Two-nucleon transfer reactions in deformed nuclei using very heavy ions

Nuclear Physics A361 (1981) 215 - 306; @ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

TWO-NUCLEON TRANSFER REACTIONS IN DEFORMED NUCLEI USING VERY HEAVY IONS + M. W. GUIDRY

and T. L. NICHOLS

Department of Physics, University of Tennessee, Knoxville, Tennessee 37916, USA t and The Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen Q, Denmark

R. E. NEESE Department of Physics, University of Tennessee, Knoxville, Tennessee 37916, USA

and J. 0. RASMUSSEN,

L. F. OLIVEIRA ++ and R. DONANGELO

it

University of California, Lawrence Berkeley Laboratory, Berkeley, California 94720, USA

Received 10 July 1980 (Revised 29 September 1980) Abstract: Semiclassical methods found to be highly accurate for inelastic scattering are applied to the calculation of rotational population signatures in heavy-ion two-neutron transfer reactions involving highly deformed targets. Basic features to be expected for such reactions are predicted, and are shown to have straightforward semiclassical interpretations. The rotational population signatures for 2-neutron transfer are shown to be quite different from those expected for the analogous inelastic scattering case. Several calculations are shown for Xe projectiles on rare-earth targets, and it is demonstrated that such reactions can provide a unique probe of nuclear structure in high angular momentum states. The extension of the general ideas employed here to l-nucleon transfer in deformed nuclei and to several other examples of transfer to highly collective states in deformed and vibrational nuclei using very heavy ions is briefly considered. Experimental possibilities are discussed, and it is concluded that relevant experiments in this virtually unexplored area are possible using sophisticated particle-y spectroscopic techniques.

1. Introduction Transfer reactions have long been used to probe nuclear structure. It is well known that one-nucleon transfer\reactions provide detailed information about nuclear single-particle structure, whileZnucleon transferreactionsprovideanalogous information about particle-particle correlations in nuclei (e.g., pairing). The majority of early transfer experiments and theory were devoted to studies using light ions [e.g., (t, p) or (p, d) reactions I)]. More recently, there has been considerable * Work supported by USDOE and the Danish Research Council. + Permanent address. tt Present address: Univenidade

Federal de Rio de Janeiro, Rio de Janeiro, Brasil. 275

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transfer

interest in transfer reactions emphasizing light heavy ions and target-projectile systems only a few nucleons removed from closed shells [e.g. 6oNi(‘80, 160)62Ni, etc. ‘)I. The emphasis on such systems can be understood in terms of the historical development of experimental and theoretical transfer techniques. The vast majority of transfer experiments have utilized some variant of charged-particle spectroscopy, while transfer theory is synonymous with the distorted-wave Born approximation (DWBA) in most applications. Neither of these tools are well-suited to the study of transfer reactions involving very heavy ions or nuclei far removed from closed shells. Heavy-ion straggling and the close spacing of collective states tax the resolution of charged-particle spectroscopy experiments, while the strong coupling of the ground state to collective states in the same systems invalidates the perturbation theory on which DWBA is based. These problems have been partially overcome with high-resolution spectrographs such as Q3D systems, and the introduction of coupled-channel calculations for transfer reactions. However, the heaviest ions for which resolutions of 50-100 keV (typical of the spacing for the strongest collective states) are attained in present systems lie roughly in the mass-20 region. Similarly, calculations based on a standard coupled-channel treatment are restricted to rather light heavy-ion systems by numerical and financial considerations. Despite these problems, the study of collective states using transfer reactions with very heavy ions is an intriguing prospect. In general, one may expect that such reactions will probe single-particle and correlation structure in nuclei as in lightion reactions, but with unique features associated with heavy ions. These features may be broadly separated into two categories. (i) Very heavy ions may lead to strong collective excitations, allowing the study of microscopic nuclear structure under the influence of the collective modes. (ii) Strong collective and single-particle excitations may imply localizations in coordinate space which are not present with light ions. Several recent experimental and theoretical developments have made it feasible to consider transfer reactions in this new region. (i) Particle-y coincidence spectroscopy techniques are being developed which use relatively low-resolution particle detection to define scattering angles and approximate energies, and high-resolution Ge(Li) detectors to identify the states excited 3, 5- 6). (ii) High-resolution spectrographs may make collective states with spacings in the few hundreds of keV range (e.g. vibrations) accessible to ions such as Ar. (iii) New theoretical methods based on semiclassical and classical-limit approximations are making the calculation of heavy-ion inelastic and transfer reactions to collective states a manageable problem 3- ‘97- “). (iv) There is now a much better understanding in transfer theory of problems such as recoil corrections and competition between successive and simultaneous particle transfer. This increased understanding of the reaction mechanism implies

M. W. Guidry et al. / Two-nucleon transfer

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a greater precision and flexibility in the use of transfer reactions as a spectroscopic probe of nuclear structure. Accordingly, we address ourselves in this paper to transfer reactions involving very heavy ions. We use the term “very heavy ion” to denote ions considerably more massive than the projectiles such as oxygen previously used in heavy-ion transfer reactions. Qualitatively, they may be characterized by having sufficient charge and mass to virtually exhaust the elastic exit channel in favor of multiple collective excitation for collisions with a collective target nucleus. Light heavyion collisions are characterized by short de Broglie wavelengths leading to localized classical trajectories for the relative ion-ion motion. Very heavy ion reactions are often characterized by the additional feature of localization in internal collective coordinates of the reacting ions because of strong collective excitations. As a practical definition we shall take very heavy ions to mean those with A X 40. Reactions with deformed targets (e.g., Xe+rare earth) are particularly good to illustrate the general statements made above about the unique features of very heavy ions. In such collisions using ions in the mass 150-200 region one obtains appreciable inelastic excitation of states with angular momentum Z - 20-30h. Classically, near the turning point in such a system it is possible to have 10-20 units of collective angular momentum in the deformed system when the transfer occurs. This is illustrated in fig. 1 for a classical calculation and fig. 2 for a semiclassical calculation +. Although all possible paths contribute, the qualitative picture that emerges in the classical limit is illustrated in fig. 3 : Strong inelastic excitation in the entrance channel, transfer of nucleons, and strong inelastic excitation in the exit channel. Therefore we may expect that two-nucleon transfer in such a reaction probes nucleon-nucleon correlations (most notably pairing) under the influence of collective angular momentum, while one-nucleon transfer yields information about the single-particle orbits in the rotating potential. As has become increasingly clear, the interplay of collective and microscopic motion is of decisive importance in determining the low-energy structure of heavy nuclei, and significant modification of microscopic nuclear structure due to collective angular momentum in the range being discussed here is well documented 16). The second unique feature attributed above to heavy-ion reactions was localization in coordinates conjugate to large quantum numbers. Two localizations have already been known for some time in reactions involving light heavy ions: (i) Simplification of the reaction mechanism by strong absorption and (ii) radial localization on classical-like trajectories for the relative motion. However, as discussed recently in connection with inelastic scattering 4), there are other localizations + We note that the angular momentum in fig. 1 has been decreased below that expected for Coulomb excitation by the Coulomb-nuclear interference. The amount of this decrease will depend on the nuclear ion-ion potential, which is not completely understood for very heavy ions. The potentials used in the calculations were chosen to be similar to those determined in ref. 4).

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Fig. 1. The time dependence of some important quantities in a heavy-ion collision. A hamiltonian with parameters similar to those listed in table 1 has been used. The collision was assumed to involve zero impact-parameter scattering with an initial orientation angle for the rotor of x0 = 30”. The radial separation r and rotor angular momentum I, are shown as a function of time, which is given in dimensionless units of the time required at asymptotic velocity to cover half the distance of closest approach for a Rutherford trajectory. Also shown is a differential transfer amplitude calculated for transfer from a pure [642]3+ 2-particle configuration using the semiclassical barrier penetration form factor described in the text. Note that most transfer occurs within about 1 fm of the turning point, and that more than half of the final collective angular momentum is in the rotor when transfer is likely to take place.

M. W. Guidry et al. 1 Two-nucleon transfer

t/10-*’

279

set

Fig. 2. Semiclassical probabilities as a function of time for multiple Coulomb excitation in a Xe+U collision. Note that there are significant probabilities for I -. 16-20 in the region near the turning point where transfer is likely to occur (region between the dashed lines). Unlike the example in fig. 1, no nuclear potential is included in this calculation.

ENTRANCE

EXIT

Fig. 3. A schematic illustration of heavy-ion transfer reactions involving deformed nuclei, as suggested by figs. 1 and 2. Although all possible paths contribute, in the classical limit we may expect paths such as that illustrated to dominate: Strong inelastic excitation in the entrance channel, particle transfer, and strong inelastic excitation in the exit channel. Accordingly, we may expect such transfer reactions to probe nuclear structure under the influence of significant amounts of collective angular momentum.

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M. W. Guidry et al. / Two-nucleon transfer

which may be of significance in heavy-ion reactions. For example, strong rotational excitation implies a classical localization in orientation angle of a deformed nucleus in a heavy-ion collision. This can be shown to lead to a situation in which inelastic excitation of particular states in a rotational band corresponds to interaction of the heavy-ion with localized zones on the surface of the deformed nucleus. For inelastic excitation this zonal localization has been exploited to investigate the deformed ion-ion interaction potential 4). For transfer reactions this leads to additional features. For single particle transfer it allows a localized angular probe of the deformed orbital lobes, while for 2-particle transfer reactions it implies probing the angular character of those orbits lying near the Fermi surface, i.e. angular localization on the deformed nuclear surface of orbits localized in momentum space near the Fermi momentum. 2. Semi&ssical

theory of 2-neuron transfer reactions for eveneven deformed nuclei

In fig. 4 we show a simplilied’coordinate system which will be used in the approximate calculations discussed in this paper. We assume no excitation of the spherical projectile, and consider only even-even nuclei. As in our previous inelastic calculations we restrict the calculation to near backward scattering in the c.m.s. (J = 0, where J is the total angular momentum, implying beam energies such that the grazing angle for optimum transfer is near 180”). We will also assume that the rotor motion and relative motion occur in the same plane. In the classical limit there is a unique relationship among incident angular momentum, angular momentum transfer and scattering angle 9, and we may write the probability for excitation of a state I in coincidence with backward projectile scattering as

B

a

Fig. 4. Coordinate system relevant to the a-particle transfer reactions discussed in this paper. The assumption is made that the collision involves only zero impact parameter, and that the scattering occurs in a plane as a result. This is rigorously correct only if the transferred cluster carries no angular momentum.

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where S~,=+oo.is the classical-limit S-matrix element for the transition 0, + Z, for an incident partial wave ii, = 0, and where we use a and /I to denote entrance and exit channel quantities. This prescription has been shown to give excellent results for heavy-ion inelastic scattering (a = /?) [refs. 3,4,8-12)]. This problem is more complicated than the inelastic situation due to the presence of different nuclei in the entrance and exit channels. In a separate publication we will describe this problem within a classical-limit coupled-channels Born approximation. In this treatment the inelastic excitation is described by the classical-limit coupled-channels methods we have successfully applied to rotational excitation, while the transfer is handled in first-order quantum mechanical perturbation theory, and account is taken of the coupling among transfer and the translational and rotational motion (recoil corrections). However, in this initial paper our concern is more with qualitative ideas than with quantitative accuracy. Accordingly, we introduce a no-recoil semiclassical approximation using classical trajectories symmetrized between entrance and exit channels, and an heuristic transfer form factor based on semiclassical arguments. This will allow the transfer theory to be developed as a transparent modification of that for inelastic scattering and will establish simple concepts which may be used in interpreting the more sophisticated calculations to follow. As we have argued in ref. ‘*), in the no-recoil limit the J = 0 S-matrix element may be written (dropping explicit reference to entrance and exit channel) ,/sin Xosin x djildX,P,(cos ~)F(j$Xo))eid’*dXo,

(2)

where Q(xo)) is a transfer form factor, the variable ji is related to the original variable x0 (fig. 4) by a canonical transformation as shown in the appendix of ref. I’), and where the phase A [defined explicitly in eq. (4) of ref. “)I is related to the classical action over the trajectory. The integral representation (2) of the J = 0 S-matrix element differs from that for rotational inelastic scattering only by the presence of the form factor F(~(x~)), and is seen to have exactly the same form as the analogous expression for rotationvibration excitation in deformed nuclei as derived in ref. l*). In writing eq. (2) we have assumed that the inelastic excitation is to be treated as a coupled-channels problem, but the transfer itself is a perturbation. As explicitly displayed in the notation, f is related to x0 by the classical equations of motion and a canonical transformation. Therefore we may expand F(~((x~)) in Legendre polynomials with argument cos 2 WXO)) = c G,(cos

ii),

L

where the orthonormality

of the Legendre polynomials implies

uL = +(2L+ 1) R F(i)P,(cos s0

2) sin jidi.

(4)

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Substituting into eq. (2) sJ=O I-0

=

~~~(~OZO~~O}~2~2~+ 1) “,/sin x0 sin? dji/dx, s0 x P,(cos

~)eidihdXo= Nt c

where sir”, is the J = 0 inelastic S-matrix element for the transition 0 --f N, and where we have used the Clebsch-Gordan series for the product of two spherical harmonics. This result may be given a simple interpretation in this no-recoil limit: the excitation amplitude involves the product of the semiclassical rotational and transfer amplitudes integrated over the orientation variable with a factor PI (cos 2) to project a good asymptotic total angular momentum I [eq. (2)]. We may then interpret the Clebsch-Gordan coefficient in eq. (5) as representing the partition of total angular momentum transfer (I) between the collective excitation (N) and the particle transfer (L). The problem has now been reduced to calculation of the transfer formfactor (defined by the coefficients a& and of inelastic S-matrix elements. The form factor is discussed in sect. 3. The evaluation of the inelastic S-matrix element has been discussed in refs. *- 12), where three methods of calculating it have been described in detail. The relationships among these methods for evaluating the classical-limit S-matrix, and the advantages and disadvantages of each approach, have been discussed for inelastic scattering in refs. lo, 1l). Similar considerations apply in the transfer problem as formulated here, since it is closely related to our previous treatments of inelastic scattering. Because none of the conclusions of this paper will depend on the excitation probabilities for highly forbidden states, nor on the inclusion of diffraction and refraction effects arising from the imaginary ion-ion potential, the calculations presented here have employed the real saddle-point metiwd of ref. *), with inclusion of deformed Woods-Saxon potentials to describe both the real and imaginary nuclear potentials. An important feature of the real saddle-point method is the approximation of the effect of the imagina~ potential W with a simple damping factor. This factor is given by :

d = expC- b @CdxoNI, where the imagina~

(61

part of the classical action is given by:

I

Wdt,

traj

with the integral over a trajectory in the real part of the potential. F$. (3) can be modified to include the effect of the imaginary potential as follows: ~(8~~)) = &8(xo))d = t: a,P&os L

ii)

(8)

M. W. Guidry et al. 1 Two-nucleon transfer

283

Then eqs. (l)-(S) remainvalid except that $7: is now the inelastic S-matrix element evaluated in the real part of the potential, and now the coeffkients aL carry information about both the transfer form factor and the imaginary potential. 3. The semiclassical transfer form factor The nuclear structure information associated with the particle transfer is contained in the transfer form factor. To first order in the transfer potential V, the form factor will involve a matrix element of the form

for a reaction a(A, B)b, where the wave functions denote internal eigenstates of the nuclei in entrance and exit channels. The transfer potential is approximately the total scattering potential minus the potential giving rise to elastic and inelastic scattering, either in the entrance channel (prior representation) or the exit channel (post representation). The elements of the T-matrix are matrix elements of the effective interaction defined by this form factor between wave functions of relative motion for the system. In DWBA these wave functions would typically be the distorted-wave eigenstates in the elastic scattering potential, corresponding to a first-order perturbation treatment of the entire problem. In the problem being discussed here they would represent semiclassical coupled-channels solutions to the inelastic scattering problem, corresponding to treating the inelastic plus elastic scattering to all orders, but the transfer itself as a perturbation. The evaluation of this form factor in a proper quantum-mechanical treatment is an involved problem which we shall deal with in a separate publication. In this paper we follow a different course and explore a semiclassical model of the transfer form factor in the no-recoil limit. In this model, the transfer will be treated as the barrier penetration of a dineutron cluster in the heavy-ion collision. All calculations will involve the same projectile, and we will concern ourselves primarily with the effect of the nuclear structure of the deformed nucleus on relative excitation probabilities. Therefore, we shall neglect structure effects associated with the (assumed spherical) projectile: in the calculations presented in this paper the projectile acts only as a potential well which can donate or accept nucleons, and which causes collective inelastic excitation of the deformed nucleus. Qualitatively, we may expect that the transfer form factor 9(x0) appearing in eq. (8) is governed by three effects, as illustrated in fig. 5. (i) The penetration effect favors polar (x N O”) collisions because the effective barrier for transfer is less in that case. (ii) The damping effect favors equatorial (x N 90”) collisions because the stronger overlap for polar collisions increases the probability that the reaction proceeds to more complex reaction channels than the ones being considered explicitly. (iii) Because of single-particle quantization in the deformed potential, the orbits available for transfer at the surface of the deformed nucleus will exhibit

284

M. W. Guidry et al. 1 Two-nucleon transfer

(A)

0 0

(8)

(Cl

Pig. 5. Illustration of the basic factors governing transfer reactions for heavy ions on deformed target nuclei. Polar collisions as in (A) favor penetration of m&eons through the effective barrier separating the ions relative to equatorial collisions as in (B). However, if the collision is too violent orientation (B) will experience less damping out of the transfer channel than (A). The penetration and damping effects compete to define an effective grazing angle. (C) representsIschematically that orbits available for transfer in the deformed potential have preferential orientations with respect to intrinsic axes.

angular localization, It follows that certain orientations of the rotor (those presenting lobes of the involved orbitals to the projectile during the collision) may be expected to be more favorable for transfer. Therefore, we may expect effects (i) and (ii) to conspire in defining an effective grazing trajectory for a particular orientation angle, while effect (iii) defines whether particular deformed single-particle orbits are likely to participate in a transfer reaction along that trajectory. Qualitatively, the product of the competing penetration and damping effects will define a pair of bands for optimum transfer which, for a given partial wave, move from the poles of the nucleus to the equator as the beam energy is increased (fig. 6). Only when these bands intersect lobes of orbitals involved in the transfer is transfer likely to occur. Since the orientation angle of the rotor is intimately connected with the amplitudes for population of rotational states by inelastic

hf. W. Guidry et al. 1 Two-nucleon transfer

285

3

X

Fig. 6. A schematic illustration in the sudden impact limit of how the damping and penetration amplitudes vary with angle as the beam energy varies. The penetration amplitude apcn[ _ &(r, 2. L’) in eq. (13)] favors polar collisions (x = 00, 180°) because the tunneling is greatest there. Conversely the damping amplitude adamP(_ exp ( - Im@)) favors equatorial collisions (x = !W) because those collisions involve less nuclear overlap, and hence less chance of depleting flux from the transfer channel into more violent ones. Thus the product ad_,apcn forms a pair of “transfer bands” which move from the tips of the nucleus to coelesce at the equator of the nucleus, and finally to disappear completely, as the beam energy is increased (horizontal striping in left- and right-hand figures). Transfer will be likely when the “transfer bands” overlap the regions of extrema in the angle-dependent form factor. For the simple schematic case of an orbital with a single pair of lobes (vertical striping on right side of fig. 6), such an overlap with the angle dependent form factor is shown in the middle right figure.

scattering, the rotational population pattern in a transfer reaction will contain information about the single-particle motion and correlations. As a first step in calculating a semiclassical approximation to F(j(x,,)) for 2-particle transfer we write the amplitude for finding a pair of particles in a &G! degenerate deformed orbital at a particular point on the deformed nuclear surface. Although this could most elegantly be handled using second quantization we defer that to a future publication and proceed, in keeping with our announced intentions, in a more heuristic manner. Assuming axial symmetry, the single-particle wave functions in the deformed potential may be written

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transferr

where the spherical coordinates in the intrinsic coordinate system referenced to the deformation symmetry axis are (r’, t?‘, q’), the Cg,,, are expansion coefficients (e.g., Nilsson coefficients) defining the unitary transformation between the spherical and deformed basis, 52 is the conserved projection of the particle angular moments on the symmetry axis (52 = A + C, where ,4 and C are the projections of the orbital and spin angular momentum of the particle, respectively), N is the principal oscillator quantum number (only approximately conserved), R,r(r’) is a radial wave function in the spherical basis and &$ is a spin function over the spin coordinate rr. Defining the antisymmetrized 2-neutron wave function for a zero-range pair (r; - V; E r’) in the standard way and integrating over the azimuthal coordinate p’ we obtain $r&‘, el) = +J$ C $FTiJ~( L’ll’

- l)“[cr( 1)/I(2)- a(2)B(1)]RNl(r’)RN,.(r’)

where we have used the notation r&n) and /3(n) to denote the spin up and spin down functions for the nth particle. Note that the Clebsch-Gordan coefficients and symmetry under parity require ’ = even and that Ir-i”/ S L’ 5 (+I’. Note also that the spin function [cl(l) assuring that the total 2-nucleon wave &2)--a(2) Kl)l is singlet (antis~met~~), function is antisymmetric since the spatial part of the zero-range 2-nucleon wave function is clearly symmetric under exchange of particles with the same isospin. Since we are interested only in relative probabilities~ we will omit the constant factors and the spin function in what follows. It is important to realize that the particles being in time-reversed (+S2) deformed orbits does not mean that the sum is restricted to L' = 0,since the angular momentum of the particles is not conserved in the deformed potential. The amplitude that the configuration defined by eq, (10) contributes to a 2particle pickup reaction populating the ground-state band of the residual deformed nucteus is defined by the amplitude that the orbit is occupied by a pair in the daughter multiplied by the amplitude for being unoccupied in the parent (and vice-versa for a stripping reaction). Introducing the pair occupation (V) and vacancy (U) amplitudes and omitting the constant and spin factors in (10) we may write the amplitude for finding a pair of particles in time-reversed deformed orbits at the intrinsic coordinates (r’, 6’) which could contribute to a 2-neutron transfer reaction populating the ground band of the daughter nucleus as

where the positive sign appfies to stripping and the negative to pickup, the index i

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Fig. 7. A schematic view of the projectile and target potential wells. In a pick-up reaction the dineutron, bound by an energy B,, in the target tunnels from R, to R,. The Q-vahte of the reaction is the difference in two-neutron binding energies for the projectile (E;,) and target (Ban). For the approximations used here to be valid, we require Q N 0, but this is not a serious problem since many heavy-ion neutron transfer ground band cross sections are likely to peak near Q = 0 due to momentum and energy conservation considerations. Since the target potential well is deformed, it will also vary with angle. In addition, since the relative separation of the projectile and target varies with time, the effective barrier between the two pot&itial wells will vary with time. Therefore this figure represents a schematic slice of the potentials at a particular time on the trajectory, and the total tunneling amplitude involves an integration of such configurations over time.

labels the deformed orbitals for which UiVi # 0, and d!!(r) d$(r’)

x

C JiGi$GiRNl(r’)RNl~(r’)(

-

is given by

l)“( lO/lOlLIO)

II’ x [(I - nrn~co)c$&~.,

-(~-(n+l)ll(n+l)~~o>C~n+~C~~~n+~].

(12)

As the potential wells in which the particles move approach each other on the trajectory, the effective barrier between the potentials is decreased as shown in fig. 7. Eq. (11) gives the amplitude for finding a pair of particles which may participate in the 2-particle transfer at the surface of the isolated deformed nucleus. The amplitude that this pair of particles appears in the other potential well when the nuclei are separated by a distance r is approximately proportional to the semiclassical (WKB) tunneling amplitude for penetration of the effective barrier I’), and the total transfer amplitude may be estimated by integrating the product of the amplitude (11) and the penetration amplitude over the classical trajectory. We define the WKB penetrability amplitude by

fis

_Mr,x, ~3 - exp - i

h

“,‘,‘JW,

RI

I

x, J%s)- &IS ,

(13)

where ~1is the dineutron reduced mass and all other quantities are defined in figs. 4 and

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M. W. Guidry et al. / Two-nucleon transfer

7. The potential V(r, x, L’, s) is the sum of the effective potentials from the two wells for the dineutron moving with angular momentum L’. The integral in (13) is to be evaluated along the least-action path for traversing the barrier. It may be approximated by adopting a path lying on the line of centers between the two ions. The differential transfer amplitude may now be written y

=

1 fJ,c C I

d~!(R,)P,,(cos

x)fp(r,

L'

2,

L),

(14

where we have made the identifications (cf. figs. 4, 7) 1x1= 10’1and R, = r’, and note that R, is determined by x and the binding energy of the two neutrons. The transfer amplitude (9(x0)) is finally given by $(X(x0)) = d %Rj,,., (y)

eiQ’dt,

(15)

where the integral is over a classical trajectory defined by the hamiltonian using quantities averaged between entrance and exit channels, and beginning with an initial orientation angle x0, and where Q is the Q-value of the reaction. The form factor (15) may be used in eqs. (2)--(5) to determine transfer probabilities. In summary, the model described here is similar to the one we have applied with considerable success to the cases of Coulomb excitation and inelastic scattering. The primary difference is that the semiclassical amplitudes for inelastic scattering in the presence of Coulomb and complex nuclear potentials will be modified by an amplitude for transfer of a dineutron cluster in the process of transferring the collective angular momentum. As we shall see, this will lead to significant modification of the population patterns expected for collective excitation alone. Although the model is quite simple, we expect that qualitative and semiquantitative predictions may be made using it. Thus it can serve as a guide for experiments and for semiquantitative interpretation of data. Inadequacies in the model are likely to lie in the crude transfer form factor rather than in the treatment of the inelastic excitation. Obviously, the effect of finite-range and recoil corrections, sequential transfer, and detailed intrinsic structure of the projectile must be included in quantitative calculations. They have been deliberately ignored in this paper because our initial aim is to explore qualitative features in as simple a manner as possible. For similar reasons the calculations have been restricted to backward scattering in the c.m.s. All of these deficiencies can be removed when required. However, we expect that the basic features outlined in this paper will remain in a more sophisticated treatment. Considerable emphasis will be placed on angular localization during the transfer process in the remainder of this paper. This localization is simplest for the backward scattering, sudden collision examples which will be used to represent basic ideas. In the more general case of a non-sudden collision and scattering to non-backward angles the localization effect is more complicated but should remain significant.

M. W. Guidry et al. / Two-nucleon transfer

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For example, in the collision of a Xe ion with a rare-earth target the target rotates 5 15” during the transfer. This leads to some smearing of the angular localization, but is unlikely to alter it appreciably. For non-backward scattering the formulas are more complicated and the collision depends on two orientation angles for the rotor instead of one. However the classical picture is still one of projectile interaction with particular zones of the nuclear surface. 4. Two-neutron transfer rotational signature We now consider application of the above formalism to some specific examples. To be definite we will consider only two-neutron pick-up from the target in the examples. Modifications for the analogous two-neutron stripping from the projectile will be obvious. Since we are concerned here with qualitative aspects of the problem, we also adopt the following approximations: (i) Calculations will be considered in the sudden limit of rotational excitation. In that case the formulas and interpretation simplify because 17= x = x0 at all times during the interaction. (ii) The WKB’ tunneling integral eq. (13) is evaluated along the line of centers rather than along the least-action path. The two do not coincide since one potential well is deformed, but the difference in the integral is not very large. Such a prescription gives a form factor which decays radially in good agreement with that from more sophisticated treatments of transfer. TABLE1 Parameters used in the calculations Nuclear ion-ion potentials R! = 1.2Ofm

ap = 0.65 fm

Ri = 1.20 fm

a, = 0.65 fm

V = 160 MeV W=

120MeV

Moments and deformation parameters Qb” = 1.2 b

QF’ = 0.80 b2

8,

/I,

= 0.30

= 0.03

Separation energy and Q-oalues Q = 0.0 MeV

S211= 13.8 MeV

Singk-particlepotsltials R, = fixed by matching condition (see text)

d = 0.65 fm

VA2”)= 104 MeV

Nilsson orbit parameters lo) u = +6 (8, - 0.3)

B* = 0

iuo, = 41A-‘13

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M. W. Cuidry ef al. / Two-nucleon transfer

R tfml

Rtfm)

10

a-

10

c505 1 % -

8-

t5213 --_ .‘c,

R (fmf6

‘12-

0

\

4

\

4-

\ \ \

6

Rlfml

8

Rtfm)

Fig. 8. The amplitudes Jr’, 0’) defined by eq. (1 I) for some configurations involving only a single rareearth orbital (single term in sum over i). The plots are contour plots in the intrinsic r’, 0’ plane and the orbitals are labeled by the standard asymptotic quantum numbers. All contours are to be muhiplied by lo-‘. The dashed line indicates the endpoints for the tunneling integral R, for various o~entations of the rotor in a ‘s4Sm(“2Xe, ‘34Xe) *?3rn collision. The angular localization of the probability for 2-nucleon transfer is clearly inplied in these figures.

(iii) Since they are widely available, we shall use Nilsson wave functions 18) (modified deformed oscillator hamiltonian) in eq. (12). However, since the oscillator potential behaves incorrectly in the surface region the effective barrier for tunneling will be defined by Woods-Saxon plus centrifugal potentials for the 2-particle motion. The depth of the 2-particle Woods-Saxon potentials is taken to be twice the energy of the Fermi surface in the Nilsson potential plus the two-particle separation energy, and the Woods-Saxon radius is adjusted so that the 2-particle Nilsson and WoodsSaxon potentials coincide at the tunneling endpoint in the deformed nucleus R,. To further simplify matters, we use coefficients from Nilsson’s original paper Is) and neglect AN = 2 couplings in the oscillator basis. None of these approximations

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are essential and they can easily be removed when a more accurate treatment is warranted. As an initial illustration of the qualitative features expected for 2-neutron transfer reactions in deformed nuclei we consider the idealized case where the deformed nucleus is assumed to have a sharp Fermi surface. Then the summation over levels i in eq. (11) is restricted to a single Nilsson orbital. Specifically, we consider the 2neutron pickup reaction ’ 54Sm(132Xe, ’ 34Xe)’ “Srn, with the Fermi surface placed at the [642]f+ and [SOS]?- Nilsson orbitals. The parameters used in the calculations are shown in table 1. In fig. 8 we show for a few orbits the quantities (10) defining the amplitude for finding the pair of nucleons at a particular set of coordinates (r’, 6’) in the intrinsic frame of the isolated deformed nucleus. The localization in orientation angle for 2-particle transfer is clearly seen. In fig. 9 the amplitudes 9(x,-J calculated using eq. (15) in the sudden limit (j = x = x0) are shown, and in fig. 10 excitation functions for the [642]3+ and [SOS]?- orbits are plotted. In these calculations we have used the real saddle-point method described in sect. 2. Accordingly, the sudden-limit amplitude 9 (x0) represents the product of a damping amplitude and transfer amplitude [cf. eq. (S)]. Some representative coefficients aL from eq. (8) are listed in table 2. The values of L for which a, is appreciable have 3 sources: (i) The values II-l’\ 5 L $ Z+l’ contributing to the 2-particle wave function in the isolated deformed nucleus (eq. 12). This corresponds to L,,, = 10, 12 for the [505]~- and [642]3+ orbitals respectively. (ii) Angular momentum

Fig. 9. The quantum number function I&) and the transfer amplitudes S(&,)) (normalized) for the examples discussed in the text involving the reaction 154Sm(132Xe,1”4Xe)1”2Smin the sudden limit.

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0

500

581

540

” 500

540

581

EL,8fMeV)

E_tMev)

INELASTIC (SUDDEN)

0 500

540

ELA8(MeV) Fig. 10. (a-b) Backward scattering excitation functions (relative probabilities for populating members of the ground-state rotational band) in the reaction ‘54Sm(‘32Xe, ‘34Xe)‘s2Sm for the case where a single Nitsson orbit contributes to the 2-particle configurations in eq. (11). All calculations are in the sudden limit for the rotational excitation. (c) Excitation functions for the analogous inelastic excitation of r5’Sm. In these calculations, all transfer probabilities have been multiplied by the same arbitrary constant relative to the inelastic case.

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TABLE2 The coefficients aL from eq. (4) for the cases considered in this paper at &, L 0

2 4 6 8 10

= 540 MeV

C64219+ (x 104)

c5051y( x 104)

A = 1 MeV

(x 104)

A = 0.5 MeV ( x 104)

8.38 2.63 -1.55 -6.65 6.5 0.34

1.68 - 14.2 10.3 -4.14 0.55 0.16

51.5 -37.2 - 13.3 -1.45 7.23 -0.45

29.0 - 19.9 -8.35 -1.96 5.67 -0.53

associated with penetration of the anisotropic barrier [eq. (13)]. (iii) Angular momentum associated with the anisotropy of the imaginary potential, the effect of which has been approximately included in uL in the real saddle-point approximation. We note that the contribution of (iii) is a consequence of the real saddle-point approximation. In either the integral or complex saddle-point methods [see ref. “)I only (i) and (ii) would contribute angular momentum, since the effect of the imaginary potential would be included in the trajectories exactly. The excitation functions in the two transfer cases shown in fig. 10 differ considerably with respect to each other and from that for inelastic scattering. This is seen more clearly in fig. 11, where the transfer excitation probabilities at a fixed energy and also the inelastic excitation probabilities for the reaction 1s8Gd(‘32Xe, 132Xe’)158Gd are compared. The conspiracy of the penetration, damping, and deformed orbital localization effects embodied in the transfer form factor to modify the rotational signature pattern from that expected for inelastic excitation is in evidence. This is qualitatively understood from considering the quantum-number function for the inelastic scattering, also displayed in fig. 9, along with the transfer amplitudes S(xJ. The quantum-number function is identical to that for inelastic scattering and has been discussed before *- 12). For the classically allowed transitions there are generally two angles x0 satisfying the semiclassical boundary conditions and the required derivatives and phases are evaluated at these angles. The probabilities for inelastic excitation are spread across a number of states up to the maximum allowed classical spin, and exhibit considerable oscillatory structure (figs. 1Oc and 11). This behavior has been quantitatively described in terms of quantum interference between the two different initial rotor orientations leading to the same final state 4*s’8-12). For example, from fig. 9 there are two initial orientations (x0 N 15”, 75”) which lead to the lO+ state. These contributions interfere constructively near Elab N 540 MeV, and destructively near Elab w 500 MeV to produce the oscillation of the lO+ probability seen in fig. 9. The rapid decrease in all inelastic probabilities for Elab 2 540 MeV is a consequence of strong absorption from the imaginary potential.

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1

transfer

INELASTIC t

Fig. 11. Excitation probabilities for ground band states at a fixed E,a, = 540 MeV for the examples described in figs. IO and 13. As discussed in the text, the diverse patterns have simple explanations in terms of the semiclassical form factor 9(x,).

In contrast to the inelastic case, the transfer excitation functions are relatively localized in energy since they are cut off by strong absorption on the upper side and exponential decay of the bound-state wave functions (tunneling factor) on the lower side. The ordering of states with respect to excitation probability (rotational signature) is also altered in the transfer cases from that for inelastic scattering. From fig. 9 it is clear that the transfer form factor may suppress orientation angles that could lead to inelastic excitation of a state. It is further clear that the manner in which the inelastic pattern is altered by the transfer form factor will be a sensitive function of the angular dependence of the form factor, and that this in turn depends on the nuclear configurations contributing to the transfer. For example, the [642]3+

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amplitude in fig. 9 is strongly peaked near the maximum of the quantum-number function (x0 N 4P) and makes considerably less contribution at those angles leading to low spins (x0 N O”, 900). This may be understood from the fact that for this case the product of the damping and penetration factors favors orientation angles broadly distributed around x0 N 50°, while from fig. 8 we see that the deformed orbital factor favors x0 m 30” primarily, and x0 N 70-900 to a considerably smaller degree. The competition of these effects generates the large peak near x0 = 400 and the smaller peak near 90” for the [642-J++ form factor in fig. 9. The net effect is to strongly suppress population of low spins and emphasize high spins in the rotational signature, as shown in figs. 1Oa and 11. We also note that the fall-off on the high-energy side of the excitation function occurs earlier and is considerably more precipitous for the [642]3+ transfer than for the other cases. This also has a simple explanation in terms of the transfer form factor. The excitation probability remaining at the highest energies in the excitation functions must come primarily from trajectories with x0 N 900 since those collisions involving smaller values of x0 will lead to more overlap and complete absorption at these energies. However, the [642]++ form factor will be small for x0 N 900 because the deformed orbital factor strongly favors x N 30“. Therefore the [642] j’ excitation functions drop rapidly for Elab k 550 MeV as the grazing orientation band defined by the tunneling and damping amplitude moves to larger angles, corresponding qualitatively to the picture depicted in the bottom diagram of fig. 6. The case of two-particle pick-up from a [505]9- configuration shown in figs. lob and 11 is considerably different. Within the approximations being used here this configuration is based on a pure shell-model orbit of unique parity, and the deformed orbital factor has only a single lobe peaked at x = 900 (cf. fig. 8). Accordingly, the form factor 9 (x0) is strongly peaked at large angles as seen in fig. 9. This has several consequences. First, consulting the quantum-number function and transfer amplitude fig. 9 we see that the form factor in this case will favor lower spins. This is confirmed by the calculations shown in figs. lob and 11. For example, at I&,, = 550 MeV the 18+ state is excited with more probability than any other state in the [624] j’ calculation, while in the [505]9- calculation ten states receive more population than it does. Conversely, as a second example, the 8+ state is strongly excited in the [SOS]ycalculation, but is hardly populated in the [642]$+ cast. Secondly, for those states which are populated in the [505]~- case only the large-angle solution to the semiclassical quantization equation will make significant contribution, and the quanta1 interference between orientation angles will be greatly suppressed. The absence of this interference structure can be seen in fig. lob and, more dramatically, in the comparison of the inelastic and [SOS]?- cases in fig, 11. Although transfer reactions are expected to be rather quanta1 phenomena, we see here an example where the addition of particle transfer to inelastic scattering leads to a more classical behavior for the rotational signature than that for inelastic scattering only.

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Finally, we observe that the excitation function for the [SOSly- example is shifted to higher energy and is broader than in the [642]$+ calculation. This has a simple explanation also in the angular dependence of the form factor F((xJ. Because it is strongly peaked at x0 N 90” for the [SOSly- case, a higher translational energy is required to initiate transfer reactions than for the [642]$+ case which gets its primary contributions from lower orientation angles. On the other hand, by the converse of the argument employed above to explain the upper cut-off of the [642]++ example, the [505]9- excitation function persists to higher energies because it gets strong contributions from equatorial collisions (x N 900). The widths of the excitation functions will depend in a complicated way on the dynamics of the scattering, but they may be expected to carry information about the angular width of the zones on the nuclear surface in which transfer is likely. As previously stated, we have formulated the transfer expressions independently of detailed shell-model structure for the sphericalcollisionpartner. This independence of l-matching conditions occurs in the limit of transfer valley width narrow compared to the nuclear radii. Such a narrow valley is probably obtained for the subbarrier transfer for rather heavy nuclei, as studied here. It should be borne in mind that shell structure effects in the spherical nucleus can be explored in later refinements of this theory. For the head-on collisions considered here, the I-matching condition at the turning point is very simple: l,R, x l,R,. If such a selectivity manifests itself, the corresponding rotational signatures will be altered by the preferential selection of certain I-components in the deformed wave function. These effects should be sought experimentally by inducing one- and twoparticle transfer reactions with different spherical projectiles.

5. Two-neutron transfer and the effect of angular momentum on pairing in deformed nuclei Residual 2-body correlations in the deformed shell-model will introduce a diffuse Fermi surface. In the simplest approximation we may consider monopole pairing forces which generate 2-particle 2-hole excitations across the Fermi surface and spread the neutron pair over the orbits lying a distance N A on either side of the Fermi surface. In the case of a finite pairing gap A the angular form factor F(x(x~)) defined in eq. (15) will involve a summation over several orbits, with the number of orbits in the sum depending on the pairing gap, the level density in the region of the Fermi surface, and the detailed structure of those levels. As figs. l-3 indicate a considerable amount of angular momentum may be present in the rotor near the turning point, where most transfer is expected to occur. From these considerations we conclude that heavy-ion TNTR should be a sensitive probe of pairing correlations under the influence of collective angular momentum in deformed nuclei.

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To illustrate some of these ideas we repeat the calculation of sect. 3 for groundband transfer, but with a BCS description of the deformed nucleus. The Nilsson levels are now occupied pairwise with an amplitude Vi, and vacant with an amplitude Vi with

1)

K2=$ [l-J*

(16)

UF+K2 = 1,

(17)

TABLE3 The BCS amplitudes UV used for the A = 1,0.5 MeV calculations UV Orbit A = l.OMeV’ C631lt+ w3x[g: c64213 + KWf+ W41?+ c5141tc5w C651lf+ [633X+ c52w Eggc5mw21t+ iI4021t+ C4@m+ [g;;;r C5321tL=‘lf+ yay;;r c41313+ [523XC5411fc41llf’ ::::;:- +

0.105 0.106 0.121 0.126 0.144 0.154 0.161 0.182 0.209 0.229 0.301 0.338 0.352 0.504 0.572 0.578 0.491 0.455 0.419 0.385 0.303 0.281 0.271 0.245 0.223 0.183 0.151 0.136 0.115 0.115

A = 0.5 MeV +

0.110 0.122 0.170 0.199 0.210 0.410 0.610 0.640 0.423 0.355 0.302 0.260 0.181 0.164 0.156 0.138 0.123

+ Levels lying approximately 44 on either side of the Fermi surface have been includeed in the calculations.

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M. W. Guidry et al. / Two-nucleon transfer

where ck is the quasiparticle energy of the kth level with respect to the chemical potential 1 (Fermi level), which has been determined by imposing conservation of average particle number. In table 3 the U V factors calculated for use in eq. (11) are shown for the cases A = 1.0, 0.5 MeV. The resulting form factors 9(x0) are shown in fig. 9, and the coefficients aL. for eq. (7) are shown in table 2. The quantity (10) defining the pair amplitude in the intrinsic frame of the isolated deformed nucleus is displayed in fig. 12 for the A = 1.O MeV case, excitation functions are plotted in fig. 13, and probability signatures at a fixed energy shown in fig. 11. Comparing figs. 10, 11 and 13 we see that there are considerable differences between the paired cases and the unpaired cases represented by the [642]++ and [505]~- examples, This is primarily due to differences in availability of nucleon pairs at the deformed nuclear surface, as can be seen by comparing figs. 8 and 11. The coherent contribution of many configurations to eq. (11) with the same sign of the U V factor has the effect of emphasizing the availability at the surface of pairs available for ground-state band transfer, and spreading them out in angle. This has two major consequences: (i) The angular dependence of the form factor S((xo) is considerably different in the paired case due to the coherent mixing of 2-particle configurations with different angular dependence (cf. fig. 9). (ii) The transfer probabilities are considerably larger in the paired cases, due to the coherent contribution of many 2-particle configurations (note the different scales in figs. 10 and 13). The paired cases exhibit excitation functions which are broader than the unpaired ones. This is due both to the wider range of angles contributing to the transfer

A =l.O

2

4

6

MeV

8

10

R (fm) Fig. 12. Same as fig. 8, but showing UY-weighted sum over orbitals for the A = 1 MeV “%rn paired case. Note the absence of sharply localized angular structure in the surface region and the larger values of f(r’, 0’) due to coherence as compared to the single-orbit examples in fig. 8.

(x

10’)

0

10

20

500

ELIB(MeV)

540 58

500

E,,,

540


58

Fig. 13. Excitation functions as in figs. IOa, b, but for the case where pairing gaps of A = 1.O,0.5 MeV have been assumed. Note that these probabilities are considerably larger than in the single-orbit configuration cases of fig. lOa, b, due to coherence effects in the two-particle transfer form factor.

PI

30

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M. W. Guidry et al. / Two-nucleon

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form factor and to the greater radial extent of the form factor for the A = 0.5, 1.0 MeV cases. Comparing the scales in figs. 10 and 13, we note that the relative probabilities are typically enhanced by about a factor of 2S40 for the A = 1.0 MeV case and about 10 for the A = 0.5 MeV case relative to the pure configurations, as a consequence of target nucleus superfluidity. In addition, the rotational signatures in the paired cases differ considerably from the unpaired ones. From fig. 11 the paired cases emphasize the highest classically allowed spins as in the [642]3+ example, but lower spins are considerably more prominent for the paired case than for the [642]$+ one. Unlike the [SOS]?- calculation, the paired examples exhibit strong interference structure. All of this is easily explained by the broad transfer amplitude in fig. 9 for the paired cases which emphasizes a wide range of spins for the excitation function and which allows significant contribution of both orientation angles leading to the same final spin. As a consequence, the paired cases exhibit rotational signature patterns with more resemblance to the inelastic excitation than the unpaired cases (cf. fig. 11). From figs. 11 and 13 we see that the rotational signature is virtually the same for the A = 0.5, 1.0 MeV cases, as could have been expected since the form factors plotted in fig. 9 for the paired examples are quite similar. This is due to the fact that for pairing gaps this large there is a strong suppression of high-l components in the 2-particle wave function, as can be inferred from the coefficients uL in table 2, or from the lack of sharply localized angular structure in fig. 12 when compared to fig. 8. As we may deduce from table 2, the angular shape of the paired form factor is dominated by the average monopole and quadrupole moments of the orbits with momenta near the Fermi momentum (those with large UP’ factors). For the A = 0.5 and 1.0 MeV examples considered here the ratio of monopole to quadrupole contributions averaged over levels lying within + A of the Fermi surface are about the same, and the angular shape of the form factor is similar for the two cases. However, the two paired cases differ in the number of orbitals i with finite values of U V making contributions to the summation in eq. (11). As a result of the increased coherence with increased A the probabilities for the A = 1.0 MeV case are about 3-5 times larger than in the A = 0.5 MeV case, as fig. 13 shows. For even smaller values of the pairing gap we may expect decreased coherence, reduced probabilities, and a tendency for the form factor to be dominated by a single orbit lying at the Fermi surface, such as in the [642]3+ and [SOSly- examples. We should also recall that in this simplified treatment of the transfer form factor the possible superfluidity of the projectile has been ignored. Considerably larger enhancement factors may occur if both collision partners are superfluid. These model calculations suggest the sensitivity of heavy-ion TNTR reactions to the pairing correlations in deformed nuclei. In a more realistic calculation the gap could be parametrized as a function of the angular momentum Zin the excitation process. Thus the angular amplitude f(r’, 0’) would become an explicit function of Z, with this dependence reflecting centrifugal and Coriolis-induced modification

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of particle-particle correlations in the deformed potential. With a sufficiently precise experimental and theoretical description of the process it should be possible to map the variation of the pairing correlations with angular momentum in deformed nuclei in a rather direct manner. We defer a detailed investigation of the effect of angular momentum on the gap and of transfer to superbands (aligned 2quasiparticle bands) until a later publication. 6. Relation to vibrational excitations, one-particle transfer, and a-transfer in deformed nuclei The idea of an angle-dependent form factor on the deformed nuclear surface has previously been introduced in a semiclassical treatment of rotation-vibration excitation I’). In that paper a form factor for K = 0 vibrational excitation appeared which was approximately proportional to P, (cos x), where Iz is the multipole order of the vibrational phonon. Since the orientation angles also define semiclassical rotational quanta that can be transferred in the collision, the excitation signatures for rotational bands built on 2” pole (K = 0) vibrational states in deformed nuclei should exhibit large fluctuations which can be traced to the action of the P,(cos x) form factor. This model of rotation-vibration excitation gives results in quantitative agreement with more conventional treatments of the same problem l’). Although we deal primarily with 2-particle transfer in this paper, one can apply analogous ideas to l-nucleon transfer reactions involving very heavy ions on deformed nuclei. We may then expect to observe deformed single-particle structure modified by the competition between Coriolis and pairing forces. For example, in a pickup reaction from an odd mass deformed nucleus (with K # 3) we may expect an angular form factor based approximately at low angular momenta on a single l-particle Nilsson orbit, which evolves into one based on Coriolis-mixed orbits at higher angular momenta. Alternatively, in l-particle pickup from an eveneven rotor, we may expect a form factor which reflects the angular momentum induced change in pair occupation amplitude Vi for the involved orbit. In either case one may expect angular surface localization due to the collective angular momentum transfer. We have argued that this angular localization leads to the possibility of segmenting the average ion-ion potential in inelastic scattering and the large U Y factor orbits in 2-particle transfer into zones on the deformed nuclear surface. By analogous reasoning we may expect the possibility of mapping the surface lobes of deformed single-particle orbitals in one-particle transfer reactions. We also note that the formalism described here could be applied in essentially the same form to describe a-particle transfer reactions. In fact, the cluster approximation would be even more valid for a-transfer due to the high binding energy of 4He. The angle-dependent form factor could be calculated macroscopically in terms of defor-

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mation changes as described in ref. *O). Alternatively, it could be formulated in terms of the geometry of neutron and proton orbits near the Fermi surface, in an extension of the approach used in sect. 5 for neutron pairs alone. Since there is no evidence as yet for experimental observation of x-particle transfer in heavy-ion scattering from deformed nuclei at Coulomb barrier energies, we do not treat this problem here. 7. Experimental possibilities and future developments The study of heavy-ion TNTR for deformed nuclei presents severe problems for traditional methods of investigating transfer reactions. These problems are associated primarily with the difficulty of obtaining sufficient particle resolution (- 100 keV) with targets of realistic thickness to resolve states for collective nuclei. An analogous situation occurs for the case of heavy-ion inelastic excitation. There it has been demonstrated that the problem may be circumvented by employing particlecoincidence techniques 3*5). .Tha t is, the heavy ion is detected with intermediate (N MeV) resolution in coincidence with y-rays detected with high (N keV) resolution. Thus, the particle is used to define incident energy and scattering angle, and the y-ray is used to define the state (and daughter nucleus). More sophisticated approaches are being developed which allow particle-y coincidence spectroscopy with very heavy projectiles 6). These methods use large solid-angle, position-sensitive gas detectors to detect scattered particles. Because they are position-sensitive, such detectors can be used to correct for the extreme Doppler broadening of y-rays emitted by nuclei recoiling from thin targets into vacuum. Thus high resolution particle-coincidence spectroscopy is possible even with very heavy projectiles. In preliminary experiments with such devices the groundstate rotational bands of 1 and 2-neutron transfer products for Xe projectiles on rare-earth targets have been seen with sufficient intensity to demonstrate that quantitative experiments are feasible 6). The essential remaining experimental problem is related to the fact that y-ray spectroscopy measures the de-excitation of states. Because of cascade feeding processes, the population pattern observed in the de-excitation will generally differ from the excitation population pattern. The two can be related in a straightforward way if the feeding pattern is known. The same situation occurs in particle-y spectroscopy of inelastic scattering but is less serious because there is good evidence that for sub-barrier scattering from welldeformed rotors the primary population is of the ground-state rotational band. In the transfer case, one is much less sure, apriori, about the feeding pattern than in inelastic scattering, since the reaction mechanism is potentially more complicated. In many cases, strong general theoretical arguments can be made favoring ground-band population. However, it is desirable to have an experimental method for distinguishing ground-band from excited-band populations if reliable nuclear structure information is to be obtained in this manner. Two techniques are possible:

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(i) Total energies and/or multiplicities of y-ray cascades could be determined using large NaI crystals (sum-spectrometer), arrays of smaller NaI crystals (multiplicity filter), or the 4~ NaI arrays (crystal balls), currently being constructed. (ii) Particle detection of the highest available resolution could be used to determine which energy ranges of the transfer product were being populated, even if the resolution is not sufficient for resolving individual states. In each of these cases it would be necessary to select the transfer channel by some method such as (a) high-resolution gating on discrete y-rays, (b) analogous gating on conversion electrons or X-rays, (c) sufficiently high mass resolution (- l-2 %). The experimental problems are therefore seen to be large but tractable. The information which the calculations suggest is available would seem to justify a serious effort to solve them. If it should happen that detailed state-by-state spectroscopy of the transfer reactions is difftcult for particular cases, we would still expect that basic nuclear structure information would survive in the integral properties of such reactions; e.g., in the cross sections for population of the transfer product integrated over a range of excitation energies. However, it is clear that more information is available in the spectroscopy of individual states, and that experimental effort should be directed toward that goal. We also note that appropriate ion-ion potentials are required to describe the transfer reaction. We have described elsewhere a new procedure by which potentials for deformed systems may be accurately determined from the particle-y spectroscopy of inelastic heavy-ion scattering 4). For the reasons discussed in refs. 3*4), we do not expect ion-ion potentials determined from particle spectroscopy experiments in which elastic, inelastic, (and sometimes transfer), reactions are not resolved to be of sufficient reliability to interpret rotational signatures for transfer reactions. Therefore, inelastic scattering data should also be collected in the transfer experiments and used to deduce ion-ion potentials by requiring a semiclassical coupledchannel fit to all inelastic excitations 4). In addition to the one- and two-particle transfer, cc-transfer, and rotation-vibration excitation experiments already discussed, there are a number of other transfer experiments with very heavy ions which might be used to exploit the most general ideas discussed here. For example, two-particle transfer to high-spin states in nuclei near the onset of permanent deformation at the beginning of the rare-earth region would be particularly interesting. The 92neutron nuclei are known to be rather good rotors at high (I - 12-16) spins, while the 90-neutron isotopes tend to be well-deformed but have quadrupole moments which increase with angular momentum. On the other hand, the M-neutron nuclei appear to have rather small deformations. A study of two-nucleon transfer among these 88-92 neutron nuclei leading to population of excited collective states should provide valuable insight into this remarkably fast onset of permanent deformation. Interesting experiments are not confined to deformed nuclei. A second class of experiments would be transfer reactions for very heavy ions scattered from strongly

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collective vibrators such as those in the Cd and Pd region. By analogy with the previous arguments, we expect that such experiments allow a study of how collective multiphonon excitation modifies the underlying quasiparticle structure of the states. Another class of experiments requiring even greater sophistication would involve transfer population of rotation-vibration bands in heavy nuclei. The octupole bands in the actinide region and the y-bands in the A h 190 region would be of particular interest, the former because of the strong collectivity and observed structure changes at higher spin observed in Coulomb excitation experiments ‘l), and the latter because of the continuing uncertainty over whether they represent stable y-deformations. In both cases, information on how the underlying quasiparticle structure is affected by collective vibration and rotation may be expected to be significant. In addition to these considerations, there is the possibility that such reactions may allow the population of high-spin states in nuclei which cannot be populated in other ways. For example, since the excitation mechanism may be largely inelastic scattering for low Q-value reactions, states coupled by large matrix elements to the ground-state may be strongly excited. Since these states could be high spin but not yrast, they might not be populated in heavy-ion, xn reactions or in radioactive decay. A final class of experiments would be those involving the collision of two strongly superfluid systems to investigate the enhancement of the collective pair mode (increase in multiparticle transfer cross section). Although there is considerable theoretical doubt as to whether a true nuclear Josephson effect exists, one still expects collective enhancement of the pair transfer mode in such systems. The degree of this enhancement will provide a test of theoretical descriptions of collective pair modes. We believe the present model to have qualitative and perhaps semiquantitative validity. It may be of use in the initial interpretation of experiments. Since virtually no data are yet available, this seems sufficient at present. A sophisticated analysis of data will require a more elaborate calculation than that sketched here. We will deal with a semiclassical coupled-channels Born approximation employing a liniterange recoil form factor in future work. Such a formalism can be based on the classical limit methods discussed here, or on more conventional semiclassical 14) or iterative quantum-mechanical “) methods. The general strengths and weaknesses of these alternatives for treating inelastic excitation have been discussed in ref. 1‘), and similar considerations could be expected to apply here. 8. Conclusions

We have begun investigation of the concept that heavy-ion transfer reactions provide a unique way of studying the effect of collective quanta on single-particle wave functions and multiparticle correlation functions in heavy nuclei. In the

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present paper a simple model has been advanced to describe two-neutron transfer for heavy-ions incident on deformed targets. Despite its simplicity we have presented arguments that this can provide at least a qualitative guide to experimentalists in a largely unknown area. The calculated rotational excitation patterns in the twoneutron transfer product are quite different from those seen in the analogous inelastic excitation. These differences are shown to have a straightforward semiclassical interpretation. We have shown that the location, width, rotational signature and magnitude of the transfer excitation function are expected to depend sensitively on the underlying nuclear structure. The extension of the general ideas employed here to l-particle transfer in deformed nuclei, and to several other examples of transfer to highly collective states in deformed and vibrational nuclei by using very heavy ions is briefly discussed. We conclude that transfer reactions with very heavy ions (A k 40) provide a unique probe of nuclear structure because they allow the study of microscopic correlation functions and wave functions under the influence of strongly excited collective modes, and because the strong collective excitation may imply localizations in coordinates conjugate to large quantum numbers which are not available by other methods, Experimental possibilities have been discussed and we conclude that sophisticated particle-y coincidence methods are a feasible way to carry out such experiments. This paper is dedicated to the memory of the late Sven Giista Nilsson. As a scientist, and as a human being, he is fondly remembered and greatly missed by those who knew and worked with him. We would like to thank Drs. D. H. Feng, T. Pinkston, R. Broglia, A. Winther, E. Maglione, I. Y. Lee, H. Massmann, D. Habs and B. Herskind for discussions concerning some of the material contained here. We thank Dr. E. Grosse for discussions and for providing the calculation in fig. 2. Two of us (M.W.G., T.L.N.) wish to thank the Danish Research Council for financial support and the Niels Bohr Institute for hospitality during part of this work. References I) B. Elbek and P. Tj$m, Adv. in Nucl. Phys. 3 (1969) 259; R. Broglia, 0. Hansen and C. Riedel, Adv. in Nucl. Phys. 6 (1973) 287, and refs. therein 2) W. R. Phillips, Rep. Prog. Phys. 40 (1977) 345 ; S. Kahana and A. J. Baltz, Adv. in Nucl. Phys. 9 (1977) 1; and refs. therein 3) M. W. Guidry, P. A. Butler, R. Donangelo, E. Grosse, Y. El Masri, I. Y. Lee, F. S. Stephens, R. M. Diamond, L. L. Riedinger, C. R. Bingham, A. C. Kahler, J. A. Vrba, E. L. Robinson and N. R. Johnson, Phys. Rev. Lett. 40 (1978) 1016 4) R. E. Neese, M. W. Guidry, R. Donangelo and J. 0. Rasmussen, Phys. Lett. 85B (1979) 201 5) M. W. Guidry et al., Nucl. Phys. A, submitted 6) D. Habs et al., unpublished 7) S.Levit, U. Smilansky and D. Pelte, Phys. Lett. 53B (1974) 39 8) H. Massmann and J. 0. Rasmussen, Nucl. Phys. A243 (1975) 155; H. Massmann, Ph.D. Thesis, University of California at Berkeley (1975), unpublished

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