ALAS from sliding rainbows

Nuclear Physics @ North-Holland

A455 (1986) 141-148 Publishing Company

ALAS FROM SLIDING

K.W.

RAINBOWS*

McVOY

Department of Physics, University of Wisconsin, Madison, Wisconsin 53706

Received

11 December

1985

Abstract: The 180” excitation function for (I +40Ca elastic scattering exhibits a series of broad maxima in the 20-50 MeV energy range. They are found to be a translation into energy language of the nuclear rainbow phenomenon described in the preceding paper, and the highest-energy maximum is shown to occur near the energy E, at which the pocket in V,(r) disappears. This fact can be used to determine the depth V, of the real part of the corresponding optical potential.

1. Introduction The existence of “anomalous large-angle scattering” (ALAS) of cr-particles from 4oCa has been known for many years, and a very similar effect for (Y+ 160 has more recently come to light ‘). A significant advance in the understanding of its origin was made in 1978, when two separate groups 2,3) found that local optical potentials (in shape roughly the square of a Woods-Saxon function) were capable of fitting the complex angular distributions for (Y+ 40Ca from 0” to 180”, over an energy range from 40 to 100 MeV. Our purpose in returning to the phenomenon here is to call attention to the fact that this anomalously large scattering dies away above a critical energy EF, and to point out that an understanding of this in terms of the dark side of a rainbow can be used to uniquely determine V,, the depth of the real part of the optical potential employed to describe this scattering. Comparison *) of the a f4’Ca optical potential with that for (Y+‘?Za made it clear that the 40Ca ALAS was connected with surface transparency, and a Brink and Takigawa decomposition of these optical scattering amplitudes ‘) into “internal” and “barrier” parts further associated the large back-angle cross section with the internal or small-l part of the amplitude. Recently Anni and Taff ara ‘) have examined the role of Regge poles in the (Y+ 40Ca as well as the (Y+ 90Zr elastic amplitudes - exactly the amplitudes which the preceding two articles 6,7) h ave shown to contain substantial nuclear rainbow effects. Anni and Taffara were especially interested in the 180” excitation functions, which, for the experimentally-determined potentials which they employed for both systems, exhibit a sequence of broad (lo-15 MeV) maxima up to a critical energy (about 75 MeV l

Supported

in part by NSF. 141

K. W. McVoy / ALAS

142

for 90Zr and 65 MeV for 40Ca), beyond which these cross sections drop monotonically with a further increase in energy. Fig. 1, taken from their work, shows the excitation function for (Y+ 4oCa. These potentials are quite surface-transparent, and their scattering amplitudes exhibit substantial resonances effects, which can profitably be viewed as the influence of the optical-potential Regge poles on the I-dependence of the (complex) phase shifts. What Anni and Taffara convincingly showed was that these slow oscillations in the 180” optical excitation functions, in exactly the ALAS energy region, can be viewed as the contributions of individual Regge poles, as fig. 1 illustrates. These poles move to the right (i.e., to higher I’s) in the complex Z-plane with increasing bombarding energy. The grazing angular momentum, I,,(E), does as well, moving faster than the poles, and a given pole makes its strongest contribution to the 180” cross section just as it is being overtaken by lgr. Physically this means that in this energy range the major contribution to the 180” cross section is always from that partial wave whose resonance is very near the top of its barrier - which is just the resonance which determines the high-l or steep side of the rainbow-dip in the semi-classical deflection function at this angle “). Eventually (at 65 or 75 MeV for the 4oCa and 90Zr potentials) the last (n = 1) resonance rises out of its pocket, so the narrow-resonance energy region comes to an end; the ALAS effect then disappears, and the 180” cross section thereafter drops monotonically. In summary, the very useful message of the Anni-Taffara analysis is that the ALAS effects, in the scattering amplitudes of the optical potentials which fit the

E (MeV) Fig. 1. Excitation

function

at 180” for the CK+@Ca potential

employed

by Anni and Taffara.

143

K. W. McVoy / ALAS

elastic

(Y+90Zr

“quasi-molecular” entire segments

and

(Y+40Ca

data,

are unambiguously

potential

resonance

resonance) effects. Not effects of individual resonances, of the lowest rotational bands in 94Mo and 44Ti.

(i.e., but of

2. The sliding-rainbow interpretation As the preceding two articles 6*7)have pointed out, these two scattering systems exhibit some of the strongest known manifestations of nuclear rainbows, and nuclear rainbows are themselves a collective manifestation of overlapping potential resonances. Ergo, it must also be possible to view the ALAS effect as some aspect of nuclear rainbows. This is indeed the case, and the relation is an exceedingly simple one. To understand it, we refer to fig. 2, which schematically shows the deflection function, e(I) = 2 dS/dl, for the real part of a deep optical potential, at an energy low enough that the rainbow angle is substantially beyond 180” (i.e., more negative than -180”); in addition the figure suggests the shape of the corresponding “rainbow type” cross section for the farside 6*7) or negative-angle component of the elastic scattering amplitude. We say “suggests” because we have allowed this cross section to continue on beyond -180”, in agreement with its deflection function.

Fig. 2. Schematic indication of a deflection function, with attendant energy E
farside Airy-type amplitude, at an than -r. The Airy pattern moves in the 180” excitation function.

The reason for doing so is that, as the bombarding energy increases, these “Airy minima”, like all other features of an angular distribution, “slide forward” by moving closer together in angle. As these minima cross -180” in their motion towards O”, they will “cause” successive minima in the 180” excitation function. The ALAS minima, in other words, are just the Airy minima of the sliding rainbow. Because there is a “last” Airy minimum and maximum in the extended angular distribution,

K. W. McVoy / ALAS

144

their passage minimum

through

180” with increasing

and maximum

cross section

bombarding

in the 180” excitation

will drop monotonically

The ALAS effect is thus limited

with a further to energies

energy will produce

function,

beyond

increase

low enough

which

a last

the 180”

in energy*. that the rainbow

angle

f& for the real part of the optical potential is beyond -180”; at these low energies all angles out to 180” are on the bright side of the rainbow, and so are classically illuminated**. Above this energy region, the bright side of’the rainbow is confined to more forward angles, leaving 180” in the rainbow shadow, where the cross section is small. Since Anni and Taffara point out that the termination of the ALAS effect occurs where the last Regge resonance moves up out of the potential pocket, i.e., near E = EF, where the pocket fills in, we conclude that the rainbow angle 8,(E) will pass through -180” for E near &, in agreement with the examples shown in the previous

two articles 6V7). V= 267.90, VI= 29.91, I’

R= 4.66, RI= 5.74, KYAI= 4.33 I “I

A= 1.461 AI=1.481 I

a

1

I

I

I

I

4HE+40CA EI..U3=50,54,56

MeV

FAR

. Y

--I

. . ..50MeV

\

5&,,

\

\

‘_ ‘. ‘...’

\

5\s\

50

100

.-

150

C.M. Angle Fig. 3. Farside moves forward

cross sections for (Y+ ?Za at 50, 54 and 56 MeV, showing how the last Airy minimum through -180” with increasing energy. The “invisible minimum” beyond 180” at 50 MeV is suggested schematically.

l Michel and Vanderpootten ‘), in their discussion of these same data, noted (i) that the slope of the envelope of the large-angle oscillations varies in a seemingly erratic way with increasing bombarding energy, and (ii) that a particularly deep dip in this envelope drifts forward with increasing energy. Both effects are simply manifestations of the forward drift of the farside Airy pattern; the dip they had in mind is the second Airy minimum forward of 0s. l * Since this rainbow is itself just overlapping resonances, an equivalent statement is that any classical illumination of large farside angles is produced by overlapping resonances.

145

K. W. McVoy / ALAS

The confirmation 180” excitation

of this “sliding

function

is provided

rainbow”

interpretation

by fig. 3, which

shows

of the minima the farside

in the angular

distributions for the Anni and Taffara potential for cx+40Ca at 50, 54 and 56 MeV. The “last” minimum in the excitation function of fig. 1 occurs at 54 MeV, and indeed fig. 3 shows that exactly at this energy the last minimum in the angular distribution passes forward through 180”; at 56 MeV it is found at 170” and the 180” excitation function has begun to rise again toward its last maximum*. As mentioned in the previous article 7), the vanishing of the ALAS in (Ylt4’Ca scattering above 60 MeV was noted some time ago by Michel and Vanderpoorten “). It was also discussed in some detail by Takigawa and Lee “) who correctly identified it as an example of the glory effect, i.e., of the passage of e(I) through -180”; whenever this happens, it is clear that all physical angles are classically illuminated. The only detail they did not comment on (because the farside cross section was not available to them) was that 0(l) passes through -180” twice, at I, as well as I,, and that the interference of these two “glories” can produce the type of oscillations in the 180” deflection function that were discussed by Anni and Taffara**.

3. Relation to Heavy-Ion Excitation This discussion

immediately

Functions at MO’?

brings to mind the large back-angle

elastic scattering

which has been seen in systems like ‘60+28Si, and in particular their oscillatory 180” excitation functions, all of which has been reviewed recently by BraunMunzinger and Barrette ‘). Might these gross-structure oscillations be due to some resonant effect similar to the above phenomenon in (Y+40Ca? The thought is an intriguing one, and indeed the Anni-Taffara mechanism of the grazing 1, I,,(E), crossing Regge trajectories as E increases, is exactly one of the resonant mechanisms suggested by Barrette et al. lo) to explain the gross structure in the 160+ *‘Si excitation function. In this context it is interesting to remark that, just as in that case, this mechanism gives an irregular sequence for the dominant l-values at successive maxima in the 180” excitation functions for (Y scattering. For instance from the Anni-Taffara graphs we extract the sequence shown in table 1. Although the comparison is enticing, it seems to us that the analysis of the heavy-ion data is simply not yet complete enough to permit reliable conclusions to be drawn regarding mechanisms for these heavier projectiles. The key to understanding the (Y+ ‘“Ca data is the availability of reasonable optical potentials which l The terminology employed here is slightly inaccurate because Anni and Taffara inadvertently used the V = 288 MeV potential of the early Michel and Vanderpoorten work 8a). These latter authors eventually showed sb) that this potential is too deep, and in effect shifts the oscillatory 180” excitation function to the right by “one notch”, thus moving its last maximum from 50 MeV to 62 MeV. Although its maxima are thus mis-labelled, the physical interpretation is unchanged, and so to avoid confusion we have employed this incorrect V = 288 MeV potential in our analysis. ** Further comments on the nearside/farside terminology in the vicinity of a glory effect are given in the appendix.

146

K. W. McVoy / ALAS TABLET E c m. 25 33 45 60

1 11.5 13.5 17 18

fit the angular distributions well, over a substantial energy range. The same cannot yet be said for the 160 +%i data, and one reason for the difficulty may be an admixture of an elastic-transfer component which, though small, is certainly expected to be larger than in (Y+40Ca; the parity-dependent potential employed by Dehnhard and collaborators ‘I) certainly suggests this. If it is indeed important at 180”, it would seem difficult to explain these data only by a “geometrical” mechanism like the rainbow*.

4. Conclusions In summary, the conclusions to be drawn from the optical fits to the elastic (Y+40Ca and (Y+“Zr data is that the large back-angle scattering is certainly a potential-resonance phenomenon, confined to the resonance energy range E < EF. In particular, the oscillations in the 180” excitation function can with equal profit be viewed as (1) a direct manifestation of grazing-l Regge poles, or (2) Airy maxima and minima an a nuclear rainbow, sliding forward through 180” with increasing energy. In any case, if the large back-angle scattering is resonance-related, it should certainly be confined to low energies - whereas the elastic-transfer mechanism may persist to considerably higher energies. On a more practical note, if the anomalously large back-angle scattering is due to a rainbow (as in the (Yf 40Ca case), its disappearance for E S=EF can be employed to determine V,, the depth of the real part of the corresponding optical potential. This is simply Because EF (the energy at which the pocket in V,(r) fills in with increasing I) clearly increases with V,. Consequently the position of the highestenergy maximum in the 180” excitation function should determine V, uniquely, provided that the shape of V(r) has been determined by other features of the angular distribution. It was basically this which allowed Michel and Vanderpoorten *b) to conclude that V, for (Y+40Ca is -180 MeV rather than 288 MeV; data below EF were incapable of distingishing between these two possibilities. This argument is clearly a translation into energy language of the angular distribution criterion which Goldberg et al, 13) employed to distinguish “deep” potentials from “shallow” ones. l As has recently been noticed I’), if the parity dependent part of the Dehnhard potential is removed, the remainder does produce a farside rainbow, suggesting that rainbows and elastic transfer can readily co-exist.

K. W McVoy / ALAS

147

Appendix The nearside/farside

terminology

glory effect. Fig. 4 attempts

becomes

a bit cumbersome

to sort this out by showing

when discussing

that farside

trajectories

a are

renamed nearside trajectories 14) as 0(I) passes through -7 (or in fact through any nr). This is merely because nearside and farside trajectories are defined and distinguished by the signs of their angular momenta normal to the scattering plane. In fig. 4, consider for definiteness the “outermost” farside trajectory, labelled Fz (whose scattering angle ~9lies between 0 and --7~), which has angular momentum directed into the page. Note that there is also a nearside trajectory, N:, (-27~ < 8 < -v, and r x p out of the page) which leads to the same physical angle, and that N; and F,’ become physically identical for 13= -7~ This is why Nr is shown as the continuation of F. in the deflection function given the same figure. Since Nr and F,’ both reach the same detector, they can interfere, and it is of course their interference which leads to the “backangle Fraunhofer oscillations” known as the glory effect. We have labelled

it here as the “outer

glory effect”, occurring

at I= I,.

f

Fig. 4. An illustration of how identifying nearside and farside trajectories by the sign of their angular momentum requires an interchange of these N and F labels when the scattering angle passes through a multiple of ?r.

In the deflection function shown, with 0,< -IT, 0(Z) also crosses -r at I= I,, of these two glories which producing an “inner glory effect”. It is the interference causes this entire backangle Fraunhofer pattern to rise and fall with a change in the bombarding energy, thus producing Anni-Taffara oscillations in the 180” excitation function.

148

It is a pleasure,

K. W. McVoy / ALAS

as usual, to thank Charles

Goebel

for being such an incisive

listener.

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