Volume39A, number4 PHYSICSLETTERS 22 May 1972

SUPERFLUIDITY IN NEUTRON STARS

G. CHANMUGAM, R.F.O’CONNELL andA.K. RAJAGOPALDepartmentofPhysicsandAstronomy,LouisianaStateUniversity, BatonRouge,Louisiana 70803,USA

Received 28 March 1972

Argumentsarepresentedto show thatthe BCS theoryof superfluidityin its original formmaynot be applicableto neutronstarmatterover awide rangeof density.

The recentdiscoveryof pulsars,which arebelievedtobe neutronstars, hasstimulatedconsiderablein-terestin neutronstarmatterand inparticularits super-fluidity [1—3]. Severalauthors [3—6]havecalculatednumerouspropertiesof superfluidneutronmattersuáhastheenergygap~, critical temperatureTc. Allof these calculations (with theexceceptionof ref. [4])haveassumedthat theuse of theBCS [7] Hamiltonian isjustified in describingsuperfluidneutronstarmatter.Thepurpose of thisletteris to point outthat theBCS theoryin its original form maynotbe applicableto neutronstarmatterat densities~ 10~’lg/cm3.

An importantparameterin theBCS theoryis thecoherence length~= F/~ whereVF is the Fermivelocity. Inmetals~ is solarge(~10—~cm) that thereare a largenumber ofoverlappingCooper pairsover arange of the order of~. Thus, if thenumberdensity ofelectronsin a metalis (~irr~)1, we have~/r

5~l0~ ina typical superconductor.On the otherhandin neutronstarmatter(for definitenesswe shalluse the resultsofØstgaard[6] we have* for theneutronFermimomen-tum kF = 0.5 fm

1, ~3.l3 MeV so that~/t~0.6.Herers correspondsto neutrons.One notesalso thatL~/EF~O.6whereas~/EF’~l03 to i0~in metals.

In orderfor the BCStheoryto be applicable,thecorrectionsto theBCS ground stateenergydueto thedifferenceH’ between theHamiltonianH and there-duced Hamiltonian Hred mustbe small. This canbeestimated inperturbation theoryand onerequiresthatW’, thecorrectionto the energydue to H’, be smallcompared withW

0 (whereW0 is thedifferencebe-tween thenormal and superfluid freeenergy). Thishasbeen estimated byBCS [7, eq. (A 14)] for metals and

* Our valuefor ~ is slightly greaterthanthevaluederivedby

Østgaard because he uses the weakcoupling approximationfor ~ whereaswe usetheexactexpression.

it was found thatW’/W0~l03, evenafterover-

estimatingW’. For our system a morerealistices-timatewould be made by carryingout the sums[7, eq. (A 14)] in the evaluationof W’ overa region2~(insteadof 6~)wide sincehere~ 1 unlikeformetals.We then findthat

w’ 31N(0)V]~

W0 1—exp(—2/N(O) I’) EF

HereN(0) is thedensityof statesperunit energyatthe Fermisurface,andV is theaveragematrix ele-mentfor pairs in ashell of thickness flw nearEF. Forneutronstarmatter,at kF = 0.5 fm—

1, one hasW ‘/W

0~1.2 which meansthat theBCS approximationbreaksdown. Thephysicalreasonfor this is thatE/r5is of orderunity which impliesvery little overlap ofCooper pairs. Infactforneutronstarmatter(whereIIW~EF)one notesthatW’/W0-eooas~/r5-+OwhileW’/W0-~Oas~/r5_~oo.

Wealsonote that W’/W0 decreaseswithkF. AtkF = 1.0fm

1, W’1W~’-0.1.Hence,the BCStheoryin its original form maynotbe applicablein neutronstarmatterwhen kF ~l.0 fm~.Thisregion fallsentirelywithin the crust of theneutronstar [8]. YangandClark[4] have approached the problem inanessentiallydiffer-ent way andassuch it is difficult to compare theirresults with theBCS approachdiscussedin this letter.Finally, we emphasizethat ourremarksare restrictedto the model of superfluidityinvolving s-waveparing.

References

[1] G. Baym, C. Pethick,D. PinesandM. Ruderman,Nature224 (1969)872.

[2] P. Sutherland,G.Baym, C. Pethick,andD. Pines,Nature225 (1970) 353.

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Volume39A, number4 PHYSICSLETTERS 22 May 1972

[3] V.L. Ginzburg,J. Stat.Phys.1(1969) 3;M. Hoffberg,A.E. Glassgold,R.W. Richardson andM.A. Ruderman,Phys.Rev.Lett. 24 (1970)775.

[4) C.H. Yang andJ~W.Clark,NuclearPhys. A174(1971)49.

[5] R. Tamagaki,Prog.Th. Phys.44 (1970)905.

[6] E. østgaard, Z. Physik 243 (1971) 79.[7] J. Bardeen, L.N.Cooperand J.R. Schrieffer,Phys.Rev.

108 (1957) 1175.[8] G. Baym, C. Pethick and P. Sutherland, Ap. J. 170 (1971)

299.

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