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PHYSICAL REVIEW 8 VOLUME 30, NUMBER 8 15 OCTOBER 1984

Crystal stability, therrs|al vibration, and vacancies

L. K. MolekoDepartment ofPhysics, University of Ottawa, Ottawa, Canada K1N 6N5

H. R. GlydeDepartment of Physics, University ofDelaware, Netvark, Delaware 19716

(Received 7 May 1984)

The stability of crystals is studied with use of a model which includes atomic vibration andthermal creation of vacancies. The vibrational dynamics is treated within an Einstein approxima-tion to the self-consistent phonon theory including cubic anharmonic contributions. The force con-.

stants in the dynamics are reduced when vacancies appear a,nd the thermal equilibrium number ofvacancies is determined by the dynamics to form an integrated, consistent model. The predicted in-

stability temperatures TI and instability volumes pl lie within 10—20% of those "observed" in corn-

puter simulations of crystal stability. The predicted instability temperatures TI lie within 20—40%of observed melting temperatures T~, depending upon the crystal. This suggests that vacancies andvibrational dynamics will induce instability not far above TM and may be largely responsible for in-

stability in real crystals. In agreement with much previous work, we find that the Lindemann melt-

ing rule holds for thermal melting in classical crystals. At quantum melting or instability, however,the Lindemann ratio 5 takes a wide range of values (0.04(5(0.35). This supports the'view thatthe Lindemann rule is an empirical expression of the vibrational amplitude at which the classicalfree energy of the solid phase exceeds that of the fluid.

I. INTRODUCTION

In this paper we explore the stability of crystals tolarge-amplitude atomic vibration and to thermal creationof vacancies. The aim is to determine whether atomic vi-bration and vacancy creation are the chief ingredients thatlead to the eventual mechanical instability of crystals. Wetest this idea by comparing the instability temperaturesand densities obtained in a model containing vibrationaldynamics and vacancies with instabilities observed incomputer simulations of identical systems. We also ex-pect that the model instability temperatures TI should notlie too far above the observed melting temperature TM ofreal crystals if the model is representative of a real crystal.The systems studied are the rare-gas crystals (RGC's), theLennard-Jones system, and the Gaussian core model(GCM) investigated by Stillinger and Weber. '

In the present model the dynamics are described by theself-consistent phonon (SCP) theory (including cubicanharmonic contributions) in an Einstein approximation.The Einstein approximation is made so that it yields thesame instability temperature as the full SCP theory.Thermal vacancies are incorporated self-consistently withthe dynamics. The vacancy formation enthalpy then de-pends upon the thermal averaging in SCP theory and theforce constants are reduced by the appearance of vacantsites in the lattice. Instability of the. crystal is identifiedwith the temperature (or density) at which the coupledequations for the dynamics and vacancies no longer yieldsolutions having a real vibrational frequency.

It is tempting to relate mechanical stability and meltingin some way. For example, Lindemann ' proposed thatcrystals melt when the rms vibrational amplitude (u )

reaches a characteristic fraction 5=(u )'i /R of the in-teratomic spacing R. There are now several accurate cal-culations' ' in a variety of crystals that find a commonvalue of 5=0.16+0.01 immediately below Tst, forthermal melting in classical crystals at least. Similarly,Hansen and Verlet"' ' observed that crystallization ofseveral fluids occurs when the height of the first peak in

the static structure factor reaches the value of&(qi)=2.90+0.05. These characteristic "rules" of melt-ing (and freezing) allude to a geometric component to thetransition which may be related to the stability of thefluid and solid phases. Or it may be that the free energyof a solid, for example, simply becomes too high relativeto that of the fluid phase when the vibrational amplitude(Lindemann ratio 5) exceeds a specific value independentof the form of the interatomic potential. The phenomenaof superheating and supercooling suggest this interpreta-tion. Indeed in the computer simulations of Hoover andRoss, ' of Street et al. ,

' and of Stillinger and Weber, ' forexample, it- was possible to take crystals to temperaturessubstantially above TM (-20%%uo) [and to instability densi-ties pt substantially below pM ( —10%)], suggesting thatTI lies significantly above T~. How far TI lies aboveT~ for real, macroscopic crystals immersed in a heat bathwithout constraints is, however, not known.

Theories of melting in three-dimensional crystals usual-ly fall into two classes, those which evaluate the free-energy differences of the fluid and solid phases and thosewhich investigate stability either analytically or by com-puter simulation. Examples of the former are the MonteCarlo (MC) study of melting in the Lennard-Jones systemby Hansen and Verlet, "' in the one-component plasma(Wigner crystal) by Slattery et al. ' and Pollock and Han-

30 4215 1984 The American Physical Society

4216 L. K. MOLEKO AND H. R. GLYDE 30

sen, ' and in Na by Stroud and Ashcroft. ' The latter

type has been the subject of intense activity in two dimen-

sions recently.The stability of crystals in three dimensions has a long

history of study. Born proposed in 1941 that crystal in-

stability (and melting) was associated with the vanishing

of the transverse elastic constants. Choquard showed

that in an Einstein (SCE) approximation to the self-

consistent harmonic (SCH) approximation there existed acritical temperature (Tr ) above which there was no self-

consistent solution yielding a real, stable Einstein frequen-

cy. This instability in the SCP theory has now been inves-

tigated in higher orders and in several different applica-tions. ' The stability of the Wigner electron solid intwo and three dixnensions has been investigated within theSCP theory. ' Plakida, Siklos, and co-workers in aseries of several papers have investigated the instability inthe SCP in several approximations and in one, two, andthree dimensions. Zubov "particularly suggests includinghigher-order anharmonic terms is important. Matsu-bara et al. and Hasegawa et a/. ' have used the SCEtheory to investigate stability of small metallic particles.Moleko and Glyde have studied the nature of the insta-bility and its detailed dependence on the approximationsmade to the SCP theory.

These SCP studies show that if the crystal volume isheld constant (as is the case in the Wigner solid) the SCHtheory has an instability at a temperature which is10—100 times the expected or observed TM. When higherorders are included, the SCP TI is reduced to (3—4) TM.When the crystal volume is allowed to expand (beyondthat observed at TM ) the TI can be reduced toTI -(1.3—1.5)TM.

Aksenov and Adkhamov et aI. have incorporatedvacancies into the SCP theory to investigate crystal stabil-

ity and the present paper follows their work very closely.However, the analytic expressions we find do not agreecompletely with theirs and we wish to explore these equa-tions numerically in specific examples. They found thatincluding vacancies reduced Tq substantially.

The stability of the fluid against forming periodic,crystal-like structures has also been investigated. Kirk-wood and Munroe, for example, derived an integralequation foi the single-particle density function, p(r), and

found that below a specific temperature the previouslyuniform solutions "jumped" to periodic crystal-like solu-

tions. Along the same lines Brout deve1oped a rnean-

field theory of the solid-fluid transition and Raveche,Kayser, and Stuart have found the emergence of period-ic density patterns in a study of integral equations forcorrelation functions normally used for the study offluids.

Ramakrishnan ' has recently developed a beautiful andspecific extension of these ideas. He derives a self-

consistency condition or equation for p(r) and expressesthe free-energy difference between the liquid and the solidin terms of p(r). The equation for p(r) depends upon thedirect correlation function c (q) as a parameter and c(q) isrelated to the structure factor by S(q)=[1—c(q)]This equation always has liquidlike solutions. However,for a critical value of c (q~ ) the equation also has a solid-

like solution [at c(q&)=0.85 in the hexagonal latticestructure considered in two dimensions]. When this criti-cal value of c (q~ ) is reached the solidlike solutions appeardiscontinuously with substantial weight of p(r) at the lat-tice points, characteristic of a first-order transition. Thefree energy of the solid phase is also lower than that ofthe liquid just beyond this critical value of c (q ~ )

[c(q&) &0.86] which identifies the solid as the equilibri-um phase. Thus at the critical value of c(q~ ) [and there-fore of the height of S(q ~)], solidlike solutions are possi-ble and the solid has a lower free energy. This suggeststhat the Hansen and Verlet freezing "rule" expresses the"structural" dependence of the point at which the free en-

ergy of the solid and fluid become equal. Similarly,Jacobs has developed a mean-field theory of melting in

which the free-energy difference of the solid and liquid isevaluated. He finds this difference vanishes at a constantvalue of 5. This suggests that Lindemann's rule is a rule

on the free energy rather than on stability of crystals.In this background, we emphasize that here we are in-

vestigating crystal stability only and compare our resultschiefly with computer simulations of TI (or pI ) with less

emphasis on comparison with melting. In Sec. II wederive the equations for the dynamics and vacancy forma-tion. In Sec. III we test the Einstein approximation madeto the dynamics and present the results for the RGC's andLennard-Jones system in Sec. IV. The results for theGCM are presented in Sec, V.

II. THEORY

A. Model

We consider a crystal of E identical atoms of mass Mon a Bravais lattice of XL sites. The crystal hasn (T)=NL(T) N thermal va—cancies where N is fixed andthe number of lattice sites, XL, increases as the number ofvacancies, n, increases with T. We assume that the crys-tal is well approximated by a sum of pairwise interactionsbetween individual atoms. The Hamiltonian is then

where T~ =p~ /2M is the kinetic energy of atom l, P(r) isthe interatomic potential, and

if site I is occupied0~= ~

0, if siteh is vacant .(2)

It is convenient to introduce a complementary occupationvariable,

0, if site L is occupiedcI —1 —g, =. '

1, if site I is vacant,(3)

where clearly c~ +o ~= 1, QI c~ =n, and the average va-

cancy concentration is

c—= (c()= gc(=1 n n

X+n

30 CRYSTAL STABILITY, THERMAL VIBRATION, AND VACANCIES 4217

In terms of cl,Nc N~

H = g Tl(1 C—I)+ 2 g 0(rI I )(1—cl)(1—cl ) .

This sets out the crystal we wish to describe.We now seek a simple model crystal to approximate the

above crystal. Using the Gibbs-Bogoliubov variationalprinciple we determine the parameters in the model crys-tal so that it best describes the real crystal. There areseveral possible choices for a model crystal, each havingadvantages and limitations. We choose a hypothetical(and somewhat artificial) model crystal in which, firstly,all the interactions are harmonic. Secondly, we assumethe model crystal also has n vacancies, but that the pres-ence of the vacancies does not change the interactions be-tween the atoms or affect the energy of the crystal. Thatis, we assume we can still represent the potential by har-monic interactions between atoms summed as if the va-cancies were not there. The model Hamiltonian is then

(N + li)!Fp ——FII —k&T ln

' —sTn .X!n! (g)

Here F~ is usual harmonic free energy corresponding toHp, the second term represents the configurational entro-

py of n randomly located vacancies, and s is the local en-

tropy change around an individual vacancy. Normal-ly, ' the last term is gn =(E+pu sT—)n, where g is thefree energy of vacancy formation. However, since we areignoring all energy changes when the vacancies are intro-duced in the model crystal, we retain only the entropyterm in Fp. When p&0 we should also keep the pU termwhere U is the volume of vacancy formation.

Using the Gibbs-Bogoliubov variational principle, thetrial Helmholtz free energy pf the real crystal is

Fr Fp + &——(H Hp ) & p

—&F,which is an upper bound to the true free energy. To findthe best 4 II(l, l') and n(T) we minimize ' Fr. At theminimum

N N

Hp= g Tl+ 2 g Pa(rl, l »

where

Pa(rI, I )=P(RI I )+ 2 u(l, l') C&(l, l') u(l, l') (7)

BFz-54+

A, n

6A

(jF+ 5n =Q.

4A(10)

is the harmonic interaction. Here the 4 p(l, l') are themodel harmonic force constants [since we use differencecoordinates u(l, l')=u(l) —u(1'), these 4 p(l, l') are de-

fined with a sign convention opposite to those of Bornand Huang ]. We seek the harmonic force constants in

(7) which best represent the real crystal containing vacan-

cies.Since in the model crystal the harmonic force constants

and the vacancies are assumed to be independent, the va-cancies will be randomly located and

&cl&0——c,& Cl C!' & 0=

& CI & 0& CI & 0 =C

where & &0 is an average over the states of Hp. TheHelmholtz free energy of the model crystal is then

Here A~II(l, l')=&u~(l, l')uII(l, l')& is the relative mean-square vibrational amplitude. For the purposes of varia-tion we may treat 4,A, and n as independent or depen-dent variables. Here it is most convenient to treat them asindependent. The coefficient of each differential in (10)must then vanish and this provides the equations ofmotion we need.

To evaluate & (H —Hp) &p over the states of Hp, we notethat the average kinetic energy of H is

cy & Tl(1 —CI ) &o= y & Tl &o&(1—

I ) &o1=1 1=1

=Ni & Tl &p(1 c)=N& Tl &—0,which cancels with the kinetic energy in Hp, so that

(1 la)

2 g l & 0(rl I') &0( ) & 4H( I I') &0( 1 c)] (1 lb)

To obtain (lib) we have used the fact that P~(r) is of fi-nite range so that

B. Vacancy concentration

To determine c (T) we use, from (10),

1=1 1'=1BF BFBn Bn Bn

+ &(H —Hp)&0 ——0

NL

=NI. (1—c)—,' g & Pz (r0 I ) & 0 . =kgTln

X+n sT+ &(H ——Hp) &0 .Bn Bc


Substituting (lib) and Bc/Bn =Nl. , we find—(e—sT)/k& T

c =n/(N+n)=e

where

(12)

and the second term represents the potential energy peratom gained (in the madel) when the atom is placed onthe surface. If p&0 we should, as noted above, replace eby h =e+pv in (12).

L L

e= —y &0("0!)&0(1—c}+ ' y &!t!H("01)&0

is the vacancy formation energy. The e is the energy re-

quired to remove an atom from the crystal and place it onthe surface. In e the first term represents the loss in po-tential when the atom is removed from within the crystal

I

C. Dynamics

To determine the dynamics we use OFT/B4 p——0 and

BF~/RA=0. To evaluate these equations it is most con-venient to use the first form of ((H —H0)& in (lla) andto expand p(rl 1 ) in a power series of the relative displace-ments u(1, 1') (r=R+u}. Then

N

((H —H0)&0= —, g e' " "lt!(Rl 1 )(1—c) ——,

' g [p(R11)+—,' 4(t, t')A(t, t')] (14)

The equation

8Fz. BFII

ac.gt, t')=

ac @.tt),+

a .@t,t')

leads to the usual relation

I

representing the perturbation. The only difference due tovacancies is that (P(r)& appears multiplied by (1—c) .Thus all higher-order SCP terms with vacancies are as be-

fore with force-constant coefficients multiplied by(1—c) . Explicitly the cubic coefficient is

A p(0, 1)=—g (A'/Mco ~)(1—e ")Q, A

Xe (q, 1L.)ep(q, k, ) cath(fico &/2keT),

(15)

which we now see holds with or without vacancies. Hereco

&and e(q, A, ) are "harmonic" frequencies and polariza-

qktion vectors given by the usual relation

co z—— pe (q, A, ) g'(1 —e ")4 p(0, 1)

D. SCEC

In the present paper we use almost exclusively an Ein-stein approximation to the SCP. In the SCH approxima-tion, the Einstein frequency is defined as the average ofthe SCH frequencies,

q, A,

(0,0)= g' 4 (0,1),3M

' 3M

Xep(q, k) .

The second equation

aFT aaA.p(t, t )

=a .,(t, t )

gives

4 p(t, t')=(V!7'Vile(rl, l')&0(1

whe~e r =R01+u01 and the average ( &z is the Einsteinapproximation to ( &0,

(17) (P(R+u) &E——(2irA) i I du e " P(R+u) . (20)

(16) where the second line follows by substituting cori from(16). Using (17) for 4 (Ol}, we obtain

1 ~, d p(r) + 2 dltl(r(1—)2 (19)' 3Ml d"' d

These are the harmonic force constants of the modelwhich best represent the real crystal, i.e., the self-consistent harmonic (SCH) force constants for a crystalhaving a vacancy concentration c. The average ( &0 in(17) can be evaluated in the usual way. ' The presentSCH theory differs from the usual one by the appear-ance of (1—c) in (17) only. Aksenov obtains a similarfactor of (1—c) in (15). However, here (15) is an internalrelation within harmonic dynamics of the model crystaland cannot contain c.

Higher-order approximations to the self-consistent pho-non (SCP) theory can be calculated in the usual way6 us-

ing perturbation theory with ((H —H0) &0 in (14)

=(fi/McoE) coth(ficoE/2k& T) (21)

is independent of l. Equations (19)—(21) with c given by(12) are the SCE approximation.

The phonon frequencies corrected for cubic anharmoni-city should be obtained from the position of the peak inthe corresponding response function. If we ignore the fre-quency dependence of the phonon lifetimes and frequency

This is obtained by making an Einstein approximation toA(0, 1) [all co &~coE in (15)] so that

qA,

l(, =(u (O, l)up(0, 1)&E——2(u (0)&H5~p


co'(SCEC) =coz+ 2coz &(coz ),where

2.36%4 [2n (coz ) + 1]

12M coE

X g (,P(R~+u)) Il —c)2

(23)

except for the factor of 2.36. Here also, 6 contains theI

shifts in this response function, the position of this peak isgiven, for phonon q, A, , by ' '

co (q, A, ) =co &+2co P(q, )(,;co),

where b is the cubic shift. As in I (Ref. 7) we define anEinstein approximation to the SCH plus cubic (SCH + C}theory as

co (SCEC)—: g [co z+2co &b(q, A, ;co)] . (22)1

qA. qA,

The first term in (22) is the SCE frequency coz given by(18). If we replace all co and co in 2co zb, (q, A,;co) by coz

qA. qA,

it was shown in I that (22) reduces to

factor (1—c) . As explained in I the factor 2.36 was in-cluded in (24) so that the SCEC frequency became unsta-ble at the same temperature in solid Ar as the fullSCH+ cubic frequencies without the Einstein approxima-tion.

The SCEC approximation consists of iterating Eqs.(19), (23), (21), and (20), with co(SCEC) obtained from (23)substituted in Eq. (21}for coz, and Eq. (12).

FrscE =FscE+~3 s

where Fscz is the Einstein approximation to Fo,

(25)

E. ISCE

To determine the equation of state of the crystals (e.g.,crystal volume as a function of T and p) we employ anEinstein approximation to the improved self-consistent(ISC) theory. ' ' The ISC theory gives at present thebest analytic description of the rare-gas crystals (RGC).It consists of the SCH free energy with the cubic termadded as an improvement. Here we seek only an approxi-mate theory and make an Einstein approximation to theISC which we call the ISCE theory. In the presence ofvacancies the ISCE free energy is

N 'FscE ,' g (P——(ro—i))z(1—c) +3kz T in[2 sinh(ficoz/2kz T)]—', iricoz coth—(ficoz/2kz T)I

+[kzT/(1 —c)][cinc —(1—c) ln(1 —c)j sTc—and

—2.36k I12[n(coz) +n(coz) j+1IEF3 ——

144co4EM'

Xg', P(ro, i) '(1—c)'dr E

(27)

III. TEST OF ISCE MODEL

In this section we test our Einstein approximation(ISCE) to the improved self-consistent (ISC) theory bystudying the change in lattice parameter, ~~, and the iso-thermal bulk modulus with temperature calculated in dif-ferent approximations. Firstly, from Fig. 1 we see thatfor Ar the ISC ~I-/Lo calculated with use of theBobetic-Barker ' potential [ISC(BB)]agrees well with theobserved value. These two curves represent the values anyother approximation should attempt to reproduce. Alsofor Ar in Fig. 1 we see the ISCE ~~/Lo calculated withthe Aziz-Chen 2 potential [ISCE(AC)] clearly lies below

is the Einstein approximation to the cubic contribution toF. Equation (27) is obtained by approximating each co~i

by coE and retaining only the leading third derivative inthe ISC cubic coefficient. We have also multiplied dd 3

by the same factor 2.36 appearing in the cubic correctionto the dynamics in the SCEC approximation. In the fol-lowing section we evaluate the lattice parameter and bulkmodulus in the RGC's as a test of the ISCE including thefactor of 2.36.

I

the ISC result at high temperature. The Aiiz-Chen po-tential is the most accurate representation of the Ar-Arinteraction presently available; the Bobetic-Barker ' po-tential is also an accurate one. For the purpose of thepresent calculations in the sohd, these two potentials areindistinguishable. Thus the difference between theISC(BB} and the ISCE(AC) in Ar represents theshortcomings of the ISCE approximation.

Also shown for Ar in Fig. 1 is the ISCE ~J-/Lo calcu-lated using the Lennard-Jones potential with parametersobtained by Horton and Leech [ISCE(L-J)]. This liesreasonably close to the observed value. Unfortunately, theLennard-Jones potential is not an accurate potential bypresent standards and the reasonable agreement of theISCE(L-J) with experiment represents a fortuitous cancel-lation of errors in the L-J potential and in the ISCE ap-proximation to the free energy.

Paskin et al. have developed an approximation to theISC which they denote the self-consistent averaged pho-non (SCAP) approximation. Their averaged phonon ap-proximation coupled with the Lennard-Jones potentialgives a &L /Lo almost identical to the ISCE(L-J) valuefound here. This suggests that the SCAP and ISCE aresimilar approximations and that in the SCAP case there isagain a cancellation of errors in the SCAP and L-J poten-tial that leads to a reasonable ~I- /Lo.

Also shown for Ar is ~I' /Lo calculated using theMorse potential [ISCE(M)] with parameters calculated byGlyde so that the SCH theory reproduces the correct lat-tice constant and isothermal bulk modulus at T=o K.


~—I

-" IS—E

I I t I ' I

~ Andean Smnmn~ ~ ~ ~~ ~

~

~~

~

Q — ' .~ Peteeon et a I.

~ Urvos et al-

15

e

O035

30-O

25-

I A ') )

20 40 6Q 80I I I I

1020 40 60 80

30—

U

E 25—

35 / f [ I [, ~o o a

0

o Korpiun Coufol--ISCE( Morse)

, o

oX,

o

15-

10-

0CL

t0 QN

Q 20 40 60 80 I00 l20

JD 20—

CP

l5-VlO

I I I I

eKorpiun et ol

0 Anderson Swensan—~—ISCE {Morse}

0

6) 0

lO I I I I I I

20 40 60 80 IOO l20CD

X40

O 0

N S5:~',

25—

20—

TEMPERATURE (K)FIG. 1. Lattice constant increase with temperature in Ar, Kr,

and Xe. The ISCE is the present improved self-consistent Ein-stein model with the Lennard-Jones, Morse, and Aziz-Chen(Ref. 42) potentials. The ISC(BB) is the ISC theory calculatedusing the Bobetic-Barker potential (Ref. 41). The observedvalues are from Refs. 45—47 (see also Ref. 48).

We also readjusted the Morse parameters to meet thesesame conditions using the present ISCE theory. Thischange moved the &I-/La curve to lie:midway betweenthe ISCE(L-J) and ISCE(M) curves in Fig. 1 for Ar.Since the change was not large, we continued to use the

l5—

i I I I I I I I

0 20 60 CO l40

TEMPERATURE (K)

FIG. 2. Isothermal bulk modulus with the same key as Fig.1. The ISC (BFW) three-body uses the Barker-Fisher-Watts po-tential plus three-body Axilrod-Teller forces. The observedvalues are of Anderson and Swenson (Ref. 49), Peterson et al.(Ref. 45), Urvas'et al. (Ref. 50), Korpiun and Coufal (Ref. 51),and Korpiun et al. (Ref. 52).


original Morse parameters. In Kr the ISCE(M) ~T. /I. o

happens to agree well with experiment, but it lies belowthe observed value in Xe.

In Fig. 2 we show Br for Ar, Kr, and Xe. Again theISC theory calculated with the Bobetic-Barker potential,now including three-body forces, ISC(BB)-3B agrees bestwith experiment in Ar. The ISCE(M) results agreereasonably well with experiment but there are cleardiscrepancies.

From Figs. 1 and 2 we see that the ISCE does not givean accurate description of the RGC's; there will be errorsin the PVT relation calculated using the ISCE(M) model.We wish to use the ISCE(M) because the ISCE is thecompanion approximation to the SCEC theory and cou-pled with the Morse potential all integrals can be doneanalytically. From Figs. l and 2 it provides a reasonableprediction of the lattice parameter (as a function of T and

p) which will be sufficiently accurate for our purposeshere. We find that vacancies have no effect on && or Bz.up to the triple point and affect the bulk thermodynamicproperties only immediately below the instability tempera-ture.

IV. ROC STABILITY

In the preceding section we determined the lattice pa-rameter a (T,P) as a function of temperature and pressureusing the ISCE free energy plus vacancies given by' (25).With a(T,P) determined, the temperature of the crystalwas increased in steps and the SCEC Einstein frequencyco (SCEC) given by (23) and the vacancy concentrationgiven by (12) was evaluated at each temperature. Eventu-ally a temperature was reached at which, during the itera-tion, the SCEC frequency continually decreased and thethermal equilibrium number of vacancies increasedwithout limit and there was no self-consistent solution inthe SCEC plus vacancy model. This temperature is de-fined as the mechanical instability temperature TI.

The role of the vacancies here is essentially to removeatoms from within the crystal to the surface and therebyreduce the force constants seen by atoms neighboring thevacant sites. In turn, when the force constants and poten-tial of interaction seen by these atoms is reduced, it be-

comes easier to create further vacancies (h is reduced).With the force constants reduced, co (SCEC) decreasesleading to larger rms vibrational amplitudes of the atoms.Significantly below TI the vibrational amplitudes and

TABLE I. Zero-pressure instability temperatures TI, Lin-demann ratio at TI(51), and vacancy concentration at TI in theRGC's.

Element

ArKrXe

TM

83115160

115155225

0.1630.1600.162

CI

2.9g10-'2.4~10 '3.5~10-'

co~(SCEC) are determined almost entirely by the SCPtheory, but the vacancies play a critical role in the actualinstability at TI. Also the value of TI is significantly de-creased by the vacancies.

The TI and the vacancy concentration immediatelybelow TI for the RGC's at zero pressure are listed inTable I. There we see that TI lies -40%%uo above the ob-served melting temperature TM and that immediatelybelow TM the predicted vacancy concentration is quitelarge ( —2—4% ). The SCEC + V Lindemann ratio,5=0.16 at TI also happens to agree with the value(5=0.16+0.01) found in accurate calculations for thermalmelting in classical crystals at TM. This agreement arisesfrom a cancellation of two errors: the Einstein approxi-mation predicts 5 to be too small and TI lies above TM.We should emphasize that the calculated T~ here is an in-stability temperature, and since it is possible to superheatcrystals (in three dimensions at least) we expect the realTI of a crystal to be above the observed TM.

In Fig. 3 we show the SCEC frequency versus crystaltemperature in Ar, Kr, and Xe. At low T, co (SCEC) fallsslowly as T increases due to thermal expansion of thecrystal. If the crystal volume were held constant,co (SCEC) would increase with T. Eventually, at TI,co (SCEC) drops suddenly from a finite value to negative(unstable crystal) values where there is no self-consistentsolution to the SCEC+ V equations. The sudden loss ofstability at TI is characteristic of a first-order transition.

In Fig. 4 we show the calculated vacancy concentrationin Ar, Kr, and Xe versus 1/T. There we see that the va-cancy concentration c remains quite small over the wholetemperature range except immediately below TI. Particu-larly, at T~ and below, the calculated c lies substantiallybelow the observed values in the RGC's. As shown inTable II the calculated vacancy formation enthalpy h at

TABLE II. Vacancy concentration c and formation enthalpy h at the observed melting temperatureTM calculated in the SCEC + V model.

ArKrXe

TM

83115160

Obs.

8901100'1100+

Calc.

85711921690

Obs.

&2X10 "3.2)& 108.2)& 10

c(TM)Calc.

3.5 &&10-4

5.6~10-44.6~ 10-'

'Schwalbe (Ref. 53).Macrander and Crawford (Ref. 57).

'Losee and Simmons (Ref. 54).Granfors et al. (Ref. 55).


0 II

ITm

-8

I

0 20 40 60 80 100 120~ 45

I I I

CD

g- 40QP

0 )I

qI I K~ I

,™ SCEC(Morse}

Losee Simm ons

0.8 I.O I.2 l.4 l.6 l.8 2.0 2.2

r 35QP

LU

O

l

I I V I

140I I I I I I

Pp 20-R

I I I I I I

0 20 60 100 O.6 0.8 I.O l.2 I .4I I

Xe

l.6 I 8I

10-

0 60 120 180 240TEMPERATURE (K)

FIG. 3. SCEC+ V Einstein frequency versus temperaturecalculated using the Morse potential.

TM lies above the values observed in Kr and Xe. This is awell-known and unresolved problem in the RGC's; modelsof vacancy creation based on pair interactions betweenatoms, such as the Morse potential used here, predictfewer vacancies than are observed and a larger h than isobserved in the heavier RGC's. The calculated values ofh shown in Table II are consistent with those obtained

-lO

0.4 0.6 0.8 I.O I.2 l.4 1.62

10/T (K }FICx. 4. SCEC+ V vacancy concentration versus inverse

temperature. The observed values are from Schwalbe (Ref. 53),Losee and Simmons (Ref. 54), and Granfors et aI. (Ref. 55).The Monte Carlo calculations are from Squire and Hoover (Ref.56). -

previously using a variety of pair potentials. We alsofind h (T) depends upon temperature. For example in Ar,h(T) drops from 857 K at TM 83 K to 705 K a——tT=110 K. Throughout we have used the observed


attach any physical significance to this increase in Tt/Tbtwith pressure since the ISCE model tends to predict anequilibrium volume that is too small and one which is in-creasingly too small as pressure increases. Hence we ex-pect the SCEC+ V model to predict a relatively morestable crystal as the pressure increases.

V. LENNARD-JONES SYSTEM

80 100 120 140 160 180 200 220

1 II

100 140 180 220 260 X)0 540 580

I I

EXPERIMFhlT——SCEC+ V {Morse)

ltingrve

instabilityc use

value of s =2kb in (12).In Fig. 5 we show the instability temperature TI in the

RGC's as a function of applied pressure. There we seethat the ratio of TI to the observed melting temperatureTbt increases slightly with pressure going from 1.4 atp=O to 1.5 at p =5 kbars in Ar, for example. We do not

t. I I

150 200 250 X)0 550 450 500 550TEMPERATURE (K j

FIG. 5. SCEC+ V instability curves compared with the ob-served melting curves. The observed values are from Crawfordand Daniels (Ref. 60) and Stryland et al. (Ref. 60).

In this section we compare the instability densities plcalculated using the SCEC+ V model with those ob-served in Monte Carlo (MC) simulations by Hoover andRoss' and by Street et al. ' The MC studies considereda system of particles in the classical limit interacting viathe Lennard-Jones potential

U(r)=4e[(o/r)' (o/r) —] .

In the SCEC+ V we consider the same system withM=40 and the potential parameters @=119.86 K and0.=3.405 A selected to simulate Ar.

In tPe MC studies the crystal phase, held at constanttemperature, was observed to break up into a fluid at den-sities p smaller than the melting density p~, determinedby Hansen and Verlet' from accurate calculations of thecrystal and fluid free energies. Equivalently, the crystalcan be superheated to temperatures TI above the meltingtemperature TM before it breaks up into a fluid. It is notclear why this is so in the MC simulation. Stillinger andWeber' suggest it is related to the latent heat that must besupplied before the crystal can melt. Since the simulationis done at constant energy it may be necessary to su-perheat (or supercompress) the crystal to a temperaturesufficiently high that after the latent heat is supplied thefluid does not drop to a temperature (or state) below theequilibrium melting temperature. In any case we take themaximum MC superheating temperature (or minimumcrystal density) as the limit of crystal stability for com-parison with the present SCEC + V model.

In Table III we list the thermodynamic melting densitypM, the MC instability density pt(MC), and the instabilitydensity pt(SCEC) calculated here using the SCEC modelwith vacancies for the I.-J system at three temperatures.From Table III we see that pi(SCEC) lies -7% and—15% below pt(MC) at temperatures T= 130 K andT=328 K, respectively. The larger difference at thehigher temperature probably reflects the increasing errorin pressure predicted by the ISCE model as T increases.The essential point is that the MC and SCEC+ V pi arevery similar, suggesting that a model including vibrationaldynamics and point defects may contain the main in-gredients that lead to crystal instability at high tempera-ture. We shall see this point is supported by comparisonswith simulations of the Gaussian core inodel in the nextsection.

UI. GAUSSIAN CORE MODEL

Stillinger and Weber' have made an interesting and ex-tensive molecular dynamics (MD) study of melting andfreezing in the "Gaussian core model" (GCM). The GCM


TABLE III. The instability densities pi found in MC simulations and in the present SCEC+ Vmodel for Lennard-Jones Ar (e= 119.86 K, o =3.405 A): p =(o lV/V).

kg T/e

1.06' 1271.17b 1402.74 3272.74' 328

Hard spheres'

1.012'1.024'1 179'1.179'1.041

q (MC)

0.8990.9331.0971.0580.943

pi(SCEC)

0.8680.8750.9400.940

p~ /pi (MC)

1.1261.0971.0751.1141.104

p~/pi(SCEC)

1.161.171.251.25

'Hoover and Ross (Ref. 16).Street et al. (Ref. 17).

'Hansen and Verlet (Ref. 14).

P(r i)j=P oepx[ (rtf—/1) ] .

By selecting appropriate temperatures in their simula-tions, Stillinger and Weber observed melting of the systemwhen it was initialized in a crystal phase and freezing intocrystals when the system was initialized as a fluid. Su-perheating and supercooling were also observed. TheGCM is especially interesting because at high density itshows waterlike properties, i.e., melting under pressure,negative thermal expansion, and negative meltingvolumes. Also the crystalline phase exists only over a fi-nite density range. At low temperature it melts at bothhigh and low densities.

The simulation studies of Stillinger and Weber in theGCM provide an excellent testing ground in a simple casefor the SCEC model of crystal stability. We may com-pare with their simulation values of stability and extendtheir stability curves into the quantum region which isinaccessible to MD.

In the GCM it is natural to introduce reduced units oflength (r"=r/l), density (p =pl ), energy (E/Po), andtime t*=t/(Ml /Po)'~ . In the classical limit and in re-duced units the equations of motions are universal and in-dependent of Po, l, and M. The density range consideredby Stillinger and Weber is 0.1&p*(1.0, with the systemcondensing into an fcc crystal at low density, p* &0.18,

I.O

0.8

~ 0.6CL

o 0.40)0

0.2

I

0 I I I I t

0 0.2 0.4 0.6 0.8 I.O 1.2Reduced length r'

I I

l.4 l.6

FIG. 6. Gaussian core-model potential.

consists of a system of particles interacting pairwise via apurely repulsive Gaussian potential,

I6

I4D I2—

IO-

8-E

C3 2.

I l I I ~ l ~ l I

FLUID

0 2 4 6 8 lO l2 l4 l6 l8 20Reduced Volume pi

FIG. 7. SCEC+ V bcc crystal instability curve for theGaussian core model compared with the molecular dynamics(MD) instability temperatures T~ and melting temperatures TMof Stillinger and Weber (Ref. 1). QL marks the SCEC + V in-stability volume in the quantum ( T=O K) limit calculated withthe GMC set to simulate Ar (M=40). H is the instabilityvolume in the harmonic approximation.

and into a bcc crystal at high density. Over most of thedensity range 0.18&p*&1.0 the crystal is bcc and, al-though we examined both phases, we discuss only thehigh-density bcc phase here. In the quantum region, how-ever, the degree of quantumness depends upon the massand M (and Po and l) must be specified. To study thequantum case we selected M=40 amu (to simulate Ar)with Po ——7074 K and l=8.332/3.44 A ' as given in Eq.(A8) of Stillinger and Weber. ' With this choice, p*=0.2corresponds to Ar under extreme pressure (5.5 Mbars).The transition from the fcc to bcc phase at p*=0.18 istherefore reminiscent of the similar fcc to bcc transitionobserved in solid helium under extreme pressure.

Before presenting the results, we note first that we mayunderstand the "pressure melting" of the GCM at highdensity readily in an Einstein model. The Einstein fre-quency coE is proportional to [(d P/dr )+(2/r)(dgldr)];see Eq. (19). From Fig. 6, dP/dr is always negative. Atsmall r, d P/dr is also negative giving a negative coE andan unstable crystal at high density. At r =l/2, d P/drbecomes positive and at r =(—', )'~ l the total force con-stant coE becomes positive. In this simple picture and as-suming nearest-neighbor forces, a positive coE and a stablebcc crystal is expected for p" & 0.707 only. This compares


TABLE IV. Reduced instability pressure PI*, temperatureTI, and vacancy concentration cI versus reduced density p .

Cl

I I I [ I I I I I

T= 75K90 a o = 2.56

p=0.280—

I I I I I I I I I I I I I

SCH

0.30.40.81.0

0.2060.451.822.82

1.40 ~10-'1.312)&10 2

8.87 X 10-'7.625 &&

10-'

6.1 ~10-'5.9 ~10-"2.75 &&

10-'"6.03)& 10

70H

Q 6C

g 50

408o 30

well with the p*&2 we find for the SCEC model withM=40. Physically, the GC potential becomes soft whenthe particles are very closely packed with r,j-l. TheWigner solid also melts under pressure because theCoulomb potential is soft. On the other hand, at lowdensity (p*&0.1) where r,z » l the GC potential actsmuch like a hard-core potential when two particles en-counter each other. If the kinetic energy (temperature) issufficiently high, we then get melting at low density in theusual way for a hard-core system.

In Fig. 7 we show the instability curve TI* Tz/$0 of-—the bcc GCM crystal calculated here using the SCECmodel as a function of reduced density p'. On the left-hand side of the figure, TI drops dramatically in the re-gion p -1 as "pressure melting" sets in. Also shown arethe values of TI* observed by Stillinger and Weber theseare the maximum temperatures to which the bcc crystalcould be "superheated. " The TJtr shown in Fig. 7 are theequilibrium melting temperatures determined by Stillingerand Weber. In the region where the instability curve isreasonably flat, the SCEC and simulation TI differ by-25%.

Two points are of interest. Firstly, since the pressure isso high at the p* considered here, the vacancy concentra-tion is negligible (see Table IV). This is because the p'U'term of h* is so large that c is negligible. Stillinger andWeber observed no apparent vacancy content. Thus thedynamics is entirely responsible for the instability in theSCEC+ V model in this case. It is interesting that wherevacancies are really absent, reasonable agreement for TIwith direct simulation can be obtained when only the vi-brational dynamics contributes.

Secondly, the TI' observed in the simulation may beonly a limit to superheating and not necessarily the trueinstability temperature of the crystal. As noted above,when the crystal melts a latent heat must be supplied.Since the simulation was done at constant energy, Stil-linger and Weber found they had to raise the temperatureof the crystal sufficiently high so that, after this latentheat was supplied and the system temperature fell, it did

20

io

0 0.2 0.O 0.6 0.8 t.0 0.8 0.6 0.4 0.2 0 0.2 0.4Reduced Nave Vector

FIG. 8. Self-consistent harmonic dispersion curves for theGCM compared with those for solid Ar using the Aziz-Chenpotential.

22 t t I

P=O.28, {MD)=-0.&8

( 8t($CEc) = o.l90

l2

lO

E 8—

not fall below the equilibrium TM at which the fluidcould exist. Hence the TI observed in the MD simulationmay be set by energy requirements and may not representthe ultimate stability temperature of the crystal at all.The true instability temperature could be higher bringingit in better agreement with the SCEC TI .

In Table V we show values of TI calculated using theSCE theory without the cubic term included. Clearly thecubic term is most important and reduces the TI substan-tially. We found that it made little difference ( —1%) toTI whether the cubic term was added as a perturbation tothe SCE or included fully in the iteration scheme.

0.20.41.0

SCE

84.5 X10-'92.5 X10-'89.12)& 10

SCEC

12.65 /1013.12 X107.625 ~ 10-'

MD'

10.4~ 108.0& 107.3X10-'

'Reference 1.

TABLE V. Reduced instability temperatures Tz for selecteddensities p* in SCE and SCEC models.

I

I I

I I I li II

2 4 6 8 IO l2 l4Reduced Temperature I(PT

FIG. 9. Lindemann ratio 5 versus temperature in the GCMmodel at p =0.2: 0, MD values of Stillinger and Weber;

—,present SCEC+ V values.


In Fig. 8 we show the SCH frequency dispersion curvesfor the GCM at p*=0.2. These curves look the same asthose for other fcc crystals, e.g., Ar.

In Fig. 9 we compare the Lindemann ratio 5(T) as cal-culated in the SCEC model and as observed by Stillingerand Weber at p*=0.2 as a function of temperature. Inboth cases 5=0.20 at the respective Ti. At TM Stillingerand Weber find 5=0.16. In the SCEC model 5 is givensimply by

52 3T[R ro'(SCEC)]

in the classical limit. From Fig. 9 we see this value isvery similar to the MD value except it is shifted to higherT (about 25% at high T*). In Fig. 10 we showco*(SCEC) versus T'.

In the quantum limit the mass M, Pa, and l must bespecified. This means that the density p* at which theGCM melts under pressure will depend on M explicitly.In Table VI we list the pressure melting p* at T=O K forM=40, 20, and 5. For M=40 the Pa and I are chosen tosimulate Ar while for M=20 and 5, Pa and I are chosento simulate Ne. For Pa and l chosen to simulate He wefound the GCM crystal unstable at all p'. Of particularinterest are the small values of the Lindemann ratio 5 atthe quantum instability in the GCM. Since the GCM isunder extreme pressure at melting the rms vibrational am-plitudes are small. Apparently, if the rms amplitude (orLindemann ratio) is small in the crystal, the crystal can"melt" with a small Lindemann ratio in the quantum lim-it. In contrast, in the Wigner solid 5 is large at quantummelting. In solid bcc helium 5 is always large so it fol-lows that 5 must be large at quantum melting in solidhelium.

The results of Table VI suggest there is no Lindemannrule for melting in the quantum limit. The 5 can take anyvalue at instability or melting. This suggests Lindemann'srule is not related to intrinsic instability in a crystal struc-ture. Rather Lindemann's rule holds only for thermalmelting in the classical limit. This suggests thatLindemann's rule reflects the structural dependence of theclassical free energy of the solid relative to that of thefluid. That is, at a critical 5 the free energy of the solidbecomes higher than that of the liquid, approximately in-

dependently of the interatomic potential.

30.8CP

Cl

0CD

u 0.7hlC3(1)

0,5

0.4—I

0 2Reduced

I

I

I

I

I

i V

6 8 I 0 I 2 I4temperature lg T

FIG. 10. SCEC+ V Einstein frequency in the GCM atp =0.2.

VII. DISCUSSION

The central result of the present paper is that the insta-bility temperature (or density) of crystals can be predictedto within 7—20% of the "observed" values using a modelwhich includes atomic vibration and vacancy formation.The observed values used for comparison are the max-imum temperature (or minimum density) to which a crys-tal can be superheated in simulation studies of melting.This suggests that atomic vibration and vacancy forma-tion together may be largely responsible for the actual in-stability of real crystals.

In the RGC's, where the vacancy concentration is signi-ficant at high temperature, including the vacancy com-ponent in the model is most important. For example, ifvacancies are not included, the instability predicted by theSCEC model for Ar increases from Tl ——115 to 145 K

TABLE VI. Instability density (pl ) and Lindemann ratio 5 at instability in the bcc and fcc GCMcrystals in the quantum (0-K) limit. The choices of Po and I simulate Ar (M=40) and Ne (M=20 and5). The quantum melting values of 5 in solid helium (bcc) and the %'igner electron solid (bcc). are5=0.35 and 5=0.26, respectively.

4020

5

GCM parameters(j)o

(10' K)

8.492.592.59

l(A)

1.1920.9640.964

6.1

3.51.7

fcc

0.0400.0860.091

2.91.70.8

bcc

0.0480.0810.094


which would remove the reasonable agreement with simu-lation. In their MC simulations of the Ar Lennard-Jonessystem, Street et al. ' observed substantial defect forma-tion near the instability point. Thus where vacancies playa role it is apparently important to incorporate them in amodel of instability. In the GCM, where the vacancyconcentration is negligible, the vibrational dynamics aloneis responsible for the instability and provides reasonableagreement with simulation values.

The SCEC model of dynamics plus vacancies we haveused contains several approximations. Firstly, we haveneglected all anharmonic terms beyond those in thelowest-order SCP approximation and the cubic term inthe second-order approximation. Zubov has suggestedthat including higher-order terms is important to predictTq accurately. Secondly we have made an Einstein ap-proximation to the dynamics throughout. This was com-pensated for somewhat by multiplying the SCEC cubicshift b, by the arbitrary factor 2.36 so that the SCECmodel predicted the same Tz as the full SCH+ cubictheory without the Einstein approximation for solid Ar.We used the same factor in the cubic term of the ISCEfree energy. It is interesting that the ISCE predicts areasonably accurate equation of state for the RGC's. TheISCE, including this factor, predicts nearly identical ther-modynamic properties for the RGC's as does the SCAPapproximation developed by Paskin et al.

Despite this, the ISCE predicts a lattice spacing a (T)which is too small at a given temperature. (See Fig. 1.)This will make the crystal more stable in the SCEC modelthan it would be if the observed a (T) were used, for ex-ample. A more accurate value of a(T) would lower Tz,probably by -5—10%, bringing it closer to the observedT~. The error in a(T,p) increases as pressure is applied,increasing this source of error at high pressure (see Fig. 5).

The model of vacancy formation is also only approxi-mate. Firstly, only pair forces have been included and, asis well known, models including only pair forces predict alarger formation enthalpy h and fewer vacancies than areobserved in the RGC's. Secondly, we have neglected anyatomic relaxation around the vacancies which would alsolower h. These approximations lead to fewer vacanciesthan a more complete model would predict, and thereforeto a higher Tz than could be expected if refinements wereincorporated. Also, we have simply used a formation en-tropy s =2k& which is consistent with observed and othercalculated values in the RGC's. The present model is notflexible enough to provide an expression for s whichwould require evaluating the local vibrational frequenciesaround the vacancy.

Removing the approximations noted above would lowerTz, and in view of this finding, an instability temperatureand density within -8—20% of that observed in simula-tions and a rz -20—50%%uo above the observed meltingtemperature seems quite reasonable.

It is interesting to compare the crystal instability curveof the GCM in Fig. 7 with the instability curve of theWigner electron (bcc) solid which we reproduce here inFig. 11. In the Wigner solid case, the instability curvewas calculated using the full SCH + cubic theorywithout making an Einstein approximation and with the

IO

4-FL UID

UJCL

QJ -Ia IO

LUI-

oO

-702-

IOO IO4 I 06

SFIG. 11. Instability curve of the Wigner electron crystal cal-

culated in the self-consistent phonon theory (SCP) including thecubic anharmonic term compared with the Monte Carlo (MC)melting lines. The MC (MELTING) is the classical melting lineof Slattery et al. (Ref. 18) given by (kzTM)=2/(I ~r, ) withI M

——178. The arrow (MC is the T=. O pressure melting r, value(r, =165) calculated by Ceperley (Ref. 61) for the charged Bosesystem (r =r,ao, ao ——0.529 172 A}. kz is Boltzmann's constantin rydberg units (1 Ry= e /2ao ——13.6048 eV). To the left of thedashed line, quantum effects are important.

cubic term included in the iteration scheme (denoted theSCP model). Vacancies were not included. The instabili-ty curves shown in Figs. 7 and 11 have the same generalshape with pressure melting taking place at small inter-paiticle spacing (r=r, ao) in each case. In the Wignersolid, pressure melting takes place only in the quantumlimit when the RMS vibrational amplitudes are large(comparable to the interparticle spacing). In the GCMcase pressure melting can take place in the classical orquantum limit, since it depends on simple derivatives ofthe potential vanishing. The de Broglie wavelength, com-pared to the interparticle spacing, is a good measure ofquantum effects. In the GCM this ratio isA, /R=0. 0041p 'z /T*'z for the potential and massselected to represent Ar. At P =1 and T*=IOX1 0where pressure melting begins, A, /R =0.04, and quantumeffects are negligible. Only at T (0.1X10 is A, com-parable to R.

In the Wigner solid, and in the classical limit, we see inFig. 11 that the instability curve calculated with the SCPmodel lies substantially above (at higher temperatures)than the melting line determined by Slattery et al. ' usingMC methods. This discrepancy suggests again that va-cancies may play an important role in thermal melting.At low temperature ( T=O K effectively) the SCP modelpredicts instability of the Wigner solid at r, =180. (Sinceexchange is neglected this corresponds strictly to a Bosesystem. ) In a MC study, Ceperley ' finds melting atr, = 165 at T=O K in the charged Bose system. This sug-gests that, as in the GCM, the SCP model may be quite

L. K. MOLEKO AND H. R. GI.YDE 30

sccUI'Rtc vfhcQ vscRQclcs 8,I'c RbscQt.Finally, we find here that there is no Lindemann rule

for melting in the quantum limit. In quantum melting 5can take any value, 0.04&5&0.35. This suggests thatLindemann's rule is not related to the absolute stability ofcrystals. However, Lindcmann's rule is found to hold ac-curately for thermal melting in the classical crystals'with 5=0.16+0.01. This suggests that at the critical 5the classical free energy of a crystal exceeds that of the

coIYcspolldlllg hquld RpploxlnlRtcly 1ndcpcndcntly of tllcinteratomic forces.

ACKNOWLEDGMENTS

%Vc tllallk DI'. S. T. C11111 fol' valualllc dlscllsslons R11dNSERC (Canada) and the University of Delaware for sup-port of this work.

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