Scattering of 25.6 MeV protons on 94Mo, 96Mo and 100Mo

Nuclear Physics A468 (1987) 247-284 North-Holland, Amsterdam

SCA’ITERING

OF 25.6 MeV PROTONS ON 94Mo, %Mo AND ‘“MO

E. FRETWURST,

G. LINDSTRGM,

K.F. VON REDEN,

V. RIECH

I. Znstitut fir Experimentalphysik, Universitiit Hamburg, W Germany S.I. VASILJEV,

P.P. ZARUBIN,

O.M. KNYAZKOV,

I.N. KUCHTINA

NIIF, Leningrad State University, USSR Received (Revised

15 December 1986 10 February 1987)

Abstract: Elastic and inelastic scattering cross sections of 25.6 MeV protons on the isotopes 94*96*‘~M~ were measured. Level energies and J” values were deduced using DWBA and coupled channels analysis. CC calculations on vibrational quadrupole one-phonon and two-phonon states prove to fit the experimental cross sections well in the anharmonic approach to first and second order. Quadrupole-octupole two-phonon states could be identified by harmonic vibrational model CC calculations. A semi-microscopic model was applied to some of the collective states, resulting in a satisfactory description of the cross sections.

E

NUCLEAR REACTIONS 94,96*100M~(p, p’), E = 25.6 MeV; measured cr( E,,, 0); deduced 94,96*‘~M~ deduced levels, J, ?r, deformation parameters. Enriched targets. model parameters. Optical model, DWBA coupled-channels analyses, macroscopic vibrational model, semimicroscopic approach.

1. Introduction

The even-mass MO isotopes with A = 92 to 106 represent a transitional region of nuclei from nearly spherical shape to static deformation. This is exhibited by analysing their level structure as measured by inelastic scattering of different projectiles and nuclear reactions as well as by the investigation of the radioactive decay of Nb and Tc isotopes, summarized in the compilations of Lederer et al. ‘) and Miillerzs3). Spectra of excited states of the isotopes 92Mo to “‘MO are typical of vibrational collective modes, whereas the unstable nuclei 1049’06M~ are more likely symmetric rotators “). In both cases collective structures are mixed with a significant amount of single-particled states. Thus the description of the level structure of these nuclei tends to be rather difficult. However, the excitation energies of states up to about 2.5 MeV in the 93*94,95,96M~ isotopes have been reproduced with fairly good success on the basis of the nuclear shell model, starting from the inert core *‘Sr (see Kumar and Bansal ‘)). On the other hand the level structure of 98,1WM~was far from as well fitted. Between calculated and measured B(E2) and B(M1) values for transitions between low-lying states of these MO nuclei considerable discrepancies are to be observed ‘). Using 03759474/87/%03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

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E. Fretwurst et al. / Scattering

of 25.6 MeVprotons

collective potential energy surfaces in the coordinates of the quadrupole deformation of even-even nuclei Gneuss and Greiner “) succeeded in calculating the collective level structure of g6Mo and ‘*MO to some degree. Significantly better results were achieved for g6,g8V’00M~ by Sambataro and Molnar ‘) applying the neutron-proton interacting boson approximation (IBA2) with the suggestion that the collective excitations are built up of the same neutron-boson configuration and of a mixture of two different proton-boson configurations (see also ref. “)). Cross section calculations of inelastic scattering to collective vibrational states of even MO nuclei were based so far on the macroscopic model of nuclear shape oscillations ‘). Several previous coupled-channels (CC) calculations on inelastic scattering data l”-i3) performed in this way succeeded in reproducing experimental data in the case of the one-phonon quadrupole- and octupole-states. In order to fit the quadrupole two-phonon triplet states exhibiting strong anharmonicity effects, the inclusion of one-phonon admixtures 14) was shown to be necessary. Recently some encouraging results for low-lying collective states have been obtained by using the IBA reduced transition matrix elements 12,15). Recently microscopic models describing the interaction of low-energy nucleons with nuclei (see e.g. ref. ‘“)) have been developed. In these models, the form factors of inelastic transitions are calculated on the basis of the effective nucleon-nucleon interaction and the nuclear matter distribution within the target nucleus. Among different existing approaches the folding model turns out to be most simple as well as most universal. It can be applied to the description of elastic and inelastic scattering on spherical and deformed nuclei using nucleons and composite particles as projectiles. Allowing for many-particle and exchange nucleon-nucleon correlations, the generalized folding model permits the density dependence of nucleonnucleon forces and the Pauli principle to be taken into account 16). Georgiev and Mackintosh 17) calculated the optical-model potential (OMP) and the form factors for inelastic transitions using a local approximation of the density matrix. However, one has to deal with the following difficulties: First, including the contributions of exchange forces to the OMP and the inelastic transition form factors requires a lengthy iteration procedure. Secondly, the formalism is not applicable to nucleon scattering on vibrational nuclei. Both difficulties could be overcome by the development of a semi-microscopic method (SMM) ‘*,i’), by which the OMP and the inelastic transition form factors can be calculated in a straightforward manner. In its improved version it includes the nucleon-phonon interaction ‘“). In our investigation of proton scattering on some even isotopes of molybdenum we pursued the following goals: The macroscopic model of nuclear shape oscillations predicts in addition to the prominent one-phonon quadrupole (QlP) and octupole excitations (OlP) and the less intensive quadrupole two-phonon (Q2P) excitations, transitions to the hexadecapole one-phonon (HlP), quadrupole three-phonon (Q3P) as well as two-phonon states coupled from the one-phonon quadrupole, octupole and hexadecapole states. With detectable intensity in proton scattering experiments

E. Fretwurst et al. / Stuttering

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249

we expect to see the hexadecapole one-phonon and the quadrupole-octupole twophonon (Q02P) states. Thus our measurements were aimed at obtaining more precise deformation parameters for the quadrupole and octupole states with the additional attempt to identify quadrupole-octupole two-phonon and hexadecapole one-phonon states. With respect to the development of the semi-microscopic model to describe collective excitations induced by nucleon-scattering, as mentioned above 20), the present investigations were also performed to test the validity and the success of this approach. Another purpose of the present investigation was to obtain additional spectroscopic information on the excited levels of the MO isotopes as well as their excitation energies, J” values and cross sections. A further aspect, which will be discussed in a different context elsewhere 21), is concerned with the parameters of the optical model potential for nuclei around A = 100. Here the anomalous mass- and energy-dependence of elastic and inelastic proton scattering cross sections in the energy range between 15 and 45 MeV for nuclei with A s 70 [refs. 22,23)],the anomalous behaviour of the optical model potential parameters, especially of its imaginary part 22924-26) are of particular interest. A coupled-channels analysis of the inelastic scattering data of a series of low-lying collective states, utilizing nuclear structure calculations with IBA and following the lines of ref. 27), is in progress now and will be reported separately. The investigation of high-lying excited states as in the case of the search for possible QO2P states requires high-resolution scattering experiments. For the comparison of the success of different model descriptions a high reliability of the measured absolute cross sections is needed. Consequently, our proton scattering experiments on the isotopes 94996*100M~ were performed with the highest possible spectral energy resolution and most attention was focussed on the accuracy in the absolute cross section determination. The following sect. 2 deals with the experimental method and the evaluation of the results. A compilation of the excited states, observed in the 94*96*‘00Mo-nuclei, including J” values and integrated cross sections is presented in sect. 3. Sects. 4 and 5 are devoted to the coupled channels analysis of the scattering cross sections using the macroscopic and the semi-microscopic models. Preliminary results have already been published elsewhere 28-31).

2. Experimental method and results Measurements of the elastic and inelastic proton scattering cross sections of 94Mo, 96Mo and looMO were performed at the isochronous cyclotron of the University of Hamburg. The analyzed proton beam with an energy of 25.6 MeV and an energy spread of AE = 10 keV was used throughout the experiments. Target thicknesses were determined both by weighing and by measurement of the energy loss of

250

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of 25.6 MeVprotons

cu-particles. This latter procedure provided additional data on the homogeneity of the target layers, which was in the order of *OS%/mm (see table 1). Metallic, self-supporting targets with a high degree of homogeneity were employed 32). Angular distributions of the scattered protons were measured with two silicon semiconductor detectors mounted on a coolable turntable in the angular range between 7.5” and 168” in steps of 2.5” to Y, resulting in about 50 different spectra for each isotope. Monitor detectors were used at *25”. The total experimental energy resolution including beam width, target contribution and detector performance amounted to about 20 keV. TABLE 1

Isotopic enrichment and thicknesses of the MO targets

Isotope

“MO %Mo lwMo

Target thickness (pg/cm*) 670 390 590

Target homogeneity (%/mm)

Isotopic enrichment (%I

0.2 0.7

88.4 96.7 93.8

In each case the absolute energy calibration was carried out using the peak positions of the well-known quadrupole and octupole one-phonon states as well as the ground state. Peak positions and the areas of all peaks were determined by the peak-search and -fit code SPECIFIT 33). Taking the possible uncertainties involved in the calibration procedure into account, an overall mean error of less than *5 keV can be attributed to the measured excitation energies even in the least accurate case. One of the main contributions to the inaccuracy of the absolute cross sections is given by the uncertainty in our knowledge of the actual target thickness. However, as will be pointed out in sect. 4.1, this difficulty could be overcome by extracting a target thickness normalization factor from the optical model fit of the elastic scattering in the very forward region. This way the absolute integrated cross sections for the elastic scattering were fixed to about f 1% . Error bars given to single points in the angular distributions result from the peak-integrating procedure and are related to statistics (see ref. “)). Due to small target contaminations, which could not explicitly be accounted for, as well as due to uncertainties in the background determination, point-to-point fluctuations of the cross sections may occur, exceeding these statistical errors. This is especially important in cases of small absolute cross sections and at high excitation energies. 3. Excited states of the 949g4100Mo-isotopes The most recent compilations of the spectroscopic data of the MO nuclei by Lederer et al. ‘) for looMo and by Miiller for 94M~ [ref. ‘)I and 96Mo [ref. ‘)I cover

E. Fretwurst et al. / Scattering of 25.6 MeVprotons

251

publications up to 1978, 1985 and 1982 respectively. They include e.g. the (p, p’)measurements of Lutz et al. lo), Awaya et al. 13) and Burger et al. 34) and in the case of 94Mo also the (t, p) data of Flynn et al. 35). In the meantime Molnar et al. “) reported results of (n, n’r) investigations on rmMo, presenting a large number of precisely measured excited states. In our own (p, p’) measurements more than 35 excited states could be identified MO. The angular distributions of all states were in each of the target nuclei 94*96*100 compared to theoretical cross section calculations: The presumably pure collective states were fitted by using the coupled channels code ECIS-79 36) (see sect. 4); the other, including single-particle states were compared to DWBA calculations, using the code DWUCK 37). Throughout this work we indicate the quality of the description of the experimental angular distributions, each consisting of N data points, by the theoretical calculations using the following goodness-of-fit parameter g:

We replaced the traditional x2 value by g to present a more instructive measure of the deviation of the theoretical curves from the experimental data. In the case of an ideal fit the g-value should approach the mean relative experimental uncertainty. A reliable approximation of the experimental integrated cross section was derived from the theoretical best-fit calculations in all cases, which led to a satisfactory description of the angular distributions. The results of the present investigation are summarized in the following tables 2, 3 and 4 showing the excitation energies, the most probable J” assignments and the integrated cross sections in the first three columns respectively. For comparison the tables show the corresponding data from the most recently published compilations. For the majority of the excited states, excellent agreement can be observed between the present investigations and results of the other quoted publications. As an exception, the levels in lmMo, quoted by Lederer et al. ‘) and taken from Awaya et al. 13) do not fit into this picture. This discrepancy is obviously due to a wrong energy calibration in the Awaya experiment, caused by an inaccurate value of the energy of the first 3- state. The problem can be solved by a linear correction. Resealing the excitation energy values reported in the Awaya paper yields a surprisingly good agreement with energies and J” values from all other sources 30). The procedure which was used to determine the J” values of the excited states, is illustrated in fig.1. Depending on the transferred L-value, the quality of the corresponding description of the measured angular distribution by the DWBA calculation as given by the goodness-of-fit parameter g leads to the most probable J” assignment. The absolute value of the quality of the corresponding best description, g = 37% in the present case, represents the order of magnitude of the mean experimental errors, as discussed at the end of the previous section.

252

E. Fretwurst et all / Scattering TABLE

of25.6 MeVprofons

2

Excited states of 94Mo Present investigation E, (keV) 0

Miiller 2,

Present investigation

J”

CTint (mb)

E, (keV)

I”

i(keV)

0”

1384 13.6 2.18 0.16 0.31 1.17

0 871.087 1573.72 1742.5 1864.27 2067.45

0’ 2+ 4+ 0+ 2’ 2+

3389

2295.2 2393.21 2423.46 2533.9 2566.8

4+ 2+ 6+ 34+

3643 3702

2+

3796

5-

3852

4+,2+

3906 3990 4005

(2’)

4095

4+ 2+

871 1573 1741 1864 2066 2121 2296 2390 2421 2534

2+ 4+ 0+ 2+ 2+ (2+, 1-j 4” 3-

0.39 14.6

2610

(4+, 5-j

1.67

2611.5 2739.81

(W (1,2)+

2768

4”, .5-

1.10

2768.1 2805.8 2835.9

(4)+ 2’, 3+ (3,4, S)-

2868

2+

0.87

2960 3012

(4+, 4-j (2+)

0.21 0.88

2870.2 2872.38 2955.85 2965.1 3011.6 3083.6

(2+) 6(+f (8+) 2+, 3+ 3-

3129.2 3163.1

(1,2)+

4138

3165.78 3171 3204 3263.8 3308.1 3320 3320.9 3339.56

(6+) 2+,3+ 4+ l-

4190

3359.77 3367.4 33751

(8+) (7-1 (5-1

3166

4+

0.26

3202 3262

1-, 2+ 2+, 2-

0.24 0.17

3368

J”

3456

3-, 4’

3532 3601

4-

4320

Uinc(mb)

0.6

0.43

0.35

0.85

(3-, 4”)

0+ 6+ 4602

Miiller ‘f E, (keV)

J”

3400.9 3448.7 3462

(53)

3512.6 3534

u,2+1 2+

(2+)

3620 3602 (:-I 3650 + 3700 0+ 3714 3793.1 2+ 3800 33805.2 3845 4+ 3867.2? 3892.7 (1,2+) 3897.3 3995? 2+ 4008.1 4032 4079 40953 4120 4140? 4174 4190.2 4223 4293 4319 4388 4436

(s) 2+ 2*, 32+ 6+

4475 4495.7 4565 4599 4636 4729

@+I

4+

4. Coupled cbaoaels analysis (macroscopic model) 4.1. ELASTIC SCATTERING

The first step of the analysis of the measured differential proton scattering cross sections consisted in the optical model fit to the elastic scattering data. It provided

E. Fretwurst et al. / Scar&ring of 25.6 MeVprotons

253

TABLE 3 Excited states of “MO Present investigation E, WV)

J”

Miiller 3, uint W)

0

0+

778 1148 1498

2+ 0+ 2+

1412 19.9 0.21 1.19

1627 1870

4+ 4+

(1.49) 1.74

2094 2234 2432

2481

32+

4+

16.6 0.44

1.53

2542 2625

4+

0.92

2734

(5_)

1.51

2807 2875

(3_) (4+, 6+)

1.11 0.27

2981 3020

1-, 2+ 4+

0.49 2.28

3088 3140 3182

3235 3287 3342

3430

(4-9 5-)

0.24

3-

1.22

(2-Y 4+) (3-9 4+)

0.3 0.22

4+

E, WV) 0 778.213 1147.86 1497.73 1625.88 1628.10 1869.53 1978.38 2095.56 2219.30 2234.51 2426.07 2438.39 2440.64 2480.98 2501.50 2540.4 2594.28 2624.3 2700.0 2734.43 2735.60 2754.93 2786.90 2790.24 2818.36 2875.37 2975.14 2978.84 2986.91 3024.60 3053.23 3087.3 3133.69 3179.0 3186.72 3202.73 3283.7 3334.97 3369.95 3416.64

0.89 3441.59 3444.7

Flynn et al. 35) J”

E, &eV)

J”

0+

0

2+ 0+ 2+

777 1146 1497 1626

2+ 0+ 2+ 2++4+

1866

4+

2228 2421

32+

2476

4+

2536

2+

2621

4+

2755

(5_)

2819 2875

4+

3020

5-

3184

4+

3241 3281

4+ 2+ 4+ 2+ 54+

(2)+ 4+ 4+

0+

(& (4)+ 3(5+) 6+ (2’, 3) (1) (3+) (3+)

(4+, 5+) (3,4+) 6+

7+ 5+ 8+

(4+, 5+)

(4+) (4+, 5+)

(8+) (4+)

3375 3418 3434

254

of 25.6MeVprotons

E. Fretwurst et al. / Scattering

3-continued

TABLE

Present investigation

Miiller 3,

E, &eV)

J”

3414 3549 3591

2+, 4+, 532+

0.29 0.25

3694

5-

0.52

3736

4+, (3-)

0.48

st

bb)

Flynn et al. 35)

E, 0-V)

J”

3473.0 3551.3

394

3781.4

(lo+)

E, WV) 3413 3556 3593 3646 3683 3709 3137

J” (2+) 5+

4+

3800 3847 0.39

(4+)

3867

(5-)

3916.1 3965 4038 4215 4280 4469

(4+) (3-) t4+, 3-) (3-)

0.44

3959

0.34 0.5 0.57

4080 4205

4+ 4+

4471 4533.4 4584.3

4595

4603 4714

both the absolute cross section normalization and a set of optical model (OMP) parameters to start the CC calculations. In those fit calculations the standard version of the optical model potential

U(r) = - wxr) - Vs.o.A2,b *0 -i

(

W,b(r)+4awWl&(r)

Ze’/ r ,

Uc(r)=

1

(Ze2/2Rc)

fx(r)=[1+exp{(r-r~“3)/a,}]-1.

potential we used

$A&, +Wr) >

,

r Z= Rc =

(3-(r/R,)2),

1-

rcA1"

r
The absolute cross section normalization, which was necessary because of the uncertainty in the target thickness measurements (see sect. 2) was carried out with emphasis on the very forward angular range (0~30”). Here the theoretical cross sections do not depend sensitively on the optical model parameters, hence a normalization could be achieved independently of the final fit parameters, which are listed in table 5. Fig. 2 shows the corresponding best fits to the experimental data.

E. Fretwurst et al. / Scattering of256

255

MeV protons

TABLE 4 Excited states of ‘O”Mo Lederer et al. ‘)

Present investigation E, (keV)

J”

0 535 694 1063 1135 1463

0+

(1605) 1770

@+I (2’)

1845 1908

3-

2032 2046 (2070) 2101 2158 (2192) 2202 2288 2342 2371 2386 2418

2+

0' 2+ 4+ 2+

ai,,

J”

E, (keV)

J”

0

0+

2+ 0” 2+ 4+

1.463 1.503 1.605 1.766 1.770

(2+)

0 535 694 1063 1136 1464 1502

0+

0.5356 0.6944 1.0637 1.1361

1.9081 1.976 2.033

3-

1907

3-, 4+

0+

2035

0+

I?, (MeV)

1526 29.0 0.33 2.20 1.49 0.3

13.9

2+

Flynn et al. 35)

2.040 2.086 2.101

(3,5) 4+ 2+

0.59 0.64

2.20 “)

(S,

0.5 0.7 0.94 1.53

2.34 “)

2+ 0+ 2+ 4+ 2+ 0’

(2+) 2082 2102

(&

Molnar et al. 4, I?, (keV) 0

535.547 695.10 1063.78 1135.94 1463.89 1504.61 1607.36 1766.48 1771.38 1908.21 1977.34 2038.01 2042.76 2086.78 2103.2

J” 0+

2+ 0+ 2+ 4+ 4+ + (2! 31

3(0+) (1-j (4+)

2186

2+ :I, 3-

1.2

2+ 3-

2.4156

(3-1

2.42 “) 2.47

(S’

(5-t

1.52 0.71 1.23 0.40 0.43 0.9 1.0

2.83 “)

4+

2931

(3-j

0.8

2.92 “)

5-

3000 3050

3.5-

0.6 0.95

4” 4+

2516 2565 2610 2658 2740 2812 2838

(4,5) 2+, 42+

2.5632 2.59 “) 2.67 “) 2.72 “)

3.03 “)

4+

(4+)

2281 2312 2334 2364 2392 2413

2518 2561 2602 2652 2733 2803 2835 2873 2923 2968 2994 3039 3065 3106

2+

2201.1 2286.5

(S, 2+, 3-

2201.1 2369.6 (2397.0) 2416.8

5+, 62+ 2+

2564.1 2580.9 2662.6 2738.0

(4+) (4+)

2822.2

(4+)

4+, .52+

(2961.2) (2996.3) 3042.2 3053.7

(2+)

(3-1

(2+) C?+c)

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E. Fretwurst et al. / Scattering

of 25.6 MeVpratons

TABLE Aontinued Present investigation E, (keV)

J”

3122 3169 3246

52+

3296

(3_)

criinr(mb)

Lederer et a& ‘) E, (MeV)

J”

0.37

3365

0.5

Flynn et al. 35) E, (keV) 3119 3148 3235 3263 3282 3306 3354 3409

J”

Molnar et al. ‘) E, (keV)

J”

2+

3421

2+

3483

2+

3.552 360.5 3666

(3,5) (3,5) 5-

3587 3557

(:)

3652 3674

596

(395)

3771

596

3445 3475 3535

3745 3791

“) Energy values obviously adopted from ref. 13).

The volume integrals of the real part &/A and the imaginary part &‘A of the OMP show a close correspondence to the respective values calculated with the parameters of Becchetti and Greenlees 38) (see ref. “‘)).

4.2. INELASTIC

SCATTERING

4.2.1. Quad~~ole-vj~rationaZ states. The coupled channels calculations were performed with the code ECIS-79 36), using the harmonic and the anharmonic vibrational model (HVM and AVM) both in first and second order. The first order HVM resulted in a very satisfactory description of the one-phonon states, coupled to the ground state: (gs-2:-3;) for the three MO isotopes (fig. 3). The OMP parameters, resulting as best-fit values in these searches are compiled in table 6. All fit calculations were performed by simultaneously searching for the depths, the radii and the diffusenesses of the real and the surface-imaginary parts of the potential as well as for the deformation parameters. Vs.o., rs.o., a~~., WO and rc were set to fixed values and the geometrical parameters of the volume and surface part of the imagina~ potential were kept identical during the search. It should be emphasized that we did not use different imaginary potentials within one coupledchannels calculation for the description of the cross sections to different excited states. Comparing the potential parameters from the elastic scattering fits to those

E. Fretwurst et al. / Scattering of 25.6 MeVprotons

o

1

z

3

4

5

251

6

7

8

TransferredL-Value --->

a.010

0.001 a.0

50.0

m0.a

150.0

CM-Angle

--->

Fig. 1. 96Mo(p, p’); E, = 3694 keV. (a) L dependence of the quality g(%) of the DWBA fit. (b) Measured angular distribution with L = 5 DWBA calculation.

of the three-state calculations we observe, according to expectation, the reduction of the depth of the imaginary potential as a consequence of increasing the number of coupled states (see tables 5 and 6). Natural relative weights of the coupled states in our CC calculations are given by the experimental errors. By arbitrarily changing these weights the fits to the individual angular distributions could be influenced. This possibility was not used throughout our calculations, instead we stuck to the natural weights, mentioned above. Consequently, the fits to the inelastic scattering cross sections are in general slightly poorer than the ground state fits. An attempt at a fit including the two-phonon triplet states by HVM in first or in second order was not successful. The calculated cross sections for some of the

258

E. Fretwurst et al. / Scattering of 25.6 MeVprotons TABLE 5 Best-fit OMP parameters

Parameter

V. [Mevl r0 [fml a0 WI 12,1/A [MeV . fm3] try) lfml WD WV1 rw lfml a, [fml 11,//A [MeV. fm3] (rw) lfml

from the elastic scattering analysis

cross section

9“Mo

=Mo

looMo

51.9 1.17 0.69 407 1.07 9.34 1.28 0.64 114 1.39

50.1 1.19 0.70 416 1.08 9.52 1.26 0.64 113 1.38

51.7 1.18 0.72 413 1.08 10.08 1.23 0.73 129 1.38

Remarks: W, = 0.25 MeV, V,,,, = 6.6 MeV, 0.85 fm, r, = 1.25 fm, Iv =I V(r) d3r,

rs.o. = 1.0 fm,

a,,,, =

(r,)=Jj V(r)? d3r/j V(r) d3r A-“3. Zw and (rw) are defined similarly.

two-phonon triplet states showed up to be too small compared to the experimental data. Therefore in a next step we used the anharmonic vibrational model (AVM) to reproduce the cross sections of the coupled level system (gs-2:-Ol-2:-4:-3;). Following Tamura 14) the observed anharmonicities were accounted for by assuming the triplet states of spin I to be composed of the pure harmonic two-phonon amplitude and a fictitious

one-phonon

amplitude: II)=sin

‘~&h)+cos

Y~I~wIJ.

The relative contribution of these two amplitudes yz. The deformations are written PZZ = Po2 sin

is determined

YI,

Po”z= Pz cos Yz*

(3) by the mixing

angle

(4) (5)

parameters /IO, p2, p4, Thus in the AVM p2r is different from po2. Six additional yo, y2 and y_, are to be considered in this approximation. In this way, however, it was possible to fit the experimental angular distributions of the above mentioned six coupled states satisfactorily. The results are shown in figs. 4, 5 and 6 and the values of the best-fit parameters are listed in table 7. It should be pointed out that the inclusion of the quadrupole two-phonon states does not yield worse fits to the ground state and the one-phonon state. Though the general fit to all states is satisfactory, some deviations with respect to the phase of the angular distributions are still to be observed (see e.g. the 0: state of 96Mo at 1148 keV (fig. 5) and the 2: state of looMo at 1063 keV (fig. 6)). Due to statistics, the quality of the fits of the

E. Fretwurst et al. / Scatteting

259

of 25.6 MeVprotons

100 WP.Pd

1.0

~

i i

0.0

50.0

100.0

150.0

a-Angle ----> Fig. 2. Elastic scattering cross sections of 25.6 MeV protons on g4*96*‘00Mo.Full lines: optical model fits.

94Mo data is generally worse than for the other isotopes. But nevertheless the deviations of the 4: state data exceed the statistical errors and are not explained so far. Concerning the 1627 keV state of 96Mo the poor 4: fit can be attributed to the experimentally non-resolved cont~butions of the 2’ state at 1626 keV [ref. ‘)I. Table 8 shows the quality of the fits for all sets of searches, measured by the goodness-of-fit parameter g (eq. (1)). Deviations between the experimental data and the fits are of the order of 10 to 30%, remarkably larger only for the 0: state.

E. Fretwurst et al. / Scattering of25.6 MeVprotons

260

870

0.0

50.0

100.0

keV - 2+

150.0 CM Angle

-----,

Fig. 3a. Measured differential cross sections of 94Mo(p, p’) with coupled channels fits on the level system (gs-2:-3;) (fuli lines).

In addition we tried still to reach a further improvement by performing second order AVM searches for the level system (gs-2f-O:-2f-4:). The resulting parameters are included in table 7. Considering the large amount of additional transition matrix elements in second order calculations, changes due to interference in the structure of the calculated angular distributions are to be expected. Improved fits can be observed for all 2: states. In some other cases, however, the first order fits are closer to the experimental data: 0z(94Mo), O~(‘ooMo) and 4:(“‘Mo).

261

E. Fretwurst et al. / Scattering of 25.6 MeVprotons

k”“!““l”““‘$

1.0

h

t&J

\ 778

2235

0 .0

50.0

keV - 2+

keV -

100.0

3-

150.0

CM Angle ---->

Fig. 3b. As fig. 3a, but for %Mo(p, p’).

The deformabilities poz and po3, evaluated in the HVM approximation coincide within <*5% with the corresponding values from AVM calculations. Table 9 presents all P-parameters resulting from first-order AVM searches. For comparison the table also shows the results of previous investigations 1oP1*S13), but only global agreement can be observed. The one-phonon amplitudes, which account for the anharmonicity of the two-phonon triplet states, are largest for the 4: states, while the 2: states show up to be almost purely harmonic.

262

E. Fretwurst et al. / Scattering of 25.6 MeV protons

1.0

0.0

50.0

100.0

150.0

CM Angle

---->

Fig. 3~. As fig. 3a, but for ‘O”Mo(p, p’).

42.2. Hexudecapole-uibrutionaI states. In addition to the quadrupole and octupole one-phonon states we looked for highly excited 4+ states to be identified as onephonon hexadecapole excitations. In 94M~ most probable candidates are the states at 2421 keV and at 2768 keV. Fig. 7 shows the DWBA calculations confirming the J” = 4+ assignment. Coupled channels calculations on the level system (gs-4+), using the vibrational model in first order, taking the parameters of the OMP from table 6 and varying only the parameters of the imaginary part and PO4are also shown in fig. 7. Similar calculations were performed for the states at 1870 keV, 2481 keV and

E. Fretwurst et al. / Scattering

of 2.5.6 ~evp~otons

263

TABLE 6 List of parameters from a CC analysis of the level system (gs-2:-3;) in the first order harmonic vibrational model Parameter V, [Meal r.

Pm1

a0 lfml I&,//A [MeV . fm’] (G) Efml WD EMeVl rw Km1 aw lfml /1,1/A [MeV. fm3] (rw) bl

“MO

%Mo

‘@@MO

51.9 1.17

50.1 1.20

50.7

0.69

0.70

0.73

407 1.07 7.87 1.29 0.67 103 1.41 0.129 0.162

422 1.09 8.18 1.27 0.65 101 1.39 0.150 0.167

425 1.09 8.84 1.26 0.70 112 1.39 0.1% 0.166

1.20

Remarks: see table 5.

3020 keV in 96Mo and for the 2104 keV state in ‘OoMo. In order to obtain the appropriate values of the deformation parameters needed for the semi-microscopic coupled-channels calculations to be presented in the following section, the 4: states, though obviously not being single-step excitations, were also calculated this way. The results of all fits are summarized in table 10. 4.2.3. Quadrupole-octupole two-phonon states. Among many further predictions of the vibrational model a quintuplet of negative-parity states l-, 2-, 3-, 4- and S composed of the coupling ‘of the strong 2f and 3; states, should be most probable. These levels are to be expected at excitation energies around the sum of the energies of the one-phonon quadrupole- and octupole-states, a region easily investigated by high-resolution (p, p’) measurements. In search for such states we used the following criteria: Coupled channels model-calculations with the coupling scheme:

have been performed utilizing the three-level-fit results documented in table 6. They provided a reliable guess for the angular distribution structures and for the order of magnitude of the absolute cross sections to be expected in case of the negativeparity quintuplet. By comparing these predictions to all measured angular distributions of appropriate excitation energy, the levels listed in table 11 were found which could possibly be explained as quadrupole-octupole two-phonon states in the three isotopes. This search failed in the case of the l- state of ‘ooh40. In order to prove this choice, we performed CC search calculations including the cross sections of these states, using the OMP parameters of table 6 and varying only the depth of the imaginary potential W,. It should be emphasized that we did not use any additional normalisation in these calculations. The results are shown in figs. 8, 9

I

1

1

.

I

I

t 1.0

b--+-7

gs -

o+

0

P

0

fi

. + -“-t,

870

1.0 k

L 0

keV - 2+

m

“mm

1573

Q

keV - 4+

m Q

w 1741

keV - 0+

*

3

*

*.

t

+t

t*

2534

0.0

50.0

100.0

keV - 3-

1

150.0 CM Angle

--->

to the ground state and the Fig. 4. Measured differential cross sections of 94Mo(p, p’) with transitions 2:, Oz, 2:, 4:, 3; excited states together with coupled channels fit calculations (full lines), using the anharmonic vibrational model to first order (AHVMl).

E. Fretwurst et

265

ni. i &after&g of25.6MeVpmtons

778

keV

-

2+

o+ 1.n

--=---A

0.0

50.0

2235

100.0

keV

-

3-

i

X50.0 CM Angle ----->

Fig. S. As fig. 4 for %Mo(p, p’).

i

E. Ftetwwst et al. / Scattering of 25.6Me~~~otons

266

535

keV

-

2+

694

keV

-

0+

1063

0.0

50.0

keV

-

2+

1135

keV

-

4+

1908

keV

-

3-

100.0

150.0 CH Angle

Fig. 6. As fig. 4 for *ooMo(p, p’).

---7

E. Fretwurst et al. / Scattering of25.6 MeVprotons T.~BLE

267

7

List of parameters from first and second order CC calculations including one- and two-phonon states using the anharmonic vibrational model g4Mo

Parameter

a0

lfml

[Z&A [MeV - fm”] (rv)

Km1

6, [MW rw lfml flw WI IZ&A (rw)

[MeV. fm3]

Cfml

lo%40

2nd order

1st order

2nd order

1st order

2nd order

51.8 1.17 0.69 408 1.07

51.9 1.17 0.65 401 1.05

50.4 1.19 0.71 420 1.09

50.8 1.19 0.66 411 1.07

50.8 1.19 0.73 424 1.09

51.8 1.18 0.70 412 1.07

7.73 1.29 0.68 102 1.41

8.70 1.27 0.43 104 1.38

8.18 1.27 0.66 102 1.39

9.04 1.25 0.62 103 1.36

8.60 1.26 0.71 112 1.40

9.34 1.24 0.68 113 1.37

1st

V, tMeV1 r. WI

“MO

order

0.131 0.161 -0.003 12 0.568 89 0.070 147

0.135 0.008 14.5 0.246 94 0.066 160

0.154 0.168 0.001 56 0.702 88 0.063 132

0.011 54

0.209

0.202 0.166

0.155

0.010

0.001 43

39 0.061

0.209

0.131

81

53

71

0.057 171

0.045 126

0.050 126

Remarks: see table 5.

and 10 and in table 11 and in some cases they clearly support our assumption, that the selected levels could be members of the quadrupole-octupole two-phonon quintuplet. (See e.g. the l- state of 94Mo and the 4- states of 96Mo and ‘ooMo.) In order to discriminate between single-step excitations and the assumed two-step processes, a detailed comparison of the different predictions for the angular distribution structures has to be carried out. Fig. 9 includes a direct-transition calculation of the 3- state (dashed line), showing that differences between the two interpretations are to be observed in the extreme forward and backward regions. Consequently a clarification of the excitation mechanism can possibly be obtained by further precision experiments on these high-lying states with emphasis on the extreme angular regions. 5. Coupled channels analysis (semi-microscopic

model)

As mentioned in the introduction CC calculations, based on a semi-microscopic model (SMM) were alternatively applied to describe the measured cross sections in some selected cases: In this approach the real part of the optical model potential and the inelastic transition formfactors were calculated using the following SMM

268

E. Fretwurst et al. / Scattering

of 25.6Me~~rotons

TABLE 8 Quality of fits from different CCA approaches, expressed by the goodness-of-fit parameter g (in %) Isotope ‘=Mo

96Mo

‘O”Mo

Ex (keV)

J”

I

II

III

IV

V

VI

0 871 1573 1741 1864 2534

0”

26.4

26.3 33.3

26.5 30.5 42.3 64.5 71.4 24.6

27.1 25.8 43.6 72.2 42.5

35.7 41.4 41.6

34.3 43.4

0 778 1148 1498 1627 2234

0+ 2+ 0” 2+ 4+ 3-

16.3

19.2 22.7 93.8 34.7 48.2 32.2

16.9 21.7 71.8 30.0 44.9

38.0 36.4

59.5 50.4

38.8

37.9 47.4

0 535 694 1063 1135 1908

0+ 2+ 0” 2+ 4+ 3-

12.3

13.1 13.6 42.8 35.4 20.3 19.3

12.9 14.2 60.8 23.9 28.5

37.9 31.3

31.7 23.2

2+ 4+ 0+ 2+ 3-

24.6 18.1 24.1

30.9 13.0 12.8

18.7

56.2

49.3 32.1

Remarks: I: OM-f& on elastic scattering cross sections. II: HVM (1st order) CCA on the level system {gs-2:-3;). III: AVM (1st order) CCA on the level system (gs-2:-Oz-2:-4:-3;). IV: AVM (2nd order) CCA on the level system (gs-2:-Og-2:-4:). V: SMM-CCA on the level system (gs-2:) resp. (gs-4:) (see sect. 5). VI: SMM-CCA on the level system fgs-2:-4:)

resp. (gs-2:-3;)

(see sect. 5).

formalism, while the standard optical model supplied spin-orbit part and the Coulomb part of the potential.

5.1. THE SEMI-MICROSCOPIC

the imaginary

part, the

MODEL

The real part of the optical model potential calculated following the procedure of ref. 39)and taking into account the non-additive contributions of the many-particle and exchange nucleon-nucleon correlations in the optical model potential and the inelastic transition formfactors:

E. Fretwurst et al. / Scattering of 25.6 MeVprotons

269

TABLE 9 Deformation parameters of the “*96*‘ooMo-isotopes, obtained by an AVM (1st order)-CC analysis Nucleus 94Mo

Transition gs+2: gs+3; 2:+0: 2:+2; 2:+4: gs+o: gs+2: gs+4:

96Mo

gs+2: gs+3; 2:+0; 2:+2; 2:+4: gs+o: gs+2: gs+4:

looMo

gs+2: gs+3; 2:+0; 2:+2; 2;+4; gs+o: gs+2; gs+4:

Present investigation

Ref. “)

Ref lo)

0.131 0.161

0.155 0.148

0.160 0.16

0.125 0.131 0.071

0.155 0.155

0.160 0.160

PO2 PO3 P20 P22 P24 Pi0

P”02 P”04

-0.001 0.013 -0.059

0.165 0.153

0.175 0.19

0.128 0.154 0.115

0.165 0.165 0.165

0.175 0.175 0.150

0.0003 0.024 -0.042

0.01 0.03 0.03

0 0.04 0.04

0.202 0.166

0.224 0.158

0.226 0.21

0.19 0.17

0.127 0.161 0.163

0.200 0.224 0.166

0.226 0.226 0.170

0.19 0.19 0.19

P20 P22 824

P;lz PL 802 PO3 P20 P22 P24 Pi0 P” 02 PL

-0.03 0.08

0.154 0.168

PO2 P 03

P” 00

-0.10 0.04

Ref 13)

0.008 0.037 -0.027

-0.03 0.06 -0.03

0 0.06 0.02

0 0.05 -0.01

The density matrix p(r, r’) was approximated by the first term of an expansion proposed by Negele and Vautherin 40). In contrast to the macroscopic model the SMM form factors of inelastic transitions f*(r) (first order) and momentum transfer:

FAlhz (second

fh(r)={l+I~~(~)K(r)}{U~030(r)+2dp~(r)p~o(r)+I,~(r)}+I,,(r)K(r)loo(r)

F&r)=

~~~,0(r)~~,0(r)+~~,,(r)K(r){~~20(r>+2~~o(r)~~,0(r)+~~,o(r)}.

order)

depend

significantly

on the angular

,

(7) (8)

In these equations U,“(r) and U,“, denote, respectively, the potential and the inelastic transition form factors, calculated using the folding model without many-particle and exchange nucleon-nucleon correlations. PO(r) is the nuclear matter density distribution, pAO(r) describe the transition densities. With (9)

270

E. Fretwrst

et aL f Scattering of 25.6 ~eVp~tons

T

F:’ “, i: t-.

“’

..\

\,

\\

_ _,_

*‘-

0.1

~

-..

+

, \

t

1.0

L _‘..,

\

c

.\ --. b ‘-.

-._,_.-'-

t

0.1

+*

2768

keV - 4t

t.

L

0.0

50.0

I

100.0

.

,

I

.

150.0 CM-Angle

---->

Fig. 7. “Mo(p, p’)-Measured differential cross sections of the transitions to the assumed hexadecapole states. Full lines: L = 4 DWBA calculations. Dashed lines: coupled channels calculations on the level system (gs-4:).

271

E. Fretwurst et al. / Scattering of 25.6 MeVprotons TABLE 10 List of parameters from a CC analysis of some 4+ states Isotope

rw (fd

aw (fm)

g(%)

& (kev)

WD (MeV)

“+Mo

1573 2421 2768

8.47 8.52 8.49

1.28 1.28 1.28

0.68 0.68 0.68

0.070 0.030 0.060

39 35 43

%Mo

1627 1870 2481 3020

9.15 9.16 9.16 9.14

1.27 1.27 1.21 1.21

0.66 0.66 0.66 0.66

0.059 0.049 0.053 0.069

45 59 30 32

“‘hilo

1135 2104

10.11 10.14

1.26 1.26

0.71 0.71

0.068 0.042

65 42

804

Remarks: see table 5.

and

E-

U,D(r)-dp:lr)-~*~oP:P:O(r)V&r) (10)

we write the exchange-integral

IAn(r) for h f 0:

andforA=O,n=Oandn#O:

(V,(r) denotes the Coulomb potential, phoo(r, s) the A-component of the density matrix, q&s) the exchange part of the effective interaction with s = Jr-r’/ and&(x) are the spherical Bessel functions.) The parameter d has been introduced - it is the only free parameter - to scale the density dependence of the nucleon-nucleon interaction. The nucleon-phonon interaction is treated in the same way as in the macroscopic vibrational model ‘). But the consideration of many-particle and exchange nucleonnucleon correlations leads to terms in the interaction, which are non-linear with respect to the phonon operators. Due to the Pauli principle the nucleon-phonon interaction depends on the energy of the incident nucleon. Nuclear structure information, which has been extracted from scattering data of electrons and r-mesons or can be calculated using nuclear models, is fed into the SMM formalism via the

E. Fretwurst et al. / Scattering of 25.4 MeVprotons

272

0.1

At

*

b

‘7%

4%Y’

3202

keV -

l-

3262

keV -

Z-

3456

keV -

3-

0.1

0.1

.

l

0.1

-I m

@,

0.01 P

0.1

-

Fig. 8. Measured angular distributions of the assumed quad~poIe-o~upole two-phonon states in 94Mo. Full lines: coupied channets calculations on the level system {gs-2:-3;-l--2--3--4--5-).

213

E. Fretwurst et al. / Scatteting of 25.6 MeVprotons

l

0.01

3287 keV - 2-

-2807 keV - 3-

0.1

0.01

50.0

100.0

150.0

e - ad --> Fig. 9. As fig. 8 for %Mo(p, p’). (For further explanation see text.)

274

E. Frerwurst et a,! / Scatiering of 25.6 MeV protons

2202

keV

-

2-

2658

keV - 4-

0.1

0.01

50.0

150.0

100.0 ad

Fig. 10. As fig. 8 for “‘M(p, p’).

-

Angle --->

275

E. Fretwurst et al. / Scattering of 25.6 MeVprotons

Quadruple-octupole two-phonon states: excitation energies, imag. potential depths and quality of fits E, (kev)

I”

94Mo

3202 3262 3456 3532 3796

12345-

7.83

142 99 102 60 82

g6Mo

2981 3287 2807 3088 2739

12345-

8.10

73 54 55 44 71

2202 2371 26.58 3050

12345-

8.78

43 54 44 63

Isotope

“‘MO

b

WV)

g&l

transition densities p,J r). Thus it is possible, to test microscopic nuclear structure models, such as the particle-phonon model 41), the random phase approximation, nuclear field theories 42) etc. In principle the formalism can be expanded to take into account the differences of proton- and neutron-dist~butions in nuclei and in the corresponding transition densities. Wence proton and neutron scattering on nuclei as well as charge exchange reactions can be described in a unique way. 5.2. COUPLED-CHANNELS

CALCULATIONS

AND RESULTS

above SMM formalism forms the basis of semi-microscopic coupled-channels calculations (SMM-CCA). Apart from the scaling factor d they do not include any free parameter. The real part of the OM potential and the inelastic transition formfactors are determined by the effective NN forces and by the properties of the nuclear structure. The parameters of the nuclear matter density Fermi dist~bution were taken from muonic X-ray transition data 43). Diflerences in the density distributions of protons and neutrons were not considered. For the transition densities we use The

ho-

dd r)

rdr.

Schmid-Wildermuth forces &), modified by including the density dependence were used as effective NN interaction. The value of the parameter d taken identical for all three isotopes, was essentially predetermined by an application of the SMM to

276

E. Fretwurst

et al / Scattering

of 2.5-6 MeVprotons

the elastic proton scattering on different nuclei 45,2QP3Q). The parameters of the imaginary part, the spin-orbit part and the Coulomb part of the OM potential were taken from the results of the macroscopic coupled-channels analysis (see sect. 4) and their values have been fixed in the course of the SMM-CCA calculations. The description of the cross sections of inelastic scattering on vibrational nuclei requires the knowledge of the dynamical deformation parameters. They may be different for protons and neutrons, used as projectiles and may depend on the type of target nucleons, with which the projectiles interact 46*47).In the present stage of our calculations we did not consider these differences. It would be meaningful only for the simultaneous description of proton- and neutron-scattering on the same nuclei due to the different sensitivity of the projectile to the proton- and neutrondistributions within the target nucleus. Therefore we used the best-fit values of p,+ from the MAC-CC analysis, as compiled in sect. 4. Our calculations using SMM-CCA were thus predetermined completely and did not contain any free parameter. This method was applied to several different level systems, consisting of the ground state and some of the collectively excited states in the MO isotopes. The results of these calculations are presented together with the experimental data in figs. 11 to 14, and will be discussed in detail below. The calculations of the elastic proton scattering in a two-channel appro~mation (gs-2:) and in a three-channel approximation (gs-2:-4:) are shown in figs. 11, 12 and 13. A good description of the elastic cross sections was achieved, using the same value of the parameter d for all isotopes. Moreover, this value agrees well with the d-value of ref. 39). A variation of this parameter by about 20% significantly improves the description of the angular dist~butions at large scattering angles “). Even in the case of the inelastic scattering exciting the 2: state, the cross sections are well reproduced by SMM-CCA. The calculations of the 4: state cross sections were performed, assuming these states to be pure one-phonon states. Owing to the neglected, but predominating two-phonon contributions to this level (see table 9) the assumption obviously turns out to be oversimplified. Two-channel (gs-4:) and three-channel (gs-2:-4:) approximations were employed, and only fair agreement with experimental data was achieved in both cases. In spite of the crude approximation for the 4: state excitation, the step from the two-channel ~lculation to the three-channel calculation apparently leads to better descriptions of the elastic scattering and the 2: state cross sections (see figs. l&l2 and 13). As will be discussed below, this observation may be explained by the fact, that especially the optical model parameters used could not be adjusted by search calculations. The SMM-CC calculations on the level system (gs-2:-4:) were performed to first as well as to second order. In second order minor improvements in the description of the structure of the angular distributions can be observed for some cases (see e.g. fig. 12). The inelastic scattering cross sections for the excitation of the octupole one-phonon 3; states were calculated in the three-channel approximation (gs-2:-3;). The agreement with the measured angular distribution is satisfactory.

2534

0.0

50.0

100.0

keV ‘- 3-

150.0 CM-Angle

---->

Fig. 11. Semi-microscopic coupled channels description of 94Mo(p, p’) differential cross sections. Ground site transition:coupling of (gs-2:) (full line); (gs-2:-4:), harmonic vibrational model first order (HVMI) (dashed tine); 2: state tnmsi~ion: coupling of (gs-2:) (full line); (gs-2:-3;) (dashed line); &s-2:-4:) (HVMI) (dashed-dotted line). 4: sin& transition: coupling of (gs4:f. 3; state transifion: coupling of (gs-2:-3 ;)*

5.

5. 2

1.0

7 8 % 3

:

h L

‘;:

a.1

-

-? %

B s

778 10.0

_

1.0

-

10.0

-

1.0

_

keV -

2+

Fig. 12. Semi-microscopic coupled channels description of 96Mo(p, p’) differential cross sections. Explanations as in fig. 11, but in addition: 2: stare ~rans~t~un:coupling of {gs-2:-4:), harmonic vibrational model to second order (HVM2) (dotted line); 4: stare ?r~~si~~~~:coupling of fgs-2:-4:), (HVMl) (dashed tine).

535

0.0

keV -

2+

1135

keV -

4+

1908

keV

3-

-

100.0

50.0

150.0 CM-Angle

Fig. 13. Semi-microscopic

coupled

channels

description

nations

of looMo(p, p’) differential as in fig. 11.

---->

cross sections.

Expla-

E. Fretwurst et al. / Scattering of 25.6 MeVprotons

280

Thus the SMM-CC calculations lead to an overall satisfactory description of the selected proton scattering cross sections including the strongest collective states (see table 8). Several improvements to the method are possible: In our calculations the parameters of the imaginary part of the OMP were taken from the MAC-CC analysis of the elastic and the inelastic scattering cross sections. The real part of the semi-microscopic optical model potential differs from the real part of the phenomenological OMP. Owing to the general theory of the optical model 48), real and imaginary parts of the potential are connected with each other. Therefore the phenomenological best-fit absorption parameters may not be the optimum values for the description of the scattering data in the semi-microscopic model. To illustrate this, we performed a SMM-CC calculation on the two-level system of 96M~, using parameters of the imaginary potential, which are different from those of the macroscopic analysis (see table 12). The results are shown in fig. 14. They demonstrate that the second set of absorption parameters leads to a substantial improvement in the description of the eleastic and inelastic angular distributions at large scattering angles and to a reduction of the associated x2; The g-values of the fits, as defined in sect. 3, decreased from 65% to 38% for the ground state and from 73% to 36% for the 2: state. (We did not perform a parameter search, since these calculations take much more time than in the case of the MAC-CC analysis.) Thus, improvements of the present SMM-CCA descriptions are to be expected by taking into account adjustments of the parameters for the imaginary part of the optical model potential and considering additionally the observed anharmonicities. Furthermore, the consideration of different proton- and neutron-distributions within the nucleus, the application of the Tassi mode149) or of microscopic calculations for the transition densities, and of the modified Slater approximation for the density matrix p(r, r’) should be essential for improvements of the SMM-CCA. It seems to be most promising to employ the isospin representation in the semi-microscopic model to include proton- and neutron-scattering as well as charge exchange reactions.

TABLE 12 Parameters of the imaginary optical potential, used in the semi-microscopic CC analysis Isotope

Level-system

94Mo “MO 96Mo “‘MO

(D-2:)

94Mo 96Mo “‘MO

(gs - 4:)

W, (MeV)

W,, (MeV)

rw (fm)

aw (fm)

0.25 0.25 1.oo 0.25

8.82 8.99 8.67 9.36

1.28 1.26 1.29 1.25

0.64 0.63 0.66 0.70

0.25 0.25 0.25

9.22 9.90 10.75

1.28 1.27 1.24

0.64 0.62 0.68

set No. 1 set No. 2

E. Fretwurst et al. / Scattering of 25.6 MeVprotons

281

L

0.0

50.0

100.0

150.0

CM-Angle ---> Fig. 14. Semi-microscopic coupled channels ling of (gs-2:). Full lines: OMP parameters

description of 96Mo(p, p’) differential cross sections. Coupset No. 1. Dashed lines: OMP parameters set No. 2 (see table 12).

282

E.

Fretwurst

et al. / Scattering of 25.6 MeV protons

6. Summary

In summary, the presented investigations of elastic and inelastic proton scattering on 94Mo, 96Mo and looMo lead to the following results: Due to the achieved excellent spectral energy-resolution level energies and J” assignments of many excited states in these nuclei could be determined. Coupled-channels calculations on the collective quadrupole one- and two-phonon states employing the macroscopic vibrational model in the anharmonic limit to first order yielded very good fits to the measured data. These calculations resulted in reliable parameters of the optical model potential as well as in deformation parameters, describing the assumed nuclear shape vibrations, without any further individual normalization. Among high-lying 4+ states possible hexadecapole one-phonon states have been found. It is suggested that some of the high-lying excited states could be interpreted as quad~pole-octupole two-phonon states. The proton scattering cross sections to the ground states, the 22,4: and 37 states could be described fairly well using coupled-channels calculations, which were based on a semi-microscopic model starting from the effective NN interaction for the real part of the OM potential and the inelastic transition form factors. Further work will concentrate on: (i) A comparative investigation of the elastic scattering data from the Mo(p, p’) measurements as well as from proton scattering on neighbouring nuclei with regard to recent theories of the optical model. (ii) The coupled-channels analysis based on the interacting boson model. A first attempt carried out for the description of (p, p’) data on Pd isotopes was rather promising 27). (iii) Developments of the semi-microscopic model with emphasis on the imaginary part of the optical model. This investigation was part of the collaboration program of the Universities of Leningrad (USSR) and Hamburg (BRD), which is financially supported by the Deutscher Akademischer Austauschdienst (DAAD) and the Universities of Leningrad and Hamburg. Many thanks are due to Prof. Dr. K. Gridnev. Without his initiative and encouragement this collaboration could not have been carried out. The assistance of the Hamburg cyclotron group is greatly appreciated for providing best accelerator performance. References 1) Table of isotopes, 7th edition, ed. CM. Lederer and V.S. Shirley (New York, 1978) 2) H.-W. Miiller, Nucl. Data Sheets 44 (1985) 277 3) H.-W. Miiller, Nncl. Data Sheets 35 (1982) 281

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