- Email: [email protected]
Determination of the intrinsic quadrupole moment of the superdeformed band in 143Eu
NUCLEAR PHYSICS A FJAEVIER
Nuclear Physics A 584 (1995) 149-158
Determination of the intrinsic quadrupole moment of the superdeformed band in 143Eu S.A. Forbes a, A. Ata9 b,c, G.B. Hagemann b, B. Herskind b, S.M. Mullins a,l, EJ. Nolan a, j. Nybergb,a, M.J. Piiparinen b'e, G. Sletten b, R. W a d s w o r t h f a Oliver Lodge Laboratory, University of LiverpooL Liverpool, L69 3BX, UK b Niels Bohr Institute, Tandem Accelerator Laboratory, University of Copenhagen, DK-4000 Roskilde, Denmark c Department of Radiation Sciences, Uppsala University, BOX 535, S-75121 Uppsala, Sweden o The Svedberg Laboratory, Uppsala University, BOX 533, S-75121 Uppsala, Sweden e Department of Physics, University of Jyvdskyld, PO Box 35, FIN-40351 Jyvdskyld, Finland f Department of Physics, University of York, Heslington, York, YO1 5DD, UK
Received 30 August 1994
Abstract The intrinsic quadrupole moment, Qo, of the superdeformed band in ~43Eu has been extracted from nuclear lifetimes, measured in a Doppler shift attenuation method experiment. Data were collected with the NORDBALL gamma-ray spectrometer following the reaction H°Pd (37C1, 4n) 143Eu at a beam energy of 160 MeV. A centroid shift analysis of the data was carried out, using the Braune stopping powers to model the slowing down of the recoil nuclei in the target and backing materials. This gave a value of Q0 = 13.0 =k 1.5 eb for the band. The result is in good agreement with the theoretical prediction of Total Routhian Surface calculations, and supports the assigned single particle configuration, involving occupation of both neutron and proton i~3/2 intruder orbitals from the Nosc= 6 oscillator shell. Keywords: NUCLEAR REACTIONS u°Pd (37C1, 4n), E = 160 MeV; measured Ey Doppler shifts. 143Eu superdeformed band deduced r, Qo, 132.
1 Present address: Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, L8S IK1, Canada. 0375-9474/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 5 - 9 4 7 4 ( 9 4 ) 00498-6
150
S.A. Forbes et aL/Nuclear Physics A 584 (1995) 149-158
1. Introduction The origin of superdeformation lies in the formation of a second energy minimum in the fission barrier, corresponding to highly deformed prolate nuclear shapes. Favourable shell corrections, as calculated from nuclear mean field models such as with the harmonic oscillator potential, stabilise these shapes. These pronounced shell gaps in the energy level spectrum at large deformations remain when considering a more realistic potential, such as in the Woods-Saxon or Nilsson model, and form a chain across a range of deformation values [ 1 ]. When taking rotation of the nucleus into consideration for high spin states, it is found that the energy level spectrum is largely unaffected, apart from the so-called intruder orbitals from higher oscillator shells. These are greatly lowered in energy and only reach the Fermi surface at large deformation. As the intruder orbitals are strongly deformation driving, it is expected that the deformation of a nucleus at high spin depends on the number of intruder orbitals occupied in the configuration. It is thus important to interpret superdeformation in terms of the characteristic intruder orbitals occupied [2,3]. The collective rotation of superdeformed nuclei gives rise to characteristic cascades of gamma-rays, with a narrow energy spacing between the transitions, indicating a large dynamic moment of inertia. Although this provides a means of identification, the superdeformed nature of the bands must be confirmed by a measurement of the deformation. This can be done by an experimental measurement of the lifetimes of the nuclear states in the superdeformed band. The reduced transition probabilities are then calculated from the measured lifetimes according to the rotational model, and the intrinsic quadrupole moment extracted from these results. The quadrupole deformation parameter/32, as defined for an axially symmetric nuclear shape, is then calculated from the measured quadrupole moment. The first cases of superdeformation were discovered in the nuclei ~32Ce [4] in the mass A ~ 130 region and 152Dy [5] in the mass A ~ 150 region. Many other superdeformed bands have now been observed in these regions, and also several in the mass A ~ 190 region. The deformation has been obtained in several cases, by measurements of the quadrupole moments in lifetime experiments. In the mass A ~ 130 region these bands are found to have a quadrupole deformation parameter of/32 ~ 0.35-0.40, corresponding to a deformed shape with a major:minor axis ratio close to 3:2 (e.g. 132Ce [6] ). The superdeformed bands in the mass A ,-~ 150 region originate from more deformed nuclei, with values of/32 " " 0 . 5 5 - 0 . 6 0 , giving axis ratios near 2:1 (e.g. 152Dy [7] ). The difference in deformation between the mass regions is expected from the position of the shell gaps at different deformations in the energy level spectrum. A highly regular rotational structure has been discovered in 143Eu [8,9] and was the first such example found in a nucleus intermediate between the mass A ~ 130 and A ,.~ 150 regions. It was thus important to measure the quadrupole moment of the band and hence compare the deformation with other mass regions. The band was found to have an intensity of approximately 1% of the 143Eu channel in the reaction [9]. The spin assignments for the band are taken from [9] based on the discovery of linking
S.A. Forbes et al./Nuclear Physics A 584 (1995) 149-158
151
transitions. It should be noted that the band was originally assigned to the nucleus 142Eu [8] but evidence since then has indicated that the correct assignment is to 143Eu [9]. Parts of this work have been published in conference proceedings [ 10]. Theoretical calculations have been performed by Chasman [ 11 ] to predict the relative excitation energy of the superdeformed well, and the depth of the superdeformed well for nuclei with 56 < Z < 70 and 72 < N < 88. The conditions to stabilise superdeformation are (i) a small relative excitation energy of the superdeformed well, to optimise the chance that the states will be populated by the energy brought in through the reaction, and (ii) a deep superdeformed well, so that tunnelling through to the normal deformed states does not occur too rapidly. These calculations had indicated that the nucleus 143Eu (N = 80, Z = 63) would be a favourable candidate. The superdeformed well is stabilised by the shell gap at deformations of f12 ~ 0.5-0.6, for neutron number N = 80, and is predicted to be yrast at a relatively low spin. In order to further investigate the systematics of the shell gaps, it is important to check the predicted deformation of this first superdeformed band found the mass A ~ 140 region.
2. Doppler shift attenuation method In a heavy-ion fusion-evaporation reaction, the recoil nuclei produced with initial recoil velocity v0, slow down in the target and backing materials and are eventually brought to rest. The Doppler shift attenuation method (DSAM) [12], is based on the fact that the lifetimes of many nuclear states are of the same order of magnitude as this slowing down time (a few tenths of a picosecond). Gamma-rays depopulating a particular excited nuclear state will be emitted after an average time t, related to the mean lifetime r of that state. At time t, the recoil will have slowed down and have an average attenuated velocity ~. Stopping theories must be used to model the slowing down of the recoils, and thus relate velocity to time, and hence lifetimes, of the states.
2.1. Centroid Shift Analysis A Centroid Shift Analysis [ 13] was performed on the DSAM data to measure the lifetimes of the states in the superdeformed band in 143Eu. If a particular state emitting a gamma-ray of known energy E0, decays in flight, the gamma-ray energy will be Doppler shifted. The shifted energy, Es, is measured from a gamma-ray spectrum taken with detectors at an angle of 0 with respect to the beam direction (assuming that the recoils are emitted in a narrow forward cone at 0°). The average attenuated velocity for this state can then be calculated from the equations:
E~= Eo 1 + - c o s 0 c
~
AE= V-EocosO, c
(1)
152
S.A. Forbes et al./Nuclear Physics A 584 (1995) 149-158
where AE = [Es - Eo[ is the Doppler shift. The Doppler Shift Attenuation Factor F ( r ) is defined as (2)
F(T) = --,
u0
where the initial recoil velocity v0 is calculated from reaction kinematics. The F ( T ) value measured for each transition in the band is plotted against the transition energy to give an F ( r ) curve. It is assumed in the calculations that the quadrupole moment does not change down the band. For a particular quadrupole moment, the lifetimes of the states in the band can be calculated from the rotational model, and a theoretical F ( z ) curve plotted. The quadrupole moment is changed in the calculations until the best fit to the experimental F ( r ) curve is found. For each chosen quadrupole moment, the computer code FTAU [ 14] was used to calculate the intrinsic lifetime of each state in the band. The slowing down processes of the recoiling nuclei in the target and backing foils were modelled using the Braune stopping powers [ 15]. This allowed the intrinsic F/(7) values to be calculated. The code FFEED [ 14] then corrects these intrinsic values to take into account in-band feeding. This is done by solving the Bateman equations [ 16] to give cumulative F(~-) values, which are plotted along with the experimentally measured values. It is assumed in these calculations that the sidefeeding lifetimes have the same time structure as the band, and that there is a finite feeding time at the top of the band of 1 fs.
3. Experimental details An experiment was performed to measure the mean lifetimes of the states in the superdeformed band in 143Eu, by the Doppler shift attenuation method. It was performed using the NORDBALL gamma-ray spectrometer [17], situated at the Niels Bohr Institute, Rise, Denmark. The array used consisted of 19 BGO (bismuth germanate) Compton suppressed HPGe (hyper-pure germanium) detectors and one LEP (Low Energy Photon) detector. These detectors were arranged in four rings, at angles of 0 = 37 °, 79 °, 101 ° and 143 ° with respect to the beam direction. The spectrometer also had a 4 ~ calorimeter consisting of 53 elements of BaF2 (barium fluoride) scintillator, used for timing information and gamma-ray sum energy and multiplicity measurements. The reaction used to populate the high spin states in 143Eu, by heavy-ion fusionevaporation, was 11°Pd (37C1, 4n)
143Eu
at a beam energy of 160 MeV. The chlorine beam was supplied by the Rise tandem accelerator and Linac booster. The thickness of the palladium target used was 900/~g/cm 2, backed with 10 m g / c m 2 of gold. The initial recoil velocity was calculated to be u o / c = 2.40%, for nuclei created in reactions midway through the target.
153
S.A. Forbes et al./Nuclear Physics A 584 (1995) 149-158
f 6°°/
l
~ 400
• •
•
•
•
•
•
•
0=37 ° •
•
•
•
200 0
6ooi •~
400
...I •
,
~2o
i 400
7
500
600
700
•
' 800
•
•
' 900
1000
1100
"
1200
1300
•
1400
1500
1600
Energy (keV)
Fig. 1. Spectra at 0 = 37° and O = 143° showing respectively the forwards and backwards Doppler shifted gamma-rays for the superdeformedband in 143Eu.The spectra are the sum of gates on shifted transitions in the band. The peaks belonging to the band are labelled by o. Other strong peaks are known contaminants from normal deformed states in 143Euand other residual nuclei created in the reaction. A total of 260 million gamma-gamma coincidence events were collected on magnetic tape with the condition that at least 3 BaF2 elements had also fired. The data were sorted offline into different gamma-gamma coincidence matrices, according to detector angle. The angles of most interest are those where the Doppler shift is greatest, furthest away from 90 ° . The detectors at 37 ° thus give the greatest forward shift, and those at 143 ° the greatest backward shift. Thus matrices created were for events at 37 ° versus each of 37 ° , 79 ° , 101 ° and 143 °, and for events at 143 ° versus each of 79 °, 101 ° and 143 °. Each of the seven matrices contained between 30-50 million events. Spectra from which to make measurements, were created by setting a gate on one axis of a matrix, to project out coincidence events. The spectra were background corrected by subtracting a fraction of the total projection at the corresponding angle. Several such spectra were added together at each angle to improve the statistics. For this analysis, the spectra from which the energy centroids of shifted peaks were measured, were created by setting gates on the shifted transitions belonging to the superdeformed band. A forwards shifted spectrum was created by summing the data for 37 °, and likewise a backwards shifted spectrum for 143 °. These spectra are shown in Fig. 1, with the shifts indicated.
4. Results The energy centroids of the Doppler shifted peaks of transitions in the superdeformed band, Es, were measured from spectra created as described above. The unshifted energies of the band, E0, were taken from [9]. From the measured Doppler shifts, the average attenuated velocities, U, for each transition were calculated according to Eq. (1). The
154
S.A. Forbes et al./Nuclear Physics A 584 (1995) 149-158
Table 1 Experimentalresults for the transitionsof energy E0 belongingto the superdeformed band in 143Eu. Tabulated are the experimentalvaluesof the average Dopplershift AE = ~ (AE37o q-AEI43 o ), averageattenuated velocity ~/c and Doppler Shift AttenuationFactor F(r)expt. The initial recoil velocitywas calculatedas vo/c = 2.40%. Calculated values of F(~-)13cb and the intrinsic lifetime 'rl3eb for a quadrupolemoment Q0 = 13 eb are also given - -
1
E0 (keV)
AE (keV)
~/c (%)
F('r)expt
F('r)13eb
rl3eb (fs)
1503 1444 1384 1325 1266 1208 1149 1091 1032 973 913 854 794 733 672 609 547
28.8(5) 27.8(5) 25.6(5) 24.5(5) 23.2(5) 21.7(5) 20.4(5) 19.0(5) 17.5(5) 16.1(5) 14.9(5) 12.8(5) 10.9(5) 9.3(5) 6.4(5) 4.5(5) 2.3(5)
2.44(3) 2.41(3) 2.32(3) 2.32(3) 2.30(3) 2.25(3) 2.22(3) 2.19(3) 2.13(3) 2.07(3) 2.05(3) 1.88(3) 1.72(3) 1.59(3) 1.20(4) 0.92(5) 0.53(5)
1.01(2) 1.00(2) 0.97(2) 0.97(2) 0.96(2) 0.94(2) 0.93(2) 0.91(2) 0.89(2) 0.86(2) 0.85(2) 0.78(2) 0.72(2) 0.66(2) 0.50(2) 0.38(3) 0.22(4)
1.00 0.99 0.99 0.98 0.97 0.96 0.95 0.93 0.91 0.88 0.84 0.79 0.72 0.62 0.49 0.34 0.16
2 2 3 3 4 5 7 9 12 15 21 30 43 64 98 161 274
484
1.0(5)
0.26(6)
0.11(5)
0.04
511
Doppler Shift Attenuation Factors F(~-) associated with each level were then calculated according to Eq. (2). The experimental F(~-) values are given in Table 1, along with the measured values of the Doppler shift and attenuated velocities. Theoretical F(~-) curves for different values of the intrinsic quadrupole moment Qo were calculated and plotted along with the experimentally obtained values, against gamma-ray energy Eo. The results are shown in Fig. 2, and the best fit to the data is Qo = 13 eb. The calculated F ( r ) values for Qo = 13 eb are given in Table 1. Also given are the corresponding intrinsic lifetimes, r, of the levels calculated for this value of Q0. The errors on the data points represent the errors in measuring the Doppler shifts from the spectra. The horizontal dotted lines centred on F(~-) = 1 give the error on the initial recoil velocity, i.e. the initial velocity spread for reactions occurring anywhere in the target, and not just at the centre as assumed in the above calculations. The main source of error on the final value of the quadrupole moment, is however due to the uncertainty in the values of the stopping powers used, which is of the order of 10%. The result obtained for the intrinsic quadrupole moment of the superdeformed band in 143Eu is thus Q0 = 13.0-4- 1.5 eb. The error combines the uncertainty in the stopping powers, and the experimental error in measuring the energy centroids.
S.A. Forbes et aL/Nuclear Physics A 584 (1995) 149-158
155
0 1.8 0".-~,,,,,,{~E,,~,~,,~,,,,, ...................................... ~"
i~,
0.6-
~" 0.4"
14eb ,,,Z///~ //~//~- 1 3 e b
0.2. j # 1 2 e b
0.0 400
,Y/, ,
I
600
800
,
I
1000
,
I
1200
,
I
1400
,
1600
Energy, E o (keV) Fig. 2. Experimental F ( r ) data points for the superdeformed band in 143Eu (with experimental error bars shown), plotted against gamma-ray energy, E0. Also shown are the theoretical F(r) curves for intrinsic quadrupole moments Q0 = 12, 13, 14 and 15 eb. The horizontal dotted lines indicate the spread in the initial recoil velocity.
The quadrupole deformation parameter, /32, can be calculated from the quadrupole moment. Using a relationship to second order in /32, assuming no hexadecapole deformation [ 18], this gives a value of the deformation parameter /32 ----0 . 6 0 -t- 0 . 0 7 .
However, it is expected that hexadecapole deformation is important for nuclei in this region, with a value of/34 = 0.05 given for 143Eu [ 19]. The quadrupole deformation parameter calculated as in [ 19] from the quadrupole moment, assuming this hexadecapole deformation, is /32 (m) = 0.52 :i: 0.05. With the inclusion of the hexadecapole deformation, a smaller value of the quadrupole deformation is obtained.
5. Discussion
The shell gaps that stabilise superdeformed shapes are spread across a range of deformation values, depending on the nucleon number [1]. In the mass A ~ 130 region, the shell gaps corresponding to Z = 58, 60 and N = 74 are predicted to be at deformations of/32 ,-~ 0.35-0.40. The shell gaps for the mass A ,-~ 150 region, for Z = 64, 66 and N = 86 are predicted to be at higher deformations of/32 ~ 0.55-0.60. The shell gaps for the mass A ~ 140 region at Z = 62, 64 and N = 80 are predicted to be at intermediate deformations of/32 "~ 0.50-0.55, similarly for the mass A ~ 190
156
S.A. Forbes et al./Nuclear Physics A 584 (1995) 149-158
region with Z = 80. The result of/32 -- 0.52 presented in this paper is in very good agreement with the calculations. Of particular interest to studies of large deformations are the intruder orbitals, as they possess a strong deformation driving effect. The occupation of intruder orbitals, which have a polarising effect on the deformed core, are predicted to form the basis for the superdeformed bands. At low spin and for typical ground state deformations, these intruder orbitals lie high in energy above the Fermi level, and are thus relatively inaccessible to experimentalists. This is why the study of high spin states is so important, as the intruder orbitals are strongly affected by rotation and are lowered in energy towards the Fermi level. They thus may become occupied at high spin, and determine the spread of deformation of superdeformed bands within each mass region. From the configurations predicted for the superdeformed bands in each mass region, it is expected that one or two Nose = 6 neutron i13/2 intruder orbitals are occupied in the mass A ~ 130 region superdeformed bands. In the mass A ~ 150 region, as many as five or six intruder orbitals can be occupied, from a combination of Nose = 7 neutron j15/2 orbitals and No~e = 6 proton i13/2 orbitals, resulting in a greater enhancement of deformation. These are also the intruder orbitals occupied in the mass A ~ 190 region. The typical measured deformation tends to increase in each mass region as the number of intruder orbitals occupied increases. The quadrupole moments and hence deformations, of many superdeformed bands have been experimentally measured, and are in general agreement with predicted values from theoretical Total Routhian Surface [ 3] calculations. Near the Fermi surface at large deformations for mass A ~ 140 nuclei, are two neutron i13/2 intruder orbitals and a proton i13/2 intruder orbital from the Nose = 6 shell. The degeneracy of two for these orbitals is removed by the rotation. The proposed single particle configuration for the superdeformed band in 143Eu involves five of these intruder orbitals [8], four neutron and one proton, represented by the notation v64~61. Total Routhian Surface calculations for 143Eu are shown in Fig. 3, based on the positive parity, positive signature configuration of the odd i13/2 proton intruder orbital. These predict that the nucleus has a very constant superdeformed shape, with /32 = 0.49-0.51, (and /34 = 0.04-0.05), indicating that there is a strong shell structure stability with respect to rotational frequency. The deformation calculated by Chasman [ 11 ] for 143Eu corresponds to /32 = 0.57. More recently, calculations have also been performed by Nazarewicz et al. [ 19] for a number of nuclei in this region. They predict the quadrupole deformation to be/32 0.50 for 143Eu. A hexadecapole deformation was considered in both these calculations. The value of the quadrupole deformation of/32 = 0.52 found from the experiment reported in this paper is thus in agreement with the predicted values from these calculations. This supports the assignment of the intruder orbitals u64~61 to the superdeformed band. The measurement of the deformation of the superdeformed band in 143Eu has shown that its deformation lies between typical values for the mass A ~ 130 region and the mass A ,-~ 150 region, as predicted by the chain systematics of shell gaps in the energy level spectrum. The measured deformation agrees well with predicted values for =
157
S.A. Forbes et al./Nuclear Physics A 584 (1995) 149-158
-ho) = 0.544 MeV
qht0 = 0.346 MeV
[32 = 0.51, [34= 0.05, ']/= l °
[32 = 0.51, [34 = 0.05, • = 1° 0.40
~ .,~.,..L..
~..,.~. ~
0.3° 0.2° " 0.10
I _~--~ '1
0 +
0.°°
0.10
0.20
0.30
0.40
0.50
0.110
0.(X) 0.00
0.10
0.20
0,30
0.40
0.50
C#3
~o) = 0.741 MeV
• ~o = 0.840 MeV
II
[32= 0.50, [34= 0.04, T = 2°
[32= 0.49, [34= 0.04, T = 3°
0
.
3
0
I
0.60
0.30
~
0.20
.vv 0.00
0.10
0.20
0.30
0.,10
I).50
0.61/
O.O0 0.00 0.10
X=[~2cos(~
,)Lr /'''1'1 '1 l 0.30 0.40 0.50 0.60 I
0.20
+30)
Fig. 3. Total Routhian Surface (TRS) calculations at different rotational frequencies, hm, for the positive parity, positive signature, proton configuration in 143Eu. The position of the lowest energy minimum at each rotational frequency is indicated by a dot (•). The superdeformed minimum is seen to become lowest in energy as the rotational frequency is increased. The deformations/32, ]~2 and y corresponding to the superdeformed minimum, are given for each frequency. the assigned intruder orbital configuration. A study o f superdeformed bands p r o v i d e s a u n i q u e o p p o r t u n i t y to access the intruder orbitals, and to map the relationship between the n u m b e r o f intruder orbitals occupied and enhanced deformation. It is important to explore further the mass A ~
140 region, to c o m p a r e it with the well studied mass
A ~ 130, 150 and 190 regions.
Acknowledgements This w o r k was supported by the U K Science and E n g i n e e r i n g Research C o u n c i l , the D a n i s h Natural S c i e n c e Research Council, the S w e d i s h Natural Science Research C o u n c i l and the A c a d e m y o f Finland. O n e o f us ( S A F ) acknowledges the receipt o f S E R C studentships during the course o f this work. We also wish to thank the staff o f
158
S.A. Forbes et al./Nuclear Physics A 584 (1995) 149-158
t h e T a n d e m A c c e l e r a t o r L a b o r a t o r y at t h e Niels B o h r I n s t i t u t e for p r o v i d i n g the b e a m a n d e x p e r i m e n t a l support.
References [ 1] [2] [3] [4] [5]
[61 [7]
18] [9]
110]
111] 1121 [131 [14] [ 151 [16] [17] [18] [ 19]
J. Dudek, W. Nazarewicz, Z. Szymanski and G.A. Leander, Phys. Rev. Lett. 59 (1987) 1405. W. Nazarewicz, Contemporary Topics in Nuclear Structure Physics (World Scientific, 1988) p. 467. R. Wyss, J. Nyberg, A. Johnson, R. Bengtsson and W. Nazarewicz, Phys. Lett. B 215 (1988) 211. P.J. Nolan, A. Kirwan, D.J.G. Love, A.H. Nelson, D.J. Unwin and P.J. Twin, J. Phys. G 11 (1985) L17. RJ. Twin, B.M. Nyak6, A.H. Nelson, J. Simpson, M.A. Bentley, H.W. Cranmer-Gordon, P.D. Forsyth, D. Howe, A.R. Mokhtar, J.D. Morrison, J.F. Sharpey-Schafer and G. Sletten, Phys. Rev. Lett. 57 (1986) 811. A.J. Kirwan, G.C. Ball, P.J. Bishop, M.J. Godfrey, P.J. Nolan, D.J. Thornley, D.J.G. Love and A.H. Nelson, Phys. Rev. Lett. 58 (1987) 467. M.A. Bentley, A. Alderson, G.C. Ball, H.W. Cranmer-Gordon, P. Fallon, B. Fant, P.D. Forsyth, B. Herskind, D. Howe, C.A. Kalfas, A.R. Mokhtar, J.D. Morrison, A.H. Nelson, B.M. Nyak6, K. Schiffer, J.E Sharpey-Schafer, J. Simpson, G. Sletten and P.J. Twin, J. Phys. G 17 (1991) 481. S.M. Mullins, R.A. Wyss, P. Fallon, T. Byrski, D. Curien, S.A. Forbes, Y.-J. He, M.S. Metcalfe, P.J. Nolan, E.S. Paul, R.J. Poynter, RH. Regan and R. Wadsworth, Phys. Rev. Lett. 66 (1991) 1677. A. Atag, M. Piiparinen, B. Herskind, J. Nyberg, G. Sletten, G. de Angelis, S.A. Forbes, N. Gj0rup, G.B. Hagemann, F. lngebretsen, H.J. Jensen, D. Jerrestam, H. Kusakari, R.M. Lieder, G.V. Marti, S.M. Mullins, D. Santonocito, H. Schnare, K. Str~ihle, M. Sugawara, P.O. Tjcm, A. Virtanen and R. Wadsworth, Phys. Rev. Lett. 70 (1993) 1069. A. Ataq, M. Piiparinen, B. Herskind, J. Nyberg, G. Sletten, G. de Angelis, R.M. Clark, S.A. Forbes, N. Gj0rup, G.B. Hagemann, F. lngebretsen, H.J. Jensen, D. Jerrestam, H. Kusakari, R.M. Lieder, G.V. Marti, S.M. Mullins, P.J. Nolan, E.S. Paul, P.H. Regan, D. Santonocito, H. Schnare, K. Str'fihle, M. Sugawara, RO. Tj0m, A. Virtanen and R. Wadsworth, Proceedings of the International Conference on Nuclear Structure at High Angular Momentum, Ottawa, 1992, Nucl. Phys. A 557 (1993) 10%. R.R. Chasman, Phys. Lett. B 187 (1987) 219. L.C. Northcliffe and R.E Schilling, Nucl. Data Tables A 7 (1970) 233. P.J. Nolan and J.E Sharpey-Schafer, Rep. Prog. Phys. 43 (1979) 1. D.J.G. Love, Daresbury Preprint DL/NUC/P296E (1990). K. Braune, Diploma Work, Max-Plank-Instittit far Kernphysik, Heidelberg (1977). H. Bateman, Proc. Camb. Phil. Soc. 15 (1910) 423. B. Herskind, Nucl. Phys A 447 (1985) 395. K.E.G. Lobner, M. Vetter and V. H6nig, Nuclear Data Tables A 7 (1970) 495. W. Nazarewicz, R. Wyss and A. Johnson, Nucl. Phys. A 503 (1989) 285.
Nuclear Physics A 584 (1995) 149-158
Determination of the intrinsic quadrupole moment of the superdeformed band in 143Eu S.A. Forbes a, A. Ata9 b,c, G.B. Hagemann b, B. Herskind b, S.M. Mullins a,l, EJ. Nolan a, j. Nybergb,a, M.J. Piiparinen b'e, G. Sletten b, R. W a d s w o r t h f a Oliver Lodge Laboratory, University of LiverpooL Liverpool, L69 3BX, UK b Niels Bohr Institute, Tandem Accelerator Laboratory, University of Copenhagen, DK-4000 Roskilde, Denmark c Department of Radiation Sciences, Uppsala University, BOX 535, S-75121 Uppsala, Sweden o The Svedberg Laboratory, Uppsala University, BOX 533, S-75121 Uppsala, Sweden e Department of Physics, University of Jyvdskyld, PO Box 35, FIN-40351 Jyvdskyld, Finland f Department of Physics, University of York, Heslington, York, YO1 5DD, UK
Received 30 August 1994
Abstract The intrinsic quadrupole moment, Qo, of the superdeformed band in ~43Eu has been extracted from nuclear lifetimes, measured in a Doppler shift attenuation method experiment. Data were collected with the NORDBALL gamma-ray spectrometer following the reaction H°Pd (37C1, 4n) 143Eu at a beam energy of 160 MeV. A centroid shift analysis of the data was carried out, using the Braune stopping powers to model the slowing down of the recoil nuclei in the target and backing materials. This gave a value of Q0 = 13.0 =k 1.5 eb for the band. The result is in good agreement with the theoretical prediction of Total Routhian Surface calculations, and supports the assigned single particle configuration, involving occupation of both neutron and proton i~3/2 intruder orbitals from the Nosc= 6 oscillator shell. Keywords: NUCLEAR REACTIONS u°Pd (37C1, 4n), E = 160 MeV; measured Ey Doppler shifts. 143Eu superdeformed band deduced r, Qo, 132.
1 Present address: Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, L8S IK1, Canada. 0375-9474/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 5 - 9 4 7 4 ( 9 4 ) 00498-6
150
S.A. Forbes et aL/Nuclear Physics A 584 (1995) 149-158
1. Introduction The origin of superdeformation lies in the formation of a second energy minimum in the fission barrier, corresponding to highly deformed prolate nuclear shapes. Favourable shell corrections, as calculated from nuclear mean field models such as with the harmonic oscillator potential, stabilise these shapes. These pronounced shell gaps in the energy level spectrum at large deformations remain when considering a more realistic potential, such as in the Woods-Saxon or Nilsson model, and form a chain across a range of deformation values [ 1 ]. When taking rotation of the nucleus into consideration for high spin states, it is found that the energy level spectrum is largely unaffected, apart from the so-called intruder orbitals from higher oscillator shells. These are greatly lowered in energy and only reach the Fermi surface at large deformation. As the intruder orbitals are strongly deformation driving, it is expected that the deformation of a nucleus at high spin depends on the number of intruder orbitals occupied in the configuration. It is thus important to interpret superdeformation in terms of the characteristic intruder orbitals occupied [2,3]. The collective rotation of superdeformed nuclei gives rise to characteristic cascades of gamma-rays, with a narrow energy spacing between the transitions, indicating a large dynamic moment of inertia. Although this provides a means of identification, the superdeformed nature of the bands must be confirmed by a measurement of the deformation. This can be done by an experimental measurement of the lifetimes of the nuclear states in the superdeformed band. The reduced transition probabilities are then calculated from the measured lifetimes according to the rotational model, and the intrinsic quadrupole moment extracted from these results. The quadrupole deformation parameter/32, as defined for an axially symmetric nuclear shape, is then calculated from the measured quadrupole moment. The first cases of superdeformation were discovered in the nuclei ~32Ce [4] in the mass A ~ 130 region and 152Dy [5] in the mass A ~ 150 region. Many other superdeformed bands have now been observed in these regions, and also several in the mass A ~ 190 region. The deformation has been obtained in several cases, by measurements of the quadrupole moments in lifetime experiments. In the mass A ~ 130 region these bands are found to have a quadrupole deformation parameter of/32 ~ 0.35-0.40, corresponding to a deformed shape with a major:minor axis ratio close to 3:2 (e.g. 132Ce [6] ). The superdeformed bands in the mass A ,-~ 150 region originate from more deformed nuclei, with values of/32 " " 0 . 5 5 - 0 . 6 0 , giving axis ratios near 2:1 (e.g. 152Dy [7] ). The difference in deformation between the mass regions is expected from the position of the shell gaps at different deformations in the energy level spectrum. A highly regular rotational structure has been discovered in 143Eu [8,9] and was the first such example found in a nucleus intermediate between the mass A ~ 130 and A ,.~ 150 regions. It was thus important to measure the quadrupole moment of the band and hence compare the deformation with other mass regions. The band was found to have an intensity of approximately 1% of the 143Eu channel in the reaction [9]. The spin assignments for the band are taken from [9] based on the discovery of linking
S.A. Forbes et al./Nuclear Physics A 584 (1995) 149-158
151
transitions. It should be noted that the band was originally assigned to the nucleus 142Eu [8] but evidence since then has indicated that the correct assignment is to 143Eu [9]. Parts of this work have been published in conference proceedings [ 10]. Theoretical calculations have been performed by Chasman [ 11 ] to predict the relative excitation energy of the superdeformed well, and the depth of the superdeformed well for nuclei with 56 < Z < 70 and 72 < N < 88. The conditions to stabilise superdeformation are (i) a small relative excitation energy of the superdeformed well, to optimise the chance that the states will be populated by the energy brought in through the reaction, and (ii) a deep superdeformed well, so that tunnelling through to the normal deformed states does not occur too rapidly. These calculations had indicated that the nucleus 143Eu (N = 80, Z = 63) would be a favourable candidate. The superdeformed well is stabilised by the shell gap at deformations of f12 ~ 0.5-0.6, for neutron number N = 80, and is predicted to be yrast at a relatively low spin. In order to further investigate the systematics of the shell gaps, it is important to check the predicted deformation of this first superdeformed band found the mass A ~ 140 region.
2. Doppler shift attenuation method In a heavy-ion fusion-evaporation reaction, the recoil nuclei produced with initial recoil velocity v0, slow down in the target and backing materials and are eventually brought to rest. The Doppler shift attenuation method (DSAM) [12], is based on the fact that the lifetimes of many nuclear states are of the same order of magnitude as this slowing down time (a few tenths of a picosecond). Gamma-rays depopulating a particular excited nuclear state will be emitted after an average time t, related to the mean lifetime r of that state. At time t, the recoil will have slowed down and have an average attenuated velocity ~. Stopping theories must be used to model the slowing down of the recoils, and thus relate velocity to time, and hence lifetimes, of the states.
2.1. Centroid Shift Analysis A Centroid Shift Analysis [ 13] was performed on the DSAM data to measure the lifetimes of the states in the superdeformed band in 143Eu. If a particular state emitting a gamma-ray of known energy E0, decays in flight, the gamma-ray energy will be Doppler shifted. The shifted energy, Es, is measured from a gamma-ray spectrum taken with detectors at an angle of 0 with respect to the beam direction (assuming that the recoils are emitted in a narrow forward cone at 0°). The average attenuated velocity for this state can then be calculated from the equations:
E~= Eo 1 + - c o s 0 c
~
AE= V-EocosO, c
(1)
152
S.A. Forbes et al./Nuclear Physics A 584 (1995) 149-158
where AE = [Es - Eo[ is the Doppler shift. The Doppler Shift Attenuation Factor F ( r ) is defined as (2)
F(T) = --,
u0
where the initial recoil velocity v0 is calculated from reaction kinematics. The F ( T ) value measured for each transition in the band is plotted against the transition energy to give an F ( r ) curve. It is assumed in the calculations that the quadrupole moment does not change down the band. For a particular quadrupole moment, the lifetimes of the states in the band can be calculated from the rotational model, and a theoretical F ( z ) curve plotted. The quadrupole moment is changed in the calculations until the best fit to the experimental F ( r ) curve is found. For each chosen quadrupole moment, the computer code FTAU [ 14] was used to calculate the intrinsic lifetime of each state in the band. The slowing down processes of the recoiling nuclei in the target and backing foils were modelled using the Braune stopping powers [ 15]. This allowed the intrinsic F/(7) values to be calculated. The code FFEED [ 14] then corrects these intrinsic values to take into account in-band feeding. This is done by solving the Bateman equations [ 16] to give cumulative F(~-) values, which are plotted along with the experimentally measured values. It is assumed in these calculations that the sidefeeding lifetimes have the same time structure as the band, and that there is a finite feeding time at the top of the band of 1 fs.
3. Experimental details An experiment was performed to measure the mean lifetimes of the states in the superdeformed band in 143Eu, by the Doppler shift attenuation method. It was performed using the NORDBALL gamma-ray spectrometer [17], situated at the Niels Bohr Institute, Rise, Denmark. The array used consisted of 19 BGO (bismuth germanate) Compton suppressed HPGe (hyper-pure germanium) detectors and one LEP (Low Energy Photon) detector. These detectors were arranged in four rings, at angles of 0 = 37 °, 79 °, 101 ° and 143 ° with respect to the beam direction. The spectrometer also had a 4 ~ calorimeter consisting of 53 elements of BaF2 (barium fluoride) scintillator, used for timing information and gamma-ray sum energy and multiplicity measurements. The reaction used to populate the high spin states in 143Eu, by heavy-ion fusionevaporation, was 11°Pd (37C1, 4n)
143Eu
at a beam energy of 160 MeV. The chlorine beam was supplied by the Rise tandem accelerator and Linac booster. The thickness of the palladium target used was 900/~g/cm 2, backed with 10 m g / c m 2 of gold. The initial recoil velocity was calculated to be u o / c = 2.40%, for nuclei created in reactions midway through the target.
153
S.A. Forbes et al./Nuclear Physics A 584 (1995) 149-158
f 6°°/
l
~ 400
• •
•
•
•
•
•
•
0=37 ° •
•
•
•
200 0
6ooi •~
400
...I •
,
~2o
i 400
7
500
600
700
•
' 800
•
•
' 900
1000
1100
"
1200
1300
•
1400
1500
1600
Energy (keV)
Fig. 1. Spectra at 0 = 37° and O = 143° showing respectively the forwards and backwards Doppler shifted gamma-rays for the superdeformedband in 143Eu.The spectra are the sum of gates on shifted transitions in the band. The peaks belonging to the band are labelled by o. Other strong peaks are known contaminants from normal deformed states in 143Euand other residual nuclei created in the reaction. A total of 260 million gamma-gamma coincidence events were collected on magnetic tape with the condition that at least 3 BaF2 elements had also fired. The data were sorted offline into different gamma-gamma coincidence matrices, according to detector angle. The angles of most interest are those where the Doppler shift is greatest, furthest away from 90 ° . The detectors at 37 ° thus give the greatest forward shift, and those at 143 ° the greatest backward shift. Thus matrices created were for events at 37 ° versus each of 37 ° , 79 ° , 101 ° and 143 °, and for events at 143 ° versus each of 79 °, 101 ° and 143 °. Each of the seven matrices contained between 30-50 million events. Spectra from which to make measurements, were created by setting a gate on one axis of a matrix, to project out coincidence events. The spectra were background corrected by subtracting a fraction of the total projection at the corresponding angle. Several such spectra were added together at each angle to improve the statistics. For this analysis, the spectra from which the energy centroids of shifted peaks were measured, were created by setting gates on the shifted transitions belonging to the superdeformed band. A forwards shifted spectrum was created by summing the data for 37 °, and likewise a backwards shifted spectrum for 143 °. These spectra are shown in Fig. 1, with the shifts indicated.
4. Results The energy centroids of the Doppler shifted peaks of transitions in the superdeformed band, Es, were measured from spectra created as described above. The unshifted energies of the band, E0, were taken from [9]. From the measured Doppler shifts, the average attenuated velocities, U, for each transition were calculated according to Eq. (1). The
154
S.A. Forbes et al./Nuclear Physics A 584 (1995) 149-158
Table 1 Experimentalresults for the transitionsof energy E0 belongingto the superdeformed band in 143Eu. Tabulated are the experimentalvaluesof the average Dopplershift AE = ~ (AE37o q-AEI43 o ), averageattenuated velocity ~/c and Doppler Shift AttenuationFactor F(r)expt. The initial recoil velocitywas calculatedas vo/c = 2.40%. Calculated values of F(~-)13cb and the intrinsic lifetime 'rl3eb for a quadrupolemoment Q0 = 13 eb are also given - -
1
E0 (keV)
AE (keV)
~/c (%)
F('r)expt
F('r)13eb
rl3eb (fs)
1503 1444 1384 1325 1266 1208 1149 1091 1032 973 913 854 794 733 672 609 547
28.8(5) 27.8(5) 25.6(5) 24.5(5) 23.2(5) 21.7(5) 20.4(5) 19.0(5) 17.5(5) 16.1(5) 14.9(5) 12.8(5) 10.9(5) 9.3(5) 6.4(5) 4.5(5) 2.3(5)
2.44(3) 2.41(3) 2.32(3) 2.32(3) 2.30(3) 2.25(3) 2.22(3) 2.19(3) 2.13(3) 2.07(3) 2.05(3) 1.88(3) 1.72(3) 1.59(3) 1.20(4) 0.92(5) 0.53(5)
1.01(2) 1.00(2) 0.97(2) 0.97(2) 0.96(2) 0.94(2) 0.93(2) 0.91(2) 0.89(2) 0.86(2) 0.85(2) 0.78(2) 0.72(2) 0.66(2) 0.50(2) 0.38(3) 0.22(4)
1.00 0.99 0.99 0.98 0.97 0.96 0.95 0.93 0.91 0.88 0.84 0.79 0.72 0.62 0.49 0.34 0.16
2 2 3 3 4 5 7 9 12 15 21 30 43 64 98 161 274
484
1.0(5)
0.26(6)
0.11(5)
0.04
511
Doppler Shift Attenuation Factors F(~-) associated with each level were then calculated according to Eq. (2). The experimental F(~-) values are given in Table 1, along with the measured values of the Doppler shift and attenuated velocities. Theoretical F(~-) curves for different values of the intrinsic quadrupole moment Qo were calculated and plotted along with the experimentally obtained values, against gamma-ray energy Eo. The results are shown in Fig. 2, and the best fit to the data is Qo = 13 eb. The calculated F ( r ) values for Qo = 13 eb are given in Table 1. Also given are the corresponding intrinsic lifetimes, r, of the levels calculated for this value of Q0. The errors on the data points represent the errors in measuring the Doppler shifts from the spectra. The horizontal dotted lines centred on F(~-) = 1 give the error on the initial recoil velocity, i.e. the initial velocity spread for reactions occurring anywhere in the target, and not just at the centre as assumed in the above calculations. The main source of error on the final value of the quadrupole moment, is however due to the uncertainty in the values of the stopping powers used, which is of the order of 10%. The result obtained for the intrinsic quadrupole moment of the superdeformed band in 143Eu is thus Q0 = 13.0-4- 1.5 eb. The error combines the uncertainty in the stopping powers, and the experimental error in measuring the energy centroids.
S.A. Forbes et aL/Nuclear Physics A 584 (1995) 149-158
155
0 1.8 0".-~,,,,,,{~E,,~,~,,~,,,,, ...................................... ~"
i~,
0.6-
~" 0.4"
14eb ,,,Z///~ //~//~- 1 3 e b
0.2. j # 1 2 e b
0.0 400
,Y/, ,
I
600
800
,
I
1000
,
I
1200
,
I
1400
,
1600
Energy, E o (keV) Fig. 2. Experimental F ( r ) data points for the superdeformed band in 143Eu (with experimental error bars shown), plotted against gamma-ray energy, E0. Also shown are the theoretical F(r) curves for intrinsic quadrupole moments Q0 = 12, 13, 14 and 15 eb. The horizontal dotted lines indicate the spread in the initial recoil velocity.
The quadrupole deformation parameter, /32, can be calculated from the quadrupole moment. Using a relationship to second order in /32, assuming no hexadecapole deformation [ 18], this gives a value of the deformation parameter /32 ----0 . 6 0 -t- 0 . 0 7 .
However, it is expected that hexadecapole deformation is important for nuclei in this region, with a value of/34 = 0.05 given for 143Eu [ 19]. The quadrupole deformation parameter calculated as in [ 19] from the quadrupole moment, assuming this hexadecapole deformation, is /32 (m) = 0.52 :i: 0.05. With the inclusion of the hexadecapole deformation, a smaller value of the quadrupole deformation is obtained.
5. Discussion
The shell gaps that stabilise superdeformed shapes are spread across a range of deformation values, depending on the nucleon number [1]. In the mass A ~ 130 region, the shell gaps corresponding to Z = 58, 60 and N = 74 are predicted to be at deformations of/32 ,-~ 0.35-0.40. The shell gaps for the mass A ,-~ 150 region, for Z = 64, 66 and N = 86 are predicted to be at higher deformations of/32 ~ 0.55-0.60. The shell gaps for the mass A ~ 140 region at Z = 62, 64 and N = 80 are predicted to be at intermediate deformations of/32 "~ 0.50-0.55, similarly for the mass A ~ 190
156
S.A. Forbes et al./Nuclear Physics A 584 (1995) 149-158
region with Z = 80. The result of/32 -- 0.52 presented in this paper is in very good agreement with the calculations. Of particular interest to studies of large deformations are the intruder orbitals, as they possess a strong deformation driving effect. The occupation of intruder orbitals, which have a polarising effect on the deformed core, are predicted to form the basis for the superdeformed bands. At low spin and for typical ground state deformations, these intruder orbitals lie high in energy above the Fermi level, and are thus relatively inaccessible to experimentalists. This is why the study of high spin states is so important, as the intruder orbitals are strongly affected by rotation and are lowered in energy towards the Fermi level. They thus may become occupied at high spin, and determine the spread of deformation of superdeformed bands within each mass region. From the configurations predicted for the superdeformed bands in each mass region, it is expected that one or two Nose = 6 neutron i13/2 intruder orbitals are occupied in the mass A ~ 130 region superdeformed bands. In the mass A ~ 150 region, as many as five or six intruder orbitals can be occupied, from a combination of Nose = 7 neutron j15/2 orbitals and No~e = 6 proton i13/2 orbitals, resulting in a greater enhancement of deformation. These are also the intruder orbitals occupied in the mass A ~ 190 region. The typical measured deformation tends to increase in each mass region as the number of intruder orbitals occupied increases. The quadrupole moments and hence deformations, of many superdeformed bands have been experimentally measured, and are in general agreement with predicted values from theoretical Total Routhian Surface [ 3] calculations. Near the Fermi surface at large deformations for mass A ~ 140 nuclei, are two neutron i13/2 intruder orbitals and a proton i13/2 intruder orbital from the Nose = 6 shell. The degeneracy of two for these orbitals is removed by the rotation. The proposed single particle configuration for the superdeformed band in 143Eu involves five of these intruder orbitals [8], four neutron and one proton, represented by the notation v64~61. Total Routhian Surface calculations for 143Eu are shown in Fig. 3, based on the positive parity, positive signature configuration of the odd i13/2 proton intruder orbital. These predict that the nucleus has a very constant superdeformed shape, with /32 = 0.49-0.51, (and /34 = 0.04-0.05), indicating that there is a strong shell structure stability with respect to rotational frequency. The deformation calculated by Chasman [ 11 ] for 143Eu corresponds to /32 = 0.57. More recently, calculations have also been performed by Nazarewicz et al. [ 19] for a number of nuclei in this region. They predict the quadrupole deformation to be/32 0.50 for 143Eu. A hexadecapole deformation was considered in both these calculations. The value of the quadrupole deformation of/32 = 0.52 found from the experiment reported in this paper is thus in agreement with the predicted values from these calculations. This supports the assignment of the intruder orbitals u64~61 to the superdeformed band. The measurement of the deformation of the superdeformed band in 143Eu has shown that its deformation lies between typical values for the mass A ~ 130 region and the mass A ,-~ 150 region, as predicted by the chain systematics of shell gaps in the energy level spectrum. The measured deformation agrees well with predicted values for =
157
S.A. Forbes et al./Nuclear Physics A 584 (1995) 149-158
-ho) = 0.544 MeV
qht0 = 0.346 MeV
[32 = 0.51, [34= 0.05, ']/= l °
[32 = 0.51, [34 = 0.05, • = 1° 0.40
~ .,~.,..L..
~..,.~. ~
0.3° 0.2° " 0.10
I _~--~ '1
0 +
0.°°
0.10
0.20
0.30
0.40
0.50
0.110
0.(X) 0.00
0.10
0.20
0,30
0.40
0.50
C#3
~o) = 0.741 MeV
• ~o = 0.840 MeV
II
[32= 0.50, [34= 0.04, T = 2°
[32= 0.49, [34= 0.04, T = 3°
0
.
3
0
I
0.60
0.30
~
0.20
.vv 0.00
0.10
0.20
0.30
0.,10
I).50
0.61/
O.O0 0.00 0.10
X=[~2cos(~
,)Lr /'''1'1 '1 l 0.30 0.40 0.50 0.60 I
0.20
+30)
Fig. 3. Total Routhian Surface (TRS) calculations at different rotational frequencies, hm, for the positive parity, positive signature, proton configuration in 143Eu. The position of the lowest energy minimum at each rotational frequency is indicated by a dot (•). The superdeformed minimum is seen to become lowest in energy as the rotational frequency is increased. The deformations/32, ]~2 and y corresponding to the superdeformed minimum, are given for each frequency. the assigned intruder orbital configuration. A study o f superdeformed bands p r o v i d e s a u n i q u e o p p o r t u n i t y to access the intruder orbitals, and to map the relationship between the n u m b e r o f intruder orbitals occupied and enhanced deformation. It is important to explore further the mass A ~
140 region, to c o m p a r e it with the well studied mass
A ~ 130, 150 and 190 regions.
Acknowledgements This w o r k was supported by the U K Science and E n g i n e e r i n g Research C o u n c i l , the D a n i s h Natural S c i e n c e Research Council, the S w e d i s h Natural Science Research C o u n c i l and the A c a d e m y o f Finland. O n e o f us ( S A F ) acknowledges the receipt o f S E R C studentships during the course o f this work. We also wish to thank the staff o f
158
S.A. Forbes et al./Nuclear Physics A 584 (1995) 149-158
t h e T a n d e m A c c e l e r a t o r L a b o r a t o r y at t h e Niels B o h r I n s t i t u t e for p r o v i d i n g the b e a m a n d e x p e r i m e n t a l support.
References [ 1] [2] [3] [4] [5]
[61 [7]
18] [9]
110]
111] 1121 [131 [14] [ 151 [16] [17] [18] [ 19]
J. Dudek, W. Nazarewicz, Z. Szymanski and G.A. Leander, Phys. Rev. Lett. 59 (1987) 1405. W. Nazarewicz, Contemporary Topics in Nuclear Structure Physics (World Scientific, 1988) p. 467. R. Wyss, J. Nyberg, A. Johnson, R. Bengtsson and W. Nazarewicz, Phys. Lett. B 215 (1988) 211. P.J. Nolan, A. Kirwan, D.J.G. Love, A.H. Nelson, D.J. Unwin and P.J. Twin, J. Phys. G 11 (1985) L17. RJ. Twin, B.M. Nyak6, A.H. Nelson, J. Simpson, M.A. Bentley, H.W. Cranmer-Gordon, P.D. Forsyth, D. Howe, A.R. Mokhtar, J.D. Morrison, J.F. Sharpey-Schafer and G. Sletten, Phys. Rev. Lett. 57 (1986) 811. A.J. Kirwan, G.C. Ball, P.J. Bishop, M.J. Godfrey, P.J. Nolan, D.J. Thornley, D.J.G. Love and A.H. Nelson, Phys. Rev. Lett. 58 (1987) 467. M.A. Bentley, A. Alderson, G.C. Ball, H.W. Cranmer-Gordon, P. Fallon, B. Fant, P.D. Forsyth, B. Herskind, D. Howe, C.A. Kalfas, A.R. Mokhtar, J.D. Morrison, A.H. Nelson, B.M. Nyak6, K. Schiffer, J.E Sharpey-Schafer, J. Simpson, G. Sletten and P.J. Twin, J. Phys. G 17 (1991) 481. S.M. Mullins, R.A. Wyss, P. Fallon, T. Byrski, D. Curien, S.A. Forbes, Y.-J. He, M.S. Metcalfe, P.J. Nolan, E.S. Paul, R.J. Poynter, RH. Regan and R. Wadsworth, Phys. Rev. Lett. 66 (1991) 1677. A. Atag, M. Piiparinen, B. Herskind, J. Nyberg, G. Sletten, G. de Angelis, S.A. Forbes, N. Gj0rup, G.B. Hagemann, F. lngebretsen, H.J. Jensen, D. Jerrestam, H. Kusakari, R.M. Lieder, G.V. Marti, S.M. Mullins, D. Santonocito, H. Schnare, K. Str~ihle, M. Sugawara, P.O. Tjcm, A. Virtanen and R. Wadsworth, Phys. Rev. Lett. 70 (1993) 1069. A. Ataq, M. Piiparinen, B. Herskind, J. Nyberg, G. Sletten, G. de Angelis, R.M. Clark, S.A. Forbes, N. Gj0rup, G.B. Hagemann, F. lngebretsen, H.J. Jensen, D. Jerrestam, H. Kusakari, R.M. Lieder, G.V. Marti, S.M. Mullins, P.J. Nolan, E.S. Paul, P.H. Regan, D. Santonocito, H. Schnare, K. Str'fihle, M. Sugawara, RO. Tj0m, A. Virtanen and R. Wadsworth, Proceedings of the International Conference on Nuclear Structure at High Angular Momentum, Ottawa, 1992, Nucl. Phys. A 557 (1993) 10%. R.R. Chasman, Phys. Lett. B 187 (1987) 219. L.C. Northcliffe and R.E Schilling, Nucl. Data Tables A 7 (1970) 233. P.J. Nolan and J.E Sharpey-Schafer, Rep. Prog. Phys. 43 (1979) 1. D.J.G. Love, Daresbury Preprint DL/NUC/P296E (1990). K. Braune, Diploma Work, Max-Plank-Instittit far Kernphysik, Heidelberg (1977). H. Bateman, Proc. Camb. Phil. Soc. 15 (1910) 423. B. Herskind, Nucl. Phys A 447 (1985) 395. K.E.G. Lobner, M. Vetter and V. H6nig, Nuclear Data Tables A 7 (1970) 495. W. Nazarewicz, R. Wyss and A. Johnson, Nucl. Phys. A 503 (1989) 285.