neutron

scattering

measurements on nickel at 1.1

T, [8]

and 2

T, [9]

revealed an enhancement of the

scattering

for small

momentum transfers similar to that observed in iron.

The present paper is concerned with detailed measu- rements

throughout

the Brillouin zone and

extending

up to 180 meV i.e.

3 kTc.

2.

Properties

of nickel at finite temperatures.

Although

band

theory gives qualitative

agreement with many of the

physical properties

observed in nickel substantial

discrepancies

exist. These diffe-

rences become more fundamental at finite tempera-

tures where band

theory

fails to account for the observed thermal variations of the

magnetization

and

susceptibility,

and for the

magnitude

of the Curie temperature. Thus fluctuations of the

magnetization density,

not contained in band

theory,

with thermal

energies

k T must dominate in

characterizing

these

properties.

It has been shown that transverse fluc- tuations

(angular

rotations) of the

magnetization density

can

destroy long

range order in itinerant

ferromagnets [10-14, 15]

at temperatures consistent with the observed Curie temperature. The mechanism has similarities to localized systems described

by

a

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004703049100

(3)

492

Heisenberg

model but with detailed differences due to the itinerant nature of the

magnetic

carriers.

Since Tc

(Stoner) [15]

is

substantially larger

than Tc

(observed) [16],

it is

generally

assumed that the

exchange splitting

does not

collapse

at Tc

(obs).

Indirect evidence in support of an

exchange split

band above

T,

in nickel is

provided by

the small

magneto-volume anomaly

at

Tr [17].

The

magnitude

of the volume

change

at

Tc

is similar to that observed in iron but it is of a different

sign,

i.e. nickel

expands

on

becoming paramagnetic [18].

A.R.P.E.S. measure- ments at 1.1

Tr [19]

have been

analysed, assuming

a temperature

independent

line

shape,

to indicate an

exchange splitting

reduced

by

50

%, although

a more

detailed

analysis [20] suggested

no

change

in the

exchange splitting

with temperature

[21].

However,

polarized

neutron measurements at 600 K suggest that the symmetry of the

magnetization density

is temperature

dependent [22].

On the basis of these data it was

proposed

that the thermal variation of the

exchange splitting

was different for the two

symmetries. Although

the static

susceptibility

deviates

from Curie-Weiss behaviour the thermal variation is

significant.

If a temperature

independent

contri- bution,

possibly

of orbital origin, is subtracted from the observed

susceptibility

the

remaining

Curie-

Weiss component has an effective moment of 1.61 JlB

yielding

a

paramagnetic

moment pp = 0.9 YB

(y2 eff pp(pp

+

2)) larger

than the

ground

state value

[23].

Several models have been

proposed

to account for the thermal

magnetic properties

of itinerant systems and all claim reasonable agreement with

experiment.

In models based on CPA calculations the parama-

gnetic phase

is described

by

disordered local moments whose

amplitude

can increase with temperature

[5, 24].

Moriya [14,

25] has shown that in the case of weak

ferromagnets

i.e. small moments and low Curie tem-

peratures, the

longitudinal

fluctuations lead to a

Curie-Weiss

susceptibility

and

large

effective moments.

Friedel et al.

[26]

have shown that if a moment is created in the

paramagnetic

state the excitation cannot be localized within a distance smaller than -

Kf- 1,

which is

large

in the case of nickel.

Cyrot

[27,

28]

has

pointed

out that for the CPA to be a valid des-

cription

of a

magnetically

disordered state

Kf- 1 must

be smaller than the interatomic

spacing.

This condi-

tion is not satisfied for nickel and it is

therefore

not

possible

to

speak

of individual

spins

on each nickel atom. In other models the itinerant nature of the

magnetic

carriers

give

rise to

spatial

correlations which

can stabilize the

exchange splitting

above

Tc [10,

11,

13].

The

paramagnetic phase

is therefore characterized

by

short range

magnetic

order

indicating

the minimum distance over which the

magnetization

can be reversed.

These models are

capable

of accounting for the

details of

magnetic scattering

function

S(Q,

E) [10]

as observed in constant E scans

[29], reproducing

the

observed

peak positions,

their shifts in temperature,

shape

and widths. Whether these

peaks

in constant E

scans are the signature of «

propagating spin

waves »

above

T,,,

as inferred

by

Lynn and Mook

[29]

for the

momentum range - 0.2 to 0.6

Å - I,

is controversial

[30-32].

Initial

polarized

neutron measurements and

pola-

rization

analysis [8]

enabled the wavevector

depen-

dence of the

paramagnetic scattering

from a natural

nickel

sample

at 1.1

T,

to be determined. These mea- surements confirmed that the

long wavelength

fluc-

tuations are indeed enhanced with a characteristic

wavelength

of 21

A.

The measurements

presented

here

have been carried out on a 6°Ni crystal in which the strong disorder

(isotope) scattering

has been suppress- ed thus

enabling

data of

good

statistical

significance

to be collected. Both constants E and

Q

measurements have been

performed throughout

the zone and up to 180 meV

(3 kT c).

On the basis of these data it is pos- sible to make a critical

comparison

with current

theories.

3.

ExperimentaL

The

sample

used was the same

single crystal

of 6°Ni

for which

preliminary

measurements have

already

been

reported [9].

The

crystal

in the form of a

cylinder

50 mm

long

and 7 mm diameter has a (

110 >

axis

parallel

to the

cylinder

axis. A boron nitride support cemented to one end of the

cylinder

was used to locate

the

crystal

at the centre of a coaxial

molybdenum

resistor of diameter 38 mm inside a furnace. Chromel- alumel

thermocouples

attached to each end of the

crystal

indicated a

stability

of better than ± 0.50 and

an absolute difference of 1° at all temperatures up to 1 273 K

(2 T c).

The calibration of the

thermocouples

was verified

by measuring

the

abrupt change

in

pola-

rization of the neutron beam diffracted from

(111)

on

warming through

the Curie temperature 631 K.

The

paramagnetic scattering

from nickel at 1.25,

1.5 and 2

Tc

was

investigated

both as a function of momentum and energy transfer

using

the

polarized

neutron

triple

axis spectrometer D5 located on the hot source of the H.F.R. at the I.L.L. Since monochro- matic

polarized

neutrons with

energies

of up to 1 eV

can be incident on the

sample

the spectrometer is

ideally

suited for

investigating

the

paramagnetic

res-

ponse of itinerant magnets. Constant

Q

scans carried

out at

large

wavevectors i.e. 0.5 qm

Q

qm, where qm represents the zone

boundary

wavevector in the 111 or 001 directions, extended up to 180 meV i.e.

~ 3

kTc.

For these measurements a fixed final neu- tron energy of 115 meV was used

together

with an

Er filter to minimize

higher

order contamination. The

integrated scattering

from the constant Q scans was

compared

with that

directly

obtained

using

fixed ener-

gy windows

corresponding

to

energies

of

kTc,

2

kTr

and 3

kTc.

The

intrinsically

poor resolution associated with short

wavelength

neutrons was

compounded by

the use of

monochromating

and

analysing crystals

with

large

lattice

spacings.

The

(111)

reflection of

Cu2MnAl (d111

= 3-45

A)

was used as both

polarizing

(4)

monochromator and

analyser.

Guide fields maintained the neutron

polarization

between monochromator and

sample

and between

sample

and

analyser.

Helm-

holz coils around the furnace enabled the neutron

polarization

at the

sample

to be either

parallel

or

perpendicular

to the

scattering

vector. The vertical

and horizontal field

required

to rotate the neutron

polarization

was some 100 Oe and had

negligible

effect on the

sample.

An r.f coil in the

incoming

beam

enabled the incident neutron

polarization

to be

reversed.

By reversing

the incident

polarization, using

the

flipper,

and

measuring

the scattered neutron count for incident neutrons

polarized perpendicular

C or

parallel C to

the

scattering

vector, a clean determina-

tion of the

paramagnetic scattering Cil

- C is

possible.

All other forms of

scattering

are not sensitive to the

relative orientation of the neutron

polarization

and

scattering

vector and therefore subtract out Correc- tions for both

incomplete polarization

of the beam

and

flipping efficiency

were made

following

the pro- cedure outlined

by

Ziebeck et al.

[33].

For neutrons

polarized parallel

and

perpendicular

to the

scattering

vector the

magnetic

response function

S(Qcv)

is related

to the observed

intensity by

where V is the volume of the

sample,

Vr the volume

of the unit cell,

NM

the number of

magnetic

atoms

per unit cell and

ki kf

the incident and final wavevec- tors. The

scattering

was

placed

on to absolute scale

using

two methods : the nuclear

spin

incoherent

scattering

from a vanadium

cylinder

of similar dimen- sions as the 60Ni

crystal

and

by comparison

with the

enhanced

scattering

observed in

polycrystalline

nickel

rod under similar

experimental

conditions and for which the normalization

procedure

is

straight

forward [34]. The agreement obtained using the two

techniques

was better than 5

%.

4. Resets.

The wavevector

dependence

of the enhanced

scattering

which occurs for wavevectors less than - 0.25 qM was

investigated

in the

[ 111

direction in the zones centred

on

(000)

and

(111).

After

correcting

for the

magnetic

form factor

[35]

the results obtained in the two zones were found to have the same wave vector

dependence.

These results, which indicate the

spatial

extent of the

ferromagnetic

correlations, are in agreement with those

previously

reported and with the results of Steinsvoll et al.

[30]

whose measurements were con-

fined to temperatures less than 1.24

T,.

At the three temperatures at which measurements were made i.e.

1.25, 1.5 and 2

Tc

the results

extrapolated smoothly

to the observed value of the static

susceptibility

at

Q = 0.

Thus the enhancemeot of the

scattering

decrea-

ses with temperature but remains substantial even at 2

Tc.

From the width

Qw

of the forward

peak

the

wavelength characterizing

the

spatial

correlations

A,

= 2 x/Qw was found to decrease from - 21 A at 1.25 Tc to - 14 A at 2

Tc.

The momentum range over which enhanced

scattering

is observed

corresponds

to the zone in which the acoustic

spin

wave propagates in the

ground

state before

entering

the Stoner conti-

num. The characteristic

wavelength À-c quoted

above

in our

opinion

is a

signature

of the

interancy

of ma-

gnetic

electrons and should not be confused with the

«

thermodynamic »

correlation «

length

». For a de-

tailed discussion the reader is referred to

[38].

The

scattering

at

large

momentum transfers i.e.

0.25 qm

Q

qm was

investigated

in the zones

centred on

(111) (220)

and

(002). Although

the magne- tic form factor

substantially

reduces the

scattering

at

large

momentum transfers it is

possible

to extend

measurements to

high

energy transfers and maintain the

polarization along

the

scattering

vector when

determining Cjj.

The energy

dependence

of the

scattering

close to the zone

boundary

is of

particular

interest since at these momenta the « on » site

pair

correlation function

( Si. Si >

is

expected

to dominate.

Whereas the

long wavelength

fluctuations characterize the

macroscopic

features of the response, the microsco-

pic

details are reflected in the energy

dependence

of (

S;. S; ).

In a local moment system as described

by

a

Heisenberg

model, the correlation function is cha- racterized

by

a thermal energy scale. For an itinerant magnet the response at the zone

boundary

is expected

to extend up to the band width i.e. some eV’s.

Although

the

Heisenberg

model is often

applied

to iron,

unjus- tifiably,

its extention to nickel is even less well founded.

At temperatures

corresponding

to 1.5 and 2

Tc

pre-

liminary

constant q scans close to the zone

boundary

in the

[111]

direction and

extending

up to 60 meV i.e.

~

kTr

revealed

significant scattering increasing slightly

with energy transfer.

Subsequent

constant q scans confirmed that measurable

scattering

existed up to at least 180 meV i.e. - 3

kTc.

The

integrated scattering

from these scans, 0.74

,uB,

was found to be in excellent agreement with that observed

using

a fixed window of

3

kTc.

These results are in contrast with those obtained

on

paramagnetic

iron in which the

scattering

close

to the zone

boundary

had fallen to a

negligible

level

by

70 meV i.e. less than energy

kT,, (iron).

At

large

wavevectors in nickel more

scattering

occurs at an

energy transfer greater than

kTc

than below. Measu-

rements at 1.25,1.5 and 2

T c

suggest that the

integrated scattering

at

large

wavevectors does not

change substantially

with temperature. This would suggest that the fluctuations at large wavevectors are

predo- minantly

quantum fluctuations,

possibly

due to por- tions of the bandstructure

being

rather flat in the

regions of (q,

w) space

probed

in the

experiment.

The observed

scattering

may be

integrated through-

out the zone to obtain an « effective »

amplitude [36] :

(5)

494

It may be seen that the

phase

space factor

Q 2 gives

a substantial

weight

to the

large

Q components which

are

predominantly

quantum fluctuations.

Consequen- gly

the thermal fluctuations which characterize the small wavevector response

give

rise to a

negligible

contribution to the

amplitude.

An

important

consi-

deration in the definition

of p

is the energy range over which

S(Qco)

has been

integrated

to obtain

S(Q).

Experimentally

the maximum energy transfers em-

ployed

were 180 meV

(3 kTc) a

value sufficient to

collect the

majority

of thermal fluctuations. The

amplitude

was determined to be 0.87 PB per nickel atom. It is also

possible

to estimate the

amplitudes corresponding

to an

integration

up to

energies

of

kTc

and 2

kTc;

these values are 0.38 PB and 0.63 PB per nickel atom

respectively.

From these values it may be seen that more

weight

lies above an energy cor-

responding

to

kTc

than below, in contrast to a local

moment system, and iron. A

plot

of

Q2 S(Q),

for the

thermal fluctuation, as a function of wavevector

yields

distinct

peaks

at wavevectors - 0.35, 0.4 and 0.45 A-1

for 1.25, 1.5 and 2

Tr respectively.

Thus fluctuations with Fourier components

corresponding

to these

wavevectors contribute

significantly

to the on site

correlation function. This behaviour is similar to that observed in iron, except for iron

[1]

the characteristic wavevector was

essentially

temperature

independent

up to 1.5

T,

and

corresponded

to - 0.4 A-1. The

peaks

observed in the

Q2 S(Q)

vs. Q

plots

for nickel

Fig. 1. - The wavevector dependence of the paramagnetic scattering from nickel at 1.25, 1.5 and 2 T,. The arrows

indicate the experimental values of the bulk susceptibility.

give

rise to characteristic

wavelengths

for the ferro-

magnetic

correlations of between 21 - 14 A

(i.e.

for 2

and 1.25

Tc)

in agreement with that obtained from

the width of the forward

peak.

The measurements of Steinsvoll et al.

[30]

were

confined to wavevectors less than 0.4 Å - 1 and to

temperatures below 1.24

T,.

At 1.16

T,

Steinsvoll

et al. obtained reasonable agreement with observation

using

a

scattering

function in which both the wave- vector and energy

dependence

had

simple

Lorentzian

forms. Thus the wavevector

dependence

of the scat-

tering

was

given by

where

M2(o)

is the static

susceptibility

in units of

MB

and K is the inverse correlation

length.

Values for K were scaled from those observed in the critical

region [37] using

Such that at 1.16

T,,

x = 0.174 A-1. Since the for- malism is based on

theory applicable

to critical

scattering

it is

important

to establish over which range of temperature and momentum it remains

applicable.

A

comparison

of the results of Steinsvoll et al. at 1.24

Tc

with those of the present

investigation

at

1.25

Tr

reveals

satisfactory

agreement over the momen-

tum range covered

by

both measurements i.e.

0.4 Å - 1. However this momentum range is too small to

provide

a sensitive test of the

appropriate

functional form. Nevertheless in the momentum range covered the results at 1.24-1.25

T, already diverge substantially

from the calculated momentum

depen-

dence

(2),

if the scaled value x = 0.232

Å - 1,

deduced

from

equation (3),

is used. A more

satisfactory

agree- ment with observation in the

restricted q region

can be

obtained

using

the same Lorentzian form but with

K = 0.16

A-1.

Thus

substantially

more short range order is observed even at 1.25

Tc

than is

predicted by

the

simple

model based on critical

scattering

and

using

values for K scaled from observations close to

Tc.

Agreement

obtained at 1.16

Tc

and below is not

surprising

since at temperatures close to

Tc

the

forward

scattering

is dominated

by large amplitude

fluctuations associated with residual

long

range order which at

T,

causes the

susceptibility

to

diverge.

Above

1.25

Tc

i.e. at 1.5 and 2

T,

the functional form

(2), using

the scaled values of x

(3), completely

fails to

account for the observed

scattering.

Over the momen-

tum range 0-0.4 Å - 1 a

satisfactory

agreement can be obtained

using

K=0.18 and 0.17 at 1.5 and

2Tc.

Within the

experimental

error it may be concluded that between 1.25 and 2

Tc,

a

characteristic qc

exists

which is

essentially

temperature

independent

with a

(6)

value K = 0.17 + 0.01 A-’. This result reflects the limited momentum range over which the

comparison

between

theory

and

experiment

was made and uncer-

lines the

importance

of short

wavelength

fluctuations in

establishing

the presence or the absence of short range order. It is evident that a Lorentzian does not

yield

a

peak

in

Q2 M’(Q)

vs.

Q plots

since

Q2 M2(Q)

tends to a constant value - K’ at

large

wavevectors.

Furthermore if the Lorentzian form of

M2(Q)

is

extended to the zone

boundary

it

yields

a level sub-

stantially

lower than that observed Thus the

amplitude

per nickel atom derived

by integrating

the Lorentzian

throughout

the zone

yields

a value less than half that obtained

experimentally

and which decreases with

increasing

temperature. This

analysis

underlines the

unsuitability

of localized models

applied

to itinerant

magnetism.

5. Conclusion.

As in the case of iron, CPA calculations and other models based on disordered local moments are unable

to account for the enhanced

scattering

observed at

small wavevectors

[7] (Fig.

2). In calculations based

on disordered local moments the response at

Q

= 0

is enhanced

by

a factor -

T/T -

0 over the level

at the zone

boundary

as

expected

for a local moment system described

by

a

Heisenberg

model. Furthermore the width of the observed forward

peak

is

significantly

smaller than that calculated

using

either CPA or a

Heisenberg

model. Thus the observed correlations are

significantly longer

range than those calculated as may be

anticipated

from the arguments of Friedel et al.

[26], Cyrot [27, 28]

and

Capellmann [38].

At 1.25 and 1.5

T,,

the temperatures at which measurements have been made in both iron and nickel, the

spatial

extent

of the correlations were observed to be of

longer

range in nickel. This result is consistent with the

larger

stiffness constant of the

spin

wave observed in the

ground

state

indicating

a

larger

Bloch wall thickness i.e. the

limiting length

scale over which the

magnetiza-

tion can be reversed.

Since more

scattering

occurs above an energy

kTc

the

large amplitude

per nickel atom 0.83 JlB derived above Tr cannot be taken as evidence for an increased

thermally

driven

amplitude.

Detailed CPA calcu- lation [24]

yield

only an

amplitude

of 0.27 /lB. However

Fig. 2. - The observed wavevector dependence of the paramagnetic scattering is shown, at 1.25 Tc, to be in good agreement with that of Steinsvoll et al. The data are com-

pared to a Lorentzian fit as proposed by Steinsvoll et al. [30]

using a value Of K (0.232

A -’ )

scaled from the critical region.

This curve does not give a satisfactory fit to the data neither

at small Q nor at large Q. Better agreement at small Q can be obtained using K: = 0.16 A-’ but the level at large Q is too low thus giving rise to a magnetic amplitude much smaller than that observed. At large Q the prediction of a CPA

model is similar to the Lorentzian prediction, as is expected

for local moment models but again the agreement with observations is poor even at small Q.

it is

possible

that

longitudinal

fluctuations do

play

an

important

role in nickel but not any conclusion

can be made on the basis of model calculations using

the static

approximation.

In order to make a

quanti-

tative

comparison

with

experiment explicit

conside-

ration of the quantum fluctuations must be made in a

model

including

the

dynamic

effects of

spin

fluctua-

tions

[38, 39].

Acknowledgments.

Discussion with M.

Cyrot

and T.

Moriya

are

gratefully acknowledged.

We are also

grateful

for the technical support

provided by

A. Perkins and P.

Agnes.

References

[1] BROWN, P. J., CAPELLMANN, H., DEPORTES, J., GIVORD, D. and ZIEBECK, K. R. A., J. Magn. Magn. Mat.

30 (1982) 243.

BROWN, P. J., CAPELLMANN, H., DEPORTES, J., GIVORD, D., JOHNSON, S. M., LYNN, J. W. and ZIEBECK, K. R. A., J. Physique 46 (1985) 827.

[2] SHIRANE, G., BONI, P., WICKSTED, J. P., Phys. Rev.

(to appear) (BNL-36586).

[3] HASEGAWA, H., J. Phys. F 13 (1983) 2655.

[4] EDWARDS, D. M., J. Magn. Magn. Mat. 36 (1983) 213.

[5] HASEGAWA, H., J. Phys. F 14 (1984) 1235.

[6] SHASTRY, B. S., EDWARDS, D. M. and YOUNG, A. P., J. Phys. C 14 (1981) L665.

[7] CHAUMET, M., JOHNSON, S. M., NEUMANN, K., THOMAS, and Ziebeck, K. R. A., J. Magn. Magn. Mat.

(1985) in press.

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