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Total and partial widths of isobaric analog resonances
V o l u m e 40B, n u m b e r 2
TOTAL
AND
PHYSICS
PARTIAL
WIDTHS
OF
LETTERS
ISOBARIC
26 June 1972
ANALOG
RESONANCES*
W. R. COKER, H. BLEDSOE ** and T. TAMURA Center for Nuclear Studies, University of Texas, Austin, Texas 78712, USA Received 3 May 1972
Total and partial widths and resonance mixing phases for dj/2 analog resonances in 87Rb, 89y, 91,93Nb ' 93,97Tc are calculated in the framework of a nonorth~o'gonal, optical-model basis in a shellmodel theory of reactions. The proton optical potential used is identical to that with which the experimental resonance parameters were extracted, for 89y and 91Nb where the most detailed and unambiguous experimental studies have been made.
Comparison of predictions of various microscopic and semi-microscopic theories of isobaric analog resonances (IAR) with experimental data has been somewhat inconclusive in the past because of ambiguity and approximation both in the calculations and the extraction of quantities for c o m p a r i s o n f r o m e x p e r i m e n t [1-3]. A r e c e n t study [4] of the 8 9 y and-91Nb ground state d5/2 IAR, in which the a n a l y z i n g power was used, together with the c r o s s s e c t i o n for e l a s t i c s c a t t e r i n g , to provide a f a i r l y m o d e l - i n d e p e n d e n t s e p a r a t i o n of d i r e c t and c o m p o u n d - e l a s t i c c r o s s s e c t i o n s , and thence r a t h e r r e l i a b l e optical and r e s o n a n c e p a r a m e t e r s , has p r o m p t e d us to c a r r y out detailed c a l c u l a t i o n s using the g e n e r a l i z e d Fano approach of Bledsoe and T a m u r a [5]. We a r e able to calculate all e n t r a n c e channel r e s o n a n c e p a r a m e t e r s : ER, F, F_, and ~bR, i.e., the r e s o nance e n e r g y , the t o t a l ~ i d t h , the p a r t i a l width, and the mixing phase [2,4]° The s e n s i t i v i t y of Fp to the magnitude and shape of the c h a r g e - e x c h a n g e potential Vl(r) has b e e n investigated by a n u m b e r of w o r k e r s [1,3,5]. The m o s t s a t i s f a c t o r y r e s u l t s have b e e n obtained with the Woods-Saxon d e r i v a t i v e f o r m for V1, and a s t r e n g t h about equal to that of the (volume) s y m m e t r y t e r m in the optical model potential. In o r d e r to have a unique and c o n s i s t e n t s t r e n g t h for the r e a l p a r t of V1, we use the definition Vp(ER) _ Vnc(Es) = ToRE(V1)
(I)
which is in the spirit of the Lane equations: Vp(ER) is the strength of the real part of the proton optical potential at E R, and VnC(Es) is * Supported in part by the U. S. Atomic Energy Commission. ** Present Address: Dept. of Physics, State Univ. of New York, StonyBrook, Long Island, New York 11790.
164
the depth of the potential binding the parent neutron at separation energy E s. Note that E R = Ac - Es, where Ac is the Coulomb displacement energy of a proton, and TO is the isospin of the target (or core) nucleus. As in ref. [5], the imaginary part of ToV1 is identical to the imaginary part of the proton optical model potential. Thus, once a proton optical model potential is given the resonance parameters are determined uniquely. We briefly summarize the relations used in the calculations reported here. If ~AX = (2To + 1)-1/2 <~bnIToVI - iWl~k (E))
(2)
where ~k(E) is an eigenstate of the proton optical model Hamiltonian, HO =K +V_ + iW+ Vc, P for proton channel ~, then the total width is given by F(E) = 2 ~
2
t~
Re(vAX) -
2
~ p ofo Im(vA,) t~
0
E -E'
dE', (3)
the level shift by . ~, ~ Re(u~/.t) A(E) =2-J P J ~ dE'-,~
im(ui2 '
(4)
and the partial width in channel k by FA ~exp(2i ~bR ) = 2~(u2tt),
(5)
where q5R is the resonance mixing phase. For a derivation and fuller discussion of these equations, consult ref. [5]. A numerically more convenient approximation to eqs. (3) and (4) was suggested in ref. [5], which amounts to replacing the energy integration by a radial integration involving the Green's function of Hp. Such an approximation neglects the discrete spectrum of Hp,
Volume 40B, number 2
PHYSICS LETTERS
26 June 1972
Table 1 Comparison of calculated and experimental partial and total widths of d5/2 IAR seen in proton elastic scattering on the targets listed. Widths are in keV. In column 5 is the spectroscopic factor S deduced from F~/S and Fexp; Sd_ • . ~. .P' ~J is the generally accepted spectroscopic factor from (d,p) experiments; F the predicted total wldth, using S; ~bR and qSR(exp) are the calculated and experimental resonance mixing phases in radians.
zs
Target
rp/S
F/s
Fe~ p
S
86Kr 88Sr 90Zr 92Mo 92Zr 96Mo
14.1 6.40 3.56 1.54 5.65 6.18
63.3 27.7 22.3 38.9 51.7 88.5
7.3 6.3 3.1 1.5 4.0 2.5
0.52 0..98 0.87 0.97 0.71 0.40
Sdp 0.56 0.79 0.89 0.84 0.55 0.42
[16] [17] [18] [19] [18,20] [21]
F
Fex p
q5R
32.9 27.1 19.4 37.7 36.7 35.4
36.0 19.0 17.0 30.0 30.0 33.0
0.17 0..12 0.11 0.08 0.11 0.13
qbR(exp) 0.15 0.01 ±0.05 0.05 ±0.05 0.17 0.0 ±0.1 0.0 ±0.1
and s u r p r i s i n g l y is found n u m e r i c a l l y to be e x c e e din g l y poor. We have t h e r e f o r e e v a lu a t e d the e n e r g y i n t e g r a l s d i r e c t l y . The c o n t r i b u t i o n of ine l a s t i c p r o t o n c h a n n e l s to F has been neglected, s i n c e for al l c a s e s c o n s i d e r e d it is s m a l l cornp a r e d to the e x p e r i m e n t a l e r r o r in F. The proton o p t i cal p o t e n ti a l used in the anal y s i s of K r e t s c h m e r and Graw [4] is that of B e c c h e t t i and G r e e n l e e s [6], e x c e p t that K r e t s c h m e r and Graw r e d u c e the s t r e n g t h of WD to 4.0 MeV to c o m p e n s a t e fo r the e x t r a c t i o n of the cornpound e l a s t i c c o n t r i b u t i o n f r o m the (p,p) e x c i t a tion c u r v e . A s i m i l a r r e d u c t i o n of WD was made in p r e v i o u s w o r k [2] for the s a m e r e a s o n . We have adopted the potential of ref. [4] for p + 88Sr, and p + 90Zr. The r e s u l t s of our c a l c u l a t i o n s a r e s u m m a r i z e d in table 1. We have a l s o p e r f o r m e d c a l c u l a t i o n s for the d5/2 IAR o b s e r v e d in p r o t o n e l a s t i c s c a t t e r i n g on 86Kr [7] 92Mo [8], 9 2 Z r [9], and 96Mo [10]. The b e s t a g r e e m e n t with F e px was obtained for 86Kr, 92Mo and 9 2 Z r with WD ~ 6-7 MeV; the table g i v e s the p a r a m e t e r s c a l c u l a t e d with WD s e t to 8 MeV. Finally, fo r 96Mo, the b e s t a g r e e m e n t was obtained with WD at its s t a n d a r d value of 12 MeV. Such a v a r i a t i o n in WD is p a r t i a l l y u n d e r standable ff one c o n s i d e r s the magnitude of the e x p e c t e d compound e l a s t i c c o n t r i b u t i o n to the c r o s s section. The (p, n) channel is well open for 86Kr, 9 2 Z r and 96Mo. It is c l o s e d for 9 0 Z r and 92Mo, and b a r e l y open for 88Sr. Thus the only p e r p l e x i n g r e s u l t is that F for 92Mo cannot a l s o be fit with WD = 4 MeV. H o w e v e r , in the a n a l y s i s of 92Mo(p, Pc) a n a l y z i n g p o w e r data r e p o r t e d in ref. [8] the total widths of the f i r s t four IAR in 93Tc w e r e all taken as 30 keV, and a s e a r c h was made only on Fp and ~R" It is l i k e l y that an e q u a l ly good fit to the ground s t a t e IAR could have been obtained with F ~ 20 keV (and thus WD ~ 4 MeV), as in the c a s e s of 9 0 Z r and 88Sr. Neutron s t r e n g t h function data a l s o f a v o r s the l o w e r value of WD f or 88Sr, 9 0 Z r , 92Mo [4,12]. The s t r o n g , a l m o s t l i n e a r dependence of F on
2o so 4o
9°Zr , ~ / / / / (Fp/S),lo
' / F / S/ , F
eSSr
60
40
, / \\\\\\,, ,\\\\\\\\\re...,px / rp/S
20 /.
o 40 so
ao
I~)
92Mo
/ o
5
15
eOt 92Zr 922
//s,r
I0
I I0
0I
t5
0
t/,
/
?
15
r
rp/S IC)
15
so ,o
20
/ 0
5
I0
15
0
, rp/S~ 5
I0
15
WP(MeV) Fig. 1. (a) Dependence of total and partial widths of the ground d5/2 IAR in 89y and 91Nb on the value used for WD in the proton optical potential. The crosshatched bars represent the experimental total widths, from ref. [4]. (b) Dependence of total partial widths of the ground d5/2 IAR in 93Tc and 93Nb on the magnitude of WD in the proton optical potential. The crosshatched bars r e present the experimental total widths, from refs. [8] and [9]. (c) Dependence of total and partial widths of the ground d5/2 IAR in 87Rb and 97Tc on the magnitude of WD in the proton optical potential. The crosshatched bars represent the experimental total widths from refs. [7] and [10].
165
Volume 40B, number 2
PHYSICS LETTERS
WD is i l l u s t r a t e d in fig. 1. It is a l s o s e e n that Fp is n e a r l y independent of WD. A s the table shows, t h e r e is good a g r e e m e n t b e t w e e n c a l c u lated and o b s e r v e d v a l u e s of Fp for all nuclei considered; calculated resonance phases are c o n s i s t e n t with those used in the data a n a l y s e s [4,7-10,13]. A p o s i t i v e phase of 0.1 r a d i a n is g e n e r a l l y [13] r e q u i r e d to fit IAR data, and all c a l c u l a t e d p h a s e s a r e of this o r d e r of magnitude. It is a l s o notable that F for 88Sr is o v e r e s t i m a t e d by 30%, e v e n with W D = 4 MeY. Since the unc e r t a i n t y in the e x p e r i m e n t a l total widths a r e probably ( c o n s e r v a t i v e l y ) 2 0 - 3 0 % - c o m p a r e , f o r i n s t a n c e , r e f s . [4] and [11] - the a g r e e m e n t is p e r h a p s g e n e r a l l y b e t t e r than one would anticipate. Had the G r e e n ' s function been used to c a l c u l a t e F, as in eq. (5.6) of ref. [5] one would have obtained, e.g., -300 keV for 90Zr. T h r e e e a r l i e r a t t e m p t s to c a l c u l a t e total widths w e r e confined to the lead region. The c a l c u l a t i o n s of Bund and B l a i r [1] o v e r e s t i m a t e d the total width of the analog of the ground s ta te of 208Pb by about a f a c t o r of 2, while the abs o r p t i o n widths f o r v a r i o u s IAR in 209Bi a r e o v e r e s t i m a t e d by f a c t o r s r a n g i n g f r o m 2 to 8 in the r e c e n t w o r k of A u e r b a c h et al. [14]. F i n a l l y , T a m u r a and B l ed s o e [5] using the G r e e n ' s function a p p r o x i m a t i o n to eqs. (3) and (4), a l s o o v e r e s t i m a t e the total width of the 0+ IAR in 208Bi by about a f a c t o r of 2.5. In s u m m a r y , we have p r e s e n t e d a s u c c e s s f u l c a l c u l a t i o n of both the total and p a r t i a l widths of IAR in 8 9 y and 91Nb, using an o p t ic a l p o t e n tial c o n s i s t e n t with that n e c e s s a r y in the e x p e r i m e n t a l a n a l y s i s with which our r e s u l t s a r e c o m p a r e d . Such c a l c u l a t i o n s have not, to our knowledge, b e e n as s u c c e s s f u l p r e v i o u s l y in any m a s s r e gi on. A c c e p t i n g the i n c r e a s e in WD as one m o v e s away f r o m m a s s 90, and the l e s s e r r e l i ability of the e x p e r i m e n t a l r e s o n a n c e p a r a m e t e r s , the a g r e e m e n t f o r Kr and Mo is a l s o i m p r e s s i v e . Since the p r e s e n t w o r k was c o m p l e t e d a l e t t e r has a p p e a r e d by S p en c e r and K e r m a n [15], in which the Lane equations a r e r e p o r t e d to be s o l v ed d i r e c t l y with a n o n - s t a n d a r d , m a n y - p a r a m e t e r optical potential, including a s p i n - s p i n t e r m and an l - d e p e n d e n t c e n t r a l potential. T o ta l and
166
26 June 1972
p a r t i a l widths and r e s o n a n c e e n e r g i e s a r e obtained f o r many ground state IAR in the A = 90 r e g i o n , including those f o r t a r g e t s 86Kr, 88Sr, 9 0 Z r and 92Mo. The total widths p r e d i c t e d a r e 14 keV for 92Mo, 9 0 Z r , 18 keV f o r 88Sr, and 24 keV for 86Kr.
References [1] G.W. Bund and J.S.B[air, Nuel. Phys. A l l 4 (1970) 384. [2] W.J. Thompson, J. L. Adams and D. Robson, Phys. Rev. 173 (1968) 975. [3] H. L. Harney and H. A. Weidenmuller, Nuci. Phys. A139 (1969) 241. [4] W. Kretschmer and G° Graw, Phys. Rev. Letters 27 (1971) 1294. [5] H. Bledsoe and T°Tamura, Nuc|. Phys. A164 (1971) 191. [6] F.D. Becchetti and G.W. Green[ees, Phys. Rev. 182 (1969) 1190. [7] C.L. Holias, H.R. Hidd[eston, V.D. Mistry, S. Sen and P . J . Riley, Phys. Rev., to be published° [8] G. Graw, in Polarization phenomena in nuclear reactions, eds. H. H. Barschall and W. Haeberli (Univ° of Wisconsin Press, Madison, 1971), p. 179-203. [9] J. L. Ellis and W. Haeber[i, in Nuclear isospin, eds. J. D. Anderson et al. (Academic Press, N.Y., 1969), p. 651-654. [10] C. F. Moore, P. Richard, C. Watson, D. Robson and J. Fox, Phys. Rev. 141 (1966) 1166. [11] G. C[ausnitzer, R. F[eisehmann, G. Graw and K. Wienhard, in Nuclear isospin, eds. J. D. Anderson e t a | . (Academic Press, N.Y., 1969), p. 629-633. [12] A. M° Lane, private communication; A. P. Jain, Nucl. Phys. 50 (1964) 157. [13] S. Darmodjo, R . Aiders, D. Martin, P. Dyer, S. All and S.A.A. Zaidi, Phys. Rev. C4 (1971) 672. [14] N. Auerbach, J. Hufner, A.K. Kerman and C.M. Shakin, Revs. Mod. Phys. 41 (1972) 48. [15] J. E. Spencer and A. K. Kerman, Phys. Letters 38B (1972) 289. [16] K. Haravu et al., Phys. Rev. C1 (1970) 938. [17] E. R. Cosman e t a [ . , Phys. Rev. 165 (1968) 1175. [18] B. L. Cohen and O. V. Chubinsky, Phys. Rev. 131 (1963) 2184. [19] J . B . Moorhead and R.A. Moyer, Phys. Rev. 184 (1969) 1205. [20] R.G. Clarkson and W. R. Coker, Phys. Rev. C2 (1970) 1108. [21] S. A. Hjorth and B. L. Cohen, Phys. Rev. 135 (1964) B920.
TOTAL
AND
PHYSICS
PARTIAL
WIDTHS
OF
LETTERS
ISOBARIC
26 June 1972
ANALOG
RESONANCES*
W. R. COKER, H. BLEDSOE ** and T. TAMURA Center for Nuclear Studies, University of Texas, Austin, Texas 78712, USA Received 3 May 1972
Total and partial widths and resonance mixing phases for dj/2 analog resonances in 87Rb, 89y, 91,93Nb ' 93,97Tc are calculated in the framework of a nonorth~o'gonal, optical-model basis in a shellmodel theory of reactions. The proton optical potential used is identical to that with which the experimental resonance parameters were extracted, for 89y and 91Nb where the most detailed and unambiguous experimental studies have been made.
Comparison of predictions of various microscopic and semi-microscopic theories of isobaric analog resonances (IAR) with experimental data has been somewhat inconclusive in the past because of ambiguity and approximation both in the calculations and the extraction of quantities for c o m p a r i s o n f r o m e x p e r i m e n t [1-3]. A r e c e n t study [4] of the 8 9 y and-91Nb ground state d5/2 IAR, in which the a n a l y z i n g power was used, together with the c r o s s s e c t i o n for e l a s t i c s c a t t e r i n g , to provide a f a i r l y m o d e l - i n d e p e n d e n t s e p a r a t i o n of d i r e c t and c o m p o u n d - e l a s t i c c r o s s s e c t i o n s , and thence r a t h e r r e l i a b l e optical and r e s o n a n c e p a r a m e t e r s , has p r o m p t e d us to c a r r y out detailed c a l c u l a t i o n s using the g e n e r a l i z e d Fano approach of Bledsoe and T a m u r a [5]. We a r e able to calculate all e n t r a n c e channel r e s o n a n c e p a r a m e t e r s : ER, F, F_, and ~bR, i.e., the r e s o nance e n e r g y , the t o t a l ~ i d t h , the p a r t i a l width, and the mixing phase [2,4]° The s e n s i t i v i t y of Fp to the magnitude and shape of the c h a r g e - e x c h a n g e potential Vl(r) has b e e n investigated by a n u m b e r of w o r k e r s [1,3,5]. The m o s t s a t i s f a c t o r y r e s u l t s have b e e n obtained with the Woods-Saxon d e r i v a t i v e f o r m for V1, and a s t r e n g t h about equal to that of the (volume) s y m m e t r y t e r m in the optical model potential. In o r d e r to have a unique and c o n s i s t e n t s t r e n g t h for the r e a l p a r t of V1, we use the definition Vp(ER) _ Vnc(Es) = ToRE(V1)
(I)
which is in the spirit of the Lane equations: Vp(ER) is the strength of the real part of the proton optical potential at E R, and VnC(Es) is * Supported in part by the U. S. Atomic Energy Commission. ** Present Address: Dept. of Physics, State Univ. of New York, StonyBrook, Long Island, New York 11790.
164
the depth of the potential binding the parent neutron at separation energy E s. Note that E R = Ac - Es, where Ac is the Coulomb displacement energy of a proton, and TO is the isospin of the target (or core) nucleus. As in ref. [5], the imaginary part of ToV1 is identical to the imaginary part of the proton optical model potential. Thus, once a proton optical model potential is given the resonance parameters are determined uniquely. We briefly summarize the relations used in the calculations reported here. If ~AX = (2To + 1)-1/2 <~bnIToVI - iWl~k (E))
(2)
where ~k(E) is an eigenstate of the proton optical model Hamiltonian, HO =K +V_ + iW+ Vc, P for proton channel ~, then the total width is given by F(E) = 2 ~
2
t~
Re(vAX) -
2
~ p ofo Im(vA,) t~
0
E -E'
dE', (3)
the level shift by . ~, ~ Re(u~/.t) A(E) =2-J P J ~ dE'-,~
im(ui2 '
(4)
and the partial width in channel k by FA ~exp(2i ~bR ) = 2~(u2tt),
(5)
where q5R is the resonance mixing phase. For a derivation and fuller discussion of these equations, consult ref. [5]. A numerically more convenient approximation to eqs. (3) and (4) was suggested in ref. [5], which amounts to replacing the energy integration by a radial integration involving the Green's function of Hp. Such an approximation neglects the discrete spectrum of Hp,
Volume 40B, number 2
PHYSICS LETTERS
26 June 1972
Table 1 Comparison of calculated and experimental partial and total widths of d5/2 IAR seen in proton elastic scattering on the targets listed. Widths are in keV. In column 5 is the spectroscopic factor S deduced from F~/S and Fexp; Sd_ • . ~. .P' ~J is the generally accepted spectroscopic factor from (d,p) experiments; F the predicted total wldth, using S; ~bR and qSR(exp) are the calculated and experimental resonance mixing phases in radians.
zs
Target
rp/S
F/s
Fe~ p
S
86Kr 88Sr 90Zr 92Mo 92Zr 96Mo
14.1 6.40 3.56 1.54 5.65 6.18
63.3 27.7 22.3 38.9 51.7 88.5
7.3 6.3 3.1 1.5 4.0 2.5
0.52 0..98 0.87 0.97 0.71 0.40
Sdp 0.56 0.79 0.89 0.84 0.55 0.42
[16] [17] [18] [19] [18,20] [21]
F
Fex p
q5R
32.9 27.1 19.4 37.7 36.7 35.4
36.0 19.0 17.0 30.0 30.0 33.0
0.17 0..12 0.11 0.08 0.11 0.13
qbR(exp) 0.15 0.01 ±0.05 0.05 ±0.05 0.17 0.0 ±0.1 0.0 ±0.1
and s u r p r i s i n g l y is found n u m e r i c a l l y to be e x c e e din g l y poor. We have t h e r e f o r e e v a lu a t e d the e n e r g y i n t e g r a l s d i r e c t l y . The c o n t r i b u t i o n of ine l a s t i c p r o t o n c h a n n e l s to F has been neglected, s i n c e for al l c a s e s c o n s i d e r e d it is s m a l l cornp a r e d to the e x p e r i m e n t a l e r r o r in F. The proton o p t i cal p o t e n ti a l used in the anal y s i s of K r e t s c h m e r and Graw [4] is that of B e c c h e t t i and G r e e n l e e s [6], e x c e p t that K r e t s c h m e r and Graw r e d u c e the s t r e n g t h of WD to 4.0 MeV to c o m p e n s a t e fo r the e x t r a c t i o n of the cornpound e l a s t i c c o n t r i b u t i o n f r o m the (p,p) e x c i t a tion c u r v e . A s i m i l a r r e d u c t i o n of WD was made in p r e v i o u s w o r k [2] for the s a m e r e a s o n . We have adopted the potential of ref. [4] for p + 88Sr, and p + 90Zr. The r e s u l t s of our c a l c u l a t i o n s a r e s u m m a r i z e d in table 1. We have a l s o p e r f o r m e d c a l c u l a t i o n s for the d5/2 IAR o b s e r v e d in p r o t o n e l a s t i c s c a t t e r i n g on 86Kr [7] 92Mo [8], 9 2 Z r [9], and 96Mo [10]. The b e s t a g r e e m e n t with F e px was obtained for 86Kr, 92Mo and 9 2 Z r with WD ~ 6-7 MeV; the table g i v e s the p a r a m e t e r s c a l c u l a t e d with WD s e t to 8 MeV. Finally, fo r 96Mo, the b e s t a g r e e m e n t was obtained with WD at its s t a n d a r d value of 12 MeV. Such a v a r i a t i o n in WD is p a r t i a l l y u n d e r standable ff one c o n s i d e r s the magnitude of the e x p e c t e d compound e l a s t i c c o n t r i b u t i o n to the c r o s s section. The (p, n) channel is well open for 86Kr, 9 2 Z r and 96Mo. It is c l o s e d for 9 0 Z r and 92Mo, and b a r e l y open for 88Sr. Thus the only p e r p l e x i n g r e s u l t is that F for 92Mo cannot a l s o be fit with WD = 4 MeV. H o w e v e r , in the a n a l y s i s of 92Mo(p, Pc) a n a l y z i n g p o w e r data r e p o r t e d in ref. [8] the total widths of the f i r s t four IAR in 93Tc w e r e all taken as 30 keV, and a s e a r c h was made only on Fp and ~R" It is l i k e l y that an e q u a l ly good fit to the ground s t a t e IAR could have been obtained with F ~ 20 keV (and thus WD ~ 4 MeV), as in the c a s e s of 9 0 Z r and 88Sr. Neutron s t r e n g t h function data a l s o f a v o r s the l o w e r value of WD f or 88Sr, 9 0 Z r , 92Mo [4,12]. The s t r o n g , a l m o s t l i n e a r dependence of F on
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WP(MeV) Fig. 1. (a) Dependence of total and partial widths of the ground d5/2 IAR in 89y and 91Nb on the value used for WD in the proton optical potential. The crosshatched bars represent the experimental total widths, from ref. [4]. (b) Dependence of total partial widths of the ground d5/2 IAR in 93Tc and 93Nb on the magnitude of WD in the proton optical potential. The crosshatched bars r e present the experimental total widths, from refs. [8] and [9]. (c) Dependence of total and partial widths of the ground d5/2 IAR in 87Rb and 97Tc on the magnitude of WD in the proton optical potential. The crosshatched bars represent the experimental total widths from refs. [7] and [10].
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Volume 40B, number 2
PHYSICS LETTERS
WD is i l l u s t r a t e d in fig. 1. It is a l s o s e e n that Fp is n e a r l y independent of WD. A s the table shows, t h e r e is good a g r e e m e n t b e t w e e n c a l c u lated and o b s e r v e d v a l u e s of Fp for all nuclei considered; calculated resonance phases are c o n s i s t e n t with those used in the data a n a l y s e s [4,7-10,13]. A p o s i t i v e phase of 0.1 r a d i a n is g e n e r a l l y [13] r e q u i r e d to fit IAR data, and all c a l c u l a t e d p h a s e s a r e of this o r d e r of magnitude. It is a l s o notable that F for 88Sr is o v e r e s t i m a t e d by 30%, e v e n with W D = 4 MeY. Since the unc e r t a i n t y in the e x p e r i m e n t a l total widths a r e probably ( c o n s e r v a t i v e l y ) 2 0 - 3 0 % - c o m p a r e , f o r i n s t a n c e , r e f s . [4] and [11] - the a g r e e m e n t is p e r h a p s g e n e r a l l y b e t t e r than one would anticipate. Had the G r e e n ' s function been used to c a l c u l a t e F, as in eq. (5.6) of ref. [5] one would have obtained, e.g., -300 keV for 90Zr. T h r e e e a r l i e r a t t e m p t s to c a l c u l a t e total widths w e r e confined to the lead region. The c a l c u l a t i o n s of Bund and B l a i r [1] o v e r e s t i m a t e d the total width of the analog of the ground s ta te of 208Pb by about a f a c t o r of 2, while the abs o r p t i o n widths f o r v a r i o u s IAR in 209Bi a r e o v e r e s t i m a t e d by f a c t o r s r a n g i n g f r o m 2 to 8 in the r e c e n t w o r k of A u e r b a c h et al. [14]. F i n a l l y , T a m u r a and B l ed s o e [5] using the G r e e n ' s function a p p r o x i m a t i o n to eqs. (3) and (4), a l s o o v e r e s t i m a t e the total width of the 0+ IAR in 208Bi by about a f a c t o r of 2.5. In s u m m a r y , we have p r e s e n t e d a s u c c e s s f u l c a l c u l a t i o n of both the total and p a r t i a l widths of IAR in 8 9 y and 91Nb, using an o p t ic a l p o t e n tial c o n s i s t e n t with that n e c e s s a r y in the e x p e r i m e n t a l a n a l y s i s with which our r e s u l t s a r e c o m p a r e d . Such c a l c u l a t i o n s have not, to our knowledge, b e e n as s u c c e s s f u l p r e v i o u s l y in any m a s s r e gi on. A c c e p t i n g the i n c r e a s e in WD as one m o v e s away f r o m m a s s 90, and the l e s s e r r e l i ability of the e x p e r i m e n t a l r e s o n a n c e p a r a m e t e r s , the a g r e e m e n t f o r Kr and Mo is a l s o i m p r e s s i v e . Since the p r e s e n t w o r k was c o m p l e t e d a l e t t e r has a p p e a r e d by S p en c e r and K e r m a n [15], in which the Lane equations a r e r e p o r t e d to be s o l v ed d i r e c t l y with a n o n - s t a n d a r d , m a n y - p a r a m e t e r optical potential, including a s p i n - s p i n t e r m and an l - d e p e n d e n t c e n t r a l potential. T o ta l and
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26 June 1972
p a r t i a l widths and r e s o n a n c e e n e r g i e s a r e obtained f o r many ground state IAR in the A = 90 r e g i o n , including those f o r t a r g e t s 86Kr, 88Sr, 9 0 Z r and 92Mo. The total widths p r e d i c t e d a r e 14 keV for 92Mo, 9 0 Z r , 18 keV f o r 88Sr, and 24 keV for 86Kr.
References [1] G.W. Bund and J.S.B[air, Nuel. Phys. A l l 4 (1970) 384. [2] W.J. Thompson, J. L. Adams and D. Robson, Phys. Rev. 173 (1968) 975. [3] H. L. Harney and H. A. Weidenmuller, Nuci. Phys. A139 (1969) 241. [4] W. Kretschmer and G° Graw, Phys. Rev. Letters 27 (1971) 1294. [5] H. Bledsoe and T°Tamura, Nuc|. Phys. A164 (1971) 191. [6] F.D. Becchetti and G.W. Green[ees, Phys. Rev. 182 (1969) 1190. [7] C.L. Holias, H.R. Hidd[eston, V.D. Mistry, S. Sen and P . J . Riley, Phys. Rev., to be published° [8] G. Graw, in Polarization phenomena in nuclear reactions, eds. H. H. Barschall and W. Haeberli (Univ° of Wisconsin Press, Madison, 1971), p. 179-203. [9] J. L. Ellis and W. Haeber[i, in Nuclear isospin, eds. J. D. Anderson et al. (Academic Press, N.Y., 1969), p. 651-654. [10] C. F. Moore, P. Richard, C. Watson, D. Robson and J. Fox, Phys. Rev. 141 (1966) 1166. [11] G. C[ausnitzer, R. F[eisehmann, G. Graw and K. Wienhard, in Nuclear isospin, eds. J. D. Anderson e t a | . (Academic Press, N.Y., 1969), p. 629-633. [12] A. M° Lane, private communication; A. P. Jain, Nucl. Phys. 50 (1964) 157. [13] S. Darmodjo, R . Aiders, D. Martin, P. Dyer, S. All and S.A.A. Zaidi, Phys. Rev. C4 (1971) 672. [14] N. Auerbach, J. Hufner, A.K. Kerman and C.M. Shakin, Revs. Mod. Phys. 41 (1972) 48. [15] J. E. Spencer and A. K. Kerman, Phys. Letters 38B (1972) 289. [16] K. Haravu et al., Phys. Rev. C1 (1970) 938. [17] E. R. Cosman e t a [ . , Phys. Rev. 165 (1968) 1175. [18] B. L. Cohen and O. V. Chubinsky, Phys. Rev. 131 (1963) 2184. [19] J . B . Moorhead and R.A. Moyer, Phys. Rev. 184 (1969) 1205. [20] R.G. Clarkson and W. R. Coker, Phys. Rev. C2 (1970) 1108. [21] S. A. Hjorth and B. L. Cohen, Phys. Rev. 135 (1964) B920.