- Email: [email protected]
Vertex constants and the problem of the nucleon-nucleon potential in the generator coordinate method
Physics Letters B 293 ( 1992 ) 13-17 North-Holland
PHYSICS LETTERS B
Vertex constants and the problem of the nucleon-nucleon potential in the generator coordinate method D. Baye a n d N.K. T i m o f e y u k Physique NuclOaire ThOorique et Physique MathOmatique, Universitk Libre de Bruxelles, C.P. 229, B-1050 Brussels, Belgium Received 26 June 1992; revised manuscript received 20 August 1992
The influence of the nucleon-nucleon potential on the value of the asymptotic normalization coefficient is investigated in the generator coordinate method. In the simple case of ~70--,160 + n, all the potentials considered give overestimated values of the vertex constant.
The generator coordinate method ( G C M ) which is equivalent to the resonating-group method ( R G M ) [ 1,2 ] is a powerful tool for calculating properties of light nuclei such as binding energies, RMS radii or transition probabilities, as well as for investigating different nuclear reactions between light ions [ 3 ]. This method is essentially free from adjustable parameters since the only input is the nucleon-nucleon ( N N ) potential, but the choice of this potential may introduce some uncertainty in the model. However a nucleus possesses another type of fundamental characteristic: the amplitude of virtual decay into two clusters. The on-shell value of this amplitude is called vertex constant and is equivalent to the coupling constant in particle physics. It is related with the asymptotic normalization coefficient in the wave function of the relative motion between the clusters [4]. The GCM is able to reproduce correctly the binding energy of two clusters and the asymptotic shape of their relative wave function. Checking its ability to provide realistic vertex constants is very important since they play a crucial role in different surface processes such as transfer reactions and low-energy radiative capture. For Is and lp shell nuclei, vertex constants are studied with shell-model wave functions in refs. [ 5-7 ]. In these works, one of the most popular NN potentials employed in GCM calculaPermanent address: Institute of Nuclear Physics, Ulugbek, 702132 Tashkent, Uzbekistan.
tions, the Volkov potential V2 [8 ], systematically overestimates the vertex constants. In this paper, we calculate with the GCM the constant for the 170--, 1 6 0 + n vertex as well as the overlap integral between the 170 and ~60 wave functions which enters into distorted-wave amplitudes [ 9 ]. We choose 170 for two reasons. First, it is well represented as an 160 cluster plus one neutron and the GCM with a single channel is thus justified. Second, an experimental value is known with good accuracy from the Coulomb-nuclear interference in the subbarrier 160 + 170 elastic scattering [ 10 ]. In R G M notations [2 ], the normalized wave function of the 160 + nucleon system can be written for angular momentum J as [ 11 ]
~JM= 17~/2~¢~oSMg~(p),
( 1)
where d is the antisymmetrization projector and p = (p, g2p) is the relative coordinate. The channel function ~o/Mis [ Yl(g2p)®ON]JMo0where 0Nthe spinisospin state of the nucleon and ~o is the 160 groundstate internal wave function. The unknown relative function gJ(p) is determined from the 17-nucleon microscopic hamiltonian. For the ground state, the orbital momentum l of the relative motion is 2 while J is 2. Let q be the Sommerfeld parameter of the bound state and x = (21tEp/h2) 1/2 its wavenumber, where/t is the reduced mass and EB is the binding energy of 1 6 0 + N . At large distances, the relative function g~[(p) takes the asymptotic form
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
13
Volume 293, number 1,2
PHYSICSLETTERSB
gf(p) ,"~CfW_,a+l/2(2tcp)/p ,
(2)
where W is a Whittaker function and Cf is the asymptotic normalization coefficient. As is well known, g{ is not a wave function for the relative motion. Indeed, g: may contain arbitrary components of forbidden states (which do not appear here for l = 2 ) and does not satisfy orthogonality properties (see refs. [2,12] and references therein). A closer approximation to a relative wave function is Rf = JV')/2g J where ~ is the RGM overlap operator which does not depend here on J. Since its exchange part is short-ranged. Xtt is close to unity at large distances and gS and ~f share the same asymptotic behaviour (2). However, ~] is uniquely defined and satisfies the orthogonality properties of a wave function. The coefficient Cf also determines the asymptotic properties of the overlap integral
If(r) = 171/2r-2(~Uf(p--r) l ~SM) .
(3)
Hence, Its =~A~tgt J = X t 1/2g~ ~s is also uniquely defined and shares the asymptotic behaviour (2). The coefficient Cf is connected with the vertex constant Gf for the 160 + N virtual decay by eq. (98) of ref. [ 4 ] as
Gf = -nl/2(h/llc) exp[ l i n ( l + q ) ] Cf .
(4)
When N is a neutron, Gf can be rewritten with eq. (87) ofref. [4] as
G f = (4n)l/2it(hc) -1 (itl
~flg{;> ,
(5)
where ~ f is the RGM (direct and exchange) potential operator [2], and it is a spherical Hankel function. If the wave functions of 160 and 160 + n are consistent with the potential, (4) and (5) are equivalent. Eq. (5) allows to calculate the vertex constant from wave functions which do not have a correct asymptotic behaviour, because -/rf is short-ranged. This property is employed in refs. [ 6,7 ] where shell-model wave functions play the role of ~s~t. However, (5) is not particularly well adapted to the GCM because it requires integral transforms of the complicated potential kernel. The 170 wave function ( 1 ) is approximated in the GCM by the finite linear combination
~SM= Z f f(R,)cbfM(R,) , n
14
(6)
22 October 1992
where • JM(Rn) is a projected Slater determinant defined in the two-centre harmonic-oscillator shell model with parameter 1.8 fm, and the coefficients f f ( R n ) are obtained numerically from a Hill-Wheeler equation [ 11,3 ]. A good accuracy is reached with 10 values Rn of the generator coordinate (from 0.8 fm to 8 fm). The overlap integral If(p) is then easily obtained from thefJ(Rn) by a Fourier transform technique [ 1 ]. The relative function gJ(p) forms the direct part of IS(p). The exchange part ISex (p) contributes mainly at small p. The Cf are obtained by comparing If(p) with its asymptotic form (2), which is reached between 6 and 7 fm. The following NN potentials are employed in our calculations. Some parameter o f each potential is modified in order to get the correct neutron separation energy 4.15 MeV in 170. ( 1 ) Different potentials of Volkov [ 8 ] reproduce the binding energies of 4He and 160 and give s-wave scattering lengths and effective ranges not too far from the singlet and triplet ones. (2) The potential of Hasegawa, Nagata and Yamamoto [ 13 ] reproduces the a and deuteron binding energies and has a realistic behaviour for the odd-state interaction. Two variants are considered. In HNY, the triplet-even strength of the gaussian with fl= 1.127 fm is modified by A, while in MHN, the Majorana parameter of the same gaussian is increased by Am. (3) The three-gaussian Minnesota potential (MN3) [ 14] gives correct n-p triplet and p-p singlet s-wave range parameters as well as fair n-p 3S1 and p-p 1So phases up to about 150 MeV. It reproduces the binding energy and RMS radius of the ~t particle. (4) The potential of Gogny et al. (GPT) [ 15 ] reproduces two-nucleon data, charge RMS radii of closed-shell nuclei in the Hartree-Fock approximation and gives reasonable saturation properties of nuclear matter but it underestimates the 160 binding energy. We increase by ,4 (i.e. about 25%) the strength of the triplet-even gaussian with fl= 1.418 fm. With all these NN potentials, the vertex constants I Gf 12 displayed in table 1 are larger than the experimental value IGexpl2=0.109+0.006 fm [10]. The overestimation reaches about 50% with V2. The lowest value I G J 12= 0.127 fm obtained with HNY and MHN is still too large by about 17%. Notice that the value 1.8 fm for the oscillator parameter, which pro-
Volume 293, number 1,2
PHYSICS LETTERS B
22 October 1992
Table 1 Asymptotic normalization coefficient C~, squared modulus of the vertex constant G[, matter RMS radius (r 2) 1/2 of 170, RMS radius (p2) 1/2of the overlap integral l'[(p) and spectroscopic factor S, calculated in the GCM with different NN potentials. NN potential V1 V2 V3 V4 HNY MHN MN3 GPT
m = 0.6025 m=0.6193 m = 0.5934 m=0.5821 d= 19 MeV Am = - 0.109 u=0.930 d=25 MeV
V2(b= 1.7) m=0.5879 V2(b= 1.9) m=0.5726
C[ (fm)
IG]I 2 (fm)
(
r 2) 1/2 (fm)
(,02) 1/2 (fm)
S
0.983 1.00 0.944 0.912 0.893 0.892 0.901 0.979
0.154 0.161 0.142 0.132 0.127 0.127 0.129 0.152
2.698 2.702 2.695 2.695 2.693 2.693 2.727 2.698
3.58 6.61 3.52 3.50 3.48 3.48 3.48 3.57
1.121 1.121 1.122 1.122 1.122 1.122 1.122 1.121
0.937 1.12
0.140 0.201
2.480 3.009
3.51 3.73
1.120 1.120
vides a realistic 160 radius, does not correspond to saturation. When the 160 energy is minimized, the different potentials underestimate the 160 radius, with oscillator parameters ranging from 1.28 to 1.50 fm. This drawback might be related to the overestimation of the vertex constant. Indeed as shown for V2 in table 1, the vertex constant is rather sensitive to the oscillator parameter. With b = 1.7 fm, the overestimation is already significantly reduced. However, we cannot employ the oscillator parameter corresponding to saturation because the vertex constant would describe neutron separation from a contracted 160 core. The neutron orbital would be unphysically distorted. A good effective potential providing the correct 160 radius at saturation is not available. All the potentials give approximately the same 170 matter RMS radius 2.70 fm in good agreement with the experimental charge RMS radius 2.7 I0 + 0.015 fm [16]. The RMS radius o f the overlap integral I](p) is more sensitive to potential choice and is larger than the experimental value /,)2,, NY / e x1/2 p = 3.45 _+0.03 fm obtained from magnetic electron scattering [ 10] (we give this value multiplied by the correction factor ( 17/16 ) 1/2 ~ 1.031 arising when the overlap integral is calculated as a function of the relative distance between 160 and n). A correlation can be seen in fig. 1 between the [G][ 2 values and (/2 2) I / 2 . A linear fit crosses the experimental rectangle defined by [ Gexp ]2 a n d / ,\ V, 2 . ,l e x1/2 p • Additional configurations might reduce the disagreement between theoretical and experimental ver-
tex constants. For V2, coupling 160+n with 13C+a reduces C] by about 1%. Studying configurations with an excited 160 core is beyond the scope of the present model. However, let us notice that shell-model calculations with V2 also overestimate the vertex constants of p-shell nuclei [6 ]. In our opinion, the overestimation is more a potential problem than a consequence o f our restricted configuration space. The conventional spectroscopic factor S [ 4 ] reads oo
S= [. dpp2I'/(p) 2 0 = E ['12n+l(gd[Unl)2 ' n
(7)
since the oscillator states u,~ are the eigenstates o f Yr. The eigenvalues/t~¢ of ~ are given b y / z 0 = # ~ = 0 , /t2= ( 1 7 / 1 6 ) 2, # 3 = 1 - 5 0 / 1 6 3 .... [ 1 ]. Since ~12 is very close to Uo2 (except in its asymptotic part), S is slightly smaller than the shell-model value #2 ~ 1.129 and is almost independent o f the potential choice. This reflects the fact that we have only one configuration in the 170 wave function. The experimental value o f Sis 1.06 + 0.11 [ 10]. The main contribution comes from the direct part I'/air(r) o f the overlap integral. It is approximately equal to 79% for each potential. The overlap integrals I[ calculated with different N N potentials differ from each other at their maxim u m by less than 12%. For the H N Y potential, I[ is presented in fig. 2. It is almost equal to the empirical single-particle wave function found from magnetic 15
Volume 293, number 1,2
PHYSICSLETTERSB
22 October 1992
V2(b=1.9)~ 0.20
N
v2
0.15 V2(b=l.7~
HNY 0.10
I
I
I
I
I
I
I
I
3.45
3.50
3.55
3.60
3.65
3.70
3.75
3.80
Fig. 1. Linear fit of the vertex constants I G]I 2 calculated with different NN potentials as a function of the RMS radii 1/2 of the corresponding overlap integrals I'[(p). The rectangle represents experimental data [ 10].
HNYI~ 0.2 ? E 0.1
2
4
6
8
Fig. 2. Comparison of the overlap integral I~ calculated with the HNY potential, and of the single-particle wave function derived from magnetic electron scattering [ 10] (multiplied by the square root of the experimental spectroscopic factor). The direct and exchange parts o f l ] are also displayed. At the scale of the figure, 1s exchange is negligible.
electron scattering [ 10] (multiplied by the experimental value of S 1/2) in the internal region but differs in the asymptotic region. The exchange contribution is negligible for I s nucleons while it reaches 10% in the nuclear interior for lp nucleons. 16
The inaccuracy of a vertex constant may have important consequences. In processes where nuclei exchange nucleons at large distances, the cross section is mainly sensitive to the tail of the nucleon boundstate wave function and therefore to the asymptotic normalization coefficient. When the internal contribution is suppressed for some reason, for example at sub-barrier energies or in the presence of strong absorption in heavy-ion induced reactions, the cross section is proportional to the squared modulus of the vertex constant. It will be directly affected by any error on this constant. For charged particles, the Coulomb repulsion cuts off the contribution from the internal region to lowenergy radiative capture. This effect is expecially important for reactions leading to weakly-bound nuclei such as ct(3He, y)TBe and 7Be(p, 7)SB at energies close to zero. For the ~(3He, 7)VBe reaction, the dependence of the astrophysical S-factor at zero energy on the NN potential has been investigated with the R G M by Kajino [ 17 ] who recommends the M H N potential. For VBe(p, 7)SB, V2 has been employed to calculate S ( 0 ) in a GCM calculation [ 18]. The model of refs. [6,7 ] suggests that the GCM S ( 0 ) might be significantly overestimated. However, the GCM de-
Volume 293, number 1,2
PHYSICS LETTERS B
scription in ref. [ 18 ] is based on a three-cluster model a n d the saturation c o n d i t i o n of the ct a n d 3He clusters is much better satisfied than for 160. The overestim a t i o n of the vertex constant might therefore be weaker in the G C M than in refs. [5,6] where shellmodel wave functions are employed which are not consistent with the N N interaction. Nevertheless, the choice of the N N potential in future G C M calculations of such reactions deserves further attention. N.K.T. would like to t h a n k Professor C. LeclercqWillain and the Universit6 Libre de Bruxelles for their hospitality. This work is supported in part by a PSle d'Attraction Interuniversitaire of the Services de P r o g r a m m a t i o n de la Politique Scientifique.
References [ 1] H. Horiuchi, Prog. Theor. Phys. Suppl. 62 (1977) 90. [2] Y.C. Tang, in: Topics in nuclear physics II, Lecture Notes in Physics, Vol. 145 (Springer, Berlin, 1981 ) p. 572. [ 3 ] D. Baye and P. Descouvemont, Proc. 5th Intern. Conf. on Clustering aspects in nuclear and suhnuclear systems (Kyoto, Japan, 1988), J. Phys. Soc. Jpn. Suppl. 58 (t989) 103.
22 October 1992
[4] L.D. Blokhintsev,I. Borbelyand E.I. Dolinskii, Sov. J. Part. Nucl. 8 (1977) 485. [5]L.D. Blokhintsev, A.M. Mukhamedzhanov and N.K. Timofeyuk, Ukr. J. Phys. 35 (1990) 341. [6] L.D. Blokhintsevet al., Izv. Akad. Nauk SSSR, Ser. Fiz. 54 (1990) 569. [7 ] A.M. Mukhamedzhanovand N.K. Timofeyuk,Sov. J. Nucl. Phys. 51 (1990) 431. [8] A.B. Volkov,Nucl. Phys. 74 (1965) 33. [9] N. Austern, Direct nuclear reaction theories (Wiley, New York, 1970). [ 10] S. Burzynskiet al., Nucl. Phys. A 399 (1983) 230. [ 11] D. Baye and P. Descouvemont, Nucl. Phys. A 407 (1983) 77. [12]D. Baye and P. Descouvemont, Ann. Phys. (NY) 165 (1985) 115. [13] A. Hasegawa and S. Nagata, Prog. Theor. Phys. 45 (1971) 1786; Y. Yamamoto, Prog. Theor. Phys. 52 (1974) 471. [ 14] D.R. Thompson, M. LeMere and Y.C. Tang, Nucl. Phys. A 286 (1977) 53. [ 15] D. Gogny, P. Pires and R. De Tourreil, Phys. Lett. B 32 (1970) 591. [ 16] H. Miska et al., Phys. Lett B 83 (1979) 165. [ 17] T. Kajino, Nucl. Phys. A 460 (1986) 559. [18 ] P. Descouvemont and D. Baye, Nucl. Phys. A 487 (1988) 420.
17
PHYSICS LETTERS B
Vertex constants and the problem of the nucleon-nucleon potential in the generator coordinate method D. Baye a n d N.K. T i m o f e y u k Physique NuclOaire ThOorique et Physique MathOmatique, Universitk Libre de Bruxelles, C.P. 229, B-1050 Brussels, Belgium Received 26 June 1992; revised manuscript received 20 August 1992
The influence of the nucleon-nucleon potential on the value of the asymptotic normalization coefficient is investigated in the generator coordinate method. In the simple case of ~70--,160 + n, all the potentials considered give overestimated values of the vertex constant.
The generator coordinate method ( G C M ) which is equivalent to the resonating-group method ( R G M ) [ 1,2 ] is a powerful tool for calculating properties of light nuclei such as binding energies, RMS radii or transition probabilities, as well as for investigating different nuclear reactions between light ions [ 3 ]. This method is essentially free from adjustable parameters since the only input is the nucleon-nucleon ( N N ) potential, but the choice of this potential may introduce some uncertainty in the model. However a nucleus possesses another type of fundamental characteristic: the amplitude of virtual decay into two clusters. The on-shell value of this amplitude is called vertex constant and is equivalent to the coupling constant in particle physics. It is related with the asymptotic normalization coefficient in the wave function of the relative motion between the clusters [4]. The GCM is able to reproduce correctly the binding energy of two clusters and the asymptotic shape of their relative wave function. Checking its ability to provide realistic vertex constants is very important since they play a crucial role in different surface processes such as transfer reactions and low-energy radiative capture. For Is and lp shell nuclei, vertex constants are studied with shell-model wave functions in refs. [ 5-7 ]. In these works, one of the most popular NN potentials employed in GCM calculaPermanent address: Institute of Nuclear Physics, Ulugbek, 702132 Tashkent, Uzbekistan.
tions, the Volkov potential V2 [8 ], systematically overestimates the vertex constants. In this paper, we calculate with the GCM the constant for the 170--, 1 6 0 + n vertex as well as the overlap integral between the 170 and ~60 wave functions which enters into distorted-wave amplitudes [ 9 ]. We choose 170 for two reasons. First, it is well represented as an 160 cluster plus one neutron and the GCM with a single channel is thus justified. Second, an experimental value is known with good accuracy from the Coulomb-nuclear interference in the subbarrier 160 + 170 elastic scattering [ 10 ]. In R G M notations [2 ], the normalized wave function of the 160 + nucleon system can be written for angular momentum J as [ 11 ]
~JM= 17~/2~¢~oSMg~(p),
( 1)
where d is the antisymmetrization projector and p = (p, g2p) is the relative coordinate. The channel function ~o/Mis [ Yl(g2p)®ON]JMo0where 0Nthe spinisospin state of the nucleon and ~o is the 160 groundstate internal wave function. The unknown relative function gJ(p) is determined from the 17-nucleon microscopic hamiltonian. For the ground state, the orbital momentum l of the relative motion is 2 while J is 2. Let q be the Sommerfeld parameter of the bound state and x = (21tEp/h2) 1/2 its wavenumber, where/t is the reduced mass and EB is the binding energy of 1 6 0 + N . At large distances, the relative function g~[(p) takes the asymptotic form
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
13
Volume 293, number 1,2
PHYSICSLETTERSB
gf(p) ,"~CfW_,a+l/2(2tcp)/p ,
(2)
where W is a Whittaker function and Cf is the asymptotic normalization coefficient. As is well known, g{ is not a wave function for the relative motion. Indeed, g: may contain arbitrary components of forbidden states (which do not appear here for l = 2 ) and does not satisfy orthogonality properties (see refs. [2,12] and references therein). A closer approximation to a relative wave function is Rf = JV')/2g J where ~ is the RGM overlap operator which does not depend here on J. Since its exchange part is short-ranged. Xtt is close to unity at large distances and gS and ~f share the same asymptotic behaviour (2). However, ~] is uniquely defined and satisfies the orthogonality properties of a wave function. The coefficient Cf also determines the asymptotic properties of the overlap integral
If(r) = 171/2r-2(~Uf(p--r) l ~SM) .
(3)
Hence, Its =~A~tgt J = X t 1/2g~ ~s is also uniquely defined and shares the asymptotic behaviour (2). The coefficient Cf is connected with the vertex constant Gf for the 160 + N virtual decay by eq. (98) of ref. [ 4 ] as
Gf = -nl/2(h/llc) exp[ l i n ( l + q ) ] Cf .
(4)
When N is a neutron, Gf can be rewritten with eq. (87) ofref. [4] as
G f = (4n)l/2it(hc) -1 (itl
~flg{;> ,
(5)
where ~ f is the RGM (direct and exchange) potential operator [2], and it is a spherical Hankel function. If the wave functions of 160 and 160 + n are consistent with the potential, (4) and (5) are equivalent. Eq. (5) allows to calculate the vertex constant from wave functions which do not have a correct asymptotic behaviour, because -/rf is short-ranged. This property is employed in refs. [ 6,7 ] where shell-model wave functions play the role of ~s~t. However, (5) is not particularly well adapted to the GCM because it requires integral transforms of the complicated potential kernel. The 170 wave function ( 1 ) is approximated in the GCM by the finite linear combination
~SM= Z f f(R,)cbfM(R,) , n
14
(6)
22 October 1992
where • JM(Rn) is a projected Slater determinant defined in the two-centre harmonic-oscillator shell model with parameter 1.8 fm, and the coefficients f f ( R n ) are obtained numerically from a Hill-Wheeler equation [ 11,3 ]. A good accuracy is reached with 10 values Rn of the generator coordinate (from 0.8 fm to 8 fm). The overlap integral If(p) is then easily obtained from thefJ(Rn) by a Fourier transform technique [ 1 ]. The relative function gJ(p) forms the direct part of IS(p). The exchange part ISex (p) contributes mainly at small p. The Cf are obtained by comparing If(p) with its asymptotic form (2), which is reached between 6 and 7 fm. The following NN potentials are employed in our calculations. Some parameter o f each potential is modified in order to get the correct neutron separation energy 4.15 MeV in 170. ( 1 ) Different potentials of Volkov [ 8 ] reproduce the binding energies of 4He and 160 and give s-wave scattering lengths and effective ranges not too far from the singlet and triplet ones. (2) The potential of Hasegawa, Nagata and Yamamoto [ 13 ] reproduces the a and deuteron binding energies and has a realistic behaviour for the odd-state interaction. Two variants are considered. In HNY, the triplet-even strength of the gaussian with fl= 1.127 fm is modified by A, while in MHN, the Majorana parameter of the same gaussian is increased by Am. (3) The three-gaussian Minnesota potential (MN3) [ 14] gives correct n-p triplet and p-p singlet s-wave range parameters as well as fair n-p 3S1 and p-p 1So phases up to about 150 MeV. It reproduces the binding energy and RMS radius of the ~t particle. (4) The potential of Gogny et al. (GPT) [ 15 ] reproduces two-nucleon data, charge RMS radii of closed-shell nuclei in the Hartree-Fock approximation and gives reasonable saturation properties of nuclear matter but it underestimates the 160 binding energy. We increase by ,4 (i.e. about 25%) the strength of the triplet-even gaussian with fl= 1.418 fm. With all these NN potentials, the vertex constants I Gf 12 displayed in table 1 are larger than the experimental value IGexpl2=0.109+0.006 fm [10]. The overestimation reaches about 50% with V2. The lowest value I G J 12= 0.127 fm obtained with HNY and MHN is still too large by about 17%. Notice that the value 1.8 fm for the oscillator parameter, which pro-
Volume 293, number 1,2
PHYSICS LETTERS B
22 October 1992
Table 1 Asymptotic normalization coefficient C~, squared modulus of the vertex constant G[, matter RMS radius (r 2) 1/2 of 170, RMS radius (p2) 1/2of the overlap integral l'[(p) and spectroscopic factor S, calculated in the GCM with different NN potentials. NN potential V1 V2 V3 V4 HNY MHN MN3 GPT
m = 0.6025 m=0.6193 m = 0.5934 m=0.5821 d= 19 MeV Am = - 0.109 u=0.930 d=25 MeV
V2(b= 1.7) m=0.5879 V2(b= 1.9) m=0.5726
C[ (fm)
IG]I 2 (fm)
(
r 2) 1/2 (fm)
(,02) 1/2 (fm)
S
0.983 1.00 0.944 0.912 0.893 0.892 0.901 0.979
0.154 0.161 0.142 0.132 0.127 0.127 0.129 0.152
2.698 2.702 2.695 2.695 2.693 2.693 2.727 2.698
3.58 6.61 3.52 3.50 3.48 3.48 3.48 3.57
1.121 1.121 1.122 1.122 1.122 1.122 1.122 1.121
0.937 1.12
0.140 0.201
2.480 3.009
3.51 3.73
1.120 1.120
vides a realistic 160 radius, does not correspond to saturation. When the 160 energy is minimized, the different potentials underestimate the 160 radius, with oscillator parameters ranging from 1.28 to 1.50 fm. This drawback might be related to the overestimation of the vertex constant. Indeed as shown for V2 in table 1, the vertex constant is rather sensitive to the oscillator parameter. With b = 1.7 fm, the overestimation is already significantly reduced. However, we cannot employ the oscillator parameter corresponding to saturation because the vertex constant would describe neutron separation from a contracted 160 core. The neutron orbital would be unphysically distorted. A good effective potential providing the correct 160 radius at saturation is not available. All the potentials give approximately the same 170 matter RMS radius 2.70 fm in good agreement with the experimental charge RMS radius 2.7 I0 + 0.015 fm [16]. The RMS radius o f the overlap integral I](p) is more sensitive to potential choice and is larger than the experimental value /,)2,, NY / e x1/2 p = 3.45 _+0.03 fm obtained from magnetic electron scattering [ 10] (we give this value multiplied by the correction factor ( 17/16 ) 1/2 ~ 1.031 arising when the overlap integral is calculated as a function of the relative distance between 160 and n). A correlation can be seen in fig. 1 between the [G][ 2 values and (/2 2) I / 2 . A linear fit crosses the experimental rectangle defined by [ Gexp ]2 a n d / ,\ V, 2 . ,l e x1/2 p • Additional configurations might reduce the disagreement between theoretical and experimental ver-
tex constants. For V2, coupling 160+n with 13C+a reduces C] by about 1%. Studying configurations with an excited 160 core is beyond the scope of the present model. However, let us notice that shell-model calculations with V2 also overestimate the vertex constants of p-shell nuclei [6 ]. In our opinion, the overestimation is more a potential problem than a consequence o f our restricted configuration space. The conventional spectroscopic factor S [ 4 ] reads oo
S= [. dpp2I'/(p) 2 0 = E ['12n+l(gd[Unl)2 ' n
(7)
since the oscillator states u,~ are the eigenstates o f Yr. The eigenvalues/t~¢ of ~ are given b y / z 0 = # ~ = 0 , /t2= ( 1 7 / 1 6 ) 2, # 3 = 1 - 5 0 / 1 6 3 .... [ 1 ]. Since ~12 is very close to Uo2 (except in its asymptotic part), S is slightly smaller than the shell-model value #2 ~ 1.129 and is almost independent o f the potential choice. This reflects the fact that we have only one configuration in the 170 wave function. The experimental value o f Sis 1.06 + 0.11 [ 10]. The main contribution comes from the direct part I'/air(r) o f the overlap integral. It is approximately equal to 79% for each potential. The overlap integrals I[ calculated with different N N potentials differ from each other at their maxim u m by less than 12%. For the H N Y potential, I[ is presented in fig. 2. It is almost equal to the empirical single-particle wave function found from magnetic 15
Volume 293, number 1,2
PHYSICSLETTERSB
22 October 1992
V2(b=1.9)~ 0.20
N
v2
0.15 V2(b=l.7~
HNY 0.10
I
I
I
I
I
I
I
I
3.45
3.50
3.55
3.60
3.65
3.70
3.75
3.80
Fig. 1. Linear fit of the vertex constants I G]I 2 calculated with different NN potentials as a function of the RMS radii
HNYI~ 0.2 ? E 0.1
2
4
6
8
Fig. 2. Comparison of the overlap integral I~ calculated with the HNY potential, and of the single-particle wave function derived from magnetic electron scattering [ 10] (multiplied by the square root of the experimental spectroscopic factor). The direct and exchange parts o f l ] are also displayed. At the scale of the figure, 1s exchange is negligible.
electron scattering [ 10] (multiplied by the experimental value of S 1/2) in the internal region but differs in the asymptotic region. The exchange contribution is negligible for I s nucleons while it reaches 10% in the nuclear interior for lp nucleons. 16
The inaccuracy of a vertex constant may have important consequences. In processes where nuclei exchange nucleons at large distances, the cross section is mainly sensitive to the tail of the nucleon boundstate wave function and therefore to the asymptotic normalization coefficient. When the internal contribution is suppressed for some reason, for example at sub-barrier energies or in the presence of strong absorption in heavy-ion induced reactions, the cross section is proportional to the squared modulus of the vertex constant. It will be directly affected by any error on this constant. For charged particles, the Coulomb repulsion cuts off the contribution from the internal region to lowenergy radiative capture. This effect is expecially important for reactions leading to weakly-bound nuclei such as ct(3He, y)TBe and 7Be(p, 7)SB at energies close to zero. For the ~(3He, 7)VBe reaction, the dependence of the astrophysical S-factor at zero energy on the NN potential has been investigated with the R G M by Kajino [ 17 ] who recommends the M H N potential. For VBe(p, 7)SB, V2 has been employed to calculate S ( 0 ) in a GCM calculation [ 18]. The model of refs. [6,7 ] suggests that the GCM S ( 0 ) might be significantly overestimated. However, the GCM de-
Volume 293, number 1,2
PHYSICS LETTERS B
scription in ref. [ 18 ] is based on a three-cluster model a n d the saturation c o n d i t i o n of the ct a n d 3He clusters is much better satisfied than for 160. The overestim a t i o n of the vertex constant might therefore be weaker in the G C M than in refs. [5,6] where shellmodel wave functions are employed which are not consistent with the N N interaction. Nevertheless, the choice of the N N potential in future G C M calculations of such reactions deserves further attention. N.K.T. would like to t h a n k Professor C. LeclercqWillain and the Universit6 Libre de Bruxelles for their hospitality. This work is supported in part by a PSle d'Attraction Interuniversitaire of the Services de P r o g r a m m a t i o n de la Politique Scientifique.
References [ 1] H. Horiuchi, Prog. Theor. Phys. Suppl. 62 (1977) 90. [2] Y.C. Tang, in: Topics in nuclear physics II, Lecture Notes in Physics, Vol. 145 (Springer, Berlin, 1981 ) p. 572. [ 3 ] D. Baye and P. Descouvemont, Proc. 5th Intern. Conf. on Clustering aspects in nuclear and suhnuclear systems (Kyoto, Japan, 1988), J. Phys. Soc. Jpn. Suppl. 58 (t989) 103.
22 October 1992
[4] L.D. Blokhintsev,I. Borbelyand E.I. Dolinskii, Sov. J. Part. Nucl. 8 (1977) 485. [5]L.D. Blokhintsev, A.M. Mukhamedzhanov and N.K. Timofeyuk, Ukr. J. Phys. 35 (1990) 341. [6] L.D. Blokhintsevet al., Izv. Akad. Nauk SSSR, Ser. Fiz. 54 (1990) 569. [7 ] A.M. Mukhamedzhanovand N.K. Timofeyuk,Sov. J. Nucl. Phys. 51 (1990) 431. [8] A.B. Volkov,Nucl. Phys. 74 (1965) 33. [9] N. Austern, Direct nuclear reaction theories (Wiley, New York, 1970). [ 10] S. Burzynskiet al., Nucl. Phys. A 399 (1983) 230. [ 11] D. Baye and P. Descouvemont, Nucl. Phys. A 407 (1983) 77. [12]D. Baye and P. Descouvemont, Ann. Phys. (NY) 165 (1985) 115. [13] A. Hasegawa and S. Nagata, Prog. Theor. Phys. 45 (1971) 1786; Y. Yamamoto, Prog. Theor. Phys. 52 (1974) 471. [ 14] D.R. Thompson, M. LeMere and Y.C. Tang, Nucl. Phys. A 286 (1977) 53. [ 15] D. Gogny, P. Pires and R. De Tourreil, Phys. Lett. B 32 (1970) 591. [ 16] H. Miska et al., Phys. Lett B 83 (1979) 165. [ 17] T. Kajino, Nucl. Phys. A 460 (1986) 559. [18 ] P. Descouvemont and D. Baye, Nucl. Phys. A 487 (1988) 420.
17