Finite-range corrections in two-nucleon transfer reactions

I

2*F

I

NHcIear P&&s Not to

be

A162 (1971) 376-384;

reproduced

by

@ North-Holland Publishing Co., Amsterdam

photoprint or microfilm without written permission from

the

publisher

FINITE-RANGE CORRECTIONS INTWO-NUCLEONTRANSFER RIZACTIONS E. ROST and P. D. KUNZ t Department of Physics and Astro~~ys~c~ Uniuersity of Colorado, Boulder, Colorado USA it Received 28 September 1970 Abstract: A method is developed for calculating finite-range corrections in two-nucleon transfer reactions which is readily adapted to current distorted-wave Born approximation calculations. Finite well wave functions are used and terms up to second order in the range are carried. The method is tested for the 48Ti(p, t)46Ti reaction at 27 MeV where the DWBA calculations are known to be very sensitive to parameter variations. Although the second-order terms are appreciable the method does appear to be converging.

1. Introduction

The extension of the direct-reaction stripping theory to two-nucleon transfer reactions has provided considerabIe information on the properties of nuclear energy levels “). However, a detailed exploitation of the two-nucleon coherence “) is often frustrated by uncertainties in the distorted-wave Born approximation (DWBA) analysis of the reaction. Probably the most drastic approximation in the usual analysis is the zero-range approximation which assumes that the transfer process takes place at the c.m. of the transferred pair, This approximation ignores both the range of the interaction and the size of the light-particle wave function [i.e. the triton in a (p, t) reaction]. Consequently, the zero-range approach is expected to be less accurate for two-nucleon transfer than for single-nucleon transfer processes. Finite-range effects may be included exactly by numeric~Iy evaluating the sixdimensional integrals that ensue “). This procedure involves rather excessive computational difficulties. Of more value at this time are approximate-correction approaches which have proven to be effective in the single-nucIeon transfer theory “). These are essentially Taylor series expansions in the range parameter which enable one to approximate the finite-range effects while keeping the formulation suitable for DWBA computer programs. The extension of such an approach to two-nucleon transfer reactions has been accomplished by Bencze and Zimanyi “) and improved upon by Chant and Mangelson ‘). In this paper we generalize the work of Chant and Mangelson so that finite-well wave functions may be used throughout; a biproduct is a simpler formulation of the problem. AIthough we specialize to the (p, t) reaction for clarity, the modi~cations + Presently at the Niels Bohr Institute, Copenhagen, Denmark. tt Work supported in part by the US Atomic Energy Commission. 376

TWO-NUCLEON

TRANSFER

377

REACTIONS

for other two-particle transfer reactions are straightforward and are discussed elsewhere ‘). Results are presented for the 48Ti(p, t)46Ti reaction which is known to be sensitive to parameter variations “) but has very sharp diffraction features. DWBA calculations are presented and second order effects are studied. 2. Finite-range approximation The transition ampli~de for the distorted-wave Born appro~mation the A(p, t)B reaction can be written as “)

treatment of

where Xi(+) and xr’-’ are the distorted waves for the incoming and outgoing particle, !J?i;:; and !P2=pm are the wave functions for the initial and final nuclei A and B, and 4, is the triton internal function. For brevity the spin labels are suppressed at this point. Eq. (1) has employed the usual direct-reaction approximation in denoting the core by &, viz., exchange processes involving core rearr~gement are ignored ’ “). The integration over the core coordinates is given by a shell-model expansion

x (JBJhWJAMA)(TB

TN,

NITA

NA>S@“T(rI,

y2),

(2)

where j, denotes the shell-model quantum numbers (n, , I, ,j,). To proceed further with the evaluation of (1) we take simple Gaussian shapes for the potential and internal function, V(x) = - V, exp (- p”x”), (3)

bt,(r1,r2,r3) = Nexp [ -t:i:j~]

=~exp[-~-~I,

(4)

r = r2-rl,

P =

r3-2

R

-$(rlfr2).

=

%

fr2h

Thus the potential has strength VOand range parameter /I-i; d is the rms radius of the triton and the normalization constant N is given by J1&12drdp = 1 which yields N = (rc” A6 3*)-%. It is convenient to further approximate the interaction as Y&+&r)+

V{p+)

w

-2Yo exp [-P’p2-Q?2r2].

(5)

A justification of this approximation is discussed by Chant and Mangelson ‘). The integration over the relative coordinate r can be accomplished using the method

378

E. ROST AND P. D. KUNZ

of Bayman expanded

and Kallio

ll).

Each component

*PL(rI

,4

of the two-nucleon

= C G(r, nn

WY”(~)Y”@)l”,

wave function

$ is

,

(6)

so that the integration over P WIII pick out the A = 0 term [see ref. rl) for details]. Substituting (2) (4) (5) and (6) into (1) and integrating over r yields the following six-dimensional integral r,i =

dr,dpXi-)* s

(rT)Fx(rT + ~p)e+pZA+)(yrT

+ w),

(7)

where rr = $(rI+rZ+r,), .s* = j?2+(342)-l, a = -1 3, Y = @--2)/A,

CL= W-1)/A.

The coordinates are referred to the c.m. of nucleus B. The form factor FX(rT+ ccp) contains all normalization factors which can be omitted here. By performing a Taylor series expansion of Xi’+’ and F, x!+)bT+C(P)

=

exp

bp

* v~rT>xi+)(~rT)~

=

exp

iw

* vrT}Fx(rT>,

F,(R)

=

(8) F,(rT

the integral Tfi =

+

w)

over p yields the result 5, ‘) y$~-~

s

drTXi-)*(rT) exp [‘i ” + “S:i

+ ‘xv’]

F,(r,)x~+)(yr,),

(9)

where

and the subscript on the Laplacian operators denote the functions they operate upon. The VT term can be evaluated using the Schrbdinger equation with optical potential Vi; the V,“; term can be manipulated by partial integration to act on xr and then treated similarly. Finally, by performing a partial wave expansion for the distorted waves x = r-‘Clul(r)Y~(P), one arrives at radial integrals I,,. = I~;‘+P’ 11’9 I(l) = 11’

z%E-3

p

?ltE-3

21’

=

dr ui’)(r)g,(r) s

1 32E2

exp

2m,

F

dr[Cf@(r)gx(r)t$‘(y)

I

2

1

(Ui(yr)-

Uf(r)-Ei+Ef)

+ Cfiu”l”‘(r)gx(r)~l”(rr)],

1

u?(v),

(loa)

Pw

TWO-NUCLEON

TRANSFER

REACTIONS

379

where

(11) and

The first term, eq. (lOa), would be equivalent to the usual finite-range form-factor correction term “) if F,(r) would satisfy a dineutron Schriidinger equation in a well U,(r). [A WKB evaluation “) of U,(r) is inaccurate in the nuclear interior where the form factor oscillates rapidly.] Another approach, used by Chant and Mangelson ‘) is to expand F,(r) in terms of harmonic oscillator functions in which case eq. (12) is readily evaluated. A method to be described applicable for an arbitrary form factor involves an expansion of F,(r) in eigenstates of V: subject to appropriate boundary conditions. It is convenient to separate the problem into two regions, viz. an inside region where the form factor oscillates rapidly and an exterior region where a WKB approach is adequate. Thus, in the interior region 0 5 r s R1 consider the expansion fk)

= C a, j& n

9,

(13)

where fx(t) is the radial part of E;(r) and the eigenvalues k, are determined imposing a Stiirm-Liouville condition

by

=

0.

r=Ri

The end-point RI is chosen to be the last extremum offX(r>. Since j&r) V,”jt(k, r> = - k,”j&

satisfies

r-1

and since jL(knr) are orthogonal functions over the interval 0 5 r g R, , we obtain

exp G or

( >

F,(r) = C a, exp (n

T)

s,(r) = C a, exp n

with a,=

R1 r2dr j&k, r)fx(r) s* R1 r2drlj#,r>l’ s0

j,(k,r)?Yb(P),

(I41

380

E. ROST

AND

P. D. KUNZ

Only six terms in the expansion are suflicient to obtain three-place accuracy for typical form factors. For large radii r > R,, a WKB-like approach was found to be satisfactory, i.e., at each r we compute

l

K2(r) = -

F,(r)

vm9

= _1

o-9

fX(r) then for r > R, we have

To first approximation

(9 1

f-Jr).

s,(r) = exp A second-order

(16)

correction is easily employed which adds a term

[s]2[YX~2(r~fx(r)) -

&7.&9= i

(17)

~"(r>fx(r)l

to g,(r) above and was found to contribute roughly a 10% effect for r near RI and much less for large r. Thus we feel confident that g,(r) is computed to an accuracy of a few percent for all r-values. The second-order term, eq. (lob), comes from terms involving V* operating on the optical potentials. This term was not evaluated by Chant and Mangelson ‘). However, it is easy to evaluate eq. (lob) in a DWBA program using simple modifications of the distorted-wave radial wave functions; derivatives of these wave functions are obtained by integrating uI(r). Although smaller than the first term, the secondorder term does give an appreciable contribution for (p, t) reactions as will be discussed below. 3. CalcuIations for specific cases calculations were performed for the 48Ti(p, t)46Ti reactions at 27 MeV incident energy for excitation of the ground state, and excited 2’ and 4+ states in 46Ti. The data were measured by Baer et cal.“) and cover the angular range 5” to 110” with good statistics. The Q-values are such that the triton energies in the outgoing DWBA

TABLE 1

Optical-model Projectile proton triton

“1 Units

V

r0

potentials a

employed W

54.6

1.12

0.78

3.2

165.4

1.16

0.752

16.4

of MeV or fm where appropriate.

in the DWBA

calculations

“)

WD --

dJ

a’

pi...

rs.o.

h.

6.0

1.32

0.568

6.2

0.98

0.75

0

1.498

0.817

0

The notation

is defined

in ref. “).

TWO-NUCLEON

TRANSFER

REACTIONS

381

channel are close to the energies of triton elastic scattering experiments I’), consequently optical parameters are available. Proton optical potentials were taken from the work of Becchetti and Greenlees 13). The parameters used are listed in table 1. The form factor in eq. (2) was obtained by assuming pickup of two neutrons in the lf+ orbital, i.e., the expansion was truncated at one term. This approximation is probably not very good since the Ti isotopes are believed to have considerable configuration mixing in low-lying states and the (p, t) cross section depends on a coherent sum of these admixtures. However, since we are primarily interested in finite-range comparisons, the pure configuration calculation approximation should be adequate for testing our finite range procedure. The two-neutron wave function specializes to $l(rl cl, r2 Q) = [#%(rr o,)@(r,

cJJJ,

J = 0,2,4,6,

where the single-neutron functions 4% are obtained by solving the Schrijdinger equation assuming a potential well and appropriate boundary conditions. In particular, a single-nucleon separation energy is needed which has often been taken to be onehalf of the two-nucleon separation energy. However, we feel that the use of the actual one-nucleon separation energy is preferable as it yields the correct limit of a coupledchannels formulation specialized to one channei 14). (The difference is one-half the pairing energy, i.e. about 1.5 MeV and is usually not too important.) The potential well is the shell-model potential as modified by the extra valence nucleon(s). We assume a Woods-Saxon potential with parameters r. = 1.25 fm, a = 0.65 fm, Iz = 25 and a depth (X 55 MeV) adjusted to give the specified separation energy. The remaining parameter needed is the finite-range parameter which appears in the form (2&)-l = [4p2 +4/(3~l~)]-*. Thus, there is a contribution both from the range of the potential PA1 and from the size of the triton A, the former being more important. We use the value given by Henley and Y u “) of 8-l = 1.6 fm but note that the value is quite uncertain. With the triton rms value A = 1.7 fm we obtain the range parameter (2&)-l = 0.69 Em. This number is about the same as the range parameters used in single-nucleon transfer reactions “). 4. Results and discussion The computer program DWUCK 16) was modified to incorporate the finite~rang~ terms in eq. (10) using the methods described in sect. 2. The first-order term, denoted as 1::’ in eq. (lOa), contains only a form factor modification which reduces to the zero-range form factor for E + co. Fig. 1 compares the zero- and finite-range form factors for a typical case. The dominant effect here is a damping of the interior which comes from an increase in the tail region combined with a decrease inside. Some modification of the maxima and minima is also found. The (p, t) di~erential cross sections which result from the zero- and finite-range descriptions are compared in fig. 2. The curves were all scaled to the data near the first maximum which is the usual procedure followed in the analysis of the nuclear

E. ROST AND P. D. KUNZ

382

stripping data, The theory is usually normalized empirically “) since the interaction potential and triton internal function assumptions in eqs. (3) and (4) are known to be extremely crude 18). The finite-range calculations were found to yield predicted cross IO’

.I,

48Ti

46Ti,

(p,t) (f 7,212

2+LEVEL

PICKUP

L= 2 FORM

FACTOR

3.

1

10-q

cl

I

-

ZERO

----

FINITE

I

RANGE RANGE

I

I

2

4

‘,

\ I

\

I 6

r (fm) Fig. 1. Form factor for the reaction 48Ti(p, t)46Ti leading to the 0.889 MeV 2’ state in 46Ti. The figure compares the zero-range resuit with that obtained using the finite-range treatment with (25)-l = 0.69 fm.

sections a factor of 0.8, 1.6 and 1.1 higher than the zero-range calculations when fitted to the data for the Of, 2’ and 4+ levels respectively. The O+ result is at first glance surprising in view of the tail enhancement (the form factors look rather similar to the one shown in fig. 1). This is explained by a disturbingly large interior contribution which tends to cancel the exterior contribution to the (p, t) cross section, This feature is a cause of the large parameter sensitivity noted by Baer et al. “) and provides a stringent test on a DWBA theory. The finite-range calculations fit the data better than the zero-range ones especially for the excitation of the 4+ level. This is gratifying since the crude (f3)’ configuration

TWO-NUCLEON

TRANSFER

REACTIONS

383

assumption is expected to be most accurate for this case [the important 17) (p+)” admixtures do not enter for this transition]. Improved agreement for the other transitions will probably require better two-nucleon wave functions and may well

48Ti

tp,t)

46Ti

(fy,#

9

Tf

27.0 MeV

PlCKUP

-

ZERO

----

Is1 ORDER FINITE

RANGE

---

2nd

ORDER

FINITE

RANGE RANGE

E, = 0 889 MeV

0’

Fig. 2. Angular distribution of tritons from the 48Ti(p, t)46Ti reaction leading to the ground state (0’) and the excited 2+ and 4” states in 4BTi. The data were taken by Baer et ul. ‘) at 27.0 MeV incident proton energy. The curves correspond to DWBA calculations with the finite-range correction ignored (solid curve), treated to first order (dashed) or treated to second order (dash-dot). Optical parameters and other calculational details are given in the text.

require improvements in the direct-reaction analyses such as those discussed by Ascuitto and ~l~ndenning 1g). The convergence of the finite-range expansion procedure may be investigated by comparing the zero-, first-, and second-order curves in fig. 2. The latter terms, while

384

E. ROST AND P. D. KUNZ

appreciable, are smafler than the first-order ones and in a direction so as to reduce the effect calculated in fkst order. Although this is no proof of convergence (the series is probably asymptotic) it does lend credence to the validity of the finite-range expansions procedure. References 1) R. A. Broglia and C. Riedel, Nucl. Phys. A92 (1967) 145 2) S. Yoshida, Nucl. Phys. 33 (1962) 685 3) N. Austem, R. M. Drisko, E. C. Halbert and G. R. Satchler, Phys. Rev. 133 (1964) B3 4) F. G. Perey and D. S. Saxon, Phys. Lett. 10 (1964) 107; P. G. A. Buttle and L. J. B. Goldfarb, Proc. Phys. Sot. 83 (1964) 701; Gy. Bencze and J. Zimanyi, Phys. Lett. 9 (1964) 246 5) W. R. Smith, Nucl. Phys. A94 (1967) 550 6) Gy. Bencze and J. Zirnanyi, Nucl. Phys. 81 (1966) 76 7) N. S. Chant and N. F. Mangelson, Nucl. Phys. A140 (1970) 81 8) H. W. Baer, J. 3. Kraushaar, C. E. Moss, N. S. P. King and R. E. L. Green, to be published 9) I. S. Towner and J. C. Hardy, Adv. in Phys. 18 (1969) 401 10) N. K. Glendenning, Ann. Rev. Nucl. Sci. 13 (1963) 191 11) B. F. Bayman and A. Kallio, Phys. Rev. 156 (1967) 1121 12) J. C. Hafele, E. R. Flynn and A. G. Blair, Phys. Rev. 155 (1967) 1238 13) F. D. Becchetti and G. W. Greenlees, Phys Rev. 182 (1969) 1113 14) P. D. Kunz and E. Rest. to be published 15) E M. Henley and D. V. L. Yu, Phys. Rev. 133 (1964) B1445 16) P. D. Kunz, unpublished 17) E. R. Flynn and 0. Hansen, Phys. Lett. 31B (1970) 135 18) R. Glover, A. Jones and J. Rook, Nucl. Phys. 81 (1966) 289 19) R. J. Ascuitto and N. K. Glendenning, Phys. Rev. C2 (1970) 415; ibid, to be published