The threshold anomaly in 16O + 208Pb scattering

Nuclear Physics A505 (1989) 84-102 North-Holland, Amsterdam

THE TI-IRESHOLD

ANOMALY

IN =O +2’*Pb SCATTERING*

I.J. THOMPSON

Physics Deparfment, Uniuersif_vof Surrey, Guildford GUZ 5XH, UK M.A. NAGARAJAN,

J.S. LILLEY

and M.J. SMITHSON

SERC Daresbury Laboratory, Warrington WA4 4AD, UK Received 6 June 1989 (Revised 7 July 1989) Abstract: The empirically observed energy dependence of the optical potential at energies close to the Coulomb barrier (the threshold anomaly) in the collisions of ‘“0 on “‘Pb is investigated within the framework of coupled-reactions-channels (CRC) calculations. The motivation is to determine the extent to which the observed threshold anomaly can be accounted for by strong coupling of the elastic to the inelastic and transfer channels. CRC calculations have been performed at a range of energies above and below the Coulomb barrier, both for the complete set of reaction channels, and also, to see the effects more simply, for individual inelastic and transfer channels. An energy-dependent local polarisation potential has been extracted. It wit1 be seen how this polarisation potential satisfies the dispersion relations, and how well it reproduces the elastic scattering and the fusion cross sections over a range of energies.

1. Introduction A systematic analysis of the elastic scattering of 160 by 208Pb over a wide range of energies has presented ‘) clear evidence for a strong energy dependence of the optical potential at energies close to the Coulomb barrier. The energy dependence of the heavy-ion optical potential had already been noted in the elastic scattering of 32S by 40Ca [ref. 2)]t and has since been observed in other systems “). The real part of the optical potential was observed to become more attractive as the energy approached the Coulomb barrier, while at the same time the imaginary potential decreased in strength. It was shown that the local energy dependence of the real and imaginary potentials are related through a dispersion relation “f. The observed decrease in the strength of the imaginary potential as the incident energy is reduced toward the Coulomb barrier can be interpreted in terms of an effective closure of non-elastic channels at the lower energies. According to this view, the observed phenomenon - termed the “threshold anomaly” - is a coupledchannels effect. In order to verify this, extensive coupled-reaction-channels (CRC) calculations were performed where several nucleon transfer and inelastic channels l

Dedicated

to the memory

0375-9474/89/$03.50 0 (North-Holland Physics

of Dr. Lionel J. Goldfarb.

Elsevier Science Publishers Publishing Division)

B.V.

85

I.J. Thompson et al. / Threshold anomaly were

included

laboratory

5,6). The

energies

They indicated

calculations

of 80 and

that the strong

channel

gave rise to the correct

thereby

improved

of ref. ‘) were

performed

for the projectile

102 MeV with the aid of the FRESCO coupling trend

the agreement

of the nonelastic

of decreasing

with experiment.

channels

large-angle Simultaneously,

‘) program. to the elastic

cross sections,

and

the calculations

also showed that coupled-channels effects enhanced the fusion cross sections at these energies, and brought them into close correspondence with the experimentally observed cross sections “). These features exhibit clearly the importance of channel couplings, but do not address the problem of predicting the energy dependence of the optical potential. It should be pointed out, however, that by inversion of the CRC results at 80 MeV, it was verified that an attractive real polarisation potential is present 9), which goes further towards establishing the coupled-channels effect as the correct interpretation of the observed increase of the real potential at this energy relative to standard (double-folded) potentials. The motivation of this paper is to investigate the coupled reaction channels effects over a range of energies around the Coulomb barrier, and to see to what extent these can reproduce the observed energy dependences. The CRC calculations referred to above ‘) did not include a-transfer channels whereas the data of Videbaek et al. “) indicated the corresponding cross sections to be large. It is therefore important to include this channel in some manner, and study its effect on the elastic scattering and fusion cross sections. We thus report fully the effects of the inclusion of the a-transfer channel (in addition to the nucleon transfer and inelastic channels) on the elastic scattering and fusion cross sections in the 160 + “‘Pb system at a large number of projectile energies. In sect. 2, we present the results of the CRC calculations at several energies between 78 and 102 MeV (lab), and compare the predictions with experimental data at the energies where they are available. We also give the decomposition reaction cross section into the total inelastic, nucleon and a-transfer cross sections. We attempt

in sect. 3 to extract

the energy

by constructing “mean” local polarisation In order to determine how these potentials being

coupled,

they are also calculated

dependence

of the total and fusion

of the optical

potentials

potentials from each CRC calculation. depend on the parameters of the states for model

calculations,

in which

either

a

single inelastic state, or a single transfer state, is coupled to the elastic channel. By artificially varying the Q-value of the coupled state, the Q-dependence of the polarisation potential has been studied. The dispersion relations can also be verified, by comparing the real part of the polarisation potential with that determined from its imaginary component by means of a subtracted dispersion relation “). Fusion cross sections are discussed in sect. 4, where also the mean spin (L) and mean-squared spin (L2) of the compound nucleus predicted by the CRC calculations are compared with some experimental data. The prediction of the barrier penetration model, where the barrier is calculated with the sum of the bare and the polarisation

LJ.

86

Thompson et al. / 77zreshold anomaiy

potentials, is compared with the predictions of the CRC calculations. The summary and conclusions are presented in sect. 5. 2. Coiffed-reaction-chancels

caIculatious

for 160 + ‘OsPb

The channels included in the CRC calculations are those expected to have significant couplings to the elastic channel, and are shown schematically in fig. 1. The single-nucleon transfer channels included correspond to neutron pickup (160, 170) and proton stripping (160, “N) reactions. The inelastic channels included were the excited 3-’ state of oxygen at 6.13 MeV, and the lowest 2’, 3- and 5- states of 208Pb.The a-transfer channels were approximated by a single (r60, ‘*C) channel with a Q-value of -20 MeV, and a form factor obtained by binding the alpha particle on ‘O*Pb with 1 MeV binding energy. The a-transfer strength was adjusted to yield a cross section of 18 mb at the projectile energy of 80 MeV, close to the measured value of Videbaeck ef al. *)_ (This approximation avoids the need for microscopic a-transfer form factors.) The nucleon transfer spectroscopic factors were taken from the results of sub-Coulomb reactions studied in Franey et al. lo). The inelastic states were treated as collective (vibrational) states and their form factors were chosen to be the derivatives of the monopole potentials with deformation lengths listed in table 1. The CRC calculations were performed with the program FRESCO ‘>. Details of the method are described elsewhere “). The optical potentials in the elastic and inelastic channels were assumed to be identical. These and the optical potentials used in the nucleon and o-transfer Couplings

I/*-15N

o+f2

c

in the’%D* *08Pb

reaction

7/p209&

oc 212

PO

o+16

Fig 1. Schematic

picture

of channels

0

0’ 208Pb

in the ‘hO+Z”IIPb CRC calculations.

87

Deformation

lengths

Nucleus

State

I60 *“‘Pb “‘Pb “‘Pb

32+ 35

for inelastic

(vibrational)

Energy

states

(MeV)

Deformation length (fm)

6.13 4.07 2.61 3.20

2.15 0.40 0.80 0.468

TABLET Optical

potentials W

Nuclei I60 + ‘70+ LSN + LZC+

60.5 78.28 78.28 78.28

*‘aPb “‘Pb 209*i *l*po

“) Woods-Saxon

rot (fm)

(MeV)

squared

0.658 0.65 0.65 0.65

1.179 1.215 1.215 1.215

10.00 “) 17.11 17.11 17.11

1.0 1.162 1.162 1.162

0.4 0.623 0.623 0.623

form factor.

1.0 0.9 0.8 $

0.7 0.6 0.5

90

120

150

180

Fig. 2. Elastic scattering angular distributions of 160 + 2o8Pb at 80 MeV: The curves are CRC predictions. The dotted line indicates the contribution when the coupling of the elastic and nonelastic channels is completely turned off. The dot-dashed curve shows the prediction when only inelastic couplings are included, the dashed line shows the effect of a further inclusion of single-nucleon transfer channels and the full line the effect finally of further including a-transfer. All of these calculations used the bare potential of table 2, with rot = 1.0 fm. The shaded line shows the effect, in the complete CRC calculations, of increasing rot to 1.4fm. The experimental data shown as full circles are from ref.‘), and the open circles are those from ref. I*).

88

Lf. Thompson ef al. / ~reshold

channels

are shown

predicted

at X0 MeV are shown

line is the prediction

in table

including

scattering

in fig. 2 for different

when the channel

shows the effect of coupling further

2. The elastic

anomaly

coupling

to the inelastic

the nucleon-transfer

of including the cr-transfer channel. figure. The decomposition of the total

is turned

channels,

channels,

cross sections The dotted

off. The dot-dashed

the dashed

line

line the effect of

and the solid line shows the effect

The experimental reaction

differential

sets of couplings.

cross

data ‘,12) are indicated section

into

inelastic,

in the nucleon

transfer, a-transfer and fusion cross sections is shown in table 3. This table lists the results of the previous calculation “), the present one with r,, = 1.0 fm (as in ref. ‘>), and a further calculation with rol = 1.4 fm which is discussed below. Experimental cross sections are listed in the last column. The contributions of the different channels to the partial-wave cross sections (for rol = 1.0 fm) are shown in fig. 3. These may be compared with the ones shown in fig. 2 of ref. ‘). In fig. 3, qransfer( L) includes both nucleon and a-transfer. The present results should be compared with those of the earlier calculations 5), which neglected a-transfer. Including this channel considerably improves the agreement between theory and experiment. The measured total reaction cross section is reproduced, and the solid curve in fig. 2 illustrates that the calculated elastic scattering distribution agrees well with the experimental data, except in the angular region between 120” and 140”. However, including the G-channel has very little effect on the fusion cross section, even though it alone accounts for almost 20% of the total reaction cross section. The observed elastic scattering cross section has been fitted by an empirical optical potential ‘). The resulting optical potential for Elab = SO MeV predicted a total reaction cross section of 120 mb, and also yielded an imaginary potential at 12.4 fm which is larger than that obtained 3 below]. In order to determine

from the CRC calculations whether a fit to the elastic

[see ref. “) and sect. scattering could be

TABLE 3 Integrated

CRC cross sections

Previous CRC (mb) [ref. ‘)I

CRC (mb) rol = 1.0 fm

total reaction fusion

82.3 37.6

101.5 36.6

inelastic transfers neutron proton transfers a-transfers

i3.9 25.2 5.6

Cross section

“) Fusion-evaporation processes not included h, See text. ‘f Nob adjusted to experiment.

at 80 MeV

CRC jmb) r0,=1.4fm

Exp. [ref. s)]

126.5 66.4

1004 10 36*4”)

(mb)

-

9.6 24.6 12.7 18 “) {expected

9.8 I 26.1 9.4 14.3 ‘) to be less than 5 mb).

24zk2 2112 19*2

I.J. Tltompson et al, / Threshold anomaly

89

Partial wave, L Fig. 3. Partial reaction cross sections: The full CRC calculation corresponding to col. 3 of table 3 (roI = 1.0 fm) gives the dotted curve for fusion, the dashed curve from inelastic excitations, and the dot-dashed curve from the transfer (nucleon + a) channels. The sum of these is the total reaction cross section (solid curve).

obtained within the CRC framework, we carried out a set of CRC calculations in which the radius of the bare imaginary potential was increased from 1.0 fm in steps of 0.1 fm to 1.4 fm. No significant changes occur for ror s 1.2 fm, whife for rot > 1.2 fm the main effect is an increase in the fusion cross section. Results for r,, = 1.4 fm, are shown as the shaded line in fig. 2. The use of the large radius ( ror = 1.4 fm) improves the fit to the elastic cross section at intermediate angles, but at the expense of a fusion cross section which is too large. The fourth column of that the total reaction cross section has risen to 126.5 mb, mainly increase of the fusion cross section to 66 mb, which is about double value. These results should be compared with the analyses of Wong

table 3 shows caused by an the measured et al. ‘I), who

attempted to fit the elastic and fusion cross sections of ref. *) simultaneously with an optical potential. They obtained a total reaction cross section of 120 mb, which is similar to the CRC value with r,, = 1.4 fm. However, their parametrisation yielded values

of the total

direct

(inelastic

plus transfer)

and fusion

cross sections

of 84

and 36 mg, respectively. It would appear, therefore, that any analysis which fits the elastic scattering distribution closely, predicts a total reaction cross section close to 120 mb, which is larger than the reported experimental value. It is unlikely, however, that the missing cross section is in the fusion channel, which has been independently determined from the fission yield *.“). If this is so, then it has to be considered that the additional absorption needed to fit the elastic scattering must be associated with additional reaction channels, and not with the bare imaginary potential which determines the fusion in CRC calculations. The later data of ref. r2) show some

1.J. Thompson et al. / Threshold anomaly

90

‘“0 f 208Pb CRC 78 MeV (x 1000)

86 MeV (x 100)

t

I

:

:

30

60

:

:

:

:

90

120

L

:

: 150

:

1 0

(deg.)

Fig. 4. Comparison of elastic cross sections with CRC predictions: The CRC calculations included couplings at the energies shown on the figure, and used a bare potential with rO, = 1.0 fm.

TABLE 4 Energy

dependence of the total reaction cross section: Comparison CRC predictions with data of ref. “) at energies available

Projectile

energy

(lab)

(TV (theory)

fMeV)

imb)

oR (exp.) (mb)

78 80 83 86 88 90 96 102

41 101 259 425 529 629 899 11134

45.7 “) loo* 10 237 + 20 440 572150 578*55 904 1147*95

“) From optical-model

fit to elastic

scattering.

of

full

I.J. Thompson et al. / Threshold anomaly

differences details

with the earlier

of the optical

91

data of ref. ‘), and there are some uncertainties

potential

and total reaction

The predictions of the CRC for the elastic with the experimental data at the additional

in the

cross sections.

cross sections are compared in fig. 4 energies of 78, 86, 96 and 102 MeV.

The agreement between theory and experiment is reasonably good at all these energies for most of the angular range measured experimentally. At 96 and 102 MeV, the CRC predicts increases in the elastic cross section at large angles not apparent in the experimental data. This possibly indicates the need to include other quasielastic channels (two- and three-particle transfer channels, for example) as one moves toward higher projectile energies. The calculations of ref. “) also exhibited a rise at large angles at 88 and 96 MeV which was attributed to deficiencies in the ingoing wave boundary condition approximation. The variation of the total reaction cross section with incident can be seen from table 4.

energy

is well reproduced

3. Channel-coupling

by the CRC calculations,

and polarisation

as

potentials

The agreement between the CRC predictions and the experimental elastic-scattering cross sections indicates that the channel-coupling effects reproduce the same behaviour which gives rise to the threshold anomaly in the conventional optical model treatment ‘). It would be desirable if this energy dependence could be extracted from the CRC calculations in a way which is more transparent and in which the detailed effects of different, individual quasi-elastic processes can be more easily perceived and compared. In the present work, this has been done using effective potentials. It is always possible to incorporate the effect of channel couplings in the elastic optical potential by means of a polarisation potential. In general, however, the formal theory gives a polarisation potential which will not only be complex

and energy-dependent,

but will also be non-local

and angular-momentum

dependent. In order to compare our calculations with the optical model, to obtain a local, L-independent polarisation potential, but the construction a potential is not unique 13). We have used the following prescription L-independent

“weighted

mean”

local polarisation

V’(r)

=

where

potential:

CLwL(r)V!(r) CL wL(r)

V’,(r) are the trivially-equivalent

L-dependent

we wish of such for an

(3.1)

’ polarisation

potentials

defined

by V’,(r)

and wL are weight

factors

chosen

= -$

L

(VSW),

as

WL(T)= ULl ULW12

(3.3)

92

IJ.

for some coefficients

et ai. ,I ~resh~ld

aL to be specified.

the poles in the trivially of r for which

~ornp~on

equivalent

anomaiy

This choice of weight factors wL( r) eliminates

potentials

V:(r)

by giving zero weight to values

uL(r) = 0.

The efIect of channel couplings will be to modify the scattering wave functions in the range of angular momenta around the grazing angular momentum. We therefore

tried uL as defined

by the following

form:

~L=(2~+l)(l-l&.12L so that U~CCa,(L),

the partial

reaction

(3.4)

cross section.

As a test of this procedure, two sets of CRC calculations were performed, in which only two channels were coupled together. The first set included the inelastic projectile excitation ‘**Pb( %, ‘60(3-))208Pb, and the second set the neutron transfer reaction z0sPb(“60, “O(g.s.))‘*‘Pbf~~). In each case, the effects of the Q-value of the reaction on the polarisation potential and on the fusion cross sections were determined. The inelastic scattering was calculated for Q = -6.13 MeV (the physical value), Q = 0 (adiabatic approximation), and Q = i-6.13 MeV (unphysical). The transfer reaction was calculated for Q = -3.23 MeV, Q = 0 and Q = +3.23 MeV. The inelastic coupling strength was determined as in the full calculations described above, whereas the transfer coupling was arbitrarily doubled to simulate the cumulative effects of several channels. The results, presented in fig. 5a-c and fig. 6a-c, show the bare potential and the resulting effective real potentials (the sum of the bare and the calculated local polarisation potentials) at a series of projectile energies. It can be seen that the effective potentials have a smooth radial dependence in the case of the physical Q-value, but exhibit shapes with strong energy dependences in the case of the unphysical Q-values. For Q
which is attractive

10

11 r (fm.)

at most energies,

12

10

11

and varies smoothly

12 r (fin.)

11

over this energy

12 r (fm.)

Fig. 5. The effective real potential (Coulomb+ nuclear) evaluated in the barrier region for ‘60+L0XPb in the two channel model with inelastic coupling to the r60(3-) state for different Q-values: (a) Q = -6.13 MeV (physical), (b) Q = 0, (c) Q = +6.13 MeV. The dotted curve shows the effective potential with no couplings and the solid line, the dashed line, and the dot-dashed line include the polarisation potential of eq. (3.1), at 70, 80 and 90 MeV respectively.

1.J. ?%omp.~on et al. / Rzreshold

93

anomaly

78

72 11.0

11.5

12.0 r (fm.)

t2.5

11.0

11.5

12.0 r (fm.)

12.5

il.0

11.5

12.0

12.5

I (fm.)

Fig. 6. Effective real potentials for transfers: Same as fig. 5, for transfer couplings to the $- state of “‘Pb (see text for details) for different Q-values: (a) Q = -3.23 MeV (physical), (b) Q = 0, (c) Q = +3.23 MeV.

range. In the case of transfer coupling, the polarisation potential at 90 MeV is repulsive at the barrier top and becomes attractive at large radii. The dependence on energy is much stronger for the adiabatic and unphysical (Q > 0) cases, where dramatic changes are seen across the energy range both in magnitude and shape of the resulting barrier. These changes imply a strong Q-dependence of the fusion excitation function. This will be discussed below. It has been shown that the general, t-dependent non-local polarisation potentials satisfy a dispersion relation between their real and imaginary parts “). However, it is not clear a priori whether the local polarisation potential defined by eq. (3.1) will necessarily do so. The solid lines in figs. 7 and 8 show the energy dependence of the real and imaginary parts of the local polarisation potential determined from eq. (3.1) and evaluated at 12.4 fm, for the inelastic and transfer coupling cases discussed above. The real potential predicted by the subtracted dispersion relation (using a reference energy of 120 MeV) is shown as a dotted line. The agreement for both physical and unphysical Q-values is gratifying, and indicates that non-locality effects are smal1, at least at the strong absorption radius (12.4 fm) used here. This lends confidence to the procedure used for generating the local polarisation potentials. A similar analysis of the polarisation potentials was performed for the complete CRC calculations. Effective potentials for 80, 86,96 and 102 MeV are shown in fig. 9. They exhibit a general decrease with energy across this region. A more detailed energy dependence over a wider range is presented in fig. 10, in which are plotted values of both the real and imaginary polarisation potentials evaluated at 12.5 fm. Also shown as the dotted curve is the variation of the real potential predicted by the dispersion relation. It can be seen from the figure that there is a strong structure in the energy dependence of both the real and imaginary parts of the polar~sation potential: they are not smooth functions of energy. At this radius, they also underprediet the magnitude obtained empirically I), but this difference can be attributed to the shape of the mean potential V”,(r) varying from the Woods-Saxon form. The

94

(bf

(4

SO

70

80

90 Et60)

100

110

(Me!?)

Fig. 7. The energy dependence of the real of inelastic coupling for different Q-values evaluated at 12.4 fm. The solid curves are The dotted lines show the predictions of a

and imaginary parts of the polarisation potential in the case (top: Q = +6 MeV, middle: Q = 0, and bottom: Q = -6 MeV), evaluated from the two-channel calculations using eq. (3.1). subtracted dispersion relation using the imaginary potentials shown.

sharp energy variations in Im VP (fig. 10) have large effects on the dispersion integral. Applying the dispersion relation to a smoothed version of Im V,(r) results in a prediction which is in close agreement with a smoothed form of Re V,(r), If the local po~arisation potentiat extracted according to eq. (3.1) is a proper representation of the coupled-channels effects, an optical-model calculation using the sum of the bare and polarisation potentials should reproduce the corresponding CRC elastic cross section. Such a comparison is shown for 80 MeV scattering in fig. 11, and it is evident that the loal polarisation potential is satisfactory at this energy. Such good agreement was not obtained, however, in the energy range 90-96 MeV, and an examination of the polarisation potential shows that it is emissive

et al. / ~res~o~~ anomalv

I.J. ~ompson

Real

95

Imaginary 0.0 -0.2

-a4

-0.6 ~

_. a 0.2

ps .y ~ ..!

-0.4

Fig. 8. Same

80

100

120

80

100 E(‘*O)

120

(MeV. lab)

as fig. 7, for transfer couplings. The top corresponds to Q = +3.23 MeV, the middle Q = 0, and the bottom to Q = -3.23 MeV.

to

at the barrier top although it becomes absorptive at smaller and larger radii. This feature strongly affects the elastic scattering. Nor was good agreement obtained for the two-channel calculations shown in fig. 5 for energies above 85 MeV. Examination of these cases showed a strong L-dependence in Vt( P) which could not be adequately represented by a mean L-independent potential. Even though the elastic scattering is not always properly fitted, the simple polarisation potential is instructive because it provides a theoretical basis for understanding the origin of the energy and radial variations of the polarisation potential near threshold, and the possible differences depending on the nature of the channel couplings. 4. Fusion cross sections The effects of channel coupling on fusion cross sections are known to be significant. In table 5 and fig. 13, we compare the CRC predictions of the total fusion cross

1.J. Thompson er al. / lkeshofd

96

anomaly

102 Mel

-2.04 12.0

12.5

13.0

13.5

1 .O

r (fm.) Fig. 9. The real polarisation with all couplings included

potential for ‘60+208Pb evaluated using eq. (3.1) from CRC calculations (corresponding to column 3 of table 3), for different energies as indicated.

sections with experimental data in the energy range from 78 to 102 MeV. The agreement between theory and experiment is seen to be good. It should be borne in mind, however, that, within the CRC formalism, the fusion cross section is defined as the difference between the total reaction cross section and the total quasi-elastic (inelastic+ transfer) cross section. In view of the fact that not all the open channels were included in the calculation, the fusion cross section predicted by the model can be larger than the observed value, and this would be more apparent at the higher energies. There has been considerable interest in the effect of the Q-value of the non-elastic channels on the fusion cross sections 14). The excitation functions of the fusion cross section in the different two-channel cases are shown in fig. 12. Except for the Q > 0 case at intermediate energies, there is a general enhancement in the fusion cross section. The onset of fusion shifts downwards in energy relative to that for the bare potential,

by an amount

which increases

as the Q-value

increases.

Also, it can be

seen that the fusion enhancement for each Q-value exceeds that of the others over a region which is located close to the effective threshold energy. For example, in the inelastic case, this occurs at 74 MeV for Q =6 MeV, 80 MeV for Q=O, and >86 MeV for Q = -6 MeV. For both Q < 0 and Q = 0, fusion is generally enhanced. However, the excitation function for Q > 0 is qualitatively different. This is particular noticeable in the case of inelastic coupling (fig. 12a), where, except near threshold, channel coupling causes fusion to decrease, and dramatically changes the slope of the excitation function. Its behaviour corresponds directly to the variations in the polarisation potential illustrated in fig. 5, where it can be seen that the real barrier for Q > 0 changes from one extreme of being reduced in both height and width at

I.J. Thompson

-l.O-

-0.8’

: 70

:

70

: 80

:

:

80

et al. / Threshold

:

: 100

: 90

:

90

anomaly

:

:

100

:

97

: 110

:

:

:

110

: 120

: 120

E(‘60) (MeV)

Fig. 10. Energy dependence of the real and imaginary parts of the polarisation potential for ‘60+208Pb full CRC calculations (see fig. 9), evaluated at 12.5 fm. The solid lines are evaluated using eq. (3.1), and the dotted line is the prediction of the subtracted dispersion relation for the imaginary potential shown.

70 MeV, to being higher and wider at 90 MeV. The same general effects are seen in the transfer coupling (fig. 12b), though they appear to be less pronounced partly because the Q-value was not varied over as wide a range as in the inelastic coupling example. The effect of channel coupling on the spin distribution of the fused nuclear system has been shown in fig. 3 for a projectile energy of 80 MeV. One of the effects of channel coupling is to increase the partial fusion cross sections for high L. This effect can also be represented in terms of an average (L)and an average (L')for fusion defined by

98

0.8

O-O,9

180

e,,(de@ Fig. 11. Calculated elastic-scattering angular distributions for ‘60+208Pb at 80 MeV. The dotted and solid curves are the same as in fig. 2. The dashed curve is the optical-model prediction of the elastic scattering using the sum of the bare and the polarisation potentials.

where O-~(L) is the partial fusion cross section and oF is the total fusion cross section. The CRC predictions for (L) and (t”} are shown in table 5. There are a few reported measurements of (L’) for the 160+208Pb system extracted from the measured fission fragment angular dist~butions 5. Above the barrier, at 90 MeV, the calculated

15). These are also presented in table value of (L*) agrees with experiment.

However, nearer threshold, CRC underpredicts the experimental values. These results suggest that the channels included in the CRC calculations are not sufficient to describe the compound-nucleus spin distributions. Finally, in table 5 we also include the total fusion cross sections, (L) and (I,*) calculated using a barrier-penetration model r6) (BPM). The barrier penetration model assumes that all the flux that penetrates the barrier (defined by the real part of the elastic potential) leads to fusion. The BPM had been applied to the analyses of the fusion cross section in the 160+z*8Pb system by assuming that the radial dependence of the real potential was the same as the bare potential, but its depth was energy-dependent 16). The need to change the radial shape of the real potential in order to obtain the correct spin distribution has been pointed out 16.17). Accordingly, we reevaluated the predictions of the BPM for the fusion cross sections by employing our local polarisation potentials which have energy-dependent geometries. The resulting fusion cross section (crF), (L) and (L’) are shown in table 5. It can be seen that while the BPM qualitatively reproduces the energy dependence of crF, (L) and (L*), it also consistently unde~redicts the CRC values. If the

99

Energy

dependence

Energy

uF (mb)

Fusion

with BPM predictions

(L)

Fusion

(L’)

78

CRC BPM Experiment

6.0 3.1 S&*0.6

8.6 1.3

104 70 170130

80

CRC BPM Experiment

36.6 28.5 36*4

9.1 8.1

107 83 200 * 20

83

CRC BPM Experiment

157 145 108*10

12.7 12.0

195 171 270 f 40

86

CRC BPM Experiment

297 270 235

16.4 15.0

314 259

88

CRC BPM Experiment

385 333 350*40

18.6 16.6

400 316

90

CRC BPM Experiment

466 448 377 * 50

20.6 19.5

488 433 430

96

CRC BPM Experiment

683 576 685 * 70

25.7 22.6

754 579

102

CRC BPM Experiment

884 846 844190

29.6 28.2

1000 902

Experimental ref. “).

absorption

Total fusion

Source

(MeV)

barrier,

mF, (L) and (L*) from CRC compared

of fusion:

fan values

(the imaginary

the BPM prediction

are from ref.“)

potential)

(at 78 MeV) and ref. s). The (L’) values

leading

to fusion

was confined

are from

within

the

for cr; would be an upper limit. The fact that ~~(BPM)

is smaller than (r,(CRC) suggests that one of the effects of channel coupling is to give rise to a fusion imaginary potential which is of larger range, and can allow for fusion within the barrier. This finding is consistent with the models of ref. ‘I) and Stelson I*). The effect of a surface fusion imaginary potential on (L) and (L*) has studied in a simple model, and will be presented in a future publication 19).

been

5. Summary and conclusions We have presented a detailed survey of elastic scattering, total reaction cross sections and fusion in the collisions of ‘08Pb by 160 in the energy range of 78 MeV102 MeV, with the aid of large coupied reaction channels (CRC) calculations.

100

70

80

90 L

100

ii0

75

80 E,*

WV)

85

6-W

Fig. 12. Energy dependence of the total fusion cross section corresponding to the two channel calculations for (a) inelastic couplings and (b) transfer couplings (see text and figs. 5 and 6). The different curves correspond to bare potential only (dotted), and for different Q values as indicated.

Calculations

which include

selected

inelastic,

nucleon

and alpha transfer

channels

reproduce the elastic scattering and total fusion cross sections reasonably well over the entire energy range. The calculated local polarisation potential (eq. (3.1)) was complex and energy-dependent, and was seen to satisfy the dispersion relations, suggesting that the non-locality effects are perhaps not too severe. The sum of the bare and polarisation potentials was able to yield elastic scattering cross sections which agreed fairly well by the CRC predictions at all energies (except in the energy

1 o-’ 70

80

90

100

%,, WW from full CRC (solid line), Fig. 13. Total fusion cross sections: Predictions for ‘60izo8Pb coupling (dotted line) Data are from ref. “) (full circles) and ref. ‘*) (open circles).

and no

1.J.

?%ompson et al. / Threshold anomaly

101

range 90-96 h&V). The polarisation potential therefore provides a link between the empirically observed threshold anomaly and the channel coupling effects. The effect on the fusion cross section of the Q-value of inelastic and transfer channels was investigated by means of model calculations. The Q < 0 channel is generally more effective in enhancing fusion, except, as the energy decreases below the barrier, when first the Q = 0 and then the Q > 0 channels give greater enhancement. Altering the Q-value alters the threshold at which coupled channels effects manifest, according to the energy at which the non-elastic channels become closed. In all cases, it should be pointed out, the effects persist at higher energies over a considerable range. The fusion cross sections over the entire energy range (W-102 MeV) were well accounted for by the CRC cafculations. The detailed analysis of the spin distributions of the compound nucleus shows that the channel coupling effects were important. At energies below the barrier, however, there remains a large discrepancy between the mean squared spin predicted by the CRC and the experimental value. If the experimental result is confirmed, this would imply that the channels included are insufficient to account for the details of the fusion cross section The comparison of the predicted fusion cross sections, (L) and (L2) by CRC and those given by the BPM (where the real barrier was calculated from the sum of the real bare potential and Coulomb plus polarisation potential) showed that the BPM consistently underpredicted the CRC results over the energy range investigated. This suggested that some part of the fusion cross section possibly arises from absorption within the barrier. This wouid simultaneously predict an increase in (t) and (L*), providing for a closer correspondence with the CRC results. The CRC calculations were performed taking into account the coupling of all non-elastic channels to the elastic channel, but neglected any mutual coupling among the non-elastic channels. The results of these calculations seem to indicate that such a model is adequate for the description of the elastic scattering and the total reaction and fusion cross sections. However, the individual non-elastic cross sections are not well represented by this model. There are indications (e.g. in ref. ““)) that the inelastic excitation of **%b to the J‘- state is influenced by coupling to the transfer channels, and more complete CRC calculations are warranted. It is of interest to investigate whether such couplings between inelastic and transfer channels will affect the partial fusion cross sections. Some preliminary calculations are being performed, and the results of these investigations will be presented in the future. References 1) J.S. Lilley, B.R. F&on, M.A. Nagarajan, 1.3. Thompson and D.W. Banes, Phys. Lett. B151(1985) 181 2) A. Baeza, B. Bilwes, R. Bilwes, J. Diaz and J.L. Ferrero, Nucl. Phys. A419 (1984) 412 3) B.R. F&on, D.W. Banes, J.S. Lilley, M.A. Nagarajan and I.J. Thompson, Phys. Lett. B162 (1985) 55; B. Bilwes, R. Bilwes, J. Diaz and J.L. Ferrero, Nucl. Phys. A449 (1984) 519; A.M. Stefanini et al., Phys. Rev. Lett. 59 (1987) 2852

102

I.J. Thompson et al. / Threshold anomaly

4) M.A. Nagarajan, C. Mahaux and G.R. Satchler, Phys. Rev. Lett. 54 (1985) 1136; C. Mahaux, G.R. Satchler and H. Ngo, Nucl. Phys. A449 (1986) 354 5) I.J. Thompson, M.A. Nagarajan, J.S. Lilley and B.R. F&on, Phys. Lett. B157 (1985) 250 6) S.C. Pieper, M.J. Rhoades-Brown and S. Landowne, Phys. Lett. B162 (1985) 43 7) I.J. Thompson, Comp. Phys. Reports 7 (1988) 167 8) F. Videbaeck, R.B. Goldstein, L. Grodzins and S.G. Steadman, Phys. Rev. Cl5 (1977) 954 9) A.A. Ioannides and R.S. Mackintosh, Phys. Lett. 8161 (1985) 43 10) M.E. Franey, J.S. Lilley and W.R. Phillips, Nucl. Phys. A324 (1979) 193 11) S.S. Hong, T. Udagawa and T. Tamura, Nucl. Phys. A491 (1989) 492 12) E. Vulgaris, L. Grodzins, S.G. Steadman, and R. Ledoux, Phys. Rev. C33 (1986) 2017 13) M.E. Franey and P.J. Ellis, Phys. Rev. C23 (1981) 787 14) R.A. Broglia, C.H. Dasso, S. Landowne and A. W&her, Phys. Rev. C27 (1983) 2433 15) R. Vandenbosch, in The many facets of heavy-ion fusion reactions, Argonne National Laboratory, ANL-PHYS-86-1 (1986), p. 155 16) M.A. Nagarajan and G.R. Satchler, Phys. Lett. B173 (1986) 29; G.R. Satchler, M.A. Nagarajan, J.S. Lilley and I.J. Thompson, Ann. of Phys. 178 (1987) 110 17) C.H. Dasso, S. Landowne and G. Pollarolo, Phys. Lett. 21 (1989) 25 18) P.H. Stelson, Phys. Lett. 8205 (1988) 190 19) G.R. Satchler, M.A. Nagarajan, J.S. Lilley and I.J. Thompson, in preparation 20) J.S. Lilley, Lecture notes in physics, vol. 317 (Springer, Berlin, 1988) p. 256