Coulomb excitation of Pd108, Pd110, Ru102 and Ru104 using oxygen ions

Nuclear Physics 32 (1962) 190---223; ~ Not to

North-Holland Publishing Co., Amsterdam

be reproduced by photoprint or microfilm without written permission from the plblisher

COULOMB EXCITATION USING

O F P d l°s, P d 11°, R u 1°2 A N D Ru 1°4 OXYGEN IONS

D. E C C L E S H A L L , B. M. H I N D S , M. J. L. Y A T E S

A.W.R.E., Aldermaston, Berks., England Received 30 O c t o b e r 1961 O x y g e n - 1 6 ions h a v i n g energies u p to 40 M e V h a v e b e e n u s e d to excite levels in P d l°s, P d tl°, R u 1°2 a n d Rut°* b y t h e direct a n d d o u b l e E2 C o u l o m b e x c i t a t i o n m e c h a n i s m s . T h e d e c a y of t h e s e levels w a s o b s e r v e d b y d e t e c t i n g :F-rays e m i t t e d f r o m t h i c k t a r g e t s either d i r e c t l y or in coincidence w i t h back, s c a t t e r e d o x y g e n ions. Collective 2 + a n d 4 + levels a t a b o u t twice t h e e x c i t a t i o n e n e r g y of t h e first e x c i t e d s t a t e h a v e b e e n o b s e r v e d in t h e f o u r nuclei, a n d in P d x0o t h e 0 + s t a t e k n o w n a t t h e s a m e e x c i t a t i o n e n e r g y as t h e 4 + s t a t e m a y also h a v e b e e n excited. Values of t h e ratio B E , ( 2 + --+ 4 + ) / B E 2 ( 0 + --> 2 + ) h a v e b e e n o b t a i n e d for t h e f o u r nuclei a n d a r e c o n s i s t e n t w i t h t h e p r e d i c t i o n s of t h e p h o n o n model.

Abstract:

1. I n t r o d u c t i o n Scharff-Goldhaber and Weneser 1) first pointed out that even medium weight nuclei exhibit a near-harmonic energy level spectrum and that a simple collective quadrupole vibrational model could account for m a n y of the observed properties. This model predicts that at twice the energy of the first excited level (one phonon state) there should be a degenerate triplet of levels (two phonon state) having spins 0, 2 and 4 and positive parity. In several even nuclei with A ~ 100 one or more levels are known which, on the basis of their excitation energies, spins and decay schemes, can be related to the two phonon levels 2). Stelson and McGowan 3) have Coulomb excited the second 2 + ( 2 ' + ) levels in a number of nuclei with A ,m 100 and have shown that the ratio of the reduced E2 transition probabilities, B E 2 ( 2 ' + ---> 2-+- ) / B E 2 (2 + --> 0 ~ - ), fluctuates about unity indicating that the second 2 + level, like the first 2 + level, has a collective character. In Cd 114 these authors also obtained evidence for the excitation of a 4 + level. They conclude that this level was formed b y double Coulomb excitation and the observed yield of ~,-rays required a collective enhancement of the B~2(4+ -> 2 + ) over the single particle value. Various modifications 1, 4-s) to the simple vibrational model have been proposed to account for the observed departures from the ideally harmonic energy level spectrum and the departures from the predicted B~2(2'+ -+ 2 + ) value. Sheline 9) has discussed the general systematics of quadrupole vibrational states in nuclei throughout the periodic table. An alternative approach, a rotational model involving non-axial symmetry 190

COULOMB EXCITATION

191

of the nuclei, has been proposed b y Davydov and others 10,11). This model can predict 2 + and 4 + levels at about twice the energy of the first 2 + level but does not allow a collective 0 + level. Davydov and Chaban 1~) have modified this model b y introducing /~-vibrations and vibration-rotation interaction, thereby admitting a collective 0 + state in the level scheme. Klema et al. 13) have extended the calculations of Davydov and Chaban. In the present experiments an attempt was made to excite the 0 + and 4 + members of the triplet expected at about twice the energy of the first excited state. With the beams of oxygen ions having energies up to 40 MeV available from the tandem Van de Graaff it was anticipated that these levels could be excited by double E2 Coulomb excitation 14-19). The second 2 + level would be excited by double E2 as well as direct E2 Coulomb excitation. The four nuclei, Ru 1°~, Ru 1°4, Pd 1°8 and Pd 11° were chosen because they have the lowest first excited states and the largest values of BE2(0+ --> 2 + ) in the mass region of interest and so promised to be the easiest to study.

2. The Experimental Arrangement The 016 ion beam from the Aldermaston Tandem Van de Graaff accelerator was used throughout the experiments. Beam energies up to 40 MeV were obtained by selecting the 4 + , 5 + or 6 + ion charge states in the 90 ° analysing magnet at an appropriate centre terminal voltage. The analysed beam passed through a 1.17 m thick concrete wall to the experimental room and was strong focussed on to a target situated about 10m from the analysing magnet. It was found that the neutron and y-ray background in the experimental room was negligible and no shielding of the y-ray detectors was required. Three types of experiment were performed: a direct measurement of the v-ray spectrum and yields from the target, a coincidence measurement between pairs of y-rays and a coincidence measurement between y-rays and oxygen ions backscattered off the target. Thick targets of the metallic separated isotopes t were used throughout and were prepared by lightly pressing 15 mg of the metallic powder on to a 0.076 mm thick lead foil. The resulting mean target thickness was about 30 mg/cm2; more than adequate to stop the oxygen ions. The lead foil was retained in a small thin-walled aluminium cup. The target chamber used for the direct y-ray and y-y coincidence measurements was a stainless steel cylindrical cup 7.5 cm long and 1.59 cm diameter, having a wall thickness of 0.127 mm. The target and holder were placed at the closed end of the cup and a guard ring, positioned close to the mouth of the cup and kept at a potential of about --100 V, suppressed secondary electrons. The y-rays were detected in 7.6 cm dia. × 7.6 cm long or 4.43 cm dia. × 5.06 cm * Separated isotopes were obtained from "The Isotope Division", O.R.N.L., Oak Ridge, Tennessee, U.S.A.

192

D. ECCLESHALL et al.

long NaI (T1) crystals mounted on EMI 9578B or RCA 6342A photomultipliers. For the coincidence measurements between r-rays and scattered oxygen ions, the oxygen ions were detected in an annular gold-silicon surface barrier counter. The incident beam passed through the centre hole of the silicon annulus on to a target about 5 cm beyond it, and was prevented from hitting the silicon by a cobalt stop. With this system it was possible to detect oxygen ions scattered through a small angular range close to 180 ° (about 171 ° to 176 ° in the laboratory system). Two target chambers were used for the experiments with the junction counter. One of these empoyed a stainless steel cup similar to the one described above with the junction counter mounted close to the mouth of the cup. This arrangement had the advantage that the charge collected at the target could be measured reliably and although this measurement was not required to interpret the data, it was valuable as a check on the stability of the junction counter and its associated circuitry. The chief disadvantage was that when y-rays were detected at about 90 ° to the beam direction unreliable corrections for ~-ray absorption had to be made. Additional information at these angles was obtained using a conventional cylindrically symmetric scattering chamber. The silicon used to make the junction counter had a resistivity of about 300 ~Q. cm and the evaporated annular gold film was about 100/~g/cm 2 thick. The performance of the counter was such that, under optimum bias conditions, the resolution (full width at half height) for 5 MeV ,c-particles was ~ 1 % . During the Coulomb excitation experiments the noise and leakage current in the junction counter increased slightly, thus causing a decrease in the bias applied to the counter because of the voltage drop across the counter load resistance. In order to minimize changes in output pulse height from this effect the pulses from the counter were fed into a charge sensitive preamplifier similar to the one described by Blankenship and Borkowski 20). A typical spectrum of oxygen ions backscattered off a thick target is shown in fig. 1. A discriminator was set to accept only pulses greater than a certain height, thereby defining the target thickness. To avoid errors resulting from a change in the resolution of the spectrum the peak to peak noise level in the counter was not allowed to rise above about 4 % of the minimmn pulse height selected by the discriminator. The experimental runs were stopped periodically to allow the counter noise to subside. It was found that recovery could be accelerated by letting air into counter assembly. Typically, using a beam of 10 l° oxygen ions per second, giving an overall counting rate in the junction counter of about 2000 per sec, and with a bias of about 20 V, it was necessary to "condition" the counter for about 10 min every 45 min. Using this procedure it was found that at a given bombarding energy and for a given target the ratio of the number of counts recorded in the junction counter to the charge collected on the target cup remained constant to within 4 ~o over a period of 24 h. This

193

COULOMB EXCITATION

variation could have been caused by small gain changes in the amplifier and discriminator. In the coincidence experiments fast and slow outputs were taken from each counter and fed into a fast-slow coincidence circuit set to a coincidence resolving time, 2% of about 24 nsec. The output of the coincidence circuit was used to IOOOO

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of oxygen

ions backscattered off a thick target of 40 MeV.

o f P d l°a a t a b o m b a r d i n g

energy

open a gate which allowed pulses from one or other of the counters to be recorded on a 100-channel kicksorter. A discriminator or single channel analyser was incorporated in one of the counter channels in order to select only a certain range of pulse heights from this counter, the number of pulses being recorded on a scaler.

3. The Coulomb Excitation Theory Arguments similar to those used b y Stelson and McGowan 3) show that only the Coulomb excitation mechanism can cause the excitation of levels in the

194

et al.

D. ECCLESHALL

targets at the bombarding energies used. The formulae used for calculating the total cross-sections which are required for the interpretation of the direct y-ray and 7-~ coincidence measurements have been taken from the review article b y Alder et al. 1~) and are the results of perturbation theory. Since in the experiments the particle trajectories are classical (~ >> 1) the total crosssection for direct E2 excitation is given b y

/Zle\ 2

=

-

(1)

,

Z1Z2e 2 ~ 0 Vl Vf '

where Z 1 is the atomic number of the incident projectile, Z 2 the atomic number of the target nuclei, vi the initial relative velocity of the target and projectile, vt the final relative velocity of the target and projectile, a the symmetrised parameter representing half the distance of closest approach in a head-on collision, mo the reduced mass of projectile, BEe the reduced electric quadrupole transition probability for E2 excitation, and ]~.e(~) tile classical E2 Coulomb excitation function. Since in the experiments the bombarding energy E is very much greater than the excitation energy A E of the excited level, we can write a

The total cross section for the double E2 excitation of a J + state through an intermediate 2 + state m a y then be written (*E2E2 = 0.0272 CjFj(}I, }=)a-=aEe(0q - -+ 2 + ) a E 2 ( 2 + --+ J + ) ,

(2)

where Cj = 21 =

1 for J = AE(O+

2 , 4 and C]----~ for J =

-+ 2 + ) ,

20. =

AE(2+

o, -+ J + ),

while F1(21, 22) has to I be calculated from the detailed expressions for double excitation given b y Alder et al. 14). The constant 0.0272 is chosen to make F j(21,22) = 1 in the sudden approximation 21 = ~2 = 0, when eq. (2) reduces to the expression used b y Newton and Stephens 17). A. C. Douglas zl) has calculated F2(21, 23) for several values of 21 and ~2 up to ~i ---- ~2 = 0.5 and these are shown in table 1. In the present experiments the cross-sections for $1 from

COULOMB EXCITATION

195

about 0.2 to 0.4 contribute significantly to the double excitation yields and ~J~l ranges from about 1.2 to 1.5. TABLE I

F](~I, ~2)

Values of s

Fo 0.0 F 2 F4 Fo 0.1 F 2 F4

0,0

O.1

0.2

0.3

0.4

0.5

1.000 1.000 1.000

0.951 0.963 0.974

0.934 0.975 "1.014

0.944 1,021 1.106

0.972 1,091 1.241

1,014 1.180 1.413

0.836 0.885 0.948

0.775 0.858 0.978

0.748 0.866 1,051

0.743 0.896 1.157

0.752 0.942 1.292

0.687 0.802 0.990

0.642 0.784 1.044

0.620 0.790 1.126

0.613 0.811 1.233

0.583 0.747 1.078

0.551 0.735 1.141

0.535 0.741 1.227

0.511 0.710 1.186

0.488 0.704 1.255

Fo 0.2 F s F~ Fo 0.3 F I F4 Fo 0.4 F~ F4 Fo 0.5 F 2 Fa

0.461 0.688 1.308

Equation (2) can therefore be used, at least for the excitation of a 4 + or 0 + level, to obtain the reduced transition probabilities, B~(2-}- -+ J + ) , which can be compared with the predictions of the simple vibrational model, viz.,

B~.(2 + --> J + ) - 2(2J+ 1) BE2(0+

2+)

25

The excitation of the second 2 + level can proceed b y direct E2 as well as double E2 excitation and there will be interference between the two processes 3, 16). Equations (1) and (2) are valid only when the total probability for excitation in a given Rutherford orbit, P(O), is small, i.e. when

p(o)-

dace (0) doR(0) << 1,

where dac~ is the differential cross-section for Coulomb excitation and da R the Rutherford cross-section.

196

D. E C C L E S H A L L et al.

At the bombarding energies used the principal contribution to P(O) comes from the excitation of the first 2 + level, for which perturbation theory gives

[Z~e\ ~ PE2(O, ~) = 4 ~ ) a-4BE2(O+ --~ 2+)d/E2(0 ' ~:) sin-4 (½0),

(4)

where dJE~(O, ~) is the classical differential E2 excitation function given by Alder et al. 14). The maximum value of eq. (4) for the bombarding energies and targets used is shown in table 2. It can be seen that in some cases P(~, ~) is as high as 25 % where perturbation theory is expected to be unreliable. However, this value at 0 = z is a maximum and in the ~,-~,coincidence measurements the y-ray yields averaged over all angles, 0, are measured. Moreover since the targets used were thick there are contributions from ions with energies lower than the bombarding energy. Both these factors lower the effective value of P, and it is estimated that the resulting thick target yields will differ from those calculated from equations (1) and (2) by no more than 5 %. In the 016-7-ray experiments we measure directly the value of P(z, ~) averaged over a range of interaction energies. Exact calculations of the excitation probabilities of the various states in an ideal vibrational nucleus have been made by Alder and Winther 15) and by Robinson 16). The probability PN for the excitation of the level or set of levels having a principal quantum number N is given by 1

PN(O, ~) = ~

PN~(o, ~) exp [--PE2(O, ~)],

(5)

where PEe(O, ~) is given by equation (4). It can be seen f o r N ----- 1 that equation (5) gives the same result as perturbation theory apart from the exponential term which represents the depopulation of the ground state. This can also be shown to be true for N = 2, in which case it must be remembered that there is a triplet of states having J ~ = 0 + , 2 + , 4 + . For the ideal vibrational nucleus and for scattering through 180 ° where only the M = 0 subst'ates of the triplet are populated, the relative excitation probabilities for the states, P~(Ja), are given by P2(4+) : P 2 ( 2 + ) : P2(0+) = 18 : 10: 7,

(6)

and this result is independent of energy. Equations (5) and (6) can thus be used to compare the experimental results with those expected on the ideal vibrational model. In the nuclei investigated the energy level spectrum is not exactly harmonic, and the second 2 + state can be excited directly. For the excitation of the 4 + and 0 ' + levels allowance can be made for the fact that ~:2~ $1 in order to obtain values of BE2(2+ -->J + ) . Douglas ~1) using perturbation theory has calculated that, for scattering through

197

COULOMB EXCITATION

180 ° a n d for the range of ~z and ~ encountered in the present experiments, one can replace P~2(n, ~) b y P,.9.(n, ~1) × P~2( n, ~,), where PE2(n, ~z) is obtained from eq. (4) b y substituting the appropriate value of ~. The resulting error on the calculated excitation probabilities is less t h a n 2 % provided perturbation t h e o r y is valid, t h a t is provided t h a t the exponential term is close to unity. 4. The

Experimental

Results

Measurements on the direct ?-ray spectra were made with the 7.6 cm × 7.6 cm NaI(T1) crystal situated at 15 cm from the target and 125 ° to the beam direction. A typical ?-ray spectrum, t h a t for a P d 1°8 target and 35 MeV oxygen ions, is shown in fig. 2, from which it can be seen t h a t only the ?-rays from the 5000

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decay of the first 2 + level and the 0.511 MeV ?-rays from positon annihilation are observed. Since the angular distribution of the ?-ray from the decay of the first excited state is to a good approximation of the form

W(O) = l + a 2 P 2 ( c o s O)

D. ECCLESHALL et al.

198

and since P2(cos 125 °) ~ 0 the measurments can be interpreted directly in terms of the ~,-ray yield integrated over all angles. Hence the total y-ray yield, Yexp (number of ),-rays emitted per incident oxygen ion), is given b y Yexp --

A pe ev Q

,

(7)

where A is the number of counts in the photopeak, e v the photopeak efficiency of the v-ray counter, e the electronic charge, Q the total charge collected on the target, and p the charge state of the oxygen ions arriving at the target. The number of counts observed, A, was corrected for y-ray absorption and for counting losses, both of these corrections being less than 5 %. The photopeak efficiencies were obtained from the calculated total efficiencies and the experimental photofractions of Heath 22). It is believed from measurements made here with Co 6° and Na 22 sources, and from measurements made b y Bell et al. 2a), that the error on the photopeak efficiencies used is less than 5 %. The charge state, p, was taken as that selected b y the analysing magnet and so the possibility of charge changing collisions occurring in the drift tubing between the anlysing magnet and target has been neglected. It is estimated that such collisions are negligible and this was confirmed experimentally since the y-ray yield from a thick target was the same for a given oxygen ion energy independent of the charge state selected to within an experimental accuracy of 4 %. The calculated ~,-ray yield is obtained from the total cross-section aE2 b y the expression Yealc

N a I ~'Em,. F dE l - t = A i-t-~JE0 aE2 kd(ox)J dE,

(8)

where N A is Avogadro's number, A the atomic weight of the target material, the total internal conversion coefficient, I the isotopic abundance of the isotope, dE/d(px) the stopping power of the oxygen ions, E 0 the bombarding energy, and Era1n a low energy limit for which the integrand in eq. (8) is essentially zero. The stopping power used to calculate equation (8) was obtained b y multiplying the stopping power for protons having the same velocity as the oxygen ions b y the square of the average charge state of the oxygen ions at this velocity. The proton stopping powers were obtained from t'he empirical formulae given b y Whaling 24) and the average charge state from the curve given b y Papineau 25). Thus the stopping powers have been extrapolated from the proton stopping powers assuming a theoretical charge and velocity dependence. Unfortunately the empirical proton stopping power formulae of Whaling 24) are stated to be valid only for velocities which correspond to oxygen ion energies of greater than about 35 MeV in the targets used. Furthermore, the use

COULOMB

EXCITATION

199

t

+ ~ .

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D. ECCLESHALL et al.

200

of a universal curve such as Papineau's in order to obtain the average charge state is likely to lead to some inaccuracy 25-29). Some confidence can be placed in the method, however, from the generally good agreement obtained between earlier experiments of the authors 19) and other workers (see Alder et al. 14)). Because the energy dependence of the cross-sections is much stronger than that of the stopping power, the integral in equation (8) is insensitive to the energy 500

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CHANNEL NUMBER F i g . 3. T h e ~ - ? c o i n c i d e n c e

spectrum

f o r p d l ° S ; Eoz* =

40 MeV.

dependence of the stopping power. For the excitation of the 0.430 MeV state in Pd l°s using 30 to 40 MeV oxygen ions the relative yields calculated using different forms of stopping power, S, normalised at the bombarding energy are Ycale = 1 for S calculated by the method described, Yoale ----- 0.93 for S oc E -°.5, Yeale = 1.04 for S = constant. The stopping power used at the bombarding energy is therefore quoted with the experimental results so t h a t the values of BE. can be corrected without

COULOMB~XCITAT~OS

201

appreciable error as more accurate data on the stopping powers become available. The results on the direct excitation of the first 2 + state in Pd 1°8 and P d n° are shown in table 2. The values of Ymle have been derived b y substituting into the expression for (~2 the single particle value of B~2 ( 0 + --~ 2 + ) defined b y Alder et al. 14) so that Yexp/Yoalo gives the enhancement of the BE2. The Iooo [--

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CHANNELNUMBER Fig. 4. The 7-7 coincidencespectrum (or Pd11°; Eoz6 = 40 MeV. values of I and a used to obtain Yeale are also shown in the table together with the value of ¢ and P~2 (~, ¢) at the bombarding energy E 0 . The errors (standard deviations) in BR2 ( 0 + --> 2 + ) and Yelp include a 5 % error on the ?-ray counter efficiency and a 4 % error on the charge state, p, as well as the errors on the determination of A. The errors in Y~to arising from the uncert ai nt y in the stopping power and from the use of perturbation theory have not been included. The results obtained are in good agreement with those obtained b y Stelson and McGowan 8o) which are also quoted in the table. The ?-y coincidence measurements from which most of the coincidence

~02

D. ECCLESHALL et al.

yields were obtained were made with a very poor counter geometry in order to smear out the angular correlations and to obtain an optimum real to random coincidence ratio. Typically, the counter from which the energy spectrum was recorded was situated at 0 ° to the beam direction and at a distance of 1 cm from the target. The second counter in which only the photopeak of the 7-ray from the 2 + --> 0 + transition was selected was placed approximately 2 cm I000

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400

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tOO

NUMBER

Fig . 5. T h e ~-~ c o i n c i d e n c e s p e c t r u m for RuZ°~; Eo16 = 40 MeV.

from the target and 135 ° to the beam direction. Although strongly anisotropic angular correlations are expected for the ~-rays following the decay of a 4 + level which has been excited by the double E2 excitation mechanism ~1), it has been calculated that the assumption of isotropy would not lead to errors of greater than 6 % in calculating the coincidence yields integrated over all angles. In this geometry, however, positon annihilation radiation was detected with good efficiency. Also, ~,-rays scattered from one crystal to the other give rise to coincidences, but the direct ~,-ray spectra showed that there were no y-rays of sufficient energy and intensity to give peaks in the gated coincidence

COULOMB

EXCITATION

~03

spectra in the region of interest. In order to check this and to obtain some information on Coulomb excited ~,-rays which could not be resolved in energy from the annihilation quanta additional experiments were performed with both counters 5 cm or more from the target, at 90 ° to each other and with lead shielding between them. Typical ~-y coincidence spectra are shown in figs. 3, 4, 5 and 6. 500

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•

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tO0

CHANNEL NUMBER

Fig. 6. The ~-~ c o i n c i d e n c e s p e c t r u m for RuZ°~; Eoz6 = 40 MeV.

Because of the low background observed in the direct y-ray spectrum the coincidence yields can be normalised to the number of counts recorded by the single channel analyser which selected a small range of pulse heights centred on the maximum of the photopeak in the gating spectrum. The ratio LC.~xpof the yield of the y-rays in coincidence with the 7-rays from the first excited state to the yield of the first excited state y-rays is then given by

Y ( J + --->2+) -

Rexp -- Y (2+ --->-0 + )

-

A & N s'

(9)

204

D.

ECCLESHALL

el al.

where A is the number of counts in a photopeak of the coincident y-ray spectrum, e7 the photopeak efficiency of the coincident y-ray, and Ns the number of counts selected by the single channel analyser. Small ( ~ 5 %) corrections have to be made to A for electronic dead-time losses and y-ray absorption and to N s for background counts in the spectrum selected by the single channel analyser. Because very few ( ~ 1 % ) of the y-rays selected in the single channel analyser are from the decay of higher excited states we m a y write for the calculated yield ratio

r'" d Eo

aI

E

'dE

where a] is the cross-section for the excitation of the state J, and e is the fraction of decays of this state proceeding by y-ray emission to the first excited state. The absolute value of the stopping power is not needed to obtain Reale and the result is insensitive to the energy variation assumed for the stopping power for the reasons discussed in connection with equation (8). The results are shown in table 3. The errors quoted for Rexp arise mainly from the uncertainty in A but include a 5 % error on the photopeak efficiency. This 5 % error is assigned on the basis of measurements made of the variation of photopeak efficiency with source to counter distance using Cs1~7 and Hg ~°3 sources, and the measurements of the absolute efficiency at the larger distances referred to earlier. Where the level excited was also directly excited by aparticles in the work of Stelson and McGowan a0) we have calculated the yield ratio for the direct E2 excitation of the level, Reale (E2), using the values of BE2 (0@ ~ 2 + ) and eBE2(O + -+ 2 ' + ) given by these authors and quoted in the table. The errors shown on Realc(E2 ) are derived from the errors on the reduced transition probabilities used. The values of Realc for double E2 excitation, Reale (El, E2), have been obtained from equations (2) and (10) by putting Cj = 1, F] = 1, e = 1, BE2(2+ --->J-t-)/BE2(O + --+ 2 + ) = 1 and using the values of BE2(0÷ ~ 2 + ) quoted in the table. Where the ?,-rays observed were not also observed by Stelson and McGowan it is reasonable to suppose that they originate entirely from double E2 excitation. It is worth noting that if the level excited is a 4 + level then, because F4(81, 22) m 1 for the range of 21 and 22 encountered in the experiments, we can write: Rexp Rea,c(E2, E2)

J a

BE2(2-+---->4 + )

= ~ BE~(O+ - ~ ~ + ) "

The 016-y coincidence measurements on Pd l°s and Pd n° were made at a bombarding energy of 40 MeV, with the y-ray counter 6 cm from the target

COULOMB

205

EXCITATION

o~

r

.~c~

~~Oo~ ..o :-:

~

0

~> ~

"~

~> 0

u.-

o~

~ ~.~ -~ ~

a 0

~÷j C~

0

c~

c~

.<

~

~:a0 m

c~_H

~.~ -~'~ ~

I

÷ ~q

~+~ °-H

<>Hi ~,~ ~ E ~ ' ~

~9

~'~

"~

.

~

~

~

206

D. ECCLESHALL et al.

and at angles 0 °, 29~ °, 45 °, 67½° and 90 ° to the beam direction. In the case of Ru xo2 and Ru 1°*, measurements were made at the same bombarding energy and target to counter distance but at one angle, 90 °, only. Typical coincidence y-ray spectra are shown in figs. 7, 8, 9 and 10. It m a y be shown that only y-rays from the palladium and ruthenium isotopes in the targets contributed to the coincidence yield. The isotopic constitutions of the targets are shown in table 4 ZOO00

1

I

I

I

I

I

I

t

I

0.430 McV

15000

.510HEY

~10000

x20

/

5000

,/""-i:../,

iO

20

30

O.620MtV

V \

o.94oM,v \%.,.~.. .... ~.....~,, •

"~-.-".. 40

:. eee

~...-. eeteoee%% e

]

I

I

]

so

eo

7o

80

•

•

"•

e•

t 90

too

CHANNEL NUMBER

Fig. 7. T h e OZ~-y coincidence s p e c t r u m for P d l°s.

together with the energies of the y-rays which were produced from each isotope in observable intensity The known light element impurities in the target would not give backscattered oxygen ions of sufficient energy to be registered in the coincidence circuitry. Also, any high energy light particles (protons, ~-particles, etc.) emitted in nuclear reactions with these impurities would not have been registered because their range in silicon would have been greater than the thickness of the depletion layer in the junction counter. No significant quantities of heavy and medium e]ement impurities were present in the targets. In the coincident spectra some of the y-ray peaks of interest are not corn-

COULOMB EXCITATION

207

pletely resolved from one another and a computer programme which fits Gaussian line shapes to the photopeaks b y a least squares method was used in order to obtain the number of counts in and the position of each photopeak 81). The peak totals at the various angles were subjected to small corrections for counting losses, 7-ray absorption and, in the case of the 7-rays from the decay of the first excited state in the main isotope, for random coincidences. For Pd 1°~ 2000

I

I

I

I

I

I

I

r

l

0.37t,MeV

1600-

0.440McV

1200

O.S45MtV BOO I

°

4O0

~ , i xlO

j,/ tO

i

2O

0.8t0HtV

•.

30

40

50

60

70

80

90

100

CHANNEl. NUMBER

F i g . 8. T h e OZ6-7 c o i n c i d e n c e s p e c t r u m for P d zl°.

and Pd zl° the corrected totals A (0) were analysed to obtain the angular distribution and the integrated peak totals A o, b y fitting the results to an expression of the form A (0) = A o [ l + a 2 P 2 ( c o s O)+a 4 P4(cos 0)].

(12)

The angular distributions m a y be simply interpreted since when the oxygen ions are scattered through 180 ° only the M = 0 substates of the excited levels are populated. This result is independent of the reaction mechanism and, in particular, is independent of the oxygen ion energy. The distributions for the

D. ECCL~SHALL 8t al;"

208

y - r a y transitions likely to be e n c o u n t e r e d in these e x p e r i m e n t s and the corrections to t h e m for the finite size of the counters and the fact t h a t o x y g e n ions are not d e t e c t e d e x a c t l y at 180 ° are discussed in the Appendix. The p r o b a b i l i t y t h a t the interaction of a b a c k s c a t t e r e d ion selected in the

500

i

i

t

0.475MeV i

i

i

i

i

i

400

w

500

o 20o

! ~ 0,6i5 HeY

j

I00

_LJ

I0

20I

~x5

I.O00

I 30

40l

14eV

i ~ T "'..'i".'-'~ " . . T o'''.." 50 60 70 80 90 I00

CHANNEL NUMBER Fig. 9. The O1~-y coincidence spectrum for Ru 1°~.

e x p e r i m e n t s gave rise to a ),-ray is given b y

P~exp --

A o

e~,Ns

(13)

where N s is the n u m b e r of b a c k s c a t t e r e d ions selected. The calculated probabili t y for the excitation of a level, Peale, averaged over the range of interaction energies defined b y the b e a m e n e r g y E o and the discriminator in the junction c o u n t e r circuit, m a y be o b t a i n e d from equations (4), (5) and (6) and the

COULOMB

EXCITATION

9~09

expression

P daR o

P~,c = I f=, Eo

I. i -I dE

F dE l -I dffR Ld(0x) ]

'

(14)

dE

where E 0 - E 1 and E o - E~ are the range of interaction energies (target thicknesses) for inelastic and elastic scattering respectively. In the experiments the 2000

I

I

I

I

I

I

I

I

i

1600

0 360MeV w

1200

£

}, 800

0 . 5 3 0 MeV

//

400

.k"./

• J 10

•

Z0

50

°°~° 40

°°

°= .I '°

-. 50

eV

°°,°~°

50

70

°°°° 80

90

100

CHANNEL NUMBER

Fig. 10. The O'S-~ coincidence spectrum for Ru 1°~.

range of interaction energies was such that, to a very good approximation, E1 = E2. In principle it is possible to obtain the target thickness from the range of backscattered ions selected and the stopping power of the oxygen ions but in practice the stopping power is not known sufficiently well, especially at the lower energies which the scattered oxygen ions have on emerging from the target. The target thicknesses were therefore obtained from the experimental value of the probability of exciting the first excited state, _Pexp(2+), b y

~10

D. ECCLESHALL et al.

comparing it with the calculated values using equation (14) and the reduced transition probabilities given by Stelson and McGowan 3o). In order to obtain Pexp(2+) it was necessary to subtract from the probability of observing the first excited state 7-ray, P~exp (2+ -~ 0+), the contribution of the cascade TABLE 4

T h e isotopic a n a l y s i s of t h e t a r g e t s

(1) Target

pdtoS

p d no

(2) Isotope

102 104 105 106 108 110

(3)

(4)

Abundance

Associated 7 - r a y s (MeV)

(%)

<

0.1 0.3 0.7 3.2 94.7 1.1

102 104 105 106 108 110

0.10 0.35 0.86 7.28 91.42

RutOS

96 98 99 100 101 102 104

0.02 0.03 0.1 0.3 1.8 97.2 0.5

RulO.

96 98 99 100 101 102 104

< <

0.513

0.510 0.430

0.03 0.02 0.08 0.12 0.27 1.32 98.16

C o l u m n 1 gives t h e m a i n i s o t o p e in t h e t a r g e t . C o l u m n 2 lists t h e m a s s n u m b e r of t h e c o n s t i t u e n t i s o t o p e s a n d c o l u m n 3 gives t h e i r a b u n d a n c e . C o l u m n 4 s h o w s t h e energies of t h e 7 - r a y s p r o d u c e d b y t h e " i m p u r i t y " i s o t o p e s a n d w h i c h a r e o b s e r v a b l e in t h e 016- 7 coincidence e x p e r i m e n t s .

decays of the higher states. The magnitude of this contribution can be obtained from P~exp( J + -+ 2 + ) corrected in turn where necessary for the v-rays arising from the impurity isotopes in the target. Fortunately, in order to obtain the

211

COULOMB EXCITATION

+~

A

~+~

¢,

'6 ,-,,i

~

c~ ~

0

~

0 .¢x~ ~o

~.~ 0

. ~

x

.,t

~s.~

0

~.~

v

0

~'~

0 ,,,-i

0

v

X

"6 ~..~

i 0

£

~

+

&

-~.+

++ ¢,1

¢N1

0

0

+.+

+.+

[.-,

~ M

o

212

D.

latter contribution to a sufficient target thickness is required.

ECCLESHALL

accuracy,

et al. only an approximate

value of the

Table 5 shows the results for Pdlos and Pdllo. The y-rays observed and the corresponding levels excited are shown in columns 2 and 3. Column 4 lists the spins and parities of these levels as determined from the angular distributions or as known from the work to which reference is made. Column 5 gives P,,, the probability

of observing

the y-ray,

Py

exp,

corrected

where necessary

for

the cascade decay of higher excited states and the y-rays originating from the impurity isotopes. In order to compare the results with theory table 5 also gives the target thickness determined from Pex,(2+). This was used to calculate the probability, P zcale, for exciting all members of the two phonon triplet of the ideal vibrational model and the probability, EP~~~JO+ -+ 2’+), for direct excitation of the second 2f level with subsequent y-ray decay via the first excited state. For the calculation of EP~~~~,(O+ A 2’+) perturbation theory was used with the values of EB,,(O+ --f 2’+) quoted in table 3. 5. Individual

5.1.

THE

NUCLEUS

Cases

Pd’“*

Typical y-y and 016-y coincidence spectra are shown in figs. 3 and 7 respectively. Apart from the 0.430 MeV y-ray from the first excited 2+ state, y-rays of 0.510f0.010 MeV and 0.62OfO.010 MeV were observed which, since they were in coincidence with the 0.430 MeV y-rays, correspond to the excitation of states at 0.940f0.012 and 1.050f0.012 MeV. The 0.430MeV y-rayobservedin the y-y coincidence spectrum can be accounted for by random coincidences and by real coincidences with the Compton tails of the spectra of the higher energy gamma-rays. The level excited at 0.940 MeV in the present experiment can be identified with the 2+ state excited by Stelson and McGowan “) at 0.941 kO.007 MeV. It is not possible from our results to obtain any reliable quantitative information on this level because in the y-y measurements it is difficult to estimate the contribution to the cascade y-ray yield arising from positon annihilation quanta. Also in the 016-y measurements a large fraction of the observed 0.510 MeV y-ray yield could be attributed to the direct excitation of the first 2+ level in the impurity isotope Pd 106.No cross-over 0.940 MeV y-ray was seen in the O16-y coincidence spectra and a limit to its yield is shown in table 5. Because of the large errors on the cascade yield it is not possible to quote a meaningful cascade to cross-over ratio for the decay of this state. The angular distribution of the 0.620 MeV y-rays in coincidence with backscattered oxygen ions is shown in fig. 11 where the points are those obtained ,experimentally and the curve is that expected for the transition 4+ (M = 0) -+

COULOMB EXCITATION

213

2 + , including the finite geometry corrections (see Appendix). A least squares fit of the angular distribution (equation (12)) to the experimental points gives for the coefficients a 2 and a 4, as = 0.30-t-0.09 and a4 ---- --0.24-t-0.09. These m a y be compared with the coefficients expected for a 4 + ( M = 0)--> 2 + transition, namely a2 = 0.434 and a 4 =---0.230. The results are therefore consistent with the excitation of a 4 + level, although tile observed distribution could be fitted b y a 2 + (M ---- 0) --> 2 + transition with an E2/M1 mixture, d, of about --3.0. Additional evidence for the excitation of a 4-t- rather than a 2 + level can, however, be obtained from the ?-y coincidence yields and the fact 1

I

i

I

I

BOO

700

600

t

o 5O0 z

400 3O0 200

i_

ioo rI O

~ 22.5

__

I 45.0

I 67.5

I 90.0

L

GAMMA COUNTER ANGLE WITH BEAM DIRECTION DEGREES

Fig. 11. T h e a n g u l a r d i s t r i b u t i o n of 0.620 MeV y - r a y s in c o i n c i d e n c e w i t h 40 MeV o x y g e n ions b a c k s c a t t e r e d off P d l°s. The c u r v e is t h a t e x p e c t e d for a 4 + (M = 0) --~ 2 + t r a n s i t i o n .

t h a t the ),-ray was not observed by Stelson and McGow'an 3) and so was probably excited by double E2 excitation in our experiments. The observed ?-~, coincidence yield is about 0.75 times the double excitation yield calculated on the assumption t hat F] = 1 and B ( 2 + - - ~ J + ) / B ( 0 4 - - > 2 + ) = 1. For the excitation of a 2 + level table 1 shows t hat F 2 ~ 0.7 at the bombarding energies used. Thus the ratio of experimental to calculated yields would give B(2+--~2'+)/B(0+ ~ 2 + ) ~ 1; that is at least 2.5 times greater than expected on the collective models so far proposed. The 2 + level at 0.940 MeV can be identified as a collective level and it is unlikely t hat a second highly collective 2 + level exists at so close an energy. Recently Wahlgren and Meinke 33) and Bunker and Starner s3,~4) have

D. ECCLESI-IALL et al.

214

shown that a level in Pd l°s at 1.05 MeV is fed b y the decay of Ag l°s. This level decays through the first excited state and is identified as a 0 + level on the basis of 7-7 angular correlation measurements a3). Both groups have also studied the decay of a long lived isomeric state in Ag 1°8 and find evidence for a level at 1.05 MeV or 1.155 MeV, depending on the temporal ordering of a pair of 7-rays. Their results are consistent with a 4 + assignment for the level and it can therefore be identified with that excited in the present experiments at 1.05 MeV. It is possible that the 0 + state at 1.05 MeV is also excited b y double excitation. For the Ole-7 coincidence experiments the ideal vibrational model predicts that the ratio of the contributions to the observed coincidences from the 0 + state and the 4 + state, Io/I 4, is 0.39. Since the 7-rays are emitted isotropically from a 0 + state, the resulting angular distribution is W(O) = Io+I4W4(0). A least squares fit of this distribution to the experimental points gives lo/14 = 0.4s+0.47. Hence the experimental results do not exclude the excitation of a 0 + level in addition to the 4 + level in the proportions predicted b y the ideal vibrational model. If a 4 + level alone is excited the 7-7 coincidence measurements give B~o ( 2 + --> 4+)/Bp.2(0+ --> 2 + ) = 0.73-4-0.10. This result is a weighted mean of the results listed in table 3 and includes the corrections for the factor Fa which was taken as unity for Rml¢ quoted in the table. If there is a contribution from the 0 + level as well as the 4 + level then, taking the ratio of the reduced transition probabilities for feeding 4 + and 0 + levels that are given b y the vibrational model, it can be shown that the ratio of the transition probabilities has to be reduced b y about 15 % from the results quoted above, i.e. BE2 ( 2 + --> 4+)/B~2(0 + --> 2 + ) = 0.62-4-0.10. For the O1e-7 coincidence measurements (table 5) the observed probability for the emission of the 0.620 MeV y-ray, Pexp(0.620), is 0.00252±0.00023 compared with the calculated probability for exciting the two phonon triplet, P2eale = 0.005534-0.00069. After correcting P2ealc for the fact that the level spectrum in Pd l°s is not harmonic (see section 3) and using relation (6) we obtain Pexp (0.620) Peep (0.620) 1.394-0.22, = 1.00i0.16. Peale(4+) Pe~c(4+ and 0 + ) If a 4 + state only is excited this result m a y be stated: B E ~ ( 2 + - + 4 + ) / BE~(0 + --> 2 + ) = 1.004-0.16. If both 0 + and 4 + states are excited in the proportions given b y relation (6) for the phonon model we obtain BE2(2+ 4 + ) / BE2(0 + --->2 + ) = 0.725t50.11. -

5.2. T H E

NUCLEUS

-

Pd n°

Specimen y-y and O18-y coincidence spectra are shown in figs. 4 and 8 respectively. The energies of the y-rays from Pd n° are 0.370 MeV, 0.440 MeV

COULOMB ~JCCITATION

215

and 0.545+0.010 MeV although, as with Pd l°s, the 0.370 MeV ~,-ray observed in the ~-~, coincidence experiments can be attributed to random coincidences and to coincidences with the Compton tails of the higher energy y-rays. No information on the levels of Pd 11° is available from studies on radioactive nuclei, but the level at 0.810 MeV can be identified with the 2 + level excited by Stelson and McGowan at 0.8134-0.007 MeV. These authors also find some evidence for a second ~-ray at 0.552 MeV which can probably be identified with the y-ray at 0.545 MeV found in the present experiment. I

I

I

I

I

BOO

700

600 o

'-~ 500

~ 400 3~

200

IOO

I

I

I

0

22.5

45.0

L 67.5

I 90.0

GAMMA COUNTER ANGLE WITH BEAM DIRECTION. DEGREES

Fig. 12. T h e a n g u l a r d i s t r i b u t i o n of 0.545 MeV y - r a y s in c oi nc i de nc e w i t h 40 Me V o x y g e n i ons b a c k s c a t t e r e d off P d ix°. T h e c u r v e is t h a t e x p e c t e d for a 4 + (M = 0) --~ 2 + t r a n s i t i o n .

From the table 3 it can be seen that the yield of 0.440 MeV y-rays in the ~-~ coincidence experiments is greater than the yield calculated assuming only direct E2 excitation of the level at 0.810 MeV [Re~ > Rcaze(E2)3. Unfortunately, the cascade to crossover ratio and E2/M1 ratio are not known for the decay of the 0.810 MeV level and it is therefore not possible to calculate exactly the y-ray yield for double E2 excitation. By taking a cascade to crossover ratio of about three and almost pure E2 decay, which are typical of the neighbouring nuclei, and using the data given by Stelson and McGowan a) we obtain Reale(E2 E2, 0.440) ~ 1.5× 10-3. This calculation also includes a correction for the factor F 2 which was taken as u n i t y for the calculations quoted in the table. Hence the experimental yield ratio Rexp(0.440 ) = (5.8+ 0.9) × 10 -3 can be accounted for by a combination of direct E2 and double

D. ECCLESHALL el al.

216

E2 excitation without including a ny interference between the two mechanisms. U nf o r tu n ately in the O16-V coincidences measurements there is an overwhelming contribution to the yield at 0.440 MeV from the direct excitation of the first excited state at 0.430 MeV in the i m p u r i t y isotope Pd l°s and it is therefore not possible to obtain any reliable quantitative information. No crossover 0.810 MeV y-ray was seen and a limit to the observed excitation probability is shown in table 5. As with Pd 1°8, owing to large errors on tile cascade y-ray yield it is not possible to set a limit to the cascade to crossover ratio. The angular distribution of the 0.545 MeV y-rays in coincidence with backscattered oxygen ions is shown in fig. 12. The points are those obtained in the experiment and the curve is t hat expected for the transition 4 + (M = 0 ) - + 2 + after the finite geometry corrections have been included. A least squares fit of the angular distribution eq. (12), to the points gi-es a ~ - 0 . 3 6 + 0 . 0 7 and a4= --0.31+0.11. The results are therefore consistent with the excitation of a 4 + level at 0.915Q:0.012 MeV. Additional analysis for a possible contribution from a 0 + level was performed although there is no previous experimental evidence for its existence. The result obtained was lo/14 ---- 0.1910.24. Hence the possibility of a contribution from a 0 + level cannot be excluded. Treating the 0.545 MeV y-ray yield at 40 MeV in the 7-7 coincidence experiments in the same way as the Pd l°s results it is found that, if only a 4 + level at 0.915 MeVis excited, BE2(2+ -+ 4 + ) / B E 2 ( 0 + --->2 + ) = 0.704-0.15; whereas, if both a 4 + and a 0 + level are excited in the ratio predicted by eq. (6), BF.2(2+ -+ 4 + ) / B E z ( O + -+ 2 + ) = 0.60-4-0.13. Similarly, from the 016-V coincidence experiments it is found t ha t the ratios of the transition probabilities are 0.904-0.17 ( 4 + alone) and 0.644-0.12 ( 4 + and 0 + ) . The O16-7 and 7-Y results are compatible whether or not one invokes the excitation of a collective 0 + state in addition to the 4 + state.

5.3. T H E

NUCLEUS

R u l°z

Typical y-ray spectra obtained from the V-V and the O16-y measurements are shown in figs. 5 and 9. Only two nuclear y-rays were seen, at 0.475 MeV and 0.625 MeV, and th e y m a y be a t t r i b u t e d to the decay of states at 0.475 MeV ( 2 + ) and 1.100 MeV. The 0.475 MeV v-ray appears in the V-V coincidence spectra mainly as a result of random coincidences and the 0.510 MeV y-ray is from positon annihilation. The 0.625 MeV y-ray was observed b y Stelson and McGowan thus identifying a second 2 + level at 1.100 MeV. These authors also show from a s tu dy of the decay of Ru 1°2 that a 4 + level exists at approximately the same excitation energy 35). T hat this 4 + level is excited in the present experiment can be seen from an analysis of the results shown in table 3 similar to that performed for the 0.810 MeV level in Pd 11°. In Rui% however, the

COULOMB EXCITATION

217

cascade to crossover ratio (1.5±0.3), E2/M1 ratio and B ( 2 + - - > 2 ' + ) / B (0 ~ 2 + ) are known, giving a final value for the double excitation at 40 MeV of the 2 ' + l e v e l of Rcalc(E2 E2, 0.630)*= (0.56 ~0.20) × 10-z. E ven allowing for possible constructive interference between the direct and double excitations, the observed coincidence yield is considerably higher than the calculated one. If the interference term can be neglected as in Pd n°, then the expected 2 ' + yield ratio of ( 3 . 3 + 0 . 4 ) × 10-3 can be subtracted from Rexp to obtain BE~" ( 2 + -+ 4+)/BE2(0 + -+ 2 + ) = 0 . 6 6 i 0 . 1 7 . Quantitative analysis of the 01~-7 experiment is not possible since a measurement was made at only one y-ray angle. 5.4. T H E

NUCLEUS

R u 1°4

As with Ru 1°2 only two prominent 7-rays, 0.360 MeV and 0.530 MeV, were observed as can be seen in figs. 6 and 10. These y-rays were also observed by Stelson and McGowan a). The similarity of the results of the y-y coincidence measurements to those of Ru 1°2 can be seen from table 3 and provides very strong evidence that two states having spins 2 and 4 and positive parity exist at about 0.890 MeV excitation. Proceeding as in the case of Ru 1°2 and assuming that the cascade to crossover ratio and E2/M1 ratio are similar to those in Ru ~02 it is found th at BE2(2+ -+ 4+)/B~.2(0+ -~ 2 + ) = 0.57+0.14. In fig. 10 it can be seen that there is some evidence for a y-ray at 0.890 MeV which can be interpreted as the crossover y-ray from the second 2 + state. Measurements at other y-ray angles would be required to obtain quantitative information on the cascade to crossover ratio for the decay of the state.

6. C o n c l u s i o n s Collective 2 + and 4 + levels have been excited in the four nuclei at about twice the excitation energy of the first excited state. In Pd 1°8 it is possible on the basis of the 016-y angular distribution evidence t hat the 0 + level in Pd 1°8 at the same excitation energy as the 4 + state is also excited and is therefore collective. In the other isotopes a collective 0 + state has not been identified b u t m a y still exist. It is predicted that this 0 + state would be excited more weakly than the 2 ' + or 4 + state and the y-ray emitted in its decay may not be resolved in energy from the other y-rays observed. Also, if the collective 0 + state were at a much higher excitation energy than the 2 ' + and 4 + states (e.g. the 0 + state in Ru 1°2 at 1.84 MeV 35)), it would not have been observed in these experiments. In Pd n° it was possible to compare the experimental and calculated yields of the cascade y-ray from the decay of the second 2 + state. It was found t hat contributions from both the direct and double Coulomb excitation mechanisms are required to account for the experimental results. The observed yields of y-rays from the 4 + states have been interpreted

D. ECCLESHALL Bt a l .

218

to obtain values of B~2(2+ -+ 4 + ) / B E 2 ( 0 + --> 2 + ) and these are given in section 5 together with the various assumptions made. The ratio B~2(2+ --> 4 + ) / B ~ e ( 0 + -+ 2 + ) can be calculated for the ideal vibrational model (equation (3)) and for the asymmetric rotor model n). A collective 0 + level is not admitted within the framework of the latter model and the value of B~2(2+ --> 4 + ) / B E ~ ( 0 + -+ 2 + ) is predicted to be between 0.49 and 0.51 (depending on the parameter 7)- If in Pd l°s only the 4 + state were excited the y-V and 0~-~ coincidence experiments give values of 0.73 40.10 and 1.00±0.16 respectively, for the ratio of the transition probabilities.

3"0t

f 2÷

2+

4, (o+)

~

4*

--

2+

4+ ~

2

÷

g ..-

L0

2+

2+

O+ RU102

0÷ Rut°4

PdIo$

2~

r] .Z+

0*

- 0+ PdllO

Fig. 13. The energy level spectra showing the 7-rays observed.

These are significantly higher than the predicted value. The same conclusion m a y be drawn from the experiments on P d n° although in R u 1°~ and R u 1°4 the errors on the quoted values of BE2(2+ -* 4 + ) / B E 2 ( 0 + --> 2 + ) are large enough to give agreement with the asymmetric rotor model. The observed coincidence yields in Pd l°s are, however, consistent with the predictions of the phonon model. Thus, if both the 0 + and 4 + levels at 1.05 MeV were excited in the proportions predicted, the values of BE2(2+ --~ 4 + ) / BE2(0 + ~ i + ) obtained in the 7-7 and Ole-y measurements are 0.62~:0.10 and 0.724-0.11 respectively, compared with the predicted value of 0.72. The results obtained for the transition probability ratio in the other isotopes agree well with the result for Pd l°s.

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219

The theory of D a v y d o v and Chaban 12) admits a collective 0 + state, b u t Klema et al. 13) have shown that it is not possible to reconcile completely the energy spectrum of Pd 1°8 with this theory if the 0 + level at 1.05 MeV is the fl-vibrational state. The reduced transition probabilities have not been calculated using this model. In Ru 1°* McGowan and Stelson ~) have calculated that the D a v y d o v and Chaban theory requires that the 0 ' + s t a t e is about 0.100 MeV higher than the 2 ' + , 4 + doublet. No r-ray corresponding to a 0 ' + -~ 2 + transition of about 0.725 MeV was observed in the experiments and it is estimated that if a 0 ' + level at 1.200 MeV exists BR,(2 + -+ O'+)/BE2(O+ -+ 2 + ) < 0.1; i.e. less than one third of the value predicted b y the ideal vibrational level. Finally the low lying energy level spectra for the four nuclei are shown in fig. 13 together with the r-ray transitions observed. We are indebted to J. Harrington-Lynn and R. A. Brown for assistance with the experiments and to Drs. A. C. Douglas and N. MacDonald for valuable discussions on the results and their interpretation. We are also grateful to Dr. K. W. Allen for helpful discussions and for encouragement during the course of the work.

Appendix THE

ANGULAR

CORRELATIONS

Litherland and Ferguson 3e) have pointed out that states resulting from nuclear reactions which have axial symmetry can be regarded as aligned states, and can be described in terms of a relatively small number of population parameters for the magnetic substates. If in a two body nuclear reaction the outgoing particle is detected in a counter at 0 ° or 180 ° relative to the incident beam, then, since the orbital magnetic quantum numbers of the incoming and outgoing particles are zero, the population of the magnetic substate of the state in the residual nucleus will be limited b y the spins of the target nucleus and of the incident and emergent particles. In fact for a spin zero particle in and out, and a spin zero target, only an M = 0 substate of the residual nucleus can be attained. It is of course impossible to place a counter at precisely 180 °, but the use of a counter symmetric about the beam axis, i.e. a circular counter with a hole in it, retains the axial symmetry of the system and allows formulae for counter Aze corrections to be developed. Since substates other than M ----- 0 will now be involved this requires a knowledge of the reduced matrix elements for the transitions involved. If the annular counter is small enough for its size corrections to be neglected, so that only the M = 0 substate is excited, no knowledge of the reaction process is required. Then the observed angular distribution of

D. ECCLESHALL et al.

220

a y-ray in a transition from a state in the residual nucleus of spin J to a state of spin I is given by

W (O) = WLL (O) + 2~WLL,(O ) +,~2WL, L, (0), where

I~/*LL'(O) •

X k

(--1)I+L+L'+~'k(J, J, O, OIkO)ZI(LJL'J, Ik) Jk 7- pk(cos 0), Jo

while 6 is the appropriate multipole mixing ratio (E2/M1 in our case), L L ' are the interfering multipolarities for the y-ray, Z 1 is the coefficient tabulated by Sharp et al.aY), (jjOOlkO) is the Clebsch Gordan coefficient, also tabulated in ref. aT), Jk/Jo is the angular distribution attenuation coefficient for the y-ray counter (see below), and Pk (cos 0) is the Legendre polynomial. For the transitions likely to be encountered in the present experiments the first y-rays emitted after excitation have the following distributions, aside from the coefficients Jk/Jo:

2 -[- ---->0 + ,

W(O) = l+{-P2(cos O)--~P4(cos 0),

0 + ~ 2+,

Isotropic,

2'+

w(o) = 1 + (½-

2+,

4 q- -+ 2 + ,

p

(cos o)

' P , ( e o s 0)

I~V(O) = 1 + 4~P2(cos O)--lSp4(cos 0). 49

It is easily shown that the second y-ray of the cascade (0+, 2 + , 4 + ) M = 0--~ 2 + - + 0 + has the same angular distributions as the first, provided = oo. The expressions for the attenuation coefficients Jk for an axially symmetric counter have been given by Rose 38) and values of Jk/Jo have been tabulated by Rutledge ag) for the 7.6 cm x 7.6 cm NaI (T1) crystals used in the experiment. However, these calculations have assumed that all the pulses produced by the interaction between a y-ray of a given energy and the crystal were detected and included in the analysis of the experimental data. As a consequence they represent lower limits if only the photopeak area is measured. In order to obtain the corrections for the finite size of the annular counter it was assumed that only the M = 0 substate is populated in the centre of mass co-ordinate system in which the Z axis is along the s y m m e t r y axis as the hyperbolic path of the scattered particles 15, 16). This assumption is certainly justified for direct E2 Coulomb excitation of the first excited state. At ~ = 0.4 (the worst case) scattering through 160 ° instead of 180 ° changes the coefficients a2 by --0.6 °/o and a 4 by --2.1 °/o where now the angular distribution is measured on the co-ordinate system described above. In the experiments the largest angle subtended by the annular counter corresponded to a scattering angle of 172.6 ° and the mean angle was 174.4 ° .

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221

It m a y be shown that the problem reduces to finding geometric coefficients, similar to those for the v-ray counter, for an "imaginary counter" in the centre of mass system of half the angular size of that actually used. From the expressions given by Rose, and assuming that the angular distribution of the excited recoiling ions is isotropic in the angular range detected, it m a y be shown that the attenuation coefficients for an annular counter detecting charged particles are given b y Jk = f~' Pk(cos/5) sin/3 d/~.

]

I

I

z

3ooo-

2000

-

o

IO00--

~

~

-t---

I

I

I

[

I

0

22.5

45.0

6715

90.0

,_

GAMMACOUNTERANGLEWITHBEAMDIRECTION.DEGREES, Fig. 14. The a n g u l a r d i s t r i b u t i o n of 0.430 MeV ~ - r a y s in c o i n c i d e n c e w i t h 30 MoV o x y g e n i ons b a c k s c a t t e r e d off P d x°8` The c u r v e is a l e a s t s q u a r e s fit of eq. (12).

The counter used in the experiment subtended half angles of 4.02 ° and 8.62 ° (lab. system) corresponding to half angles Yl and y~ of 1.73 ° and 3.71 ° for the imaginary counter. This gives J~/Jo = 0.996 and JJJo = 0.987. Because only the photopeak area in the gamma spectrum was measured it was thought desirable to measure the attenuation coefficients directly. An experimental measurement yields the product of the attenuation factors for the y-ray and annular counters. Such a measurement was made of the y-rays from the first excited (2+) state of Pd z°8 at an incident beam energy of 30 MeV (fig. 14). The low energy was chosen to remove any effective contribution from cascade y-rays from higher states. The following results were obtained: a s = 0.600-~-0.013,

a 4 = --1.035i0.010.

222

D. ECCLESHALL et al.

Theory gives a 2 = ~ and a, = _ 1 # for an unattenuated distribution. The measured combined attenuation coefficients are thus

Jz/JolsxfJJol, = 0.8404-0.018,

JJJola×JJJolr = 0.6044-0.006.

The table of Rutledge gives for the 7-counter JJJo ~- 0.830 and JJ]o = 0.520. The ratio (JJJo)/(J2/Jo) given by Rutledge may be seen to lie almost on a universal curve for a 7.6 cm × 7.6 cm NaI(T1) crystal at different distances and for different 7-ray energies. By assuming a similar universal curve through the alues of experimental attenuations measured, combined attenuation coefficients were obtained which were used in subsequent analysis. ~i ~ r attenuation factors may be introduced by multiple scattering of the bomb~:iding ions, or by the precession of the angular momentum vector of the excited nucleus in the atomic electric and magnetic fields. For the former effect the r.m.s, scattering angle was calculated to be of the order of a degree, and a lower limit of 10 -1° sec to the lifetimes for which the latter effect is significant has been suggested by Alder et al. la). Both these effects have therefore been ignored in the calculations. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26)

G. Scharff-Goldhaber and J. Weneser, Phys. Rev. 98 (1955) 212L D. Strominger, J. M. Hollander and G. T. Seaborg, Rev. Mod. Phys. 30 (1958) 585 P. H. Stelson and F. K. McGowan, Phys. Rev. 121 (1961) 209 L. Wilets and M. Jean, Phys. Rev. 102 (1956} 788 B. J. Raz, Phys. Rev. I f 4 (1959) 1116 T. T a m u r a and L. G. Komai, Phys. Rev. Letters 3 (1959) 344 G. R. Satchler, Comptes Rendus du Congrbs International de la Physique NucMaire, 1958 (Dunod, Paris, 1959) p. 768 M. Jean, Nucl. Phys. 21 (1960) 142 R. K. Sheline, Rev. Mod. Phys. 32 (1960) 1 A. S. Davydov and G. F. Filippov, Nucl. Phys. 8 (1958) 237 A. S. Davydov and V. S. Rostovsky, Nucl. Phys. 12 (1959) 58 A. S. Davydov and A. A. Chaban, Nucl. Phys. 20 (1960) 499 E. D. Klema, C. A. Mallman and P. Day, Nucl. Phys. 25 (1961) 266 K. Alder, A. Bohr, T. Huus, B. Mottelson and A. Winther, Rev. Mod. Phys. 28 (1956) 432 K. Alder and A. Winther, Mat. Fys. Medd. Dan. Vid. Selsk. 32, No. 8 (1960) D. W. Robinson, Nucl. Phys. 25 (1961) 459 J. O. Newton and F. S. Stephens, Phys. Rev. Letters 1 (1958) 63 B. Elbeck, G. Igo, F. S. Stephens and R. lY[. Diamond, Proc. Sec. Conf. on R e a c t i o n s b e t w e e n Complex N u c l e i , G a t l i n b u r g (John Wiley and Sons, Inc., 1960) B. M. Adams, D. Eccleshall and M. J. L. Yates, Proc. Sec. Conf. on R e a c t i o n s b e t w e e n Complex Nuclei, Gatlinburg, (John Wiley and Sons Inc., 1960) J. L. Blankenship and C. J. Borkowski, I.R.E. Trans N S 8 (1961) 17 A. C. Douglas, p r i v a t e c o m m u n i c a t i o n R. L. Heath, I.D.O. 16408 (July, 1957) unpublished F. R. Bell, R. C. Davis and N. H. Lazar, O.R.N.L. 1975, September 1955, p. 72, unpublished W. Whaling, H a n d b u c h der Physik (Springer-Verlag, Berlin, 1958) Vol. 34, p. 193 A. Papineau, Comptes Rendus 242 (1956) 2933 I-L IZ. Heckman, B. L. Perkins, W. G. Simon, F. M. Smith and W. H. Barkus, Phys. Rev. 117 (] 960) 544

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27 28 29 30 31 32 33 34 35 36 37

P. G. Roll and F. E. Steigert,Nucl. Phys. 17 (1960) 54 P. G. Roll and F. E. Steigert,Phys. Rev. 120 (1960) 470 L. C. Northcliife,Phys. Rev. 120 (1960) 1744 P. H. Stelson and F. K. McGowan, Phys. Rev. 110 (1958) 489 M. J. L. Yates, unpublished (1961) M. A. Wahlgren and W. W. Meinke, Phys. Rev. 118 (1960) 181 M. E. Bunker and J. w. Stamer, Bull. Am. Phys. Soe. 5 (1960) 253 M. E. Bunker, privatecommunication F. K. M c G o w a n and P. H. Stelson, Phys. Rev. 123 (1961) 2131. A. E. Litherland and A. J. Ferguson0 Can. J. Phys. 39 (1961) 788 W. T. Sharp, J. M. Kennedy, B. J, Sears and M. G. Hoyle, Chalk River Report C.R.T. 556

38 39

M. E. Rose, Phys. Rev. 91 (1953) 610 A. R. Rutledge, Chalk River Report C.R.P. 851 (1959)

(1953)