Semiclassical description of multipair transfer processes in heavy ion collisions with superfluid systems

Nuclear Physics A474 (1987) 240-252 North-Holland, Amsterdam

SEMICLASSICAL PROCESSES

DESCRIPTION OF MULTIPAIR IN HEAVY ION COLLISIONS SUPERFLUID SYSTEMS

TRANSFER WITH

P. LOTTI and A. VITTURI Dipartimento

di Fisica “Galileo

Galilei”,

Universitci di Padova and INFN,

C.H.

Padoua, Italy

DASSO

Sektion Physik der Universitiit The Niels Bohr Institute,

Miinchen, Am Coulombwall 1, D-8046 Garching, BRD and University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark Received

13 July 1987

Abstract: Pair transfer processes involving a superfluid system are studied in terms of phase space distributions constructed in a product representation which blends both ordinary- and gauge-space degrees of freedom. The time evolution of these distributions is followed by solving a collection of classical equations of motion, the quanta1 fluctuations being accounted for by the sampling of all possible initial orientations of the (undetermined) intrinsic system in gauge space. The excitation of the pairing rotational degree of freedom - corresponding to a variation in the number of particles - is induced by a gauge-deformed ion-ion potential, as befits the superfluid character of the target. Different orientations leading to the same final mass transfer may produce (in the case of large pairing deformation) interference effects in the final population of the members of the pairing rotational band. We also discuss how this approach can be used to describe the effect of pair transfer modes on sub-barrier fusion processes.

1. Introduction Dynamical studies of the pairing interaction in nuclei are based on two-particle transfer processes in collisions with both light and heavy ions. These reactions provide

a direct indication

of the degree

of collectivity

induced

by the pair field.

Quantitatively, the magnitude of these effects can be measured in terms of the enhancement of the associated cross sections as compared to the characteristic strength of uncorrelated transitions. In the case of superfluid systems the coherence displayed by the pairing modes leads to two-particle correlations between the ground states of neighbouring nuclei which are strong enough to make the probabilities of transferring more than a single pair quite appreciable ‘). It is therefore possible to think of experimental conditions (for example in the collision of two superfluid nuclei) in which many pairs of identical nucleons are exchanged between the colliding partners. It has been speculated ‘) that this may result in the presence of oscillations in the final population of the different mass partitions. Evidence for this kind of fluctuations has not yet 0375-9474/87/$03.50 0 (North-Holland Physics

Elsevier Science Publishers Publishing Division)

B.V.

P. Lotti et al. / Semiclussical description

been found

in direct reaction

processes.

241

Some of these features,

however,

be present in recent cold fission data3). The existence of large two-particle transfer

couplings

conditions

this class of reactions.

the formalism

that is used to describe

seem to

in the superfluid

case

At least all

the channels corresponding to the ground states of the nuclei with A, A f 2, A f 4, . . . particles must be included in the calculation. In addition, a microscopic description based on the transfer of individual nucleons requires the inclusion of a large number of intermediate states in the odd-mass nuclei. This scheme has been attempted in the past (cf. for example ref. “)) leading to calculations of a high degree of complexity. Alternatively, the study of two-particle phenomena can be eased by exploiting a close analogy with the particle-hole situation in the case of inelastic excitations. Correlations of the latter kind are responsible for the deviations (either static or dynamic) of the nuclear surface from a spherical shape. In the particle-particle case pairing correlations are also assumed to produce static or dynamic deformations, this time in an abstract space spanned by a gauge angle 4 [ref. ‘)I. The number-ofparticles operator fl can be considered as the generator of infinitesimal rotations in this space. It turns out that the intrinsic hamiltonian of a superfluid system, expressed in terms of the gauge coordinates, has the same form as the intrinsic hamiltonian of a deformed system with a symmetry axis. This analogy makes it possible to consider multipair transfer reactions in one-to-one correspondence with inelastic processes where the nucleus is cranked to states of high angular momentum. An essential element in this approach to multipair transfer processes is the characterization of the interplay between the ordinary reaction variables and those in the abstract space in which the motion across the mass partitions takes place. A link between the two sets of degrees of freedom can be established “) by ascribing a deformation to the nuclear shape in gauge space. This deformation describes the finite content in number of particles which is present in the intrinsic state. A gauge-deformed density

can then be identified,

whose

Fourier

expansion

~(r, 4) = C PAN(r) eeiAN’ AN

directly yields the couplings to the different members of the pairing By introducing the parameterization of the nuclear shape 6, R(4)=&

(

rotational

band.

I+$3os(21) >

in the ion-ion potential, one obtains an interaction term which couples the ordinary spatial variables with the gauge coordinates. The hamiltonian which describes the reaction process thus contains:

242

l? Lotti et ai. / Semiclassical

deseri~~~on

(i) A relative motion component, whose initial value depends on the conditions in which the reaction takes place (i.e. impact parameter, bombarding energy in the c.m. system,

etc.).

(ii) An intrinsic component related to the structure of the superfluid target and therefore expressed in gauge coordinates; this term includes the number of particles in the target and specifically accounts for the evolution in time of the transfer process. (iii) The coupling-interaction term, which connects the ordinary variables (i.e. relative distance, linear and angular momenta, orientation angles, etc.) with the gauge coordinates. In our approach we consider this hamiltonian as describing a classical system and solve the corresponding equation of motion. A similar problem has also been considered in ref. ‘). The relevant equations are listed in sect. 2, and the results of numerical calculations illustrate the applications presented in sect. 3. In sect. 4 we discuss the way of extending the probability distributions for values of AN (number of transferred particles) beyond the classically maximum allowed value AN,,,. In sect. 5 we also apply the formalism to describe the effect of the pair transfer modes on sub-barrier fusion processes. A summary and conclusions are given in sect. 6.

2. Formalism Following the discussion tonian of the form

where the different

in the previous

terms on the right-hand

VO

V “““p=l+exp[(r-R(~))/a]+

section

we introduce

a classical

hamil-

side,

Z1Z2e2 r

’

describe the relative motion, intrinsic motion and coupling interaction respectively. Note that the conjugate momentum to the gauge angle, p+, corresponds to the variable AN representing variations in the number of particles in the system. From this hamiltonian the following equations of motion can be derived,

I? Lotti et al. / Semiclassical

description

243

This set of equations is integrated numerically for the ordinary space variables in a polar reference frame which has its x-axis aligned along the beam direction. For these coordinates we then use the fixed initial conditions, r(t=O)=co,

p*(t=O)=

cp(t=O)=r,

Insofar

-xQz,

p,(t=O)=hC.

as the gauge space variables

are concerned

we take the original

value

of

values

in

AN to be zero, i.e. p$,(t=O)=O

but allow for the gauge the interval

angle

to assume

with equal

weight

all possible

This procedure, which reflects its consistency with the uncertainty principle, is based on the idea that the quanta1 fluctuations which determine the possible outcome of the motion in the combined space are essentially due to the intrinsic variables. The information obtained solving the equations of motion by sampling the initial value of the gauge angle &, is somehow equivalent to following the evolution of the system in phase space through its Wigner transform. This situation is thus similar to the one encountered deep inelastic

in the interplay collisions.

between

For a discussion

relative

and intrinsic

of this subject,

motion

variables

in

cf. e.g. ref “).

3. Applications To provide a concrete illustration we have considered the reaction 40Ca + ‘16 Sn. From the binding energies of the even tin isotopes a moment of inertia 9 = 5 h* MeV’ is extracted. A bombarding energy EC,,. of 120 MeV and an impact parameter p = 3 fm were chosen, conditions which lead to a near-grazing collision. A characteristic value of &, = 18 was assumed to define the strength of the coupling. In fig. 1 we show the final values of p+, (i.e. AN) as a function of the initial orientation of the rotor in gauge space. The ensemble of results roughly spans an interval from -2.5 to 2.5, a range of values that readily gives an idea of the pair-transfer processes that are classically allowed for these bombarding conditions (zero and plus or minus one pair). Basic symmetries resulting from the assumed deformation of the density become apparent in the plot. Another interesting aspect

244

k? Lotti et al. / Semiclassical

description

Fig. 1. Final values of pd (i.e. the final mass transfer AN) obtained in the classical approach as a function of the angle Co characterizing the initial orientation of the rotor in gauge space. The calculation refers to the case of the reaction ?Za+ ‘%I, at a center-of-mass bombarding energy of 120 MeV and an impact parameter p = 3 fm. For the ion-ion potential we have adopted the parameterization of ref. ‘I), while a value p, = 18 was taken for the pairing deformation parameter.

of this curve are the pronounced plateaus at AN = 0 which result for initial orientations $rr S 4 ,s& and &i-s4 0~ $r. This feature reflects the sharp exponential character of the coupling. Indeed, there is a strong asymmetry between the gain in transfer amplitude obtained by bringing the surfaces closer together by a certain quantity and the loss resulting from pulling the surfaces away by the same amount. Note that in the quoted intervals cos (24,) s 0. This introduces a qualitative difference with what it is obtained for long-range couplings such as Coulomb excitation in the case of an ordinary rotor. We obtain a classical probability for the different multi-particle transfer processes by converting the uniform distribution in & according to P(AN)

= p(&)

*CL d(AN)

(Given

the multi-valued

character

of the function

_-th&!-

27r d(AN) &(AN),

*

this expression

is under-

stood to include a summation over the separate fragments of the curve which lead to the same value of AN.) The resulting function P(N) for our example is displayed in fig. 2. Note the characteristic rainbows that appear at the edges of the classically allowed values of AN. A simple prescription for the projection into actual eigenstates of the number of particles can be implemented by identifying the areas subtended under the curve P(N) in intervals centered on even AN’s as the probabilities of transferring such a number of pairs. This leads in our case to the values P(0) = 0.52 and P(2) = P( -2) = 0.24. These numbers can be compared with the corresponding quantities obtained in a semiquantal approach by integrating along the classical trajectory the radial form factor for two-particle transfer. This corresponds to the AN = 2 term in the Fourier expansion of V,,,,(r, 4). One gets in this way a value of P(52) equal to 0.10, thus

245

P Lotti et al. / Semiclassical description 0.0

I

I

0.6z

, ,

av 0.40.20.0

I -4

-2

0

2

4

AN Fig. 2. Classical probability the function AN(&)

P(AN) associated with the different final mass partitions, displayed in fig. 1. For details of the calculation cf. caption

obtained from to fig. 1.

showing an order-of-magnitude consistency between the semiquantal and the semiclassical descriptions of the process. Both calculations have been done with reaction Q-values which follow from the adopted expression of the intrinsic energies as a function of A. This, naturally, establishes a symmetry between the cross sections obtained for the addition and removal of pairs. As noted before, the classically allowed values of a given AN receive contributions from different initial orientation angles, eventually merging at the maximum allowed an improved prescription for projecting into value AN,,,,, . From this observation number-of-particles eigenstates can be implemented. To this end, one can associate to each contribution an amplitude 9-‘o) with a complex phase e’@“, obtained by evaluating along the trajectory the classical action integral

In fig. 3 the action integral is shown as a function of the initial orientation &. For illustration the individual contributions from the ordinary and variables are separately shown in the interval 0 G 4. < 2~. Note that while the action @J,is periodic with respect to shifts +o+ +o+ nn-, the same is not true

angle gauge radial in the

case of @+, which depends on the origin chosen to measure the angle +o. This however does not affect the calculation of the probability for transferring an integer number of pairs of particles. The multiple values of the phases associated with the full range -cc s +os + cc is just what gives rise to the selection rule AN = even. Consequently, the drawing of a continuous curve P(AN) for all values of AN is entirely dependent on the choice of interval a G 4os b and is at its worst if one chooses the narrowest possible interval, i.e. lb -al = r. The new probability distribution is compared in fig. 4a with the previous result obtained in the pure classical approach (namely without interference). Since one of the functions is continuous while the other is discrete, what should be comparable is the sum of the ordinates corresponding to the quanta1 results with the area under the classical curve.

P. Lotti et al. / Semiclassical description

246

Fig. 3. Action integral Q, as a function of the initial orientation angle &. In the upper part is displayed the contribution @, = -IT: r@, dt coming from the radial variables using as reference scale the value for & = 0. In the lower part is shown the other contribution @+ = - j:C;: ##+, dr associated with the gauge variables. For details of the calculation cf. caption to fig. 1.

In order to investigate the features of the distribution P(AN) in a case in which many pairs of particles are exchanged we have arbitrarily increased the pairing deformation parameter & by a factor of 10, leaving unaffected the trajectory of relative motion. This leads to a simple scaling of the values of AN,,, and of the gauge action integral, which now becomes the dominant ingredient in determining the phase differences. The associated probabilities for different final channels are shown in fig. 4b, again compared with those obtained ignoring the interference. The distribution has now developed marked oscillations around the pure classical results. This pattern is quite similar to the ones displayed by the final spin population in the ineiastic excitation of a rotor and in ref. ‘) within this context. Note that we have constructed our probability distributions P(AN) from the information provided by only one impact waves.

parameter.

A more correct

treatment

should

sample

all partial

4. Treatment of the classically forhidden region The action integrals Sp defined in the previous section are pure real numbers. - although essential Consequently, the phases e’@” calculated in the applications to determine the interference patterns - did not introduce any attenuating effects. The action integrals provide, however, a way to extend the formalism into the classically forbidden region. Along the lines of the present contribution, this could

I? Lotti et al. / Semicl~ssicul

247

~escripfio~

AN

0.10 &, = 180 0.085 z

0.06 I 0.04 -

AN Fig. 4. Semiclassical probability dist~bution P(AN) (full dots) associated with for two values of the pairing deformation parameter & The dashed line has eye. In both cases the continuous curves refer to the prediction of the pure without any interference mechanism. For details of the calculation cf.

be achieved

by extending

the integration

of the classical

different mass partitions, been drawn to guide the classical picture, namely caption to fig. 1.

equations

of motion

into

a more general regime in which the dynamical variables are allowed to assume complex values. The imaginary parts of the action integrals Q, then provide the exponential damping factors in the probabilities for transfer processes with AN > AN,,.,,, . One can avoid the introduction of complex equations of motion and implement, instead, a simpler prescription. Indeed, an accurate estimation of the complex action integrals can also be made by extrapolating into the forbidden region the results obtained with the real trajectories. A straightforward analytic continuation in the complex plane of the curve AN(&) yields the intersections of the surface with the physically meaningful (though classically unallowed) multipair transfer processes. For example, in the curve shown in fig. 1, the aim would be to find the complex values of sbO which yield AN = 4,6, . . . . These solutions, when injected in the corresponding analytic continuation of the curve @(C&J, produce the attenuation factors. This scheme is concretely illustrated in fig. 5. In (a) the location of the roots of the equation AN( &,) = 0,2 and 4 is shown for the case of & = 18. One can see that as the number of transferred particles increases, the two real roots coalesce (as

-

0. ... ..... .2. . .

. . 2. . . . . . . . . . 9

(a>

1

;I

0

IO3

lo"

2 .. AN

(b) 4,=18

I

-

i

76'r-

“‘, I

I

AN

40

50

u 30

--I

/I,=180

(4

Fig. 5. (a) Values of the initial orientation angles in the complex plane $a which lead to even values of AN, both in the classically allowed and forbidden regions. Also the intersections for other values of AN, in steps of 0.2, are shown. A value of p, = 18 is taken. (b) Semiclassical probability distribution P(AN) associated with the different mass partitions. For the classically forbidden values of AN(AN > AN,,,,,) the prescription outlined in sect. 4 has been used. A value of p,, = 18 is taken. (c) Same as (b), but for /3, = 180

e-g o.o-

0.1

I

249

t? Lotti et al. / Semiclassical description

anticipated)

and then move into complex values. Of the pair of solutions for each meaningful ones are those lying in the upper plane. AN > AN,,,,, the physically

Figs. 5b, 5c, extend the calculations the same values of the parameters.

of P(AN)

shown

previously

in figs. 4a, 4b for

5. Effects in the fusion channel Two-particle transfer processes have been mentioned as one of the possible mechanisms responsible for the observed enhancements of sub-barrier fusion cross sections over predictions of the simple barrier penetration models. Quantitative analyses of these effects have been performed with calculations which explicitly take into account the way in which open reaction channels affect the tunneling phenomena ‘l). Within our approach the effective coupling of the entrance channel to the pair transfer modes is simulated by the procedure of averaging over the initial orientation of the superfluid system in gauge space. The reasons for the actual enhancement of the transmission coefficient can be easily visualized by plotting the collection of different effective barriers which result from the ensemble of initial conditions. Examples for three relevant values of &, are shown in fig. 6a. It is seen that some values of &, lower and other values increase the potential barrier with respect to the unperturbed one (which is obtained for &,=&T). One can define as contributing to fusion those partial waves for which the maximum of the effective barriers (i.e. the ones displayed in fig. 6a plus the centrifugal term) is below the bombarding energy E. For a given potential, this sharp cut-off prescription leads u !. One example of such distribution is shown to the familar triangular distribution by the dashed curve in fig. 6b. This corresponds to the 40Ca+ “%n reaction at an energy E = 140 MeV, calculated in the no-coupling situation characterized by 4. = $T. When the average over the initial values of 4. is taken into account the edge of the distribution smooths out, as shown by the full-drawn curve, obtained with pP = 18. This example illustrates how effects due to pair transfer modes can be put into correspondence with those created by couplings to collective surface modes 12). To obtain the fusion cross section as a function over the set of initial orientations 40, using u(E)=&

where,

for the excitation

functions

o(E, 40) = d(4o)

a(E, a(E,

of energy

40)

Wo

one can again average

,

40) we use the simple

Wong formula

a(40) -ln(l+exp(E~~~Po))). E

This expression, through the parameter E( +o), incorporates thickness of the barriers and therefore takes into account barrier penetration which are associated with the radial

into the picture the finite the quanta1 effects in the motion. The predictions,

’ 6

I 8

r (fm)

i 10

1 12 1 14

0a j

1 0 20

40

l(k)

a0

loo

Fig. 6. (a) Ion-ion potential as a function of distance for selected values of the gauge orientation angle @a. A value fip = 18 is assumed. (b) Partial wave fusion cross section o, obtained in the sharp cut-off model by averaging over all orientations of the gauge angle (bo.The bombarding energy is taken to be equal to 140 MeV. The dashed curve gives the prediction in the absence of coupling to the pair mode. (c) Fusion cross section as a function of the bombarding center-of-mass energy evaluated according to the procedure discussed in the text. The dashed curve indicates the result in the absence of coupling

6OL

I

8

loo

2, 120

E

~140

160

k? Lotti et al. / Semiclassical description

therefore,

also extend

from the coupling

to energies

well below the lowest effective barriers

to the pair transfer

251

resulting

mode.

Due to the exponential character of a(E, &,), the averaging procedure leads to a net increase in the fusion cross section. The coupled cross section for our example (solid line) is compared in fig. 6c with the pure unperturbed value (dashed line). Note that in these calculations we have implicitly used the sudden limit, in which the excitation energy of the coupled mode is neglected. As it is known from the case of inelastic excitations, the use of this approximation tends to overestimate the cross section enhancement. Thus, this example should be regarded only as a qualitative illustration

of these effects. 6. Summary and conclusions

In this paper we have investigated the problem of multiparticle transfer in heavy ion collisions from a novel point of view. Exploiting a macroscopic parameterization for the pair transition density in superfluid systems we have constructed an explicit expression for the interaction term which combines the coordinates and momenta in ordinary space with those in gauge space describing collective motion across the mass partitions. The identification of a hamiltonian in the combined set of variables opens the possibility of solving the associated classical equations of motion and thus of implementing a semiclassical treatment of the process. In addition to its simplicity this approach benefits from the clear intuitive content characteristic of semiclassical descriptions. Besides the regime of classically allowed motion (with its rather straightforward projection into eigenstates of the number of particles) it has also been shown how to extend the formalism into the classically forbidden region by analytic continuation techniques. This procedure is particularly relevant since we have found that for the typical grazing conditions prevalent in heavy-ion reactions the transfer of several pairs of particles becomes rapidly inhibited. Because of this reason we concluded that oscillations in the population of the neighbouring systems as a function of number of particles - though interesting - can hardly be expected for typical values of the pair deformation parameter. The characteristic range of the coupling (of the order of a nuclear diffuseness parameter) makes the nature of the problem significantly different from the semiclassical treatment of Coulomb inelastic excitations. This can be appreciated from the shape of the function AN(&) displayed in fig. 1, which is qualitatively different from the sinusoidal shape expected for the long-range Coulomb coupling (cf. e.g. ref. lo)). Finally we have indicated how the sampling of the initial conditions over the product phase space in ordinary- and gauge-variables can be used to understand and quantitatively study - the role of pairing degrees of freedom in the fusion process at energies close or below the electrostatic barrier. In this context the presence of the factor 3A in the definition of an effective deformation parameter

252

P. Lotti et al. / Semiclassical

“/I$,” = /?,/3A makes this mechanism collective inelastic excitations.

We are pleased

to acknowledge

description

less important

discussions

when competing

with A. Winther.

with strongly

This work was

supported in parts by grants from the Bundesministerium fiir Forschung und Technologie (06 LM 177 II), the Danish Ministry of Education and Fondazione A. Della Riccia. One of us (P.L.) would also like to thank the Niels Bohr Institute of Copenhagen for the kind hospitality.

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