Are the K-quantum number selection rules
conserved in the order to chaos transition region?
G. Benzoni1 , A. Bracco1 , S. Leoni1 , N. Blasi1 , F. Camera1 , C. Grassi1 , B. Million1 ,
A. Paleni1 , M. Pignanelli1 , E. Vigezzi1 , O. Wieland1 , M. Matsuo2 , T. Døssing3 , B.
Herskind3 , G.B. Hagemann3 , J. Wilson3 , A. Maj4 , M. Kmiecik4 , G. Lo Bianco5 , C.
Petrache5 , M. Castoldi6 , A. Zucchiati6 , G. De Angelis7 , D. Napoli7 , P.
Bednarczyk4,8 , D. Curien8 .
1
Dipartimento di Fisica, Universitá di Milano and INFN Sez. Milano, Via Celoria
16, 20133 Milano, Italy
2
Graduate School of Science and Technology, Niigata University, Niigata 950-2181,
Japan
3
The Niels Bohr Institute, Blegdamsvej 15-17, 2100, Copenhagen, Denmark
4
Henryk Niewodniczanski Institute of Nuclear Physics, 31-342 Krakow, Poland
5
Dipartimento di Fisica, Universitá di Camerino and INFN sez. Perugia, Italy
6
INFN sez. Genova, Genova, Italy
7
Laboratori Nazionali di Legnaro, via Romea, Legnaro (PD), Italy
8
Istitute de Recherches Subatomic,F-67037 Strasbourg Cedex 2, France
1
Abstract
The validity of the selection rules on the K-quantum number in rapidly rotating warm
nuclei is investigated through the analysis of the variance and covariance of the spectrum
fluctuations of the γ-cascades feeding into low-K and high-K bands in
163 Er.
Low-K bands
are found to be fed by a much larger effective number of cascades than high-K bands. The
covariance between pairs of gated spectra shows that the cascades feeding low-K bands are
different from those feeding high-K bands. The data are compared to simulated spectra
obtained using energy levels and transition probabilities calculated with the band mixing
model including the residual interaction and a term representing the effect of the K-quantum
number on the rotational energy. The K-selection rules within the angular momentum region
30h̄ ≤ I ≤ 40h̄ for the excited rotational bands forming the ridge structures (with internal
energy up to 1.2 MeV) are found to be conserved while at higher internal energy (up to ≈
4 MeV) evidence for partial conservation has been found.
PACS numbers: 21.10.Re, 21.60.-n, 23.20.Lv, 24.60.Lz, 25.70.Gh, 27.70+q
Keywords: K-quantum number, compound states, warm rotating nuclei, band crossings, residual interaction, quasi-continuum gamma spectra.
Corresponding author:
Angela Bracco
Dipartimento di Fisica, Universitá di Milano
Milano,Italia 20133
e-mail: [email protected]
fax: +39 02 50317 487
2
The conditions under which K, the projection of the aligned nucleonic spins on the
symmetry axis in deformed nuclei, is a good quantum number remain a topic of much
current interest, as testified by the extensive experimental work on high-K isomers
[1]. The study of nuclei with high values of the K-quantum number is interesting
both from the point of view of the decay out from these configurations but also in
connection with their feeding, which allow to investigate the validity of the associated
selection rules at high excitation energies. In fact, as it was stated by B.Mottelson [2]
the question of K-quantum number violation in thermally excited states is a key issue
in the study of the transition between ordered and chaotic many-nucleon motion
caused by the residual interaction and high level density. This problem has been
addressed by studying the γ-decay from neutron resonances at energy U ≈ 8 MeV
[3, 4] and by studying the γ transitions of quasi-continuum nature emitted by nuclei
formed in fusion reaction, which is probing the energy region extending up to ≈ 4
MeV [5, 6]. Since the violation or persistence of the K-quantum number depends
on the thermal excitation energy, it becomes particularly interesting to focus the
attention where the order-to-chaos transition is predicted to take place, namely at U
≈ 1-2 MeV [7, 8]. In this way one can investigate whether or not the selection rules,
valid for the description of the quantal systems, are conserved in the chaotic region.
In this paper we present a new study of the warm rotational motion in the
163
Er nu-
cleus based on the measurement of the γ-transitions forming quasi-continuum structures in γ − γ spectra and populating specific configurations with different values of
the K-quantum number. The data were analyzed with the fluctuation and covariance
analysis technique [9, 10, 11]. This particular nucleus has been previously studied [5]
and it represents a good case for further and more detailed investigations of the validity of selection rules in the order-to-chaos transition region since possible conservation
of the K-quantum number has been suggested. Two are the novelties of the present
work. The first is the more detailed experimental investigation of the warmest part of
the decay and the second, very important for the physics discussion, is that, for the
first time, a direct and rather realistic comparison between experiment and theory is
made. The latter is based on simulated spectra constructed using recent calculations
3
on this specific nucleus [12], that can be treated in the same way as the data.
The experiment was carried out using the EUROBALL array at the IReS Laboratory
(France), employing the reaction
18
O +
150
Nd, at Ebeam = 87, 93 MeV. The
150
Nd
target was made of a stack of two thin foils for a total thickness of 740 µg/cm2 .
The corresponding maximum angular momentum reached in the reaction has been
calculated to be 40 and 45 h̄, respectively. Energy-dependent time gates on the Ge
time signals were used to suppress background from neutrons. A total of ≈ 3×109
events of triple and higher Ge-folds were finally obtained, with
162,163
Er as main
evaporation residua. The data have been sorted into a number of γ − γ matrices
in coincidence with specific γ-transitions of the
163
Er nucleus [13]. First, a matrix
collecting the entire decay flow of 163 Er (named total ) has been constructed by gating
on the three cleanest low spin transitions. In addition, seven matrices gated by
transitions belonging to the low-K bands (K=5/2) labeled A, B, E and F in ref.
[13], and by the high-K bands (K=19/2) labeled K1, K2 and K4 in ref. [13] have
been sorted, together with their corresponding two-dimensional (2D) background.
For each 2D spectrum all known peak-peak and peak-background coincidences have
been subtracted using the Radware software [14]. The separately gated matrices have
also been added together in one low-K (A+B+E+F) and one high-K (K1+K2+K4)
matrix. Figure 1 (left column) shows example of cuts perpendicular to the Eγ1 = Eγ2
diagonal, 60 keV wide, in the total, low-K and high-K γ-γ matrices, at the average
transition energy (Eγ1 + Eγ2 )/2 = 900 keV.
The fluctuations of counts in each channel of the selected measured 2D spectra,
expressed as variance and covariance, are evaluated by the program STATFIT [10]
and stored into 2D spectra. One additional option is applied: all pairs of resolved
transitions are removed in the triangular sector Eγ1 ≥ Eγ2 with the proper intensity
from the γ − γ spectra, before the fluctuations are extracted, since the fluctuations
are severely affected by the low lying intense transitions [10]. Because each rotational
Eγ -cascade on the average contributes one count in each
moments are evaluated over sectors of
2
4h̄2
× 4h̄= ,
=
4
4h̄2
=
interval, the statistical
corresponding to 60 keV×60 keV
intervals for rare earth nuclei around
163
Er.
From the fluctuation spectra we first extract the effective number of decay paths,
(2)
which eventually feed into the gate-selected band. The number of decay paths Npath
having two γ transitions with energies lying in a chosen 60 keV ×60 keV window in
the γ − γ coincident spectrum is obtained from the simple expression
(2)
Npath =
µ2
µ1
N
× P (2)
−1
(1)
where N is the number of events, while µ1 and µ2 are the first and second moments of
the distribution of counts, all evaluated in a =4 h̄2 × =4 h̄2 sector of the γ − γ matrix. The
superscript (2) indicates that the number of paths was derived using up to the second
moment of the path detection probability, while the P (2) factor takes into account for
the finite resolution of the detector system [9, 10].
The number of paths obtained from the analysis of the first ridge of the 2D matrices
gated by the different bands is found in average ≈ 10 for both low-K and high-K
configuration, and correspond in both cases to a total number of about 20 discrete,
yet unresolved bands, when the number of paths for the individual configurations
are added together taking into account the relative intensity of the selected configuration and the fraction of common paths (measured by the covariance coefficient
lately discussed) [11]. Such results are shown in the bottom part of figure 2 by open
circles (low-K) and squares (high-K), and are consistent with the analysis of the ridge
structure of the total spectrum (triangles), which gives all together ≈ 45 discrete
rotational bands in the
163
Er nucleus, before rotational damping sets in [15].
In contrast to the results of the ridge analysis the number of paths obtained by analyzing the valley region is found to depend significantly on the nuclear configuration.
This result is shown in the top part of figure 2 together with the number of paths
deduced from the total Eγ1 × Eγ2 spectrum. As the valley is probing the region in
which the rotational bands are strongly mixed, this result intuitively suggests that
the mixing process is indeed different for high-K and low-K states. This could be
5
due to the combined effect of both different level densities and distribution of B(E2)
probabilities (namely, rotational damping widths). In fact, while the level density for
high-K states is ≈ 3 times lower than for the low-K ones [12], the rotational damping
width has been measured to be ≈ 30% reduced for high-K states [16].
To provide a better understanding of the problem of mixing of states with different quantum numbers we have studied the correlations in fluctuations between two
spectra associated to low-K and high-K quantum numbers. These correlations are
expressed by the covariance of counts, defined as
µ2,cov (A, B) ≡
1 X
(Mj (A) − M̃j (A))((Mj (B) − M̃j (B))
Nch j
(2)
where M (A) and M (B) refer to spectra gated by transitions from two different bands,
A and B. The sum is over a region spanning Nch channels (in this case 15 × 15) in
the two-dimensional 60 keV×60 keV window, and M̃ denotes an average spectrum,
(which in our case is obtained by the routine STATFIT as a numerical smoothed 3rd
order approximation to the 2D spectrum). To normalize the covariance and thereby
determine the degree of correlation between the two spectra, the correlation coefficient
r(A, B) is calculated:
µ2,cov (A, B)
r(A, B) ≡ q
(µ2 (A) − µ1 (A))(µ2 (B) − µ1 (B))
(3)
Here, µ2 denotes the second moment defined for the same region Nch , related to
the expression for the covariance by µ2 (A) = µ2,cov (A, A). The first moment µ1 is
the average of M over the region Nch . The subtraction of the first moments in the
denominator of eq. 3 corrects for the direct contribution to µ2 from counting statistics,
which is linear in the number of counts. The more interesting fluctuations are due
to the nature of the finite number of transitions available to each cascade, and their
contribution to µ2 is quadratic in the number of events.
Figure 3 shows the average values of the correlation coefficient r extracted from the
6
covariance analysis of the ridge and valley structures of γ-γ coincidence spectra of
163
Er. In the case of the ridge analysis, r is found to be positive and of the order of 0.2
for configurations with similar low-K values (fig. 3a), while it is approximately zero
for combinations of low-K and high-K spectra (fig. 3b). This shows that there are
basically no cross-transitions between the ≈ 15 bands feeding high-K states and the
≈ 15 bands feeding low-K states. Such results can be interpreted as an effect of the
persistence of a strong K selection rule governing the cold rotational bands forming
the ridges. In the case of γ transitions populating the valley region the data suggest
a different picture. The correlation coefficient is positive both in the case of similar
K configurations and for the combination of low-K with high-K, and in particular
is larger (but not yet 1) in this second case, pointing to a possible persistence of
selection rules on K also in this regime.
To discuss the present experimental results and their physical implications on the
problem of the K-quantum number violation in thermally excited states, we have
performed simulations using band mixing calculations including both the residual
interaction and a term that takes into account the angular momentum carried by the
K-quantum number [12]. The calculations were made employing 4000 np-nh basis
states with the lowest excitation energies to diagonalize the Hamiltonian for each
spin and parity Iπ . The truncation corresponds to a cut off of approximately 4 MeV.
An important feature of the band mixing model including a J2z term (to produce levels
with high values of the K-quantum number) together with a surface delta residual
interaction is that the onset of K-mixing is found to take place at around U = 1.5
MeV while the onset of damping (mixing of all configurations, independent of the
K-quantum number) occurs at around U = 1 MeV.
The simulated γ − γ spectra of interest were constructed by a Monte Carlo code,
successfully employed to study rotational damping in different region of mass and
deformation [17, 18]. The code provides a realistic description of the experimental
γ-decay, being based on the competition between E1 and E2 transitions and on the
levels and E2 transition probabilities microscopically calculated for the specific case of
7
the 163 Er nucleus [12]. Each γ-cascade is started from initial values of internal energy
and spin randomly chosen from a two-dimensional entry distribution of Gaussian
shape, with centroids and widths reproducing the experimental conditions of the
163
Er experiment previously discussed (i.e. < U > = 4 MeV, F W HMU = 4 MeV,
< I > = 44 h̄, F W HMI = 20 h̄). While for the E2 transition probability the
band mixing model values were used, for the E1 transition probability we have used
the calculated level density and a GDR strength function corresponding to a prolate
nucleus with quadrupole deformation β = 0.25 and rotating collectively. In addition,
an exponential quenching factor that takes into account the difference in K-quantum
number between the initial and final states has been used. Such factor is analogous
to the one employed in the analysis of the E1 decay-out from isomeric states [19, 20].
The choice of these parameters for the entry points and E1 transition probability
resulted in a good reproduction of the measured intensities of both low spin yrast
transitions and ridge structures [17, 21].
The right column of figure 1 shows examples of 60 keV wide projections perpendicular
to the Eγ1 = Eγ2 diagonal of simulated γ-γ matrices collecting all cascades (Total,
panel d)) and cascades finally feeding into low-K (namely K≤ 8, panel e)) and high-K
(K> 8, panel f)) bands. The projections are taken at the average transition energy
(Eγ1 +Eγ2 )/2 = 900 keV, as in the corresponding experimental spectra (left column of
figure 1). The asymmetry in the spectra is connected to the discrete lines subtraction
which is here performed only in the Eγ1 ≥ Eγ2 region. It is worth noticing that in
the simulated matrices only the yrast and the first excited band (for each parity and
signature configuration) have been subtracted from the coincidence spectra, resulting
in a more pronounced ridge structure than in the data, where all discrete transitions
known from the level scheme have been removed (see right part of the spectra).
The fluctuations of counts in each channel of the selected 2D spectra, expressed as
variance and covariance, are evaluated also for the simulated data and the results
obtained for the ridge and valley structures are shown with lines in figures 2 and 3.
It is remarkable how well the band mixing model predicts both the number of paths
8
and the correlation coefficients from the covariance analysis. In particular, in the case
of the fluctuation analysis, shown in figure 2, both the number of low-K and high-K
discrete bands and the total number of unresolved discrete states are well reproduced
by the simulation. For comparison, the total number of discrete bands, as directly
extracted from the bands mixing calculations (BM) by counting the number of bands
branching out to less than 2 states [15], is also given (thin solid line). In the case of
the valley, probing the warmer region of strongly interacting bands, the simulation
calculations also give almost an order of magnitude difference between high-K and
low-K states, at least up to Eγ ≈ 1000 keV (corresponding to spin 34h̄ and heat
energy ≤ 2 MeV), while for higher values of transition energies the number of high-K
states become closer to the total number of states, as also shown by the data.
The simulation calculations well reproduce also the average correlation coefficient
both for the ridge and valley structures, as shown in figure 3, suggesting the following
picture. The fact that in the case of similar K values the r correlation coefficient
is small (20% at most) indicates a limited cross talk between states with similar K,
as found in the analysis of both ridges and valleys. This is related to the limited
number of E1 transitions emitted in the cascade (3 to 4 transitions at most out of
≈20 γ’s), which allow to change parity configuration. On the contrary, the covariance
analysis between low-K and high-K states gives very different results in the case of
the ridge and valley structures: while in the case of the cold discrete bands K is a
good quantum number, as testified by the almost 0 correlation coefficient found from
the ridge covariance (panel b)), a gradual weakening of selection rules on K is found
at increasing internal energies, as shown by the large positive values of r obtained
from the valley covariance (panel d)). In fact, if K were still strongly conserved up in
the region of rotational damping, one would expect at most a 20% cross talk via E1
decay (which would be also hindered between states with a large difference in K), as
also measured among states with similar K (panel c)). The fact that higher values
of r are found instead indicates, on the contrary, a weakening of selection rules on K:
cascades finally feeding into specific low-K or high-K states may pass through states
with similar K values up in the region of the strongly interacting bands, therefore
9
implying a progressive weakening of selection rules along the decay. This is consistent
with the results obtained by the fluctuation analysis of the valley region (presented in
figure 2 b), which shows a gradual approach of low-K and high-K data at the highest
values of transition energies (namely spins/heat-energies). The present picture well
agrees with the theoretical predictions from the band mixing model on
163
Er [12],
showing that the onset of K-mixing takes place around a heat energy of U ≈ 1.5 − 1.8
MeV, while the statistical limit of strong K-mixing is approached only above 2 MeV.
This dependence of K-mixing on excitation energy is related to the K-selectivity of
the matrix elements of the adopted two-body residual interaction, which is found to
well reproduce the data.
We have discussed the onset of K-mixing in rapidly rotating warm nuclei by means of a
comparison of high statistics experimental data on 163 Er and recently developed band
mixing calculations for the specific nucleus. The experimental results from the fluctuation and covariance analysis on γ-γ coincidence spectra and the good agreement
between data and theory lead to the conclusion that K selection rules are partially
conserved up to the region of excitation energy 2.5-3 MeV, as probed in average by
the quasi-continuum transitions in the spin region 30-40 h̄, sampled by the present
experiment. In particular, K-selection rules are rather well conserved up to approximately 1 MeV, as deduced by the analysis of the ridge structure which is formed
by discrete unresolved transitions among excited rotational bands. At higher internal
energy (U≤ 2 MeV), probed by the continuous transitions forming the valley, a partial
conservation of the K selection rules is found, as shown by both the fluctuation and
covariance analysis. In particular, the covariance analysis between low-K and high-K
data gives rather large and positive values of the r correlation coefficient, but not yet
corresponding to the fully mixed regime (expected to produce a value of r equal to
1). For a better understanding of the thermal energy dependence of the K-mixing
problem it will be interesting to investigate in future works high K-bands of larger
internal energies.
10
Figure 1: 60 keV wide projections perpendicular to the Eγ1 = Eγ2 diagonal of experimental and simulated 2D spectra of
163
Er (left and right panels, respectively), at
the average transition energy < Eγ >=900 keV. The spectra collect either the total
γ-decay flow (panel a) and d)) or the γ-decay in coincidence with low-K (panel b) and
e)) or high-K (panel c) and f)) specific configurations. The reduced intensity observed
in the Eγ1 ≥ Eγ2 region of the spectra is due to the subtraction of all discrete lines
known from the level scheme, in the case of the experimental data, and of the yrast
and first excited bands, in the case of the simulation.
Figure 2: panel a): The number of decay paths extracted from the fluctuation analysis of the ridge structure of
163
Er. The open circles (squares) refer to the number
of unresolved rotational bands populating the ridges of γ-γ matrices gated by low-K
(high-K) configurations, while the full triangles give the number of discrete paths
obtained from the total matrix. The corresponding values for simulated spectra are
shown by dotted, dashed and full lines (low-K, high-K and total, respectively). The
total number of discrete bands directly extracted from the bands mixing (BM) calculations is also given for comparison (thin solid line). panel b): The number of
strongly interacting bands obtained from the fluctuation analysis of the valley region
is shown by open circles (squares) for low-K (high-K) gated spectra, while the full
triangles show the results obtained from the total γ-γ matrix. The dashed and solid
lines give the theoretical expectations for total and high-K cascades, as obtained from
simulated spectra.
Figure 3: The results of the covariance analysis on ridge (bottom panels) and valley
(top panels) structures of 163 Er. Panels a) and c) show by open squares the correlation
coefficient r obtained experimentally by averaging over pairs of γ-γ spectra gated by
low-K configurations, while the correlation coefficient obtained from the experimental
analysis of the low-K versus the high-K matrices is shown by full circles in panels b)
and d). The theoretical values, as obtained from the covariance analysis of simulated
spectra, are represented by dashed and solid lines, respectively.
11
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12
[21] A. Bracco et al., to be published.
13
Simul
c)
100
100
50
High K
High K
Counts [x100]
f)
Counts [x100]
200
0
e) 300
b)
600
200
400
Low K
100
Low K
Counts [x100]
Counts [x100]
Data
a)
d) 400
600
200
Total
Total
400
-100
0
100
(Eγ1- Eγ2) [keV]
-100
0
100
(Eγ1- Eγ2) [keV]
Fig. 1
Counts [x100]
Counts [x100]
800
100000
b)
N(2)path
10000
1000
Exp Total
Exp Low K
Exp High K
Sim Tot
Sim High K
100
Valley
10
100
80
Exp Total
Exp Low K
Exp High K
Ridge
a)
path
N
(2)
BM
60
Simul
40
20
0
600
Simul
700
800
900
1000
Eγ [keV]
Fig 2
1100
1200
Average Low-K
c)
1,0
0,5
0,5
0,0
0,0
1.00
1.00
<r>
<r>
Valley
1,0
0.75
a)
Ridge
0.75
0.50
0.50
0.25
0.25
0.00
0.00
-0.25
600
Low K - High K
1,5
1,5
700
800
900
1000
1100
1200
-0.25
600
d)
Valley
b)
Ridge
700
Eγ [keV]
800
900
Eγ [keV]
Fig 3
1000
1100
1200