LETTEI~]~ AL I~UOVO CIMEI~/TO
VOL.
19,
N. i 0
9
Luglio
1977
Sub-Coulomb Transfer Reaction between H e a v y Ions.
Analytical Expressions.
J. CUGNO~
Universitd de LiJge, Physique s
Thdorique
lnstitut de Physique au Sart Tilamn - B-4000 T~i~ge 1, Belgique
(rieevuto 1'1 Aprile 1977)
The particle transfer between heavy ions under the Coulomb barrier plays an
interesting role in recent studies of nuclear as well as atomic physics. I t is usually
studied in the frame of the DWBA theory (1). The complexity of the formalism requires
a numerical integration. Hence, the influence of some physical parameters on the
energy dependence as well as the angular dependence of the cross-section or of the
optimum Q-value is not clear. I t is then desirable to obtain accurate approximations allowing to calculate the DWBA integral in a closed analytical form. The
WKB approximation of the wave functions for the relative motion almost fits this
desire. Recently (2), another method has been proposed, which lies on a suitable
approximation of the relative motion wave function (which is described below) a n d
on neglecting the radial variation of the form factor. The latter approximation is satisfactory for the charged-particle transfer case, where the effective domain of integration
is very limited. Here, we go beyond this limitation and still get analytical form for the
transition probability.
Our starting point will be the D W B A integral in the prior representation (1)
(1)
I = fzI' (r) I'(r) z~(r) dr,
where the Xl and Z2 are the relative-motion wave functions in the entrance and exit
channels respectively, and where r is the relative distance between the ions or between
the cores, since at this stage, we neglect recoil effects. The q u a n t i t y F(r) is the form
factor which is often written as
(2)
F(r) ----_Vexp [-- qr/~]
r
(~) P . J . k .
BUTTLE a n d L. J . B. GOLDFARB: N u c l . P h y s . , 1 7 6 A , 299 (1971).
(2) J . CUGNON, D. GARDES a n d R . D x SILVJ~IRA: tO b e p u b l i s h e d (1977) a n d E u r o p e a n Conference on
Nuclear P h y s i c s w i t h H e a v y I o n s (Caen, 1976), c o m m . no. 28.
361
362
J. CUG~O~
I n order to have an analytic expression for the integral (1), we invoke the fact that
most of the contribution to the integral (1) comes from a relatively narrow domain
of values of r near the classical t u r n i n g points (1). Hence, we construct the functions Z~,
i = 1, 2 b y approximating the ion-ion potentials i n channels 1 and 2 by constant gradient potentials in the region of interest, it is to say b y approximating the potential
energy curves by their tangents. We choose the tangential point as the one corresponding to the largest of the closest distances of approach in the two channels. We
denote b y ~ the associated relative distance. This choice is motivated b y the classical
situation, where the transfer mainly takes place at R. We neglect the variation of r -~
in the form factor and make it equal to R -x, since its variation is not i m p o r t a n t on the
effective domain of integration.
The transition probability is given b y
co
C3)
(p.q p;)~ a2(p' ) @ d p '
~'=N~
2
where the a / s are the Fourier transforms of the Z's. For constant gradient potentials,
the wave functions aflp) are given, with the right normalization, in ref. (s)
(4)
a~(p)
-
-
1
_ _
(E-- U~)p-- ~
exp
,
where U1 and Us are the w~lue of the potential-energy curves r
and r
at point R,
and where/~1 and F 2 arc minus their gradient at this point. We assume for the time
being t h a t the largest of the closest distances of approach corresponds to channel 2.
I n other words, E = /72. Hence, eqs. (3)-(5) yield
(5)
a~
N~
i
UDp]"
fdp' q~ -t- (p'--p)~
q
The second integrand is peaked at p'_~ p. Hence, we expand the argument of the
exponential in power series of ~ = p ' - - p and stop it at the first order. We obtain,
after integration on ~,
+m
(6)
~o = 2~8~2-~-1F2/~2
exp [i(~p + flp3)__rp2]
,
(a) L. D. L~N~)~Uand E. lYI.LIFSHITZ:Quantum Mechanics (Oxford, 1965), chapt. III.
SUB-COULOMB TRANSFER REACTION BETWEEN
363
HEAVY ION8 ETC.
w h e r e we h a v e p u t
(7)
~
~
1 (E--
= __ _ _
1(,
~)
,
# = ~ F~-~
1)
~-~
,
~ - - 2F~/~2
T h e f u n c t i o n i n t h e i n t e g r a l is a n a l y t i c i n t h e w h o l e c o m p l e x p l a n e p = x + iy. I n
o r d e r to t r a n s f o r m t h e i n t e g r a l i n eq. (6), t h a t we call I ( a , 8, 7), we i n t e g r a t e a l o n g
t h e c o n t o u r s s h o w n i n fig. l . I t c a n b e seen t h a t t h e i n t e g r a t i o n a l o n g t h e l a t e r a l p a r t s
-~/3p
H-
--~/3fl
a)
)
b)
Fig. I. - Contours of integration used for the evaluation of integral (6) according to the sign of 8:
a) fl>O, b) f l < 0 .
of t h e c o n t o u r is v a n i s h i n g w h e n t h e s e p a r t s are p u s h e d t o w a r d s i n f i n i t y . A l i t t l e a l g e b r a
leads to t h e f o l l o w i n g r e s u l t s
is)
I(~, #, ~,) = exp ~
( 9 ~ r + 273)
If
--lID
["
dp exp ~ (r ~ + 3 ~ / +
]
i#p8 .
T h e r e m a i n i n g i n t e g r a l is a r e p r e s e n t a t i o n of t h e A i r y f u n c t i o n (see (4)). H e n c e
(9)
I ( ~ , fl, ~') = ( ~
exp
[3(3~fl + 7 ~) - - 7 3] a * ~3fl(3fl)t "
I t c a n b e of i n t e r e s t t o m e n t i o n t h e f o l l o w i n g a s y m p t o t i c f o r m s
0o)
I ( ~ , fl, 7) =
[-,]'
i3~'~
~)
exp
]
[3(3~r + F ~) - - r 2] 9
9exp
2~
( 3 ~ + r2)! ,
(4) 1~[. ABRAMOWITZand I. A. STEGUN: Handbook el Mathematical Functions (New York), chapt. 10.
364
~. cu~o~
which is valid if 3aft + 7~>> ~ (fla)t, and
I(~,fl, 7 ) = 2
(11)
-
I
:r
(_3o~-)j
] [
89
exp 2 ~ [ 3 ( 3 a f l + ? ~ ) - - T z ]
]
9
9cos
(-- ~'~-- 3zefl)~-- ~t ,
which holds for 73+ 3aft< 0 and ]3aft + 72[ >> 5(f14)t,
I n the 7-+ 0 limit, eq. (12) reduces to the result obtained in (2)
2~
(12)
I(zr fl, 0) = ( ~
~g
A i (3fl)h "
Because of the behaviour of the Airy function, ~ "~ 0 plays, in the 7----0 case, the
role of an effective threshold for the reaction. I n other words, the reaction yield is
drastically reduced for positive values of a/(3fl)~. According to eq. (7), it is clear that
the effective threshold corresponds to a crossing point of the energy curves in the entrance
a n d in the exit channel. The variation of the form factor (? # 0) changes this effective
threshold.
The method we have sketched here can be extended to / # 0 transfer case9 Furthermore, it is possible to use it together with semi-classical approximations to derive formulae for the differential cross-section and for the o p t i m u m Q-value as a function of
angle and energy. This will be shown in detail in a subsequent paper (s) as well as the
comparison with the experiment and with other methods (e.s) in the same spirit. I n the
following, we just give a few typical remarks concerning the neutral-particle transfer
case.
Let Z~Z s be the product of the charges and let us consider an exothermic reaction
Q>o.
I n that case, F I = F ~ = - F = Z 1 Z ~ e 2 B -~, a n d the distance :R is given by
1~ = (Z1Z2)e2E -1, where E is the incident energy. The coefficients cr fl, 7 (see eq. (10))
are now
(13)
r = ~--~,
fl =
6 (1 1)
--
,
r = 2F/~1
We define the optimum Q-value as the Q-value for which II[ ~ (or to) is m a x i m u m ,
or, in other words, for which ~1II2/~ = 0. Looking at eq. (9), and considering t h a t fl
is always small for neutron transfer, one can see t h a t large values of 7 can shift the
m a x i m u m of lit ~ significantly from the m a x i m u m of the Airy function towards the positive value of 3aft + ?~. Approximating the Airy function as in eq. (13), we find t h a t
QoDt -----0. This corresponds to the classical result. However, if 7 is small, the effect
of the q u a n t u m nature of the relative motion can be important. Indeed, in t h a t case,
the m a x i m u m of o~2 lies in the vicinity of the m a x i m u m of the Airy function. Approxim a t i n g the latter b y a Gaussian, we find
(14)
(5)
(e)
(~)
(0
Qop~
J. CUGNON: to be p u b l i s h e d .
Z. E. SWITKOWSKI, R. •. ~VIELAND a n d A. WINTItER: Phys. Rev. Left., 33, 840 (1974).
TER-]VIARTIROSYA.'r ~urn. ~ksP. Teor. Fiz., 29, 713 (1955).
W. E . FRAHI~: Nucl. Phys., 2 7 2 A , 413 (1976).
SUB-COULOMB TRANSFER REACTION B E T W E E N HEAVY IONS ETC.
365
All these three terms are due to q u a n t u m effects. The last term arises from the difference of relative motion between the entrance and the exit channels. That is the only
term given in ref. (2). The first two terms originate from the form factor. The first
one looks like the expression proposed in ref. (e). However, it contains the influence of
the sign t**--/~1, which changes according as the particle is transferred from the heavy
or from the light ion. This aspect is absent from the formula provided in ref. (~) in the
neutron transfer case.
Another interesting point is the energy dependence of o) at low energy, a question
which has an interest of the astrophysical point of view. The relevant values of Q are
the positive ones. Otherwise, ~o is drastically reduced. Since fl is small, ~2 + 3aft is
expected to be positive and large enough in order that expression (10) is valid. We
expand in power series of 3afl/? ~, and get, to the second order,
Q~~1
where ~ is the Sommerfeld parameter. This expression is slightly different from the
one proposed in ref. (s), b u t it also predicts that the transfer reaction yield can be
larger than the compound reaction yield at low energies, provided that the Q-value is
not very different from the o p t i m u m value.
I n conclusion, we have worked out analytical expressions of the reaction yield for
heavy ions induced transfer under the Coulomb barrier. They have been obtained
under acceptable simplifying assumptions. We have discussed some aspects of the
neutron transfer case. We t h i n k t h a t our formulae could be useful for qualitative
discussions of several aspects of the transfer reaction. Among these, let us mention
the Q-dependence of the cross-section.