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Radial dependence of the P1 interaction in the schematic model
Volume 33B, n u m b e r 7
RADIAL
PHYSICS
LETTERS
DEPENDENCE OF THE THE SCHEMATIC
7 D e c e m b e r 1970
Pl INTERACTION MODEL
IN
S. F U J I I a n d K. M O R I T A
Department of Physics and Atomic Energy Research Institute, College of Science and Engineering, Nihon University, Chiyoda-ku, Tokyo, Japan Received 5 October 1970
E l e c t r o - e x c i t a t i o n s of jTr= 1-, T = 0 levels in self-conjugate closed shell nuclei are calculated, a s suming that the radial p a r t of the s e p a r a b l e potential of YI" ]I1 type has one node or m o r e and, making a r e m o v a l of the spurious mode in the s c h e m a t i c model.
In t h i s n o t e we a t t e m p t to a p p l y t h e s c h e m a t i c m o d e l [1] to s t a t e s o f J ~ = 1 - a n d T = 0 i n selfconjugate closed shell nuclei with some considerat i o n s . F o r e x c i t a t i o n s of t h e s e s t a t e s t h e t r a n s i t i o n c h a r g e d e n s i t y i s p r o p o r t i o n a l to t h e t r a n s i tion matter density Pl, =<~Pl, ~ 6 ( r - r i ) ~ o ~ , (1) i as far as isospin impurities are ignored. From t h e c o n d i t i o n t h a t t h e c e n t r e of m a s s ( c . m . ) r e m a i n s f i x e d t h r o u g h o u t t h e e x c i t a t i o n , it i s s h o w n t h a t t h e r a d i a l p a r t of P l m u s t h a v e o n e n o d e o r more and so the transition charge density as w e l l [2]. Now l e t u d e n o t e t h e v a r i a t i o n of n u c l e a r f i e l d from the static potential in the time-dependent Hartree-Fock a p p r o x i m a t i o n [1]. u i s a p p r o x i m a t e l y p r o p o r t i o n a l to Pl
f V( r , Jr') P l ( r ' ) d r ' ,
[ 5]
JIll/ i//I
16'
IF1'
/:
16"
(2)
if we d r o p t h e d e p e n d e n c e of t h e t w o - b o d y f o r c e V on spin and isospin and the exchange terms arising from anti-symmetrization. For the shape v i b r a t i o n s , f o r i n s t a n c e , Rowe [3] h a s p o i n t e d out a similarity between the transition matter d e n s i t y a n d t h e v a r i a t i o n of n u c l e a r f i e l d . S u p posing this similarity for our case also and putting
g = k f ( r l ) f ( r 2 ) ~_. (_)m y ~
f rom
o from [ 1 0 ] , ~
(/'1) y l m ( i ' 2 )
,
(3)
m
we e x p e c t t h a t f ( r ) m a y h a v e a n o d e . Internal excitation modes are separated from c . m . m o t i o n in t h e f o l l o w i n g m a n n e r . F o r s i m p l i f i c a t i o n we u s e t h e h a r m o n i c o s c i l l a t o r w a v e
0
1.0 q (f")
2D
Fig. 1. F o r m f a c t o r s for e l e c t r o - e x c i t a t i o n s of JY= 1-, T = 0 levels in 160. The solid and dash-dotted c u r v e s are i F c ] 2 and..]FTI 2 calculated for 7.12 MeV level where b = 1.76 fm is chosen. The e x p e r i m e n t a l data [5,10] are those for this level. The dotted curve is '']FcI 2 obtained for 47.69 MeV level. functions_ to d e s c r i b e s i n g l e p a r t i c l e s t a t e s . L e t b~a denote the creation operator for a particle on s i n g l e p a r t i c l e l e v e l " A " a n d a h o l e o n " a " c o m b i n e d to J ~ = 1 - , M ( p r o j e c t i o n ) = 0 a n d T = 0. T h e p a r t i c l e - h o l e p a i r s "A, a " c a n b e n u m b e r e d w i t h i n t e g e r i ' s (1 ~< i < N ) . T h e n we i n t r o d u c e a l i n e a r t r a n s f o r m a t i o n of 453
Volume 33B, number 7
PHYSICS
LETTERS
i
-
=
- -
)tN~l
¢ ,rom [ I t ]
o from
7 December 1970
j=l
[12]
z_eJ
(wc'TF)2
(8)
J
is obtained. H e r e F is the c o l u m n m a t r i x with e l e m e n t s of (9)
Fi=~2j A+ 1C(JAlja;
,6'
N-I I .o
o
yj : - N c ~
2.0
Fig. 2. Form factors for electro-excitations of j?r = 1-, T = 0 levels in 40Ca. The solid and dash-dotted curves are .,tFcI 2 and ,,IFTI 2 calculated for 6.95 MeV level where b = 2.08 fm is chosen. The experimental data [11,12] are those for this level. In, ~this connection see the text also. The dotted curve is ]Fc~2 obtained for 27.36 MeV level. ¢
i~=1 cijb~. ,
(j = 1,2 . . . . . . . N)
I
,
Wkj ( w c ' T F ) k ,
(10)
k =1 ~k + w
q (f")
sj =
,
and C 'T is the t r a n s p o s e d m a t r i x of the N × ( N - l ) m a t r i x C' that is d e r i v e d f r o m C by d r o p p i n g the N t h column. The EJ's a r e the e i g e n v a l u e s of c'TEc ' w h e r e E i ~ the d i a g o n a l m a t r i x c o n s t i t u t e d by the u n p e r t u r b e d e n e r g i e s of p a r t i c l e - h o l e p a i r s . T h i s d i a g o n a l i z a t i o n is m a d e with the m a t r i x W a s W C ' T - E c ' W T =E'. F o r x j a n d y j w e get N-1 1 xj = N e k=l ~ (we'TF) k , (101
iFr
N
½0½) f R A R a f ( r ) r 2 d r
(4)
w h e r e N c is the n o r m a l i z a t i o n c o n s t a n t . We c o n s i d e r not only l]~co e x c i t a t i o n s but a l s o 3fiw to c a l c u l a t e the e i g e n m o d e s f o r 160 and 4 0 C a b e c a u s e the e l e c t r o n s c a t t e r i n g d a t a [5] f a v o u r the a d m i x t u r e of high e n e r g y c o n f i g u r a t i o n s [2]. F o r f ( r ) we a s s u m e
f(r) = ~ Kj(r/R) 2j-1 exp[-(r/R) 2] , j=l
(11)
with the c o n d i t i o n , cO
w i t h the b o s o n a p p r o x i m a t i o n . H e r e the cij's c o n s t i t u t e an N × N o r t h o g o n a l m a t r i x C with
ciNcc 2f2JA A + 1
(A]Ipl[a) .
f 0 (5)
JA i s the total a n g u l a r m o m e n t u m of "A" and (Allplla) the r e d u c e d m a t r i x e l e m e n t [4] of n u c l e o n momentum. Since c~ represents c.m. motion, we put the c r e a t i o n o p e r a t o r f o r o u r v i b r a t i o n a s N-1 N-1 Q~ = ~-~ s~ j~__lyjsj (6)
j=l xj
+
and s o l v e [H, Q~] = w q ~ ,
(7)
by " l i n e a r i z a t i o n " and d r o p p i n g a s p u r i o u s c o u p l ing of i n t e r n a l m o d e s to c . m . m o t i o n . A s a r e s u l t the d i s p e r s i o n f o r m u l a ,
454
f(r)r 3 dr = 0 .
(12)
T h i s c o n d i t i o n f o r the f o r c e m a y be too r e s t r i c t i v e hut the f i n i t e n e s s of the f o r c e r a n g e is c o n s i d e r e d to s o m e e x t e n t by t a k i n g R / b = 1.3 w h e r e b is the oscillator range parameter. For single particle e n e r g i e s of 1 6 0 the e n e r g y s p e c t r u m c a l l e d (1) in [6] is c h o s e n . In e a s e of 4 0 C a the n e u t r o n e n e r g i e s adopted by G i l l e t and S a n d e r s o n [7] a r e u s e d f o r 2s, l d , 2p and I f l e v e l s w h i l e the l o c a t i o n of o t h e r l e v e l s is c a l c u l a t e d f r o m the h a r m o n i c o s c i l l a t o r p o t e n t i a l with s p i n - o r b i t coupling. H e r e we c h o o s e t / ~ o = 13.1 M e V , K = 0 . 0 7 a n d / ~ = 0 . 0 in N i l s s o n ' s n o t i o n and t a k e 35.3 MeV f o r the e n e r g y d i f f e r e n c e b e t w e e n l d 3 / 2 and 3P3/2. T h e c a l c u l a t e d e i g e n e n e r g i e s a r e shown in t a b l e 1. The C o u l o m b and t r a n s v e r s e f o r m f a c t o r s ( [ F c I 2 and ] F T ] 2) f o r e l e c t r o - e x c i t a t i o n s of 7.12 MeV l e v e l in 1 6 0 and 6.95 MeV l e v e l in 4 0 C a , o b t a i n e d f r o m the B o r n a p p r o x i m a t i o n a r e
PHYSICS
Volume 33B, number 7
Table 1 Eigenenergies (MeV) o f J ~ = 1- and T = O in 160 and 40Ca. F o r the potential p a r a m e t e r s , k = -887.6 MeV, K1 = 1.0, g-2 = - 0 . 4 and Ki = 0 (j >13) for 160 and k = - 1 0 7 . 2 MeV, g l = 1.0, g2. = - 3 . 2 , K3 = 0.8 and Kj = 0 (j >~ 4) for ~UCa a r e chosen.
lI ¢
/ I
2
-,
I-I
7 D e c e m b e r 1970
LETTERS
\ /
/
,,o¢ o
-2
160
40Ca
7.12 15.36 16.91 22.33 47.69 56.80 61.32 62.86 65.64 67.17 70.13
6.95 9.63 11.57 13.26 13.54 14.59 19.54 27.36 35.35 37.56 37.85 38.08 38.63 38.73 39.68 40.12 4O.27 40.49 40.60 41.18 42.96 45.84
-5 Fig. 3. Radial dependences of the t r a n s i t i o n m a t t e r density for the lowest level of our model (solid curve) and the c o r r e s p o n d i n g variation of n u c l e a r field (dotted curve) in 160 and 40Ca. The units of ordinate are a r b i trary. d i s p l a y e d in f i g s . 1 a n d 2. H e r e I F c ] 2 a n d F T [ 2 a r e d e f i n e d b y d a / d ~ = a M o t t . ( I F c [ 2 + (~ + + t a n 2 ½0)]FT]2). T h e c e n t r e of m a s s c o r r e c t i o n [8] a n d t h e n u c l e o n s i z e e f f e c t [9] h a v e b e e n considered. The experimental data for these levels are quoted from [5,10-12]. (Eisenstein et al. [12] i n t e r p r e t e d t h e i r d a t a n e a r 6 . 9 5 M e V a s a c o m p l e x of 2 + a n d 3 - f o r m f a c t o r s b u t t h e i r estimation for the contribution from I- is not m o d e l - i n d e p e n d e n t . ) F r o m f i g s . 1 a n d 2 we f i n d it p e r m i s s i b l e to c o m p a r e o u r I F c ] 2 w i t h t h o s e experimental data directly. The agreement with e x p e r i m e n t i s m o d e r a t e l y good f o r t h e d e p e n d e n c e on m o m e n t u m t r a n s f e r q b u t t h e d i f f r a c t i o n m a x i m a a r e too l a r g e . T h i s d i s c r e p a n c y m a y b e p a r t l y r e m o v e d b y c o n s i d e r i n g t h e a d m i x t u r e of 3 p - 3 h c o m p o n e n t s [13,14]. In t h i s c o n n e c t i o n i t h a s b e e n o b s e r v e d [11] t h a t t h e f o r m f a c t o r s f o r 5.9 M e V l e v e l of 4 0 C a i s a b o u t t e n t i m e s a s s m a l l a s 6 . 9 5 M e V l e v e l . F u r t h e r t h e e f f e c t of i s o s p i n i m p u r i t i e s s h o u l d b e i m p o r t a n t a t q ~ co.
R a d i a l d e p e n d e n c e s of P l a n d u g i v e n b y (2) f o r t h e l o w e s t l e v e l s of 1 6 0 a n d 4 0 C a a r e s h o w n in fig. 3. T h e b e h a v i o r of P l s e e m s to r e p r e s e n t to c o m p r e s s i b l e n a t u r e of o u r v i b r a t i o n s . It a l s o i s s e e n t h a t t h e n e g a t i v e p a r t of u a t t r a c t s p a r t i c l e s and the positive part repels them. Thus the radial b e h a v i o r of u i s c o m p a t i b l e w i t h t h e o n e of P l , though the potential parameters should be further investigated by comparing theory with experiment more carefully. Comparing the first diffraction maximum with I F C }2 c a l c u l a t e d f o r o t h e r l e v e l s o f J ~ = 1 - a n d T = 0 n e a r t h e s a m e q, we c a n i n d i c a t e t h a t t h e e x c i t a t i o n of t h e l o w e s t l e v e l of o u r m o d e l i s distinctly stronger than other levels except for t h e 4 7 . 6 9 M e V l e v e l i n 1 6 0 a n d 27.36 M e V l e v e l in 4 0 C a . O u r C o u l o m b f o r m f a c t o r s f o r t h e s e l e v e l s a r e d i s p l a y e d i n f i g s . 1 a n d 2. T h u s it m a y b e s a i d t h a t 7.12 M e V l e v e l in 1 6 0 a n d 6 . 9 5 M e V l e v e l in 4 0 C a c o m p r i s e a l a r g e p o r t i o n of t h e c o m p r e s s i b l e m o d e . W e w o u l d l i k e to e x p r e s s s i n c e r e t h a n k s to P r o f e s s o r Y. T o r i z u k a , P r o f e s s o r M. O y a m a d a a n d D r . K. I t o h f o r m a k i n g t h e i r d a t a a v a i l a b l e t o u s b e f o r e p u b l i c a t i o n . We t h a n k P r o f e s s o r K. I z u m o , D r . B. I m a n i s h i a n d M r . M. Obu f o r helpful discussions.
455
Volume 33B, number 7
PHYSICS
References [1] G. E. Brown, Unified theory of nuclear models and f o r c e s (North-Holland, A m s t e r d a m , 1967). [2] S.Fujii, Prog. Theor. Phys.42 (1969) 416. [3] D . J . R o w e , Phys. Rev. 162 (1967) 866. [4] M . E . R o s e , Elementary theory of angular m o m e n tum (John Wiley and Sons, New York, 1957). [5] Y. Torizuka, M. Oyamada, K. Nakahara, K. Sugiyama, Y. Kojima, T. Terasawa, K.Itoh, A. Yamaguchi and M. Kimura, Phys. Rev. L e t t e r s 22 (1969) 544. [6] S.Fujii, Nuel. Phys.67 (1965) 592. [7] V.Gillet and E . A . S a n d e r s o n , Nucl. Phys. A91 (1967) 292. [8] L . J . T a s s i e and F . C . B a r k e r , Phys. R e v . l l l (1958) 940.
456
LETTERS
7 December 1970
[9] R.S.Willey, Nucl. P h y s . 4 0 (1963) 529. [10] J. C. B e r g s t r o m , W. Bertozzi, S. Kowalski, X. K. Maruyama, J . W . Lightbody J r . , S . P . Fivozinsky and S. P e n n e r , Phys. Rev. L e t t e r s 24 (1970) 152. [11] K.Itoh, M.Oyamada and Y.Torizuka, to be published. [12] R.A. Eisenstein, D.W. Madsen, H. Theissen, L. S. Cardman and C.K. Bockelman, Phys. Rev. 188 (1969) 1815. [13] G. E. Brown and A . M . G r e e n , P h y s i c s L e t t e r s 15 (1965) 168. [14] W . J . G e r a c e a n d A . M . G r e e n , Nucl. Phys. A l l 3 (1968) 641.
RADIAL
PHYSICS
LETTERS
DEPENDENCE OF THE THE SCHEMATIC
7 D e c e m b e r 1970
Pl INTERACTION MODEL
IN
S. F U J I I a n d K. M O R I T A
Department of Physics and Atomic Energy Research Institute, College of Science and Engineering, Nihon University, Chiyoda-ku, Tokyo, Japan Received 5 October 1970
E l e c t r o - e x c i t a t i o n s of jTr= 1-, T = 0 levels in self-conjugate closed shell nuclei are calculated, a s suming that the radial p a r t of the s e p a r a b l e potential of YI" ]I1 type has one node or m o r e and, making a r e m o v a l of the spurious mode in the s c h e m a t i c model.
In t h i s n o t e we a t t e m p t to a p p l y t h e s c h e m a t i c m o d e l [1] to s t a t e s o f J ~ = 1 - a n d T = 0 i n selfconjugate closed shell nuclei with some considerat i o n s . F o r e x c i t a t i o n s of t h e s e s t a t e s t h e t r a n s i t i o n c h a r g e d e n s i t y i s p r o p o r t i o n a l to t h e t r a n s i tion matter density Pl, =<~Pl, ~ 6 ( r - r i ) ~ o ~ , (1) i as far as isospin impurities are ignored. From t h e c o n d i t i o n t h a t t h e c e n t r e of m a s s ( c . m . ) r e m a i n s f i x e d t h r o u g h o u t t h e e x c i t a t i o n , it i s s h o w n t h a t t h e r a d i a l p a r t of P l m u s t h a v e o n e n o d e o r more and so the transition charge density as w e l l [2]. Now l e t u d e n o t e t h e v a r i a t i o n of n u c l e a r f i e l d from the static potential in the time-dependent Hartree-Fock a p p r o x i m a t i o n [1]. u i s a p p r o x i m a t e l y p r o p o r t i o n a l to Pl
f V( r , Jr') P l ( r ' ) d r ' ,
[ 5]
JIll/ i//I
16'
IF1'
/:
16"
(2)
if we d r o p t h e d e p e n d e n c e of t h e t w o - b o d y f o r c e V on spin and isospin and the exchange terms arising from anti-symmetrization. For the shape v i b r a t i o n s , f o r i n s t a n c e , Rowe [3] h a s p o i n t e d out a similarity between the transition matter d e n s i t y a n d t h e v a r i a t i o n of n u c l e a r f i e l d . S u p posing this similarity for our case also and putting
g = k f ( r l ) f ( r 2 ) ~_. (_)m y ~
f rom
o from [ 1 0 ] , ~
(/'1) y l m ( i ' 2 )
,
(3)
m
we e x p e c t t h a t f ( r ) m a y h a v e a n o d e . Internal excitation modes are separated from c . m . m o t i o n in t h e f o l l o w i n g m a n n e r . F o r s i m p l i f i c a t i o n we u s e t h e h a r m o n i c o s c i l l a t o r w a v e
0
1.0 q (f")
2D
Fig. 1. F o r m f a c t o r s for e l e c t r o - e x c i t a t i o n s of JY= 1-, T = 0 levels in 160. The solid and dash-dotted c u r v e s are i F c ] 2 and..]FTI 2 calculated for 7.12 MeV level where b = 1.76 fm is chosen. The e x p e r i m e n t a l data [5,10] are those for this level. The dotted curve is '']FcI 2 obtained for 47.69 MeV level. functions_ to d e s c r i b e s i n g l e p a r t i c l e s t a t e s . L e t b~a denote the creation operator for a particle on s i n g l e p a r t i c l e l e v e l " A " a n d a h o l e o n " a " c o m b i n e d to J ~ = 1 - , M ( p r o j e c t i o n ) = 0 a n d T = 0. T h e p a r t i c l e - h o l e p a i r s "A, a " c a n b e n u m b e r e d w i t h i n t e g e r i ' s (1 ~< i < N ) . T h e n we i n t r o d u c e a l i n e a r t r a n s f o r m a t i o n of 453
Volume 33B, number 7
PHYSICS
LETTERS
i
-
=
- -
)tN~l
¢ ,rom [ I t ]
o from
7 December 1970
j=l
[12]
z_eJ
(wc'TF)2
(8)
J
is obtained. H e r e F is the c o l u m n m a t r i x with e l e m e n t s of (9)
Fi=~2j A+ 1C(JAlja;
,6'
N-I I .o
o
yj : - N c ~
2.0
Fig. 2. Form factors for electro-excitations of j?r = 1-, T = 0 levels in 40Ca. The solid and dash-dotted curves are .,tFcI 2 and ,,IFTI 2 calculated for 6.95 MeV level where b = 2.08 fm is chosen. The experimental data [11,12] are those for this level. In, ~this connection see the text also. The dotted curve is ]Fc~2 obtained for 27.36 MeV level. ¢
i~=1 cijb~. ,
(j = 1,2 . . . . . . . N)
I
,
Wkj ( w c ' T F ) k ,
(10)
k =1 ~k + w
q (f")
sj =
,
and C 'T is the t r a n s p o s e d m a t r i x of the N × ( N - l ) m a t r i x C' that is d e r i v e d f r o m C by d r o p p i n g the N t h column. The EJ's a r e the e i g e n v a l u e s of c'TEc ' w h e r e E i ~ the d i a g o n a l m a t r i x c o n s t i t u t e d by the u n p e r t u r b e d e n e r g i e s of p a r t i c l e - h o l e p a i r s . T h i s d i a g o n a l i z a t i o n is m a d e with the m a t r i x W a s W C ' T - E c ' W T =E'. F o r x j a n d y j w e get N-1 1 xj = N e k=l ~ (we'TF) k , (101
iFr
N
½0½) f R A R a f ( r ) r 2 d r
(4)
w h e r e N c is the n o r m a l i z a t i o n c o n s t a n t . We c o n s i d e r not only l]~co e x c i t a t i o n s but a l s o 3fiw to c a l c u l a t e the e i g e n m o d e s f o r 160 and 4 0 C a b e c a u s e the e l e c t r o n s c a t t e r i n g d a t a [5] f a v o u r the a d m i x t u r e of high e n e r g y c o n f i g u r a t i o n s [2]. F o r f ( r ) we a s s u m e
f(r) = ~ Kj(r/R) 2j-1 exp[-(r/R) 2] , j=l
(11)
with the c o n d i t i o n , cO
w i t h the b o s o n a p p r o x i m a t i o n . H e r e the cij's c o n s t i t u t e an N × N o r t h o g o n a l m a t r i x C with
ciNcc 2f2JA A + 1
(A]Ipl[a) .
f 0 (5)
JA i s the total a n g u l a r m o m e n t u m of "A" and (Allplla) the r e d u c e d m a t r i x e l e m e n t [4] of n u c l e o n momentum. Since c~ represents c.m. motion, we put the c r e a t i o n o p e r a t o r f o r o u r v i b r a t i o n a s N-1 N-1 Q~ = ~-~ s~ j~__lyjsj (6)
j=l xj
+
and s o l v e [H, Q~] = w q ~ ,
(7)
by " l i n e a r i z a t i o n " and d r o p p i n g a s p u r i o u s c o u p l ing of i n t e r n a l m o d e s to c . m . m o t i o n . A s a r e s u l t the d i s p e r s i o n f o r m u l a ,
454
f(r)r 3 dr = 0 .
(12)
T h i s c o n d i t i o n f o r the f o r c e m a y be too r e s t r i c t i v e hut the f i n i t e n e s s of the f o r c e r a n g e is c o n s i d e r e d to s o m e e x t e n t by t a k i n g R / b = 1.3 w h e r e b is the oscillator range parameter. For single particle e n e r g i e s of 1 6 0 the e n e r g y s p e c t r u m c a l l e d (1) in [6] is c h o s e n . In e a s e of 4 0 C a the n e u t r o n e n e r g i e s adopted by G i l l e t and S a n d e r s o n [7] a r e u s e d f o r 2s, l d , 2p and I f l e v e l s w h i l e the l o c a t i o n of o t h e r l e v e l s is c a l c u l a t e d f r o m the h a r m o n i c o s c i l l a t o r p o t e n t i a l with s p i n - o r b i t coupling. H e r e we c h o o s e t / ~ o = 13.1 M e V , K = 0 . 0 7 a n d / ~ = 0 . 0 in N i l s s o n ' s n o t i o n and t a k e 35.3 MeV f o r the e n e r g y d i f f e r e n c e b e t w e e n l d 3 / 2 and 3P3/2. T h e c a l c u l a t e d e i g e n e n e r g i e s a r e shown in t a b l e 1. The C o u l o m b and t r a n s v e r s e f o r m f a c t o r s ( [ F c I 2 and ] F T ] 2) f o r e l e c t r o - e x c i t a t i o n s of 7.12 MeV l e v e l in 1 6 0 and 6.95 MeV l e v e l in 4 0 C a , o b t a i n e d f r o m the B o r n a p p r o x i m a t i o n a r e
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Volume 33B, number 7
Table 1 Eigenenergies (MeV) o f J ~ = 1- and T = O in 160 and 40Ca. F o r the potential p a r a m e t e r s , k = -887.6 MeV, K1 = 1.0, g-2 = - 0 . 4 and Ki = 0 (j >13) for 160 and k = - 1 0 7 . 2 MeV, g l = 1.0, g2. = - 3 . 2 , K3 = 0.8 and Kj = 0 (j >~ 4) for ~UCa a r e chosen.
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,,o¢ o
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160
40Ca
7.12 15.36 16.91 22.33 47.69 56.80 61.32 62.86 65.64 67.17 70.13
6.95 9.63 11.57 13.26 13.54 14.59 19.54 27.36 35.35 37.56 37.85 38.08 38.63 38.73 39.68 40.12 4O.27 40.49 40.60 41.18 42.96 45.84
-5 Fig. 3. Radial dependences of the t r a n s i t i o n m a t t e r density for the lowest level of our model (solid curve) and the c o r r e s p o n d i n g variation of n u c l e a r field (dotted curve) in 160 and 40Ca. The units of ordinate are a r b i trary. d i s p l a y e d in f i g s . 1 a n d 2. H e r e I F c ] 2 a n d F T [ 2 a r e d e f i n e d b y d a / d ~ = a M o t t . ( I F c [ 2 + (~ + + t a n 2 ½0)]FT]2). T h e c e n t r e of m a s s c o r r e c t i o n [8] a n d t h e n u c l e o n s i z e e f f e c t [9] h a v e b e e n considered. The experimental data for these levels are quoted from [5,10-12]. (Eisenstein et al. [12] i n t e r p r e t e d t h e i r d a t a n e a r 6 . 9 5 M e V a s a c o m p l e x of 2 + a n d 3 - f o r m f a c t o r s b u t t h e i r estimation for the contribution from I- is not m o d e l - i n d e p e n d e n t . ) F r o m f i g s . 1 a n d 2 we f i n d it p e r m i s s i b l e to c o m p a r e o u r I F c ] 2 w i t h t h o s e experimental data directly. The agreement with e x p e r i m e n t i s m o d e r a t e l y good f o r t h e d e p e n d e n c e on m o m e n t u m t r a n s f e r q b u t t h e d i f f r a c t i o n m a x i m a a r e too l a r g e . T h i s d i s c r e p a n c y m a y b e p a r t l y r e m o v e d b y c o n s i d e r i n g t h e a d m i x t u r e of 3 p - 3 h c o m p o n e n t s [13,14]. In t h i s c o n n e c t i o n i t h a s b e e n o b s e r v e d [11] t h a t t h e f o r m f a c t o r s f o r 5.9 M e V l e v e l of 4 0 C a i s a b o u t t e n t i m e s a s s m a l l a s 6 . 9 5 M e V l e v e l . F u r t h e r t h e e f f e c t of i s o s p i n i m p u r i t i e s s h o u l d b e i m p o r t a n t a t q ~ co.
R a d i a l d e p e n d e n c e s of P l a n d u g i v e n b y (2) f o r t h e l o w e s t l e v e l s of 1 6 0 a n d 4 0 C a a r e s h o w n in fig. 3. T h e b e h a v i o r of P l s e e m s to r e p r e s e n t to c o m p r e s s i b l e n a t u r e of o u r v i b r a t i o n s . It a l s o i s s e e n t h a t t h e n e g a t i v e p a r t of u a t t r a c t s p a r t i c l e s and the positive part repels them. Thus the radial b e h a v i o r of u i s c o m p a t i b l e w i t h t h e o n e of P l , though the potential parameters should be further investigated by comparing theory with experiment more carefully. Comparing the first diffraction maximum with I F C }2 c a l c u l a t e d f o r o t h e r l e v e l s o f J ~ = 1 - a n d T = 0 n e a r t h e s a m e q, we c a n i n d i c a t e t h a t t h e e x c i t a t i o n of t h e l o w e s t l e v e l of o u r m o d e l i s distinctly stronger than other levels except for t h e 4 7 . 6 9 M e V l e v e l i n 1 6 0 a n d 27.36 M e V l e v e l in 4 0 C a . O u r C o u l o m b f o r m f a c t o r s f o r t h e s e l e v e l s a r e d i s p l a y e d i n f i g s . 1 a n d 2. T h u s it m a y b e s a i d t h a t 7.12 M e V l e v e l in 1 6 0 a n d 6 . 9 5 M e V l e v e l in 4 0 C a c o m p r i s e a l a r g e p o r t i o n of t h e c o m p r e s s i b l e m o d e . W e w o u l d l i k e to e x p r e s s s i n c e r e t h a n k s to P r o f e s s o r Y. T o r i z u k a , P r o f e s s o r M. O y a m a d a a n d D r . K. I t o h f o r m a k i n g t h e i r d a t a a v a i l a b l e t o u s b e f o r e p u b l i c a t i o n . We t h a n k P r o f e s s o r K. I z u m o , D r . B. I m a n i s h i a n d M r . M. Obu f o r helpful discussions.
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References [1] G. E. Brown, Unified theory of nuclear models and f o r c e s (North-Holland, A m s t e r d a m , 1967). [2] S.Fujii, Prog. Theor. Phys.42 (1969) 416. [3] D . J . R o w e , Phys. Rev. 162 (1967) 866. [4] M . E . R o s e , Elementary theory of angular m o m e n tum (John Wiley and Sons, New York, 1957). [5] Y. Torizuka, M. Oyamada, K. Nakahara, K. Sugiyama, Y. Kojima, T. Terasawa, K.Itoh, A. Yamaguchi and M. Kimura, Phys. Rev. L e t t e r s 22 (1969) 544. [6] S.Fujii, Nuel. Phys.67 (1965) 592. [7] V.Gillet and E . A . S a n d e r s o n , Nucl. Phys. A91 (1967) 292. [8] L . J . T a s s i e and F . C . B a r k e r , Phys. R e v . l l l (1958) 940.
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[9] R.S.Willey, Nucl. P h y s . 4 0 (1963) 529. [10] J. C. B e r g s t r o m , W. Bertozzi, S. Kowalski, X. K. Maruyama, J . W . Lightbody J r . , S . P . Fivozinsky and S. P e n n e r , Phys. Rev. L e t t e r s 24 (1970) 152. [11] K.Itoh, M.Oyamada and Y.Torizuka, to be published. [12] R.A. Eisenstein, D.W. Madsen, H. Theissen, L. S. Cardman and C.K. Bockelman, Phys. Rev. 188 (1969) 1815. [13] G. E. Brown and A . M . G r e e n , P h y s i c s L e t t e r s 15 (1965) 168. [14] W . J . G e r a c e a n d A . M . G r e e n , Nucl. Phys. A l l 3 (1968) 641.