Spectroscopy in the second and third minimum of actinide nuclei

Progress in Particle and Nuclear Physics PERGAMON

Progress in Particle and Nuclear Physics 49 (2002) 325-402 http://www.elsevier.com/locate/npe

Spectroscopy in the Second and Third Minimum of Actinide Nuclei P.G. THIROLF and D. HABS Sektion Physik, Ludwig-Marimilians-Universitiit, Miinchen. Germany Maier-Leibnitz-Laboratorium, Garching, Germany

Abstract Progress in our understanding of the nuclear structure in the second and third minimum of the potential energy surface is reported emphasizing recent achievements via complementary experimental approaches. Detailed spectroscopic results of collective properties are obtained in the second and third minimum as well a.s for spin dependent fission properties. A novel approach could be established to determine the depth of the potential wells. Empirical collective level systematics developed for normal deformed nuclei were tested upon their predictive power for extremely deformed configurations. A phenomenological approach based on a valence correlation scheme was applied to the second potential well in order to predict B vibrational phonon energies and magic nucleon numbers. Perspectives for future experimental efforts will be discussed.

0146-6410/02/$ - see front matter 0 2002 Elsevier PII: SOl46-6410(02)00158-8

Science

BV. All rights reserved.

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Contents 1

Introduction

327 '

S p e c t r o s c o p y in t h e S e c o n d M i n i m u m

331

2.1

Fission isomeric ground state rotational band in 24°Pu . . . . . . . . . . . . . . . . . . .

331

2.2

Measurements of quadrupole m o m e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . .

334

2.3

Spectroscopy of collective excitations in the second m i n i m u m of 24°pu . . . . . . . . . .

335

2.4

2.3.1

7 spectroscopic studies

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

2.3.2

Conversion electron experiments . . . . . . . . . . . . . . . . . . . . . . . . . . .

340

2.3.3

Combined analysis o f T - and conversion electron data . . . . . . . . . . . . . . .

345

2.3.4

Discussion of rotational and vibrational spectroscopic properties

349

335

Excitation and decay of the superdeformed 236I U . . . . . . . . . . . . . . . . . . . . . . 2.4.1

Back-Decay to the first m i n i m u m

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

2.4.2

~/ spectroscopy in the second m i n i m u m of 236U . . . . . . . . . . . . . . . . . . .

352 352 355

2.5

7 decay towards the second m i n i m u m of 239 U . . . . . . . . . . . . . . . . . . . . . . . .

358

2.6

Predictions of phenomenological collective level and lifetime systematics . . . . . . . . .

360

2.6.1

Predictions derived f r o m a valence correlation scheme . . . . . . . . . . . . . . .

360

2.6.2

Extension of the Grodzins systematics . . . . . . . . . . . . . . . . . . . . . . . .

363

2.7

2.8

Multi-phonon fl vibrational excitations in 24°pu

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

366

2.7.1

Transmission resonances in the prompt fission probability . . . . . . . . . . . . .

367

2.7.2

Re-investigation of the transmission resonances in 24°p u

372

2.7.3

Experimentally resolved intermediate structure . . . . . . . . . . . . . . . . . . .

374

2.7.4

Ground state excitation energy in the second m i n i m u m

375

2.7.5

Discussion of rotational and vibrational properties . . . . . . . . . . . . . . . . .

377

2.7.6

Spin dependent fission properties

379

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

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

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

Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

S p e c t r o s c o p y in t h e T h i r d M i n i m u m

381

384

3.1

Expected spectroscopic properties in the third m i n i m u m

3.2

Experimental evidence f o r a third m i n i m u m in Th isotopes . . . . . . . . . . . . . . . .

384

3.3

Identification of hyperdeformed states in U isotopes

388

3.4

Spectroscopy in the third well of 234 U . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1

3.5

4

.........

Depth of the third m i n i m u m

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

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

Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Conclusions

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

384

391 394 395

396

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1

327

Introduction

Studies of fission isomerism in the actinide mass region flourished in the 1960's and 1970's after its first observation by Polikanov et al. in 1962 [1], while in the decade after the conference on fission in Jiilich (1979) [2] rather few new experimental results on fission isomers in the second, superdeformed minimum of the potential energy surface were reported. However, during this period theoretical and experimental progress was achieved establishing a (hyperdeformed) third potential minimum [3]. After the first observation of high-spin superdeformation in the rare earth region in 1986 by Twin and coworkers [4] spectroscopic studies concentrated on establishing and unveiling this phenomenon in various mass regions [5, 6]. Subsequent experimental efforts to search also for hyperdeformed nuclear structures in the high-spin regime failed so far. Spectroscopic investigations of nuclei in the vicinity of the second and third minimum of the potential energy surface in actinide nuclei appeared at the end of the millennium with advanced experimental techniques, leading to remarkable new insights into the structure of extremely deformed nuclei. A comprehensive review of the new experimental results achieved since the last review article on this field by Metag, Habs and Specht [7] will be given in this paper with special emphasis on results recently achieved with complementary spectroscopic techniques in Munich and Heidelberg. For a more detailed discussion of fission isomer properties and the double-humped fission barrier in general the reader is referred to previous review articles by Vandenbosch [8] and Bjcrnholm and Lynn [9]. The appearance of a second minimum in the potential energy surface of actinide nuclei, giving rise to the existence of fission isomers, was first described in a macroscopic-microscopic theoretical approach by Strutinsky as the superposition of microscopic shell corrections to the nuclear binding energy, varying periodically with deformation, onto the unstructured macroscopic part of the deformation energy described by the liquid drop model [10]. The result is a double-humped fission barrier, as displayed for 24°Pu in Fig. 1 as a function of the nuclear deformation.

1. minimum

2. minimum

236U prompt fission

isomeric (delayed)

fission

t 0

!

de'or~nation | ~

~ ~ C~

Figure 1: Potential energy surface as a function of the nuclear deformation. The appearance of the double-humped fission barrier on top of the unstructured macroscopic liquid drop potential energy (dashed curve) due to the superposition of microscopic shell corrections gives rise to the existence of the superdeformed fission isomer. As an example we give the values of 236/U, where the barrier heights are YEA = (5.6 -I- 0.3)MeV and FIB = (5.7 t- 0.3)MeV [7] and where the energy of the ground state in the second minimum lies at En = 2.81MeV above the ground state of the first minimum.

The isomeric ground state can either decay via delayed fission or by 7 decay into the normal deformed first minimum. Specht and coworkers could show that fission isomers are strongly deformed shape isomers when identifying the ground state rotational band in the second minimum of 24°Pu [11]. Final proof that fission isomers indeed represent superdeformed nuclear shapes was achieved by the

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328

determination of their quadrupole moments via lifetime measurements of rotational states, resulting in to an axis ratio of 21 for typical values of the intrinsic quadrupole moment Qe -30 eb, corresponding prolate deformed nuclei [7]. Fig. 2 displays a cut through the potential landscape along the fission path (similar to the curve shown in Fig. 1). In addition to the already discussed appearance of the second minimum here also a third well is drawn in the fission barrier, as theoretically predicted and experimentally observed in the Thorium isotopes [3]. Going from the normal deformed first minimum to the second well the quadrupole deformation parametrized by pz increases from an axis ratio of about 1.3:1 to the superdeformed axis ratio of 2:1, while in the hyperdeformed third minimum with an axis ratio of 3:l reflexion-asymmetric octupole deformed nuclei (characterized by ps) can be found, as illustrated in the upper part of Fig. 2.

axis ratio:

2:l

3:l

P!??P!lii!~??. ,______________ _.__.._

9

1.

2.

3.

0.25

0.6

0.9

_. _. _.

-

I.2

\

I 0

-9

Figure 2: Schematic overview of the multiple-humped fission barrier in light actinide isotopes together with the corresponding nuclear shapes. Lower part: cut through the potential energy surface along the fission path, revealing a superdeformed (SD) second minimum at an axis ratio of 2:l and a hyperdeformed (HD) third minimum at an axis ratio of 3:l. The energy and location of the saddle points and minima (except for the third one) were taken from Ref. [12] while the energy of the third minimum was taken from Ref. [13]. The whole energy scale was slightly shifted in order to reproduce the energy of the ground states in the first well. In the upper part the corresponding nuclear shapes are displayed as a function of the quadrupole and octupole degrees of freedom. From [14].

A whole ‘island’ of fission isomers was identified in the actinide region (2 = 92 - 97, N = 141- 151) with presently 34 experimentally observed fission isomers, as indicated by Fig. 3. Half-lives range from 5 ps to 14 ms, thus being shorter than those for spontaneous fission from their respective nuclear ground state by 24 to 30 orders of magnitude. The exclusive appearance of fission isomers in the actinide region can be understood within the concept of the double-humped fission barrier. Shell corrections to the nuclear binding energy, typically amounting to several MeV, can only lead to significant structures in the fission barrier like the formation of a second well, if the smoothly varying part of the deformation energy is small. This occurs in the actinide mass region due to an almost cancellation between the

329

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surface tension and the Coulomb force. Moreover, fission isomeric states are concentrated in nuclei close to the superdeformed magic neutron number N = 146 [15]. Here the shell corrections are more pronounced, leading to the longest half-lives for fission from the isomeric state of even-even nuclei. With increasing proton number the outer barrier is lowered due to the decrease in the macroscopic liquid drop part of the fission barrier, while at the same time its maximum is shifting towards smaller deformations. Therefore fission lifetimes become too short for experimental observation of fission isomers for Z > 97. In the lighter actinides the inner barrier gets more and more penetrable, allowing for 7 decays back to states in the normal deformed first minimum. Additionally indicated in Fig. 3 is the region between the Thorium and Uranium isotopes, where a third potential well has already been experimentally identified. For the lighter actinides the outer barrier becomes thicker and is split in half. 98 Cf

Z

9.5 ns 600 ns "~ f

97 Bk 55 ns 96Cm

95Am 94 Pu

30n~f

5n~f

~sf

.~f

15ns

l~sf

0..~sf ~ _ s f

~f

34as l.l~t s I~- 6 a s 3ns -I.3 MeV 0.3~lS)Me~l 3(3)MeV 203 keV 3.~.~f ll.~OnsfI~ 0 ~ f ~.~,f 3~sf

93 Np

40 lls f----~ f

92 U

,{

91

820 ns 2 ns ~'f "----~f 180ns >lOOns -1.3MeV ~1.3 MeV 5_~f 4~f .<~p.~f l ~ f -~'fl4ms 5.5~sf 30ns .~sf

---'~flms --'--'~f640ns~ : ~ f

50ns ----'~f3"6ns --"~f60ns 3 ~ s f

90ns,f 151

116 ns

> Ins

f

195 ns

149

> 250ns t 148

Pa

150

N

147

9O ~ 139

140

141

142

143

144

1,15

146

Figure 3: Part of the nuclear chart giving the half-lives of all fission isomers presently known. Two values for the same nucleus indicate spin isomeric states in the second minimum. Isotopes, were the existence of a third minimum in the potential energy surface has been experimentally established have been marked by the index 'HI'.

With the aim in view of performing a detailed spectroscopy in the second potential well, Fig. 4 reminds of the collective vibrational structure to be expected below the pairing gap. Above the ground state rotational band collective quadrupole and octupole surface vibrational excitations are expected similar to the first minimum. On top of every vibrational excitation a rotational band occurs. Besides the superdeformed fission isomers in the actinide region, superdeformation is a general phenomenon in nearly all other mass regions at high angular momenta [5, 6]. For those nuclei a lowering of the potential in the region of superdeformation occurs for high spins (up to ,~ 60h) due to the additional rotational energy. Since the depth of the potential well in the region of high spin superdeformation continuously decreases with lower angular momenta, the decay from the second into the first minimum ('decay-out') typically occurs for angular momenta between about 10h and 30h. In contrast to this the double humped fission barrier for shape isomers in the actinides already exists at low angular momenta, allowing to reach the ground state of superdeformed nuclei through a population of the second minimum.

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330

o&pole

quadrupole

I

I P

1 ;_

I

I

z

p”

-5 -

ci

4+ 3+ -2+ -4+ K’= 2+

%=,I-

T =;

K’-

_4-

-7 f=

4‘ T

5-

CT K9Q2-

-6 -4+ -2+ -0+

K%o+ Figure 4: Schematic overview of the band structure of collective surface vibrations in an even-even nucleus with an arbitrary energy scale for the vibrational states. From [44].

Spectroscopic studies in fission isomers are extremely difficult from the experimental point of view due to the low production cross sections in the presence of the background component from prompt fission dominating by 5 to 6 orders of magnitude. However, they offer unique advantages for clarifying the nuclear structure of strongly deformed nuclei compared to more conventional spectroscopy in the first minimum. The very different energies for rotational and vibrational excitations due to the low angular momenta in fission isomers allow for a clear separation between the two, in contrast to the highspin regime, where Coriolis mixing effects often create a complex spectroscopic situation preventing unambiguous signatures of collective structures. Thus fission isomers offer the unique possibility of detailed spectroscopic studies on collective vibrational excitations in highly deformed nuclei, while no equivalent information has been obtained so far in the high-spin superdeformation regime. The paper contains two major sections: the first one deals with spectroscopic studies in the second minimum. After a review of results on general properties of fission isomers the discussion concentrates on the prototype isotope for spectroscopic studies in the second well, 240fPu. Experiments on lowlying collective vibrational excitations will be presented (Sect. 2.3) as well as recent results obtained for multi-phonon ,B vibrational resonances at excitation energies about 1 MeV below the fission barrier top (Sect. 2.7). Spectroscopic studies on the decay of the fission isomer in 236U will be reviewed in Sect. 2.4 including a revised interpretation of these data based on the information obtained in the 240fPu investigations. An application of phenomenological collective level systematics in the second minimum will be discussed in Sect. 2.6. In the second main section recent progress in our understanding of the nuclear structure in the third minimum will be discussed, ranging from the identification of hyperdeformed resonances to detailed studies of excited rotational bands, closing with the perspective of a new experimental approach to the nuclear shape in the third well. Here the picture has changed compared to earlier reviews, showing a larger depth of the third minimum.

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Spectroscopy

331

in the Second Minimum

Advances in experimental techniques in recent years allowed for major improvements of our knowledge on the nuclear structure in the extremely deformed second minimum. Long-standing questions could New results will be presented in this be answered using complementary experimental techniques. section that span the whole energy range from low-lying collective vibrational states to multi-phonon p vibrational excitations just below the barrier top. In the first section the identification of the ground state rotational band in the second minimum of 240Pu by Specht et al. [ll] is reminded as the first observation of a collective structure in the second well, indicating qualitatively the strong deformation of fission isomers. Quantitative proof of fission isomers being superdeformed shape isomers was achieved by measurements of quadrupole moments. The second section reviews recent measurements in Americium isotopes using laser spectroscopic techniques. Advanced y-spectroscopy techniques allowed for detailed investigations in 240fP~ complemented by a combined analysis with conversion electron data as presented in Sect. 2.3. The subsequent Sect. 2.4 including results on the y back decay from the second to reviews y spectroscopic studies in ‘?J the first minimum as well as a revised interpretation of the collective level structure in view of the General aspects of nuclear structure properties information obtained from the 240/Pu investigations. in the second well will be discussed in Sect. 2.6 in the framework of phenomenological collective level systematics. Recent progress in the understanding of p vibrational multi-phonon excitations in 240Pu up to excitation energies about an MeV below the barrier top will be presented in Sect. 2.7. A concluding section will discuss perspectives of future experimental efforts.

2.1

Fission isomer%

ground state

rotational

band in 240Pu

It took a decade from the first observation of isomeric fission by Polikhanov et al. in 1962 [l] to the identification of fission isomers with strongly deformed shape isomers in a pioneering experiment by Specht and coworkers in 1972 [ll, 71. They exploited the correlation between the moment of inertia deduced from the identification of rotational states and the underlying nuclear deformation. Conversion electron spectroscopy was performed due to the highly converted decay of low-energy rotational states in high-Z nuclei. Choosing the 23sU(cr,2n) reaction because of its high cross section of 10 pb [16] for the population of the fission isomer in 240Pu , Specht et al. measured delayed coincidences between conversion electrons and fission fragments at the Garching Tandem accelerator. In order to suppress effectively the dominant background from prompt fission they used a recoil shadow technique by placing the target into the hole of an annular Si detector thus providing a selective trigger on delayed fission fragments. The energy spectrum of low-energy conversion electrons preceding fission was measured in an iron-free magnetic /I spectrometer capable of handling the enormous electron background resulting from inner electron shell ionization. In order to demonstrate the difficulties of such experiments Fig. 5 displays the differential cross section for electron production as a function of the energy. The dominant part of the cross section is accounted for by the b electron cross section which steeply drops off with the electron energy. The curve shown has been calculated with the ‘binary encounter approximation’ for the system 1s0+208Pb at a beam energy of 90 MeV [17, 181. This cross section scales with 2,” (charge of the projectile) for constant Eproj. /Mpr,,j. and can therefore easily be scaled for 11.3 MeV deuterons. The remaining uncertainties (20 MeV beam energy in the experiment, different target) can be neglected for the subsequent considerations. For comparison the cross sections for conversion electrons from the rotational bands in the 1. and 2. minimum have been added, together with the expected region for vibrational excitations. The 2+ -+ O+ transition in the second minimum is dominated by a I2 orders of magnitude stronger b electron production, while the higher-lying ground state rotational band members could only be identified via their L,M and N conversion lines. It is only in the electron energy region beyond about 300 keV where vibrational excitations are expected that the b-electron background component drops to an amount comparable with the cross section for the vibrational excitations. The conversion electron energy spectrum preceding isomeric fission of 240Pu, corrected for random coincidences, is shown in Fig. 6. The interpretation of the most prominent lines as LI1, LiII and M,N conversion lines belonging to three E2 transitions within the K=O+ ground state rotational band in the

332

t! G. Thirolf; D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

Rotational band

I

5

10

I

I

I

50

III,

I

100

Electron - Energy

500

1000

[ keV]

Figure 5: Differential cross section for the production of electrons in the reaction d -+ 238U, comparing the intensity of the d electron spectrum and the conversion lines in the first and second minimum together with the expected region of vibrational excitations. The curve for the d electrons has been calculated with the ‘binary encounter approximation’ [17, 181. From [19, 201.

second minimum is indicated as well. The final result on ground state rotational bands in 240Pu both for the first and second minimum is illustrated in Fig. 7. The energy of the lowest 2+ -+ O+ transition in the second minimum not observed due to the 6 electron background was estimated from the rotational sequence. Fitting the observed transition energies with a Harris parameterization in the angular momentum

EI = AI(I + 1) + BI’(I

+ l)*

resulted in values of A = (3.343 & 0.003) keV and B = (-0.28 f 0.04) eV. The rotational parameter A = ftz/20 directly reflects the moment of inertia of the rotational band, while the second order non-adiabaticity parameter B describes the affection of the band by rotation-vibration interactions. The extracted value of A in the second minimum, found to be reduced by more than a factor of two compared to the one known in the normal deformed first minimum (A = 7.156 keV [21]) for the first time provided direct experimental evidence for the large deformation of fission isomers. Hence it was justified to identify fission isomers with strongly deformed shape isomers. Quantitatively the superdeformed character of fission isomers with an axis ratio of 2:l was determined by measurements of the quadrupole moment [7], in good agreement with theoretical values [22, 231.

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II G. Thirolf; D. Habs / Prog. Part, Nucl. Phys. 49 (2002) 325-402

30

40

60

50 ELECTRON

ENERGY

70

80

90

100

110

I keV 1

Figure 6: Energy spectrum of conversion electrons preceding isomeric fission in 240Pu from the pioneer experiment by Specht et al. [ll]. Random coincidence events have been subtracted. At specific energies longer measurements were performed resulting in reduced error bars. From [ll].

h

-__-_~-_--_--___-__~)

*4oPU

0

prompt fission

0

deformation

Figure 7: Rotational bands built upon the ground state and fission isomeric state of 240Pu [ll]. Excitation energies and spins of the rotational levels are indicated. The rotational spacings are not drawn to scale, but enlarged by about a factor of 4 compared to the fission barrier heights. From [7].

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2.2

Measurements

of quadrupole moments

In contrast to the analysis of the moment of inertia, where the nuclear deformation can only be extracted in a model dependent way from rotational band transition energies, the measurement of quadrupole moments of fission isomeric states can provide more quantitative information. The nuclear deformation can be deduced from the lifetimes of rotational excitations, which are determined by the quadrupole moment. For a given parameterization of the nuclear shape, the deformation of an isomeric state can be inferred from the measured quadrupole moment. Quadrupole measurements in the second minimum became feasible with the development of the ‘charge plunger technique’ [24,25] that exploits the impact of nuclear transitions on the atomic shells. For a detailed description of the method the reader is referred to the article by Metag et al. [7]. This method was used to determine the nuclear quadrupole moment of the fission isomeric states in 23gPu and 236,238U.The results are listed in Tab. 1 together with all quadrupole moments measured so far in fission isomers. The quadrupole moment for the 36 ps fission isomer in 236Pu was obtained from a different approach, in which electromagnetic transition rates were deduced from the branching ratio for spontaneous fission to electromagnetic decays of the rotational states as inferred from the measured angular distribution of isomeric fission fragments [26]. The values for the Americium isotopes have been obtained by the Maim group from a determination of the difference S < r2 > of the nuclear mean square radii between the fission isomer and the ground state, measuring the optical isotope shift and hyperfine structure with resonance ionization laser spectroscopy in a buffer gas cell [27, 28,29,30]. The deformation parameters pz and the intrinsic quadrupole moments Qe listed in Tab. 1 have been deduced from the nuclear parameters employing a charge distribution from a deformed Fermi model [23] and alternatively using the droplet model [31]. Table 1: Quadrupole moments and deformations of fission isomers and nuclear ground states in the first minimum. c/a denotes the axis ratio of a prolate spheroid. Nucleus

Isomer

Qo(eb) 32rt5

cl@2

1.9 fO.l 29zt3a 1.8 *o.l” 37t1’4c 2.0 f0.3” 36fs4’ 2.0 f0.1” 33.95 a = 0.690* 35.5+1.0,t, f 1.2h,,, p2 = 0.678(51)h 34.45 /32 = 0.694J a [32] * [22] c [26] ’ [23] e [24]

Ground

Qctic(eb) 33.1* 33.3* 34.8b,36.7d

I

J [29] 9 [33’I

stateg

Qo(eb)

c/a

10.81f0.11 10.84f0.07 (11.27f0.15) (11.3hO.5)

1.29f0.02 1.29f0.02 1.31f0.02 1.31f0.05

PI

The quadrupole moments of the fission isomers exceed those of the respective nuclear ground states in the first minimum [33] by about a factor of three, in good agreement with theoretical values for the second well [22, 231. The corresponding nuclear deformation is expressed in Tab. 1 in terms of the axis ratio c/u of a prolate spheroid amounting to about 2:l for the (superdeformed) second minimum and ~1.3:1 for the (normal deformed) first well. Within experimental uncertainties the quadrupole moments of fission isomers compiled in Tab. 1 are in agreement with the expected trend of increasing values in the series of U-Pu-Am isotopes due to the general A dependency of the quadrupole moment and due to the increase of the deformation in the well of the shell correction energy as a function of N and Z, respectively [9]. A remarkable result determined in the isotope shift measurements of the Mainz group in the Americium isotopes was the small change of the intrinsic quadrupole moments c~Q;;~Z~,~~‘* = 0.63(8)eb [28] and bQ~~4JS242J= -0.leb [29], respectively, demonstrating the stability of the superdeformation in the second potential minimum.

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PG. Thirog D. Habs / Prog. Parr. Nucl. Phys. 49 (2002) 325-402 2.3

Spectroscopy

of collective

excitations

in the second

minimum

Experimental insight into the structure of superdeformed nuclei started already 30 years ago. Recently it was again this prototype isotope N = 146 where for the first time a detailed spectroscopy of collective minimum was achieved [34, 351, as it will be presented in the following

2.3.1

y spectroscopic

o~~*~PzL

with the fission isomer in ‘*‘Pu with its magic neutron number vibrational states in the second subsections.

studies

A renaissance of fission isomer spectroscopy started with the availability of newly developed, high efficiency Germanium detectors, which were used in an experimental campaign at Heidelberg, continuing earlier studies with the NaI crystal ball spectrometer [36, 371. Excited states in ‘*‘fPu were populated via the 23sU(a,2n)-reaction at E, = 25 MeV by using a self-supporting 238U-target of 1.7 mg/cm2. The The target thickness was chosen to stop the 240Pu reaction products but not the fission fragments. pulsed o-beam (pulse distance 74 ns, pulse width < 1 ns) was provided by the Tandem accelerator of the Max-Planck-Institut fiir Kernphysik in Heidelberg [34]. The fission fragments were detected by eight segmented position sensitive Parallel Plate Avalanche Counters (PPACs, see bottom part of Fig. 8) arranged in two pyramids covering a solid angle of z 80 % of 47r. In order to optimize the online trigger separation between isomeric fission, signalling the population of 240fPu and prompt fission events, which were a factor of up to lo5 more frequent, a close detector geometry was built to minimize the time-of-flight (TOF) of the fission fragments. The pulse heights allowed for an unambiguous separation of fission fragments from scattered beam particles. Six (four) Germanium CLUSTER detectors of the German EUROBALL collaboration [38] were used in two experiments to measure y-rays with high energy resolution. To achieve the highest full energy peak efficiency the six (four) CLUSTER detectors, comprising 42 (28) individual detectors, were arranged as near as possible to the target position. Fig. 8 gives an impression of the y- and fission fragment detector arrangement around the target. Averaging over the two beam times the total full energy peak efficiency of the setup was determined to be z 8.5 % (at 800 keV), and the overall detector energy resolution was measured to be 2.3 keV (at 800 keV). The close distance between detectors and target of 11.5 cm was tolerable because the interesting events are characterized by low y-multiplicities (N, 5 10) [39] and no Doppler broadening occurred. y rays were registered requiring a coincidence between two fission fragments in opposite PPAC’s and suppressing prompt fission events by a hardware condition on the faster of the two PPAC-signals with respect to the beam pulse. During the analysis y transitions in the second minimum were selected by requiring a isomeric fission event in coincidence with a prompt y ray. To obtain a clean y-spectrum, a prompt time window was applied to reduce the background caused by neutrons and y-rays emitted in flight from the isomeric fission products. A remaining background contribution was subtracted using the y-spectrum observed in coincidence with prompt fission events, which is expected to be of the same overall shape as the isomeric fission spectrum. The resulting y-spectrum is shown in Fig. 9. The high-resolution experiment was preceded by a preparatory y-experiment to investigate the feasibility of the 238U(cu,2n)-reaction for T-spectroscopic measurements [37, 391. Instead of the highresolution Ge-detectors the Heidelberg-Darmstadt Crystal Ball spectrometer, a 47r array consisting of 162 individual NaI(T1) modules [40], was employed. The particle detector setup and the trigger scheme were comparable in both experiments. The high full-energy peak efficiency of the crystal ball (- 60 % at E-, = 800 keV) supplied valuable complementary information on the y cascades in 240fP~. Especially, the y-sum energy E-,,s,, provided information about the excitation energy of states populated after neutron emission. Fig. 10 displays the resulting y-sum energy spectrum together with the strongest of the observed y-cascades. Note that the sum energies of the cascades were deduced from the individual y-energies not taking into account possible losses of excitation energy via conversion electron decays. The intensities quoted are efficiency-corrected y-yields per isomeric fission event. About 20 % of the isomeric fission events are not preceded by prompt (53.5 ns) y-decays of excited states in the second well. Since the rotational in-band E2 transitions are converted for spins < lOh, the y-lines observed in

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336

CLUSTER detector

PPAC:

fiberglass frames //

Figure 8: Experimental setup of the y spectroscopy experiment on 240fP~ performed in Heidelberg [39]. Eight Parallel Plate Avalanche Detectors for the detection of fission fragments were positioned around the target and surrounded by 6 Germanium CLUSTER detectors [38] for the y ray detection (only four of them are included in the drawing). Bottom part: one of the PPAC detector modules with its segmented anode foil. From [34].

Fig. 9 above E., x 150 keV have to be assigned to the -y-decay of intrinsically excited superdeformed configurations. Moreover, aa the production of Pu K X-rays by atomic processes is negligible, the observed Pu K X-ray lines have to be attributed to the K-conversion of transitions in the second minimum of 240Pu with energies larger than the K-binding energy of 121.8 keV. The y-spectrum (Fig. 9) is dominated by a single line at (786.1 f 0.1) keV, unambiguously assigned to 240’Pu [34]. Its efficiency-corrected y-intensity of (37 & 2) YO, measured relative to the number of isomeric fission events, is ten times larger than the intensity of any other transition observed. The high intensity of this transition may be explained by the decay of the bandhead of an excited rotational band (furtheron called a-band) to the ground state band (labelled ‘g’), assuming fast in-band decays of the excited band members and slow inter-band decays. Including the constraints due to the Alaga rule [41, 421, a consistent and unambiguous description of the strong y transitions is obtained by assuming the following scenario (illustrated by the level scheme displayed in Fig. 18): The existence of a K” = 2- rotational band with a bandhead energy of 806.2 keV (a-band), and by assigning the 786.1 keV line to the 2; - 29’ transition and the two weaker y-lines at 758.9 keV and 805.4 keV to the 3; - 49’ and 3; - 2: transition, respectively. The energy difference of the latter two lines is in perfect agreement with the 4: + 2: transition energy of 46.72 f 0.09 keV known from the conversion electron experiment in [7, 111. Moreover, the resulting energy difference

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-50

, 760

I,,

760

1

600

I,,

620

I,

640

..I

660

Figure 9: Background-subtracted y-ray spectrum for 240fP~ resulting from the 238U(a,2n) reaction at E L1 =25 MeV). The thin lines enclosing the zero line mark the statistical la fluctuation expected due to the subtracted background spectrum. The assignments are discussed in the main text. From [34].

between the I,” = 2- and II = 3- state of 19.4 * 0.3 keV corresponds to a moment K” = 2- a-band which agrees within 5% with that of the ground state band.

of inertia of the

Because of the K-hindrance of the El transitions between a K” = 2- band and the K” = O+ gsband, the members of the K” = 2- a-band are expected to decay predominantly by in-band transitions. Assuming a spin population and a quadrupole moment for the a-band similar to that of the ground state band, the in-band decays are found to proceed more likely via unstretched AI = 1 EZ-transitions with negligible Ml contribution rather than by stretched E2’s with AI = 2. The feeding of the band head via stretched and unstretched E2 transitions from the higher-lying members of the a-band results in an anisotropy of a2 = 0.05 - 0.1 for the 786 keV transition, in good agreement with the measured angular distribution W(0,) = 1 +a&(cos 0,) with a2 = 0.04 f0.08. The splitting of the total intensity of 41% carried by the in-band transitions of the a-band on stretched and unstretched transitions is also the reason why these transitions could not be observed in the conversion electron measurement of Specht et al. [ll]. From the inter-band transition intensity of the 3- state its half-life can be estimated to be = 130 ps. This value corresponds to a [AK] = 2 hindrance factor of B(E1),,U./B(El)eZp x 3 . 105, in agreement with measured El-hindrance factors for ]AK] = 2 transitions in the rare earth and

338

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counts / 50 keV

Figure 10: y sum-energy spectrum of 240fP~ (right panel) measured with NaI resolution in a preparatory experiment [39]. The decay scheme deduced from this measurement (left panel) summarizes the strongest of the observed y-cascades. From [34].

actinide region, which cover a range of 2 . lo5 - 2 . lo8 [43]. Moreover, the measured intensity ratio a gr ees with the Alaga rule value of 1.6 for an El transition when Z(3, -+ 2,f)/Z(3, -+ 4;) = 1.45f0.35 allowing for a small K=l admixture in the 3- state. A conclusive proof of the K” = 2- assignment for the a-band was obtained in the conversion electron experiments discussed in Sect. 2.3.2, where the conversion coefficient of the 786.1 keV transition was measured to be ok = (4.6 f 1.7) . 10m3 [35]. This value established directly the El character of the 786.1 keV y-transition and fixed the negative parity assignment of the 806.2 keV state. Due to their fast intra-band decays, higher spin states of the K” = 2- rotational band do not manifest themselves via inter-band transitions. However, a second rotational band (c-band) located above the K” = 2- band is populating the higher-lying members of the band and thus allows for an extension of the level scheme: supported by the coincidence relations observed in the preparatory crystal ball measurement, most of the y-lines observed in the energy region around 550 keV (see inlet in Fig. 9) can be attributed to transitions feeding the a-band. Levels of the c-band are depopulating into consecutive levels of the a-band in two sequences of Z, + Z, and Z, -+ (Z, + 1) transitions, which allow to deduce the a-band levels up to I” = 66. These transitions confirm the spacing between the two lowest levels of the a-band and the extension of the band expected for an A-parameter of A, = 3.26(2) keV. The c-band can be assigned to a K = l(2) rotational band with a band head at 1344.4kO.5 (1360.9f0.3) keV (see level scheme Fig. 18). Although there is evidence for direct decays of c-band states into the ground state band (weak y-lines at 1341 keV, 1320 keV, 1356 keV and 1380 keV can be assigned to the 2, + 29’, 3, + 49’, 4, -+ 4: and 6, + 69’ transitions) the spin assignment of the c-band levels has to be regarded as tentatively when considering only the y spectroscopy data. This ambiguity will be removed when discussing the results from conversion electron measurements in Sect. 2.3.2. Moreover, the overall y-intensity of 13 % carried by the c-+a transitions agrees with the value of 12 % deduced in the crystal ball experiment. In the energy region from 750 to 870 keV (lower part of Fig. 9) additional lines show up beside the a-band decays, which can be grouped in three sequences. The strongest ones are obviously due to I: + Z,’ transitions from the decay of the even spin members of an excited rotational band (bband). Regarding the possible K and parity values of the b-band, a positive parity assignment is rather unlikely: For a K” = O+ band, the O+ + 29’ transition would be expected at 808 keV, but is not observed; moreover, all other Zb + (I - 2)9 and I,, + (I + 2), transitions expected for decays of a positive parity band are not seen either. On the other hand, the assumption that the Zb + Z,

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decays are of El type, i. e. that the initial states have negative parity, is readily explaining the absence Moreover, the anisotropy of az = 0.65(15) of competing decays to other members of the gs-band. measured for these transitions agrees well with the value of az M 0.7 expected for pure dipole radiation. Therefore, the most likely assignment for the b-band is K A = l- with a bandhead at (836.0k0.5) keV. A K” = 2- assignment with a band head at (846.8kO.3) keV cannot completely be excluded, because the 1; -+ (I + l),’ and 1; + (I - 1); El-decays of the odd spin members of the b-band result in only marginally significant y-lines (see Fig. 9). Approximately 14 % of the observed y flux proceeds through this band, again in agreement with the finding in the crystal ball experiment (see Fig. 10). Although the evidence is weak, it seems worthwhile to mention a candidate for a K" = O- octupole band: a y-line observed at (251.5f0.2) keV with an intensity of 1.3% can be combined with part of the 554.8 keV line to the bandhead energy of the a-band of (806.2fO.l) keV. Furthermore, a weaker y-line observed at (216.5f0.5) keV can be combined with a 569.2 keV component of the broad 570.3 keV line structure to form another decay cascade of the 2- band head of the a-band leading to the 2+ member of the gs-band. The 555 and 569 keV lines are also associated with c-band decays, however, their intensities show significant deviations from the smooth behaviour of the other c-band decays and justify the partitioning. This can be illustrated with Fig. 11, where the y intensities of transitions connecting the c- and a-band are displayed. I

3-

.e

2-

GJ

1 1 0+ga 0’

1 -

Ic-Ia

+ ..‘.

.;-

.I..

.__j ‘..,

‘.

‘.., ‘,f

1

angular momentum Iti]

2

3

4

5

6

1, [%I

angular momentum Ifi]

Figure 11: Experimentally observed y ray intensities for transitions between the c- and a-band. Shifts of data points to positions marked by hatched symbols originate from a partitioning of the measured intensity between two transitions as discussed in the text.

With the reasonable assumption that the lower-energy transitions are first in the decay of the 2, state, the two cascades lead to intermediate states at (554.7f0.2) keV and (589.7f0.5) keV, respectively. Their spacing of (35.0f0.6) keV and their feeding via the 2- state suggests a l- and 3- spin assignment for the two levels as expected for a K” = O- rotational band with a moment of inertia very similar to that of the gs-band. Summarizing the results from the y spectroscopy experiments it can be stated that the strongest non-rotational excitations in 240fP~ observed with y spectroscopy are characterized by K" = 2- and K" = l-(2-) (Fig. 18). This is in contrast to expectations (441 that the lowest-lying intrinsic excitation modes in the second minimum can be assigned to collective surface vibrations of the /3- and y- or K" = O- octupole type. No evidence was found for the /I- and y-vibration, and only a weak signal for a K" = O-octupole vibration, even though from investigations of transmission resonances in isomeric and prompt fission of 240Pu [45] the bandhead energies of the quadrupole vibrations are expected to be in the same energy range as the observed ones. Complementary conversion electron studies reviewed in the following Sect. 2.3.2 could confirm this expectation for the p vibrational band by their identification in the electron spectra. The weak evidence of a K" = O- octupole band head at an energy of 554.7 keV is in agreement with results deduced from fission resonances in 240Pu [46], which constrained the excitation energy connected with the (K" = O-)octupole phonon to (750 f 200) keV, but the population of this band by only 2.0 % via the a-band appears to be surprisingly low. Altogether, y spectroscopy with state of the art experimental detection techniques marked a breakthrough in the spectroscopy of fission isomeric collective excitations. Nevertheless, the need for com-

PG. Thirolf; D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

340

elementary ambiguous excitations 2.3.2

experimental information from conversion determination of spins and multipolarities via EO decays.

Conversion

electron

electron studies became evident, allowing for an as well as for a search for collective quadrupole

experiments

The experiments were performed at the Garching Accelerator Laboratory [35, 471. Conversion electrons preceding the fission decay of the 3.8 ns isomer in 240Pu were measured in coincidence with isomeric fission. Excited states in the second minimum of 240Pu were populated by the 238U(Cu,2n) reaction at the same beam energy E, =25 MeV as used in the 7 ray experiment [34]. In order to achieve an efficient suppression of the dominant background from prompt fission events, the same recoil shadow technique as in the work of Specht et al. (111 was used. The 238U target (30 pg/cm2 on a 10 /-lg/cm2 C-backing, diameter 2.5 mm) was placed inside the 4 mm hole of an annular Si detector (100 pm thick, 450 mm2 active area). cr-induced prompt fission reactions happened at the target position in the shadow of the sensitive detector area, while 240Pu nuclei populated in an isomeric state recoiled from the target and decayed in flight in front of the annular detector with an average half-length of 1.8 mm (corresponding to the isomeric half-life of 3.8 ns), as illustrated in Fig. 12b). Thus only fragments from isomeric fission were detected (R =0.14/4~), while prompt fission events were quantitatively suppressed. b)

238

U target

II

annular detector 10 mm

Figure 12: a) Experimental setup for the conversion electron measurements. Only one of the three Mini-Orange spectrometers placed symmetrically around the beam axis at an angle of 50” is shown. b) Enlarged drawing of the annular fission fragment detector setup. From [35].

Electrons preceding the isomeric fission decay and therefore corresponding to conversion electrons promptly emitted by the 240fP~ recoil nucleus were measured with three Mini-Orange spectrometers as indicated in Fig. 12a). Mini-Orange spectrometers [48, 49, 501 combine the excellent energy and time resolution of a cooled Si(Li) detector with the high efficiency and energy selectivity of a magnetic MiniOrange (MO) filter. Data were obtained from experiments with two sets of Mini-Oranges adjusted for an optimized transmission from 400-600 keV and 500-700 keV, respectively. The MO filter consisted of 6 (5) wedge shaped permanent Nd2Fe14B magnets, which transported and focused the electrons with a * well defined transmission curve, while at the same time suppressing the copious low-energy b electrons. Moreover, y rays from the target were blocked by a 20 mm long cylindric Pb absorber (radius 6 mm), positioned in the-center of the MO. The optimum transmission efficiency of the MO filter was achieved with symmetric object ‘and image distances, a field strength of the toroidal field of 90 & 5 mT (for the 6-wedge MO) and a minimum thickness of the magnets. A transmission value of 2.3 % at 626 keV was measured with a single Si(Li) detector of 300 mm2 active area (3 mm thickness). The statistically most prominent feature in the electron spectrum in Fig. 13 is a broad structure between 620 and 650 keV, which is resolved into six individual lines with line widths of about 4.0 keV.

II G. ThiroQ D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

341

As the y-ray

spectrum displays no significant transitions at the corresponding energies, a conversion coefficient of (YK 2 0.3 is deduced for these e- lines, which is only compatible with a dominating EO multipolarity. This structure was interpreted as being due to the decay of a rotational band built on top of the lowest P-vibrational phonon.

380

400

420

440

460

480

500

520

electron energy [keV]

540

electron energy [keV]

Figure 13: Conversion electron spectrum in coincidence with delayed fission from 240fPu (from [35]). The summed transmission efficiency (weighted average for the two sets of magnets) of the three Mini-Orange spectrometers is represented by the solid line. Below: Enlarged parts of the electron spectrum with fitted line spectra and transition assignments (as discussed in the main text). From [35].

The spin assignment of the 1s + Jg sequence is based on the intensity pattern of the individual lines, which decreases in a regular way (see bottom panels of Fig. 13), similar to the intensity distribution observed in the same reaction for the ground state rotational band of the second minimum of 240fPu [ll] and to the lowest P-vibrational bands observed in the second minimum of the neighbouring isotopes 236,238U[51]. Fig. 14 shows the conversion electron spectra obtained in coincidence with the exhibiting widths approximately twice as large as isomeric fission of 236f,238fU. P rominent structures, the experimental resolution, were interpreted by Goerlach et al. [51] due to their energy spacing and relative intensities as K and L conversion-electron lines associated with average transition energies of 685 and 645_keV, thus establishing the phonon energies of the first excited @ vibrational bands for these shape isomers to be h~(~~*fU)=647.8 keV and tWp( 236fU)=686.0 keV, respectively. Moreover, similar sequences of closely spaced but resolved EO transitions from K” = O+ bands have also been observed in the first minimum of 234U and 240Pu after (o,2n) reactions [52] as well as in conversion electron studies following Coulomb excitation using an (a, ok) reaction on 238U [53].

342

PG. Thimlf; D. Habs/Prog.

Parr. Nucl. Phys. 49 (2002) 325-402

*3*U(d,pn)23W

20

E,=2QMeV

i

5 5

I

K685ksV

2W(d,pn)236’U

100

200

300

400 electron

500 energy

600

LM

700

, @J,

600

900

[k&l

Figure 14: Energy spectra of conversion electrons in coincidence with isomeric fission decay of the shape isomers 236fU and 238fU, establishing the phonon energies of the first excited @ vibrational bands &~~(~~s’U)=647.8 keV and Aws( 236fU)=686 keV. The insets show the time spectra of the coincidences for the specified region of electron energies. From [51].

Further support for the interpretation of the characteristic structure in the electron spectrum of Fig. 13 as being decays from a K * =O+ p vibrational band comes from a comparison of the relative intensities of individual electron transitions in 240Pu and 234U The angular momentum distribution of the transitions in 240fPu nicely coincides with the rather similar angular momentum distributions for the lowest P-vibrational band in the first minimum of 234U and 240Pu as well as for the distribution obtained from the ground state rotational band in the second well of 240Pu and a theoretical angular momentum distribution calculated in the semi-classical approach of [54], as can be seen in Fig. 15. In all these cases the population maximum occurs around 2fi, while angular momenta >lOA are populated with intensities below the detection limit. Usually the multipolarity of a converted transition can be determined by the K/L ratio. However, for all of the three fission isomers summarized in Table 2 an unambiguous assignment could not be achieved via this method due to the limited experimental statistics. Only pure E2 transitions seem to be inconsistent with the experimental K/L ratios. Nevertheless, as included in Table 2, multipolarities other than EO require unrealistic or at least unusually large population probabilities of the decaying level to be consistent with the observed intensities. This is illustrated in Table 2 by comparing the total number of y- and conversion electron transitions per isomeric fission event N,,, = N, + N, required to explain the observed intensities of K lines. For the intensity carried by the p band the situation is as expected: the lower the energy fiw, of the first /?-phonon the stronger the feeding. For 238fU with &.+=647.8 keV a value of 4.3 % relative to the isomer decay and for 236fU with fiw,=686 keV a value of 2.4 % has been observed 1511, while for 240fPu with fuJB=770 keV only 2.0 % were measured [35]. In view of all these arguments the most likely multipolarity for the observed transitions in 240fPu is EO. Their interpretation as the decay of the lowest P-vibrational band is illustrated in Fig. 16 together with the similar situation in 236fU and 238fU, respectively. In comparison with the ground state rotational band the level spacing in the excited K" = O+ bands is increasing less with spin J. Hence the degeneracy of the Is +I, transitions is eliminated, allowing to resolve the individual transitions in the case of 240fPu as shown in Fig. 13. The resulting p phonon energy ?iwp = (769.9 f 1.0) keV falls

PG. Thirolf; D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

80

2

: QSband

60

A :p

.zf,gs40 a, E ._

343

(2. min.) [I 1 I

band (2. min.)[35]

0

: p band (1. min.)

V

: p band (1. min.)‘[55]

[55]

20

0 0

2

4

6

8

angular momentum

10

[h

1

Figure 15: Experimental angular momentum distributions for the ground state rotational band (2. minimum) and /3 vibrational bands of 240Pu (1. and 2. minimum) and 234U (1. minimum). The lowest data point for the ground state band comprises the unobserved intensity of the O+ and 2+ state. Also shown is a theoretical distribution (open squares and connecting solid line) calculated for 240Pu with the semi-classical approach of [54]. Consistently in all cases the maximum lies around 2 fi and angular momenta > 10h are populated below the detection limit.

within the expected margins of 600-800 keV deduced from transmission resonance measurements [46]. The resulting level scheme of the p band is included in the overall level scheme for the second minimum of 240Pu as discussed in the next section and displayed in Fig. 18. Potential y-ray transitions from the ,B band to the ground state band J$ -+ (J - 2): may be covered up by otherwise assigned transitions. However, the discussion of the branching ratios between EO transitions and E2 y-ray transitions given in the following section will show that none of the E2 y transitions is expected to be visible in the y-ray spectrum.

PG. Thirolf; D. Habs / Prog. Part, Nucl. Phys. 49 (2002) 325-402

344

Table 2: Phonon energies, population intensities (normalized to delayed fission) and K/L ratios for the first excited /3 vibrational bands in the fission isomers 236f!238fU [51] and 240fP~ [35]. K,/DF denotes the experimental intensity of K conversions relative to the number of isomeric fission events (DF). In addition the bottom part of the table gives the theoretical K/L ratios for the most important multipolarities aX together with the total number of electron and y transitions Nttt = (iV, + N,) per isomeric fission event required to explain the observed intensities of the K-lines [45, 551. 238f u

hwa K, /DF K/Le3P ox EO El E2 Ml M2

236fU

647.8 keV 3.5 % 4.8 f 1.2

(K)theor 5.3 5.7 2.9 5.2 4.2

236fU

0.043 5.50 2.00 0.35 0.14

236f u

240f pu

686 keV 2.0 % 4.7 f 1.4

770 keV 1.7 % 5.0 f 1.3

(A$ + N)/DF 0.024 3.70 1.25 0.23 0.095

240f

0.020 4.70 1.40 0.28 0.13

p"

Figure 16: Decay pattern of the first excited p vibrational bands in the second minimum for 240f Pu. While for the U isotopes unresolved EO transitions were interpreted in analogy to the decay scheme of states in the first minimum (from Ref. [51]), the decay pattern for 240fP~ was based on the observation of individual EO transitions (from Ref. [35]). The competing E2 transitions are suppressed by the conversion coefficient and therefore are not drawn in the figure.

236fU, 238fU and

l?G. Thiro& D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402 2.3.3

Combined

analysis

of y- and conversion

345

electron data

Both the prompt y-ray spectrum from Fig. 9 and the conversion electron spectrum shown in Fig. 13, respectively, have been measured with the identical (a,2n) reaction and the same trigger condition for isomeric fission of 240fPu, thus allowing for a combined analysis of y and electron data as indicated by the combined display in Fig. 17. Since conversion electron transition energies differ from the corresponding y-ray transitions by the K-binding energy of 121.8 keV, the electron spectrum in Fig. 17 has been shifted by EK to allow for a direct comparison. The electron spectrum contains a much smaller random background contribution compared to the y-ray spectrum because converted transitions from fission fragments occur at a significantly reduced rate and only electrons from the target area are focused onto the Si(Li) detectors. In general it should be noted that the properties of 240fPu as being an ideal rigid rotor can be used advantageously for the construction of the level scheme. Assuming a smooth variation of the moment of inertia already the definite identification of only a few low-spin members of a rotational sequence allows rather accurate predictions for the higher-lying members of that rotational band. Thus even transitions with limited statistical significance can rather reliably be placed into the level scheme.

E, fkevl Figure 17: Combined view of y-spectroscopy (upper part) and conversion electron data (lower part) from the second minimum in 240Pu. For better comparison the conversion electron spectrum has been shifted by the electron binding energy of EK = 121.8 keV.

For by far the strongest y-ray transition in 240fPu at 786.1 keV a conversion coefficient of ok = (4.6 f 1.7) . 10m3 was obtained after averaging over the two sets of experiments. A comparison with theoretical conversion coefficients (561 shows that the experimental value agrees well with the theoretical El conversion coefficient of 5.9. 10b3. This proved the El character of the 786.1 keV transition, a result that was used in [34] to deduce conclusively the spins and parities of a rotational band built on top of a

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state at 806.2 keV (a-band band.

in the level scheme, Fig. 18), and to support

its interpretation

as a X” = 2-

c band K&l20 %


- IO’ -10

ps

gsband

Figure 18: Level scheme of the second minimum of 240Pu. The accuracy of the Dopplercorrected transition energy is given in round parentheses 0, the absolute y-ray intensity (in %) with respect to the isomeric fission decay in rectangular parentheses [] and the dominant multipolarity together with the absolute electron intensity (in %) in round parentheses (). Transitions with electron energies below 100 keV were detected in Ref. [ll]. Levels with energies in parentheses were introduced in Ref. [35] by a smooth extrapolation of the moment of inertia. Intensity values containing the variables [U],[X],[Y] indicated transitions that appear twice in the level scheme, with the variables denoting the distribution of the measured intensity between the two transitions. From [35].

The high intensity of the 786.1 keV transition was explained by the decay from the band head of the a-band to the ground state band assuming fast rotational in-band transitions and slow inter-band decays. From the branching ratio of the decay of the 3--state of this band to the ground state band via El interband transitions a lifetime for the band head of -130 ps could be estimated 1341as already discussed in Sect. 2.3.1. Except for the 786 keV transition (37%) all other y-ray transitions in 240fPu have absolute intensities of less than 5 % with respect to the isomer decay and thus only EO transitions or transitions with a multipolarity larger than 2 are expected to result in detectable lines in the conversion electron

347

F!G. Thirolj D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

spectrum. However, transitions

with high multipolarity can be excluded; due to their longer lifetimes suppressed by the competing fission decay. Thus the lines observed in the expanded parts of Fig. 13 will be very likely due to ED transitions, i.e. new and complementary information is obtained from the conversion electron spectroscopy, exceeding the information obtained from the y-ray spectroscopic studies. Thus the combined interpretation of y ray and conversion electron transitions in 24o/Pu enables for the first time a detailed spectroscopy in the second minimum leading to the level scheme for 240@u as displayed in Fig. 18. Table 3 lists the properties of the transitions identified in the conversion electron spectrum together with the corresponding y lines and t,he resulting conversion coefficients for the strongest lines. In the last column the interpretation of the transitions in view of the level scheme displayed in Fig. 18 is listed. While a negative parity for the a- and b-band was already established in Ref. [34] as a result of the y spectroscopic studies, this property could not be determined conclusively for the c-band. Taking into can be expected account also the conversion electron data, (JC + J,) and (JC --f Jb) EO transitions under the assumption of a negative parity for the c-band. For the transitions connecting the c-band and b-band (JC + Jb) this expectation could be nicely confirmed up to the spin 6tL, thus proving the negative parity of the c-band. After subtracting the &contribution of the strong group of EO lines around 411 keV the 1; -+ 1; transition is below the detection limit, while the 2; --f 2, line remains clearly visible. Note that also in the y-ray measurement [34, 391 the indications for the l- level of the c-band were rather weak. Due to the different moments of inertia of the c-band and b-band these EO-transition energies are spread out over a larger energy range. Assuming a smooth variation of the moments of inertia with spin also candidates for the 7, --t 76 to 10, --t 106 EO transitions could be identified. On the other hand, the st.rong (JC + J,) E2/Ml transitions from the c-band to the K” = 2- band at 806 keV show only an EO admixture of at most 10 %. Strong evidence for candidates with a direct decay sequence to the ground state band J, + Jg and J, -+ (J -t 1)9 could be identified in the high energy part of the y-ray spectrum [57] which are included in the level scheme of Fig. 18. The newly placed transitions between the four bands of N.ef. [34] strongly support the level scheme and the parity assignments. Although the direct experimental evidence for the l- bandhead of the c-band (band head energy 1344.5 keV) is still weak, the strong E0 transitions to the K” =l- b-band favour the K” = lassignment for the c-band. In the conversion electron spectrum an additional bunching of strong EO transitions (280.3 keV [1.7%]; 289.7 keV [2.3%]; 296.0 keV [l.O%]) corresponding to nuclear transition energies around 411 keV is observed. They connect bands with rather similar moments of inertia. Due to their rather large intensities they cannot feed into the P-band or the proposed O- octupole band. A direct feeding into the ground state band is excluded from our knowledge of collective excitations obtained in transmission resonance measurements. Therefore, these EO transitions most probably lead into one of the negativeparity bands (a-, b- or c-band). Similar to the other negative-parity bands this new band is expected to decay also by El transitions to the ground state band. The EO transitions with transition energies around 411 keV can be arranged in the level scheme by introducing additional Kn = l- ban& wit,h band head energies of 1246 keV and 936 keV [57]. The EO transitions between the new bands and the b-band or between the c-band and the new bands would be in the region of 411 keV. Band heads are expected at these energies from former measurements with NaI detectors [34, 391. Their expected interband transitions to the ground state band can be tentatively identified as weak y-lines in the high energy part of the high resolution y-ray spectrum [39, 571. Since these two bands are rather tentative they were not included in Fig. 18. they are strongly

In the second minimum of 236!238Usimilar lower-lying EO transitions with transition energies around 400 keV have been observed as well [51]. Although the electron detection efficiency is continuously decreasing below 400 keV, no prominent structures seem to appear in the energy range down to -100 keV. Thus the conversion electron experiment from Gassmann et al. [35] spans an energy range down to the highest energies studied in the experiment by Specht and coworkers [ll]. Within the accuracy of the calibration the conversion electron spectrum of Fig. 13 shows similar structures like the spectrum of Ref. [ll] above 90 keV. In the level scheme

of Fig. 18 the lO+ level at 364.5 keV has been added

to the ground

state

F!G. Thirog D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-i-402

348

Table 3: Electron and y ray energies, tation of the individual transitions.

E,

Intensity in % 1.7 2.3 1.0 0.7 0.4 0.4 5 0.1 5 0.1 0.3 0.7(3) -A A

in keV 280.3 289.7 296.0 393.0 (15) 398.6 (14) 407.2 (12) 413.4 416.8 421.8 (13) 433.0 434.7 436.3 439.1 (13) 450.0 (14) 460.4 (12) 473.3 476.2 487.5 492.2 501.0 506.5 515.8 523.1 617.8 625.8 630.7 636.4 643.2 648.1 657.1 664.3 677.9 683.4 693.9 699.0 702.6 704.9 714.2 724.2 730 737 742.7 770.3

0.1 0.2 0.7 0.7(2) -B B 0.3 < 0.1 0.2 0.3 0.2 0.3 0.12 0.19 0.20 0.36 0.49 0.33

(15) (15) (12) (12) (14) (10) (13) (12) (14) (13) (12) (11) (11) (10)

0.2 0.16

0.2 (1)

-L

0.12 (1) 0.12 (1)

intensities

[ntensity

E, in keV 402.1 411.5 417.8 514.8 520.4 529.0 535.2 (2) 538.6 (2) 544 554.8 (2) 556.5 558.1 (3) 561.0 570.3 582.9 (2) 595.1 598.0 609.3 614.0 622.8 628.3 637.8 644.9 739.6 747.6 752.5 758.2 765.0 769.9 778.9 786.1 799 805.4 815.7 820.8 824.4 826.7 836 846 852 859 864.5 892.1

and conversion

in % IO.1 3.4 IO.1 5 0.8

2.5 (3) 1.5 (2) 3.6 2.7 (3) 1.0 1.8 3.5 1.1

coefficients

QK

2 17 5.75 L 10 (+ L-line (414.7keV))

I 0.04 5 0.07 D.26f0.11

(3) (3) (5) (3)

together

Interpretation transitions

(2)

(1) (2) (3) (2) (3) (3)

::3” 1.7 1.8 2.6 3.2 0.5 0.6 0.5 0.5

2 1 1 1 1 1

0.6 0.95 0.3 0.25 2.5 1.7

(5.5i1.7).10-3

(3) (3) (3) (4) (4)

+ --f + -+ + -+ + -+

23435526-

(c+b) (c-tb) (c-tb) (c-ta) (c--ta) (c-tb) (c-ta) (c+b)

347586-

+ -+ --f -+ --f -+

347586-

(c-+a) (c+a) (c+b) (c-+a) (c+b) (c+a)

7-

(c-ta)

9- --f 9-

(c-ib)

lo- --t lo- (c+b) lo+ -+ lo+ p t gs 8’ -+ 8+ (B --f gs) 6+ --t 6+ (/? -+ gs) 4+ -+ 4+ (P--f gs) 2f --f 2+ (/I --f gs) o+ + o+ (p + gs) 5- -+ 6+ (b+gs) El 2- -+ 2+ (a+gs) 38642-

L-line (753 keV) L-line (765.0 keV)

of the

23424526-

7- i

5 0.2 IO.2 IO.7 1.6 (3) I 0.2 I 0.2 1.2 (3) 36.6 (3)

with the interpre-

--t --t --f -+ +

2+ 8+ 6+ 4+ 2+

(a-tgs) (b+gs) (b--tgs) (b-tgs) (b-tgs)

349

I? G. Thimlf; D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

band by extrapolation of the moment of inertia known from the lower members of the band. From the spin population of the p band a sizeable intensity for the lO+ level can still be expected compared to the 8+ level. However, for the 10 + + 8+ transition K-conversion is possible. Therefore, most of the intensity occurs at very low electron energies and the intensity of the L lines is reduced by about a factor of 3 below the detection limit of Ref. [ll] and could consequently not be observed in their measurement, Above 97 keV K conversion lines with accompanying weaker L lines determine the pattern of the electron spectrum. With the known level spacings of the a-band and the high intensity of this band, it is intriguing to identify the expected L- and M-lines of the (5- -+ 4-), (5- --t 33) and (4- + 22) transitions of the (K” = 22) rotational band to corresponding structures in the spectrum of Specht et al. [II]. Their energies and intensities are included in the level scheme of Fig. 18. It seems worthwhile to re-investigate this low-energy region with the improved modern experimental techniques. rotational

2.3.4

Discussion

of rotational

and vibrational

spectroscopic

properties

For 240fP~ a clear separation between rotational and vibrational excitations has been observed. Considering the rotational excitations first, in order to compare the rotational energies of the collective bands observed in y and conversion electron spectroscopy to those of the ground state band of 240fPu, the sequences were fitted with a power series in angular momentum [42] E(K, I) = EK + AI(I + 1) + B12(I + 1)2 + b(K, l)(-1)‘+‘1(1+

l)(Az + &I(1

+ 1))

(2)

The parameters resulting from the fit of Eq. (2) to the observed bands are compiled in Table 4 together with corresponding values for rotational bands in the normal deformed minimum of 240Pu and 23*U [55, 581 as well as for the ground state and p vibrational bands observed in the superdeformed second minimum of the Uranium isotopes 236,238U[51]. The different values for the rotational parameter A of the ground state bands in 240Pu and 240fP~ (similar to 236,236/U and 238,238fU) reflect the different deformations in the normal deformed first and superdeformed second minimum [59]. The second order non-adiabaticity parameter B for the ground state band and b-band as well as the staggering coefficient A2 are found to be smaller as compared to the normal deformed bands. However, the higher rigidity against rotation, which is observed for ground state bands in superdeformed actinide nuclei (601, is not prevailing for excited superdeformed configurations. Being inversely proportional to “,h; _%yaymeters, the dynamical moments of inertia defined as 02(J) = iE(J)-,(J-l)l and @z(J) = I ( )45-2 ( )’ ’ respectively, have been extracted from the rotational bands in 240fP~ as shown in Fig. 19. The similarity of the moment of inertia of the a-band to that of the ground state band in 240fP~ indicates that these bands are based on superdeformed configurations with very similar deformations. The smaller moment of inertia for the c-band compared to the ground state band suggests a deformation between that of the normal deformed and superdeformed minimum. For the p vibration an increase of the moment of inertia of about lo-20 % above the ground state value can be deduced. Only the B band exhibits a considerable variation of the moment of inertia with the rotational frequency. For the lowest spins the /3 band has a moment of inertia close to that of the rigid body (Oris/t22 M 200 MeV-’ [7]), while it gradually approaches the value of the ground state rotational band at higher spins. In general all rotational bands with K 2 0 may be subject to a signature splitting caused by the Coriolis interaction between the two (fK) components of the intrinsic wave function. In well-deformed even-even actinide nuclei, however, only the K” = l- bands show a significant staggering between even and odd spin states, which can be well described by the corresponding signature term included in Eq. (2). This seems to be also true for the superdeformed K” = l- b-band of 240fP~ (band head energy 836 keV), an observation which provides further evidence for its K” = l- assignment. The b-band shows a rather constant moment of inertia of the same size as those of the ground state , a- and P-band when considering the even and odd spin members separately. However, deducing the moment of inertia for subsequent spins leads to a strongly alternating behaviour of O2 with spin J. There is no obvious reason for the anomaly of the odd-even staggering at spin 3fi and 4fL. The admixture of different K components in the intrinsic wave functions of strongly deformed nuclei is of general interest. Since we expect steeply down-sloping Nilsson orbitals with high j quantum

F! G. Thirolf; D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

350

Table 4: Rotational energy parameters formed second minimum. Nucleus T4OfPU

K”

(et) -0.28(4) -

(2) -

(:G) -

806.2

3.26(2) 3.07(l)

1344~(~;3osO.9) 769.9

4.20(3) 2.69

-1.7(6) 0.34(16) 4.3(8) 4.36

-70(6) -

0.20(13) -

o+

0.0

7.09

-2.58

-

-

0, 1-

587.1 936.1

5.13 6.02

1.51 -1.11

411.

-1.63

7.42 5.26 6.36 6.52

-3.26 0.87 -22.7 8.01

760. -

-22.3 -

o+ 0;

I;;-) Of

ssaU

Of

0.0

0, 12-

668.9 930.8 1128.7

[341 [71 [341 [34]

%+

686.0

3.0(l)

-

-

-

[511

0;

0.0

3.27(3)

-

-

-

[71 _

236U

0+

0.0

7.49(2)

-3.34(9)

-

-

2S6fu

0,+

647.8

3.1(l)

-

-

-

[511

0';

0.0

3.36(l)

-

-

-

[611 1

=nf u

first and superde-

[Ref.] (k:V) 3.343(3) (3.5)

2,

24OPu

E(K,I=K) (keV) 0.0 (547.7)

for 240Pu and 236,238Uin the normal deformed

numbers and low K values as a function of deformation, strong Coriolis matrix elements at the Fermi surface of the second minimum leading to K mixing are expected, even for low spin levels of even-even nuclei. While good agreement with the Alaga predictions of the branching ratios for the K” = 2band could be found, i.e. a pure ground state- and a-band, too small branching ratios between spindecreasing transitions J + (J - 1) and spin-increasing transitions for the El transitions between the b-band and the ground state band have been observed. A small admixture of K” = O- components to the K” = l- bands may cause a drastic increase in the branching ratios between spin-increasing and spin-decreasing transitions, because usually the AK = 0 transitions are faster than the AK = 1 transitions [43]. Therefore, the variations of the moments of inertia and the branching ratios provide evidence for K mixing of the K” = I- wave functions of the second minimum. In the case of 240fPu this K mixing obviously manifests itself in the decay of the c-band, where the sequence of EO transitions between the K” = l- c- and b-bands can be interpreted as a ,Dvibration built on the K” = l- -state at 836.0 keV, while at the same time the weakness of the ED admixture in the (J, + J,,) E2/Ml transitions from the c-band to the K” = 2- band at 806 keV favours the interpretation of these transitions as a y-vibration built on the K” = 2- band. Next collective vibrations will be discussed, especially the ,0 vibrations. Besides the first P-vibrational phonon at 769.9 keV in 240fP~ identified in the conversion electron studies discussed in Sect. 2.3.2 and 2.3.3, multiple /3 phonons were observed as O+-transmission resonances at excitation energies of 4.5 MeV, 5.1 MeV and 5.5 MeV [46, 621. Considering the excitation energy of the ground state of the second minimum of (2.25 f 0.2) MeV [9, 621 these resonances correspond to the excitation of 3, 4 and 5 /3 vibrational phonons as will be discussed in more detail in Sect. 2.7. The equivalent picture holds for the second minimum of 236U: Here the energy of the lowest P-vibrational phonon of 686 keV was also determined by conversion electron spectroscopy [51] and the K” = O+ transmission resonances were observed at 4.2 MeV, 5.2 MeV and 5.8 MeV corresponding to an average P-phonon energy of Rwa M 700 keV [46]. Taking into account the excitation energy of the second minimum in 236U of 2.814 MeV [63] these resonances can be assigned to the excitation of 2, 3 and 4 p phonons. Thus it seems that the slightly anharmonic higher P-vibrational excitation can be identified rather clearly in the second well of the actinide nuclei, while in general the character of the low-lying K” = O+ bands of deformed nuclei in the first minimum is rather questionable and mixtures of /I bands with double y bands [64, 651, double

351

F!G. Thirolt; D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

. 200-

___ _______________-__------

rigid rotor ff\

7 2 “r: ‘0

t&150-

* p= 100

*“I-I---*

1‘ band, “c-band”

\/L*

,,I,IIIII~, 9

2345678 -

: $:Tdband, “a-band” l ground state band

.-’

.C’./-.-.

10

20@!

l $=

I- band, “b-band”

* odd spins * even spins

2

3

4

5

6

7

8

9 10

Figure 19: Dynamic moments of inertia observed in 240/Pu. From [35].

02 as a function

of spin J for the collective

bands

octupole bands, 2-quasi-particle-states or pairing vibrations occur [SS]. Closely-spaced O+ bands in the first minimum of actinides point to a complex situation [52], while in the second minimum the strong EO transitions indicate a pronounced collective /? vibration with axial symmetry [66, 671. The EO transitions of the p band decay in the second minimum of 240Pu are probably the fastest EO transitions observed until now. Since the Jo + Jg transitions successfully compete with the rotational in-band transitions Jp + Jp - 2 even at high spins, their partial EO lifetimes have to be significantly faster than the typical rotational ljfetime of about 10 ps, which leads to an estimate of the EO-transition

strength $(EO) = (too’~~~~r~‘09’ ( of larger than 40.10p3, assuming Y-(EO)
strength

p*(EO) is proportional

to e

and @, where /?o describes

the

deformed equilibrium shape [66]. Applying this relation, the common enhancement of p”(EO) in the second minimum is readily explained and $(EO) values of about 2000 . 1O-3 are predicted. In the vibrating deformed rotor model the moment of inertia is proportional to the mass parameter B and to pi, which seems to be approximately fulfilled when comparing the corresponding values in the first and second minimum. In order to estimate the E2-transition strength between the p and ground state band the harmonic quadrupole vibrator model was used, where the vibrational energy is given by hw = h(g)‘j2 and the E2-transition strength by B(E2,Ol -+ 2;) 0: (C . B)-‘1’ with C denoting the stiffness of the potential. For the mass parameter B, besides an oscillatory behaviour, a general decrease is expected with increasing deformation and in particular for 240Pu a decrease by a factor of 1.7 is calculated going from the first to the second minimum [68]. From a comparison of the vibrational energies in the first and the second minimum of 240Pu, tiw= 1089 keV [52] and tuJ= 769.9 keV [35], respectively, together with the theoretical reduction of the mass parameter it can be concluded that the stiffness parameter C is about a factor of 3.4 smaller in the second minimum, which leads to an increase of the B(E2,0,+ -+ 28+) value by a factor of 2.4. For 236,238Uthe corresponding &vibrational O+ states in the first minimum with strong EO transitions are located at 919.2 keV and 996.7 keV, respectively, leading to estimates of similarly enhanced B(E2,0,+ + 2;) values in the second minimum. Therefore, the

PG. Thirolt; D. Habs /Prog. Part. Nucl. Phys. 49 (2002) 325-402

352

vibrational quadrupole transitions between the p and the ground state bands in the second minimum are more enhanced than in the first minimum. In the first minimum of the actinides the measured B(E2,OJ -+ 2;) values amount to about 1.2 single-particle units, which leads to in contrast 3 singleparticle units for the corresponding values in the second minimum. Using this value the partial life time for E2 y-ray transitions from the b band to the ground state band is estimated to be 3 PS, which is a factor of 10 longer than the partial lifetime estimated for the EO transitions. Therefore, the expected absolute y-ray transition intensities for the /3 band to the ground state band in 240P~ are below the detection limit of 0.1%. The predominant observation of decays of negative-parity states (98%) within the second minimum can be explained by a special filtering action of the inner and outer fission barrier. Competing decay paths like fission and back decay into the normal deformed first minimum may distort or even inhibit the feeding of superdeformed levels with respect to their parity and/or K-quantum number. For example, the P-vibration is the doorway state for the back decay and fission. In the excitation energy range higher than 1 MeV competing decays out of the second minimum are thus expected for states build on several P-phonons, as indeed observed when studying transmission resonances in 23gPu(d,p)240Pu [62]. Thus, only negative-parity states will decay predominantly via y-ray decay within the second minimum in the energy range where the dominant population of the second well via the ((r,2n) reaction occurs. This decay will occur either by an El transition directly to the ground state band or by a cascade of collective E2-EO transitions to lower negative-parity bands. Within the second minimum the /3 vibrations are more collective compared to the first minimum. Due to its sensitivity to enhanced (AJ = 0) EO transitions conversion electron spectroscopy is especially well suited to select these /3 vibrational components. Thus, even the EO transitions on top of the negative-parity octupole band heads may be interpreted as showing contributions from a first quadrupole phonon. The analysis of these more collective b vibrations may be very useful to understand the ambiguous character of low-lying Ot bands in deformed nuclei. Theoretical calculations of intrinsic excitations in the second minimum of 240Pu are rather sparse. Results obtained by Nakatsukasa et al. within the cranked shell model extended by the random-phaseapproximation predict the lowest octupole vibrational band for several superdeformed Hg nuclei to be the K” = 2- band [69]. In a preliminary extension of these calculations to the second minimum of 240P~, the lowest octupole bands are predicted to be a K” = 2- and K” = l- band [69], in accord with our experimental finding. Results by Soloviev [70], obtained within the framework of the quasiparticlephonon nuclear model, predicted collective vibrational states with K” = O+, 2+, O- as the lowest excited states at z 800 keV excitation energy, while the energy of the first K” = 2- state was calculated to be around = 1150 keV.

2.4

Excitation

and decay

of the superdeformed

236f U

Besides the fission isomer in 240Pu discussed in the previous section also the 120 ns fission isomer in 236U has been the object of extensive studies using y- and conversion electron spectroscopy. In the following subchapter the discussion of the so far only unambiguous identification of a fission isomeric y decay branch in 236fU by the Heidelberg group [71, 631 will be used to present also experiments performed to search for the y decay in 238fU and from a potential isomer in 233Th. The second subchapter will be dedicated to a review of the y spectroscopic results obtained in the second minimum of 236U including an interpretation of the collective level structure based on a comparison with the situation found in 24O’Pu. 2.4.1

Back-Decay

to the first minimum

Besides the isomeric fission decay through the outer barrier electromagnetic decay paths through the inner barrier back into normal deformed configurations represent an alternative decay mode for fission isomers. Evidence for a y-decay branch from a fissioning isomer was first published by Russo et al. [72], who reported on an E2 decay from the ground state of 238fU with a transition energy of 2.514 keV to the 2+ member of the ground state band at 44.9 keV and possibly, on an El decay with E, =1.878 keV

I? G. Thiro& D. Habs / Prog. Part. Nucl. Phys. 49 (2002) m-402 to the first excited

353

l- state, however without observing the coincident decay of the I- state. They deduced a 7 branching ratio with respect to the isomeric fission channel of r~-,/r~f e40, assuming that about half of the y-decay channels have escaped observation. This ratio has to be compared with ~18 [IS], obtained by comparing the measured half-life of 238Uwith the estimate by Metag of rF-,/rFf the systematics for partial fission half-lives. The experimental results of Ref. [72] have been questioned in subsequent investigations [73, 741 but were supported by results obtained by Steinmayer et al. [75]. Moreover, also the monopole transition of the 238U shape isomer to the ground state in the first well has been observed (with a relative intensity of 0.3%) in a conversion electron experiment by Kantele et al., resulting in an excitation energy of the second minimum En = 2558 keV [76]. However, in a careful reinvestigation at the Darmstadt-Heidelberg crystal ball, a 4~ NaI spectrometer (771, the experimental evidence for the y back decay in 238U reported by Russo et al. and Steinmayer et al. could not be confirmed [78]. Using the same 238U( d 3pn) reaction as in Ref. [72, 74, 751 the high granularity and efficiency of the crystal ball in contrast to the previous experiments using a high resolution detector in principle allowed to measure the so far unobserved decay intensity via 2- and 3-y cascades. In addition a particle telescope allowed for an unambiguous identification of the proton ejectile, a feature also not included in the previous experiments. However, no evidence for a y transition from the shape isomer back to the first well was found at a cross section limit of da/dR M lOpb/sr. The obtained sensitivity allowed for a comparison with the cross sections and branching ratios published by Russo et al. [72] and Steinmayer et al. [75], resulting in a discrepancy within at least la (781. Thus we are still left with a rather unclear situation with reasonable doubts about the validity of the published results on the y back decay in 238U. The extreme sensitivity of the crystal ball spectrometer for rare y decays was also used to search for the theoretically expected y decay of superdeformed states in the second minimum of Thorium isotopes. With lower proton number the penetrability of the inner barrier rises, while there is almost no change of the outer barrier as a function of Z [79]. Thus the branching ratio between y decay and isomeric fission favours the y decay in light actinide nuclei, which is already more likely in 236U by a factor of 6.5 and 18 for 238U, respectively. In the Thorium isotopes therefore only the y back decay of fission isomers to the first minimum should be accessible to experimental observation. High-precision measurements of the fission cross section in 233Th following the (d,pf) reaction resulted in resonance structures that were explained by a pronounced second and even third minimum in the potential energy surface (as will be discussed in detail in Sect. 3.2). However, despite extensive search no fission isomer could be found in Thorium isotopes, only weak evidence for a delayed y decay in 233Th was reported in Ref. [SO, 811. Confirmation of this finding failed in a remeasurement by the same group [78], with a cross section limit for the y decay with excitation energies between 2 and 4 MeV (and an isomeric halflife 2ns< tl12 ~1~s) that covered the energy range predicted for the position of the second minimum in 233Th by theory and measurements of the prompt fission cross section. Reminding the low binding energy of B, = 4.79 MeV in 233Th, decay channels competing with the isomer population are not restricted to prompt fission and 7 decay to the first minimum. However, in contrast to the situation in even Thorium isotopes the barrier parameters in the odd Thorium isotopes are not known with the precision needed for a reliable calculation of the population of the second minimum in the (d,p) reaction, Thus the question of the existence of shape isomers in Thorium isotopes still remains open. The only unambiguous identification of the y back decay of a fission isomer so far was obtained in two series of experiments on the decay of 236fU performed at the Darmstadt-Heidelberg crystal ball. From the semi-empirical procedure of Metag [15] a value of IF7/rFf r 5 & 2 was expected and similar values of 6 to 8 were estimated from measurements of isomeric fission cross sections 182, 83, 841. Using the 235U(d,p) reaction Schirmer et al. [71] succeeded to identify the y back decay into the first minimum by analyzing the ‘missing energy’, defined as the difference of the initial excitation energy in 236U, determined by the proton energy, and the prompt y sum energy. The excitation energy of the shape isomeric ground state was determined as E(OT,) =2.75(l) MeV with rFY/lYFf = 8 f 3. With better statistics and improved precision Reiter et al. [78, 631 simultaneously measured both decay paths of the fission isomer in 236U, y decay and isomeric fission, thus allowing for an unambiguous assignment of y cascades to the transition from superdeformed configurations to the ground state deformation, while also allowing for a precise extraction of fission barrier properties. 5 y-decay cascades were observed

I! G. Thirolf; D. Habs /Prog. Parr. Nucl. Phys. 49 (2002) 325-402

354

that could be attributed to the population of dipole states in the first minimum, as indicated in Fig. 20 together with their relative intensities. In addition an upper limit of
2814

(1161x)

1807 1604

688

(3.8 ns)

2+ 0+

Figure 20: y back decay of the shape isomer in 236U at 2814 keV through five El cascades. An upper limit for the E2 transition to the 2+ state at 45 keV is also given. The upper limit for the isomeric EO decay derived from the data of Ref. 1851was added to the figure taken from [78].

The y decay of the fission isomer was found to proceed exclusively via El radiation, since the spin of the ground state in the second minimum could be unambiguously determined as O+ by the angular correlation of the decays. An E2 decay of the isomer into the 2+ rotational state at 45 keV, which previously was reported to carry the main intensity in the y decay of 238fU [72, 751, could be excluded with an upper limit of 53%. The excitation energy of the shape isomer was determined to be Err =2.81(2) MeV, with all cascades connected with the decay of known states in the first minimum of 236U. Thus the first y ray in each cascade could be used for the determination of the energy of the ground state in the second well. The branching ratio between y back decay and isomeric fission amounts to l?~~/r~f =6.5(8), in agreement with earlier estimates [82, 83, 841 and with significantly improved accuracy compared to [71]. In addition the ratio of delayed to prompt fission and the isomeric ratio, respectively, could be deduced as

Add, pfdel) = Ad4 pf) Ao(d,

PIIMin.)

Aa(d,

P1Min.J

=

4.9(2). 10-5

(3)

(3.0 f 1.2) . 10-d

(4)

A indicates that cross sections were derived for an angular range of proton detection from B = 119” 159”. The partial half-life could be derived from the ratio between y decay and isomeric fission as tiff = (880595) ns. This value together with the half-life of prompt fission in the first minimum, approximated by the period of the /3 vibration (Z’lf =4.10-*’ s) could be used to calculate the penetrability of the outer barrier Ps according to trrf = tIf/PB as PB = (4.5 f 0.4). lo- 15. In a parabolic parameterization of the fission barrier this leads to a curvature of the outer barrier h~ = (0.51 f 0.04) MeV in good agreement with the values quoted by Bjornholm and Lynn of fiwe = (0.60 f 0.06) MeV [9].

I? G. Thirolf; D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402 2.4.2

y spectroscopy

in the second minimum

of m

3.55

u

Triggering on either of the two decay branches of the fission isomer, y back decay and isomeric fission, respectively, several identical prompt cascades feeding the isomer were observed as a function of the y multiplicity in the experiment of the Heidelberg group [63, 781. Fig. 21 displays the resulting y spectra for the prompt multiplicities Np = 2 and Np = 3 measured with NaI detector resolution using the Darmstadt-Heidelberg crystal ball spectrometer. The spectra show prominent lines at E, = 280 keV and E, = 780/830 keV, respectively.

r

Figure 21: Prompt y ray energy spectra from transitions in the second minimum of 236U as a function of the multiplicity of the y cascades following the 235U(d,p) reaction (Ed = 11MeV). The y rays were detected with the NaI crystal ball spectrometer. From [78].

Using yy coincidence matrices it could be shown that several individual transitions in the energy region above 1.2 MeV were in coincidence with these transitions. Moreover the high y efficiency of the crystal ball spectrometer offered the additional unique possibility to analyze correlations between individual transition energies and the sum energy of the cascades, allowing to propose a decay scheme for the strongest transitions in the second minimum. The lowest-lying y decaying states were placed at excitation energies of E” =290 keV, 830 keV and 1110 keV, as shown in Fig. 22. The level at 830 keV dominantly populates the ground state rotational band and to a small extend also the level at 290 keV. The level at EZ = 1110 keV decays via E-, = 280 keV and E7 = 7801830 keV to the lower levels. A remarkable result is the self coincidence of the 780/830 keV structure, requiring a level at E” = 1610 keV. A comparison of the expected electron signal with data from a conversion electron measurement in 236fU [51] clearly indicated an El character for the 280 keV transition while excluding E2/Ml for the 7801830 keV lines. No indication was found for a y decay of the /3 vibrational band at E = 685 keV [51] in 236fU. This agrees with the expectation that the large deformation in the superdeformed second minimum should lead to strong EO matrix elements resulting in a predominant decay via conversion electrons. From a combined analysis of excitation functions for prompt and delayed fission probabilities and calculations based on a doorway state model, vibrational energies in the second minimum were determined as 300-350 keV for the octupole vibration and 7OO~t150 keV for the y vibration [46] in 236fU. Thus in Ref. [63, 781 the strong 830 keV transition was interpreted as the band head of the y vibration in the second well of 236U. However, in view of the more recent results obtained on the collective structure in I14’fPu as discussed in Sect. 2.3 a revised interpretation has to be considered for 236fU,

356

PG. Thim& D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

Figure 22: Level scheme of excited collective states in the second minimum of 236U as derived from y transitions following the 235U(d,p) reaction at Ed = 11 MeV. The levels of the ground state rotational band were derived from Ref. [61], where a rotational parameter A = 3.36(l) keV was reported for 236fU. From [78].

Starting point is a stunning similarity of the y ray spectra obtained from 236fU and 240fP~ using in both experiments the NaI detectors of the Darmstadt-Heidelberg crystal ball spectrometer. In 236fU the already discussed strong doublet structure at 780/830 keV was measured (Fig. 21), while a remarkably similar y ray spectrum for 240fPu was observed in [37], also revealing a doublet at 785/827 keV as its most prominent structure as shown in Fig. 23. In the subsequently performed high-resolution study in 240fP~ the 7851827 keV structure emerged as only one single intense transition at 786.1 keV [34] that was identified as the decay of a K” = 2- band head. It is intriguing to expect a similar behaviour also for 236fU in a future high-resolution experiment. This expectation is supported by the exclusion of an Ml/E2 character of the 7801830 keV structure in 236fU as discussed above, thus giving strong evidence for an El multipolarity similar to the situation in 240fPu. In both nuclei the y ray spectra feature a dominant transition around 800 keV with an intensity of 14.4% for 236fU (following a (d,p) reaction) (781 and 36.6% for 240fPu (following an (o,2n) reaction) [39], respectively. A comparison of the angular momentum distribution of the 235U(d,p) and the 238U(cr 2n) reaction as reported by Borggreen et al. [61] exhibits about a factor of 0.6 lower spins populated in case of the (d,p) reaction. Thus the rotational band built upon the potential K” = 2band head in 236fU cannot accumulate intensity in the band head via fast intra-band transitions as effectively as in 240fPu thus explaining the lower decay intensity in case of 236fU. For the angular distribution of the 786 keV transition in 240fPu an anisotropy ratio of $$$ = 0.98f0.04 was found, compatible with isotropic emission [39]. This was explained by a feeding of this K” = 2band head predominantly via fast unstretched AI = 2 E2 intra-band rotational transitions. In Ref. [78] an angular correlation coefficient as = -0.096 * 0.138 was measured for the energy region covering the 780/830 keV structure (750-850 keV) in 236fU, also revealing an isotropic characteristic that might be explained by an analogous feeding mechanism. Moreover, an anisotropy of a2 = -0.459 f 0.175 was observed for the El transition at 280 keV (781, demonstrating the capability of the crystal ball spectrometer to identify angular anisotropies. Within the time resolution of NaI detectors the 780 keV transition in 236/U was found to be prompt with r < 2 ns, completely in agreement with the expected lifetime of N 150~s derived for the K” = 2- band head in 240fPu [35]. From a more general point of view the argument given in Sect. 2.3.4 to explain the predominant population of excited states with negative parity in the second minimum of 240fPu via a filtering action of the fission barrier should hold in a similar way also for 236fU. In conclusion of all the arguments given above an interpretation of the 830 keV transition in 236fU corresponding to a K” = 2- octupole band head similar to the situation in 240fPu appears highly

PG. Thimlj D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

200

400

600

800

1000

1200

1400 1600

1800

357

2000

E.,[keVI

Figure 23: Prompt y ray energy spectra from transitions in the second minimum of 240Pu as a function of the multiplicity of the y cascades following the 238U(a,2n) reaction (E, = 25MeV). From [37].

probable,

however, the assignment

-

of K” = l- presently

cannot

be excluded.

1610

(Oct. -

Oct.)

w. - 8)

0' -

(1 2-j 830 ' act. (K’ =l-,2-) 665

P(K"=o+)

(1.)-

290

Oct.(K’=

o-) 0

236fU

gs (K"=O+)

Figure 24: Proposed scheme of collective surface vibrational formed second minimum of 236U, derived from y spectroscopy measurements (511.

excitations in the superde[63] and conversion electron

Fig. 24 summarizes the proposed interpretation of the level scheme as the manifestation of collective surface vibrations, in particular the 290 keV state representing the band head of the octupole vibration, while the 830 keV transition is identified with the decay of a K” = 2-(l-) band head similar to the situation observed in 240fF’~. The observation of self coincidences of the strongest transitions was attributed to the decay of two-phonon states as indicated in Fig. 24. Closing the section on the 7 spectroscopy in the second minimum of 236U, it should be pointed out that in the second minimum of 238U also a single strong y line at 616 keV was observed [36], which from analogy of the depopulation pattern again probably corresponds to the deexcitation of the Kn = 2octupole vibration.

p: G. Thimlj D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

358

2.5

y decay

towards

the second

minimum

of 23gU

Spectroscopic information on y ray transitions in the second minimum of the odd-N isotope 23gU was provided by a neutron capture experiment performed at the linac-based neutron time-of-flight spectrometer GELINA at Gee1 [86, 87, 881. y ray spectra with high energy resolution were obtained for s-wave resonances between 600 and 800 eV. In previous 23sU(n,f) experiments the s-wave resonance at E, =721.6 eV exhibited an unusually large fission width [89] and a capture width considerably smaller compared to neighbouring resonances [go], both indicating a nearly pure class-II characteristic. This class-II state partially populates via y decay the ground state in the second minimum. In Ref. [86] Oberstedt et al. showed that a possibly populated shape isomer in 23gU should have a half life larger than 0.25~s decaying to a considerably large extent via y decay back to the normal deformed ground state. Moreover, when considering the general lifetime systematics when going from an even-even nucleus (e.g. 23*/U with r rj2 =195 ns) to an odd-even nucleus with an increase by some 3-5000 times 191, an even much longer half life in the order of hundreds of microseconds can be expected. The population of such a shape isomer in slow neutron-capture reactions proceeds through y decay of this class-11 state, which is mainly located in the second well. As a consequence additional y transitions should appear in the y spectrum besides the usual y transitions from pure class-1 capture resonances. Such additional y transitions in the energy range between 450 keV and 1.5 MeV were identified in [87,88] for the E,, =721.6 eV resonance and attributed to the y decay towards the shape-isomeric ground state in 23gU. While these y transitions show a strong intensity correlation for the resonances around 720 eV with the strength of the corresponding class-II admixture, well-known class-1 y transitions from the y cascade within the first potential well reveal a strong anti-correlation. This behaviour is illustrated in Fig. 25, where the relative peak area 1+, of six candidates for y transitions towards the shape-isomeric ground state in 23gU is displayed as a function of the neutron resonance energy in comparison with the calculated theoretical estimate I -,,,r(theo). The anti-correlation of class-1 states is clearly visible in the behaviour of the 472.8 keV transition,

. 477.8 ke” A -.‘.-_.-. 549.8 keV

0.8 -

+ .‘. ... 597.9 keV 4 _.__.._ .._._605.6 keV * _-------- 63’2.5 keV

0.6 -_

0 -

L,, Theo)

750

760

0.4 -

700

710

720

730

740

770

Figure 25: Relative peak area I-, as a function of the neutron resonance energy corresponding to y transitions measured from the class-11 s-wave neutron resonance in 23gU at E, =721.6 eV [88]. The measured values are compared to estimated values I.,,I(theo) based on resonance parameters (open circles). The peak areas were normalized to the total peak areas summed over the four major neutron resonances within the intermediate structure around E,, =720 eV. From [88].

In this way 11 y transitions

could be assigned

to the decay of the s-wave neutron

resonance

at

359

P.G. Thirolj D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

keV towards the ground state in the second minimum. The resulting proposed level scheme is shown in Fig. 26. Two transitions could not be placed within the decay scheme. It shows transitions with dominant El, Ml and E2 multipolarity where higher transition energies predominantly originate from the neutron capture state. Directly populated levels should exhibit spin values <5/2, while for upper levels negative parities can be expected due to the dominance of El transitions. Since neither y-y coincidences nor angular correlations have been measured, the spin and multipolarity assignments indicated in Fig. 26 should be considered as very tentative. However, the highest transition energy of 3107 keV indicates the population of the ground state in the second minimum of 23gU at an excitation energy of EII = 1700.0 f 2.3 keV, consistent with results from measured neutron resonance spacings of class-I and class-11 type (EII = 1.9 f 0.3 MeV [9]).

4807.0

S,=

\bfJ

4806.26 S,+

721.6 eV

I : : :

3107.0

I : : : _=_.=_.=-,_

==== =” ==== = ====

I I

(l/2- ,3/2-)

549.8(1.1)[4.4]

I

I A

‘7 477WO.4) [I631 (El)

+

Y

239f

U

163O:S

3/2(l/2+,5/23

60X6(0.5) [I I .8] (El)

1

1700.0(2.3)

;;;cg

1083.4

I I

J

112+

I I

Vlj2>250ns)

477.8

(3/2- ,5/2-,112‘)

174.0 0.0

(l/2-,312-) (S/2+,3/2+)

%I

42.5 0.0

E*

Figure 26: Proposed level scheme above the shape-isomeric ground state in 23sU according to [88]. The intensities indicated in square brackets correspond to the efficiency-corrected peak areas relative to the total area summed over all transitions. Multipolarity and spin assignments may be considered as tentative.

E G. Thirog D. Habs / Prog. Part. Nucl. Phys. 49 (2002)325-402

360

2.6

Predictions

of phenomenological

collective level and lifetime systematics

In view of the wealth of detailed spectroscopic information now available for specific nuclei in the superdeformed second well it is intriguing to try to incorporate some of these properties discussed in the previous sections in a broader scope of systematic trends that will be discussed in the following two sections.

2.6.1

Predictions

detived from

a valence

correlation

scheme

Empirical systematic trends across nuclear shells have already been studied in the first minimum in the framework of ‘valence correlation schemes’ (VCS). Introduced into the literature by Zamfir et al. [91, 92, 931 and Casten et al. [94], a VCS is an approach that attempts to account for the systematic behaviour of nuclear observables in terms of a simple dependence on the numbers of valence nucleons. VCSs are useful tools to predict unknown properties and to highlight nuclei deviating from the expected patterns, Successful phenomenological parameterizations were developed in the first minimum for the energies of the first 3- octupole excitations in non-doubly magic nuclei (A>30) [91] and in a similar way for the energies of the 2: y vibrational band heads in non-magic Z>20 nuclei [92]. Underlying is the microscopical interpretation of vibrational states as due to particle-hole excitations produced by a vibrational field acting on a pair-correlated system of nucleons in the valence major shell [42]. Thus the corresponding excitation energies simply depend on their collectivity, which in turn depends on the occupation of certain favoured shell model orbits, i.e. on the proton and neutron number, here represented by the sum of the valence nucleon numbers, Nt = N,, + N,, where each number is counted relative to the nearest closed shell (valence particles or valence holes, depending on which is smaller). The maximum energy for the vibrational states is expected at or near the minimum number of active particles, i.e. near closed shells. As more valence nucleons are added, more possibilities for allowed matrix elements open up, resulting in larger collectivity and lower values of the excitation energies. The minimum energy will occur, within this simplified picture, midway between magic nucleon numbers, where the largest concentration of allowed orbit pairs near the Fermi surface is located. Hence a characteristic and smooth variation of vibrational band head energies is expected as a function of Nr. It should be noted that the Nt parameter is directly related to the idea of F-spin [95] (F = ;N,) introduced by the IBA model, being a useful label for low-lying positive-parity levels in even-even nonclosed-shell nuclei [96]. Applying the concept of the Nt systematics to the actinide mass region results in the data shown in Fig. 27, where the excitation energies of the p and y vibrational band heads (O* and 2+ states, respectively) in the first minimum (taken from Ref. [97]) are plotted versus the number of valence nucleon pairs (Nr + N,,)/2. The systematics follows a characteristic trend with two almost linear branches and a distinct minimum. According to the expectation already discussed, the minimum energies for both collective vibrational band heads show up exactly midway between closed shells, here between the spherical “‘Pb (2 = 82, N = 126) and the next higher-lying deformed shell closure near 2 = 100 and N = 152, the latter derived from spectroscopic studies of heavy elements [98]. In case of the O+ levels this systematics even allows to distinguish between states with equal quantum numbers but different character of the wave function. This behaviour, reflecting the complex structure of O+ states, is illustrated in Fig. 28. Levels with strong EO transitions (e.g. in 232Th,232~234~238U, 240Pu) are located on a curve similar to the one displayed in Fig. 27 (labelled ‘A’), while O+ states decaying mainly with weak or no EO decay components group along separate systematic curves (‘B’, ‘C’). Given the clear systematic behaviour of p and y vibrational phonon energies in the first minimum ’ as visible in Fig. 27, it is tempting to transfer this approach also to the second minimum. It should be noted that N = 146 already represents a deformed shell closure, so only the proton contribution has to be considered when calculating the Nt parameter in this case. Certainly much less experimental information is presently available in the second well, however, when considering the p band head energies, the U isotopes, differing from each other by one valence nucleon pair, reveal a rising trend from 238fU to 234fU, while the Pu isotopes show the opposite behaviour of a higher value of E(Og) for 240fPu compared to 238fPu. For deformed nuclei there are gaps in the single particle Nilsson energies that act like deformed shell

R G. Thirolf D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

361

1. minimum: t

0 P$+ 3Y2 7

6

9

e

10 II 12 I3 I4 I5 16 I7

Z=82 Nd26

I Y g +o” i5

900 800 700 mr 600

i

5oog

t

7

8

e

9 10 II I2 I3 I4 15 I6 I7

Figure 27: Systematics of the experimental excitation energies of the ,?J and y vibrational band heads in the first minimum of actinide isotopes plotted versus the number of valence nucleon pairs. The data are taken from [97].

closures. Having in mind the systematics found in the first minimum, a consistent arrangement can be found when assigning 2 = 78(106) as the lower (higher) deformed magic proton number, consistent with the shell correction energy minima calculated in [99]. In this case the data points can be arranged in the Nt scheme as shown in Fig. 29 around similar curves as in the first minimum with a distinct minimum at midshell. It should be emphasized that the data points originating from direct spectroscopic studies of the B vibration in 240fPu [35] and 236f~238fU[20] result in highly accurate values of the first p phonon energy that justify the curve drawn in Fig. 29. On the other hand the additional experimental values emanating from transmission resonance studies [46] complement these findings reflecting the general trend in good agreement with the precise spectroscopic data points within their larger uncertainties. Thus the rather simple empirical Nt systematics for the first time offers a possibility to predict /? and y vibrational phonon energies in the second minimum. Moreover, from the characteristic systematic trends visible in Fig. 27 and Fig. 29 additional predictions could be derived for new magic numbers in strongly deformed nuclei. Thus it has been shown that the VCS concept is a valuable tool to describe global signatures of nuclear structure and their evolution in the normal- as well as in the superdeformed minima of the potential energy surface from a minimum of experimental data, facilitating predictions of unknown properties by interpolation. .

PG. Thin?& D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

362

n Ra VTh

Au 0

Pu

0

Cm

0

Cf

600 E

I 6

8

I

I

10

I 12

I

I 14

I

I 16

I

I 18

I

I 20

N, =

I

I 22

( Np + N, )/2

Figure 28: Excitation energies of O+ states in the first minimum of actinide isotopes (from [97]) plotted versus the number of valence nucleon pairs. States decaying with strong (‘A’) and weak EO decay contributions (‘B’), respectively, are found to group along different systematic curves.

2. minimum: y band

4

5

4

5

6

7

8

910

6

7

8

910

CNp+ NJ/2

Figure 29: Systematics of the experimental excitation energies of the p and y vibrational band heads for fission isomers in actinide isotopes plotted versus the number of valence nucleon pairs. The data are taken from [20, 35, 461.

f?G. Thimlf; D. Habs /Prog. Part Nucl. Phys. 49 (2002) 325-402 2.6.2

Extension

363

of the Grodzins systematics

Having observed the obvious systematic trends for collective excitations identified in the first and SUCcessfully transferred to the second minimum as discussed in the previous section, it was tempting to search for additional indications of systematic similarities between the normal deformed and superdeformed potential well. A very successful empirical systematics in the first minimum was first developed by Grodzins [loo], correlating the reduced electric quadrupole transition probabilities B(E2) t from the ground state to the first excited 2+ state in even-even nuclei with its excitation energy Ez+, resulting in a characteristic A and 2 dependence. The corresponding functional dependence will be introduced following the description given in the recent compilation of electric quadrupole transition probabilities by Raman and coworkers [loll. B(E2) t values are basic experimental quantities that do not depend on nuclear model assumptions, in contrast to the nuclear deformation parameter p. Assuming a uniform charge distribution out to the distance R(6’, 4) and zero charge beyond, p is related to B(E2) T by /3 = (4r/3ZR2)[B(E2) 0

t /e’]l/’

According to the global systematics, the knowledge of the excitation energy E(keV] of the first 2+ state is enough to predict the value of the corresponding lifetime rr(ps) and, hence, the B(E2) t (ezb2) value. Within the framework of the hydrodynamical model with irrotational flow, Bohr and Mottelson [102, 103, 1041 have derived simple expressions for the rr values. They derived 77 z 1.40. ~cI’~E-~Z-~A”~

(6)

for collective rotations of axially symmetric nuclei. The Em4Ze2 dependence in the above expression was adopted by Grodzins [loo] in his empirical fits for all even-even nuclei, but he replaced A’/3 by A. When the exponents of E and A were allowed to vary it was already found by Raman and coworkers [105, 1061 that the best global fit to the data compiled in Ref. [107] was obtained with

7-r= 1.25. 10’4E-4Z-2A0.6g When converted

to B(E2)

t, this expression

(7)

led to

B(E2) 1‘= 3.26E-‘Z2A0.‘jg

(8)

It was also shown by Raman et al. [106] that the l/E dependence is more important than the exact A dependence. Fixing the exponent of A as -f instead of -0.69, the revised best fit to the data of Ref. [107] was found to be [108] E(E2)

t= 2.6E-‘Z2A-2/3

.

(9)

Thus having established the functional relationship between E and A and E and rT, respectively, the analysis of the most recent adopted 7-Tvalues (excluding those for closed-shell nuclei) resulted in rY = (1.59 f 0.28) . 1014E-4Z-2A0.6g The corresponding

predictions

for B(E2) B(E2)

(10)

1‘ and /3 are given by

?‘= (2.57 + 0.45)E-1Z2A-2’3

(11)

and /3 = (466 f 41)E-1’2A-1

.

One of the simplest theoretical models to understand the trends in the Shell Asymptotic Nilsson Model (SSANM), which is based on the ansatz it can be in a single shell’ [108, 109, 1101. In the version of the SSANM [llO], the B(E2) t values (in units of e2b2) are expressed via the intrinsic

(12) B(E2) f values is the Single‘A nucleus is as deformed as developed in Refs. 11091 and quadrupole moment Q. by

PG. Thimlf; D. Habs /Pmg.

364

Part. Nucl. Phys. 49 (2002) 325-402

5

B(E2)

t=

-k&o? 167r

C&o# ‘4

(13)

The fairly good global description of a wide range of nuclei by the presented improved Grodzins systematics can be illustrated by reminding that about 90% of 300 measured B(E2) 1‘values from nuclei in the first minimum with deformations ranging from near-spherical to /? N 0.3 can be described by this systematics within a factor of two [loll. H owever, when focusing on the local properties in the region of the actinides, the picture of a rather constant behaviour of the product between E(2+) and B(E2) t expected within the Grodzins systematics changes into a rising trend when plotted versus the intrinsic quadrupole moment Q. as shown in Fig. 30 (normalized to 238U). The circular data points represent 25 actinide nuclei with 2 288 and E(2+) <200 keV. Two data points were added in Fig. 30 from recent which will be discussed spectroscopic studies in 252,254N~[ill, 1121 (shown as triangles in parentheses), separately at the end of the section. An even more significant evidence for the rather linear increase of the Grodzins systematics with the quadrupole moment in the actinides is revealed when adding the data points from the shape isomers in the Uranium isotopes 236,238U,which are the only fission isomers where 2: energies and quadrupole moments have been measured. These data have been added as diamond data points to Fig. 30, nicely fitting into the systematic trend already revealed by the less deformed nuclei.

4 2 ‘m N”

2500

-

2000

-

1500

-

c? ST c

CT

si +-

1000 -

500

-

fii 0,

i

, 0

, 5

,

10

/,/,,I

,,,,,,

15

20

25

30

35

40

Q. fehl Figure 30: Local Grodzins systematics (normalized to 238U) in the region of the actinides (25 nuclei with 2 288 and E(2:) <200 keV: circles). Triangles: data for 252,254Nofrom extrapolated E(2:) energies [112]. A discussion of these data points (therefore put in parentheses) is given in the text. Diamonds: data from fission isomers 236f,238fU.

A linear fit to the data results in the dotted curve added to Fig. 30, allowing to parameterize dependency of the Grodzins systematics (normalized to 238U) on the quadrupole moment Qe as E(2+)B(E2)

V3 = (66.77 f 4.44) . Q. - (171.47 f 45.86) . .(2/92)'(238/A)

the

(14)

An explanation of this remarkable behaviour may be found in the reduction of the pairing interaction when approaching the rigid rotor limit at large nuclear deformations. While the presentation of the Grodzins systematics in Fig. 30 allows to identify the linear trend as a function of the quadrupole moment, from an experimentalists point of view the correlation between the 2+ excitation energy and the quadrupole moment Qc as displayed in Fig. 31 (for the same data as included in Fig. 30 and normalized to 238U) is of larger p ractical use, allowing to deduce the quadrupole moment Qo from a measured value for the 2: transition energy. Exploiting Eq. (13) the fit curve in Fig. 30, corresponding to Eq. (14), has been converted into

l?G. Thirolf; D. Habs / Prog. Parr. Nucl. Phys. 49 (2002) 325-402

2’3 = (66.77 f 4.44)/Q,, - (171.47 &45.86)/Q;

E(2+) . (Z/92)2(238/A)

365

(15)

and added to Fig. 31. Two theoretical calculations of the moment of inertia as a function of deformation for 240Pu using Nilsson and Woods-Saxon potentials [59] have been converted into the functional dependence between the E(2+) energy and the quadrupole moment in order to include them to Fig. 31 (solid and dashed curve). .

actinides

A 252.2MNo l 236f,238fu

i

\’ -

ioo-

woods-Saxon ht. Interpolation

S 3 e T. 8 a

IO\ 0

5

/

5

I

,

10

9,

I,

15

I

20

(

25

(

30

I,

I,

35

40

Q. Ml Figure 31: Correlation between the measured excitation energy of the first 2+ state and the intrinsic quadrupole moment Q. normalized to the Z and A dependence of the Grodzins systematics and to 23aU The same data as in Fig. 30 are displayed. The linear fit to the data of Fig. 30 has been converted according to Eq.( 15) and added as the dotted curve. The solid and dashed curves represent calculations of the moment of inertia for 240Pu as a function of deformation [59] (see text).

Interestingly the deviation of the experimental data from the linear interpolation for quadrupole moments in the range of lo-15 eb to somewhat larger 2+ energies also occurs in the theoretical calculations which take the change of the pairing interaction with deformation into account. The two data points included in Fig. 30 and Fig. 31 for the two isotopes 252,254N~have to be distinguished from the other data points shown, since they reflect extrapolated rather than directly measured information. The excitation energies in the ground state rotational bands of the No isotopes have been measured for I >_ 6fL, while the transition energies for the 2: states were deduced from an extrapolation performing a Harris fit to the lowest six visible y rays according to the prescriptions given in Refs. [113, 114, 1151. Subsequently B(E2) t va 1ues and quadrupole deformations were derived using the prescriptions given by Grodzins [loo] and Raman [loll as presented above. It should be noted that the determination of their quadrupole moments in Ref. [112] was based on a local extrapolation without taking into account the systematical trend as illustrated in Fig. 30. Using instead an interpolation based on Eq. (14) may suggest smaller values for the quadrupole moments compared to the ones given in Ref. [112]. The slightly smaller deformation of 254N~ reported earlier [ill] may reflect this tendency.

PG. ThirolJ; D. Habs /Prog. Part. Nucl. Phys. 49 (2002) 325-402

366

2.7

Multi-phonon

p vibrational excitations

in 240Pu

Multi-phonon states in the second minimum carry important information on collective properties of these highly deformed configurations. In contrast to the first minimum, multiphonon P-vibrations at higher excitation energies can be investigated in the second minimum as transmission resonances in the prompt fission probability, since the P-vibrations manifest themselves as doorway states to fission. Moreover, direct decay spectroscopy of vibrational and rotational states with standard techniques as discussed in Sect. 2.3.1 and 2.3.2 is not possible due to the too short lifetimes of these highly excited states against fission in the excitation energy region around 1 MeV below the barrier top. A few members of the vibrational series were systematically observed in actinide isotopes just below the fission barrier 191. Due to the damping of the vibrational motion their large fission width is distributed over many compound states in their vicinity. While the fine structure due to compound levels in the first minimum (class-1 states) so far has not been experimentally resolved, it has been achieved to resolve, besides the gross structure of the p-vibrational transmission resonances, the intermediate structure due to superdeformed class-11 compound states due to their lower level density. Studies of fission resonances with high energy resolution can therefore give spectroscopic information on superdeformed compound states with specific spin J” and spin projection K at excitation energies below the fission barrier. The more one reaches with efficient detector systems into the subbarrier regions of vibrational damping, the better one should be able to observe isolated levels with small widths in the fission channel. The overall structure of the transmission resonances observed in 240Pu (displayed in Fig. 32 from the experiment by Just et al. [llS]) at an excitation energy with respect to the ground state in the first minimum E*=5.1 MeV [117, 118, 119, 1201 and at E*=4.6 MeV [45, 46, 116, 1211 was repeatedly interpreted as one of the best examples for damped vibrational resonances.

3

4

5

6

7

a

Excitation Energy in *OPu (MeV)

Figure 32: Prompt fission probability and isomeric population probability of 240Pu. The solid and dashed lines correspond to doorway state model calculations [116].

In the experiment of Gllssel et al. [119] on 240Pu the E*=5.1 MeV resonance group was investigated with a high energy resolution of 3 keV, and with the aid of a detailed model description [119] all experimentally resolved class-II states being predominantly populated in the (d,p)-reaction could be identified as (K’,J)=(0+,2) states, while the O+ states were not resolved due to their broad fission width. Until now, the lower E’ =4.6 MeV resonance group was studied only with modest resolution [116, 1211. Extending the model description to this group narrow fission widths for all spin values of superdeformed class-II states and a reduced level density can be expected. Therefore the E* =4.6 MeV vibrational resonance was reinvestigated with good statistics and high energy resolution and for the first time individual K” =O+ superdeformed rotational band members with spin and parity of J”=O+, 2+, 4+

PG. Thiro& D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

could be identified 2.7.1

367

in this energy region [62].

Tmnsmission

resonances

in the prompt jission probability

In the picture of the double-humped fission barrier the excitation and decay process of excited nuclear levels populated in direct reactions can be separated in two steps. In the first step the direct reaction predominantly populates compound states in the first minimum, which then decay via various open channels. The fission decay channel can be expressed as a tunnel process (transmission) through the double barrier, as schematically indicated in Fig. 33. Transmission resonances in the prompt fission probability will occur whenever a class-I compound state energetically coincides with a vibrational excitation in the second minimum. This can be seen schematically in Fig. 33 for the situation of three vibrational states in the second well. The overlap between the narrow, closely-spaced states in the first minimum with the broader and wider spaced levels in the second minimum results in a selection of the fissioning states. The right-hand part additionally illustrates the resulting fission cross section with the different transmission resonance groups.

L

\

/

\_ Deformation

Figure 33: Schematical description of the occurrence of transmission resonances in prompt fission probability through a double humped fission barrier. An enhancement the prompt fission probability can be observed at excitation energies where Compound clear states populated in the first minimum energetically coincide with vibrational states the second well.

the in nuin

A fairly reasonable description of the measured fission probabilities can be obtained within the ‘doorway-state model’ [9, 1221, where the compound states in the first minimum couple to p vibrational excitations in the second well that act as doorway states in the fission process. The approach outlined in the following section, going back to Lynn et al. [122], is based on a simple model of the fission process through a double-humped barrier, but nevertheless allows for detailed statements on the resonant structures in the fission probability. Starting from the total Hamiltonian H, which can be separated into H = Ho + Hi + Hip, eigen functions In > and eigen values E, for the part Ha, describing the degree of freedom of the deformation along the fission path and eigen functions ]m > and eigen values E,,, of the component Hi , describing the remaining intrinsic degrees of freedom of the system for a given deformation follow as: Hp]pn >= E,lPn >

06)

H;lm >= E,]m

(17)

>

PG. ThiroK D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

368

In the model Hia describes the coupling between the p degrees of freedom and the remaining ones, allowing to expand the eigen functions (c > of the total system into products of ]m > and ]Pn >: ]c >= Ccm,&n

> ]Pn >

(18)

m,n According to the preferred compound states, respectively,

localization in the first or in the second minimum can be classified as follows: (,Bln >,

States in the first (second) Description

of the

fission

]bIIn

minimum probability

>,

]cln >

]cZln > .

oder

are referred to as class-1 (class-II) without

the /3 vibrational

and

v-4 states.

damping

The schematic illustration of the character of damped and undamped transmission resonances in terms of the fission width and fission probability, given in Fig. 34 [123] IS intended to provide an overview of the model describing the fission process via a transmission resonance as outlined in the subsequent section. undamped transmission resonance

Pil

fission probability

E. damped transmission resonance

r” P” E” Figure 34: Damped and undamped transmission resonances. The first two rows show the Lorentzians of the fission width for class-1 states and the fission probability for an undamped transmission resonance, where the significantly wider distribution of the fission probability is clearly visible. The third sketch displays the strength function of the B vibration in case of damping into class-11 states. As can be seen from the fourth panel, the maximum values for the class-1 fission widths in each class-11 resonance group are constant, while the widths Schematically a constant level of the groups vary with the strength of the p vibration. distance DI for states in the first and DI, for states in the second minimum was assumed. Similar to the fission widths also the fission probabilities shown in the bottom panel exhibit approximately a Lorentzian shape, however significantly broader. From [123].

For a pure and undamped transmission resonance, not distributed onto the neighbouring intrinsic states, be described as [119]

DI

Pfl = 2~

where the strength of the p vibrational state is the fission probability Pfr of a class-1 state can

rArB/r?I

(E - Es)’ + a. W2

(20)

369

P.G. Thirolj D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

where the width W of the resonance

in the fission probability

is given by

DI describes the Here IT1 (rrlr) represents the y decay width in the first (second) minimum. level distance between class-1 states in the first well and J!$ stands for the excitation energy of the /3 vibrational state. The widths rA and l?s describing the tunneling of a state through the inner and outer fission barrier, respectively, can be determined using the transmission probabilities PA,* through a single barrier with height EA and E B, respectively, calculated by Hill and Wheeler [124] rA,E

=

PA-B

’ -hIir

=

{l+

- E)/FLWA,B)}-~. 2

ezp(2n(EA,s

(2‘4

.

tzwlr corresponds to the p phonon energy in the second minimum, while fiwA,s denote the curvature The experimentally accessible fission cross of the parabolic parameterization of the fission barrier. section u can be obtained from the cross section oCr for the formation of a compound state ]cl > in the first minimum and the fission probability C = UC1. Pfr

(23)

The expression (20) introduced for the fission probability Pfr results from the ratio between the fission width If1 of the class-I state and the sum of fission and y width If1 + rrl, since below the neutron binding energy the neutron decay width I’,, and the decay width In for direct fission through the double barrier without interaction with states in the second minimum can be neglected:

rD rfr rjr + rrr + rD+ r, z rjr + rrr ’ rfr

Pfr =

The fission width vibrational state

If1 can be derived

+

from the transmission

coefficient

(24) Tf, in the vicinity

of a B (25)

and the usual dependence

between

transmission

T and decay width I via the level distance

r=T.; resulting

D (26)

in T-

Dr

l fr= 2%

rAb

(E -

q#

+ t.

r;

.

(27)

rrr =

widths through the inner and outer barrier rA + rS + r711reflects the sum of the tunneling and the width of the y decay in the second minimum. In Eq. (20) l?11 contributes only to the first term of the width W of the fission probability Pf,, showing that the resonance width W in the fission probability is significantly broadened compared to the width (I II ) included in Eq. (27) for the fission width Ifr. Description

of the fission probability with damping

Taking into account the distribution of the p vibrational strength onto the neighbouring intrinsic states leads to damped transmission resonances. Given the situation that the level distances in the second minimum are much larger than the decay widths of the compound states (non-overlapping resonances: ICI1 << Drr), the fission width rjr can be expressed according to [119] as

rjr = -Dr

.

FdlFdr

27r (E - Ec,r)2 + a . r,:,

(28)

P. G. Thidj

370

D. Habs / Pmg. Part. Nucl. Phys. 49 (2002) 325-402

with

i; Cl1 and -rell represent

the decay widths

of the compound

states

through

the inner and outer barrier,

respectively. When determining the decay widths rcll and ferr via rA and rn the distribution of the vibrational strength onto the neighbouring states has to be taken into account. With the vibrational component c&r of the compound state wave function in the second minimum [42]

rw

&I

C&f= 7jg where rw corresponds as follows.

*

E@)* + a. rL

(E -

to the width of the coupling

(damping

width),

FC1r and fcrr and be expressed (31)

(32) However, it has to be taken into account that the vibrational state not only couples to the class-II compound states (l?w), but it can also tunnel through the barriers (rA,rB), thus requiring to use the total width rv,l instead of rw in the denominator of Eq. (31) and (32)

rvII=rW+rA+rB.

(33)

Finally this leads to rArW

DII

&11 = -

’

27T

DII T:c11= SF . (E Similar to the situation

of an undamped

-

(34)

+ + .r&

(E-ED)*

rBrw Es)’ +

resonance

(35)

a. r$,l .

the decay width leads to the fission probability

PfI = DI Pfl = . 27~

where the half width given by

LiL

irrI

(36)

(E - E,II)~ + i . W,,z,

We11 of the individual

class-11 resonance

groups in the fission probability

is

The fission probabilities of the class-1 states within the class-11 resonance groups (Eq. 36) show the same Lorentzian shape as the fission widths of class-1 states within the class-11 resonance groups (Eq. 28), however they exhibit a significantly broader distribution, as can be seen from the width of the distribution Wcrl > rcll. For the general situation without restriction to DII >> rcll and Drl >> Wcllr respectively, the following more generalized expression has be to used for the fission width rfr (Eq. 27)

rf, = C n

With the simplifying assumption the sum can be explicitly calculated

fd II r,,=-$.-.

I .

i!clrnrcIln -

27r (E - Ec,&

of equidistant as:

states

DI

&I

(38)

+ + I’&,, ’ ]cl1n > with (&I,,

= Ecrro + nD11) [125],

sinh(~rcItlDrI)

coW~r,nlDrr)

- cos[27+ - Em)/%]

.

(3%

371

PG. ThiroEf;D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

The fission probability

P, follows as: ryID,IrcI, DI

cosh(rr,I,/D,I)

-

cos[27G

-

-%,Io)/~IIl

(46)

sinh(d’,II/DIr)

~&I,

In contrast to Eq. (36) Pfr in the general situation cannot be expressed any more as a Lorentzian distribution with a width W,lr. In case it is experimentally not possible to resolve the class-11 states (as in the early transmission resonance experiments with semiconductor detectors, e.g. [46, 117, 121]), only an average fission probability < Pf > can be measured: =i. This results in the following expression

DI

=-------

~&I

r 7,

DII

r$I+

In case of non-overlapping resonances average fission probability expressed by

(DII >>

over the class-11 states:

&&II -Jo--71

inserting

rw

(E - EB)’ + a . r;,,

whereas in case of overlapping class-II states large, thus simplifying Eq. (42) as

averaged

coth

r,,,)

DI 2?7

dE

for the fission probability

rCIIDI rcII. ro2 yI II

i%II

=--

PJI

/

DII

.-

fcIl,

fell

DI

(42)

DII

and ICI1 leads to the

rArB

(43)

r,dv

with DII << rcIl the argument

of coth will become

LILI

=,_

(44)

rcI,FcII+rc,Iry,DI,/D, L Inserting

2

ICI,, ICI, and ICI, in this case allows to express the averaged %

DI

fission probability

as:

~W~Ab?&I~I,

-. 2~

(45)

(E - Ep)2 + t. WB2

with

r,,,+~.rw.r.- DI 77

rArB

112

.

rII

7,

In this situation the fission probability again exhibits a Lorentzian shape. For DII >b rclI the fission width of the individual class-II resonances Wclr can be expressed function of < Pf > 2 < Pf 9 WC{, = Ak . arccos 1 7T l< Pf >2 1 (

(46)

as a

(47)

As already seen before, in this non-overlapping scenario we find no longer a Lorentzian shape of the fission probability, whereas in case of D 1I << PC11the fission probability shows the Lorentzian shape originally predicted by Bondorf 11261. This discrepancy can be explained reminding that Bondorf calculated the fission probability from averaged fission widths: < =

<

rf,

>

r,, > + < rr, >

(43)

I? G. Thirolf; D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

372

while the exact procedure prior to averaging

requires

to determine

the fission probabilities

of the individual

class-1 states

(49)
However, Bondorf was aware of this inaccuracy discussed together with experimental

<

fission probability

< while for a damped

resonance

rJ1

of a damped

PJ

rJI

<

=

The maximum

by

>mas=$.

>

+

ryl >

.F.

with Drr >> lYc,l finally can be calculated

with DII << rci, the maximum

of the transmission

factor F, which will be

{ grArB$}-1’2 fission probability

(51) can be derived as

PAPS (PA

Re-investigation

>

<

resonance

< Pf Ano&

2.7.2

a

resonances

+

PB)

in 240Pu

The experiment to reinvestigate the transmission resonance structure of 240Pu was carried out at the MeV) with a Munich Accelerator Laboratory employing the 23gPu(d,pf)240Pu reaction (Qcs=4.309 deuteron beam of Ed = 12.5 MeV, and using an enriched (99.9 %) ~30 @g/cm2 thick target of 23sP~203 on a 30 pg/cm2 thick carbon backing [62]. Protons were measured in coincidence with the fission fragments. The excitation energy I? of the 240Pu compound nucleus could directly be deduced from the kinetic energy of the protons, which were analyzed by a Q3D magnetic spectrograph [127] set at Or,ab = 130” relative to the incoming beam (n =lO msr). The position in the focal plane was measured by a light-ion focal-plane detector of 1.7 m active length using two single-wire proportional counters surrounded by etched cathode foils [128]. dipole 2

dipole 1

beam

Figure Munich parallel proton

target

focal plane detec or (17mlang)

35: Experimental setup for the transmission resonance experiments performed at the Q3D magnet spectrograph. The spectrograph is equipped with two position sensitive plate avalanche counters for the detection of fission fragments in coincidence with ejectiles registered in a light-ion focal plane detector [128].

373

l?G. ThiroK D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

Fission fragments were detected in two position-sensitive two wire planes (with delay-line read-out) corresponding experimental setup used is displayed in Fig. 35.

avalanche detectors (PSAD) [14,62] having to horizontal and vertical directions. The

Thus the spatial positions of the fragments and their angular correlation with respect to the recoil axis could be determined. The fission detectors covered the range of rj = O”-100” relative to the recoil axis with a solid angle coverage of 24% of 47r (without double-counting of fission fragments). The time resolution between the proton detector and the fission detectors was measured to be 4 ns FWHM after correcting for the time-of-flight in the spectrograph. The energy calibration was taken from the energy resolution was measured “sPb(d,p) reaction (Q~s=1.710&0.015 MeV [129]). Th e experimental to be ~7 keV for the calibration lines detected in the focal plane close to the investigated region in 24OPu. The measured proton energy spectrum from the (d,p) reaction in coincidence with the fission fragments (after subtraction of random coincidences) is shown in Fig. 36a) in terms of the excitation energy of the compound nucleus 240Pu. The spectrum is proportional to the product of the fission probability and the known smoothly-varying (d,p) cross section, which shows no fine structure [117].

4.3

4.4

4.5

4.6

4.1

4.8

4.9

5

5.1

5.2 E* (MeV)

Figure 36: a) Proton Hunyadi et al. [62]. Specht et al. [117]. Ref. [62] (solid line)

coincidence spectrum of the 23gPu(d,pf) reaction recently measured by b) Proton coincidence spectrum at Ed = 12.5 MeV and 0 =140” from For comparison also spectra folded with a resolution of 17 keV from and [119] (dashed line) are shown. From [62].

In Fig. 36b) the Munich measurement is compared with previous high resolution measurements of the same reaction: The (d,pf)-spectrum of Fig. 36b) was measured with a resolution of 17 keV by Specht et al. [117]. It is nicely reproduced in all fine structures after folding the spectrum from [62] with the reduced resolution and applying a global shift of 120 keV to higher energies on the experimental data of Ref. [117]_to account partially for the proper Q-value. When folding the more recent measurement by Ghissel et al. (1191, which was performed at Ed =12.5 MeV and 0 =125” with an experimental resolution of 3 keV, only a small overall shift of 12 keV was necessary. In this experiment the protons were detected in the Q3D using a multi-wire proportional chamber with digital single wire readout. Severe losses of efficiency of up to 50% become apparent in Fig. 36b) for the measurement of Ref. [119] in the lower half of the 5.1 MeV resonance.

I! G. ThiroK D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

314 Experimentally

2.7.3

resolved

intermediate

structure

Looking at the overall features of Fig. 36a) Hunyadi et al. [62] observed for the two transmission resonances at 4.6 MeV and 5.1 MeV the known gross structures with a damping width of about 200 keV.

4.4

4.5

4.4

4.5

4.6

4.1

4.8

4.9

5

5.1

5.2

4.9

5

5.1

5.2 ;

0

_-_--

> 2

150

q

Ioil

s 5

50

3

0

4.7

48

4.6 /47------b.8 ____----

4.9

%\

5

5.1

5.2

E’ (MeV)

Figure 37: a) Proton spectrum of the reaction 23gPu(d,pf) measured with 7 keV resolution (Hunyadi et al. [62]). Also shown is a fit of the data with rotational K” =O+ bands. The O+, 2+, 4+ picket fences show the rotational bands used in the fit. In the upper part of the 5.1 MeV resonance the fit (thin line), is based on the fitted positions of the rotational bands from Fig. 37d). b),c) The full dots represent the a2 (a4) fission angular distribution coefficients determined in the same measurement, while the open symbols are the a2 (a4) coefficients taken from GE&se1 et al. [119]. The thick full line results from the fit to spectrum a). The thin lines correspond to the theoretical a2 (a4) coefficients for different spin values j of the transferred neutron. d) Proton spectrum of the reaction 23gPu(d,pf) from Gllssel et al. [119] measured at 3 keV resolution together with a fit of the spectrum using rotational bands as indicated. From [62].

Besides these well-known features the proton spectrum exhibits a rich structure of individual transitions. The resolved intermediate structure of the lower resonance shows a regular pattern of well-resolved triplets, with a weaker lower and upper peak which are separated by 19 keV and 43 keV, respectively, from the stronger central peak. These separations are very close to the O+-2+ and 2+-4+ separation energies (20 keV, 46.6 keV) of the K” = O+ ground state rotational band in the second minimum [ll], strongly suggesting these structures as being due to excited K” = O+ rotational bands in the superdeformed minimum of 240Pu. Thus, for the lower resonance at 4.6 MeV, in the intermediate structure for the first time series of “pure” resonance states with K” =Of and a spins 0+,2+,4+ with rotational energy spacings as known from the ground state band in the second minimum [ll, 1181 have been observed. This is in contr_ast to the excitation energy region above N 5 MeV, where Hunyadi et al. [62] could confirm the bunching bf peaks ‘without any systematic trend’ [118] observed earlier [117, 1191. The 5.1 MeV resonance region in the spectrum of Glhsel et al. [119], where sufficient overlap with the spectrum of Hunyadi et al. could be observed, was fitted with K”=O+ rotational bands assuming a Lorentzian line shape for the band members. The position and the amplitude of each band were treated as individual parameters, and a common rotational parameter was used. For the intensity ratio of the

F?G. Thin$

D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

315

band members the value of Gllssel et al. [119] was accepted as starting value for the fit procedure. The lit nicely reproduced the experimental data with 13 rotational bands (see Fig. 37d), whose band-head positions, as fixed parameters, were also used to describe the same structure in the work of Hunyadi et al. [62]. Satisfactory description of the experimental data was achieved again by fitting the line width and the relative amplitudes of the bands, as it is shown in Fig. 37a) with a thin curve. The 4.6 MeV group and the lower part of the 5.1 MeV group obtained in the experiment of Hunyadi et al. was also fitted by the same procedure, however, the rotational parameters AZ/20 and the Iz+/Is+ and Id+/&+ intensity ratios could be separately determined for the bands with prominent band heads at E*=4434, 4526, 4625, 4685, 4703 and 4733 keV. In order to prove the spin and K-assignments of the observed compound levels in the second minimum the corresponding fission fragment angular correlations with respect to the recoil axis were analyzed, describing them in the usual way with coefficients a2 and a4 of Legendre polynomials. In Fig. 37 the measured proton coincidence spectrum (Fig. 37a) is shown together with the coefficients a2 (Fig. 37b) and a4 (Fig. 37~). For comparison also the spectrum of GlLsel et al. [119] is shown in Fig. 37 together with their a2 and a4 coefficients marked by open circles in Fig. 37b) and Fig. 37~). The horizontal lines in Fig. 37b) and Fig. 37~) show the theoretical a2 and a4 coefficients for different values of the angular momentum j of the neutron transfered in the (d,p) reaction, leading to 240Pu states with K=O and J= (l/2 f j[. Due to the low ground state spin (J: = l/2+) of 23gPu, in 240Pu only low total angular momenta J are populated. O+ states can easily be identified in the fission fragment angular correlations through their isotropic emission characteristics. The large and positive a2 and a4 coefficients point to a dominant K=O character of both resonances. From 240Pu(y,f) data [130] a “ suppression of the K” = O- channel by more than two orders of magnitude relative to the K” = O+ channel” was deduced [119]. For the lower 4.6 MeV resonance a K” = O- contribution was discussed as well, however, as will be described below, a reasonable description of the fission probability could be achieved assuming a K’ = O+ channel [45, 46, 1161. 2.7.4

Ground

state

excitation

energy in the second

minimum

From the experimental findings discussed in the previous section it seems that a complete spectroscopy of K”=O+ bands is possible in the regions of vibrational damping. In order to check this completeness and the consistency of the observed level density with the excitation energy in the second minimum, a statistical analysis of the level distances was performed using the band head energies. The statistical distribution of the ratio of experimental and calculated average O+ level distances, using the back-shifted Fermi-gas formula in a parameterization by Rauscher et al. [131], was generated. The shape of the resulting distribution, displayed in Fig. 38a), was successfully approximated by a Wigner-type distribution, as it is expected for repelling states with the same angular momentum and parity. A similar analysis was performed in Ref. [119] and [132]. The x2-value of the Wigner fit to the level density (displayed in Fig. 38b) was minimized by varying the back-shift term of the Fermi-gas formula around the expected ground state energy of the second minimum. The best fit was obtained for an energy of EII =2.25f0.20 MeV (statistical uncertainty). Fig. 39 illustrates this novel method to determine the potential depth of the second minimum in 240Pu. The experimentally observed O+ level distances in the transmission resonance groups of 240Pu (full circles) are compared to calculations of the level density in the first minimum according to different models. The solid line in the left part of the figure corresponds to the level distances in the first minimum calculated in the framework of the back-shifted Fermi gas model in the parameterization by Rauscher et al. (1311, also used to create the distribution of Fig. 38. In order to reproduce the experimentally observed O+ level distances this curve has to be shifted by 2.25 MeV, corresponding to the ground state energy in the second minimum of 240Pu For comparison also level distances in the first minimum calculated with the Bethe formula and within the ‘constant temperature’ formalism in a parameterization by v.Egidy et al. [133] are shown.

l?G. Thiro(t; D. Hubs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

376

lo:

a)

_ b) Ep2.25 MeV

0-

sl.2MeV

+-

I

Figure 38: (a) Distribution

of the level distances

D$+,

of all multi-phonon

K= O+ bands

observed in 240Pu (621, divided by the level distances D$CIrca,c calculated within the ‘backshifted Fermi gas model’ in a parameterization by Rauscher et al. [131]. A value of En =2.25 MeV for the excitation energy of the ground state in the second well of 240Pu was used. The solid line shows a fit with a Wigner distribution. (b) x2 values of the fits to the experimental O+ level density distribution as a function of the ground state excitation energy Eu in the second well. A parabolic fit to the region between 1.9 and 2.6 MeV results in a minimum x2 at En =2.25(20) MeV [62]. From [14].

-

2200

Rauscher

5400

3800 E* (keV)

Figure 39: Distance of the J” = O+ levels in 24”Pu as a function of the excitation energy. The solid line in the left part of the figure corresponds to the level distances in the first minimum calculated in the framework of the back-shifted Fermi gas model in a parameterization by Rauscher et al. [131]. In order to reproduce the experimentally observed O+ level distances (full circles) this curve has to be shifted by 2.25 MeV, corresponding to the ground state energy in the second minimum of 240Pu. For comparison also level distances in the first minimum calculated with the Bethe formula and within the ‘constant temperature’ formalism in a parameterization by v.Egidy et al. [133] are shown. From [14, 571.

l? G. Thirog D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

a, E

b [136]

2.6

[91 d [137]

c

::;

* el 2.0 w 1.8 1.6

317

t

ab

cd

e

e [621

Figure 40: Excitation energy of the ground state in the second minimum of 240Pu, ewerimentally determined by extrapolated excitation functions (Ref. a-d) and by a statistical K= of level density analysis [62]. The dashed line corresponds to the average value of E;, from Ref. a-d with the shaded area representing its error margin.

The value of E,r =2.25&0.20 MeV extracted for the excitation energy of the ground state in the second minimum of 240Pu is in good agreement with the fission isomer energy obtained from the wellknown method of extrapolated excitation functions of various experiments, listed in Fig. 40 (see e.g Ref. [134] and references therein). Thus having proven the reliability of this novel method, an excellent tool exists to address the question of the depth of the (hyperdeformed) third minimum of the potential surface, where so far no experimental information existed. This will be discussed in Sect. 3.4.1. 2.7.5

Discussion

of rotational

and vibrational

properties

Based on the detailed structure resolved in the transmission resonance groups as shown before, resulting spectroscopic properties of the vibrational and rotational bands can be discussed. In the conversion electron measurements of Gassmann et al. [35] the phonon energy of the first P-vibrational excitation could be determined as hwp =769.9 keV. With respect to the excitation energy of the superdeformed ground state (E 11 =2.25&0.20 MeV), determined by the statistical analysis of the level distances in the previous section, the vibrational resonances centered around 4.6 and 5.1 MeV could be attributed to three and four fl-phonon excitations, respectively. In Fig. 41 all presently known vibrational band heads in the second minimum of 240Pu are shown. The already discussed spectrum of low-lying collective octupole excitations is apparent, where the energetic positions of the band heads as well as their collective vibrational structure are found to be in reasonably good agreement with theoretical predictions by Soloviev et al. obtained within the framework of the quasiparticle-phonon nuclear model [70]. M oreover, an almost harmonic spectrum of quadrupole phonons was observed. A reduced energy difference between the vibrational states of (0.5-0.6) MeV is expected at higher energies, because the potential well opens up at the top of the barrier. The observation of subsequent P-vibrations in transmission resonances can provide a unique possibility to study slightly anharmonic vibrational series at large nuclear deformations in a more convenient way than in the first minimum, where the high level density causes a complete damping of the /Cvibrations. Summarizing the spectroscopic studies on 240fPu the detailed spectroscopic information obtained in conversion electron and y spectroscopy for the lowest phonons was extended to the third and fourth P-vibrational phonon by transmission resonance spectroscopy. It will be challenging to reach the intermediate levels between 1.5 and 2.0 MeV in the second minimum at the limit of both methods, because then common levels could be observed and the scaling laws for the fission probability and the barrier penetrabilities could be tested independently. In addition individual dynamical moments of inertia reflecting both the nuclear deformation and collective structures of the excitations, could be extracted for the first time in the high excitation energy region of the 4.6 MeV resonance group for the three well separated K==O+ bands with band head energies of E*=4434, 4526 and 4625 keV. For the other rotational bands of this resonance group, as for the energy region above E* ~5 MeV only an average moment of inertia could be determined, respectively.

t! G. Thirolj D. Habs /Prog. Part. Nucl. Phys. 49 (2002) 325-402

378 inner barrier EA=5.8MeV

Quadrupole phonons: (K==O+)

ss& 4.625 4.526 2.38(20) 4.434

outerbarrier E,=5.45MeV

\ 3h9 OCtUpOk phooons:

(-3.85)(-1.6)'...." 2h9

K=l1.345KCI-

3.02(20)0.77

;I:; Km

P_bandhW B

K==C 0.555-

3.7"s 225(20) E*

(MeV)

0 %I*

240fpu

\4

WeV)

Figure 41: Overview of vibrational excitations in the second minimum of 240Pu. The lowlying collective band heads have been observed in Ref. [34] and [35], while the higher-lying multi-phonon B vibrations resulted from transmission resonance studies [46, 621. From [62].

$200 zz Z180 P ?60

_______._

____

__

1

- K’=O+,, /i

cl

140 KG& IOA

I____.SDrigidrotorvalue

___--__--__-A

K’=2-

H&i+++-K==O&

!I KX=O;hw

B

Figure 42: Moments of inertia of rotational bands in the second minimum of 240Pu as a function of the excitation energy En with respect to the ground state in the second minimum. Subscripts ‘e’ and ‘0’ denote moments of inertia that have been extracted for the even- and odd-spin members of the respective rotational band. From [62].

319

P. G. Thirolf; D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

The resulting moments of inertia are shown in Fig. 42 together with those for the superdeformed ground state rotational band [ll] and the low-lying collective excitations in the second minimum that were observed in recent y-ray spectroscopy [34] and conversion electron experiments [35], respectively. The moments of inertia of the ground state band in the second well and the higher phonon bands show a surprising agreement, while larger variations arise for the low-lying excited superdeformed bands. The moments of inertia of the fission isomeric ground state band could be reproduced by recent cranking model calculations yielding O/fi* =155 MeV-’ [138]. One might have expected that the moments of inertia for the bands built on the third and fourth B-vibrational phonon might be closer to the rigid body moment of inertia because of the reduced pairing at higher excitation energies. However, this is not observed experimentally. Perhaps the selection of bands with strong P-vibrational components results in weaker non-collective contributions. While in the first minimum K” = O+ vibrations show strong admixtures of pairing vibrations we expect much less admixtures in the second minimum as a significantly larger collectivity of P-vibrations in the second minimum was observed [35]. Therefore pairing vibrations and blocking could be reduced, which might explain the observed preservation of the moment of inertia of the superdeformed ground state band. 2.7.6

Spin

dependent

fission properties

Aiming at a deduction and interpretation of the spin-dependent fission probabilities, relevant properties will be discussed in the following section, which are compiled in Table 2.7.6, analogous to Ref. [119]. The relative fission cross sections of(J”) were obtained from the fit of the spectra in Fig. 37. In the maximum of the 5.1 MeV resonance we obtain (taking an average of the values from our measurement and that of Ref. [119]) a dominance of the 2+ states with of = 0.66 * 0.05, which nicely agrees with the value of 0.67 f 0.06 obtained by Britt et al. [139]. Resonance:

J”

0+ 2+ 4+

af(J”) 0.19(3) 0.66(5) 0.15(5)

4J”)

0.024 0.14 0.08

(&(J”))

0.79(12) 0.47(3) 0.19(6)

DI

DII

(all K)

(K” = 0+)

(keV) 0.175 0.039 0.027

(keV) 21 21 21

Resonance:

J”

0+ 2+ 4+

4-J”) 0.31(6) 0.59(6) 0.10(5)

4-J”) 0.015 0.12 0.07

&(J”))

0.042(5) 0.0098(g) 0.0029(13)

5.1 MeV

Wcrt

wJ=)

(Tf(J”))

(keV) 215 7.5 2.6

8.3.10-4 3.7.10-3 5.4.10-3

3.0. 1O-3 3.3. 10-a 1.3.10-a

WClI

qI(J=)

(Tf(Jn))

(keV) 2.2 0.6 0.2

2.3. 1O-4 l.l.1o-3 l.5.1o-3

1.0. 1O-5 1.1.10-5 0.4.10-5

4.6 MeV

DI

DII

(all K)

(K” = O+)

(keV) 0.51 0.11 0.08

(keV) 96 96 96

Table 5: Spin dependent properties of the two b-vibrational fission resonances in 240Pu around E’ =4.6 MeV and 5.1 MeV, respectively. ar(J*) (cQ(J”)) is the normalized relative fission (compound) cross section. (P,(Y)) represents the average spin dependent fission probability, while Dr is the (calculated) level spacing for all K-values in the first minimum and Drr the (experimentally determined) level spacing in the second minimum (K” = Of). WC11is the width (FWHM) of the fission probability of class-II compound levels. In the last two columns the y-transmission coefficient T,I in the first well and the effective transmission coefficient @(Jr)) are given (see text).

The relative (d,p)- compound cross sections CXJJ”) at Ed = 12.5 MeV were obtained in DWBA calculations for deformed nuclei, where the final Nilsson orbitals were distributed over the compound nuclear levels by strength functions [119]. They agree quite well for the 5.1 MeV resonance with values

380

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of aC(J”) calculated by Andersen et al. [140] for Ed = 13.0 MeV. The excitation energy dependence of q(J”) for states in the 4.6 MeV resonance was taken from Ref. [140]. For the energy- and spin-averaged fission probabilities at the maxima of the two resonances values = 2.0. low3 could be deduced in Ref. [46]. While the of (Pf(5.1 MeV)),, = 0.10 and (Pf(4.6 MeV)),, upper resonance could be well reproduced in a theoretical description assuming K”=O+, for the lower resonance also a contribution from a O- resonance was discussed. However, a satisfactory description of this resonance could also be achieved with pure K” =O+ characteristics, assuming a fragmentation of the third b phonon over states with energies of 4.2 MeV (30%), 4.5 MeV (45%) and 4.7 MeV (25%). Using (P,), the average spin dependent fission probability (Pf(J”)) can be calculated by (Pf(JX)) = (P,) . cr~(J”)/ac(J”). The level spacings DI given in the fifth column of Tab. 2.7.6 are calculated with the standard back-shifted Fermi gas formula [131]. However, for the level spacing in the second minimum, DII, the restriction to K” = O+ bands, in contrast to earlier publications [118, 1191, causes O+, 2+ and 4+ levels to have the same level density. The numbers given in Table 2.7.6 are experimentally determined by the fit procedure described earlier. For the FWHM Wcrr of the fission probability of class-11 compound levels the relation WC,1 = is deduced

Jk .arcos T

in Ref. [119] under the assumption

1-

WdJV2 I- P’AJ’V

that PC11 < DII. As long as (Pf(J”))

is smaller than

For larger values of (Pf(J”)) the class-II levels overlap m this width W cl1 can be determined. too strongly. This relation was used in [62] to calculate the W,II widths given in Table 2.7.6. A Wclrvalue for the O+ levels in the centre of the 5.1 MeV resonance cannot be calculated, however, within the experimental error of (Pf(Jm)) a value of 215 keV is estimated. Therefore, a fit with K” = O+ bands in the upper resonance still appears reasonable. The energy and spin dependence of the averaged fission probability (Pf(J”)) can be explained by: (i) a rather complete K mixing in the first well, (ii) a K conservation in the second well and (iii) the competition between y decay in the first well and fission through the double-humped barrier. The processes can be described by a width r or by a transmission coefficient T, which are connected by: T = (271/D) . r, where D is the level spacing of states of the appropriate spin and parity. The y-“transmission coefficient” T,I(J”) = (2r/DI( J”)) . I’$” increases with spin and excitation energy, while the decay width Py is M 23 meV at 5 MeV [119] and changes little with excitation energy (10 meV/MeV [141]), spin J, spin projection K and parity [141]. The double-humped fission barrier predominantly selects K” = O+ states for transmission resonances out of all possible K values for a given J value in the first well. For the lower 4.6 MeV resonance the K components with K # 0 decay predominantly by y decay and the decrease of (Pf(J*)) for K” =O+ states with increasing spin J is approximately proportional to 1/ (25 + 1). For the upper 5.1 MeV resonance the decay by fission of K” = O+ states is faster than the y-decay (or comparable). Thus the spread of (Pf(J”)) with spin is reduced at the upper resonance, explaining also the smooth increase of the Is+/Ie+ ratio with excitation energy. When comparing measured fission probabilities with theory, the averaging over unresolved class-1 states causes some complications because e.g. for the averaged fission probability one first has to calculate the probability for individual class-1 states before averaging: (Pf(J”)) = (rfr(J”)/ (I?,l(J*) + r&J”))) = (F . (rfI)) / ((lY,l) + (ryI)). Here F is a fluctuation factor, which may be smaller than 1 by up to 30% [126; 1411. In Ref. [123] a formula for the averaged fission probability (Pf)mz at the center of a P-vibrational resonance was deduced under the additionally fulfilled conditions (0, >> I’71 and DII > WcII):

Here PA and Ps are the penetrabilities outer barrier (EB = 5.45 MeV,tiws=0.6

of the inner barrier (EA = 5.8 MeV, AwA=0.82 MeV) and the MeV) [119]; tiq=O.77 MeV is the ,!Sphonon energy in the sec-

ond minimum; rw = y/(m is the damping width into class-11 compound states; y (- 200 keV) denotes the observed resonance width and T71 describes the y-transmission coefficient. The obtained dependence of Pf on the barrier penetrabilities o( dm differs from 0: PA . PB/~(PA + PB)~ for an

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381

undamped resonance [142] and from 0: PA . Pe/ (PA + PB), when averaging over the fission width and the y-width separately [126]. The experimental fission probabilities (Pf(J”)) are nicely reproduced by for resonance energies of 5.1 MeV and 4.5 MeV. Experimentally it is the theoretical value of (Pf),,, difficult to determine the exact positions of the p vibrations, because of centroid shifts introduced by weighting the measured resonance intensities with the steep slope of the barrier penetrabilities. Although the resonance centroid obtained by averaging over the measured intensity results in (E’) =4.6 MeV, an improved value of 4.5 MeV can be obtained by taking into account the fragmentation of the resonance strength and the weighting with the absolute fission probability. The fission probabilities (P,( J”)) are strongly spin dependent, but the effective transmission (Tf(J”)) = (Pf(J”)) .2’,1(J”)/(l - (P,(Y))) through the barriers turns out to be rather spin independent. This is expected, because the barrier heights, to first order, vary little for different J members of a rotational band with K” = O+. Also the observed reduction of the transmission Tf for spin 4 compared to spin 0 by a factor of x 2.4 for both resonances can be explained, using the Hill-Wheeler formula [124] for the energy dependent penetration probabilities PA and PB. For the 4+ states in the first minimum about 142 keV are bound in rotational motion. Taking into account the change of the moments of inertia for the two barriers and the curvatures AWAand hwg for the parabolic barriers one obtains for the change of dm the same factor of 2.4. This introduction of an effective transmission coefficient (Tf) is reasonable for Z’f < T,r < 1, even when averaging over class-1 states, because (Pf) = (Tf/ (T, + T-,1)) N (Tf/T,r) 21 (Z’f)/(T,r). At the top of the barrier, for overlapping class-II states (Tj/ (T, + T,I)) and (Tf)/ ((Tf) + (T-,1)) lead to the same results (1231. However, even in the transition region the effective transmission (Tf) shows a remarkable scaling behaviour in Table 2.7.6, where the values for the upper resonance are obtained from the lower resonance by multiplying with a factor of 300. (Tf) seems to be a very useful quantity and the fluctuation factor F [126] seems to show only small variations. 2.8

Perspectives

In view of the remarkable similarity between the y ray spectra observed with moderate energy resolution in 236fU and “‘fPu as discussed in Sect. 2.4.2 it seems worthwhile to repeat the study of 236fU with a modern highly efficient, high-resolution Germanium detector array like,MINIBALL 11431. From the analogy to 240fPu a revised assignment of the measured collective vibrational structure was proposed for 236fU calling for experimental confirmation. In a high-resolution measurement the band head spins and paritiks could be determined. Finding excited collective bands in 236fU corresponding to the structure identified in the high-resolution study of 240fPu would provide strong evidence for the consistency of the proposed level scheme. In addition to the spin and K dependency of the moment of inertia studied in 240fPu experimental information from 236fU would allow to consider also the A and Z dependency. Motivated by the findings of neutron capture measurements reporting the existence of an isomeric state in 23gU at an excitation energy of 1700f2.3 keV with K” 5 5/2+ decaying with a half life of tlj2 > 0.25~s predominantly to the ground state in the first minimum [SS], it appears worthwhile to search for this y decay in a 238U(d,p)23gU reaction. Similar to the experiments reported in Sect. 2.4 on the shape isomer in 236U the energy conservation can be exploited, where the excitation energy introduced in the (d,p) reaction reappears in the sum of the y ray energies of the prompt deexcitation towards the isomeric ground state and the isomeric energy of the y decay to the first well. Together with the use of a pulsed deuteron beam this could be exploited for a significant reduction of background contributions. Furthermore when using the (d,p) reaction close to the Coulomb barrier the reduced excitation energy for the even-even target allows to drastically reduce the fission yield. In this way the accidental background hampering the identification of the isomeric decay can be significantly suppressed. Furthermore theoretical advances in the description of collective vibrations and their moments of inertia appear highly desirable, allowing to compare the presented systematics of collective excitations in even-even nuclei with predictions. Theoretical progress in the understanding of the nuclear structure of strongly deformed heavy nuclei critically depends on experimental information on single-particle properties. So far spectroscopic studies on fission isomers concentrated on even-even isotopes, while only sparse data is available for odd-N fission

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382

isomers. Therefore it is planned to study the fission isomer in the N=143 isotope 237Pu produced in the 235U(cq2n) reaction at 24 MeV [144] with conversion electron and y spectroscopy, for the latter using the high-resolution and highly efficient MINIBALL spectrometer [143]. In 237Pu the existence of two fission isomers is known [144, 145, 1461, the lowest energy state in the second minimum has a halflife of llOf12 ns while a second isomer with a halflife of 112Ort80 ns is located about 300 keV above the short-lived one. From their relative population it could be concluded that the longer lived isomer has a spin several units larger than the shorter lived isomer, however, an unambiguous determination of the spins in 237Pu was not yet possible [147]. Nevertheless spin assignments of most likely I = 11/2 were derived for the excited, long-lived state and I = 5/2 for the short lived ground state in the second minimum of 237Pu from a statistical model analysis of the reaction [148]. isomeric fission yield ratio of the two isomers obtained in the 23gPu(y,2n) photonuclear However, g-factor measurements for the long-lived isomer resulted in a value of g = -0.45(3), unusually large for a deformed odd-N nucleus [146, 1471, that could only be explained by a K = l/2 structure.

237f

h

Figure 43: Expected nuclear structure in the second minimum of 237Pu based on the two known fission isomers. Known properties are indicated such as halflives (144, 145, 1471, spin assignments [148] and g-factors [146], while in addition the expected structure of rotational and p vibrational bands (assuming a /l phonon energy of 600 keV) has been included.

Fig. 43 summarizes our present knowledge of the nuclear structure in the second minimum of 237Pu, including the expected structure of rotational bands built on the isomeric ground states and ,0 vibrational bands assuming a p phonon energy of 600 keV. The expected structure of the rotational band in 237/Pu is consistent with the experimental results for the 2.6 ns isomer in 23gfPu as reported by Backe et al. [149]. Moreover, early theoretical calculations of the single-particle energy levels for neutrons at the large deformation in the second minimum resulted in partly contradicting predictions (see Ref. [148] and references therein), clearly calling for improved theoretical calculations based on new conclusive experimental information from the identification of specific Nilsson orbitals. The measurements will also be of interest to determine the parameters of the Nilsson model for heavy and superheavy nuclei in the first minimum. Due to the steep sloping-down of low-K orbitals with deformation orbitals of the next higher shell in the first minimum occur at the Fermi surface for actinides in the second minimum.While for odd-neutron nuclei the next magic number N = 184 is unanimously predicted, for odd-proton nuclei different predictions of the next magic proton number N = 114,120 and 126 are given. Therefore also a study of an odd-proton fission isomer would be very helpful to determine the forces of the shell model. For an odd-Z, even-N fission isomer to be studied in highresolution y spectroscopy 23gAm with a half life of 163 ns seems to be the best candidate. It can again be produced by the (q2n) reaction using a 237Np target (ti/z = 2.1 * 10%). This reaction also was

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383

chosen to measure the first and only g-factor of an odd-Z, even-N fission isomer. For the ground state of 23gfAm a K = R = 7/2 value was obtained [150]. Another topic of general interest is the question of transitions linking the superdeformed second potential minimum with the normal deformed first well. A high-resolution y spectroscopic study of the back decay in 236fU would allow for a significantly increased precision determining the excitation energy of the isomeric ground state and the application of the Ritz combination principle would provide an additional proof of the proposed five El cascades. Transitions between nuclear states with a large deformation difference may have large transition probabilities for EO decays. Such an EO decay has been found in the decay of the 238U fission isomer back into normal deformed states [76], while the corresponding search in 236fU was unsuccessful 1851. This may have been caused by the inaccurate knowledge of the fission isomeric ground state excitation energy, which at the time of Ref. [85] was assumed to be 2.3 f 0.3 MeV, compared to the nowadays best value of EJI = 2.814(17) MeV [63, 781. With 236fU representing the only case where the y back decay into the normal deformed first minimum has unambiguously been identified (Sect. 2.4.1) it will be worth to repeat the search in the correct energy region with an improved experimental technique. Preparing for an experiment to search for the EO back decay in 236fU an efficient Mini-Orange setup has been developed [151, 1521 that may allow to identify the high energetic conversion electrons from the from the isomeric ground state in 236fU.

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3

Spectroscopy in the Third Minimum

Besides the experimental studies on superdeformed states in fission isomers as discussed in the Previous section, the spectroscopy of high-spin states from strongly deformed nuclei has been a forefront topic of nuclear structure research for almost the last two decades. Triggered by the observation of highof high-spin spin superdeformed bands in is2Dy [4], extensive efforts succeeded in the identification superdeformation with an axis ratio of 2:l in various mass regions [5, 61. Encouraged by this success, the search was extended to hyperdeformed states with an axis ratio of 3:1, located in a third minimum of the potential energy surface. Early evidence for hyperdeformation in “‘Dy was reported by GalindoUribarri et al. [153]. Discrete transitions have been tentatively assigned to a hyperdeformed band in ls2Dy by Viesti et al. [154]. LaFosse et al. [155] made a more definite observation of candidates for a hyperdeformed band in 147Gd, an evidence that could not be confirmed by further studies of the same group [156]. Despite all efforts, no discrete hyperdeformed levels have been identified so far. In the actinide region a third minimum in the potential energy surface was predicted already thirty years ago by Moller et al. [157]. Experimentally the third minimum is indicated from a microstructure in the resonances found in the light actinides using the (n,f),(t,pf) and (d,pf) reactions (see e.g. Refs. [3, 9, 158, 159, 160, 1611). Other evidences for the third minimum are the observation of asymmetric angular distributions of light fission fragments around 232Th [162] and the analysis of the slopes of the near-barrier photofission cross sections of actinides [163]. A status report summarizing the experimental knowledge on the third minimum in the fission barrier of actinide isotopes until those recent experimental findings subject to the main parts of this section can be found in the article by Blons [3]. Following two introductory sections on the specific nuclear properties expected in the third minimum and their first experimental identification in Thorium isotopes, two subsequent sections deal with recent experimental progress in the understanding of the third minimum in actinide isotopes: the identification of hyperdeformed rotational bands in U isotopes (Sect. 3.3) and the spectroscopy of hyperdeformed rotational bands in 234U leading to the first measurement of the potential depth of the third well (Sect. 3.4). AS a perspective to further experimental activities new approaches to determine the nuclear shape in the third minimum (Sect. 3.5) will be presented.

3.1 Expected spectroscopic

properties

in the third minimum

A typical property of nuclei in the third well is their reflection asymmetric shape. An examination of the potential energy surface within the framework of the triple-humped fission barrier reveals the existence of two third wells at positive and negative values of the octupole deformation coordinate ps. For a reflection asymmetric shape the nuclear wave function no longer has a well-defined parity, but definiteparity wave functions can nonetheless be constructed from linear combinations of the original wave functions, leading to different eigenvalues E+ and E-. The rotational bands will thus have opposite parities but the same moment of inertia 0. The energy splitting AE+_ =E&-E,, depends on the transmission coefficient of the barrier between the two third wells. Thus the experimental observation of pairs of rotational bands with the same K but opposite parity provides a conclusive signature for the identification of the third minimum. These octupole-deformed rotational bands have been observed also in the first minimum in the actinide region (see e.g. [164]). The different implications of the octupole deformation have been reviewed by Butler and Nazarewicz [165]. Calculations expect the appearance of the third minimum in the potential barrier at deformation parameters p2 z 0.90 and /J z 0.35 [lo, 13, 166, 167, 171, 172, 173, 1741. This indicates that nuclei in the third well are much more elongated compared to the second minimum, resulting in a significantly increased moment of inertia 0 or reduced inertia parameter h2/20, respectively. Hyperdeformed nuclear states are expected to exist at F?/20 z 2 keV [173], compared to the value of N 3.3 keV found in the second minimum [II].

3.2

Experimental

evidence for a third minimum

in Th isotopes

Searching for an experimental identification of the third minimum in the potential energy surface, the nucleus 231Th played a key role as being the most extensively studied testground. Its 720 keV resonance

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385

was studied in high-resolution experiments using the TOF technique [158] as well as data from (n,f) reactions [177, 1781. All analyses of these data from different groups [178, 179, 180, 1811 agreed on the necessity of using two rotational bands with opposite parities. The Saclay group was able to fit simultaneously all the experimental data from energy and angular distribution measurements [180], describing the E, =720 keV resonance as a superposition of two rotational bands of opposite parity with K=1/2. The deduced rotational parameters were (fi2/20)+

=

1.9 f 0.3 keV; a+ = 0.2 * 0.2

(53)

(tL2/20)-

= 2.1 f 0.1 IceV; a- = 0.3 f 0.2

(54)

EiK =8.5 keV. Additional experimental data were obtained at Saclay using the and nE+,_ =EiK(d,pf) reaction [185], thus introducing more angular momentum into the compound nucleus. As expected, higher-J members of the rotational sequences, not populated in the (n,f) reaction, appear quite accurately on the predicted curve. This can be seen in Fig. 44, where the experimental status of 231Th [3] is summarized. The results from (n,f) studies, where the K=7/2 bands are not fed, were compared to the (d,pf) data at forward angles, where the K=7/2 contribution is negligible. All expected members of the two K=1/2 rotational bands could be identified from J=1/2 up to J=13/2, thus strongly indicating the presence of a third minimum in the 231Th fission barrier.

,010

5850

5900

EXCITATION ENERGY (keV)

Figure 44: Comparison of the experimental and calculated 230Th(n,f) cross section with the experimental and calculated 230Th(d,pf) probability. From [3].

Theoretical progress also enabled purely microscopic calculations for the potential energy surface. Selfconsistent mean-field calculations within the Hartree-Fock-Bogolyubov (HFB) approach by Berger et al. [171] for the fission barrier in 230Th clearly exhibit a third minimum at large quadrupole deformations. Besides the non-relativistic HFB approach potential energy surfaces for fission in recent years have also been calculated within the relativistic mean-field model (1821, approximately reproducing the doublehumped fission barrier of 240Pu. Even the triple-humped fission barrier of 232Th could be described by the relativistic calculations with a quality comparable to the non-relativistic models [174], as illustrated by Fig. 45, where the potential energy surface of 232Th calculated with different relativistic mean-field parameterizations is compared to non-relativistic calculations and experimental values. While early shell-correction calculations [12] as well as HFB [171] and relativistic mean-field calculations predicted rather shallow third minima (-0.2-0.5 MeV), shell-correction calculations by Cwiok et al. [13] predicted much deeper hyperdeformed third minima (--l-3 MeV), in better agreement with recent experimental findings that will be discussed in Sect. 3.4.1. In Ref. [13] Cwiok et al. performed systematic calculations of potential energy surfaces of the even-even

386

E!G. Thirolf; D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

0.5

1.0

2.0

1.5

3.0

3.5

0 Dls

*=Th

IO

A YE+WS ,’ :

l

D

I

g

2.5

. ~~~,(O-‘~ l

...‘.+ ...._...._ p .--‘.,,,

:

.....

4.0 -

exp.’

exmZ

..

p.JL,

NL-SH 0

50

100

150

200

Q

250

300

350

bl

Figure 45: Potential energy surface of 232Th calculated with different relativistic mean-field parameter sets (PL-40: [168], NLl: [169], NL-SH: (1701). Barrier heights as obtained in a non-relativistic Hartree-Fock calculation with the Gogny Dls force (1711, in a macroscopicmicroscopic calculation (‘YE+WS’: Yukawa-plus-exponential and Woods-Saxon) [13] and experimental values are shown for comparison. exp’ is taken from Ref. 191,exp* from Ref. [183]. From [174].

Rn,Ra,Th and U isotopes using the shell correction approach with the axially deformed average WoodsSaxon potential [175], while for the macroscopic part the Yukawa-plus-exponential mass formula of (176) was used. The energies of the local minima in the potential energy surface were obtained by performing a multi-parameter minimization of the total energy with respect to /j’s-Dr. It was emphasized that this model does not contain any free parameter adjusted to the properties of the hyperdeformed configurations. According to these calculations third minima appear in many actinide nuclei, characterized by very large elongations (/3s N 0.9) and significant reflection asymmetry, 0.35 <_ 8s < 0.65. In the nuclei around 234U the hyperdeformed minimum splits into two distinct minima with very different values of ,&,(A =3-7). Fig. 46 shows the resulting potential energy curves for Th, U and Pu isotopes as a function of the quadrupole deformation ,&. The energy and location of the saddle points and minima (except for the third one) were taken from Ref. [12], while for the energy of the third minimum the values calculated by Cwiok et al. [13] were taken for the less reflection-asymmetric hyperdeformed minimum (/33 M 0.4). The deepest third minima are predicted for the Thorium isotopes, while the lowering of the outer barrier with increasing proton number prevents the formation of pronounced third minima beyond Uranium. The analysis of experimental data in Thorium isotopes resulted in values for the moment of inertia of about 20/ti2 close to 500 MeV-‘. For comparison moments of inertia have been calculated using Nilsson and Woods-Saxon potentials as a function of deformation 11781. For large deformations & N 0.9 the results are close to the rigid-body value. As illustrated in Fig. 47, the experimental values were found to be consistent with the moment of inertia at the hyperdeformed (axis ratio 3:1) third minimum.

t? G. Thiro[f; D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

0.0 0.2 0.4 0.5 0.1 1.0

0.0

0.2

0.4

0.6

P2

p2

0.1

1.0

0.0 0.2 0.4 0.6 0.1 1.0

A

0.0

0.2 0.4

0.6

0.11 1.0

a

Figure 46: Potential energy as a function of the quadrupole deformation parameter for Th, U and Pu isotopes. The energy and location of the saddle points and minima (except for the third one) were taken from Ref. [12], while the energy of the third minimum for the less reflection asymmetric hyperdeformed minimum (pa = 0.4) was taken from Ref. [13]. The energy scale was slightly shifted in order to reproduce the energy of the ground states in the first well [184].

600 -

0 . . l -

F-

2

z g;

‘.

- - - -;-

Rigid

Nilsacm ~x~.Saxo”

4w-

I -

,............’

*WI-

Figure 47: Moment of inertia 0, calculated as a function of deformation for 240Pu [178] (reflection symmetric shapes). For 232Th [173] values calculated at the third minimum (octupole deformed shape) are also shown. The experimental values of 0 at the first and second minimum for 240P~ [ll] and at the third minimum for 231Th [3] are denoted as crosses. From [3].

387

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388

3.3

Identification

of hyperdeformed

states in U isotopes

Based on modest energy resolution measurements, subbarrier resonances around 5 MeV excitation energy have been observed in various actinide nuclei [9, 158, 1801. They were interpreted as resulting from a coupling of compound states in the first potential well to vibrations of the fission degree of freedom of the second minimum, as already discussed in Sect. 2.7.1. Aiming at the experimental manifestation of hyperdeformed states, Blons et al. [3, 161) measured the subbarrier resonances in the fission probability of Th and U isotopes with high energy resolution. While they were able to describe the microstructure of the resonances for the Thorium isotopes 230~231~233Th as originating from hyperdeformed rotational bands, rather puzzling results were encountered for the Uranium isotopes [185]. Already earlier Goldstone et al. [121] proposed that the observed transmission resonances in 234U and in 236U might be due to states in the second minimum, consisting of a fission vibration coupled to low-lying states in the second well such as rotations and K” = O+, O-, 2+, I-... collective vibrations. Blons et al. [185] were able to resolve the fine structure of the resonances in 234U, measuring also the fission fragment angular distribution. However, the results obtained were not consistent with the angular distributions calculated with the assumptions of single pure J resonances, thus urging for clarification. If the excitation energy is higher than the inner barrier of the second well, but lower than the barriers of the third well, then the transmission resonances corresponding to class-11 and class-111 states could in principle be distinguished according to their width. The width of class-11 resonances should be much broader due to the strong coupling to class-1 states, while the damping of class-III resonances is hindered by the inner barriers of the third minimum. Accordingly the best energy region for studying the hyperdeformed states by measuring transmission resonances in the actinides is indicated by the shaded energy regions in Fig. 48, illustrating the 232,234,236U potential barriers derived from the barrier parameters given in [12].

0.0

0.2

0.4

0.6

0.6

1.0

Figure 48: Potential energy as a function of the quadrupole deformation parameter for selected U isotopes (subset of Fig. 46). The energy and location of the saddle points and minima (except for the third one) were taken from Ref. [12], while the energy of the third minimum for the less reflection asymmetric hyperdeformed miriimum (/?s r 0.4) was taken from Ref. (131. The energy scale was slightly shifted in order to reproduce the energy of the ground states in the first well. The best excitation energy region for studying hyperdeformed states is marked by the horizontally shaded regions. From [186].

236U was experimentally studied by the Debrecen group, using their 103 cm isochronous cyclotron for the 235U(d,pf)236U reaction at Ed = 9.73 MeV. The energy of the outgoing protons was analyzed by a split-pole magnetic spectrograph [187] set to 140’ relative to the beam axis. The position and energy of the protons were analyzed by two position sensitive Si detectors in the focal plane, resulting in an energy resolution of less than 20 keV. Fission fragments were detected by two position-sensitive

389

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avalanche detectors. Protons were measured in coincidence with fission fragments. spectrum is shown in Fig. 49a) as a function of the excitation energy.

5.2 5.3 5.4 5.5 5.6

0

2

1

3

The resulting

proton

4

E’ (MeV) 25

P’ ’ ’ ’ -

’

I

K=O

--

.___..

K=l

4

\

__-_

K=2

_.

K=3

K=4 __._ K=5

:.-.___./2.u_y._ I 0

0

20

60

I

80

Figure 49: (a) Proton spectrum measured for 236U in coincidence with fission fragments as a function of the excitation energy, compared to the fitted rotational band; (b) the results of the x2 analysis; the theoretical predictions for the hyperdeformed (HD) and superdeformed (SD) rotational parameters are indicated by arrows; (c) experimental fission-fragment angular distribution for the 5.47 MeV resonance with respect to the recoil axis (z axis) compared to calculated distributions for different K values. From [186].

Resonances with a width (FWHM) of about 30 keV were found at 5.28, 5.37 and 5.47 MeV, similar to those observed by Goldstone et al. with modest energy resolution of 60-70 keV [121]. These resonances do not appear to be appreciably damped, although they are located above the top of the inner barrier. Thus they cannot be associated with superdeformed states in the second minimum. Already here an indication for the presence of a rather deep third minimum (as predicted by Gwiok et al. [13]) was found, which will be confirmed by dedicated studies on 234U discussed in the following section. Assuming that each of the resonances contains a rotational band similar to those found by Blons et al. [161] in Thorium, the energy spectrum was fitted using Gaussians to describe the different band members, following the same procedure of Blons et al. [161] with widths fixed by the experimental resolution. The non-resonant part of the fission probability was taken into account as an exponential background contribution. The relative excitation probabilities of the rotational band members were taken from Back et al. [ISO], derived from a DWBA analysis of the (d,p) reaction at the same beam energy. For the fitting procedure the band head energy, the absolute intensity of the band and its rotational parameter (identical for all three resonances) were treated as free parameters. Assuming the observation of a rotational band built on an excited state rather than on the ground state in the third minimum, the K value of the band head

I? G. Thim& D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

390 was varied during

the fit. The best fit was obtained for K=4 as shown in Fig. 49a). With K=7/2 for the 235U target and most likely I=0 transfer at the low beam energy (9.73 MeV) used in the experiment, the excitation of predominantly K= 3 and 4 states can be expected in agreement with the fit result. The calculated x2/F values are displayed in Fig. 49b) as a function of the rotational parameter fi2/28. The predicted values for hyper- (HD) and superdeformed (SD) states are indicated by arrows. From the best fit a rotational parameter of hZ/2B = 1.6’::,” keV was extracted in good agreement with the value of (fi”/20),,,, = 2.0 keV predicted for a hyperdeformed state [173]. Fission fragment angular distributions were generated for the 5.47 MeV resonance as well as for the continuous non-resonant part of the fission probability, normalized to the known (d,pf) angular distribution [188]. The angular distribution is displayed in Fig. 49c). All distributions were fitted with even Legendre polynomials up to fourth order. For the non-resonant region the resulting angular coefficients were found to be in good agreement with published data [139, 189, 1901. Measured and calculated angular distributions were compared for different K values in order to extract information on the K value of the hyperdeformed rotational band. From the comparison the most probable K value of the band is K=4 (see Fig. 49c), consistent with the analysis of the proton spectrum.

h2/2s(keV)

%(de,)

Figure 50: (a) Fission probability of 234U as a function of the excitation energy. The full curve is a result of the fit with fi2/20 = 2 keV according to the deepest x2 minimum; (b) the results of the x2 analysis; (c) fission fragment angular distributions for the 4.86 MeV, 4.81 MeV and 4.97 MeV resonances in 234U compared to the theoretically calculated curves assuming different K values for the rotational bands. From 11861.

Krasznahorkay et al. 11861 reanalyzed the results of Blons et al. [185] for 234U in a similar way, assuming that each of the resonances observed in Fig. 50a) consisted of a rotational band. The energy of the band heads, the absolute intensity of the bands and a rotational parameter (h2/20) were fitted to the experimental data. The resulting inertia parameter was found as fi2/28 =1.8$:$ keV, which is the average of the x2 minima and again agrees well with the prediction for hyperdeformed states. The angular distribution of the fission fragments was also calculated for different K values and compared to the experimental data (see Fig. 50~). Good agreement was obtained, supporting that each of the peaks contained a complete rotational band with K= 1 or 2 rather than one definite state with well-defined J’ as assumed by Blons et al. [185]. The target K value of 5/2 implied the excitation of K= 2 or 3 states assuming l=O transfer.

EG. Thirolf; D. Habs /Prog. Part. Nucl. Phys. 49 (2002) 325-402

3.4

Spectroscopy

391

in the third well of 234 U

In the work already discussed in Sect. 3.3 Krasznahorkay et al. [lSS] reanalyzed the fission resonances in 234U measured by Blons et al. [185] and showed that the unresolved peaks around E* = 4.9 MeV could be interpreted as hyperdeformed states in the third well of the potential barrier. The aim of a subsequent work was to study the 233U(d,pf)234U reaction with better energy resolution than Blons et al. (1851, to resolve the hyperdeformed rotational bands and, from the level densities, to estimate the depth of the third minimum. In order to investigate the hyperdeformed bands in 234U the excitation energy was chosen between the energy of the inner and outer barriers of the second well, i.e. between 4.5 and 5.2 MeV [186]. In this energy range the widths of the superdeformed resonances in the second well are expected to be much broader than those of the hyperdeformed states due to the strong coupling to the normal deformed states. The widths of the hyperdeformed states due to the higher outer barriers of the third well remain below the actual experimental resolution of -5 keV. The experiment on 234U was carried out with a Ed = 12.5 MeV deuteron beam of the Munich Tandem setup was accelerator, using enriched (99 %) M 30 pg/cm2 thick targets of 233U. The experimental identical to the one used for the transmission resonance studies in 240Pu [62] discussed in Sect. 2.7.2. The energy of the outgoing protons was analyzed by a Q3D magnetic spectrograph, while fission fragments were detected by two position-sensitive avalanche detectors. Protons were measured in coincidence with fission fragments. The obtained proton - fission fragment coincidence spectrum is shown in the bottom part of Fig. 51 as a function of the excitation energy.

et al.

AA

200

Krasznahorkay

et al.

Figure 51: Part of the proton spectrum measured by Krasznahorkay et al. [132] in coincidence with fission fragments from the 233U(d pf) reaction (bottom panel) compared to the results from Blons et al. [185] (upper panel). From [132].

Comparing the low-energy part of the proton spectra from the recent measurement of Krasznahorkay et al. [132] with the earlier data of Blons et al. [185] presented in the upper part of Fig. 51 it can be concluded that the energy resolution has considerably been improved and that a fine structure of the peaks can clearly be seen. Experimentally the very large quadrupole and octupole moments of the hyperdeformed states should manifest themselves by the presence of alternating parity bands with very large moments of inertia [185]. Assuming overlapping rotational bands with the same moment of inertia, inversion parameter [164] and intensity ratio for the members in a band, the spectrum was fitted using simple Gaussians for describing the different band members in the same way as in Ref. [186]. The result of the fit is shown as an overlay to the experimental spectrum in Fig. 52a).

392

I?G. Thiro& D. Habs /Pmg.

3 200 z 175 150 125 100 75 50 25 0 4800 3 O" 2.5 2 1.5 1 0.5 0 -0.5 -1 4800

5200

5000

, 5000

Part. Nucl. Phys. 49 (2002) 325-402

5400

, , / , 5200

5400 F

(keV)

Figure 52: (a) Part of the measured proton energy spectrum fitted with 24 rotational bands with a common rotational parameter. The spectrum was divided into two parts at E= 5150 keV for the fitting procedure; (b) Experimental fission fragment angular distribution coefficients as a function of the excitation energy compared to the calculated values using K=l (upper curve) and K=3 (lower curve) for all of the bands. The K=O curve is very close to the one for K=l while the K=2 curve falls in between K=l and K=3. Both are not shown. From [132].

The obtained widths of the peaks show the experimental energy resolution up to about 5.3 MeV. Above this excitation energy the peaks get increasingly broader due to the increasing fission width when approaching the top of the fission barrier (see Fig. 1 in Ref. [186]). The relative intensities of the members of the rotational bands have been taken from Back et al. [191] and are given in Table 6. These values have been calculated by DWBA for 235U(d,p). These relative intensities depend sensitively on the distribution and width of the single particle states involved in the (d,p) process at the given excitation energy. These relative intensities have been adopted as initial parameters for the fitting procedure. The final values are shown in Table 6. The jump in the relative intensities going from J=2+ to 3-, as predicted by Back et al. [191] might be a consequence of the alternating parities within the rotational bands, although the fit was actually not very sensitive to the relative intensity of the 2+ state. Increasing this intensity by a factor of two worsens the x2 by less than 8%. After fixing the relative intensities of the band members, two specific parameters for each band were used during the fitting procedure: the energy of the band head and the absolute intensity of the band. A common rotational (tL2/26’) an d inversion splitting (AE+_) parameter was adopted for each band. Although the statistics in the second part of the spectrum is better, the density of the states is about two times larger, making the determination of the rotational parameter more uncertain. The rotational parameter was therefore determined for both parts separately and the weighted average was calculated. The resulting parameters were i?/26 = 2.2 f 0.2 keV and AE+_ = 0’:: keV. Using

393

PG. Thirolj D. Habs /Prog. Par!. Nucl. Phys. 49 (2002) 325-402

Table 6: Relative intensity ratios of the rotational-band members populated in the 233U(d,pf) reaction. The calculated values marked by a) and b) are taken from Ref. [191] for rr = + and -, respectively. The underlined values represent the relative intensities for an alternating parity, octupole rotational band. The accepted values were obtained from a fit of the 5.1 MeV region of the energy spectrum. J

0

I

2

3

4

5

6

7

8

a) b) adopted Ref. [192]

0.04 0.08 0.00 0.47

0.04 0.23 0.03 0.60

0.11 0.50 0.10 1.00

0.50 1.00 I.00 0.58

0.85 1.19 0.59 0.22

0.69 1.00 0.37 0.15

0.69 0.92 0.32 0.11

0.50 0.23 0.06 0.04

0.23 0.42 0.06 0.07

alternatively relative intensities reported by Bjornholm et al. [192] for low-lying rotational bands in the 233U(d,pf) reaction at I3 MeV resulted in almost as a good fit as above with: fi2/20 = 1.5 f 0.5 keV and AE+_ = 0 f 15 keV. The somewhat different values obtained by assuming these two very different sets of relative intensities indicates the sensitivity of the rotational parameter for the assumed relative intensities. Finally Krasznahorkay et al. (1321 report as the final value of the rotational parameter its weighted average of the two analyses: ft2/20 = 2.1 f 0.2 keV. The rotational parameter agrees well with the value corresponding to a hyperdeformed nuclear shape and to the value previously obtained by the same authors [186]. The value for the inversion splitting parameter AE+_ M 0 is consistent with the small inversion parameters obtained by Blons el al. [185] for the Thorium isotopes. It is noteworthy to mention that the analysis reported in [132] is not sensitive to the parity of the states. In case of K # 0 no distinction is possible between quadrupole and octupole rotational bands. Assuming the observation of rotational bands built on excited states, it was tried to vary the K value of the band head during the fitting procedure. However, the result of the fit was found to be insensitive to the K value when it was varied between 0 and 2, because the relative intensity of the members of the band with J 5 2 is much less compared to the intensity of the J = 3 line (see Table 6). In case of the quadrupole scenario K = 1 was used for all bands. Fission-fragment angular distributions were generated as a function of the excitation energy, normalized to the known (d,f) angular distribution [188] and fitted with even Legendre polynomials up to fourth order. The a2 angular distribution coefficient is shown in Fig. 52b) as a function of the excitation energy. In order to obtain information on the spins and K values of the observed rotational bands and to check the assumptions for the energy spectrum fit, the angular distribution coefficients of the fission fragments were calculated for different K values and compared to the experimental ones (solid lines in Fig. 52b). Assuming quadrupole rotational bands with intensities peaking at J” = 2’ even the gross structure of the measured angular distribution coefficients could not be explained. The calculated a2 coefficients were always too low compared to the experimental ones. This can be understood since u2 = -0.20 for J = 2 and 0.03 for J = 3 while increasing with J. In order to explain the experimentally measured values of a2 = 0.5 an intensity distribution peaking at J values larger than 2 has to be assumed, most likely at J = 3 as assumed before.

394

3.4.1

PG. 7’hirolj D. Habs /Prog. Part. Nucl. Phys. 49 (2002) 325-402

Depth of the third minimum

Recent progress in the spectroscopy of heavy and superheavy elements also triggered refined theoretical calculations, resulting in a depth of the third minimum that is predicted to be much larger (AE x 3 MeV [13]) than believed earlier [12]. Until recently no direct experimental information was available on the depth of the third minimum. Fortunately, the method of a statistical level density analysis established in the second minimum of 240Pu [62] as presented in Sect. 2.7.4 provides a tool that could be applied also to the situation of 234U the density of the J = 3 states has been the hyperdeformed third well. Further investigating determined from the experimental data discussed in the previous section. The average distance of the two closest neighbours of a given state is shown in Fig. 53. The level spacing distribution is close to a Wigner distribution [193] but the mixing-in of some Poisson type distribution cannot be excluded. The density of J = 3 states has been calculated as a function of the excitation energy using the backshifted Fermi-gas description with parameters determined by Rauscher et al. [131]. In order to estimate the depth of the third well the experimentally obtained and calculated values were compared. It was assumed that the same parameterization of the level density formula is valid in the third well, as was determined by Rauscher et al. [131] by fitting the level densities in the first well of the potential barrier. This assumption is based on the finding of Glassel et al. [119] that the level density of the 2+ states lying in the second well of 240Pu is the same as the level density in the first well. The shell-correction energy used to determine the level density parameter “u” has been taken from the work of Mijller et al. [194] while the spin cutoff-parameter o was determined by using the rigid rotor rotational parameter suggested by Rauscher et al. [131]. The result of the comparison is shown in Fig. 53.

lC233 , ,

,E;’WV:

5

Figure 53: Average distances of the J=3 levels in the third minimum of 234U as a function of the excitation energy. The solid curves show calculated values by different formulae (see text), the circles correspond to experimental values. From [132].

The calculated curve initially had to be shifted by 2.7 MeV to reproduce the experimental values. According to this comparison the ground state in the third well is shifted up by 2.7 MeV, which also means that the “microscopic correction” C(N,Z) in Eq. (14) of Ref. 11311 should be modified by this energy and the level distances should be recalculated. Doing this in a recursive way resulted in a final value of 3.1 MeV for the energy of the ground state in the third well (curves connected with an arrow “Rauscher” in Fig. 53). Taking into account the shell and nucleon pairing correlation effects Mughabghab and Dunford (1951 calculated and fitted the spin cutoff parameter as a function of the atomic mass and found large

PG. Thirolf, D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

395

deviations from the one obtained with the rigid rotor rotational parameter, however in the region of the actinide nuclei these deviations do not significantly affect the accuracy of the level density parameterization. Also two other formulas were used to estimate the level distances, which were parameterized by yon Egidy et al. [133]. They used a constant temperature level density formula and the Bethe formula for the back-shifted Fermi gas model. The theoretical curves (also shown in Fig. 53) were calculated with these two formulas and parameters determined by von Egidy et al. [133] by fitting the low-lying level scheme (E* _< 1.5 MeV)of 234U. From the uncertainties of the calculated and measured level distances the error of the energy determination was estimated to be 0.4 MeV. Cwiok et al. [13] predicted two different hyperdeformed minima for 2a4u with very different/3x (A = 3 - 7) values. One of them has an octupole deformation parameter of/~a ,~ 0.4 and a minimum of E, II = 3.5 MeV while the other is more reflection-asymmetric and has an octupole deformation parameter of /3a ~ 0.6 and a minimum of E1tl = 2.7 MeV. The experimental value of EIII = 3.1 4- 0.4 MeV obtained in [132] is between the two predicted values with an error bar that overlaps with both theoretical values. At this moment there is no further information on the/~a of this nucleus. Recently Shneidman et al. [196] discussed the relationship between di-nuclear systems and nuclei in highly deformed states. They calculated the potential energy of di-nuclear systems as a function of the mass asymmetry ratio and found a few global minima. Assuming a l°°Zrq-134Te clusterization for 2a4U the potential energy of the system was also calculated as a function of the distance between the two cluster components, resulting in a depth of the potential keeping the two clusters together to be 3.4 MeV. This value agrees well with the depth of the third minimum determined experimentally by Krasznahorkay et al. [132], pointing towards a cluster structure in the hyperdeformed third minimum. 3.5

Perspectives

Besides the reflection-asymmetric, octupole deformed nuclear shape manifesting itself via alternatingparity bands little is known so far experimentally on the nuclear shape in the third minimum. In light c~-particle nuclei the similarity between hyperdeformed and cluster-type states, i.e. quasimolecular states, has been studied by several authors (see e.g. [13] and references therein). In the actinide region Cwiok et al. [13] calculated the hyperdeformed minima within the shell correction approach. They showed that the density distribution at the third minimum resembles a di-nucleus consisting of a nearlyspherical nucleus around the doubly magic la2Sn and a complementary well-deformed nucleus from the neutron-rich A~100 region as shown schematically in Fig. 54 for 2a~U.

(hyperdeformed) 3. minimum 236 U (132=0.85, ~3 =0"6)

p [fm] 10--

5--

0--

-5 - -10 -15

I

I

I

I

I

I

I

I

10

5

0

5

10

15

20

25 Z [fm]

Figure 54: Schematical display of the shape of 236U at the hyperdeformed third minimum configuration. The density distribution calculated with the shell correction approach by Cwiok et al. [13] resembles a di-nucleus consisting of a nearly-spherical nucleus around the doubly magic la2Sn and a complementary well-deformed nucleus from the neutron-rich A~100 region.

R G. Thiro& D. Habs / Prog. Part. Nucl. Phys. 49 (2002) 325-402

396

Such a cluster structure would be a drastic manifestation of nuclear shell structure at very large deformations. The special role of the ‘32Sn structure in the fission process has been stated before in the context of mass distributions of fission fragments [197, 1981 and the analysis of cold fission data [199]. Due to the proximity of the hyperdeformed minimum to the fission saddle point the di-nuclear structure should manifest itself in a significantly more pronounced fragment mass distribution in coincidence with a hyperdeformed fissioning system. the fission fragRecent experiments have been performed in Debrecen and Munich [200] investigating resonances ment mass distribution from the 235U(d,pf) reaction in coincidence with hyperdeformed previously identified at 5.28, 5.37 and 5.47 MeV [186]. Preliminary evidence of a sharpening of the fragment mass distribution requires further confirmation. Similar evidence was also found in first experiments using monoenergetic neutrons produced in a 7Li(p,n) reaction followed by neutron induced fission of 231Th [184]. Both theoretical predictions and preliminary experimental evidence will trigger further experimental efforts to clarify a potential cluster structure in the third minimum of actinide nuclei, thus giving unprecedented direct access to the nuclear shape in the hyperdeformed third well.

4

Conclusions

The detailed high-resolution spectroscopy in the second minimum confirmed the schematic picture (Fig. 4) of the collective rotations and vibrations, however, a much more detailed knowledge on the variations of the moments of inertia and vibrational energies was obtained. Although the investigation of fission isomers is hampered by the small production cross sections, new highly efficient detector arrays promise significant progress in the near future. Spectroscopy of fission isomers has the big advantage that the rotational bands can be followed down to the band heads. The bands show a clean separation of vibrational and rotational excitations due to the large moments of inertia, while at the same time Coriolis mixing of these low spins is rather small. The very regular rotational bands allowing good predictions of the decay patterns are very helpful in constructing the level schemes and deducing spin values. The next important step will be the study of single-particle orbitals at the deformation of the second minimum, for and odd-neutron as well as for an odd-proton isomer. Then basic properties of the nuclear forces with respect to deformation like surface effects and pairing can be determined for the Nilsson level schemes. The Nilsson levels are also the basic input for calculations of the shell correction with the Strutinsky method. Since many levels from higher shells at spherical deformations are sloping down as a function of deformation, their localization in the second minimum is also useful for our understanding of heavy and superheavy nuclei. Several new and unexpected phenomena were observed for nuclei in the second minimum, e.g. a predominant population of negative parity states at lower excitation energies, which is due to the filtering action of the inner and outer barrier during deexcitation. This population pattern is quite different from the first minimum. It will be of interest to look for similar changes of the population pattern for other superdeformed areas in the chart of nuclei at high spins. Also the relation between the moment of inertia and the quadrupole moment for larger deformations established here may be carried over to other areas of superdeformation, then studying in addition the spin dependence. Presently the third minimum is only studied via transmission resonances. Recent measurements prove that the depth of the third minimum is much deeper (l-3 MeV) than assumed before (0.2 - 0.5 MeV). Still the small width and height of the outer and inner barrier of the third minimum result in very short decay times by fission and back decay. Since it is known from the fission decay of isomers in the second minimum that the lifetimes of odd-odd isomers are prolonged by about a factor of lo6 compared to even-even nuclei, lifetimes of odd-odd isomers in the third minimum may reach into the ns to ps range. A measurement of such lifetimes would result in a further proof of the larger depth of the third well. These longer lifetimes would enable to observe y and conversion electron transitions within the third minimum, thus allowing to carry over the very successful high resolution spectroscopy from the second to the third minimum.

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Acknowledgements The authors want to thank all members of the experimental collaborations from groups in Munich, Debrecen, Heidelberg, Bonn and Paris working together to enable the physics progress reported here. Especially we acknowledge fruitful collaboration and stimulating discussions with A. Krasznahorkay, D. Schwalm and H. Hiibel. The work of the Munich and Debrecen groups presented here has been supported by DFG under HA 1101/6-l and 436 UNG 113/129/O, the Nederlandse Organisatie voor Wetenschapelijk Onderzoek (NWO), the Hungarian Academy of Sciences under No. 119 and the Hungarian OTKA Foundation No. T23163 and N26675.

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