The influence of the real nuclear potential on the heavy-ion elastic cross section

["-'-2"[N

t

Ne.. &,or Pkvsics.

A239 (1975) 134-.-156; !~'},~.NorlkdIo~*la*;d P.~bdiehm# Co. Am, teed:~,m

Not to be reproduced by pbo~orrinl or microfilm ~i~hoe~ ~ri~te-r~!¢~m~i~k~e~f~m ~he t'~ubti:a~e~

THE INFLUENCE OF THE REAl, NtJCLEAR POI"EN[IAI. ON THE HEAVY-ION ELASTIC CROSS SE(7IION N. ROWLEY Institut de P/o sique Nucldaire, Division de Pk.;,sique Tkdcvique ~, 9M06 Or.vay, F?~mce

Received 30 May 1974 ',Revised 31 July i974) Abstract: Using :1 simple but reasonably realistic form for the lor~g-range p,.tr,: of the real m~¢lcusnucleus potential some simple approximations are d,.ri~cd A~r the nuclear paarl,; of Oie WKB pha=e sh;ft and classical deflection function for ~he ela5"ic ~,ca{termg of ~wo l}eavy i,.~:>,, q'l}a formulae are presented in the form of tmiversal Am. tion5 ~ h~:h :dlow a~~r~p.idc,,,d~Ia~o~ ot ~hc approximate "rainbow" at.gle and angular momentt~m, l'hc ;vm~c:ical s~mplic~ty of d_~i~h:nm of the ndclear phase allows a rapid determination of the a,.,"c' 9<~:.ding cross :.cctk)r., t,~ng ~b.e partial wave expaw;on. The cross sections show a sh~f~ t< '.e ~uar~er-pomt Io fbrward anees with respect to the corresponding cross section in ~t=* "b ~ ~: teal nuclear potee, tiat is ignored. This suggests that critical angular momenta ~., 'tuatev, t~m..~ the usual qv'~ter.point recit~ will generally be . ,,o be dominated by absorptio~ rather than by the real poten~ia!.

I. lntrodt~c~:ion

Several p u b l i c a t i o n s [e.g. refs. ~- 4)! have s t u d i e d the effec~ o f ~he real p a r t ~,i"the n u c l e a r p o t e n t i a l on the elastic cross section u s i n g s e m i c l a s s i c a l technique:< T h e semiclassical t h e o r y o n l y r e q u i r e s k n o w l e d g e o f the cIassicat d e f l e c t i o n ;unctio[t lb'. the p r o b l e m a n d this is often a p p r o x i m a t e d b3 s o m e p a r a m e t r i s e d [i:~rm }"or a s~rongly a t t r a c t i v e n u c l e a r p o t e n t i a l the cfas~icat dcflectioT~ functiot~ ~ilI e x h i b i t a "'rain° b o w " 5) (a p o i n t w h e r e the deflection f u n c t i o n is q a t i o n a r y a ~ & ~espect to the a n g u l a r m o m e n t u m ) . In this case the p a r a m e t r i s a t i o n o f F o r d a n d Wheele~ a) ( o r s o m e m o d i f i e d f o r m o f this ) m a y be useful. T h e o ~ e c t o f the p r e s e n t article, howe',er~ is to r a k e a s i m p l e ( b u t not t o o unrealistic) m o d e l for the reaI p a r t o f t h e n v c l e a r p o t e n t i a l b e t w e e n t ~ o nuclei a n d va use this m o d e l to d e r i v e d i r e c t l y a s i m p l e a p p r o x i m a t i o n for the n u c l e a r p a r t o f the W K B p h a s e shift w h i c h will t h e n e n a b l e the r a p i d e v a l u a t i o n o f ti, e e l a s t i c c r o s s section by use o f the p a r t i a l wave e x p a n s i o n , T h e s e p a r a t i o n o f ti:e p h a s e i n t o n u c I e a r a n d C o u l o m b c o m p o n e n t s will a l l o w us to use the e x a c t C o u ! o m b p h a s e s in o u r c a l c u l a t i o n s a n d will t h u s a v o i d ~he a d d i t i o n a l a s s u m p t i o n s o f the m e t h o d o f s t a . Laboratoire associ6 au CN RS. 134

ELASTIC (?ROSS SECTION

~35

l~onary phase umally found in semiclass~cal calculatim~s of scattering cross sections. The separation will also allow us to s,oo explicitly the effect of the real nuclear po,'ential by evaluating the corresponding cross sections with zero nuclear phases. From the '~/KB nuclear phase it is simple to calculate the nuclear contribution to he classical defl,:ction fimction and again we know the Coulomb contribat!e~ exact!y. Fhe separation n t o two components assures that at large values of the angul ~r menentum t the ~!eflection function just has its Rutherford value. This wi~! , ~ i',,,~ r u e for a deflection function such as that of Ford and Wheeler which, in order- ~ give a good fit ~'~ the region of the rainbow angle, may deviate qui~e strongly from the Coulomb branch at large /-value., 2). ![n order to improve this sittmiion cme ~ould have to fit the nuclear branch of the deflection function as well as po-,s{b!c and then lit this on to the Coulomb brmz':h at some appropriate point. A further advantage of the present method is that we shall be able to write bo~h 1he nuclear phase and nuclear deflection as universal functions. This will enabi,: u~ to easily evaluate the rainbow angle and acgular momentum and wi!l also demor,tra~e the relative importance of the nuclear potential in different reactio[~s, We shai~ ~ee that in this respect the important quantity is just the wave number k of the relative motion. 2. The nuclear potential and the critical angular momentum In general the nuciear potential will cont fin both real and Hnaginary par,:~, t~or ~arge separations the real part of the potential is strongly attractive and wilt, there!\~re, give rise to a barrier when added to the Coulomb potential. For an incident energy E .~bove this Coulomb barrier, partial waves with small/-values wi':lt be able te surmount the potential barrier and penetrate deep into the nuclear interior, It is ass~,~med that in this region the imaginary nuclear potential is large and that all such partia~ waves are essentia!ly absorbed from the elastic scattering channel (this does ;~ot necessarily imply compound nucleus formation but simply that the nuclei wil~ undergo sonic non-elastic event). For sufficiently large values of l, however, the add[tion,.~d repulsive centrifugal term in the potential makes the height of the barrier g:ea~er than E and we assume that such partial waves are essentially reflected. It has been shown 6) that the Coulomb barrier usually occurs at rather ktrge separations of the two ions and we shall assume that in this region the absorptive part of the nuclear potential is negligible 7). The transition between reflectim~ ;rod absorption will then be determined by the barrier: penetration factor. We shal~ ~s~: the formula of Ford et al. 8) to give an estimate of the number of partial wave~, o:er which this transition takes place. For each incident energy E the nuclear potentia~ determines a critical angaktr momentum t, for which the ileight of the barrier in the total poter.,tial is just E~ We shall see that the transition from small absorption to almost total absorpiion ~ake, place over a few/-values around l,. Since this transition will generally be rapid, 1hc~

136

N. ROWLEY

if the effec: o f the real nuclear potential on the partial waves 1 > i, is s m a l l the corresponding, cross sections should be rather similar to the so-ca~qed sharp-oat-off cross sec ...... ~Jf Blair 9). However we shall show that if' ~he wave ntmtber o f the reaction is large then a large number o f partial waves t > 1,, are appreciably affected by the nuclear potential and it is the effect o f such partial ~aves that we shall consider here. The real nuclear potential may be represented in several simple way~: S a x o n - W o o d s optical potentials, folded potentials < 7, to), etc. However one feature c o m m o n to ahnost all ion-ion potentials is that for large separations their dependence is exponential. As we haw: noted above, the barrier occurs at rather large separation-, and since we only need to know the potential for separado:~s beyond the barrier .,e ~haH take it to have just an exponential t\~rm. This potential will be bad in the nuclear interior (being much toc~ deep) but since all partial waves which penetrate riffs region are assumed to be abaorbed this ~i]i not affect our discussion. We shall, therefore, ',ake the nuclear potentia! l/s te be ?i-,cn by

l,~

=

-

|'~, exp ( - r/i'),

(1)

where r is the separation of the nuclei, Vo is a p, ,;ti~e comdant and the surly, co thickness T is a constant. A value of T o f a b o m 0.6 fm has [~::en |kmnd ") to be reasonable for the folded potcmial and we shall use this ~ alu,: her< Since the cridca angular nlomentum I~ essentially detern>ir~es the position at which absorption cuts off the lower partial waves it will also roag'. 4, determine ~he posidon o f the fall-off of the cross section from its RutkerPord va!ue ~~). We have, theteik)re, a reasonable idea of its value from the experimei~m} cro~,, -ecth.)n aad ,h~dl write our formulae in such a way, that ( parameIrises Vo, i.e. ~ = -- t},(()exp(.~;-;TL

(2)

[see eqs. (39), (40), (41)]. For smaller separations of the nuclei the po~cn~mI ~iil bc ~ome,aLa~ flatter thaa exponential and if the barrier occurs in such a tegim~ it may be desir~ble to us;' a~, effective surface thickness larger than the value quoted above. Ho,~ever our aim is ~o study the relative importance of the nuclear potential from r,":ction ~o reaction raffler than to obtain a fit to any given reaction; since we shah see that the dominant quand~3 in this respect is the wave number (which may change appreciably from reaction t,., reaction), the use of a fixed value o f T will give us a good qualilatb, e idea of the~,e variations.

3. The WKB phase and classical defiec=tio~l function In all semiclassical calcalations of the ion-ion cro~s sectkm the vaiidib: o f the W K B approximation is assumed. The reason for the usefulness o f this assumption is iSsimple relation between the W K B phase ,~'~B and the classical deflection fu~ctior~

ELASTIC CROSS SECT~Or',~

U~7

O1 = 2 &~WKR 81

(31~

[ref '*")t

The catculalions then often proceed by taking some parametrised t'or~ i~; O~ ~ deriving the phase from this by use o f e q . (3) [refs. l, z)]. The use of the m e S ~ *,~" stationary phase then tells us that the main co~atribt~tion to the cross sectio~ at ~~ angle 0 should come from partial waves which would classically be scattered i~ t[~,: region of the angles + 0-2~zn. It is possible then to talk about diff,:rent contribt~i~:~,, to th~ scattering amplitude coming from different "branches" L z, .s) of the det]ec~i~,,~ function. However the use of a parametrised deflection function takes an approximate i'~,.,~ for the pI~,ase %r all partial waves including waves not affected by the nuclear p,~,~ temial. In other words we must throw away our exact knowledge of the CottloH~b phases in order to take inlo account the effect of the nuclear potential on a relative!2, small number of partial ~aves. In ref. 2) it has been suggested that in order to co~,, pensate for this effect at forward angles, the contribution to the scattering amplit table coming from the Coulomb branch of the deflection function may be replaced by just the Coulomb amplitade for all partial waves. It is clear however that if the contribution coming from the absorbed partial waves (see sect. 8) at the angle con~ sidered is large, such an approximation will not be good. We shal:,' explicitly ex~r:,:*t this contribution in our calculations and show that for small values of the wa'~e number, the forward angle oscillations are dominated by the term. We shall take the opposite course from the deflection function repre~e~ati(~ an8 shall derive directly an approximate form for the nuclear part of the WKB phase~ With the assumed potential it is po~;sible to derive the approximate phase in the form of a universal function which may be used directly to calculate the cross sectio~ b> the partial wave expansion. While we shall not use the cla~;sical deflection function to calculate an~ cro,,o section;, it is sti!l an interesting quantity since it gives a clear pictork~l representatk~ of how stro~.gly the nuclear potential affects the scattering. With the prese~'~ ~ * ~ of the phase it is possible to derive ,~=iversal functions for the nuclear part v,f ~ c classical deflation function and its derivative. These will enable a rapid es~ima~io~ of the rainbow angle (the angle for which the deflection function has a maximt,~ val~e) and the corresponding angular momentum without having to actually cak~;I,:~c the deflection function.

4. Validity of the WKB phase The WKB phase for the motion of a particle with angular momentum I i~ a po~ teat~al V(r)is defined t3, 14) by =

f; o

[k,(,-)-k]d,-,

138

N, R O W L E Y

v'here the instama~ :xms wave number kz(r ) i>, give~ by k,(,-) = [12m(/:2~-. V ( r ) - 7 ~(r))~/&

(5)

and the inili , wavenumber k is

where E is the c.m. er, e,gy of the sy:stem and m b, h~ reduc~:d ma-~. Tile distance r¢~ is the classical turning point for the rnotion and is, therefore, the ~ar~,est r ~ l of the equation e - V(,')- J ",(,') = O. (7) where ;#'~(r) is the centrifugal potential % ( , ' ) = h2l(l+ 1)/2,,,,': ~ h~'t~/2,,,-~.

(8)

Since we shall only consider cases where l~ is large we choose the at~ove approximation for its simplicity rather than the more exact and more c,.mven~io~aI Ibrm containing (1+_})2. The validity of the WKB phase depends o~ two a>ump~om,. Firstly that Oec potential V varies rather slowly over the de Broglie wa, e~ ,,,~~b ~0r the motion and secondly that the e~ect of barrier penetration i~ small. The first condition may be expressed as V'! ...... i < k.

Vl

(9)

where V' is the derivative of ~.he potential V. Since we :~re ? n y concernea with the motion on the far side of the barrier, V' is always less th~r~ the derivative of the Coulomb potential, since in this region the nuclear |orce has n )t begun to dominate~ Since the barrier occurs at large separations there is little overlap of the charge dis. tributio as at this point and we may write the Coulomb potentiaI ~: as |.~. = Z~ Zzea~; -,

(10)

where Z~ e and Z ~ e :,~re the charges of the two nuc!.ei, ln~crtmg )'~.: into eq. (9) gives

kr >:l.

(ti)

Two quantities which are useful in the analysis c f Coulomb scattering are the Coulomb parameter ~l and the half-distance of closest approach in a head-on collisiom a. These quantities are given by the equations tl = Z~ Z2 e~'/h~,

(12)

a = Z t Z 2 eZ/2I:: ",

(13

where r is the relative velocity of ti~e ions at large separatk:ms,

ELASTIC CROSS SECTION

139

In terms of these quantities we may write the wave number k as k = q/,~.

(14)

The distance of closest approach ;'1 (in the Coulomb field) for an angular momentum I is given by r, = a{I +[1 +(l/q)2]~} = a(l +cosec (½0,)),

(15}

where 0 t is defined by i = q c o t (½03,

(1,b~

and is just the deflection function in the Coulomb field, i.e. 0 l = O~

=

2 arctg(q/I).

(17}

tf condition (11 ) is satisfied at the barrier position %, it will be satisfied at all larger separations. We shall see that the barrier position is very little different from r,: (the subscript c will be used to replace the subscript l~) and we may rewrite eq. ( l I L using eq. (t4), as a/rc < q.

(IS)

q(1 + cosec (~0¢)) >.'.- 1.

(t 9)

Using eq. (15)one then has

It is impmtant to note thin in eqs. (t5), (16), (19) and in what follows that 0~ is just a convenient method of expressing functions of l and does not denote a particuIar scattering angle (just 0). In particular 0¢ will not necessarily be tile quarter~point O~ of our cross sections. We see the condition (19) on the Coulomb parameter is not ,'ery restrictive smcc cosec (½ O~) > I.

5. Barrier penetration The second condition on the WKB phase is that the barrier penetration l~lctor should be small. The reason for t>is is that if the penetration is large tae phase (4) must be corrected in order to match the wave functions across the Cou!omb barrier s), If the top of the Coulomb barrier is parabolic we may use the expression of refl ~} tk~r :he barrier penetration factor TE (above and below the barrier) as a function of the incident energy E {2(h T e = [1 + exp (2n(E-Eb)//w9)]-' in eq.(20)

h~o = ( - V"lm)~, where V ~ is the curvature of the top of the barrier, and E b is the barrier height, The critical angular m o m e n t u n Ic is just the value of I for which the barrier height

140

N, ROWLEY

is equal to tho incident energy and is determined by the equado.qs 6} ~ ( " 0 + S "~("0 + ::,0",,) ~ E,

(22)

.<~ (i'{ + 7 " + l';~),, = 0.

(23])

~jl o

Sin~.e the nuclear potential is exponemial lhe nuclear co~m'ibution ~o V"' i:, g,'eater than the Coulomb plus centriftigal conlribution by a factor of abotll/'t,:¢f, Tl;tcreR)re V"(#.b) ~ I~'(%) = 7l ( 1%' + 7 " <),,.

(24)

Since % will not be much different from r~ we obtain 2E .... sin (~0,.)/(1 + sin (~0~)):':. Ta

-V"~

h,o ~ 2-~(a sin

(½0<)i7")~i(i++:!n (,!,9,)),

(25)

(26)

Although tl,e energy/=7 is fixed in eq. (20) the: bareier J, iF'~t :], change-. "t°h*:change in Eb as one goes from ( to 1 is roughly

AEt, = 2 , ~ ( t - I

'(i,,

{27)

Therefore as a f~ nction of l the barrier penetration ! ] may be written

h = [1 +exp ( ( I - l~)iAfw)]-'

(28)

where

'(7¢

a,,,, = 274

,, (i +sin (~e~,))~ ~ec (~¢).

(29)

For energies high above the Coca!omb barrier 0, becomes small and the angular factor in eq. (29) becomes unity. We then obtain the formula given b,, Brink *e'). Since the barrier position ( ~ re) does not change much from rea~:lioa to reaction 6) (being of the order ofro(A~+A~), where ro ~ 1.4 fin arid A~ and el~ are ttle nuclear masses) {hen Atw can easily be estimated from the quarter-point of the elastic cross section. For 77c~ v 0.06 we obtain Alw z 0.65(1 +sin (½0~))~ sec (½O~).

(30)

So long as 0+ is not too large (i.e. E not too near the Co~,~!m'nb barrier) then Aa~, is fairly small, being about 0.9, 1.2, t.8 ['or quarter~poi ats of 6t;i~, 90:, 120 ~respectively. The distance Ar from the top of the barrier to the distance of closest approach for the partial wave lc + t is given by

(AdTf"

~- 2 c:,s (½0o)al,,7:

(3t)

ELASTIC CROSS S E C T I O N

l,iI

The ratio Ar/T must be small for the barrier to be considered parabolic, therel~,)ce ~e need qT/a cos (½0¢) = kT sec ()-0¢) 7; 2. (32) We shall see that the quantity on the left of this inequality wi~! occur elsewhere i~ the work. The inequality is satisfied in the cases we shall consider. Since A~w is small the WKB phase should be reasonably good for aH b~t a &~;v partial waves in the region of I¢. However if only a small number of partial wave~ above l~ are appreciably affected by the real nuclear potential the use of tk~:. WK~ phase is bad. In this connection it should also be noted that ~in this case, semicla~<,~cx~ calculations of the cross section using a classical deflection function are rather dubio~>~ We shall see later, however, that for sufficiently large values of the wave rmmber of the reaction the number of partial waves affected by the real potential is m~a~b greater than the number affected by barrier penetration. We can in this case obtain a reasonable idea of the effect of such partial waves by using the WKB phase ;rod b3 usieg a sharp cut-offofthe partial waves below l~. In terms of the reflection coeflicier~s q~ (i.e. the moduli of the scattering matrix) v,e shall take th = t fbr I > 1~ and *L, = O for I < 1¢. 6. Evaluation of the phase and deflection function The WKB phase has been defined in eq. (4). We shall, however, not use this ftt~l expression since we know exactly the Coulomb phases 6c [ref. 1,:~)] 1

(~3)

6 c -- 6oc+ ~ arctg(q/j), j=l

where 6oc

arg (F(1 + iq)).

~'~~

We shall, then, just calculate the nuclear part of the WKB phase as the difference between expression (4) with the n~clear potential and without the nuclear potential ~e~ =

r

2,,,(e-

Vc - 7 l ) )1 ' d r -

f.

. . . . . . . ( 2 r e ( E - tc - 7 .~))-~dr , -ii

. .

(~

,<.

The distance r~ is given by eq, (15) while ro is given by

E = vd,'o)+ t 7(ro) + vN(,'o),

t:36)

Since the nuclear potential varies rapidly and is only important in the regio~ of ~tte distance of closest approach we can replace I/c +'¢"t by its first-order expait~ixm about r~, Vc(r) + 't',(r) = E + (1~ + 7,"-;),,(r- r,). (37) The nuclear potential is given from eq. (23) as VN = ' T ( ~ + "¢";),b e- 4'-' ",. r.

(38 }

142

N, R O W L E Y

Since the barriv~ position will not be much d/fl?rent from G ~e may repki;~ V~'.+ '~....¢ in this equa':on by ds value at r~ to ~)btain the nuclear potenti~d I,;~ =

T(I.~ + ~'";). e- ~' -'~"

(39)

~

= - l~; e- ~' " """ ;

(40)

where I/.

2TE

.

.

.

.

.,

. . . . . . . . sm (~O:)l(l-~- sin (~0~W.

(41)

a

This is then the equation which enables one to write Vo in_ ~erms o f l~ (i.e, 0:) (see sect. 2). To obtain eq. (41) we have simply used eqs. (8), (10), (15) in eq. (39i The nuclear potential written in such a form was first u~ed by Blair ~"v) Since only partial waves in the region o f t~ m°e st?~ ngty afl;2-cted by ~he tmclear force we may also replace Vc+ ;(-"~in eq. (37) by its v~:lue at q . In,,~crting thi~ and eq. (39) into eq. (36) one then obtains for ro r t - r o = T c " ~'~-''~:;'

(42)

For 1 = I¢ u e have r~ = r b and, thercibre, as a!~,;~ fo~r, ~ b-' Pk~ir., rb = r.-7:

(43)

fl = e~i~-.,~- !,

(44)

fi = (,.,_ ,.o)/~r:

(45)

- ~. :: T.

(46)

Eq. (42) thet~ becomes where

I f / i s not too different f r o m t¢ we may write

.x. = -z: i ( t - t~i/T = (I-- t j / ~ . (J~

ilc

~lTla

:os

f47)

where •J =

(~0<) :-~- I;'/" ~,.-c (~0,;).

t4S)

This is just the quantity we foun,+ in incqu,'lity ( .T. >) ' ~v) as lhe . anti is given by. B~a.~r width in/2space corresponding to a distance .F in configuration space. Using the above approxi:r.,ation for Vc+ Y'~ tnd making the substitution r-~'~ = y T

(49)

in eq. (35) one obtains where

T ' ~ (1 + s i n (½G))"

(sl)

ELASTIC CROSS S E C T I O N

= 2.

-,lj

143

e

)--y-]dyj..

(52)

Given an~ experimental cross section we can immediately obtain an idea of {t;e importance o f the real nuclear potential from q, k and 0~ (approximately the experimental quarter-point) since ~ and A are determined once these quantities are specified. For reactions with similar quarter-points the importance of the potential wil! i~crease with k and q (i.e. for reactions between heavier ions). While eq. (44) is explicit for x it is, unfortunately, implicit for [I. Since each :,..~t{~c of the angular momentum corresponds to a different value of x we must lind fi (r) numerically. However, the solution may be evaluated ;ather simply. If the softltio~ for the partial wave It is fll(xt), let the solution for thc partial wave 11 + 1 be tL The~l, since fl will not change much from one partial wave to the next,

where 6 i~, small. If we then calculate the quantity

132,

fie = e t q - x - ' =:, 8e~ ~ f l ( l + 6 ) ,

(£i)

/L, =//0 +& -t0,

(5:9

/3 = ½[(1 + f l , ) - ( ( 1 +flj)2_4//z)~].

(55)

t!hen

This procedure may then be iterated t:9 give the desired accaracy. Only about two iterations ~-e usually required to give fl to an accuracy of about 1 in !0 ~, Of courw for l = / ~ we know that fl = 1 and we shall start with this value for l = ( + 1 Fig. 1 shows *he behaviour of/l as a function ofx. For large values of.,.., we have fl = e - ~ - i

(57)

We can see that the function [1 decrea~,es very rapidly near 1 = l,. For example fi:~:' A -~ 3,/~ is already0.38 for l = /¢+1. Even for A ~ 40, f l i s a b o u t 0 J 9 f o r / = /.:+ i. This suggests that an expansion of I(/?) in terms of[I may be good.

0 72",

/~ = e×p (:3- x -I )

C 50 0 25~

000

05

110

1.5

2.0

Fig, 1, The l~,mclior~ ~(x) (s~;e eq, (44)) which giv~.s the change in the distance of ciosest approach due to the nuclear potential {eq, (45)) as a ~ n c t i o n of the angular momentum (eq. 47 q

144

N. R O W L E Y

I

[

,

l

"\'%\

r) 25[-

..... i

ZN

'

Fig. 2. t he cxacI expression (eq. (52)) and series (eq. {58)) for l{fl) are compared, -]-hr W K B m~.~.ar phase i'; j u q propcrtionat 1o lf/J~

From eq. (4~) we obtmn up to Rmrth order i_n [L

l(fi) = f i { ~ - ½ ( 2 - , / 2 ) f i - - ½ ( 2 x / 2 - 1 - , - \

3)t3'-~.~

3-6,~2-~7ii¢s~.

(5~)

Fig. 2 shows the exact function l(fl) obtained by nup,~erica~ integration of eq. (52) and the value given by the ;,eries (58), for the entire range ot fi-value~ (one to zero). We see in fact that h e error in the series is only about 3 for ¢~ = 1 (i.e. [ = !~). For l = l~ + 1 the error is entirely negligible even for A ~ ~,G The main errors we have in such a model, then, come fi~n neglecting d~e barrier penetration. Since this only strongly affects the partial wa~ cs ne~r 1,, and ~inc~: lhe phase varies rapidly in this region s) (and will not therefotc give any coherent addition of the error) we may expect our model to give a good idea of the gross effect of ti,e nuclear potential. The results may, however, not be reliabte if the number o f partial waves strongly affected by the nuclear potential ( ~ A) is nol greater than the number affected by barrier penetration ( -~ Jut,). tlmsevcr in thi~ c:a,z ~he effec~ of ~i;.-e real potential will be small anyway. From eqs (3), (44), (47), (48) one can easily derive the nuclear part of the classical deflection function O~ and its derivative using the relation

We obtain =

J(in,

-

A

(60)

ELASTIC CROSS SECTION r

I45

r

& ) N : _ L!: I ×~ .3 ~( ,

~00

I

J(:~)

[

I I

r L .......................

0

J~

o5

IT0

x

~5

J

2,0

Fig. L l h e umvers;~l functions J and K which give the nuclear deflection and its derivative are ~,ho~,'m ag functions o f .r (see eqs, 1"47), (60), (61)). The vert~cat axis is logarithmic.

f ~

e - 3,

J(fl) = n-~/3 o ivZi}-(i-Ce~J'ii ~ . dy,

_ _~_J(/0 + _~-~-/3z ( ~ K(B) - 1-/~

2(1-/0Jo

(62)

e-~'(l-e-~') ( y - / / ( 1 - e - Y ) ) ~-dy.

((',3)

The thctors of ~ appear in eqs. (51 ), (62), (63) so that L J and K all tend to fi as I3 tends to zero. In fig. 3, J and K are both shown as functions of x. We see that tbr any reaction the functions L J and K are all universal functions of ~ and [1, This prc, perty is useful when considering the extent to which the nuclear force wilt affec~ the scattering.

7. The rainbew angle and angular momentum The above expressions for the phase and deflection function only relate to the nuclear potential. However from the way we have defined the phase in eq. (35) we can obtain the total phase and deflection function by the simple addition of the well known Coutomb quantities defined in eqs. (17), (33). Fig. 7 shows the Coulomb deflection function O c and total deflection function O~

6)t = o C + OF,

(64)

for ~1 = 18.I,/¢ = 24, a = 3.8 fm, T = 0.6 fro. Also shown is the Ford-Wheeler t}

146

N. R O W L E Y

deflection fi~r~ction O w' which gives the best fit in ,he region of the rainbow angte 0,, of

..... o,-,,

[,o(;-' ,,Uii?J _

~,;-,.-~e 0~ = 63.03, p = 4.5 :>, /~ = 24.0, /~ := 26.75. It i~ evident that the first two partial waves of the nuclear branch and ~.he ('o~,lomb branch for O ! 5 0 are given rad~er badly. Either one of these branches can only be hnpro,.ed at the e×pa3se o f the other. As mentioned previously one could improve the fit by titfi~g ¢"qb' ,-,::~. the nuclear branch and then joining the deflection function on ;o the Cot lo ~ branch. There is, however, no unique prescription for doing this arid it is okviously mo~e satisfactory to have a universal function which tends automatically |o the C3ulomb branch for large I. From eqs. (60). (61) it is easy to calculate the rainkow angh: 0~ and rainbow angular momentum 1~ for a given/~. The rainbow angle is determined by '~ (O}: + O7) (?I

0

(66)

Since Otc varies rather slowly compared with O~ w,' ~r~ay rav it~ derivative at/~ into eq. (66) to obtain an estimam of & and hence i,. ~ :rag e~,. (61) gives

K(x,)

=

,":~T~ ~ sin (-½0,.)(1 + :~,~ (~0¢)) ~ ...................... ',,r,',,} cos i ~

:~

,,

(67)

and then i, = t~+x,~t = l,:+kT(.v, sec (.~¢)),

(6s)

where the subscript r refers to t~. The value o f % can be read from tig. 3 once re, flax b.'en e~atuated. This them gives

0~ = o~'C._ O~'.

(69)

...............

(70)

where

= :z2

(71)

and J(x~) can also be taken from fig. 3. We see from eqs. (67), (70) that x~ and O~ are just funcuons of (I)r,) and 0,. Table 1 shows values of x~ sec (½ 0¢) and O~ evaluated numerically (using 5g. 3) for T/r¢ = 0.06 and for a range o f values of0~. The factor sec (~ 0¢) has been included with % since it appears in the expression for l~. Note tha~ for a given pair of ions the number of partial waves affected by the real nuclear potential is greater for larger energies (hence smaller values of 0¢) since k increases with E and since x,~ .,.ec (} 0,) increases as 0¢ becomes smaller.

ELASTIC CROSS SECTION

!47

"lTn~Le 1 Evaluation of '°rainbow" quantities O~ (deg)

K(x~)

x,sec(~0~)

d(x0

O, ~ (deL~

20 40 60 80 If~ 120 140 160

0.05 0.12 0.24 0,45 0.85 1.7 4.2 17.9

2,2 1.5 I.l 0.74 0.51 0.34 0.18 0.12

0.04 0.11 0.19 0.31 0.48 0.70 1.0 1.6

1.3 3A) 4o8 6,5 i-~,? '<~,~ 8.,~; 7,0

For a ralio T/r~ -: 0.06, eqs. (67), (70) have been ~;olved with the aid of the functions A"and J g~¢a in fig. 3. The solution has been found for various values of 0¢ and yields the rainbo~ angular m ~ mentum (see eq. (68)1 and rainbow angle (eq. (69)).

8. The rainbow and sharp-cut-off cross sections

In order to see explicitly the effect of the real nuclear potential we sMIl calcu!ate the cross section both including and omitting the real nuclear phases, if we omit ti~c real nuclear phase (and hence the nuclear deflection) the corresponding detleclien function wilt be just the Coulomb deflection function cut off below l~. Theretk)~,: in semiclassical terms there is only cme branch which contributes at each scattering angle. Since the corresponding cross section will be just the sharp-cut-off cro~ section of Blair s) we shall labcl it sharp cut-off. We shall refer to the cross sectioa including the real nuclear phases as the rainbow cross section since ils deilcctio~ function contains a "rainbow". Note however that even this cross sectiot~ wil~ bc calculated using the sharp cut-off of the reflection coefficient defined in sect. 5~ As I tend:s to I¢ the classical rainbow deflection function tends to mir, us infinity a~d thus/~ is an orbitting partial wave. However we shall see that generally only this oac partial wave has a negative scattering angle; the deflection o~" partial ~aves ! > ~ i,; reduced by the nuclear potential but remains positive. The effect of this single orbiting partial wave on the gross structure of the cross section should, then, be sm~!!. On the other hand a fairly large number of partial waves in the region of l, !~avc ~t reduced (but still positive)scatteringangle and we shall see thai: this leads to a paliiag of the cross section in the region of 0~ to more forward angles. This effect may t~e~ lead to errors in l~ if deduced from ,:he experimental quarter-point 0 i usi~g ~I:.~e relation t~ = rt cot (ZOo).

~ ~

The nuclear potential gives rise to a maximum in the classical deflection functioa at the rainbow angle 0~, The flatness of the deflection function in ti~is region sho~tkl then (on semiclass~cal grounds) lead to an enhancement of the cr~,~s section reiativ~'

148

N. ROWLEY

to the R~_~thcrfbrd cross sectio,?. In fact the clas:sica[ cro~s scclion -~) at this point would be infinite. TLe degree of the enhancement witI depend on ~he t~umber ofpartiaI wave. in the region of 0~ and we may expect, therefore, thal this e~Mncement will he >.rge for la,'ge values o f k {e.go for the high energies ~eeded l~Z~rreactions between very heavy ior"). The semicla.:ficat evaluation of the ~,,catle,'mg cro:,'~ :,ection hing¢~ on ~he at>. proximation of stationary phase. This approxhaia6on i~ unlikely to ~ very gaod ~).~r the steep nuclear branch of the defleciion function. We ~hatL thereli:~r¢, calcutaie the scattering amplitude by use of the partial wave expansion (using the sharp-cut-aft of the reflection coefficient defined in sect. 5)

(73)

f(0) = ~-77:~. (21 + 1 )P,(~'os 0)e~,,,,, ~ " ' , + : ,,," 2ikl[

= f c ( 0 ) - 2it~ ~£o ( 2 / + 1)P~(cos 0)e :'a'~ + 2i~. 7 " ¢2t¢. t)G(cos 0}e:'"~(e ~"~'° o,- 1)

(74) = k:(0)-k~(0) + Mo).

(75)

In eqs. (~-l.), (75)j~: is the Coulomb arnpli~.ude ~

1 ~ (2,'+ l)&(cos 0)e >'~'¢

fc(O) = 2ik

........... ----!!........... exp ( - 2i,~ In (sin t ~0}))e 2'a~'~, 2k sin 2 (½0)

(76)

(where 3 c is givee by eq. (34)) and ihe angular momentum 1...... is chosen .so that 3~,#: 1

for

1 > i~,~,.

(77)

i.e. in such a way that the effect of the nuclear pode~ltiat on i~ighc~ partiat ~,~ves is negligible. In optical model calculations t,,.,~ is usual}y chosen by imposing some numerical conditio~ on the scattering matrix. For exampie Hodgson ~'*) g:ves the following condition for [ > /,,~

1-Re (V,e:~'% <_" ,,:.

(78)

where e is small. In order to satisfy such a condition we must have both

l - ~ h < ~,

(79)

26~ < (2e) ~.

(80)

if the imaginary part of the nuclear potential is small for separations gre:,ter *i~art

ELASTIC CROSS S E C T I O N

149

the barrier, then r/t is determined by the barrier penetration. Using eq. (28) ~ror large /-values we obtain ~e - " - t'~~/~" (81 } and eq. (79) becomes

!f:2)

lm,~ > I~+Aae(ln e - l - I n 2). lrtserting our real phase into eq. (80) (using eq. (57) for large 1) we obtain

or

lr,,~ > l~+A(½1n c ~+tn ~-1.35).

(84}

Hodgson ~9) sugge!~ts the value of 10 - a tbr e and eqs. (82), (84) then become

t,n~ > l~+8.5A~e,

(85)

lm~ > l~+A(3.26+ln ~).

(86)

So long as the imagirary optical potential is small beyond the barrier it is easy to calculate analogous conditions [or any numerical restriction on the scattering matrix. Since we have a fairly explicit form for the WKB nuclear phase then using re.cure rence relations for Pz (cos 0) [ref. 2o)! and 6c, the calculation of t)~,r rainbow cross sections is very rap~d and we can easily increase l~a~ until the desired cc::,'ergence in the cross section is reached even if the number of partial waves involved is large. TaNe 2 sho'~s the values of k, q and 1~ for ~he three model reactions we shall study below (labelled l, I], HI). Also shown are the quantities A, ~ and Al~p (derived from eqs. (48), (51), (29)using these values of k, ~l and I¢) and the value of 1,,~ givel~ by eq. (86). TABLE 2 Details of model reactions and values of lm,~ Reactior"

;: (fro- ~)

~i

t~

A

x

~l~,

,'......

t

4.7

18.t

24

3.6

t.36

1.0

3"

l[

6,2

31.0

26

5.8

i.61

1.4

48

tli

32.5

209.0

120

39.2

7,45

2. I

326

The model reaction is completely specified by k, rt and l~. The values of A, x, ~,lea and l~m are de,ermined by eqs, (48), (51), (29) a n d (86) respectively. The value of T i s taken as 0.6 fin i~ each case~

The cross sections obtained using these values of lma.~are tbund to change by le~;s than abot~t t '}i; ever the angular range~ shown in figs. 4-6 even if the values of l~,.,~ are doubled. Condition (85) is of little interest here since 'we are applying a sharp
150

N. ROWLI:~Y

1

.~#

b, i0 ~b

t,]l

r_ << "+~ ~ !*,...]

" , "',

I

~ "~','b'trl "~. . . . . . . . . i ........~'....................... [ tt., 2............t ~-,,~,z,~,2?.., ,;i O, 70 '9 ~;"

/

L_

30

)0

@CF1~

Fig. 4. T h e s h a r p - c u t - o f f a n d r a i n b o w c r o s s sectiol~ are ~:ompared ftor a s m ~ ii ~ l , . m o f ~L ,

I ' " "-"

q "-.,!0

q D 5 [e~,

0 + "]3

_:'i,. 1~8

.

!.0

0.5

-----

s+hor'p <_ul oft

\<.,\

60

40

80

(7~

74o

I~0

ek;[~, Fig, fi, See c a p t i o n to fi::;. 4

2 Or-- ................................................................... tl

•

....

,'t

;

',/v

t

k'*'k

e.-~zJ2

~-Y'J2

', \

0&

,

i

I

\

-

t

~"3

100

9,

0c

!70

G,tm

Fig, 6. T h e s h a r p - c u t - o f f a n d r a i n b o w c r o s s s e c t i o n s for a large ",,'atue o f ,] arc corr~f~arcd,

dition would have been satisfied for the values of 1~,~ given if we had taken the barrier penetration into account. Eqs. (74), (75) show very explicitly the various contributions Go the scatterirlg at~,p!itude. The amplitude fc is the Coulomb amplitude due to aft partial waves; .~; ; the amplitude arising from the absorption of lhe first 1<-~1 partial wavc.~; anti fN is just the additional amplitude we obtain by the inclusion of the ,e~.al

ELASTIC CROSS SECTION

m~clearpotcntiaL The

151

rainbow cross section as defined at the beginning of this secti :,4~

is then just ~r = I f c - - f , b , + J ~ l 2

(8?)

and the sharp-cut-off cross section (corresponding simply to the, cut-off Coulomb deflection function) is a = lye-J~,b,I 2.

~SS)

Both cross sections will be evaluated by the partial wave expansion and givc~ ~v, ! h~;ir ratios to the Rutherford cross ~ection a~, [ref. ~6)] a2 ~'R

¢ sin 4 (½0)

•

!

e

9. Discussion of the results

Figs. 4,-,6 show the sharp-cut-off and rainbow croas sections for ~'l~e r e a c t i o ~ I, II, l~I defined in table 2. Figs. 7-9 show the corresponding Coulomb and rainbow deflection functions. In physical te_~ms the quantities defined in table 2 might co~respond ~o the reactions (l) (II)

160+S4Feat

52MeV,

160~q-ll6sfl at 66 MeV,

(III) 84Kr+ZS2Th at 500 MeV. I

"1

d \ ¢-'.'~4"-~ f

~

~c

lr

\

.....

30

35

Ford ~wheeier Bt ~o

40

1

45

Fig, 7, ~[he cut-off Coulomb dellectio~ function is compared with the rainbow deflection fts[~ctioo (smai~ ,J). ANc showp is the Ford-Wheeler parametrisation which gives the best fit in th,e regioa of d~e rainbow angle.

N. RO'WLEY

152

~<,p~,f .... ,.

i/

\,

"-..>:

701

<~"%,. _

............................................................................................. : ic

~r

30

5,5

~,2

!

,:,L

'~:

Fig. 8, See the first iu~{ o f ~ ~. . . . . ip'i.:>a ~o {ig. 7,

It must b~ stressed, howe~er, that no fit to experir.~er~tai data is implied. Our ;-ira is simply to discuss the relative importance o f the aJclea; potential under dif~%rent conditions. We see from figs, 7 .• 8 that fo," the values of Imp. d,:iined by cq. ¢8 ~~'a ) the rai~bo~ and Coulomb deflection functions (as we might expe
.o

\.

120

110

Ic

125

It-

i35

~,~5

i

~55

~~

Fig. 9. C o m p a r i s o n o f the c u t - o f f C o u l o m b a n d r a i ~ b , ~ ~ debit:calm, i u n c d o ~ , s s~o~v~; s b ~ ~-"~.k~rge number o f partial waves may be alE~cted by :he nuclear p o l e n d a ! if :~ is ~arg~

ELASTIC CROSS S E C T I O N

I5~

appears at angles smaller than 0¢ and the peak of the cross sectiot~.s always dizpk~ys some degree of enhancement. . ! ,Most of the small oscillations seen in figs. 4 and 5 (frequency -~ ~,/I~1, artd to some extent the larger oscillations, will be damped from the experimenta~ cross secticms by absorptive efli~cts a~) which we have not taken into account here. Absorbfive efii~c~s may also affect the position of the ex!- erimental quarter-point but this effect :~bc<:}dbe fifirly small so long as the cut o f f o f ( ' e rh takes place fairly rapidly arotmd Table 3 shows the error obtained in 1¢ if the rainbow cross sections tbr the ~b~vc cases are interpreted as sharp-cut-off cross sections. The errors are typically of t}~.e order of I0 ~ . Also shown in table 3 are the shift of the quarter-point and the mac
I II ItI

Sharp c ~t-off

Rainbow

~o

to

%

~,4.2 l(0.0 120.2

24 26 120

690 96 0 113.5

AI~

AtjL

.t0,

~q"

2.2 1.9 18.0

0.12 0.07 0.15

5.2 4.0 6,7

5.(} 62 6,6

7o 26.2 27.9 138.0

The error Ale in the cri ical angular m o m e n t u m ]'~ obtained by interpreting the rainbow cross sev~iv~ as a sharp cut-off is shown. The shift o f the quarter-point is seen to be of the same order" a~ ,*b.e nuclear deflection at the r a i - h w angle. All angles are in degrees.

Elsewhere 22) we shall define a deflection function in a manner rather simiku to cq° (3) but using phases coming from an optical model code (using realistic cc.mplex potentials found to fit experimental dat~,.) rather than WKB phases i.e.

"~/~o;'

Ol+½ ~ ~"k'/+ t

_

a~.').

~'or energies sufficiently high above the Coulomb barrier this deflection t~nctio~ exhibits a rainbow at~d is fitted rather well by eqs. (60), (64) for t > 1~+ 1. Aiso ~he optical model cross section di~pl~tys the shift in the quarter-point described i~bovc~ However for energie~ near the Coulorab barrier (hence large values 07 0;) file deflc¢~ tion function no longer displays the classical rainbow, the cut off of the ~ is very slow and the shiti of the quarter-point to smaller angles is not obtained. It should also be mentioned here that if one or both of the nuclei we are consider~a~ is strongly deformed then the quarter-point of the cross section may actmli~y be slurred towards larger angles 23). For the smaR values of A in reactions I, II we see (figs. 4, 5) that th~ large o~cib

154

N. ROWLEY

lations in ihe forward angle cros~ section (and even the sma!{ o~ci~tatio.~s m the tY~IIoff regic ~0 are very little different in the sharpr reaction 111 (very large d ), thai the rainbow and sharp-cut-offcross sections are very different. We. see (fig. 9) at large number of partial waves affected by the nuclear potential; the e~;hmcement of the eros, :,:,?lion is enormous, (r*/~R)~ ~ ~ 1.9 as opposed to about 1 35 for lthe sharp cu>of!', ~md the oscillations at forward angles arc almost exactly ~~at ,i~,,hase with tile ~ha~p < J > off oscillations. It is interesting to note that th,~se os, ,:a~,ms (now dominated b j j ~ ) have the same frequency as bet'ore. Also note tha: !.he ?eak is nearer to 0: than .~n the case of small ~. Since such a large number of partial waves contribute to this enhancement it will obviously persist even if the barrier pene,,ration is correctly taken into account. While such a large peak has not been observed expc 2mentally, indirect e~'idence of its existm:ce ma2 be found in experiments perlbr~n:~d with a 500 MeV k~'pton beam on very heavy targets an). It has been shown 2a) that for the large values of r/e~ countered in such experiments the elastic cross sections will be smeared out by the excitation of the low-lying states of the target and projectile. Despite this smearing the exF,erimenta~ l~eaks fbr the above reactions are large (being about 1.6 for the z-*aTh or :°sPb target). Such high peaks cannot possibly be explained by a sharp-cut-off cross ~ection which on!y ha,; a peak ofabou~ ~.35 beli)re it is smeared out. We thus have very, s{rong indirect evidence for the enimncement we have seen above in such reactions.

1~. Summary With the use of a simplified real nuclear potential we have derived simple ~ayodel expressions for the nuclear parts of tile WKB phase shift and classical deflection function for the elastic scattering of two heavy ions. The expressions were written in the form of univer,;al functions which allowed a simple discussion of tile retative importance of the nuclear potential under different conditions. The separatiol of the phase into nuclear and Coulomb components allowed us to use the partial wave expansion for the cross sectic~ and thus enabled a direct comparison of the rainbow

ELASTIC CROSS SECTION

155

cross section (real nuclear potential included) and sharp-cut-off cross :~ection (real nuclear potential neglected). The two important effects of the real nuclear potential were to slightly ~;Jaift the quarter-point of th~ cross section towards forward angles and to produce an e~~ha~:e~ ment of the large peak in the cross section. For very large wave numbers (he,ace ve~'3heavy ions) it was showJ~ that a large enhancement of the peak in the c~'os~ :~ectio~ is possible and indirect evidence for such an effect was mentioned. For small wave numbers the oscillations in the cross section at tbrwavd :~:h.', were found to be domin~3ted by the absorption and may not, there~bre, be de,.~,z,:ih,e~ in terms of interference between the Couiomb and nuclear parts of a dellectio~ fimo~ fiord. The effect of barrier pmetration was shown to affect only a few partial w=~ves i~ the region of the critical angular momentum. A correct treatment of these patti~! waves should produce some damping of the elastic cross sections we derived, The~'e-fore for small values of A the enhancement of the cross sections may not be sig~i~ctmL However since the scatter ng angle of all the partial waves in the region ofl~ is r educed~ the shift to forward angles of the fall-off region of the cross section should be a re~I effect so long as the partial waves l
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) tl) t2) ! 3) 14) 15) 16)

K, ¢V, Ford and 3. A. Wheeler, Ann. of Phys. 7 (1959) 287 R. da Sitveira, Phys. Lett. 45B (1973) 211 R. A, Malfliet, S, Landowne and V. Rostokin, Phys. Lett. 44B (1973) 238 R, A. BrogIia, S. Landowne and A. Winther, Phys. Lett. 40B (1972) 293 M. V. Berry, Proc. Phys. Soc. 89 (1966) 479 D. M. Brink and N. Rowley, NucI. 2hys. A219 (1974) 79 R. A. Broglia and A. Winther, Phys. Rep. C4 (1972) 153 K. W. Ford, D. L. Hili, M. Wakano and J. A. Wheeler, Ann. of Phys. 7 (1959~ 239 J. S. Blair, Phy~:. Rev. 95 (1954) 1218; 108 (1957) 827 G, Michaud and E. W. Vogt, Phys. Rev. C5 (I972) 350 W. E. Frab,n, Phys. Rev. Lett. 26 (1971) 568; Ann. of Phys. 72 (1972) 524 R. G, Newton, Scattering theory of~iaves andpartictes (McGraw-Hill, 1966) p. 574 E, Merzb~cher, Quantum mechanics (Wiley, New York, 1961) p. 1t 2 D~ Bohm, Quantum theory (Prentice Hail, t951) p. 265 D. M. Brink, Lecture notes on heavy-iot~ reactions, Orsay (1972) L. D. Landau and I~. M. Lifshitz, Quantt:m mechanics (Pergamon, Oxford, 1965) ~, 4]!.~

156

N. R O W L E Y

17) J, So BlaiL iv Lectures on Theoretical Ph~.~ies, t:-oL 6'C. ed~ P. D Km~z, ~). A~ l.ir~] a~d W. ft. Britten (Ur,'versity of Colorado Press, Bmdde,, t966j~ 18) L, D, Lar a u and E. M. Lifshitz, Mechanics (Pergamor~, Oxt):~rd, 1960) p. 49 19) P. E. IA. dgson, The optical model o]elastic s~,~tterit~.£~ {O~ford Un~ver~;d? th'e~% ~9~:~3~p. 42 20) M /. ,;~mowitz and I. A. Stegun, Hamlbook ~/° matltem~zt~'cul fut~ct~on.~ {Do;e~ New York, 1965) p. 13 21) W. E. Frahn and R. |1. Ventner~ Arm. of Ph~,,~ ~4 (I'~:~6~ 24~ 22) N. Rowley and E. P!agnot. work in progre~,*.; 23) N, Ro~,\ley, Nucl. Phys, A219 (1974) 93 24) P. Colombani, J. C. Jacmart, N, Poff6, M. Riou. C. S~df~a~-~ a~.~d J. Tys, P!'~?s. t , ~ . -t2ti ~.19~2) 197