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Antisymmetrized deuteron-nucleus scattering
Volume 46B, number 1
PHYSICS LETTERS
3 September 1973
ANTISYMMETRIZED DEUTERON-NUCLEUS SCATTERING N. AUSTERN University of Pittsburgh*, Pittsburgh, Pennsylvania 15260, USA and Brookhaven National Laboratory, Upton, New York 11973, USA
Received 26 June 1973 A practical procedure for calculatingantisymmetrizationeffects is outlined. It is seen that these effects are neither large nor unusual. A proper theory of deuteron-nucleus scattering should be antisymmetrized with respect to exchange of the neutron and proton of the deuteron with the constituent nucleons of the target nucleus. A practical procedure for calculating antisymmetrization effects is outlined in the present article. It is seen that these effects are neither large nor unusual [ 1]. The scattering can be described with an antisymmetrized product wave-function, xP(0p.., np, On... nn)
j---lp
/'=In
× q~(lp...np, In...nn) tk(0p, On), provided target-nucleus excitations are ignored. Here N is a normalization constant, and • is the wavefunction of the target nucleus. For simplicity • is assumed to be a single determinant of closed shells. The relative wavefunction ~k(0p, On) is to be determined. A Schrodinger-like equation for @(0p, On) is obtained if the symmetric many-body Hamiltonian is diagonalized in the space spanned by eq. (1), This is done, in the familiar fashion, by requiring (¢, [ E - H ] 4) = 0.
(2)
Eq. (2) is greatly simplified if we assume ~k(0p, On) is orthogonal to each of the occupied neutron and proton orbitals in ~. Under this assumption, {tp + t + V(p, n) + U(p) + U(n) + e - E ) if(p, n) = O, (3) where U(p) and U(n) are Hartree-Fock potentials * Supported by the National Science Foundation.
for the proton-nucleus and neutron-nucleus interactions, respectively. The notation in eq. (3) has been somewhat simplified. Also for simplicity U(p) and U(n) are assumed to be identical, and target nucleus recoil is ignored. The relation between eqs. (2), (3) is well known [2]. Eq. (2) contains a variety of complicated exchange terms, which drop out under the condition of orthogonality that leads to eq. (3). The complete eq. (2) is satisfied both by the orthogonal relative function if(p, n), and also by an assortment of superfluous functions, based on the bound orbitals in ~. The simplified eq. (3) does not have these superfluous solutions, however eq. (3) must be used with an associated orthogonality constraint. One way to combine eq. (3) with the orthogonality constraint is to left multiply by projection operators that exclude occupied orbitals from the set of intermediate states coupled to ~kby the reduced Hamiltonian operator. This step yields ( 1 - P p ) ( 1 - P n ) {tp +tn + V(p, n)
(4)
+ Up + U + e - E } t~(p, n) = 0
where Pp, for example, is P =~ I~/(p))(~bi(P')l. i=lp
(5)
The ~bi(p) are the occupied proton orbitals of the target nucleus. The projection operators in eq. (4) of course commute with the Hartree-Fock single-particle terms in the reduced Hamiltonian. Eq. (4) therefore becomes {tp + tn + Up + Un + e - E } ~ (p, n) = --( 1 - Pp) (1 - Pn) V(p, n) qJ (p, n).
(6) 49
Volume 46B, number 1
PHYSICS LETTERS
Eq. (6) is of the expected Bethe-Goldstone form, as has been noted before [ 1]. It is important that the counter terms Pp VqJ, Pn V~, PpPn VqJ, in eq. (6) are all of short range; this follows because V is of short range in r p - r n and Pp, Pn contain only bound orbital in the variables rp, r n. Therefore the counter terms cause no essential change in the way eq. (6) is used to discuss scattering. Eq. (6) can be converted into a variety of equivalent integral equations [ 1]. In the present article this equation is discussed as it stands. The counter terms in eq. (6) weaken the neutronproton interaction in the nuclear interior, hence they contribute to deuteron breakup and they modify the elastic scattering. These terms possess much of the helpful structure of separable potentials, because the projection operators Pp, Pn are linear combinations of factorable terms. Other easy simplifications of the counter terms are caused by the short range of V(p, n). The single particle potentials on the left-hand side of eq. (6) also cause scattering the breakup. The importance of the counter terms must be judged by comparison with these single-particle potentials. Let us introduce relative and center of mass variables
r= rp - rn, n = ½(rp + rn) , and approximate V(p, n) by V(p, n) = V(r) ~ -D6(r),
(7)
where D is the volume integral of -V(r). Then (Pp+ P ) V(r) ~(r, R)
%)+
(0, R)
(8)
3 September 1973
For the left-hand side this integral converges if we use the usual factored approximation for ~(r, R).] The projection operators in eq. (8) have little effect on the estimate of strength. The doubly-projected counter term in eq. (6) reduces in a similar fashion
P P V(r) ~(r, R)
(9)
* * . ~._ D .~. i(rp)¢)j(rn) f ¢)i(R)¢/(Rl~(O,R)d3R t,]
This term has the structure of a separable potential multiplying if(0, R). The strength of this separable potential seems small, as before, and this potential has no remarkable dependence on (rp-rn), such as might emphasize breakup effects. Because the double-projected term involves overlap of bound and continuum wavefunctions, it drops off with increasing bombarding energy. The singly-projected terms vary much more slowly with energy. The above discussion of the counter terms thus shows that they cause little change in the familiar three-body picture of deuteron scattering. In a sense this only reduces eq. (6) to a problem previously not solved, however it does provide orientation. Calculations based on eqs. (8), (9) are planned. One immediate step will be to project eq. (6) on to the deuteron ground state, to obtain an antisymmetrized equivalent of the Watanabe folding model [3]. What is note. worthy at the present stage is that major new effects [ 1] are not expected. Use of a finite-range separable interaction for V(r), instead of eq. (7), would not affect this conclusion.
1
Each term in the sum in eq. (8) has essentially the structure of a local potential multiplying the wavefunction qs(0, R); this is the same general structure as in the expression [U(rp) + U(rn) ] qJ(r, R) on the left-hand side of eq. (6). Qualitatively similar effects may be expected from these two expressions. However, eq. (8) is inherently much weaker than the other expression. This may be seen by substituting numbers, or simply by noting that eq. (8) arises from one nucleon-nucleon interaction, whereas U(p) and U(n) arise from the interactions of p and n with all A-nucleons of the target nucleus and therefore tend to be A-times stronger. [The "strength of a short-ranged interaction term in eq. (6) is probably best measured by its volume integral in the six-dimensional configuration space. 50
I am grateful to Dr. C.M. Vincent for many discussions of this work. A preliminary article was prepared at the Brooldaaven National Laboratory in the summer of 1972. I am grateful to Dr. C. Dover for discussions at that time, and to Brookhaven for their usual warm hospitality. [ 1 ] Recent discussions of these effects are given by W.F. Junkin and F. Villars, Ann. Phys. 45 (1967) 93; R.C. Johnson and P.J.R. Soper, Phys. Rev. C1 (1970) 976; P.J.R. Soper, Ph.D. thesis University of Surrey (1969) unpublished; B.L. Gambhir and J.J. Griffin, Phys. Rev. C5 (1972) 1856. [2] H. Feshbach, Ann. Phys. 23 (1963) 47; S.Saito, Prog. Theoret. Phys. 41 (1969) 705; N. Austern, Direct nuclear reaction theories (Wiley-Interscience 1970) ch. 10. [3] S. Watanabe, Nucl. Phys. 8 (1958) 484.
PHYSICS LETTERS
3 September 1973
ANTISYMMETRIZED DEUTERON-NUCLEUS SCATTERING N. AUSTERN University of Pittsburgh*, Pittsburgh, Pennsylvania 15260, USA and Brookhaven National Laboratory, Upton, New York 11973, USA
Received 26 June 1973 A practical procedure for calculatingantisymmetrizationeffects is outlined. It is seen that these effects are neither large nor unusual. A proper theory of deuteron-nucleus scattering should be antisymmetrized with respect to exchange of the neutron and proton of the deuteron with the constituent nucleons of the target nucleus. A practical procedure for calculating antisymmetrization effects is outlined in the present article. It is seen that these effects are neither large nor unusual [ 1]. The scattering can be described with an antisymmetrized product wave-function, xP(0p.., np, On... nn)
j---lp
/'=In
× q~(lp...np, In...nn) tk(0p, On), provided target-nucleus excitations are ignored. Here N is a normalization constant, and • is the wavefunction of the target nucleus. For simplicity • is assumed to be a single determinant of closed shells. The relative wavefunction ~k(0p, On) is to be determined. A Schrodinger-like equation for @(0p, On) is obtained if the symmetric many-body Hamiltonian is diagonalized in the space spanned by eq. (1), This is done, in the familiar fashion, by requiring (¢, [ E - H ] 4) = 0.
(2)
Eq. (2) is greatly simplified if we assume ~k(0p, On) is orthogonal to each of the occupied neutron and proton orbitals in ~. Under this assumption, {tp + t + V(p, n) + U(p) + U(n) + e - E ) if(p, n) = O, (3) where U(p) and U(n) are Hartree-Fock potentials * Supported by the National Science Foundation.
for the proton-nucleus and neutron-nucleus interactions, respectively. The notation in eq. (3) has been somewhat simplified. Also for simplicity U(p) and U(n) are assumed to be identical, and target nucleus recoil is ignored. The relation between eqs. (2), (3) is well known [2]. Eq. (2) contains a variety of complicated exchange terms, which drop out under the condition of orthogonality that leads to eq. (3). The complete eq. (2) is satisfied both by the orthogonal relative function if(p, n), and also by an assortment of superfluous functions, based on the bound orbitals in ~. The simplified eq. (3) does not have these superfluous solutions, however eq. (3) must be used with an associated orthogonality constraint. One way to combine eq. (3) with the orthogonality constraint is to left multiply by projection operators that exclude occupied orbitals from the set of intermediate states coupled to ~kby the reduced Hamiltonian operator. This step yields ( 1 - P p ) ( 1 - P n ) {tp +tn + V(p, n)
(4)
+ Up + U + e - E } t~(p, n) = 0
where Pp, for example, is P =~ I~/(p))(~bi(P')l. i=lp
(5)
The ~bi(p) are the occupied proton orbitals of the target nucleus. The projection operators in eq. (4) of course commute with the Hartree-Fock single-particle terms in the reduced Hamiltonian. Eq. (4) therefore becomes {tp + tn + Up + Un + e - E } ~ (p, n) = --( 1 - Pp) (1 - Pn) V(p, n) qJ (p, n).
(6) 49
Volume 46B, number 1
PHYSICS LETTERS
Eq. (6) is of the expected Bethe-Goldstone form, as has been noted before [ 1]. It is important that the counter terms Pp VqJ, Pn V~, PpPn VqJ, in eq. (6) are all of short range; this follows because V is of short range in r p - r n and Pp, Pn contain only bound orbital in the variables rp, r n. Therefore the counter terms cause no essential change in the way eq. (6) is used to discuss scattering. Eq. (6) can be converted into a variety of equivalent integral equations [ 1]. In the present article this equation is discussed as it stands. The counter terms in eq. (6) weaken the neutronproton interaction in the nuclear interior, hence they contribute to deuteron breakup and they modify the elastic scattering. These terms possess much of the helpful structure of separable potentials, because the projection operators Pp, Pn are linear combinations of factorable terms. Other easy simplifications of the counter terms are caused by the short range of V(p, n). The single particle potentials on the left-hand side of eq. (6) also cause scattering the breakup. The importance of the counter terms must be judged by comparison with these single-particle potentials. Let us introduce relative and center of mass variables
r= rp - rn, n = ½(rp + rn) , and approximate V(p, n) by V(p, n) = V(r) ~ -D6(r),
(7)
where D is the volume integral of -V(r). Then (Pp+ P ) V(r) ~(r, R)
%)+
(0, R)
(8)
3 September 1973
For the left-hand side this integral converges if we use the usual factored approximation for ~(r, R).] The projection operators in eq. (8) have little effect on the estimate of strength. The doubly-projected counter term in eq. (6) reduces in a similar fashion
P P V(r) ~(r, R)
(9)
* * . ~._ D .~. i(rp)¢)j(rn) f ¢)i(R)¢/(Rl~(O,R)d3R t,]
This term has the structure of a separable potential multiplying if(0, R). The strength of this separable potential seems small, as before, and this potential has no remarkable dependence on (rp-rn), such as might emphasize breakup effects. Because the double-projected term involves overlap of bound and continuum wavefunctions, it drops off with increasing bombarding energy. The singly-projected terms vary much more slowly with energy. The above discussion of the counter terms thus shows that they cause little change in the familiar three-body picture of deuteron scattering. In a sense this only reduces eq. (6) to a problem previously not solved, however it does provide orientation. Calculations based on eqs. (8), (9) are planned. One immediate step will be to project eq. (6) on to the deuteron ground state, to obtain an antisymmetrized equivalent of the Watanabe folding model [3]. What is note. worthy at the present stage is that major new effects [ 1] are not expected. Use of a finite-range separable interaction for V(r), instead of eq. (7), would not affect this conclusion.
1
Each term in the sum in eq. (8) has essentially the structure of a local potential multiplying the wavefunction qs(0, R); this is the same general structure as in the expression [U(rp) + U(rn) ] qJ(r, R) on the left-hand side of eq. (6). Qualitatively similar effects may be expected from these two expressions. However, eq. (8) is inherently much weaker than the other expression. This may be seen by substituting numbers, or simply by noting that eq. (8) arises from one nucleon-nucleon interaction, whereas U(p) and U(n) arise from the interactions of p and n with all A-nucleons of the target nucleus and therefore tend to be A-times stronger. [The "strength of a short-ranged interaction term in eq. (6) is probably best measured by its volume integral in the six-dimensional configuration space. 50
I am grateful to Dr. C.M. Vincent for many discussions of this work. A preliminary article was prepared at the Brooldaaven National Laboratory in the summer of 1972. I am grateful to Dr. C. Dover for discussions at that time, and to Brookhaven for their usual warm hospitality. [ 1 ] Recent discussions of these effects are given by W.F. Junkin and F. Villars, Ann. Phys. 45 (1967) 93; R.C. Johnson and P.J.R. Soper, Phys. Rev. C1 (1970) 976; P.J.R. Soper, Ph.D. thesis University of Surrey (1969) unpublished; B.L. Gambhir and J.J. Griffin, Phys. Rev. C5 (1972) 1856. [2] H. Feshbach, Ann. Phys. 23 (1963) 47; S.Saito, Prog. Theoret. Phys. 41 (1969) 705; N. Austern, Direct nuclear reaction theories (Wiley-Interscience 1970) ch. 10. [3] S. Watanabe, Nucl. Phys. 8 (1958) 484.