Time for occurrence of heavy-ion reactions

ANNALS

OF PHYSICS

189, 352-362 (1989)

Time for Occurrence

of Heavy-Ion

TAKEHIKO Department Ohya

Reactions

SUZUKI

of Physics, Shizuoka University, 836 Shizuoka, Japan

Received July 14, 1988; revised September 23, 1988

Representation of time at which reactions occur in heavy-ion collisions is developed in terms of the wave packet description. Its application to a quasi-elastic scattering reveals that the reaction takes place earlier with increasing scattering angles. This quantum-mechanical trend deviates appreciably from the result obtained by classical trajectories. 0 1969 Academic Press, Inc.

1. INTRODUCTION Despite the sufficiently small wavelength in heavy-ion collisions the concepts based on classical trajectories commonly encounter the problem associated with quantum features. Angular distributions of nucleon transfer exhibiting a single peak, for instance, must be accounted for as a consequence of the quantummechanical complementarity [l]. When an incident nucleus with asymptotic wavenumber k of relative motion passes through the region with radial width d outside the target and therein nucleon transfer takes place, the uncertainty relation gives the width of the peak as A0 - (kd) ~ ‘. The maximum of the angular distribution do/d0 at the “classical grazing angle” is generated by full contribution of many partial waves and hence does not correspond to any single grazing trajectory. The wavelength should be compared with d, not with the sum of nuclear radii R when one discusses the plausibility of classical pictures. The size of d is associated with the asymptotic form of the transferring nucleon wavefunction denoted e-‘ld/r, and is then provided by the nucleon binding energy. Although kR g 1 one does not necessarily have kd 9 1. Properties of the wavefunction in this reaction system also indicate the inapplicability of classical trajectories 121. Now time-dependent descriptions of heavy-ion collisions have been devised and have particularly revealed aspects of deeply inelastic collisions with respect to reaction times and so forth [3]. Classical versions are often utilized and provide qualitative results. In view of the above remark, however, it is to be expected that quantum representation of the time-evolution of reactions should also exhibit features different from classical ones. While interaction or transit times have been used extensively in classical and quantum manners, it appears that the time at which a reaction occurs is conventionally classically considered without quantum 352 OOO3-4916/89 $7.50 Copyright 0 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.

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examination. In a classical and rough picture, for example, quasi-elastic nucleon transfer is assumed to take place at the distance of closest approach on the Rutherford trajectory, and then the time when the reaction is induced is thereby related to a single corresponding scattering angle. However, since the angular distribution is in practice produced by contribution of many partial waves, as stated above, correspondence between the time and the angle is anticipated to be different. Further, in contrast with the classical picture there exists no quantum method of using a single wave packet continuously describing the relative motion between colliding heavy-ions for the transition from an initial state to a given final one. That is, while the classical approach is based on the notion that nuclei move along trajectories ranging from the initial state to the final one, an incident wave packet leads to production of new wave packets for various final channels during collisions. Then the plausibility of the classical time remains questionable, which provokes some quantum investigations expected to display features other than classical ones. In this paper representation of the time at which reactions take place is developed on the basis of the wave packet theory. In Section 2 the time for occurrence of reaction is found to be provided in terms of the transition matrix element. In Section 3 this is applied to a quasi-elastic heavy-ion transfer reaction, and a classical analogue of the time is also presented. Comparison between quantum and classical times is made in Section 4.

2. OCCURRENCE OF REACTIONS

In quantum scattering theory wave packets for final channels are generated by an incident wave packet due to some interaction. Accordingly, in this sense, in terms of an arising wave packet we develop a quantum representation of the time at which reactions occur. The time is to be obtained through differentiation of the phase of the off-shell transition matrix element with respect to incident wavenumber. Suppose that the total Hamiltonian H is given as H= Ho, + V,,

H,, = H,, + A,,

c=i, f,

(2.1)

where H,, is the Hamiltonian for relative motion composed of the kinetic energy and hermitian distorting potential, h, the intrinsic Hamiltonian of two colliding nuclei, and I’, the hermitian residual interaction. The decomposition of H is made in accordance with initial and final fragments denoted c = i and c = f, respectively. Then the relevant stationary state equations are written as (H-E(k))

Y"+'(k)=0

,

(Hrc -g)

@$“(k)=O, (2.2)

(h, - 44, = 0,

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TAKJZHIKO

SUZUKI

where k is the incident wave vector. The intrinsic energy E, is taken to vanish for the ground state in c=i, and then E(k) = fi2k2/2Mi, k= Ikl. By use of the full stationary state Y (‘j(k), the total system in the wave packet description is represented as y(v(t)= j d3kfa(q') e-'E(k')"h y" + '(k'),

(2.3)

where q’ = k’ - k and k is hereafter taken to be an average incident wave vector. The amplitude a(q’) is here real, and hence the center of a free packet would, at t = 0, reach the origin of the relative coordinate between two nuclei [4]. In order to investigate emergence of wave packets during collisions we employ an expression X(t) = \ d3k’a(q’) e -iE(k’)l/fr(E(+)(k’)_Hor)-l

v,y(+)(k’),

which stands for scattered packets due to the interaction (2.4), X(t) obeys ifi; it follows that Z(t)=

IX(t))(X(t)l

X(t) = &X(t)

(2.4)

Vr. Since, from (2.3) and

+ VfY((t),

(2.5)

satisfies

ifi -$a0 = C&r, Z(t)1 + Vf I v~)>
- Ix(t)>< Yv(t)l Vf.

(2.6)

This is an equation by which behaviours of arising wave packets will be described. The product (4, IX(t)) denotes the amplitude of the state 4, in the packet X(t) and is the wave packet for relative motion asymptotically corresponding to some given final channel 01. The index for the incident channel is omitted throughout for brevity. Let Z,(t) be the scalar product taken as

Z&J = <-WI MA

IJW> =TrGA IZWI A).

(2.7)

While Z,( t = - co) = 0, Z,(t) is considered to gradually increase with the passage of time and finally becomes a nonvanishing constant. Then the time derivative of Z,(t) is, as a function of time, to have a bump during the period of interaction as explicitly shown later and is here regarded as a measure of emergence of the wave packet. Now from (2.5) or (2.6) one has (2.8)

In order to further rewrite Eq. (2.8), Eqs. (2.3) and (2.4) are used with the assumption of the completeness relation for @I-) so that

OCCURRENCE

;Z,(t)

=;

OF HEAVY-ION

355

REACTIONS

f d3p d3k’ d3k”a(q’) a(q”)

x exp f (AE’ - AE”) I] T,*(p, k’) T,(p, k”) x [(Iv-)+AE’)-‘-(W;+‘+AE”)-‘I,

(2.9)

where W;*‘=E”‘(k)-E, -h2p2/2M,, AE’=E(k’)-E(k)=~U.q’+fi2q’2/2Mi with u = Ak/Mi, and similarly AE” = E(k”) - E(k) with k” = k + q”. In Eq. (2.9), T, is the off-shell matrix element given as T,(p, Fi)= (@I-‘(p)cj,I Then an alternative

V,l!?‘(+‘(ii)),

i-i=k’,

k”.

(2.10)

expression of (2.9) reads (2.11)

where g,(t) = f d3k’a(q’)

T,(p, k’)ePidE”lfi,

(2.12)

A, =(!y+~)-‘q!g!-‘-(!!$!-‘;+

_...

(2.13)

The series expansion in (2.13) corresponds to that in powers of the energy spreads in the denominators in (2.9). We now proceed to obtain an explicit and approximate expression of (2.11). Let the matrix element (2.10) be written as T, = IT, 1 eiqn’, q, being real. Then it is assumed to be given in such a way that TAP, k’) = I TAP, k’)l expCh(p,

WI

= ITAP, k)l expCi{v,(p, k)+q’.Vrl,(~, = TAP, k) expCiq’ .VV,(P,

k)Il

k)l,

(2.14)

where V is the gradient with respect to k alone. It also follows that V& =- 1 VT,(P, k) 2i T,(p, k) --*‘* > ’

(2.15)

which is not associated with the time delay because the energy conservation is here not taken into account. Keeping only the first term in (2.13) and bearing (w~-))-‘-(wh+))-l=2ni6(W,) in mind, Eq. (2.11) turns out to be

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TAKEHIKO

SUZUKI

U*>

(2.16)

.VqJp, k) -dE’r/h}].

(2.17)

x ITAP, WI* MC

P,

where K,(t, p, k) = j d3k’a(q’) exp[i{q’ The amplitude

a(q) is now chosen to be a(q’) = (2~‘/n)~‘” exp( -p*q’*),

(2.18)

which leads to

IKAt,

P,

k)l* = Pw*/~)~‘* exp{ -P*S*A~~)), b = /A4 + (fit/2Mi)*e

S=bAp,k)-ut,

(2.19)

The present approach is based on the notion that there is no sufficiently localized wave packet centered and moving on a classical trajectory. Therefore, we may from the beginning adopt a packet with sufficient width so that the spreading rate is negligible during collisions, i.e., b N p4 or AE’1: fiu .q’. On writing qol(p, k)= q,(p, k, 0), where 8 is the angle between the directions p and k, S* in (2.19) is decomposed as S*=F+G, F= ~*(f-uVq&*)*=

u~(~-u-’

~rjol/~k)*,

G = (Vyla)* - (u 4&/u)’

= k-*(i&/iM)*,

(2.20)

u = lul,

by means of

b, ={(aka?01 >+k-‘cot0 Now the time at which reaction takes place is to be defined in terms of K,(t, p, k). From (2.19) and (2.20), IK,(t, p, k)( * in (2.16) is, as a function of time, found to have a bump the width of which depends on the packet size to be arbitrarily chosen in principle. Since p b R the spread in time leads to an ambiguity in the definition of the time. Then the time is taken to be independent of the size

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REACTIONS

and is presumed to correspond to the maximum value of I&(& p, and k. Hence it is provided through F = 0 and is denoted Ta =--,184, u ak

357

k)/* for given p (2.21)

where the relation E(k) = E, + r’i2p2/2Mr due to the delta function in (2.16) is incorporated after the differentiation with respect to k in view of the remark concerning Eq. (2.15). Note that the formula (2.21) is derived by the assumption that, as previously mentioned, the center of the free wave packet would reach the origin at t = 0. It is evident that if the center is taken to pass the origin, say at t = i, r, on the left-hand side is replaced by t, - i in accordance with substitution of t - i for t in (2.3) or (2.20). Obviously, t, depends on the scattering angle 8, and further from (2.16) the angular weight w(e) is expressed as (2.22)

w(e) a ITAP, k)12ev{-W~*)~

with E(k) = E, + fi2p2/2M,. While the time delay (or advance) is defined in terms of the wave packet motion for t + co or in the asymptotic region and is given by differentiating with respect to k the phase q,(p, k) on the energy-shell for E(k) = E, + fi2p2/2Mr [4], the time in (2.21) is concerned with the wave packet behaviour during the period of interaction or in the reaction region. According to our procedure it is obtained in such a way that the phase q,Jp, k) with p independent of k is differentiated with respect to k first, and then the energy-conservation relation is incorporated. Hence ?a is related to the time delay. Let Z, and rja be, respectively, the time delay and the phase on the energy-shell. Then, bearing in mind q, = qa (p = k,, k, ~9) with k, = CWWfi*)(E(k) - dll’*, we have, for the choice of i = 0,

1 art, f, z--=7, 24ak

1 ah +---, 4 ah

If the distorting potential in H,, is, for simplicity, Z, is readily given as [4] r, = t, -r/u,,

u, =-.%i Mf

(2.23)

not taken to be a long range one, (2.24)

where t, is the time when the center of the outgoing wave packet reaches some given radius r in the asymptotic region. Therefore, the second term in (2.23) turns out to be (2.25) the implication of which is evident. Thus our manipulation connects the occurrence time with the time delay and thereby also provides a significance of the derivative of the phase with respect to the final channel wavenumber.

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TAKEHIKOSUZUKI

3. QUASI-ELASTIC

SCATTERING

In order to simply exhibit a feature of z, in (2.21), we adopt the quasi-elastic scattering for nucleon transfer. The matrix element (2.10) in a simple form will be obtained in a manner analogous to the DWBA type calculation for the strong Coulomb field [S]. The state !P(+ ) is replaced by 4,s; + ) where &, is the ground state of hi and 6: + ) is the distorted wave due to a nuclear optical plus Coulomb potential. Since the reaction takes place dominantly at distances larger than the sum of nuclear radii $ +) is therein taken to be the Coulomb wavefunction. Damping of 6: + ) due to strong absorption is taken into account, for simplicity, by incorporation of the angular momentum cut-off in partial wave expansion of T,(p, k). Assuming @S+) to be the Coulomb wavefunctions the (off-shell) matrix element reads TAP, k) =$

1

(21+

1) exp{i(rr,(p)

z,(P, k)

+ a,(k))}

P,(COS

o),

(3.1)

IZL

where L is the cut-off angular momentum, and we consider vanishing momentum transfer and so forth. The Coulomb phase shift is written as a,(k) =n(log(n’+

i2)‘/2 - l} + itan-‘(n/f),

angular (3.2)

where n + (n2 + p)l12 $1, I= I + f, and n = 442, Z2e2/fi2k with M = Mi = Mr. Then it follows that for L + 1

a,(k)= CL(k)+ with i = L + 4, and a,(p) is similarly given by

written. Let Z,(p, k) be assumed to be real and

Z,(p, k) N i-1’2ZLe-Y(‘-L),

y=z

1

1 1 -+z 7 (P >

(3.4)

as an analog of the WKB calculation [5]. Using the asymptotic form of P,(cos 0) and keeping only the dominant term, T,(p, k) becomes TAP, k) = C(p, k, B)ei6

1

exp[ (i(8-- 0) - y}(Z- L)],

IhL

6 = aL(p) + o,(k) +; - I%,

C(p, k, 0) being real. Replacing summation

B= (e(p) + ww 2 ’

by integration,

(3.5)

one has

T,(~,k)=C(p,k,e)e’~{y+i(e-8)}-~.

This is highly simple, and 1T,(p, k)12 on the energy-shell, denoted p = k for the quasi-elastic scattering, explicitly

(3.6)

which is hereafter displays physical

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significance. The angular width AtI = y = (kd))’ originates from the uncertainty relation mentioned in Section 1. The number of partial waves contributing to T,(p, k) in (3.5) is of the order of an angular momentum spread Al- l/y, which readily leads to A0 Al N 1. Putting particularly 8 = 0 in (3.5), the maximum value of the angular distribution do/de cc sin 0 1T,(p, k)12 proves to arise from the full contribution of the partial wave amplitudes of the same sign. In this sense 0 is not the classical grazing angle corresponding to the single grazing angular momentum L. This simply and explicitly indicates that the notion of the classical trajectories is incapable of generating the angular distribution. Incidentally, the above tendency of the angular width decreasing with the increasing incident energy might have been suggested first in observed results of Ref. [6]. Now it is, therefore, naturally inferred that the time z, in (2.21) also considerably differs from that classically corresponding to the distance of closest approach. From (2.15) and (3.6) we derive the formula of t,. First note that

a6 (e - e)(aypk) + r(as/ak) !?!$U.Qa =-.g u (e-e)*+Y2 ’ where p = k is not yet taken into account. Then assuming that i=

{(kR)2-2n(kR)}“2,

R = r&4 ;‘3 + FI;‘~),

(3.8)

it follows that

(0 - e) - n iog(d + Ly

where evidently we have 0 = 8(k) and y = (kd) -I for p = k. The value of 0 is given by the observed angular distribution do/de, and then one obtains that of L through i=n .cot(8/2) which is equivalent to the “quantum” formula in (3.3), not the classically derived one. The expression (3.8) appears to be a classical one but usually provides the value of R nearly independent of k although somewhat large compared with the conventional one. Then L and R thus obtained are regarded as quantities for the parametrization associated with strong absorption, and not directly related to the grazing trajectory [ 11. Incidentally we briefly refer to the intensity w(e) in (2.22). By use of (3.6), G in w(e) proves to be (3.10)

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Since /A% R, we have G/(2p*)= {(L/k)+d}*/(2~*)< 1 for 0=8. Therefore, w(0) is described dominantly by 1r,( p, k)lj = k. For the purpose of contrasting r, with that obtained by the classical picture we consider the Rutherford trajectory given by

r(cp)=-;c$~{sin(~-f)-sin(~)~P1,

(3.11)

where rp and 0 are the azimuthal and (classical) scattering angles, respectively. The distance of closest approach is then written as D = (n/k)( 1 + cosec(8/2)) at which nucleon transfer is assumed to take place. Let t = rE be the time when the projectile reaches r = D. Then one has for 7c > to 7,

D dr fl =to- s(10 (P2-2W-I -=to +2E(k) 4r1

1

+ n log p - n + (p2 - 2np - l’*)l’* @*+

E(k)----

pp2

Z,Z,e*

r

I2

)

l/2

'

h*l’*

2Mr2

u*

)I

(3.12)

)

p = kao,

where a, is chosen so that a, = r (cp = cpo) and a, cos ‘p. = ut, < 0 as shown in Fig. 1. This corresponds to the motion of a free wave packet associated with (2.3) in such a way that a classical free particle with vanishing angular momentum I’=0 would arrive at the origin r =0 at t=O. That is, the time origin for the classical motion is taken to coincide with that for the quantum one on the basis of the free particle motion.

FIG. 1. A classical Rutherford trajectory given in (3.11). While an incident nucleus travels along the trajectory and passes the point P at f = f ,,, a free particle moving with velocity u passes the point Q at [ = to and then reaches the origin 0 at t = 0.

OCCURRENCE OF HEAVY-ION REACTIONS

361

4. RESULTS AND REMARKS The times r, in (3.9) and r, in (3.12) for the reaction i9’Au(r4N, 13N) 198A~ at E,, = 120 MeV [6] which we here assume to be quasi-elastic for exemplification are exhibited as a function of 0 in Fig. 2. In (3.12), t,, is divergent when I, + - co. Then we put t, = - lOR/u. For 8 + 0, we have not z, -+ 0, but z, + -n/(ku) independent of choice of I,. Now r, increases with increasing values of 0 because of deceleration due to the Coulomb repulsion and the longer path to r = D. Then an appreciable deviation is found out to indicate inapplicability of the classical picture. Correspondence of z, to 0 results from the many relevant partial waves. A salient feature consists in that the reaction takes place earlier for larger scattering angles in contrast with the classical one. In the formula (3.9) the first and second terms in square brackets are leading ones and are simply expressed by the angular momentum L used for the sharp cutoff in the expansion (3.1). Introduction of a smooth cut-off into (3.1) yielding some contributing partial waves, does not have much effect on comparison between 7cl and 7r since many other Coulomb partial waves with I> L are predominant therein. The third term is related to the expression (3.4), but gives rise to only a slight dent around 8 = 8. As 8 --t 0, 7rx approaches 7, corresponding to higher angular momentum, which might be suggestive of the correspondence principle. However, it is to be noted that behaviours of the wave packet at least around the nuclear periphery fail to have a t (1 o-22 s)

FIG. 2. The solid curve indicates TV, and the dashed curve represents zc. The angular distribution arbitrarily normalized is given by the dotted curve. We adopt y = 7.0 x 10v2 = 4.0” and 0 = 41.0”. The latter leads to L = 76 and kR = 110.3 or r0 = 1.60 fm. du/dtl

595/189/2-9

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close resemblance to the classical motion. The difference between r, and rc should be noted particularly in the vicinity of 0 = 0 where the reaction probability is concentrated. If the binding energy of transferring nucleon were taken to be smaller, the width d would be greater; that is, the reaction zone would be more expanded. Then, from (3.9), the dent around 8 = 0 would be deeper, which implies that the projectile should enter the region earlier to induce the reaction. In addition to interaction or transit times widely considered, manners of occurrence of reactions will also serve for time-dependent descriptions. While classical and/or statistical methods have been applied to time-dependent descriptions of deeply inelastic collisions (DIC) [3], our version would provide the time for occurrence of high excitations in DIC and disclose stages of the damped collisions. Transition amplitudes for DIC have been devised on the energy-shell [7]. Investigations of the time demand the off-shell matrix elements at the outset. Attempts to extract classical representations from quantum mechanics have been made so far [S]. Specifically for collisions of composite particles, however, quantum transitions associated with intrinsic degrees of freedom yield new scattered wave packets for various channels and thereby give rise to the discontinuity between wave packet motions before and after interactions. The discontinuity naturally arises even in elastic scattering. Hence the conception of an imaginative single wave packet motion throughout nuclear processes as a classical analogue should be renounced although existence of such a wave packet appears to be implicitly assumed for the classical picture. Behaviours of the quasi-elastic scattering discussed before with respect to the uncertainty relation may be described, without referring to change in internal states, in terms of the analogue of waves passing through the zone with some width on the assumption of a negligible difference between the initial and the final wavelengths. Nevertheless, the uncertainty relation argued therein should be recognized to inhere in the transition amplitude involved in a newly generated wave packet. The present utilization of the standard wave packet theory suggests an instance in which classical versions in heavy-ion collisions disaccord with the limit of quantum mechanics due to the sufliciently small wavelength. REFERENCES 1. T.

SUZUKI,

2. N. AUSTERN,

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Berlin, 1980;

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to the Theory of Heavy-

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(1987), 143. Theory,” Wiley, New York, 1964; N. AUSTERN, “Direct Nuclear Reaction Theories,” Wiley, New York, 1970. 5. For example, see W. E. FRAHN in “Heavy-Ion, High-Spin States and Nuclear Structure,” Vol. I, pp. 157-253, IAEA, Vienna, 1975. 6. J. A. MCINTYRE, T. L. WAI-TZ., AND F. C. JOB=, Phys. Reo. 119 (1960), 1331.

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7. K.

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MICHEL,

Phys. Rev. C 36

DIETRICH AND CH. LECLERCQ-WILLAIN, Ann. Phyx (N.Y.) 109 MCGRATW, AND D. R. DEAN, Phys. Reo. C 30 (1984), 887. For example, see R. ZUCCHINI, Ann. Phys. (N.Y.) 159 (1985), 199.

(1977),

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LEE,

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