Renormalization of the optical potential due to pairing correlations

2•.E ]

Nuclear Physics 3 4 (1962) 476---482, (~) North-Holland Publzsh,ng Co., Amsterdam Not to be reproduced by photoprtnt or macrofflm without written permassmn from the pubhsher

R E N O R M A L I Z A T I O N OF T H E O P T I C A L P O T E N T I A L D U E TO PAIRING CORRELATIONS JAMES E

YOUNG *

Institute /or Theoretwal Physws, Unwers, ty o/ Copenhagen, Denmark R e c e i v e d 9 M a r c h 1962 T h e m o d i f i c a t i o n s of t h e optical p o t e n t i a l o w i n g to t h e p r e s e n c e of p a i r i n g correlations h a v e b e e n e s t i m a t e d S u p p r e s s m n of a p a i r of quasi-particles, t e , a v a c u u m f l u c t u a t i o n , ren o r m a h z e s t h e real p a r t of t h e o p t m a l p o t e n t i a l T h i s is a n e x c e e d i n g l y s m a l l effect g i v e n b y t h e ratio (gap p a r a m e t e r ) l / ½ × (energy of optical m o d e l s t a t e ) G e n e r a h z a t a o n of t h e c o n c e p t of a p a m n g b o u n d s t a t e leads to t h e n o t i o n of low-lying, finite hfetxme s t r u c t u r e s m t h e s p e c t r u m of t w o quasx-partmle excltataons T h e s e s t a t e s split off f r o m t h o s e describing t h e s c a t t e r i n g of a quasi-particle p a i r I t is s h o w n t h a t t h e Schraeffer c n t e r m n (the e n e r g y dzHerence of t w o s t a t e s m t h e single quasx-partmle s p e c t r u m s h o u l d be approxam a t e l y e q u a l to t h e excltataon e n e r g y for a collective state) for t h e e x i s t e n c e of a b o u n d quaslp a r t m l e p a i r m a y be u n d e r s t o o d in a simple w a y A n adlabatac a p p r o x i m a t a o n is t h e u n d e r l y i n g basis of t h e p h y s i c a l r e s u l t B o t h t h e real a n d i m a g i n a r y p a r t s of t h e optical p o t e n t i a l are r e n o r m a h z e d here.

Abstract"

1. I n t r o d u c t i o n The positive frequency part of the one-particle Green's function G carries the A + 1 spectrum of excitations It is this which is of interest when we add a nucleon to the A-particle ground system This IS a physical situation in elastic scattering The complex self-energy 2: emerging in the simplest of the Green's function descriptions is the optical potential Uo(so) One IS entitled to ask how the optical potential is mochfled b y the posslblhty of correlations in the nuclear medium These m a y be present before the nucleon IS added, thus presenting to it an altered density distribution Alternately, the nucleon m a y Itself participate in, or estabhsh, a correlated motion through its interactions The correlations arising to influence optical potential are, for example, the hole-particle vibrations, the quadrupole deformation and other, high-frequency, shape excitations of the A-particle system Apart from these there also exist pairing correlations which are believed to be of the type, namely B C S , arising in the theory of superconductivity Pmrlng correlations can well be imagined to be important in their influence upon the optical potential These as we know manifest themselves most ? O n leave f r o m t h e U m v e r s l t y of C a h f o r m a , Los A l a m o s Scientific L a b o r a t o r y , N A S -N R C P o s t d o c t o r a l Fellow 476

~NORMALIZATION

OF

THE

OPTICAL

POTENTIAL

477

strongly for the last few nucleons in a given nuclear system As it IS also true t h a t the added nucleon is in states e >-- eF where such correlations are appreciable, we might hope to encounter measurable renormahzatlons to the optical potential The developments in pairing theories have taken two lines in their applications to the properties of matter In the first of these it IS sought to discuss the effects of p a m n g correlations in systems at zero temperature Such are, by definition, in a state of thermodynamic equlhbrlum Alternately, one can inquire as to the quenching of correlative effects in extended systems at finite temperature Such systems have been described in terms of ensemble or thermodynamic Green's functions The flmte nucleus is often characterlzed as a self-binding system which is said to be m equilibrium at zero pressure and temperature In the absence of the residual inter-nucleon forces, it is possible to Identify the linearly independent motions A variational treatment, e g , the Hartree-Fock (H-F) method, for the energy of a real, interacting system will define a new set of Independent motions In the H-F method, dlagonalization of the residual force for the ground state energy shows t h a t the H-F distribution over occupied states is depleted The equilibrium occupation number function taken over these states shows a loss of probablhty The "missing" probability is carried by two-hole and two-particle excitations, for example (It is possible to think of the redistribution of probablhty as a finite temperature effect ) The Brueckner theory i) contains, in the third-order contributions to single-particle energles, lust such effects Alternately, the canomcal transformation techniques build, in the presence of short range attractive forces, a new set of Independent motions, those of the quasi-particles Again there is a depletion of occupied states to estabhsh these motions The excitations we cited above, of twohole and two-particle character, are not entlrely dlagonahzed by the canonical method When these are dlagonallzed with respect to the quasi-particle motions, a single question is asked Is the probability distribution, the excitation spectrum of independent motions, modihed in such a way as to change the previously existing thermodynamic properties of the system? Generally, the answer IS no The depletion of states is small or the occupation number alteration is slight Insofar as this remains true for finite nuclei, there being problems away from closed shells, the zero-temperature Green's function m a y be used to depict the pairing interaction This premise forms the basis of subsequent discussions

2. Single-Particle Propagator (i and Two-Body t-Matrix for B.C.S. Theory The dynamical behavlour of G is determined by the two-particle Green's function G~ That in turn will here be obtained from the non-gauge-lnvariant

478

j

E

YOUNG

factormatlon of G3 utilized b y Kadanoff and Martin (K-M) 2) In the notation of these authors the dynamical equation for G is, in a 4-dimensional, spacetime, matrix notation ~ - I ( 1 - - i ) V G 0 - - - I ' ) - - G 0 - - - I ' ) ] =-

,VL+-(11 ',

1' 1+),

~-1 ( l - - i ) = Go-~(1 - - i ) - - W G ( i - - 1+)

(1)

The pairing interaction V is taken to be that yielding the Gor'kov factorlzatlon 3), namely

/-w(t-t'), V(1--1') = t

0,

PF--o D < p(1), p(l') < pF+O D,

otherwise

In an interval of 2osu about the Fermi momentum PF, this interaction has constant m a t n x elements Thedenslty-densltycorrelatlonfunctlon, L+-(12, 1'2') for a pair of fermlons In conlugate spin states IS defined as L+-(12, 1'2') = G2+-(12,

l'2')--G(1,

l')G(2, 2')

The only other undefined quantity appearing up to this point is the unperturbed propagator Go, about which we shall have more to say The pair correlation function satisfies the equation L + - 0 2 , 1'2') = --3 f

di~(1--i)G(2--i)V[L+-(ii,

l ' 2 ' ) + G ( i , I')G(i, 2')]

Such m a y be approximated b y the equation L+-(12, 1'2') = --3 f

dI~(1--I)G(2--i)VL+-(ii,

1'2')

(2)

The validity of the approximation rests upon the infinite separation of primed and unprlmed coordinates Physically, this simply says that correlated pairs are stable against perturbations save in the case that these have zero frequency It is necessary to include the following details 4) for the sake of completeness Starting from the statement L + - ( l l , 22) =

--AZlV 9",

It IS readily established that L+-(12, 1'2') = F ( 1 - - 2 ) ~ ( 1 ' - - 2 ' ) , where _~(1--2)

=

A f did(1--i)G(2--i)

and ~ is the time-reversed form of F The gap parameter being given as A =

we finally determine the one-particle propagator as

G-l(1--1 ') = ~ - l ( 1 - - 1 ' ) + z ] ~ ( l ' - - l ) A

(3)

RENORMALIZATION

OF

THE

OPTICAL

POTENTIAL

479

If G o IS the Hartree-Fock (H-F) propagator, G is interpreted as the same type propagator where however, the zero of energy, the Ferml energy eF, IS shlfted by the p a m n g potentlal Propagation m the presence of the palrmg lnteractlon is described through G A t-approxlmatlon Is sought next Such has already been dlseussed in another context by B a y m and Kadanoff 5) The defining equation is

V(l*--2*)G2+-(l*2 *, 12) ---- f d l " d 2 " t ( l * 2 * , l " 2 " ) G ( l " - - l ) G ( 2 " - - 2 ) (4) This equation has the soluhon t=

V/(I+~VdG),

(5)

whlch then corrects the quasi-particle propagation b y a contribution arising from the formation of a bound state The ladder one constructs from eq (5) is asymmetric One side of it describes quasl-partlcle propagation according to the B C S Green's function, the other propagatmn b y the shafted H - F function It is straightforward to obtain the self-energy operator X, corresponding to t of eq (5), as =

Ad3

The physical facts in the formalism m a y be extracted b y going over to an energy and angular momentum representataon of the varmus operators Given that d,(e) = (e--e,) -1, e~ = gj--#, /, = gF--,VG(1--1+), the B C S propagator IS determined as G~(e) = ( e + ~ l ) ~ _ E 2 ,

E , ~ = e~2+A 2

Here, g~ is the energy reqmred to add a particle to the ~-th orbital in the absence of pairing and /~ xs the chemical potential shifted from the Fermi energy gv b y the pairing lnteractmn The well-known gap-equataon, O~ ½v

s/L2+A

' -

=

being extracted from IVj(e), we find the weak-couphng result for particle number m j-space to be N , = £2,(1--e,/E) (6) As we noted earher, this is not the only source of a loss of probability from occupied states The t-matrix is now evaluated in an orbital angular and energy representatmn, with the usual LS-I] transformatxon being amphed The pairing

480

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force factors into multlpoles m the new representation, thus

2') = & (7)

Y.f 34

The e n e r g y - d e p e n d e n t function is given b y the two algebraically equivalent forms following

VL(e)

*~X IV~d

{

1

y ~--S3--S 4

e+E,_e3+,rI i_e_e3_E 4

-

-

Eaq-eal

2E 4 A

Ea--e4 O(e--E,) I e_E4_ss+,~ ? ~ ] '

Ea--e4 1--O(e--E4)

(Ta)

(7b)

In the above, 0 is the u m t step functlon vanishing for negatlve arguments, and the s u m is over unoccupied states F o r an a t t r a c t i v e potential one has 2L ---- --[Xz[, 2L being the s t r e n g t h of the L - t h multlpole of the pairing force It is s t r a i g h t f o r w a r d to show from (7b) t h a t the pairing t-matrix is hermltlan, 1 e , exhibits no pole s t r u c t u r e T h e form (7a) is Included because it suggests a physical i n t e r p r e t a t i o n of a more general process t h a n t r e a t e d thus far Particularly, in the two-particle p r o p a g a t o r we m a y expect to find states at low excitation, which we shall describe as b o u n d or correlated There also a p p e a r the states describing the m o t i o n of two ind e p e n d e n t excitations We shall call these scattering states In the B C S theory, the two kinds of states p l a y no distinct role

3. Generalization of Pairing B o u n d States The nucleon b o u n d states m a y be seen if we add a particle, as in elastic scattering, to a nucleus which has a quasi-particle present T h e states formed in two particles are analogous to excltons We ask for the pairing insertions in the nucleon line leading to self-energy diagrams This determines the ren o r m a h z a t l o n of the nucleon-optical potential due to p a m n g correlations We discuss two situations A nucleon a d d e d to an even target In Its g r o u n d state, e g , n + N 1 a°, breaks a pair The particle and quasi-particle now present in l n t e r m e d l a t e states t h e n scatter to final states, restoring the pair and leaving a particle in the optical model state We always t h i n k of Interactions between one species, e g , n e u t r o n s In c o m p u t i n g the H - F potential, 1 e the optical potential, for the a d d e d nucleon from the B r u e c k n e r theory, we have already t a k e n into account most of the self-energy-described here T h a t which remains,

RENORMALIZATION O F

THE O P T I C A L

POTENTIAL

481

owing specifically to pairing forces, m a y be crudely estimated A ground-state pair in a vacuum fluctuation scatters to the optical model single-particle state In the latter state the nucleons rescatter and de-excite Although the pairing interaction IS almost zero in the optical model state (?~oD ~ 4 MeV in the NI Isotopes, a 2 MeV neutron has 9 MeV of excitation in these isotopes) we allow this potential to rescatter the nucleons This estmlate gives 2~ ~ A*/2B for the self-energy of a nucleon in a state of excitation energy B The contribution is small ( A m 1 5 MeV in the N1 isotopes) and renormahzes the real part of the optical potential A new situation appears in the scattering of a nucleon from an odd-mass Isotope, e g , n-kN159 Again we think of one species, e g , the protons forming a single closed shell. The self-energy processes described above still occur In addition to these, the added and target nucleons m a y scatter into intermediate orbltals within which they now pair. These states m a y be considered as having unperturbed energaes computed as two-hole, two-particle excitations from the ground state of the even A + 1 system Thus the Intermediate states of two excitations m a y be spoken of as two quasi-particle states Here, the selfenergy for the added nucleon has not been Included m the optical potential and m a y be computed as follows The unperturbed elgenfunctlons In two nucleons are taken to be

F ':l

~ , j(1, 2) = ~ Lm--m[ 02 ~'m(1)v~"(2)

(8)

The energy of the two-particle state is/~. and since we deal with one species of nucleon, 0 __< m __<] is required Finally, the time-reversal symbol has been used in the wave function of nucleon "2", and it is considered that k~(1, 2) is totally antlsymmetrlc Assuming a zero range two-nucleon force, we chagonallze this with respect to the basis of (8) The result is to give a set of two-particle esgenvectors 7tv, y having eIgenenergies e~ The higher states . . can be made to correspond with those in the A + 1 spectrum of excitations For the lower states it is generally necessary to Introduce a specificity force to achieve the correspondence e), e g , t h a t producing quadrupole deformation In perturbation theory, the self-energy of the added particle in state k is computed, in lowest order, as

z(k) =

X k ' l , 1'

(0Iv

1

- vl0). Eo--h+*~

(9)

The symbols here have the following meaning [0) is the antlsymmetnzed product of wave functions for the added particle in state k and unpaired nucleon In state 1 (primes on state symbols denote intermediate states), E 0 = E I + E ~ IS the sum of the lmtlal energy of ground-state target eo ---- d o + E 1 (d o = O) and that of the Incident nucleon Ek, while h ---- h(1)+h(2)+VD(1, 2) is the two-particle HamlltonIan (VD is that part of the two-particle interaction

482

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dlagonallzed in the pairing process) and finally, V IS the remamlng nondiagonal interaction (where VD+V IS not included in the H-F potential) If It should be possible to identify Independent excitations in mtermechate states as we have claimed, the energy denominators of (8) are E ' I + E ' . - - E 1 E k On the other hand the states of the A + I system, e~(A+l), yield denominators E . ~(A + 1) + e 0(A + 1) -- E k - e 0(A) Here Ex IS the excitation energy in the A + 1 system To the extent that the lowest state of positive energy in the A + I system is that eo(A)+E'k, the denominator is given by

Ex~q- E'k--Ek ----- - - ( h ) + E o

(10)

This means that the Intermediate states k' carrying a large effect due to pairing he close to the Fermi surface of the A particle system Moreover, we see that the self-energy insertion in the nucleon hne lmphes Ex ~(A + 1) ~ Ex ~ ( A - - l ) for collective excitations Then, finally, the presence of two quasiparticles with energy dz//erence (Ex ~ ~ E ' I - - E 1, or Ex ~ ~ Ek--E'k) equal to the excitation of a collective state determines the self-energy in the singleparticle spectrum The low-lying states T~, 1, associated with Ex v, in the even systems A + 1 and A--1 are collective The supposition that Ex~(A--1) is to be inferred from Ex~(A+I) is analogous to the averages lmphed in apphcatlons of pairing theories The foregoing Intuitive ideas of a two-particle bound state are substantiated by some computations due to Schrleffer ~) This author finds that (10) holds for the renormahzatlon of the quasi-particle spectrum in B C S theory We would also point out that the two-particle bound state formed in a single colhslon, that of Brueckner, Eden and Francis s), finds its natural explanation here The properties of such states have been discussed from another point of view by Shaw 9) We thank B Mottelson for communicating the work of J Schrleffer while our analysis was in progress, and L P Kadanoff for his discussions on pairing theory The criticisms of G E Brown and P W Anderson ehmlnated a serious error in an earlier computation Professor Nlels Bohr is especially thanked for the hospitality provided by the Institute for Theoretical Physics References K L L V G L J K G

A Bruecknerand D T Goldman, Phys Rev 117 (1960)207 P K a d a n o f f a n d P C Martin, Phys Rev 124 (1961)670 P Gorkov, J E T P 34 (1958) 735, Sovmt Physms (JETP) 7 (1958) 505 Ambegaokar and L P Kadanoff, 2ffuovo Clm 22 (1961) 914 Baym and L P Kadanoff, Phys Rev 124 (1961)287 S Klsshnger and R A Sorensen, Mat Fys Medd Dan Vxd Selsk 32, No 9 (1960) C Schrleffer, preprmt, Oct 1961 A Brueckner, R J Eden and N C Francxs, Phys Rev 99 (1955)76 L Shaw, Annals of Physics 8 (1959) 509