The tensor force and collective excitations in nuclear matter

1.B:I.C [

Nuclear Physics A205 (1973) 398--412; (~) North-Holland Publishing Co., Amsterdam

Not to be reproduced by photoprint or microfilmwithout written permission from the publisher

THE TENSOR FORCE AND COLLECTIVE

EXCITATIONS

IN NUCLEAR MATTER DAVID M. CLEMENT Department of Physics, University of California, Los Angeles, Calif. 90024 t Received 16 November 1972 Abstract: The formalism for the particle-hole Green function is developed to handle the nucleon-

nucleon tensor potential in order to investigate both the ring correlation energy and collective excitations in nuclear matter. The formalism is exact for direct ring diagrams. An approximation for exchange ring diagrams is also introduced. The one-pion-exchange potential with cut-off radius is used throughout. No collective excitations are found coming from the tensor force.

1. Introduction

Ring d i a g r a m s describe the scattering o f particle-hole pairs in a m e d i u m . Their i m p o r t a n c e as a class was first studied in the metallic electron gas p r o b l e m where divergent s e c o n d - o r d e r results in the c o r r e l a t i o n energy were observed due to the longrange C o u l o m b potential. However, when the entire series o f ring d i a g r a m s was s u m m e d by G e l l - M a n n a n d Brueckner t) a n d by H u b b a r d 2), finite results were obt a i n e d *. H u b b a r d ' s a p p r o a c h , a n d the one which we follow, was to develop the G r e e n function technique. One o f the chief virtues o f this a p p r o a c h , f r o m o u r p o i n t o f view, is t h a t in the ring a p p r o x i m a t i o n a clear s e p a r a t i o n can be m a d e in the c o r r e l a t i o n energy between the c o n t r i b u t i o n s f r o m single-particle m o d e s a n d collective m o d e s o f excitation. Thus without a separate calculation one o b t a i n s k n o w l e d g e o f the collective m o d e s o f the system. O u r objective is to find the effect o f the t e n s o r force on such excitations in nuclear m a t t e r using, in particular, the o n e - p i o n - e x c h a n g e p o t e n t i a l (OPEP). The effect o f central potentials has been investigated by A m u s y a 6), D o v e r a n d L e m m e r 7), a n d G l a s s g o l d et al. s). Collective excitations in a F e r m i system are observed experimentally as a r e s o n a n t response to a w e a k external p e r t u r b a t i o n whose F o u r i e r c o m p o n e n t couples to the density fluctuation o f the system. This density fluctuation is essentially the particlehole G r e e n function. Regardless o f the actual range o f the t w o - b o d y p o t e n t i a l in the p r o b l e m , the collective b e h a v i o r is a long-wavelength p h e n o m e n o n , i.e., involves small transfers o f m o m e n t u m f r o m the external field to the system. The r e a s o n is t h a t the c o o p e r a t i v e b e h a v i o r o f m a n y fermions can be i n d u c e d only i f the wavelength o f the t Work supported by the National Science Foundation. Pines' book on the many-body problem 1o) provides a concise review of this subject. Fetter and Walecka's text 11) works out many of the details. We will refer to the latter extensively in developing our formalism in sects. 2 and 3. 398

NUCLEAR-MATTER TENSOR FORCE

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external perturbation is much larger than the interparticle separation. For this reason we refer to these collective phenomena as induced long-range correlations. In the nuclear-matter problem the contribution of ring diagrams to the ground state correlation energy has been calculated by Dahlblom and his collaborators to fourth order for both the direct a) and exchange 4) diagrams. They employ the OPEP potential with a cut-off radius. They find that the series converges rapidly, and that thirdand fourth-order diagrams contribute on the order of an MeV attraction to the correlation energy at normal density. The third-order ring diagrams have importance in their own right since they also represent the lowest-order expansion of the three-body cluster energy. The Green function formalism, which we develop in sects. 2 and 3 to handle the tensor force, enables us to obtain a compact expression for the correlation energy. We advertise that the formalism sums the direct ring diagrams exactly, and further, that as a calculational device it is more efficient than a term-by-term summation such as Dahlblom's 3, 4) since the final expression only involves a double integral. The technique is valid for any local central or tensor potential, and the test for applicability is whether the Fourier transform can be written only as a function of the momentum transfer. The advantage such a potential has in our formalism is that the integral equation for the Green function in the ring approximation is factorable and that each mode is associated with a definite momentum transfer which is uncoupled from the others. We also develop an averaging procedure which takes into account exchange ring diagrams in the correlation energy. It turns out that the approximation is accurate to 10 ~ for the second-order exchange diagrams, but somewhat less accurate for the third- and higher-order ones. Our standard of comparison here is Dahlblom and Kouki's exact calculation 4). Results and conclusions are given in sect. 4.

2. Particle-hole Green function Let the Hamiltonian of a system of many fermions be H = H o + Ha, where H o = ~, e~ N(C*~ C~),

(1)

H1 = ½Z (~BI Vh,a)N(C2 C~ C*pCa).

(2)

=#76

Here the C are the ordinary fermion creation and destruction operators, and N represents the normal ordering with respect to the unperturbed non-degenerate Hartree-Fock ground state ~ 0 or H0. The expectation value of the interaction Hamiltonian H a in the exact ground state ~k0 can be written in terms of the particle-hole Green function G~raa(z) = - i(~'o1ZEC~(z)C,(z)C~(O)Ca(O)]l~'o), (3)

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where

C~('c) = e'ta~/hC=et -m~/n, and T represents the time ordering of the fermion operators. Then ( H 1 ) = ½i ~ (ctflJ I / J y f i ) D ~ ( z ~ 0+),

(4)

D~,rp,('r) = G=rlj~(z)-(o)G:,ra,(r )

(5)

where is the density fluctuation. Here, (0)G is the particle-hole Green function for the noninteracting system. Its presence in eq. (4) is needed to remove from ( H 1 ) the firstorder interaction energy already contained in the single-particle energies e~ of Ho. The ground state correlation energy is obtained by the formal device [ref. ~1), pp. 69-70] of introducing a linear interaction strength 2 for adiabatically turning on the interaction H 1 (2), i.e., H(2) = H o + H , ( 2 ) = H o + 2 H ' , . Then it is easy to show that the ground state energy Eo(2) satisfies dE° - (0o(2)1 d2

H1(2--~)10o(2)), z

and therefore the correlation energy Ec, or energy shift, is just

E c = ½i 2 (~fljVJy5) a,8},~

fo

d2(a)D=,t3~(z ~ 0+),

(6)

where all of the interactions in (X)D now have the form 2I,'. Both E¢ and D(x) have a diagrammatic linked cluster expansion in terms of Goldstone, or time-ordered, graphs. Some ring diagrams for the correlation energy are h-ht+O

Q

o (a)

Q

(c)

o (b)

h-h%Q

(d)

Q

(e)

---Q--e,

Q (f) Fig. 1. Some second- and third-order Goldstone ring diagrams for the correlation energy.

NUCLEAR-MATTER TENSOR FORCE

401

represented in fig. 1. For our purposes, however, it is more convenient to work with the Fourier transforms of G(z) and D(Q, e.g.,

(~),

D~,(,) = =

dze

D~,a~(Q.

(7)

Once all the time integrations are performed G(~o) and Dqo) have a diagrammatic expansion in terms of Feynman graphs, i.e., where the intermediate Fermion lines can propagate both as holes and as particles. The Green function a(¢o) satisfies Dyson's integral equation which in operator form reads G(¢o) = (o)G(¢o)+ (o)G(oJ)F(eJ)G(~o),

(8)

where F is a sum of irreducible vertex parts. If only the first-order vertex part - the two-body interaction V - is kept, the resulting approximation [ref. ~~), pp. 154-167] is known as the random-phase approximation (RPA). After inserting G = ( o ) G + D into eq. (8) the integral equation then simplifies to RPA RPA D~p,(¢o) = h - 1 (o)Ga??0t(o)) ~ (~"~] V]fl rZ -- ~fl I )[(o)G6,fl,fl6((o) +D6,8,fl~(fD)],

p,,~,

(9)

where V O(F-- c00(7 -- F)

(o)6~p~(o)) = h6~6ar L h o g - ( ~ - ~ ) + i t /

-

~O(ot~- F)O(F ~ J- ~,) ]

.

'

(10)

here 0(x) is the step function and F refers to the Fermi level. The poles of G(og) yield the excitation energies E, of the system [ref. 11), p. 559]. If one wants to use the RPA prescription for G(o)) in order to obtain these eigenvalues, then discard the inhomogeneous term (o)G(a)) in eq. (8). This is valid only when the poles of (o)G and G do not coincide. Then one obtains the RPA eigenvalue equation

= {Ofi,-F)O(F-oO-O(F-~,)O(~-F)}

~., (,~'~lVl~'~z-~'>ava,a~(oJ).

p'6"

(11)

The G, ra6(¢o) represent the familiar X and Y amplitudes depending on whether (a~,) represents a particle-hole or hole-particle pair. 3. Ring correlation energy in nuclear matter

We specialize immediately to nuclear matter where the single-particle wave functions are plane waves, normalized to unity in a box of volume O. Our notation for the single-particle states is (rl~) = fm-÷eS*''laaTa), 0~ ~- a, O'a~ Ta,

a -- a,

~ ~ a~ --O'a~ ~Ta~

(o0 -- a,,'r a.

(12)

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D.M. CLEMENT

The single-particle spectrum will be of the form e~ = ½;~*a2 + 0 ( ~ - F)A, 1,

}.* = h2/M *,

O(~-F) = O,

a > kF,

(13)

a < kF,

which allows for an energy gap A between occupied and unoccupied states as well as an effective mass M * . 3.1. PLANE-WAVE TWO-BODY MATRIX ELEMENTS The expressions which we develop for the correlation energy are valid for a local central or tensor potential which we represent as

V(r) = ~¢ E ALgL(r)( ~t'" yL(p)).

(14)

L=O, 2

Here • contains the isospin dependence, ~.L the spin dependence, and E L and yL are spherical tensors of rank L. The dot product is the usual one for spherical tensors. Plane-wave two-body matrix elements of eq. (14) would be "~,.J~a~bMs "~.-~¢ycCYdM"$ ~*..FZ'a.CbMT ~...."r c Z ' d M T ST

× (TMTlYlTMr)

Z

(SmslZLISm's) " vL(kca--k~b),

(15)

L=0, 2

Kab

= a+b,

Kcd = C+ d,

2kab = a - b ,

2kcd = C-- d.

The Fourier transform is defined as

vL(Q) = f d3rei~'rALgL(r)Y~(P) = 47ziLyL(~.)ALIL(Q),

(16)

tL(Q) = j o~r2dr YL(r)jL(Qr).

(17)

where

By recoupling the Clebsch-Gordan coefficients and applying the Wigner-Ekhart theorem to the spin matrix elements, we can define coupled particle-hole matrix elements, both antisymmetrized and non-antisymmetrized,

<~/~lVl#>

/ x [ S " ] ~ ( - 1)s'+u's

E r-S"S"L ""M"sM"'S~

L=o, 2 S a = ( - t ) t-*~-*',

L

vv

tt

IF~(SL Tt t , abcd) [(A)F,(S T ~ , abcd),

IS"] = 2S"+1,

(18)

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where --uFL('q"T",~ -IYL[~"T"

' abcd)t = [LS,,]-½(_I) abcd)J

(A)--~,W -- ,

x W(½½½½; T T " ) ( T M r [ J I T M r )

TM

Z A sr

kS"

i 2

S"

i) (--

I)T[T]

¢zt k~'v"'kb" c"-"; t[v~(k~a- K,,b)--( -- 1)s+ rvS(k~a + k,,b)].

(19)

Here W is a Racah coefficient, A is a normalized 9-j symbol, and S"T" refer to the coupled spin and isospin of the ph pair, both of which are conserved by a central or tensor force. We consider two special cases: (i) Spin-isospin independent central potential (L = 0):

F(ST, abcd) = 46so 6To v(kcn- k.b), (a)F(ST, abed) = 45so 5ro v(kca- k~b)-- v(kca + k.b);

(20)

(ii) OPEP potential with cut-off radius G (L = 0 or 2): e-~'[

V(r) = Voz, ' ' 2 --

a~ "a2+S,2(P)

( 3 ~)] 1+ - +

x

x = pr,

O(r-r¢),

x

/l = 0.7 fm -1,

Vo = 3.488 MeV.

(21)

This may be written in the form of eq. (14) with ,~¢ = z, • z2,

"uwz= [al xa2]~ '

AL = (6LO+SmX/~)X/12~Vo,

(22)

gz(r) = htLt)(ix)O(r--r~).

(23)

v~(Q) = 4rCX/12nVo(SLO--SL2x/Yy)Y~(O)IL(Q),

(24)

The Fourier transform is with IL(Q) = - e-ur~#Q {Q cos (Qrc)+l~Q2+#2sin(Qrc) _3~L 2 (1 + #r=)j, (Qr¢) t #:ro

(25)

J "

The F matrix elements are

F°(ST, abcd) = - 16~zVo bs~ 6rx Io([kcd--k.b[), (a)Fo°(ST, abed) = - 16n Vo{gs~ 5T~ Io([kca-- kab[) -

9 ( - 1)s+ r W(½½½½; SI)W(½½½½; T1)Io([kca + k.b[)},

rE(ST, abcd) = 16~/8n/51/o 5sl 5rx 12([k¢a- kabl)y2(k~a- kab), (a)FuZ(ST, abed) = 16rrx/8~z/5go 5s~ {5T~ [2([kcd -- kabl)~2(kea-/Cab) -

-

3(-- l)rW(½½½½; 7"1)12(1ken+ k.bl) Y2,,(kca+ kab)}.

(26)

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D.M. CLEMENT

3.2. C O U P L E D P A R T I C L E - H O L E G R E E N F U N C T I O N A N D T H E C O R R E L A T I O N E N E R G Y

Both the particle-hole Green function G and the density fluctuation D may be given in coupled representations, for example, SS', TT" DM~M's,M~M'T(a cbd) = ~ C .~cus,..o÷cMT,~.,~du,s ~½s r½~T r ~ S ' C~3~/~,TSrS~D~rt~~ .

(~/~)

(27)

Inserting eq. (18) for the coupled matrix elements into eq. (6) the correlation energy is •

1

E~ = ~ Z JK.bK~dZ [ S T L ] ~ ( - 1)s+ r f d2 2 0 .bcd STL .J o x [FL(ST. abcd) x [t~)Dss" rr(acbd, z ~ 0+)] L' o]o,

(28)

where the vector coupling is S + S = L, T + T = O, L + L = O. Since E~ is clearly a sum of scalar functions, the z-axis may be chosen along any convenient momentum vector. We will always choose this to be the direction of momentum transfer Q. 3.3. R I N G C O R R E L A T I O N E N E R G Y F O R D I R E C T D I A G R A M S

The RPA integral equation for D in the coupled representation is obtained by multiplying eq. (9) by the necessary Clebsch-Gordan coefficients and inserting the coupled tA)F matrix elements. The result is

DSS',rr' M s M ' s , M T M ' T ~"a C o"a", tn)

~

h-'I2-'(o)G~a(CO)(--1)S+"'~[S]÷ Z,~M~M,,sn I'~S-'-SL L

x {JKo~o~'~TT"6MT~'T(-- I~M'~+M~ ~Lt~'r cdab)(o)Gdbbd(O~) "1 ~a)-~,~'*, -[- E

(~K od,Kab,

L SS, TT" MTM'T(d t b t bd, ¢o)}. (A)F,(ST, cd t ab t )DM"sM's,

(29)

b'd"

Now, of the four momenta appearing in (29), only three are independent because of momentum observation. Suppose these three independent momenta are taken to be a, d, and the momentum transfer Q = c-a = b-d.

(30)

Then the direct part of ta)Fu( L ST, cdab) would involve v~(Q). Further, if the exchange term is neglected, the RPA integral equation is then factorable. Taking Q along we get

DSS, rr MT~T(acbd, 09) = h-l~2-1to)G .... (°9)(--1)S+Us[s] ~: ~, "~msMsOtS Ms~s, ' --sz F~(ST, cdab) L=0, 2

× ( ( - 1) s +

+ )2 •-'Us~s, * * tarOTS.. . . . .

CO)}. (31)

d'

Now multiply both sides by ~_~, solve for ~ D , and reinsert into D. The result is

D SS, M ~T,T uT~T(acbd, o~) = h- lf2- x(_ 1)M~+MT co)G.... (09) w)Gabbd(Co)f(TSMsQ) (32) 1-J(Qco)f(TSMsQ)

NUCLEAR-MATFER TENSOR FORCE

405

where f ( T S M s Q ) = ( - 1)S+Ms[S] ~ E L=0,

SSL Fo(ST, L Cr~s~tso cdab),

(33)

2

J(Qog) = h-112-1~. (o)G.cc.(¢o) = h-l(2n) -a Idaa(o)G.cca(aJ).

(34)

a

Expressions for J(Qto) are given in the appendix. Inserting D into the correlation energy [eq. (28)], converting the momentum sums to integrals via

ff

2~K°~K~.~(fR)a(2n)-9(4n QRdQ d3a abcd

dad,

(35)

and replacing ~_~d~o ~ 2~d~o, we obtain for the direct contribution to the ring correlation energy __

E,,a

f , Qo

2ihO E I T ] | (2n) a rSMs Jo

pl

(22dQ| din| ffoo

dR [RJ(Q~o)f(TSMsQ)] 2 Jo ~Z 1-RJ(Qog)f(TSMsQ )"

Jo

(36)

The R-integration yields f ~ dR (RJ f ) 2 2 1-RJf

-

it-l-In

tc = i ( x l - a r c t a n xi ) +real part, \ tG/

(37)

where x, the dielectric function, is defined as (38)

tc(TSM s Qog) = x~ + itq = 1 - J(Qog)f(TSM s Q).

Since E~ is real we only need the imaginary part of eq. (37). The final result for the direct ring correlation energy per particle is er, d --

E~'d A

3~*kE f°~x2dx f ~ d y ~. IT] {arctan tq(TSMsQ°9) 16n Jo Jo TSMs x~(TSMsQog)

tq(TSMsQog)} .

(39) The integration variables x and y are defined as x = Q/kF,

(40)

y = 2hco/g*k 2.

For the local spin-isospin independent central potential [eq. (20)],

r/

er d = x2dx dy arctan-- - t q ' 4rig Jo Jo ~

} ,

g = 4,

(41)

which is identical in form with the electron gas result of Hubbard 2) where the singleparticle degeneracy g is 2 instead of 4. Each of the terms in the ring diagram series can be recovered ifeq. (39) is expanded in powers of V. The second-order term would be *

2

oo

oo

( 2 ) _ 3;t~kvf x2dx f d2 ~, [T]~i(TSMsQ~o)(I_x~(TSMsQa~)). er, a 16n Jo Jo STMs

(42)

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3.4. CONTRIBUTIONS OF COLLECTIVE AND SINGLE-PARTICLE MODES TO THE RING CORRELATION ENERGY

It is clear from eqs. (6) and (7) that the contribution to the ring correlation energy comes from the residues of the poles of the density fluctuation D. These poles are of two types [cf. eq. (32)]. The first are the single-particle excitations (ho9 = ep-eh) into the particle-hole continuum, i.e., the poles of the unperturbed ph propagator t0)G(to ), For fixed momentum transfer Q = p - h these excitations are bounded by X < 2,

0 ~--- V ~ X']-½X 2,

½X2--X < V <-- x + k x 2,

X > 2,

(43)

where Q

h~-A

kF

~*k~

"

/ J'/,¢, c,°'~" x

/ 2

Fig. 2. The particle-hole continuum and collective excitations. The shaded region between the line ab and x + (½x2) represents the contribution of collective excitations to the ground state correlation energy from above the particle-hole continuum.

These regions are shown in fig. 2. Such excitations are usually strongly damped but nonetheless make the largest contribution to the ring correlation energy. The second contribution comes from collective excitations outside the ph continuum. In the ring approximation these states are not damped. The boundary defining these [line ab in fig. 2] is obtained from the dispersion relation t G(Qog) = 1-Jr(Qog)f(Q) = 0

(44)

for each of the multipoles T S M s. However, the entire shaded region between the line ab and the ph continuum contributes to the correlation energy because of the adiabatic switching on of the interaction H 1(2). This also manifests itself in the expression for the correlation energy [eq. (39)] through the term arctan(xi/G) which requires careful treatment. Outside the ph continuum tq is identically zero (see appendix). Then arctan(xi/G) is either zero (when xr > 0) or +rr (when xr < 0). The sign of rc t The dispersion relation is equivalent to the RPA eigenvalue equation (11 ).

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is resolved by looking at how/~i goes to zero. Noting J~ < 0 the sign o f f then becomes the determining factor. These arguments are summarized as follows: For ~ = 0: arctan~Cl -- = { Kr

! --

~c~>O ~c~ < 0,

f > 0

x~<0,

f<0.

(45)

Above the ph continuum Jr is positive. Therefore, for collective solutions to exist, the ph matrix elements f must be repulsive. 3.5. A P P R O X I M A T E

TREATMENT

OF EXCHANGE

RING

DIAGRAMS

We have obtained an exact expression for the direct ring correlation energy in nuclear matter because the R P A integral equation is factorable, i.e., the direct matrix element depends only on Q. Suppose for the moment that some average matrix element (A)F could be defined that still had this property. Define

f(TSMsQ) = (_I)S+Ms ~ C~sMs SSL o (A)Fo(ST, --L cdab),

(46)

L=0,2

g(TSMsQco) = 1-J(Qcg)f(TSMsQ).

(47)

Then the ring correlation energy including exchange is

e,

-

2ih£2

2 [ ]

A(2x) 3 rsMs

T

fo°Q2dfo°dOf(TSMsQ) f f(TSMsQ) o

-

-

-

(2J(Q°9)f(TSMsQ))2 2 1-2J(Qco)f(TSMsQ )

rT]f_(TSMsQ)

_ 3 *kg [' x2dxf dy y 16re Jo do r~s- - f(TSMsQ )

x{arctan ~I(TSMsQa)) K,(TSMsQo))

ffI(TSMsQo))}. (48)

We describe here the averaging procedure to obtain (A)F. The part of this matrix element depending on m o m e n t u m variables is VL / I ) ] / I~S+TvL[k ' /~k~--/--k ] ,uk ab t

k cdJ" x

We want to replace the second v~ L by an average depending only on Q. To get some idea of the exchange matrix elements involved we turn to the second-and third-order Goldstone graphs shown in fig. 1. We see there that are two types of exchange vertices. The first is a backward-going exchange vertex and appears for the first time in second order. Working out the kinematics we require an average tbr L

h + O),

where h and h' are hole momenta. The second is a forward-going exchange vertex and appears for the first time in third order when the middle interaction is exchanged. Here we require an average for L r). v.(h-h

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D . M . CLEMENT

The fact that there are two kinds of exchange vertices is unfortunate because we can only handle one in our formalism. Since we have to make a choice we will average the backward exchange vertex. That will at least treat the second-order exchange diagram properly, and since there are more backward than forward exchanges in third order, it will probably not be too bad an approximation in third order as well. The averaging procedure will be simply to average over the hole momenta, h, h', i.e., e~(Q) =

Qf~2f.f d3hd~h'v~.(h-h ' +Q),

f2v = fRdah, where the hole momentum integrations are restricted by the particle lines still attached to the vertices, i.e., to h < kr, Ih+QI > kF. For Q > 2kF there is no restriction on the angle integration ~i. We make the further approximation therefore of allowing unrestricted angle integrations over/i, fi'. This is probably not too serious an error since the correlation energy weights the contributions by QZ [cf. eq. (48)], thus favoring large Q anyway. In summary, we replace vL(kab+kcd)i n (A)F by

~L(Q) = (f2F)-2f fh, h,
(49)

where

iL(Q) = f or2dr (~Fr J,(kFr)) 2gz(r)Jz(Qr)•

(50)

Note the density dependence of the average exchange matrix element here. 3.6. S U M M A R Y OF F O R M U L A S FOR THE OPEP POTENTIAL

The double integrals for er and er, a, eqs. (48) and (39), were performed numerically. For reference we write down the functionf(TSMsQ) in the S = T = 1 mode which enters into the dielectric function ~ = l -Jr: f ( 1 1 M s Q) = - 16n Vo{[Io(Q)- ¼[o(Q)] + ( - 1)Ms(1 +

5MsO)[I2(Q)+ ½i2(Q)] }.

(51)

The integrals 1z are given by eq. (25) while i z are obtained by numerically integrating

To obtain f,

i L =fo~r2dr( 3 jl(kFr))ZhLX)(ix)O(r--rc)jL(Qr). (52) kFr set iL to zero. The phase space integral J(Q~o) is given in the appendix. 4. Results and conclusions

Results for the ring correlation energy using the OPEP potential with cut-off radius r o = 1 fm are presented in table 1 for different densities. At k r ---- 1.4 f m - t we report

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Dahlblom and Kouki's results 4) as well. The contributions of either the central or tensor components of OPEP calculated separately are also given. The higher-order diagrams beyond second order are labeled Ae. We make the following observations based on table 1: TABLE 1 R i n g c o r r e l a t i o n e n e r g y p e r p a r t i c l e , e = e d q - e c , f o r t h e O P E P p o t e n t i a l ( c u t - o f f r a d i u s 1 fro) f o r second and higher order k z ( f m - 1)

e¢(2)

Aed

--4.92 --9.65 (--9.3) -- 15.40

- - 1.45 --2.03 (--2.2) - - 1.66

--0.24 - - 1.16

--0.22 --0.68

--4.34

- - 1.38

--4.86 --9.58 (--9.2) - - 15.32

--1.46 --2.03 (--2.2) - - 1.66

--0.32 --1.30 (--1.14) --4.69

--0.23 --0.68 (--1.08) - - 1.53

ea~2)

,de~

central and tensor: 1 1.4 2 tensor only: 1 1.4 2 central only: 1 1.4

--0.059 - - 0.071 (--0.070) -- 0.088

0.009 0.006 (0.0006) 0.003

0.003 0.004

--0.001 0

0.006

- - 0.001

H e r e d e = e - - e t2). T h e p a r a m e t e r s f o r t h e s i n g l e - p a r t i c l e s p e c t r u m a r e d = 0, ( M * / M ) D a h l b l o m a n d K o u k i ' s r e s u l t s 4) a r e g i v e n in p a r e n t h e s e s . E n e r g i e s a r e in M e V .

= 0.646.

(i) The ring correlation energy is attractive and increasing in magnitude monotonically with density. The second-order direct diagram etd2) exhibits roughly a kF~ density dependence while the higher-order direct ring diagrams go roughly as k~. The effective density dependence might be somewhat reduced if the s.p. spectrum were altered at each density. The higher-order exchange diagrams A e e do not increase in magnitude as rapidly with density, but in any case no tendency toward saturation is observed in the ring diagrams. (ii) The OPEP central potential by itself contributes less than - 0 . 1 MeV to the correlation energy. We note, however, that the interference between central and tensor force is not insignificant in the higher-order ring diagrams. The tensor OPEP contributes - 2 . 0 MeV to A e at k r = 1.4 fm -~ when acting alone, but with the central force present this is decreased in magnitude to - 1 . 8 MeV. (iii) Our results compare favorably with Dahlblom and Kouki 4). In principle the second-order direct diagram etd2) should agree exactly (we use the same s.p. spectrum as they use), but we note a 3 % or 4 % discrepancy. We have taken their results for e (3) and e ~4) and added them together to compare with our Ae. For the tensor force only, they obtain - 1 . 1 4 MeV for e~a)+e~4) while we get - 1 . 3 MeV for the entire

410

D.M. CLEMENT

series. Dahlblom and Kouki also estimate the total fifth-order contribution at - 0 . 1 3 MeV. Adding, say, 2 of this to etaa)+e~ 4) would yield excellent agreement with our complete result. This certainly indicates convergence of the ring series. Their second-order exchange ring diagram e~2) of - 2 . 2 MeV agrees with our average result - 2.0 MeV to 10 ~o. That is very encouraging, but there are two disturbing points to note. First, our higher-order exchange results (Aee = - 0 . 6 8 MeV) are only 3 of theirs (e(ea)-t-ee = - 1 . 0 8 MeV). Quite likely the trouble lies in third-order ex~change diagrams such as fig. I f which has not been handled properly even in an average sense, since we have assumed the middle interaction line in this diagram to have the same average as the backward exchange line. Secondly, our e~2) result for the central part of OPEP is a factor of 10 larger than theirs. This is probably because our averaging procedure for the exchange matrix element is only valid at large Q whereas the central potential has significant matrix elements only for small Q and this leads to an over-estimate. TABLE 2 D e p e n d e n c e o f ring correlation energy on energy gap A between occupied and unoccupied states A

0 10 20

e c2)

Ae

--13.52 --10.68 --9.90

--1.84 --1.35 -- 1.07

T h e parameters are: rc -- 1 fm, k~- = 1.4 fm -1, M * / M

=

0.646. Energies are in MeV.

We have also evaluated the ring correlation energy as a function of the energy gap between occupied and unoccupied states. The results for the full OPEP potential are shown in table 2. The variation in e (z) as A changes from 0 to 20 MeV is about a 30 ~o decrease in magnitude. In numerically integrating eqs. (39) and (48) for the correlation energy we also search for collective solutions down to Q/kr = 0.03. We find none. We conclude, therefore, that the tensor force does not give rise to collective solutions, at least when only the OPEP potential is used. Therefore only the s.p. excitations contribute to the correlation energy. The negative result can be understood by the fact that collective excitations are essentially long-wavelength phenomena, i.e., involve small Q. In contrast, contributions to the correlation energy come from intermediate Q, and in fact are weighted by Q2 [cf. eqs. (39) and (48)]. Therefore a central force, which has its largest matrix elements for small Q, contributes least to the correlation energy; the tensor force, on the other hand, has its largest matrix elements at intermediate Q and dominates the correlation energy. Compare, as an extreme example, the Fourier transform of a central Yukawa potential with that of the tensor OPEP potential without any cut-off radius. The former goes a s ( Q 2 .~_~A2)- ! while the latter goes a s Q Z ( Q 2 _ b / ~ 2 ) - 1 . Such a tensor potential has little opportunity to build up collective solutions at small Q.

NUCLEAR-MATTER TENSOR FORCE

411

However, collective solutions can occur from the central part of OPEP, though for smaller Q/kv than we have investigated. A sufficient condition for a collective solution is that G(o9 = Ogph(max)) < 0, where hc%h(max ) is at the top of the ph continuum. This follows from the fact that the limiting value of rcr is 1 for large co and thus a sign change in G is indicated. Replacing Q by zero in f ( 1 1 M s Q), eq. (51), and evaluating Jr from the expression in the appendix for A = 0, and in the limit of small Q, f = -- 16nVo 10(0) = 16nVo e-"r°# - 3(1 + #r~), j

_

kv { _ l + l n 2 k v l 4n2,~ * Q l'

then the condition for a solution to exist is - Q - < 2 e x p V- 4n2"~*

kF

1].

[- k v f

Using M * / M = 0.646, k F = 1.36 f m - 1 , and r c = 1 fro, this relation requires Q/k F < 0.01. Because the only non-zero direct matrix elementsf(TSMs) for the OPEP potential are for T, S = 1, 1, only this channel contributes to the correlation energy. However the matrix elementsf are non-zero in all channels. Since the exchange matrix elements in f a r e much smaller than the direct ones (in the S, T = 1, 1 channel), our conclusions regarding the tensor potential and collective excitations are unchanged. To the extent that one can ignore surface effects 9) in finite nuclei, one would conclude that the tensor force would have minimal effect on collective excitations such as the giant dipole resonance. The author would like to thank Dr. C. W. Wong for suggesting this problem and for several useful discussions, particularly in connection with exchange ring diagrams. Also, the author would like to acknowledge a useful conversation with Dr. C. Dover, and to thank Dr. S. Moszkowski for critically reading the manuscript.

Appendix We summarize here the phase-space integral J(e

o) = h - l e - I

Z

....

a

The real and imaginary parts of J = Jr + / J i are separated by means of the identity (x + irl)-t = P ( x - 1) T in6(x).

412

D . M . CLEMENT

In terms of dimensionless variables

Q

v+-

kF

hog+ A ~*k~'

t h e i n t e g r a l s f o r J , are 4/r2,~ *

--Jr

= -1+

k~

1 (v_-v+)+

1

F

1 + - - (1 2x

(I__[(V_Ix)__(½X)2])II.I [ _l +_ [(~../x)-

½x] :-[(v_ix)-½x]

[(v+/x)+(½x)] 2) In 1-- [(v+/x)+½x] 1 1+ [(v+/x)+½x]

I'

x>2,

4~2~ * - -

Jr

k¢

= -l+

1 (v_ - v + ) +

v_ In

2x

2x

+ - 1- [ 1 - ( v Z / x 2 ) - ¼ x

2] In

2x 1

+~[1

(v /x)2 El-(v-/x)]

2-¼x z

_ v+ In

(v+lx)Z

2x

, [l +(v+lx)]2-+x~ i

!

1-[(v_/x)+kx]

1 --[(v+lx)--½x] 2 2) - a x, 2 -(v+/x

] In

1 + [(v +/x) - ½x] 1 + [(v_/x) + ½x]

x<2.

'

F o r d = 0 this r e d u c e s t o 4n2~ * 1 - J , = -- 1 + - - (1 --

kF

2x

[(v/x) --½x] 2) In

1 + [(v/x)--½x] [ 1-E(~/x)-½xl

_ 1 (1 2x T h e i n t e g r a l Ji is

{

1-[(v_/x)-½x]

~*x Ji 7C2kF

2

l-[(v_/x)-kx] ~ 2V_ 0

[(v/x)+½x] 2) In

x <= 2, x > 2,

1 + [(v/x) + -

x-kx'

ix] ! [

1 [(v/x) + ½x]

< v_ <= x + ~ x ~

½x2-x <=v_ <=x+½x ~ 0 < v_ < x--½x 2

X < 2, otherwise.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11)

M. Gell-Mann and K. Brueckner, Phys. Rev. 106 (1957) 364 J. Hubbard, Proc. Roy. Soc. A243 (1957) 336 T. Dahlblom, K. Fogel and A. TOrn, Nucl. Phys. 56 (1964) 177 T. Dahlblom and T. Kouki, Nucl. Phys. A175 (1971) 45 S. Hatano, Prog. Theor. Phys. 24 (1960) 418 M. Amusya, Nucl. Phys. 56 (1964) 305 C. Dover and R. Lemmer, Phys. Rev. 165 (1968) 1105 A. GIassgold, W. Heckrotte and K. Watson, Ann. of Phys. 6 0959) l W. Brenig, Nucl. Phys. 22 (1961) 14 D. Pines, The many-body problem (Benjamin, New York, 1961) A. Fetter and J. Walecka, Quantum theory of many-particle systems (McGraw-Hill, New York, 1971)