Volume 37B, number 2
PHYSICS LETTERS
NUCLEAR OF
GIANT
DIPOLE
15 November 1971
EXCITATION
AND QUADRUPOLE
RESONANCES
J. BANG and P. D. KUNZ
Niels Bohr Institute, University of Copenhagen, Denmark Received 15 September 1971 Cross sections for nuclear excitation of giant dipole and quadrupole resonances are calculated in DWBA, using a macroscopic form factor. Agreement with existing data is obtained. The depence of the matrix elements on mass number is exposed, leading to a prediction of comparatively small cross sections for heavier nuclei.
E x c i t a t i o n of giant dipole r e s o n a n c e s t a t e s in a n u m b e r of light and medium nuclei was s e e n a l r e a d y in the i n v e s t i g a t i o n s by M a r l s , T y r e n and c o - w o r k e r s [e.g. 1] by impinging protons. The p r o t o n e n e r g i e s w e r e ~ 180 MeV and the c r o s s s e c t i o n s s t r o n g l y f o r w a r d peaked. F o r this r e a s o n , and b e c a u s e of the background, e s t i m a t e s of total c r o s s s e c t i o n s w e r e difficult; s e e , h o w e v e r , fig. 1. With a c c e l e r a t o r s a v a i l a b l e now, a b r o a d e r study of t h e s e p r o c e s s e s as w e l l as of excitation of giant quadrupole r e s o n a n c e s by c h a r g e d p a r t i c l e s should be p o s s i b l e . We t h e r e f o r e think that it could be of s o m e u s e to p r e s e n t h e r e s o m e e s t i m a t e s of the c r o s s s e c t i o n to be e x p e c t e d for such i n e l a s t i c p r o c e s s e s . In the c a l c u l a t i o n s , the u s u a l DWBA f o r m a l i s m for i n e l a s t i c s c a t t e r i n g with excitation of c o l l e c t i v e s t a t e s is adopted. T h e i n t e r a c t i o n s b e t w e e n p r o j e c t i l e and t a r g e t is expanded in p r o d u c t s of s p h e r i c a l t e n s o r functions of the v a r i a b l e s of the two s y s t e m s :
coupl--
;t=l,2 ~'=0, 1
(;)
F o r the c o l l e c t i v e v a r i a b l e a , we s h a ll u s e the definition 128
i
i
d__g_~ d~ mb
C)
1
~
~'~
~
b)
(1)
The f o r m f a c t o r f~t(r) is, following the a r g u m e n t s of Zim~tnyi et al. [2], of the t y p e r ~ f , f = (1 + exp ((r-R)/a)) "1 w h e r e the p a r a m e t e r s in this W o o d s - S ax o n f a c t o r should roughly c o r r e s p o n d to t h o s e of the n u c l e a r m a t t e r d i s t r i bution. The a b s o l u t e c r o s s s e c t i o n s depend on the n u c l e a r m a t r i x e l e m e n t s K <11 a I 0 >, the s i z e of which can be e s t i m a t e d in the following way
[3].
In the giant dipole c a s e , e.g., a is the d i s t a n c e b et w een the c e n t r e s of m a s s of p r o t o n s and n e u t r o n s , and 6p I the c o r r e s p o n d i n g density d i s p l a c e m e n t , Pn - Pp, which for s e l f - c o n s i s tency r e a s o n s is roughly p r o p o r t i o n a l for the p o t en t i al which it g e n e r a t e s
10
0
2'5
~°
50
- - 7 5
Fig. 1. Cross sections for giant dipole excitation in 40Ca(p,p~), a), b) E p = 182MeV, c) Ep=40MeV. Experimental points (Ep = 182 MeV) fromq'yren et al. [1].
PHYSICS
Volume 37B, number 2 l
I
15 November 1971
LETTERS
etc., but for estimates like eq. (6) we may just take P0 to be a function of r, say P0 ~ f ( h e r e and in the following we limit ourselves to spherical nuclei). F r o m the volume potential of the shell model, and isovector density dependent part can be distinguished
l
mb/sr 16'
V = V O + t . T V1/2A = V0 + ¼(2tz) Pl V1/P0, V1 ~ 100 M e V this determines K, since by eqs.(4) and (6) V1 6Pl VI'4nA a(2tz)r Ylu 6V1 = 4 - ( 2 t z ) P0 4(r2)PoA
16'
10-,i
(8) (9)
so, from (3) 61 = Vl~A/(r2) po
(10)
The m a t r i x elements of a a r e obtained from its m i c r o s c o p i c r e p r e s e n t a t i o n
zi(2tz)i
o~10 =~-
10,3
(11)
Z
SO
h-2 [[H, a l 0 ] , a l 0 ] = - ~ [ [ ~
hi, a l 0 ] , a l 0 ]
~23 -
- 4~MA
(12)
On the other hand I
50
~o
I
I
100
150
o,]Jo>=-2
=-2 ~ E i
i
Fig. 2. C r o s s sections for giant dipole excitation in 208pb(p,p) h) E p - 1 8 MeV, i) E p = 2 7 M e V .
[ o>
(0[q [i) (i[ o~I0)
If the dipole strength is concentrated in one state, I1), with energy ffWl, we t h e r e f o r e have
(01o~[1) ( l lot [0) = 3h-/w 16~MA 601 ~ 6V1 = K a F M
(3)
or
501 = K a(2tr)r YI~ PO/A
(4)
~ f dT Por2 Ylu * Ylg a = g-~Ks -4hA
fr2 Po
dr
(5)
- 4~AM
The densities p a r e o p e r a t o r s in the nuclear coordinates
1
Po(r) =•
A
~ 5(r-ri) i=l
(7)
(10a)
eq. (12) by
(0[[[H, ~20], 0~20]I 0) 5k-2(r2) (6)
(14)
Quite s i m i l a r a r g u m e n t s can be used to find the coupling constant and matrix elements for the isovector quadrupole mode. Only, apart from trivial changes, (10) is h e r e r e p l a c e d by K21~'=l = VI~A/(r4)pO
or
K = 4~A/(r 2) p 0
(13)
(12a)
F o r the i s o s c a l a r quadrupole oscillation mode with approximately the s a m e frequency, we a s sume the s a m e spatial form of o~2~ , but instead of (4)
5p 0 = K a Y 2 ~ r
aP0 { \ - fl Y2 ~ r ~r ]
~
(4b) 129
Volume 37B, number 2 ,o"
PHYSICS LETTERS
Table 1 Optical parameters, conventional notation and units
I
mb/sr
,d ~
15 November 1971
\ ",
\
f-~ \
I\/,,
/
k)_
curve
V
rR
a,b c h,i,j,k 1
16 41 63 I00
1.18 1.17 1.17 1.37
aR
W
0.55 8 0.75 5.3 0.75 0 0.60 46
4WD 0 7.2 48 0
rI 1.51 1.32 1.32 1.37
aI 0.55 0.51 0.82 0.60
~
Ell ~
v
72A -1"3/ MeV
E21 ~ 123A -1/3 MeV E20
ld'
58A -1/3 MeV
(16)
F o r the product of coupling constants and n u c l e a r m a t r i x e l e m e n t s , we obtain in the 3 c a s e s , in a p p r o p r i a t e u n i t s , for p r o t o n s
,,
X = I , T = I : ½R 2 ~ 50A-1/3 MeV fm
',,l) o
I
t
lOO
15o
1 = 2 , ~ ' = 0 : ~ R 3
(17)
oo
Fig. 3. Cross sections for giant quadrupole excitation j) in 208pb(p,p'), Ep = 27 MeV, k) Ep = 18 MeV, 1) in 208pb (o/. ¢z~) E - 43 MeV. - i . e . s u r f a c e o s c i l l a t i o n s - , giving
K = 4~r/5
(6b)
The r e l a t i o n between 5 p and 6V for a s u r f a c e o s c i l l a t i o n gives
5 V = K ~ Y20 r ~ Vo/B r
(gb)
F o r a h a r m o n i c potential V0 = ½ m w 2 r
2
(15)
(3), (9) and (16) then lead to K2,~ =0 = 4nm o)2A/5
(lOb)
The f r e q u e n c i e s of the different modes may be calculated, e.g., in RPA, or just taken from e x p e r i m e n t s . This leads to the a p p r o x i m a t e r e lations
130
The s t r e n g t h s of these r e s o n a n c e s a r e of c o u r s e s p r e a d over a c o n s i d e r a b l e i n t e r v a l ; this m u s t p a r t i c u l a r l y be the case for the dipole. In t h e s e c a l c u l a t i o n s , however, we a s s u m e that all the s t r e n g t h is c o n c e n t r a t e d in one state. The p a r t i c u l a r c a s e s we show a r e for the dipole state at 15 MeV in 208pb and the i s o s c a l a r quadrupole s t a t e at 10MeV in the s a m e nucleus, and for the dipole state at 18MeV in Ca. We test two different a s s u m p t i o n s for the r a d i u s and diffuseness of the form factor. We calculate c r o s s s e c t i o n s for both 18,27,40 and 182 MeV p r o t o n s and 43 MeV alpha p a r t i c l e s . The optical potential p a r a m e t e r s a r e given in table 1, the c r o s s sections in figs. 1-3. The form f a c t o r s were calculated with the following p a r a m e t e r s : curve b) : k =4.89, a = 0 . 6 5 ; c u r v e s a), c ) : R = 4 . 3 7 5 , a = 0 . 8 0 , c u r v e s h),i), j),k), 1):R = 7.50, a =0.65 (all in fro). The authors a r e grateful to P r o f e s s o r B. Mottelson for suggesting this p r o b l e m and for many i m p o r t a n t i n f o r m a t i o n s .
References [1] H.Tyren. Th.A.J.Maris, Nucl. Phys. 4 (1958) 662. [2] J. Zim~inyi. I. Halpern. V.A. Madsen, Phys. Letters 33B (1970) 205. [3] A. Bohr and B. Mottelson, lecture notes.