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Nuclear ground state properties of 16O by an extension of the Eden-Emery method
Volume 18, number 3
PHYSICS LETTERS
NUCLEAR GROUND STATE PROPERTIES BY A N E X T E N S I O N OF THE EDEN-EMERY
1 September 1965
OF 160 METHOD
*
A. D. MacKELLAR ** and R. L. BECKER Oak Ridge National Laboratory, (gak Ridge, Tennessee
Received 22 July 1965
The use of a t w o - n u c l e o n r e a c t i o n o p e r a t o r t p e r m i t s c a l c u l a t i o n s of n u c l e a r p r o p e r t i e s e m ploying available s t r o n g n u c l e o n - n u c l e o n p o t e n t i a l s v which account well for the o n - t h e - e n e r g y shell data on the t w o - n u c l e o n s y s t e m s . B r u e c k n e t ' s a p p r o x i m a t i o n is the analogue of the H a r t r e e - F o c k a p p r o x i m a t i o n in which v is r e p l a c e d by t. It s e e m s likely that w h e n e v e r shell effects a r e i m p o r t a n t the r e a c t i o n m a t r i x m a y differ c o n s i d e r a b l y f r o m that a p p r o p r i a t e to an infinite medium. Thus it would be d e s i r a b l e to have r e action m a t r i c e s c a l c u l a t e d s e p a r a t e l y for i n dividual nuclei. The f i r s t work in this d i r e c t i o n was that of Eden et al. [1]. The method used by Eden et al. has s e e m e d r a t h e r r e s t r i c t e d b e c a u s e it a p p r o x i m a t e s the single p a r t i c l e potential by a h a r m o n i c potential plus a spatially c o n s t a n t t e r m which d i f f e r s f r o m state to state. However, for the ground state of 160 the h a r m o n i c o s c i l l a t o r eigenfunctions a r e thought to provide a good a p p r o x i m a t i o n to the occupied single p a r t i c l e s t a t e s . Eden et al. made c a l c u l a t i o n s with a n u m b e r of G a m m e l - T h a l e r potentials. With the one used by B r u e c k n e r and G a m m e l [2] in the o r i g i n a l n u c l e a r m a t t e r c a l c u l a t i o n s , which we r e f e r to as the B r u e c k n e r G a m m e l - T h a l e r potential, Eden et al. obtained a binding e n e r g y p e r n u c l e o n in 160 of 9.4 MeV, r e a s o n a b l y close to the e x p e r i m e n t a l value of 8.0 MeV. B r u e c k n e r et al. [3] u s i n g the B r u e c k n e r - G a m m e l r e a c t i o n m a t r i x e l e m e n t s , obtained only 2.0 M e V / n u c l e o n in 160. On the other hand, they obtained an r . m . s , r a d i u s of 2.56 fm in good a g r e e m e n t with the e x p e r i m e n t a l value of 2.75 ± 0.05 fro, w h e r e a s Eden et al. obtained only about t h r e e - f o u r t h s of this, 1.8 fro. We have i n v e s t i g a t e d the a p p r o x i m a t i o n s c o n tained in the c a l c u l a t i o n of Eden et al., and have * Research sponsored by the U.S.Atomic Energy Commission under contract with the Union Carbide Corporation. ** Oak Ridge Graduate Fellow from Texas A and M University. 308
g e n e r a l i z e d the method to r e m o v e those a p p r o x i m a t i o n s which a p p e a r e d the m o s t s e r i o u s . C a l c u l a t i o n s for 160 have been p e r f o r m e d with the B r u e c k n e r - G a m m e l - T h a l e r potential in o r d e r to d e t e r m i n e the quantitative i m p o r t a n c e of the i m p r o v e m e n t s in the c o m p u t a t i o n a l method, and then with the m o r e r e c e n t H a m a d a - J o h n s t o n pot e n t i a l [4], which gives a b e t t e r account of the t w o - n u c l e o n data. We r e t a i n all t e r m s in the n u c l e o n - n u c l e o n potential. The r e t e n t i o n of the two-body s p i n o r b i t potential is i m p o r t a n t b e c a u s e , even though it is of s h o r t r a n g e , it has a depth just outside
l -5.125 zo uJ
CURVE' /
--6.25
/
o
0.3
0.4 0.5 0.6 O, OSCILLATORRANGEPARAMETER(fro-I)
Fig. 1. Comparison of three approximations to the ground state binding energy of 160, employing a simplified Brueckner-Gammel-Thaler nucleon-nucleon interaction.
Volume 18, number 3
P HYSI C S L E T T E RS
the h a r d c o r e which is appreci~tbly g r e a t e r than the c e n t r a l potential. The t e n s o r f o r c e is handled a c c u r a t e l y by solving coupled equations. T h i s gives r i s e to the m a i n i n c r e a s e in c o m p l e x i t y in the p r e s e n t c a l c u l a t i o n . E d e n et al. t r e a t e d the t e n s o r f o r c e in an a p p r o x i m a t i o n which n e g l e c t s the r e a c t i o n back on the d o m i n a n t component of the wave function and hence o v e r - e s t i m a t e s the m a t r i x e l e m e n t s . I n p e r t u r b a t i o n t h e o r y this r e a c t i o n would f i r s t o c c u r in second o r d e r , but in the n u c l e o n - n u c l e o n potential the t e n s o r t e r m is not s m a l l . With t e n s o r f o r c e s , the t r i p l e t (S = !) r e l a t i v e B e t h e - G o l d s t o n e wave functions for / ~ j a r e exp r e s s i b l e in t e r m s of s p i n - a n g l e functions ~ l S as
3d/nlj(r) = ciJjll (~)Unlj(r) + QJj,I ~2, l(r)Wnlj (r)" (1) The coupled B e t h e - G o l d s t o n e e q u a t i o n s for Unl" and Wnlj a r e , in the c a s e of the r e l a t i v e l s sta~e with the c e n t e r of m a s s in its lowest state for example,
-u"(x) + Ix 2 " ;t + Vg(X)]U(X ) + JgVr(X)W(X ) = 2 ~o = n=0 ~ Vno(X)f ox' ~n°(x')[Vc(X')U(X') + o
1 September 1965
Fig. 1 gives a c o m p a r i s o n with the c a l c u l a t i o n s of E d e n et al. u s i n g the s a m e potential, the B r u e c k n e r - G a m m e l - T h a l e r potential s i m plified by o m i t t i n g the t e n s o r force except in r e lative s s t a t e s and by dropping the two-body s p i n o r b i t force. The methods used to achieve e i g e n "value s e l f - c o n s i s t e n c y a r e d e s c r i b e d by Eden et al. The value of the o s c i l l a t o r r a n g e p a r a m e t e r at which s e l f - c o n s i s t e n c y is found is denoted by a o. C u r v e 1 was c a l c u l a t e d in the s a m e way as the c u r v e of E d e n et al. except for the coupled equation t r e a t m e n t of the t e n s o r force. Selfc o n s i s t e n c y o c c u r s at a s m a l l e r value of a c o r r e s p o n d i n g to a 17% i n c r e a s e in n u c l e a r r a d i u s ; the p r e d i c t e d binding e n e r g y is r e d u c e d by about 1.5 MeV p e r nucleon. A v e r y s i m p l e kind of v e l o c ity dependence of the s i n g l e - p a r t i c l e potential is contained in the method of E d e n et al. through a s p a t i a l l y c o n s t a n t t e r m which differs f r o m state to state. C u r v e 2 d i f f e r s f r o m c u r v e 1 in taking into account the dependence of the c o n s t a n t t e r m in the single p a r t i c l e potential in r e l a t i v e c o o r d i n a t e s on the state of the c e n t e r - o f - m a s s motion
(2)
+,4~'VT(X')W(X')] - w"(X) + [X2 + 6
X2
- h + Vc(X) +
' - 2VT(XHw(x) ÷ ~ - VT(X}u(x} = ~
~z-=0
; ~on~(x)
o
~' x
-=oZ='>==~-3.t25
x %~.(x,) {[re(X,) - 2VT(X')]w(x'l ~ 4g VT(X')"(x')} • H e r e x = ~r, where ~ = (Moo/~i)~ is the o s c i l l a t o r r a n g e p a r a m e t e r , q~nl(X) is an o s c i l l a t o r r a d i a l eigenfunction, Vc(X) is the c e n t r a l potential, VT(X) is the r a d i a l p a r t of the t e n s o r potential, and is the u n p e r t u r b e d r e l a t i v e e n e r g y in u n i t s of {if00. When the potential has a h a r d c o r e one obt a i n s as the g e n e r a l i z a t i o n of Bethe and Golds t o n e ' s [5] t r e a t m e n t for a h a r d c o r e plus c e n t r a l potential, for x,--< Xc,
z =
HAMAOA- JOHNSTON.
-s.aso ms 2 1 6 2
j// /
'
vC(X)u(x) + ~ VT(X)w(x) = u'(Xc)5(x-xc) + gl(X) , [Vc(X) - 2VT(X)]W(X) + ~VT(X)U(X) =
(3) -12.5
= w'(xe)6(X-xc) + g2(x) . The t e r m s gl(X) and g2(x) w~re found to be n e g l i gibly s m a l l in t e s t c a s e s , and w e r e neglected t h e r e a f t e r . F o r x >/x C the coupled i n t e g r o - d i f f e r e n t i a l equations w e r e r e d u c e d to a set of coupled d i f f e r e n t i a l equations together with a set of l i n e a r a l g e b r a i c equations.
0.3
0.4 0.5 a , OSCILLATOR RANGE PARAMETER (fm"|)
0.6
Fig. 2. Average binding energy per nucleon in 160, as calculated using the full Brueckner-Gammel-Thaler and Hamada-Johnston nucleon-nucleon interactions. The values of Rr.m.s, and E/A at the value of ot for which Hartree-Fock selfconsistency is most nearly satisfied a r e indicated. 309
Volume 18, number 3 .
PHYSICS
LETTERS
of the p a i r , t h e r e b y m a k i n g the s e l f - c o n s i s t e n c y condition m o r e n e a r l y satisfied. T h i s has little effect on the r . m . s , r a d i u s but i n c r e a s e s the binding e n e r g y by over an MeV p e r nucleon. Fig. 2 gives a c o m p a r i s o n of c a l c u l a t i o n s for the full B r u e c k n e r - G a m m e l - T h a l e r and 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 s , in which all r e l e v a n t coupled equations w e r e s o l v e d without decoupling a p p r o x i m a t i o n s , and the c e n t e r of m a s s d e p e n d e n c e of the single p a r t i c l e potential in r e l a t i v e c o o r d i n a t e s was included. At self c o n s i s t e n c y the full B r u e c k n e r - G a m m e l - T h a l e r potential gives about the s a m e binding as the simplified B r u e c k n e r - G a m m e l - T h a l e r potential (curve 2, fig. 1), about 1 M e V / n u c l e o n m o r e than e x p e r i m e n t , while the H a m a d a - J o h n s t o n pot e n t i a l gives about 0.7 M e V / n u c l e o n too little. The r . m . s , r a d i u s of the shell model ground state obtained f r o m e i t h e r of these p o t e n t i a l s is about half way between the e x p e r i m e n t a l value and the value of E d e n et al. I n c l u s i o n of the c o r r e l a t i o n s in the wave function i m p l i c i t in the B r u e c k n e r a p p r o x i m a t i o n will r a i s e the r . m . s , r a d i u s s o m e what [6], as will a d m i x t u r e s from h i g h e r s h e l l s in the single p a r t i c l e wave functions, and z e r o point v i b r a t i o n s [7]. The single p a r t i c l e e n e r g i e s for n e u t r o n s obt a i n e d u s i n g both the full B r u e c k n e r - G a m m e l T h a l e r and H a m a d a - J o l m s t o n i n t e r a c t i o n s a r e g i v e n in table 1. Values obtained by B r u e c k n e r et al. [3], u s i n g the B r u e c k n e r - G a m m e l - T h a l e r potential, and e x p e r i m e n t a l e n e r g i e s a r e also given. In obtaining the binding e n e r g y our p r o t o n e n e r g i e s include a Coulomb c o n t r i b u t i o n of 4.3 M e V / n u c l e o n , taken from the p e r t u r b a t i o n c a l c u l a t i o n of Eden et al.; this a g r e e s quite c l o s e l y with the r e s u l t s of the m o r e e l a b o r a t e t r e a t m e n t of B r u e c k n e r et al. [3].
1 September 1965
Table 1 160 single particle energies, p~-p~ doublet splitting, and total binding energy per nucleon, in MeV, and r.m.s, radius, in fm. Intern
Present calc Calc. of ref. 3 HJ
i s½ (n) lp~ (n) I p½ (n) 1 p½ (p)
lp½ - l p i
Experiment
I
BGT
BGT
BGR *
-59.0 -66.6 -30.9 -35.6
-37.0
-44.3 -19.0
-21.7
-22.3 -31.5 -18.0 -27.2 8.6 4.1
-10.6 - 6.4
-14.9 -10.7
-15.6
BE/A
r.m.s, radius
-12.1
4.1
-7.3 -9.2 2.16 2.11
-
2.0
-
2.56
4.4
6.1"*
8.0 2.40 2.57+0.0f -
* The contribution of the hard core to all reaction matrix elements was arbitrarily reduced to 0.825 of the v a l u e calculated. ** Excitation energy of the first 3- state in 150. We wish to thank Dr. T. A. Welton for many h e l p ful d i s c u s s i o n s . R e f e~'ence8 1. R.J. Eden and V. J. Emery, Proc. Roy. Soc. (London) A248 (1958) 266; R.J.Eden, V.J.Emery and S.Sampanthar, Proc. Roy.Soc.(London) A253 (1959) 177, 186. 2. K.A. Brueckner and J, L.Gammel, Phys. Rev. 109 (1958) 1041. 3. K.A.Brueckner, A.M.Lockett and M. Rotenberg, Phys. Rev. 121 (1961) 255. 4. T. Hamada and I. D. Johnston, Nucl. Phys. 34 (1962) 383. 5. H.A. Bethe and J. Goldstone, Proc. Roy. Soc.(London) A238 (1957) 551. 6. Present authors, to be published. 7. G.E.Brown and G.Jacob, Nuclear Phys.42 (1963) 177.
* * * * *
MUON
CAPTURE
AND ELECTRIC
DIPOLE
EXCITATION
FOR
LITHIUM
A . LODDER and C . C . JONKER Natuurkundig Laboratorium van de Vrije Universiteit, A rasterdam, The Netherlands
Received 23 July 1965
The total muon c a p t u r e p r o b a b i l i t i e s for 6Li and 7Li w e r e m e a s u r e d by E c k h a u s e [1], with the result 310
APC(6Li)
= 6100
+
1400
sec -1
AP~C(7Li) = 1800 • 1100 sec -1
PHYSICS LETTERS
NUCLEAR GROUND STATE PROPERTIES BY A N E X T E N S I O N OF THE EDEN-EMERY
1 September 1965
OF 160 METHOD
*
A. D. MacKELLAR ** and R. L. BECKER Oak Ridge National Laboratory, (gak Ridge, Tennessee
Received 22 July 1965
The use of a t w o - n u c l e o n r e a c t i o n o p e r a t o r t p e r m i t s c a l c u l a t i o n s of n u c l e a r p r o p e r t i e s e m ploying available s t r o n g n u c l e o n - n u c l e o n p o t e n t i a l s v which account well for the o n - t h e - e n e r g y shell data on the t w o - n u c l e o n s y s t e m s . B r u e c k n e t ' s a p p r o x i m a t i o n is the analogue of the H a r t r e e - F o c k a p p r o x i m a t i o n in which v is r e p l a c e d by t. It s e e m s likely that w h e n e v e r shell effects a r e i m p o r t a n t the r e a c t i o n m a t r i x m a y differ c o n s i d e r a b l y f r o m that a p p r o p r i a t e to an infinite medium. Thus it would be d e s i r a b l e to have r e action m a t r i c e s c a l c u l a t e d s e p a r a t e l y for i n dividual nuclei. The f i r s t work in this d i r e c t i o n was that of Eden et al. [1]. The method used by Eden et al. has s e e m e d r a t h e r r e s t r i c t e d b e c a u s e it a p p r o x i m a t e s the single p a r t i c l e potential by a h a r m o n i c potential plus a spatially c o n s t a n t t e r m which d i f f e r s f r o m state to state. However, for the ground state of 160 the h a r m o n i c o s c i l l a t o r eigenfunctions a r e thought to provide a good a p p r o x i m a t i o n to the occupied single p a r t i c l e s t a t e s . Eden et al. made c a l c u l a t i o n s with a n u m b e r of G a m m e l - T h a l e r potentials. With the one used by B r u e c k n e r and G a m m e l [2] in the o r i g i n a l n u c l e a r m a t t e r c a l c u l a t i o n s , which we r e f e r to as the B r u e c k n e r G a m m e l - T h a l e r potential, Eden et al. obtained a binding e n e r g y p e r n u c l e o n in 160 of 9.4 MeV, r e a s o n a b l y close to the e x p e r i m e n t a l value of 8.0 MeV. B r u e c k n e r et al. [3] u s i n g the B r u e c k n e r - G a m m e l r e a c t i o n m a t r i x e l e m e n t s , obtained only 2.0 M e V / n u c l e o n in 160. On the other hand, they obtained an r . m . s , r a d i u s of 2.56 fm in good a g r e e m e n t with the e x p e r i m e n t a l value of 2.75 ± 0.05 fro, w h e r e a s Eden et al. obtained only about t h r e e - f o u r t h s of this, 1.8 fro. We have i n v e s t i g a t e d the a p p r o x i m a t i o n s c o n tained in the c a l c u l a t i o n of Eden et al., and have * Research sponsored by the U.S.Atomic Energy Commission under contract with the Union Carbide Corporation. ** Oak Ridge Graduate Fellow from Texas A and M University. 308
g e n e r a l i z e d the method to r e m o v e those a p p r o x i m a t i o n s which a p p e a r e d the m o s t s e r i o u s . C a l c u l a t i o n s for 160 have been p e r f o r m e d with the B r u e c k n e r - G a m m e l - T h a l e r potential in o r d e r to d e t e r m i n e the quantitative i m p o r t a n c e of the i m p r o v e m e n t s in the c o m p u t a t i o n a l method, and then with the m o r e r e c e n t H a m a d a - J o h n s t o n pot e n t i a l [4], which gives a b e t t e r account of the t w o - n u c l e o n data. We r e t a i n all t e r m s in the n u c l e o n - n u c l e o n potential. The r e t e n t i o n of the two-body s p i n o r b i t potential is i m p o r t a n t b e c a u s e , even though it is of s h o r t r a n g e , it has a depth just outside
l -5.125 zo uJ
CURVE' /
--6.25
/
o
0.3
0.4 0.5 0.6 O, OSCILLATORRANGEPARAMETER(fro-I)
Fig. 1. Comparison of three approximations to the ground state binding energy of 160, employing a simplified Brueckner-Gammel-Thaler nucleon-nucleon interaction.
Volume 18, number 3
P HYSI C S L E T T E RS
the h a r d c o r e which is appreci~tbly g r e a t e r than the c e n t r a l potential. The t e n s o r f o r c e is handled a c c u r a t e l y by solving coupled equations. T h i s gives r i s e to the m a i n i n c r e a s e in c o m p l e x i t y in the p r e s e n t c a l c u l a t i o n . E d e n et al. t r e a t e d the t e n s o r f o r c e in an a p p r o x i m a t i o n which n e g l e c t s the r e a c t i o n back on the d o m i n a n t component of the wave function and hence o v e r - e s t i m a t e s the m a t r i x e l e m e n t s . I n p e r t u r b a t i o n t h e o r y this r e a c t i o n would f i r s t o c c u r in second o r d e r , but in the n u c l e o n - n u c l e o n potential the t e n s o r t e r m is not s m a l l . With t e n s o r f o r c e s , the t r i p l e t (S = !) r e l a t i v e B e t h e - G o l d s t o n e wave functions for / ~ j a r e exp r e s s i b l e in t e r m s of s p i n - a n g l e functions ~ l S as
3d/nlj(r) = ciJjll (~)Unlj(r) + QJj,I ~2, l(r)Wnlj (r)" (1) The coupled B e t h e - G o l d s t o n e e q u a t i o n s for Unl" and Wnlj a r e , in the c a s e of the r e l a t i v e l s sta~e with the c e n t e r of m a s s in its lowest state for example,
-u"(x) + Ix 2 " ;t + Vg(X)]U(X ) + JgVr(X)W(X ) = 2 ~o = n=0 ~ Vno(X)f ox' ~n°(x')[Vc(X')U(X') + o
1 September 1965
Fig. 1 gives a c o m p a r i s o n with the c a l c u l a t i o n s of E d e n et al. u s i n g the s a m e potential, the B r u e c k n e r - G a m m e l - T h a l e r potential s i m plified by o m i t t i n g the t e n s o r force except in r e lative s s t a t e s and by dropping the two-body s p i n o r b i t force. The methods used to achieve e i g e n "value s e l f - c o n s i s t e n c y a r e d e s c r i b e d by Eden et al. The value of the o s c i l l a t o r r a n g e p a r a m e t e r at which s e l f - c o n s i s t e n c y is found is denoted by a o. C u r v e 1 was c a l c u l a t e d in the s a m e way as the c u r v e of E d e n et al. except for the coupled equation t r e a t m e n t of the t e n s o r force. Selfc o n s i s t e n c y o c c u r s at a s m a l l e r value of a c o r r e s p o n d i n g to a 17% i n c r e a s e in n u c l e a r r a d i u s ; the p r e d i c t e d binding e n e r g y is r e d u c e d by about 1.5 MeV p e r nucleon. A v e r y s i m p l e kind of v e l o c ity dependence of the s i n g l e - p a r t i c l e potential is contained in the method of E d e n et al. through a s p a t i a l l y c o n s t a n t t e r m which differs f r o m state to state. C u r v e 2 d i f f e r s f r o m c u r v e 1 in taking into account the dependence of the c o n s t a n t t e r m in the single p a r t i c l e potential in r e l a t i v e c o o r d i n a t e s on the state of the c e n t e r - o f - m a s s motion
(2)
+,4~'VT(X')W(X')] - w"(X) + [X2 + 6
X2
- h + Vc(X) +
' - 2VT(XHw(x) ÷ ~ - VT(X}u(x} = ~
~z-=0
; ~on~(x)
o
~' x
-=oZ='>==~-3.t25
x %~.(x,) {[re(X,) - 2VT(X')]w(x'l ~ 4g VT(X')"(x')} • H e r e x = ~r, where ~ = (Moo/~i)~ is the o s c i l l a t o r r a n g e p a r a m e t e r , q~nl(X) is an o s c i l l a t o r r a d i a l eigenfunction, Vc(X) is the c e n t r a l potential, VT(X) is the r a d i a l p a r t of the t e n s o r potential, and is the u n p e r t u r b e d r e l a t i v e e n e r g y in u n i t s of {if00. When the potential has a h a r d c o r e one obt a i n s as the g e n e r a l i z a t i o n of Bethe and Golds t o n e ' s [5] t r e a t m e n t for a h a r d c o r e plus c e n t r a l potential, for x,--< Xc,
z =
HAMAOA- JOHNSTON.
-s.aso ms 2 1 6 2
j// /
'
vC(X)u(x) + ~ VT(X)w(x) = u'(Xc)5(x-xc) + gl(X) , [Vc(X) - 2VT(X)]W(X) + ~VT(X)U(X) =
(3) -12.5
= w'(xe)6(X-xc) + g2(x) . The t e r m s gl(X) and g2(x) w~re found to be n e g l i gibly s m a l l in t e s t c a s e s , and w e r e neglected t h e r e a f t e r . F o r x >/x C the coupled i n t e g r o - d i f f e r e n t i a l equations w e r e r e d u c e d to a set of coupled d i f f e r e n t i a l equations together with a set of l i n e a r a l g e b r a i c equations.
0.3
0.4 0.5 a , OSCILLATOR RANGE PARAMETER (fm"|)
0.6
Fig. 2. Average binding energy per nucleon in 160, as calculated using the full Brueckner-Gammel-Thaler and Hamada-Johnston nucleon-nucleon interactions. The values of Rr.m.s, and E/A at the value of ot for which Hartree-Fock selfconsistency is most nearly satisfied a r e indicated. 309
Volume 18, number 3 .
PHYSICS
LETTERS
of the p a i r , t h e r e b y m a k i n g the s e l f - c o n s i s t e n c y condition m o r e n e a r l y satisfied. T h i s has little effect on the r . m . s , r a d i u s but i n c r e a s e s the binding e n e r g y by over an MeV p e r nucleon. Fig. 2 gives a c o m p a r i s o n of c a l c u l a t i o n s for the full B r u e c k n e r - G a m m e l - T h a l e r and 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 s , in which all r e l e v a n t coupled equations w e r e s o l v e d without decoupling a p p r o x i m a t i o n s , and the c e n t e r of m a s s d e p e n d e n c e of the single p a r t i c l e potential in r e l a t i v e c o o r d i n a t e s was included. At self c o n s i s t e n c y the full B r u e c k n e r - G a m m e l - T h a l e r potential gives about the s a m e binding as the simplified B r u e c k n e r - G a m m e l - T h a l e r potential (curve 2, fig. 1), about 1 M e V / n u c l e o n m o r e than e x p e r i m e n t , while the H a m a d a - J o h n s t o n pot e n t i a l gives about 0.7 M e V / n u c l e o n too little. The r . m . s , r a d i u s of the shell model ground state obtained f r o m e i t h e r of these p o t e n t i a l s is about half way between the e x p e r i m e n t a l value and the value of E d e n et al. I n c l u s i o n of the c o r r e l a t i o n s in the wave function i m p l i c i t in the B r u e c k n e r a p p r o x i m a t i o n will r a i s e the r . m . s , r a d i u s s o m e what [6], as will a d m i x t u r e s from h i g h e r s h e l l s in the single p a r t i c l e wave functions, and z e r o point v i b r a t i o n s [7]. The single p a r t i c l e e n e r g i e s for n e u t r o n s obt a i n e d u s i n g both the full B r u e c k n e r - G a m m e l T h a l e r and H a m a d a - J o l m s t o n i n t e r a c t i o n s a r e g i v e n in table 1. Values obtained by B r u e c k n e r et al. [3], u s i n g the B r u e c k n e r - G a m m e l - T h a l e r potential, and e x p e r i m e n t a l e n e r g i e s a r e also given. In obtaining the binding e n e r g y our p r o t o n e n e r g i e s include a Coulomb c o n t r i b u t i o n of 4.3 M e V / n u c l e o n , taken from the p e r t u r b a t i o n c a l c u l a t i o n of Eden et al.; this a g r e e s quite c l o s e l y with the r e s u l t s of the m o r e e l a b o r a t e t r e a t m e n t of B r u e c k n e r et al. [3].
1 September 1965
Table 1 160 single particle energies, p~-p~ doublet splitting, and total binding energy per nucleon, in MeV, and r.m.s, radius, in fm. Intern
Present calc Calc. of ref. 3 HJ
i s½ (n) lp~ (n) I p½ (n) 1 p½ (p)
lp½ - l p i
Experiment
I
BGT
BGT
BGR *
-59.0 -66.6 -30.9 -35.6
-37.0
-44.3 -19.0
-21.7
-22.3 -31.5 -18.0 -27.2 8.6 4.1
-10.6 - 6.4
-14.9 -10.7
-15.6
BE/A
r.m.s, radius
-12.1
4.1
-7.3 -9.2 2.16 2.11
-
2.0
-
2.56
4.4
6.1"*
8.0 2.40 2.57+0.0f -
* The contribution of the hard core to all reaction matrix elements was arbitrarily reduced to 0.825 of the v a l u e calculated. ** Excitation energy of the first 3- state in 150. We wish to thank Dr. T. A. Welton for many h e l p ful d i s c u s s i o n s . R e f e~'ence8 1. R.J. Eden and V. J. Emery, Proc. Roy. Soc. (London) A248 (1958) 266; R.J.Eden, V.J.Emery and S.Sampanthar, Proc. Roy.Soc.(London) A253 (1959) 177, 186. 2. K.A. Brueckner and J, L.Gammel, Phys. Rev. 109 (1958) 1041. 3. K.A.Brueckner, A.M.Lockett and M. Rotenberg, Phys. Rev. 121 (1961) 255. 4. T. Hamada and I. D. Johnston, Nucl. Phys. 34 (1962) 383. 5. H.A. Bethe and J. Goldstone, Proc. Roy. Soc.(London) A238 (1957) 551. 6. Present authors, to be published. 7. G.E.Brown and G.Jacob, Nuclear Phys.42 (1963) 177.
* * * * *
MUON
CAPTURE
AND ELECTRIC
DIPOLE
EXCITATION
FOR
LITHIUM
A . LODDER and C . C . JONKER Natuurkundig Laboratorium van de Vrije Universiteit, A rasterdam, The Netherlands
Received 23 July 1965
The total muon c a p t u r e p r o b a b i l i t i e s for 6Li and 7Li w e r e m e a s u r e d by E c k h a u s e [1], with the result 310
APC(6Li)
= 6100
+
1400
sec -1
AP~C(7Li) = 1800 • 1100 sec -1