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Superdeformed nuclei
NUCLEAR PHYSICS A
Nuclear Physics AS57 (1993) 17038~ North-Holland, Amsterdam
Superdeformed
Nuclei1
Sven Aberg Department
of Mathematical
Physics,
Lund Institute
of Technology,
P 0 Box 118,
S-221 00 Lund, Sweden
Abstract The structure of superdeformed (SD) nuclei is discussed, and some open questions are raised. Octupole vibrations built on SD and hyperdeformed (HD) shapes are described in an analytically solvable model, and structure effects on low-lying vibrational states, as well as the deformation cranked Nilsson-Strutinsky
splitting
of the giant resonance
calculations
states
are discussed.
From
it is predicted that HD nuclei in the 15sDy region
have a strong, static octupole deformation,
and the structure
The nucleus ‘&Gd is found to be a particularly
of such nuclei is discussed.
favoured candidate
for a hyperdeformed
pearshape.
1. INTRODUCTION The last few years the study of superdeformed nuclei has evolved to a broad research area by itself. In this paper I shall first review some of the present knowledge, and then discuss the role of the octupole degree of freedom at extreme deformations. A more complete discussion of SD nuclei can be found in the review papers [l, 21 as well as in several of the contributions The existence
to this volume.
of superdeformed
high-spin nuclei was predicted
fore the discovery [5], and in fig. 1 a large-scale
calculation
[3, 41 several years be-
from 1982 is summarized
[6].
For each (even-even)
nucleus between $Seaa and ‘A:Pois4 the predicted size of (super) deformation is shown together with the angular momentum value at which it is calculated to become yrast. We particularly note very favourable (i.e. low spin-value and large deformation) SD regions around lg2Hg, 15’Dy and, at somewhat smaller deformations, around 13*Ce. Several SD nuclei have later on been discovered in exactly these mass regions, nicely confirming the predictions.
This agreement
shows that we have a rather good understanding
between experiment
and theory
of how the nuclear shell structure
or
shell energy varies with neutron and proton numbers, as well as with angular momentum and quadrupole deformation. The SD region with largest deformations (similar to those of low-spin fission isomers) around 152Dy has been extended by the recently discovered SD band in 143E~ [7]. The calculations give good hope in finding the “connecting” SD nuclei like europium isotopes with N=81-84. Even more exciting is maybe the predicted ‘Work supported
by the Swedish
Natural
Science Research
Council
03759474/93/$06.00 0 1993 - Elsevier Science Publishers B.V.
All rights reserved.
18~
S. Aberg 1 Superdeformed nuclei
Figure 1: Part of the nuclide chart showing the lowest spin value, I,, for which even-even nuclei are calculated to obtain a large deformation, E > 0.35. 1, is easily obtained from the digit, as e.g. 6 means 1, = 65 etc. The corresponding deformation can corresponds to 0.35 5 s < 0.45, The calculation was performed
shape of the “superdeformation” is denoted by a letter and the be read out from the inserted plan over the (&,7)-plane. E.g. a b to 0.45 5 E < 0.55 and c to 0.55 5 E < 0.65, all with 171< 5’. with the MO potential neglecting pairing, and ~4 was included
only in the macroscopic contribution to the potential energy[b]. regions of new, yet not discovered, SD nuclei around 78Se, “Ru
and lssRu, see fig. 1.
1 .l Energy sequences The main part of our present knowledge about the structure of SD nuclei comes from measured r-ray sequences. By plotting the inverse of the difference between two consecutive r-ray energies (multiplied by four), giving the ,7(‘) moment of inertia, versus the r-ray energy, i.e. twice the rotational
frequency, it has been found that different nuclei
and different SD bands in one and the same nucleus may show different patterns. In fig. 2 we show the SD bands of the three isotones r5’Gd, “‘Tb and “‘Dy. The different g(2) behaviour has been explained [8] as being due to a different number of high-N orbitals occupied, ~6’, ~6~ and ~6~~ respectively, i.e. 2, 3 or 4 neutrons in the lowest N=6 orbitals (emerging from i13/2). This high-N scheme has been found to give a consistent picture of all observed SD bands in the A=150 region. In the A=190 region on the other hand, Jc2) is found to increase vs w,,~ in a quite similar way for each observed SD band [2]. In this context it is interesting to return to the yet unobserved region of SD nuclei around “Ru, in which nuclei are predicted to become yrast at angular momentum I z 35, see fig. 1. In fig. 3 calculated single-particle energies are shown versus rotational frequency at the appropriate
quadrupole
deformation.
The high-N orbitals
[550 l/2] and [541 3/2]
S. Aberg I Superdeformed nuclei
0.4
0.5
0.6
0.7
0.4
0.5
Frequency Figure
2:
the three
Measured N=86
and calculated
isotones
[12] Jc2)
150Gd, lslTb
1 0.7
0.6
hw moments
and lszDy.
19c
1 0.4
s 0.5
8 0.6
(MeV) of inertia
The neutron
for superdeformed
configuration
while the proton
configurations
are assigned
as 7r6’, n63 and ~6~, respectively.
are close
Fermi
for protons
as well as for neutrons
to the
energy
and a similar behaviour expected.
as in the “‘Dy
bands
is assigned (From
in
as v7l
[9].)
for nuclei around s’Ru, moment of inertia is thus
region of the Jc2)
However, it is now protons and neutrons in the same N=5 orbitals rather than
the proton N=6 and neutron interesting
1 0.7
N=7 orbitals
which are active.
effects coming from residual neutron-proton
From studies of the observed J(‘)
behaviour
This structure
may show
interactions.
it has been suggested that unpaired as
well as paired bandcrossings occur in SD nuclei. For example, in 146J47Gd there are many reasons to believe that the observed “bump” in J@) is related to an unpaired band crossing between the two specific neutron orbitals [651 l/2] and [642 5/2] [ll]. The very
5.2-
4.s:~$~i,~+-
ew$.___;__
‘0.
4.6-
*_______
4.4-
._______*..__.._*:=;;==~*::~~r~~~:r~rrrr+-------+‘-----_,>____8~0~-g~ .-. -o=====?~+ -+-+-___+
l _______
._______
0 ._--_
--
O..i====
-----
$lr-__T
9;=;l;h&~E:~z_
9 -.-__
* _
4.2-
w/w, Figure
3: Neutron
single-particle
energies
vs rotational
frequency
at the fixed superdeformation,
E = 0.52, ~4 = 0.05. Note the high-N orbital [550 l/2] just below the Fermi surface for N=44.
(From [lo].)
2oc large J(‘)
S. Aberg I Superdeformed nuclei values at the lowest frequencies
seen in 14’Gd and lsoGd have, on the other
hand, been explained [12] as being due to a paired bandcrossing, namely between a SD g-band and a SD s-band built by the alignment of a pair of neutrons in the N=6 orbital. Also the increase of Jr(‘) versus w,,~, seen for more or less all SD bands in the A=190 region, has been explained [13] as coming from paired bandcrossings, ~7~ and 7r72 (i.e. the SD AB and ab crossings). It is the increasing part of the expected J(‘) bump, caused by these two subsequent bandcrossings, one then is supposed to see. A problem is however that the expected turn over and decrease of Jc2) has not been seen, but instead each new experimental point shows an increase of ,7t2) [14]. F rom the calculations one expects a total alignment of about 8ti from the two bandcrossings (151, while the present data implies an alignment of at least about 106. With the recently observed transitions connecting
the SD band with normaldeformed
It is of course states in 143Eu [7], a better spin assignment have been made possible. our hope to see the connecting transitions also in other SD nuclei, partly in order to get better spin assignments of SD states, but also to address the most intriguing question of understanding
how the ordered
SD band mixes with the presumably
surrounding of normaldeformed states. Additional nuclear structure information been obtained
fully chaotic
about SD states in the A=150 region have [16] w h ere a rather consistent picture
through the analysis of Ragnarsson
emerges between observed
and calculated
bands through
comparisons
of relative align-
ments between different nuclei or different SD bands.
1.2 Electromagnetic properties The final proof that the observed y-ray energies really correspond
to SD shapes was
the measurement
moment
of the quadrupole
moment
[17].
The quadrupole
has now
been measured for several SD nuclei, and the variation in the A=150 region can be well understood by the high-N classification scheme discussed above. I.e. the occupation of high-N orbitals
not only has an important
influence on the Jc2)
moment of inertia,
also it is connected to large single-particle quadrupole moments. Through the study of intensity functions for r-rays as functions of rotational
but
frequency,
one has in particular learned about the process of decay out of the SD band. Already from the first calculations of the decay pattern [18] i t was clear that standard theory gave a too smooth depopulation curve. It now seems that a rather strong variation of the pairing deformation in the barrier region between SD and ND shapes is needed to understand the sudden depopulation pattern [19]. In fact, in selfconsistent pairing calculations the effective pairing gap in the barrier region is found to vary rather strongly with angular momentum [19]. However, several open questions still remain to be answered about how the decay precedes out of SD bands, as e g what is the influence of specific SD configurations on the decay out of the band[20]. An interesting observation [21] is that th e intensity curve looks quite different for lg3Hg than for most other nuclei in the Hg-region, see fig. 4. In ref. [22] this could be understood if the lowest SD bands in lg3Hg are connected with strong Ml-transitions. Since Ml
21c
S. Aberg I Superdeformed nuclei
Superdeformation in the Hg isotopes. I
: ._ ZP
,
.2 12 I
%h E- .I
.I
/I/ (f-f+ I
I
__I H ,o)
300450600750
300 450 500 750
300450500750 E
Y
300 450 600 750
300 450 600 750
300 450
wo 750
WW
Figure 4: Observed E2 r-ray intensities in SD bands in Hg-isotopes. The thin lines correspond to calculated r-ray intensities assuming no Ml-strength (dashed lines) and B(M1) % l.O& (solid lines). No decay back to normaldeformed states is included, but the calculated drop of r-ray EZ-intensity is because of Ml-transitions and converted E2- and Ml-transitions. (From [21] and [22].) transitions at low spins between strongly coupled SD bands have small y-ray energies they are strongly converted.
This results in an effective drop in r-ray intensity, although
the total intensity in the SD band may remain constant.
This suggested scenario could
explain not only the r-intensity function (see fig. 4) but would also provide an explanation why gating on one SD band (band three) gave transitions in the other band (band one), as well as why the linking transitions could not be seen. In the first experimental study [13] transitions seemed to go from band three to band one, but not from band one to band three, and the bands were suggested to be connected with El-transitions. Recent experiments
[23] show however that transitions occur in both directions,
i e what the
Ml-scenario would suggest. Further support for the Ml scenario is provided by a recent Oak Ridge experiment [24] w here an enhanced x-ray intensity was seen when gating on the lowest SD band as compared to gating on transitions between normaldeformed states. This is exactly what would be expected if the drop of intensity in SD band in lg3Hg is due to converted (Ml) e1ectrons.l One should notice that if the r-ray intensity is substantially larger than the calculated ‘After the Tokyo conference Sharpey-Schafer et a1[25].
the Ml-scenario
was finally confirmed
in a Eurogam
experiment
by J.F.
22c
S. iberg I Superdeformed nuclei
lower curves in fig. 4, no Ml-transitions
(or very weak Ml-transitions)
can take place. We
have thus obtained an estimated measure of Ml-matrix elements by using only the r-ray intensity curves! On the other hand an observed drop of intensity could either correspond to a depopulation of the SD band, or to strongly converted Ml-transitions. The latter may however not occur in the Dy-region since the drop of -y-ray intensity there occurs at much higher spins (around spin 25) where Ml’s have no chance to compete with stretched E2’s. If we adopt the Ml-scenario intensity
and try to determine
B(Ml)-values
from measured r-ray
functions
(see fig. 4), we find for the different SD bands that the even isotopes have decay curves which remind about weak or no Ml-transitions; the intensity drop is most certainly caused by a decay out of the SD band. For the odd Hg-isotopes it is found that the SD bands in “‘Hg and in lglHg correspond to weak B(Ml)-values, while the lowest bands in lg3Hg correspond to strong B(Ml)-values (B(M1) M 1~;). The behaviour of the second excited SD band in lglH may also indicate strong Ml-transitions. A simpleminded theoretical estimate can begmade for the B(Ml)-values for deformation aligned orbitals [22]. (R o t a t ion aligned orbitals are here not interesting are strongly decoupled, leading to a minor feeding of the unfavoured band.) high spins (I >> K) and asymptotic B(M1,
a1 = 1) M
quantum number ([Nn,A]
since they Assuming
n) we get
O.O21(R f 3.2)‘,~~%, neutrons O.O37(Q f 2.6)‘/.~$, protons
(1)
where the plus-signs refer to (s,) = C = R - A = l/2, and the minus-signs to C = -l/2. By studying a Nilsson diagram with single-particle energies plotted versus rotational frequency for SD nuclei in the Hg-region (fig. 5), we find that orbitals with expected large B(Ml)-values, say B(M1) > 0.5&, are the neutron orbitals [624]9/2, [512]5/2 and [505]11/2, and the proton orbital [514]9/2. These neutron orbitals are expected to be populated in N=113 (as lg3Hg) and 115 (and [505]11/2 in N=105) nuclei but not in the lowest SD bands in N=109 and 111 nuclei (as isgJglHg), supporting the above analysis of observed intensity curves. The proton orbital [514]9/2 is expected to be occupied by an odd particle in Z=83 and we thus predict a smooth y-ray intensity drop-off in Bi isotopes, presumably
starting
gives the strongest
B(Ml)-value
at an even higher frequency for this single-particle
greatest interest to get more detailed experimental
than in lg3Hg since our estimate
orbital (z 1.8,~:).
information
It is of course of
about magnetic
properties
of SD nuclei to see whether the extreme deformation causes changes e.g. of the magnetic polarisation (cf the contribution by Hamamoto [26]).
2. OCTUPOLE
VIBRATIONS3
In this section we shall use a simple model to describe octupole vibrations rotating SD and HD nuclei. Similar work has been presented in [28] and [29]. 3The work presented in this section was done in collaboration
with FL Nazmitdinov
[lti].
in non-
23~
S. Aberg f Superdeformednuclei
-6
*_w-----__
-7
--
r-
_
_ c B,= 0.465, p,=o.055,r=oe
0.2
0.0
0.4
0.6
h w (Me’/) Figure
5:
Calculated
shown vs the rotational with the important levels are indicated
neutron
and proton
cranked
Woods-Saxon
single-particle
energies
are
frequency
for SD lMHg. Quasiparticle diagrams are shown as inserts aligning trajectories labelled. The parity and signature (7r, cx) of individual in the following way: (F, o) = (+, l/2) - full lines; (+, -l/2) - short-dashed
lines; (-, l/2) -long-dashed
lines; (-,-l/2)
- dot-long-dashed
2.1 The model Consider a residual octupole-octupole harmonic oscillator:
interaction
w = T + VHo - f $
lines. (From 1131.)
added to the pure (non-rotating)
X&E;“+&‘)
(21
K-O
where T is the kinetic energy operator, vu0 = $2
[LQ: (3” -I- Y2) -I- t&2]
,
(3)
24c
S. berg
i.e. the axially symmetric
harmonic
are expressed in double-stretched
/ Superde&ormednuclei
oscillator potential,
coordinates
(2” = 2
and the octupole operators
. z, etc). The operators
as well as
the coupling constants,
result from requiring selfconsistency between shape and potential in the vibrations [30,31]. Before we discuss the solutions of eq. (2) let us discuss the deformation dependence of the free response of the four operators
i
N N
01 00 03 02
z(4zZ-CC2-y2) 2 (2.22 XYZ 2 (&?T” - - 3y2) 3z2 .- 3y2)
(6) If we write the unperturbed
particle-hole %h
=
%
energies created with these operators +&wz$ml-fLWl
as (7)
we see immediately that for example for K = 3 only two types of excitations are possible, namely corresponding to (m,,mL) = (0, I) and (0,3). The deformation dependence of these particle-hole
excitation
energies,
as well as for the corresponding
K = 0,l and 2 are shown in fig. 6. In particular
energies for
we note (see eq. 6) that the 3fiw excita-
tion in deformed nuclei splits up into two K = 0 and two K = 1 energies,
but only one
each for K = 2 and 3. We will come back to these high-lying energies later on.
2.2 Low-energy
solutions
Eq. (2) is now solved in the RPA. This can be done analytically for closed-shell configurations [31, 271. In fig. 7 the energy of the lowest-lying solutions at the 2:l deformation for K = 0,1,2 and 3 are shown versus the number of occupied closed shells, NF. At the 2:l deformation two different types of closed shells emerge. As can be seen in the single-particle diagram of fig. 8 odd values of NF correspond to a total of 4, 16, 40, . . .particles. These closed shells can be viewed as the sum of two equally sized spherical closed-shell nuclei, 2+2, 8+8, 20+20 etc, while the combinations of two neighbouring spherical closed-shell nuclei, 2+8, 8+20, 20+40, etc corresponds to the even NF-values 2, 4, 6, etc, see fig. 8. The viewing of SD closed-shell nuclei (in the HO) as consisting of two spherical nuclei placed side by side can be justified not only from fig. 8 and from the nice geometrical picture (see also [32]), but also by studying the fusion of two spherical nuclei. With diabatic conditions on single-particle levels one obtains for the fused system the 2:l configuration through Harvey’s prescription [33] (only quanta in the z-direction may change) together with the Pauli principle, see e.g. [34]. From the geometrical point
S. herg
nuclei
t S~erd~o~d
2%
Unpettnrbed Energies (KS” MeV) 200
’
I
I
I
I
I
1K=O
I I
0
t
0.2
1
0.4
I
I
t-
0.6
0.8
I
E
Figure 6: Unperturbed H.O. energies for the different octupole operators, lY’s~, are shown as functions of quadrupole deformation. Each state is denoted by the number of excited quanta in the z- and perpendicular directions, (m,, ml), see eq.(7). (From [27].) Energy
2:l
def.
7
Np
3
4
5
6
Z-N=
16
28
40
60
80
8
9
10
110
140
182
Figure 7: Low-lying octupole vibrational states in the 2:l deformed pure H.O. are shown versus the number of tilled shells, NF. The energy is expressed in units of Fw,. Note how even ATF-numberscorrespond to lowest energies for all K-modes.
26c
S. Aberg I Superdeformed nuclei
t -1.0
1:2
2:s
4
I -0.5
I:, i 0
05
b~_ w-w3 i3
ID
Figure 8: Single-particle spectrum for the axially symmetric pure H.O. potential.
The arrows
mark the deformations corresponding to integer ratios of the main axes (or oscillator frequencies). Thick solid and dashed lines have the quantum number (n~,n,) = (N,O) and (N - 1, l), respectively, where N is the main oscillator quantum number. (From [30].) of view it seems natural is more soft against lowest-lying
that the case corresponding
octupole
vibrational
to two unequal spheres (even ~VF)
vibrations.
Indeed, the RPA calculation shows that the mode with even NF is always lowest in energy, irrespective of
K-quantum number, see fig. 7. At the 3:l deformation one can distinguish
between
three different
types of closed
shells, where NF = 5,8,. 20+20+20,
. . correspond to three equal spherical magical numbers 8+8+8, . . , NF = 4,7,. . . correspond to two large spheres and one smali, i.e. 8+8+2, . . . , and finally NF = 3,6,. . . correspond to one large and two small spheres,
20+20-t-8, i.e. 8+2$2, 20+8+8, . . . (cf [32]). W e h ave obtained analytical solutions for all 3 x4 different combinations of closed shells and K-quantum numbers [27]. In fig. 9 the lowest-lying solutions are shown versus shell filling number for the K = 0 mode. We see a three-fold staggering with the asymmetric combinations lowest in energy. This is the case for the other K-modes as well.
2.3 Giant octupole resonances In fig. 10 all solutions, including the high-lying ones, are shown for the 2:l deformation.
S. Aberg I Superdeformed nuclei
Exicitation
27c
energy
2.0
3:l
def.
K=O
1.6
0.8 0.4
1
0.0
J
7
9
Shell Figure
11
13
number
9: Similar to fig. 7 but for the 3:l deformation
and only showing K = 0 states.
The shell
number NF = 3 corresponds to N = 2 = 12 particles and NF = 13 to N = 2 = 180 particles. The three different kinds of closed shells are symbolically illustrated by the combination of large and small spheres. def.
2:l Excitation
energy
6.0 5.0 4.0 3.0 2.0 1.0 0.0 5.0 4.0 3.0 2.0 1.0 0.0 0
2
4
6
8
IO
Shell number
Figure
12
14
0
2
4
6
8
IO
Shell number
10: Similar to fig. 7 but showing all solutions.
I2
14
16
S. Aberg I Superdeformed nuclei
28c
Cluadrupole splitting of giant resonances
I
140 -r
,
I
I
I
I
I
1
I
a
I
I
I
1
I
5 r” 120 9 ?5 % 100 E IJJ 806 ‘Z s -2 60w 40
J
I 0 0.0
0.2 0.1 Deformation,
0.3
I 0.4
E
Figure 11: The deformation dependence of the high-lying parts of the giant octupole resonance (2’ = 0), giant quadrupole resonance (2’ = 0) and the giant dipole resonance (T = 1). Solid lines mark K = 0,dashed K = 1, dotted K = 2 and dot-dashed K = 3. (From [27].) The staggering seen for the low-lying states (fig. 8) is not present for the high-lying, resonance
type of states.
Furthermore,
giant-
we see that for these solutions the energy scales
as A-‘i3 for NF > 4, i.e. for particle numbers larger than about 20. By this observation, which also applies to the 3:l (and 1:l) de f ormation, we may obtain a simple description of how the (isoscalar)
giant octupole resonance (GOR)
splits up in deformed nuclei [27], and
in fig. 11 the GOR which emerges from the 3fLw excitation energy mode, is compared to the giant dipole and giant quadrupole resonances in quadrupole deformed nuclei. While the two latter resonances split up in (L+ 1) components to one K-component,
the GOR
(i.e. 2 and 3), each corresponding
splits up in 6 components
instead
of the 4 one might
have expected. The explanation comes from the structure of the octupole operators. It is easily seen from eq (6) that th e unperturbed 3/iw energy is found to split up into two components both for K = 0 and for K = 1 (fig. 6). Our general knowledge of giant resonances indicates that one may get a fairly good understanding in the pure H 0 of how e.g. the centroid energy splits up in deformed nuclei. The description made above may therefore be quite applicable. For the low-lying solutions, on the other hand, nuclear structure plays a more important role and we expect quite strong deviations from the HO result in more realistic models. After the study of this analytically solvable model we therefore now turn to more realistic calculations.
S. Aberg I Superdeformed nuclei
3. SUPER-
AND
HYPERDEFORMED
PEARSHAPED
29c
NUCLEI
In this section we shall study the possible existence of static octupole deformed nuclei with approximate microscopic
axis ratia of 2:l (subsect.
3.1) and 3:l (subsect.
3.2).
In subect.
3.3 a
analysis of the Jc2) moment of inertia is performed.
3.1 Superdeformations In the upper part of fig. 12 the shell energy is plotted as a function of the neutron number at the 2:l deformation for two values of the octupole deformation parameter, ~3 = 0 and ss = 0.15, at the constant
rotational
frequency w,,~ = 0.06 &.
spin-orbit force the magic numbers at the 2:l deformation
Due to the
are of course changed compared
to the pure HO. Still we see how every second magic number, i’V = 32,64 and 116, are favoured by octupole (Ysc) deformations, while nuclei with magic numbers inbetween, N=44
and 86, are octupole
closed shell is connected
stable.
The result from the HO study that every second
to a lowered octupole vibrational
valid also in these more realistic calculations,
energy (see fig. 7) thus remains
and also at high spins.
By combining suitable proton and neutron numbers we see that it is difficult to find combinations where both the proton number and the neutron number favour octupole deformations (possible except for N = 2 = 32). S o we have to stick to nuclei where either the protons or the neutrons favour 2:l deformed pearshapes. A few selected potential-energy surfaces in the (E, cs)-plane are shown in fig. 13. Since the calculations are performed for high spins the pairing interaction is ignored. In the cranking prescription with es # 0 the only good quantum
number is simplex, S$J = s$, where S = II. R=(A), ll is the parity
’
~-0.6
Figure 12: The shell energy deduced from the cranked Nilsson model, expressed in units of LJ = 41A-‘13 MeV, is plotted vs the neutron number for 2:l (E = 0.6,&d = 0.06) and 3:l (8 = 0.9,~~ = 0.14) shapes at the fixed rotational frequency o = 0.06 $0. Solid lines are used for reflection symmetric shapes (ES = 0) and dashed lines for c3 = 0.15. (From [35].)
:
_~~~~~~~
j ,::t -0.2
0.4
0.6
0.8
0.2
0.4
0.6
cl.8
1.0
1.0
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1 .a
Ed%)
f&t%)
Figure 13: Potential-energy surfaces in the fs,ss)-plane v&d at IX = 40’ for 15sDy, lssJssEg and zwRn. A free minirnisation has implicitly been performed oi the +degree of freedom, while t-s is included according to ss = -0.47~(2 - c&s. (Prom f35j.f operator and R+(x) spin-parity
rotates the nucleus 180” around the x-axis, the rotation axis.
sequence . . . , 39-, 40+, 41-,
42+, . . . thus corresponds
sequence . . . , 39+, 40-, 41+, 42-, . . . corresponds
to a = -1.
The
to s = +1 while the
The energy surfaces shawn
in fig. 13 are all valid for I” = 40+, i.e. for s = 1. Our favourite superdeformed nucleus 152Dy is indeed seen to be quite soft in the octupole direction. Octupofe softness for 2:1 deformations is also found in the Hg-region, and ZWRn, which is certainly quite unstable against fission, is seen to have a static octupole deformation with E = 0.6 and us = 0.07.
3.2 ~yperdefor~a~ions It is difficult to find any SD nucleus that has a strong octupole deformation, and where the barrier in the es-direction exceeds say f MeV. But if we turn to hyperdeformed, 3:l deformed nuclei, that corresponds to c-values around 0.9, deep and rigid minima are indeed seen in IMEg as well as in 2wRn, see fig. 13. The shell energy diagram for 3:1 shapes based on the cranked Nilsson model (lower part of fig. 12) shows very strong deviations from the estimates made above within the pure HO (fig. 9), reflecting the fact that shell gaps are not so well defined in realistic calculations at the 3:l deformation, The strong favouring of octupole deformation at N w 82 (or 2 x 82) and N x Il.8 is responsible for the octupole minima in 19*Hg and 2ooRu. For these nuclei the competition with fission is very strong, and it seems extremely difficult experimentally to reach the BD octupole minimum. From rotating liquid-drop calculations we know that the highest angular momentum that can be reached before f&Son sets &I, is obtained in the mass
3ic
S. &erg / Superdeformed nuclei
0
6000 4000 2000 Angular momentum, I2
0
2000
6000
4000
Angular momentum, I*
Figure 14: The energy of lowest SD and HD configurations are compared for 15sDy and 146Gd and plotted versus the square of the angular momentum. region A=120-150 [36]. A combination of the minima at N x 62 and 2 % 82 (the shell energy diagram for protons looks similar to fig. 12) of the ~3 = 0.15 curve (lower part of fig. 12) thus seems to be the most promising case for a nucleus with a hyperdeformed pear shape. In fig. 14 the total energy for the lowest HD and SD configurations
are shown as
functions of angular momentum
(squared) for 146Gd and *52Dy. In both cases the hyperdeformation corresponds to E x 0.93, cs M 0.13, while the (reflection symmetric) superdeformation is somewhat larger for ls2Dy (E M 0.58) than for 146Gd (E w 0.52). The shape of the hyperdeformed
pearshaped nuclei ls2Dy and 146Gd is shown in fig. 15.
Through 7 - 7 correlation measurements a hyperdeformed band(s) was recently proposed to be seen by Galindo-Uribarri et al at Chalk R iver [37]. In the present calculation the ND minimum of is2Dy is found to become yrast at about spin 90 while the crossing with the SD band in 146Gd occurs “already” at spin 80. {Notice that the calculation of the HD in 152Dy presented in 1371did not include the important octupole degree of freedom.) Fig. 16 exhibits potenti~-energy O+, 40f, 60+ and 40-.
-
Figure 15: Eg = -0.056).
surfaces in the f.s,s3)-plane
valid for ‘*Gd
at 1” =
The SD minimum becomes yrast at around spin 50, and, as was
_.-_-_ irli
Approximate shape of HD 1s2Dy or 146Gd (c = 0.93, 63 = 0.s2,
&4
=
0.
13,
32c
S. Aberg f Superdefiirmed rurclei
0.4
0.6
0.8
1.0
Figure 16: Similar to fig. 13 but for selected spin-p~ity combinations of ‘Wd. O+, 40+, 60+ and 40-.
The SD minimum becomes yrast at around spin 50, and, as was
seen in fig. 14, the HD configuration eg. 40-)
becomes yrast at I z 8Ofi. The s = -1 sequence (as
is calculated to come almost 1 MeV higher in excitation energy than the lowest
s = 1 band.
This feature can be understood
fig. 17, valid at the relevant hyperdeformation. survive to the highest rotational frequencies.
from the single-particle
diagrams shown in
The 0.5-1.0 MeV gaps at Z=64 and N=82
This suggests 146Gd to be one especially favoured candidate for HD, with one band pushed down in energy by almost one MeV, reminding about the SD band in lszDy. The barrier between the “two” octupole deformed minima (es = f0.12) at c M 0.9 is found to be 2.5-5.0 MeV {for I” = 60s) depending on what path is chosen in the (E,E~) space. Due to the high spins and low level-density the pairing correlations are expected to be weak implying a large mass parameter. The parity splitting, determined by tunneIing between the “two” octupole minima, is thus expected to be very small, i.e. we expect e.g. the spin-parity
sequence . . . , 59-,
60+, 61-,
62+,
. . . to follow a smooth rotational band. The competition with fission certainly makes the population of the HD discrete states very difficult. In the present calculations we unfortunately cannot calculate the full fission barrier due to an insufficient shape parametrization. The feeding of the HD minimum is, however, in a similar way as for SD 15aDy, expected to be favoured by one state being pushed down in energy. But also the calculated very low-lying GDR component with a peak energy at about E,,, = 4 - 5 MeV at this extreme quadrupole deformation, is expected to help the population of the low-lying HD band. By assuming that these features are able to overcome the competition with fission we shall discuss the possibility to remain in the HD band. The energy curves between the SD and HD minima for I” = 60” and 80f are shown in
S. &erg I Superd&med nuclei
33c
PROTONS
w/w* NEUTRONS
5.8
Figure 17: Single-particle levels for protons (upper part) and neutrons (lower part) at the hyperdeformation E = 0.93, Es = 0.12, ~4 = 0.13, cs = -0.056, as obtained from the cranked Nilsson potential. Solid lines indicate simplex s = i and dashed lines s = 4. The two lowest “N=7” proton orbits are indicated. Since parity is mixed the given high-N notation is however quite approximate. fig. 18. Compared to the barrier between SD and normal-deformed states (see e.g. [38]) the barrier between HD and SD states is certainly smaller. But, as was discussed above, we expect a large mass parameter. Furthermore, the decay from the HD band occurs to SD excited states which have a much lower calculated level density than normal deformed states [39] to which the SD states decay. Also favouring the decay within the HD band is the expected very large quadrupole moment, Qo M 35 eb, giving rise to ~(~2)-tr~sitions with about 10 kWu. These three factors, large mass parameter, low SD level density aad large B(E2)-values, are all very much expected to favour the decay within a low-lying hyperdeformed rotational band.
34c
S. Aberg / Superdeformed nuclei
Energy (MeV)
T----t
Figure 18: Energy curves between SD and HD minima in 146Gd at 1” = 60+ and I” = 80+. The energy is explicitly minimized with respect to es and ~4 while ~5 is taken as a given function of B and ~3. The SD minimum corresponds to 6s = 0 while the HD minimum is octupole deformed with ~3 = 0.12. The excitation energy &D - &D changes from 8 MeV to 1 MeV, and the HD barrier changes from 1.5 MeV to 3 MeV as the angular momentum changes from 60 to 80.
255 0*4 OJ 0.6 0#7 0,s 0,9 E(4 64’Es)
1
3.3 Moments of inertia As discussed in subsection 1.1 the high-N orbits play an important role in determining the ,7(2) moment of inertia of SD bands. To investigate the g(2) properties of HD bands we show in fig. 19 the alignment, {j5), and the contribution to the J(2)-moment of inertia from the two lowest-lying proton N=7 orbitals at four different deformations: Norma1 deformation (& = 0.30,~s = 0), superdeformation (s = 0.57,~ = 0) and two hyperdeformations (E = 0.93 with ~~ = 0 and ~3 = 0.12). The contribution to J@) from orbital Y is calculated as
(f-3) where (hNila*on- wj, ) Iw;v) = e; Jw;~)
(9)
By defining gzi,= in this way the total Jc2) is simply obtained by adding the contributions from all occupied states $2) = c $$‘. (10) YE0C.z This way of calculating J’$& differs from the previous way [S] where the derivatives
do, du
I
8
“‘I”
I
+__---
6-
e---_‘.__.._
-t-
I I
-4
I
1 -___----
1
I
I
I
I
. . . . . . .-?-
___
& =0.93&,
=o
-*m... I I
& =0.93E, I I
-0.12 , I
t
I
50
40 u, J wb..v 30
(MeV-‘) 20 +rc-----
10 n
O.to2
0
%J
043
0.;4
;
0,
O.b2 w,J~~
*.b4
o.bs
Figure 19: Angular momentum component along rotation axis, < j, > (upper diagrams), and $21 orb,v moment of inertia (lower diagrams) for the two lowest proton N=7 configurations, are shown versus rotational frequency for the four deformations specified in the upper left figure. The two figures on the left-hand side are valid for the lowest x7 configuration, and the two figures on the right-hand side are valid for the next lowest r7 configuration. In the octupole deformed case the next lowest configuration changes smoothly with w,,t from N M 7 to N w 6 (see fig.17) explaining the rather small < j, >-values. nume~cally that gzj,,
were calculated as differences.
One advantage with the present approach is
now is a local quantity which is more easy to follow vs w,.+
It also makes
possible a microscopic other orbitals,
analysis of the origin of changes in &&, from interactions with see refs. [40, 411. A s is usually done in the cranked Nilsson model, all
moments of inertia and rotational frequencies shown below are renormalized to obtain rigid-body rotation on the average [42]. At normal deformations the angular momentum vector of a nucleon in a high-N orbital quickly aligns, and obtains a value close to the maximum values 7.5 and 6.5, respectively for the two lowest-lying N=7 orbitals. The corresponding J@) contributions decrease drastically from the huge value of about 50 R2MeV-r to values below 5 at rotational frequencies, w,.,~ 2 250 keV. Also at the 2:l deformation
(E = 0.58) a rather drastic spin alignment occurs accom-
36c
Moment
S. Aberg I Superd@ormed nuclei
of
Ioertla
2oor-----t 80
‘&Gd
-
lS2DY
-J(l)
hyperdeformed
Figure 20: Microscopically calculated J’(l) and .7c2) moments of inertia at the HD 146Gd and 15aDy. The rigid-body values are shown for comparison, panied by a decrease in the Jt_“b’ moment of inertia. different behaviour
of ,$,!i
for the lowest-lying
analysis of SD configurations. alignment J$i-values
As discussed in subsection
1.1 the
high-N orbits provide a good tool in the
If we now turn to the HD cases shown in fig. 19 the spin
is seen to occur much smoother than for the smaller deformations, and the are consequently smaller. But in particular we note that the orbit-specific
variation of 322
with w,.,~ does not seem to occur at the hyperdeformation.
The disap-
pearance of the high-N or high-j specific characters at very large deformations is due to the larger mixing between the high-j “intruder” orbitals and deformation aligned orbitals. We thus expect smaller differences of Jt2) between different HD bands, and in particular the high-N character, important at SD shapes, seems to be of less importance. In fig. 20 the J(2l- moment of inertia of HD 152Dy and r4’Gd are shown vs w,,t. As could be expected from the discussion above it is seen to be rather independent of the rotational
frequency.
moment of inertia.
It is furthermore The Jt2)
tion, s = 1, but is more or less identical J(2)-value
extracted
very similar to J(i)
as well as to the rigid-body
moment of inertia is shown for the lowest-energy to the first excited s = -1
from the +y- y coincidence
experiment
configura-
configuration.
The
by the Chalk River group
agrees very well with the calculated values, and the observed [37], $22 M 130tL2MeV-‘, ridge in the 7 - 7 plot was suggested to be related to HD is2Dy. The observed rotational frequency interval 0.45 < wrof < 0.75~eV, would however correspond to extremely large spin values, 78 < I < 98fi..
4. SUMMARY Several properties of SD nuclei are now rather well understood, such as the underlying shell structure of observed mass regions, the high-N scheme, some hints about specific orbitals around the Fermi surface (e g from band-crossings), etc. Still several problems remain to be solved, like the continued rise of Jr(2) in the Hg-region, pairing properties
S. Aberg I Superdeformed nuclei
37c
of SD nuclei, etc, and also if new predicted regions of SD nuclei really exist. A problem of specific interest is the decay out of SD bands. We showed that an oberved decrease in the T-ray intensity in some cases could be due to strong interband Ml-transitions, and some neutron and proton orbitals in SD nuclei around lg2Hg were discussed as good candidates for giving rise to sufficiently strong Ml-transition matrix elements. The recent measurement of Ml-transitions between SD bands in lg3Hg [25] confirmed the predictions, and in fact implies that
we now face a new era of SD spctroscopy
questions e.g. about magnetic Octupole
vibrational
where interesting
properties in SD nuclei may be answered.
states built on SD and HD shapes were described
solvable model (HO + O-O-force).
in a simple
Th e 1ow-lying SD states can be devided into two classes
of closed shell, corresponding to two spherical nuclei put beside each other of equal or unequal size. Octupole vibrations built on superdeformed states of the latter type was found to come lowest in energy, and for K = 0 the same effect was seen in NilssonStrutinsky calculation in terms of a lower shell energy for static reflection asymmetric shapes. The simple model gave similar types of structure types of asymmetric
shapes.
In the realistic calculation
at HD but now with two different such a three-periodic
variation of
the shell energy could however not be seen, since the relatively small HO gaps at the 3:l deformation are replaced by a more complicated structure in realistic potentials. The high-lying model.
(3L)
oc t u p o1e vibrational
states are nicely described
The energies have a simple A- ‘I3 dependence
by the simple
and split up in 6 components
due
to the coupling with the quadrupole deformed field. The total deformation splitting of the GOR is found to be considerably larger than for the GDR or GQR. Octupole correlations were found to be very large at hyperdeformed shapes, where deep stable minima in the potential-energy surfaces for several A M 150 nuclei. The nucleus ‘ZGdss was found to be a particularly favourable case for a hyperdeformed pearshaped nucleus. Both proton number Z=64 and neutron number N=82 are connected with gaps in the single-particle situation
energy diagram of 700-1000
thus reminds about what is calculated
being pushed down in energy compared band in laGd competition
keV at all rotational
frequencies.
The
for SD ls2Dy and results in one band
to the first excited band.
The hyperdeformed
is calculated
to become yrast at 1, M 8OtL, and in the population stage the to fission is expected to be very serious. The calculated &value is however
quite uncertain, in particular due to uncertainties in the macroscopic model. For example, the used liquid-drop model does not account for nuclear finite-range interactions that is expected to lower 1,. A lowering of I, is also expected if the shape parametrization is improved, e.g. by freely including the ts and cs degrees of freedom. We conclude that several properties
of superdeformed
nuclei are now rather well un-
derstood. But we have also pointed out that many questions remain to be answered, and that new interesting features of nuclei with exotic shapes are to be revealed. We thus eagerly await result from the new generation
of r-ray detecter
systems.
S. Aberg 1 Superdeformed nuclei
38c
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]
S. Aberg, H. Fl ocard and W. Nazarewicz, Annu. Rev. Nucl. Part. Sci. 40 (1990) 439. R.V.F. Jansses and T.L. Khoo, Annu. Rev. Nucl. Part. Sci. 41 (1991) 321. R. Bengtsson et al, Phys. Lett. 57B (1975) 301. K. Neergtd and V.V. Pashkevich, Phys. Lett. 59B (1975) 218. P.J. Twin et al Phys. Rev. Lett. 57 (1986) 811. S. Aberg, Phys. Ser. 25 (1982) 23. A. Atq et al, subm. to Phys. Rev. Lett. T. Bengtsson, I. Ragnarsson and S. Aberg, Phys. Lett. 208B (1988) 39. P.J. Twin, in M.S. Hussein et al, editors, Int. Nucl. Whys. Conf., Sao Paulo, World Scientific, 1990, p.373. I. Ragnarsson, Proc Workshop on the Science of Intense Radioactive Ion Beams, Los Alamos (1990) p. 199. G. Hebbinghaus et ai, Phys. Lett. 240B (1990) 311. W. Nazarewicz, R. Wyss and A. Johnson, Phys. Lett. 225B (1989) 208. M.A. Riley et al, Nucl. Phys. A512 (1990) 178. M. Carpenter et al, these proceedings. T. Bengtsson, private comm. 1990. I. Ragnarsson, these proceedings. M.A. Bentley et al, Phys. Rev. Lett. 59 (1987) 2141. I. Ragnarsson and S. Aberg, Phys. Lett. 180B (1986) 191. Y.R. Shimizu et al, Phys. Lett. B274 (1992) 253. B.R. Mottelson, these proceedings. M.A. Riley, priv. comm. P. Semmes, I. Ragnarsson and S. Aberg, Phys. Rev. Lett. 68 (1992) 460. P. Failon et ai, preprint 1992, subm. to Phys. Rev. Lett. D. Cullen et al, preprint 1992, to be publ. J.F. Sharpey-Schafer et ai, to be publ. I. Hamamoto, these proceedings. R. Nazmitdinov and S. Aberg, Phys. Lett. B289 (1992) 238. T. Nakatsukasa, S. Mizutori and K. Matsuyanagi, Prog. Theor. Phys. 87 (1992) 607. S. Mizutori, these proceedings. Aa. Bohr and B.R. Mottelson, Nuclear Structure, Vol II, New York, Benjamin (1975). H. Sakamoto and R. Kishimoto, Nucl. Phys. A501 (1989) 205. W. Nazarewicz and J. Dobaczewski, Phys. Rev. Lett. 68 (1992) 154. M. Harvey. Proc. 2nd Int. Conf. on Clustering Phenomena in Nuclei, College Park 1975, USDERA report ORO-4856-26, p. 549. S. Aberg, in: Nuclear structure and heavy-ion reaction dynamics (Notre Dame 1990), Inst. Phys. Conf. Ser. No 109, Ch. 6, ed. R.R. Betts and J.J. Kolata (1991) p. 143. J. Hijller and S. Aberg, Z. Phys. 336 (1990) 363. S. Cohen, F. Plasil and W.J. Swiatecki, Ann. Phys. 82 (1974) 557. A. Galindo-Uribari et al, to be publ. T. Bengtsson et ai, Phys. Scripta 24 (1981) 200. S. Aberg, Nucl. Phys. A477 (1988) 18. J.-y. Zhang and S. Aberg, Nucl. Phys. A390 (1982) 319. S. Aberg, to be publ. G. Andersson et al, Nucl. Phys. A268 (1976) 205.
Nuclear Physics AS57 (1993) 17038~ North-Holland, Amsterdam
Superdeformed
Nuclei1
Sven Aberg Department
of Mathematical
Physics,
Lund Institute
of Technology,
P 0 Box 118,
S-221 00 Lund, Sweden
Abstract The structure of superdeformed (SD) nuclei is discussed, and some open questions are raised. Octupole vibrations built on SD and hyperdeformed (HD) shapes are described in an analytically solvable model, and structure effects on low-lying vibrational states, as well as the deformation cranked Nilsson-Strutinsky
splitting
of the giant resonance
calculations
states
are discussed.
From
it is predicted that HD nuclei in the 15sDy region
have a strong, static octupole deformation,
and the structure
The nucleus ‘&Gd is found to be a particularly
of such nuclei is discussed.
favoured candidate
for a hyperdeformed
pearshape.
1. INTRODUCTION The last few years the study of superdeformed nuclei has evolved to a broad research area by itself. In this paper I shall first review some of the present knowledge, and then discuss the role of the octupole degree of freedom at extreme deformations. A more complete discussion of SD nuclei can be found in the review papers [l, 21 as well as in several of the contributions The existence
to this volume.
of superdeformed
high-spin nuclei was predicted
fore the discovery [5], and in fig. 1 a large-scale
calculation
[3, 41 several years be-
from 1982 is summarized
[6].
For each (even-even)
nucleus between $Seaa and ‘A:Pois4 the predicted size of (super) deformation is shown together with the angular momentum value at which it is calculated to become yrast. We particularly note very favourable (i.e. low spin-value and large deformation) SD regions around lg2Hg, 15’Dy and, at somewhat smaller deformations, around 13*Ce. Several SD nuclei have later on been discovered in exactly these mass regions, nicely confirming the predictions.
This agreement
shows that we have a rather good understanding
between experiment
and theory
of how the nuclear shell structure
or
shell energy varies with neutron and proton numbers, as well as with angular momentum and quadrupole deformation. The SD region with largest deformations (similar to those of low-spin fission isomers) around 152Dy has been extended by the recently discovered SD band in 143E~ [7]. The calculations give good hope in finding the “connecting” SD nuclei like europium isotopes with N=81-84. Even more exciting is maybe the predicted ‘Work supported
by the Swedish
Natural
Science Research
Council
03759474/93/$06.00 0 1993 - Elsevier Science Publishers B.V.
All rights reserved.
18~
S. Aberg 1 Superdeformed nuclei
Figure 1: Part of the nuclide chart showing the lowest spin value, I,, for which even-even nuclei are calculated to obtain a large deformation, E > 0.35. 1, is easily obtained from the digit, as e.g. 6 means 1, = 65 etc. The corresponding deformation can corresponds to 0.35 5 s < 0.45, The calculation was performed
shape of the “superdeformation” is denoted by a letter and the be read out from the inserted plan over the (&,7)-plane. E.g. a b to 0.45 5 E < 0.55 and c to 0.55 5 E < 0.65, all with 171< 5’. with the MO potential neglecting pairing, and ~4 was included
only in the macroscopic contribution to the potential energy[b]. regions of new, yet not discovered, SD nuclei around 78Se, “Ru
and lssRu, see fig. 1.
1 .l Energy sequences The main part of our present knowledge about the structure of SD nuclei comes from measured r-ray sequences. By plotting the inverse of the difference between two consecutive r-ray energies (multiplied by four), giving the ,7(‘) moment of inertia, versus the r-ray energy, i.e. twice the rotational
frequency, it has been found that different nuclei
and different SD bands in one and the same nucleus may show different patterns. In fig. 2 we show the SD bands of the three isotones r5’Gd, “‘Tb and “‘Dy. The different g(2) behaviour has been explained [8] as being due to a different number of high-N orbitals occupied, ~6’, ~6~ and ~6~~ respectively, i.e. 2, 3 or 4 neutrons in the lowest N=6 orbitals (emerging from i13/2). This high-N scheme has been found to give a consistent picture of all observed SD bands in the A=150 region. In the A=190 region on the other hand, Jc2) is found to increase vs w,,~ in a quite similar way for each observed SD band [2]. In this context it is interesting to return to the yet unobserved region of SD nuclei around “Ru, in which nuclei are predicted to become yrast at angular momentum I z 35, see fig. 1. In fig. 3 calculated single-particle energies are shown versus rotational frequency at the appropriate
quadrupole
deformation.
The high-N orbitals
[550 l/2] and [541 3/2]
S. Aberg I Superdeformed nuclei
0.4
0.5
0.6
0.7
0.4
0.5
Frequency Figure
2:
the three
Measured N=86
and calculated
isotones
[12] Jc2)
150Gd, lslTb
1 0.7
0.6
hw moments
and lszDy.
19c
1 0.4
s 0.5
8 0.6
(MeV) of inertia
The neutron
for superdeformed
configuration
while the proton
configurations
are assigned
as 7r6’, n63 and ~6~, respectively.
are close
Fermi
for protons
as well as for neutrons
to the
energy
and a similar behaviour expected.
as in the “‘Dy
bands
is assigned (From
in
as v7l
[9].)
for nuclei around s’Ru, moment of inertia is thus
region of the Jc2)
However, it is now protons and neutrons in the same N=5 orbitals rather than
the proton N=6 and neutron interesting
1 0.7
N=7 orbitals
which are active.
effects coming from residual neutron-proton
From studies of the observed J(‘)
behaviour
This structure
may show
interactions.
it has been suggested that unpaired as
well as paired bandcrossings occur in SD nuclei. For example, in 146J47Gd there are many reasons to believe that the observed “bump” in J@) is related to an unpaired band crossing between the two specific neutron orbitals [651 l/2] and [642 5/2] [ll]. The very
5.2-
4.s:~$~i,~+-
ew$.___;__
‘0.
4.6-
*_______
4.4-
._______*..__.._*:=;;==~*::~~r~~~:r~rrrr+-------+‘-----_,>____8~0~-g~ .-. -o=====?~+ -+-+-___+
l _______
._______
0 ._--_
--
O..i====
-----
$lr-__T
9;=;l;h&~E:~z_
9 -.-__
* _
4.2-
w/w, Figure
3: Neutron
single-particle
energies
vs rotational
frequency
at the fixed superdeformation,
E = 0.52, ~4 = 0.05. Note the high-N orbital [550 l/2] just below the Fermi surface for N=44.
(From [lo].)
2oc large J(‘)
S. Aberg I Superdeformed nuclei values at the lowest frequencies
seen in 14’Gd and lsoGd have, on the other
hand, been explained [12] as being due to a paired bandcrossing, namely between a SD g-band and a SD s-band built by the alignment of a pair of neutrons in the N=6 orbital. Also the increase of Jr(‘) versus w,,~, seen for more or less all SD bands in the A=190 region, has been explained [13] as coming from paired bandcrossings, ~7~ and 7r72 (i.e. the SD AB and ab crossings). It is the increasing part of the expected J(‘) bump, caused by these two subsequent bandcrossings, one then is supposed to see. A problem is however that the expected turn over and decrease of Jc2) has not been seen, but instead each new experimental point shows an increase of ,7t2) [14]. F rom the calculations one expects a total alignment of about 8ti from the two bandcrossings (151, while the present data implies an alignment of at least about 106. With the recently observed transitions connecting
the SD band with normaldeformed
It is of course states in 143Eu [7], a better spin assignment have been made possible. our hope to see the connecting transitions also in other SD nuclei, partly in order to get better spin assignments of SD states, but also to address the most intriguing question of understanding
how the ordered
SD band mixes with the presumably
surrounding of normaldeformed states. Additional nuclear structure information been obtained
fully chaotic
about SD states in the A=150 region have [16] w h ere a rather consistent picture
through the analysis of Ragnarsson
emerges between observed
and calculated
bands through
comparisons
of relative align-
ments between different nuclei or different SD bands.
1.2 Electromagnetic properties The final proof that the observed y-ray energies really correspond
to SD shapes was
the measurement
moment
of the quadrupole
moment
[17].
The quadrupole
has now
been measured for several SD nuclei, and the variation in the A=150 region can be well understood by the high-N classification scheme discussed above. I.e. the occupation of high-N orbitals
not only has an important
influence on the Jc2)
moment of inertia,
also it is connected to large single-particle quadrupole moments. Through the study of intensity functions for r-rays as functions of rotational
but
frequency,
one has in particular learned about the process of decay out of the SD band. Already from the first calculations of the decay pattern [18] i t was clear that standard theory gave a too smooth depopulation curve. It now seems that a rather strong variation of the pairing deformation in the barrier region between SD and ND shapes is needed to understand the sudden depopulation pattern [19]. In fact, in selfconsistent pairing calculations the effective pairing gap in the barrier region is found to vary rather strongly with angular momentum [19]. However, several open questions still remain to be answered about how the decay precedes out of SD bands, as e g what is the influence of specific SD configurations on the decay out of the band[20]. An interesting observation [21] is that th e intensity curve looks quite different for lg3Hg than for most other nuclei in the Hg-region, see fig. 4. In ref. [22] this could be understood if the lowest SD bands in lg3Hg are connected with strong Ml-transitions. Since Ml
21c
S. Aberg I Superdeformed nuclei
Superdeformation in the Hg isotopes. I
: ._ ZP
,
.2 12 I
%h E- .I
.I
/I/ (f-f+ I
I
__I H ,o)
300450600750
300 450 500 750
300450500750 E
Y
300 450 600 750
300 450 600 750
300 450
wo 750
WW
Figure 4: Observed E2 r-ray intensities in SD bands in Hg-isotopes. The thin lines correspond to calculated r-ray intensities assuming no Ml-strength (dashed lines) and B(M1) % l.O& (solid lines). No decay back to normaldeformed states is included, but the calculated drop of r-ray EZ-intensity is because of Ml-transitions and converted E2- and Ml-transitions. (From [21] and [22].) transitions at low spins between strongly coupled SD bands have small y-ray energies they are strongly converted.
This results in an effective drop in r-ray intensity, although
the total intensity in the SD band may remain constant.
This suggested scenario could
explain not only the r-intensity function (see fig. 4) but would also provide an explanation why gating on one SD band (band three) gave transitions in the other band (band one), as well as why the linking transitions could not be seen. In the first experimental study [13] transitions seemed to go from band three to band one, but not from band one to band three, and the bands were suggested to be connected with El-transitions. Recent experiments
[23] show however that transitions occur in both directions,
i e what the
Ml-scenario would suggest. Further support for the Ml scenario is provided by a recent Oak Ridge experiment [24] w here an enhanced x-ray intensity was seen when gating on the lowest SD band as compared to gating on transitions between normaldeformed states. This is exactly what would be expected if the drop of intensity in SD band in lg3Hg is due to converted (Ml) e1ectrons.l One should notice that if the r-ray intensity is substantially larger than the calculated ‘After the Tokyo conference Sharpey-Schafer et a1[25].
the Ml-scenario
was finally confirmed
in a Eurogam
experiment
by J.F.
22c
S. iberg I Superdeformed nuclei
lower curves in fig. 4, no Ml-transitions
(or very weak Ml-transitions)
can take place. We
have thus obtained an estimated measure of Ml-matrix elements by using only the r-ray intensity curves! On the other hand an observed drop of intensity could either correspond to a depopulation of the SD band, or to strongly converted Ml-transitions. The latter may however not occur in the Dy-region since the drop of -y-ray intensity there occurs at much higher spins (around spin 25) where Ml’s have no chance to compete with stretched E2’s. If we adopt the Ml-scenario intensity
and try to determine
B(Ml)-values
from measured r-ray
functions
(see fig. 4), we find for the different SD bands that the even isotopes have decay curves which remind about weak or no Ml-transitions; the intensity drop is most certainly caused by a decay out of the SD band. For the odd Hg-isotopes it is found that the SD bands in “‘Hg and in lglHg correspond to weak B(Ml)-values, while the lowest bands in lg3Hg correspond to strong B(Ml)-values (B(M1) M 1~;). The behaviour of the second excited SD band in lglH may also indicate strong Ml-transitions. A simpleminded theoretical estimate can begmade for the B(Ml)-values for deformation aligned orbitals [22]. (R o t a t ion aligned orbitals are here not interesting are strongly decoupled, leading to a minor feeding of the unfavoured band.) high spins (I >> K) and asymptotic B(M1,
a1 = 1) M
quantum number ([Nn,A]
since they Assuming
n) we get
O.O21(R f 3.2)‘,~~%, neutrons O.O37(Q f 2.6)‘/.~$, protons
(1)
where the plus-signs refer to (s,) = C = R - A = l/2, and the minus-signs to C = -l/2. By studying a Nilsson diagram with single-particle energies plotted versus rotational frequency for SD nuclei in the Hg-region (fig. 5), we find that orbitals with expected large B(Ml)-values, say B(M1) > 0.5&, are the neutron orbitals [624]9/2, [512]5/2 and [505]11/2, and the proton orbital [514]9/2. These neutron orbitals are expected to be populated in N=113 (as lg3Hg) and 115 (and [505]11/2 in N=105) nuclei but not in the lowest SD bands in N=109 and 111 nuclei (as isgJglHg), supporting the above analysis of observed intensity curves. The proton orbital [514]9/2 is expected to be occupied by an odd particle in Z=83 and we thus predict a smooth y-ray intensity drop-off in Bi isotopes, presumably
starting
gives the strongest
B(Ml)-value
at an even higher frequency for this single-particle
greatest interest to get more detailed experimental
than in lg3Hg since our estimate
orbital (z 1.8,~:).
information
It is of course of
about magnetic
properties
of SD nuclei to see whether the extreme deformation causes changes e.g. of the magnetic polarisation (cf the contribution by Hamamoto [26]).
2. OCTUPOLE
VIBRATIONS3
In this section we shall use a simple model to describe octupole vibrations rotating SD and HD nuclei. Similar work has been presented in [28] and [29]. 3The work presented in this section was done in collaboration
with FL Nazmitdinov
[lti].
in non-
23~
S. Aberg f Superdeformednuclei
-6
*_w-----__
-7
--
r-
_
_ c B,= 0.465, p,=o.055,r=oe
0.2
0.0
0.4
0.6
h w (Me’/) Figure
5:
Calculated
shown vs the rotational with the important levels are indicated
neutron
and proton
cranked
Woods-Saxon
single-particle
energies
are
frequency
for SD lMHg. Quasiparticle diagrams are shown as inserts aligning trajectories labelled. The parity and signature (7r, cx) of individual in the following way: (F, o) = (+, l/2) - full lines; (+, -l/2) - short-dashed
lines; (-, l/2) -long-dashed
lines; (-,-l/2)
- dot-long-dashed
2.1 The model Consider a residual octupole-octupole harmonic oscillator:
interaction
w = T + VHo - f $
lines. (From 1131.)
added to the pure (non-rotating)
X&E;“+&‘)
(21
K-O
where T is the kinetic energy operator, vu0 = $2
[LQ: (3” -I- Y2) -I- t&2]
,
(3)
24c
S. berg
i.e. the axially symmetric
harmonic
are expressed in double-stretched
/ Superde&ormednuclei
oscillator potential,
coordinates
(2” = 2
and the octupole operators
. z, etc). The operators
as well as
the coupling constants,
result from requiring selfconsistency between shape and potential in the vibrations [30,31]. Before we discuss the solutions of eq. (2) let us discuss the deformation dependence of the free response of the four operators
i
N N
01 00 03 02
z(4zZ-CC2-y2) 2 (2.22 XYZ 2 (&?T” - - 3y2) 3z2 .- 3y2)
(6) If we write the unperturbed
particle-hole %h
=
%
energies created with these operators +&wz$ml-fLWl
as (7)
we see immediately that for example for K = 3 only two types of excitations are possible, namely corresponding to (m,,mL) = (0, I) and (0,3). The deformation dependence of these particle-hole
excitation
energies,
as well as for the corresponding
K = 0,l and 2 are shown in fig. 6. In particular
energies for
we note (see eq. 6) that the 3fiw excita-
tion in deformed nuclei splits up into two K = 0 and two K = 1 energies,
but only one
each for K = 2 and 3. We will come back to these high-lying energies later on.
2.2 Low-energy
solutions
Eq. (2) is now solved in the RPA. This can be done analytically for closed-shell configurations [31, 271. In fig. 7 the energy of the lowest-lying solutions at the 2:l deformation for K = 0,1,2 and 3 are shown versus the number of occupied closed shells, NF. At the 2:l deformation two different types of closed shells emerge. As can be seen in the single-particle diagram of fig. 8 odd values of NF correspond to a total of 4, 16, 40, . . .particles. These closed shells can be viewed as the sum of two equally sized spherical closed-shell nuclei, 2+2, 8+8, 20+20 etc, while the combinations of two neighbouring spherical closed-shell nuclei, 2+8, 8+20, 20+40, etc corresponds to the even NF-values 2, 4, 6, etc, see fig. 8. The viewing of SD closed-shell nuclei (in the HO) as consisting of two spherical nuclei placed side by side can be justified not only from fig. 8 and from the nice geometrical picture (see also [32]), but also by studying the fusion of two spherical nuclei. With diabatic conditions on single-particle levels one obtains for the fused system the 2:l configuration through Harvey’s prescription [33] (only quanta in the z-direction may change) together with the Pauli principle, see e.g. [34]. From the geometrical point
S. herg
nuclei
t S~erd~o~d
2%
Unpettnrbed Energies (KS” MeV) 200
’
I
I
I
I
I
1K=O
I I
0
t
0.2
1
0.4
I
I
t-
0.6
0.8
I
E
Figure 6: Unperturbed H.O. energies for the different octupole operators, lY’s~, are shown as functions of quadrupole deformation. Each state is denoted by the number of excited quanta in the z- and perpendicular directions, (m,, ml), see eq.(7). (From [27].) Energy
2:l
def.
7
Np
3
4
5
6
Z-N=
16
28
40
60
80
8
9
10
110
140
182
Figure 7: Low-lying octupole vibrational states in the 2:l deformed pure H.O. are shown versus the number of tilled shells, NF. The energy is expressed in units of Fw,. Note how even ATF-numberscorrespond to lowest energies for all K-modes.
26c
S. Aberg I Superdeformed nuclei
t -1.0
1:2
2:s
4
I -0.5
I:, i 0
05
b~_ w-w3 i3
ID
Figure 8: Single-particle spectrum for the axially symmetric pure H.O. potential.
The arrows
mark the deformations corresponding to integer ratios of the main axes (or oscillator frequencies). Thick solid and dashed lines have the quantum number (n~,n,) = (N,O) and (N - 1, l), respectively, where N is the main oscillator quantum number. (From [30].) of view it seems natural is more soft against lowest-lying
that the case corresponding
octupole
vibrational
to two unequal spheres (even ~VF)
vibrations.
Indeed, the RPA calculation shows that the mode with even NF is always lowest in energy, irrespective of
K-quantum number, see fig. 7. At the 3:l deformation one can distinguish
between
three different
types of closed
shells, where NF = 5,8,. 20+20+20,
. . correspond to three equal spherical magical numbers 8+8+8, . . , NF = 4,7,. . . correspond to two large spheres and one smali, i.e. 8+8+2, . . . , and finally NF = 3,6,. . . correspond to one large and two small spheres,
20+20-t-8, i.e. 8+2$2, 20+8+8, . . . (cf [32]). W e h ave obtained analytical solutions for all 3 x4 different combinations of closed shells and K-quantum numbers [27]. In fig. 9 the lowest-lying solutions are shown versus shell filling number for the K = 0 mode. We see a three-fold staggering with the asymmetric combinations lowest in energy. This is the case for the other K-modes as well.
2.3 Giant octupole resonances In fig. 10 all solutions, including the high-lying ones, are shown for the 2:l deformation.
S. Aberg I Superdeformed nuclei
Exicitation
27c
energy
2.0
3:l
def.
K=O
1.6
0.8 0.4
1
0.0
J
7
9
Shell Figure
11
13
number
9: Similar to fig. 7 but for the 3:l deformation
and only showing K = 0 states.
The shell
number NF = 3 corresponds to N = 2 = 12 particles and NF = 13 to N = 2 = 180 particles. The three different kinds of closed shells are symbolically illustrated by the combination of large and small spheres. def.
2:l Excitation
energy
6.0 5.0 4.0 3.0 2.0 1.0 0.0 5.0 4.0 3.0 2.0 1.0 0.0 0
2
4
6
8
IO
Shell number
Figure
12
14
0
2
4
6
8
IO
Shell number
10: Similar to fig. 7 but showing all solutions.
I2
14
16
S. Aberg I Superdeformed nuclei
28c
Cluadrupole splitting of giant resonances
I
140 -r
,
I
I
I
I
I
1
I
a
I
I
I
1
I
5 r” 120 9 ?5 % 100 E IJJ 806 ‘Z s -2 60w 40
J
I 0 0.0
0.2 0.1 Deformation,
0.3
I 0.4
E
Figure 11: The deformation dependence of the high-lying parts of the giant octupole resonance (2’ = 0), giant quadrupole resonance (2’ = 0) and the giant dipole resonance (T = 1). Solid lines mark K = 0,dashed K = 1, dotted K = 2 and dot-dashed K = 3. (From [27].) The staggering seen for the low-lying states (fig. 8) is not present for the high-lying, resonance
type of states.
Furthermore,
giant-
we see that for these solutions the energy scales
as A-‘i3 for NF > 4, i.e. for particle numbers larger than about 20. By this observation, which also applies to the 3:l (and 1:l) de f ormation, we may obtain a simple description of how the (isoscalar)
giant octupole resonance (GOR)
splits up in deformed nuclei [27], and
in fig. 11 the GOR which emerges from the 3fLw excitation energy mode, is compared to the giant dipole and giant quadrupole resonances in quadrupole deformed nuclei. While the two latter resonances split up in (L+ 1) components to one K-component,
the GOR
(i.e. 2 and 3), each corresponding
splits up in 6 components
instead
of the 4 one might
have expected. The explanation comes from the structure of the octupole operators. It is easily seen from eq (6) that th e unperturbed 3/iw energy is found to split up into two components both for K = 0 and for K = 1 (fig. 6). Our general knowledge of giant resonances indicates that one may get a fairly good understanding in the pure H 0 of how e.g. the centroid energy splits up in deformed nuclei. The description made above may therefore be quite applicable. For the low-lying solutions, on the other hand, nuclear structure plays a more important role and we expect quite strong deviations from the HO result in more realistic models. After the study of this analytically solvable model we therefore now turn to more realistic calculations.
S. Aberg I Superdeformed nuclei
3. SUPER-
AND
HYPERDEFORMED
PEARSHAPED
29c
NUCLEI
In this section we shall study the possible existence of static octupole deformed nuclei with approximate microscopic
axis ratia of 2:l (subsect.
3.1) and 3:l (subsect.
3.2).
In subect.
3.3 a
analysis of the Jc2) moment of inertia is performed.
3.1 Superdeformations In the upper part of fig. 12 the shell energy is plotted as a function of the neutron number at the 2:l deformation for two values of the octupole deformation parameter, ~3 = 0 and ss = 0.15, at the constant
rotational
frequency w,,~ = 0.06 &.
spin-orbit force the magic numbers at the 2:l deformation
Due to the
are of course changed compared
to the pure HO. Still we see how every second magic number, i’V = 32,64 and 116, are favoured by octupole (Ysc) deformations, while nuclei with magic numbers inbetween, N=44
and 86, are octupole
closed shell is connected
stable.
The result from the HO study that every second
to a lowered octupole vibrational
valid also in these more realistic calculations,
energy (see fig. 7) thus remains
and also at high spins.
By combining suitable proton and neutron numbers we see that it is difficult to find combinations where both the proton number and the neutron number favour octupole deformations (possible except for N = 2 = 32). S o we have to stick to nuclei where either the protons or the neutrons favour 2:l deformed pearshapes. A few selected potential-energy surfaces in the (E, cs)-plane are shown in fig. 13. Since the calculations are performed for high spins the pairing interaction is ignored. In the cranking prescription with es # 0 the only good quantum
number is simplex, S$J = s$, where S = II. R=(A), ll is the parity
’
~-0.6
Figure 12: The shell energy deduced from the cranked Nilsson model, expressed in units of LJ = 41A-‘13 MeV, is plotted vs the neutron number for 2:l (E = 0.6,&d = 0.06) and 3:l (8 = 0.9,~~ = 0.14) shapes at the fixed rotational frequency o = 0.06 $0. Solid lines are used for reflection symmetric shapes (ES = 0) and dashed lines for c3 = 0.15. (From [35].)
:
_~~~~~~~
j ,::t -0.2
0.4
0.6
0.8
0.2
0.4
0.6
cl.8
1.0
1.0
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1 .a
Ed%)
f&t%)
Figure 13: Potential-energy surfaces in the fs,ss)-plane v&d at IX = 40’ for 15sDy, lssJssEg and zwRn. A free minirnisation has implicitly been performed oi the +degree of freedom, while t-s is included according to ss = -0.47~(2 - c&s. (Prom f35j.f operator and R+(x) spin-parity
rotates the nucleus 180” around the x-axis, the rotation axis.
sequence . . . , 39-, 40+, 41-,
42+, . . . thus corresponds
sequence . . . , 39+, 40-, 41+, 42-, . . . corresponds
to a = -1.
The
to s = +1 while the
The energy surfaces shawn
in fig. 13 are all valid for I” = 40+, i.e. for s = 1. Our favourite superdeformed nucleus 152Dy is indeed seen to be quite soft in the octupole direction. Octupofe softness for 2:1 deformations is also found in the Hg-region, and ZWRn, which is certainly quite unstable against fission, is seen to have a static octupole deformation with E = 0.6 and us = 0.07.
3.2 ~yperdefor~a~ions It is difficult to find any SD nucleus that has a strong octupole deformation, and where the barrier in the es-direction exceeds say f MeV. But if we turn to hyperdeformed, 3:l deformed nuclei, that corresponds to c-values around 0.9, deep and rigid minima are indeed seen in IMEg as well as in 2wRn, see fig. 13. The shell energy diagram for 3:1 shapes based on the cranked Nilsson model (lower part of fig. 12) shows very strong deviations from the estimates made above within the pure HO (fig. 9), reflecting the fact that shell gaps are not so well defined in realistic calculations at the 3:l deformation, The strong favouring of octupole deformation at N w 82 (or 2 x 82) and N x Il.8 is responsible for the octupole minima in 19*Hg and 2ooRu. For these nuclei the competition with fission is very strong, and it seems extremely difficult experimentally to reach the BD octupole minimum. From rotating liquid-drop calculations we know that the highest angular momentum that can be reached before f&Son sets &I, is obtained in the mass
3ic
S. &erg / Superdeformed nuclei
0
6000 4000 2000 Angular momentum, I2
0
2000
6000
4000
Angular momentum, I*
Figure 14: The energy of lowest SD and HD configurations are compared for 15sDy and 146Gd and plotted versus the square of the angular momentum. region A=120-150 [36]. A combination of the minima at N x 62 and 2 % 82 (the shell energy diagram for protons looks similar to fig. 12) of the ~3 = 0.15 curve (lower part of fig. 12) thus seems to be the most promising case for a nucleus with a hyperdeformed pear shape. In fig. 14 the total energy for the lowest HD and SD configurations
are shown as
functions of angular momentum
(squared) for 146Gd and *52Dy. In both cases the hyperdeformation corresponds to E x 0.93, cs M 0.13, while the (reflection symmetric) superdeformation is somewhat larger for ls2Dy (E M 0.58) than for 146Gd (E w 0.52). The shape of the hyperdeformed
pearshaped nuclei ls2Dy and 146Gd is shown in fig. 15.
Through 7 - 7 correlation measurements a hyperdeformed band(s) was recently proposed to be seen by Galindo-Uribarri et al at Chalk R iver [37]. In the present calculation the ND minimum of is2Dy is found to become yrast at about spin 90 while the crossing with the SD band in 146Gd occurs “already” at spin 80. {Notice that the calculation of the HD in 152Dy presented in 1371did not include the important octupole degree of freedom.) Fig. 16 exhibits potenti~-energy O+, 40f, 60+ and 40-.
-
Figure 15: Eg = -0.056).
surfaces in the f.s,s3)-plane
valid for ‘*Gd
at 1” =
The SD minimum becomes yrast at around spin 50, and, as was
_.-_-_ irli
Approximate shape of HD 1s2Dy or 146Gd (c = 0.93, 63 = 0.s2,
&4
=
0.
13,
32c
S. Aberg f Superdefiirmed rurclei
0.4
0.6
0.8
1.0
Figure 16: Similar to fig. 13 but for selected spin-p~ity combinations of ‘Wd. O+, 40+, 60+ and 40-.
The SD minimum becomes yrast at around spin 50, and, as was
seen in fig. 14, the HD configuration eg. 40-)
becomes yrast at I z 8Ofi. The s = -1 sequence (as
is calculated to come almost 1 MeV higher in excitation energy than the lowest
s = 1 band.
This feature can be understood
fig. 17, valid at the relevant hyperdeformation. survive to the highest rotational frequencies.
from the single-particle
diagrams shown in
The 0.5-1.0 MeV gaps at Z=64 and N=82
This suggests 146Gd to be one especially favoured candidate for HD, with one band pushed down in energy by almost one MeV, reminding about the SD band in lszDy. The barrier between the “two” octupole deformed minima (es = f0.12) at c M 0.9 is found to be 2.5-5.0 MeV {for I” = 60s) depending on what path is chosen in the (E,E~) space. Due to the high spins and low level-density the pairing correlations are expected to be weak implying a large mass parameter. The parity splitting, determined by tunneIing between the “two” octupole minima, is thus expected to be very small, i.e. we expect e.g. the spin-parity
sequence . . . , 59-,
60+, 61-,
62+,
. . . to follow a smooth rotational band. The competition with fission certainly makes the population of the HD discrete states very difficult. In the present calculations we unfortunately cannot calculate the full fission barrier due to an insufficient shape parametrization. The feeding of the HD minimum is, however, in a similar way as for SD 15aDy, expected to be favoured by one state being pushed down in energy. But also the calculated very low-lying GDR component with a peak energy at about E,,, = 4 - 5 MeV at this extreme quadrupole deformation, is expected to help the population of the low-lying HD band. By assuming that these features are able to overcome the competition with fission we shall discuss the possibility to remain in the HD band. The energy curves between the SD and HD minima for I” = 60” and 80f are shown in
S. &erg I Superd&med nuclei
33c
PROTONS
w/w* NEUTRONS
5.8
Figure 17: Single-particle levels for protons (upper part) and neutrons (lower part) at the hyperdeformation E = 0.93, Es = 0.12, ~4 = 0.13, cs = -0.056, as obtained from the cranked Nilsson potential. Solid lines indicate simplex s = i and dashed lines s = 4. The two lowest “N=7” proton orbits are indicated. Since parity is mixed the given high-N notation is however quite approximate. fig. 18. Compared to the barrier between SD and normal-deformed states (see e.g. [38]) the barrier between HD and SD states is certainly smaller. But, as was discussed above, we expect a large mass parameter. Furthermore, the decay from the HD band occurs to SD excited states which have a much lower calculated level density than normal deformed states [39] to which the SD states decay. Also favouring the decay within the HD band is the expected very large quadrupole moment, Qo M 35 eb, giving rise to ~(~2)-tr~sitions with about 10 kWu. These three factors, large mass parameter, low SD level density aad large B(E2)-values, are all very much expected to favour the decay within a low-lying hyperdeformed rotational band.
34c
S. Aberg / Superdeformed nuclei
Energy (MeV)
T----t
Figure 18: Energy curves between SD and HD minima in 146Gd at 1” = 60+ and I” = 80+. The energy is explicitly minimized with respect to es and ~4 while ~5 is taken as a given function of B and ~3. The SD minimum corresponds to 6s = 0 while the HD minimum is octupole deformed with ~3 = 0.12. The excitation energy &D - &D changes from 8 MeV to 1 MeV, and the HD barrier changes from 1.5 MeV to 3 MeV as the angular momentum changes from 60 to 80.
255 0*4 OJ 0.6 0#7 0,s 0,9 E(4 64’Es)
1
3.3 Moments of inertia As discussed in subsection 1.1 the high-N orbits play an important role in determining the ,7(2) moment of inertia of SD bands. To investigate the g(2) properties of HD bands we show in fig. 19 the alignment, {j5), and the contribution to the J(2)-moment of inertia from the two lowest-lying proton N=7 orbitals at four different deformations: Norma1 deformation (& = 0.30,~s = 0), superdeformation (s = 0.57,~ = 0) and two hyperdeformations (E = 0.93 with ~~ = 0 and ~3 = 0.12). The contribution to J@) from orbital Y is calculated as
(f-3) where (hNila*on- wj, ) Iw;v) = e; Jw;~)
(9)
By defining gzi,= in this way the total Jc2) is simply obtained by adding the contributions from all occupied states $2) = c $$‘. (10) YE0C.z This way of calculating J’$& differs from the previous way [S] where the derivatives
do, du
I
8
“‘I”
I
+__---
6-
e---_‘.__.._
-t-
I I
-4
I
1 -___----
1
I
I
I
I
. . . . . . .-?-
___
& =0.93&,
=o
-*m... I I
& =0.93E, I I
-0.12 , I
t
I
50
40 u, J wb..v 30
(MeV-‘) 20 +rc-----
10 n
O.to2
0
%J
043
0.;4
;
0,
O.b2 w,J~~
*.b4
o.bs
Figure 19: Angular momentum component along rotation axis, < j, > (upper diagrams), and $21 orb,v moment of inertia (lower diagrams) for the two lowest proton N=7 configurations, are shown versus rotational frequency for the four deformations specified in the upper left figure. The two figures on the left-hand side are valid for the lowest x7 configuration, and the two figures on the right-hand side are valid for the next lowest r7 configuration. In the octupole deformed case the next lowest configuration changes smoothly with w,,t from N M 7 to N w 6 (see fig.17) explaining the rather small < j, >-values. nume~cally that gzj,,
were calculated as differences.
One advantage with the present approach is
now is a local quantity which is more easy to follow vs w,.+
It also makes
possible a microscopic other orbitals,
analysis of the origin of changes in &&, from interactions with see refs. [40, 411. A s is usually done in the cranked Nilsson model, all
moments of inertia and rotational frequencies shown below are renormalized to obtain rigid-body rotation on the average [42]. At normal deformations the angular momentum vector of a nucleon in a high-N orbital quickly aligns, and obtains a value close to the maximum values 7.5 and 6.5, respectively for the two lowest-lying N=7 orbitals. The corresponding J@) contributions decrease drastically from the huge value of about 50 R2MeV-r to values below 5 at rotational frequencies, w,.,~ 2 250 keV. Also at the 2:l deformation
(E = 0.58) a rather drastic spin alignment occurs accom-
36c
Moment
S. Aberg I Superd@ormed nuclei
of
Ioertla
2oor-----t 80
‘&Gd
-
lS2DY
-J(l)
hyperdeformed
Figure 20: Microscopically calculated J’(l) and .7c2) moments of inertia at the HD 146Gd and 15aDy. The rigid-body values are shown for comparison, panied by a decrease in the Jt_“b’ moment of inertia. different behaviour
of ,$,!i
for the lowest-lying
analysis of SD configurations. alignment J$i-values
As discussed in subsection
1.1 the
high-N orbits provide a good tool in the
If we now turn to the HD cases shown in fig. 19 the spin
is seen to occur much smoother than for the smaller deformations, and the are consequently smaller. But in particular we note that the orbit-specific
variation of 322
with w,.,~ does not seem to occur at the hyperdeformation.
The disap-
pearance of the high-N or high-j specific characters at very large deformations is due to the larger mixing between the high-j “intruder” orbitals and deformation aligned orbitals. We thus expect smaller differences of Jt2) between different HD bands, and in particular the high-N character, important at SD shapes, seems to be of less importance. In fig. 20 the J(2l- moment of inertia of HD 152Dy and r4’Gd are shown vs w,,t. As could be expected from the discussion above it is seen to be rather independent of the rotational
frequency.
moment of inertia.
It is furthermore The Jt2)
tion, s = 1, but is more or less identical J(2)-value
extracted
very similar to J(i)
as well as to the rigid-body
moment of inertia is shown for the lowest-energy to the first excited s = -1
from the +y- y coincidence
experiment
configura-
configuration.
The
by the Chalk River group
agrees very well with the calculated values, and the observed [37], $22 M 130tL2MeV-‘, ridge in the 7 - 7 plot was suggested to be related to HD is2Dy. The observed rotational frequency interval 0.45 < wrof < 0.75~eV, would however correspond to extremely large spin values, 78 < I < 98fi..
4. SUMMARY Several properties of SD nuclei are now rather well understood, such as the underlying shell structure of observed mass regions, the high-N scheme, some hints about specific orbitals around the Fermi surface (e g from band-crossings), etc. Still several problems remain to be solved, like the continued rise of Jr(2) in the Hg-region, pairing properties
S. Aberg I Superdeformed nuclei
37c
of SD nuclei, etc, and also if new predicted regions of SD nuclei really exist. A problem of specific interest is the decay out of SD bands. We showed that an oberved decrease in the T-ray intensity in some cases could be due to strong interband Ml-transitions, and some neutron and proton orbitals in SD nuclei around lg2Hg were discussed as good candidates for giving rise to sufficiently strong Ml-transition matrix elements. The recent measurement of Ml-transitions between SD bands in lg3Hg [25] confirmed the predictions, and in fact implies that
we now face a new era of SD spctroscopy
questions e.g. about magnetic Octupole
vibrational
where interesting
properties in SD nuclei may be answered.
states built on SD and HD shapes were described
solvable model (HO + O-O-force).
in a simple
Th e 1ow-lying SD states can be devided into two classes
of closed shell, corresponding to two spherical nuclei put beside each other of equal or unequal size. Octupole vibrations built on superdeformed states of the latter type was found to come lowest in energy, and for K = 0 the same effect was seen in NilssonStrutinsky calculation in terms of a lower shell energy for static reflection asymmetric shapes. The simple model gave similar types of structure types of asymmetric
shapes.
In the realistic calculation
at HD but now with two different such a three-periodic
variation of
the shell energy could however not be seen, since the relatively small HO gaps at the 3:l deformation are replaced by a more complicated structure in realistic potentials. The high-lying model.
(3L)
oc t u p o1e vibrational
states are nicely described
The energies have a simple A- ‘I3 dependence
by the simple
and split up in 6 components
due
to the coupling with the quadrupole deformed field. The total deformation splitting of the GOR is found to be considerably larger than for the GDR or GQR. Octupole correlations were found to be very large at hyperdeformed shapes, where deep stable minima in the potential-energy surfaces for several A M 150 nuclei. The nucleus ‘ZGdss was found to be a particularly favourable case for a hyperdeformed pearshaped nucleus. Both proton number Z=64 and neutron number N=82 are connected with gaps in the single-particle situation
energy diagram of 700-1000
thus reminds about what is calculated
being pushed down in energy compared band in laGd competition
keV at all rotational
frequencies.
The
for SD ls2Dy and results in one band
to the first excited band.
The hyperdeformed
is calculated
to become yrast at 1, M 8OtL, and in the population stage the to fission is expected to be very serious. The calculated &value is however
quite uncertain, in particular due to uncertainties in the macroscopic model. For example, the used liquid-drop model does not account for nuclear finite-range interactions that is expected to lower 1,. A lowering of I, is also expected if the shape parametrization is improved, e.g. by freely including the ts and cs degrees of freedom. We conclude that several properties
of superdeformed
nuclei are now rather well un-
derstood. But we have also pointed out that many questions remain to be answered, and that new interesting features of nuclei with exotic shapes are to be revealed. We thus eagerly await result from the new generation
of r-ray detecter
systems.
S. Aberg 1 Superdeformed nuclei
38c
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