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Effective tensor interaction in nuclei
Volume 18, number 1
PHYSICS LETTERS
4r(n+½) M n = - n ' ~ . J 4 n + ~ tan 6 n. ~
(1~) / ~ = 14.4 M e V . i n the l a s t row. The r e l e v a n t phase shifts w e r e taken f r o m ref. 12. Table 2 shows the p a i r i n g [15, 16] m a t r i x e l e m e n t s [ = 0 for the (s, d ) - s h e l l . C o m p a r i n g the f i r s t two rows one can see that the s e p a r a t i o n d i s t a n c e d = 1.1 fm r e p r o d u c e s the s a m e r e s u l t s a s the m o r e c o r r e c t t r e a t m e n t in the f i r s t row, where d v a r i e s with n. F o r comp a r i s o n , the m a t r i x e l e m e n t s of [ a r e given. T h e r e is a definite c o r r e c t i o n to the v a l u e s u s e d before. However, other u n c e r t a i n t i e s like lack of s e l i - c o u s i s t e n c y i n Ho, o v e r s i m p l i f i e d two~3ody potential, core e x c i t a t i o n s [13] and the m i x i n g of d e f o r m e d s t a t e s [14] a r e likely to i n troduce c o r r e c t i o n s of the s a m e size. In fact, the c a l c u l a t i o n of B e r t s c h [13] shows that r e n o r m a l i zation of the force through i n c l u s i o n of o n e - p a r ticle, o n e - h o l e e x c i t a t i o n s l o w e r s the g r o u n d state slightly m o r e than i n t r o d u c t i o n of the state dependence [5] r a i s e s it. Thus, one should e i t h e r leave both effects out, o r include b e t h ; i n c l u s i o n of only one will lead to l a r g e r e r r o r s . In the case of 180, we have dealt with only the s i n g l e t force, s i n c e we r e s t r i c t our c o n s i d e r a tions to r e l a t i v e S - s t a t e s . In 18F the t r i p l e t force would also e n t e r in r e l a t i v e S - s t a t e s , and the tens o r force would s e e m to s i g n i f i c a n t l y c o m p l i c a t e the a r g u m e n t , s i n c e s e c o n d o r d e r t e r m s v l TQ v l '
where v ! i s now m a i n l y t e n s o r force, a r e no longer s m a l l here. However, it can be shown [17] that these s e c o n d - o r d e r t e r m s lead to an effective
EFFECTIVE
TENSOR
1 August 1965
c e n t r a l force v e r y s i m i l a r to the KK force [3], and a r e n o r m a i i z a t i o n of the t e n s o r p a r t of v l. Thus, the above d i s c u s s i o n can be extended to the case of t e n s o r force s i m p l y by adding a longr a n g e t e n s o r force, the r e n o r m a l i z e d t e n s o r p a r t of v l. The author would like to thank P r o f e s s o r G.E. Brown for d i s c u s s i o n of the content of this paper. 1. K. Kumar, Perturbation theory and the nuclear many-body problem (Amsterdam, North-Holland Publ.Comp., 1962). 2. S.A.Moszkowski and B.L.Scott, Nucl. Physics 29 (1962) 665. 3. A. Kallio and K. Kolltveit, Nucl. Physics 53 (1964) 87. 4. T. Engeland and A. Kallio, Nucl. Physics 59 (1964) 211. 5. R.K.Bhaduri and E.L.Tomusiak, preprint (to be published). 6. Ref. 1., p.196. 7. T.A. Brody and M. Moshinsky, Tables of transformation brackets (Monngrafias Del Instltuto De Fisica, Mexico, 1960). 8. A.Messiah, Quantum mechanics, Vol. 1 (Amsterdam, North-Holland Publ. Comp., 1961) p.405. 9. L.C.Gomes, J.D.Walecka and V.F. Weisskopf, Ann. Phys. 3 (1958) 241. 10. A. Erd~lyi, Higher Transcendental Functions, Vol. II, New York (1953) p. 199. 11. A.M.Green, A.Kallio and K.Kelltveit, Physics Letters 14 (1965) 142. 12. T. Hamada and I.D.Jolmston, Nucl. Physics 34 (1962) 382. 13. T. Engeland, Nucl. Physics, to be published; G. Bertsch, to be published. 14. G.E. Brown, Proc. Paris Conf. on Nuclear structure (1964). 15. A.Kallio, Ann.Acad.Scient.Fenn., Ser.A, VI Physics 163, Helsinki (1964). 16. T.T.S.Kuo, E.Baranger and M. Baranger, Prec. 1965 Spring Meeting at Washington, D.C. 17. T.T.S.Kuo, to be published.
INTERACTION
IN N U C L E I
*
T. T. S. KUO and G. E. BROWN P a l m e r Physical Laboratory, Princeton University, Princeton, New J e r s e y R e c e i v e d 25 June 1965
E m p l o y i n g the Moszkowski-Scott s e p a r a t i o n method [1], s e v e r a l s h e l l model s t u d i e s u s i n g * This work was supported by the U. S. Atomic Energy Commission and the Higgins Scientific Trust Fund. This report made use of the Princeton Computer Facilities supported in part by the National Science Foundation Grant NSF--GP579. 54
h a r d core r e s i d u a l i n t e r a c t i o n have been r e c e n t l y r e p o r t e d by Kallio and Kolltveit [2] and o t h e r s [3]. In these c a l c u l a t i o n s , a s i m p l e r e s i d u a l i n t e r action i s used. Namely, it i s c e n t r a l and acts i n r e l a t i v e S state alone. An advantage of u s i n g this i n t e r a c t i o n is its s i m p l i c i t y in calculation, be-
Volume 18, number 1
PHYSICS LETTERS
cause the n u c l e a r r e a c t i o n m a t r i x G is well app r o x i m a t e d by Vl, the long r a n g e p a r t of the potential. This s i m p l i c i t y , however, d i s a p p e a r s when t e n s o r force i s p r e s e n t [4] as in m o r e r e a l i s t i c potentials; in p a r t i c u l a r , in both the Hamacla-Johnston and the Yale potential [5, 6]. Hence, before u s i n g these " r e a l i s t i c " i n t e r a c tions in s h e l l - m o d e l p r o b l e m s , it is n e c e s s a r y to f i r s t u n d e r s t a n d how to handle the t e n s o r int e r a c t i o n s . In this note, we shall r e p o r t on t h i s a s p e c t for the H a m a d a - J o h n s t o n i n t e r a c t i o n . It will be seen that the 3S 1 state t e n s o r c o n t r i b u t i o n can be a c c u r a t e l y a p p r o x i m a t e d by the c l o s u r e r e l a t i o n with an effective e n e r g y d e n o m i n a t o r . In e v a l u a t i n g the shell model m a t r i x e l e m e n t s , we shall f i r s t t r a n s f o r m to the c e n t r e of m a s s and r e l a t i v e c o o r d i n a t e s [7]. The m a j o r task is then to evaluate the m a t r i x e l e m e n t
M= (~ONL gOnllGi~ON,L,~On, I,)Sj , (1) q~NL and ~anl are r e s p e c t i v e l y the c e n t r e
where of m a s s and r e l a t i v e o s c i l l a t o r wave functions, S is the total spin and G is the n u c l e a r r e a c t i o n m a t r i x defined by G ~o = V ~ o = V ~ .
(2)
Here ~b i s the c o r r e l a t e d wave function which v a n i s h e s inside the h a r d core, V the i n t e r a c t i o n . G satisfies
G= V - V~ G ,
(3)
where Q is the P a u l i o p e r a t o r and e the e n e r g y d e n o m i n a t o r (positive definite). F o r s i m p l i c i t y , we s h a l l a s s u m e eq. (3) is independent of the cent r e of m a s s v a r i a b l e s *. To evaluate the m a t r i x e l e m e n t of G, we follow the Moszkowski-Scott s e p a r a t i o n method and w r i t e down the following expansion:
GmGF8 + Vl - Vl ~ov l - 2 C s ~
QV l - G s -Q- e l G s + (4) - G ~1- 1--~C Ske e o / s"
Here the s u b s c r i p t s s and l stand for s h o r t - r a n g e and l o n g - r a n g e r e s p e c t i v e l y . The s u p e r s c r i p t F i n d i c a t e s that Q/e in eel. (3) has b e e n r e p l a c e d by 1/eo, e o being the f r e e - s p a c e e n e r g y d e n o m i n a t o r . We need to d e t e r m i n e the s e p a r a t i o n d i s t a n c e * As discussed by Wong [8], eq. (3} in fact depends on the centre of mass variables. We think that this dependence can be taken away to large extent by some averaging procedure. The uncertainties so caused should be unimportant compared with other uncertainties inherent from the shell model.
1 August 1965
dnl
for the case of the t r i p l e t - e v e n HamadaJohnston i n t e r a c t i o n ; dnl is defined so that the diagonal m a t r i x e l e m e n t of GF v a n i s h e s . We shall c o n s i d e r j u s t the 3S 1 state b e c a u s e s t a t e s of l >/2 hardly feel the h a r d core. With the p r e s e n c e of t e n s o r force, S wave and D wave a r e coupled. Denoting the c o r r e l a t e d S and D wave by u and w, one has
a s F % 0 = g' = ~ + w ,
(s)
where q~n0 is the u n p e r t u r b e d 381 wave function. The u and w can be obtained by i n t e g r a t i n g , outward f r o m the core, the coupled Schroedinger equations, with e n e r g y eigenvalue equal to that of ~an0. The boundary conditions a t the core r a dins r c a r e obviously U(rc) = 0 and w~rc) = 0. At the s e p a r a t i o n distance" dno , we r e ~ a i r ~ U~dn0) = ~o.n(d.r,) and u' (ann) = ~0'n0(e..). To determi,ne dn"0~, o'~"e m o r e bou~adary e o n ~ i o n i s s t i l l n e e d e d . This will be provided by the boundary condition on w.
Let us define the defect wave function X by X = ~ - e/ .
(6)
Since o r i g i n a l l y t h e r e is no D wave in ~0n0, one s e e s at once f r o m eq. (5) that w is j u s t -X in the D channel. × s a t i s f i e s ex = QV~ as is obvious f r o m eqs. (2) and (3). This equation i s v e r y difficult to solve as it stands. However, the n a t u r e of its solution can be e a s i l y v i s u a l i z e d through the r e f e r e n c e s p e c t r u m a p p r o x i m a t i o n for nuc l e a r m a t t e r [9, 10]. Under this approximation, X/, the defect wave function i n t h e /th channel, satisfies [:~_
/(/+1)
72
m,VlXl
.maV~Ol.
(7)
Here m* is the effective m a s s , y2 is a p a r a m eter. It is u s u a l l y l a r g e and positive so that the m a i n effect of the P a u l i o p e r a t o r Q is r e p r o d u c e d , i.e. ×l decays exponentially outside the r a n g e of V. For our p r e s e n t case, we have Vs i n s t e a d of V i n eCl. (7). F o r t >dno , Vs v a n i s h e s by definition. Hence for r > d n o , we s i m p l y have
xl ~ - ~ ( r r )
= -rrh20)
0v~),
where h9(1) i s the exponentially decaying Hankel f u n c t i o n - I l l ] . Thus the r e m a i n i n g b o u n d a r y condition on w is
(8) The tin0 so d e t e r m i n e d will ensm-e the v a n i s h ing of (qn0 G~F q,,n). We s t i l l need the value of V2. However,°no p~-ecise value of it is available. 55
Volume 18, number 1
PHYSICS LETTERS
1 August 1965
GORE
2
/i
"7 x
H2(Tr)
I
lJ I
rc
.B
dno r
Fig. 1. Determination of the separation distance.
F o r t u n a t e l y , we have found the s e p a r a t i o n calculation i s v e r y i n s e n s i t i v e to 72 a s s e e n f r o m table 1, where we parametriT.e 2A i n s t e a d of 72. They a r e r e l a t e d by ~,2 = 2 Am/~ 2. The r a n g e of A c o v e r s that of ref. 10. Fig. 1 s u m m a r i z e s how the v a r i o u s q u a n t i t i e s a r e d e t e r m i n e d . Both d and w(d) have been found to be quite i n s e n s i t i v e to the o s c i l l a t o r p a r a m e t e r v (= m ~ / ~ . That they depend so little on 2 A is p a r t l y due to the slow v a r i ation o f ~ ' 2 / J F 2 i n the r e g i o n of i n t e r e s t . In the following, we s h a l l u s e a state i n d e p e n d e n t d and w(d), n a m e l y d = 1.08 fm and w(d) = 0.1. F o r the 8S 1 state, the f i r s t n o n - v a n i s h i n g t e n s o r t e r m in eq. (4) i s -VT! (Q/e) VTl , w h e r e VT! i s the long r a n g e p a r t of the t e n s o r i n t e r a c t i o n . To evaluate this t e r m , the m a i n difficulty i s the t r e a t m e n t of Q/e. As usual, one w r i t e s e = E(a) + E(b) - E¢) - E(m), where E{l) and E(m) a r e the s e l f - c o n s i s t e n t e n e r g i e s of the i n i t i a l s t a t e s . We c o n s i d e r h e r e the i n t e r a c t i o n of two of the l a s t p a r t i c l e s , and t h e r e f o r e take E(l) +E(m) = - 2 0 MeV as a t y p i c a l value. The E(a) and E(b) are the e n e r g i e s of the i n t e r m e d i a t e s t a t e s above the F e r m i s u r f a c e . It i s well known that they should be evaluated off the e n e r g y shell. But, a s is c l e a r f r o m the d i s c u s s i o n by Bethe [12] one m a y well
n =1 2 4 56
50 d[fm] w(d) 1.07 1.12 1.19
0.09 0.11 0.11
d
100
1.08 1.12 1.20
5
Fig. 2. Plot off(k) for (2010[ VTl QVT[[2010>. V=0.29 frn-2 a n d d = 1 . 0 8 f m .
take i n t e r m e d i a t e s t a t e s s i m p l y as f r e e p a r t i c l e s , m a k i n g a s m a l l c o r r e c t i o n for the (small) potential in i n t e r m e d i a t e s t a t e s l a t e r . We s h a l l thus use plane wave i n t e r m e d i a t e s t a t e s with E(a) and E(b) b e i n g s i m p l y the k i n e t i c e n e r g i e s . Using plane wave i n t e r m e d i a t e state s u g g e s t s that it would be s u i t a b l e to u s e the n u c l e a r - m a t t e r angle a v e r a g e P a u l i o p e r a t o r [13]. That is, Q ~ , K , kf) is taken to be = 0
if
k 2 + ¼K2 < k 2
Q(k, K, kf) = 1
if
k - ½K > kf
_ k 2 + ¼K 2 . k 2
kK
150
w(d)
d
0.08 0.10 0.10
1.08 1.13 1.21
w(d) 0.08 0.10 0.10
otherwise.
(9)
Here k and K a r e r e s p e c t i v e l y the r e l a t i v e and c e n t r e of m a s s m o m e n t u m , kf i s the F e r m i m o m e n t u m . The a n g l e - a v e r a g e a p p r o x i m a t i o n has been shown to be a c c u r a t e in n u c l e a r - m a t t e r c a l culations [13]. The following e x p r e s s i o n s a r e then obtained.
(NL.ll Vrl Q vrzI NL. s: =/I(k) f(k) = k 2 f K2 dK d~Kd~ k
Table 1 3S1 separation distance for Hamada-Jolmston interaction. v = 0.29 fm -1 (corresponds to mass number 40). 2A[MeV]
4
k If"]
(RKIVrllNLnl) 2 × Q(kukO x e(k,h') '
(lO)
with e{k,K) = (k2 +¼K2) ~i2/m + 20. The i n t e g r a tion over s p a t i a l v a r i a b l e s i s understood. The r a n g e of K i n t e g r a t i o n i s 0 to 2 kf. The e x p r e s s i o n was e v a l u a t e d n u m e r i c a l l y on the P r i n c e t o n IBM-7094 computer. F o r the p r e s e n t , we s h a l l
Volume 18, number 1
PHYSICS LETTERS
1 August 1965
Table 2 Diagonal 3S1 reduced matrix elements for Hamada-Johnston interaction v = ~.29 fm -2, bf = 1.3 fro-'1.,d = 1.08 fm and w(d) = 0.1.
.
I
II
HI
L
-VTQVTl
- csQVTl
Ll
z+n+m
K-K
0
1 2 3
4 2 0
-3.32 -3.17 -3.13
249 261 264
-2.00
-2.67
-8.87
-8.17
1
0 1 2
4 2 0
-3.65 -3.64 -3.65
222 222 222
-1.94
-2.07
-7.66
-7.68
2
0 1
2 0
-2.92
227 225
-1.71
-1.53
-6.18
-6.32
-2.95
0
0
-2.07
246
-1.43
-1.12
--4.62
-5.02
3
take k f a s 1.3 fm - 1 . This can c e r t a i n l y be i m p r o v e d upon. A b e t t e r c h o i c e would be kf~o) w h e r e p i s the l o c a l d e n s i t y . Such a c a l c u l a t i o n i s being planned. The i n t e g r a n d f(k) of eq. (10) i s p l o t t e d in fig. 2 f o r s e v e r a l v a l u e s of kf. It i s i n t e r e s t i n g to s e e that f(k) p e a k s s h a r p l y a r o u n d 2.2 fm - 1 . This m e a n s the c o n t r i b u t i o n to the s e c o n d - o r d e r t e n s o r term predominantly comes from intermediate states of e n e r g y a r o u n d 200 MeV. This i s m a i n l y due to the p r o p e r t y of the F o u r i e r t r a n s f o r m of VTl (r) and the p r e s e n c e of the P a u l i o p e r a t o r . V a l u e s of the d i a g o n a l m a t r i x e l e m e n t s of VT1 (Q/e)VTl a r e t a b u l a t e d in t a b l e 2. The o f f - d i a g o n a l o n e s a r e found to be v e r y s m a l l , a r o u n d 0.1 MeV. One s e e s that they a r e r a t h e r i n s e n s i t i v e to NL. This i n d i c a t e s that e x c e p t for v e r y l a r g e NL, w h e r e the p a i r of n u c l e o n s a r e c l o s e to the s u r f a c e m o s t o f t h e t i m e , u s i n g a c o n s t a n t kf m a y be adequate. The s h a r p p e a k i n g of f(k) s t r o n g l y s u g g e s t s that u s i n g c l o s u r e with an effective e n e r g y d e n o m i n a t o r i s a good a p p r o x i m a t i o n ; n a m e l y
V Q ( Tl~ VTl)
--.1-z~--(V21). ~eff
(11)
C a l c u l a t i o n s do show that this is t r u e . As shown b y table 2, e e l f ~ 240 MeV wouid p r a c t i c a l l y r e p r o d u c e the o r i g i n a l n u m b e r s for a l l c a s e s . As shown by fig. 2, e e f f i s only w e a k l y dependent o n kf. Some kind of a g r e e m e n t was a l s o o b t a i n e d f o r v v a l u e s c o r r e s p o n d i n g to m a s s n u m b e r 16 and 116. We thus a r r i v e at the s i m p l e approximation -
Q VTl-~ VTl -~
-
-
e eff
(12)
2
8V2TI(r)
2VTI(r)S12 +
,
e elf
[MeV]
v a l i d for t r i p l e t s t a t e s ; h e r e VTl(r) i s the r a d i a l p a r t alone, S12 i s the t e n s o r o p e r a t o r . Note that now VTl(Q/e)VTl a c t s a s a f i r s t o r d e r effective f o r c e . To obtain eel. (12), the r e l a t i o n S 2 2 = 8 - 2S12, v a l i d for t r i p l e t s t a t e s , h a s beefi used. The c r o s s t e r m -2Gs(Q/e)VTI i s s i m p l y given, u n d e r the presen{ t r e a t m e n t , by (13) (-2Gs Q
VTl)
: 2~/~ ~
( J f 2 ~ r ) l Vrl(r) t~no(r)).
This i n t e g r a l was a l s o found to be i n s e n s i t i v e to v a l u e s of 7. Thus 7 c o r r e s p o n d i n g to 2A = 50 h a s been u s e d and r e s u l t s a r e given in table 2. The column u n d e r Vcl in the t a b l e i s the c o n t r i b u t i o n f r o m the l o n g - r a n g e c e n t r a l H a m a d a - J o h n s t o n i n t e r a c t i o n . Thus, e x c e p t the P a u l i and d i s p e r sion c o r r e c t i o n s , the t o t a l 3S 1 c o n t r i b u t i o n s of the H a m a d a - J o h n s t o n i n t e r a c t i o n can be taken a s the s u m of -2Gs(Q/e)VTl , Vcl and the a v e r a g e of -VTl(Q/e)VTl. They a r e c o m p a r e d with the c o r r e s p o n d i n g r e d u c e d m a t r i x e l e m e n t s of the K - K i n t e r a c t i o n , the v a l u e s of which a l s o do not include the P a u l i and d i s p e r s i o n c o r r e c t i o n s in the l a s t two c o l u m n s of the table. They turn out to be so c l o s e to each other. F i n a l l y , by s c a l i n g with the r e s u l t s of Scott [4], we m a y e s t i m a t e our P a u l i and d i s p e r s i o n c o r r e c t i o n s to be a r o u n d + 1 MeV. Adding this to the n u m b e r s given in table 2, the total H a m a d a Johnston 3S 1 r e d u c e d m a t r i x e l e m e n t s can be e s timated. One of the a u t h o r s (T.K.) w i s h e s to thank D r s . C. W. Wong, L. Z a m i c k and A. M. G r e e n for helpful d i s c u s s i o n s . 1. S.A. Moszkowski and B. L. Scott, Ann. Phys. 11 (1960) 65.
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PHYSICS LETTERS
4r(n+½) M n = - n ' ~ . J 4 n + ~ tan 6 n. ~
(1~) / ~ = 14.4 M e V . i n the l a s t row. The r e l e v a n t phase shifts w e r e taken f r o m ref. 12. Table 2 shows the p a i r i n g [15, 16] m a t r i x e l e m e n t s [ = 0 for the (s, d ) - s h e l l . C o m p a r i n g the f i r s t two rows one can see that the s e p a r a t i o n d i s t a n c e d = 1.1 fm r e p r o d u c e s the s a m e r e s u l t s a s the m o r e c o r r e c t t r e a t m e n t in the f i r s t row, where d v a r i e s with n. F o r comp a r i s o n , the m a t r i x e l e m e n t s of [ a r e given. T h e r e is a definite c o r r e c t i o n to the v a l u e s u s e d before. However, other u n c e r t a i n t i e s like lack of s e l i - c o u s i s t e n c y i n Ho, o v e r s i m p l i f i e d two~3ody potential, core e x c i t a t i o n s [13] and the m i x i n g of d e f o r m e d s t a t e s [14] a r e likely to i n troduce c o r r e c t i o n s of the s a m e size. In fact, the c a l c u l a t i o n of B e r t s c h [13] shows that r e n o r m a l i zation of the force through i n c l u s i o n of o n e - p a r ticle, o n e - h o l e e x c i t a t i o n s l o w e r s the g r o u n d state slightly m o r e than i n t r o d u c t i o n of the state dependence [5] r a i s e s it. Thus, one should e i t h e r leave both effects out, o r include b e t h ; i n c l u s i o n of only one will lead to l a r g e r e r r o r s . In the case of 180, we have dealt with only the s i n g l e t force, s i n c e we r e s t r i c t our c o n s i d e r a tions to r e l a t i v e S - s t a t e s . In 18F the t r i p l e t force would also e n t e r in r e l a t i v e S - s t a t e s , and the tens o r force would s e e m to s i g n i f i c a n t l y c o m p l i c a t e the a r g u m e n t , s i n c e s e c o n d o r d e r t e r m s v l TQ v l '
where v ! i s now m a i n l y t e n s o r force, a r e no longer s m a l l here. However, it can be shown [17] that these s e c o n d - o r d e r t e r m s lead to an effective
EFFECTIVE
TENSOR
1 August 1965
c e n t r a l force v e r y s i m i l a r to the KK force [3], and a r e n o r m a i i z a t i o n of the t e n s o r p a r t of v l. Thus, the above d i s c u s s i o n can be extended to the case of t e n s o r force s i m p l y by adding a longr a n g e t e n s o r force, the r e n o r m a l i z e d t e n s o r p a r t of v l. The author would like to thank P r o f e s s o r G.E. Brown for d i s c u s s i o n of the content of this paper. 1. K. Kumar, Perturbation theory and the nuclear many-body problem (Amsterdam, North-Holland Publ.Comp., 1962). 2. S.A.Moszkowski and B.L.Scott, Nucl. Physics 29 (1962) 665. 3. A. Kallio and K. Kolltveit, Nucl. Physics 53 (1964) 87. 4. T. Engeland and A. Kallio, Nucl. Physics 59 (1964) 211. 5. R.K.Bhaduri and E.L.Tomusiak, preprint (to be published). 6. Ref. 1., p.196. 7. T.A. Brody and M. Moshinsky, Tables of transformation brackets (Monngrafias Del Instltuto De Fisica, Mexico, 1960). 8. A.Messiah, Quantum mechanics, Vol. 1 (Amsterdam, North-Holland Publ. Comp., 1961) p.405. 9. L.C.Gomes, J.D.Walecka and V.F. Weisskopf, Ann. Phys. 3 (1958) 241. 10. A. Erd~lyi, Higher Transcendental Functions, Vol. II, New York (1953) p. 199. 11. A.M.Green, A.Kallio and K.Kelltveit, Physics Letters 14 (1965) 142. 12. T. Hamada and I.D.Jolmston, Nucl. Physics 34 (1962) 382. 13. T. Engeland, Nucl. Physics, to be published; G. Bertsch, to be published. 14. G.E. Brown, Proc. Paris Conf. on Nuclear structure (1964). 15. A.Kallio, Ann.Acad.Scient.Fenn., Ser.A, VI Physics 163, Helsinki (1964). 16. T.T.S.Kuo, E.Baranger and M. Baranger, Prec. 1965 Spring Meeting at Washington, D.C. 17. T.T.S.Kuo, to be published.
INTERACTION
IN N U C L E I
*
T. T. S. KUO and G. E. BROWN P a l m e r Physical Laboratory, Princeton University, Princeton, New J e r s e y R e c e i v e d 25 June 1965
E m p l o y i n g the Moszkowski-Scott s e p a r a t i o n method [1], s e v e r a l s h e l l model s t u d i e s u s i n g * This work was supported by the U. S. Atomic Energy Commission and the Higgins Scientific Trust Fund. This report made use of the Princeton Computer Facilities supported in part by the National Science Foundation Grant NSF--GP579. 54
h a r d core r e s i d u a l i n t e r a c t i o n have been r e c e n t l y r e p o r t e d by Kallio and Kolltveit [2] and o t h e r s [3]. In these c a l c u l a t i o n s , a s i m p l e r e s i d u a l i n t e r action i s used. Namely, it i s c e n t r a l and acts i n r e l a t i v e S state alone. An advantage of u s i n g this i n t e r a c t i o n is its s i m p l i c i t y in calculation, be-
Volume 18, number 1
PHYSICS LETTERS
cause the n u c l e a r r e a c t i o n m a t r i x G is well app r o x i m a t e d by Vl, the long r a n g e p a r t of the potential. This s i m p l i c i t y , however, d i s a p p e a r s when t e n s o r force i s p r e s e n t [4] as in m o r e r e a l i s t i c potentials; in p a r t i c u l a r , in both the Hamacla-Johnston and the Yale potential [5, 6]. Hence, before u s i n g these " r e a l i s t i c " i n t e r a c tions in s h e l l - m o d e l p r o b l e m s , it is n e c e s s a r y to f i r s t u n d e r s t a n d how to handle the t e n s o r int e r a c t i o n s . In this note, we shall r e p o r t on t h i s a s p e c t for the H a m a d a - J o h n s t o n i n t e r a c t i o n . It will be seen that the 3S 1 state t e n s o r c o n t r i b u t i o n can be a c c u r a t e l y a p p r o x i m a t e d by the c l o s u r e r e l a t i o n with an effective e n e r g y d e n o m i n a t o r . In e v a l u a t i n g the shell model m a t r i x e l e m e n t s , we shall f i r s t t r a n s f o r m to the c e n t r e of m a s s and r e l a t i v e c o o r d i n a t e s [7]. The m a j o r task is then to evaluate the m a t r i x e l e m e n t
M= (~ONL gOnllGi~ON,L,~On, I,)Sj , (1) q~NL and ~anl are r e s p e c t i v e l y the c e n t r e
where of m a s s and r e l a t i v e o s c i l l a t o r wave functions, S is the total spin and G is the n u c l e a r r e a c t i o n m a t r i x defined by G ~o = V ~ o = V ~ .
(2)
Here ~b i s the c o r r e l a t e d wave function which v a n i s h e s inside the h a r d core, V the i n t e r a c t i o n . G satisfies
G= V - V~ G ,
(3)
where Q is the P a u l i o p e r a t o r and e the e n e r g y d e n o m i n a t o r (positive definite). F o r s i m p l i c i t y , we s h a l l a s s u m e eq. (3) is independent of the cent r e of m a s s v a r i a b l e s *. To evaluate the m a t r i x e l e m e n t of G, we follow the Moszkowski-Scott s e p a r a t i o n method and w r i t e down the following expansion:
GmGF8 + Vl - Vl ~ov l - 2 C s ~
QV l - G s -Q- e l G s + (4) - G ~1- 1--~C Ske e o / s"
Here the s u b s c r i p t s s and l stand for s h o r t - r a n g e and l o n g - r a n g e r e s p e c t i v e l y . The s u p e r s c r i p t F i n d i c a t e s that Q/e in eel. (3) has b e e n r e p l a c e d by 1/eo, e o being the f r e e - s p a c e e n e r g y d e n o m i n a t o r . We need to d e t e r m i n e the s e p a r a t i o n d i s t a n c e * As discussed by Wong [8], eq. (3} in fact depends on the centre of mass variables. We think that this dependence can be taken away to large extent by some averaging procedure. The uncertainties so caused should be unimportant compared with other uncertainties inherent from the shell model.
1 August 1965
dnl
for the case of the t r i p l e t - e v e n HamadaJohnston i n t e r a c t i o n ; dnl is defined so that the diagonal m a t r i x e l e m e n t of GF v a n i s h e s . We shall c o n s i d e r j u s t the 3S 1 state b e c a u s e s t a t e s of l >/2 hardly feel the h a r d core. With the p r e s e n c e of t e n s o r force, S wave and D wave a r e coupled. Denoting the c o r r e l a t e d S and D wave by u and w, one has
a s F % 0 = g' = ~ + w ,
(s)
where q~n0 is the u n p e r t u r b e d 381 wave function. The u and w can be obtained by i n t e g r a t i n g , outward f r o m the core, the coupled Schroedinger equations, with e n e r g y eigenvalue equal to that of ~an0. The boundary conditions a t the core r a dins r c a r e obviously U(rc) = 0 and w~rc) = 0. At the s e p a r a t i o n distance" dno , we r e ~ a i r ~ U~dn0) = ~o.n(d.r,) and u' (ann) = ~0'n0(e..). To determi,ne dn"0~, o'~"e m o r e bou~adary e o n ~ i o n i s s t i l l n e e d e d . This will be provided by the boundary condition on w.
Let us define the defect wave function X by X = ~ - e/ .
(6)
Since o r i g i n a l l y t h e r e is no D wave in ~0n0, one s e e s at once f r o m eq. (5) that w is j u s t -X in the D channel. × s a t i s f i e s ex = QV~ as is obvious f r o m eqs. (2) and (3). This equation i s v e r y difficult to solve as it stands. However, the n a t u r e of its solution can be e a s i l y v i s u a l i z e d through the r e f e r e n c e s p e c t r u m a p p r o x i m a t i o n for nuc l e a r m a t t e r [9, 10]. Under this approximation, X/, the defect wave function i n t h e /th channel, satisfies [:~_
/(/+1)
72
m,VlXl
.maV~Ol.
(7)
Here m* is the effective m a s s , y2 is a p a r a m eter. It is u s u a l l y l a r g e and positive so that the m a i n effect of the P a u l i o p e r a t o r Q is r e p r o d u c e d , i.e. ×l decays exponentially outside the r a n g e of V. For our p r e s e n t case, we have Vs i n s t e a d of V i n eCl. (7). F o r t >dno , Vs v a n i s h e s by definition. Hence for r > d n o , we s i m p l y have
xl ~ - ~ ( r r )
= -rrh20)
0v~),
where h9(1) i s the exponentially decaying Hankel f u n c t i o n - I l l ] . Thus the r e m a i n i n g b o u n d a r y condition on w is
(8) The tin0 so d e t e r m i n e d will ensm-e the v a n i s h ing of (qn0 G~F q,,n). We s t i l l need the value of V2. However,°no p~-ecise value of it is available. 55
Volume 18, number 1
PHYSICS LETTERS
1 August 1965
GORE
2
/i
"7 x
H2(Tr)
I
lJ I
rc
.B
dno r
Fig. 1. Determination of the separation distance.
F o r t u n a t e l y , we have found the s e p a r a t i o n calculation i s v e r y i n s e n s i t i v e to 72 a s s e e n f r o m table 1, where we parametriT.e 2A i n s t e a d of 72. They a r e r e l a t e d by ~,2 = 2 Am/~ 2. The r a n g e of A c o v e r s that of ref. 10. Fig. 1 s u m m a r i z e s how the v a r i o u s q u a n t i t i e s a r e d e t e r m i n e d . Both d and w(d) have been found to be quite i n s e n s i t i v e to the o s c i l l a t o r p a r a m e t e r v (= m ~ / ~ . That they depend so little on 2 A is p a r t l y due to the slow v a r i ation o f ~ ' 2 / J F 2 i n the r e g i o n of i n t e r e s t . In the following, we s h a l l u s e a state i n d e p e n d e n t d and w(d), n a m e l y d = 1.08 fm and w(d) = 0.1. F o r the 8S 1 state, the f i r s t n o n - v a n i s h i n g t e n s o r t e r m in eq. (4) i s -VT! (Q/e) VTl , w h e r e VT! i s the long r a n g e p a r t of the t e n s o r i n t e r a c t i o n . To evaluate this t e r m , the m a i n difficulty i s the t r e a t m e n t of Q/e. As usual, one w r i t e s e = E(a) + E(b) - E¢) - E(m), where E{l) and E(m) a r e the s e l f - c o n s i s t e n t e n e r g i e s of the i n i t i a l s t a t e s . We c o n s i d e r h e r e the i n t e r a c t i o n of two of the l a s t p a r t i c l e s , and t h e r e f o r e take E(l) +E(m) = - 2 0 MeV as a t y p i c a l value. The E(a) and E(b) are the e n e r g i e s of the i n t e r m e d i a t e s t a t e s above the F e r m i s u r f a c e . It i s well known that they should be evaluated off the e n e r g y shell. But, a s is c l e a r f r o m the d i s c u s s i o n by Bethe [12] one m a y well
n =1 2 4 56
50 d[fm] w(d) 1.07 1.12 1.19
0.09 0.11 0.11
d
100
1.08 1.12 1.20
5
Fig. 2. Plot off(k) for (2010[ VTl QVT[[2010>. V=0.29 frn-2 a n d d = 1 . 0 8 f m .
take i n t e r m e d i a t e s t a t e s s i m p l y as f r e e p a r t i c l e s , m a k i n g a s m a l l c o r r e c t i o n for the (small) potential in i n t e r m e d i a t e s t a t e s l a t e r . We s h a l l thus use plane wave i n t e r m e d i a t e s t a t e s with E(a) and E(b) b e i n g s i m p l y the k i n e t i c e n e r g i e s . Using plane wave i n t e r m e d i a t e state s u g g e s t s that it would be s u i t a b l e to u s e the n u c l e a r - m a t t e r angle a v e r a g e P a u l i o p e r a t o r [13]. That is, Q ~ , K , kf) is taken to be = 0
if
k 2 + ¼K2 < k 2
Q(k, K, kf) = 1
if
k - ½K > kf
_ k 2 + ¼K 2 . k 2
kK
150
w(d)
d
0.08 0.10 0.10
1.08 1.13 1.21
w(d) 0.08 0.10 0.10
otherwise.
(9)
Here k and K a r e r e s p e c t i v e l y the r e l a t i v e and c e n t r e of m a s s m o m e n t u m , kf i s the F e r m i m o m e n t u m . The a n g l e - a v e r a g e a p p r o x i m a t i o n has been shown to be a c c u r a t e in n u c l e a r - m a t t e r c a l culations [13]. The following e x p r e s s i o n s a r e then obtained.
(NL.ll Vrl Q vrzI NL. s: =/I(k) f(k) = k 2 f K2 dK d~Kd~ k
Table 1 3S1 separation distance for Hamada-Jolmston interaction. v = 0.29 fm -1 (corresponds to mass number 40). 2A[MeV]
4
k If"]
(RKIVrllNLnl) 2 × Q(kukO x e(k,h') '
(lO)
with e{k,K) = (k2 +¼K2) ~i2/m + 20. The i n t e g r a tion over s p a t i a l v a r i a b l e s i s understood. The r a n g e of K i n t e g r a t i o n i s 0 to 2 kf. The e x p r e s s i o n was e v a l u a t e d n u m e r i c a l l y on the P r i n c e t o n IBM-7094 computer. F o r the p r e s e n t , we s h a l l
Volume 18, number 1
PHYSICS LETTERS
1 August 1965
Table 2 Diagonal 3S1 reduced matrix elements for Hamada-Johnston interaction v = ~.29 fm -2, bf = 1.3 fro-'1.,d = 1.08 fm and w(d) = 0.1.
.
I
II
HI
L
-VTQVTl
- csQVTl
Ll
z+n+m
K-K
0
1 2 3
4 2 0
-3.32 -3.17 -3.13
249 261 264
-2.00
-2.67
-8.87
-8.17
1
0 1 2
4 2 0
-3.65 -3.64 -3.65
222 222 222
-1.94
-2.07
-7.66
-7.68
2
0 1
2 0
-2.92
227 225
-1.71
-1.53
-6.18
-6.32
-2.95
0
0
-2.07
246
-1.43
-1.12
--4.62
-5.02
3
take k f a s 1.3 fm - 1 . This can c e r t a i n l y be i m p r o v e d upon. A b e t t e r c h o i c e would be kf~o) w h e r e p i s the l o c a l d e n s i t y . Such a c a l c u l a t i o n i s being planned. The i n t e g r a n d f(k) of eq. (10) i s p l o t t e d in fig. 2 f o r s e v e r a l v a l u e s of kf. It i s i n t e r e s t i n g to s e e that f(k) p e a k s s h a r p l y a r o u n d 2.2 fm - 1 . This m e a n s the c o n t r i b u t i o n to the s e c o n d - o r d e r t e n s o r term predominantly comes from intermediate states of e n e r g y a r o u n d 200 MeV. This i s m a i n l y due to the p r o p e r t y of the F o u r i e r t r a n s f o r m of VTl (r) and the p r e s e n c e of the P a u l i o p e r a t o r . V a l u e s of the d i a g o n a l m a t r i x e l e m e n t s of VT1 (Q/e)VTl a r e t a b u l a t e d in t a b l e 2. The o f f - d i a g o n a l o n e s a r e found to be v e r y s m a l l , a r o u n d 0.1 MeV. One s e e s that they a r e r a t h e r i n s e n s i t i v e to NL. This i n d i c a t e s that e x c e p t for v e r y l a r g e NL, w h e r e the p a i r of n u c l e o n s a r e c l o s e to the s u r f a c e m o s t o f t h e t i m e , u s i n g a c o n s t a n t kf m a y be adequate. The s h a r p p e a k i n g of f(k) s t r o n g l y s u g g e s t s that u s i n g c l o s u r e with an effective e n e r g y d e n o m i n a t o r i s a good a p p r o x i m a t i o n ; n a m e l y
V Q ( Tl~ VTl)
--.1-z~--(V21). ~eff
(11)
C a l c u l a t i o n s do show that this is t r u e . As shown b y table 2, e e l f ~ 240 MeV wouid p r a c t i c a l l y r e p r o d u c e the o r i g i n a l n u m b e r s for a l l c a s e s . As shown by fig. 2, e e f f i s only w e a k l y dependent o n kf. Some kind of a g r e e m e n t was a l s o o b t a i n e d f o r v v a l u e s c o r r e s p o n d i n g to m a s s n u m b e r 16 and 116. We thus a r r i v e at the s i m p l e approximation -
Q VTl-~ VTl -~
-
-
e eff
(12)
2
8V2TI(r)
2VTI(r)S12 +
,
e elf
[MeV]
v a l i d for t r i p l e t s t a t e s ; h e r e VTl(r) i s the r a d i a l p a r t alone, S12 i s the t e n s o r o p e r a t o r . Note that now VTl(Q/e)VTl a c t s a s a f i r s t o r d e r effective f o r c e . To obtain eel. (12), the r e l a t i o n S 2 2 = 8 - 2S12, v a l i d for t r i p l e t s t a t e s , h a s beefi used. The c r o s s t e r m -2Gs(Q/e)VTI i s s i m p l y given, u n d e r the presen{ t r e a t m e n t , by (13) (-2Gs Q
VTl)
: 2~/~ ~
( J f 2 ~ r ) l Vrl(r) t~no(r)).
This i n t e g r a l was a l s o found to be i n s e n s i t i v e to v a l u e s of 7. Thus 7 c o r r e s p o n d i n g to 2A = 50 h a s been u s e d and r e s u l t s a r e given in table 2. The column u n d e r Vcl in the t a b l e i s the c o n t r i b u t i o n f r o m the l o n g - r a n g e c e n t r a l H a m a d a - J o h n s t o n i n t e r a c t i o n . Thus, e x c e p t the P a u l i and d i s p e r sion c o r r e c t i o n s , the t o t a l 3S 1 c o n t r i b u t i o n s of the H a m a d a - J o h n s t o n i n t e r a c t i o n can be taken a s the s u m of -2Gs(Q/e)VTl , Vcl and the a v e r a g e of -VTl(Q/e)VTl. They a r e c o m p a r e d with the c o r r e s p o n d i n g r e d u c e d m a t r i x e l e m e n t s of the K - K i n t e r a c t i o n , the v a l u e s of which a l s o do not include the P a u l i and d i s p e r s i o n c o r r e c t i o n s in the l a s t two c o l u m n s of the table. They turn out to be so c l o s e to each other. F i n a l l y , by s c a l i n g with the r e s u l t s of Scott [4], we m a y e s t i m a t e our P a u l i and d i s p e r s i o n c o r r e c t i o n s to be a r o u n d + 1 MeV. Adding this to the n u m b e r s given in table 2, the total H a m a d a Johnston 3S 1 r e d u c e d m a t r i x e l e m e n t s can be e s timated. One of the a u t h o r s (T.K.) w i s h e s to thank D r s . C. W. Wong, L. Z a m i c k and A. M. G r e e n for helpful d i s c u s s i o n s . 1. S.A. Moszkowski and B. L. Scott, Ann. Phys. 11 (1960) 65.
57