Local modes, phonons, and mass transport in solid $^4$He

10 Pages·0.2 MB·English

by  N. Gov

Local modes, phonons, and mass transport in solid 4He.
N. Gov and E. Polturak
Physics Department,
Technion-Israel Institute of Technology,
Haifa 32000, Israel
Weproposeamodeltotreatthelocalmotionofatomsinsolid4Heasalocalmode. Inthismodel,
thesolidisassumedtobedescribedbytheSelfConsistentHarmonicapproximation,combinedwith
anarrayoflocalmodes. Weshowthatinthebccphasetheatomiclocalmotionishighlydirectional
and correlated, while in the hcp phase there is no such correlation. The correlated motion in the
bcc phase leads to a strong hybridization of the local modes with the T1(110) phonon branch,
9 which becomes much softer than that obtained through a Self Consistent Harmonic calculation,
9 in agreement with experiment. In addition we predict a high energy excitation branch which is
9 importantforself-diffusion. Boththehybridizationandthepresenceofahighenergybrancharea
1 consequenceofthecorrelation, andappearonlyinthebccphase. Wesuggestthatthelocalmodes

n can play the role in mass transport usually attributed to point defects (vacancies). Our approach
a offers a more overall consistent picture than obtained using vacancies as the predominant point
J defect. In particular, we show that our approach resolves the long standing controversy regarding
8 the contribution of point defects to thespecific heat of solid 4He.

2 PACS: 67.80-s,67.80.Cx,67.80.Mg
v
I. INTRODUCTION bythespecificheatdata. Severalattemptsweremadeto
3
2 reconcile this controversy by using the assumption that
4 Atomic motion in solid 4He was extensively studied vacancies in solid 4He are delocalized and occupy an en-
2 over the years, both theoretically and experimentally1. ergy band13,12,8. In this approach,the magnitude of the
1 The quantum nature of the solid comes into play mainly specific heat is divided by the bandwidth, so that it can
8
through the large amplitude of zero point motion of the be made small through the choice of a suffficiently large
9
/ atoms in the lattice, an effect especially evident in the bandwidth. However,inordertoobtainarealisticmagni-
at solidatits lowestdensity. Inthis paper, we shalldiscuss tudeofthespecificheatonehastochoseanunphysically
m only these low density solids ( 21 cm3 molar volume). largebandwidth, sothatthis approachisunsuccessfulin
- Calculationsofthe phononspe∼ctrumwithinthe selfcon- settling the controversy8.
d sistentharmonicapproximationseemto agreefairly well In this paper, we propose a somewhat different de-
n with experiment in the hcp phase2, while in the bcc scriptionof the solid, one which treats the motion of the
o phase, only the energy of the T (110) phonon is over- atom inside it’s potential-well as a ”local mode”. Lo-
c 1
: estimatedbyafactoroftwo3. Correctionsresultingfrom calmodesasexcitationsinanharmoniccrystalsweredis-
v the localmotionofthe atomsduetoquantumeffectsare cussedpreviouslybySieversandTakeno14,andbyFlynn
Xi treated using correlated basis wave functions1. A prob- and Stoneham15. We show below that the other degrees
lemwhichsofarwasnotaddresedwithinthesemodelsis of freedom are largely decoupled from this local motion.
r
a that of mass transport, arising from atomic exchange at As such, this mode can be assumed to be an indepen-
low temperature. At higher temperatures, this exchange dentdegreeoffreedomofthesolid. Theexcitedstatesof
is assumed to occur with thermally activated point de- thislocalmodecanbethermallyactivated,andseemsto
fects, traditionally taken as vacancies. With point de- reproduceallthephysicaleffectsusuallyattributedtova-
fects, some additional complexity is to be expected, as cancies. Moreover,we showthatthis picture hasnoneof
here one deals with objects of a size similar to that of the inconsistencies connected with vacancies. The prop-
the zero point amplitude of the atom. The existence erties of the local modes are different in the bcc and
of thermally activated non-phononexcitations is well es- the hcp solid phases. In the bcc phase, we show that
tablished experimentally by several different techniques, itisenergeticallyadvantegousforthelocalmotionofthe
suchasNMR4,5,6,7,massdiffusion8,9,ionmobility10,and different atoms to be correlated. The new ground-state
X-ray diffraction11. However,the properties of point de- has several new features, among them a softened trans-
fects deduced from these various experiments have so verse phonon mode and a high-energy excitation branch
far led to controversialconclusions. For example, taking which opens an additional channel for atomic diffusion.
theexperimentallydeterminedactivationenergiesatface In contrast, the triangular symmetry of the hcp lattice
value, the calculateddensity of vacanciesis so largethat frustratesthe possibilityoflong-rangeorderingofthe lo-
their contribution to the specific heat should be compa- calmodes,which thereforeremainuncorrelated. In both
rable to that of the phonons12,8. This is not borne out phases the local modes are highly directional due to the
anisotropic potential seen by an atom in the crystal, a
1
constraint which we show leads to a small contribution K.E. 34K, V 40K, E 6K.
total
h i≃ h i≃− h i≃−
to the specific heat of the solid. Based on the above calculation, we shall assume that
the atoms have a local-mode that is highly directional
along one of the directions equivalent to (100), and of
II. LOCAL MODES IN BCC 4HE energy 6 10K.Experimental evidence for the existence
−
of such a ”local mode” comes from NMR measurements
Calculationsoftheground-stateenergyofbcc4Heusu- ofthelinewidth(1/T2)inbcc3He-4Hemixturecrystals5.
ally employ variational wavefunctions that account for Themotionalnarrowingofthe NMRresonanceline with
the short-range correlations between the atoms1. Since temperature shows thermally activated behaviour with
we want to focus here on the nature of the local motion anactivationenergyof 7 1K5,4. At temperatures above
±
of the atom inside the lattice, we will take a different 1K, the NMR line in the solid becomes narrower than
approach. In this approach we would like to isolate the that of the liquid, indicating that atomic motion in the
lowest energy excited state of the atom inside its poten- solid is faster than in the liquid. At the same time, the
tial well, and treat it as a local excitation of the lattice. diffusion coefficient of the solid remains several orders
This excited state consists of a local oscillatory motion of magnitude smaller than that of the liquid5, indicating
of the atom along a particular direction and produces thatthisrapidmotionisofalocalnature. Weproposeto
an oscillating electric dipole, similarly to the usual Van- identify this rapid motion as associatedwith the excited
der Waals interaction. Unlike in the case of the Van-der stateofanatominthe well,withanactivationenergyof
Waals interaction, in which the dipolar fluctuations are 7 1K.
±
random,weshowthatinthebccsolidtheselocaldipoles Thehighlydirectionalmotionofthe atomleadstothe
are correlated and a new ground-state of lower energy creation of a local (oscillating) electric dipole in the di-
is created. We therefore begin by investigating the po- rectionofthemotion. Thislocalelectric-dipoleiscreated
tential well of an atom in the bcc lattice. This can be duetoinertia,astheelectroniccloudcanbethoughtofas
done in the following way: Using the standard helium being slightly displaced relative to the ion. This electric
pair-potential υ(r)1 and taking the atoms as stationary dipoleduetomixingofthe s and p electronic-levelsof
| i | i
we can map the potential-well near an atom along any the 4He atom. The amount of mixing can be estimated
direction in the lattice. In Fig.1 we plot this potential from perturbation theory as
along the main directions (100),(110) and (111). It is
ψ = s +λ p E ψ E ψ s E s λ2 p E p
clear that in the directions normal to the cube’s faces | i | i⇒ 0 ≃h | | i−h | | i≃ h | | i
(i.e. (100),(010)etc.) the confining potentialwellis very λ2 7/2.46 104 0.00284,λ 0.0168 (1)
⇒ ≃ · ≃ ≃
wide with a pronounced double-minimum structure.
where s and p stand for the ground-state and first
We also plot in Fig.2 the lowest two energy levels of a excited-st|aite ofth|ei4He atom, λ is the mixing coefficient
one-dimensional Schrodinger’s equation for a 4He atom
and p E p 2.46 104 K is the excitation energy of
solved in each of the potential wells. The energy differ- the fihrst| a|toim≃ic excite·d-state16. The estimated mixing is
encebetweenthesetwolevelsislowestinthe(100)direc-
small and the magnitude of the induced dipole moment
tion,oftheorderof10K. Theatomicdisplacementisde-
is therefore
scribedbyamixingbetweenthe lowesttwoenergylevels
inthepotentialwell,whichcorrespondstoamotionwith µ =e ψ x ψ 2eλ s x p e 0.03˚A (2)
amplitude of 1˚A (in the (100) direction). Although | | h | | i≃ h | | i≃ ·
the above trea∼tment assumes that other atoms are sta- where s x p 0.9˚A . The estimation of the mix-
h | | i ≃
tionary,it indicates the directions along which there will ing λ and the dipole-moment µ serves to set an upper
| |
be large amplitude local motion in the solid. It is evi- bound on the magnitude of this effect. Since the atoms
dent that the motion will be highly directional, with the possess an oscillatory electric-dipole moment they have
largest amplitude along the (100) direction, while in the long-rangedipole-dipole interactions. The instantaneous
other directions the amplitude is much smaller. Allow- dipolar interaction energy is given by
ing other atoms to move as well, so that they get out of
3cos2(µ (r r )) 1
erpeaodcthuencoettiahthleerw’svilawlluaeeyff,eoccftaitnvheeolynelnybeerscgooyfmteleenvesthhlaedlilpffooewtreeernn.tcieaF,loswrineecllxeaatmnhde- Edipole =−|µ|2Xi6=0" |r·0−0r−i|3 i − # (3)
ple, a calculation which allows opposite atoms to move where the sum is over all the atoms in the lattice, ri
slightly, reduces the energy difference in the (100) direc- beingthe instantaneouscoordinateofthei-thatom. For
tion to almost 6K. oscillating dipoles with random phases, the average in-
In order to obtain meaningful numerical values, we stantaneous interaction energy summed over the lattice
scaled the potential υ(r) so that the calculated one- would be zero. However, the energy of a dipolar array
dimensional potential energy and kinetic energy, aver- canbe made lowerby correlatingthe phases ofthe oscil-
aged over the different directions, reproduce the known latingatoms. Thelowestinteractionenergyarrangement
kinetic, potential and total energy of the bcc phase1: of the dipoles in the bcc lattice is such that they are os-
cillating with the same phase. Since the direction of the
2
dipole shows the instantaneous direction of the motion to a single phonon mode, justifying our assumption that
or displacement, such a state is just a uniform motion the local modes can be treated separately to a good ap-
or translation of the entire lattice. This arrangement is proximation. Thus, the only elementary excitations of
therefore unphysical, and we have to look for symmet- the dipole array would be in the (110) direction, in the
ricarrangementswithrespecttothenumberofup/down form of the T (110) phonon. We shall now calculate the
1
dipoles. The two arrangements shown in Fig.3 are the dispersion relation of such an excitation by a mean-field
two’antiferroelectric’configurationsalongthesymmetry solution of an effective Hamiltonian.
axesofthe crystalwith individualdipolesorientedalong TheHamiltoniantreatmentofinteractinglocalexcita-
the (001)direction, and a zero totaldipole moment. For tionswasdevelopedoriginallybyHopfield18fortheprob-
thesetwophysicalpossibilitiesthesumin(3)withaunit lemofexcitons ina dielectric material. The localexcita-
dipole is given in Fig.3. Thus, the ground state in our tionsaretreatedasbosonsandtheeffectiveHamiltonian
picture has the atoms executing this local oscillation in describing their behavior is19
a correlated fashion, as shown in Fig.3a along one par-
ticular direction. Similar correlated motion exists in the H = (E +X(k)) b †b + 1
loc 0 k k
other two orthogonaldirections. 2
k (cid:18) (cid:19)
X
+ X(k) b †b† +b b (5)
k −k k −k
III. ELEMENTARY EXCITATIONS OF THE Xk (cid:16) (cid:17)
DIPOLE GROUND-STATE
whereb †,b arecreation/anihilationoperatorsofthe lo-
k k
cal mode, and X(k) is the interaction matrix element
The ground state of the dipoles described in the pre-
given above (4). The Hamiltonian H (5) which de-
ceding section will be affected by the excitations of the loc
scribes the effective interaction between localized modes
lattice, namely phonons. In fact, our basic assumption
canbediagonalizedusingtheBogoliubovtransformation
in which the local motion can be separated from these
β =u(k)b +v(k)b† . Thetwofunctionsu(k)andv(k)
other degrees of freedomneeds justification. The oscilla- k k −k
are given by:
tory atomic motion induced by the phonons will modu-
late the relativephases ofthe dipoles. Let us look atthe
1 E +X(k) 1 E +X(k)
ground state of the dipoles concentrating on oscillations u2(k)= 0 +1 ,v2(k)= 0 1
2 E(k) 2 E(k) −
oriented along the (001) direction. We now need to con- (cid:18) (cid:19) (cid:18) (cid:19)
sideronly phononswhichwill modulate the localmotion (6)
alongthisdirection. Inthebccstructure,only3phonons
fulfillthiscondition: L(001),T(100)andT (110). Letus Where E(k), the energy spectrum of the diagonalized
1
calculatetheenergyofthedipolararraywhenmodulated Hamiltonian is:
along these 3 directions. For a modulation along some
directionk ,the dipolarinteractionenergyis givenby17: E(k)= E0(E0+2X(k)) (7)
p
3cos2(µ (r r )) 1 The ground-state wavefunction of the local-modes is
X(k) = µ2 · 0− i − given by20:
−| | Xi6=0" |r0−ri|3 #
v
exp[2πik·(r0−ri)] (4) |Ψ0i= exp ukbk†b†−k |vaci (8)
k (cid:18) k (cid:19)
Atk =0theinteractionmatrixX(k)isjustthedipolar Y
energy (3). WeshowinFig.4X(k)inthe(110)directionforaunit
In Fig.4 we plot the value of X(k), the energy of the dipolemoment. Wenowneedtofixthesizeofthedipole
dipolararraymodulatedbytherelevantphonons:L(001), moment µ in order to calculate the energy spectrum.
| |
T(100), and T1(110), for dipole moment µ =1. We see We would like, according to our definition ot the local
| |
that for a modulation by L(001) and T(100) the peri- mode, that the energy cost of flipping the direction on
odicity of X(k) is over a full unit-cell, that is twice the a single dipole out of the ground state to be E . This
0
periodicity of these phonons. Since symmetric functions conditionisequivalenttodemandingthat2 X(k=0) =
| |
of periodicities π/a and 2π/a are orthogonal, the wave- E . We also see from (7) that in order for the dipoles
0
functions of the phonons and of the dipole-excitations to have a gapless mode at k 0 we must have that:
→
are orthogonal along these directions. The dipole array X(k = 0) = E /2. Using this condition, the value of
0
−
cannot therefore be excited by any phonon along these E = 7K (see previous section) determines the size of
0
twodirections. RegardingthemodulationbytheT1(110) the dipole-moment as: µ e 0.01˚A. This value is
| | ≃ ·
mode, here the periodicity of X(k) is the same as that indeed smaller than our previous estimation, which was
of the T1(110) phonon, which can therefore couple to an upper bound on the size of dipole moment (2).
the dipole array. We conlude therefore that the coupling From its very definition, the phase modulation in the
of the local modes to the lattice excitations is limited (110) direction of the atomic motion with energy E(k)
3
(Eq.7) should be just the T (110) phonon. In Fig.5 we where a are Bose operators and ε(k) is the SCH-
1
compare the experimental values of T (110) taken from phonon spectrum.
1
neutron scattering with the calculated E(k). The agree- The local-mode
ment is excellent for all k. From (7) and Fig.4 we find
that at the edge of the Brillouin zone the energy E(k) Hl0oc = E0bk†bk (11)
of the phonon is just the bare energy of the local mode, k
X
E , since X(√2π/a) = 0 and the dipoles have changed
0 The part describing the first order (dipolar) coupling
fromtheconfigurationofFig.3atoFig.3b. Werecallthat
between the phonons and the localized modes is 18
the value ofE that we usedwas takenfrom NMR data.
0
The agreement between these two independent determi-
H = (λ(k)b +µ(k)a )(a †+a ) (12)
nations of E , that from NMR and that from neutron c k k k −k
0
k
scattering,emphasizestheself-consistencyofourdescrip- X
tion. We stressthatthe valueofE0 is the onlyempirical λ(k)bk† µ(k)ak† (ak+a†−k)
− −
input into the calculation, while the functional behav-
(cid:16) (cid:17)
ior is completely defined by the lattice structure and the where, in the dipolar approximation (and a cubic lat-
dipolar form of the interactions. tice) the two functions λ and µ are given by λ(k) =
woOuludr bmeoddiefflerinendticathteasntohbattaionneldyftrhoemTth1(e11S0C)Hphcoanlcoun- iE0 −32Xǫ((kk)) 12 and µ(k) = −E032Xǫ((kk)), with X(k) the
lation, because it is the only excitation which couples dipo(cid:16)le matri(cid:17)x element (4). This is just the standard
to the local motion. Indeed, in the experiment3 this is coupling Hamiltonian of an atom to a transverse pho-
ton field, which is replaced here by the T (110) SCH
the only phonon branch which is not described well by 1
phonon. ThetotalHamiltonianH doesnotinvolvequar-
the SCH calculation. The fact that the SCH calculation
tic terms and can be diagonalized using the canonical
works rather well for other directions, is consistent with
transformation18
our picture in which there are no elementary excitations
of the dipole array in these directions.
α =Aa +Bb +Ca† +Db†
1 k k −k −k
α =Ba +Ab +Da† +Cb† (13)
2 k k −k −k
IV. HARMONIC AND LOCAL-MODE
where these operators describe the two branches of
HYBRIDIZATION
the hybridized energy spectrum. The transformation
functions A(k),B(k),C(k),D(k) can be written down
An equivalent way to describe the mutual influence
explicitly18. The corresponding dispersion relation is
between the local modes and the lattice can be done by
taking the interaction between the dipoles as resulting ǫ2(k) 6 X(k)
fromanexchangeofvirtualharmonictransversefluctua- =1 (14)
tionsofthelattice. Ananalogouscase,thatofexcitonsin E2(k) − E01 E(k) 2
− E0
a dielectric, was treated by Hopfield18 (in that case, the
(cid:16) (cid:17)
interaction is mediated through the exchange of virtual which describes two energy branches E and E . The
1 2
photons). The two excitations, the local mode and vir- equivalenceofthisandthetreatmentintheprevioussec-
tual harmonic phonon, are then hybridized through the tion is due to the use of the same dipole interaction ma-
same dipolar interaction matrix X(k), which we used in trix X(k) in both. With X(k), we can solve (14) to find
the direct interaction picture. In our case, the natural the two energy branches:
choice for the mediating virtual phonon is the T (110)
1
phonon, as calculated by the SCH method. The use of E1 =ε(k)/2
the phononcalculatedby the SCH method is important, E =2E (15)
2 0
since this calculation is largely independent of the local
degrees of freedom described by the local mode. The We see that the energy of the lower branch, E1 is half
motivation for using this approachis that it allows us to ofthatoftheSCH-phononforallk. ComparingtheSCH
obtain an additional branch of the excitation spectrum calculation and the experimental results for the T1(110)
which has observable consequences. phonon, we find that there is indeed a constant, k in-
The Hamiltonian describing the local-mode and SCH- dependent ratio between them. This ratio turns out to
phonon hybridization is18 be 1.7 for all momenta, close to the predicted ratio of
∼
2 (15) (Fig.6). This small discrepancy is within the un-
H =H +H0 +H (9)
0 loc c certainty of the SCH calculations and the experimental
data. Despite this small discrepancy we find the agree-
where the three components of (9) are:
ment very satisfying. Our picture of a hybridization of
The SCH-phonon
a dipole array with the harmonic lattice can therefore
H = ε(k)a†a (10) account for the main effect of the local motion. The up-
0 k k
perbranch,E ,adispersionlessexcitationwithenergyof
k 2
X
4
14K (Fig.6), was possibly observed in inelastic neutron VI. ATOMIC SELF-DIFFUSION
scattering experiment21, in which a strong feature was
observed at an energy transfer of 1.4meV. This feature We have seen that in the first excited state of the
was interpreted by Glyde1 in terms of an interference atom in its potential well, the atom oscillates with a
effect between phonon modes, which is not inconsistent comparatively large amplitude ( 1˚A). This motion
with our hybridization picture. Another possible way to can have an effect on the atomic ∼exchange rate which is
observethisexcitationwouldbebyRamanscattering. In governed by the overlap of the wave functions of neigh-
addition, this energy branch is important in the process bouring atoms. At low temperature, the atomic ex-
ofmassdiffusion, andwillbe discussedinthis contextin change,orselfdiffusion,occursbytunnelingandissmall
a following section. ( D 10−9cm2/sec in the hcp phase8). Thermally ac-
0
We would like to point out that local modes can tivate∼d diffusion is traditionally attributed to vacancies.
arise also in classical crystals, as a result of a large Wewouldliketoshowthatthermallyexcitedlocalmodes
anharmonicity14. As shownby these authors,these local can produce self diffusion at rates which compare favor-
modes can assume many of the roles of vacancies. There ably with experimental data.
are severalsimilarities between this work and ours; first, Thetheoreticalcalculationofthezero-temperatureex-
the treatment in non-perturbative; second, in a simple change rate of atoms in solid Helium is a long-standing
cubic lattice, they also find that there are two excitation problem23. It is extremely difficult to perform accurate
branches. These similarities occur despite the different theoreticalcalculationsofthiseffectduetothesmallness
basic assumptions of the classicalmodel and the present of the exchange frequency as compared with the Debye
work, which uses the quantum properties of the crystal frequency. In these calculations, the atoms participating
as the starting point. in the exchange are treated as tunneling along a one-
dimensional closed path. At high densities, it is found
thattheeffectivepotentialbarrierfortunnelingismainly
V. LOCAL MODES IN HCP 4HE
duetothepotentialenergyassociatedwiththeelasticde-
formation of the crystal during the atomic exchange. At
We now turn to discuss the effects arising from the lower densities it is more difficult to calculate the height
local motion in the hcp structure. Repeating the calcu- of the tunneling barrier, but empirically it turns out to
lation done in section II, we plot in Fig.7 the potential be of the order of the kinetic energy of the atoms (30K
wellalongthe maindirectionsofthe hcpcrystal,namely to40K).Thus,inordertoobtainaroughestimateofthe
(1000),(0100)and (0001). We also plot in Fig.7 the low- exchangerate,itissufficienttoviewtheproblemasthat
est two energy levels of a one-dimensional Schrodinger’s ofaone-dimensionaltunnelingoffreeparticlesthrougha
equation solved in each of the potential wells. The en- squarebarrier. Sincewe’redealingwithlowdensitysolid,
ergy difference between these two levels is lowest in the the height of the barrier should be comparable with the
(0100) direction, of the order of 16K. Similarly to what kinetic energy of the atoms. We would like to compare
wasstatedforthebccstructure,thecalculationindicates the exchangerateofatomsinthe groundstatewiththat
the direction along which there will be large amplitude of atoms in the excited state of the local mode.
local motion. We begin with the hcp phase, looking at the (0100)
As in the bcc phase, evidence for the existence of a direction, that of the largest amplitude atomic motion.
”local mode” comes from NMR measurements6,22 of the We take the atom in the ground-state with it’s kinetic
linewidth (1/T2) with energy 14K. We propose there- zero-point energy (Ekin 10K) and adjust the bar-
≃ ≃
fore that the NMR experiments measure the energy of rier height to reproduce the experimental rate of ex-
the bare local mode. change at low temperatures. We take the width of the
In contrast to the bcc phase, we do not expect the lo- barrier to be a =3.6˚A, i.e. the nearest neighbour dis-
cal electric-dipoles in the hcp phase to have long-range tance. We find that a barrier height of 44K gives a
order. Thisisduetothefactthatthereisgeometricfrus- transmission probability Γ 10−7, and a self diffusion
tration against an ’antiferroelectric’ order in a triangu- coefficient DG (E /¯h)a≃2Γ 10−9cm2/sec , which
0 ≃ kin ∼
lar lattice. We calculated the dipolar interaction energy is consistent with the experimental results. The bar-
forseveralsimpledipolearrangementspreservingthenet rier height found in this estimation is consistent with
zerodipolemoment,anddidnotfindanyarrangementin the empirical value, of the order of the total kinetic en-
which this energy was negative. Comparisonof the SCH ergy of the atoms in the solid. We now turn to ther-
calculationofthephononsspectrumwithexperimentre- mally activated self diffusion. In the first excited state
veals that there is an overall good agreement, with no of atoms in the well, the total energy of the atomis 24K
exceptions, such as found for the T1(110) phonon in the (10K from the ground state + 14K for the first excited
bccphase. Thus,weconcludethatthereisnolongrange level of the local mode). The barrier height remains
order within the dipole array like in the bcc phase, and unchanged, but the atom can now tunnel from one of
the localmodesinthe hcpsolidarelargelyuncorrelated. the lobes of the excited-state wavefunction (see Fig. 2),
which are approximately 1˚A from its equilibrium posi-
5
tion. Since it tunnels into an identical excited state, the VII. CONTRIBUTION OF LOCAL MODES TO
effective width of the barrier is now a=3.6-2=1.6˚A. We THE SPECIFIC HEAT
thus find a transmission probability in the excited state
Γ 10−2, and a prefactor of the diffusion coefficient One of the controversial issues with vacancies in solid
DE0≃≃ (Ekin/¯h)a2Γ ∼10−4cm2/sec. This means in our 4Heisthatbasedonthemeasuredvaluesoftheactivation
model that as atoms are thermally excited out of the energy, their estimated contribution to the specific heat
ground state, the diffusion coefficient would increase ex- is comparable to that of the phonons12,8. Yet, there in
epxopneernitmiaelnlytawl ivtahluteesm8paerreatDuE0re≈as1D0−E04ecxmp(2−/sEec0,/kaTnd).tThhaet ntroibeuxtpioenrimofetnhtealloecvaidlemnocdeefsorprite.seLnettedushecroentsoidtehrethspeeccoifinc-
of the activation energy E0=13.9K, which is very close heat.
to the value of E0 =14K we took from NMR as describ- Since the energy of the local mode is different in each
ing the energy of the local mode. We therefore conclude direction of the lattice, the highest possible contribution
that exchange of atoms occupying a thermally activated to the specific heat would come from the excitation of
localmodecanaccountforselfdiffusionatarateusually localmodesalongthedirectionswheretheexcitationen-
attributed to vacancies in the hcp phase. ergy is lowest. Because of the restricted solid angle in
Therateofselfdiffusioninthebccphaseisanorderof whichthesemodesareactive,thefractionofphasespace
magnitude larger than in the hcp phase8. In this phase, occupied by them would be correspondingly small, and
the directionoflargestlocalatomic motionis (100). Re- their contributionto the specific heatwouldbe muchre-
peating the above calculation of the tunneling rate for duced compared with that obtained previously. In these
thiscase,wehavetheground-statekineticzero-pointen- previous estimations, the point defect was assumed to
ergy similar to the hcp (Ekin 10K), and barrier width occupy phase space uniformly, thus yielding a large con-
along the (100) direction of a≃=4.12˚A. For a tunneling tribution. Toobtainaquantitativeestimateoftheabove
rateatlowtemperatureswhichwouldbehigherbyanor- effect we calculate the ratio between the contribution to
derofmagnitudecomparedwiththehcpphaseweneeda thespecificheatfromdirectionallocalmodesandtheto-
barrierheightof 30K,which is againclose to the total talexperimentalspecificheatofthebcc24andhcp25solid
∼
kineticenergyofthe atominthe ground-state( 34K)1. phases. Weusedtheenergiescalculatedabovefortheex-
∼
In order to look at thermally activated self diffusion, the citationenergiesalongthe principaldirectionsofthe bcc
modelusedaboveforthehcpphaseisinapplicable,since and hcp lattice, and a simple linear interpolation for the
the energy of the excited state E2 =2E0 14oK (15) is excitationenergiesintheintermediateorientations. This
≃
associated with more than one atom. In order to under- ratio is plotted against temperature in Fig.10. For the
standthe physicalnatureofthisexcitationweplotnb(r) hcp phase, the maximum contribution is less than 1%,
(Fig.8),thespatialextentofthedensityofatomsexcited while the bcc phase it is 5%. The decrease of this ra-
by this branch of the local-modes in the (110) direction. tio near T=1.65K is due t∼o the premelting enhancement
This is the Fourier transform of the k-space density of of the total specific heat24. These values fall within the
the localized-modes in the upper branch (13): error bars of the specific heat data, and explain why no
thermally activated contribution is seen in solid 4He.
n (k)= b†b = C(k)2 (16)
b k k 2 | |
D E
WhatisseeninFig.8isthatthefunctionnb(r)extends VIII. CONCLUSION
over two unit cells in real space, in the (110) direction.
Since the energy is twice the energy of the bare local-
Inthisworkwehaveproposedanewapproachtotreat
mode, it can be interpreted qualitatively as two atoms thelocalbehaviourof4Heatomsinthebccandhcpsolid
excited in adjacent unit cells, each having an energy
phases. We treat the excitations of atoms inside their
E =7K. This corresponds to a ”flip” of the dipole mo-
0 potential well as local modes. The anisotropy of the
mentofthesetwoatoms. InFig. 9weshowthatwiththis
potential renders these modes highly directional. Due
excitation,thereare4atomsoscillatinginsuchaway,as
to the symmetry of the bcc phase we propose that the
to reducethe potentialbarrierforthe exchangeofatoms
local-mode is hybridized with the harmonic density fluc-
1and2. Thistype ofatomicexchangeresemblesphonon
tuations (SCH). The hybridization is described by the
assistedtunneling,wherethe correlated”breathing”mo-
dipole-dipole interaction and the spectrum of the hy-
tionofthe4atomsislocallyequivalenttoaphononexci-
bridized T (110) phonon is calculated. An additional
1
tationatthe edgeofthe Brillouinzone. Thistype ofself
excitationbranchisidentifiedanditisthisbranchwhich
diffusion was considered in the past15, and was recently
seems to control the anomalously large self-diffusion in
found to be consistent with experiment in bcc 4He9 as
the bcc solid. In the hcp phase, the symmetry does not
thedominantmasstransportchannel,withanactivation
allowforcorrelationsoftheselocalmodes. Consequently,
energy of 14.8K8,9. Thus, altough quantitative calcula-
thereisnohybridizationwiththephonons,andthether-
tions of the rate of difusion are outside the scope of this
mally activated self diffusion in this phase is controlled
paper,boththemechanismandtheactivationenergyare
by the energy of the bare local mode.
in accord with experiment.
6
The directionality of the localmodes means that their Sons,Inc.
contributiontothe specific-heatofthesolidisnegligible. 21E. B. Osgood, V. J. Minkiewicz, T. A. Kitchens, and G.
Wethereforedemonstratedthatthelocalmodeapproach Shirane, Phys. Rev. A5,1537(1972).
can describe experimental data coming from neutron- 22I.Schuster,E.Polturak,Y.Swirsky,E.J.SchmidtandS.G.
scattering, NMR and diffusion experiments within one Lipson, J. Low Temp. Phys. 103, 159 (1996).
physical model, while at the same time resolving a long- 23M.Roger,J.H.Hetherington,andJ.M.Delrieu,Rev.Mod.
standing discrepancy concerning the specific heat con- Phys. 55, 1(1983).
tribution of point defects. The physical picture of our 24J. K. Hoffer, W. R. Gardner, C. G. Waterfield, and N. E.
Phillips, Jour. Low Temp. Phys. 23, 63(1976)
proposed local mode is very different from the classical
25W.R.Gardner,J.K.Hoffer,andN.E.Phillips,Phys.Rev.
picture of a vacancy. We therefore see no reason to con-
sider point defects in solid 4He as vacancies, as they can A7, 1029(1973).
be consistentlytreatedasnaturalexcitationsofthe solid
lattice, without having to physically remove atoms from
it.
Acknowledgements
This work was supported by the Israel Science Foun-
dationandbytheTechnionVPRfundforthePromotion
of Research.
1H.R.Glyde, ’Excitations in Liquid and Solid Helium’,Ox-
ford Series on Neutron Scattering in Condensed Matter,
1994.
2V. J. Minkiewicz, T. A. Kitchens, F. P. Lipschultz, R.
Nathans, and G. Shirane, Phys. Rev. 174, 267(1968).
3V. J. Minkiewicz, T. A. Kitchens, G. Shirane, and E. B.
Osgood, Phys. Rev. A8, 1513(1973).
4A. R. Allen, M. G. Richards, and J. Schratter, Jour. Low
Temp. Phys. 47, 289(1982).
5I. Schuster, Y. Swirsky, E.J. Schmidt, E. Polturak, and
S.G. Lipson, Europhys. Lett. 33, 623(1996).
6D. S. Miyoshi, R. M. Cotts, A. S. Greenberg, and R. C.
Richardson, Phys. Rev. A2, 870(1970).
7V.N. Grigor’ev, Low. Temp, Phys. 23, 2(1997).
8N. E. Dyumin, N. V. Zuev, V. V. Boiko, and V. N.
Grigor’ev, Low Temp. Phys. 19, 696(1993).
9I.Berent,andE.Polturak,Phys. Rev. Lett.81,846(1998).
10A.J.Dahm,inProgress inLowTemp.Phys.D.F.Brewer,
ed.,vol. X, p.73, North Holland(Amsterdam,1986).
11R.O. Simmons, J. Phys. Chem. Solids, 55, 895(1994).
12C. A. Burns and J. M. Goodkind, Jour. Low Temp. Phys.
95, 695(1994).
13A. F.
Andreev, in Progress in Low Temp. Phys. D. F. Brewer,
ed.,vol. VIII,p.69, North Holland(Amsterdam,1982).
14S. Takeno and A. J. Sievers, Sol. State Commun. 11,
1023(1988).
15C. P. Flynn and A. M. Stoneham, Phys. Rev. B1,
3966(1970).
16H.E. White, ’Introduction to Atomic Spectra’, McGraw-
Hill, New York (1934).
17W. R.Heller and A.Marcus, Phys. Rev. 84, 809(1951).
18J.J. Hopfield, Phys. Rev. 112 (1958) 1555.
19P.W. Anderson,’Concepts In Solids’, 1963, p.132-148.
20K. Huang, ’Statistical Mechanics’,1987, John Wiley and
7
50
(a)
30
al (K) 10 E0
enti −10
ot
P
−30
(111)
−50
−10
al (K) −20 E0
enti −30
ot
P
−40 (b)
(110)
−50
−10
K)
al ( −20
enti −30 E
Pot 0
−40
(100)
−50
−2 −1 0 1 2
O
Distance from equilibrium (A)
FIG.1. Thepotential-wellofanatominbcc4Healongdif-
ferent directions. The energy difference E0 between the low-
est two energy levels (dashed lines) are: (111)- 59.5K, (110)- FIG. 3. The two ’anti-ferroelectric’ arrangements of the
27.6K, (100)- 10.6K. dipoles lying along the major axes of the bcc phase. The
atomshavingthesamedipolemomentshavethesameshade.
The sum of the dipole-dipole interaction (Eq.3) for a unit
0.6 dipole-mo ment are: (a) -0.08, (b) 0.0 (˚A−3).
0.4
−1oΑ)
2 ψ|(
|
0.2
00
−10
K) −20
al (
nti
e
ot −30
P
−40
−50
−2 −1 0 1 2
Distance along (100) direction Ao
FIG.2. Theprobability distribution in the(100) potential
well(showninthelowerpart). Thegroundstatewavefunction
isthesolidlinewhilethedashedlineisthefirstexcitedstate.
8
0 FIG. 5. The experimental data [3] (solid circles) for the
T1(110) phonon compared with the calculation (Eq.7) (solid
L(001)
−1 line). Also shown is the energy of the bare local-mode
K) E0=7K.
k) ( −2
X(
−3
15
−E/2
0
−40 0.5 1.0 π/a E2=2E0
0
−1 T(100)
K) SCH
k) ( −2 10
X(
−3
−E/2 K)
0−40 0.5 1.0 π/a ogy (
0 ner E0
E
−1 5 E1
K)
k) ( −2
X(
T(110)
−3 1
−E/2
0
−4
0 0.5 1.0 π/a’
Wavevector Ao−1
0
FIG.4. The calculated interaction matrix X(k) (Eq.4) as 0 0.5 1
a function of the wavevector k, for the three phonon modes Wavevector Ao−1
thatcouldaffectadipolararray. Thedipolemomenthasbeen FIG. 6. The SCH calculation [1] (dashed line) compared
normalized to give a gapless mode: X(k=0)= E0/2. The withtheexperimentalresults[3](solidcircles)fortheT1(110)
unit-cell dimensions (˚A): a=4.12/2, a′ =a√2. − phonon. Solid line is our calculated spectrum (Eq.7). Also
shownisthepredictionofthehybridizationmodel(Eq.15)for
thehigh-energy branch E2 =2E0.
8
E
0
6
K)
k) ( 4
E(
2
0
0 0.5 1
Wavevector Aoo−1
9
0
−10
al (K) −20 E0
enti −30
ot
P
−40 (0001) 2 2
−50 1 (110)
−2 −1 0 1 2
0
−10
al (K) −20 E0 2 2
nti
ote −30
P
−40 (1000)
−50
−2 −1 0 1 2
0
−10
FIG.9. Schematic picture of an excitation of the high en-
K)
al ( −20 E ergybranch2E0,astwoadjacentlocalmodesalongthe(110)
otenti −30 0 direction. The filled circles represent atoms that execute the
P breathing motion, allowing atoms 1 and 2 (open circles) to
−40 (0100)
exchangeplaces more easily.
−50
−2 −1 0 1 2
Distance from equilibrium (Ao)
FIG. 7. The potential-well of an atom in hcp 4He along
bcc
different directions. The energy difference E0 between the hcp
lowest two energy levels (dashed lines) are: (0100)- 16.0K, 0.05 0.05
(1000)- 21.0K, (0001)- 23.0K.
0.35
al)
ot
T
C(v
o−1nsity (A) 0.25 ocal mode)/ 0.025 0.025
e L
n d C(v
o
ati
p
u
c
c
o
e
d
o 0.15
m
−
al
c
o
d l
e
z
ali
m
or 0.05 00 0.5 1 1.55 1.6 1.65 0
N
Temperature (K)
FIG.10. Theratioofthecontributionofthelocalmodeto
thetotal specific heat in thehcp and bcc phases [24,25]. For
−0.050 5 10 clarity,thetemperaturescaleforthebccphasewasexpanded.
Distance along the (110) direction (Ao)
FIG. 8. The normalized Fourier transform of the k-space
density of the local-mode occupation in the high-energy
branch E2 (Eq.16), along the(110) direction.
10