The Q-value dependence of heavy-ion transfer reactions

I

2*N

I

Nuclear Physics

Al78

(1971) W-64;

@

North-Holland

Publishing

Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

THE Q-VALUE DEPENDENCE

OF HEAVY-ION TRANSFER REACTIONS

K. ALDER Institute

and D. T~UTMA~~

of Theoretical Physics,

University

of Basel, Switzerland

Received 26 May 1971 Abstract: The dependence of transfer reactions between heavy ions on the Q-value has been investigated. Neutral or charged particles may be exchanged between projectile and target. The results have been obtained with the semiclassical theory of transfer reactions.

It is well known that transfer reactions between heavy ions are suitably well described by the distorted wave Born approximation. The rather complicated calculations for the general case ‘) can be considerably simplified by the restriction that the bombarding energies are well below the Coulomb barrier “). Although most of the theories are only justified for neutron transfer reactions the formalism can also be used approximately for the transfer of charged particles when the asymptotic behaviour of the bound state wave function is described by an equivalent function for neutral particles with an adjusted binding energy “). A further simplification can be obtained if in the reaction a+A --+ (a-n)+(A+n),

(1)

with the masses a, n and A and the corresponding the Coulomb parameters:

charge numbers Z,, Z, and Z,

e2 Z,Z,aA

‘li = 2

k,(a+A)

and

e* (Z,-Z,)(Z,+Z,)(a-n)(A+-n)

qf = -

h2

k&+A)

,

(2)

are large compared to unity. In this case, the process can be described almost exactly [refs. 4*‘)I in the framework of the semiclassical transfer theory (SCM). For the total cross section we get then

with (4)

HEAVY-ION TRANSFER

61

The function BA(e,p, 4,6) which describes the differential connected with the radial integrals by &(&>PY5,q

= c I ?a, I,(+h O)b,&, P

P, 4, 412,

cross section “) is

(5)

where I,,&,

p, r, 6) =

=$zy)(ip(t +E cash w))lp(l +E cash w) s0 x cos 5’~sinh w + 5w + p arctan

Furthermore,

we introduce the following symbols: t = ttf-tit

5’ = (f+S = (ki-k$,,

(74

with k;=_-Ak,. A-f-n

(74

The distance ac is defined by a, =

r_ -

vi+%,

k

k,+k:

and represents half the distance of the closest approach in a head-on collision. Finally, the dimensionless parameter p is defined by P = lcc2C

(9)

where K is connected with the binding energy Et, of the transferred neutral particle or an adjusted binding energy for the transfer of charged particles by Jc=

’ 2nA [I&l --. ? n-i-A h2

The transferred orbital angular momentum is denoted by /2 and plays a similar role as the multipole order in electromagnetic transitions. The parameter x2 describes the strength of the reaction and contains the spectroscopic factors, the normalization of the wave function, finite-range corrections and kinematical factors. Therefore, the SCTT not only describes the transfer process very easily and nearly exactly for q >> 1 but also gives a simple parameterization of the process in terms of dimensionless simple physical parameters. The cross section depends very strongly on the parameter p, which is mainly determined by the binding energy Ebr and on 5 and 6 which are determined mainly by the Q-value of the reaction.

62

K. ALDER

AND D. TRAUTMANN

Since the main contribution of the integrand in eq. (6) originates from the region w M 0 and E = 1 (W = 0 corresponds to the classical turning point and E is connected with the scattering angIe 9 by E = sin-i$!J we can express the phase in eq. (6) as function of c and 6 by S’csinh w+tw m (t’-t-l)w = (25fS)w.

(10)

Obviously, the functions I,,, and Hn have a maximunl for 5 ++S z 0, if the parameter p is kept constant. In the general case we thus define an effective g-value by 5&f = 5 t-f-4

(11)

where a depends on p and ;1and must be of the order of unity. It is determined by the condition that the maximum of H, is given by teff = 0 for constant values of p and 1. Fig. 1 shows the dependence of a on p and ;i. As can be seen a reaches unity only asymptotically for large values of p and 3,. For physically reasonable cases with x O-4 we have therefore to take into account the correct g-dependence P w l-3and1 in eq. (11) for more exact investigations. Now let us consider the whole process as a pure fictitious energy transfer with an effective Q-value QefP Following the theory of Coulomb excitation ‘) we get:

(12)

1.*< 15

2

2.5

3

3.5 s

Fig. 1. The parameter CLis illustrated as function of p for different multipole orders i.

HEAVY-ION

TRANSFER

63

The value of Q,rr is then given by

(13) which means that the cross section has a maximum value for p = constant if reff = 0 or Qeft = 0. Substituting in eq. (13) the expressions for ri and 6 [see eqs. (2) and (7)] we get for the best Q-value with a maximal cross section:

Q_~ max _ -

Ei

$B

(s+Js2+4t)2-1,

(14)

with r = -a--n

s = (&-1)(1-t),

(~,-z”)(z*+zll)

a

-

&3?4

(15)

In the limiting case where a = 1, eq. (14) reduces to the following simple expression for QmaX:

=E, 1 _ z"(z*+z"-za)n-t-4 Q max ‘Ai

z,z,

___ A

(16)

I *

In refs. **‘) it has been suggested to introduce an effective Q-value defined by

Qerr = Q+(udr)- Uf(r))r=2.c

(17)

where Ui(r) and Uf(r) are the Coulomb potentials in the entrance and exit channel, respectively. This expression (16) for Qeff deviates even for a = 1 from the correct TABLE 1 For 11 selected transfer reactions the ratio Q,../E, Reaction

is computed

according

to eqs. (16) and (17)

Qlll,,/E~ Q,& h. (1711 [es.Cl@1

1

0

0.0085

2

0.1759

0.2935

0

0.1681

0

0.0042

3 4

‘$J(d,

5

1zO(iBe, tBe)‘iO

0

0.0630

6

‘;N(‘$N,

0

0.0720

7

‘fO)z$a

0

0.0504

8

40Ca(1i0, 20 i$a(‘zO,

‘tO)$Ca

0

0.0252

p)‘;;U

9

“g;Pb(‘;O,

10

l6gO(l:B,

11

197A~(11B 79 5

‘;N)‘$N

‘;O)‘;;Pb ‘iBe)‘iF

’ l”Be)lztHg 4

0

0.0048

-0.0973

-0.0433

-0.1895

-0.1875

64

R. ALDER

AND D. TRAUTMANN

formula (13). In table 1 the ratio of Q,,, for K = 1 to the incoming energy in the c.m. system is computed accord& to eqs. (16) and (X7) for a number of seIected transfer reactions, In fig. 2 the ratio Qm,/Ei is plotted as a function of the parameter c1for the same transfer reactions as indicated in table 1. It is thus seen that the correct value of

Fig. 2. The ratio Q,.JEs isplatted as a function of u for a numb% of selected transfer reactions. The numbers 1 to 11 refer to the reactions indicated in table 1.

Q_% may differ considerably if it is computed by eq. (17). Therefore, eqs. (12)-(15) have to be used for the exact calculations and for the determination of the best Qvalues for which the reaction takes place. References 1) T. Kamuri and W. Yoshida, Nucl. Phys. Al29 (1969) 624 2) 3) 4) 5) 6) 7) 8) 9)

P. J. A. Buttle and L. J. B. Goldfarb, Nucl. Phys. 78 (1966) 409 P. J. A. Buttle attd L. J. 3. Gold&b, Nucl. Phys. A115 (1968) 461 D. Trautmann aad K. Alder, Helv. Phys, Acta 43 (1970) 363 K. Alder and D. Trautmann, Ann. of Phys. 66 (1971) 884 K. Alder, R. Morf, M. Pauli and D. Trautmann, to be published K. Aider, A. Bohr, T. Hulls, B. Moftetson and A. Winther, Rev. Mod. Pbys. 28 (1956) 432 R. A. Broglia and A. Winther, to be published R. M. Diamond, A. M. Poskanzer, F. S. Stephens, W. J. Swiatecki and ID. Ward, Phys- Rev. Lett. 20 (1968) 802