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Physica B 353 (2004) 296–304

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Critical phenomena at diamagnetic phase transitions

A. Gordona,�, N. Logoboyb,c, W. Jossc

aDepartment of Mathematics and Physics, Faculty of Science and Science Education, University of Haifa-Oranim,

Oranim-KiriatTivon, Tivon 36006, IsraelbSchool of Practical Engineering, Ruppin Academic Center, Physics and Engineering Research Institute, Emek Hefer, 40250, IsraelcGrenoble High Magnetic Field Laboratory, Max Planck Institut fur Festkorperforschung and Centre National de la Recherche

Scientifique, B.P. 166X, F-38042, Grenoble Cedex 9, France

Received 3 October 2004; accepted 6 October 2004

Abstract

The critical phenomena near diamagnetic phase transitions and in the domain phase are considered. Possibilities of

the soft-mode existence and the static critical effect in susceptibility related to the appearance of Condon domains in

aluminium are discussed. The constructed phase diagram and the temperature behaviour of the susceptibility give the

value of the phase-transition temperature observed in aluminium. We present the temperature dependence of the

magnetisation in each domain being the order parameter of the diamagnetic phase transition and confirming the

softening of the orbital magnon-mode in aluminium. Calculations of the spectral density of excitations at diamagnetic

phase transitions are made in order to explain the critical growth of the helicon damping observed in aluminium. The

temperature dependence of the damping coincides with that measured in aluminium. The calculated magnetic-induction

splitting due to Condon domains turns out to be close to that estimated in muon rotation spectroscopy experiment.

r 2004 Elsevier B.V. All rights reserved.

PACS: 71.10.Ca; 71.70.Di

Keywords: Electron gas in quantising magnetic fields; De Haas-van Alphen effect; Condon domains; Diamagnetic phase transitions

1. Introduction

Oscillations of the thermodynamic quantities inan external magnetic field are the result of the

e front matter r 2004 Elsevier B.V. All rights reserve

ysb.2004.10.010

ng author. Tel.: +972 4 9838844; fax:

.

ss: algor@math.haifa.ac.il (A. Gordon).

oscillations of the density of states and the factthat the magnetic field quantises the energy levels(Landau quantisation) [1]. Many properties of theelectron gas in normal metals are periodic func-tions of magnetic fields as successive Landau levelssweep through the Fermi level due to an increaseof the external magnetic field—for instance,oscillations of magnetisation (de Haas-van Alphen

d.

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A. Gordon et al. / Physica B 353 (2004) 296–304 297

effect-dHvA) [1]. The magnetic field changes thedensity of states and consequently also the internalenergy of the electron gas, and when 4pwB41(wB ¼ qM=qB is the differential magnetic suscept-ibility, M is the oscillatory part of the magnetisa-tion, B is the magnetic induction), the‘‘realignment’’ of the density of states, connectedwith the change of magnetic induction occurringduring the stratification into phases, becomesenergetically ‘‘convenient’’ resulting in formationof Condon domains [1,2]. This instability causedby magnetic interactions among conduction elec-trons is known as diamagnetic phase transition [2].The transformation occurs in each cycle of dHvAoscillations when the reduced amplitude of oscilla-tions a ¼ 4pwB (the ratio of the amplitude to theperiod of dHvA oscillations) approaches unity [1].The reason for this collective effect is that anelectron is subject to the magnetic inductioninstead of the magnetic field (the Shoenberg effect)[1]. Condon domains were predicted by Condon inRef. [3] and discovered in silver by means ofnuclear magnetic resonance [4], and in berylliumand white tin by means of muon spin–rotationspectroscopy [see the recent review [5]]. Thedomain formation at diamagnetic phase transi-tions was extensively theoretically studied [see therecent review [6]]. The temperature dependence ofthe magnetic-induction bifurcation due Condondomains was satisfactory reproduced in quasi-two-dimensional and three-dimensional metals [6].Diamagnetic phase transition has been recentlyshown to be associated with instability of a three-dimensional electron gas against a temperature-dependent magnon soft mode, the frequency ofwhich vanishes at the phase-transition point.Temperature and sample size softening of themagnon mode of an orbital nature were consid-ered in quantising magnetic fields under conditionsof the strong dHvA effect in bulk metals andconfined metallic samples [7]. A considerablegrowth of the helicon damping in aluminium hasbeen recently related to critical slowing down ofthe electron relaxation time [8,9].Static critical effects like the temperature beha-

viour of the static susceptibility provide a deeperinsight into the physical nature of the aboveinstability of an electron gas in normal metals

and improve the understanding of the underlyingphysics of this unusual sort of ‘‘magnetism’’. Inspite of the fact that Condon domains have beenalready detected in silver, beryllium, aluminium,white tin and lead, much information lacks aboutthem. For instance, few experimental data areavailable concerning relation of the domainformation to critical static and dynamic phenom-ena, which are usually characteristic of phasetransitions. Domain and domain wall sizes havenot been measured. Thus, characteristic magneticlengths of the system are unknown. Knowledge onthe spatial structure of the domains is still lacking.The available experimental data have not beenalways interpreted in a proper way or theirinterpretation is absent at all, like in the interestingpaper on Condon domains and helicons inaluminium [8].The goal of this research is description of some

critical phenomena near diamagnetic phase transi-tions and in the domain phase. We discuss apossibility of the soft-mode existence and the staticcritical effect in susceptibility related to appear-ance of Condon domains in aluminium. Wecalculate the spectral density of excitations atdiamagnetic phase transitions in order to explaincritical growth of the helicon damping observed inaluminium. The work is partly motivated byunexplained results obtained in aluminium inRef. [8]. We show that the study of heliconresonance provides a monitor for dynamic criticalphenomena. We calculate the magnetic-inductionbifurcation due to Condon domains in aluminium,which coincides with that estimated in experimenton muon rotational spectroscopy.

2. Model

The equation of state describing magneticoscillations is the Lifshitz–Kosevich–Shoenbergequation (see Eq. (2)) [1]. However, resonancemeasurements [3,6] show that the magnetic-induc-tion bifurcation due to Condon domains exhibitstemperature dependence with a critical indextypical for the mean-field theory [5,6]. In berylliumthe temperature dependence of the magnetic-induction splitting in the domain phase, describing

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by the ða � 1Þ1=2 type curve, was detected [5,6].Thus, the order parameter of the diamagneticphase transition is proportional to ða � 1Þ1=2 andapproaches zero when the reduced amplitude a

tends to unity. Diamagnetic phase transitions insilver and beryllium were therefore shown to be ofsecond order in temperature [5,6], and it turnedout that the symmetry-breaking order parametershould grow according to ðT c � TÞ

1=2 or ða � 1Þ1=2

on cooling in this case. Ferromagnetic, antiferro-magnetic, superconducting and various otherkinds of broken-symmetry transitions behave inmore or less the same way. In the case of acontinuous symmetry group, the low-frequencycollective excitations, the so-called soft modes, arepresent. In isotropic magnets magnon modessoften as temperature approaches its phase-transi-tion value from below. The ða � 1Þ1=2-type beha-viour of the order parameter of the diamagneticphase transition resembles the situation in ferro-electrics, in which the soft mode takes place at thephase transition. Diamagnetic phase transitionshave been recently shown to be resulted frominstability with respect to the soft mode of anorbital (non-spin) nature [7]. Thus, magnon-like(flipping) and nonlinear (domain-wall-like) excita-tions exist in metals undergoing diamagnetic phasetransitions.The oscillator part of the thermodynamic

potential density can be written neglecting allharmonics in the Lifschitz–Kosevich formula, butone, the first harmonic approximation [1,10]:

O ¼1

4pk2a cosðbÞ þ

1

2a2 sin2ðbÞ

� �; (1)

where H is the magnetic field inside the material,b ¼ kðB � HÞ ¼ kðhex þ 4pMÞ; (k ¼ 2pF=H2; F isthe fundamental oscillation frequency, hex ¼

Hex � H ; is the small increment of the magneticfield H and the external magnetic field Hex; all thecomponents of vectors are taken along thedirection of the magnetic induction). In the firstharmonic approximation the magnetisation isfound from the implicit equation of state [1]:

4pkM ¼ a sin½kðhex þ 4pð1� nÞMÞ�; (2)

where a is the reduced amplitude of oscillations:a ¼ 4pkA ¼ 4pðqM=qBÞB¼H ¼ 4pwB [1], A is the

first harmonic amplitude, n is the demagnetisationfactor. If the magnetic interaction is strongenough, a state of lower thermodynamic potentialcan be achieved over part of an oscillation cycle bythe sample breaking up into domains, for whichthe local value of magnetisation alternates in signfrom one domain to the next. Close to the phase-transition temperature (which is found from theequation aðTc;HÞ ¼ 1) we can present the thermo-dynamic potential per unit volume as an expansionin powers of magnetisation:

O ¼ 2pð1� aÞM2 þ8

3p3k2aM4: (3)

Accordingly, we arrived at the Landau-typethermodynamic potential density [6,11]

O ¼ �A

2M2 þ

B

4M4; (4)

where A ¼ 4pða � 1Þ; B ¼ ð32=3Þp3k2a: In the caseof the breaking up into domains these equationsare valid over the range of domain existence in thedHvA cycle. According to [10], the temperatureand field dependence of the amplitude is

aðT ;HÞ ¼ a0ðHÞlT

sinhðlTÞexp½�lðHÞTD�; (5)

where l � 2p2kBmcc=e_H ; mc is the cyclotronmass. The limiting amplitude a0ðHÞ � aðlT !

0;HÞ is the combination of temperature-indepen-dent factors in the Lifshitz–Kosevich formula [10]�a0 ¼ ðHm=HÞ

3=2; kB is the Boltzmann constant, e

is the absolute value of the electron charge, c is thelight velocity, mc is the cyclotron mass, _ is thePlanck constant, TD is the Dingle temperature andHm is the limiting field, above which diamagneticphase transition does not occur at any tem-perature. Close to the phase transition,A ¼ ðqa=qTÞT¼Tc

ðTc � TÞ; ða � 1Þ ¼ ðqa=qTÞT¼Tc

ðT c � TÞ [6].

3. Static critical phenomena in aluminium

Condon domains have been recently observed inaluminium by muon rotation spectroscopy [12]thereby justifying results of the earlier work [8], inwhich the influence of the diamagnetic phase

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Fig. 1. Temperature–magnetic field phase diagram is con-

structed in aluminium at TD ¼ 0:1K: Phase-transition tem-perature Tc is equal to 1.2K in applied magnetic field 4.8T. It

coincides with that observed in Ref. [8].

A. Gordon et al. / Physica B 353 (2004) 296–304 299

transition on the characteristics of helicon waveswas studied.The susceptibility to external magnetic field wHex

is different from the differential susceptibilitywBð4pwB ¼ aÞ and is given by [1]:

wHexðH ;TÞ ¼

aðH;TÞ

4p½1� ð1� nÞaðH;TÞ�; (6)

where aðTÞp1: In order to find the phaseboundary separating homogeneous and domainphases we use

aðTc;Hex;TDÞ ¼ 1 or aðT ;Hc;TDÞ ¼ 1; (7)

where Tc is the phase-transition temperature andHc is the phase-transition field, in which theCondon ordering takes place, TD is the Dingletemperature. As is seen from Eq. (7), the suscept-ibility wHex

equals to 1=4pn at the diamagneticphase-transition temperature or phase-transitionmagnetic field. If the sample is of a needle-type, thedemagnetisation factor n 1; and the suscept-ibility wHex

tends to infinity as n tends to zero. Forthe needle-type geometry of a sample, the mea-sured susceptibility wHex

ðH;TÞ plays the role of thedivergent susceptibility in the case of second-orderferromagnetic phase transitions as the temperaturereaches T c or the magnetic field reaches Hc: Thus,the needle-type geometry is especially convenientfor searching for drastic temperature and mag-netic-field effects in susceptibility signalising theonset of a diamagnetic phase transition. Eq. (7)describes the temperature–magnetic field phasediagram determining the locus of points showingthe series of the phase-transition temperatures.The phase transition of the second order with thewell-defined transition temperature and magneticfield takes place only at discrete values of magneticinduction. The envelope of these points serves asthe diamagnetic phase-transition boundary.Since the recent direct experiment on muon

rotation spectroscopy [12] confirms domain for-mation in aluminium reported in Ref. [8] we cananalyse the data of Ref. [8] in terms of Condondomains. Results of Ref. [8] show the heliconresonance frequency modulated by dHvA oscilla-tions. The authors of Ref. [8] observed unusuallylarge amplitude of the sample resonant–frequencyoscillations originated from the extremal sections

of hole-Fermi surface as functions of the appliedmagnetic field. They observed the studied effect inaluminium at all magnetic-field directions, forwhich the oscillations due to the hole-Fermisurface of the second zone are large enough. Thephase transition into the domain phase seemed tooccur at T ¼ 1:121:2K at a given magnetic fieldH ¼ 4:8T: High-frequency dHvA oscillations areindeed related to the second-zone hole surface [13].The oscillation-activated frequency is equal to F ¼

4:36� 104 T [13]. Measurements reported in Ref.[8] were performed at the temperature range0.45–4.2K in a magnetic field up to 6T. All theeffects related to Condon domains were obtainedin two extremely pure samples with Dingletemperatures at the range 0.05–0.1K.We present the phase diagram for aluminium at

the experimentally relevant range of temperaturesand fields in Fig. 1 ðTD ¼ 0:1KÞ: For its construc-tion we use Eqs. (5) and (7) and see thatthe diamagnetic phase transition occurs atT ¼ 1:121:2K and H ¼ 4:8T in accord withexperiment.In Fig. 2 the temperature dependence of the

susceptibility is plotted in the homogeneous phasefor the sample, in which the diamagnetic phasetransition was mainly studied (Eq. (6)). Thedemagnetisation factor of the sample is 0.64. Asis expected from the theory, we obtain the

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Fig. 2. Temperature dependence of the susceptibility wHex;

multiplied by 4p; is shown in aluminium revealing its maximalvalue at the phase-transition temperature T c ¼ 1:2K: Thedashed line denotes the approximate relation 4pwHex

¼ 1=n [1,

Eq. (6.50), p. 272] Its application over the entire domain phase

is questionable.

A. Gordon et al. / Physica B 353 (2004) 296–304300

maximal value of the susceptibility 4pwHex¼

1=n ¼ 1:57 at T ¼ 1:2K: The dashed line denotesthe approximate relation 4pwHex

¼ 1=n [1, Eq.(6.50), p. 272] Its application over the entiredomain phase is questionable. From the theore-tical point of view, the physics underlying theprocess of magnetisation in the domain phase hasnot been understood. According to the Shoen-berg’s consideration [1], the susceptibility is equalto 1=4pn over the entire range of the domainexistence. An apparent flaw in this conclusion isthat since the magnetisation is no longer uniform,due to the subdivision into domains, the use of ademagnetisation factor is not justified. At firstglance, so long as the domain spacing is smallcompared to the dimensions of the specimen, anysubstantial element of the surface can be treated asa region of the uniform magnetisation, and theapproximation seems to be justified. However, thisconsideration excludes a hysteresis effect or atemperature dependence of the susceptibilityknown from ferromagnetism and ferroelectricity.Thus, the Shoenberg’s relation between the sus-ceptibility and the demagnetisation factor needsdeeper investigation. Since the demagnetisationfactor is not a well-defined quantity we omit it inthe further use of Eq. (2). We shall show that the

susceptibility in the domain state should bestrongly temperature-dependent.The critical behaviour of the susceptibility on

the side of the homogeneous phase may take placealso in the domain phase exhibiting temperaturebehaviour of the soft-mode type [7]. The behaviourof the unexplained satellite in the helicon reso-nance spectrum, the frequency of which decreaseson approaching the phase-transition temperaturein aluminium, resembles the typical soft-modebehaviour and may evidence in favour of non-spinmagnon-mode softening [7]. However, there are nodata enough for a quantitative analysis. Inaddition, a possible interaction between thehelicon and magnon modes disturbs to separatethe temperature behaviour of the soft mode fromthat of the helicon mode. We can state two things:(1) the satellite attributed by us to the soft modeappears exactly at the phase-transition tempera-ture and the temperature dependence of itsfrequency resembles the tendency characteristicof the soft-mode behaviour. However, we shall usetwo additional arguments to elucidate this point.In order to study the soft-mode problem more

carefully, we can examine the temperature depen-dence of the magnetisation in the centre of dHvAoscillation (h ¼ 0), which shows the temperaturedependence of the order parameter. At h ¼ 0 thetwo minima have the same free energy correspond-ing to the ‘‘remnant’’ (h ¼ 0) magnetisation [6]

Mr ¼1

2pk

3ða � 1Þ

2a

� �1=2: (8)

Eqs. (5) and (8) give the temperature depen-dence of the magnetisation in each domain. Wecan present the temperature dependence of themagnetisation in two ways: (1) using the equationof state (2), (2) taking its approximate form,which is the result of the expansion of the sinein Eq. (2) according to Eqs. (3) and (4) (Eq. (8)).In Fig. 3 we present a plot of the reducedmagnetisation m ¼ 4pkM as a function oftemperature according to Eq. (2) (curve 1)and to Eq. (8) (curve (2)) and by using para-meters of the aluminium sample measuredin Ref. [8] TD ¼ 0:1K; F ¼ 4:36 � 108 T; �F ¼

3:76 eV; T ¼ 0:65K; Zc ¼ 1:3; DH ¼ 5:5G: It is

Þ1=2

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Fig. 3. Temperature dependence of the reduced magnetisation

in the domain phase, m ¼ 4pkM; in the centre of the period ofdHvA oscillations in aluminium: curve 1 was calculated by Eq.

(2) and curve 2 was calculated by using Eq. (8). It is seen that

the magnetisation tends to zero at Tc ¼ 1:2K:

Fig. 4. Temperature dependence of the reduced magnetisation

m calculated at kh ¼ 0 (curve 1) and kh ¼ p=2 (curve 2) fromEq. (2).

A. Gordon et al. / Physica B 353 (2004) 296–304 301

seen from Fig. 3 that the phase transition is equalto T c ¼ 1:2K: In the two cases the magnetisationexhibits therefore the temperature dependence,which is typical for the order parameter of thephase transition, indicating a possibility ofthe existence of the soft mode. It follows fromthe comparison between curves 1 and 2 that theirtemperature dependence is strongly different farfrom the phase-transition temperature and is thesame in its environment. Near the diamagneticphase transition we can present the temperaturedependence of the reduced magnetisation asfollows:

m ¼ 61=2� � a � 1

a

� �1=2¼ 61=2� � qa

qT

� ���������T¼Tc

ðT c � T

¼ ð6Þ1=2lLðaÞðTc � TÞ1=2;ð9Þ

where LðaÞ ¼ j coth a� ð1=aÞj is the Langevinfunction (see, for instance, Ref. [14]), a ¼ lT c:Now we calculate the bifurcation of the

magnetic induction due to two Condon domainsat T ¼ 0:65K by using the data in Fig. 3 andtaking into account that the magnetic-inductionsplitting DB ¼ 4pDM ¼ 8pMr: The splitting is 5Gfor curve 1 and 4G for curve 2. The calculatedbifurcation DB coincides with the line splitting inaluminium estimated in experiment [12] as 4G,

which is too small to be resolved by the presentday muon rotation spectrometer. At a � 1 themaximal magnetic-induction splitting is given byDB ¼ 4H2=F ð3=2Þ1=2: It increases as the secondpower of magnetic field.Since the frequency of the helicon resonance in

aluminium was measured in several points of theperiod of oscillations, it is worthwhile to calculatethe temperature dependence of the magnetisationat a definite field declination from the centre of theperiod of oscillations h. Using Eq. (2) we calculatethe temperature dependence of the reduced mag-netisation m at kh ¼ 0 and kh ¼ 2p=DH

� h ¼ p=2:

In Fig. 4 we present the temperature dependenceof the reduced magnetisation calculated at kh ¼ 0(curve 1) and kh ¼ p=2 (curve 2) from Eq. (2). It isseen from curve 2 that the declination field smearsout the phase transition because the magnetisationin each domain is non-zero at all the temperatures.Consequently, the phase-transition temperatureis not well-defined. Perhaps, the complicatedtemperature dependence of the satellite in alumi-nium [8] is related to the fact that measurementswere made outside the centre of the period ofoscillations.A generalised formula for the susceptibility

to external field is required to perform amore detailed analysis of the diamagneticphase transition outside the centre of the period

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A. Gordon et al. / Physica B 353 (2004) 296–304302

of oscillations [1]:

wHexðH ;TÞ ¼

aðH ;TÞ cos ðkh þ mÞ

4p½1� aðH;TÞ cos ðkh þ mÞ�: (10)

In Fig. 5 we present the temperature dependenceof 4pwHex

in the inhomogeneous phase (curve 1)and in the homogeneous and domain phases(curves 2 and 3). The susceptibility exhibits asingularity at the phase transition (curve 1) inaluminium on approaching the phase transitionfrom below. This plot is complementary to thegraph in Fig. 2 and demonstrates the criticalphenomenon pointed out in the homogeneousphase. Curves 2 ðkh ¼ p=16Þ and 3 ðkh ¼ p=2Þ arecalculated by using Eq. (10). They show thetemperature dependence of the susceptibility inthe homogeneous and domain phases outside thecentre of the period of oscillations at n ¼ 0:Declinations from the centre of the period ofoscillations smear out the phase transition anddemonstrate a maximum instead of singularity,which displaces from the phase-transition tem-perature at kh ¼ 0 (curve 2) and disappears incurve 3. All the three curves reveal the diamagneticbehaviour with the susceptibility ð4pwHex

Þ tendingto �1 at very low temperatures, as expected fromthe theory [1,6].

Fig. 5. Temperature dependence of the susceptibility wHex;

multiplied by 4p; is shown in aluminium: kh ¼ 0 (curve 1), kh ¼

p=16 (curve 2) and kh ¼ p=2(curve 3) (Eq. (10)).

4. Critical dynamics at diamagnetic phase

transitions

The striking temperature growth of the helicondamping was related in Ref. [9] to critical slowingdown of the electron relaxation time. However, thetemperature dependence of the helicon dampingwas obtained on the basis of the relationshipbetween critical indices within the framework ofthe mean-field approximation. We will justify thisresult and extract it from the spectrum ofmagnetisation fluctuations at diamagnetic phasetransitions.Adding the inhomogeneous term

K=2ðqM=qxÞ2; K ¼ r2c=4; where K is the inhomo-geneity coefficient, rc is the cyclotron radius, to thethermodynamic potential (3), (4), and using thetime-dependent Ginzburg–Landau equation [11]

qM

qt¼ �G

qOqM

(11)

for the overdamped case, we obtain the follow-ing equation:

qM

qt¼ G K

q2Mqx2

þ AM � BM3

� �; (12)

G is the Landau–Khalatnikov transport coeffi-cient. By using the time-dependent Ginzburg–Lan-dau equation we see that the local rate ofdisplacement of the order parameter is linearlyproportional to the local thermodynamicforce presented by the functional derivative ofthe thermodynamic potential density. Theconstant of proportionality, the kinetic coefficientG; is the response coefficient, which defines atime scale for the system. This approach,based on the thermodynamic potential includingthe term with the sixth power of magnetisation,was also used for the description of motionof interphase boundaries at first-order dia-magnetic phase transitions under temperaturechanges [6].Passing to the moving frame, s ¼ x2vt;

where v is the velocity in the direction x, weobtain

KGd2M

ds2þ v

dM

ds� Gð�AM þ BM3Þ ¼ 0: (13)

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A. Gordon et al. / Physica B 353 (2004) 296–304 303

Particular solutions to Eq. (13) are as follows:

M ¼ð3=2aÞ1=2 a � 1ð Þ

1=2=2pk

1þ exp � x � vtð Þ=D� � ; (14)

where

D ¼rc

4

2

p a � 1ð Þ

� �1=2(15)

and

v ¼ 3Grcpða � 1Þ

2

� �1=2: (16)

Eq. (14) describes the interface between the‘‘magnetised’’ and the homogeneous phases thusgiving the wall of a cluster of the domain phase.Now we consider the influence of critical

fluctuations of magnetisation on the helicondamping. The spectrum of the Mq—fluctuationsin the ‘‘harmonic’’ approximation (B ¼ 0; see Eq.(4)) is given by Ref. [15]

I~qðoÞ ¼Z þ1

�1

dt expð�iotÞhMð~q; 0ÞMð�~q; tÞi

¼kBT

4pjða � 1Þj 1þ q2r2c=4� 2tq

1þ ðotqÞ2

!

¼2GðkBTÞ

t�2q þ o2; ð17Þ

where q is the wave number and

tq ¼1=G

4pja � 1j 1þ q2r2c=4� : (18)

Eq. (17) can be rewritten as follows.

IqðoÞ ¼2kBT

ow00ðo; qÞ; (19)

where the imaginary part of the dynamic suscept-ibility is given by

w00ðq;oÞ ¼ wðq; 0Þotq

1þ ðotqÞ2; (20)

and the static q-dependent susceptibility is givenby

wðq; 0Þ ¼1

4pjða � 1Þj 1þ q2r2c=4� : (21)

The condition ðotqÞ2� 1 leads to the intensity

of magnetisation fluctuations (Eq. (17)), which

does not contain critical singularities. This casedescribes the dynamics of precursor clusters of theordered phase, the profile of which is given by Eq.(14) [15]. When the condition ðotqÞ

2 1 is

fulfilled, critical singularities in the phase-transi-tion dynamics are revealed. The temperaturedependence of the helicon damping caused byinstability of an electron gas at the diamagneticphase transition should be related to the dampingof the intensity of magnetisation fluctuationsdetermined by 1=tq: Under conditions of experi-ment [8] q ¼ p=L; where L is the sample widthequal to 1.2mm for the two used samples, andhence the condition q2r2c=4 1 is fulfilled for longwave lengths. Consequently, the damping isproportional to (a–1). Then the comparison withexperimental data made in Ref. [9] is justified andgives a good agreement between theory andexperiment for the temperature dependence ofthe helicon damping.Size-dependent effects on the phase transition

are not significant under conditions of experiment[8] because the samples are thick. The minimalsample-width Lmin; beyond which the Condonordering becomes unstable, is determined byLmin ¼ 9rc=2ðlTcÞ

3=2 [16]. In the magnetic fieldused in Ref. [8] the critical size Lmin ¼ 1:3mm; i.e.,Lmin=L � 10�3: Size-dependent effects in all staticand dynamic phenomena in aluminium would beappeared at a sample thickness starting fromabout 10 mm.

5. Summary

We considered some critical phenomena neardiamagnetic phase transitions and in the domainphase. We discussed a possibility of the soft-modeexistence and static critical effect in susceptibilityrelated to appearance of Condon domains. Theconstructed phase diagram and the temperaturebehaviour of the susceptibility give the value of thephase-transition temperature observed in alumi-nium. We presented the temperature dependenceof the magnetisation in each domain being theorder parameter of the diamagnetic phase transi-tion and confirming the mentioned phase-transi-tion temperature. We calculated the spectral

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A. Gordon et al. / Physica B 353 (2004) 296–304304

density of excitations at diamagnetic phase transi-tions in order to explain the critical growth of thehelicon damping observed in aluminium. Thetemperature dependence of the damping coincideswith that measured in aluminium.

Acknowledgements

We are indebted to I.A. Itskovsky and V.S.Egorov for useful discussions. We express ourdeep gratitude to P. Wyder for his interest in thiswork and his permanent inspiring influence on thisfield of research. One of us (A.G.) thanks theCentre for Computational Mathematics andScientific Computation of the University of Haifafor support.

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