PHYSICS REPORTS (Section C of Physics Letters) 4, no. 4 (1972) 153- 198. NORTH-HOLLAND PUBLISHING COMPANY

SEMICLASSICAL THEORY OF HEAVY ION REACTIONS

R.A.BROGLIA and A.WINTHER 9”he Niels Bohr institute3 University of Copenhagen,

Copenhagen, Denmark

Received 22 May 1972

Contents:

I, Introduction 2. The equations of motion

2.1. Channel wave functions and cross sections 2.2. Non-orthogonal&! of the basis 2.3. Antisymmctrization 2.4. Coupled equations for transfer

3. Nuclear matrix elements 3.1, Evaluaticn of the overlap matrix 3.2. Llastic and inelastic matrix clcmcnts 3.3. Transfer matrix clcmcnts

155 156 156 159 162 164 168 168 174 176

4. Solutions of the equations of motion 180 4.1. Unitarity and absorption 180 4.2. Dctcrmination of the orbit 182 4.3. First order perturbation theory 188 4.4. Second order perturbatiaa theory 1911

Acknowledgements 193 Appendix A 194 Appendix B 196 References 198

The appllcatil,n of the semiclassical theory [ 1 1 of direct reactions among heavy ions is discussed. Prescriptions arc given to USC this picture also for bombarding cncrgics in which the nuclear forces play an important role in defining the trajectory of rciativc motton of tlw tw colliding Ions.

The competition bctwcerl dlffcrent proccsw lcadmp to the same final channel and the mtlucnce of channels which art’ sUctn& coupled to the final channel are discussed in the t’ramcwork of the semlclassicat coupled channel equations of mot:,c)n. The possibility of including, within the semiclassical scheme, effects of antisymmetrization, recoil and non-orthogonality of: the basis vectors arc also discussed.

: M’SK S REPORTS (Section C of Pt1YSICS LETTERS) 4, No. 4 (197-j 153 198.

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I. Introduction

In the semiclassical description of reactions between heavy ions one utilizes the fact that the wavefength associated with the relative motion of the centers of mass of the colliding nuclei is very small. Thus for collisions below the Coulomb barrier the wavelength should be compared to the distance of closest approach, while for collisions above the barrier it may be compared to the sum of the nuclear radii. Such a comparison is shown in fig. 1. It is seen that the ratio of the rete- vant distance D to the wavelength 3c is typically of the order 1 02. We may thus form wave~a~kets of dime?tsion small compared to D but still containing suf~~ient~y many waves to ensure a we’ll- defined ntonle~l turn.

0-I 0 100 200 if 300 600 500 600 E (revs

Eb 1~ 270MeV

Fig. 1. The ratio between the characteristic length D and the wavelen@h in the relative motion for argon projectiles on mercury as a functton of bambarding energy in MaV. For entxgics below ihe Coulomb barrier tEba270 MeV), D is taken to be the dis-

tancc of closest approach in a tocad on collision, while for E > f?b, II is taken to be the sum of the nuclear radii.

suffice to msure the ap~~icab~ity of ciassical physics for the elastic s~att~rin~~ If in the s~at~~r~~~ pmc-ess there is a transfer of etler~y AE, mass Am, charge AZ and annular t~~~~~~~~~~ At the ~~~a

dition ( t 2) must be s~l~p~~~et~ted by further ~Qt~ditiu~s. In principle such ~~~~~ss~s could be de- scribed in a cfassical picture provided one would knaw the time at which thf: tra~~f~~ takes gta,;e,

Since, however, in the transfer process only few quanta are involved it is not possible tu d~~~~~ this instant within the coillision time. We must thercfme in addition to f I.2) require

Al --j-g I )

In the semiclassical description the two reactions leading from A,a to B,b namely (A+B,a+b) and (A+b,a+B) are incoherent. In the center of mass system they are described by collisions, where the polar and azimuthal scattering angles are 6 and Cp or n-2 and #+n respectively (see

It is mted thu- I- the in~~~aIi%~~s A > a and B > b i% follows that c > d. Since the transfer cross ~~ti~~ USllai~~ &Y . ~4th the mass of the transferred cluster the exchange reaction can often be neglected except X ri e c and B * b.

E-l& 2. C'lil\SK.i~l dcwiption af the CtirCct reaction :$,A - b,B and the cxchanpc reaction a,.4 -4 B,b leading to the same fin.il * rate.

whme $k and $t are the int~nsi~ wa~~f~~~%~~ns of the two nuclei a. and A, in the states m and n r~~~~~ti~~l~~ 3, and fA bein the ~~~~sp~~din~ intrinsic esovlinates. The phase 6, is defined by

while the classical relative velocity V, i,s defined by

where E, is the channel energy

E,=EZ+E,n = “$2 + m,c2 ) (2.9)

E$ and kg being thae total energies of the nuclear states n and m of the nuclei A and a r~s~~~~t~~~~y. The coefficients c,(t) describe the amplitude that the system at time t is in the state r, Thus r7t ti~ll~ c= - 00 wz have

~41~re G indicates the entrance channel. For ;1 given classical tra.jcctvry the ~r~b~bi~ity p, tarot tfle _ +stenl after the reat:tion has taken place (F -t-b is foernd in channd s is given by

Ps = kg” )I2 .

From this ~robab~~i~y one may evaluate the cross section for the r~~~tio~” 0 I+ s by ~~~t~~~y~~~ - _

(2.l1) with the classical cross section for the orbit leading ta the amplitude c,. For an orbit with impact parameter p and bombarding<energy corresponding to channel s the cross section in ther. center of mass sytem is given by

ere 8 is the scatlering angle in the center of mass system while @ is the azimuthal angle d~fi~~n~ the scattering plane. The connection bctwcen tht” scaitering angle and the impact ~~ra~~t~r is

* In the total wavefunction we Icave out the function dclxribing the shape af the w~v~p~ck~t definmg the tciiativc matim of the nuclear centers sf ma$s. In a semiclassical description fhe only effect of thcsc functions is to give a r~iati~n ~tw~~n the c~S* &al and the quantal variables, resulting in the condicon (2.16) below.

found by solving the classical ?quations of motion using the potential US(R) between the two ions.

Since we use a different potentI& and energy of relative motion far each channel we define the overate cross sections for the twc) channels r and s by

he relative velocity at la distances in channel s, i.e. vS = v,(=j). on which is linear in A~/~~ and NC/E. As loug as such corrections on the cross section they can hardly be considered significant and

hem in the fol~owin~.

The scparstion of 6, -- 6, itlto o 2nd y is somewhat arbitrary but it is noted that the approxi- mation of neglecting recoil effects by setting

tbB =

c = a -- d, = b - d,

C=A--d,=B-d,, (2.40)

UatiQt~~ f ) and (2.42) are ~~n~ra~~zati~ns elf the eqs. (2.24) to (2.26) and reduce tol g ia seE I to zero. In the ~r~s~~~ situation the fundamental eq. (2.16) for the semi-

ical ~~~r~~irnati~l~ clads to the f~~~~~~n~ expression for o,, defined in,(2.20):

(2.43)

Thd” ~~~~~~~~ti~~~ (ml’ the ~v~rl~~ matrix now yracceds exactly analogous to the evaluation of the

c3 the total overlap arc in general smaller thau those d. Except for the ease where the charge tratrsfcr is

case the direct term (2.30) does no. exist, the contribu- utuaE transfers (2.40) form a rapidly decreasing the ~undi~i~ns (1.3) will be violated and the use of

t~r~~n~d from the condition that the total wave- ctrie amun~ identical particles. tion of antisymm~t~zati~n is fulfilled by multi;

(2.44)

(2.49

effect of antisymmetrization is a trivial renumbering of the particles which lead to matrix elemcn~ consistinq of in identical contributions. For scatterings close to or above the Coulomb barrier, ‘v++crc the tails. of the wavefunctions in different channels overlap, interference between dirferent terms in’ ?,$, and z$I)~ will occur. These terms are exactly the different overlaps discussed

e ~er~a~n~~g $o incre&n& coilrplex ~~~~~~-.~~~~S~~r~~ ri: R15: IXkiXM3 signs of th3ZSZ %33n;E-~re--- ~~~~~~~ed by (ZA4).

, - r .

It is noted that although the mare-complex mutial traf\sfer ~~~t~but~~n~ usually WIII~S~ s-& s~n~~~i~ss~~al way we use to evahmte them braks down, this is not true if we ~~t~~~ha cleans in i with a corresponding number of nuczlteons in A, The latter situation corres exchange reaction (2.2) which may be important if a * A, As we have can again be included in a semiclassical approximation, but we have here to use a diff~y~nt ~~a~~~~~ orbit and the contribution from exchange transfer will contribute incoherently to the ~~~~~b~a I tiow from the direct transfers,

In the following WC shall therefore mostly neglect the a~~tisyn~~netri~ation.

2.4. chdyleb ~~i~~~~o~ls fur fruns~iv The overlap (2.17) defines the geometry of our non~rth~g~na~ basis, From here WC may d~~~~

the vectors

where I$ is the interaction between the two nuclei in channel r. if r = A,a and s = B,b where a = 5 + d and B = A + d we have

In the smiclasskal approximation, the coordinate F bA is $jvcn by (2.35). This means that if ane! es recoil effects also the potential (2.5 1 ) would depend on thf* rclafiv~ ~~or~~it~~t~ rder cjf

the transferred cluster.

ffone considers in rhe sum (2.46) only the states r = sy i.e. the gound ;tnG excited states af the ~~n~~urat~on b,B, the overlap matrix clc,? and consequently flSSe NT diagor ;r.i and cd, coincides with I&. For the direct term (2.30) also the phases o and “y vanish. This is :;een explicitly for CJ from eq. (2.20) while fcrr 7 it can be seen by rewriting (2.18) in the form t I ]

(2.53)

~~~1~ the s~co~~d coi~d~tio~~ is funneled if the action integral measured in units of St is a small num- r, ix.

(2.53)

mitsis clcrix~~t at !hc distdnw tit‘ closest q.qmxdi. bit with a ante of closest approdch of

(2.56)

\ f? h’ R.A.Brogtia ad A. Winther, Semicbid theory of heavy ion reuctibm

~~~la~~tat~ve~y one can further estimate x by defining the average value r*,-r - U, in the region of the overlap i.e. we write

XS * Q&Xv IQ,,\ 7

the potent ial

f 2.59)

Iwhere rMev is i” measured in MeV, while CI! is the overlap (2.17). This relation shows that the con- ditions (2.53) and (2.54) are not independent.

If both of these conditions are fulfilled, one may treat the transfer process in first order per, t\ii$ation theory. possibly taking into account at the same time inelastic processes more ~orre~ti~, The coupled eqs. f2.48) in this case reduce to the equations

(astir) for A =+ B and a + b E =

!X 0 for A = B and a=b

is small in first order. To this order the matrix p is equal to

/3, = 8(w) - E,, .

The corresponding ~3, vector is given by

167

whcru thy projection apcrator P is defined as

(2.66)

m’n’

In this st;‘c‘tiolj we disouss the evaluation of the nu&+ar matrix elements which must be known it’t order to solve the semictassical equations of motion. These matrix elements are csmntot~ tom ~:tit?ns Mow t1nd above the Coulomb barrier and are alsa in all essentials 1’7 1 the same as those entering m a quantw\ mechanical description sf the prowss e.g. in terms of DWBA,

and

and

where

I3.5)

femm r? set of orthogonal functions in cd and the angles. It should be noticed that the radial furk- tions tl in (3.3) and (3.4) are not normnIized to unity but give rise to modified form factors.

Tlhe expansions (3.3) and (3.4) are especially appropriate for single nucleon transfer where 2 = f and where only ane term appears since the single particle orbital angular momentum I = A ES ~e~e~j~~ed by the total angular momentum j =S I and the relative parity of a and b or A and 8 respectively, For several particle transfer it may be more convenjent [ 7,f 1 ] to express th.e function ~~~~ expficitiy in terms of the coordinates ttb 9 rgb, . . . of each of the transferred nucleons, and write it as a iine,ar combination of products of the type (3.3).

I~~rting the expansiors (3.I)-(3.5) in (2.3 1) one may perform the integration over cd to fired

The ovcrkq (2.30) is most easily evahlated in an intrinsic f 1, LI 9 3) coordinate system with 3-axis ~~~~~g the vector q(t) such that the velocity u(t) is in $1~ 1,3-plane. Transforming the spherical ~~ar~o~~i~s to this system through the Eulerian angles 12 ,( f) one finds

R.A.~rog!iff and A. Win t?wr, Smiciassical tkeory of hearvy ion rafctims

m,*m, 2MR

mA+m@

-Xi-- ??

f-E t The geeomctrical connection between rdA$ rdb and CR(t) as it follows from eq. (2.39) and Fig, 3, The coordin;: tc system usrd fc+i’the o+afuatmn of the inttgrai (3.83 is indicated together with the vector kd(r). The factor s in front of rdsw is givm by

s = (m,+rnb)~m~+mg)~m,+mg)lt4rn,m~~.

R.A . Bmgiia and A. Win ther, Semiclassical theory of heavy ion reactions 171

k,(t) ’ $a = k,(t) rdA sin 6A cos q + k,(t) rdA CR(t)) (3.10)

alld performing the integration over 9 one obtains

I -- __

CA&t) = 27tiP’&2J+1)(2X+1) C (--I)J’+x+h XAA’ [” AAl I5 J’ J *

where .$(s) is the Bessel function of order CL’. A convenient way to evaluate (3.1 1) may be to use elliptical coordinates defined by

where

t?lA + !?I B m, + mb

Pl = Itl _I. _- . . . . - .-

‘dA( _ = -_-I_

2 PI? B 92 C3.13?

The uset’u1ness of the elliptical coordinates is associated with the fact that in a power series ex- p:Moll of JPr the product of the trigonometric functions reduces to a rational function (81 tn $ and q since

a11d G1lc.t’ ill + M’ + p’ is an even number. TIN recoil effect which was described by the phase CT in eq. (2.30) is now corltaincd in tk two

kkpc”ndcnt factors of eq. (3.1 1). The order of magnitude of the recoil effects can bu estimated by evahlating the wacenumber k, at time t = 0 where the overlap is maximum. One finds for a

R\~therford orbit

J-

-” __-

k,(O) a 0.22 ~G&l,” 7r .- t9 - 17._-_ tg . . 4___ fmmt ,

R

where kMcV is the bombarding energy in MeV and 19 is the center of tnass scattering angle. whik tl, ~tnd .*1, art‘ the mass numbers of the transferred cluster and the projectile a respectively. It ;s sect) t’rom expression (3.1 1 ) that the rc:coii effects arc only expected to be small for reac‘tions I.ll*nc.. .Q :.. ..I, ._. .- _#__,‘.A_ r*“ilC’ v IS Ll\.)X 1u 'I 8G C:rq,GC3.

Still the overlap functions C,, can be evaluated to a rather good accuracy by neglecting the f3~~1 1imctiatl &caLlsc tlzc error introduced 1s quaJratic in the argument and sitlct the m;rin overlap comes from a region close to the 3-axis (sin @A < 1). Within the same approximatioln otw

lI\ay substitlrte cos 9, in the exponent by unity. Assuming that the overlap c’on~~s from a rather

small region of r&,-values around k’& we allay extract the exponential function and write the overlap function in the form

172 R.A. Bro_alia and .A. Winther, Semiclassical theory of heavy ion reactions

where

(3.15j

(3.16)

In the following we shall assume that for distances q(t), where the reaction takes place,

BW~USC of the symmetry of the sum (3.15) under a change of sign of M o.le finds that P&+X+\ I 1t nust be even. This parity selection rule is only fulfilled if one neglects the terms with pq -+ 0.

In the following we shall include recoil effects only to the extent that it was done in ey. (3.16), but neglect terms in (3.7) with p’ f 0. Under these circumstances one may simplify the eva’,ua- lion of C,,(a~ by introducing the momentum representation [ 1 11. In this representation the rnodi-

cd formfactor 14 (.,XIJ(~d,A) is represented by

g,(CI) = x (_ 1 )j’. z jh+A’-12 _ _... ---. -._-. __----. ----- _ J4n(29+i)(ZX+l)(~A+l)r~At+l)

zi 1.2’

(3.20)

tration purposes we shall mainly consider the special case of the transfer of neutral this case one can often to a good accuracy give explicit expressions for the integr:il in

lLA.BroRJia and A. Winther, Semidassical theory of heavy ion reactions 173

eq. (3.20). In order to see this, it is noticed that transfer processes t&e place at distances 35(t) be- tween the centers of mass larger than the sum of the nuclear radii. “I’he overlap receives the lnain contribution from a region between the surfaces of the two nuclei where the radial wavefunctiol~s in (3.3) and (3.4) are proportional to

W) - exp f-KT) ,

where K is related to the binding energy of the cluster. The integrand in (3.8) which is approxi- mately proportional to the product of two functions of the type (3.21) would have its maximum at the surface of the nucleus where the exponential decay is steepest. From this estimate it seems likely that the detailed behaviour of the cluster wavefunction is unimportant in the nucleus out- side which the wavefunction decays more slowly.

Thus if we assume that u$?&~ (ri in (3.3) decays more slowly outside the nucleus b, than uBfi) (r) decajrs outside nucleus A we may substitute u$?$ (hX)J -

(r) by its assymptotic form outside b. Considering a neutral cluster we may therefore assume

(3.22)

where k,,,(x) is a spherical Bessel fu>lction of imaginary argument as defined in Appendix A, while N is a normalization determined by the boundary condition at the nuclear surface. The quantity K$$ is related to the binding energy BzFi

i! of the cluster in state I= in the nucleus a (in state m)

forming the nucleus b (in state pl. It is de ined by

TIIC Hank&transform (3.18) of the tail of the wavefunction is given by

-A’- I p*

.-_

k2 + K2

whcrc K is given by (3.23). Inserting this result in (3.20) we find

- -___ _ .______. _ _^ . __ ___ _ _ .__ _______

$x(q) = h’,(K% 7; J4?r(201)(2X+1)(2A+l)(?iZ’+l)~$k$ x A‘?’

(3.23)

11-1 trying to generalize the formula (3.25) to charged particle transfer one may expsnd tht’ Whittaker functions w~~ich constitute tlw tail of the radial wavthnctions in tcrlns of t!ltl t‘unctions k,(r) (we Appendix A), and apply the result (3 -25 ) t’or each term. In using this rnctlmd OIW Mets the difficulty that the tails of the wavefunctions arc much steeper than the function (3.2 1 ).

(3.33)

where Vnl is the real part of the optical potential generated by the nucleus a. Let us first consider the matrix elements (3.33) diagonal in the target nucleus i.e. the monopofe-

manopole interaction between the two ions. This ion-ion potential can be written in the form [ 2]

(3.34)

where p: is the nucleon density in nucleus A, the distance between tf>e centers of mass of the two nueiei being r. For the actual estimates of rfza and p: we use the common parametrizations*

IJ,(r) vP = CI _----m_--

I + exp ((r--R,)la,} (3.39

Wih

ps = 0.17 fn? , R; = 1 .a4 Afd” , ad = 054 fm . (3.38)

The values given Ear the parameters are average:: estimates for heavy nu&i and low rclat:ve vdocitirs. Inserting (3.35) and (3.37) in (X34j c3nc may write

(3.39)

*SeQQ. A.Boirr and B.Mottt?lson, Nuclear structure, Vol. I (Benjamin, New York, 19691 pp. 158 and 23’7.

S = -$~l$p,a, * - 3 MeV fme2. ( 3.441

The result (3.43) far the morlopole-mot70poIt interaotion can be used ako for ~va~l~~t~n~ the ~Q~d~a~~~~ matrix elements corr~s~~~:di~~~ to vibrztional or rotational modes of the nuoktr density. To obtain the formfactor for the excitation af surface vibr~~ioi~al states in the nucl~s A ene wouJd thus substitute the radius parameter R, in (3.42) by

and

‘The cluster wavefunctions p are normalized such that

137

(3 -49)

(3.50j

(3.5 1) n

and similar for 9 atb) The quantity pB denotes the density distribution of the cluster d in the nu- . 9 cleus B in the state q.

Writing the interaction VaA in the form (2.49)-(2.5 1) and using the above expansions we find the following expression for the transfer matrix element

and

(3.53)

The r?lain contribution to the matrix element comes from the interaction V& for 11’ = II. The oh- terms in the first summation in (3.5 2) containing the nondiagonsl part of l/d* are small

The matrix element (3.55) can be evaluated in the same way as the overlap (3.7). If recoil ef- fccts are only taker! into account through the phase 6 one finds in analogy to (3.15) th;at the first

where

In (3.59) and (3.60) we have assumed that the interaction I&, has only diagonal matrix ele- nlents between tie intrinsic states x defined in (3.5) i.e. we define an eSkcl:ive potential i$, by the expression

(3.61 f

Far single particle transfer the assumption amounts to the neglect of sgh-flip in the transfer prwxti, while for severa. partklc transfer it imyks in addition a +.xns~ rvation of al symnxtrics in the cluster, One may avoid this approximation [ 111 by writing the two cluster wavefunctions ~~~~icitly in terms of the ~wavefunctions of each of the nu,cleons in d referred to the center of the mass of A and b respectively, and writing the interaction VP& as a sum of interactions of each nucleon with the core A.

The matrix element (3.60) can conveniently be expressed in the momentum representation Caere in al~alo~y to eq. (3.20) one finds

(3.62)

In this expression we have used the form (3.57) of the interaction potential, and have introduced the binding energy B,d of the cluster d in the state 2:.

,\n especial\y simpfc fsrm of the matrix element is ob%ained for transfer of a neutral cluster N$~R CIIIC n~ay negltxt tht‘ contribution to the integral coming from the inside of nu::leus b, TJti-

liZiIqz6LW -‘) one finds the expression given tin ref. [ 9 ] i.e.

ste that in this ap~roxis~at~on it is possible to separate f,(%!) into a spectroscopic amplitude tomes a fu~~~~~~tor depending on!y on binding energy and total angular rn~rn~~ntum transfer.

tn this st_‘cfim we shall give a number f3r one and two neutron transfer. While

of iilustrations ;>n the solution of the equations of tnotiw the semiclassical theory was especially designed to deal

with reactions below the C’aulomb barrier, *we Gall here also discuss the extension of the theory ia ttighrr borr,barding t’nc‘cgies.

For reactions below the Coulomb barrier we assume that no reactions leading into compound sites take plain and that the only reactions are inelastic scattering and transfer (tunneling) 06’ a t‘2as, ~~~~~~~o~~~ betw~n ttlc two systems.

ait that this equation contains interference terms between different channels is due to the nonorthogonaiity of the basis states 3/,(r) in the expansion (2.8). If one enforces the eq. (4.1 1 to be fulfilled for ;f finite number of states this implies that the states which are left out are ortitro- gunaf to ali those intruded.

hxforming the time derivatives and utilizin‘g the coupled eqs. (2.Li8) one finds that (4.1 ) iilqJ/ies at the following symmetry relations must hc satisfied

(4.2)

~~~~i~a represents the socalled post-prior relation in the semiciassical description. I[t mily be y by writing the difference (4.2) in the form

R.A.Broglia and A. Winther, Semiclassical theory of heavy ion reactions 181

= (E~“_E~A )(tibB,GaA) + (J/bB,(T~b-cJh4’.L/bB-UaA),J/aA) . = (pLf*-aA

a )(+bB,$aA) + ((&.j+ &,B)$bB,$a”) - (J/bB,(T,,+UaA)J/“A) (4.3)

where we have used

V aA = vbA + H,(s,) - HA(&) -- &A - l/d 3 (4*4)

alld the corresponding relations for VbB and T&,. Performing the differentiations with respect to r aA and rbB on tke channel wavefunctions one should disregard the second derivatives of tia\llA and ot’ $b~B , :m.x the sma!lncss of such terms compared to the other ternts is essential to the sernicl;Gc,il description. The result can be shown to be identical to the time derivative of the

overlap cntcring in (4.2). One should here remember that the overlap besides dcpcnding explictly OII r through the phase i> also depends implicitly on I in virtue of the relation (2.16).

It is apparent from (4.2) that one does not have the freedom of choosing either the intzractiorm Vs -- Us or the interaction Vr - U, in the transfer matrix elements without introducing further modifications in the equations. Only at times t = f=~ where the overlap clt,, vanishes there is a complctc post-prior symmetry. Mc return to this question in section 4.3 below in connection with a treatrncnt of the perturbation theory.

Tile problem of including absorption into the equations t.2.48) is similar to the probleur trcatcd in rd. [ 15 ] of the Coulontb excitation of particle unstable states. The absorption rcprcscnts the pqulation of states which arc not. or cannot, explicitly bc included in the coupled cqu:it;ons. The negkct of sonic of the reaction channels will violate the unitarity equations (4.! ), md ~31 in gt~r~l lead to a system of intcgro-differcntial coupled equations. Assuming th:rt tht‘ rtitc of pcq~r- Miorl of compound states is independent of the cncrgy with which they ;1rc popul~~tcd, ills ktmds in the intcgroiiift‘crcntial equations have no iimc delay, and the ~ouplcci cquatiorrs WI be writtcu in the form

where WS is a complex interaction in the channel s, wit!1 both diag~,nal and non-diagonal m&ris dmt31ts.

111 the following WC sha!l neglect the real part of ict’,. The imagintiry potcrktl thus introduc~cd has no ct’t’c~f on the &ssic;rl trajectory, which is only‘dctcrmincd by the real part t’;. Ttlc CffCCt of tk ;lbsorption is quivalcnt tcj a position dcpenclcnt nncm free path [ 161.

l(‘lltl imaginary potential can by cstin atcd by observing th;rt it is proportiotInl to the 0Wr:;Q’ 0t‘ the densities of the two niiclci i.e.

The rn;ltrix clcmcnts of W can bc cstimatcd making use of the results in section 3.2. For the I~loI~~~~~ole-mo~lopolc matrix clcmcnt OIW thus finds for values of raA > R, + R,A

From the conservation of energy and angular momentum one finds

(4x3)

wlrile E is the bombarding energy in the center of mass system. From this equation one readily finds the follov(ing connection between @ and p, i.e.

WO

9=n--2 dW ^-

0 1 -- w2 --u&l/w l/l7

where W, is the first positive zero of the denominator. Eq e (4.16) can be evaluated analytically if we neglect

leads to the well known formula

6 = 2 arctg(n,lp)

(4.16)

the nuclear interaction. In this case (4.16)

(4. I 7)

where a, is half the distance of closest approach in a head on collision in the channel r i.e.

lf nuclear attraction is included one finds for large distances according to (3.43)

U&r) = 2a,E/r - Sr2 esp ( --(r--R )/cJ,~ (4.19)

and ~~~r~~si~~ (4.16) must be evaluated numerically. Typical behaFricurs of the function (4.16) w given in ref. [ 161 together with the corresponding Gross section; which are compared to the quanta results, The tr~e~tories to be used in the coupled equations (2.48) are defined by the fact that the

wav~~~~kets in all channels should have an optimum overlap in the region of the interaction (see ref. [ I I). In the present paper we shall assume that this is accomplished by choosing all orbits to be sirWar i.e. to have the same symmetry axis and the same scattering angle 8. hlthough it is ~~~~~i~~able that a better choice could be made we shall see that below the Couiomb barrier the ~r~~~ilt prescription leads to the equations obtained from the exact quantum mechanical couplej dxmxl equations of Coulomb excitation in the limit of short wav~iength [ 171 l

To &scribe the hyperbolic motion below the Coulomb bwrier we choostz the i’ocal coardinatc SYSTEM where the z-axis is perpendicular to the plane of the orbit and where the x-axis bisects the bugle between the assyml~totae (see e.g. ref. [ 181). In terms of the parameter NJ, the relative posi- tien vector R,(t) = (X,, Y,.Z,) has the components

Xr = a, (cash w,+E) , Y, = (I, (e2 - 1 )lj2 sinh w, , z,=o. (4.3)~

f 4.29

We may compare the present results with the equations wsed in the serniclasslcaf. description of ~~~l~~n~b e&citation [ 191. For Am = AZ = 0, as is the case in Couiomb excitation, the parameter [’ is equal to 6 to first order in AE/E, and the parametric form of the coupled eqs. Q.48) are t~~~~for~ of the same form as those given in refs. [ X 71 and [ 191, It should be noted however that the ~aran~eters \ic’, defined in (4.23) are slightly different for each p;iir of nuclear states. Although this is in complete agreement with the derivation of the semiclassical equations given in ref. [ 17 ] (see lot. cit. eq. (25)) this effect has not been taken into account in numerical calculations until IlOW,

The abcave considerations are only strictly valid for trajectories where the nuclear attraction can be cnmpletcly neglected ix, eithw for b~n~bar~in~ energies wet1 teiow the Coulonlb barrier UT

er ~~~~~~s~ for s~~t~~~n~ angles well below the angle co~es~ond~ng to the grazing co& sian. En ~t~~t~~~~ wkwe the effects of the nuclear attraction on the orbit is not too violent one may estimate the scattering angle by a siassicral perturbation theory using the hyperbolic orbit as the ~~npert~~rbed trajectory. In the perturbation throw the change A6 in the center of mass ~catte~n~ angle is given by

(4.36)

__ “.-_.- w +?*37)

Si~~ce the observed scattering angle 9 is given by

tY=r 1

arc sm - + A8 E

(4.38)

c2 = a/a, (4.39)

E* = l/sin p . (4.41)

t” function (4.38) is Clustrated in fig. 5 for different values of the parameter c2. These curves can be used to estimate the Coulomb orbit which has the same impact para~neter as the actual orbit.

The validity of the classical perturbation theory depends on the magnitude of the integral (4.X). In m t QS cases. the standard potential (3.43) wound lead to a rainbow angle in the elastic s~at~~r~n~ cross section. in the neighborhood of which the perturbation theory breaks down. We CWW 3 gr~ing cohision as the Coulomb trajectory for which the integr 11 (4.34) satisfies the rela- tror;r

(4.4%)

ere LI is a dimensionless number of the order of magnitude of say 0.1. Using the same approxi- atiorrs as those leading to (4.37) one finds that eq. (4.42) can be written as

(4.43)

‘s = 1.30 fm .

ZI-I order to derive (4.43) we have used the expansion

(4.44)

*here 12 is a typical value of Q& 1 +Q. t or given bombarding conditions one can estimate cJ from (4.43) and the actual scattcrinp

e fop c be read off from fig. 5.

RA .&oglia ,a& A. Winther, Semiclassicd fheory of heavy ion rewions

10’

to’

16

lf-

I I -1

j

I

187

Fig. 5. Correction to the scattering angle due to the nuclear attraction in first order perturbation theory. The CUIWS indicate the diffcrmc between the observed “eccentricity” E‘~ = 1 /sin (s/2) and the eccentricity in the Coulomb orbit in units of EC 1 where

c-1 is defined in eq. (4 4G), and for different values of ~2 defined in eq. (4.39).

For the evaluation of nuclear reactions it is important to know the distance: of oht‘st ~tpprc-;lcl~ b for ,1 trzljcctory of given center of mass scattering angle 9. This distance can be calculated from tk irrll !ct parameter p by means d the relatioxl

‘l7-w com~cction between p and 19 i?. given by (4.16). In the following scctiotls wt) shall estimate cross sections for nuchr reactions based OII

Coulomb trajectories. Such results can be ~rscd for dtuations where the niickar attrxtb~l disturbs the hyperbolic mbit if one buses ti Coulomb trajectory with an escentricity co corresl)ondit% to the

obscrvcd scatterins angle and with an effective center of mass energy E,, whit11 sati.sfics the rt-

\a1 ion

kz, Z,e’

X,, (I +q-J = h . I1.47)

4.3. F-sb m.kT pmrbCItioi1 rheor?: 11~ this section we shall make a number of approximations which are similar to those of finite

raf~m distorted wave Borll approsimation (DWBA). These approximations wili enable us to give c”syliclit espressions for cross seotions which can be compared to the results of DWBA and to e,y- perimcnts where the inelastic processes are not expected to be too important.

Trtxating transfer and inelastic processes in first order only, the coupled eqs. (2.66 j, (2.67) can be solved csplicitly inserting at the right hand side CL:= 6,,6,, where the index 0 indicates the %Tround states. We assume again that a = L‘ b + d and B = A + d and find that the transfer amplitu& 3f timt i = += is given by

t 41.48)

where AE is the Q-value i.e.

AZ= “c” t EB - Ea - E" . 9 m n (4.49)

tt’ integral (4.48) displays ;i pcrfeci post-prior syn:IMry. This can be shown \* ~~~i~~~~rl~ the relation (4.2) and the fact that the overlap OL,, vanishes for I -j too, i.e.

(4.50)

In the remainder of this section we shall consider the special case of the single neutron transfer. r&ng recoil effects one finds according to (3.59) and (3.63)

(4.51)

orbital integral

III deriving (4.5 1) we have utilizec.1 the relation

which hold:: for h + A + A’ even. The differential cross section for transfer is given by

For unpolarized target and phojectile tke transfer probability is equal to

189

(4.53)

(4.54)

(4.55)

(4.57)

cietermines the re!ative probability of transfer 3s a function of Q-value md smttkng angle, bvhile the othrr factors depend on the nuclear rnatri:x ekments. it is noted that in tk cross st’c’tioll there is 110 intcrfercnce between different values of S, J’, and the angular mon~entzrn~ trmskr ?L

The orbital integmls S,, can be evaluated by inllroducinp the yarmctriztition ot‘ the h~pt’rboli;: motion ;IS discussed an the previous section. One 1’Ms in terms of the par;rrntJt~r w

p /(h)~ [l(b) = Ilr,lY

TlPE\ i\tKit’ltitieS 43, and u,, are defined in (4.24) and (4.25 >. The integrals S,, depkd on fhe choice #of coordinate, system as spherical tensors of rank X.

They arqnost easily evaluated in the focal coordinate system used in section 2 where

~~s~r~ing the ~~~ran~etr~~atio~~ (4.28) in (4.53) and setting 0 = I in (4.33) it is seen that the expres- n (4.5 1 p for the transfer amplitude: is identical to the expression given in ref. [ 31, which was

derived by performing the semiclassical limit of DWBA. Due to the fact that the phase

is an odd funcltion t>f \tl while the other factors in SAr are even functions of w the orbital integrals iis tile above coordinate system are ail real functions of the four parameters CF, E, g’ and p. Since ~~~~~h~~~~ore ahe nuclear matrix element is real (if the radial functions are chosen to be real) it is SCXEP that the first order transfer amplitude is purely imaginary.

The orbital integrals can be evaluated analy tica.liy for t = [’ = 0 and e = 1 (&=n). One finds in the above usttd coordinate system

(4.62)

er~~atio~~ of these formulae and their generalization to slighter values of X is given in Appen- . Xote that the quantity 2Ti defined in (4.57) is given by

in the general case a rrlativety simple approximate formula for S the explicit c:xpression for k, in (4 53) expanding the integrand

hr can be obtained by inscrGn&

IK given in .~~p~~~dix B. For 2p( 1 +E) 2 A me obtains i;? powers of \o. Such formckte

Firstly we natice that the Q-value dependence is given by the factor exp (--d2/(2~tg). The ~ax~urn in the arn~~~tud~ c for a given scatte~n~ angle c~cucs for a Q-value detrained by

(4.67)

The steepness with which the cross sectors d~~r~a~es as the (&v&e d~~at~~ from the o~t~rnum value is det~rrn~~~d by pc in such a way that the Q-value dependence is mush stringer far IooseIy bound neutrons than far tightly bound neutrons. The dependence can be ~~~us~ated by the simple case of ba~~~vards slatterns where the cross section is simply ~ru~~rt~~na~ to

Mdw~ ri,s = 4 b?f +vs)* We have here used the elation (4.32) and (4.33) and neglected ir to obtain

where k, is the average wavenumber in the relative motion. The dependence (4.68) of the cross caption on the difference between t e distances of closest a~~~ua~h in e~~tranc~ and exit channel [ 121 is ~~~it~ dramatic because of the large valets of q i~~v~~v~~ 5n heavy ion ~~acti~~s.

l%r ~~Qnvanisl~~~~ Q~values the amplitude depends strongly on the magnetic ~ua~tu~~ ~~nlb~r fl such that far AE > 0 the smallest value of EJt = -X gives the largest ~~ntr~buti~n. This is espe-

e values of k and F)E. i-e, for forward scattering angles where the highest valrte of h ~i~nsistent wi the insular momenta gives the largest ~~ntr~buti~n. For given values of iz and ,+I” in (4.52) this means lthat h = A + A’ except if J r= A - 4 and J’ = A’ .- i in which t-ase

the transfer from a state of ant~para~le~ spin to a~~~th~r state of anti- u~~fa~~~red compared to the transfer to a state of parallel spin.

Thu ~~~ativ~ rna~nitud~ of the twa transfers J = A + iY J’ = A’ + 1 and J =: A -I- $, J” = A” _ 5 wkre itr both cases X = A + A’ is, apart m spe~tr~s~o~i~ factors, given by the ratio of the 3--- i s~~~~~s in (4S 2). One finds the f~~ll~w result

N (A’+ frhtf)h,-h 1 J ._“I_ -..-_- $& P ‘~~~~~~~~~~

z r2 dr

1__- __._ . . ..I_-.%“. -_--. --p-

~~~~~~~~~~,-~ ~~~ 1~~~~~~~~~~~ r2 dr ’

~4.~~~

1

As arl illustration of a hither or&r prmms WC consider “rhe ~~rnpetit~~~ between first and nsfer processes in the kwo rieUtK?ll transfer reactj~n arn~n~ heavy ions.

‘fk first order tw~mi~eutr~n transfer was d~;~~~ssed in the previ~~~s section. Xn seamed order ~~~~~ are rna~ly ~orn~~et~t~ve rea~ti~~~s leadii~~ t\, the same final states. PLO obvious one is the in-

met also in connection with the coupled eqs. (2.661, (2.67), where the interaction responsible for the transfer had to be modified, when inelastic channels were treated explicitly. In a similar way 8s in i2.66) one znay write the first two terms in (4.75) as the matrix element of an effective

.tr~~~~~~~~~~~~t!~r.~~~~ b_v, _ .a I 1X .a.*/ _I_ $. ,~ \* B ““7 - P_ z ~1 . . Cd “? ‘.2 * *%;;,z ” “_“>r: _“x; -’ ,“?_ :,; _ *T -.,.$ w.. c-” -T- Iyx _ __ ._” / ;y “‘a’.” =*:~-,._I.,-~,.~~~~~? p -.--“=r . _, -_ I ._ __ “)‘.’ I,x. l”p’TI T,_I-_i_I^_ _^ -” _.

( I

‘;, 7 -*. f ,_.;- _ L _ - =.- _ 5 ~‘ 1 _

ajrl _ (~~,) ___ ~~~~~ICI’P,fV,*_(:~~~~~~~. 1 _ c.,~~~~~~:i~~~~~~~~~~~~-~ --“:- -2 I ,

v --I-- *II -,

(4’7f) ; j

fF QF

%r is seen from (4.76) that the inclusion of a larger number of intermediate states has the effect sf gradually reducing the simultaneous transfer.

as was discx 4szed in the previous section the first order time integrals are real quantities if one USES the focal coordinate systerm, i.e. the amplitude to first order is purely imaginary. The second order contribution is, in the same system, a complex quantity3 the real part being equal to half the product of the two first order integrals corresponding to the transitions (a& + (f,F) and (f,F) -+ (b,B). This real p;lrt does not interfere with the first order contribution to the transfer probability.

While the real part of the second order amplitude vanishes exponentially for increasing energies cf the intermediate states, the imaginary part vanishes only inversely proportional to this inter- r,lcdiate Q-value. Thus one may for large values of (EFF -&&/A where r is the collision time (2.57) write the second order amplitude as

(4.7”)

06

fi?: c 1 1 _ _I _-I____.._- 8 EfF - E,, s

FF -cm

in the form of a single time integral as a first order amplitude. The effect of these virtual transitions via high excited states can therefore be included as a

cOrrection to the interaction in first order perturbation theory. The nonorthogonality corrections ~CJ the first order interaction (4.76) also receive a contribution from the high intertnediate states.

En order to have a c’onsistent description of the transfer of tightly bound clusters in terms of a 0~ step process this contribution should be cancelled by the virtual transitions (4,771.

Acknowledgements

Tfic authors would like to thank K.Alder, P.R.Christensen, TKammuri, SLandowne, J .Lind- hard. R.Liotta, GMorrison, B.Nilsson, P.$iemens, and D.Trautmann for discussions. Une #of us WV wishes to thank the members of the Physics Department at Florida State University, Tallahnssce, for the hospitality extmckd to him during a stay, where part of the present work was performed.

;wi N are the us :a1 Bessel and Hankel functions *. In complete analogy to these definitions w shalt w the symbols i and k to denote the spherical Bessel and Wankef functions of imaginary

1 1 _ i,t-‘) = -,_ *C’ _( __ f ,‘_ t’“‘__

we

m=O (22 jrn

(A31

Y ~~~~~t~~Rs (A3 1 and (A4) constitute fundamental real solutions of the Schriidinger eyrxation for constant negative energy They have the following assymptotic behaviour for z -+ 0”

-1 L-2 i,tz) * --^-ll-_

f G-l )!! I.+ ___2L__- + ---- ..!l_e.- + 1!(ZI-t3) 2!(21+3)(21+5) I’. )

(A71

The frtnctio~ i,(z) can most easily be evaluated by the recurrence relation

utilizing

sinh 2 i,(t) =7-, cash z

i_,(z) = _2. (Al3

Similarly .Q(L) can be found from

utilizing

k,(z) = k,(z) = ne+/22 .

The prodwt of k, and a s~~~t~rica1 harmonics satisfy an addition tlttc‘orenl*

(A171

(A20)

(AX)

-4~ 211 application of the latter Lorrnulae we ?~ote that the Whittaker functiorl which constitute the tail of the wave function for a charged particle can be expanded in :erms of k functions. Thus one finds

(AZ)

(AZ) rK’>Kj.

Whittaker function and & in power in (A221 is normalized such that for

tiz result can be obtained by using (AXI) expressing the tegratin_e term by term. The Whittaker function roaches the function li,( Kr).

Appendix B

me according to (4.58)

R.A.Bmgiiia and A. Winther, Semi~las~i~d theory of heavy ion reactions 197

1,,@.p,M’) = j-W dwk,(p( l+e cash WI 0

-^-

X p(l+e cash w) cos &*-- 1 sinh w t’e sinh w+&~+p arctg ---+--- l e+cosh w

(B2)

Inserting in (B2) the explicit expression (A@ for the spherical Hankel function k, one finds

J I xci =.!n exp {_p( l+e)} c _.lh?-eX my ___._ _ ._ ____

m M?-d! [2p( l+e)]m

with

w

J - s exp I--pe(cosh \v--- 1 )+i[% sinh w+if\v+ip@(rv)} ( I+E)‘” mp - dw --- - ..:. ._ ._._ ..___ __^_ . . _ __. . .____“______._. ____ _.._ __ ___.____.__ _ ___ -_ _ _ ___

-0 (e cash w+ 1 )m

and

(B3)

(B4)

tB5)

The integral (B4) CM be evaluated explicitly for [ = t’ = 0 and 9 = n @=I). One finds inde- psrldc~a af p

where I!’ is the Whittaker function defined in ref. [ 201. In fact the Whittaker function CM be tx- ~wsstxf in terms of tiankel functions of imaginary argument. Using recursicx~ relations ;rmong tl~es~ functions one may finally write the orbital integral (B3) in the t’orm 1221

whm [I!] is the 4argcst integer <rz. In the general case one may estimate Jmcl by expanding the exponent in (WI) to second order in

10. One finds

with

One 1x1~ thus write the sum (B3) in the form

R.A.B@& and A. Winthm, &mkdassical themy of heavy ion mactfons

cases it may hc sufficient to include only the term with m = 0 in which case one finds the e formula (4.65) for S,,. While the expression (B8) is correct to an accuracy of about < I .,C the espressioll(4.65) has a more limited region of vaIidity i.e. p( l+e) > 1

R..&BrogIia and ~.Winther, Nuci. Phys. Al 82 (1972) 112. P.E,Ho&son, Nucleer reactions and nuclear structure (Claredon Press, Oxford, 197 1) and references therein. f)i.Tnutmann and KAlder, Helv. Phys. Acta 43 (1970) 363. $_Austem. Direct nuclear reaction theories (Wiley, New York, 1970). A.~scuitto. N.Clendennin_g and BSbrensen. Phys. Lett. 34B (1971) 17. T_Tamura. D&s, R.A.Broglia and S.Landowne, Phys. Rev. Lett. 25 (1970) 1503. R,&Br+is, T.Kanmwi. R.Liotta, A.Winther and B.Nilsson, NucI. Phys.. to be published. R.SMuIIIken, C.A.Rieke, D.Orloff and H.Orloff, J. Chem. Phys. 17 (1949) 1248. PJ.A.ButtIe and LJ.B.GoIdfarb, Nucl. Phys. 78 ( 1966) 409. W.T.Pinkston and C.R.Sa tchler, NucI. Phys. 72 (1965) 64 1.

fts. Suet. Phys.. to be pubIished. .J..A.Buttie and L.J.B.Goidfasb, Nucl. Phys. 176 (1971) 299.

K-.&l&i a;md .%.Winther. Nucl. Phys. Al 32 ( 1969) 1. K_&Ider 2nd A.Winther, Electromagnetic excitations of heavy ions (North-Ho:. .nd, Amsterdam) to be published. H.X.!%eidenmiiller and A.Winther. Ann. of Phys. 66 ( 197 1) 2 18. R.:1.Broglia. S.Landowrte and .%Winther, Phys. Lett. 40B (197:!) 293.

K..tider a;lC H.K.A.Pauli, Nucl. Phys. 128 (1969) 193. K.lUder. .A.Hohr, f.Huus, BMottelson and A.Winther, Rev. Mod. Phys. 28( 1356) 432. A.lh-‘inther and J-de Boer, in Coulomb excitation, eds. KAIder and A.Winther (Academic Press, New York, 1966). H@cr transcendental functions I-111 and Tables of integral transforms I-II, ed. A.Erdelyi (McGraw Hill, 1954). M.Danos and L.C.Masimon, 5. Math. Phys. 6 (1965) 766.

er. R.Morf, XP.Pauli and D.Trautmann, Nucl. Phys., to be published.