VOLUME 32, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 22APRIL1974

7S. Kullander et al.t Nucl. Phys. A173, 357 (1971). 8We are indebted to Dr. J. Knoll for the use of his

DWIA program and to Dr. H. M. Hofmann for using his program for precision cheeks.

9C. D. Epp and T. A. Griffy, Phys. Rev. C 1, 1633 (1970).

10J. Knoll, thesis, University of Freiburg, 1971 (un­published); R. D. Viollier and K. Alder, Helv. Phys. Acta 44, 77 (1971).

ML. R. B. Elton and A. Swift, Nucl. Phys. A94, 52 (1967). The potential depths were slightly adjusted to reproduce the (Em) values of Table I.

We wish to report some early encouraging r e ­sults of a continuum-shell-model calculation of elastic neutron scattering from leO in which exci­tations of up to four-particle, four-hole (4p-4h) states in the target nucleus are included. The need for such a calculation ar ises because the existing descriptions of the system 160 + w are in­complete. JR-matrix fits to neutron cross-section data and their reduction into the various partial waves such as recently reported by Johnson,1

though extremely useful in providing a schemati-zation of experimental data, shed no light on the dynamics of the compound system. In view of the sensitivity of the cross section to the internal structure of the total system (target + projectile), as suggested by the numerous peaks in the reso­nance curve, a microscopic description would be most desirable. Unfortunately, the microscopic models which have been employed so far are in­capable of accounting for the compound reso­nances in the spectrum.

There have been several attempts to calculate the neutron phase shifts in terms of forces that were successful in predicting bound-state proper­ties of nuclei. Dover and Van Giai2 use the Har-tree-Fock field derived from Skyrme's effective interaction in the no-polarization approximation. MacKellar, Reading, and Kerman3 use forces due to Davies, Krieger, and Baranger and Tabakin

12A. E. Glassgold and P. J. Kellogg, Phys. Rev. 109, 1291 (1958).

13H. Hiramatsu et al. [Phys. Lett. 44B, 50 (1973)] r e ­port 20% missing strength in the £-shell region.

14L. D. Miller, Phys. Rev. Lett. 28, 1281 (1972). 15H. S. Kohler, Nucl. Phys. 88, 529 (1966). 16L. R. B. Elton, Phys. Lett. 25B, 60 (1967). 17The consequences on Hartree-Fock theories are dis­

cussed in more detail by G. J. Wagner, in Lecture Notes in Physics, edited by U. Smilansky et al. (Spring­er, Berlin, 1973), Vol. 23.

t8U. Amaldi, J r . , et al., Phys. Lett. 25B, 24 (1967).

with perturbative corrections to the matrix ele­ments. The results of there calculations are very similar to the results obtained by one of us,4

who assumed the 160 nucleus to be a closed core, and similar to the result represented by the dashed line in Fig. 1(b). The nonresonant back­ground phase shifts and the resonance in the d3/2

partial wave near 1 MeV are well reproduced. In view of the rather simple configurations em­ployed by the above authors, their failure to pre­dict the numerous peaks in the experimental reso­nance curve is not surprising.

Following Brown's suggestion in 1964 of the possible interpretation of the excited states in leO in terms of rotational bands, a number of model calculations5,8 have appeared. The r e ­sults unanimously point to the importance of con­figurations consisting of four holes in the leO core and four particles in the upper orbits. For the first excited J*=0 + state at 6.05 MeV, the 4h-4p configurations carry a strength of about 85%. This finding is in excellent agreement with an earlier calculation by Unna and Talmi7 who used simple shell-model arguments and the known energy-level separations in neighboring nuclei to deduce that excitation of two nucleons in leO from the p1/2 to s1/2 orbit requires an ener­gy of at least 13.8 MeV while excitation of two neutrons and two protons requires only 6.9 MeV.

Continuum—Shell-Model Description of Compound Resonances in the System 160 +nf

Jacob George and R. J. Philpott Department of Physics, The Florida State University, Tallahassee, Florida 32306

(Received 21 January 1974

Bound and continuum properties of the system leO+« are obtained from a microscopic model by means of an extended /^-matrix formalism. The model includes configurations arising from excitation of up to four particles from the closed 160 core and provides a useful dynamical description of the observed low-energy spectrum.

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VOLUME 32, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 22 APRIL 1974

[0)

s+ r r 5-2 2 2 2

-4.0 -2.0 -0.0 0.2 1.0 2.0 3.0

E c m ( M e V ) — 4.0

2 2 2 2

-4.0 -2.0 -0.0

FIG. 1. (a) Bound and (b) continuum spectrum of the system uO+n calculated using the continuum shell model compared with (c) bound and (d) continuum ex­perimental results represented by Johnson's R-matrix fit (Ref. 1). Two-body matrix elements are from Zuker (Ref. 10) and from True (Ref. 11). Values of single-particle parameters are listed in Table I. The dashed curve in (b) is the result of including only configura­tions arising from leO ground state coupled to a neu­tron. The full curve in (b) represents the complete calculation.

Since the first excited 0+ state occurs low enough in energy to be of importance in low-energy phe­nomena, we have a strong indication that any pro­cess which is sensitive to the nuclear structure of 160 cannot be adequately described if 4p-4h configurations are omitted. Our present results confirm this view.

For our calculation we have employed a recent reformulation8 of the continuum shell model with­in the framework of the ^-matrix theory. Since a detailed account of this model has been report­ed,8 '9 it is not necessary to describe it here. However, we should emphasize that in the vicini­ty of the target nucleus, the scattered nucleon is treated on the same footing as the bound nucleons and the Pauli principle is correctly taken into ac ­count.

To start with, we have included all configura­

tions arising from a maximum of four holes in the lp1/2 shell and five particles in the 2s1/2 and ld5/2 shells. This part of our model space is the same as that employed in shell-model calcula­tions for 160 by Zuker, Buck, and McGrory6

(ZBM) and for 18F and 180 by Zuker.10 For the two-body matrix elements of the residual interac­tion we have used the same set as given in Ref. 10. In our model, the coupling to the one-nucle­on continuum is achieved by expressing part of the total wave function in terms of the wave func­tion of the target nucleus coupled to the single-particle wave function (still retaining antisym­metry) which is taken to be an expansion in har­monic-oscillator basis states. This results in a larger shell-model space (than in ZBM). We have used a central Gaussian two-body force from True11 to calculate two-body interaction matrix elements not given in Ref. 10. This interaction has the form

V(r) = V0(1.6PTE + PSE)exp(-r2/r0%

where PTE and PSB are the triplet even and sin­glet even projection operators, respectively, and V0 = - 32.5 MeV, r0 = 1.65fm. The harmonic-oscillator parameter v=mw/ti was chosen to be 0.32 fm"2 in accordance with electron-scattering results.12 '13

The average potential experienced by the neu­tron due to the 160 core was represented by a Woods-Saxon potential of the form

U(r) = -Vcfc(r)+V H LSmvc

2 r dr l ^ r . 5 .

w h e r e / j ( r ) s { l + exp[( r - /2 < ) /aJ}" 1 and tf^r^173, with r f = 1.25 fm and ^ = 0.65 fm. VLS was kept constant at 6.31 MeV and Vc was used as an ad­justable parameter (variations limited to the range 45-55 MeV). The potential depth Vc9 and a constant addition to the single-particle ener­gies, As p>, were adjusted separately for each set of quantum numbers J* T in order to fit the binding energies and to obtain optimum agree­ment with the observed positions of the r e so ­nances. Their values are given in Table I. In the following discussion we shall show that these rather artificial changes are only of secondary importance and reflect deficiencies in the resid­ual interaction or model space.

Our results (Fig. 1) verify the compound char­acter of the resonances in the system leO + n. With few exceptions, a one-to-one correspon­dence can be established between all experimen­tal and theoretical peaks in the energy region

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VOLUME 32, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 22 APRIL 1974

TABLE I. The Woods-Saxon well depths and single-particle energies for the various partial waves.

Partial wave (MeV)

As.p. <MeV)

Sl/2 Pitt Pill ^3/2 dm ft/2 fin

45.0 52.0 52.0 52.0 55.7 52.0 52.0

- 1 . 5 - 0 . 8 - 1 . 2 - 0 . 8 - 0 . 5 - 1 . 2 - 1 . 1

Single-particle energies

€s = -0 .70 MeV+As.p. e*i/2 = ~ 3 ' 4 5 M e V + A s . P . ed5/2= 0.05 MeV + A s .p .

shown. Thus, even the limited configuration space we employ here (which allow holes only in the lp1/2 orbit) possesses enough degrees of free­dom to predict all the observed peaks in the ex­perimental resonance curve. The positions of the theoretical peaks match fairly well with ex­periment and the nonresonant background is well reproduced. The result obtained when the cor re ­lated leO ground state is coupled to a single par­ticle in the no-polarization limit is shown by the dashed line. Only the nonresonant background and the single-particle peak near 1 MeV are r e ­produced in this approximation. The contrast with the full calculation is striking.

Considering that we have put together a two-body interaction piecemeal and that the number of orbits included in our calculation is rather limited, we are not surprised that some of the theoretical widths disagree with experiment. A closer study of these discrepancies has led us to some interesting and potentially useful observa­tions. In the first place, we have found that the components of the residual interaction which pro­duce correlations in the conventional shell-model (ZBM) space have by far the most pronounced ef­fect on the widths as well as the positions of the resonances. Figure 2 compares results obtained for the d3/2 partial wave when ZBM matrix ele­ments are employed, and when they are not, to the experimental data. The numbers which ap­pear on the calculated peaks refer to the specific eigenstates of a conventional shell-model calcula­tion which dominate these resonances. It is clear

FIG. 2. The effect of different two-body forces on the calculated dy2 resonances in the system l eO+«: (a) Calculation employs Gaussian (Ref. 11) matrix ele­ments supplemented by matrix elements from ZBM (Ref. 10); (b) calculation employs Gaussian (Ref. 11) matrix elements throughout, (c) Experimental data as fitted by Johnson (Ref. 1). In (a) and (b) the single-particle resonances are identified. Labels on the com­pound-resonance peaks correspond to the eigenstates in a conventional shell-model calculation.

that while the single -particle peak is relatively unaffected by the change in the matrix elements, the peaks of a more compound character are con­siderably altered in both width and position.

Secondly, according to our preliminary results, the calculated widths of d5/2, f5/2, f7/2, andp1/2

partial waves are of the same order of magnitude as the experimental widths, while the p3/2 widths disagree by an order of magnitude. The relative inaccuracy in the p3/2 partial wave suggests that excitations from the l/>3/2 orbit are important. In short, the observed resonance structure prom­ises to provide a sensitive test of the model space as well as the residual interaction.

In conclusion, we have pointed to the incom­pleteness of the existing theoretical description of the system lsO + w, have presented arguments based on previous studies of leO energy-level spectrum suggesting the importance of configura­tions due to excitation of four particles from the closed core, and have demonstrated by a con­tinuum-shell-model calculation including such configurations that our model indeed possesses the degrees of freedom required to provide a use -ful microscopic description of the resonance

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V O L U M E 32, N U M B E R 16 P H Y S I C A L R E V I E W L E T T E R S 22 A P R I L 1974

structure at low energies. A novel feature of the model is that it allows the calculation of both bound and continuum states from the same forces. In the light of the demonstrated sensitivity of the resonance structure to the residual interaction and to the dimensionality of the shell-model space, the choice of these components of the model deserves more detailed investigation. We are currently considering these questions and are also investigating the physical content of the resulting wave functions.

We gratefully acknowledge the use of the Oak Ridge National Laboratory-University of Roches­ter shell-model code14 and a structure-coeffi­cient tape kindly prepared for us by Edith Hal-bert .

t W o r k supported in pa r t by the National Science Foundation under Gran t s No. NSF-GP-15855 , No. N S F -

The reactions

K-p-K°n, (1)

K~d-K°nn, (2)

K+d-K°pp (3)

were observed with the Argonne effective-mass spectrometer for If N 1.2 GeV2 at 3, 4, and 6 GeV/e. The spectrometer, consisting of a large dipole magnet surrounded by magnetostrictive wire spark chambers, has been described pre­viously.1 Only the incident beam particle and the pions from the K° decay were detected.

This experiment is the first detailed study of the relative energy dependence of the two charge-exchange reactions, with data at three energies for both K" and K+ reactions. The numbers of events are shown in Table I. The data sample for

GJ-367 , and No. NSF-GU-2612. 1 C. H. Johnson, Phys . Rev. C 1_, 561 (1973). 2C. B . Dover and N. Van Giai , Nucl. P h y s . A177, 559

(1971). 3A. D. MacKel la r , J . F . Reading, and A. K. Kerman ,

P h y s . Rev. C 3 , 460 (1971). 4R. J . Phi lpot t , Phys . Rev. C _5, 1457 (1972). 5 P . F e r e n m a n , in Cargese Lectures in Physics,

edited by M. Jean (Gordon and Breach , New York, 1969), Vol. 3 , and r e f e rences the re in .

6A. P . Zuker , B. Buck, and J . B. McGrory , P h y s . Rev. Let t . 2 1 , 39 (1968).

7I . Unna and I. T a l m i , Phys . Rev. 112, 452 (1958). 8R. J . Phi lpot t , Phys . Rev. C _7, 869 (1973). 9R. J . Phi lpot t , Nucl. P h y s . A208, 236 (1973).

10A. P . Zuker , P h y s . Rev. Let t . 23 , 983 (1969). n W . W. T r u e , Phys . Rev. ^30 , 1530 (1963). 12H. Cranne l , Phys . Rev. 148, 1107 (1966). 13I. Sick and J . S. McCarthy, Nucl. P h y s . A150, 631

(1970). 1 4J. B . F r e n c h , E . C. Halber t , J . B . McGrory , and

S. S. M. Wong, in Advances in Nuclear Physics, edited by M. Bar anger and E . Vogt (P lenum, New York, 1969).

Reaction (3) contains more than twice the num­ber of events collected by all previous experi­ments above 2 GeV/c.

Reactions (2) and (3) are closely related by line reversal, with differences between them due to interference of amplitudes with opposite C parity in the t channel. The dominant exchanges are thought to be the p and A2 trajectories, having opposite C parity. If these trajectories are de­generate, as suggested by the p and A2 masses and by duality arguments, the amplitudes would be 90° out of phase, and the two cross sections would be identical.2 Our data show that this sim­ple model is inadequate since the K+ cross sec­tions are consistently higher than the K" cross sections.

The basic trigger for the experiment was an incident K*, an interaction in the 20-in. target,

Systematic Study of K** Charge Exchange from 3 to 6 GeV/c*

R. Diebold, D. S. Ayres, A. F. Greene,t S. L. Kramer, A. J. Pawlicki, and A. B. Wicklund Argonne National Laboratory, Argonne, Illinois 60439

(Received 19 F e b r u a r y 1974)

Differential c r o s s sec t ions for K~p^K°n and K+n-* K°p have been m e a s u r e d at 3 , 4 , and 6 GeV/c using a data sample of 6000 events . Cont ra ry to s imple exchange -degene r ­ate m o d e l s , the r a t io of K+ to A" c r o s s sec t ions was found to be approximate ly 1.35, with l i t t le dependence on e i ther s o r t. Both r eac t ions show a shallow dip near the f o r ­ward d i rec t ion , suggest ing the impor tance of spin-fl ip ampl i tudes .

904