LETTEI~]~ A L I~UOVO CIMEI~/TO VOL. 19, N. i 0 9 Luglio 1 9 7 7
S u b - C o u l o m b Transfer R e a c t i o n b e t w e e n 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 h e a v y ions under the Coulomb barrier plays a n interesting role i n recent studies of nuclear as well as atomic physics. I t is usually studied in the frame of the D W B A 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 o p t i m u m Q-value is n o t clear. I t is t h e n desirable to o b t a i n accurate approxi- mations allowing to calculate the D W B A integral in a closed analytical form. The W K B a p p r o x i m a t i o n of the wave functions for the relative motion almost fits this desire. Recently (2), another m e t h o d has been proposed, which lies on a suitable approximation of the relative m o t i o n wave function (which is described below) a n d on neglecting the radial variation of the form factor. The latter approximation is satis- factory for the charged-particle transfer case, where the effective domain of integration is very limited. Here, we go b e y o n d this l i m i t a t i o n a n d still get analytical form for the t r a n s i t i o n p r o b a b i l i t y .
Our starting p o i n t 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 a n d Z2 are the relative-motion wave functions i n the entrance a n d exit channels respectively, and where r is t h e relative distance between the ions or between the cores, since a t 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) ---- _V exp [-- 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 N u c l e a r 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.
3 6 2 J. CUG~O~
I n order to have an a n a l y t i c expression for the integral (1), we invoke the fact t h a t most of the c o n t r i b u t i o n 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 a n d 2 b y c o n s t a n t gra- dient potentials in the region of interest, it is to say b y a p p r o x i m a t i n g the potential energy curves b y their tangents. We choose the t a n g e n t i a l p o i n t as the one corre- sponding 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 m a i n l y takes place a t R. We neglect the v a r i a t i o n of r -~ i n the form factor a n d make i t equal to R -x, since its v a r i a t i o n is n o t i m p o r t a n t on t h e effective domain of integration.
The transition p r o b a b i l i t y is given b y
C3) co
~'=N~
(p. p;)~ a2(p' ) @ d p '
q 2where the a / s are the F o u r i e r transforms of the Z's. F o r constant gradient potentials, the wave functions
aflp)
are given, with the right normalization, in ref. (s)(4)
1
a~(p)
-- _ _ e x p(E-- U~)p-- ~
,
where U 1 a n d U s are the w~lue of the potential-energy curves r a n d r at p o i n t
R,
a n d where/~1 a n d F 2 arc m i n u s 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 i n t e g r a n d is peaked a t p'_~ p. Hence, we expand the a r g u m e n t of the e x p o n e n t i a l i n power series of ~ =
p ' - - p
and stop it a t the first order. W e obtain, after integration on ~,+m
(6) ~o = 2~8~2-~-1F2/~2 exp
[i(~p + flp3)__rp2]
,
S U B - C O U L O M B T R A N S F E R R E A C T I O N B E T W E E N H E A V Y I O N 8 E T C . 363 w h e r e w e h a v e p u t
1
1(,
1)
= __ _ _ , , ~ - - 2F~/~2(7)
~
~
(E--
~ )
#
= ~ F~-~
~-~
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 + i y . I n o r d e r t o 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 w e c a l l I ( a , 8, 7), w e 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 s e e n 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) f l > 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 a r e 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 l e a d s t o t h e f o l l o w i n g r e s u l t s
If ["
]
is)
I(~,
#, ~,) = exp ~ ( 9 ~ r + 273) dp exp ~ (r ~ + 3 ~ / + i#p8 .--lID 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, ~') = ( ~ e x p [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 0 o ) I ( ~ , fl, 7) =
[ - , ] '
i 3 ~ ' ~ ~ ) e x p [3(3~r + F ~) - - r 2] 9]
9 exp 2 ~ ( 3 ~ + r2)! ,3 6 4
~. c u ~ o ~
which is valid if 3aft + 7~>> ~ (fla)t, a n d
-
] [
]
I
:r
89
(11)
I(~,fl, 7 ) = 2 ( _ 3 o ~ - ) j
exp 2 ~ [ 3 ( 3 a f l + ? ~ ) - - T z ] 99 cos (-- ~'~-- 3zefl)~-- ~t , which holds for
73+ 3aft< 0
a n d ]3aft + 72[ >> 5 ( f 1 4 ) t ,I n the 7-+ 0 limit, eq. (12) reduces to the result obtained in (2) 2 ~ ~g
(12)
I(zr fl, 0) =
( ~ A i(3fl)h "
Because of the behaviour of t h e Airy function, ~ "~ 0 plays, i n the 7----0 case, the role of a n 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 t h a t the effective threshold corresponds to a crossing p o i n t of the energy curves in the entrance a n d in the exit channel. The v a r i a t i o n of the form factor (? # 0) changes this effective threshold.
The method we have sketched here can be extended to / # 0 transfer case9 F u r t h e r - more, it is possible to use it together with semi-classical approximations to derive for- mulae for the differential cross-section a n d for the o p t i m u m Q-value as a function of angle a n d energy. This will be shown in detail in a subsequent paper (s) as well as the comparison with the experiment a n d 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 c a s e .
Let
Z~Z s
be the product of the charges and let us consider an exothermic reaction Q > o . I n t h a t case, F I = F ~ = - F = Z 1 Z ~ e 2 B-~,
a n d the distance :R is given by1~ = (Z1Z2)e2E -1,
where E is the incident energy. The coefficients cr fl, 7 (see eq. (10)) a r e n o w6 (1 1)
(13) r = ~--~, fl = - - , r = 2F/~1
We define the o p t i m u m 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), a n d considering t h a t fl is always small for n e u t r o n 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 t h e posi- tive value of 3aft + ?~. A p p r o x i m a t i n g 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 n a t u r e of the relative motion can be i m p o r t a n t . 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. Approxi- m a t i n g the latter b y a Gaussian, we find(14) Qop~
(5) J . CUGNON: t o b e p u b l i s h e d .
(e) Z. E . SWITKOWSKI, R . • . ~VIELAND a n d A. WINTItER: P h y s . Rev. Left., 33, 840 (1974). (~) TER-]VIARTIROSYA.'r ~urn. ~ksP. Teor. Fiz., 29, 713 (1955).
SUB-COULOMB T R A N S F E R R E A C T I O N B E T W E E N H E A V Y IONS E T C . 3 6 5 All these three terms are due to q u a n t u m effects. The last t e r m arises from the differ- ence of relative motion between the entrance a n d the exit channels. T h a t 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 n e u t r o n transfer case.
Another interesting point is the energy dependence of o) a t low energy, a question which has a n 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 a n d large enough in order t h a t expression (10) is valid. We expand in power series of
3afl/? ~,
a n d 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 t h a t the transfer reaction yield can be larger t h a n the compound reaction yield at low energies, provided t h a t the Q-value is n o t 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 i n d u c e d transfer u n d e r the Coulomb barrier. T h e y have been obtained under acceptable simplifying assumptions. We have discussed some aspects of the n e u t r o n transfer case. We t h i n k t h a t our formulae could be useful for qualitative discussions of several aspects of t h e transfer reaction. A m o n g these, let us m e n t i o n the Q-dependence of the cross-section.
