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Coulomb displacement energies from isobaric analogue state studies in deformed nuclei
Volume27B, number 10
PHYSICS LETTERS
COULOMB DISPLACEMENT ENERGIES FROM ANALOGUE STATE STUDIES IN DEFORMED
14 October 1968
ISOBARIC NUCLEI
Y. CASSAGNOU, P. FOISSEL, J. GASTEBOIS, C. LEVI, W. MITTIG and L. PAPINEAU Service de Physique Nucl~aire ~ B a s s e Energie, Centre d'Etudes Nucl~aires de Saclay, France
Received 5 August 1968
Coulomb displacement energies for isobaric analogue states of some low-lying states of the deformed nuclei 171yb, 173yb, 175yb and 177yb deduced from the analysis of the excitation functions determined by elastic scattering of protons are found lower than the displacements calculated with the usual formulas. This effect is attributed to the deformation of the nucleus. In the l a s t few y e a r s , Coulomb d i s p l a c e m e n t e n e r g i e s have been d e t e r m i n e d f r o m B r e i t - W i g n e r a n a l y s i s of i s o l a t e d r e s o n a n c e s o b s e r v e d in proton e l a s t i c s c a t t e r i n g . In deformed n u c l e i , the spacings of the analogues of low-lying states of odd-even isotopes a r e u s u a l l y s m a l l e r than the total widths of the r e s o n a n c e s . F o r example, for the odd-even y t t e r b i u m isotopes, level s p a c i n g s a r e as s m a l l as 50 keV [1,2], while the total widths of i s o b a r i c analogue r e s o n a n c e s a r e of the o r d e r of 100-150 keV. A c c u r a t e d e t e r m i n a t i o n of the r e s o n a n c e e n e r g i e s then a p p e a r s to be difficult. N e v e r t h e l e s s , a complete a n a l y s i s of both e l a s t i c s c a t t e r i n g and i n e l a s t i c s c a t t e r i n g exciting the 2 +, 4 + and 6 + s t a t e s of the g r o u n d - s t a t e r o t a t i o n a l band shows [3] that s o m e of the analogue r e s o n a n c e s o b s e r v e d in e l a s t i c s c a t t e r i n g e x c i t a tion functions can be c o n s i d e r e d as p r a c t i c a l l y isolated. Briefly, qualitative a r g u m e n t s a r e the c o n s i d e r a t i o n of o r b i t a l m o m e n t a (and p e n e t r a tion factors), and of p a r t i a l r e d u c e d widths (i.e. s p e c t r o s c o p i c factors). M o r e o v e r , as, i s o b a r i c analogue s t a t e s a r e e s s e n t i a l l y seen in e l a s t i c s c a t t e r i n g through the i n t e r f e r e n c e t e r m between the r e s o n a n c e and the potential s c a t t e r i n g , one can use the fact that this i n t e r f e r e n c e v a n i s h e s for a given o r b i t a l m o m e n t u m l at the angles where the c o r r e s p o n d i n g L e g e n d r e p o l y n o m i a l Pl(COS 0) goes to zero. This gives in p a r t i c u l a r the p o s s i b i l i t y to get f r o m the excitation function m e a s u r e d at 141 ° the r e s o n a n c e e n e r g i e s of I = 1 r e s o n a n c e s even in the p r e s e n c e of an o v e r l a p p i n g I = 3 r e s o n a n c e . In this l e t t e r , we r e p o r t the r e s u l t s obtained by fitting the excitation functions of e l a s t i c s c a t t e r i n g of protons on 170yb, 172yb, 174yb, 176yb at 141 ° l a b o r a t o r y angle. A s i n g l e - l e v e l B r e i t W i g n e r type f o r m u l a has been applied to the I = 1
r e s o n a n c e s . Obtained r e s o n a n c e e n e r g i e s a r e l i s t e d in table 1, with the deduced Coulomb d i s p l a c e m e n t energy AEcexp (the most r e c e n t B(n) values have been used; they a r e often in g r e a t d i s a g r e e m e n t with older values). It can be easily seen that: 1°) for the s c a t t e r i n g of protons on 170yb and 176yb, analogue state spacings a r e , within the e x p e r i m e n t a l e r r o r s , the s a m e as the c o r r e s ponding p a r e n t state spacings; 2 °) the Coulomb d i s p l a c e m e n t energy is p r o p o r t i o n a l to Z A -1~, such that t h e r e is no isotope shift; 3 °) the obtained values show a s y s t e m a t i c t r e n d to be lower than those which can be c a l c u lated by the u s u a l s e m i - e m p i r i c a l f o r m u l a s [4-6] which were e s t a b l i s h e d f r o m data on s p h e r i c a l n u c l e i . The d i f f e r e n c e s between the e x p e r i m e n t a l values and the calculated d i s p l a c e m e n t s AE~.W,A given by the A n d e r s o n - W o n g f o r m u l a [4] (which has been e s t a b l i s h e d f r o m Q(p, n) m e a s u r e m e n t s and consequently r e q u i r e s no f u r t h e r calculation to obtain AEc) a r e of the o r d e r of 150 keV. T h e r e is also evidence of such a difference between the 181Hf - 181Ta Coulomb d i s p l a c e m e n t energy d e t e r m i n e d with the r e s o n a n c e energy of the ~-[503] analogue state [7] and the s e m i - e m p i r i c a l one (table 1). In a very crude approximation, and as a tentative explanation, this effect could be a c counted for by the e l e c t r i c quadrupole t e r m app e a r i n g in the expansion of the charge of an e l l i p soidal n u c l e u s . This leads to a r e l a t i o n between the actual Coulomb d i s p l a c e m e n t energy AEc(5) and the d i s p l a c e m e n t for the s a m e n u c l e u s a s s u m e d s p h e r i c a l [11] AEc(5 ) = A E c ( 0 ) ( 1 - ~ 6 2 ) , where 5 is the d e f o r m a t i o n . F o r the low-lying s t a t e s of the y t t e r b i u m i s o topes, one can take 5 ~ 0.3, and 631
V o l u m e 2 7 B , n u m b e r 10
PHYSICS
LETTERS
14 October 1968
Table 1 Coulomb energy d i s p l a c e m e n t s for deformed states. Target
Parent
Eex c
B(n)
Eli~b
AE exp
nucleus
state
(MeV)
(MeV)
(MeV)
(MeV)
AE AW
170yb
1-(~-[521]) ~-(~-[510])
0 (L~95
10.530 ± 0.020 11.510 ± 0.020
17.084 ± 0.032* 17.064 ± 0.032
17.21
6.616 ± 0.012 [2]
172yb
~-(~-[510])
1.073
6.370 + 0.012 [2]
11.780 + 0.020
17.008 ± 0.032
17.14 ± 0.10
174yb
~-(~-[510])
0.552
5.820 ± 0.012 [2]
11.720 ± 0.020
16.921 ± 0.032
17.07 ± 0.10
176yb
~-(½-[510]) ~-{~-[501])
0.379 1.362
5.562 ± 0.012 [2]
11.720 ± 0.020 12.730 ± 0.020
16.837 ± 0.032 16.858 ± 0.032
17.00 ± 0.10
180Hf
~-(½-[503])
0.670 [7]
5.700 ± 0.001 [9]
12.270 ± 0.050
17.232 ± 0.050
17.38 ± 0.10
(MeV) ±
0.10
* The same value h a s been obtained at Copenhagen [10]. A g c ( 0 ) - A E c ( 5 ) ~ 140 k e V . It s e e m s t h a t s o m e o t h e r e f f e c t s m a y b e s t u d i e d on t h e b a s i s of t h e r e l a t i v e s h i f t s of t h e a n a l o g u e s t a t e s [8]. 1 °) D i f f e r e n t q u a d r u p o l e m o m e n t s a s s o c i a t e d with the parent states, can introduce shifts in Coulomb energies. 2 °) W h e n t h e a d d e d p r o t o n o c c u p i e s a d i f f e r e n t Nilsson orbit, it has a different orientation with r e s p e c t to t h e d e f o r m e d C o u l o m b f i e l d , w h i c h also leads to slightly different Coulomb displacem e n t e n e r g i e s w i t h i n a s e q u e n c e of s t a t e s [12]. 3° The centrigual stretching (or coupling to the beta-vibrational band) has the effect that the charge radius is a little different for each spin v a l u e w h i c h l e a d s to s l i g h t l y d i f f e r e n t m o m e n t s of i n e r t i a f o r t h e a n a l o g u e r o t a t i o n a l s t a t e s . Coulomb energy displacements may then be differe n t f o r t h e s t a t e s of a g i v e n r o t a t i o n a l b a n d . A l l t h e s e e f f e c t s a r e e x p e c t e d to b e of t h e o r d e r of o n e o r a f e w t e n s of k e V ( a p a r t f r o m screening factors due to pairing which may obs c u r e s o m e of t h e s e e f f e c t s ) , i . e . of a n o r d e r of magnitude smaller than the observed one. As they may simultaneously occur, a detailed study h a s n o t y e t b e e n p o s s i b l e w i t h i n t h e a c c u r a c y of these experiments. These results, and those obtained by the
632
Heidelberg group for the isobar analogue states i n t h e N = 82 r e g i o n [13] s h o w t h a t in o r d e r to g e t e v i d e n c e f o r s h e l l o r d e f o r m a t i o n e f f e c t s on t h e Coulomb displacement energies in heavy nuclei, one must extend accurate isobaric analogue state s t u d i e s t o o t h e r n u c l e i of t h e r a r e e a r t h r e g i o n a n d of t h e t r a n s i t i o n r e g i o n s ( w h e r e o n e m a y hope to measure changes in deformation through t h e i r e f f e c t s on C o u l o m b e n e r g y ) .
1. J . G a s t e b o i s , t h e s i s , P a r i s (1968}, to be published. 2. D . G . B u r k e et al., Dansk. Vid. Selsk. Mat. Fys. Medd. 35 n ° 2 (1966). 3. Y. Cassagnou et al., to be published. 4. J . D . A n d e r s o n et al., Phys. Rev. 138B (1965} 615. 5. D . D . L o n g et al., Phys. Rev. 149 (1966) 906. 6. P. Von Brentano et al., J a h r e s b e r i c h t Heidelberg, 1965, p. 30. 7. F . A . R i c k e y et al., Bull. Am. Phys. Soc. 12 (1967) 1195. 8. S.G. Nilsson, L e c t u r e s in t h e o r e t i c a l physics, Vol. VIII C, p. 208, University of Colorado P r e s s . 9. F . A . R i c k e y , Thesis. F l o r i d a State University (1966). 10. S.Whineray, private communication. 11. M.H. M a c F a r l a n e , I s o b a r i c spin in nuclear physics, T a l l a h a s s e e (1966) p. 399. 12. C . J . V e j e , private communication. 13. J. Self et al., J a h r e s b e r i c h t Heidelberg 1967, p. 59.
PHYSICS LETTERS
COULOMB DISPLACEMENT ENERGIES FROM ANALOGUE STATE STUDIES IN DEFORMED
14 October 1968
ISOBARIC NUCLEI
Y. CASSAGNOU, P. FOISSEL, J. GASTEBOIS, C. LEVI, W. MITTIG and L. PAPINEAU Service de Physique Nucl~aire ~ B a s s e Energie, Centre d'Etudes Nucl~aires de Saclay, France
Received 5 August 1968
Coulomb displacement energies for isobaric analogue states of some low-lying states of the deformed nuclei 171yb, 173yb, 175yb and 177yb deduced from the analysis of the excitation functions determined by elastic scattering of protons are found lower than the displacements calculated with the usual formulas. This effect is attributed to the deformation of the nucleus. In the l a s t few y e a r s , Coulomb d i s p l a c e m e n t e n e r g i e s have been d e t e r m i n e d f r o m B r e i t - W i g n e r a n a l y s i s of i s o l a t e d r e s o n a n c e s o b s e r v e d in proton e l a s t i c s c a t t e r i n g . In deformed n u c l e i , the spacings of the analogues of low-lying states of odd-even isotopes a r e u s u a l l y s m a l l e r than the total widths of the r e s o n a n c e s . F o r example, for the odd-even y t t e r b i u m isotopes, level s p a c i n g s a r e as s m a l l as 50 keV [1,2], while the total widths of i s o b a r i c analogue r e s o n a n c e s a r e of the o r d e r of 100-150 keV. A c c u r a t e d e t e r m i n a t i o n of the r e s o n a n c e e n e r g i e s then a p p e a r s to be difficult. N e v e r t h e l e s s , a complete a n a l y s i s of both e l a s t i c s c a t t e r i n g and i n e l a s t i c s c a t t e r i n g exciting the 2 +, 4 + and 6 + s t a t e s of the g r o u n d - s t a t e r o t a t i o n a l band shows [3] that s o m e of the analogue r e s o n a n c e s o b s e r v e d in e l a s t i c s c a t t e r i n g e x c i t a tion functions can be c o n s i d e r e d as p r a c t i c a l l y isolated. Briefly, qualitative a r g u m e n t s a r e the c o n s i d e r a t i o n of o r b i t a l m o m e n t a (and p e n e t r a tion factors), and of p a r t i a l r e d u c e d widths (i.e. s p e c t r o s c o p i c factors). M o r e o v e r , as, i s o b a r i c analogue s t a t e s a r e e s s e n t i a l l y seen in e l a s t i c s c a t t e r i n g through the i n t e r f e r e n c e t e r m between the r e s o n a n c e and the potential s c a t t e r i n g , one can use the fact that this i n t e r f e r e n c e v a n i s h e s for a given o r b i t a l m o m e n t u m l at the angles where the c o r r e s p o n d i n g L e g e n d r e p o l y n o m i a l Pl(COS 0) goes to zero. This gives in p a r t i c u l a r the p o s s i b i l i t y to get f r o m the excitation function m e a s u r e d at 141 ° the r e s o n a n c e e n e r g i e s of I = 1 r e s o n a n c e s even in the p r e s e n c e of an o v e r l a p p i n g I = 3 r e s o n a n c e . In this l e t t e r , we r e p o r t the r e s u l t s obtained by fitting the excitation functions of e l a s t i c s c a t t e r i n g of protons on 170yb, 172yb, 174yb, 176yb at 141 ° l a b o r a t o r y angle. A s i n g l e - l e v e l B r e i t W i g n e r type f o r m u l a has been applied to the I = 1
r e s o n a n c e s . Obtained r e s o n a n c e e n e r g i e s a r e l i s t e d in table 1, with the deduced Coulomb d i s p l a c e m e n t energy AEcexp (the most r e c e n t B(n) values have been used; they a r e often in g r e a t d i s a g r e e m e n t with older values). It can be easily seen that: 1°) for the s c a t t e r i n g of protons on 170yb and 176yb, analogue state spacings a r e , within the e x p e r i m e n t a l e r r o r s , the s a m e as the c o r r e s ponding p a r e n t state spacings; 2 °) the Coulomb d i s p l a c e m e n t energy is p r o p o r t i o n a l to Z A -1~, such that t h e r e is no isotope shift; 3 °) the obtained values show a s y s t e m a t i c t r e n d to be lower than those which can be c a l c u lated by the u s u a l s e m i - e m p i r i c a l f o r m u l a s [4-6] which were e s t a b l i s h e d f r o m data on s p h e r i c a l n u c l e i . The d i f f e r e n c e s between the e x p e r i m e n t a l values and the calculated d i s p l a c e m e n t s AE~.W,A given by the A n d e r s o n - W o n g f o r m u l a [4] (which has been e s t a b l i s h e d f r o m Q(p, n) m e a s u r e m e n t s and consequently r e q u i r e s no f u r t h e r calculation to obtain AEc) a r e of the o r d e r of 150 keV. T h e r e is also evidence of such a difference between the 181Hf - 181Ta Coulomb d i s p l a c e m e n t energy d e t e r m i n e d with the r e s o n a n c e energy of the ~-[503] analogue state [7] and the s e m i - e m p i r i c a l one (table 1). In a very crude approximation, and as a tentative explanation, this effect could be a c counted for by the e l e c t r i c quadrupole t e r m app e a r i n g in the expansion of the charge of an e l l i p soidal n u c l e u s . This leads to a r e l a t i o n between the actual Coulomb d i s p l a c e m e n t energy AEc(5) and the d i s p l a c e m e n t for the s a m e n u c l e u s a s s u m e d s p h e r i c a l [11] AEc(5 ) = A E c ( 0 ) ( 1 - ~ 6 2 ) , where 5 is the d e f o r m a t i o n . F o r the low-lying s t a t e s of the y t t e r b i u m i s o topes, one can take 5 ~ 0.3, and 631
V o l u m e 2 7 B , n u m b e r 10
PHYSICS
LETTERS
14 October 1968
Table 1 Coulomb energy d i s p l a c e m e n t s for deformed states. Target
Parent
Eex c
B(n)
Eli~b
AE exp
nucleus
state
(MeV)
(MeV)
(MeV)
(MeV)
AE AW
170yb
1-(~-[521]) ~-(~-[510])
0 (L~95
10.530 ± 0.020 11.510 ± 0.020
17.084 ± 0.032* 17.064 ± 0.032
17.21
6.616 ± 0.012 [2]
172yb
~-(~-[510])
1.073
6.370 + 0.012 [2]
11.780 + 0.020
17.008 ± 0.032
17.14 ± 0.10
174yb
~-(~-[510])
0.552
5.820 ± 0.012 [2]
11.720 ± 0.020
16.921 ± 0.032
17.07 ± 0.10
176yb
~-(½-[510]) ~-{~-[501])
0.379 1.362
5.562 ± 0.012 [2]
11.720 ± 0.020 12.730 ± 0.020
16.837 ± 0.032 16.858 ± 0.032
17.00 ± 0.10
180Hf
~-(½-[503])
0.670 [7]
5.700 ± 0.001 [9]
12.270 ± 0.050
17.232 ± 0.050
17.38 ± 0.10
(MeV) ±
0.10
* The same value h a s been obtained at Copenhagen [10]. A g c ( 0 ) - A E c ( 5 ) ~ 140 k e V . It s e e m s t h a t s o m e o t h e r e f f e c t s m a y b e s t u d i e d on t h e b a s i s of t h e r e l a t i v e s h i f t s of t h e a n a l o g u e s t a t e s [8]. 1 °) D i f f e r e n t q u a d r u p o l e m o m e n t s a s s o c i a t e d with the parent states, can introduce shifts in Coulomb energies. 2 °) W h e n t h e a d d e d p r o t o n o c c u p i e s a d i f f e r e n t Nilsson orbit, it has a different orientation with r e s p e c t to t h e d e f o r m e d C o u l o m b f i e l d , w h i c h also leads to slightly different Coulomb displacem e n t e n e r g i e s w i t h i n a s e q u e n c e of s t a t e s [12]. 3° The centrigual stretching (or coupling to the beta-vibrational band) has the effect that the charge radius is a little different for each spin v a l u e w h i c h l e a d s to s l i g h t l y d i f f e r e n t m o m e n t s of i n e r t i a f o r t h e a n a l o g u e r o t a t i o n a l s t a t e s . Coulomb energy displacements may then be differe n t f o r t h e s t a t e s of a g i v e n r o t a t i o n a l b a n d . A l l t h e s e e f f e c t s a r e e x p e c t e d to b e of t h e o r d e r of o n e o r a f e w t e n s of k e V ( a p a r t f r o m screening factors due to pairing which may obs c u r e s o m e of t h e s e e f f e c t s ) , i . e . of a n o r d e r of magnitude smaller than the observed one. As they may simultaneously occur, a detailed study h a s n o t y e t b e e n p o s s i b l e w i t h i n t h e a c c u r a c y of these experiments. These results, and those obtained by the
632
Heidelberg group for the isobar analogue states i n t h e N = 82 r e g i o n [13] s h o w t h a t in o r d e r to g e t e v i d e n c e f o r s h e l l o r d e f o r m a t i o n e f f e c t s on t h e Coulomb displacement energies in heavy nuclei, one must extend accurate isobaric analogue state s t u d i e s t o o t h e r n u c l e i of t h e r a r e e a r t h r e g i o n a n d of t h e t r a n s i t i o n r e g i o n s ( w h e r e o n e m a y hope to measure changes in deformation through t h e i r e f f e c t s on C o u l o m b e n e r g y ) .
1. J . G a s t e b o i s , t h e s i s , P a r i s (1968}, to be published. 2. D . G . B u r k e et al., Dansk. Vid. Selsk. Mat. Fys. Medd. 35 n ° 2 (1966). 3. Y. Cassagnou et al., to be published. 4. J . D . A n d e r s o n et al., Phys. Rev. 138B (1965} 615. 5. D . D . L o n g et al., Phys. Rev. 149 (1966) 906. 6. P. Von Brentano et al., J a h r e s b e r i c h t Heidelberg, 1965, p. 30. 7. F . A . R i c k e y et al., Bull. Am. Phys. Soc. 12 (1967) 1195. 8. S.G. Nilsson, L e c t u r e s in t h e o r e t i c a l physics, Vol. VIII C, p. 208, University of Colorado P r e s s . 9. F . A . R i c k e y , Thesis. F l o r i d a State University (1966). 10. S.Whineray, private communication. 11. M.H. M a c F a r l a n e , I s o b a r i c spin in nuclear physics, T a l l a h a s s e e (1966) p. 399. 12. C . J . V e j e , private communication. 13. J. Self et al., J a h r e s b e r i c h t Heidelberg 1967, p. 59.