Coupled-channel analysis of proton inelastic scattering by 48Ti

Nuclear Physics A231 (1974) 365-375; Not to be reproduced by photoprint

COUPLED-CHANNEL

ANALYSIS

or

@ North-Holland Publishing Co., Amsterdam

microfilm without written permission from the publisher

OF PROTON

INELASTIC

SCATTERING

BY 48Ti + H. F. LUTZ, D. W. HEIKKINEN, W. BARTOLINI and I. D. PROCTOR Lawrence Livermore Laboratory, Livermore, California 94550 Received 12 November 1973 (Revised 11 June 1974) Abstract: Proton elastic and inelastic scattering to the first four excited states of 48Ti has been mea. sured at bombarding energies of 16.5, 18,20,22, and 24 MeV. These data have been analyzed using a coupled-channel calculation with a vibrational phonon model. The optical model parameters were chosen to vary smoothly with bombarding energy and the deformation parameters were chosen to be independent of bombarding energy. It was found that the amount of onephonon states admixed in the two-phonon states is considerable.

E

NUCLEAR REACTIONS 48Ti(p, p), (p, p’), E = 16.5, 18.0, 20.0, 22.0, 24.0 MeV; measured a(&,, e), o(E,,, 8). 4*Ti deduced levels, deformation parameters pn. Enriched target. Coupled-channel calculation.

1. Introduction

Coupled-channel calculations ‘) in which one employs a vibrational phonon model to describe the nuclear states have been successful in treating inelastic scattering. We have previously applied this type of analysis to inelastic proton scattering from the even-mass isotopes of cadmium 2), molybdenum 3), germanium 4), and titanium “). In the calculation of the excitation of the O+-2+-4+ group of states lying above the first 2+ state in an even-mass nucleus one employs, in addition to the usual optical model parameters, ten coupling parameters when one-phonon admixtures are permitted in the predominately two-phonon states. It seems only reasonable to question the value of nuclear structure information obtained with so many parameters. In this paper we report a study of the excitation of the O’-2+-4+ triplet of 48Ti using proton inelastic scattering at bombarding energies of 16.5, 18, 20, 22, and 24 MeV. In the analysis of these data we constrained ourselves to use an optical model whose parameters varied smoothly with bombarding energy and a set of coupling parameters that were independent of bombarding energy. 2. Experimental procedure Proton beams with energies varying from 16.5 MeV to 24 MeV were provided by the Livermore cyclograaff. This accelerator consists of an isochronous cyclotron with, t Work performed under the auspices of the US Atomic Energy Commission. 365 October

1974

366

H. F. LUTZ et al.

nominal energy of 14.7 MeV injecting into a model EN tandem Van de Graaff. The target was a self-supporting metallic film enriched to 99 % 48Ti. It had an area1 density of 1.1 mg * cme2. Scattered particles were observed with lithium-drifted silicon detectors cooled by means of thermoelectric devices to minimize leakage currents. Detectors with thicknesses up to 5000 pm were used in the present experiment. No particle identification was necessary since we only analyzed low-lying states in 48Ti. Signals from the detectors were processed by conventional electronics and stored in a small digital computer. Spectra were written on magnetic tape and then processed in a large digital computer with a code that fitted a sum of Gaussian shaped peaks to each spectrum and extracted the information necessary to deduce differential cross sections. 3. Results and analysis The experimental results are displayed in figs. 1-5. Each figure shows the angular distributions for a given level at the five bombarding energies used in this experiment. The curves in the figures are discussed later, but it should be noted that all five angular distributions at a particular bombarding energy were calculated simultaneously. The optical model parameters for the 0~-2:-0~-2~-4~ calculations are shown in table 1. The values of the geometric parameters defining the Woods-Saxon real and derivative Woods-Saxon imaginary potentials are ordinary, with the possible exception of the diffuseness parameter that is somewhat smaller than usual. The geometric parameters do not change as a function of bombarding energy. The values shown in table 1 are the result of a number of parameter variation studies made with the constraint, mentioned in the introduction, that we seek a set that varied smoothly with energy. First, the elastic differential cross sections were analyzed with an ordinary optical model code. These studies indicated that the values for the imaginary part of the potential and the diffuseness should be decreased from their ordinary optical model values when used in the coupled-channel code. Finally, parameter studies were made with the coupled-channel code using visual inspection of the quality of the fits to the angular distributions for the ground and first-excited states. In addition to the somewhat small value of the diffuseness parameter, one should note the slow decrease of the imaginary part of the optical model potential as a function of bombarding energy. Indeed, W, is almost constant between 20 and 24 MeV. In the calculatian of the excitation of the levels involved in the Ol-2:-O:-2:-4: coupling the ten parameters are /3e2, p21,, &, and Bb;, where I’ is the spin of the predominately two-quadrupole-phonon state, which takes on the values 0, 2, and 4. For the sake of clarity we have schematically illustrated the levels and couplings in fig. 6. The quantities /?b; measure the amount of one-phonon state admixed into the predominantly two-quadrupole-phonon state. It has been found in the present study

48Tih P)

367

Fig. 1. Elastic scattering data at the five bombarding energies of the present experiment. The curves are the results of the coupled-channel calculation for elastic scattering. At each bombarding energy the calculation simultaneously generates fits to the elastic data and the first four excited states of 48Ti. The optical model parameters are listed in table 1.

and in most previous studies that one-phonon admixtures are necessary since a pure two-quadrupole phonon description badly underestimates the cross sections. Following the notation of Deye et al. “) we describe wave functions for one- and two-quadrupole-phonon states with admixtures as one-phonon

: Y = p1 cP1+ql Gp,+r, G3,

two-phonon

: Y’ = pz Q1 + q2 Gz + r2 c#$,

(1)

368

H. F. LUTZ et al.

t i

t

4

2Lf MeV +t tt +t tt

-....i-.

w C. M.

RNGLf

Ipo

180

I DEGREE3 I

Fig. 2. Experimental data and theoretical fits to the first excited state of 48Ti (Jx = 2+). Deformation parameters determined by fitting the inelastic cross sections are listed in table 2.

where the p-values are the one-phonon amplitudes, the q-values are the two-phonon amplitudes and the r-values are additional contributions that insure normalization of the wave functions but do not contribute to the coupling between levels. The normalization condition implies p~+q~+r~

= 1 for n = 1,2.

(3)

The relations between the coupling parameters and the amplitudes are given by the

48TiCp,P)

369

Fig. 3. Angular distributions for the first excited O* state in =*Ti. The analysis permitted this state to be described as a mixture of two-phonon and one-phonon states. The wave function deduced from the deformation parameters are given in table 3.

equations MP142+Pz41)

= P21’9

(4)

Po2P2

=

B&3

(5)

802242

=

NW)*>

and the assumption that p1 x 1, which merely states that the one-quadrupole description is valid for the first excited state.

(6)

phonon

H. F. LUTZ et al.

370

2921 keV

?

t +t 16.5 nev

+ttt t

t

++t++++t ‘t++t+

\/_ t t

18 wev

t t+

t

t

++tt + t+

‘.“I t 20 tlev +t

t

+ ttt

t t+ t t + t

t

t \I

22 tlev t ++t

<

t+tt

*

t

1

Fig. 4. Angular distributions and fits from the coupled-channel calculation for the second 2+ state in “sTi. This state has a considerable admixture of one-phonon component.

of the first 2+ The parameter BOz is determined mainly by fitting the magnitude state. The p-values for the two-phonon states are determined by the values of fib;, necessary to fit the experimental data and the relation P2 = PaPo2*

The values of Pzr, depend on the q-values and p02, P2I’

=

92Po2.

(7)

371

48Tih P)

C. M. ANGLE ( DEGREES 1

Fig. 5. Experimental

data and theoretical fit for P = 4+ state. TABLE 1

Optical model parameters

EdMeW 16.5 18.0 20.0 22.0 24.0 ry=rw=rG=1.25fm.

used in 00+-21 +-O++ -Zz + -42+ coupled-channel V(MeV)

%W.W 12.5 11.4 11.0 10.9 10.8

46.3 45.8 45.2 44.6 44.0 a = 0.60 fm.

aw = 0.47 fm.

calculation VXMeV 7.0 8.0 8.0 8.0 8.0

H. F. LUTZ er al.

372

t

4;

.

l

.

2f 2 0+ 2

t B

t

A B

20

. 4

22

B 24

Fig. 6. Schematic representation of deformation parameters (B) used in coupled-channel analysis. The double primed quantities represent single-phonon admixtures in thepredominantly two-phonon states.

The values of &

are not taken as independent but are given by /%I, = &*B,r)*.

(9

These relations and assumptions greatly limit the parameter space over which a search should be performed. We evaluated the relations for q-values running from 0.5 to 1.0 in steps of 0.1 and then optimized the fits for each value by adding a onephonon contribution. The r-values were determined from the normalization condition. Visual inspection of the quality of fits to the data was used to choose the optimum set of coupling parameters. These are listed in table 2. The amplitudes of the wave functions corresponding to these coupling parameters are listed in table 3. Because of the length of each individual calculation we were not able to perform an exhaustive search as is customary in, say, optical model fits to elastic scattering data. We cannot TABLE2 Summary of p-parameters used in coupled-channel calculation with admixed wave functions Level

Quantity

Value

one-quadrupole 2+

B02

0.24

two-quadrupole 0+

k B”o0

2f

;:A

4+

B24

PO2

0.19 0.21 -0.02 0.14 0.18 0.05 0.19 0.21 0.04

373

48Tib, P) TABLE 3

Amplitudes

of wave functions determined from the p-parameters

given in table 2

Pll

?z= l;Z=2 n=2;z=o n=2;1=2 n=2;1=4

0.98 -0.08 0.21 0.17

5 0.21 0.8 0.6 0.8

0.54 0.77 0.58

even exclude the possibility that different combinations of coupling parameters and optical parameters would result in fits as good as the ones that we have obtained. Under these circumstances we have not assigned probable errors to our results and consider them to be qualitative indicators of the wave functions. The parameters listed in tables 1 and 2 were used to generate the theoretical fits to the experimental data shown in figs. 1-5. The fits to the elastic angular distribution as a function of energy are quite good. In fig. 2 the angular distributions for the excitation of the first 2+ state at 983 keV are plotted. There is somewhat more structure in the data than the theoretical fits but the general behavior of the angular distributions as a function of bombarding energy is certainly reproduced. The fits to the excited O+ state at 3000 keV are shown in fig. 3. This is the most weakly excited transition in the present work and there could be some contamination from compound nucleus processes. The integrated cross section for the excited O+ transition decreases by a factor of two between 16.5 MeV and 24 MeV. The quality of the theoretical fits improves with increasing bombarding energy and is quite good for 22 and 24 MeV. The fits to the second 2+ state at 2421 keV are shown in fig. 4. Qualitatively these angular distributions are similar to those of the first excited state except that their magnitude is reduced by more than a factor of ten. The quality of the fits to this level is quite good with the possible exception of the 24 MeV case where the theoretical curve is somewhat too small. The data for the 4+ state at 2295 keV are shown in fig. 5. The calculation had the most difficulty fitting this transition. In general the theoretical curve is greater than the data in the 60-80” region but underestimates the back angle data. The values of the deformation parameters determine the amplitudes of the onephonon and two-phonon wave functions given in table 3. The value for p1 is very close to unity implying that the one-quadrupole-phonon is a good description of the first excited state. On the other hand the two-phonon-like wave functions are badly admixed with r-values that are comparable to the q-values. This indicates that the two-quadrupole-phonon description is not adequate for the wave functions of the O+-2+-4+ group of states. For inelastic scattering, however this description may be adequate because of the enhanced effect of the collective part of the wave functions on the cross sections. Deye et al. “) have summarized the work of the Oak Ridge group 7-1o) in deducing the wave functions for lo6Pd, 1°*Pd, li”Pd, liZCd, l14Cd, l16Cd and lz4Te. These

374

H. F. LUTZ

et al.

ate nuclei which are believed to be good examples of the vibrational model. It is of interest to compare our results for 48Ti with their results. Deye et al. “) find, in almost every case, the r-values in the two-phonon-like wave functions are the largest amplitudes. Single-phonon admixtures are necessary in all but one excited Of state and three 4” states. Moreover, the sign of the p-value for every excited O+ state is negative. We thus see that qualitatively the wave functions for 48Ti and the vibrational nuclei in the cadmium region are quite similar. The nucIeus 48Ti falls in the middle of the fS shell and its shell model structure has been calculated by McCullen et al. 11) and Ginocchio et al. “- 14). McCullen et al. I*) pointed out that 48Ti is its own cross conjugate nucleus: the wave functions are either even or odd under the interchange of protons and neutron holes. This property is sometimes called the signature of the wave functions. The E2 transition matrix is predicted to be inhibited for transitions between states with the same signature. Lawson “) considered the E2 matrix elements in 48Ti and came to the conclusion that the pure f+ configuration with an effective charge is not adequate to fit the decay data. He finds that the Of state at 3 MeV is not an (fS)8 configuration but is predominately (red+)-’ (rtfa)4 (vf+)-‘. Bardin et al. 16) have quite recently studied the y-ray decay of 48Ti. If we restrict ourselves to the levels studied in the present paper and do not consider the excited O+ state, then the signature rule is followed. The only case of inhibition, however, occurs in the transition between the second 2’+ state and the ground state. This transitionis also inhibited in the collective vibrational model. It thus seems, at least for the transitions studied in the present work, that the self-conjugate structure of 48Ti has no apparent effects on the inelastic scattering. From the wave functions that we have deduced we can calculate certain ratios of reduced transition probabilities, which can then be compared with the y-ray studies of Bardin et al. l6 ). These ratios are B(E2; 2’ + 0) B(E2;2+0)

1--P: =-f P:

B(E2; I’ -+ -_ 2) = 2qq. B(E2; 2 -+ 0)

(10)

(11)

The values are tabulated in table 4. The y-ray data imply that the single-quadrupolephonon amplitude of the first excited state is 0.96 instead of the value 0.98 found in the present analysis. The two experiments agree on the amplitudes of the two-quadrupole-phonon components of the Ot and 2” states. There is disagreement for the 4+ state, which we had difficulty in fitting. The overall agreement is quite reasonable. In summary, we have measured the excitation of the low-lying one- and twoquadrupole-phonon states in 48Ti at energies between 16.5 and 24 MeV by inelastic proton scattering. These data were analyzed using a coupled-channel calculation in which the optical parameters varied smoothly as a function of energy and the de-

4*Ti(P,P)

375

TABLE4 Ratios of reduced transition probabilities deduced from the wave function amplitudes and compared with the y-ray data of Bardin et al. 16) Ratio

Present experiment

B(E2; 2’ + 0) B(E2; 2 + 0)

0.043

0.095k24.2

B(E2; 0’ + 2) B(E2; 2 -+ 0)

1.28

1.04 &19.0X

B(E2; 2’ -+ 2) B(E2; 2 -+ 0)

0.12

B(E2; 4’ -+ 2) B(E2; 2 + 0)

1.28

y-ray data %

043 . +I=%

- 80.1yo

0.60 rt29.0 %

formation parameters were independent of energy. The theoretical predictions were reasonably successful in accounting for the experimental data. The phonon mode wave functions that were deduced from the deformation parameters exhibited considerable one-phonon admixtures in the predominately two-phonon states. Finally, the self-conjugate structure of 48Ti has no apparent effects on the inelastic scattering. References 1) T. Tamura, Rev. Mod. Phys. 37 (1965) 679; Oak Ridge National Laboratory Report no. ORNL4152, 1967 (unpublished) 2) H. F. Lutz, W. Bartolini and T. H. Curtis, Phys. Rev. 178 (1969) 1911 3) H. F. Lutz, D. W. Heikkinen and W. Bartolini, Phys. Rev. C4 (1971) 934 4) T. H. Curtis, H. F. Lutz and W. Bartolini, Phys. Rev. Cl (1970) 1418 5) H. F. Lutz, W. Bartolini, T. H. Curtis and G. M. Klody, Phys. Rev. 187 (1969) 1479 6) J. A. Deye, R. L. Robinson and J. L. C. Ford, Jr., Nucl. Phys. A180 (1972) 449 7) R. L. Robinson, J. L. C. Ford, Jr., P. H. Stelson and G. R. Satchler, Phys. Rev. 146 (1966) 816 8) R. L. Robinson, J. L. C. Ford, Jr., P. H. Stelson, T. Tamura and C. Y. Wong, Phys. Rev. 187 (1969) 1609 9) P. H. Stelson, J. L. C. Ford, Jr., R. L. Robinson, C. Y. Wong and T. Tamura, Nucl. Phys. A119 (1968) 14 10) M. N. Rao, P. H. Stelson, R. L. Robinson and J. L. C. Ford, Jr., Nucl. Phys. Al47 (1970) 1 11) J. D. McCullen, B. F. Bayman and L. Zamick, Phys. Rev. 134 (1964) B515 12) J. N. Ginocchio, Phys. Rev. 144 (1966) 952 13) J. N. Ginocchio and J. B. French, Phys. Lett. 1 (1963) 137 14) J. N. Ginocchio, Nucl. Phys. 63 (1965) 449 15) R. D. Lawson, Nucl. Phys. Al73 (1971) 17 16) T. T. Bardin, J. A. Becker and T. R. Fisher, Phys. Rev. C7 (1973) 190