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Nilsson model for 57Fe
V o l u m e 20, n u m b e r 6
PHYSICS
NILSSON
LETTERS
MODEL
1 A p r i l 1966
FOR
57Fe
P. C. S O O D
Department of Physics, University of Alberta, Edmonton, Canada R e c e i v e d 18 F e b r u a r y 1966
The low lying levels in 57Fe are interpreted in terms of the Nilsson model. The energy scale is chosen to correspond to the single particle level separations in the spherical limit. Good agreement for energy levels is obtained for the deformation parameter 71= 3 which corresponds to the eccentricity parameter 5 = 0.25 in agreement with the value derived from the B(E2) and quadrupole moment for the first excited state.
T h e r e has been evidence that some of the i r o n isotopes may show r o t a t i o n a l s p e c t r a and e a r l i e r we p r e s e n t e d [1, 2] the r o t a t i o n a l model i n t e r p r e t a t i o n for l e v e l s i n 56Fe. Lawson and Macf a r l a n e [3] had applied this model to 57Fe a few y e a r s ago, but f r o m c o m p a r i s o n with the scanty e x p e r i m e n t a l i n f o r m a t i o n available at that t i m e they concluded that the single p a r t i c l e r o t a t i o n a l model does not work s a t i s f a c t o r i l y . The past two y e a r s have brought about a v a s t i n c r e a s e in the e x p e r i m e n t a l data [4, 12] for this n u c l e u s and at this stage a p r o p e r e x a m i n a t i o n is called for. The e x p e r i m e n t a l level s p e c t r u m up to 1.7 MeV excitation e n e r g y is shown in fig. l a . The spin p a r i t y a s s i g n m e n t s for the five lowest levels have been uniquely d e t e r m i n e d in the r e c e n t high r e s o l u t i o n d e t e c t o r e x p e r i m e n t s [10-12]. The s p i n of 1.27 MeV l e v e l was f i r s t given [13] as ~, but l a t e r e x p e r i m e n t s [7, 8] proved this to be ina c c u r a t e ; t h e m o r e ~ r o b a b l e a s s i g n m e n t [~/] for this l e v e l is ½ _ T h e ~- c h a r a c t e r i s t i c for 1.63 MeV level is well e s t a b l i s h e d [8]. In this paper we look for an i n t e r p r e t a t i o n of these l e v e l s in t e r m s of the N i l s s o n model [14]. In this model f i r s t we have to d e t e r m i n e the ene r g y s c a l e by s p e c i f y i n g X ~ o. This is done by c o m p a r i n g the s i n g l e p a r t i c l e level s e p a r a t i o n s in the s p h e r i c a l limit. F o r this n u c l e u s we have [15] AE(p~ - p l ) ~ 1.56 MeV. 2
(1)
MeV ,~3
3~-
1.3s-,2z 12o--
MeV
[NnzA],O,
,.65
~z~]3rz-
~5
[~l]l,'i
~-'''""" [3o3]-,'/zu7
i.Ol--
o.zo~
~-
-
3~o.3z ........
o.367
s/g-
[z~2]~K= 3/2-
--5/z-
oJ3c
2
3/'2-
F r o m table l(a)° of N i l s s o n [14] we find that it is equal to 3.0 X~5~o and this gives us X~ o ~ 0.52 MeV.
-
o.o~4
3~-i/z ~' EXPT.
o
.[~l~ T.EO.
- - I / 2 K
=112-
(2)
With this energy scale the Nilsson energy levels f o r the deformation p a r a m e t e r 7/= 3 are shown in fig. l b . The t h e o r e t i c a l basis f o r this value of
Fig. 1. The low l y i n g e n e r g y l e v e l s i n 5 7 F e (a) e x p e r i m e n t a l l e v e l s c h e m e (b) s i n g l e ~ a r t i c l e o r b i t a l s in t h e N i l s s o n m o d e l f o r W = 3 a n d X~O~o = 0.52 MeV. On t h e right are the postulated rotational bands.
647
Volume 20, number 6
PHYSICS LETTERS
7/ is giv en l a t e r in this paper. The l a b e l l i n g on the r i g h t in the f i g u r e c o r r e s p o n d s to the a s y m p totic quantum n u m b e r s . The c o r r e s p o n d i n g o r b i t a l s in N i l s s o n ' s o r i g i n a l n o m e n c l a t u r e a r e ( f r o m ground s t a t e upwards) nos. 20, 16, 10, 17 and 19 r e s p e c t i v e l y . R e m e m b e r i n g that no p a r a m e t e r s have been adjusted, the a g r e e m e n t with the e x p e r i m e n t is s e e n to be r e m a r k a b l e . All the def o r m e d s i n g l e p a r t i c l e o r b i t a l s e x c e p t ~- (which c o m e s f r o m the d i f f e r e n t m a j o r s h e l l in the s h e l l m o d e l picture) s e e m to have been identified. It has been s u g g e s t e d [10] that 1.01 MeV and 1.20 MeV l e v e l s m ay a r i s e f r o m the coupling of a phonon e i t h e r to the ground s t a t e or a low lying e x c i t e d state. Having i d e n t i f i e d t h e s e i n t r i n s i c s t a t e s one ma y then c o n s i d e r the o th e r low lying l e v e l s as m e m b e r s of the r o t a t i o n a l bands s u p e r i m p o s e d on t he s e i n t r i n s i c s t a t e s ; this is shown on the r i g h t in fig. 1. Two c l o s e lying bands with K = ½and K = ~- a p p e a r in this p i c t u r e . As is w e l l known, the i n t e r n a l s t r u c t u r e of t h e s e bands will be s t r o n g l y influenced by band m i x i n g due to C o r i o l i s i n t e r a c t i o n ; this f e a t u r e will be i n v e s t i g a t e d in a s e p a r a t e study. However, t h e s e bands ma y be u s e d h e r e to d e t e r m i n e the d e f o r m a t i o n p a r a m e t e r as follows. Granting that the t h r e e l o w e s t l e v e l s a r e m e m b e r s of a K = ½- band, we can u s e the B(E2) f o r the 5- 1- ~ t r a n s i t i o n o r the q u a d r u p o l e m o m e n t Q of the f i r s t e x c i t e d s t at e ( e x t e n s i v e l y s t u d ie d by M ~ s s b a u e r s p e c t r o s c o p y ) to d e t e r m i n e the i n t r i n s i c q u a d r u p o l e m o m e n t Qo and h e n c e the e c c e n t r i c i t y p a r a m e t e r 5 and the r e l a t e d p a r a m e t e r 7/. The r e l e v a n t e x p r e s s i o n s a r e [14, 17] B(E2; i---f) =
5
e2 Q2 I ( / i 2 g o I /i 2
IlK) t 2 ;
3K2- /(I+1). Q =Qo (f+1)(2/+3) ' Qo = 0.8
ZR2o5(1 + 0.5 5)
(3)
(4) ;
= -~[1 - ~5 2- ~31-~27~ ~
(5)
(6)
Taking the e x p e r i m e n t a l value [18] of B(E2) = 0.037 e 2 × 10 -48 c m 4 we get
t q o t = 1.36 b ,
(7)
which g i v e s the q u a d r u p o l e m o m e n t of the ~sta te as 0.27 b in a g r e e m e n t with the e x p e r i m e n t a l r e s u l t s [19]. T h i s value for Qo c o r r e s p o n d s to the eccentricity parameter *****
648
1April 1966 5 ~ 0.25.
(8)
This result, combined with × ~ 0.08 found suitable for light nuclei [20], gives y ~ 3 in agreement with the value empirically arrived at in this study. It may also be mentioned that Chasman [21] has obtained agreement with the experimental magnetic moment for this nucleus by using modified (towards asymmetry of the order 7 = -20 °) Nilsson Hamiltonian with 0.2 < 5 < 0.3. Thus we conclude that the low lying levels of 57Fe can be quite satisfactorily described in terms of the Nilsson model. Further implications of such a conclusion are being investigated.
References 1. R.K. Gupta and P. C. Sood, Progr. Theor. Phys. 31 (1964) 509. 2. P.C. Sood and R. K. Gupta, Indian Jl. Pure and Applied Phys. 2 (1964) 301. 3. R.D. Lawson and M. H. Maefarlane, Nuel. Phys. 24 (1961) 18. 4. V.P. Bochin, K.I. Zherebtsova, V.S. Zolotarev, V. A. Komarov, L.V.Krasnov, V.F. Litvin, Yu. A. Nemilov, B.G.Novatskyand Sh. Piskorzh, Nucl. Phys. 51 (1964) 161. 5. H.J.Bjerregaard, P.F.Dahl, O.Hansen and G. Sidenius, Nuel.Phys.51 (1964) 641. 6. A. Sperduto and W. W. Bueehner, Phys. Rev. 134 (1964) B142. 7. J.C.Legg and E.Rost, Phys. Rev. 134 (1964) B752. 8. L.V. Groshev, A. M. Demidov, G.A.Kotelnikov and V. N. Lutsenko, Nucl.Phys. 58 (1964) 465. 9. G.A.Bartholomev and J. F. Vervier, Nuel.Phys. 50 (1964) 209. 10. O.C.Kistner and A.W.Sunyar, Phys. Rev. 139 (1965) B295. 11. J.M. Mathiesen and J. P. Hurley, Nucl. Phys. 72 (1965) 475. 12. G.D. Sprouse and S. 8. Hanna, Nucl. Phys. 74 (1965) 177. 13. G.A. Bartholomew and M. R. Gunye, Bull. Am. Phys. Soe. 8 (1963) 367. 14. S.G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29 No. 16 (1955). 15. J . P . Schiffer, Revs. Mod. Phys. 36 (1964) 1065. 16. A.K. Kerman, Mat. Fys. Medd.Dan. Vid. Selsk. 30 No. 15 (1956). 17. A. K. Kerman, in Nuclear reactions, eds. P. M. Endt and M.Demeur (North-Holland Publ. Comp., Amsterdam, 1959) Vol. I, ch.X. 18. R.C. Ritter, P.H. Stelson, F.K. MoGowan and R.L.Robinson, Phys.Rev. 128 (1962) 2320. 19. G.H. Fuller and V.W. Cohen, Nuclear moments (Nuclear Data Sheets, Appendix 1, May 1965) 109. 20. B.R. Mottelson and 8.G.Nilsson, Mat. Fys. Skr. Dan.Vid.Selsk. 1 No. 8 (1958). 21. R.R. Chasman, Phys. Rev. 129 (1963) 2113.
PHYSICS
NILSSON
LETTERS
MODEL
1 A p r i l 1966
FOR
57Fe
P. C. S O O D
Department of Physics, University of Alberta, Edmonton, Canada R e c e i v e d 18 F e b r u a r y 1966
The low lying levels in 57Fe are interpreted in terms of the Nilsson model. The energy scale is chosen to correspond to the single particle level separations in the spherical limit. Good agreement for energy levels is obtained for the deformation parameter 71= 3 which corresponds to the eccentricity parameter 5 = 0.25 in agreement with the value derived from the B(E2) and quadrupole moment for the first excited state.
T h e r e has been evidence that some of the i r o n isotopes may show r o t a t i o n a l s p e c t r a and e a r l i e r we p r e s e n t e d [1, 2] the r o t a t i o n a l model i n t e r p r e t a t i o n for l e v e l s i n 56Fe. Lawson and Macf a r l a n e [3] had applied this model to 57Fe a few y e a r s ago, but f r o m c o m p a r i s o n with the scanty e x p e r i m e n t a l i n f o r m a t i o n available at that t i m e they concluded that the single p a r t i c l e r o t a t i o n a l model does not work s a t i s f a c t o r i l y . The past two y e a r s have brought about a v a s t i n c r e a s e in the e x p e r i m e n t a l data [4, 12] for this n u c l e u s and at this stage a p r o p e r e x a m i n a t i o n is called for. The e x p e r i m e n t a l level s p e c t r u m up to 1.7 MeV excitation e n e r g y is shown in fig. l a . The spin p a r i t y a s s i g n m e n t s for the five lowest levels have been uniquely d e t e r m i n e d in the r e c e n t high r e s o l u t i o n d e t e c t o r e x p e r i m e n t s [10-12]. The s p i n of 1.27 MeV l e v e l was f i r s t given [13] as ~, but l a t e r e x p e r i m e n t s [7, 8] proved this to be ina c c u r a t e ; t h e m o r e ~ r o b a b l e a s s i g n m e n t [~/] for this l e v e l is ½ _ T h e ~- c h a r a c t e r i s t i c for 1.63 MeV level is well e s t a b l i s h e d [8]. In this paper we look for an i n t e r p r e t a t i o n of these l e v e l s in t e r m s of the N i l s s o n model [14]. In this model f i r s t we have to d e t e r m i n e the ene r g y s c a l e by s p e c i f y i n g X ~ o. This is done by c o m p a r i n g the s i n g l e p a r t i c l e level s e p a r a t i o n s in the s p h e r i c a l limit. F o r this n u c l e u s we have [15] AE(p~ - p l ) ~ 1.56 MeV. 2
(1)
MeV ,~3
3~-
1.3s-,2z 12o--
MeV
[NnzA],O,
,.65
~z~]3rz-
~5
[~l]l,'i
~-'''""" [3o3]-,'/zu7
i.Ol--
o.zo~
~-
-
3~o.3z ........
o.367
s/g-
[z~2]~K= 3/2-
--5/z-
oJ3c
2
3/'2-
F r o m table l(a)° of N i l s s o n [14] we find that it is equal to 3.0 X~5~o and this gives us X~ o ~ 0.52 MeV.
-
o.o~4
3~-i/z ~' EXPT.
o
.[~l~ T.EO.
- - I / 2 K
=112-
(2)
With this energy scale the Nilsson energy levels f o r the deformation p a r a m e t e r 7/= 3 are shown in fig. l b . The t h e o r e t i c a l basis f o r this value of
Fig. 1. The low l y i n g e n e r g y l e v e l s i n 5 7 F e (a) e x p e r i m e n t a l l e v e l s c h e m e (b) s i n g l e ~ a r t i c l e o r b i t a l s in t h e N i l s s o n m o d e l f o r W = 3 a n d X~O~o = 0.52 MeV. On t h e right are the postulated rotational bands.
647
Volume 20, number 6
PHYSICS LETTERS
7/ is giv en l a t e r in this paper. The l a b e l l i n g on the r i g h t in the f i g u r e c o r r e s p o n d s to the a s y m p totic quantum n u m b e r s . The c o r r e s p o n d i n g o r b i t a l s in N i l s s o n ' s o r i g i n a l n o m e n c l a t u r e a r e ( f r o m ground s t a t e upwards) nos. 20, 16, 10, 17 and 19 r e s p e c t i v e l y . R e m e m b e r i n g that no p a r a m e t e r s have been adjusted, the a g r e e m e n t with the e x p e r i m e n t is s e e n to be r e m a r k a b l e . All the def o r m e d s i n g l e p a r t i c l e o r b i t a l s e x c e p t ~- (which c o m e s f r o m the d i f f e r e n t m a j o r s h e l l in the s h e l l m o d e l picture) s e e m to have been identified. It has been s u g g e s t e d [10] that 1.01 MeV and 1.20 MeV l e v e l s m ay a r i s e f r o m the coupling of a phonon e i t h e r to the ground s t a t e or a low lying e x c i t e d state. Having i d e n t i f i e d t h e s e i n t r i n s i c s t a t e s one ma y then c o n s i d e r the o th e r low lying l e v e l s as m e m b e r s of the r o t a t i o n a l bands s u p e r i m p o s e d on t he s e i n t r i n s i c s t a t e s ; this is shown on the r i g h t in fig. 1. Two c l o s e lying bands with K = ½and K = ~- a p p e a r in this p i c t u r e . As is w e l l known, the i n t e r n a l s t r u c t u r e of t h e s e bands will be s t r o n g l y influenced by band m i x i n g due to C o r i o l i s i n t e r a c t i o n ; this f e a t u r e will be i n v e s t i g a t e d in a s e p a r a t e study. However, t h e s e bands ma y be u s e d h e r e to d e t e r m i n e the d e f o r m a t i o n p a r a m e t e r as follows. Granting that the t h r e e l o w e s t l e v e l s a r e m e m b e r s of a K = ½- band, we can u s e the B(E2) f o r the 5- 1- ~ t r a n s i t i o n o r the q u a d r u p o l e m o m e n t Q of the f i r s t e x c i t e d s t at e ( e x t e n s i v e l y s t u d ie d by M ~ s s b a u e r s p e c t r o s c o p y ) to d e t e r m i n e the i n t r i n s i c q u a d r u p o l e m o m e n t Qo and h e n c e the e c c e n t r i c i t y p a r a m e t e r 5 and the r e l a t e d p a r a m e t e r 7/. The r e l e v a n t e x p r e s s i o n s a r e [14, 17] B(E2; i---f) =
5
e2 Q2 I ( / i 2 g o I /i 2
IlK) t 2 ;
3K2- /(I+1). Q =Qo (f+1)(2/+3) ' Qo = 0.8
ZR2o5(1 + 0.5 5)
(3)
(4) ;
= -~[1 - ~5 2- ~31-~27~ ~
(5)
(6)
Taking the e x p e r i m e n t a l value [18] of B(E2) = 0.037 e 2 × 10 -48 c m 4 we get
t q o t = 1.36 b ,
(7)
which g i v e s the q u a d r u p o l e m o m e n t of the ~sta te as 0.27 b in a g r e e m e n t with the e x p e r i m e n t a l r e s u l t s [19]. T h i s value for Qo c o r r e s p o n d s to the eccentricity parameter *****
648
1April 1966 5 ~ 0.25.
(8)
This result, combined with × ~ 0.08 found suitable for light nuclei [20], gives y ~ 3 in agreement with the value empirically arrived at in this study. It may also be mentioned that Chasman [21] has obtained agreement with the experimental magnetic moment for this nucleus by using modified (towards asymmetry of the order 7 = -20 °) Nilsson Hamiltonian with 0.2 < 5 < 0.3. Thus we conclude that the low lying levels of 57Fe can be quite satisfactorily described in terms of the Nilsson model. Further implications of such a conclusion are being investigated.
References 1. R.K. Gupta and P. C. Sood, Progr. Theor. Phys. 31 (1964) 509. 2. P.C. Sood and R. K. Gupta, Indian Jl. Pure and Applied Phys. 2 (1964) 301. 3. R.D. Lawson and M. H. Maefarlane, Nuel. Phys. 24 (1961) 18. 4. V.P. Bochin, K.I. Zherebtsova, V.S. Zolotarev, V. A. Komarov, L.V.Krasnov, V.F. Litvin, Yu. A. Nemilov, B.G.Novatskyand Sh. Piskorzh, Nucl. Phys. 51 (1964) 161. 5. H.J.Bjerregaard, P.F.Dahl, O.Hansen and G. Sidenius, Nuel.Phys.51 (1964) 641. 6. A. Sperduto and W. W. Bueehner, Phys. Rev. 134 (1964) B142. 7. J.C.Legg and E.Rost, Phys. Rev. 134 (1964) B752. 8. L.V. Groshev, A. M. Demidov, G.A.Kotelnikov and V. N. Lutsenko, Nucl.Phys. 58 (1964) 465. 9. G.A.Bartholomev and J. F. Vervier, Nuel.Phys. 50 (1964) 209. 10. O.C.Kistner and A.W.Sunyar, Phys. Rev. 139 (1965) B295. 11. J.M. Mathiesen and J. P. Hurley, Nucl. Phys. 72 (1965) 475. 12. G.D. Sprouse and S. 8. Hanna, Nucl. Phys. 74 (1965) 177. 13. G.A. Bartholomew and M. R. Gunye, Bull. Am. Phys. Soe. 8 (1963) 367. 14. S.G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29 No. 16 (1955). 15. J . P . Schiffer, Revs. Mod. Phys. 36 (1964) 1065. 16. A.K. Kerman, Mat. Fys. Medd.Dan. Vid. Selsk. 30 No. 15 (1956). 17. A. K. Kerman, in Nuclear reactions, eds. P. M. Endt and M.Demeur (North-Holland Publ. Comp., Amsterdam, 1959) Vol. I, ch.X. 18. R.C. Ritter, P.H. Stelson, F.K. MoGowan and R.L.Robinson, Phys.Rev. 128 (1962) 2320. 19. G.H. Fuller and V.W. Cohen, Nuclear moments (Nuclear Data Sheets, Appendix 1, May 1965) 109. 20. B.R. Mottelson and 8.G.Nilsson, Mat. Fys. Skr. Dan.Vid.Selsk. 1 No. 8 (1958). 21. R.R. Chasman, Phys. Rev. 129 (1963) 2113.