Correlation energy of 16O

I,A,cl

Nuclear Phystcs A I 0 0 (1967) 241--247, (~) North-Holhmd Pubhshmg Co, Amsterdam Not to be reoroduced by photoprmt or microfilm w~thout written permission from the pubhsher

CORRELATION G

E

ENERGY OF

160

BROWN

NORDITA, Copenha,qen and C

W

WONG

Department oJ Theorettcal Physics, O:~[old Uml erstt),, England Received 12 May 1967 Abstract The c o r r c l a h o n energy, i e the zero-point energy of v i b r a t i o n s and p a m c l e - h o l e exc~tatmns, is calculated for ~60 This involves choosing a model space consisting of 0s, 0p, Is, 0d, lp, Of levels an d then defining an effectwe force for this model space m t e rms of a r e a c h o n matrix so calculated that only i n t e r m e d i a t e states outside ol the model space are e mpl oye d Planewave m t e r m e d m t e states are used as m t e r m e d m t e states The effecttve H a m d t o n m n is then d l a g o n a h z e d m the model space, using quasl-boson c o m m u t a t t o n relations This is e qui va l e nt to s u m m i n g the ring d i a g r a m s m the graphical e x p a n s i o n of the energy w~th some exchange included

1. Introduction In calculating effective interactions m nuclei from the t w o - b o d y force m free space within the f r a m e w o r k o f the Brueckner-Bethe theory, one of the m o s t difficult p r o b lems is to h a n d l e correctly the s p e c t r u m o f i n t e r m e d i a t e particle energies j u s t a b o v e the F e r m i surface This is difficult, b o t h because the particle self energy is to be evalu a t e d off the energy shell and because it is often desirable to include rescattermg correcttons m the form o f t h r e e - b o d y interactions in thts self energy 1) P r o p o s e d values for this self energy near the F e r m i surface are still in a state o f oscillation Sprung, B h a r g a v a and D a h l b l o m 2) certamly overdtd the a t t r a c t i o n in m t e r m e d l a t e states a n d were forced to m o d i f y their spectrum, j o m m g it on s m o o t h l y to the hole spectrum A recent i m p r o v e d t r e a t m e n t o f the t h r e e - b o d y terms 3) along wtth mcluslon o f effects o f the tensor force 4) p r o m i s e to gtve less a t t r a c t i o n It is, in any case, clear f r o m these works that the s p e c t r u m far a b o v e the F e r m i surface m a y be a p p r o x i m a t e d by t h a t foi free particles, t e ek = k2/2m, b o t h m nuclear m a t t e r a n d in finite nuclet W e take the view then, as indeed we have In the past s,6), that one o u g h t to evaluate the reactton m a t r i x G using plane-wave m t e r m e d I a t e states, a l t h o u g h the initial states or holes are t a k e n in a self-consistent well, here a p p r o x i m a t e d by an oscillator potential The use o f different r e p r e s e n t a t i o n s for initial and m t e r m e d m t e states causes some difficulties with the exclusion principle, b u t these were t a k e n care o f rtgorously in ref 6) The use o f plane waves is u n d o u b t e d l y not g o o d for states j u s t a b o v e the F e r m i 241 July 1967

242

G

E

BROWN1 A N D C

~V ~ ¢ O N G

sea, therefore we p r o p o s e now to exclude such states as i n t e r m e d i a t e states in defining the reaction m a t r i x G and then d l a g o n a h z e a p p r o x i m a t e l y the H a m i l t o m a n in a space including these states m o r e c o i r e c t l y In o t h e r words, we calculate effective interactions in a m o d e l space, s u m m i n g over I n t e r m e d i a t e states outsMe o f this space a n d then w o r k with the effective i n t e r a c t i o n m o u r m o d e l space Such Ideas weie i n t r o d u c e d by Eden and F i a n c l s 7) and treated m o r e Iagorously by Bloch and H o r o w l t z 8) a n d by B r a n d o w 9) A c o m p l e t e discussion is to be f o u n d in B r a n d o w ' s p a p e r O u r m o d e l space as the space o f 0s, 0p, Is, 0d, l p , Of orbitals O b t a i n i n g an effective m t e l a c t l o n to use within this m o d e l space, we shall a p p r o x i m a t e l y d l a g o n a h z e the l e s u l t l n g H a m l l t o m a n using the q u a s i - b o s o n a p p r o x i m a t i o n This IS equivalent to Including b o t h direct a n d exchange terms in s e c o n d - o r d e r p e r t u r b a t i o n theory, d~rect b u b b l e g r a p h s to all orders, t o g e t h e r with some o f the h i g h e r - o r d e r exchange terms 2. Reaction matrix elements Using o s c d l a t o r states for the Initial particles with oscillator c o n s t a n t ha) = 13 5 M e V a n d p l a n e - w a v e i n t e r m e d i a t e states, reaction m a t r i x elements G were calculated from

c = v-v-Oc, e

v~here V is the potential, Q the P a u h o p e r a t o r excluding certain i n t e r m e d i a t e states a n d e the energy d e n o m i n a t o r , which is t a k e n here to be

e

~

---

2m

~

---

2m

--~1--~2,

(l 1)

where e 1 a n d e2 are the energies o f the two particles in their imtial states W e calculated G's for two cases, with P a u h o p e r a t o r s Q which projected out o f possible p l a n e - w a v e I n t e r m e d i a t e states the 0s, 0p orbxtals a n d which p r o j e c t e d o u t the 0s, 0p, ls, 0d, l p a n d Of orbltals These two cases are referred to as the A = 16 and A = 80 cases, lespectlvely, since the filled orbltals w o u l d c o r r e s p o n d to these nuclei W e show in table 1 the G - m a t r i x elements c o r r e s p o n d i n g to these two cases, calculated using the local density a p p r o x i m a t i o n ( L D A ) for the P a u h ope~ a t o r and global a p p r o x i m a t i o n o f ref 6) In the latter a p p r o x i m a t i o n the P a u h o p e r a t o r is h a n d l e d exactly In the table, the energy e 1 +e2 = - 9 0 M e V c o r i e s p o n d s to the Interaction o f two 0s particles, e l + e 2 = - 6 3 MeV to the interaction o f a 0s a n d 0p particle, a n d e 1 ± e 2 = - 3 6 M e V to the i n t e r a c t i o n o f two 0p particles A l t h o u g h the case o f two paiticles Interacting in the Is, 0d shell does n o t enter into the binding energy, it was included to indicate the general m a g n i t u d e o f the d o u b l e c o u n t i n g involved when one uses the G - m a t r i x elements calculated for A = 16 in processes such as that shown m fig 1, as was d o n e by K u o and B i o t i n 5) In the case o f the 3S 1 case, this d o u b l e

243

CORRELATION ENERGY OF 160 TABLE l

Reaction matrix elements calculated ~lth Pauh operators corresponding to A - - 1 6 a n d A -- 80 ]nitml state energy 1~ e2 (MeV) 8

~So case

--36 --63 --90 -- 8

3Sx case

--36 --63 --90

In

Quantum numbers of relatwe motion n N L

G-matrix elements MeV A ~ 16 A = 80 -- -LDA Global LDA Global appro\ approx

0 1 2 0 1 0 0

1 1 0 1 0 0 0

2 0 0 0 0 1 0

--6 --4 --1 --6 -- 3 --5 --5

0 1 2 0 1 0 0

1 1 0 1 0 0 0

2 0 0 0 0 1 0

--9 70 - - 6 88 --3 29 --7 70 --5 18 6 41 5 82

66 39 73 07 84 65 40

--6 --4 1 --5 --3 --5 --5

50 23 77 91 84 57 38

--6 --4 --1 5 --3 --5 --5

24 08 52 79 69 53 32

-6 06 4 04 1 68 5 73 --3 79 --5 53 --5 35

--9 02 - - 6 26 3 35

--8 --5 --1 --6 --3 --5 --5

12 24 95 37 67 56 04

--6 --4 --2 --5 --3 --5 --5

5 23 --5 84

86 66 33 94 94 45 18

An oscillator constant corresponding to ho) = 13 5 MeV was used In the two cases where the global approximation was not calculated for A = 16, the L D A was used for b o t h A = 16 and A = 80 G-matrix elements c o u n t i n g as s e e n t o b e a p p r e c i a b l e , a l t h o u g h it is b e t t e r t o i n c l u d e it t h a n t o l e a v e o u t the renormahzatlons

o f t h e t y p e s h o w n m fig

1 completely

od~od of~,~

of

Fig I A process r e n o r m a h z m g matrix elements of the type ((0d, 0d) J, T I G](0d, 0d) J, T) which involves some double counting If the G-matrix element for A = 16 is used One should use the A = 80 G-matrix element here, where Of states are expllcttly projected out of allowed intermediate states

3. A p p r o x i m a t e

diagonalization of the effective Hamiltonian

in t h e e n l a r g e d m o d e l s p a c e

Gaven the effective H a m t l t o n l a n

H = ZT~a~ak+½Z(k, k

k2[ V] k3 k~)a~+2 akl~ ak3 ak.,

(2)

w h e r e all p l a n e - w a v e s t a t e s o r t h o g o n a l t o t h e 0s, 0 p , I s, 0 d , l p , Of o r b l t a l s m a k i n g u p t h e m o d e l s p a c e a r e i n c l u d e d a s l n t e r m e d ~ a t e s t a t e s i n c a l c u l a t i n g G, o u r j o b is n o w

244

O

E

BRO\¥N

AND

(

Vv

WONO

to dnagonahze a p p r o x i m a t e l y t f ~nthm this m o d e l space W e d o this by t l u n c a t l n g H, k e e p m g only these m a m x elements o f G going into the a s s u m e d self-consistent energies a~ a n d into p a m c l e - h o l e m a m x elements, I e

Ill

n

t

/

+~ ~ ('n'71Glu)a+.aZa,aj m,n z 1

+~g

(U]Olmn)aj + a, + area,,,

m, tl,

(3)

,,~

where m, n refer to partxcles, l , j to holes ( p a m c l e s referring to Is, 0d, lp, Of states, holes to 0% 0p states m the case o f ~°O) H e r e each matrax element o f G is a s s u m e d to be a p p r o p r i a t e l y ant~symmetllzed, a l t h o u g h for snnphclty this is not shown This H a m l l t o n l a n is n o w d l a g o n a h z e d using the q u a s l - b o s o n a p p r o x i m a t i o n , whlcla conSlStS o f wewlng b,.+ = a + a, as a b o s o n o p e r a t o r , assigning to it the c o m m u t a t i o n relatmns [b .... b +] = o.,.0 u (3 1) Now, one can easily cast H into the f o r m *

H = ~,.Z.,(b,+,bm,) . B ....., s

A,.,i.s

b.j

'

I,J

where

A . . . . ~ = c,.,6,.,,6,~+(mjJGlln),

(4 1)

B,~, .~ = (mnlGltj),

(4 2)

Ho = - ½ Z e . , , - ~ Z (mdGI,n,) m

But 1

z~

m 9

=

-½Yr

A

(4

3)

t

o) ,.°, t

'

.8 .... ~

A ..... ~.~bL!

I

= 2',.,Z.,IGr(b,.+, bm,)

(X; - ~j-)

b,,j '

z, j , P

Xv

where the ( - r , , ) are the c o l u m n vectors o f the r a n d o m phase a p p r o x , m a t l o n , eq (5) being easily o b t a i n e d by inserting the unit o p e r a t o r c o m p o s e d o f these c o l u m n t This and the subsequent development were suggested to one of the authors (G E B ) some years ago by D J Thouless

CORRELATION

ENERGY

245

OF 160

and row vectors before the last column vector Now, for positive E n Xp) + - rp = G ,

(b"+"' b'°3

(5 1)

where Q~ is just the creation operator for the nth excitation, whereas for negative It lS the annihilation operator, so that

Ep,

H ~- 4 2 IEpI[QpQ + +O+Qp]+Ho P

IEplQ,+Q p - =l

= p

E [ emt + ( m ' l G l m 0 - -

~ IEpl]

m i

(5 2)

p

The correlation or zero-point energy is then = -½{Y~c.,,+ ~ ( m d G i m 0 -

Eco,r

m, I

m, i

= -½{TrA-

~ IE.I} p

Z[Epl} P

= - ½ Z (G-G), "

(5 3)

P

where E 1 are the elgenvalues of A and EpI1 are the positive etgenvalues of the matrix A B (-B-A), I e the E.1 are the elgenvalues calculated in the Tamm-Dancoff approxlmanon and the E~l those calculated in the RPA Inclusion of angular momentum and lsospln leads to the statistical factor (2J+ 1)(2T+ 1) for each excitation of angular momentum J and isospm T Thus, the final formula is g ....

-

L X (2J+l)(2T+l)(Elsrp - E HJ r . , ~,

(6)

p J,T

the lower suffix p referring to the pth vibranon of angular momentum J and lsospln T

4. D i s c u s s i o n and results

The procedure carried out here to evaluate Ecorr IS just that of Sawada et al 11) for the electron gas, but adopted here fore the finite system and with the Important difference that here the antisymmetrIzed G-matrix elements are used The formahsm thus corresponds to evaluating the contribution to the energy coming from the sum of ring diagrams shown in fig 2 Since antlsymmetrlzed G-matrices are used, those

+

~

+'"

F~g 2 G r a p h s showing the contribution of ring d m g r a m s to the energy Is, 0d, lp, Of, hole states are 0s, 0p

Particle states here are

246

G

E

BROWN

AND C

Vv' W O N G

exchange graphs are mctuded which can be o b t a i n e d by a n t l s y m m e t r l z l n g in a n y given interaction The s e c o n d - o r d e r processes are, consequently, all included, b u t only certain o f the h i g h e r - o l d e r d i a g r a m s are In fact, most o f the c o n t r i b u t i o n to £~orr comes from the s e c o n d - o r d e r term, and xt is s o m e w h a t o f a luxury to keep the h i g h e r - o r d e r terms, b u t usually one wants to w o r k o u t the vibrations, anyway, so t h a t one has all necessary r e f o r m a t i o n O m intention was to evaluate Ecorr from calculation o f all v i b r a t i o n s using the H a m a d a - J o h n s t o n potential However, in the case o f the o d d - p a l l t y states, c o m p l e t e results for 1 hm excitations were available only for the K a l l l o - K o l l t v e l t potential, as calculated by MavromatJs, M a r k i e w l c z and G r e e n 12) In cases ~ h e r e they could be c o m p a r e d , they were n o t much different f i o m results with the H a m a d a - J o h n s t o n potentml These gave a c o n t r i b u t i o n o f 2 t 2 MeV to the correlation energy All evenp a r i t y excitations involving u n p e r t u r b e d excltat)ons o f 2he) were calculated in b o t h the T a m m - D a n c o f f and r a n d o m phase a p p r o x i m a t i o n s using the u n p e r t m b e d energies o f Jolly 13), n a m e l y 0sa, 0 M e V , 0p÷, 27 M e V , 0p_~_, 33 MeV, 0d~, 44 M e V , l s , , 45 M e V , 0d~, 49 M e V , 0fl, 64 M e V lp~, 69 M e V 0fl, 71 M e V , lp~, 72 MeV when m e a s u r e d ielatlve to the 0s~ state These are, o f course, n o t well k n o w n , b u t Jolly has some a r g u m e n t s fo~ this choice E v e n - p a r i t y excltatmns c o n t i l b u t e a total o f 21 4 M e V to the c o r r e l a t i o n energy Thus, the total c o n t r i b u t i o n f r o m excitations of u n p e r t u r b e d energy 1ho) and 2he) is 42 6 M e V In calculating these, the original G - m a m x with exclusion principle fo~ A = 16 has been used rathm t h a n that for A = 80, as we should have, so this figure is a bit t o o Imge (Every term in E~o~r o f e q (6) has the same sign, so t h a t decreasing the m a g m t u d e s o f any terms will decrease E~or~ ) O n the other hand, excitations o f u n p e r t u r b e d energy 3hco have been left out If one calculates the b i n d i n g eneigles fo~ 160 in the global a p p r o x i m a t i o n f r o m the G - m a t r i x elements o f table I for Pauh principles c o r r e s p o n d m g to A = 16 a n d A = 80, one finds a p o t e n t i a l energy o f 279 9 M e V in the f o r m e r case, 254 2 M e V in the lattei case, glwng a difference o f 25 7 M e V Thus, the net correction IS 42 6 - 2 5 7 -- 16 9 M e V ol a b o u t 1 MeV/pa~t~cle b i n d i n g Because o f inaccuracies a n d omissions refel red to above, this is n o t ~e~y accurate, b u t we believe that it represents r o u g h l y the size o f the effect The m a i n reason that this c o l r e c t l o n is so small is t h a t the Jolly u n p e r t u r b e d energies for excitations up two shells are large ~ 40 MeV, so t h a t the even-parity vlbrat)ons d o n o t c o n t r i b u t e m u c h to E .... On the o t h e r hand, they give the T = 1, 0 + and 2 + v i b r a t i o n s at energies where they seem to be observed m inelastic electron scattering, and the T = 0, 2 + v i b r a t i o n at 27 MeV, which is a b o u t the lowest it could c o m e w i t h o u t having been observed The c o n t r i b u t i o n to E~o~r f r o m o d d - p a n t y states is a b o u t the same as u n p u b l i s h e d results o b t a i n e d some years ago by K a l h o The conclusions from this w o r k are as follows (l) The d o u b l e counting levels involved in using a G - m a t r i x calculated w i t h o u t

CORRELATION ENERGY OF 160

247

projecting out higher oscillator states, such as the Of state in fig 1 to introduce renormahzatlons as shown In fig 1 is not serious It can be corrected for m a stra~ghtforward way - by using G-matrices calculated with exclusion of the larger space from the intermediate plane-wave states (11) Corrections to be binding energy from using better particle spectra close to the Fermi surface are probably not large, ~ 1 MeV In 1 6 0 Our calculations relating to point (u) were rather rough, for reasons mentioned, but it would be straightforward to improve on all points, except for the uncertainty m unpertubed even-parity excitations We would like to thank Alpo Kalho and A M Green for several helpful conversations We are very grateful to A P Shukla, who carried out calculations of the evenparity vibrations Some o f the calculations were carried out at Princeton, where the work was supported by the U S Atomic Energy Commission This work made use of the Princeton Computing Facilities, supported m pair by the National Science F o u n dation G r a n t N S F - G P 579

References 1) H A Bethe, Phys Rev 138 (1965) B804 2) D W L Sprung, P C Bhargava and T K Dahlblom, Phys Lett 21 (1966)538, P C Bhargava and D W L Sprung, Ann of Phys 42 (1967) 222 3) H A Bethe, preprlnt 4) T Dahlblom, private communication 5) T T S Kuo and G E Brown, Nuclear Physics 85 (1966) 40 6) C W Wong, Nuclear Physics Agl (1967) 399 7) R J Eden and N C Francis, Phys Rev 97 (1955)1367 8) C Bloch and J Horowltz, Nuclear Physics 8 (1958) 91 9) B H Brandow, Revs Mod Phys, to be published 10) D J Thouless, Nuclear Physics 22 (1961) 78, G E Brown, Unified theory of nuclear models (North-Holland Publ C o , Amsterdam, 1964) chapt V 11) Sawada, Brueckner, Fukuda and Brout, Phys Rev 108 (1957) 507 12) Mavromatls, Markiewlcz and Green, to be published 13) H P Jolly, Phys Lett 5 (1963)289