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Spectra of odd-odd nuclei
Volume 26B, number 7
PHYSICS LETTERS
SPECTRA
OF
ODD-ODD
4March 1968
NUCLEI
V. M. OSADCHIEV and M. A. TROITSKI
[. V.Kurehatov Atomic Energy Institute, Moscow, USSR Received 24 January 1968
For spectra of odd-odd nuclei with odd particle in a state of low angular momentum, an equation for amplitude of interaction is written in a channel where the coordinate dependence of the residual interaction is not essential.
The s p e c t r a of n u c l e i with double c l o s e d s h e l l c o r e plus one p a r t i c l e and one hole can be found f r o m the equation
s t a t e s of an e x t r a p a r t i c l e and an e x t r a hole, and eq. (1) y i e l d s
IM = ( n l _n2)(12IMIFstI12IM)xllI~. ( W s + e l - e 2 ) X12
IM
(2)
(¢os + e 1 - e2)X12 = (n I - n 2) ~ ( 1 2 I M I F I v ~ I M ) x uIM ,
(1) w h e r e the a m p l i t u d e F , i r r e d u c i b l e in the p a r t i c l e hole channel, i s a r e s i d u a l i n t e r a c t i o n [1]; ×/M is the wave function of a p a r t i c l e and a hole coupled t o g e t h e r to a s t a t e with a n g u l a r m o m e n t u m I, e u the e i g e n v a l u e s of the s i n g l e p a r t i c l e H a m i l t o n i a n and n u the occupation n u m b e r s of the c o r e . T h e p r o b l e m c o n s i s t s of the c h o i c e of a r e s i d u a l i n t e r a c t i o n F . F o r the c a s e when the p a r t i c l e and the hole a r e coupled to low a n g u l a r m o m e n t u m (I << A½, w h e r e A i s the a t o m i c n u m b e r ) and t h e i r quantum n u m b e r s slightly differ, the s o l u tion of eq. (1) does not s u b s t a n t i a l l y depend upon the c o o r d i n a t e f o r m of the block F . One can take any f o r m with c h a r a c t e r i s t i c r a n g e of the o r d e r of the d i s t a n c e b et we e n n u c l e o n s in n u c l e i . Of c o u r s e , d e l t a - f u n c t i o n f o r c e s will be the s i m p l e s t a p p r o x i m a t i o n to use. H o w e v e r f o r l a r g e I 1 (I ~ A~) the solution of eq. (1) is s e n s i t i v e to the r a n g e of F . T h e p r o b l e m r e q u i r e s the i n t r o d u c tion of many m o r e c o n s t a n t s , including i n t e r a c tion with the e f f e c t i v e r a d i u s , t e n s o r f o r c e s etc. In this note we s k e t c h our p r o c e d u r e f o r high a n g u l a r m o m e n t u m s t a t e s [2]. Let the i n t e r a c t i o n of the p a r t i c l e and the hole coupled to a s t a t e of high a n g u l a r m o m e n t u m be weak (~ e F ( 2 j + 1)/Afli, BI>2 >> 1, the F e r m i e n e r g y e F = 40 MeV) as c o m p a r e d with d i f f e r e n c e A I b e t w e e n the l e v e l s of the s a m e s y m m e t r y in the s h e l l - m o d e l s c h e m e . This i s v a l i d f o r the ground and f i r s t e x c i t e d s t a t e s of o d d -o d d nuclei (AI ~ e F ( 2 j + 1)/A). Then the e i g e n v a l u e ws of eq. (1) is c l o s e to the s h e l l m o d e l d i f f e r e n c e e 2 - e l , w h e r e 2 and 1 a r e
T h e s t a t i c a m p l i t u d e F st is c o n s t r u c t e d f r o m the block F by taking into acco u n t the p a r t i c l e - h o l e e x c i t a t i o n of the c o r e (121M 1Fst1121M) = (12IMI F 112IM) +
(3)
n v -np + ~ (12IM] F! VI~IM) ~ (vIzlM[ r st ] 121M) . As is s e e n f r o m eq. (2), the p a r t i c l e and the hole a r e not s p r e a d o v e r s i n g l e p a r t i c l e l e v e l s , i . e . t h e r e is the p e r t u r b a t i o n t h e o r y f o r wave function (B72 << 1). We would h o w e v e r e m p h a s i z e that u s ing p e r t u r b a t i o n t h e o r y to evaluate the e n e r g y eqs. (1) and (3) i s not c o r r e c t , b e c a u s e the a m plitude F st a r i s e s f r o m the s u m m a t i o n o v e r many s i n g l e p a r t i c l e l e v e l s . Th e equation f o r the s t a t i c a m p l i t u d e F st m ay be w r i t t e n in any of t h r e e channels (fig. 1). F o r
L Fig. 1. n u c l e i with one p a r t i c l e and one hole plus the c o r e the hole u su al l y has a s m a l l s i n g l e p a r t i c l e a n g u l a r m o m e n t u m J l and the p a r t i c l e h as a l a r g e one. It is c l e a r that the an g u l ar m o m e n t u m which passles through the I and J channels i s l a r g e (~A~), but f o r the p a r t i c l e - h o l e L - c h a n n e l we h a v e s m a l l v a l u e s L = 0 - 2jl (for i n s t a n c e , the 421
Volume 26B. number ‘7
PHYSICS
LETTERS
4 March 1968
ground states of 2%i, 208T1: j, = $, j2 = t, I= 5+, J= 4+, 5+, L = O+, li). Therefore instead of evaluating the matrix element (1211 l?st / 121) in the direct I-channel we shall recouple the channels
6+ =:-_
5+ -
(12ZMI rst/ 121M) = T (-)jl+j2+1+L+1(2L +l) x
X
j, j, I .j2
j2
(4)
L
(11Lm 1Iq22Lm)
.
I i
Evaluating (L / Tst / L) first according to an equation similar to eq. (3) in which one can take any range-de endence of residual interaction we obtain (II rst PI) by summation over all possible L. We use the delta-function form of particlehole interaction [l]
-!?LF dCF
= f +gm~+(f’+g’aa)w
,
wheref,g are strength constants and dn/deF a dimensionless factor. An example for 208~1 shows how our procedure works (fig. 2). The ground and first excited states are the 5+, 4+ states respectively. Eq. (2) yields for them the unperturbed wave function [P3S;n2g$]l. The latter was obtained numerically in ref. [3]. From eqs. (2) and (4) one can see the construction of the (5+, 4+) doublet: zero-order corresponds to the shell-model, the matrix element (L = 0 / rst 1L = 0) shifts the degenerate (%g) level and the matrix element (L = 1 1Pt / L = 1) removes the degeneracy of the state with various I values. Thus we get the correct order of levels. Moreover the evaluation of the multiplet splitting is straightforward now [ 21. The low-lying levels in odd-odd nuclei are studied in many papers. We shall confine our discussion to some of them. Kim and Rasmussen in their calculations of the particle-hole spectra for 208Bi, 208~1 [3] have used the particle-particle interaction recoupling the block F irreducible in the particle-hole channel I. The particle-particle interaction was t&en from the spectra of nuclei with two particles above the magic core.
*****
422
619 492 473 328
Fig. 2. Strictly speaking the amplitude F is not connected with the amplitude irreducible in the particleparticle channel which determines the spectra of two particles above the core [4]. To use the pp interaction one has to write down the equation for rst in the particle-particle J-channel. The particle-hole spectra were evaluated also by Birbrair et al. [5] with delta-forces in eq. (1). For the case of large angular momenta th(:ir model forces are not correct. We are grateful to Dr. A. B. Migdal, A. A. Lushnikov, E. E. Saperstein, V. A. Khodel for discussion.
References 1. A. B. Migdal, The theory of the finite Fermi-sys-
2.
3. 4. 5.
tems (Wiley 1967); A.B.Migdal, Nucl. Phys. 57 (1964) 29. Some detailed papers and the results of calculation of low-energy levels for 206Bi, 208Pb, 208T1, 9oY, 66Y. 4GK are published in Nucl. Phys. (Soviet) 6 (1967) 961. Y.E.Kim and J.O.Rasmussen, Phys. Rev. 135 (1964) B44. E. E. Saperstein and M.A. Troitski, Nucl. Phys. (Soviet) 1 (1965) 400. 13.L. Birbrair, G. M.Bukat and V. N.Guman, Nucl. Phys. (Soviet) 1 (1965) 971; 4 (1966) 473.
PHYSICS LETTERS
SPECTRA
OF
ODD-ODD
4March 1968
NUCLEI
V. M. OSADCHIEV and M. A. TROITSKI
[. V.Kurehatov Atomic Energy Institute, Moscow, USSR Received 24 January 1968
For spectra of odd-odd nuclei with odd particle in a state of low angular momentum, an equation for amplitude of interaction is written in a channel where the coordinate dependence of the residual interaction is not essential.
The s p e c t r a of n u c l e i with double c l o s e d s h e l l c o r e plus one p a r t i c l e and one hole can be found f r o m the equation
s t a t e s of an e x t r a p a r t i c l e and an e x t r a hole, and eq. (1) y i e l d s
IM = ( n l _n2)(12IMIFstI12IM)xllI~. ( W s + e l - e 2 ) X12
IM
(2)
(¢os + e 1 - e2)X12 = (n I - n 2) ~ ( 1 2 I M I F I v ~ I M ) x uIM ,
(1) w h e r e the a m p l i t u d e F , i r r e d u c i b l e in the p a r t i c l e hole channel, i s a r e s i d u a l i n t e r a c t i o n [1]; ×/M is the wave function of a p a r t i c l e and a hole coupled t o g e t h e r to a s t a t e with a n g u l a r m o m e n t u m I, e u the e i g e n v a l u e s of the s i n g l e p a r t i c l e H a m i l t o n i a n and n u the occupation n u m b e r s of the c o r e . T h e p r o b l e m c o n s i s t s of the c h o i c e of a r e s i d u a l i n t e r a c t i o n F . F o r the c a s e when the p a r t i c l e and the hole a r e coupled to low a n g u l a r m o m e n t u m (I << A½, w h e r e A i s the a t o m i c n u m b e r ) and t h e i r quantum n u m b e r s slightly differ, the s o l u tion of eq. (1) does not s u b s t a n t i a l l y depend upon the c o o r d i n a t e f o r m of the block F . One can take any f o r m with c h a r a c t e r i s t i c r a n g e of the o r d e r of the d i s t a n c e b et we e n n u c l e o n s in n u c l e i . Of c o u r s e , d e l t a - f u n c t i o n f o r c e s will be the s i m p l e s t a p p r o x i m a t i o n to use. H o w e v e r f o r l a r g e I 1 (I ~ A~) the solution of eq. (1) is s e n s i t i v e to the r a n g e of F . T h e p r o b l e m r e q u i r e s the i n t r o d u c tion of many m o r e c o n s t a n t s , including i n t e r a c tion with the e f f e c t i v e r a d i u s , t e n s o r f o r c e s etc. In this note we s k e t c h our p r o c e d u r e f o r high a n g u l a r m o m e n t u m s t a t e s [2]. Let the i n t e r a c t i o n of the p a r t i c l e and the hole coupled to a s t a t e of high a n g u l a r m o m e n t u m be weak (~ e F ( 2 j + 1)/Afli, BI>2 >> 1, the F e r m i e n e r g y e F = 40 MeV) as c o m p a r e d with d i f f e r e n c e A I b e t w e e n the l e v e l s of the s a m e s y m m e t r y in the s h e l l - m o d e l s c h e m e . This i s v a l i d f o r the ground and f i r s t e x c i t e d s t a t e s of o d d -o d d nuclei (AI ~ e F ( 2 j + 1)/A). Then the e i g e n v a l u e ws of eq. (1) is c l o s e to the s h e l l m o d e l d i f f e r e n c e e 2 - e l , w h e r e 2 and 1 a r e
T h e s t a t i c a m p l i t u d e F st is c o n s t r u c t e d f r o m the block F by taking into acco u n t the p a r t i c l e - h o l e e x c i t a t i o n of the c o r e (121M 1Fst1121M) = (12IMI F 112IM) +
(3)
n v -np + ~ (12IM] F! VI~IM) ~ (vIzlM[ r st ] 121M) . As is s e e n f r o m eq. (2), the p a r t i c l e and the hole a r e not s p r e a d o v e r s i n g l e p a r t i c l e l e v e l s , i . e . t h e r e is the p e r t u r b a t i o n t h e o r y f o r wave function (B72 << 1). We would h o w e v e r e m p h a s i z e that u s ing p e r t u r b a t i o n t h e o r y to evaluate the e n e r g y eqs. (1) and (3) i s not c o r r e c t , b e c a u s e the a m plitude F st a r i s e s f r o m the s u m m a t i o n o v e r many s i n g l e p a r t i c l e l e v e l s . Th e equation f o r the s t a t i c a m p l i t u d e F st m ay be w r i t t e n in any of t h r e e channels (fig. 1). F o r
L Fig. 1. n u c l e i with one p a r t i c l e and one hole plus the c o r e the hole u su al l y has a s m a l l s i n g l e p a r t i c l e a n g u l a r m o m e n t u m J l and the p a r t i c l e h as a l a r g e one. It is c l e a r that the an g u l ar m o m e n t u m which passles through the I and J channels i s l a r g e (~A~), but f o r the p a r t i c l e - h o l e L - c h a n n e l we h a v e s m a l l v a l u e s L = 0 - 2jl (for i n s t a n c e , the 421
Volume 26B. number ‘7
PHYSICS
LETTERS
4 March 1968
ground states of 2%i, 208T1: j, = $, j2 = t, I= 5+, J= 4+, 5+, L = O+, li). Therefore instead of evaluating the matrix element (1211 l?st / 121) in the direct I-channel we shall recouple the channels
6+ =:-_
5+ -
(12ZMI rst/ 121M) = T (-)jl+j2+1+L+1(2L +l) x
X
j, j, I .j2
j2
(4)
L
(11Lm 1Iq22Lm)
.
I i
Evaluating (L / Tst / L) first according to an equation similar to eq. (3) in which one can take any range-de endence of residual interaction we obtain (II rst PI) by summation over all possible L. We use the delta-function form of particlehole interaction [l]
-!?LF dCF
= f +gm~+(f’+g’aa)w
,
wheref,g are strength constants and dn/deF a dimensionless factor. An example for 208~1 shows how our procedure works (fig. 2). The ground and first excited states are the 5+, 4+ states respectively. Eq. (2) yields for them the unperturbed wave function [P3S;n2g$]l. The latter was obtained numerically in ref. [3]. From eqs. (2) and (4) one can see the construction of the (5+, 4+) doublet: zero-order corresponds to the shell-model, the matrix element (L = 0 / rst 1L = 0) shifts the degenerate (%g) level and the matrix element (L = 1 1Pt / L = 1) removes the degeneracy of the state with various I values. Thus we get the correct order of levels. Moreover the evaluation of the multiplet splitting is straightforward now [ 21. The low-lying levels in odd-odd nuclei are studied in many papers. We shall confine our discussion to some of them. Kim and Rasmussen in their calculations of the particle-hole spectra for 208Bi, 208~1 [3] have used the particle-particle interaction recoupling the block F irreducible in the particle-hole channel I. The particle-particle interaction was t&en from the spectra of nuclei with two particles above the magic core.
*****
422
619 492 473 328
Fig. 2. Strictly speaking the amplitude F is not connected with the amplitude irreducible in the particleparticle channel which determines the spectra of two particles above the core [4]. To use the pp interaction one has to write down the equation for rst in the particle-particle J-channel. The particle-hole spectra were evaluated also by Birbrair et al. [5] with delta-forces in eq. (1). For the case of large angular momenta th(:ir model forces are not correct. We are grateful to Dr. A. B. Migdal, A. A. Lushnikov, E. E. Saperstein, V. A. Khodel for discussion.
References 1. A. B. Migdal, The theory of the finite Fermi-sys-
2.
3. 4. 5.
tems (Wiley 1967); A.B.Migdal, Nucl. Phys. 57 (1964) 29. Some detailed papers and the results of calculation of low-energy levels for 206Bi, 208Pb, 208T1, 9oY, 66Y. 4GK are published in Nucl. Phys. (Soviet) 6 (1967) 961. Y.E.Kim and J.O.Rasmussen, Phys. Rev. 135 (1964) B44. E. E. Saperstein and M.A. Troitski, Nucl. Phys. (Soviet) 1 (1965) 400. 13.L. Birbrair, G. M.Bukat and V. N.Guman, Nucl. Phys. (Soviet) 1 (1965) 971; 4 (1966) 473.