- Email: [email protected]
The isotopic spin dependence of the nuclear optical model potential
Volume 5, number 1
PHYSICS LETTERS
Rdfdrences
1) N. Crayton et al., Revs.Modern Phys. 34 (1962) 186. 2) G. Baumann, H. Braun et P. CUer, Compt. rend. 254 (1962) 3839.
1 June 1963
3) R.D. Lawson et M. Rotenberg, Nuovo Cimento 17 (1960) 449. 4) G. Baumann, H. Braun et P. Crier, Compt. rend. 256 (1963) 918.
*****
OF
THE
THE ISOTOPIC SPIN NUCLEAR OPTICAL
DEPENDENCE MODEL POTENTIAL
J. DABROWSKI and A. SOBICZEWSKI Institute for Nuclear Research, Warsaw Received 6 May 1963
The i n t e r e s t in the isotopic spin dependence of the "o" on the t a r g e t nucleon "i", Moi , s u m m e d o v e r optical potential has i n c r e a s e d essentially since the all the t a r g e t nucleons 7). Since o b s e r v a t i o n of the excitation of the analog s t a t e in MOi=( ~ 3Moi+~ 1Moi)+ (3Moi- 1Moi) t o ' t i , (3) the (p, n) r e a c t i o n by Anderson et al. 1). As suggested by Lane 2) the (p,n) r e a c t i o n can be d e s c r i b e d w h e r e 3M, 1M a r e s c a t t e r i n g amplitudes in i s o by the i s o t o p i c - s p i n dependent optical potential: topic triplet and isotopic singlet s t a t e s r e s p e c t i v e l y , we obviously get for the optical potential V the f o r m V = V o + VI A - l t T , (1) of eq. (1). As the e l e m e n t s of the s c a t t e r i n g ampliw h e r e t, T a r e the isotopic spin o p e r a t o r s of the tude M can be e x p r e s s e d with the help of nucleons c a t t e r e d nucleon and the t a r g e t nucleus r e s p e c nucleon p h a s e shifts, one eventually gets f o r m u l a e tively. f o r Vo and V1 as functions of nucleon-nucleon p h a s e In a r e c e n t a r t i c l e Hodgson 3) collects the r e s u l t s shifts. of all kinds of e s t i m a t e s of Re V1. One is faced with T h e s e f o r m u l a e together with their detailed quite a r a n g e of values of Re V1 between 15 and 200 derivation have been given in ref. 8). We have apMeV. The lower limit ~robably can be put equal plied them to calculate Vo_.and V1 with the help of z e r o . P e r e y and Buck ~), in their extensive analthe new Yale p h a s e shifts 9) . We r e s t r i c t o u r s e l v e s y s i s of neutron elastic s c a t t e r i n g , found s a t i s f a c t o r y to the following r e m a r k s concerning our calculation: a g r e e m e n t with all the available e x p e r i m e n t a l data 1. We have c o n s i d e r e d the c a s e of nuclear m a t t e r on medium and heavy nuclei f r o m 1 to 25 MeV withwith k F = 1.38 f m -1 (r o = 1.1 fro). Thus our r e s u l t s out the V1 p a r t of V. Buck 5), in his analysis of 11.1 should be c o m p a r e d with phenomenological values MeV proton s c a t t e r i n g on s e p a r a t e d isotopes of Zn, of Vo and V1 in the center of the nucleus. does not find a c l e a r c o r r e l a t i o n between Re V and By applying the f o r m u l a e of r e f . 7, 8) one o~ = (N - Z ) / A . Cohen 6 ) in his a n a l y s i s of the loca- gets V^, V. as functions of E, the kinetic energy tion of neutron single p a r t i c l e levels f r o m stripping of the s c a t t e r e d nucleon inside of the nuclear m e d i r e a c t i o n s , s a y s that none of the r e s u l t s d i s c u s s e d in um. To get the energy Eo of the s c a t t e r e d nucleon his p a p e r would be qualitatively a l t e r e d if the s y m outside of the t a r g e t nucleus we have applied the m e t r y energy c o r r e c t i o n w e r e either doubled in equation : magnitude or neglected completely. It is probably E o = E + R e Vo . (4) s a f e to say that the phenomenological e s t i m a t e s give f o r Re V1 : 3. The m o m e n t a of the t a r g e t nucleons have been neglected. 0 <~Re V1 5 2 0 0 M e V . (2) 4. In calculating I m V we applied the Goldberger We f e e l then that it is d e s i r e a b l e to have a theot r e a t m e n t of the Pauli principle, explained in r e t i c a l derivation of V1 that could be at l e a s t a guide r e / . 10). in the phenomenological e s t i m a t e s . Such a t h e o r e t i 5. Among the p h a s e shifts given in ref. 9) we cal derivation is c o m p a r a t i v e l y s i m p l e at high e n e r have chosen YLAM ( T = 1), and YLAN 3 M ( T = 0) g i e s , say > 50 MeV. In this energy r a n g e the optical for l ~< lma x = 4. In calculating the OPE contrimodel potential is given by the f o r w a r d s c a t t e r i n g bution we have used: g2/tic = 14, mlr - m~o = amplitude f o r s c a t t e r i n g of the incoming nucleon 135.04 MeV/c 2. 87
Volume 5, n u m b e r 1
PHYSICS
LETTERS Vo
°
o
Vp = Vo - ~ ( V l + 8esy m ~ ) .
.
~
"*° 1
,
.
E . LrtcvJ
\\\
Fig. 1. Calculated values of the optical model parameters. The r e s u l t s are shown in fig. 1. Because of the approximations inherent in the Watson theory of the optical model only the part of the c u r v e s for E_ 50 MeV is expected to be theoretically reliable. The main f e a t u r e s of our r e s u l t s a r e : a) Re V1 ~ 40 MeV for 0 ~ E o ~ 100 MeV which is substantially less than the majority to the phenomenological estimates. b) Re V1 < 0 for g o ~ 200 MeV which c l e a r l y is the result of the increasing role of the repulsive c o r e in nucleon-nucleon interaction at higher energies. At these energies, however, we do not have any phenomenological estimate to c o m p a r e with. c) Im V1 ~ 20 MeV for E o ~ 50 MeV. Here we do not have any other estimate to c o m p a r e with. Hence in any phenomenological estimates of V1 one should keep in mind that Re V1 is expected to change sign at about 200 MeV, and that V1 has an imaginary part of the s a m e o r d e r of magnitude as the real part. The following difficulty in estimating the magnitude of V1 f r o m proton 'scattering should be mentioned. Whereas the optical potential Vn for neutrons, according to eq. (1) is v = vo +
(5)
the nuclear optical potential Vp for protons is *****
88
1 June 1963
(6)
The last t e r m appears because the kinetic energy of a proton inside of the nucleus is diminished by the a v e r a g e Coulomb repulsion, as pointed out by Satchler i1). The s y m m e t r y energy Csvm enters into eq. (6) because for stable nuclei ~ e Coulomb energy is exactly balanced by the s y m m e t r y energy. The r e a l part of eq. (6) has been discussed bef o r e (see e.g. ref. 3)). We notice only that our c u r v e Re Vo in fig. 1 gives BRe Vo/B E^ = 0.2 which should be c o m p a r e d with the value' 0.29 and 0.28 found by Buck 5 ) f r o m neutron and proton scattering respectively in the energy range 0 - 20 MeV. In the case of the imaginary part of eq. (6) 8Im Vo/8 E o is expected to be negative and it may happen that Im V1 + 8 e s y m 8Ira Vo/BE o b e c o m e s negative. For instance at E 0 ~ 100 MeV the curve I m Vo in fig. 1 gives aim Vo/a E o ~ -0.05. This together with the empirical value of e ~vrn = 48 MeV gives 8 e s y m aim Vo/8 E o ~ -20 MeV~w--hich is about equal in magnitude to our calculated Im V1. At lower energy one should expect l a r g e r values of ]SIm Vo/8 Eol. This indicates that the t e r m p r o portional to ~ in the imaginary part of the proton nuclear optical potential probably has, especially at lower energies, the s a m e sign as that of the neutron potential. The authors wish to e x p r e s s their thanks to P r o f e s s o r Breit, P r o f e s s o r Cohen and Dr. Buck for sending their r e s u l t s p r i o r to publication.
References 1) J.D.Anderson, C.Wong and J.W.McClure, Phys. Rev. 126 (1962) 2170. 2) A.M. Lane, Nuclear Phys. 35 (1962) 676. 3) P.E.Hodgson, Phys. Letters 3 (1963) 352. 4) P.G.Perey and B.Buck, Nuclear Phys. 32 (1962) 353. 5) B.Buck, Phys.Rev.(1963), in press. 6) B. L. Cohen, in press. 7) W. B. Riesenfeld and K. M. Watson, Phys. Rev. 102 (1956) 1157. 8) F. E. Bjorklund, B.A. Lippmann and M. J. Moravcsik, Nuclear Phys. 29 (1962) 582 ; 40 (1963) 690. 9) G. Breit et al., Phys.Rev. 128 (1962) 826; M.H.Hull Jr. et al., Phys.Rev. 128 (1962) 830. 10) J.D~browski and A.Sobiczewski, Acta Physica Polonica 20 (1961) 243. 11) G.R.Satchler, Phys.Rev. 109 (1958) 429.
PHYSICS LETTERS
Rdfdrences
1) N. Crayton et al., Revs.Modern Phys. 34 (1962) 186. 2) G. Baumann, H. Braun et P. CUer, Compt. rend. 254 (1962) 3839.
1 June 1963
3) R.D. Lawson et M. Rotenberg, Nuovo Cimento 17 (1960) 449. 4) G. Baumann, H. Braun et P. Crier, Compt. rend. 256 (1963) 918.
*****
OF
THE
THE ISOTOPIC SPIN NUCLEAR OPTICAL
DEPENDENCE MODEL POTENTIAL
J. DABROWSKI and A. SOBICZEWSKI Institute for Nuclear Research, Warsaw Received 6 May 1963
The i n t e r e s t in the isotopic spin dependence of the "o" on the t a r g e t nucleon "i", Moi , s u m m e d o v e r optical potential has i n c r e a s e d essentially since the all the t a r g e t nucleons 7). Since o b s e r v a t i o n of the excitation of the analog s t a t e in MOi=( ~ 3Moi+~ 1Moi)+ (3Moi- 1Moi) t o ' t i , (3) the (p, n) r e a c t i o n by Anderson et al. 1). As suggested by Lane 2) the (p,n) r e a c t i o n can be d e s c r i b e d w h e r e 3M, 1M a r e s c a t t e r i n g amplitudes in i s o by the i s o t o p i c - s p i n dependent optical potential: topic triplet and isotopic singlet s t a t e s r e s p e c t i v e l y , we obviously get for the optical potential V the f o r m V = V o + VI A - l t T , (1) of eq. (1). As the e l e m e n t s of the s c a t t e r i n g ampliw h e r e t, T a r e the isotopic spin o p e r a t o r s of the tude M can be e x p r e s s e d with the help of nucleons c a t t e r e d nucleon and the t a r g e t nucleus r e s p e c nucleon p h a s e shifts, one eventually gets f o r m u l a e tively. f o r Vo and V1 as functions of nucleon-nucleon p h a s e In a r e c e n t a r t i c l e Hodgson 3) collects the r e s u l t s shifts. of all kinds of e s t i m a t e s of Re V1. One is faced with T h e s e f o r m u l a e together with their detailed quite a r a n g e of values of Re V1 between 15 and 200 derivation have been given in ref. 8). We have apMeV. The lower limit ~robably can be put equal plied them to calculate Vo_.and V1 with the help of z e r o . P e r e y and Buck ~), in their extensive analthe new Yale p h a s e shifts 9) . We r e s t r i c t o u r s e l v e s y s i s of neutron elastic s c a t t e r i n g , found s a t i s f a c t o r y to the following r e m a r k s concerning our calculation: a g r e e m e n t with all the available e x p e r i m e n t a l data 1. We have c o n s i d e r e d the c a s e of nuclear m a t t e r on medium and heavy nuclei f r o m 1 to 25 MeV withwith k F = 1.38 f m -1 (r o = 1.1 fro). Thus our r e s u l t s out the V1 p a r t of V. Buck 5), in his analysis of 11.1 should be c o m p a r e d with phenomenological values MeV proton s c a t t e r i n g on s e p a r a t e d isotopes of Zn, of Vo and V1 in the center of the nucleus. does not find a c l e a r c o r r e l a t i o n between Re V and By applying the f o r m u l a e of r e f . 7, 8) one o~ = (N - Z ) / A . Cohen 6 ) in his a n a l y s i s of the loca- gets V^, V. as functions of E, the kinetic energy tion of neutron single p a r t i c l e levels f r o m stripping of the s c a t t e r e d nucleon inside of the nuclear m e d i r e a c t i o n s , s a y s that none of the r e s u l t s d i s c u s s e d in um. To get the energy Eo of the s c a t t e r e d nucleon his p a p e r would be qualitatively a l t e r e d if the s y m outside of the t a r g e t nucleus we have applied the m e t r y energy c o r r e c t i o n w e r e either doubled in equation : magnitude or neglected completely. It is probably E o = E + R e Vo . (4) s a f e to say that the phenomenological e s t i m a t e s give f o r Re V1 : 3. The m o m e n t a of the t a r g e t nucleons have been neglected. 0 <~Re V1 5 2 0 0 M e V . (2) 4. In calculating I m V we applied the Goldberger We f e e l then that it is d e s i r e a b l e to have a theot r e a t m e n t of the Pauli principle, explained in r e t i c a l derivation of V1 that could be at l e a s t a guide r e / . 10). in the phenomenological e s t i m a t e s . Such a t h e o r e t i 5. Among the p h a s e shifts given in ref. 9) we cal derivation is c o m p a r a t i v e l y s i m p l e at high e n e r have chosen YLAM ( T = 1), and YLAN 3 M ( T = 0) g i e s , say > 50 MeV. In this energy r a n g e the optical for l ~< lma x = 4. In calculating the OPE contrimodel potential is given by the f o r w a r d s c a t t e r i n g bution we have used: g2/tic = 14, mlr - m~o = amplitude f o r s c a t t e r i n g of the incoming nucleon 135.04 MeV/c 2. 87
Volume 5, n u m b e r 1
PHYSICS
LETTERS Vo
°
o
Vp = Vo - ~ ( V l + 8esy m ~ ) .
.
~
"*° 1
,
.
E . LrtcvJ
\\\
Fig. 1. Calculated values of the optical model parameters. The r e s u l t s are shown in fig. 1. Because of the approximations inherent in the Watson theory of the optical model only the part of the c u r v e s for E_ 50 MeV is expected to be theoretically reliable. The main f e a t u r e s of our r e s u l t s a r e : a) Re V1 ~ 40 MeV for 0 ~ E o ~ 100 MeV which is substantially less than the majority to the phenomenological estimates. b) Re V1 < 0 for g o ~ 200 MeV which c l e a r l y is the result of the increasing role of the repulsive c o r e in nucleon-nucleon interaction at higher energies. At these energies, however, we do not have any phenomenological estimate to c o m p a r e with. c) Im V1 ~ 20 MeV for E o ~ 50 MeV. Here we do not have any other estimate to c o m p a r e with. Hence in any phenomenological estimates of V1 one should keep in mind that Re V1 is expected to change sign at about 200 MeV, and that V1 has an imaginary part of the s a m e o r d e r of magnitude as the real part. The following difficulty in estimating the magnitude of V1 f r o m proton 'scattering should be mentioned. Whereas the optical potential Vn for neutrons, according to eq. (1) is v = vo +
(5)
the nuclear optical potential Vp for protons is *****
88
1 June 1963
(6)
The last t e r m appears because the kinetic energy of a proton inside of the nucleus is diminished by the a v e r a g e Coulomb repulsion, as pointed out by Satchler i1). The s y m m e t r y energy Csvm enters into eq. (6) because for stable nuclei ~ e Coulomb energy is exactly balanced by the s y m m e t r y energy. The r e a l part of eq. (6) has been discussed bef o r e (see e.g. ref. 3)). We notice only that our c u r v e Re Vo in fig. 1 gives BRe Vo/B E^ = 0.2 which should be c o m p a r e d with the value' 0.29 and 0.28 found by Buck 5 ) f r o m neutron and proton scattering respectively in the energy range 0 - 20 MeV. In the case of the imaginary part of eq. (6) 8Im Vo/8 E o is expected to be negative and it may happen that Im V1 + 8 e s y m 8Ira Vo/BE o b e c o m e s negative. For instance at E 0 ~ 100 MeV the curve I m Vo in fig. 1 gives aim Vo/a E o ~ -0.05. This together with the empirical value of e ~vrn = 48 MeV gives 8 e s y m aim Vo/8 E o ~ -20 MeV~w--hich is about equal in magnitude to our calculated Im V1. At lower energy one should expect l a r g e r values of ]SIm Vo/8 Eol. This indicates that the t e r m p r o portional to ~ in the imaginary part of the proton nuclear optical potential probably has, especially at lower energies, the s a m e sign as that of the neutron potential. The authors wish to e x p r e s s their thanks to P r o f e s s o r Breit, P r o f e s s o r Cohen and Dr. Buck for sending their r e s u l t s p r i o r to publication.
References 1) J.D.Anderson, C.Wong and J.W.McClure, Phys. Rev. 126 (1962) 2170. 2) A.M. Lane, Nuclear Phys. 35 (1962) 676. 3) P.E.Hodgson, Phys. Letters 3 (1963) 352. 4) P.G.Perey and B.Buck, Nuclear Phys. 32 (1962) 353. 5) B.Buck, Phys.Rev.(1963), in press. 6) B. L. Cohen, in press. 7) W. B. Riesenfeld and K. M. Watson, Phys. Rev. 102 (1956) 1157. 8) F. E. Bjorklund, B.A. Lippmann and M. J. Moravcsik, Nuclear Phys. 29 (1962) 582 ; 40 (1963) 690. 9) G. Breit et al., Phys.Rev. 128 (1962) 826; M.H.Hull Jr. et al., Phys.Rev. 128 (1962) 830. 10) J.D~browski and A.Sobiczewski, Acta Physica Polonica 20 (1961) 243. 11) G.R.Satchler, Phys.Rev. 109 (1958) 429.