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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�
The paramagnetic response of nickel
athigh 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
neutronscattering
measure-ments have shown that the
paramagnetic phase
ofiron is characterized
by
strongspatial
correlations and asuppression
of the thermal fluctuations with shortwavelengths [1,
2]. The enhancedlong
wave-length
fluctuations have lowenergies
and contributesignificantly
to the free energy,characterizing
thethermal
properties
of iron aboveTc.
The responseintegrated
over all thermalenergies (2 KTc) yields
anamplitude
per iron atom of1.55 PB
much smaller than the ground state value of 2.2 MB. Thus themagni-
tude, wavevector and energydependence
of thescattering
isincompatible
with the concept of ther-mally
disordered atomic moments(close
to theground
state
value)
and with thepredictions
of CPA calcu- lations [3-7].A local moment model based on a
single
site CPAhas been
proposed
for nickel[5]
andS(Q, cv)
is pre- dicted to be similar to that of iron except that theamplitude
of the moment may increase with tempe-rature. A disordered atomic moment
description
forthe
paramagnetic phase
of nickel issurprising
sincethe
ground
state moment 0.61 pB involves less thanone electron
(or
hole). In order to test thevalidity
of such
hypothesis preliminary
neutronscattering
measurements on nickel at 1.1
T, [8]
and 2T, [9]
revealed an enhancement of the
scattering
for smallmomentum transfers similar to that observed in iron.
The present paper is concerned with detailed measu- rements
throughout
the Brillouin zone andextending
up to 180 meV i.e.
3 kTc.
2.
Properties
of nickel at finite temperatures.Although
bandtheory gives qualitative
agreement with many of thephysical properties
observed in nickel substantialdiscrepancies
exist. These diffe-rences become more fundamental at finite tempera-
tures where band
theory
fails to account for the observed thermal variations of themagnetization
andsusceptibility,
and for themagnitude
of the Curie temperature. Thus fluctuations of themagnetization density,
not contained in bandtheory,
with thermalenergies
k T must dominate incharacterizing
theseproperties.
It has been shown that transverse fluc- tuations(angular
rotations) of themagnetization density
candestroy long
range order in itinerantferromagnets [10-14, 15]
at temperatures consistent with the observed Curie temperature. The mechanism has similarities to localized systems describedby
aArticle published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004703049100
492
Heisenberg
model but with detailed differences due to the itinerant nature of themagnetic
carriers.Since Tc
(Stoner) [15]
issubstantially larger
than Tc(observed) [16],
it isgenerally
assumed that theexchange splitting
does notcollapse
at Tc(obs).
Indirect evidence in support of an
exchange split
band above
T,
in nickel isprovided by
the smallmagneto-volume anomaly
atTr [17].
Themagnitude
of the volume
change
atTc
is similar to that observed in iron but it is of a differentsign,
i.e. nickelexpands
on
becoming paramagnetic [18].
A.R.P.E.S. measure- ments at 1.1Tr [19]
have beenanalysed, assuming
a temperatureindependent
lineshape,
to indicate anexchange splitting
reducedby
50%, although
a moredetailed
analysis [20] suggested
nochange
in theexchange splitting
with temperature[21].
However,polarized
neutron measurements at 600 K suggest that the symmetry of themagnetization density
is temperaturedependent [22].
On the basis of these data it wasproposed
that the thermal variation of theexchange splitting
was different for the twosymmetries. Although
the staticsusceptibility
deviatesfrom Curie-Weiss behaviour the thermal variation is
significant.
If a temperatureindependent
contri- bution,possibly
of orbital origin, is subtracted from the observedsusceptibility
theremaining
Curie-Weiss component has an effective moment of 1.61 JlB
yielding
aparamagnetic
moment pp = 0.9 YB(y2 eff pp(pp
+2)) larger
than theground
state value[23].
Several models have been
proposed
to account for the thermalmagnetic properties
of itinerant systems and all claim reasonable agreement withexperiment.
In models based on CPA calculations the parama-
gnetic phase
is describedby
disordered local moments whoseamplitude
can increase with temperature[5, 24].
Moriya [14,
25] has shown that in the case of weakferromagnets
i.e. small moments and low Curie tem-peratures, the
longitudinal
fluctuations lead to aCurie-Weiss
susceptibility
andlarge
effective moments.Friedel et al.
[26]
have shown that if a moment is created in theparamagnetic
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 amagnetically
disordered stateKf- 1 must
be smaller than the interatomic
spacing.
This condi-tion is not satisfied for nickel and it is
therefore
notpossible
tospeak
of individualspins
on each nickel atom. In other models the itinerant nature of themagnetic
carriersgive
rise tospatial
correlations whichcan stabilize the
exchange splitting
aboveTc [10,
11,13].
Theparamagnetic phase
is therefore characterizedby
short rangemagnetic
orderindicating
the minimum distance over which themagnetization
can be reversed.These models are
capable
of accounting for thedetails of
magnetic scattering
functionS(Q,
E) [10]as observed in constant E scans
[29], reproducing
theobserved
peak positions,
their shifts in temperature,shape
and widths. Whether thesepeaks
in constant Escans are the signature of «
propagating spin
waves »above
T,,,
as inferredby
Lynn and Mook[29]
for themomentum range - 0.2 to 0.6
Å - I,
is controversial[30-32].
Initial
polarized
neutron measurements andpola-
rization
analysis [8]
enabled the wavevectordepen-
dence of the
paramagnetic scattering
from a naturalnickel
sample
at 1.1T,
to be determined. These mea- surements confirmed that thelong wavelength
fluc-tuations are indeed enhanced with a characteristic
wavelength
of 21A.
The measurementspresented
herehave been carried out on a 6°Ni crystal in which the strong disorder
(isotope) scattering
has been suppress- ed thusenabling
data ofgood
statisticalsignificance
to be collected. Both constants E and
Q
measurements have beenperformed throughout
the zone and up to 180 meV(3 kT c).
On the basis of these data it is pos- sible to make a criticalcomparison
with currenttheories.
3.
ExperimentaL
The
sample
used was the samesingle crystal
of 6°Nifor which
preliminary
measurements havealready
been
reported [9].
Thecrystal
in the form of acylinder
50 mm
long
and 7 mm diameter has a (110 >
axisparallel
to thecylinder
axis. A boron nitride support cemented to one end of thecylinder
was used to locatethe
crystal
at the centre of a coaxialmolybdenum
resistor of diameter 38 mm inside a furnace. Chromel- alumel
thermocouples
attached to each end of thecrystal
indicated astability
of better than ± 0.50 andan absolute difference of 1° at all temperatures up to 1 273 K
(2 T c).
The calibration of thethermocouples
was verified
by measuring
theabrupt change
inpola-
rization of the neutron beam diffracted from
(111)
onwarming through
the Curie temperature 631 K.The
paramagnetic scattering
from nickel at 1.25,1.5 and 2
Tc
wasinvestigated
both as a function of momentum and energy transferusing
thepolarized
neutron
triple
axis spectrometer D5 located on the hot source of the H.F.R. at the I.L.L. Since monochro- maticpolarized
neutrons withenergies
of up to 1 eVcan be incident on the
sample
the spectrometer isideally
suited forinvestigating
theparamagnetic
res-ponse of itinerant magnets. Constant
Q
scans carriedout at
large
wavevectors i.e. 0.5 qmQ
qm, where qm represents the zoneboundary
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 usedtogether
with anEr filter to minimize
higher
order contamination. Theintegrated scattering
from the constant Q scans wascompared
with thatdirectly
obtainedusing
fixed ener-gy windows
corresponding
toenergies
ofkTc,
2kTr
and 3
kTc.
Theintrinsically
poor resolution associated with shortwavelength
neutrons wascompounded by
the use of
monochromating
andanalysing crystals
with
large
latticespacings.
The(111)
reflection ofCu2MnAl (d111
= 3-45A)
was used as bothpolarizing
monochromator and
analyser.
Guide fields maintained the neutronpolarization
between monochromator andsample
and betweensample
andanalyser.
Helm-holz coils around the furnace enabled the neutron
polarization
at thesample
to be eitherparallel
orperpendicular
to thescattering
vector. The verticaland horizontal field
required
to rotate the neutronpolarization
was some 100 Oe and hadnegligible
effect on the
sample.
An r.f coil in theincoming
beamenabled the incident neutron
polarization
to bereversed.
By reversing
the incidentpolarization, using
the
flipper,
andmeasuring
the scattered neutron count for incident neutronspolarized perpendicular
C orparallel C to
thescattering
vector, a clean determina-tion of the
paramagnetic scattering Cil
- C ispossible.
All other forms of
scattering
are not sensitive to therelative orientation of the neutron
polarization
andscattering
vector and therefore subtract out Correc- tions for bothincomplete polarization
of the beamand
flipping efficiency
were madefollowing
the pro- cedure outlinedby
Ziebeck et al.[33].
For neutronspolarized parallel
andperpendicular
to thescattering
vector the
magnetic
response functionS(Qcv)
is relatedto the observed
intensity by
where V is the volume of the
sample,
Vr the volumeof the unit cell,
NM
the number ofmagnetic
atomsper unit cell and
ki kf
the incident and final wavevec- tors. Thescattering
wasplaced
on to absolute scaleusing
two methods : the nuclearspin
incoherentscattering
from a vanadiumcylinder
of similar dimen- sions as the 60Nicrystal
andby comparison
with theenhanced
scattering
observed inpolycrystalline
nickelrod under similar
experimental
conditions and for which the normalizationprocedure
isstraight
forward [34]. The agreement obtained using the twotechniques
was better than 5
%.
4. Resets.
The wavevector
dependence
of the enhancedscattering
which occurs for wavevectors less than - 0.25 qM was
investigated
in the[ 111
direction in the zones centredon
(000)
and(111).
Aftercorrecting
for themagnetic
form factor
[35]
the results obtained in the two zones were found to have the same wave vectordependence.
These results, which indicate the
spatial
extent of theferromagnetic
correlations, are in agreement with thosepreviously
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 resultsextrapolated smoothly
to the observed value of the static
susceptibility
atQ = 0.
Thus the enhancemeot of thescattering
decrea-ses with temperature but remains substantial even at 2
Tc.
From the widthQw
of the forwardpeak
thewavelength characterizing
thespatial
correlationsA,
= 2 x/Qw was found to decrease from - 21 A at 1.25 Tc to - 14 A at 2Tc.
The momentum range over which enhancedscattering
is observedcorresponds
to the zone in which the acoustic
spin
wave propagates in theground
state beforeentering
the Stoner conti-num. The characteristic
wavelength À-c quoted
abovein our
opinion
is asignature
of theinterancy
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
atlarge
momentum transfers i.e.0.25 qm
Q
qm wasinvestigated
in the zonescentred on
(111) (220)
and(002). Although
the magne- tic form factorsubstantially
reduces thescattering
at
large
momentum transfers it ispossible
to extendmeasurements to
high
energy transfers and maintain thepolarization along
thescattering
vector whendetermining Cjj.
The energydependence
of thescattering
close to the zoneboundary
is ofparticular
interest since at these momenta the « on » site
pair
correlation function
( Si. Si >
isexpected
to dominate.Whereas the
long wavelength
fluctuations characterize themacroscopic
features of the response, the microsco-pic
details are reflected in the energydependence
of (S;. S; ).
In a local moment system as describedby
a
Heisenberg
model, the correlation function is cha- racterizedby
a thermal energy scale. For an itinerant magnet the response at the zoneboundary
is expectedto extend up to the band width i.e. some eV’s.
Although
the
Heisenberg
model is oftenapplied
to iron,unjus- tifiably,
its extention to nickel is even less well founded.At temperatures
corresponding
to 1.5 and 2Tc
pre-liminary
constant q scans close to the zoneboundary
in the
[111]
direction andextending
up to 60 meV i.e.~
kTr
revealedsignificant scattering increasing slightly
with energy transfer.
Subsequent
constant q scans confirmed that measurablescattering
existed up to at least 180 meV i.e. - 3kTc.
Theintegrated scattering
from these scans, 0.74
,uB,
was found to be in excellent agreement with that observedusing
a fixed window of3
kTc.
These results are in contrast with those obtainedon
paramagnetic
iron in which thescattering
closeto the zone
boundary
had fallen to anegligible
levelby
70 meV i.e. less than energykT,, (iron).
Atlarge
wavevectors in nickel more
scattering
occurs at anenergy transfer greater than
kTc
than below. Measu-rements at 1.25,1.5 and 2
T c
suggest that theintegrated scattering
atlarge
wavevectors does notchange substantially
with temperature. This would suggest that the fluctuations at large wavevectors arepredo- minantly
quantum fluctuations,possibly
due to por- tions of the bandstructurebeing
rather flat in theregions of (q,
w) spaceprobed
in theexperiment.
The observed
scattering
may beintegrated through-
out the zone to obtain an « effective »
amplitude [36] :
494
It may be seen that the
phase
space factorQ 2 gives
a substantial
weight
to thelarge
Q components whichare
predominantly
quantum fluctuations.Consequen- gly
the thermal fluctuations which characterize the small wavevector responsegive
rise to anegligible
contribution to the
amplitude.
Animportant
consi-deration in the definition
of p
is the energy range over whichS(Qco)
has beenintegrated
to obtainS(Q).
Experimentally
the maximum energy transfers em-ployed
were 180 meV(3 kTc) a
value sufficient tocollect the
majority
of thermal fluctuations. Theamplitude
was determined to be 0.87 PB per nickel atom. It is alsopossible
to estimate theamplitudes corresponding
to anintegration
up toenergies
ofkTc
and 2
kTc;
these values are 0.38 PB and 0.63 PB per nickel atomrespectively.
From these values it may be seen that moreweight
lies above an energy cor-responding
tokTc
than below, in contrast to a localmoment system, and iron. A
plot
ofQ2 S(Q),
for thethermal fluctuation, as a function of wavevector
yields
distinct
peaks
at wavevectors - 0.35, 0.4 and 0.45 A-1for 1.25, 1.5 and 2
Tr respectively.
Thus fluctuations with Fourier componentscorresponding
to thesewavevectors contribute
significantly
to the on sitecorrelation function. This behaviour is similar to that observed in iron, except for iron
[1]
the characteristic wavevector wasessentially
temperatureindependent
up to 1.5
T,
andcorresponded
to - 0.4 A-1. Thepeaks
observed in theQ2 S(Q)
vs. Qplots
for nickelFig. 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 characteristicwavelengths
for the ferro-magnetic
correlations of between 21 - 14 A(i.e.
for 2and 1.25
Tc)
in agreement with that obtained fromthe width of the forward
peak.
The measurements of Steinsvoll et al.
[30]
wereconfined to wavevectors less than 0.4 Å - 1 and to
temperatures below 1.24
T,.
At 1.16T,
Steinsvollet al. obtained reasonable agreement with observation
using
ascattering
function in which both the wave- vector and energydependence
hadsimple
Lorentzianforms. Thus the wavevector
dependence
of the scat-tering
wasgiven by
where
M2(o)
is the staticsusceptibility
in units ofMB
and K is the inverse correlationlength.
Values for K were scaled from those observed in the criticalregion [37] using
Such that at 1.16
T,,
x = 0.174 A-1. Since the for- malism is based ontheory applicable
to criticalscattering
it isimportant
to establish over which range of temperature and momentum it remainsapplicable.
A
comparison
of the results of Steinsvoll et al. at 1.24Tc
with those of the presentinvestigation
at1.25
Tr
revealssatisfactory
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 theappropriate
functional form. Nevertheless in the momentum range covered the results at 1.24-1.25
T, already diverge substantially
from the calculated momentumdepen-
dence
(2),
if the scaled value x = 0.232Å - 1,
deducedfrom
equation (3),
is used. A moresatisfactory
agree- ment with observation in therestricted q region
can beobtained
using
the same Lorentzian form but withK = 0.16
A-1.
Thussubstantially
more short range order is observed even at 1.25Tc
than ispredicted by
the
simple
model based on criticalscattering
andusing
values for K scaled from observations close toTc.
Agreement
obtained at 1.16Tc
and below is notsurprising
since at temperatures close toTc
theforward
scattering
is dominatedby large amplitude
fluctuations associated with residual
long
range order which atT,
causes thesusceptibility
todiverge.
Above1.25
Tc
i.e. at 1.5 and 2T,
the functional form(2), using
the scaled values of x(3), completely
fails toaccount for the observed
scattering.
Over the momen-tum range 0-0.4 Å - 1 a
satisfactory
agreement can be obtainedusing
K=0.18 and 0.17 at 1.5 and2Tc.
Within the
experimental
error it may be concluded that between 1.25 and 2Tc,
acharacteristic qc
existswhich is
essentially
temperatureindependent
with avalue K = 0.17 + 0.01 A-’. This result reflects the limited momentum range over which the
comparison
between
theory
andexperiment
was made and uncer-lines the
importance
of shortwavelength
fluctuations inestablishing
the presence or the absence of short range order. It is evident that a Lorentzian does notyield
apeak
inQ2 M’(Q)
vs.Q plots
sinceQ2 M2(Q)
tends to a constant value - K’ at
large
wavevectors.Furthermore if the Lorentzian form of
M2(Q)
isextended to the zone
boundary
ityields
a level sub-stantially
lower than that observed Thus theamplitude
per nickel atom derived
by integrating
the Lorentzianthroughout
the zoneyields
a value less than half that obtainedexperimentally
and which decreases withincreasing
temperature. Thisanalysis
underlines theunsuitability
of localized modelsapplied
to itinerantmagnetism.
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 atsmall wavevectors
[7] (Fig.
2). In calculations basedon disordered local moments the response at
Q
= 0is enhanced
by
a factor -T/T -
0 over the levelat the zone
boundary
asexpected
for a local moment system describedby
aHeisenberg
model. Furthermore the width of the observed forwardpeak
issignificantly
smaller than that calculated
using
either CPA or aHeisenberg
model. Thus the observed correlations aresignificantly longer
range than those calculated as may beanticipated
from the arguments of Friedel et al.[26], Cyrot [27, 28]
andCapellmann [38].
At 1.25 and 1.5T,,
the temperatures at which measurements have been made in both iron and nickel, thespatial
extentof the correlations were observed to be of
longer
range in nickel. This result is consistent with thelarger
stiffness constant of the
spin
wave observed in theground
stateindicating
alarger
Bloch wall thickness i.e. thelimiting length
scale over which themagnetiza-
tion can be reversed.
Since more
scattering
occurs above an energykTc
the
large amplitude
per nickel atom 0.83 JlB derived above Tr cannot be taken as evidence for an increasedthermally
drivenamplitude.
Detailed CPA calcu- lation [24]yield
only anamplitude
of 0.27 /lB. HoweverFig. 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
thatlongitudinal
fluctuations doplay
an
important
role in nickel but not any conclusioncan be made on the basis of model calculations using
the static
approximation.
In order to make aquanti-
tative
comparison
withexperiment explicit
conside-ration of the quantum fluctuations must be made in a
model
including
thedynamic
effects ofspin
fluctua-tions
[38, 39].
Acknowledgments.
Discussion with M.
Cyrot
and T.Moriya
aregratefully acknowledged.
We are alsograteful
for the technical supportprovided 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.