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Pauli anisotropy in high-energy deuteron scattering
Volume 61B, number 1
PHYSICS LETTERS
1 March 1976
P A U L I A N I S O T R O P Y IN H I G H - E N E R G Y D E U T E R O N S C A T T E R I N G ~ N. AUSTERN Department of Physics, University of Pittsburgh, tS"ttsburgh, Pennsylvania 15260, USA
Received 15 November 1975 Antisymmetrization of the deuteron-nucleussystem leads to an effective three-body interaction that is quadrupole in the neutron-proton displacement vector. This.can cause anisotropic deuteron breakup or polarization of scattered deuterons. The effect is largest at about 100 MeV. A relation to work by Ioanides and Johnson is indicated.
An alternative derivation of the spin-dependent effects discussed in the preceding article by Ioanides and Johnson [1] is provided by the orthogonalized deuteron-nucleus scattering analysis of Pong and Austern [2, 3]. In this approach an antisymmetrized manybody theory of the deuteron-nucleus system is reduced to an effective three-body Schr6dinger equation of Bethe-Goldstone form [2, 3] [E-E °- tp-t n-up-
mation [3]. Then the effective Schr6dinger equation becomes [E-E °-tp-t
n - U p - Un - V]~(rp, rn)
- VoP(R ) exp(-r2//32) [$(rp, rp) + $(r n, rn) ] , = -VoP(R)exp(-r2//32 )
Un - V(r)]~(rp, rn) (1)
= ( Q - 1) V(r)
~(rp, rn).
Here the target nucleus is considered stationary at the origin of coordinates; tp, t n, Up, Un are kinetic and potential energy operators for the individual proton and neutron; V(r)is the proton-neutron interaction, I with r = rp - rn, R - 7(rp + rn). The projection operator Q enforces orthogonality. It projects out any singleparticle states that are occupied in the target nucleus. Thus Q - 1 = - P p - Pn +PpPn,
(2)
where Pp, Pn project on to the occupied proton and neutron single-particle states. It is shown in ref. [3] that the effect of the PpPn term is much smaller than that of Pp + Pn, hence it is disregarded. The Pauli counter term on the RHS ofeq. (1) is conveniently reduced by introducing zero-range approximation for the interaction, V(r) ~. VoS(r ).
(3)
The projection operators are expressed as density matrices, and are factorized by the local density approxi¢t Supported by the National ScienceFoundation.
Here p ( R ) is the number density of nucleons, the nonlocality range/3 ~ 1 fro, and the simplification Z -- N has been used• The shift operators give a convenient representation of the r dependence of the counter term. In matrix elements, integration by parts on the coordinate R enables the shift operators to operate to the left. In applications to scattering, the wavefunction qJ(R, R ) is usually associated with a rather well-defined local momentum K, and this allows the replacement VR ~ iK in the shift operators. Alternatively, in scattering matrix elements K might refer to the momentum in the outgoing wavefunction, generally with about the same value. The momentum K is used henceforth. The shift operators are now conveniently expanded in partial waves• Because the range/3 is small, only the first two nonvanishing terms are taken, giving [ E - E ° - tp - t n - Up - U n - 1I] ff(rp,rn) - 2 VoP(R ) exp(-r2//3 2) •
1
X [1o(TKr) - 5]2({Kr)P2(I('~)] ~ ( R , R).
(5)
Volume 61B, number 1
PHYSICS LETTERS
Ref. [3] considers the contribution of the monopole term to the deuteron optical potential. This term contributes about 15 MeV at low bombarding energy. It becomes smaller as the energy rises, both because fo decreases and because the oscillations tend to average out in matrix elements. The quadrupole term in eq. (5) leads to new effects. For this term to be appreciable it is necessary to go to high enough energies so that J2 can have good overlap with the factor exp(-r2//32). Best overlap occurs in the vicinity of 100 MeV, where ½K~ ~ 2. At this energy the magnitude of the quadrupole term is comparable to that of the monopole term at low energy. The quadrupole term can affect the angular distribution in the deuteron breakup cross section. The quadrupole Pauli counter term yields an interesting polarization effect in the matrix element for deuteron elastic scattering, where its overlap with the D state part of the deuteron wavefunction plays a role. The P2 factor in eq. (5) is largest if r is aligned with the momentum K. Since the deuteron wavefunction favors alignment between r and the deuteron spin axis, the net effect is a tendency for the quadrupole term to cause alignment of the spin with respect to K. Since the D state has an amplitude of approximately 25 per-
1 March 1976
cent, the net effective strength of the polarization-dependent scattering potential may be about 4 MeV. The polarization effect occurs because the proton and neutron encounter the target nucleus material at different locations, hence with different phases. This phase difference is a function of the orientation of the protonneutron displacement vector. Similar polarization effects in the Ioanides-Johnson adiabatic analysis [I ] are derived by considering the nuclear medium as the cause of a spin-dependent internal potential that affects the relative motion of the proton and neutron. I am grateful to Dr. C.M. Vincent and Dr. W.S. Pong for discussions about this work. I am grateful to Dr. R.C. Johnson for his continuing friendly cooperation.
References [1] A.A. Ioanides and R.C. Johnson, Phys. Lett. 61B (1976)4 [2] N. Austern, Phys. Lett. 46B (1973) 49. [3] W.S. Pong, Ph.D. thesis, University of Pittsburgh (1974) unpublished; W.S. Pong and N. Austern, Annals of Physics, in press.
PHYSICS LETTERS
1 March 1976
P A U L I A N I S O T R O P Y IN H I G H - E N E R G Y D E U T E R O N S C A T T E R I N G ~ N. AUSTERN Department of Physics, University of Pittsburgh, tS"ttsburgh, Pennsylvania 15260, USA
Received 15 November 1975 Antisymmetrization of the deuteron-nucleussystem leads to an effective three-body interaction that is quadrupole in the neutron-proton displacement vector. This.can cause anisotropic deuteron breakup or polarization of scattered deuterons. The effect is largest at about 100 MeV. A relation to work by Ioanides and Johnson is indicated.
An alternative derivation of the spin-dependent effects discussed in the preceding article by Ioanides and Johnson [1] is provided by the orthogonalized deuteron-nucleus scattering analysis of Pong and Austern [2, 3]. In this approach an antisymmetrized manybody theory of the deuteron-nucleus system is reduced to an effective three-body Schr6dinger equation of Bethe-Goldstone form [2, 3] [E-E °- tp-t n-up-
mation [3]. Then the effective Schr6dinger equation becomes [E-E °-tp-t
n - U p - Un - V]~(rp, rn)
- VoP(R ) exp(-r2//32) [$(rp, rp) + $(r n, rn) ] , = -VoP(R)exp(-r2//32 )
Un - V(r)]~(rp, rn) (1)
= ( Q - 1) V(r)
~(rp, rn).
Here the target nucleus is considered stationary at the origin of coordinates; tp, t n, Up, Un are kinetic and potential energy operators for the individual proton and neutron; V(r)is the proton-neutron interaction, I with r = rp - rn, R - 7(rp + rn). The projection operator Q enforces orthogonality. It projects out any singleparticle states that are occupied in the target nucleus. Thus Q - 1 = - P p - Pn +PpPn,
(2)
where Pp, Pn project on to the occupied proton and neutron single-particle states. It is shown in ref. [3] that the effect of the PpPn term is much smaller than that of Pp + Pn, hence it is disregarded. The Pauli counter term on the RHS ofeq. (1) is conveniently reduced by introducing zero-range approximation for the interaction, V(r) ~. VoS(r ).
(3)
The projection operators are expressed as density matrices, and are factorized by the local density approxi¢t Supported by the National ScienceFoundation.
Here p ( R ) is the number density of nucleons, the nonlocality range/3 ~ 1 fro, and the simplification Z -- N has been used• The shift operators give a convenient representation of the r dependence of the counter term. In matrix elements, integration by parts on the coordinate R enables the shift operators to operate to the left. In applications to scattering, the wavefunction qJ(R, R ) is usually associated with a rather well-defined local momentum K, and this allows the replacement VR ~ iK in the shift operators. Alternatively, in scattering matrix elements K might refer to the momentum in the outgoing wavefunction, generally with about the same value. The momentum K is used henceforth. The shift operators are now conveniently expanded in partial waves• Because the range/3 is small, only the first two nonvanishing terms are taken, giving [ E - E ° - tp - t n - Up - U n - 1I] ff(rp,rn) - 2 VoP(R ) exp(-r2//3 2) •
1
X [1o(TKr) - 5]2({Kr)P2(I('~)] ~ ( R , R).
(5)
Volume 61B, number 1
PHYSICS LETTERS
Ref. [3] considers the contribution of the monopole term to the deuteron optical potential. This term contributes about 15 MeV at low bombarding energy. It becomes smaller as the energy rises, both because fo decreases and because the oscillations tend to average out in matrix elements. The quadrupole term in eq. (5) leads to new effects. For this term to be appreciable it is necessary to go to high enough energies so that J2 can have good overlap with the factor exp(-r2//32). Best overlap occurs in the vicinity of 100 MeV, where ½K~ ~ 2. At this energy the magnitude of the quadrupole term is comparable to that of the monopole term at low energy. The quadrupole term can affect the angular distribution in the deuteron breakup cross section. The quadrupole Pauli counter term yields an interesting polarization effect in the matrix element for deuteron elastic scattering, where its overlap with the D state part of the deuteron wavefunction plays a role. The P2 factor in eq. (5) is largest if r is aligned with the momentum K. Since the deuteron wavefunction favors alignment between r and the deuteron spin axis, the net effect is a tendency for the quadrupole term to cause alignment of the spin with respect to K. Since the D state has an amplitude of approximately 25 per-
1 March 1976
cent, the net effective strength of the polarization-dependent scattering potential may be about 4 MeV. The polarization effect occurs because the proton and neutron encounter the target nucleus material at different locations, hence with different phases. This phase difference is a function of the orientation of the protonneutron displacement vector. Similar polarization effects in the Ioanides-Johnson adiabatic analysis [I ] are derived by considering the nuclear medium as the cause of a spin-dependent internal potential that affects the relative motion of the proton and neutron. I am grateful to Dr. C.M. Vincent and Dr. W.S. Pong for discussions about this work. I am grateful to Dr. R.C. Johnson for his continuing friendly cooperation.
References [1] A.A. Ioanides and R.C. Johnson, Phys. Lett. 61B (1976)4 [2] N. Austern, Phys. Lett. 46B (1973) 49. [3] W.S. Pong, Ph.D. thesis, University of Pittsburgh (1974) unpublished; W.S. Pong and N. Austern, Annals of Physics, in press.