TWO STEP PROCESSES IN TRANSFER REACTIONS

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TWO STEP PROCESSES IN TRANSFER REACTIONS

E. Rost

To cite this version:

E. Rost. TWO STEP PROCESSES IN TRANSFER REACTIONS. Journal de Physique Colloques,

1974, 35 (C5), pp.C5-61-C5-65. �10.1051/jphyscol:1974534�. �jpa-00215697�

JOURNAL DE PHYSIQUE

Colloque C5, supplément au no 11, Tome 35, Novembre 1974, page C5-61

TWO STEP PROCESSES IN TRANSFER REACTIONS

E. ROST

Service de Physique Théorique, CEN-Saclay, France and University of Colorado, Boulder, Colorado, USA

Résumé.

L'inclusion des processus en deux étapes dans l'analyse d'ondes distordues d'expé- riences réaction directe est discutée. On démontre que de tels effets peuvent être importante pour les ions lourds, pour les projectiles aux hautes énergies, et pour l'analyse des transitions faibles. Le mécanisme de la réaction en deux étapes peut brouiller l'extraction d'information spectroscopique d'une expérience, mais dans certains cas, la nature cohérente des processus

une et deux étapes peut révéler une information sur la phase des amplitudes spectroscopiques qui autrement ne seraient pas accessibles.

Abstract.

The inclusion of two step processes in the distorted waves analysis of direct-reaction experiments is discussed. It is shown that such effects can be important for heavy ions, for high energy projectiles and for the analysis of weak transitions. The two step reaction mechanism can confuse the extraction of spectroscopic information from expriment but in certain cases the cohe- rent nature of the one-step, two-step interference can provide spectroscopic phase information not otherwise available.

First of all, 1 will limit this talk to direct reactions which are defined

to be those involving only a few degrees of freedom. An important subset of these involve only one degree of freedom, usually a single- particle or collective coordinate, and are analyzed using the distorted wave Born approximation (DWBA).

If the first-order term is sufficient, we speak of a one- step process even if the interaction strength is arbi- trary (or renormalized). 1 will discuss some cases where one term is not sufficient either because of selection rules or angular distribution anomalies requiring coherent interference terms.

There are several important topics that 1 will omit.

One is the formal problem of understanding the DWBA series, i. e. which series and what convergence, if any. Such questions are especially relevant for two- step formulations in transfer reactions and have been discussed in the literature [ 2 ] . 1 will also not discuss the very important work on two-step resonant inelastic scattering which Amos has reported on in the confe- rence. Finally 1 will restrict the subject material to selected examples covering a variety of projectiles and energies.

The initial theory of two step processes in transfer reactions was outlined a decade ago by Penny and Satchler [3] who considered the transfer reaction a + A

b + B in the presence of inelastic excita- tions A' in the target and B' in the residual nucleus.

The transition amplitude is

T

y,, 11 ^{xi.!;, < bB' 1 V 1 aA' > dr, dr, (1)}

A',,

which reduces to the usual expression if A'

A and B'

B. The generalized distorted waves x:Afil and can be solved via coupled Schrodinger equations given a small number of couplings A

A' and B

B' (this is usually accomplished using a collective mode]).

Alternatively the functions x:AfLr (or X&,L,) may be further approximated for A # A' leading to second- order DWBA terms.

It is instructive to estimate the importance of a single two-step term, say A

A'

B. This can be done by expanding Xai,it in a DWBA series and eva- luating the Green's function at an appropriate ave- rage energy. Expressing the result in terms of cross sections, Schaeffer R. (private communication) has obtained the expression

where hk is a typical momentum and

are the one-step cross sections evaluated at their maximum value (i. e. a forward angle for light pro- jectiles, the grazing angle for heavy ions). For strong collective transitions A

A', the factor in brackets is in the range 0.1 t o 1 and increases with the energy and mass of the projectile. If

is considerably larger than

the two-step term will dominate ; if

and

are comparable, the coherent inclu- sion of the two-step term may cause a significant effect.

This will be illustrated in the examples which follow.

Two step processes in single nucleon transfer reac- tions are often observed for light nuclei where selection

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974534

C5-62 E. ROST

rules prevent or inhibit the direct process. This was pointed out by Dupont and Chabre [4] for the I2C(d, 3He)11B reaction at 28 and 50 MeV. The 712- level at 6.76 MeV is excited with appreciable strength, much greater that that obtainable from direct transfer of the small If,,, admixture in the 12C ground state.

However a simple model which uses the inelastic excitation to the 2' level as an intermediate process enabled them to explain the experimental data in terms of the measured

2+ inelastic scattering and 0'

312- proton pickup reactions. In this calcula- tion the spectroscopic factor for the 2'

712- pickup (the second step) was taken to be the same as that for the Of

312- pickup, as would be the case in an extreme weak coupling model for llB.

One can refine the calculation to allow for mixed one and two step processes by using a more detailed model of the parent and daughter nuclei. For example, we can use the deformed, shell-mode1 calculation of Clegg [ 5 ] to express a given l l B state J in terms of its 12C parent 1 coupled to a p-shell proton hole (with j

312 or 112)

The sum over 1 is truncated to include only the ground state Of and 4.43 MeV 2' state of 12C, viz. the direct and simplest two-step process. The values for

are given in table 1. They show, for example, that the ground state of l l B will be populated primarily by the direct process, the 512- level by the indirect two- step process, and the 112- level will be fed coherently by both processes. Thus the pickup reaction on 12C will measure one step, two step, and mixed processes for the different levels of the llB (or the mirror llC) nucleus.

Structure coejjîcients cdj (deJned in eq. ( 3 ) ) for low-lying llB or l l C levels

State J 12C parent 1

hole

-

312 (ground)

312 - 0.906

2

112 0.422

112 O

112 0.546

2

312 0.837

512 2

312 0.958

312 (excited)

312 0.172

2

312 0.922

112 2

312 - 0.895

The connection of the

structure coefficients with the experimental data requires specification of the distortions, the inelastic transitions, and the pro- jectile-interaction overlap (in the zero-range approxi- mation the later is characterized by a simple number Do). We will regard the structure of 12C and llB

(or llC) as known and use it to study the reaction mechanism occurring ; of course the process can be turned around to test models of nuclear structure if the reaction mechanism is understood.

A favorable case is provided by the 52 MeV 12C(d, 3He)11B reaction measured at Heidelberg [6].

In this case the optical potentials are known, the inelastic (d, d') scattering to the 2' level of 12C is well described by the collective model and the zero- range approximation with Do

- 170 MeV fm3I2 has been established [7]. Thus we have a parameter- free theory to compare with the data in figure 1 and the results are quite encouraging. It is seen that the flat angular distribution of the 512- level is explained as is the overall absolute yield (to within about 50 %).

The 112- level at 2.14 MeV is fed coherently by both one and two step processes which add distructively in agreement with the data. This also causes a some- what different angular distribution which might be incorrectly interpreted as a j-dependent DWBA effect. Finally, the 112- strength is different than that which is obtained by assumption of simple pli, pickup. This illustrates the danger of extracting weak spectroscopic factors with an oversimplified DWBA analysis.

.. \

MeV

MeV

MeV

I

10 20 30 40 50 60 0c.m.

FIG. 1. - Differential cross sections for the lzC(d, 3He)lBl reaction at 52 MeV incident energy. The distorted waves theory

is described in the text.

What happens for other energies and other projec- tiles ? An extreme example was provided me this year by the Saturne synchrotron which measured the 12C(p, d)llC reaction at 700 MeV incident energy [SI.

This energy is far higher than ever before used for a

TWO STEP PROCESSES IN TRANSFER REACTIONS

nuclear transfer experiment of this kind and it seemed interesting to apply a « conventional » analysis to see whether specific high-energy results would emerge.

Some modifications were required ; in particular the optical potentials had to be calculated from jîrst principles using the impulse approximation [9] with measured nucleon-nucleon scattering phase shifts and the electron scattering density distributions. This procedure is rather uncertain for the deuteron.

Another serious problem is caused by the large momentum mismatch which requires us to examine more closely the projectile interaction overlap. This can be studied by using the plane wave Born approxi- mation for a one-step process which gives for the cross section

69

^{const. (D: +}

^.

Here Do and

are Fourier transforms of

for S and D deuteron components and F i s the Fourier transform of the transferred nucleon [Il]*. Figure 2 plots these transforms over a large range of momentum and also shows how they Vary for the limited angular range of the Saturne experiment. It is seen that D

(DO + 0;)'" is rather flat and could be taken constant for the region involved if the Born approxi- mation were valid. Then putting distortion back in one can use the usual zero-range DWBA or coupled- channel programs with only an adjusted value of D.

This approximation neglects any explicit distorted wave effects of the deuteron D-state or other effects involving specific high-energy properties (e. g., the N*

nucleon isobars) which are obliterated in order to easiIy treat the absorption and nucleus structure effects. The latter are described in the Clegg model [5]

so that the theory can be applied without arbitrary parameters (the overall D value of 60 MeV fm3I2 is in the expected range in Fig. 2).

The comparison of the theory with the data is given in figure

The relative cross sections corresponding to the five levels measured are rather well accounted for considering the uncertainties in the reaction model used. The relative two-step yield is well explained and comparison with figure 1 shows the increased impor- tance at high energy, in agreement with eq. (2). The extra structure obtained for the direct process is not surprising in view of the simplifying assumptions concerning the deuteron D-state. The most interesting disagreement is at forward angles for the 112- transi- tion which, 1'11 remind you, is fed coherently by inter- fering one and two step mechanisms. Evidently some- thing is wrong in the relative phase here but it wilI be necessary to do better with the distortions and deuteron interna1 wave function before attributing this effect to specific exotic processes. This example does illus- trate, however, the possible exploitation of the coherent nature of the mixed one and two step transitions.

'k ( p , d ) l l ~

700 MeV

FOURIER TRANSFORMS

I

1 O" I O ' 20" 30' 8c.m.

FIG. 2. ^- Fourier transforms of the neutron and deuteron wave

11 1

functions ; K = ka -

,

^,-

I L L

tum variables appropriate for the Born approximation. The lower figure gives these functions over the angular range appropriate to the 700 MeV W ( p , d) experiment as indicated by the dotted lines in the upper figures. The value Dexp = 60 MeV fm31-2 shown on the right has been used to normalize the data and is roughly equal to an average value of (DO f D2)1/2 over this angular

range.

1 do not wish to spend much time on heavy-ion transfer reactions since Mme Mallet-Lemaire will treat these in detail but 1 will mention some work of Roussel and Pougheon [Il] at Orsay. I t is well known that heavy ion reactions near the Coulomb barrier have rather dramatic kinematic or window effects, many of which can be understood with simple semi- classical models. The model can be generalized to include two-step transitions and in some cases the experimental parameters can be designed so as to emphasize particular two-step processes. Application to the 19F(160, 15N)"Ne 4' level which is one-step forbidden (assuming no g-shell in "Ne) yielded a cross section comparable with the direct ground state yield for 46 MeV incident energy. Hopefully one can exploit the heavy-ion windows to tune in the processes of interest for the study of the reaction mechanisms. Also this experiment when compared with 19F(3He, d)20Ne experiments [12] again shows the relatively greater importance of two-step processes for heavy ions as suggested by eq. (2).

The extension of the direct reaction theory to two-

*

nucleon transfer is quite straightforward and there

FIG. 3. - Differential cross sections for the iowest five levels of 11C excited by the 12C(p, d)llC reaction at 700 MeV.

already exists considerable work done by Glendenning, Ascuitto and others [13] including inelastic (or core- excitation as it is often called) p,rocesses in two- nucleon transfer reactions. Some of this work will be discussed in this conference by Seltz and Mallet- Lemaire. The results for light ions (e. g.

t)) show that two-step effects can be appreciable for deformed nuclei, although for spherical targets considerable spectroscopy can be obtained via one-step DWBA.

For heavy ions, however, large effects in relative yields (Mallet-Lemaire finds a factor 7) can occur when one and two-step terms interfere.

For the last part of my talk 1 will consider another type of two-step process, viz. that involving alternate particle partitions. Such processes were introduced [14]

a few years ago to explain puzzling features in the 48Ca(3He, t ) 4 8 S ~ reaction which could be explained by invoking a pickup-stripping mode with 47Ca(g. S.) + a in the intermediate state.

The extension of the direct reaction theory to include the

processes is quite straightforward and eq. (1) becomes

x Gy

< ^Y I

1

> xi+) + non-orthog., (5) where a denotes the a + A partition, etc.

Inelastic effects have been ignored and the second distorted wave Born approximation has been employed although these simplifications can be removed without

difficulty. The non-orthogonality term can often (but not always) be avoided by suitable choice of post or prior representations of the interactions V (see [15] for details). As applied in practice only the few

channels with appreciable stripping (pickup) strengths are included in eq. (5).

The estimation of eq. (2) is still appropriate and in the new notation becomes

For strong stripping (pickup) transitions (- 10 mblsr), a typical

is of the order of a few tenths of a millibarn, by no means negligible. Such strong transitions occur for spherical nuclei (like 48Ca) where the truncation to a few

is feasible. In other cases the effects of such two-step processes may wash out. This would be fortunate since deformed and vibrational nuclei already suffer from two-step core excitation effects in the analysis of transfer reaction data.

It is tempring to apply eq. (5) to two-nucleon transfer reactions, e. g. to consider the p

d + t successive pickup process in the (p, t) reaction. Unfortunately this requires either a full finite-range calculation or else explicit treatment of the non-orthogonality terms in eq. (5) [16]. A simpler reaction to study is the (p, n) reaction including the p

d

n pickup- stripping mode and 1 will discuss some Boulder work by Rickertsen and Kunz [17] on the analysis of (p, n) reactions to isobaric analog states of spherical nuclei.

In the energy region studied (22 to 40 MeV) the

d) pickup data are well known so that a choice of a few y was easily made. Because the final state is an analog of the target, the (d, n) stripping is trivially related to the (p, d) pickup and thus the two-step p

d

n term, by itself, can be calculated without any adjustable parameters. Two-step cross sections were foùnd to be roughly comparable in magnitude with the data with rather featureless angular dis- tributions. Of course the (p, n) reaction can proceed via a one-step, charge-exchange mode which is usually taken to be of the form

where g(r) is taken to be of Yukawa form and V, is the isospin strength of the effective interaction.

In the calculations of Rickertsen and Kunz [18], the value of V, is the only adjustable parameter whether the two-step p

d

n term is included or not.

The results of the calculation are .rather surprising.

I t was found that a value V, x 34 MeV was required

when the two-step process was included as compared

to the V, x 19 MeV value without. Thus the one

and two-step processes interfere distructively for

the (p,

analog state transitions. The angular dis-

tributions obtained with the mixed processes are

TWO STEP PROCESSES IN TRANSFER REACTIONS C5-65

vastly superior in their fit to the data 1181 and certain anomalies are explained by the coherent interference between the two processes. Finally the extracted values of Y, are considerably more constant for different targets a,nd incident energies. Looking at their results from another point of view we could say that a simple microscopic model analysis without two-step terms extracts a renormalized effective interaction and that the extraction is less accurate.

The energy dependence of the one-step effective interaction is an unfortunate effect which occurs because of the neglect of the two-step pickup-stripping terms.

Another effect of two-step processes is to modify selection rules. For the analog state (p, n) transitions, which are O +

O+, the direct process requires AS

O since the parity rule forces AL

O. Differen- tial analyzing powers were measured at Saclay [19]

for the 60Ni(p, n)60Co and 120Sn(p, n)120Sb analog state transitions using 24.5 MeV polarized protons.

The results show rather large, structured analyzing powers which cannot be reasonably explained with a conventional microscopic model theory. A prelimi- nary calculation [20] including the two-step pickup- stripping mode was quite encouraging although a detailed fit to the Saclay data was not obtained. Since the two-step pickup-stripping mode does not require AL

one obtains AS

1 terms in the transition and the coherent effect on the analyzing powers are dramatic. This result suggests caution in extracting spin-dependences of effective interactions without considering explicitly the important multi-step pro- cesses which contribute.

My final example will again deal with a selection rule effect, this time involving medium-heavy ions in

the 26Mg(6Li, 6He)26A1 reaction. For the analog state which is

we have a transition

O + + 1 + -+O+ +

which is rigorously forbidden in a single step process involving a microscopic effective interaction. The process can proceed, however, via a pickup-stripping process. The experiment of Duhm

al. [21] measured this reaction at 32 MeV incident energy and found an oscillatory angular distribution characteristic of L

1 transfer and of magnitude about 10 pb/sr. Their two-step calculation using 7Li(3/2-) + 25Mg(5/2+) and 7Li(l/2-) + 25Mg(5/2+) as intermediate parti- tions gave a very nice fit to the magnitude and shape of the analog state yield. The authors also concluded on the basis of relative yields that the single-step allowed transitions contain sizeable two-step contri- butions, a conclusion in accord with the (p, n) work described above.

For my own conclusion 1 believe that two-step processes should be explicitly taken into account in many direct reactions, especially at high energy or with heavy ions. This is especially crucial if there are selection rules or strong interfering processes contri- buting. However, even in situations where the single- step transfer is large, the explicit two-step contri- butions can substantially change the effective interaction or spectroscopic amplitudes extracted from the data and may even confuse us in the inter- pretation of angular distributions. It is important now to carefully calculate these processes for a large variety of direct reactions so as to know how to correct for (and when to ignore) their effects and, in a more optimistic sense, how to exploit these processes in studying nuclear structure.

References

[l] AUSTERN, N., Direct Nuclear Reactions (Wiley, New York) 1970.

[2] GREIDER, K. and DODD, L., Phys. Rev. 146 (1966) 67 ; ROBSON, D., Phys. Rev. C 7 (1973) 1.

[3] PENNY, S. K. and SATCHLER, G. R., Nucl. Phys. 53 (1964) 145.

[4] DUPONT, Y. and CHABRE, M., Phys. Lett. 26B (1968) 362.

[5] CLEGG, A. B., Nucl. Phys. 38 (1962) 353.

[6] HINTERBERGER, F . et al., Nucl. Phys. A 106 (1968) 161.

[7] BASSEL, R. H., Phys. Rev. 149 (1966) 791.

[8] BAKER, S. D. et al., to be published.

[9] KERMAN, A. et al., Ann. Phys. (N. Y.) 8 (1959) 551.

[IO] ELTON, L. R. B. and SWIFT, A., Nucl. Phys. A 94 (1967) 52.

[ I l ] ROUSSEL, P. and POUGHEON, F., Phys. Rev. Lett. 30 (1973) 1223.

1121 SIEMSSEN, R. H. et al., Phys. Rev. 140B (1965) 1258.

[13] ASCUITTO, R. J. et al., Phys. Lett. 34B (1971) 17 ; KING, C. H. et al., Phys. Rev. Letf. 29 (1972) 71 ; ASCUITTO, R. J. and GLENDENNING, N. K., Phys. Lett. 45B

(1973) 85.

[14] TOYAMA, M., Phys. Lett. 38B (1972) 147 ;

SCHAEFFER, R. and BERTSCH, G. F., Phys. Lett. 38B (1972) 159.

[15] UDAGAWA, R. et al., Phys. Rev. Lett. 25 (1973) 1507.

[16] KUNZ, P. D. and ROST, E., Phys. Lett. 47B (1973) 136.

[17] RICKERTSEN, L. D. and KUNZ, P. D., Phys. Lett. 47B (1973) 11.

[18] CARLSON, J. D. et al., Phys. Rev. Lett. 30 (1973) 99.

[19] Moss, J. M. et al., Phys. Rev. C 6 (1972) 1968.

[20] RICKERTSEN, L. D., Univ. of Colorado progress report (1973) unpublished.

[21] DUHM, H. H. et al., Phys. Lett. 48B (1974) 1.