Lifetime measurements of states in 110Pd

Nuclear Physics A503 (1989) 575-588 North-Holland, Amsterdam

LIFETIME MEASUREMENTS OF STATES IN t1°Pd* B. KOTLINSKI, D. CLINE, A. BACKLIN I and D. CLARK 2

Nuclear Structure Research Laboratory, University of Rochester, Rochester, NY 14627, USA Received 2 May 1989 Abstract: Lifetimes of the 27,47,67,2;,4;,0;,2; and 4; states in II°Pd have been measured using the recoil distance method. The measured lifetimes are compared with the E2 and M1 matrix elements measured in a separate Coulomb excitation experiment. The lifetime measurements agree, within the "'5-10% experimental errors with the values obtained by Coulomb excitation. This agreement confirms the accuracy of the complicated analysis technique used to extract the matrix elements from the Coulomb excitation data. The measured E2 matrix elements can be understood within a macroscopic quadrupole collective model involving mixing of four quasi-rotational bands which have different shapes.

E

NUCLEAR REACTIONS lIOPdeHNi,58Ni), 190 MeY; measured recoil distance decay curves following Coulomb excitation, p y-coin; deduced lifetimes for eight states, E2 matrix elements; compared with theory.

1. Introduction

For a long time, the structure of the Pd and Cd nuclei was commonly assumed to have a vibrational character. A primary reason for this assumption was the occurrence of a closely spaced 0+, 2+ ,4+ triplet (two-phonon triplet in the vibrational model) at about twice the energy of the, presumed one-quadrupole phonon, 27 state. On the other hand, the measurement of the large static quadrupole moment of the first 2+ state in 114Cd [ref. I)] raised questions about the applicability of the simplest vibrational model, which predicts a static quadrupole moment of zero. A set of Coulomb excitation experiments and the present lifetime measurements were performed to study the structure of IIOPd. The results of the Coulomb excitation experiment are published separately 2). This paper describes the lifetime experiment, performed using the recoil distance technique, and also includes a comparison of the results with the values obtained from the Coulomb excitation measurements. The Coulomb excitation study of llOPd was performed using 16 0 , 58 Ni and 208Pb ions beams (a detailed description is given in Hasselgren et al. 2)). Deexcitation *

Work supported by the National Science Foundation. Department of Radiation Science and The Svedberg Laboratory, Uppsala University, S-75121, Sweden. 2 Eastman Kodak Company, Rochester NY, USA. 1

0375-9474/89/$03.50 © Elsevier Science Publishers B.Y. (North-Holland Physics Publishing Division)

576

B. Kotlinski et at. / Lifetime measurements

gamma rays were detected in coincidence with ions scattered over a wide range of scattering angles. The E2 and Ml matrix elements then were determined by a least-squares fit to the measured gamma-ray intensities, using the Rochester Coulomb-excitation least-squares search code "GOSIA" 3). This semi-classical, coupled-channels code allows a model-independent least-squares fit of the matrix elements to the data. The complicated analysis allowed the determination of 110 E2 matrix elements for the low-lying states, including static moments and signs of many interference terms plus 9 Ml matrix elements. This, almost complete, set of E2 matrix elements for the low-lying levels showed that the level spectra can be arranged into four collective bands based on 07,2;,0; and O~ band heads. These bands have both level spacings and inband B(E2) values close to the rotational model predictions. Coulomb excitation with heavy ions leads to a very complicated dependence of the data on the many matrix elements involved in the large number of states excited. It is necessary to use a random set of matrix elements as starting values in the least-squares search to ensure that the solution is model independent and the process has to be repeated several times using different sets of starting matrix elements to test the uniqueness of the best fit set of matrix elements. The numerical procedure of fitting 119 parameters (Ml and E2 matrix elements) to a few hundred gamma-ray intensities, using a steepest descent method, can lead to a local minimum. For example in the case of the 4~ level, two sets of matrix elements involving this level were found to fit the experimental data. In addition to the possible ambiguities in the fitting procedure, there are a number of possible sources of systematic error that are not included explicitly in the Coulomb-excitation analysis. Some of them, such as quantal corrections, can be estimated for a few states. Others, like the dipole polarization can be roughly estimated using approximate formulas. This paper reports the direct measurement of the lifetimes of 8 low-energy levels, performed in order to check the accuracy of the Coulomb excitation results, the numerical methods and various approximations used, and as a way to resolve the 4~ level ambiguity. Recoil-distance measurements are capable of providing lifetimes with >- 10% accuracy. Thus a comparison of these lifetimes with the comparableprecision values derived from the Coulomb excitation results serves as a sensitive test of the accuracy of matrix elements derived from heavy-ion induced multiple Coulomb excitation experiments.

2. Experimental method The mean lifetimes for low-lying levels in 110Pd are of the order of picoseconds. The most reliable experiment method for this time range is the recoil distance method (ROM). This technique has been described in many publications 4.5). The principle of our experiment is sketched in fig. 1. A 190 MeV 58Ni beam from the Rochester Tandem was used to Coulomb excite an enriched 670 fLg/em" thick

577

B. Kotlinski et al. / Lifetime measurements

NI BEAM

n

Ge(Li)

Y TARGET ANNULAR VELOCITY SI DETECTOR SHIFTER Fig. 1. Experimental setup for the lifetime measurement in tlOPd.

IIOPd target. This rolled, self-supporting target was glued with Epoxy to an aluminium frame and then stretched flat with a steel insert, to form a highly reflective surface. The backscattered 58Ni ions were detected in an annular Si detector placed at a distance of 17 mm from the target, and covering an angular range from 145° to 170°. The corresponding target nuclei recoiled into a forward cone between 2.4° and 8.7°, with an average velocity of f3 = 0.0525. The deexcitation gamma rays, emitted in coincidence with backscattered 58Ni ions, were detected by a 10% efficiency (compared to a 7.62 diameter by 7.62 length NaI at 25 em for a 1.33 MeV gamma ray) Ge(Li) detector, placed at 0° with respect to the beam axis and at a distance of 5.6 em from the target. Another 8% efficiency Ge(Li) was placed at 29° to check the gamma-ray angular distribution. Both detectors had a 1.9 keV resolution at 500 keV. In most RDM experiments, the stopper is thick enough to stop the recoils. The finite time of slowing down (""'3 ps) results in Doppler-broadened peaks in the y-ray spectrum. In our experiment, a thin velocity shifter foil was used, which slows down, but does not stop the recoiling nuclei, and thus changes the Doppler-shifted energy of the gamma transitions. As a result, two peaks are observed for each transition in the spectrum: a "fast" and a "slow" one, and Doppler-broadening of line shapes is no longer a problem. A 2.3 mg/ crrr' thick movable Ni velocity shifter was mounted between the target and the 0° Ge(Li). It was stretched on a steel frame similar to the target frame and had a highly reflective surface. The thickness of the velocity shifter was chosen after careful consideration of the recoil velocity and the transition energies of the deexcitation gamma rays in 110Pd. The thickness was sufficient to slow down the recoils from f3 = 0.0525 ± 0.0020 to f3 = 0.037 ± 0.002, resulting in an 8 keV separation of the "slow" and "fast" components for a 500 keV gamma transition. It is important in this type of experiment to accurately measure the initial recoil velocity (v). The velocity was measured by replacing the slowing-down foil with a 45.0 mg/ ern" thick Ni foil to stop the recoils completely and thus the relative Doppler shift could be determined. Then the recoil velocity was calculated from the observed Doppler shift by averaging over the finite solid angle of the Ge(Li) detector. The measured velocity was f3 = 0.0525 (20) i.e. with a 4% error.

578

B. Kotlinski et al. / Lifetime measurements

A laser light was used in order to adjust the target and velocity-shifter surfaces to be parallel. The process of comparing the position of the reflected laser beam at 2 m from the target allowed us to set the shifter foil parallel to the target foil with an accuracy of 0.014°. This corresponded to =0.3 urn difference in the target-shifter distance over the beam spot area. Special care was taken to keep the scattering chamber clean since the size of dust particles is of the order of 1-10 urn, which is comparable to the smallest target-shifter distance; and this could cause electric contact between them. In the recoil distance apparatus design at Rochester 6), the target-shifter distance can be changed by a conventional differential micrometer or by an "Inchworm". Both push a parallelleafspring kinematic stage with the shifter frame mounted on it. The "Inchworm" is a commercially available device" that uses three piezo-electrical crystals to move a metal rod in fine =6 nanometer steps over a range of 25 millimeters, with the speed being variable between 0.6 mm/ sand O.OI!Lm/s. It also includes an optical measurement of the position with an accuracy of ±2!Lm. The capacitance of the target-shifter system was measured in order to find the distance more precisely than given by the optical encoder. The target and the velocity shifter foils act as a capacitor, with the capacitance being a function of the distance between them. For small distances (much smaller then the surface size), the capacitance should approximately follow the formula for two infinite, parallel, conductive plates that is it should behave like -11 d. During our experiment, a pulser triggered by the Ge(Li) detector was connected to the target, and the shifter foil was then connected through a preamplifier and amplifier to a 4K ADC. Observation of the pulser's peak position allowed us to set the distance and to watch its stability during the experimental runs with 1 u.m accuracy. It should be mentioned that, for small distances and a very intense 58Ni beam, the capacitance technique failed, due to secondary electrons coupling both foils. It was necessary to turn off the beam to eliminate this problem. The measurements were performed at 15 distances, from 1O!Lm to 1.5 mm. An electrical contact occurred below 10!Lm and thus this was the smallest usable distance. The capacitance measurement was calibrated using the micrometer, and the zero distance was determined by extrapolating this calibration for small distances where C -11 d. 3. Analysis In the present recoil-distance experiment, 9 levels in 1IOPd were excited and 14 v-transitions were seen. The observed transitions are shown by arrows between the levels in fig. 2. Unfortunately, four of these transitions, marked with dashed arrows, were too weak to be used in the lifetime analyses. The level scheme adopted is from Hasselgren et at. 2). Gamma spectra for three distances are shown in fig. 3. For most * Burleigh Instruments, Inc.

B. Kotlinski et at. / Lifetime measurements

579

8~----.I I I I

2

I I

6~

'">

I

(])

~ W

o

c' 0+ GROUND "GAMMA' 2 BAND BAND BAND

0+ 3 BAND

Fig. 2. Level spectrum of IIOPd ned in the lifetime analysis. The arrows show the gamma transitions observed during the experiment.

transitions, both "fast" and "slow" peaks can be seen. In two cases, the "slow" and "fast" components of close lying transitions formed unresolved doublets. The intensity ratio for a large target-shifter distance, where only "fast" peaks exsited, was then used to unfold the components. Energy and angle straggling acting on the excited target recoil nuclei passing through the shifter foil caused the "slow" peaks to be about 20% wider than the "fast" ones. Experimental y-ray intensities were not corrected for y-rays emitted during the slowing-down process in the velocity shifter. The time required for the recoils to pass through the shifter was below 0.15 ps and the lost v-intensity was below 4% for all cases. A quantity R is defined as the ratio of the "fast" to the "total" ("slow" + "fast") intensities for each transition If

R=--I s + I, To a first-order approximation R has a simple relation with mean lifetime T In R = -d / VT, where d is the distance and v the recoil velocity. Fitting a straight line to the experimental values of In R as a function of d determines T. There are several effects that distort this simple picture: (i) cascade feeding into the level of interest, (ii) different solid angles for "slow" and "fast" peaks, (iii) different Ge(Li) efficiency for "slow" and "fast" peaks,

B. Kotlinski et at. / Lifetime measurements

580

~ Z ::J

o U

la' 10° - f - - - - -I- , - - , -I- - , - - -Ir - - - - -I , - - - - -I - , - - -I- - ' ' -

a

100

200

300

400

500

600

CHANNEL Fig. 3. Sample 'loPd gamma spectra from the present experiment in which 1IOPd nuclei are Coulomb excited by a 190 MeV 58Ni beam.

(iv) the deorientation effect, that is the attenuation of the nuclear alignment, which results in a time-dependent angular distribution of the emitted v-rays. The computer code ORACLE 7) was used for the final analysis of the present data. This code makes corrections for all of the above mentioned effects and then fits the lifetimes to the corrected data. As part of the input, the code requires the relative initial population of the levels and the gamma-transition angular distribution parameters. These were calculated with the semi-classical Coulomb excitation code using matrix elements from ref. 2). The following modifications were made to the code "ORACLE" for the present case: (1) Since we were using a thin velocity shifter foil, the code had to be adapted for cases where the corrections for the "slow" peak were calculated in the same way as for the "fast" one, but using a smaller recoil velocity. (2) Attenuation coefficients 02 and 04, related to the finite size of the y detector, were added. (3) The model describing the attenuation of the nuclear alignment, the deorientation effect, was changed as described below.

B. Kotlinski et al. / Lifetime measurements

581

1

TABLE

The parameters for the nuclear deorientation model Parameter a)

Standard

Case A

Case B

3.0 0.02 0.0345 3.5

3.0 0.001 1.0

3.0 1.0 0.1 10.0

Z/A

Z/A

Z/A

6.0 0.6

6.0 0.6

i,

r (pS-I) A* (pS-l) 7c

(ps)

g K(MGs) x

O.oI

6.0 0.6

H) The meaning of the parameters is explained in ref. 10).

The attenuation of the nuclear alignment due to the deorientation effect is a complicated process for which no adequate theory presently exists. Therefore, semi-empirical approximate models have to be used to estimate this effect. The original version of the program ORACLE used the Abragam-Pound 8) model to describe the deorientation effect. The time dependence of the attenuation coefficient, 0k( t) in this model has an exponential decay. Our analysis of Coulomb excitation experiments 9) showed that the "two state-model" of Brehn and Spehl 10) was in better agreement with the experimental time-integrated angular distribution data. The formula used by this model is given in ref. 10). For large time t, Ok(t) approaches a constant "hard core" value, and both the magnitude and the slope of the fast decaying part depend on a set of adjustable parameters. The values of the parameters were adopted from ref. 9,10), and are listed in table 1 as the "standard" set. The "adjustable" parameters used for estimating the deorientation effect have large uncertainties and therefore fits were made using several sets of these parameters in order to check the influence of the assumed G; on the lifetimes. Two such sets are listed as case A and case B in table 1. Lifetimes for 3 levels were then determined using the different O 2 and 0 4 dependences, including also the extreme cases G; = 0 and Ok = 1. As shown in table 2, in most cases, the different deorientation-effect parameters change the extracted lifetimes by less than the experimental error. This

TABLE

2

Variation of the measured lifetimes in IloPd using different deorientation model parameters In

2; 2; 47

7

(ps)

(25.6±4.7) (13.1 ±6.1) (5.9±6.8)

") Abragam-Pound model g).

Ok=O

G;

-4.1% 6.0% 1.4%

-3.4% 5.2% 1.4%

=

1

Case A

Case B

A_p a )

0.7% 3.0% 1.0%

-1.6% 0.7% -2.2%

1.7% -1.5% -0.3%

582

B. Kotlinski et al. / Lifetime measurements

is in agreement with the conclusions from earlier work 5), where the deorientation effect was etimated to produce less than a 5% change in the lifetimes extracted.

4. Results

Mean lifetimes of 8 levels in lIoPd have been measured. The quality of the experimental data can be seen in figs. 4, 5 and 6, where decay curves corresponding to the fitted mean lifetimes together with the experimental data points are shown for all 8 analyzed gamma transitions. The data points are already corrected for the effects discussed previously, therefore the decays are represented by straight lines on the In R versus distance plots.

10 08

............

<,

"~

0.6

I

04

I

......... 0.2

-L

1 0.8 0.6 0:

04

-L

-L

-L

-L

-L

200 400 600 800 1000 1200

'.,'.

,\"4 I

I

0.2

04

0.2

100 150 200 250 300 d [fLm] Fig. 4. Decay curves of the 1IOPd ground band gamma transitions. All points correspond to corrected data (see text).

B. Kotlinski et al. / Lifetime measurements

583

The measured lifetimes are listed in table 3. The experimental errors quoted include both the error of the fit and the error in the measurement of the velocity. Most of these errors are larger than 5%, and thus exceed the possible uncertainty arising from the de orientation parameters as indicated in table 2. A possible source of systematic error is connected with the correction for the cascade feeding. The feeding parameters (population probability) and the angular distribution parameters were calculated using a Coulomb excitation code 11), with matrix elements that agree well with our experimental lifetimes. Moreover, tests were made by changing the population parameters for the side feeding by an unrealistically large value of ± 100% and this influenced the extracted lifetimes by less than the error bars quoted in table 3. The angular distribution parameters (statistical tensors) were varied by ±10%, which changed the fitted lifetimes by a negligible amount of 0.3%.

Olfc-

---.L

---.L

---.L

---.L

---.L

---.L

_

. 50 100 150 200 250 300 1.0,. 0.8 ......................

<, ·~2:-2+

0.6 0::

I~I

04

I~I <,

0.2

011.0 0.8 06

---.L

---.L

---.L

---.L

---.L

---.L

_

100 200 300 400 500 600

"q, "\,

f~

04

1""']4+-2+ ~2

0.2

50

100 150 200 250 300

d[fLm] Fig. 5. Decay curves of the l111Pd gamma transitions deexciting the 0; , 2; and 4; states. All points correspond to corrected data (see text).

B. Kotlinski et al. / Lifetime measurements

584

a:: -

-

-

-

\4+-2+ \

I

50

3

3

I

I

I

I

200

250

300

I

100

150

d [fLm] Fig. 6. Decay curves of the llOPd gamma transitions deexciting the 2; ,4; states. All points correspond to corrected data (see text).

TABLE

3

Results of the lifetime measurements in llOPd and comparison with Coulomb excitation data l~

Coulomb excitation results Hasselgren et al. 2) (ps)

Recoil distance results this work (ps)

63.2± 1.9

65.6±2.5 5.9±OA 2.0±O.2 25.6 ± 1.2 7A±O.8 11.4±1.0 13.1 ±O.8 3.2±OA

6.5:,:gi 2.o:,:gf 26.8:':Yi 7.8:,:g~

15.3 :':?~ 17.5:':~1 2.7:,:g~

B. Kotlinski et al. / Lifetime measurements

585

Table 3 also includes lifetimes based on the Coulomb excitation data from Hasselgren et al. 2). As mentioned in the Introduction, these lifetimes were calculated from the M1 and E2 matrix elements extracted by fitting to the Coulomb excitation data. The recoil-distance and Coulomb excitation results differ by more than one standard deviation only for the 0; and 2; states. The code ORACLE permits only two external cascades decaying to a given level while in the case of the 27 level, there are at least four such feedings. Simulation of the effect of four feedings produced approximately a ± 1.5 ps effect. As mentioned in the Introduction, there are several possible sources of error in the analysis of the heavy-ion multiple Coulomb excitation data. The main ones are listed below: (i) The final set of E2 matrix elements corresponds to a local solution found by the numerical procedure used to fit the experimental gamma yields. (ii) Physical effects not taken explicitly into account during the analysis (or included only approximately) - dipole polarization contribution to the excitation cross section through the virtual excitation of the GDR 12) _ relativistic corrections 13) - quantum mechanical corrections to the semi-classical approximation used by GOSIA - E4 and higher order multiple excitations - the deorientation effect. Table 4 shows, for Coulomb excitation of the ground band of lI°Pd by 190 MeV 58Ni, estimates of the changes in the excitation probabilities due to quantal corrections to the semi-classical code, calculated using ECIS 14), switching off the dipole polarization correction and including a large E4 deformation. The corrections are TABLE

4

Excitation probabilities for ground-band states in IloPd excited by 190 MeV 58Ni scattered at 1650 Excitation probabilities I~

a) ") case C)

GOSIA

scis-:

No dipole polarization ")

E4 included in GOSIA C)

0.412 0.226 0.061 0.0086 0.00075

+1.6% +0.8% +5.8% +6.0% +9.1%

-1.9% +3.1% +6.4% +9.4% +13.9%

-1.9% -2.7% 0.0% +4.2% +12.6%

EelS, Quantum mechanical coupled-channel code 14) Approximate formula from ref. 17) is used in GOSIA for the dipole polarization, in this Po = 0, to simulate no virtual dipole excitation. Rigid Rotor model with 04 = 1.0 e' b' used, it gives (01IE4114) = 0.1 e b',

586

B. Kotlinski et al. / Lifetime measurements

around 5-10%, which is comparable with the precision of the Coulomb excitation data. The equality of both sets of lifetimes, shown in table 3, proves that the abovementioned effects influence the Coulomb excitation data by less than 10%. A similar comparison, for the excited states in 232 Th, has been made previously by Guidry et al. 15). Since only one band (the ground band), was excited in their case, the number of couplings was small and the analysis of the Coulomb excitation data was unambiguous. For IIOPd, 119 matrix elements have been determined from the analysis of the Coulomb excitation data, making the analysis much more complicated and a confirmation of the results through an independent direct lifetime measurement prudent.

5. Conclusion

The results of the present experiment and the Coulomb excitation study by Hasselgren et at. 2) have changed the model interpretation of the lloPd level spectrum. Instead of referring to llOPd as a vibrational nucleus with level energies distorted from a simple harmonic model by introducing anharmonicity, we would rather speak about four collective quasi-rotational bands, based on the 07 , 2~ , O~ and 0; levels. The anharmonic vibrational model cannot explain the decay properties of the O~ state. A simple mixing calculation 16) can reproduce the experimental quadrupole moments of the 27 and 2~ states as well as the 2; ~ 27 decay. But no mixing coefficient, within the same model, will give the correct O~ ~ 27 decay. The rotational model allows the low-lying states to be arranged into four bands. The present data combined with the Coulomb excitation results 2) imply that the ground-state and 2~ bands have a triaxially deformed intrinsic shape with Qo = 2.85 e- band 'Y = 16°. The data imply a larger deformation of Qo = 3.71 e- b for the O~ band. The in-band matrix elements for the three bands, the 07, 2~ and o~, are shown in fig. 7. The smooth curve shows the rotational model predictions. The structure of IloPd and a detailed comparison of the measured matrix elements with predictions of the geometrical and algebraic models is discussed extensively in the paper by Hasselgren et al. 2). Direct measurement of the lifetime of the 4; level was crucial for the final analysis of the Coulomb excitation data. Fitting the matrix elements to the Coulomb excitation data gave two equally probable solutions, one corresponding to a 4; lifetime equal to 1.2 ps and the other to 3.0 ps. Results presented here clearly point to the second choice. Generally it should be stressed that an excellent agreement between the present recoil distance masurement and the Coulomb excitation results of Hasselgren et at. 2) shows that the method used to extract the E2 properties from the Coulomb excitation gamma yields is correct and reliable. That is, the use of the semi-classical

B. Kotlinski et al. / Lifetime measurements

587

TRANSITION IN - BAND MATRIX ELEMENTS 4 GROUND BAND

3 2 .Q


r-;

C\J I

H C\J

W

H

v

o ------------

:t:~~~O:I~;85

__

4

~

O2 BAND

3

.

~---------0

2/

J

y=16° 0

=3.71

O'---------------~

2

4

6

8

10

12

I Fig. 7. Experimental in-band E2 matrix elements for the three bands discussed in the text. The smooth curve shows the rotational model predictions.

Coulomb-excitation code GOSIA to fit 119 matrix elements to several hundred Coulomb-excitation data gives a correct answer and does not introduce significant systematic errors. References I) J. de Boer, J. Stokstad, R.G. Symons and A. Winther, Phys. Rev. Lett. 14 (1965) 564 2) L. Hasselgren, J. Srebrny, c.v. Wu, D. Cline, T. Czosnyka, L.E. Svensson, A. Backlin, C. Fahlander, L. Westerberg, R.M. Diamond, D. Habs, HJ. Korner, F.S. Stephens, C. Baktash and G.R. Young, Nucl. Phys., submitted 3) T. Czosnyka, D. Cline and c.v. Wu, Bull. Amer. Phys, Soc. 28 (1983) 745 4) N. Anyas-Weiss, R. Griffiths, N.A. Jelley, W. Randolph, J. Szucs and T.K. Alexander, Nucl. Phys. A201 (1973) 513 5) M.W. Guidry, RJ. Sturm, N.R. Johnson, E. Eichler, G.D. O'Kelley, N.C. Singhal and R.O. Sayer, Phys. Rev. e13 (1976) 1164 6) D. Cline, Nuclear Structure Research Laboratory Annual Report 1981, University of Rochester (l981)p236 7) RJ. Sturm and M.W. Guidry, Nucl. Instr. Meth 138 (1976) 345

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B. Kotlinski et al. / Lifetime measurements

8) A. Abragam and RV. Pound, Phys, Rev. 92 (1953) 943 9) c.Y. Wu, Ph. D thesis, Nuclear Structure Research Laboratory Report, University of Rochester, UR-NSRL-275 (1983) 10) R Brenn, H. Spehl, A Weckherlin, H.A. Doubt and G. van Middelkoop, Z. Phys. A281 (1977) 219 11) A Winther and 1. de Boer, in Coulomb Excitation, ed. K. Alder and A Winther (Academic Press, New York, 1966) p.303 12) O. Hausser, AB. McDonald, T.K. Alexander and A.1. Ferguson, Nuci. Phys. A212 (1973) 613 13) M.P. Fewell, Nuci. Phys. A425 (1984) 373 14) 1. Raynal, Phys. Rev. C23 (1981) 2571 15) M.W. Guidry, P.A. Butler, P. Colombani, I.Y. Lee, D. Ward, RM. Diamond and F.S. Stephens, Nuci. Phys. A266 (1976) 228 16) 1. van der Laan, Ph.D. thesis, University of Amsterdam (1986) 17) K. Alder and A. Winther, in Electromagnetic excitations, (North-Holland, Amsterdam, 1975) p. 340