Continuum shell-model calculation for 208Pb

Volume 34B, number 2

PHYSICS LETTERS

CONTINUUM

SHELL-MODEL

CALCULATION

1 February 1971

FOR

208pb

*

R. F. B A R R E T T ** Institut f~r Theoretische Physik der Universitttt Frankfurt, Robert-Mayer-Strasse 8-10, Frankfurt a. M., Germany

P. P. DELSANTO Department of Physics, University of Puerto Rico, Mayaguez, Puerto Rico 00708, USA Received 3 December 1970

Results of a continuum shell-model calculation of the total photoabsorption cross section for 208pb are presented and compared with the results of experiments and of bound-state calculations.

T h e o r e t i c a l t r e a t m e n t s of the v - r a y a b s o r p tion c r o s s section for 208pb within the f r a m e work of the shel. model have to date only been made using b o u n d - s t a t e techniques [ 1 - 3 ] . This m e a n s that the continuum c h a r a c t e r of the nuc l e a r wave functions has been neglected. Such calculations can only indicate the likely position of r e s o n a n c e s in the c r o s s - s e c t i o n s and give an idea of t h e i r dipole strength. Bound-state calculations can however give no idea of the widths of the r e s o n a n c e s , which a r e i m p o r t a n t in the c o m p a r i s o n of the r e s u l t s of t h e o r e t i c a l calculations with those of e x p e r i m e n t . In this l e t t e r , we p r e s e n t the f i r s t r e s u l t s of a continuum s h e l l - m o d e l calculation for the p h o t o - a b s o r p t i o n c r o s s section of 208pb. An ext e n s i v e d i s c u s s i o n of all the r e s u l t s that one can i n f e r from such a calculation, including the p a r tial c r o s s - s e c t i o n s and a n g u l a r d i s t r i b u t i o n s will be given in a l a t e r publication. We have used the eigenchannel method of t r e a t i n g the continuum developed by Danos and G r e i n e r [4,5], and d i s c u s s e d at length in a review a r t i c l e by B i e d e n h a r n et al. [6]. The r e l i a b i l i t y and a c c u r a c y of the method have been r e c e n t l y investigated, both n u m e r i c a l l y and a n a l y t i c a l l y , and shown to be very s a t i s f a c t o r y in r e a l i s t i c s c a t t e r i n g p r o b l e m s [7]. In these c a l c u l a t i o n s , we have p a r a m e t r i s e d a Woods-Saxon potential with s p i n - o r b i t and Coulomb t e r m s to r e p r o d u c e * This work has been supported by the Bundesministerium fur Bildung und Wissenschaft and by the Deutsche For schungsgemeinsehaft. ** Postdoctoral fellow of the Alexander yon Humboldt Stiftung. 110

the known s i n g l e - p a r t i c l e and s i n g l e - h o l e energy level scheme of 208pb [8]. A "matching" r a d i u s of 17.0 fm was chosen to divide the configuration space into i n n e r and outer r e g i o n s , the i n n e r r e gion being r e g a r d e d as the r e g i o n in which all n u c l e a r i n t e r a c t i o n takes place. At the matching r a d i u s , "natural" boundary conditions a r e applied which d i s c r e t i s e the continuum and provide a b a s i s on which a n o r m a l s h e l l - m o d e l calculation may be p e r f o r m e d in t h e i n n e r region. The eigenchannel technique is used to find eigenphases at which the s h e l l - m o d e l h a m i l t o n i a n in the i n t e r n a l region and the S - m a t r i x can be s i m u l taneously diagonalised. F o r this calculation, we have r e s t r i c t e d our s h e l l - m o d e l t r e a t m e n t in the i n n e r region to a s t a n d a r d l p - l h calculation with r e s i d u a l i n t e r actions between the configurations. The configur a t i o n s (1-) c o n s i d e r e d a r e listed in table 1. F o r the p a r t i c l e - h o l e i n t e r a c t i o n , a z e r o r a n g e force w i t h a Soper exchange m i x t u r e is assumed~ i.e.,

Vii = V o S ( r i - r j ) [ % + a ~ ( a i. aj)] with Vo = - 1 2 2 0 MeV, a o = 0.865, and act = 0.135. The depth of the r e s i d u a l i n t e r a c t i o n , Vo, was chosen to be the s a m e as used by Balashov et al. [1] in an e a r l i e r b o u n d - s t a t e calculation. The r e s u l t s of the calculation of the total photoabsorpt.ion c r o s s section for 208pb a r e shown in fig. l a . B e c a u s e of the l a r g e Coulomb b a r r i e r in 208pb, the (7,P) c r o s s ,section is supp r e s s e d at these e n e r g i e s , and the c r o s s - s e c t i o n shown is in effect due to the (~, n) reaction. F o r c o m p a r i s o n , the e x p e r i m e n t a l (7,n) c r o s s - s e c tion d e t e r m i n e d by Bell et al. [9] i s shown in

Volume 34B, number 2

PHYSICS

1 February 1971

LETTERS

Table 1 1- configurations considered in the eigencharmel calculation for 208pb Neutrons .-1

3.0

Protons

'13/2 h11/2

h~/2 i11/2

ii~/2 J13/2

-i h11/2 i13/2

i1~/2 J15/2

hl~/2 g9/2

h9}2

i11/2

g7}2 h9/2

h9}2

g7/2

g7}2

f7/2

h9}2 g9/2

g7)2

f5/2

f5}2 d3/2

d3}2

f5/2

f5}2 g7/2

d3}2 Pl/2

p;}2 Sl/2

s;}2 .1/2

P;}2

d5}2

20 1.0

lO

OE

i.&

o2

o

Sl/2

10

P3/2

15

lc

P3}2 d5/2 f;~2

d5}2 f7/2

d5/2

f }2 g7/2 g9/2

t

4.o

2o

c.

o

,

..t

. . . .

,

10

fig. lb, and the r e s u l t s of a bound-state calculation including the R.P.A. by Kuo et al. [3] a r e shown in fig. 1 c. Kuo et al. obtained the position of the giant dipole resonance to be low by s e v e r a l MeV c o m pared with experiment. They used h a r m o n i c o s cillator wave functions to d e s c r i b e the singlep a r t i c l e and single-hole wave functions for lead, but concluded that a better a g r e e m e n t with exp e r i m e n t would have been obtained if wave functions in a Woods-Saxon well had been used. In fig. la, it is seen that the position of the giant dipole resonance in our calculation, in which Woods-Saxon continuum wave functions a r e used, is in better a g r e e m e n t with experiment, but is still too low by 500 keV. It is possible that this a g r e e m e n t could have been improved by optimising the depth of the residual interaction, Vo, but the amount of computing time required f o r this second calculation would have been prohibitive. Although the e n e r g i e s of the states found by

,

I.

15

Excitation energy (MeV)

Fig. 1. a) Total photo-absorption cross-section of 208pb calculated from the eigenchannel reaction theory. b) Experimental photoneutron cross-section of 208pb due to Bell et al. [9]. e) Results of a bound-state calculation (including R.P.A.) for 208pb photo-absorption cross-section by Kuo et al. [3]. Kuo et al. are lower than the c o r r e s p o n d i n g e n e r gies obtained in the continuum calculation, it can be seen that their prediction of two m a j o r peaks with associated minor peaks is in good a g r e e m e n t with the r e s u l t s shown on fig. la. The relative positions of the peaks are also in reasonable agreement. As a r e s u l t of the influence of the high Coulomb and centrifugal b a r r i e r s on the s i n g l e - p a r t i c l e states in the c a s e of lead, we find below 10 MeV a number of n a r r o w r e s o n a n ces c a r r y i n g little dipole strength. These c o r respond to the many weak states found in this region in the c o r r e s p o n d i n g bound-state calculation. 111

Volume 34B, number 2

PHYSICS

In the e i g e n c h a n n e l c a l c u l a t i o n f o r l e a d , the f u l l width half m a x i m u m of the m a i n d i p o l e r e s o n a n c e was c a l c u l a t e d to be 700 keV c o m p a r e d with an e x p e r i m e n t a l v a l u e of 4.05 MeV, and the h e i g h t w a s found to be 3.2 b a r n s c o m p a r e d with an e x p e r i m e n t a l v a l u e of 640 m b a r n s [9]. It is t y p i c a l of c o n t i n u u m s h e l l - m o d e l c a l c u l a t i o n s p e r f o r m e d so f a r that the p r e d i c t e d r e s o n a n c e s a r e h i g h e r and n a r r o w e r t h a n t h o s e e x p e r i m e n t a l l y o b s e r v e d [6]. It is b e l i e v e d that the e f f e c t of c o n f i g u r a t i o n s not i n c l u d e d in the c a l c u l a t i o n ( e s s e n t i a l l y 2p-2h c o n f i g u r a t i o n s ) would be to s p r e a d the d i p o l e s t r e n g t h , thus r e d u c i n g the h e i g h t of the p e a k s and b r o a d e n i n g t h e m . T h i s e f f e c t has b e e n o b t a i n e d in the c o u p l e d - c h a n n e l c a l c u l a t i o n s of B u c k and Hill [10] and of M a r a n g o n i and S a r u i s [11], w h e r e the p r e s e n c e of h i g h e r c o n f i g u r a t i o n s h a s b e e n s i m u l a t e d by the i n c l u s i o n of an i m a g i n a r y a b s o r p t i v e t e r m in the s i n g l e - p a r t i c l e p o t e n t i a l . The i m p o r t a n c e of 2 p - 2 h c o n f i g u r a t i o n s in the l i g h t e r n u c l e i is w e l l known [12]. F o r e x a m p l e , G i l l e t et a l . [ 1 3 ] have i n c l u d e d 2 p - 2 h c o n f i g u r a t i o n s in an 1 6 0 c a l c u l a t i o n to e x p l a i n the p r e s e n c e of fine s t r u c t u r e in the giant r e s o n a n c e . In the c a s e of h e a v y n u c l e i , the a d d i t i o n a l s p r e a d i n g width of the giant r e s o n a n c e due to the a d m i x t u r e of the m a n y 2 p - 2 h c o n f i g u r a t i o n s to the l p - l h d o o r w a y s t a t e s h a s b e e n e s t i m a t e d by D a n o s and G r e i n e r [14]. They obtain a v a l u e b e t w e e n 0.5 and 2.5 MeV. M o r e o v e r , the c o u p l i n g of the p a r t i c l e - h o l e s t a t e s to c o l l e c t i v e s t a t e s ( s u r f a c e q u a d r u p o l e and o c t u p o l e v i b r a t i o n s ) [15] m a y a l s o c o n t r i b u t e to a s p r e a d i n g of the giant resonance. The i n t e g r a t e d c r o s s s e c t i o n up to 16 MeV, ~(E)dE, was c a l c u l a t e d to be 3.9 MeV b a r n s . It can be c o m p a r e d w i t h the e x p e r i m e n t a l v a l u e s of f ~ 8 ~(E)dE = 2.91 MeV b a r n s o b t a i n e d by H a r v e y et al. [16] and f ~ 8 . 5 ~(E)dE = 4.0 MeV b a r n s o b t a i n e d by G o r y a c h e v et al. [17]. The c l a s s i c a l s u m r u l e y i e l d s a v a l u e of 2.98 MeV barns. In the e x p e r i m e n t a l d e t e r m i n a t i o n of the (7, n) c r o s s - s e c t i o n due to B e l l et al. [9] (fig. l b ) , i n t e r m e d i a t e s t r u c t u r e is found in the r e g i o n n e a r 11 MeV. T h i s i s not s e e n in the e a r l i e r m e a s u r e m e n t s of the L i v e r m o r e g r o u p [16], but w a s a l s o found in m e a s u r e m e n t s of 2 0 8 p b p h o t o -

f016

112

LETTERS

1 February 1971

n e u t r o n s p e c t r a by M c N e i l l and c o - w o r k e r s [18]. A c o m p a r i s o n of f i g s . l a and l b s h o w s that a lp-lh continuum calculation successfully exp l a i n s the g r o s s f e a t u r e s of the p h o t o - a b s o r p t i o n c r o s s s e c t i o n s , s u c h a s the p o s i t i o n of the giant r e s o n a n c e , and m a y a l s o e x p l a i n , at l e a s t in p a r t , s o m e of the s t r u c t u r e o b s e r v e d by B e l l et al. In v i e w of the p r o n o u n c e d s h e l l - s t r u c t u r e e x h i b i t e d by l e a d , s u c h a c a l c u l a t i o n should be r e g a r d e d a s a f i r s t , n e c e s s a r y s t e p b e f o r e the i n c l u s i o n of any h i g h e r o r d e r c o n f i g u r a t i o n s in a s h e l l - m o d e l t r e a t m e n t of 208pb. The a u t h o r s would l i k e to thank P r o f e s s o r W. G r e i n e r f o r a c r i t i c a l r e a d i n g of the m a n u s c r i p t .

References [1] v. v. Balashov, V. G. Shevchenko and N. P. Yudin, Soviet Physics J E T P 14 (1962) 1371. [2] V. Gillet, A. M. Green and E. A. Anderson, Nucl. Phys. 88 (1966) 321. [3] T. T. S. Kuo, J. Blomqvist and G. E. Brown, Phys. Letters 31B (1970) 93. [4] M. Danos and W. Greiner, Phys. Rev. 146 (1966) 708. [5] H. G. Wahsweiler, W. Greiner and M. Danos, Phys. Rev. 170 (1968) 893. [6] L. C. Biedenharn, M. Danos, P. P. Delsanto, W. Greiner and H.G. Wahsweiler, Rev. Mod. Phys., to be published. [7] P. P. Delsanto, M. F. Roetter and H. G. Wahsweiler, Z..Physik 222 (1969) 67. [8] D. A. Bromley and J. Weneser, Comm. Nucl. Part. Phys. 2 (1968) 151. [9] H. Bell, R. Berg~re, P. Carlos and A. Veyssi~re, Compte Rendus Acad. Sc. Paris 269 (1964) 216. [10] B. Buck and A. D. Hill, Nucl. Phys. A95 (1967) 271. [11] M. Marangoni and A.M. Saruis, Nucl. Phys. A132 (1969) 649. [12] V. Gillet, Proc. Int. Conf. on Properties of nuclear states, Montreal (1969) p. 483. [13] V. Gillet, M.A. Melkanoff and J. Raynal, Nucl. Phys. A97 (1967) 631. [14] M. Danos and W. Greiner, Phys. Rev. 138 (1965) B876. [15] D. Drechsel, J. B. Seaborn and W. Greiner, Phys. Rev. 162 (1967) 983. [16] R. R. Harvey, J. T. Caldwell, R. L. Bramblett and S. C. Fultz, Phys. Rev. 136B (1964) 126. [17] B.I. Goryachev, V.S. Ishkhanov, I. M. Kapitonov and V. G. Shevchenko, J E T P Letters 7 (1968) 161. [18] K. G. McNeill, J.W. Jury and J. S. Hewitt, Can. J. Phys. 48 (1970) 950.