ON THE USE OF REGGE-POLE ANALYSES TO EXTRACT POSSIBLE QUASI MOLECULAR ROTATIONAL BANDS FROM ELASTIC HEAVY ION CROSS SECTIONS

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ON THE USE OF REGGE-POLE ANALYSES TO EXTRACT POSSIBLE QUASI MOLECULAR

ROTATIONAL BANDS FROM ELASTIC HEAVY ION CROSS SECTIONS

K. Mc Voy

To cite this version:

K. Mc Voy. ON THE USE OF REGGE-POLE ANALYSES TO EXTRACT POSSIBLE QUASI

MOLECULAR ROTATIONAL BANDS FROM ELASTIC HEAVY ION CROSS SECTIONS. Journal

de Physique Colloques, 1971, 32 (C6), pp.C6-119-C6-124. �10.1051/jphyscol:1971615�. �jpa-00214833�

JOURNAL DE PHYSIQUE

Colloque C6, sul~/)l&rnent au

1 1-1 2, Tome 32, Novembre-DPcembre 1971, page C6- 1 19

ON THE USE OF REGGE-POLE ANALYSES

TO EXTRACT POSSIBLE QUASI MOLECULAR ROTATIONAL BANDS FROM ELASTIC HEAVY ION CROSS SECTIONS

K. W. Mc VOY (*)

Indiana University Bloomington, Indiana

Rksume.

Si une bande de rotation s'etend au-dessus du seuil d'emission de particule, ses membres sont des resonances plut6t que des etats lies. S'ils sont en outre suffisamment larges pour sc recouvrir, leurs phases et amplitudes relatives determinent la facon dont ils interferent entre eux dans les sections efficaces de diffusion et de reaction. On montre qu'un p61e de Regge perniet une parametrisation commode et Cconomique de ces amplitudes.

Abstract.

If a rotational band extends above a breakup threshold, its mcmbers will bc reso- nances rather than bound states. If in addition they arc sulliciently broad to overlap, their relative phases and amplitudes will determine the way in which they interfere with one another in scattering and reaction cross sections. It is pointed out that a Regge pole provides a convenient and econo- mical parametrization of these anlplitudes.

Although heavy-ion excitation functions often have abundant fine structure, which is unquestionably due to compound-nucleus resonances, the energy- averaged cross sections at energies above the Coulomb barrier seldom if ever exhibit structure which can be identified with a resonance in a single (Jn) state.

Many elastic excitation functions do, however, show substantial gross-structure maxima (which are surely not individual resonances because they move in energy as the scattering angle changes). We wish to examine here the possibility that they are caused by the combined influence of diffraction and a high-lying rotational band of quasi-molecular states in the compound nucleus

which is formed during the scattering.

In 12C +

scattering, for instance, these would typically be rotational states with angular momenta between I0 and 16, at excitation energies of 30 to 40 MeV in 2 4 ~ g . They would be resonances rather than bound states, of course, which would be so weak, broad and overlapping that none of them would be visible as individual peaks in the excitation functions.

In spite of this weakness of the individual levels, we wish to show that it is possible for all those resonances which overlap a t a given energy to act coherently, in such a way as to produce angular distributions and excitation functions with the observed strong gross structure, and further that the most economical mathematical description of such a phenomenon is obtained by parametrizing the partial-wave analysis in

(*)

On leave of absence from the University of Wisco~c;n.

terms of a direct-channel Regge pole. AIthough we shall concentrate on elastic scattering, which has been most thoroughly studied so far, if such resonances exist they will couple to inelastic channels as well, and the same type of analysis can equally well be applied to reaction cross sections.

The two characteristics of the ion-ion potential which are most important for determining the pro- perties of quasi-molecular states are (a) the Coulomb barrier (which peaks a t a separation somewhat less than the sum of the radii of the two ions), which is responsible for the formation of the resonances, and (b) the strong absorption inside the barrier, describing their decay into inelastic channels, which broadens them greatly. Because of the great strength of this absorption at energies above the barrier, no significant resonances can occur deep inside the compound system, i. e., in I-values substantially smaller than kR. If two heavy ions can form quasi- molecular states at all, they must be

peripheral

i. e., at 1 -- k R . It is this simple fact which imme- diately determines which partial waves have any chance of resonating, and how these resonating I-values increase with the bombarding energy.

Clearly, significant resonances can occur only if the real potential extends out to somewhat larger radii than the imaginary potential. In wave terms, a potential of this sort is equivalent to a black sphere surrounded by a thin jacket of glass. Radiation incident on it will be absorbed by the inner black region (producing diffraction) and refracted by the glass jacket ; under favorable circumstances this refraction

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971615

C6- I20

K.

MC VOY

can resonantly trap the waves, and the angular distribution of the scattered radiation will be due to the joint effects of the difrraction and resonant phenomena. The most clearly-separated situation seems to be an angular distribution whose envelope exhibits a strong forward rise (due to diffraction), a central minimum, and a smaller rise at backward angles, which is due to the overlapping resonances, i. e., to the Regge pole.

In partial-wave terms, the central absorption can be described by the Blair smooth-cutoff model, which makes the reflection coefficient

I S, I small for small I, with a rise to unity (no scattering) for 1 near

kR ; the resonances, if present, will cause a deviation from this smooth rise. An apparent example of this is shown in figure 1, which displays the Argand diagrams (Im (S,) vs Re (S,) for increasing I, at fixed energies) as well as the r],-curves, from the phase-shift analysis of elastic a-a scattering performed by Darriulat et al. [I]. The complex S, are seen to exhibit a descend ing-phase loop in the complex plane as I increases, which passes near the origin just before heading for the point (I, 0). This pass near the origin produces a distinctive dip in the r],-curve, which is perhaps the most readily recognizable sign of the influence of a Regge pole on the partial-wave decomposition.

FIG. 1 .

Argand diagrams and

curves for elastic

scat- tering at three laboratory energies. The numbers on the Argand

diagrams are I-values.

A direct-channel Regge pole is simply a pole in the I-dependence of S,. It was first pointed out by Regge that an S, which describes potential scattering can be given a meaning at complex as well as real 1'

and that any bound state or resonance in the L-th partial wave which produces a pole in the energy-dependence of S,(E) (i. e., a Breit-Wigner energy-denominator), will equally well produce a pole (at a complex 1 near Id) in its I-dependence at fixed E.

Under special circumstances such an I-pole can dominate this I-dependence. When this occurs, S, is given by a resonant or Breit-Wigner approximation in I, which is conveniently written [2]

Here B(1) is the

background

I-dependence, which si conveniently taken as the Blair smooth-cutoff func- tion in the case of strong absorption. Eq. (1) then provides a simple generalization of this smooth- cutoff parametrization, which includes in addition the effects of a group of resonances centered in I at 1 = Lo, and extending over the I-range A1 z ?.

Lo will of course increase, roughly like Lo

k R , if the resonances always occur at the nuclear surface.

Physically the states in such a set are identical except for their a n ~ u l a r momenta, like the Is, lp, Id, etc. states in a potential well, and so form exactly what is meant by a rotational band. They could of course be inserted into their respective partial waves

((

by hand

as a sequence of traditional Breit-Wigner expressions ; this would not only require a great many parameters, but in addition there would be no way of knowing the relative amplitudes of the resonances, which are essential when the states overlap. The main point of the Regge-pole approximation is that when only a single rotational band (i. e., a single Regge pole) is involved, the form of the I-dependence of these amplitudes is uniquely determined as that given by eq. (1). Consequently all the amplitudes of the band are parametrized by a single I-pole rather than by a set of energy-poles.

The Coulomb barrier of the heavy-ion potential is ideal for generating such bands of quasi-molecular resonances. Several examples are seen in figure 2, which shows the transmission coefficients 7 , ( E ) = 1

q:

for the

+ ¹⁶⁰ optical potential of Maher et al., [3] but with the imaginary potential depth reduced to 0.5 MeV in order to emphasize the resonance effects. The resonances occur as maxima in the trans- mission coefficients, which clearly move to higher energy as I increases, simply because of the added rotational energy. The sequences are labelled by the number of internal nodes in the real part of their radial wave functions, and each such sequence is a rotational band, described by a single Regge pole moving appropriately as the energy increases.

Since each level is marked by a maximum in the energy-dependence of T,(E), it must clearly also correspond to a minimum in the energy-dependence of q,(E), as noted in the a-a data. This same minimum can be seen even more clearly in the [-dependence of

at fixed E, as shown by the insert in figure 2, where the minima are labelled by the same radial quantum numbers. Alternatively, this type of evidence for rotational bands can be shown by a contour map of T,(E) in the I-E plane, as in figure 3. This one is for the 12C + I2C potential obtained by Gobbi et al. [4]

(employing the correct imaginary part in this case).

Although the small imaginary potential used in

figure 2 permitted the appearance of several rotational

ON THE USE OF REGGE-POLE ANALYSES TO EXTRACT POSSIBLE C6-121

FIG. 2.

Transmission coefficients for the

-k

optical potential of ref. 131, with W

MeV.

RIDGES INDICATE ROTATIONAL BANDS r . 0 9 (REGGE TRAJECTORIES)

-

FIG. 3.

Contour map of

in the I-E plane, for the 12C + 12C optical potential of ref.

bands, if this imaginary potential is increased to the empirically-determined value, Fig. 4 shows that all the high-n resonances become so broad that they vanish into the background, leaving only the n = 0 band, which is lowest in energy, as a distinctive reso- nant effect in the cross section. It is this fortunate and somewhat fortuitous circumstance which makes the I-Regge-pole approximation applicable in this case.

FIG. 4. -

curve and Argand diagram for

scatter- ing at

- 26 MeV. The open circles are from an optical potential given in ref. [a], and the curve and black dots are

the fit achieved with the parametrization of eq. (1).

Since eq. (1) is a parametrization of a partial-wave analysis, it can only be used to fit angular distributions.

In the well-studied

-t

case, however, the experimental angular distributions vary erratically with energy because of the fine-structure in the cross section, and must be energy-averaged before a simple parametrization like eq. (1) can be employed. As a means of doing this smoothing, we have employed as

((

data

the cross sections computed from an optical potential which was itself fitted to the energy-depen- dence of the original data. We have, however, altered the cross sections by

symmetrizing the amplitude (as the identity of the two nuclei requires), in order to see clearly what the large-angle cross sections would be if not masked by the Coulomb scattering.

Figure 5 shows the results of fitting eq. (1) to these

((

data

at energies between 18 and 32 MeV ; above 18 MeV (where a I-pole approximation begins to be adequate) the fit is convincingly good, and improves with increasing energy. This argues very strongly that these (complex) phase shifts show strong evidence for the existence of an n = 0 rotational band in this particular potential, with angular momenta in the 14 to 20 range for scattering energies between 20 and 30 MeV. This set of phases is one set which fits the data ; if they are the correct ones, they imply the existence of the rotational states in 32S, at the excitation energies between 35 and 45 MeV.

A similar fit was attempted to "C + 12C

data

generated by the potential of Gobbi et al. 141, with the less encouraging results shown in figures 6 and 7.

The apparent reason for the poor fit, however, can be

seen from the p,-curves : the Fermi function employed

for the background factor B(I) insisted that p, be very

small for small l, whereas the actual q,'s d o not drop

C6-122

Mc VOY

32 MeV

1d4

i d 5

C.M. SCATTERING A N G L E I

FIG. 5.

Fits of the Regge parametrization to

data

taken from ref. 1261.

a o o - m m uJ u, si w a w a 6 m'm w w

dl

c.m. SCATTERING ANGLE 1

FIG. 6.

Fits of the Regge

to

1ZC

a

data,, (the optical potential of ref. [41), at ECm

25 and 27 MeV.

The data are shown as dots in the first and third columns. In the second column, the dashed curve is the angular distribution obtained from eq. 1 by using the same parameters for the

background, and removing the pole factor. FIG. 7.

Same as Fig. 6 for E,,,,

29, 31 and 33 MeV.

ON THE USE OF REGGE-POLE ANALYSES TO EXTRACT POSSIBLE

below 0.1. It is encouraging to note that in spite of this handicap, the search did manage to get the q,-dip (which marks the I of the resonance) at the correct /-value at each energy. This is probably because this I-value uniquely determines the angular spacing between the back-angle

diffraction minima

in the angular distribution. This is quite an important point, for if the back-angle rise is experimentally accessible (and not due to a transfer process), it enables one to identify the resonating I-values at each energy without any analysis at all.

Although attempts to apply a Regge parametriza- tion to the partial-wave analysis of heavy-ion data are only in their infancy, it is clear that in those cases in which it does fit the data, it implies the existence of a high-lying rotational band in the ((compound nucleus)).

If such fits are achieved, the most obvious questions they will raise concern the I-dependence of the reso- nance energies, and the question of how this band is related to lower bands in the same nucleus.

We have been considering such questions recently in collaboration with A. Arima, G. Scharff-Goldhaber and A. K. Kerrnan, but since the evidence for the exis- tence of any such quasi-molecular bands is extremely tentative, we shall confine ourselves here to a few speculative remarks on 12C + I2C scattering.

If we interpret the dips in the q, 's from the Gobbi potential [4] as indicating the presence of quasi-mole- cular states in 24Mg, we can readily obtain the I-dependence of their energies. The result, shown in figure 8, indicates that El is very nearly proportional

FIG. 8.

Ground state and

molecular

rotational bands in Z4Mg, the latter extracted from the optical potential fit of ref.

The hatched region is intended to indicate the

approximate widths of the states in the band.

to I(I + 1). This is perhaps not too surprising, for the strong absorption a t small separations of the two 12C centers acts much like a hard core, forcing the wave function (in this channel) out from the origin. The resonant states can therefore only exist in a narrow

channel

(the

glass jacket

referred to above) between perhaps 4.5 and 5.5 fermis in the 12C + 12C case, and so cannot

stretch

as they are rotated faster and faster. Consequently the moment of inertia of the system remains fairly constant at 3

p ~ where p ~ , = M / 2 is the reduced mass of the system, and this constancy of 3 through-out the band makes the band energies proportional to l(1 + 1). Incidentally, the reason that the band seems to stop below I z 8 is that the lower states are below the Coulomb barrier, where their elastic widths become very small, so the states become practically invisible in the I2C + 12C channel.

This potential, like that for

+ 160, is so shallow (I4 MeV) that it has no bound states. It would thus seem to ignore the existence of 24Mg altogether- or at least assert that none of its bound states contain a significant amount of the 12C + 12C configuration, for the n

0 levels shown are the yrast levels for this potential.

The 2 4 ~ g ~ t a t e ~ most likely tocontain the I2C + 12C configuration are its collective rotational levels. In the ground-state band, states are known only u p through J

8, but if they are fitted to the VMI model [ 5 ] (G. Scharff-Goldhaber, to be published), they pro vide the extrapolation to higher spins shown 'n figure 5. If this extrapolation is taken seriously, it indicates that the ground-state and molecular bands are entirely distinct, and that the ground-state levels must indeed have no significant 12C + 12C compo- nent if the Gobbi potential is the deepest one which will fit the data.

However, it is interesting to note that the 12C + 12C threshold lies just above the J

8 level, so that if any further members of the ground-state band are found, their 12C + 12C content can be estimated from their decay rate into this channel. It is worth noting that if this (or any other strong-interaction decay) is large, it will be difficult to see E 2 y-ray transitions between the states, making it harder to identify members of the band.

In summary, even though it appears that rotational quasi-molecular states at energies above the Coulomb barrier are probably too broad and overlapping to be visible as individual bumps in heavy-ion excitation- functions, there is reason to think that the states may lie on rotational bands, and that the coherent effect of an entire band on a cross section may be large enough to see via a Regge-pole analysis.

If this possibility is realized, heavy-ion physics could

become a source of very interesting information on

high-lying rotational bands, and it will become espe-

cially important to explore all possible means of

exciting them.

K .

Mc VOY

References

DARRIULAT (P.), IGO (G.), PUCH (H. G.) and HOLM-

GREN

(H. D.), PhyS. Rev., 1965, 137, B315.

COWLEY (A. A.) and HEYMAN (G.), NucI. Phys., 1970, A 146, 465 ; M c VOY (K. W

Phys. Rev., 1971, C 3, 1104.

[3] MAHER (J. V.), SACHS (M. W.), SIEMSSEN (R. H.), WEIDINGER (A.), and BROMLEY (D. A.), Phys. Rev., 1969, 188, 1665.

[4] GOBBI (A.), WIELAND (R.), REILLY (W.), CHUA (L.), SHAPIRA (D.), SACHS (M. W.), A S C U I ~ O (R.) and BROMLEY (D. A.), Proc. of the Sixth Inst. Conf. on Heavy lon Physics, Dubna, 1971 (to be published).

[5] MARISCOTTI (M. A. J.), SCHAKFP-GOLDHABER (G.), and BUCK (B.), Phys. Rev., 1969,178, 1864 ; SCHARFF- GOLDHAUER (G.) and GOLDHABEK (A. S.), Phys.

Rev. Letters, 1970, 24, 1349.