Superconductivity in the fullerenes

ourcalculation(theneglectoflattice

dynam-icsandthe useofperfect crystallinelattices). Both plasma models may be compared to the Monte Carlo simulations of Hubbard andDeWitt (9) (at 10.5 Mbar),who used

the Lindhard dielectric function to screen thebare Coulomb interaction between the ions (dashed curve in Fig. 3 labeled Lind-hard). TheLindhard resultsareverycloseto both the OCP-LM and ion-sphere curves butveryfar fromourcalculations.

Ourphase-separation temperatures were calculated forapressure of 10.5 Mbar but arevirtuallyindependentof pressure in the rangefrom5 to20Mbar. However,atlow pressures hydrogenis amolecularinsulator and becomes a molecular metalnear2Mbar (5). Van den Bergh and Schouten (17),

usingpairpotentialsfittolow-pressure

ex-periments, found that forpressures upto1 Mbarhelium will becompletelymiscible in insulating, molecularhydrogen for temper-atures above approximately 2500 K. The miscibilityof helium in metallic, molecular hydrogen is unknown. Our calculations do not applydirectly tothis molecular phase,

but inanalogywith themetallizationofpure helium we expect that theprecise pressureat which molecularhydrogenmetallizes will be unimportant withregardtothemiscibility. Rather,weexpectthat thephase-separation

temperatures will increase monotonically fromvandenBergh and Schouten's values

at 1 Mbar to the values we calculate at

sufficiently highpressure,wherehydrogenis an atomic (rather than molecular) metal. Additional TE calculations for alloys con-taining molecular hydrogen are needed in order to accurately determine the behavior of themiscibilitygap below5 Mbar.

The primary conclusion obtained from theresults ofourTEcalculationsis that it is

crucialtotreatthe electronicstructure

accu-ratelyinordertoobtainthecorrect

thermo-dynamics for

hydrogen-helium

mixtures at

megabar pressures. The electronic energy makes a large contribution to the phase-separation temperature. This temperature could be small only if we have severely underestimated the thermal contribution. Barring this circumstance,our phase-separa-tiontemperature of15,000+-

3,000

Kfora 7% helium mixture confirms that the fluid interiorof Saturn hasatleastpartially

phase-separated, because the maximum tempera-ture in the fluid is estimated to be only 10,000 K. The estimated temperatures in the fluid interior of Jupiter range from 10,000 K nearthesurface to20,000 K at the central core (1). Thus our calculation predicts that phase separation has also be-gun inJupiter. Inview ofthis prediction, thefactthat the currentlysuccessful evolu-tionary models of Jupiter do not need to

15NOVEMBER1991

invoke phaseseparationmayindicate a fail-ure of these models. Alternatively, phase

separation may haveoccurredtoolatein the

evolution ofJupitertoprovideasignificant internal energy source up to the present time. Ineithercase,newevolutionary

calcu-lations are needed to resolve this dilemma and to confirm that the inclusion of phase separation leads to a consistent model of Saturn.

REFERENCES AND NOTES

1. A. S. Grossman, J. B.Pollack, R. T.Reynolds,A. L. Summers, H. C.Graboske,Icarus42,358 (1980); D. J.Stevenson, Annu. Rev. Earth Planet. Sci. 10, 257 (1982).

2. D. J. Stevenson and E. E. Salpeter, Astrophys. J. Suppl. Ser.35,221 (1977);ibid., p. 239. 3. R. A.Handl,B. J. Conrath, V. G. Kunde, J. C.

Pearl,J.A.Pirraglia,Icarus53,262 (1983). 4. R.Smoluchowski,Nature215,691 (1967); E. E.

Salpeter, Astrophys.J.181, L83 (1973). 5. T. W. Barbee m,A. Garcia,M. L.Cohen,J. L.

Martins,Phys. Rev. Lett.62, 1150 (1989). 6. D. A.Young,A. K.McMahan,M.Ross, Phys.Rev.

B24,5119 (1981).

7. D. J.Stevenson,ibid.12,3999(1975).

8. E. L.Pollockand B. J.Alder,Phys. Rev. A15,1263 (1977); J.P. Hansen,G. M.Torme,P.Vicillefosse, ibid. 16, 2153 (1977).

9. W. B.Hubbard andH. E. DeWitt,Astrophys. J. 290,388(1985).

10. J. J.MacFarlane,ibid.280, 339 (1984). 11. See, forexample, R.0.Jonesand0.Gunnarsson,

Rev.Mod. Phys. 61, 689 (1989).

12. See,forexample, J.DeVreeseEd., Electronic Struc-ture,Dynamics, and QuantumStructural Propertiesof Condensed Matter(Plenum,NewYork, 1985). 13. J. E.Klepeis, K. J. Schafer,T. W. BarbeeIII, M.

Ross,Phys. Rev.B, in press.

14. B.Brami, J. P. Hansen, F.Joly, Physica A95,505 (1979); S. Ichimaru, H.Iyetomi,S. Ogata,

Astro-phys.J. 334, L17(1988).

15. G. S.Stringfellow,H. E. DeWitt, W. L.Slattery, Phys. Rev. A 41,1105(1990).

16. E. E.Salpeter, Aust. J. Phys. 7, 373 (1954). 17. L. C. vanden Bergh and J. A.Schouten, J. Chem.

Phys. 89, 2336(1988).

18. We aregratefultoM. vanSchilfgaardeforproviding the linearmuffin-tin orbital (LMTO) computer codes used in the calculations.Wealsoacknowledgehelpful discussions with A. K. McMahan. This work was performed undertheauspices oftheU.S. Department of Energyby the Lawrence Livermore National Lab-oratoryunder contract W-7405-ENG48.

8July1991;accepted22August 1991

Superconductivity

in

the

Fullerenes

C.

M.

VARMA,

J. ZAANEN, K. RAGHAVACHARI

Intramolecularvibrations strongly scatter electrons near the Fermi-surface in doped

fullerenes. A simple expression for the electron-phononcoupling parameters forthis caseisderived and evaluated by quantum-chemicalcalculations. The observed

super-conductingtransition temperatures andtheirvariation with lattice constants can be

understoodonthis basis. To test the ideasandcalculationspresentedhere, we predict thathigh frequency

H.

modes acquire a width of about 20% of their frequency in

superconductive fullerenes, and soften by about 5% compared to the insulating

fiulerenes.

T HEEXCMIINGDISCOVERY OF SUPER-conductivityinmetallic

fullerenes

(1) leads us to inquire whether the clas-sicmechanismfor superconductivity,

name-ly, effective electron-electron attraction via

theinteractionofelectrons with vibrations of theions, isapplicablehereaswell. Asso-ciated with this isthe question of whether the direct electron-electron repulsion in Fullerenescansuppress conventional singlet

pairing.Inthispaper we exploit thespecial nature of cluster compounds to derive a

particularly simple expression for electron-vibrational coupling from whichparameters ofthesuperconducting state of

fullerenes

are easily calculated. Further, we present argu-ments why the effective repulsions in

fullerenes are no different than in conven-tional metals.

The lattice vibrations couple to the elec-tronic states of metallic

filllerenes

in two

AT&TBellLaboratories, MurrayHill,NJ 07974.

ways:by causing fluctuations in the hopping rate ofelectrons from one molecule to the other and by causing fluctuations in the electronic structure of a single molecule. The covalent interactions thatsplit the molecular states,which form the bands in the metallic

state,

areoveranorderofmagnitudelarger than the inter-molecular covalency. This is reflected in the intra-molecular splitting

Wintra=

20 eV(2, 3)comparedtothe width of the

tlu

bands which is

Winter

= 0.6eV (4,

6

,

tig

/ > A

L=5

/

tlu

\ D

h~~U

Fig. 1. Artists conception of theelectronic struc-ture of thefullerenes,asinferred from photoemis-sionspectroscopiesandelectronic structure calcu-lations. A denotes the alkali s level.

electron-phonon coupling(5).

Photoemission studies (7), in combina-tion with electronic structure calculations (2-4), have revealed aclear picture of the electronic structure of the fullerenes. The deviation from spherical symmetry of the icosahedra affects the 11 L = 5 spherical

harmonicstates(3),sothatthey splitintoa fivefold degenerate

h.

and two threefold degenerate

tl

and

t2.

manifolds,

respective-ly. As shown in Fig. 1, the

h.

stateslie -2

eV below the tl states and are occupied.

The L = 6-derived

tag

states lie

-1

eV above the

tl

states,andthelatterare occu-pied on doping. The unoccupied alkali s states are roughly degenerate with the

tig

states (7). So one may confine attention

simplytothethreefolddegeneratetl states. Thesetransformasx, y,andz,andsotheir

degeneracy is split byanyquadrupolar de-formation that makes the cartesian axes

in-equivalent.Withrespectto

superconductiv-ity these quadrupolar Jahn-Teller modes, havingHg symmetry, are theonly relevant

ones. This follows from the fact that in icosahedral symmetry

tluXtiu=Ag+Tig+Hg

(1)

The

T1g

mode is asymmetric so that it cannotcouple,and the

Ag

modes donotlift the degeneracy (although they change the localenergylevel).

Thevibrational modes oftheC60 mole-cules havebeen calculated. There are eight

such

Hg

modes, each of which is fivefold

degenerate.Soof the174vibrational modes ofC60, 40 can, in a static fashion

(Jahn-Teller effect) or, more importantly, in a

dynamic fashion (dynamic Jahn-Teller

ef-fect) affectthe tl. electronic states. Let us now consider the two limiting cases. The intramolecularprocessescaneither leadto a 990

pairing energy much larger than the inter-molecular transferintegrals,orthereverse is true. The latter is found more appropriate for thefullerenes,but theformer is interest-ingtodiscuss first because of itsconceptual simplicity. Consider two molecules with av-erage charge H. The Hamiltonian for the coupled problem is a 3 x 3 matrix with elements

Hy

=E8b +

Y

hymI)QmA

+Hub (2) wherei= 1,2, 3 labels the degenerate states of

tl

symmetry,

Qm,,

are the normal co-ordinates of the m-th

Hg

modewith degen-eracy Ii = 1...5.

H,,ib

is theHamiltonian of thevibrational modes withfrequencies

w,,m

ThisJahn-Tellerproblem, involvinga three-fold degenerate electronic state interacting with afivefold degenerate mode, has been

worked out some time ago (8) and the coupling matrix is foundtobe

1 (Qms- Qr,4 -

V3Qm,l

-

V3Qm,2\

;gml 5/,Qm,i Qms+ /Q-n,4-Q^3

\-V3Qm,2 - f3Qm,3

whereg,,, is thecharacteristic

displacementofthe m-thmode, by microscopic calculations.

In the strong-coupling limil willsplitasindicated inFig. 2 le of the vibration andFig.2rig] phase (8). Now if we calcul reduction in this limit (negle(

point motion) and deduce th4 tron-electron interactionthrou]

Un=Ea+i +En1

-where

ER

is the ground sta

0

9

v

+Q

Fig.2.Thesplittingpattern of th

Hgdistortionfor thetwodiffere radialdisplacementQ.

n

Fig.3.Thelocalelectron-electron interaction(U) as afunctionofparticle number (n), inducedby theJahn-Tellerinteractions (Kisthespring con-stant). A negative U is found for an average occupation (n) of3electrons,and apositiveUfor 2 or 4electrons.

-2Qm, (5 ) averagechargen, wefind the behavioras in Fig. 3 (9). The actual relevant electron-energyperunit electron interactionparameteris in this

lim-tobeevaluated it, of course

U;

=

U;

+

Uc,

where Ul

e2IR

= 0.5 to 1 eV (£ ~ 4) comes from

t, the

tl

levels directelectron-electron interactions.

U;

and ft foronephase

Ue

havein generaldifferent highfrequency ht for the other cutoffs.

late the energy This picture, which could yield an

effec-cting the zero- tive attractiveinteraction, would beagood

c effective elec- starting point for furtherdiscussion if-0;

gh

>>

W.

the conduction electron bandwidth 2Eii (4) (and the

phonon

frequency).

One would 2E; (4) then have localpairing (10) anda supercon-ate energy for ducting transition temperature of O(W2/

101),

providedastaticJahn-Tellerdistortion with a possible phase difference from one molecule to the other, although a simple staggeredcharge densitywave is frustrated

inthefcc-lattice(11)were nottooccur.For

IUZI

<

W,

as we find

below,

the above

9

adiabatic picture is not valid-an electron runs awayfromamoleculebefore it devel-ops pairing correlation with another

elec--- trononthesamemolecule. Then the

physics

wedescribed above shouldbe incorporated g/2 intothetraditional way ofconsidering

elec-91 tron-phonon scattering in metallic bands

_________ (6).

The40intramolecularHg modes are ex-pected to be nearly dispersionless in the solid state. For this case, the electron-phonon Hamiltonian is particularly simple,

-Q

andmaybewritten as

_Q

i)

He-ph

=

h,

hK'k'(mp)1C

+KkJK'a

etl levelbythe

KkKa~li

ntphases of the

IQkk$k'm,

+

X (Qmji,k-k' + Q mpa-k) (5) SCIENCE, VOL. 254

Herethescatteringmatrix element

hKCKoko(mI.)

=

A*Kkhy$m

g)

AK'ky

(6)

where

A,,.

are the elements of the linear transformation from molecular levels to band states-k is themomentumandK are theband indices.

Fors-wavesuperconductivity,inweakto intermediate coupling, one is interested in

time-reversedstates near theFermi-energy.

Soonemayconfineattentiontointra-band

pairing.Forthepresent case,the

dimension-less electron-phonon coupling constant X has a particularly simple form, which is easilyderived from the

general

expressions

(12, 13)

N(0)2

A=2

-I'

(7) mp. M(Rm

I2mL

=

N(0)-2

J dSkK KK' VkK X |-|

hkKkK(mu)12

(8)

Vk OK

where the integrals are over the Fermi-surface. Using the fact that the degeneracy

of the

tl1

levels is not lifted in a cubic environment (14)wefind

~~

I

2

~~~(9)

6 m

MWm

We can now calculate the

superconduc-tive transition temperature TC throughthe

approximate solution of the

Eliashberg

equation, given the intra-molecular defor-mationpotentials

gm.

These, and the vibra-tionalfrequencies,werecalculatedusingthe

quantum-chemical MNDO semi-empirical

technique (15). This method has been

suc-cessfully used previouslyon awidevariety of

100 80 0 60~~~~ 40 0.2 20 0 5 10 15 N(0)

(eV-'[Ce0]

1)

Fig.4. T, (inkelvin)as afunction of physically likely electronic densities of orbital statesN(O) andCoulombpseudopotentials (ii*).

Table 1. Experimental (20) and calculated frequencies

(meWXP,

(,m,

in cm-1) and deformation potentials (g,B, in

eV/A),

and thecorresponding couplingconstants Xm forN(0) = 10perspin per eV perC60moleculefor theeight

Hg

modesofa

CjO

molecule.

Wm p 273 437 710 744 1099 1250 1428 1575

Wm 263 447 711 924 1260 1406 1596 1721

gm 0.1 0.1 0.2 0.0 0.6 0.2 1.8 1.2

Xm 0.03 0.01 0.01 0.0 0.06 0.0 0.34 0.11

carboncompounds containing five- and six-membered rings and is known to be reliable

for structures, relative energies and

vibra-tional frequencies (16). For example, the

MNDO bond-lengthsinC60 1.47 and 1.40

A

(17) are in excellent agreement with the besttheoretical (18) and experimental (19) estimates. Weevaluated thecompletematrix

offorceconstants and calculated the associ-ated normal coordinates for all 174 vibra-tions inpureC60. InTable 1 the resultsfor theeightRamanactive

Hg

modefrequencies

areshownwhichhaveameandeviation of only 10% from experiment (20) (Table 1). Wecalculated the deformation

potentialsgm

bya frozen-phonontechnique. The energy of a

CG0

molecule willdependonthe ampli-tude of a frozen-in

Qm,5

photon as E =

-gmQm.s

+

KisQ~,/2

(K,.

is the

spring

constantin Eqs. 2 and 3). For each of the

eight

Hg

phonons we selected this compo-nent

(-3z2

-

r2),

and we distorted the

negatively chargedC60moleculealong these

normalcoordinates. The initial slope of the energyasafunction of the distortion ampli-tude yields then the deformation potentials. InTable 1the results for thedeformation potentials are summarized, together with the coupling constants for the individual

modes Xm) as calculated from Eqs. 7 and 9 with the observed phonon frequencies in

C60and the phononrenormalization calcu-lated below. We find that the two highest modes near 1428 and 1575

cm-'

are the most strongly coupled. The reason is that these high-lying modes involve bond-stretching, compared to the bond-bending

characterizing the lower frequency modes. The formerleads to the maximum change in energyforagiven (normalized) distortion.

Assuming asquare density of states and a

bandwidth, which is

-0.6

eVaccording to bandstructure calculations, N(0) would be

-5

orbital statesper electron volts per C60 molecule. However, the band structure cal-culations (21) give figures for

N(0)

which aremorethan twiceaslarge and experimen-tally

No

- 10 to20

[HI:,,

HC2, normal state

susceptibility

(22)].

From Table 1 we find that the overallX=

IXmX

isin therange of -0.3 to 0.9, and together with the large range in frequency for which the interac-tions are attractive, high

Ta's

are to be

expected. The large electron-phonon

cou-15NOVEMBER1991

pling also leads to a strong decrease in the phonon frequencies and a corresponding

increase in their linewidth = Ymm. It is

knownthat (24)

Aoom Xm Ym

-

S-;

WM

rN(O)jAw,,mi

(10)

Sowepredictadiminutionof about 5% in thefrequency of the high

Hg

modes anda linewidth increase of order 300

cm-",

com-paredtopureC60.

We nowpresent an approximate calcula-tion of

T,.

For X < 1, McMillan (13) has

given a very good (23) approximate solution totheEliashberg equations(12)

Wav [

1.04(1

+

X)

1

TC

j-

_1.2exp E _XA-

_g*(

.(+.2)₊

_0.62A)

(1

1)

Mostof thecoupling strength is in the two

highest lyingmodes,sothe usual averageof

the phonon frequencies (23)

(w1lg)

is not appropriate. We find

{X

Xmln[Iwm(1

- Xm)]} (12) abetterapproximation. Eq. 11

includes

the

Coulomb pseudo-potential parameter ,u*. Because the approximations such as those

due to Migdal (25) do not workfor elec-tron-electroninteractions,it isimpossible to estimate p.* (incontrast toX).Traditionally u* is usedas afittingparameter in compar-ing Tc, A(0) and the tunneling spectra to theory.Forinstance, forPb p.* =0.12 (23).

[L*

foraFermi-level in a well-isolated band is smaller than the screened repulsion parameter

g.

byafactor [1 + gin

(WIwav)]

(26),where Wis the smaller of EF and wp, the plasma frequency.

EF/Wav

in the Fullerenes is smaller

than,

say in Pb, by

_

102.

However,if one notes that the actual

electronicstructureof metallic

C60

is a lad-der of bands of width

--1

eV, spread out over20 eV,andseparated from each other by energies also of order1eV, andconsiders thecalculation of p.* in this situation, one concludesthatp.*isclose in value tothat of awide band metal.

We present inFig.4 Tcversus

N(0)

for variousvalues of p.* (27).

Tc

has a particu-larlysimplerelation to

N(0)

andtherefore to the nearest-neighbor C60 distance d in the

Fullerenes, because the other factors are

intramolecular and do not depend on d. Figure 4 shows that Tc ~ N(0), in

agree-ment with a recent compilation of lattice

constants, calculated densitiesofstates and Tc's (21). Given the physical fact we used that mostofthecouplingisintramolecular, our estimateofXfrom the

H.

modes should be asgood as the determination of vibration frequencies, that is, good to about 10%. One worry is that ourcalculation of

g,

is based on the deformations of a

Cj0

mole-cule, whereas the more appropriate calcula-tionwould have aneutralizing background. TheMigdal approximation fordetermining

Tc is onlygood in our case to

(wa5/EF)

-1/5.

Forlow density ofstates obtainable by smalldoping, we expect theCoulomb inter-actions to dominate. Inthatcasethe intra-molecular Hund's rule coupling (owingto orbitaldegeneracy) plus the almost empty

band usually favors ferromagnetism (28). This may be thesimplereasonfor therecent observationofferromagnetismin the

com-poundTDAE1C60 (29).

Note added inproof: InrecentRaman mea-surements Duclos et al. (30) find that the twohighest frequency

Hg

phonons, which we find couple most strongly to the elec-trons(Table 1), areclearlyseenin

C60

and K6C60 butdisappearin thesuperconducting

compound K3C60. This is consistent with our prediction based on Eq. 10 for their linewidth.

REFERENCES AND NOTES

1. A.F. Hebard et al.,Nature350,600(1991);M.J. Rosseinskyetal.,Phys.Rev.Lett.66,2830(1991). 2. R.C.Haddon,L. E.Brus, K.Raghavachari,Chem. Phys. Lett. 125, 459 (1986); M. Ozaki andA. Takahashi,ibid.127,242(1986).

3. S.Satpathi,ibid. 130, 545 (1986).

4. S.Saito and A.Oshiyama,Phys.Rev.Lett.66,2637

(1991).

5. J. L.Martins, N.Trouillier,M.Schnabel, preprint (University of Minneapolis, Minnesota, 1991). 6. S.Barisic,J.Labbe,J.Friedel,Phys.Rev.Lett.25,

419(1970);C. M.Varma,E. I.Blount,P.Vashista, W.Weber,Phys. Rev. B19,6130(1979);C. M. Varma and W.Weber, ibid.,p. 6142.

7. C.-T. Chenetal., Nature, in press;J.H.Weaveret al.,Phys.Rev.Lett.66,1741(1991);M. B.Jostet al.,Phys.Rev.B, in press.

8. M.C.O'Brien,J.Phys.C:SolidStatePhys.4,2524 (1971).

9. Infact,instrongcouplingoneshouldconsider the truen-particlestatesinstead ofsimple productstates asassumed in thetext.Thisleadsto anincreaseof phononphasespace,which howeveraffectsthe2, 3, and 4particlestatesinsimilar ways. Note that in this case,the lowspinstateswould beconsidered,and

Jahn-Tellerinteractions thusgiveriseto anegative Hund's rulecoupling.

10. C. M.Varma, Phys.Rev.Lett.61,2713(1988);R. Micnas,J.Ranninger,S.Robaskiewicz,Rev.Mod. Phys. 62,113 (1990).

11. F.C.Zhang,M.Ogata, T.M.Rice,preprint (Univer-sity ofCincinnati,Cincinnati,OH, 1991).Thispaper discusses thepossibilitythat the alkali-ionvibrations may leadto anintramolecularelectron-electron attrac-tion. Wefind thisunlikelybecausetheyscreentheir repulsive interactions withoutscreeningtheattractive interactions.Besides,such vibrations haveafrequency

of 0(200 cmn'),much smaller than the molecular vibrations we consider and thus are unlikely to be of importance indeterminingT.

12. G. M. Eliashberg,Zh.Eksp. Teor. Fiz 38, 966 (1960); ibid. 39, 1437 (1960) Sov. Phys.-JETh 11, 696 (1960); ibid. 12, 1000 (1961).

13. W. L.McMillan, Phys. Rev. 167, 331 (1968). 14. Inthederivation of Eq. 9 we missed the factor 5/6

which was kindly pointed out to us by M. Lannoo. 15. M.J.S. Dewar and W. Thiel, J. Am. Chem. Soc. 99,

4899(1977); ibid., p. 4907.

16. M. D.Newton and R. E. Stanton, ibid. 108, 2469 (1986).

17. R. E.Stanton and M. D. Newton, J. Phys. Chem. 92, 2141 (1988).

18. M. Haesen, J.Alimlof, G. E. Scuseria, Chem. Phys. Lett. 181, 497 (1991); K. Raghavachari and C. M. Rohifing,J.Phys. Chem., in press.

19. C. S. Yannoni et al., paper presented at the Materials Research Society Meeting, Boston, 1990. 20. D.S.Bethune,Chem.Phys. Lett. 179, 181 (1991). 21. R.M.Fleming,Nature352, 787 (1991). 22. A. P.Ramirez, M. J. Rosseinsky, D. W. Murphy,

R.C.Haddon,in preparation.

23. P. B.Allen and R. C. Dynes, Phys. Rev. B 12, 905 (1975).

24. C. 0.Rodriguezetal., Phys. Rev. B 42, 2692 (1990). 25. A. B.Migdal, Zh. Eksperim. i Teor. Fiz. 34, 1438

(1958);Sov.Phys.-JETP7, 996(1958). 26. P.Morel and P. W. Anderson, Phys. Rev. 125,

1263(1962);P. W.Anderson,inpreparation. 27. Anindependent calculationofthe electron-phonon

coupling in the Fullerenes has been carried out by M. Schluter et al. (in preparation) using a tight-binding method. Our results differ in some ways from theirs.They find the major contribution from the -400 cm- and -1600 cm-' H phonons,

where thelatter contribution is about half that of theformer. Intheir calculation of Tc they use an expressionaveraging over all the phonon frequencies irrespective of their coupling constants.

28. We envisage a competitionbetween Hund's rule cou-plings and phonon-induced attractive interactions. We note theinteresting possibility discussed by S. Chakra-varty and S. Kivelson(Europhys. Lett., in press) that in a Hubbard modelforC60dusters, the effective elec-tron-electron interaction is repulsive for small U; that is, the Hund's rule isobeyed, but for larger U it may change sign due to configuration interactions. 29. P. M. Allemand etal., Science 253, 301 (1991). 30. S. J. Duclos, R. C. Haddon, S. Glarum, A. F.

Hebard, K. B. Lyons, in preparation.

31. Wethank B. Batlogg, R. C. Haddon, S. J. Duclos, I. I. Mazin, M. F. Needels, T. T. M. Palstra, A. P. Ramirez, and M. Schluter forhelpful discussions. Weparticularly thank M.Lannoo forcarefully going through the manuscriptand pointing out several compensatingnumerical errors. J.Z. acknowledges financial support by the Foundation of Fundamental Research on Matter(FOM), which is sponsored by the NetherlandsOrganization for the Advancement of Pure Research(ZWO).

23August 1991;accepted3 October 1991

Long-Term History of Chesapeake

Bay

Anoxia

SHERRi

R.

COOPER

AND

GRACE S. BRUSH

Stratigraphicrecords fromfour sediment corescollected along atransect across the

Chesapeake Bay near themouth of the ChoptankRiver were usedto reconstruct a 2000-yearhistory ofanoxia andeutrophicationin the ChesapeakeBay. Variationsin

pollen, diatoms, concentrationoforganiccarbon,nitrogen,sulfur, acid-soluble iron,

andanestimateofthedegree of pyritizationof iron indicate that sedimentation rates, anoxic conditions andeutrophicationhaveincreasedintheChesapeakeBaysince the timeofEuropean settlement.

SINCE THE OCCURRENCE OF ANOXIA

was first reported in the Chesapeake Bay in the 1930's (1), the relative importance of climate versus eutrophication and otheranthropogenicinfluences onthe extent and duration of anoxic events has been debated(2-6). Large-scale monitoring

of the chemical and physical properties of the Chesapeake Bay began as recently as 1984 (7). Information on the long-term

historyof anoxia in the Bay is available from

stratigraphic records, accessed through

pa-leoecologicalmethods (8). In this report, we describe datafrom thestratigraphic records toevaluateconditions over time in the me-sohaline (moderately brackish) section of theChesapeake Bay.

Four sediment cores were collected in May 1985alonga transect across the Ches-apeake Bay from the Choptank River to PlumPoint,Maryland (Fig. 1). This region of the Bay iscurrentlyanoxic atleastpart of

Department ofGeographyand Environmental Engineer-ing, Johns Hopkins University, Baltimore,MD 21218.

each yearSurface salinities for this area of the Bay average between8and 15 parts per thousand (ppt) in spring and fall, respec-tively; salinities in the bottom water are consistently higher and not usually less than15 ppt (9). Thecoresranged in length from 114 to 160 cm, and have adiameter of5.7cm.The sediment in allcores was a

relativelyuniform mix of gray silt andclay,

undisturbed by mixing or bioturbation. Eachcore was cut into 2-cm intervals. The bottom sediments were dated by the car-bon-14 technique and the other samples weredatedonthe basisofpollen horizons and pollen concentration techniques (10). Theagricultural pollenhorizon was recog-nizedbyasharp increase in concentration ofragweed polleninrelationtooakpollen

and was dated as A.D. 1760 on the basisof historical records of land use in the area. Several controls were used to check the accuracy of both dating methods. For ex-ample,asedimentsampletaken from above the agricultural pollenhorizon inanearby

core could not be dated by radiocarbon