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Charge exchange in final-state interactions of (e,e′ pp) reactions
NUCLEAR PHYSICS A ELSEVIER
Nuclear Physics A585 (1995) 618-626
Charge exchange in final-state interactions of (e,e' pp) reactions C. Giusti, F.D. Pacati Dipartimento di Fisica Nucleate e Teorica, Universit~ di Pavia, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, Paoia, Italy Received 17 November 1994
Abstract
The charge-exchange mechanism due to the isospin-dependent part of the optical potential is investigated for the (e,e'pp) reaction and added to the direct one-step knockout. A small contribution is found, which is practically independent of the kinematics and of the different ingredients of the calculation.
1. Introduction The two-nucleon knockout reactions are able to explore the two-body aspects of nuclear structure, in the same way as the quasi-free (e,e'p) knockout reactions allow one to investigate the single-particle properties [1,2]. The triple-coincidence cross section of exclusive (e,e'2N) reactions involves the two-hole spectral density function and, after integrating over the energy of the residual nucleus, gives access to the two-body density [3]. Theoretical approaches to two-nucleon knockout were recently proposed and applied to (e,e'2N) and (T,2N) reactions [4,5]. From the experimental point of view, a systematic investigation of these reactions has so far been hindered by the difficulty of measuring exceedingly small cross sections and of performing a triple coincidence. A few pioneering triple-coincidence measurements of the (e,e'pp) cross section on 12C have only recently been carried out at NIKHEF [6] and result in reasonable agreement with the theoretical predictions of Ref. [4]. Many more data are expected in the future, when this kind of experiments will be fully developed with the 100% duty-factor facilities. The emission of two protons is of particular interest, as it appears very well suited to explore dynamical short-range correlations (SRCs). In fact, in this case, 0375-9474/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 5 - 9 4 7 4 ( 9 4 ) 0 0 7 9 9 - 3
C. Giusti, F.D. Pacati / Nuclear PhysicsA585 (1995) 618-626
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the effect of meson-exchange currents (MECs) and isobar states is sensibly reduced. However, the emission of a proton-proton pair is expected to be much lower than the emission of a proton-neutron pair. In photoreactions the ratio of (y,pp)/(3,,pn) absorption has been estimated to be ~ 0.07 (Ref. [7]) or ~ 0.02 (Ref. [8]) and, moreover, it has been suggested that a part of the two-proton emission strength may arise from (y,pn) events in which the outgoing neutron undergoes a charge-exchange final-state interaction [8]. It is thus important to evaluate the relevance of such a two-step charge-exchange mechanism for (e,e' pp) and (T,PP) reactions, and to separate it from the direct knockout. The aim of this paper is to calculate the contribution to this process produced by the optical potential through its isospin dependence. The calculation does not include all of the possible charge-exchange mechanisms, as only the analogue states are considered in the intermediate configuration. It is anyhow important to evaluate this effect, before exploring more complicated mechanisms. The theoretical framework is outlined in Section 2. Results are presented and discussed in Section 3 and some conclusions are drawn in Section 4.
2. Formalism
The two-step charge-exchange mechanism due to the isospin-dependent part of the optical potential has been evaluated for the (e,e'pp) reaction and added to the direct one-step knockout process, using as a starting point the approach of Ref. [4]. The main ingredients are the matrix elements of the charge-current operator taken between initial and final nuclear states. Under the assumption of a directknockout mechanism, they can be written as [4]
J~(q) = f~,f* (r 1, r2)J~(r,
r l , r2)~ti(rl,
r2)e iq'r dr dr 1 dr 2.
(1)
According to standard many-body techniques, the two-nucleon overlap integral I~i is written as the product of the pair function of the shell-model and a correlation function of Jastrow-type [9], which incorporates SRCs. Only the central part of the correlation function is considered in the calculation. The spatial part of the shell-model wave function is given by the product of two uncoupled single-particle shell-model wave functions belonging to the same shell. Antisymmetrization is fulfilled by means of suitable spin- and isospin-projection operators [4]. A sum over all the different pairs in the same shell is performed, which experimentally corresponds to detecting all the pairs of nucleons coming out of the considered shell. The nuclear current J " is the sum of a one-body and a two-body part [4]. The two-body component is derived from the effective lagrangian of Ref. [10], performing a nonrelativistic reduction of the lowest-order Feynman diagrams. Only the seagull diagrams and the diagrams with intermediate-isobar configurations are included in the calculations. In this approximation MECs do not contribute to the direct (e,e'pp) cross section and only affect the exchange contribution. The
620
C. Giusti, I.D. Pacati /Nuclear Physics A585 (1995) 618-626
pion-in-flight terms, here neglected, produce a destructive interference with the seagull ones [5], which should reduce the (e,e'pn) cross section. Thus, with this approximation, the calculated charge-exchange effect should be overestimated. In the final-state wave function ~t only the interaction of each one of the nucleons with the residual nucleus is considered, while the mutual interaction between the two outgoing nucleons is neglected. The scattering state is then written as the product of two single-particle states, eigenfunctions of a phenomenological optical potential, where, for simplicity, the spin-orbit term has been neglected. The optical potential is given by the sum of a central and an isospin-dependent term, i.e. [12] 1 Vopt = V o + ~ ,
" T V 1,
(2)
where T and a- are the isospins of the target and of the considered nucleon, respectively. The isospin-dependent part of the optical potential allows the scattering between analogue intermediate states of the nucleus. This means that a pn pair interacts with the virtual photon, and, in the final state, the neutron is transformed into a proton by the interaction simulated through the optical potential. Even a part of the protons are tranformed into neutrons by the same final-state interaction, but, being neutrons, they are not detected in the asymptotic state. The balance between the two effects gives the calculated cross section, which results in an enhancement of the direct (e,e'pp) cross section. Also nn pairs, where both neutrons are transformed into protons, are considered; however, this contribution always turns out to be negligible. The explained mechanism does not give the global effect of a two-step (e,e'pn)(n,p) reaction, as all the intermediate states which are not analogous to the considered one are not included in the calculation. Moreover, as the mutual interaction between the outgoing nucleons is neglected, the requirement of involving analogue states is only approximately fulfilled by imposing the antisymmetrization. The above mechanism requires the appropriate boundary conditions. The distortion in the final state implies the incoming-spherical-wave condition, where the plane wave gives the asymptotic state of the outgoing particle, while the spherical wave describes the particle wave function inside the nucleus, after the direct interaction. As in this region we can have both a proton and a neutron, the spherical wave must contain both the isospin components. On the contrary, as only protons are detected in the asymptotic state, the plane wave does include only protons. Therefore, the required asymptotic behaviours are: e -ipr gp ~ e ipz + f~p-)( O ) - - , r e-ipr g, ~f{n-)(O) - - , r
(3)
C. Giust~ F.D. Pacati ~Nuclear Physics A585 (1995) 618-626
621
where the scattering amplitudes are related to the ones with the outgoing-spherical-wave boundary conditions by
f<-)(O) =f~+)* (~" - 0).
(4)
The final-state wave function ~bf can thus be expanded as ~/f(rl, '1"1, r2, I"2) = E ( - 1 ) ,l.,rt
8~-n+ $,r'n gr(rl)Cl(~rl)gr,(r2)ot('f2).
(5)
The scattering wave functions are obtained from a set of two coupled Lane-type equations [11-13], which are decoupled, as shown in Ref. [12], by equating the effect of the Coulomb potential to the Coulomb-energy difference between the two channels.
3. Results
The case of two nucleons emitted from the p-shell of 160 was considered, as in Ref. [4]. An average missing energy Es of 30 MeV was assumed in the calculations. The results are obtained by integrating the nine-fold differential cross section over the energy of one of the emitted particles (E~). We adopted, as an example, the correlation function obtained from a variational calculation on 160 [14], with the hard-core NN interaction of Kallio and Kolltveit (KK) [15]. Other realistic correlations do not qualitatively change the present findings about the relevance of the considered charge-exchange mechanism. The single-particle bound states were chosen as harmonic-oscillator functions with b = (h/mto) 1/2 = 1.663 fm. This choice is consistent with the adopted correlation function [14]. More realistic choices do not indicate any significant difference. The phenomenological optical potentials of Refs. [16,17], without the spin-orbit term, were used for the scattering wave functions. The value of the isospin-dependent component is V1 -- 50 MeV in Ref. [16] and V1 = 38 MeV in Ref. [17]; both of them are consistent with the low-energy value given in Ref. [12]. Before comparing with the strength of the central part of the optical potential, one has to remember the factor 1/A in Eq. (2). The energy dependence of the optical potential produces wave functions in the initial and in the final states which are not mutually orthogonal. The spurious contribution, due to this lack of orthogonality, is subtracted from the transition amplitude. It turns out anyhow to be negligible in all of the situations considered here. Three different coplanar kinematics have been considered. Two of them are symmetrical, i.e. the two nucleons are emitted in opposite directions with respect to the momentum transfer q, at equal energies and angles. In the first one, the energy and momentum transfer to and q are fixed, and, changing the nucleonscattering angle y, we explore the cross section at different values of the recoil momentum of the residual nucleus PB, which, when final-state interactions are
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C. Giust~ F.D. Pacati / Nuclear Physics A585 (1995) 618-626
neglected, is opposite to the total momentum of the initial pair of nucleons. In this situation the shape of the resulting cross section is expected to be driven by the momentum distribution of the initial pair of nucleons, which should be mostly coupled to L = 0. In the second kinematics the energy transfer and thus the outgoing-proton energies are changed, keeping the recoil momentum and the momentum transfer fixed. In this case we mainly explore the dependence of the cross section on the correlation function. The third kinematics, which is not symmetrical in the laboratory frame, aims at investigating the dependence of the cross section on the angle Yc.m. between the outgoing protons and the momentum transfer in the center-of-mass frame of the outgoing nucleons and the exchanged virtual photon, at fixed recoil momentum. We chose in all of the cases an incident electron energy of E 0 = 500 MeV and a momentum transfer q = 500 M e V / c . In the first kinematics we explore the cross section up to P s = 350 M e V / c , with to = 200 MeV. In the second kinematics we fix PB = 150 M e V / c and vary to. In the third kinematics we keep PB = 0 and to = 200 MeV. The angular distribution of the cross section in the first kinematics is shown in Fig. 1. The expected s-wave shape distribution is confirmed. The effect of charge exchange, calculated with the optical potential of Ref. [17], turns out to be very small. The other choice of the optical potential gives similar results. The current corresponding to pion-in-flight diagrams, not included in these calculations, should not change the present findings, as in this approach charge exchange has practically the same effect on the different components of the nuclear current.
I0 I--
16 8
E v
-9
lO
-10
lO
-11
lO
-12
10
i
20
I
40
,
I
60
i
I.
80
(°)
Fig. 1. Differential cross section for the 160(e,e'pp) reaction in coplanar symmetrical kinematics, with E 0 = 500 MeV, t o - 200 M e V and q = 500 M e V / c , versus ~/. Optical potential from Ref. [17]. The dot-dashed line gives the charge-exchange contribution.
C. Giusti, F.D. Pacati / Nuclear PhysicsA585 (1995) 618-626
I0 8
623
f
E 1 (~9
-10 10
-11 10
-12 10
I
,
,
,
IO0
,
I
,
,
,
150
,
I
200
(MV) Fig. 2. Differential cross section for the 160(e,e'pp) reaction in coplanar symmetrical kinematics, with E 0 = 500 MeV, q = 500 M e V / c and PB = 150 M e V / c , versus to. Optical potential as in Fig. 1. The dot-dashed line gives the charge-exchange contribution.
I0
I138
E -9
I0 -10 I0 -11 I0 -12
10
~.-..°__°~°
I 20
,
I
40
,
I
50
,
I 80
(°)
Fig. 3. Differential cross section for the 160(e,e'pp) reaction in coplanar kinematics, with E 0 = 500 MeV, to = 200 MeV, q = 500 M e V / c and PB = 0, versus the scattering angle Yl of the first outgoing proton. Optical potential as in Fig. 1. The dot-dashed line gives the charge-exchange contribution.
624
C Giusti, F.D. Pacati ~Nuclear Physics A585 (1995) 618-626
x 1-~0 ~_
0.3
E 0.2
\\
0.1 " " "" " - .'..--7-.... ;.7.. , 7.7... .
0
I 1 O0
,
,
,
,
I 150
,
, '
,
,
I 200
(MeV) Fig. 4. Differential cross section for the 160(e,e'pp) reaction in coplanar symmetrical kinematics with E 0 --- 500 MeV, 3' = 90* and P e = q = 600 MeV/c, versus ~o. T h e dashed (dotted) line gives the result of the direct o n e - s t e p knockout for the optical potential of Ref. [17] (Ref. [16]). T h e solid (dot-dashed) line includes the effect of charge exchange.
The cross section in the second kinematics is shown in Fig. 2, versus the energy transfer ~0. The result is slowly increasing with ~o, while the charge-exchange correction is slowly decreasing and always less than 1%. In Fig. 3 the charge-exchange effect is shown in the third kinematics, where the center-of-mass angle Yc.m. is varied. Even in this situation, no significant contribution is found. Larger effects can be obtained in kinematics where the recoil momentum is very high and the total cross section is very small. In Fig. 4 is shown, as an example, the case of a symmetric kinematics where the recoil momentum is PB = 600 MeV/c. The effect is still less than 20%. Moreover, in this particular situation, more complicated mechanisms, such as three-body processes [19,20], are likely to be effective. The present approach for calculating the charge-exchange contribution has also been applied to the (y,pp) reaction and gives similar results as for (e,e'pp).
4. Conclusions In this paper we have evaluated the two-step charge-exchange contribution due to the isospin dependence of the optical potential on (e,e'pp) reactions. In the
C. Giusti, F.D. Pacati /Nuclear Physics A585 (1995) 618-626
625
considered kinematical region (to < 200 MeV and PB < 350 M e V / c ) the effect of this mechanism turns out to be very small and practically independent of the kinematics and of the choice of the correlation function. An optical potential with a larger isospin-dependent component could give a larger but not qualitatively different effect. A much more important contribution of this charge-exchange mechanism was found in Ref. [13] for the (y,p) reaction, where the optical potential has a very large influence on the calculated cross sections. On the contrary, only a small contribution was found for the (e,e'p) reaction [18], where the effect of the optical potential is not very large and of similar relevance as on two-nucleon emission processes. This result is due to the fact that (y,p) reactions at medium energies probe the high-momentum components of the bound-nucleon wave functions, while (e,e'p) and (e,e'pp) reactions in the considered kinematical regions only explore the low-momentum components. The present results do not completely exclude a sizable contribution of the two-step (e,e'pn)(n,p) reaction, but, in this case, more complicated mechanisms than the one due to an isospin dependence of the optical potential must be advocated. In fact, the mechanism considered here only takes into account the intermediate state which is analogous to the final state of the residual nucleus. Three-body mechanisms have been found to play an important role in photoreactions with two-nucleon emission in 3He [19,20]. However, these effects seem active at high-energy transfers (to > 250 MeV) [19] and at high recoil momenta (PB > 300 M e V / c ) [20]. The kinematical regions which have been explored here always concern lower-energy transfers (~o < 200 MeV). The recoil momentum is small in two of the considered kinematics, where PB = 0 and PB = 150 MeV/c, while in the first kinematics it ranges from zero up to 350 MeV/c. Therefore, the results presented in this paper appear to be reliable in the considered kinematical regions.
References [1] S. Frullani and J. Mougcy, Adv. Nucl. Phys. (1984) 1. [2] S. Boffi, C. Giusti and F.D. Pacati, Phys. Reports 226 (1993) 1. [3] W. Kratschmer, Nucl. Phys. A 298 (1978) 477; K. Gottfried, Nucl. Phys. 5 (1958) 557; Ann. of Phys. 21 (1963) 29; Y.N. Srivastava, Phys. Rev. 135 (1964) B612; S. Boffi, Two-nucleon emission reactions, eds. O. Benhar and A. Fabrocini (ETS Editrice, Pisa, 1991) p. 87. [4] C. Giusti and F.D. Pacati, Nucl. Phys. A 535 (1991) 573; C. Giusti, F.D. Pacati and M. Radici, Nucl. Phys. A 546 (1992) 607; S. Boffi, C. Giusti, F.D. Pacati and M. Radici, Nucl. Phys. A 564 (1993) 473; C. Giusti and F.D. Pacati, Nucl. Phys. A 571 (1994) 694. [5] J. Ryckebusch, M. Vanderhaeghen, L. Machenil and M. Waroquier, Nucl. Phys. A 568 (1994) 828; J. Ryckebusch, L. Machenil, M. Vanderhaeghen V. Van der Sluys and M. Waroquier, Phys. Rev. C 49 (1994) 2704.
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[6] A. Zondervan, PhD thesis, Vrije Universiteit, Amsterdam (1992); L.J.H.M. Kester et al., Nuci. Plays A 553 (1993) 709c; L.J.H.M. Kester, PhD thesis, Vrije Universiteit, Amsterdam (1993); L.J.H.M. Kester et al., Proc. Eighth Mini-Conf., eds. E. Jans, L. Lapik6s, P.J. Mulders and J. Oberski (Amsterdam 1994) p. 58; L.J.H.M. Kester et al., Phys. Rev. Lett., in press; A. Zondervan et al., to be published. [7] M. Kanazawa et al., Phys. Rev. C 35 (1987) 1828. [8] I.J.D. MacGregor et al., Nucl. Phys. A 533 (1991) 269. [9] R. Jastrow, Phys. Rev. 98 (1955) 1479; F. Iwamoto and M. Yamada, Prog. Theor. Phys. 17 (1957) 543. [10] R.D. Peccei, Phys. Rev. 176 (1968) 1812; 181 (1969) 1902. [11] A.M. Lane, Nucl. Phys. 35 (1962) 676. [12] P.E. Hodgson, Nuclear reactions and nuclear structure (Clarendon, Oxford, 1971). [13] S. Boffi, F. Capuzzi, C. Giusti and F.D. Pacati, Nucl. Phys. A 436 (1985) 438. [14] J.W. Clark, The many-body problem: Jastrow correlations versus Briickner theory, Lecture notes in physics, Vol. 138 eds. R. Guardiola and J. Ros (Springer, Berlin, 1981) p. 184. [15] A. Kallio and IC Kolltveit, Nucl. Phys. 53 (1964) 87. [16] A. Nadasen et al., Phys. Rev. C 23 (1981) 1023. [17] M.M. Giannini, G. Ricco and A. Zucchiatti, Ann. of Phys. 124 (1980) 208. [18] C. Giusti and F.D. Pacati, Nucl. Phys. A 504 (1989) 685. [19] G. Audit et al., Phys. Rev. C 44 (1991) R575; G. Audit et al., Phys. Lett. B 312 (1993) 57. [20] T. Emura et al., Phys. Rev. C 49 (1994) R597.
Nuclear Physics A585 (1995) 618-626
Charge exchange in final-state interactions of (e,e' pp) reactions C. Giusti, F.D. Pacati Dipartimento di Fisica Nucleate e Teorica, Universit~ di Pavia, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, Paoia, Italy Received 17 November 1994
Abstract
The charge-exchange mechanism due to the isospin-dependent part of the optical potential is investigated for the (e,e'pp) reaction and added to the direct one-step knockout. A small contribution is found, which is practically independent of the kinematics and of the different ingredients of the calculation.
1. Introduction The two-nucleon knockout reactions are able to explore the two-body aspects of nuclear structure, in the same way as the quasi-free (e,e'p) knockout reactions allow one to investigate the single-particle properties [1,2]. The triple-coincidence cross section of exclusive (e,e'2N) reactions involves the two-hole spectral density function and, after integrating over the energy of the residual nucleus, gives access to the two-body density [3]. Theoretical approaches to two-nucleon knockout were recently proposed and applied to (e,e'2N) and (T,2N) reactions [4,5]. From the experimental point of view, a systematic investigation of these reactions has so far been hindered by the difficulty of measuring exceedingly small cross sections and of performing a triple coincidence. A few pioneering triple-coincidence measurements of the (e,e'pp) cross section on 12C have only recently been carried out at NIKHEF [6] and result in reasonable agreement with the theoretical predictions of Ref. [4]. Many more data are expected in the future, when this kind of experiments will be fully developed with the 100% duty-factor facilities. The emission of two protons is of particular interest, as it appears very well suited to explore dynamical short-range correlations (SRCs). In fact, in this case, 0375-9474/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 5 - 9 4 7 4 ( 9 4 ) 0 0 7 9 9 - 3
C. Giusti, F.D. Pacati / Nuclear PhysicsA585 (1995) 618-626
619
the effect of meson-exchange currents (MECs) and isobar states is sensibly reduced. However, the emission of a proton-proton pair is expected to be much lower than the emission of a proton-neutron pair. In photoreactions the ratio of (y,pp)/(3,,pn) absorption has been estimated to be ~ 0.07 (Ref. [7]) or ~ 0.02 (Ref. [8]) and, moreover, it has been suggested that a part of the two-proton emission strength may arise from (y,pn) events in which the outgoing neutron undergoes a charge-exchange final-state interaction [8]. It is thus important to evaluate the relevance of such a two-step charge-exchange mechanism for (e,e' pp) and (T,PP) reactions, and to separate it from the direct knockout. The aim of this paper is to calculate the contribution to this process produced by the optical potential through its isospin dependence. The calculation does not include all of the possible charge-exchange mechanisms, as only the analogue states are considered in the intermediate configuration. It is anyhow important to evaluate this effect, before exploring more complicated mechanisms. The theoretical framework is outlined in Section 2. Results are presented and discussed in Section 3 and some conclusions are drawn in Section 4.
2. Formalism
The two-step charge-exchange mechanism due to the isospin-dependent part of the optical potential has been evaluated for the (e,e'pp) reaction and added to the direct one-step knockout process, using as a starting point the approach of Ref. [4]. The main ingredients are the matrix elements of the charge-current operator taken between initial and final nuclear states. Under the assumption of a directknockout mechanism, they can be written as [4]
J~(q) = f~,f* (r 1, r2)J~(r,
r l , r2)~ti(rl,
r2)e iq'r dr dr 1 dr 2.
(1)
According to standard many-body techniques, the two-nucleon overlap integral I~i is written as the product of the pair function of the shell-model and a correlation function of Jastrow-type [9], which incorporates SRCs. Only the central part of the correlation function is considered in the calculation. The spatial part of the shell-model wave function is given by the product of two uncoupled single-particle shell-model wave functions belonging to the same shell. Antisymmetrization is fulfilled by means of suitable spin- and isospin-projection operators [4]. A sum over all the different pairs in the same shell is performed, which experimentally corresponds to detecting all the pairs of nucleons coming out of the considered shell. The nuclear current J " is the sum of a one-body and a two-body part [4]. The two-body component is derived from the effective lagrangian of Ref. [10], performing a nonrelativistic reduction of the lowest-order Feynman diagrams. Only the seagull diagrams and the diagrams with intermediate-isobar configurations are included in the calculations. In this approximation MECs do not contribute to the direct (e,e'pp) cross section and only affect the exchange contribution. The
620
C. Giusti, I.D. Pacati /Nuclear Physics A585 (1995) 618-626
pion-in-flight terms, here neglected, produce a destructive interference with the seagull ones [5], which should reduce the (e,e'pn) cross section. Thus, with this approximation, the calculated charge-exchange effect should be overestimated. In the final-state wave function ~t only the interaction of each one of the nucleons with the residual nucleus is considered, while the mutual interaction between the two outgoing nucleons is neglected. The scattering state is then written as the product of two single-particle states, eigenfunctions of a phenomenological optical potential, where, for simplicity, the spin-orbit term has been neglected. The optical potential is given by the sum of a central and an isospin-dependent term, i.e. [12] 1 Vopt = V o + ~ ,
" T V 1,
(2)
where T and a- are the isospins of the target and of the considered nucleon, respectively. The isospin-dependent part of the optical potential allows the scattering between analogue intermediate states of the nucleus. This means that a pn pair interacts with the virtual photon, and, in the final state, the neutron is transformed into a proton by the interaction simulated through the optical potential. Even a part of the protons are tranformed into neutrons by the same final-state interaction, but, being neutrons, they are not detected in the asymptotic state. The balance between the two effects gives the calculated cross section, which results in an enhancement of the direct (e,e'pp) cross section. Also nn pairs, where both neutrons are transformed into protons, are considered; however, this contribution always turns out to be negligible. The explained mechanism does not give the global effect of a two-step (e,e'pn)(n,p) reaction, as all the intermediate states which are not analogous to the considered one are not included in the calculation. Moreover, as the mutual interaction between the outgoing nucleons is neglected, the requirement of involving analogue states is only approximately fulfilled by imposing the antisymmetrization. The above mechanism requires the appropriate boundary conditions. The distortion in the final state implies the incoming-spherical-wave condition, where the plane wave gives the asymptotic state of the outgoing particle, while the spherical wave describes the particle wave function inside the nucleus, after the direct interaction. As in this region we can have both a proton and a neutron, the spherical wave must contain both the isospin components. On the contrary, as only protons are detected in the asymptotic state, the plane wave does include only protons. Therefore, the required asymptotic behaviours are: e -ipr gp ~ e ipz + f~p-)( O ) - - , r e-ipr g, ~f{n-)(O) - - , r
(3)
C. Giust~ F.D. Pacati ~Nuclear Physics A585 (1995) 618-626
621
where the scattering amplitudes are related to the ones with the outgoing-spherical-wave boundary conditions by
f<-)(O) =f~+)* (~" - 0).
(4)
The final-state wave function ~bf can thus be expanded as ~/f(rl, '1"1, r2, I"2) = E ( - 1 ) ,l.,rt
8~-n+ $,r'n gr(rl)Cl(~rl)gr,(r2)ot('f2).
(5)
The scattering wave functions are obtained from a set of two coupled Lane-type equations [11-13], which are decoupled, as shown in Ref. [12], by equating the effect of the Coulomb potential to the Coulomb-energy difference between the two channels.
3. Results
The case of two nucleons emitted from the p-shell of 160 was considered, as in Ref. [4]. An average missing energy Es of 30 MeV was assumed in the calculations. The results are obtained by integrating the nine-fold differential cross section over the energy of one of the emitted particles (E~). We adopted, as an example, the correlation function obtained from a variational calculation on 160 [14], with the hard-core NN interaction of Kallio and Kolltveit (KK) [15]. Other realistic correlations do not qualitatively change the present findings about the relevance of the considered charge-exchange mechanism. The single-particle bound states were chosen as harmonic-oscillator functions with b = (h/mto) 1/2 = 1.663 fm. This choice is consistent with the adopted correlation function [14]. More realistic choices do not indicate any significant difference. The phenomenological optical potentials of Refs. [16,17], without the spin-orbit term, were used for the scattering wave functions. The value of the isospin-dependent component is V1 -- 50 MeV in Ref. [16] and V1 = 38 MeV in Ref. [17]; both of them are consistent with the low-energy value given in Ref. [12]. Before comparing with the strength of the central part of the optical potential, one has to remember the factor 1/A in Eq. (2). The energy dependence of the optical potential produces wave functions in the initial and in the final states which are not mutually orthogonal. The spurious contribution, due to this lack of orthogonality, is subtracted from the transition amplitude. It turns out anyhow to be negligible in all of the situations considered here. Three different coplanar kinematics have been considered. Two of them are symmetrical, i.e. the two nucleons are emitted in opposite directions with respect to the momentum transfer q, at equal energies and angles. In the first one, the energy and momentum transfer to and q are fixed, and, changing the nucleonscattering angle y, we explore the cross section at different values of the recoil momentum of the residual nucleus PB, which, when final-state interactions are
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C. Giust~ F.D. Pacati / Nuclear Physics A585 (1995) 618-626
neglected, is opposite to the total momentum of the initial pair of nucleons. In this situation the shape of the resulting cross section is expected to be driven by the momentum distribution of the initial pair of nucleons, which should be mostly coupled to L = 0. In the second kinematics the energy transfer and thus the outgoing-proton energies are changed, keeping the recoil momentum and the momentum transfer fixed. In this case we mainly explore the dependence of the cross section on the correlation function. The third kinematics, which is not symmetrical in the laboratory frame, aims at investigating the dependence of the cross section on the angle Yc.m. between the outgoing protons and the momentum transfer in the center-of-mass frame of the outgoing nucleons and the exchanged virtual photon, at fixed recoil momentum. We chose in all of the cases an incident electron energy of E 0 = 500 MeV and a momentum transfer q = 500 M e V / c . In the first kinematics we explore the cross section up to P s = 350 M e V / c , with to = 200 MeV. In the second kinematics we fix PB = 150 M e V / c and vary to. In the third kinematics we keep PB = 0 and to = 200 MeV. The angular distribution of the cross section in the first kinematics is shown in Fig. 1. The expected s-wave shape distribution is confirmed. The effect of charge exchange, calculated with the optical potential of Ref. [17], turns out to be very small. The other choice of the optical potential gives similar results. The current corresponding to pion-in-flight diagrams, not included in these calculations, should not change the present findings, as in this approach charge exchange has practically the same effect on the different components of the nuclear current.
I0 I--
16 8
E v
-9
lO
-10
lO
-11
lO
-12
10
i
20
I
40
,
I
60
i
I.
80
(°)
Fig. 1. Differential cross section for the 160(e,e'pp) reaction in coplanar symmetrical kinematics, with E 0 = 500 MeV, t o - 200 M e V and q = 500 M e V / c , versus ~/. Optical potential from Ref. [17]. The dot-dashed line gives the charge-exchange contribution.
C. Giusti, F.D. Pacati / Nuclear PhysicsA585 (1995) 618-626
I0 8
623
f
E 1 (~9
-10 10
-11 10
-12 10
I
,
,
,
IO0
,
I
,
,
,
150
,
I
200
(MV) Fig. 2. Differential cross section for the 160(e,e'pp) reaction in coplanar symmetrical kinematics, with E 0 = 500 MeV, q = 500 M e V / c and PB = 150 M e V / c , versus to. Optical potential as in Fig. 1. The dot-dashed line gives the charge-exchange contribution.
I0
I138
E -9
I0 -10 I0 -11 I0 -12
10
~.-..°__°~°
I 20
,
I
40
,
I
50
,
I 80
(°)
Fig. 3. Differential cross section for the 160(e,e'pp) reaction in coplanar kinematics, with E 0 = 500 MeV, to = 200 MeV, q = 500 M e V / c and PB = 0, versus the scattering angle Yl of the first outgoing proton. Optical potential as in Fig. 1. The dot-dashed line gives the charge-exchange contribution.
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C Giusti, F.D. Pacati ~Nuclear Physics A585 (1995) 618-626
x 1-~0 ~_
0.3
E 0.2
\\
0.1 " " "" " - .'..--7-.... ;.7.. , 7.7... .
0
I 1 O0
,
,
,
,
I 150
,
, '
,
,
I 200
(MeV) Fig. 4. Differential cross section for the 160(e,e'pp) reaction in coplanar symmetrical kinematics with E 0 --- 500 MeV, 3' = 90* and P e = q = 600 MeV/c, versus ~o. T h e dashed (dotted) line gives the result of the direct o n e - s t e p knockout for the optical potential of Ref. [17] (Ref. [16]). T h e solid (dot-dashed) line includes the effect of charge exchange.
The cross section in the second kinematics is shown in Fig. 2, versus the energy transfer ~0. The result is slowly increasing with ~o, while the charge-exchange correction is slowly decreasing and always less than 1%. In Fig. 3 the charge-exchange effect is shown in the third kinematics, where the center-of-mass angle Yc.m. is varied. Even in this situation, no significant contribution is found. Larger effects can be obtained in kinematics where the recoil momentum is very high and the total cross section is very small. In Fig. 4 is shown, as an example, the case of a symmetric kinematics where the recoil momentum is PB = 600 MeV/c. The effect is still less than 20%. Moreover, in this particular situation, more complicated mechanisms, such as three-body processes [19,20], are likely to be effective. The present approach for calculating the charge-exchange contribution has also been applied to the (y,pp) reaction and gives similar results as for (e,e'pp).
4. Conclusions In this paper we have evaluated the two-step charge-exchange contribution due to the isospin dependence of the optical potential on (e,e'pp) reactions. In the
C. Giusti, F.D. Pacati /Nuclear Physics A585 (1995) 618-626
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considered kinematical region (to < 200 MeV and PB < 350 M e V / c ) the effect of this mechanism turns out to be very small and practically independent of the kinematics and of the choice of the correlation function. An optical potential with a larger isospin-dependent component could give a larger but not qualitatively different effect. A much more important contribution of this charge-exchange mechanism was found in Ref. [13] for the (y,p) reaction, where the optical potential has a very large influence on the calculated cross sections. On the contrary, only a small contribution was found for the (e,e'p) reaction [18], where the effect of the optical potential is not very large and of similar relevance as on two-nucleon emission processes. This result is due to the fact that (y,p) reactions at medium energies probe the high-momentum components of the bound-nucleon wave functions, while (e,e'p) and (e,e'pp) reactions in the considered kinematical regions only explore the low-momentum components. The present results do not completely exclude a sizable contribution of the two-step (e,e'pn)(n,p) reaction, but, in this case, more complicated mechanisms than the one due to an isospin dependence of the optical potential must be advocated. In fact, the mechanism considered here only takes into account the intermediate state which is analogous to the final state of the residual nucleus. Three-body mechanisms have been found to play an important role in photoreactions with two-nucleon emission in 3He [19,20]. However, these effects seem active at high-energy transfers (to > 250 MeV) [19] and at high recoil momenta (PB > 300 M e V / c ) [20]. The kinematical regions which have been explored here always concern lower-energy transfers (~o < 200 MeV). The recoil momentum is small in two of the considered kinematics, where PB = 0 and PB = 150 MeV/c, while in the first kinematics it ranges from zero up to 350 MeV/c. Therefore, the results presented in this paper appear to be reliable in the considered kinematical regions.
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