Statistical concepts applied in
structure studies of warm nuclei
Rositsa Chankova
Dissertation submitted to the degree of
Doctor Scientiarum
Department of Physics
Faculty of Mathematics and Natural Sciences
University of Oslo
June 2006
Acknowledgments
I express my gratitude to my supervisor Professor Magne Guttormsen for
being so continuously supportive of this work, and for his extensive involvement and feedback throughout the whole process.
I am indebted to Andreas Schiller for his suggestions and fruitful discussions and for his contributions to the papers of this thesis, and also to Gary
Mitchell for proof-reading it.
I would like to acknowledge all the people who collaborated and otherwise contributed to this work throughout the course of the project and shall
mention most of them.
From the Oslo Cyclotron group these were Sunniva Siem, Finn Ingebretsen, John Rekstad and Cecilie Larsen.
I am also very grateful to the people outside of Oslo: T. Lönnroth, U. Agvaanluvsan, E. Algin and Alexander Voinov. They took a lot of shifts at the
Cyclotron Lab and contributed to the successful accomplishment of the experiments.
Thanks to all the other members of the Oslo group for their support, and
to the Lab engineers Eivind Atle Olsen and John C. Wikne for maintaining
the Cyclotron in good condition to perform experiments.
I would like to thank my family for their encouragement and for always
believing in me, and especially to my daughter Dejana for her patience at
my being away from home such a long time.
Financial support from the Norwegian Government Scholarship (Quota
Programme) is acknowledged. My special thanks to Michele Nysæter for the
efficient handling of the administrative and financial part of this project.
ii
Contents
1 Introduction
1.1 Motivation . . . . . . . . . . . . . . . . . . . . .
1.2 Experimental technique and methods . . . . . .
1.3 Level density and thermodynamic properties . .
1.4 Models of the radiative strength function (RSF)
1.5 Survey of the papers . . . . . . . . . . . . . . .
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2 Papers
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2.1 Free energy and criticality in the nucleon pair breaking process 18
2.2 Thermal properties and radiative strengths in 160,161,162 Dy . . . 25
2.3 Large enhancement of radiative strength for soft transitions in
the quasicontinuum . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4 Radiative strength functions in 93−98 Mo . . . . . . . . . . . . . 41
2.5 Level densities and thermodynamical quantities of heated 93−98 Mo
isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.6 Microcanonical entropies and radiative strength functions of
50,51
V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3 Summary and future perspectives
3.1 Fine structures in the level density and phase transitions . .
3.2 Radiative strength function and resonance structures . . . .
3.2.1 Local enhancement of the RSF at low γ-ray energies
3.2.2 Large enhancement of the RSF at low γ-ray energies
3.3 Future plans . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
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71
72
75
75
77
80
81
82
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1
1.1
Introduction
Motivation
Nuclei are many-particle systems whose behavior is governed by quantum
mechanics. However, the present knowledge about one of the most important interactions in nuclei, the strong interaction, is still limited. The forces
between the protons and neutrons are complicated and cannot be written
down explicitly in full detail. In the absence of a comprehensive nuclear
theory, to obtain further insight into the wide range of nuclear properties, a
number of approximate nuclear models has been constructed.
One of the most successful models in nuclear physics is the shell model [1],
and its extension to deformed nuclei, developed by Nilsson [2]. Within these
models, each nucleon moves independently in a nuclear potential (mean field)
which is caused by all of the other nucleons. Spherical nuclei have filled major
shells and as a result clear shell gaps at the magic numbers 2, 8, 20, 28, 50,
82 and 126 are produced. With the proton or neutron number at the magic
numbers, high separation energies are observed. For deformed nuclei, the
single particle energies are a function of the deformation parameter ε2 .
To describe nuclear properties at high temperature H. Bethe introduced
the Fermi gas model [3]. The nucleus is treated as a gas of non-interacting
fermions confined to the nuclear volume, and shell effects and pair correlations are neglected. Subsequently, the model has been modified by including residual interactions between the nucleons. In the low excitation region
long-range pair correlations play an important role and are roughly described
within the so-called back-shifted Fermi gas model [4].
A long-standing problem in experimental nuclear physics has been to
observe the transition from strongly paired states at zero temperature to
unpaired states at higher temperatures. At low excitation energy nuclear
structure depends on the residual long-range two-body interaction. The consequences of this interaction is the forming of J = 0 nucleon pairs, the
so-called Cooper pairs, where nucleons are moving in time reversed orbitals.
Thermal and rotational breaking of these nucleon pairs as a function of temperature T and angular frequency ω gives abrupt structural changes, such as
increased level density and rotational-spin alignment.
Pairing correlations have been successfully described by the BardeenCooper-Schrieffer (BCS) theory of superconductivity [5]. The sharp phase
transition of pairing correlations for infinite Fermi system of electrons in the
superconducting metal leads to the discontinuity of the heat capacity at the
critical temperature, which indicates a second-order phase transition. Consequently, the BCS theory was used in investigating thermodynamic properties
1
of nuclear pairing of warm nuclei [6, 7, 8, 9]. For a finite Fermi system such
as a nucleus, thermal and quantal fluctuations beyond the mean field become
large. The fluctuations wash out the discontinuity of the heat capacity in the
mean-field approximation, and as a result, an S-shape is formed [10]. The
quenching of pairing correlations has been obtained in recent theoretical approaches: the shell model Monte Carlo (SMMC) calculations [11, 12, 13], the
finite-temperature Hartree-Fock-Bogoliubov theory [14], and the relativistic
mean-field theory [15].
At low excitation energies the properties of discrete nuclear levels are
known for most stable nuclei from direct measurements. At higher excitation energies the density of nuclear levels rapidly increases, and individual
levels cannot be resolved experimentally. Instead, statistical models of the
properties of excited nuclei are invoked, and the nucleus is described in terms
of the energy, spin, and parity dependence of the level density. Numerous
theories exist for calculating nuclear level densities, typically having been
derived from thermodynamics and statistical mechanics arguments. All include a rapid (approximately exponential) increase in the level density with
increasing excitation energy. More sophisticated models, such as that of Ignatyuk [16], account for the dependence of the level density on shell effects
and rotational and vibrational collective effects which enhance the level density. One reason for the profusion of different level density theories is the
lack of experimental information to constrain them. The density at very low
excitation energies up to ∼ 2 MeV is studied in detail using spectroscopy
and counting of known, discrete levels [17], and the density at the region
around the neutron separation energy is known from neutron resonance measurements [18]. However, at other energies the experimental information is
sparse.
The radiative strength function (RSF) is a key for understanding nuclear
reaction rates in areas ranging from astrophysics to radiochemical diagnostics. Unresolved transitions in nuclear de-excitation processes are best described by statistical properties such as the RSF. However, the RSF shows
an additional Eγx dependency with x = 1 − 2 for γ energies in the 4 − 8 MeV
region. This feature is interpreted by Axel [19] as due to the collective giant
electric dipole resonance (GEDR), which represents the essential mechanism
for the γ-decay. Further studies reveal fine structures in the RSF, which in
the rare-earth region are commonly denoted as pygmy resonances [20, 21].
For lighter nuclei an unexpected large enhancement in the RSF was observed
at low γ energies [22]. It is clear that the present situation needs new experimental results.
The group at the Oslo Cyclotron Laboratory (OCL) has developed a
method to extract simultaneously the level density and the radiative strength
2
function from primary γ spectra [23]. The method is a further development
of the sequential extraction method described in [24, 25] and has been extensively tested in different regions of the nuclear chart in the last 15 years. The
level density and the radiative strength function reveal essential information
on nuclear structure. The level density is closely connected to the entropy
of the system at a certain excitation energy. When the entropy is known,
thermodynamic quantities such as temperature and heat capacity can be extracted. These quantities depend on the statistical properties of the nuclear
many-body system and may reveal additional information about pair correlations and phase transitions. The fine structures observed in the level density
enable us to obtain experimental values for the critical temperature of the
pair-breaking process.
In this work, nuclei from different parts of the nuclear chart has been
investigated. The nuclear level density is expected to have a smooth behavior
with respect to mass (A) and atomic number (Z), due to the liquid drop like
properties of the nucleus. It also has a quantum mechanical dependence
which exhibits an oscillatory behavior with respect to A and Z due to shell
effects. The latter arise from the finite size of the nucleus. The nuclear levels
of the individual nucleons in an average potential are bunched at certain
energies, leading to the shell structure. Hence, a nucleus with a closed shell
is somewhat more stable and harder to excite than a slightly heavier or
lighter nucleus. Interesting effects from the increasing single particle energy
spacings and from the change from spherical to deformed shapes can be
expected when approaching closed shells. The entropy differences between
odd-mass and even-even nuclei are also influenced by this situation.
The well-deformed rare-earth region appears to be ideal for studying nuclear properties without pronounced shell effects as a function of temperature. The single-particle Nilsson scheme displays almost uniformly distributed single-particle orbitals with both parities. The level-density parameter
in this mass region is rather constant, which can be explained by the very
uniform single particle level spacing. When shell effects are removed, the
level-density parameter shows the expected a ∝ A behavior [18].
3
Figure 1: Left: Raw (upper panel), unfolded (middle panel) and folded (lower
panel) γ spectra of 50 V. Right: Total, unfolded γ spectrum (upper panel),
second and higher generations γ spectrum (middle panel) and first-generation
γ spectrum (lower panel) of 50 V.
1.2
Experimental technique and methods
The experiments were carried out at the Oslo Cyclotron Laboratory by bombarding various targets with 3 He ions with beam currents of ∼ 2 nA for
1–2 weeks. The self-supporting targets with thicknesses of ∼ 2 mg/cm2 are
enriched to ∼ 95%. The particle-γ coincidences were measured with the
CACTUS multi-detector array. The charged ejectiles were detected by eight
particle telescopes placed at an angle of 45◦ relative to the beam direction.
An array of 28 collimated NaI γ-ray detectors with a total efficiency of ∼15%
surrounded the target and particle detectors.
The γ-ray spectra are recorded as a function of the initial excitation energy of the residual nucleus. This is accomplished by utilizing the known
reaction Q-values and kinematics. Using the particle-γ coincidence technique, each γ ray can be assigned to a cascade depopulating a certain initial
excitation energy in the residual nucleus. The data are therefore sorted
into total γ-ray spectra originating from different initial excitation-energy
bins. Each spectrum is then unfolded with the NaI response function using a Compton-subtraction method which preserves the fluctuations in the
original spectra and does not introduce further, spurious fluctuations [26].
In Fig. 1 (left panel), a typical γ spectrum taken from the 50 V coincidence
matrix is shown. The upper panel shows the raw γ spectrum, the middle
4
Figure 2: Left: Charged ejectile spectra for 93−98 Mo in coincidence with
γ-rays, labelled by the product nuclei. The arrows indicate the neutron
separation energy Bn . Right: γ-ray multiplicity Mγ (E) versus excitation
energy Eγ . The individual spectra are labelled by the product nuclei. Solid
and dashed lines represent (3 He,α) and (3 He,3 He′ ) reactions, respectively.
panel shows the unfolded spectrum, and the lower panel shows the folded
spectrum with the response functions. The top and bottom panels are in
excellent agreement, indicating that the unfolding method works very well.
From the unfolded spectra, a primary-γ matrix P (E, Eγ ) is constructed
using the subtraction method of Ref. [27]. The basic assumption of the Oslo
method, discussed in detail in Refs. [23, 28], is that the γ-ray energy distribution from any excitation energy bin is independent of how the states in this
bin have been populated. This assumption is valid for statistical γ decay,
which only depends on the γ-ray energy and the number of accessible final
states. Since the decay branchings are properties of the levels and do not
depend on the population mechanisms the assumption is trivially fulfilled if
one populates the same levels with the same weights within any excitation
energy bin. This is illustrated in Fig. 1, where the total, unfolded γ spectrum, the second and higher generations γ spectrum and the first-generation
spectrum of 50 V are shown. The first-generation spectrum is obtained by
subtracting the higher-generation γ rays from the total γ spectrum.
The (3 He,3 He′ γ) and (3 He,αγ) reactions have very different reaction mechanisms. This is demonstrated in Fig. 2, left part, where the particle spectra
5
in coincidence with γ rays show very different yields and peak structures. In
order to test whether the number of γ-rays per cascade depends on the reaction mechanism, the average γ-ray multiplicity h Mγ (E) i = E/ hEγ i as
a function of excitation energy E has been evaluated. The average γ-ray energy hEγ i is calculated from γ spectra selected at a certain energy E. In spite
of the different reaction mechanisms, the two reactions give similar results,
as seen from the right part of Fig. 2. This gives support to the applicability
of the Oslo method for both reactions.
The first generation (or primary) γ-ray matrix can be factorized according
to the Brink-Axel hypothesis [19, 29] as
P (E, Eγ ) ∝ ρ(E − Eγ )T (Eγ ),
(1)
where ρ is the level density and T is the radiative transmission coefficient.
The ρ and T functions can be determined by an iterative procedure [23]
through the adjustment of each data point of these two functions until a
global χ2 minimum with the experimental P (E, Eγ ) matrix is reached. It
has been shown [23] that if one solution for the multiplicative functions ρ
and T is known, one may construct an infinite number of other functions,
which give identical fits to the P matrix by
ρ̃(E − Eγ ) = A exp[α(E − Eγ )] ρ(E − Eγ ),
T̃ (Eγ ) = B exp(αEγ )T (Eγ ).
(2)
(3)
Consequently, neither the slope nor the absolute values of the two functions
can be obtained through the fitting procedure. Thus the parameters α, A
and B remain to be determined.
The parameters A and α can be determined by normalizing the level
density to the number of known discrete levels at low excitation energy [17]
and to the level density estimated from neutron-resonance spacing data at
the neutron binding energy E = Bn [18]. Since the experimental level-density
data points reach up to an excitation energy of only E ∼ Bn − 1 MeV, the
extrapolation is performed with the back-shifted Fermi-gas model [30, 31]
√
exp(2 aU )
ρBSFG (E) = η √
,
(4)
12 2a1/4 U 5/4 σI
where a constant η is introduced to adjust ρBSFG to the experimental level
density at Bn . The intrinsic excitation energy is estimated by U = E −
C1 − Epair , where C1 = −6.6A−0.32 MeV and A are the back-shift parameter
and mass number, respectively. The pairing energy Epair is based on pairing
gap parameters ∆p and ∆n evaluated from even-odd mass differences [32]
6
following the prescription of Dobaczewski et al. [33]. The level density
parameter is given by a = 0.21A0.87 MeV−1 .
The level density is assumed to have the standard energy and spin dependent parts
2J + 1 −(J+1/2)2 /2σ2
ρ(E, J) = ρ(E)
e
,
(5)
2σ 2
where σ is the spin cut-off parameter and an equal number of positive and
negative parity states is assumed. The spin cut-off parameter is calculated
as a function of the excitation energy by
σ = σ0 1 +
E − Bn 4(Bn − ∆)
(6)
where σ0 is the spin cut-off parameter at the neutron binding energy calculated according to [30]. This formula has the advantage that σ(E) remains
finite for all excitation energies and therefore no additional assumption for σ
below ∆ is necessary.
The absolute normalization of T is given by the determination of parameter B of Eq. (3). The experimental data on the average total radiative width
hΓγ i of neutron resonances at Bn is used for this purpose. The assumption is
that the γ-decay in the continuum is dominated by E1 and M 1 transitions.
For initial spin I and parity π at Bn , the width can be written in terms of
the transmission coefficient by the following [34]:
X
1
hΓγ i =
2ρ(Bn , I, π) I
f
Z
0
Bn
dEγ BT (Eγ )ρ(Bn − Eγ , If )
(7)
In reference [35], a detailed description of the calculation of the integral of Eq. (7) is given. Methodical difficulties in the primary γ-ray extraction prevent determination of the functions T (Eγ ) and ρ(E) in the interval
Eγ < 1 MeV and E > Bn − 1 MeV, respectively. In addition, the data at
the highest γ-energies, above Eγ ∼ Bn − 1 MeV, suffer from poor statistics.
However, the contribution of the extrapolations of ρ and T to the calculated
radiative width in Eq. (7) does not exceed 15% [35], thus the errors due to a
possibly poor extrapolation are expected to be of minor importance.
7
104
103
Level density, MeV
-
1
102
101
100
0
2
0
2
4
6
8
10
12
14
4
6
8
10
12
14
104
103
102
101
100
Excitation energy, MeV
Figure 3: Comparison of the nuclear level density extracted from neutron evaporation spectra (full circles) with discrete levels (upper panel) and
with nuclear level density (open circles) obtained from Oslo-type experiment
(lower panel).
Recently the nuclear level density has been measured independently by a
different kind of experiment; details are given in [36]. The 56 Fe level density
obtained from neutron evaporation spectra in the 55 Mn(d, n)56 Fe reaction is
compared to the level density extracted from the 57 Fe(3 He,αγ)56 Fe reaction
by the Oslo-type technique. This is demonstrated on Fig. 3. In spite of the
fact that the two methods use different underlying assumptions, different nuclear reactions and different mathematical techniques to extract the nuclear
level density, a fairly consistent result has been obtained.
8
1.3
Level density and thermodynamic properties
Employing statistical and thermodynamic concepts in the investigations of
mesoscopic systems such as atomic nuclei, is an area of active research. One
of the important aspects of these studies involve phase transitions. The distinguishing characteristic of a phase transition is an abrupt change in one or
more physical properties with a small change in a thermodynamic variable.
Phase transitions can be either dramatic - first-order, or smooth - secondorder. The first-order phase transitions are those that involve a latent heat
and quantities such as entropy and energy exhibit jumps in their temperature dependence. Second-order phase transitions have a discontinuity in the
second derivative of the free energy, and are characterized by a steady change
of some order parameter which vanishes at the transition temperature.
In the present investigation, thermodynamic studies have been performed
within the microcanonical and canonical statistical ensembles. The temperature is introduced in slightly different ways in the microcanonical statistical
ensemble (as a property of the system itself) and in the canonical statistical
ensemble (as imposed by a heat bath).
In the microcanonical ensemble theory, the important parameter of the
nucleus is the excitation energy E, which is conserved, since the system is
completely isolated. The multiplicity of states Ω(E) is directly proportional
to the level density and a spin-dependent factor (2hJ(E)i + 1), as
Ω(E) ∝ ρ(E) · (2hJ(E)i + 1),
(8)
where hJ(E)i is the average spin at excitation energy E. The experimentally measured level density in this work does not correspond to the true
multiplicity of states, since the (2J + 1) degeneracy of magnetic substates
is not included. If the average spin of levels hJi at any excitation energy
is known, this problem can be solved by multiplying an energy-dependent
factor (2hJ(E)i + 1) times the experimental level density. However, few experimental data exist on the spin distribution. A multiplicity Ω(E) based on
the experimental level density is defined as:
Ω(E) ∝ ρ(E),
(9)
and a pseudo-entropy based on the experimental level density, without the
(2J + 1) degeneracy, is utilized in the present work:
S(E) = kB ln Ω(E),
(10)
where Boltzmann’s constant is set to unity (kB = 1) for simplicity, and
Ω(E) = ρ(E)/ρ0 . The normalization denominator ρ0 is adjusted to give
9
S = ln Ω ∼ 0 in the ground state bands of the even-even nuclei in order to
fulfill the third law of thermodynamics: S(T → 0) = S0 . The fluctuations
in level spacings which are typical for small systems will make the entropy
sensitive to thermal changes. Small statistical fluctuations in the entropy S
may give rise to large contributions to the temperature T , which is defined
within the microcanonical ensemble as
−1
∂S
T (E) =
.
(11)
∂E
The heat capacity can be obtained by differentiating the temperature:
−1
∂T (E)
.
(12)
CV (E) =
∂E
V
The extraction of the microcanonical heat capacity CV (E) gives large fluctuations which are difficult to interpret [37]. Therefore, the heat capacity
has been calculated within the canonical ensemble where the energy of the
system may fluctuate but the temperature remains constant.
In order to analyze the criticality of low temperature transitions, we investigate the probability P of a system at a fixed temperature T to have the
excitation energy E, i.e.,
P (E, T ) = Ω(E) exp (−E/T ) /Z(T ),
where the canonical partition function is given by
Z ∞
Ω(E ′ ) exp (−E ′ /T ) dE ′ .
Z(T ) =
(13)
(14)
0
Lee and Kosterlitz have shown [38, 39] that for a fixed temperature T in
the vicinity of a critical temperature Tc of a structural transition the function A(E, T ) = − ln P (E, T ) will exhibit a characteristic double-minimum
structure at energies E1 and E2 . For the critical temperature Tc , one finds
A(E1 , Tc ) = A(E2 , Tc ). It can easily be shown that A is closely connected to
the Helmholtz free energy and the previous condition is equivalent to
Fc (E1 ) = Fc (E2 ),
(15)
which can be evaluated directly from the experimental data. Fc is a linearized
approximation to the Helmholtz free energy at the critical temperature Tc
according to
Fc (E) = E − Tc S(E).
(16)
10
Figure 4: Linearized Helmholtz free energy (data points with error bars) for
162
Dy at the critical temperature for the breaking of the first pair (upper
panel), for the breaking of further pairs (center panel) and for the breaking
of pairs in 161 Dy (lower panel) [40].
The critical value Tc for a potential phase transition is defined for the case in
which Eq. (16) exhibits a double-minimum structure where both minima are
equally deep. A double-minimum structure is typically caused by a locally
convex entropy. An illustration of this is shown in Fig. 4 where the condition
Fc (E1 ) = Fc (E2 ) = F0 is also fulfilled.
Using the method of the linearized Helmholtz free energy, a possible phase
transition in the low excitation region is most likely associated with the
breaking of the first pair in even-even nuclei.
In the higher excitation region further steps for transitions to higher quasiparticle regimes are washed out. The smearing in energy of the depairing
process in the presence of unpaired quasiparticles prohibits the emergence
of significant structures in Fc (E) for higher energies, and therefore does not
suggest a phase transition for these cases.
11
1.4
Models of the radiative strength function (RSF)
The radiative strength function is considered as a measure for the average
electromagnetic properties of nuclei and is fundamental for understanding
nuclear structure and reactions involving γ-rays.
The concept of radiative strength functions (RSF) was introduced in the
fundamental work of Blatt and Weisskopf [41]. The corresponding modelindependent definition of the RSF for the γ-ray transitions with energy Eγ
is given by
fXL =
Γi
Eγ2L+1 Di
.
(17)
where L is the multipolarity of the transition, X refers to the electric or
magnetic character of the transition, Γi is the partial radiative width and Di
is the level spacing.
Several models have been developed for the γ-ray strength functions fXL .
The theories behind the models are complicated, and are not presented here.
However, the resulting strength functions can be written in simple analytical forms. Various E1 and M 1 strength models that have been tested are
outlined below.
Experimentally, the main information on the γ-ray strength function has
been obtained from the study of photoabsorption cross-sections [42]. In the
Brink and Axel approach [19, 29], the E1 strength function is determined
by the properties of the giant electric dipole resonance (GEDR) around its
resonance energy, typically Eγ ∼ 10 − 15 MeV, by
fE1 (Eγ ) =
1
σE1 Eγ Γ2E1
,
2 2
3π 2 ~2 c2 (Eγ2 − EE1
) + Eγ2 Γ2E1
(18)
where σE1 , ΓE1 , and EE1 are the cross section, width, and the centroid of
the GEDR determined from photoabsorption experiments. However, serious
lack of information persists at lower γ-ray energies. It has been assumed that
the tail of the Lorentzian describing the GEDR determines the E1 strength
function at these energies. However, the experimental data on E1 RSF below
2 MeV show that the extrapolation of the GEDR to low energies fails to
describe the experimental values of the E1 strength function that indicates
a finite value of fE1 in the limit Eγ → 0.
As a result, a model for the E1 strength function was developed by Kadmenskii, Markushev and Furman (KMF) [43] which takes into account the
energy and temperature dependence of the GEDR width. Today, this model
and its empirical modifications [34] are frequently used in the description of
12
experimental data but at the same time the model needs additional experimental verification.
The E1 strength in the KMF model is given by
fE1 (Eγ ) =
0.7σE1 Γ2E1 (Eγ2 + 4π 2 T 2 )
1
,
2 2
)
3π 2 ~2 c2
EE1 (Eγ2 − EE1
(19)
where T is the temperature of the nucleus. We adopt the KMF model with
the temperature T taken as a constant to be consistent with our assumption
that the radiative strength function is independent of excitation energy. The
possible systematic uncertainty caused by this assumption is estimated in
Paper IV to have a maximum effect of 20% on the RSF. The width of the
GEDR is a sum of energy and temperature dependent parts
ΓE1 (Eγ , T ) =
ΓE1 2
(Eγ + 4π 2 T 2 ).
2
EE1
(20)
At T > 0, the KMF model gives a non-zero limit for Eγ = 0. The KMF
model is applicable only for the low-energy tail of the GEDR, since the model
diverges at Eγ ∼ EE1 . The giant dipole resonance is split into two parts for
deformed nuclei. Therefore, a sum of two strength functions each described
by the above equations is used.
The E1 strength function does not solely govern the γ-ray emission for
lower γ-ray energies. Other multipolarities, especially the M 1 strength function, play important roles as well. Experimental information on the γ-ray
strength of M 1 transitions is scarcer than for E1. It is commonly assumed
that the M 1 strength is well described by the Weisskopf model [41], where
the dipole γ-ray strength function is energy independent. However, some experiments indicate the existence of an M 1 giant resonance originating from
spin-flip excitations in the nucleus [44]. Also, the analysis of γ-ray spectra
from (n,γ) reactions [45] indicates that the use of the M 1 giant dipole resonance model gives a better fit to the experimental data than the Weisskopf
model.
The Lorentzian of M 1 radiation RSF, based on the existence of M 1 giant magnetic dipole resonance (GMDR), related to the spin-flip transition
between ℓ ± 12 single particle states
fM 1 (Eγ ) =
1
σM 1 Eγ Γ2M 1
2
2
2 2
3π 2 ~2 c2 (Eγ2 − EM
1 ) + Eγ ΓM 1
(21)
is adopted.
Although of minor importance, the E2 radiative strength fE2 has also
been included. Here, the Lorentzian E2 radiative strength
13
fE2 =
σEγ2 Γ2
1
,
5π 2 ~2 c2 Eγ2 (Eγ2 − E 2 )2 + Eγ2 Γ2
(22)
is used but with different resonance parameters and an additional factor
3/(5Eγ2 ). The resonance parameters for the E1, M 1, and E2 resonances are
taken from the compilation of Refs. [18, 42].
The total radiative strength function is taken to be a sum of E1, M 1,
and E2 radiative strength functions. In the conclusion section, these models
are compared to the experimental findings.
14
1.5
Survey of the papers
Paper I
Unique experimental information on level densities for eight rare earth
nuclei, i.e., 171,172 Yb, 166,167 Er, 161,162 Dy, and 148,149 Sm is utilized to extract
thermodynamic quantities in the microcanonical ensemble. The linearized
Helmholtz free energy is used to obtain the critical temperatures of the depairing process. In the even isotopes at excitation energies E < 2 MeV, the
Helmholtz free energy F signals for the transition from zero to two quasiparticles. For E > 2 MeV, the odd and even isotopes reveal a surprisingly
constant F at a critical temperature Tc ∼ 0.5 MeV, indicating the continuous melting of nucleon Cooper pairs as a function of excitation energy. The
clear absence of a double-minimum structure in Fc for this process is at variance with the presence of a first-order phase transition in the thermodynamic
sense.
Paper II
The level densities and radiative strength functions (RSFs) in 160,161 Dy
have been extracted using the (3 He,αγ) and (3 He,3 He′ γ) reactions, respectively. The entropy of 161 Dy follows parallel to the entropies of the even-even
160,162
Dy systems, assigning an entropy of ∼ 2 to the valence neutron. The
evolution of the probability density function with temperature is presented
for 160,161 Dy. The widths of these distributions increase anomalously in the
T = 0.5 − 0.6 MeV region. This feature of local increase in the canonical heat
capacity is a fingerprint of the depairing process. The gross properties of the
RSF are described by the giant electric dipole resonance. The RSFs show
a pygmy resonance superimposed on the tail of the giant dipole resonance.
The RSF at low γ-ray energies is discussed with respect to temperature dependency. Resonance parameters of a soft dipole resonance at Eγ ∼ 3 MeV
are deduced.
Paper III
Radiative strength functions (RSFs) for the 56,57 Fe nuclei below the neutron separation energy have been obtained from the 57 Fe(3 He, αγ)56 Fe and
57
Fe(3 He,3 He′ γ)57 Fe reactions, respectively. An enhancement of more than
a factor of ten over common theoretical models of the soft (Eγ . 2 MeV)
RSF for transitions in the quasicontinuum (several MeV above the yrast line)
has been observed. This enhancement cannot be explained by any present
theoretical model. The total RSF has been decomposed into a KMF model
for E1 radiation, Lorentzian models for M 1 and E2 radiation, and a power
law to model the soft pole. In a second experiment, two-step cascade intensities from the 56 Fe(n, 2γ)57 Fe reaction have been measured. Statistical-model
calculations based on separated RSFs from the decomposition of the exper15
imental total RSF and on experimental level densities from the Oslo-type
experiment have been performed. TSC intensities with soft primary transitions from the 56 Fe(n, 2γ)57 Fe reaction confirm the enhancement.
Paper IV
Radiative strength functions (RSFs) in 93−98 Mo have been extracted using
the (3 He,αγ) and (3 He,3 He′ γ) reactions. The RSFs are U-shaped as a function
of γ energy with a minimum at around Eγ = 3 MeV. The minimum values
increase with neutron number due to the increase in the low-energy tail of
the giant electric dipole resonance with nuclear deformation. The unexpected
strong increase in strength below Eγ = 3 MeV, here called soft pole, has been
established for all 93−98 Mo isotopes. The soft pole is present at all initial
excitation energies in the 5 − 8 MeV region. The multipolarity of the soft
pole radiation is unknown and there is still no theoretical explanation for
this very interesting phenomenon.
Paper V
Level densities for 93−98 Mo have been extracted using the (3 He,αγ) and
(3 He,3 He’γ) reactions. Data have been analyzed by utilizing both the microcanonical and the canonical ensemble. Structures in the microcanonical
temperature are consistent with the breaking of nucleon Cooper pairs. The
S-shape of the heat capacity curves found within the canonical ensemble is
interpreted as consistent with a pairing phase transition. A simple model
for the investigation and classification of the pairing phase transition in hot
nuclei has been employed and qualitative agreement with experimental data
has been achieved. Using the saddle-point approximation the experimental level densities of even-even and odd-even systems have been reproduced.
Estimates for the critical temperature of the pairing-phase transition yield
Tc ∼ 0.7–1.0 MeV.
Paper VI
The level densities and radiative strength functions (RSFs) of 50,51 V have
been extracted using the (3 He,αγ) and (3 He,3 He′ γ) reactions, respectively.
From the level densities microcanonical entropies have been deduced. The
entropy carried by the neutron hole in 50 V is estimated to be ∼ 1.2 kB , which
is less than the quasi-particle entropy of ∼ 1.7 kB found in rare-earth nuclei.
The high γ-energy part of the measured RSF fits well with the tail of the
giant electric dipole resonance. A significant enhancement over the predicted
strength in the region of Eγ . 3 MeV is seen. A similar enhancement has
also been seen in the iron and molybdenum isotopes (Paper III and Paper
V) which has not been given any theoretical explanation thus far.
16
2
Papers
Paper I M. Guttormsen, R. Chankova, M. Hjorth-Jensen, J. Rekstad, S.
Siem, A. Schiller, D.J. Dean,
Free energy and criticality in the nucleon pair breaking process
Phys. Rev. C68, 034311 (2003)
Paper II M.Guttormsen, A.Bagheri, R.Chankova, J.Rekstad, S.Siem, A.
Schiller, A. Voinov,
Thermal properties and radiative strengths in 160,161,162 Dy
Phys. Rev. C68, 064306 (2003)
Paper III A. Voinov, E. Algin, U. Agvaanluvsan, T. Belgya, R. Chankova,
M. Guttormsen, G. E. Mitchell, J. Rekstad, A. Schiller, S. Siem,
Large enhancement of radiative strength for soft transitions in the quasicontinuum
Phys. Rev. Lett. C93, 142504 (2004)
Paper IV M. Guttormsen, R. Chankova, U. Agvaanluvsan, E. Algin, L.A.
Bernstein, F. Ingebretsen, T. Loennroth, S. Messelt, G.E. Mitchell, J.
Rekstad, A. Schiller, S. Siem, A.C. Sunde, A. Voinov, S. Ødegård,
Radiative strength functions in 93−98 Mo
Phys. Rev. C 71 044307 (2005)
Paper V R. Chankova, M. Guttormsen, U. Agvaanluvsan, E. Algin, L.A.
Bernstein, F. Ingebretsen, T. Loennroth, S. Messelt, G.E. Mitchell, J.
Rekstad, A. Schiller, S. Siem, A.C. Sunde, A. Voinov, S. Ødegård,
Level densities and thermodynamical quantities of heated 93−98 Mo isotopes
Phys. Rev. C 73 034311 (2006)
Paper VI A.C. Larsen, R. Chankova, M. Guttormsen, F. Ingebretsen, T.
Loennroth, S. Messelt, J. Rekstad, A. Schiller, S. Siem, N.U.H. Syed,
A. Voinov, S.W. Ødegård,
Microcanonical entropies and radiative strength functions of 50,51 V
Phys. Rev. C (accepted for publication)
Paper VI might change slightly in the process of publication.
17
2.1
Free energy and criticality in the nucleon pair breaking process
PHYSICAL REVIEW C 68, 034311 ~2003!
Free energy and criticality in the nucleon pair breaking process
M. Guttormsen,* R. Chankova, M. Hjorth-Jensen, J. Rekstad, and S. Siem
Department of Physics, University of Oslo, N-0316 Oslo, Norway
A. Schiller
Lawrence Livermore National Laboratory, L-414, 7000 East Avenue, Livermore, California 94551, USA
D. J. Dean
Physics Division, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, Tennessee 37831, USA
~Received 19 September 2002; published 16 September 2003!
Experimental level densities for 171,172Yb, 166,167Er, 161,162Dy, and 148,149Sm are analyzed within the microcanonical ensemble. In the even isotopes at excitation energies E,2 MeV, the Helmholtz free energy F
signals the transition from zero to two quasiparticles. For E.2 MeV, the odd and even isotopes reveal a
surprisingly constant F at a critical temperature T c ;0.5 MeV, indicating the continuous melting of nucleon
Cooper pairs as function of excitation energy.
DOI: 10.1103/PhysRevC.68.034311
PACS number~s!: 21.10.Ma, 24.10.Pa, 25.55.Hp, 27.70.1q
I. INTRODUCTION
One of the most spectacular pairing phase transitions in
nature is the transition from a normal to a superconducting
phase in large electron systems. The transition is triggered at
low temperature by massive pairing of two and two electrons
into spin J50 pairs, so-called Cooper pairs @1#.
For atomic nuclei, the pairing phase transition is expected
to behave differently. First of all, the nucleus is an isolated,
few body system with two species of fermions. Surface effects are prominent and the coherence length of nucleons
coupled in Cooper pairs is larger than the nuclear diameter.
Furthermore, there are non-negligible energy spacings between the single-particle orbitals. All these facts make the
nucleus an inherently small system. Also, other types of residual interactions than pairing are of importance. The influence of these peculiar constraints on the nucleus has been
investigated theoretically for a long time @2–5#, however,
only limited experimental information is available to describe the nature of pairing within the nucleus.
The Oslo group has developed a method to derive simultaneously the level density and g -ray strength function from
a set of primary g -ray spectra @6#. The method has been well
tested and today a consistent dataset for eight rare earth nuclei is available. In the present work we report for the first
time on a comprehensive analysis of the evolution of the
pairing phase transition as a function of the nuclear excitation energy.
tor array using the pickup ( 3 He, ag ) reaction on 172,173Yb,
167
Er, 162,163Dy, and 149Sm targets and the inelastic
( 3 He, 3 He’g ) reaction on 167Er and 149Sm targets. The
charged ejectiles were detected with eight DE –E particle
telescopes placed at an angle of 45° relative to the beam
direction. Each telescope comprises one Si front and one
Si~Li! back detector with thicknesses of 140 and 3000 m m,
respectively. An array of 28 NaI g -ray detectors with a total
efficiency of ;15% surrounds the target and particle detectors. From the reaction kinematics, the measured ejectile energy can be transformed into excitation energy E. Thus, each
coincident g ray can be assigned to a g cascade originating
from a specific energy E. These spectra are the basis for the
extraction of level density and g -strength function as described in Ref. @6#. Several interesting applications of the
method have been demonstrated, see, e.g., Refs. @7–10#.
The level densities for 171,172Yb, 166,167Er, 161,162Dy, and
148,149
Sm are shown in Fig. 1. The level densities are normalized at low excitation energies where ~almost! all levels
are known, and at the neutron binding energy B n where the
level density can be estimated from neutron-resonance spacings. The spin window populated in the reactions is typically
I;2\ –6\. Already, three general comments can be made to
II. EXPERIMENTAL LEVEL DENSITIES
Level densities for 171,172Yb, 166,167Er, 161,162Dy, and
Sm have been extracted from particle-g coincidences.
The experiments were carried out with 45-MeV 3 He projectiles accelerated by the MC-35 cyclotron at the University of
19
Oslo. The data were recorded with the CACTUS multidetec148,149
FIG. 1. Experimental level densities for the nuclei 171,172Yb,
Er, 161,162Dy, and 148,149Sm. The data are taken from Refs.
@7,8,10#.
166,167
*Electronic address: magne.guttormsen@fys.uio.no
0556-2813/2003/68~3!/034311~6!/$20.00
68 034311-1
©2003 The American Physical Society
PHYSICAL REVIEW C 68, 034311 ~2003!
M. GUTTORMSEN et al.
these data: ~i! above 2 MeV excitation energy, all level densities are very linear in a log plot, suggesting a so-called
constant-temperature level density, ~ii! the level densities of
the odd-even isotopes are larger than for their neighboring
even-even isotopes, and ~iii! the even-even isotopes show a
strong increase in level density between 1 and 2 MeV, indicating the breaking of Cooper pairs.
It should be noted that the transitions considered here are
low temperature phenomena. The 171,172Yb, 166,167Er, and
161,162
Dy nuclei have well deformed shapes, and various calculations in this mass region @11–14# indicate that the transition from deformed to spherical shape occurs at much
higher temperatures than the temperature at which the first
pairs break. However, for nuclei closer to the N582 shell
gap, e.g., 148,149Sm, the coexistence between deformed and
spherical shapes at low temperatures cannot be excluded, as
discussed in Ref. @15#.
III. FREE ENERGY AND CRITICAL TEMPERATURE
The statistical microcanonical ensemble is an appropriate
working frame for describing an isolated system such as the
nucleus. In this ensemble the excitation energy E is fixed, in
accordance with the observables of our experiments. The microcanonical entropy is given by the number of levels V at E
S ~ E ! 5k B ln V ~ E ! ,
~1!
where the multiplicity V is directly proportional to the level
density r by V(E)5 r (E)/ r 0 . The normalization denominator r 0 is adjusted to give S;0 for T;0, which fulfills the
third law of thermodynamics. Here, we assume that the lowest levels of the ground state bands of the 172Yb, 166Er,
162
Dy, and 148Sm nuclei have temperatures close to zero,
giving on the average r 0 52.2 MeV21 . In the following, this
value is used for all eight nuclei and Boltzmann’s constant is
set to unity (k B 51).
In order to analyze the criticality of low temperature transitions, we investigate the probability P of a system at the
fixed temperature T to have the excitation energy E, i.e.,
2E/T
P ~ E,T,L ! 5
V~ E !e
Z~ T !
,
~2!
where the canonical partition function is given by Z(T)
5 * `0 V(E 8 )exp(2E8/T) dE8. Implicitly, the multiplicity of
states V(E) depends on the size of the system, denoted by L.
Often, it is more practical to use the negative logarithm of
this probability A(E,T)52ln P(E,T), where in the following
we omit the L parameter. Lee and Kosterlitz showed @16,17#
that the function A(E,T), for a fixed temperature T in the
vicinity of a critical temperature T c of a structural transition,
will exhibit a characteristic double-minimum structure at energies E 1 and E 2 . For the critical temperature T c , one finds
A(E 1 ,T c )5A(E 2 ,T c ). It can be easily shown that A is
20
closely connected to the Helmholtz free energy and the previous condition is equivalent to
F c ~ E 1 ! 5F c ~ E 2 ! ,
~3!
FIG. 2. Schematic representation of the entropy S in units of the
single-particle entropy s ~top panel! for even-even ~solid line!, oddmass ~dash-dotted line!, and odd-odd ~dashed line! nuclei. For the
purpose of the figure, the steps in entropy are drawn slightly staggered in energy. Lower panel: linearized Helmholtz free energy F c
at the critical temperature T c of even-even, odd-mass, and odd-odd
nuclei. All energies are measured in units of the pairing gap parameter D. The dotted lines indicate the situation if additional levels are
included below the steps in entropy.
a condition which can be evaluated directly from our experimental data. Here, it should be emphasized that F c is a linearized approximation to the Helmholtz free energy at the
critical temperature T c according to F c (E)5E2T c S(E),
thereby avoiding the introduction of a caloric curve T(E).
The free-energy barrier at the intermediate energy E m between E 1 and E 2 is given by
DF c 5F c ~ E m ! 2F c ~ E 1 ! .
~4!
Now, the evolution of DF c with increasing system size L
may determine the order of a possible phase transition
@16,17#. These ideas have, e.g., recently been applied to analyze phase transitions in a schematic pairing model @18#.
Figure 2 displays a schematic description of the entropy
for even-even, odd-mass, and odd-odd nuclei as function of
excitation energy. In the lower excitation energy region of
the even-even nucleus, only the ground state is present, and
above E;2D the level density is assumed to follow a
constant-temperature formula. It has been shown @19,20# that
the single-particle entropy s is an approximately extensive
thermodynamical quantity in nuclei at these temperatures.
The increase in entropy at the breaking of the first proton or
neutron pair, i.e., at E52D, is roughly 2s in total for the two
newly created unpaired nucleons. The requirement F c (E 1 )
5F c (E 2 ), at temperature T c , gives E 2 2E 1 5T c @ S(E 2 )
2S(E 1 ) # . Thus, with the assumed estimates above, we obtain the relation D5T c s, which may be used to extract the
034311-2
PHYSICAL REVIEW C 68, 034311 ~2003!
FREE ENERGY AND CRITICALITY IN THE NUCLEON . . .
FIG. 3. Same as previous figure in the case of an even-even
nucleus but for unequal proton and neutron fluids. Curves are given
for the neutron fluid alone ~dashed lines with pairing gap parameter
D and single-particle entropy s), the proton fluid alone ~dotted lines
with 1.1D and 0.9s), and the composite system ~solid lines!.
critical temperature for the pairing transition. Adopting typical values of D51 MeV and s52k B @19,20#, we obtain T c
50.5 MeV. For the odd-mass case, one starts out with one
quasiparticle which gives roughly one unit of single-particle
entropy s around the ground state. The three quasiparticle
regime appears roughly at E52D with a total entropy of 3s.
The region between E50 and 2 MeV is modeled with a step
at ;2 MeV, however, in real nuclei the level density is almost linear in a log plot for the whole excitation energy
region due to the smearing effects of the valence nucleon. In
the case of an odd-odd nucleus, one starts out with two units
of single-particle entropy. The two valence nucleons represent a strong smearing effect on the level density and the
modeled step structure in entropy at E52D for the onset of
the four quasiparticle regime is completely washed out.
In the higher excitation region, further steps for transitions to higher quasiparticle regimes are also washed out due
to the strong smearing effects of the already present unpaired
nucleons. The slope of the entropy with excitation energy is
determined by two competing effects: the quenching of pairing correlations which drives the cost in energy lower for the
breaking of additional pairs and the Pauli blocking which
reduces the entropy created per additional broken pair. The
competing influence of both effects is modeled by a
constant-temperature level density with the same slope for
all three nuclear systems and a slightly higher critical temperature. In this region of the model, there are infinitely
many excitation energies where the relation F c (E 1 )
21
5F c (E 2 ) is fulfilled.
The breaking of proton or neutron pairs are thought to
take place at similar excitation energies due to the approximate isospin symmetry of the strong interaction. It is indeed
FIG. 4. Linearized Helmholtz free energy at the critical temperature T c . The constant level F 0 connecting the two minima is indicated by lines.
commonly believed that the pairing gap parameter D and,
thus, the critical temperature T c for the breaking of Cooper
pairs, are approximately the same for protons and neutrons.
Furthermore, interactions between protons and neutrons will
certainly wash out any differences in behavior between the
proton and neutron fluids. In Fig. 3, the influence of differences in proton and neutron pair breaking is investigated
within our schematic model. Here, we assume that neutrons
breakup at 2D creating an entropy of 2s. The protons are
assumed to breakup at 10% higher excitation energy ~since
Z,N) creating 10% less entropy ~due to the larger proton
single-particle level spacing!. The entropy of the total system
of either proton or neutron pair breaking gives
S ~ E ! 5ln@ e S p (E) 1e S n (E) # 2ln 2,
~5!
where the last term assures that S50 in the ground state
band. The requirement F c (E 1 )5F c (E 2 ) gives T (n)
c 5D/s for
neutrons ~as in Fig. 2! and T (p)
c 51.1D/0.9s51.22D/s for
protons. In the combined system of both neutrons and pro(p)
tons, a value of T c 5(T (n)
c 1T c )/251.11D/s is deduced
from Fig. 3. Thus, typical fluctuations in the pairing gap
parameter and the single-particle entropy for neutrons and
protons give only small changes in the extracted critical temperature.
IV. EXPERIMENTAL RESULTS
In order to experimentally investigate the behavior for
even isotopes, linearized free energies F c for certain temperatures T c are displayed in Fig. 4. The data clearly reveal
two minima with F c (E 1 )5F c (E 2 )5F 0 , which is due to the
034311-3
PHYSICAL REVIEW C 68, 034311 ~2003!
M. GUTTORMSEN et al.
general increase in level density around E;2 MeV, as schematically shown in Fig. 2. For all nuclei, we obtain E 1 ;0
and E 2 ;2 MeV which compares well with 2D. We interpret
the results of Fig. 4 as the transition due to the breaking of
the very first nucleon pairs. The deduced critical temperatures are T c 50.47, 0.40, 0.47, and 0.45 MeV for 172Yb,
166
Er, 162Dy, and 148Sm, respectively.
Recently @7#, another method was introduced to determine
the critical temperatures in the canonical ensemble. Here, the
constant-temperature level density formula for the canonical
heat capacity C V (T)5(12T/ t ) 22 was fitted to the data in
the temperature region of 0–0.4 MeV corresponding to excitation energies between 0 and 2 MeV, and the fitted temperature parameter t was then identified with the critical
temperature T c . Since a constant temperature level density
formula implies a constant linearized Helmholtz free energy
F c (E) ~provided t 5T c ), this former method is almost
equivalent to the present method, i.e., of identifying the temperature T c for which the linearized Helmholtz free energy is
on average constant. Therefore, it is not surprising that the
extracted critical temperatures T c 50.49, 0.44, 0.49, and 0.45
MeV for the respective nuclei using the older method
@7,8,10# coincide well with the critical temperatures presented in this work. However, while the previous method
was based on an ad hoc assumption of the applicability of a
constant-temperature level density formula, the present
method has a much firmer theoretical foundation.
The height of the free-energy barrier should show a different dependence on the system size L according to the order of a possible phase transition @16,17#. The barriers deduced from Fig. 4 yield DF c ;0.5–0.6 MeV, values which
seem not to have any systematic dependence on the mass
number A within the experimental uncertainties. Even with
better data, an unambiguous dependence of the barrier height
on the system size would be unlikely when using A as a
measure for the parameter L since the relevant system size
for the very first breaking of Cooper pairs might be characterized by only a few valence nucleons. Another complicating interference is that other properties of the nuclear system
which might influence the onset of pair breaking also change
with mass number, e.g., deformation, pairing gap, and locations of single-particle levels around the Fermi surface.
In the schematic model of Fig. 2, we would expect a
free-energy barrier of DF c 52D;2 MeV at E52D
;2 MeV. However, the data are more consistent with the
dotted lines of Fig. 2 indicating a smoother behavior around
the expected steps due to the existence of collective excitations such as rotation and b , g , and octupole vibrations between 1 and 2 MeV for the even-even nuclei, and due to the
increasing availability of single-particle orbitals for the odd
nucleon in the case of odd nuclei. Thus, we expect the centroid of the barrier to be shifted down in energy with a corresponding proportional reduction of the barrier height, and
an inspection of Fig. 4 indeed shows that the free-energy
barrier is 0.5–0.6 MeV at ;0.6 MeV excitation energy for
22
the even-even nuclei. A similar analysis of the odd isotopes
is difficult to accomplish since there seems not to be any
common structures. Here, the unpaired valence neutron
smears out the effects of the depairing process too much to
FIG. 5. Linearized Helmholtz free energy at the critical temperature T c . The fitted constant level F 0 is indicated by lines.
be visible in the present data. However, it has been attempted
in Ref. @8# to interpret the structure in the level density of
167
Er around 1 MeV in terms of a first-order phase transition.
The smearing effect is expected to be even more pronounced for the breaking of additional pairs. Figure 5 shows
the linearized Helmholtz free energy for all eight nuclei investigated, but at slightly higher critical temperatures than in
Fig. 4. The critical temperature T c is found by a least x 2 fit
of a constant value F 0 to the experimental data. The fit region is from E52 MeV up to 5 MeV and 7 MeV for the odd
and even isotopes, respectively, giving normalized x 2 values
in the range from 0.5 to 2.5. Here, instead of a doubleminimum structure, a continuous ‘‘minimum’’ of F c is displayed for several MeV. This observation allows us to conclude that the further depairing process cannot under any
circumstances be interpreted as an abrupt structural change
in the nucleus typical for a first-order phase transition. The
constant lines of Fig. 5 visualize how surprisingly well F 0
fits the data: the deviations are typically less than 100 keV.1
The ongoing breaking of further Cooper pairs overlapping in
excitation energies above 2 MeV is therefore contrary to
what is found in the schematic model of Ref. @18#. This is
probably due to strong residual interactions in real nuclei,
such as the quadrupole-quadrupole interaction, which were
not taken into account in the model calculation. Thus
~nearly!, all excitation energies above 2 MeV will energetically match with the costs of breaking nucleon pairs. Here,
1
This fact might also settle the discussion in Ref. @8# and discard
the possible interpretation of the many negative branches of the
microcanonical heat capacity observed in Fig. 8 of this reference as
indicators of separate first-order phase transitions.
034311-4
PHYSICAL REVIEW C 68, 034311 ~2003!
FREE ENERGY AND CRITICALITY IN THE NUCLEON . . .
of the mass number A being one higher or lower than the
neighboring even isotope. We also observe that since the
148,149
Sm nuclei are not midshell nuclei, they show less entropy, reflecting the lower single-particle level density when
approaching the N582 shell gap. By evaluating the oddeven difference d S5S odd2S even , we find d S;2 for all four
isotopes, as shown in the lower panel of Fig. 6. This means
that excited holes and particles have the same degree of freedom with respect to the even-mass nuclei.
V. CONCLUSION
FIG. 6. Experimental entropy evaluated in the microcanonical
ensemble at excitation energy E54 MeV and temperature T c . In
the lower panel the odd-even difference d S5S odd2S even is displayed for the four isotopes.
all excess energy goes to the process of breaking pairs. Since
the gain in entropy dS is proportional to dE, the microcanonical temperature, T(E)5(dS/dE) 21 , remains constant
as function of excitation energy, and the level density displays a straight section in the log plot.
At higher excitation energies than measured here, the
pairing correlations vanish and the system behaves more like
a Fermi gas. Here, the free energy will indicate the closing
stage of the depairing process by increasing F c , with F c
.F 0 . However, in this regime also shape transitions and
fluctuations as well as the melting of the shell structure may
play a role and give deviations from a simple Fermi gas
model with r }exp(2AaE), a being the level density parameter. Unfortunately, these very interesting phenomena cannot
be investigated with the present experimental data.
The fitted value F 0 contains information on the entropy of
the system at T c through S5(E2F 0 )/T c . In Fig. 6, we have
compared the entropy for the various nuclei at an excitation
energy E54 MeV, an energy where all nuclei seem to ‘‘behave’’ equally well ~see Fig. 5!. Figure 6 also shows that the
odd-mass nuclei display generally higher entropy regardless
@1# J. Bardeen, L.N. Cooper, and J.R. Schrieffer, Phys. Rev. 108,
1175 ~1957!.
@2# Mitsuo Sano and Shuichiro Yamasaki, Prog. Theor. Phys. 29,
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23
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~1980!.
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@6# A. Schiller, L. Bergholt, M. Guttormsen, E. Melby, J. Rekstad,
Unique experimental information on level densities for
eight rare earth nuclei is utilized to extract thermodynamic
quantities in the microcanonical ensemble. The linearized
Helmholtz free energy is used to obtain the critical temperatures of the depairing process. For a critical temperature just
below T c ;0.5 MeV, we observe a structural transition of
even nuclei in the E5022 MeV region due to the breaking
of the first nucleon pair. Unfortunately, it was not possible to
use the development of the barrier height DF c with the size
of the system L to conclude on the presence of a thermodynamical phase transition and its order. The critical temperature for the melting of other pairs is found at slightly higher
temperatures. Here, we obtain a surprisingly constant value
for the linearized Helmholtz free energy, indicating a continuous melting of nucleon Cooper pairs as function of excitation energy. The conspicuous absence of a doubleminimum structure in F c for this process is at variance with
the presence of a first-order phase transition in the thermodynamical sense. The entropy difference between odd and
even systems is found to be constant with respect to excitation energy and is consistent with the expected values of the
single-particle entropy in these nuclei.
ACKNOWLEDGMENTS
Financial support from the Norwegian Research Council
~NFR! is gratefully acknowledged. Part of this work was
performed under the auspices of the U.S. Department of Energy by the University of California, Lawrence Livermore
National Laboratory under Contract No. W-7405-ENG-48.
Research at Oak Ridge National Laboratory was sponsored
by the Division of Nuclear Physics, U.S. Department of Energy under Contract No. DE-AC05-00OR22725 with UTBattelle, LLC.
and S. Siem, Nucl. Instrum. Methods Phys. Res. A 447, 498
~2000!.
@7# A. Schiller, A. Bjerve, M. Guttormsen, M. Hjorth-Jensen, F.
Ingebretsen, E. Melby, S. Messelt, J. Rekstad, S. Siem, and
S.W. O
” degård, Phys. Rev. C 63, 021306 ~2001!.
@8# E. Melby, M. Guttormsen, J. Rekstad, A. Schiller, S. Siem, and
A. Voinov, Phys. Rev. C 63, 044309 ~2001!.
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034311-5
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Schiller, and A. Voinov, Phys. Rev. C 65, 044318 ~2002!.
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2.2
Thermal properties and radiative strengths in 160,161,162 Dy
PHYSICAL REVIEW C 68, 064306 (2003)
Thermal properties and radiative strengths in
160,161,162
Dy
M. Guttormsen,* A. Bagheri, R. Chankova, J. Rekstad, and S. Siem
Department of Physics, University of Oslo, N-0316 Oslo, Norway
A. Schiller
Lawrence Livermore National Laboratory, L-414, 7000 East Avenue, Livermore, California 94551, USA
A. Voinov
Frank Laboratory of Neutron Physics, Joint Institute of Nuclear Research, 141980 Dubna, Moscow region, Russia
(Received 9 July 2003; published 17 December 2003)
The level densities and radiative strength functions (RSFs) in 160,161Dy have been extracted using the
s3He, agd and s3He, 3He8gd reactions, respectively. The data are compared to previous measurements on
161,162
Dy. The energy distribution in the canonical ensemble is discussed with respect to the nucleon Cooper
pair breaking process. The gross properties of the RSF are described by the giant electric dipole resonance. The
RSF at low g-ray energies is discussed with respect to temperature dependency. Resonance parameters of a soft
dipole resonance at Eg ,3 MeV are deduced.
DOI: 10.1103/PhysRevC.68.064306
PACS number(s): 21.10.Ma, 24.10.Pa, 25.55.Hp, 27.70.1q
I. INTRODUCTION
The well-deformed rare earth region appears to be ideal
for studying statistical properties of nuclei as a function of
temperature. The single particle Nilsson scheme displays almost uniformly distributed single particle orbitals with both
parities. However, the low-temperature thermal properties of
these nuclei are only poorly known. The main reason for this
is the lack of appropriate experimental methods.
The Oslo Cyclotron group has developed a method to
extract first-generation (primary) g-ray spectra at various initial excitation energies. From such a set of primary spectra,
nuclear level density and radiative strength function (RSF)
can be extracted [1,2]. These two functions reveal essential
nuclear structure information such as pair correlations and
thermal and electromagnetic properties. In the last couple of
years, the Oslo group has demonstrated several fruitful applications of the method [3–11].
The subject of this work is to perform a systematic and
consistent analysis of the three 160,161,162Dy isotopes. Since
the proton number sZ=66d and the nuclear deformation sb
,0.26d are equal for these cases, we expect to find the same
electromagnetic properties. Furthermore, the underlying uniform distribution of single particle Nilsson states should
from a statistical point of view give similar level densities
for 160Dy and 162Dy. The present dataset also allows us to
check the results using the s3He, agd and s3He, 3He8gd reactions for one and the same residual nucleus.
In Sec. II an outline of the experimental procedure is
given. The thermal aspects of the level density and RSF are
discussed in Secs. III and IV, respectively. Finally, concluding remarks are given in Sec. V.
II. EXPERIMENTAL METHOD
coincidences for 160,161,162Dy were measured with the CACTUS multidetector array. The charged ejectiles were detected
with eight particle telescopes placed at an angle of 45°
relative to the beam direction. An array of 28 NaI g-ray detectors with a total efficiency of ,15% surrounded the target
and particle detectors. The following five reactions were uti161
Dys3He, agd160Dy, 161Dys3He, 3He8gd161Dy,
lized:
162
3
161
Dys He, agd Dy, 162Dys3He, 3He8gd162Dy,
and
163
Dys3He, agd162Dy. The three latter reactions have been reported earlier [3,4,7]. The reaction spin windows are typically I,2–6 ". The self-supporting targets are enriched to
,95% with thicknesses of ,2 mg/cm2. The experiments
were run with beam currents of ,2 nA for 1–2 weeks.
The experimental extraction procedure and the assumptions made are described in Refs. [1,2], and references
therein. For each initial excitation energy E, determined from
the ejectile energy, g-ray spectra are recorded. These spectra
are the basis for making the first-generation (or primary)
g-ray matrix [12], which is factorized according to the
Brink-Axel hypothesis [13,14] as
PsE, Egd ~ rsE − EgdTsEgd.
Here, r is the level density and T is the radiative transmission coefficient.
The r and T functions can be determined by an iterative
procedure [2] through the adjustment of each data point of
these two functions until a global x2 minimum with the experimental PsE, Egd matrix is reached. It has been shown [2]
that if one solution for the multiplicative functions r and T is
known, one may construct an infinite number of other functions, which give identical fits to the P matrix by
The experiments were carried out with 45-MeV 3He
ions at the Oslo Cyclotron Laboratory. Particle-g
26
*Electronic address: magne.guttormsen@fys.uio.no
0556-2813/2003/68(6)/064306(10)/$20.00
s1d
r̃sE − Egd = A expfasE − Egdg rsE − Egd,
s2d
T̃sEgd = B expsaEgdTsEgd.
s3d
Consequently, neither the slope nor the absolute values of
the two functions can be obtained through the fitting pro68 064306-1
©2003 The American Physical Society
PHYSICAL REVIEW C 68, 064306 (2003)
M. GUTTORMSEN et al.
FIG. 1. Average experimental spin distributions (data points
with error bars) compared to Eq. (6). The data include 130 nuclei
along the b-stability line in the A=150–170 mass region.
cedure. Thus, the parameters a, A, and B remain to be
fixed.
The parameters A and a can be determined by normalizing the level density to the number of known discrete levels
at low excitation energy [15] and to the level density estimated from neutron-resonance spacing data at the neutron
binding energy E=Bn [16]. The procedure for extracting the
total level density r from the resonance energy spacing D is
described in Ref. [2]. Since our experimental level density
data points only reach up to an excitation energy of E,Bn
−1 MeV, we extrapolate with the back-shifted Fermi-gas
model [17,18]
rBSsEd = h
exps2ÎaUd
12Î2a1/4U5/4sI
,
s4d
where a constant h is introduced to fix rBS to the experimental level density at Bn. The intrinsic excitation energy
is
estimated
by
U = E − C1 − Epair,
where
C1
= −6.6A−0.32 MeV and A are the back-shift parameter and
mass number, respectively. The pairing energy Epair is
based on pairing gap parameters D p and Dn evaluated from
even-odd mass differences f19g according to Ref. f20g.
The level density parameter is given by a
= 0.21A0.87 MeV−1. The spin-cutoff parameter sI is given
by sI2 = 0.0888aTA2/3, where the nuclear temperature is described by
T = ÎU/a.
s5d
FIG. 2. Level densities estimated from neutron resonance level
spacings at Bn. The data are plotted as a function of intrinsic excitation energy Un =Bn −C1 −sDp +Dnd. The unknown level density for
160
Dy (open circle) is estimated from the line determined by a least
x2 fit to the data points.
gsE, Id =
2I + 1
2sI2
expf− sI + 1/2d2/2sI2g,
s6d
which is normalized to oI gsE, Id , 1. Figure 1 compares
gsE, Id to the spin distributions of levels with known spin
assignments f15g for nuclei along the b-stability line with
A = 150– 170. Although these data are incomplete and include systematical errors,1 the agreement is gratifying and
supports the expressions adopted for sI and g.
Unfortunately, 159Dy is unstable and no information exists
on the level density at E=Bn for 160Dy. Therefore, we estimate the value from the systematics of other even-even dysprosium and gadolinium isotopes. In order to bring these
data on the same footing, we plot the level densities as a
function of intrinsic energy U. From the systematics of Fig.
2, we estimate for 160Dy a level density of rsBnd=s9.7±2.0d
3106 MeV−1. Figure 3 demonstrates the level density normalization procedure for the 160Dy case.
The level densities extracted from the five reactions are
displayed in Fig. 4. The data have been normalized as prescribed above, and the parameters used for 160,161,162Dy in
Eq. (4) are listed in Table I. The level densities for the three
reactions previously published [3,4,7] deviate slightly since
we here have used updated and newly recommended data
[15,16]. The results obtained with the very different reactions
s3He, ad and s3He, 3He8d, are almost identical, except for the
level density of the ground state band in 162Dy. Here, the
27
In cases where the intrinsic excitation energy U becomes
negative, we put U = 0, T = 0, and sI = 1. The spin distribution of levels swith equal energyd is given by f17g
1
One typical shortcoming of these compilations are that high spin
members of rotational bands are over-represented compared to low
spin band heads.
064306-2
PHYSICAL REVIEW C 68, 064306 (2003)
THERMAL PROPERTIES AND RADIATIVE STRENGTHS…
FIG. 3. Normalization procedure of the experimental level density (data points) of 160Dy. The data points between the arrows are
normalized to known levels at low excitation energy (histograms)
and to the level density at the neutron-separation energy (open
circle) using the Fermi-gas level density (line).
s3He, 3He8d reaction overestimates the level density, as has
been discussed previously [4].
III. LEVEL DENSITY AND THERMAL PROPERTIES
The level densities of 160Dy and 162Dy are very similar,
however, 161Dy reveals several times higher level densities.
In a previous work [6], it was claimed that the entropy for
the excited quasiparticles is approximately extensive. To investigate this assumption further, we express the entropy as
SsEd = kB ln VsEd,
s7d
where Boltzmann’s constant is set to unity skB = 1d. The
multiplicity V is directly proportional to the level density
by VsEd = rsEd/r0. The ground state of even-even nuclei
represents a well-ordered system with no thermal excitations and is characterized with zero entropy and temperature. Therefore, the normalization denominator r0 is adjusted to give S = ln V , 0 in the ground state band region.
This ensures that the ground band properties fulfill the
third law of thermodynamics with SsT → 0d = 0. The same
extracted r0 is used for the odd-mass neighboring nuclei.
Figure 5 shows the entropies S for the two new reactions
reported in this work, i.e., the s3He, agd160Dy and
s3He, 3He8gd161Dy reactions. The results for the other reactions are very similar and are therefore not discussed here.
The entropy of the 161Dy nucleus is seen to display an almost
constant entropy excess compared to 160Dy. The difference,
28
DS,2, represents the entropy carried by the valence neutron
outside the even-even 160Dy core (or hole coupled to the
162
Dy core). It is an interesting feature that this difference is
almost independent of excitation energy and therefore, of the
FIG. 4. Normalized level densities for 160,161,162Dy. The filled
and open circles are measured with the s3He, ad and s3He, 3He8d
reactions, respectively.
number of quasiparticles excited in dysprosium, thus manifesting an entropy of Sqp ,2 assigned to each quasiparticle.
The concept of temperature in small systems has been
discussed extensively in the literature. Traditionally, temperature is introduced in slightly different ways in the microcanonical statistical ensemble [as a property of the system
itself by means of T=sdS/dEdV−1] and in the canonical statistical ensemble (as imposed by a heat bath). The temperatureenergy relations for rare earth nuclei (the caloric curves) derived within the two statistical ensembles display in general a
very different behavior since the nuclei under discussion are
064306-3
PHYSICAL REVIEW C 68, 064306 (2003)
M. GUTTORMSEN et al.
TABLE I. Parameters used for the back-shifted Fermi-gas level density.
Nucleus
160
Dy
161
Dy
162
Dy
a
Epair
(MeV)
a
sMeV−1d
C1
(MeV)
Bn
(MeV)
1.945
0.793
1.847
17.37
17.46
17.56
–1.301
–1.298
–1.296
8.576
6.453
8.196
D
(eV)
rsBnd
s106 MeV−1d
h
27.0(50)
2.4(2)
9.7(20)a
2.14(44)
4.96(59)
1.57
1.19
0.94
Estimated from systematics.
essentially discrete systems [3]. The microcanonical temperature can, e.g., yield violent fluctuations as a function of
excitation energy giving mostly unphysical results such as
negative heat capacities (decreasing temperature with increasing excitation energy) and even negative branches of
temperature. Also the canonical caloric curve has shortcomings since it is defined by means of the canonical partition
function, which gives a too smooth excitation energy as a
function of temperature. However, it seems evident that the
statistical concept of temperature needs averaging over a sufficient number of levels in order to avoid violent fluctuations.
For these reasons, we would like to defer the discussion of
caloric curves to another occasion [21] and instead focus on
the probability to find the system at an excitation energy for
a given temperature.
The probability that a system at fixed temperature T has
an excitation energy E, is described by the probability density function2
pTsEd =
VsEd e−E/T
,
ZsTd
s8d
where the canonical partition function is given by
oi DE VsEide−E /T .
ZsTd =
i
s9d
The moments mn of E about its mean value kEl are defined
by mn =ksE−kEldnl. Thus, the second and third moments become
m2sTd = kE2l − kEl2 ,
s14d
m3sTd = kE3l − 3kE2lkEl + 2kEl3 .
s15d
These two moments are connected to the heat capacity and
skewness of pTsEd according to
CV = m2/T2 ,
s16d
g = m3/m3/2
2 ,
s17d
respectively. We also identify the standard deviation of
the energy distribution as sE = Îm2.
Figure 6 shows the probability density functions for 160Dy
and 161Dy. Below T,0.6 MeV, the distribution is mainly
based on experimental data, but at higher temperatures the
influence of the somewhat arbitrary extrapolation of the level
density by Eq. (4) will be increasingly important. The most
interesting temperature region is around T=0.5–0.6 MeV,
where the Cooper pair breaking process is the strongest. At
this point, the even-even and odd-even nuclei behave differ-
The experimental excitation energies Ei have energy bins
of DE. In principle, the sum runs over all energies from 0
to `, and we therefore use Eq. s4d to extrapolate to the
higher energies. The energy distribution function pTsEd
has a moment of the order n about the origin given by
kEnl =
oi DE Eni pTsEid.
s10d
It is easy to show that the various moments also may be
evaluated by the differentiation of ZsTd:
T2 dZ
,
Z dT
s11d
T 4 d 2Z
+ 2TkEl,
Z dT2
s12d
kEl =
kE2l =
kE3l =
T 6 d 3Z
+ 6TkE2l − 6T2kEl.
Z dT3
29
s13d
2
The temperature T is in units of MeV.
FIG. 5. Experimental entropy for
064306-4
160,161
Dy.
PHYSICAL REVIEW C 68, 064306 (2003)
THERMAL PROPERTIES AND RADIATIVE STRENGTHS…
FIG. 7. Experimental (left) and theoretical (right) excitation energy kEl, heat capacity CV, and skewness g of the pT distribution as
a function of temperature T. The model parameters [22] used are
«p =«n =3a/p2 =0.19 MeV, Dp =Dn =0.7 MeV, r=0.56, Arigid
=7.6 keV, and "vvib =0.9 MeV.
FIG. 6. Observed probability density functions for 160,161Dy.
The right panels show the case where the experimental data of
161
Dy are replaced by a constant temperature level density, see text.
ently; 160Dy shows a broader distribution than 161Dy. This is
due to the explosive behavior of r for E.Epair
=1.5–2 MeV in even-even nuclei. Roughly, the number of
levels for the breaking of neutron or proton pairs increases
by a factor of exps2Sqpd,55 giving totally ,110 times more
levels.
Figure 4 shows that the level density of 161Dy is almost
linear in a log plot as a function of excitation energy and thus
30
follows closely the constant-temperature expression
C expsE/Tcd with Tc =0.545 MeV. In the right panels of Fig.
6, we have tested the consequences of replacing the experimental level density by this constant-temperature approxima-
tion. In the excitation energy region up to ,6 MeV (the region accessible to our experiment), pTsEd is then proportional
to expsE/Tc −E/Td according to Eq. (8). At the critical temperature T=Tc a plateau emerges which results in a broad
distribution and a consequently high heat capacity, see Eq.
(16). However, from Fig. 6 it is clear that the exact value of
the heat capacity will depend on the extrapolation of the
level density at energies above E,6 MeV.
The various experimental moments are best evaluated
from pTsEd, since the multiplicity V is directly known from
the measured level densities. The left panels of Fig. 7 show
the corresponding values of average excitation energy kEl,
heat capacity CV, and the skewness g of pT as a function of
temperature T. These key quantities characterize pTsEd, and
thereby reveal the thermodynamic properties of the systems
studied. In the right panels these functions are compared to
predictions evaluated in the canonical ensemble. The model
[22] applied here treats the excitation of protons, neutrons,
rotation, and vibration adiabatically with a multiplicative
partition function
s18d
Z = ZpZnZrotZvib ,
n
where the various energy moments kE l are evaluated
from Eqs. s11d–s13d.
064306-5
PHYSICAL REVIEW C 68, 064306 (2003)
M. GUTTORMSEN et al.
The qualitative agreement between model and experiments shown in Fig. 7 indicates that our model describes the
essential thermodynamic properties of the heated systems.
The heat capacity curves show clearly a local increase in the
T=0.5–0.6 MeV region, hinting the collective and massive
breaking of nucleon Cooper pairs. This feature was recently
discussed in Ref. [23], where two different critical temperatures were discovered in the microcanonical ensemble using
the method of Lee and Kosterlitz [24,25]: (i) The lowest
critical temperature is due to the zero to two quasiparticle
transition and (ii) the second critical temperature is due to
the continuous melting of Cooper pairs at higher excitation
energies. The first contribution is strongest for the even-even
system s160Dyd, since the first broken pair represents a large
and abrupt step in level density and thus a large contribution
to the heat capacity. In 161Dy, the extra valence neutron
washes out this step. The second contribution to CV is present
in both nuclei signalizing the continuous melting of nucleon
pairs at higher excitation energies. This second critical temperature appears at ,0.1 MeV higher values.
The skewness g reveals higher order effects in the pTsEd
distribution. For a symmetric energy distribution, g is zero.
Figure 7 shows positive values indicating distributions with
high energy tails, as is confirmed by Fig. 6. The 160Dy system shows a strong signal in g around T,0.2 MeV. This
signal is connected with the high energy tail of the pTsEd
distribution into the E.2D excitation region with high level
density.
IV. RADIATIVE STRENGTH FUNCTION AND ITS
RESONANCES
The slope of the experimental radiative transmission coefficient TsEgd has been determined through the normalization of the level densities, as described in Sec. II. However, it
remains to determine B of Eq. (3), giving the absolute normalization of T. For this purpose we utilize experimental
data [16] on the average total radiative width kGgl at E=Bn.
We assume here that the g decay taking place in the continuum is dominated by E1 and M1 transitions and that the
number of positive and negative parity states is equal. For
initial spin I and parity p at E=Bn, the expression of the
width [26] reduces to
kGgl =
1
4prsBn, I, pd
oI
f
E
Bn
dEgBTsEgdrsBn − Eg, I f d,
0
s19d
where the summation and integration run over all final
levels with spin I f which are accessible by dipole sL
FIG. 8. Unnormalized radiative transmission coefficient for
Dy. The lines are extrapolations needed to calculate the normalization integral of Eq. (19). The arrows indicate the fitting regions.
160
= 1d g radiation with energy Eg. From this expression the
normalization constant B can be determined as described
in Ref. f10g. However, some considerations have to be
made before normalizing according to Eq. s19d.
Methodical difficulties in the primary g-ray extraction
prevents determination of the functions TsEgd and rsEd in the
interval Eg ,1 MeV and E.Bn −1 MeV, respectively. In addition, the data at the highest g energies, above Eg ,Bn
−1 MeV, suffer from poor statistics. For the extrapolation of
r we apply the back-shifted Fermi-gas level density of Eq.
(4). For the extrapolation of T we use a pure exponential
form, as demonstrated for 160Dy in Fig. 8. The contribution
of the extrapolation to the total radiative width given by Eq.
(19) does not exceed 15%, thus the errors due to a possibly
poor extrapolation are expected to be of minor importance
[10].
For 160Dy, the average total radiative width at Bn is unknown. However, the five 161–165Dy isotopes exhibit very
similar experimental values of 108s10d, 112s10d, 112s20d,
113s13d, and 114s14d meV [16], respectively. It is not clear
how to extrapolate to 160Dy, but here the average value of
kGgl=112s20d meV has been adopted.
The radiative strength function for L=1 transitions can be
calculated from the normalized transmission coefficient by
TABLE II. Parameters used for the radiative strength functions.
Nucleus
160
Dy
Dy
162
Dy
161
a
E1E1
(MeV)
s1E1
(mb)
G1E1
(MeV)
E2E1
(MeV)
s2E1
(mb)
G2E1
(MeV)
EM1
(MeV)
sM1
(mb)
GM1
(MeV)
kGgl
(meV)
12.47
12.44
12.42
204.6
206.0
207.5
3.22
3.21
3.20
15.94
15.92
15.90
204.6
31
5.17
5.14
5.12
7.55
7.54
7.52
1.51
1.51
1.51
4.0
4.0
4.0
112(20)a
108(10)
112(10)
206.0
207.5
Estimated from systematics.
064306-6
PHYSICAL REVIEW C 68, 064306 (2003)
THERMAL PROPERTIES AND RADIATIVE STRENGTHS…
rough inspection of the experimental data of Fig. 9 indicates
that the RSFs are increasing functions of g energy, generally
following the tails of the giant electric dipole resonance
(GEDR) and giant magnetic dipole resonance (GMDR) in
this region. In addition, a small resonance around Eg
,3 MeV is found, the so-called pygmy resonance. These
observations have been previously verified for several welldeformed rare earth nuclei [3,10].
Also in the present work we adapt the KadmenskiŽ,
Markushev, and Furman (KMF) model [27] for the giant
electric dipole resonance:
f E1sEgd =
2
0.7sE1GE1
sEg2 + 4p2T2d
1
.
2 2
3 p 2" 2c 2
EE1sEg2 − EE1
d
s21d
Since the nuclei studied here have axially deformed
shapes, the GEDR is split into two components GEDR1
and GEDR2. Thus, we add two RSFs with different resonance parameters, i.e., strength sE1, width GE1, and centroid EE1. The M1 radiation, which is supposed to be governed by the spin-flip M1 resonance f10g, is described by
f M1sEgd =
sM1EgG2M1
1
.
3p2"2c2 sEg2 − E2M1d2 + Eg2 G2M1
s22d
The GEDR and GMDR parameters are taken from the
systematics of Ref. f16g and are listed in Table II. The
pygmy resonance is described with a similar Lorentzian
function f py as described in Eq. s22d. Thus, we fit the total
RSF given by
f = ksf E1 + f M1d + f py ,
FIG. 9. Normalized RSFs for 160,161,162Dy. The filled and open
circles are measured with the s3He, ad and s3He, 3He8d reactions,
respectively.
fsEgd =
1 TsEgd
.
2p Eg3
s20d
The RSFs extracted from the five reactions are displayed
in Fig. 9. The data have been normalized with parameters
from Tables I and II. Also here, the present results deviate
slightly from the three datasets previously published
f3,4,10g. The present RSFs seem not to show any clear
32
odd-even mass differences, and again the s 3He, ad and
s 3He, 3He8d reactions reveal similar results.
The g decay probability is governed by the number and
the character of available final states and by the RSF. A
s23d
to the experimental data using the pygmy-resonance parameters spy, Gpy, and Epy and the normalization constant
k as free parameters.
In previous works [3,10,11], the temperature T of Eq. (21)
was also used as fitting parameter, assuming that a constant
temperature could describe the data. The fitting to experimental data gave typically T,0.3 MeV which is about the
average of what is expected in this energy region. The use of
a constant temperature approach is consistent with the BrinkAxel hypothesis [13,14], which is utilized in order to separate r and T through Eq. (1).
However, experimental data indicate that the RSF may
depend also on how the temperature changes for the various
final states. Data from the 147Smsn,gad144Nd reaction [28]
indicate a finite value of f E1 in the limit Eg →0. Furthermore,
in our study of the weakly deformed 148Sm, where no clear
sign of the pygmy resonance is present, the RSF also flattens
out at small g energies [11]. In the 56,57Fe isotopes it has
been reported [29] that the RSF reveals an anomalous enhancement for g energies below 4 MeV. Also the 27,28Si isotopes show a similar increase in the RSF below 6 MeV [30].
We should also mention that the extracted caloric curve
kEsTdl of Fig. 7 indicates a clear variation in T for the excitation energy region investigated. Figure 10 shows indeed
that the strength of the tail of the GEDR, using the model of
Eq. (21), is strongly temperature dependent. Therefore, from
these considerations, we find it interesting to test the conse-
064306-7
PHYSICAL REVIEW C 68, 064306 (2003)
M. GUTTORMSEN et al.
and multiply these two functions with each other to simulate
a primary g-ray matrix. Then Eq. (1) is utilized in order to
extract r and T. It turns out that the output r is almost identical with the input. Also T is correctly extracted, except for
small deviations of ,15% for g energies below 1 MeV.
Thus, the mentioned inconsistency should not cause severe
problems.
If we assume that the RSF depends on the temperature of
the final states, it also depends on the primary g-ray spectra
chosen. Usually these spectra are taken at initial excitation
energies between E1 ,4 and E2 ,8 MeV. Thus, the average
temperature of the final states E f populated by a g transition
of energy Eg is given by
kTsEgdl =
FIG. 10. Radiative GEDR strength function of the KMF model
calculated for various temperatures.
quences by including a temperature dependent RSF in the
description of the experimental data.
However, there is an inconsistency between such an approach and our extraction of the RSF using the Brink-Axel
hypothesis through Eq. (1). The consequences have been
tested in the following way: We first construct a typical level
density and a temperature dependent transmission coefficient
33
FIG. 11. Average temperature kTl of the final state (solid line)
and standard deviation sT (dashed line) for the temperature distribution as a function of g energy in 160Dy, see text.
1
E2 − E1
E
E2−Eg
dE f TsE f d,
s24d
E1−Eg
where TsE f d = ÎsE f − C1 − Epaird/a is the schematic temperature dependency taken from Eq. s5d. Figure 11 shows kTl
and the standard deviation sT = ÎkT2l − kTl2 for states populated by a g transition of energy Eg in 160Dy. The temperature goes almost linearly from 0.6 MeV to zero, giving an
average of 0.3 MeV consistent with earlier constant temperature fits f3,10,11g. The standard deviation is relatively
large, sT , 0.1 MeV, indicating that one should not replace
T by kTl in Eq. s21d but rather calculate kf E1sEgdl analog to
the evaluation of kTsEgdl in Eq. s24d.
Figure 12 shows fits to the experimental RSFs obtained
from the s3He, ad160Dy and s3He, 3He8d162Dy reactions. The
FIG. 12. The experimental RSFs for 160,162Dy (data points) compared to model predictions using a temperature dependent GEDR
(solid line). The GEDR and pygmy resonance (solid lines) are the
most important contributions to the total RSF. The total RSFs using
a fixed temperature of T=0.30 MeV (dashed line) and T
=0.55 MeV (dash-dotted line) give lower strengths for Eg ,1 MeV.
064306-8
PHYSICAL REVIEW C 68, 064306 (2003)
THERMAL PROPERTIES AND RADIATIVE STRENGTHS…
TABLE III. Fitted pygmy-resonance parameters and normalization constants.
Reaction
s3He,ad160Dy
s3He, ad161Dy
s3He, 3He8d162Dy
s3He, ad162Dy
s3He, 3He8d162Dy
Temperature
dependence
Epy
(MeV)
spy
(mb)
Gpy
(MeV)
k
ÎU f /a
2.68(25)
2.63(17)
2.67(21)
2.73(12)
2.68(8)
2.72(9)
2.86(7)
2.80(5)
2.84(5)
2.74(22)
2.69(14)
2.71(17)
2.61(8)
2.59(5)
2.61(6)
0.27(11)
0.33(7)
0.26(8)
0.42(9)
0.44(5)
0.37(6)
0.40(4)
0.43(3)
0.37(3)
0.28(12)
0.36(7)
0.30(9)
0.28(4)
0.37(2)
0.30(3)
0.90(47)
1.57(40)
1.02(42)
0.95(24)
1.26(19)
0.90(18)
0.90(12)
1.26(11)
0.90(10)
0.78(34)
1.32(31)
0.84(29)
0.98(18)
1.36(14)
1.04(13)
1.06(12)
0.95(12)
0.76(8)
1.31(11)
1.34(10)
1.00(6)
1.27(5)
1.30(5)
0.95(3)
1.02(11)
0.96(11)
0.75(7)
0.93(4)
0.84(4)
0.66(3)
0.30 MeV
0.55 MeV
ÎU f /a
0.30 MeV
0.55 MeV
ÎU f /a
0.30 MeV
0.55 MeV
ÎU f /a
0.30 MeV
0.55 MeV
ÎU f /a
0.30 MeV
0.55 MeV
approach using a varying temperature, kf E1l, is displayed as
solid lines. Alternative fits have been made using fixed temperatures of T=0.30 MeV (dashed lines) and 0.55 MeV
(dash-dotted lines). These temperatures are typical average
values found in the canonical and microcanonical ensembles,
respectively. The GEDR contribution to the total RSF using a
varying temperature is seen to give an increased strength for
Eg ,1 MeV, which the 162Dy case seems to support. However, the 160Dy case supports the approach with a fixed temperature of T=0.30 MeV. The T=0.55 MeV approach represents a compromise at low g energies, but gives a too small
slope in the Eg ,4–7 MeV region. Unfortunately, the RSFs
in the Eg ,1 MeV region are experimentally difficult to measure. Here, a strong g-decay intensity from vibrational states
may not have been properly subtracted in the primary g-ray
spectra. Thus, the present data are not conclusive regarding
the existence of enhanced radiative strength at low g energies.
In Table III, we have summarized the fitted parameters for
the pygmy resonance and the normalization constant k.
Separate fits are performed for three cases: (i) varying temperature, constant temperatures of (ii) T=0.30 MeV and (iii)
T=0.55 MeV. The too small slope of the RSF with fixed T
=0.55 MeV is revealed in a ,30% reduction of the fitted k
parameter. All the investigated dysprosium nuclei show similar pygmy-resonance parameters except for the width Gpy,
which gets significantly higher for the case of T=0.30 MeV.
It turns out that Gpy depends strongly on the slope of the
GEDR strength function in the Eg =3 MeV region.
V. SUMMARY AND CONCLUSIONS
The present comparison between level densities and RSFs
obtained with various reactions gives confidence to the Oslo
method. The entropies of 161Dy follow parallel the even-even
160,162
Dy systems, assigning an entropy of ,2 to the valence
neutron. The evolution of the probability density function
with temperature was presented for 160,161Dy. The widths of
these distributions increase anomalously in the T
=0.5–0.6 MeV region. This feature of local increase in the
canonical heat capacity is a fingerprint of the depairing process. Also the skewnesses of these distributions are studied
showing the variation in the high energy tails as a function of
temperature. A simple canonical model is capable of describing qualitatively the various thermodynamic quantities.
The five RSFs studied reveal very similar structures for
all isotopes studied, as is expected since they all are considered to have the same deformation. The RSFs show a pygmy
resonance superimposed on the tail of the giant dipole resonance. We have tested the consequences of introducing an
RSF with varying temperatures in the GEDR case, which
gives an enhanced strength at lower g energies. Our data are
not conclusive in determining whether such effects are real
or not.
ACKNOWLEDGMENTS
Financial support from the Norwegian Research Council
(NFR) is gratefully acknowledged. Part of this work was
performed under the auspices of the U.S. Department of Energy by the University of California, Lawrence Livermore
National Laboratory under Contract No. W-7405-ENG-48.
A.V. acknowledges support from NATO under Project No.
150027/432 given by the Norwegian Research Council
(NFR).
34
064306-9
PHYSICAL REVIEW C 68, 064306 (2003)
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35
064306-10
2.3
Large enhancement of radiative strength for soft
transitions in the quasicontinuum
VOLUME 93, N UMBER 14
PHYSICA L R EVIEW LET T ERS
week ending
1 OCTOBER 2004
Large Enhancement of Radiative Strength for Soft Transitions in the Quasicontinuum
A. Voinov,1,2,* E. Algin,3,4,5,6 U. Agvaanluvsan,3,4 T. Belgya,7 R. Chankova,8 M. Guttormsen,8 G. E. Mitchell, 4,5
J. Rekstad,8 A. Schiller,3,† and S. Siem8
1
Frank Laboratory of Neutron Physics, Joint Institute of Nuclear Research, 141980 Dubna, Moscow region, Russia
2
Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA
3
Lawrence Livermore National Laboratory, L-414, 7000 East Avenue, Livermore, California 94551, USA
4
North Carolina State University, Raleigh, North Carolina 27695, USA
5
Triangle Universities Nuclear Laboratory, Durham, North Carolina 27708, USA
6
Department of Physics, Osmangazi University, Meselik, Eskisehir, 26480 Turkey
7
Institute of Isotope and Surface Chemistry, Chemical Research Centre HAS, P.O. Box 77, H-1525 Budapest, Hungary
8
Department of Physics, University of Oslo, N-0316 Oslo, Norway
(Received 26 April 2004; published 29 September 2004)
Radiative strength functions (RSFs) for the 56;57 Fe nuclei below the separation energy are obtained
from the 57 Fe3 He; 56 Fe and 57 Fe3 He; 3 He0 57 Fe reactions, respectively. An enhancement of more
than a factor of 10 over common theoretical models of the soft (E & 2 MeV) RSF for transitions in the
quasicontinuum (several MeV above the yrast line) is observed. Two-step cascade intensities with soft
primary transitions from the 56 Fen; 257 Fe reaction confirm the enhancement.
DOI: 10.1103/PhysRevLett.93.142504
PACS numbers: 25.40.Lw, 25.20.Lj, 25.55.Hp, 27.40.+z
Unresolved transitions in the nuclear -ray cascade
produced in the decay of excited nuclei are best described
by statistical concepts: a radiative strength function
(RSF) fXL E for a transition with multipolarity XL
and energy E , and a level density Ei ; Ji for initial
states i at energy Ei with equal spin and parity Ji yield
the mean value of the partial decay width to a given final
state f [1]:
2L1 =E ; J :
XL
i i
if E fXL E E
(1)
Most information about the RSF has been obtained from
photon-absorption experiments in the energy interval 8–
20 MeV, i.e., for excitations above the neutron separation
energy Sn . Data on the soft (E < 3–4 MeV) RSF for
transitions in the quasicontinuum (several MeV above
the yrast line) remain elusive. The first data in the statistical regime were obtained from the 147 Smn; 144 Nd
reaction [2]. They indicate a moderate enhancement of
the soft E1 RSF compared to a Lorentzian extrapolation
of the giant electric dipole resonance (GEDR). For spherical nuclei, in the framework of Fermi-liquid theory, this
enhancement is explained by a temperature dependence
of the GEDR width [3], the Kadmenski-MarkushevFurman (KMF) model. However, the experimental technique requires the presence of sufficiently large widths
and depends on estimates of both and total radiative
widths in the quasicontinuum below Sn .
The sequential extraction method developed at the
Oslo Cyclotron Laboratory (OCL) [4] has enabled further
investigations of the soft RSF by providing unique data
for transitions in the quasicontinuum with sufficient aver37
aging. For deformed rare-earth nuclei, it has been shown
that the RSF can be described in terms of a KMF GEDR
142504-1
0031-9007=04=93(14)=142504(4)$22.50
model, a spin-flip giant magnetic dipole resonance
(GMDR), and a soft M1 resonance [5,6]. In this work,
we report on the first observation of a strong enhancement
of the soft RSF in 56;57 Fe over the model predictions. This
enhancement has been found in Oslo-type experiments
[7] and is confirmed independently by two-step cascade
(TSC) measurements. To our knowledge, at present there
exists no theoretical model which can explain an enhancement of this magnitude.
The first experiment, the 57 Fe3 He; 3 He0 57 Fe and
57 Fe3 He; 56 Fe reactions, was carried out with 45MeV 3 He ions at the OCL. Particle- coincidences were
measured by eight Si particle telescopes at 45 with a
kinematically dominated energy resolution of 250 keV
and by an array of 28 NaI(Tl) 500 500 detectors with a
solid-angle coverage of 15% of 4 and an energy resolution of 6% at 1.3 MeV. The reaction spin window was
I 2–6h.
Primary- matrices P were obtained by a subtraction method [8] for excitation-energy windows of
4 –10.2 MeV and 3–7.6 MeV for 56 Fe and 57 Fe, respectively. These matrices were factorized into a level density
and total RSF f E (summed over all multipolarities)
according to the Brink-Axel hypothesis [9] by
PE; E / E E f E E3 :
(2)
More details on the experiment and data analysis, including the normalized level densities of 56;57 Fe, are given in
[10], and references therein.
RSFs are brought to an absolute scale by normalizing
them to the average total radiative width h i of neutron
resonances [5]. The error of the absolute normalization is
estimated to be 20%. For normalization, the assumption
of equal amounts of positive- and negative-parity states at
any energy below Sn is made. The violation of this as 2004 The American Physical Society
142504-1
VOLUME 93, N UMBER 14
PHYSICA L R EVIEW LET T ERS
sumption for low excitation energies introduces a systematic error to the absolute normalization in the order of
4%. In the case of 56 Fe, also the value of h i has to be
estimated from systematics. However, branching ratios
needed for the subsequent analysis of TSC measurements
are independent of the absolute normalization of the total
RSF and are consequently not affected by the above
assumptions. The normalized RSFs in 56;57 Fe are displayed in Fig. 1. To ensure that the total RSFs do not
depend on excitation energy, we have extracted them also
from two distinct partitions (in excitation energy) of the
primary- matrices. The striking feature of the RSFs is a
large strength for soft transitions, which has not been
observed in the case of rare-earth nuclei, where we used
the same analysis tools [5].
The soft transition strength constitutes an enhancement of more than a factor of 10 over common RSF
models recommended in compilations [11]. To our knowledge, no other model can, at present, reproduce the shape
of the total RSF either. A schematic temperature dependence of the RSF is taken into account in the KMF
model. It is, however, insufficient to describe the data.
Phenomenologically, the data are well described as a sum
of a renormalized KMF model, Lorentzian descriptions
of the GMDR and the isoscalar E2 resonance, and a
FIG. 1. Upper left panel: Total RSF f of 57;56 Fe (solid and
open circles, respectively); Lorentzian (dashed line) and KMF
model (dash-dotted line) descriptions of the GEDR. Upper
right panel: Fit (solid line) to 57 Fe data and decomposition
into the renormalized E1 KMF model, Lorentzian M1 and E2
models (all dashed lines), and a power law to model the large
enhancement for low energies (dash-dotted line). Open symbols are estimates of the E1 (circle) and M1 (square) RSF from
hard primary- rays [21]. Lower panels: Total RSF in 56 Fe (left)
38
and 57 Fe (right) for different excitation-energy windows indicated in the figure. Open circles and squares are offset by a
factor of 2 and 0.5 with respect to their true values.
142504-2
week ending
1 OCTOBER 2004
power law modeling the large enhancement at low energies:
A
B
f K fE1 fM1 2 2 2 E E2 fE2 :
(3)
3 c h
The parameters for the RSF models are taken from systematics [11]. The fit parameters for 57 Fe are K 2:12,
A 0:477 mb=MeV, and B 2:32 (E in MeV).
However, the good description of the enhancement by a
power law should not prevent possible interpretations as a
low-lying resonance or a temperature-related effect.
To ensure that the observed enhancement is not connected to peculiarities of the nuclear reaction or analysis
method, a TSC measurement based on thermal neutron
capture has been performed to confirm the findings. It has
been shown that TSC intensities from ordered spectra can
be used to investigate the soft RSF [12,13]. The TSC
technique for thermal neutron capture has been described
in [14]. It is based on multiplicity-two events populating
low-lying levels. Here, we will give only a brief description of some of the details.
The TSC experiment, i.e., the 56 Fen; 257 Fe reaction,
was performed at the dual-use cold-neutron beam facility
of the Budapest Research Reactor (see [15,16], and references therein). About 2 g of natural iron was irradiated
with a thermal-equivalent flux of 3 107 cm
2 s
1 cold
neutrons for 7 days. Single and coincident rays were
registered by two high-purity Ge detectors of 60% and
13% efficiency at a distance of 8 cm from the target and
with an energy resolution of several keV. They were
placed at 62.5 with respect to the beam axis in order
to minimize the effect of angular correlations.
TSCs populating discrete low-lying levels in 57 Fe produce peaks in the summed-energy spectrum shown on the
left panel of Fig. 2. Gating on the unresolved doublet of
the 1=2
ground state and the 3=2
first excited state at
14 keV yields the TSC spectrum on the right panel of
Fig. 2. Spectra to other final levels were not investigated
due to their lower statistics and higher background. The
TSC spectrum is compressed to 250-keV-wide energy
FIG. 2. Left panel: Summed-energy spectrum. Peaks are
labeled by the spin and parity of the final levels. SE and DE
denote single- and double-escape peaks. Right: Efficiencycorrected and background-subtracted TSC spectrum gated on
the unresolved doublet of the ground and first excited state. The
spectrum is compressed into 250-keV-wide energy bins. Error
bars include statistical errors only.
142504-2
VOLUME 93, N UMBER 14
bins. When the sequence of the two transitions is not
determined experimentally, cascades with soft (discrete)
secondary transitions are registered in the TSC spectrum
as peaks on top of a continuum of cascades with soft
primary transitions. Absolute normalization of TSC spectra is achieved by normalizing to five strong, discrete
TSCs for which absolute intensities of their hard primary
transitions and branching ratios for their secondary transitions are known [17]. The estimated error of the normalization is 20%. In the following, the smooth part of
the TSC spectrum will be investigated in more detail.
In order to separate cascades with soft primary and soft
secondary transitions in the TSC spectra, we use the fact
that the spacing of soft, discrete secondary transitions in
regions of sufficiently low level density is considerably
larger compared to the detector resolution. Thus, soft
secondary transitions will reveal themselves as discrete
peaks. On the other hand, soft primary transitions will
populate levels which are spaced much closer than the
detector resolution and will hence create a continuous
contribution. Separation of soft primary and secondary
transitions is therefore reduced to a separation of individual peaks from a smooth continuum (by, e.g., a fitting
procedure) in the appropriate energy interval [13].
The spin of the compound state in 57 Fe populated by
s-wave neutron capture is 1=2 . Thus, in the excitationenergy region 0.55–1.9 MeV, there are only three levels
which can be populated by primary E1 transitions: the
1=2
level at 1266 keV, the 3=2
level at 1627 keV, and the
3=2
level at 1725 keV. All other levels have spins 5=2
and higher and can be populated only by transitions with
M2=E3 and higher multipolarity. Assuming that transitions of such high multipolarities have a negligible
contribution to the TSC spectrum, we do not take them
into account in the further analysis. TSCs to the ground
and first excited states involving the three abovementioned levels as intermediate levels can easily be
identified from their corresponding peaks in the TSC
spectrum. Their contribution to the TSC spectra is subtracted. The remaining, continuous TSC spectrum in the
specified energy range can be assigned to TSCs with soft
primary transitions. This smooth part of the TSC
spectrum is used to test the soft RSF obtained from the
Oslo-type experiment. Estimations based on the known
level density in 57 Fe [10] show that soft primary transitions in the energy interval 0.55–1.9 MeV populate 150
levels. Assuming that primary and secondary transitions
fluctuate according to a Porter-Thomas distribution, we
estimate systematic intensity uncertainties to be 25%
for this energy interval. Finally, also the midpoint of the
TSC spectrum, where energies of primary and secondary
transitions are equal (and hence known), has been used in
the subsequent analysis. For other energy intervals, the
39
determination of the sequence of the two transitions in
TSCs is subject to large uncertainties; thus, they are
unsuitable for the present analysis.
142504-3
week ending
1 OCTOBER 2004
PHYSICA L R EVIEW LET T ERS
In the present analysis, the intensity of ordered TSCs
between an initial and final state is calculated on the basis
of the statistical model of decay from compound states:
Iif E1 ; E2 X
0
XL E2 XL
im E1 mf
; (4)
Em ; Jm
m
i
XL;XL0 ;J
m
where E1 and E2 are the energies of the first and second
transition in the TSC which are connected by Ei Ef E1 E2 . im and mf are partial decay widths and i and
m are total decay widths of the initial and intermediate
(m) levels, respectively. The average values of these
widths can be calculated from the RSF by Eq. (1).
Summing in Eq. (4) is performed over all valid combinations of multipolarities XL and XL0 of transitions and
of spins and parities of intermediate states. Thus, TSC
spectra depend on the same level density and RSFs which
are extracted from the Oslo-type experiment; see, e.g.,
Eqs. (2) and (3).
Statistical-model calculations with experimental values for the level density and the total RSF have been
performed assuming the decomposition of f according
to Eq. (3) and a standard spin-parity distribution for
intermediate states [18]. Four calculations were performed: one by neglecting the third term in Eq. (3), i.e.,
without the soft pole of the RSF, and the other three under
the assumption of E1, M1, and E2 multipolarity, respectively, for this term. In Fig. 3, results are compared to
experimental data for energies where ordering of TSCs
can be achieved. The calculation without the soft pole
does not reproduce the data at all. The experimental TSC
intensity integrated over the 0.5–2.0 MeV energy region
exceeds the calculated one by a factor of 4.8(13). For
calculations under the assumption of E1, M1, and E2
multipolarities for the soft pole, this factor is reduced
to 1.3(4), 1.0(3), and 1.4(4), respectively. Thus, any
multipolarity is acceptable. Since the two lowest data
points require an extrapolation of the total RSF below
FIG. 3. Experimental TSC intensities (compressed to 250keV-broad energy bins) for cascades with soft primary rays and at the midpoint of the spectrum (data points with error
bars). Error bars include statistical and systematic uncertainties
due to Porter-Thomas fluctuations. Lines are statistical-model
calculations based on experimental data for the level density
and f , neglecting (solid line) and assuming E1 (dashed line),
M1 (dash-dotted line), and E2 (dotted line) multipolarity for
the soft pole of the RSF.
142504-3
VOLUME 93, N UMBER 14
PHYSICA L R EVIEW LET T ERS
1 MeV energy, we have performed calculations with
different extrapolations including a resonance and an
exponential description of the enhanced soft transition
strength, avoiding the pole for E ! 0. For these extrapolations the experimental TSC intensity for the lowest energy is not so well reproduced as before. Finally, we
have performed calculations where the ratio of the
negative-parity levels to the total number of levels decreases linearly from 90% at 2.2 MeV to 50% at
7:6 MeV excitation energy. As expected, TSC intensities with soft primary rays are rather insensitive to
this variation as well.
In conclusion, an enhancement of more than a factor of
10 of soft transition strengths (a soft pole) in the total
RSF has been observed in Oslo-type experiments using
the 57 Fe3 He; 56 Fe and 57 Fe3 He; 3 He0 57 Fe reactions. This enhancement cannot be explained by any
present theoretical model. The total RSF has been decomposed into a KMF model for E1 radiation, Lorentzian
models for M1 and E2 radiation, and a power law to
model the soft pole. In a second experiment, TSC intensities from the 56 Fen; 257 Fe reaction were measured.
Statistical-model calculations based on RSFs and level
densities from the Oslo-type experiment were performed.
These calculations can reproduce the experimental TSC
intensities with soft primary rays only in the presence
of the soft pole in the total RSF. The uncertainties due to
Porter-Thomas fluctuations of TSC intensities do not
allow us to draw definite conclusions about the multipolarity of the soft pole. For better selectivity, averaging
over many initial n resonances will be needed. The satisfying reproduction of the experimental TSC data constitutes support for the physical reality of the soft pole,
independent from the Oslo-type experiment. It should be
noted that this support was gained by using a different
nuclear reaction, a different type of detector, and a different analysis method. Finally, as further supporting evidence, we would like to mention that preliminary results
on a chain of stable Mo isotopes also indicate the presence
of a soft pole in the total RSF [19], while in the case of
27;28 Si, the Oslo method was able to reproduce the total
RSF constructed from literature data on energies, lifetimes, and branching ratios available for the complete
level schemes [20].
Part of this work was performed under the auspices of
the U.S. Department of Energy by the University of
California, Lawrence Livermore National Laboratory
under Contract No. W-7405-ENG-48. Financial support
from the Norwegian Research Council (NFR) is gratefully acknowledged. Part of this work was supported by
the EU5 Framework Programme under Contract
No. HPRI-CT-1999-00099. G. M., U. A., and E. A. acknowledge support by U.S. Department of Energy Grant
No. DE-FG02-97-ER41042. Part of this research
40
was sponsored by the National Nuclear Security
Administration under the Stewardship Science
142504-4
week ending
1 OCTOBER 2004
Academic Alliances program through DOE Research
Grants No. DE-FG03-03-NA00074 and No. DE-FG0303-NA00076. We thank Gail F. Eaton and Timothy P. Rose
for making the targets.
*Electronic address: voinov@ohiou.edu
†
Electronic address: schiller@nscl.msu.edu
[1] G. A. Bartholomew, E. D. Earle, A. J. Ferguson, J.W.
Knowles, and M. A. Lone, Adv. Nucl. Phys. 7, 229 (1973).
[2] Yu. P. Popov, in Proceedings of the Europhysics Topical
Conference, Smolenice, 1982, edited by P. Oblozinský
(Institute of Physics, Bratislava, 1982), p. 121.
[3] S. G. Kadmenski, V. P. Markushev, and V. I. Furman, Yad.
Fiz. 37, 277 (1983) [Sov. J. Nucl. Phys. 37, 165 (1983)].
[4] A. Schiller et al., Nucl. Instrum. Methods Phys. Res.,
Sect. A 447, 498 (2000).
[5] A. Voinov et al., Phys. Rev. C 63, 044313 (2001).
[6] A. Schiller et al., nucl-ex/0401038.
[7] E. Tavukcu, Ph.D. thesis, North Carolina State
University, 2002.
[8] M. Guttormsen, T. S. Tveter, L. Bergholt, F. Ingebretsen,
and J. Rekstad, Nucl. Instrum. Methods Phys. Res., Sect.
A 374, 371 (1996); M. Guttormsen, T. Ramsøy, and J.
Rekstad, Nucl. Instrum. Methods Phys. Res., Sect. A 255,
518 (1987).
[9] D. M. Brink, Ph.D. thesis, Oxford University, 1955; P.
Axel, Phys. Rev. 126, 671 (1962).
[10] A. Schiller et al., Phys. Rev. C 68, 054326 (2003).
[11] P. Obložinský, IAEA Report No. IAEA-TECDOC-1034 ,
1998.
[12] A. Voinov, A. Schiller, M. Guttormsen, J. Rekstad, and S.
Siem, Nucl. Instrum. Methods Phys. Res., Sect. A 497,
350 (2003).
[13] S. T. Boneva, V. A. Khitrov, A. M. Sukhovoj, and A.V.
Vojnov, Nucl. Phys. A589, 293 (1995).
[14] S. T. Boneva et al., Fiz. Elem. Chastits At. Yadra 22, 479
(1991) [Sov. J. Part. Nuclei 22, 232 (1991)]; Fiz. Elem.
Chastits At. Yadra 22, 1433 (1991) [Sov. J. Part. Nuclei
22, 698 (1991)]; F. Bečvář et al., Phys. Rev. C 52, 1278
(1995).
[15] P. P. Ember, T. Belgya, J. L. Weil, and G. Molnár, Appl.
Radiat. Isot. 57, 573 (2002).
[16] T. Belgya et al., in Proceedings of the Eleventh
International Symposium on Capture Gamma-Ray
Spectroscopy and Related Topics, Pruhonice, 2002,
edited by J. Kvasil, P. Cejnar, and M. Krtička (World
Scientific, New Jersey, 2003), p. 562.
[17] Data extracted using the NNDC online data service from
the ENSDF database.
[18] M. Guttormsen et al., Phys. Rev. C 68, 064306 (2003).
[19] R. Chankova, Ph.D. thesis, Oslo University, 2004 (to be
published).
[20] M. Guttormsen et al., J. Phys. G 29, 263 (2003).
[21] J. Kopecky and M. Uhl, in Proceedings of a Specialists’
Meeting on Measurement, Calculation and Evaluation
of Photon Production Data, Bologna, Italy, 1994, edited
by C. Coceva, A. Mengoni, and A. Ventura [Report
No. NEA/NSC/DOC(95)1], p. 119.
142504-4
2.4
Radiative strength functions in
93−98
Mo
PHYSICAL REVIEW C 71, 044307 (2005)
Radiative strength functions in 93−98 Mo
M. Guttormsen,1,∗ R. Chankova,1 U. Agvaanluvsan,2 E. Algin,2,3,4,5 L. A. Bernstein,2 F. Ingebretsen,1 T. Lönnroth,6
S. Messelt,1 G. E. Mitchell,3,4 J. Rekstad,1 A. Schiller,2 S. Siem,1 A. C. Sunde,1 A. Voinov,7,8 and S. Ødegård1
1
Department of Physics, University of Oslo, N-0316 Oslo, Norway
Lawrence Livermore National Laboratory, L-414, 7000 East Avenue, Livermore, California 94551
3
North Carolina State University, Raleigh, North Carolina 27695
4
Triangle Universities Nuclear Laboratory, Durham, North Carolina 27708
5
Department of Physics, Osmangazi University, Meselik, Eskisehir, 26480 Turkey
6
Department of Physics, Åbo Akademi, FIN-20500 Turku, Finland
7
Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701
8
Frank Laboratory of Neutron Physics, Joint Institute of Nuclear Research, 141980 Dubna, Moscow Region, Russia
(Received 26 November 2004; published 20 April 2005)
2
Radiative strength functions (RSFs) in 93−98 Mo have been extracted using the (3 He,αγ ) and (3 He,3 He′ γ )
reactions. The RSFs are U shaped as function of γ energy with a minimum at around Eγ = 3 MeV. The
minimum values increase with neutron number because of the increase in the low-energy tail of the giant electric
dipole resonance with nuclear deformation. The unexpected strong increase in strength below Eγ = 3 MeV, here
called soft pole, is established for all 93−98 Mo isotopes. The soft pole is present at all initial excitation energies
in the 5−8-MeV region.
DOI: 10.1103/PhysRevC.71.044307
PACS number(s): 24.30.Gd, 24.10.Pa, 25.55.Hp, 27.60.+j
I. INTRODUCTION
The γ decay of nuclei at high excitation energy tends to
follow certain statistical rules. The dominating γ -transition
driving factors are the number of accessible final states and
the γ -ray transmission coefficient. The largest uncertainties
are connected to the latter factor. In the description of this
factor Blatt and Weisskopf [1] included an Eγ2L+1 dependency,
where L is the angular momentum transfer in the transition.
In their definition of the radiative strength function (RSF),
this simple energy dependence was divided out. With such
a definition, the single-particle RSF (Weisskopf ) estimates
become independent of γ -ray energy. Various concepts of
RSFs and γ decay in the continuum are outlined in the reviews
of Bartholomew et al. [2,3].
It has been well known that the RSF is not at all constant
but shows an additional Eγx dependency with x = 1−2 for
γ energies in the 4−8-MeV region. Axel [4] argued that
this feature is because of the collective giant electric dipole
resonance (GEDR), which represents the essential mechanism
for the γ decay. However, the situation is more complex.
Further studies [5–7] reveal fine structures in the RSF, which
are commonly called pygmy resonances. This name does not
refer to specific structures: the E1 pygmy resonance in the
Eγ = 5−7 MeV region of gold to lead nuclei could be because
of neutron skin oscillations [8], whereas bumps in the 3-MeV
region of rare earth nuclei are now determined to be of M1
character [9,10]. The electromagnetic character and measured
strength of the latter pygmy resonance is compatible with the
scissors mode [11]. Recently [12,13], the RSF picture of iron
isotopes has been further modified by the observation of an
42
anomalous increase in strength at γ energies below 4 MeV.
∗
Electronic address: magne.guttormsen@fys.uio.no
0556-2813/2005/71(4)/044307(7)/$23.00
It is clear that in the present situation, new experimental results
are urgently needed.
The stable molybdenum isotopes are well suited as targets
for the study of nuclear properties when going from spherical
to deformed shapes. In this work we perform a systematic
analysis of the RSFs of the six 93−98 Mo isotopes. The
RSFs depend on the dynamic properties of electric charges
present within these systems (Z = 42). Because the nuclear
deformation varies from spherical shapes (β ∼ 0) at N = 51 to
deformed shapes (β ∼ 0.2) at N = 56, we expect to observe
effects because of shape changes. Furthermore, these nuclei
reveal weak GEDR tails at low Eγ , making them interesting
objects in the search for other weak structures in the RSF.
The Oslo Cyclotron group has developed a sensitive tool
to investigate RSFs for Eγ below the neutron binding energy
Sn . The method is based on the extraction of primary γ -ray
spectra at various initial excitation energies Ei measured in
particle reactions with one and only one charged ejectile.
From such a set of primary γ spectra, nuclear level densities
and RSFs can be extracted [14–16]. The level density reveals
essential nuclear structure information such as thermodynamic
properties and pair correlations as functions of temperature.
These aspects of the molybdenum isotopes will be presented in
a forthcoming work. Various applications of the Oslo method
have been described in Refs. [17–21].
II. EXPERIMENTAL METHOD
The particle-γ coincidence experiments were carried out at
the Oslo Cyclotron Laboratory for 93−98 Mo using the CACTUS
multidetector array. The charged ejectiles were detected with
eight particle telescopes placed at an angle of 45◦ relative to
the beam direction. An array of 28 NaI γ -ray detectors with
044307-1
©2005 The American Physical Society
M. GUTTORMSEN et al.
PHYSICAL REVIEW C 71, 044307 (2005)
a total efficiency of ∼15% surrounded the target and particle
detectors.
In the present work, results from eight different reactions
on four different targets are discussed. Results from two of
those reactions have been reported earlier. The beam energies
for the different reactions are given in parentheses:
1.
2.
3.
4.
5.
6.
7.
8.
parity π at Sn , the width can be written in terms of the
transmission coefficient by the following [27]:
1
Ŵγ =
2ρ(Sn , I, π ) I
f
Sn
dEγ BT (Eγ )
0
× ρ(Sn − Eγ , If ),
94
Mo(3 He,αγ )93 Mo (new, 30 MeV)
Mo(3 He,3 He′ γ )94 Mo (new, 30 MeV)
96
Mo(3 He,αγ )95 Mo (new, 30 MeV)
96
Mo(3 He,3 He′ γ )96 Mo (new, 30 MeV)
97
Mo(3 He,αγ )96 Mo (reported in [12,21], 45 MeV)
97
Mo(3 He,3 He′ γ )97 Mo (reported in [12,21], 45 MeV)
98
Mo(3 He,αγ )97 Mo (new, 45 MeV)
98
Mo(3 He,3 He′ γ )98 Mo (new, 45 MeV).
(4)
94
The targets were self-supporting metal foils enriched to
∼95% with thicknesses of ∼2 mg/cm2 . The experiments were
run with beam currents of ∼2 nA for 1–2 weeks. The reaction
spin windows are typically I ∼ (2−6)h̄.
The experimental extraction procedure and the assumptions
made are described in Refs. [14,16] and references therein. For
each initial excitation energy Ei , determined from the ejectile
energy and reaction Q value, γ -ray spectra are recorded. Then
the spectra are unfolded using the known γ -ray response
function of the CACTUS array [22]. These unfolded spectra
are the basis for making the first-generation (or primary) γ -ray
matrix [23], which is factorized according to the Brink-Axel
hypothesis [4,24] as follows:
P (Ei , Eγ ) ∝ ρ(Ei − Eγ )T (Eγ ).
(1)
Here, ρ is the level density and T is the radiative transmission
coefficient.
The ρ and T functions can be determined by an iterative
procedure [16] through the adjustment of each data point of
these two functions until a global χ 2 minimum of the fit to
the experimental P (Ei , Eγ ) matrix is reached. It has been
shown [16] that if one solution for the multiplicative functions
ρ and T is known, one may construct an infinite number of
other functions, which give identical fits to the P matrix by the
following:
ρ̃(Ei − Eγ ) = A exp[α(Ei − Eγ )] ρ(Ei − Eγ ),
T̃ (Eγ ) = B exp(αEγ )T (Eγ ).
(2)
where the summation and integration run over all final levels
with spin If , which are accessible by γ radiation with energy
Eγ and multipolarity E1 or M1.
A few considerations have to be made before B can
be determined. Methodical difficulties in the primary γ -ray
extraction prevents determination of the functions T (Eγ )
in the interval Eγ < 1 MeV and ρ(E) in the interval E >
Sn − 1 MeV. In addition, T (Eγ ) at the highest γ energies,
above Eγ ∼ Sn − 1 MeV, suffers from poor statistics. For the
extrapolation of ρ we apply the back-shifted Fermi gas level
density as demonstrated in Ref. [20]. For the extrapolations
of T we use an exponential form. As a typical example, the
extrapolations for 98 Mo are shown in Fig. 1. The contribution
of the extrapolations of ρ and T to the calculated radiative
width in Eq. (4) does not exceed 15% [18]. The experimental
widths Ŵγ in Eq. (4) are listed in Table I. For 94 Mo,
this width is unknown and is estimated by an extrapolation
based on the 96 Mo and 98 Mo values.
The total radiative strength function for dipole radiation
(L = 1) can be calculated from the normalized transmission
coefficient T by the following:
f (Eγ ) =
1 T (Eγ )
.
2π Eγ3
(5)
The RSFs extracted from the eight reactions are displayed in
Fig. 2. As expected, the RSFs do not seem to show any oddeven mass differences. The results obtained for the (3 He,α)
and (3 He,3 He′ ) reactions populating the same residual nucleus
reveal very similar RSFs. Also for 96 Mo two different beam
energies have been applied, giving very similar RSFs. Thus,
the observed energy and reaction independency gives further
confidence in the Oslo method.
(3)
Consequently, neither the slope (α) nor the absolute values of
the two functions (A and B) can be obtained through the fitting
procedure.
The parameters A and α can be determined by normalizing
the level density to the number of known discrete levels at low
excitation energy [25] and to the level density estimated from
neutron-resonance spacing data at the neutron binding energy
Sn [26]. The procedure for extracting the total level density ρ
from the resonance energy spacing D is described in Ref. [16].
Here, we will discuss only the determination of parameter B
of Eq. (3), which gives the absolute normalization of T . For
43
this purpose we utilize experimental data on the average total
radiative width of neutron resonances at Sn Ŵγ .
We assume here that the γ decay in the continuum is
dominated by E1 and M1 transitions. For initial spin I and
III. DESCRIPTION OF THE RADIATIVE
STRENGTH FUNCTIONS
An inspection of the experimental RSFs of Fig. 2 reveals
that the RSFs are increasing functions of γ energy for Eγ >
3 MeV. This indicates that the RSFs are influenced by the tails
of the giant resonances. As follows from previous work, the
main contribution (about 80%) is because of the electric dipole
resonance (GEDR). The magnetic resonance (GMDR) and the
isoscalar E2 resonance are also present in this region.
If the GEDR is described by a Lorentzian function, one
will find that the strength function approaches zero in the limit
Eγ → 0. However, the 144 Nd(n,γ α) reaction [29] strongly
suggests that fE1 has a finite value in this limit. Kadmenskiı̆,
Markushev, and Furman (KMF) have developed a model [30]
044307-2
RADIATIVE STRENGTH FUNCTIONS IN 93−98 Mo
PHYSICAL REVIEW C 71, 044307 (2005)
FIG. 1. Measured level density ρ (upper
panel) and radiative transmission coefficient T
(lower panel) for 98 Mo. The straight lines are
extrapolations needed to calculate the normalization integral of Eq. (4). The triangle in the upper
panel is based on resonance spacing data at Sn .
describing this feature for the electric dipole RSF:
2
2
Eγ + 4π 2 T 2
0.7σE1 ŴE1
1
fE1 (Eγ , T ) =
.
2 2
3π 2h̄2 c2
EE1 Eγ2 − EE1
rare earth nuclei [13,18–20] assuming a constant temperature
parameter T in Eq. (6) (i.e., one that is independent of excitation
energy). In this work we assume that the temperature depends
on excitation energy according to Eq. (7), which gives an
increase in the RSF at low γ energy [20].
The GMDR contribution to the total RSF is described
by a Lorentzian. This approach is in accordance with numerous experimental data obtained so far [26]. However,
the experimental data scatter and the resonance parameter
values are uncertain. This is also true for the E2 resonance.
The Lorentzian description of the M1 and E2 contributions
are given in Ref. [17]. The resonance parameters for the
E1, M1, and E2 resonances are taken from the compilations of
Refs. [26,32] and are listed in Table I.
The enhanced RSF at low γ energies has at present no
theoretical explanation. Recently, the same enhancement has
(6)
The temperature T depends on the final state f and for
simplicity we adapt the schematic form
(7)
T (Ef ) = Uf /a,
where the level density parameter is parametrized as a =
0.21A0.87 MeV−1 . The intrinsic energy is estimated by
Uf = Ef − C1 − Epair with a back-shift parameter of C1 =
−6.6A−0.32 MeV [31]. The pairing energy contribution Epair
is evaluated from the three-point mass formula of Ref. [33].
Although the KMF model has been developed for spherical
nuclei, it has been successfully applied to 56,57 Fe and several
TABLE I. Parameters used for the radiative strength functions. The data are taken from Ref. [26]. The
E1 resonance parameters for the even Mo isotopes are based on photo absorption experiments [32], and the
parameters for the odd Mo isotopes are derived from interpolations.
Nucleus
93
Mo
Mo
95
Mo
96
Mo
97
Mo
98
Mo
94
a
EE1
(MeV)
σE1
(mb)
ŴE1
(MeV)
EM1
(MeV)
16.59
16.36
16.28
16.20
16.00
15.80
173.5
185.0
185.0
185.0
187.0
189.0
4.82
5.50
5.76
6.01
5.98
5.94
9.05
9.02
8.99
8.95
8.92
8.89
σM1
(mb)
0.86
1.26
1.38
441.51
1.58
1.65
Estimated from systematics.
044307-3
ŴM1
(MeV)
EE2
(MeV)
σE2
(mb)
ŴE2
(MeV)
Ŵγ (meV)
4.0
4.0
4.0
4.0
4.0
4.0
13.91
13.86
13.81
13.76
13.71
13.66
2.26
2.24
2.22
2.21
2.19
2.17
4.99
4.98
4.97
4.96
4.95
4.93
160(20)
170(40)a
135(20)
150(20)
110(15)
130(20)
M. GUTTORMSEN et al.
PHYSICAL REVIEW C 71, 044307 (2005)
been observed in the iron isotopes [12,13]. We call this
structure a soft pole in the RSF and choose a simple power law
parametrization given by the following:
fsoftpole =
1
3π 2h̄2 c2
AEγ−b ,
(8)
where A and b are fit parameters and Eγ is given in
MeV.
Previously, a pygmy resonance around Eγ ∼ 3 MeV has
been reported in several rare-earth nuclei [18–20]. The electromagnetic character of the corresponding RSF structure is now
established to be of M1 type [9,10] and is interpreted as the
scissors mode. Deformed nuclei can in principle possess this
collective motion, and, for example, 98 Mo with a deformation
of β ∼ 0.18, could eventually show some reminiscence of
the scissors mode. Data on 94 Mo [34] and 96 Mo [35] show
a summed M1 strength to mixed symmetry 1+ states around
∼3.2 MeV on the order of ∼0.6µ2N . This is about one order
of magnitude lower than the M1 strength observed in welldeformed rare-earth nuclei using the present method. This M1
strength is deemed too weak to cause a visible bump in our
RSFs above 3 MeV.
We conclude that a reasonable composition of the total RSF
is as follows:
f = κ(fE1 + fM1 + fsoftpole ) + Eγ2 fE2 ,
FIG. 2. Normalized RSFs for 93−98 Mo. The filled and open circles
represent data taken with the (3 He,α) and (3 He,3 He′ ) reactions,
respectively. The filled triangles in 93,95 Mo are estimates of E1 RSF
of hard primary γ rays [28] . The solid and dashed lines are fits to the
RSF data from the two respective reactions (see text).
where κ is a normalization constant. Generally, its value
deviates from unity for several reasons; the most important
reasons are theoretical uncertainties in the KMF model and
the evaluation of B in Eq. (4). We use κ, A, and b as free
parameters in the fitting procedure, and the results for the
eight reactions are summarized in Table II.
In Fig. 3 the various contributions to the total RSF of 98 Mo
are shown. The main components are the GEDR resonance
and the unknown low-energy structure. We observe that the
E1 component exhibits an increased yield for the lowest γ
energies because of the increase in temperature T. However,
this effect is not strong enough to explain the low-energy
upbend.
Figure 2 shows the fit functions for all reactions and
gives qualitative good agreements with the experimental data.
The fitting parameters κ, A, and b are all similar within the
uncertainties. It should be noted that the soft pole parameters
TABLE II. Soft pole fitting parameters and integrated strenghts. The B values are only lower estimates (see
text).
Reaction
(3 He,α)93 Mo
(3 He,3 He′ )94 Mo
(3 He,α)95 Mo
(3 He,3 He′ )96 Mo
(3 He,α)96 Mo
(3 He,3 He′ )97 Mo
(3 He,α)97 Mo
(3 He,3 He′ )98 Mo
κ
A
(mb/MeV)
0.44(4)
0.36(2)
0.39(2)
0.36(1)
0.32(4)
0.38(3)
0.45(5)
0.52(4)
0.37(7)
0.48(5)
0.48(6)
0.60(4)
0.47(14)
0.47(7)
0.30(10)
0.22(7)
(9)
b
2.6(3)
2.5(2)
2.6(2)
3.2(2)
45 2.7(6)
2.4(3)
2.2(5)
2.1(5)
044307-4
B(E1↑)
(e2 fm2 )
B(M1↑)
(µ2N )
B(E2↑)
(103 e2 fm4 )
0.021(5)
0.023(3)
0.024(4)
0.022(2)
0.019(7)
0.025(5)
0.020(8)
0.018(7)
1.9(4)
2.1(3)
2.2(3)
2.0(2)
1.7(6)
2.3(4)
1.9(7)
1.6(6)
14(3)
16(2)
16(2)
16(1)
13(4)
16(3)
13(5)
12(4)
RADIATIVE STRENGTH FUNCTIONS IN 93−98 Mo
PHYSICAL REVIEW C 71, 044307 (2005)
FIG. 3. Experimental radiative strength function of 98 Mo compared to a model description, including GEDR, GMDR, and the
isoscalar E2 resonance. The empirical soft pole component is used
to describe the low energy part of the RSF.
coincide with the description of the 57 Fe nucleus [13] having
A = 0.47(7) mb/MeV and b = 2.3(2).
The RSFs for Eγ > 3 MeV when going from N = 51 to 56
increase by almost a factor of 2 and this can be understood from
the corresponding evolution of nuclear deformation. Following
the onset of prolate deformation the GEDR will split into two
parts, where 1/3 of its strength is shifted down in energy and
2/3 up. Photoneutron cross sections [32] show no splitting into
two separate bumps; however, the observed increase in width
ŴE1 as a function of neutron number (see Table I) supports
the idea of a splitting, which is a well-known feature in other
more deformed nuclei. Figure 2 demonstrates that the adopted
widths describe very well the variation of the RSF strength as
function of mass number.
To investigate whether the prominent soft pole structure
is present in the whole excitation energy region, we have performed the following test. Assuming that the level density from
Eq. (1) is correct, we can estimate the shape of the strength
functions starting at various initial excitation energies using
the following:
FIG. 4. RSFs for 96,98 Mo at various initial excitation energies.
The soft pole is present for all Ei . The solid lines display the RSFs
obtained in Fig. 2.
constant is only roughly known through the following estimate:
1 N (Ei )P (Ei , Eγ )
f (Eγ , Ei ) =
.
2π ρ(Ei − Eγ )Eγ3
(10)
N (Ei ) =
Actually, f (Eγ , Ef ) would have been the proper expression
46
to investigate, but because of technical reasons we chose
f (Eγ , Ei ), which is equivalent to investigating f (Eγ , Ef )
because in our method Ef and Ei are uniquely related
by Ef = Ei − Eγ . One problem is that the normalization
Ei
0
dEγ ρ(Ei − Eγ )T (Eγ )
,
Ei
0 dEγ P (Ei , Eγ )
(11)
with Ei < Sn . However, for the expression f (Eγ , Ei ) we
are interested only in the shape of the RSFs, and an exact
normalization is therefore not crucial. The evaluation assumes
044307-5
M. GUTTORMSEN et al.
PHYSICAL REVIEW C 71, 044307 (2005)
that eventual temperature-dependent behavior of the RSF is
small compared to the soft pole structure.1
In Fig. 4, the RSFs for 96,98 Mo are shown at various initial
energies Ei . For comparison, the figure also includes the global
RSFs (solid lines) obtained with the Oslo method (Fig. 2).
Within the error bars the data support that the soft pole is
present in all the excitation bins studied.
The origin of the soft pole cannot be explained by any
known theoretical model. One would therefore need to know
the γ -ray multipolarity as guidance for theoretical approaches
to this phenomenon. Rough estimates of the reduced strength
can be obtained from the following:
1 L (2L + 1) [(2L + 1)!!]2
B(XL ↑) =
(h̄c)2L+1
8π
L+1
3 MeV
dEγ fXL (Eγ ).
(12)
×
1 MeV
In the evaluation, we have integrated the soft pole between
1 and 3 MeV. Thus, the estimates listed in Table II for the
reactions studied give only a lower limit for the respective
B(XL ↑) values. The correct result will of course depend on
the functional form of fsoftpole (Eγ ) below 1 MeV; however,
no experimental data exist in this region and any assumption
here would be highly speculative. There seems to be no
clear dependency of the B values on mass number or nuclear
deformation.
With the assumptions above, we get in the case of an E1 soft
pole an average B(E1 ↑) value of 0.02 e2 fm2 , which is 0.07%
of the sum rule for the GEDR. Assuming an M1 soft pole,
we get roughly B(M1 ↑) ∼ 2.0µ2N , which is 3−4 times larger
than the observed strength to mixed symmetry 1+ states around
3 MeV [34,35]. Provided the soft pole has E2 multipolarity
we obtain finally a B(E2 ↑) value around 15000 e2 fm4 ,
which is 5–15 times larger than the ones for the excitation to the
1
Simulations using the KMF model with fixed temperature in
the T ∼ 0.8 MeV region indicate a maximum 20% effect from
temperature dependence of the RSF.
[1] J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics
(Wiley, New York, 1952).
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Can. J. Phys. 48, 687 (1970).
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47
[8] N. Ryezayeva, T. Hartmann, Y. Kalmykov, H. Lenske,
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A. Shevchenko, S. Volz, and J. Wambach, Phys. Rev. Lett. 89,
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first excited 2+ states in the even molybdenum isotopes. Thus,
we cannot exclude any of these multipolarities, since neither
of them would yield unreasonably high transition strengths.
Moreover, we would like to point out that the observed soft
pole resides on top of the tails of giant resonances. Thus,
the transition strength included in the soft pole has to be
added to the strength in the giant resonance tail of the correct
multipolarity to give the summed transition strength.
IV. SUMMARY AND CONCLUSIONS
As expected, the observed RSFs reveal very similar shapes
because they all refer to isotopes with the same nuclear charge.
When going from N = 51 to 56 the RSF increases by almost
a factor of two for Eγ > 3 MeV, which can be understood
from the change of nuclear deformation. With the onset of
deformation, the increasing resonance GEDR width ŴE1 is
responsible for the increasing strength.
An enhanced strength at low γ energies is observed, which
is equally strong for all isotopes and excitation energies
studied. A similar enhancement has also been seen in the
iron isotopes. The multipolarity of the soft pole radiation is
unknown and there is still no theoretical explanation for this
very interesting phenomenon.
ACKNOWLEDGMENTS
Financial support from the Norwegian Research Council
(NFR) is gratefully acknowledged. Part of this work was
performed under the auspices of the U.S. Department of
Energy by the University of California, Lawrence Livermore
National Laboratory, under Contract W-7405-ENG-48. A.V.
E.A, U.A, and G.E.M acknowledge support from the National Nuclear Security Administration under the Stewardship
Science Academic Alliances program through Department
of Energy Research Grants DE-FG03-03-NA00074 and DEFG03-03-NA00076 and U.S. Department of Energy Grant
DE-FG02-97-ER41042.
[9] A. Schiller, A. Voinov, E. Algin, J. A. Becker, L. A. Bernstein,
P. E. Garrett, M. Guttormsen, R. O. Nelson, J. Rekstad, and
S. Siem, preprint, nucl-ex/0401038.
[10] M. Krtic̆ka, F. Bec̆vár̆, J. Honzátko, I. Tomandl, M. Heil,
F. Käppeler, R. Reifarth, F. Voss, and K. Wisshak, Phys. Rev.
Lett. 92, 172501 (2004).
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N. Lo Iudice, H. Maser, H. H. Pitz, and A. Zilges, Phys. Rev. C
58, 184 (1998).
[12] E. Tavukcu, Ph.D. thesis, North Carolina State University, 2002.
[13] A. Voinov, E. Algin, U. Agvaanluvsan, T. Belgya, R. Chankova,
M. Guttormsen, G. E. Mitchell, J. Rekstad, A. Schiller, and
S. Siem, Phys. Rev. Lett. 93, 142504 (2004).
[14] L. Henden, L. Bergholt, M. Guttormsen, J. Rekstad, and
T. S. Tveter, Nucl. Phys. A589, 249 (1995).
[15] E. Melby, L. Bergholt, M. Guttormsen, M. Hjorth-Jensen,
F. Ingebretsen, S. Messelt, J. Rekstad, A. Schiller,
044307-6
RADIATIVE STRENGTH FUNCTIONS IN 93−98 Mo
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
PHYSICAL REVIEW C 71, 044307 (2005)
S. Siem, and S. W. Ødegård, Phys. Rev. Lett. 83, 3150
(1999).
A. Schiller, L. Bergholt, M. Guttormsen, E. Melby, J. Rekstad,
and S. Siem, Nucl. Instrum. Methods Phys. Res. A 447, 498
(2000).
U. Agvaanluvsan, et al., Phys. Rev. C 70, 054611
(2004).
A. Voinov, M. Guttormsen, E. Melby, J. Rekstad, A. Schiller,
and S. Siem, Phys. Rev. C 63, 044313 (2001).
S. Siem, M. Guttormsen, K. Ingeberg, E. Melby, J. Rekstad,
A. Schiller, and A. Voinov, Phys. Rev. C 65, 044318
(2002).
M. Guttormsen, A. Bagheri, R. Chankova, J. Rekstad,
S. Siem, A. Schiller, and A. Voinov, Phys. Rev. C 68, 064306
(2003).
A. Schiller et al., Phys. Rev. C 68, 054326 (2003).
M. Guttormsen, T. S. Tveter, L. Bergholt, F. Ingebretsen, and
J. Rekstad, Nucl. Instrum. Methods Phys. Res. A 374, 371
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Methods Phys. Res. A 255, 518 (1987).
D. M. Brink, Ph.D. thesis, Oxford University, 1955.
[25] Data extracted using the NNDC On-Line Data Service from the
ENSDF database.
[26] P. Obloz̆inský, IAEA Report No. IAEA-TECDOC-1034, 1998.
[27] J. Kopecky and M. Uhl, Phys. Rev. C 41, 1941 (1990).
[28] J. Kopecky and M. Uhl, in Proceedings of a Specialists’
Meeting on Measurement, Calculation and Evaluation of Photon
Production Data, Bologna, Italy, 1994, edited by C. Coceva,
A. Mengoni, and A. Ventura [Report No. NEA/NSC/
DOC(95)1], p. 119.
[29] Yu. P. Popov, in Proceedings of the Europhysics Topical
Conference, Smolenice, 1982, edited by P. Obložinský (Institute
of Physics, Bratislava, 1982), p. 121.
[30] S. G. Kadmenskiı̆, V. P. Markushev, and V. I. Furman, Yad. Fiz.
37, 277 (1983) [Sov. J. Nucl. Phys. 37, 165 (1983)].
[31] T. von Egidy, H. H. Schmidt, and A. N. Behkami, Nucl. Phys.
A481, 189 (1988).
[32] Samuel S. Dietrich and Barry L. Berman, At. Data Nucl. Data
Tables 38, 199 (1988).
[33] J. Dobaczewski, P. Magierski, W. Nazarewicz, W. Satuła, and
Z. Szymański, Phys. Rev. C 63, 024308 (2001).
[34] C. Fransen et al., Phys. Rev. C 67, 024307 (2003).
[35] C. Fransen et al., Phys. Rev. C 70, 044317 (2004).
48
044307-7
2.5
Level densities and thermodynamical quantities of
heated 93−98 Mo isotopes
PHYSICAL REVIEW C 73, 034311 (2006)
Level densities and thermodynamical quantities of heated 93−98 Mo isotopes
R. Chankova,1,∗ A. Schiller,2 U. Agvaanluvsan,2,3 E. Algin,2,3,4,5 L. A. Bernstein,2 M. Guttormsen,1 F. Ingebretsen,1
T. Lönnroth,6 S. Messelt,1 G. E. Mitchell,3,4 J. Rekstad,1 S. Siem,1 A. C. Larsen,1 A. Voinov,7,8 and S. Ødegård1
1
Department of Physics, University of Oslo, N-0316 Oslo, Norway
Lawrence Livermore National Laboratory, L-414, 7000 East Avenue, Livermore, California 94551, USA
3
North Carolina State University, Raleigh, North Carolina 27695, USA
4
Triangle Universities Nuclear Laboratory, Durham, North Carolina 27708, USA
5
Department of Physics, Osmangazi University, Meselik, Eskisehir, 26480 Turkey
6
Department of Physics, Åbo Akademi, FIN-20500 Turku, Finland
7
Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA
8
Frank Laboratory of Neutron Physics, Joint Institute of Nuclear Research, RU-141980 Dubna, Moscow, Russia
(Received 30 June 2005; published 16 March 2006)
2
Level densities for 93−98 Mo have been extracted using the (3 He,αγ ) and (3 He,3 He′ γ ) reactions. From the level
densities thermodynamical quantities such as temperature and heat capacity can be deduced. Data have been
analyzed by utilizing both the microcanonical and the canonical ensemble. Structures in the microcanonical
temperature are consistent with the breaking of nucleon Cooper pairs. The S shape of the heat capacity curves
found within the canonical ensemble is interpreted as consistent with a pairing phase transition with a critical
temperature for the quenching of pairing correlations at Tc ∼ 0.7−1.0 MeV.
DOI: 10.1103/PhysRevC.73.034311
PACS number(s): 21.10.Ma, 24.10.Pa, 25.55.−e, 27.60.+j
I. INTRODUCTION
Level density is a characteristic property of many-body
quantum mechanical systems. Its precise knowledge is often
a key ingredient in the calculation of different processes,
such as compound nuclear decay rates, yields of evaporation
residues to populate exotic nuclei, or thermonuclear rates in
astrophysical processes.
Measurements of experimental nuclear level density are
an important prerequisite for thermodynamical studies of
atomic nuclei. Level density is directly connected to the
multiplicity of states, i.e., the number of physical realizations
of the system at a certain excitation energy. The entropy is
a fundamental quantity and a measure of the disorder of the
many-body system. Within the microcanonical ensemble it is
defined as the natural logarithm of the multiplicity of states.
When the entropy is known, thermodynamic quantities such
as temperature and heat capacity can be extracted. These
quantities depend on the statistical properties of the nuclear
many-body system and may reveal phase transitions.
Pairing correlations are one of the fundamental properties
of nuclei and have been successfully described by the BardeenCooper-Schrieffer (BCS) theory of superconductivity [1].
By using the BCS theory the thermodynamical properties
of nuclear pairing were investigated in the study of warm
nuclei [2–5]. In the case of a finite Fermi system such as
the nucleus, statistical fluctuations beyond the mean field
become important. The fluctuations smooth out the sharp
phase transition, and then the pairing correlations do not
vanish suddenly but decrease with increasing temperature.
The quenching of pairing correlations has been obtained in
50
recent theoretical approaches: the shell-model Monte-Carlo
∗
Electronic address: rositsa.chankova@fys.uio.no
0556-2813/2006/73(3)/034311(12)/$23.00
(SMMC) calculations [6–8], the finite-temperature HartreeFock-Bogoliubov theory [9], and the relativistic mean-field
theory [10]. Experimental data on the quenching of pair
correlations are important as a test for nuclear theories. A
long-standing problem in experimental nuclear physics has
been to observe the transition from strongly paired states at
zero temperature to unpaired states at higher temperatures.
A signature of the pairing transition at finite temperature
might be a local increase in the heat capacity as a function
of temperature [11]. Recently [12,13], fine structures in the
level densities in the 1-to 7-MeV region were reported, which
are probably because of the breaking of individual nucleon
pairs and a gradual decrease of pair correlations.
The group at the Oslo Cyclotron Laboratory (OCL) has
developed a method to extract simultaneously the level density
and the radiative strength function from primary γ spectra
[14]. The method is a further development of the sequential
extraction method [15,16]. The Oslo method has been tested
in the rare-earth mass region that led to many interesting
applications [12,17–19]. To make quantitative judgments of
the applicability of the method, the Oslo Cyclotron group
has extracted the level density and radiative strength function
(RSF) of the very light 27,28 Si nuclei, where these quantities are
known. Excellent overall agreement was found [20]. Subsequently, another extension has been made to the intermediate
nuclei 56,57 Fe and 96,97 Mo, and it has been shown that the
method can be applied in this intermediate mass region where
the level density is still relatively low [21,22]. All of these
successful applications have motivated us to employ the Oslo
method to study medium-heavy nuclei in the vicinity of closed
shells.
The naturally occurring isotopes of molybdenum span one
of the larger isotopic ranges and are well suited as targets for
the study of nuclear properties, such as the effect of changing
from spherical to deformed shapes. When approaching closed
034311-1
©2006 The American Physical Society
PHYSICAL REVIEW C 73, 034311 (2006)
R. CHANKOVA et al.
shells, the nuclear structure changes significantly, and one
expects this to influence the level densities and radiative
strength functions.
The even-even 92 Mo has a filled N = 50 neutron shell [23].
It is essentially a spherical nucleus and vibrations are primarily
governed by the proton core. As the mass increases from 94 Mo
to 100 Mo, neutrons fill the 2d5/2 and 1g7/2 subshells. Moving
away from the N = 50 shell closure, pairing and quadrupole
interactions cause a more collective behavior in the heavier
Mo isotopes. The character of the isotopes changes rapidly
from that of the essentially spherical 92 Mo to nuclei making a
transition from collective vibrators to the deformed rotors of
the unstable 104 Mo and 106 Mo isotopes [24]. The transitional
nature of molybdenum isotopes away from N = 50 has been
the focus of several efforts as described in Ref. [25] and
references therein.
Around closed shells, effects from the increasing singleparticle energy spacings can be expected. These will also
influence the entropy difference between odd-mass and eveneven nuclei. Therefore, a statistical description of the transition
from closed shells to deformed nuclei is of great interest.
In this work, a unique and consistent investigation of the
six 93−98 Mo isotopes is performed to determine experimentally
the level density from the ground state to the neutron binding
energy. The Oslo method also determines the RSFs of the
molybdenum isotopes studied; these are presented in an earlier
article [26].
II. EXPERIMENTAL METHODS
The experiments were carried out at the Oslo Cyclotron
Laboratory by bombarding 94,96,97,98 Mo targets with 3 He ions.
In the present work, results from eight different reactions on
four different targets are discussed. These are the following
six reactions that are the subject of the present investigation:
Mo(3 He,αγ )97 Mo (45 MeV)
Mo(3 He,3 He′ γ )98 Mo (45 MeV)
96
Mo(3 He,αγ )95 Mo (30 MeV)
96
Mo(3 He,3 He′ γ )96 Mo (30 MeV)
94
Mo(3 He,αγ )93 Mo (30 MeV)
94
Mo(3 He,3 He′ γ )94 Mo (30 MeV) together with the
reactions
(vii) 97 Mo(3 He,αγ )96 Mo (45 MeV)
(viii) 97 Mo(3 He,3 He′ γ )97 Mo (45 MeV)
(i)
(ii)
(iii)
(iv)
(v)
(vi)
98
98
which have been reported earlier [21,22]. The self-supporting
targets with thicknesses of ∼2 mg/cm2 are enriched to
∼95%. The experiments were run with beam currents of
∼2 nA for 1–2 weeks. The particle-γ coincidences were
measured with the CACTUS multidetector array. The charged
ejectiles were detected by eight particle telescopes placed at an
angle of 45◦ relative to the beam direction. An array of 28 NaI
γ -ray detectors with a total efficiency of ∼15% surrounded
the target and particle detectors.
51
For each initial excitation energy, the γ -ray spectra are
recorded as a function of the initial excitation energy of
the residual nucleus. This is accomplished by utilizing the
known reaction Q values and kinematics. Using the particle-γ
coincidence technique, each γ ray can be assigned to a
cascade depopulating a certain initial excitation energy in
the residual nucleus. The data are therefore sorted into total
γ -ray spectra originating from different initial excitationenergy bins. Each spectrum is then unfolded with the NaI
response function using a Compton-subtraction method which
preserves the fluctuations in the original spectra and does
not introduce further, spurious fluctuations [27]. From the
unfolded spectra, a primary-γ matrix P (E, Eγ ) is constructed
using the subtraction method of Ref. [28].
The basic assumption of this method is that the γ -ray energy
distribution from any excitation energy bin is independent
of whether states in this bin are populated directly via the
(3 He,α) or (3 He,3 He′ ) reactions or indirectly via γ decay from
higher excited levels following the initial nuclear reaction.
This assumption is trivially fulfilled if one populates the same
levels with the same weights within any excitation-energy bin,
because the decay branchings are properties of the levels and
do not depend on the population mechanisms. The assumptions
behind this method have been tested extensively by the Oslo
group and have been shown to work reasonably well [29].
The (3 He,3 He′ γ ) and (3 He,αγ ) reactions exhibit very
different reaction mechanisms. This is demonstrated in
Fig. 1, where the particle spectra in coincidence with γ rays
show indeed very different yields and peak structures.
The (3 He,αγ ) pick-up reaction reveals a cross section dominated by high ℓ neutron transfer. Here, the direct population of
the residual nucleus takes place through one-particle-one-hole
components of the wave function. Such configurations are not
eigenstates of the nucleus, but they are rather distributed over
virtually all eigenstates in the neighboring excitation-energy
region. Thus, the neutron-hole strength for single-particle
levels away from the Fermi energy is distributed over a rather
large range of background states.
However, the inelastic scattering (3 He,3 He′ γ ) reaction is
known to populate mainly collective excitations with a slightly
lower spin window. Collective excitations built on the ground
state give rise to rather pure eigenfunctions and their strength
is less spread over other eigenfunctions of the nucleus in the
neighboring excitation-energy region.
To test if the number of γ rays per cascade depends on the
two types of reactions, we have evaluated the average γ -ray
multiplicity
Mγ (E) =
E
,
Eγ (1)
as a function of excitation energy E. The average γ -ray
energy Eγ is calculated from γ spectra selected at a certain
energy E.
Figure 2 shows the γ -ray multiplicity versus excitation
energy. Despite the different reaction mechanisms, the two
reactions give similar results. In particular, the multiplicities
(solid and dashed lines) of 96 Mo and 97 Mo are equal within
their error bars, which gives support to the applicability of the
Oslo method for both reactions.
The experimental extraction procedure and assumptions of
the Oslo method are given in Refs. [14,29] and references
therein. The first generation (or primary) γ -ray matrix that is
obtained as described above can be factorized according to the
034311-2
PHYSICAL REVIEW C 73, 034311 (2006)
LEVEL DENSITIES AND THERMODYNAMICAL . . .
FIG. 1. Charged ejectile spectra for 93−98 Mo in coincidence with
γ -rays, labeled by the product nuclei. The arrows indicate the neutronseparation energy Bn .
Brink-Axel hypothesis [30,31] as
P (E, Eγ ) ∝ ρ(E − Eγ )T (Eγ ),
(2)
where ρ is the level density and T is the radiative transmission
coefficient.
The ρ and T functions can be determined by an iterative
procedure [14] through the adjustment of each data point
of these two functions until a global χ 2 minimum with the
experimental P (E, Eγ ) matrix is reached. It has been shown
[14] that if one solution for the multiplicative functions ρ and
T is known, one may construct an infinite number of other
52
functions, which give identical fits to the P matrix by
ρ̃(E − Eγ ) = A exp[α(E − Eγ )] ρ(E − Eγ ),
T̃ (Eγ ) = B exp(αEγ )T (Eγ ).
FIG. 2. γ -ray multiplicity evaluated by Eq. (1) versus excitation
energy. The individual spectra are labeled by the product nuclei.
Solid and dashed lines represent (3 He,α) and (3 He,3 He′ ) reactions,
respectively.
(3)
(4)
Consequently, neither the slope nor the absolute values of the
two functions can be obtained through the fitting procedure.
Thus, the parameters α, A, and B remain to be determined.
The parameters A and α can be determined by normalizing
the level density to the number of known discrete levels at low
excitation energy [32] and to the level density estimated from
neutron-resonance spacing data at the neutron-separation energy E = Bn [33]. The procedure for extracting the total level
density ρ from the resonance energy spacing D is described in
Ref. [14]. Because our experimental level-density data points
reach up to an excitation energy of only E ∼ Bn − 1 MeV,
034311-3
R. CHANKOVA et al.
PHYSICAL REVIEW C 73, 034311 (2006)
FIG. 3. Normalization procedure of the experimental level density (data points) of 97 Mo.
The data points between the arrows in the upper
panel are normalized to known levels at low
excitation energy (histograms). In the lower
panel they are normalized to the level density
at the neutron-separation energy (open triangle)
using a Fermi-gas extrapolation (line).
we extrapolate with the back-shifted Fermi-gas model [34,35]
√
exp(2 aU )
ρBSFG (E) = η √
,
(5)
12 2a 1/4 U 5/4 σI
where a constant η is introduced to adjust ρBSFG to the
experimental level density at Bn . The intrinsic excitation
energy is estimated by U = E − C1 − Epair , where C1 =
−6.6A−0.32 MeV and A are the back-shift parameter and
mass number, respectively. The pairing energy Epair is based
on pairing-gap parameters p and n evaluated from evenodd mass differences [36] following the prescription of
Dobaczewski et al. [37]. The level-density parameter is given
by a = 0.21A0.87 MeV−1 . The spin-cutoff parameter σI is
given by σI2 = 0.0888aT A2/3 , where the nuclear temperature
is given by the following:
T = U/a.
(6)
In cases where the intrinsic excitation energy U becomes
negative, we set U = 0, T = 0, and σI = 1.
Figure 3 demonstrates the level-density normalization
procedure for the 97 Mo case. The experimental data points are
normalized according to Eq. (3) by adjusting the parameters
A and α such that a least χ 2 fit is obtained in between the
arrows. For the lower excitation region (see upper panel),
one should take care only to include a fit region where it
is likely that (almost) all levels are known. In practice, this
53
means that the level density should not exceed ∼50 levels
per MeV. At the higher excitation region (lower panel),
the Fermi-gas extrapolation connects seamlessly the
highest-energy data points with the level-density value
determined from neutron-resonance spacing at Bn . Generally,
the resulting normalization is not very dependent on the choice
of the theoretical extrapolation function (Fermi gas) or the
chosen fit region (∼4.5 to ∼7 MeV).
Unfortunately, no information exists on the level density
at E = Bn for 94 Mo. Therefore, we estimate this value from
a systematics of other Mo isotopes where information on
the level density at Bn exists. In Fig. 4 we plot all the
known data points from the Mo isotopic chain. The oddand even-mass molybdenum nuclei fall into two groups, both
showing a decreasing level density as function of excitation
energy. This behavior is rather counterintuitive because in
a given nucleus the level density increases exponentially
with excitation energy, and for neighboring nuclei one would
naively expect quite similar level-density curves. Two effects
combine to result in the negative slope of the data points: (i) the
decrease of single-particle level density when approaching the
N = 50 shell gap resulting in a decrease of the level density in
general and (ii) the increase of the neutron-separation energy
with decreasing neutron number. For the negative slope to
emerge, both effects have to be rather precisely of the same
size for each step along the Mo isotopic chain. We have
found no good physical explanation for this to happen, but
we employ this fortuitous coincidence to estimate ρ(Bn ) =
(6.2 ± 1.0)104 MeV−1 for 94 Mo from our phenomenological
systematics.1 The splitting of data points between even and
odd Mo isotopes must not be interpreted solely as because
1
This value also agrees within a factor of 2 with the systematics of
Ref. [38].
034311-4
PHYSICAL REVIEW C 73, 034311 (2006)
LEVEL DENSITIES AND THERMODYNAMICAL . . .
FIG. 4. Level densities at the neutron-separation energy. The
unknown level density of 94 Mo (open circle) is estimated from the
slope of the data points of the odd-mass molybdenum isotopes.
of the effect of the pairing-energy shift of the level-density
curves; the difference in the magnitude of Bn between neutronodd and -even isotopes also affects the magnitude of this
splitting.
III. LEVEL DENSITY AND FINE STRUCTURES
OF THE ENTROPY
The present knowledge on level density is concentrated
in mainly two regions; the low-excitation region up to
∼2 MeV, studied in detail using spectroscopy and counting
of known, discrete levels [39] and the region around the
neutron-separation energy studied by experiments on neutron
resonances [40]. Almost nothing is known of the level density
in between the above-mentioned regions, but it is possible to
determine quite reliably two anchor points of the level density.
Figure 5 shows the extracted anchor points (filled data
points) for nine molybdenum isotopes together with the level
density deduced from known discrete levels (solid lines).
The upper anchor point is simply determined from neutronresonance data. The lower anchor point, which is the average
value of three data points, is determined such that a straight line
on a logarithmic plot, going through the upper anchor point,
provides an upper bound of the level-density distribution of
known levels. The algorithm is iterative and treats all nuclei
similarly to ensure that the results are comparable. The straight
54
line connecting the lower and upper anchor points is defined
by the constant temperature formula
ρ(E) = CeE/τ
(7)
FIG. 5. Level density of nine molybdenum isotopes. The histograms represent levels from spectroscopy [39]. A straight line is
drawn from these levels to the level density at the neutron-separation
energy that is determined by average neutron-resonance spacings.
The line represents the constant-temperature level-density formula
(see text).
with τ −1 = (ln ρ2 − ln ρ1 )/(E2 − E1 ) and C = ρ1 exp(−E1 /
τ ). Details are given in Ref. [41]. Provided that all the levels
are measured at the excitation energy of the lower anchor
point, we find from the plots of Fig. 5 that the temperature-like
parameter τ drops from 1.05 MeV for the spherical 93 Mo to
about 0.72 MeV for the well-deformed 101 Mo nucleus. This
figure also illustrates the excitation energy where one would
expect the appearance of missing levels in spectroscopic work,
typically if the density of levels exceeds 50 MeV−1 .
The level densities ρ(E) extracted from the eight reactions
are displayed in Fig. 6. The data have been normalized as
prescribed above, and the parameters used for 93−98 Mo in
Eq. (5) are listed in Table I. We find that the normalization
constant η drops by one order of magnitude when going from
deformed to spherical nuclei. This means that the spherical
93
Mo has about ten times lower level density than predicted
by a global Fermi-gas model. As mentioned earlier, this effect
is one of the reasons for the negative slope of the data points
in Fig. 4.
Our experimental data interpolate between the previously
known lower anchor point at ∼2 MeV and about 1 MeV
below the upper anchor point at ∼7 MeV. For the energy
interval between ∼6 and ∼7 MeV, we rely on models [34,35].
Despite the fact that the final extrapolation of the level density
up to the nucleon-separation energy is model dependent, this
affects only the average slope of the level density and does
not affect the fine structure. This enables us to observe fine
structures in the level density that are thought to reflect the
breaking of individual pairs. In an earlier work, we showed how
034311-5
R. CHANKOVA et al.
PHYSICAL REVIEW C 73, 034311 (2006)
also how these unpaired nucleons around the Fermi energy
can increase the cost in energy to break up further nucleon
pairs because of the blocking effect of the Pauli principle [42].
Our goal in the present work is to obtain experimental values
for the critical temperature of the pair-breaking process. On
the way, we also investigate some other thermodynamical
properties, in particular the entropy, when going from spherical
to deformed nuclei. The generalization of the concept of
temperature for a small system is not straightforward and
has been discussed extensively in the literature (see, e.g.,
Ref. [42] and references therein). Traditionally, temperature
is introduced in slightly different ways in the microcanonical
statistical ensemble (as a property of the system itself) and
in the canonical statistical ensemble (as imposed by a heat
bath). The temperature-energy relations for rare-earth nuclei
(the caloric curves) derived within the two statistical ensembles
display in general a different behavior because the nuclei under
discussion are essentially discrete systems [13].
To avoid the shortcomings imposed by the above-mentioned
statistical ensembles, a new approach for the caloric curves
based on the two-dimensional probability distribution P (E, T )
has been proposed [42,43]. This approach bypasses the wellknown problem of spurious structures such as negative temperatures and heat capacities in the microcanonical ensemble.
Conversely, more structures in the new caloric curve are
evident than in the canonical caloric curve. However, this new
method is still not well settled and we will defer the discussion
of caloric curves to another occasion.
Within the microcanonical ensemble the experimentally
obtained level density corresponds to the partition function
that is simply the multiplicity of accessible states. Thus, the
entropy S of the system within the energy interval E and E + δ
is determined by the following:
S(E) = kB ln (E),
FIG. 6. Normalized level densities for 93−98 Mo. The open and
filled circles are data from the (3 He,α) and (3 He,3 He′ ) reactions,
respectively.
a simple single-particle-plus-pairing model can qualitatively
explain the emergence of such fine structures [21]. Moreover,
we have in the past investigated how pairing correlations are
weakened in the presence of already unpaired nucleons, but
(8)
where (E) = ρ(E)/ρ0 and the Boltzmann constant is set to
unity (kB ≡ 1) for simplicity.2 To fulfill the third law of thermodynamics; namely S → 0 when T → 0, the normalization
denominator is set to ρ0 = 1.5 MeV−1 . Entropy as a function of
energy can be defined and measured for small and mesoscopic
systems as well as for large systems. However, fluctuations in
2
More precisely, multiplicity (E) is proportional to ρ(E) (2J
(E) + 1), where J (E) is the average spin of levels with excitation
energy E. However, in the present work, we neglect the weak
excitation-energy dependence of J (E).
TABLE I. Parameters used for the back-shifted Fermi-gas level density.
Nucleus
98
Mo
Mo
96
Mo
95
Mo
94
Mo
93
Mo
97
a
Epair (MeV)
a (MeV−1 )
2.080
0.995
2.138
1.047
2.027
0.899
11.33
11.23
11.13
11.03
10.93
10.83
C1 (MeV)
−1.521
−1.526
−1.531
55
−1.537
−1.542
−1.547
Bn (MeV)
D (eV)
ρ(Bn ) (104 MeV−1 )
η
8.642
6.821
9.154
7.367
9.678
8.067
75
1050
105
1320
—
2700
9.99
3.10
7.18
2.50
6.20a
1.27
0.87
0.65
0.46
0.34
0.25
0.08
Estimated from systematics (see text).
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LEVEL DENSITIES AND THERMODYNAMICAL . . .
FIG. 7. Experimental entropy for 93,94 Mo (upper panel) and
Mo (lower panel) as function of excitation energy E.
97,98
level spacings that are typical for small systems will make the
entropy sensitive to exactly how the energy interval between
E and E + δE is chosen. Thus, Eq. (8) is useful only if (E)
is a sufficiently smooth function, i.e., for the case that its first
derivative exists. Small statistical fluctuations in the entropy
S may give rise to large contributions to the temperature T,
which is defined within the microcanonical ensemble as
∂S −1
T (E) =
.
(9)
∂E
Figure 7 shows the entropy deduced within the microcanonical ensemble for 93,94 Mo (upper panel) and 97,98 Mo (lower
panel). The entropy curve plotted on a linear scale is essentially
identical to the level-density curve on a logarithmic scale. In
general, the most efficient way to create additional states in
atomic nuclei is to break J = 0 nucleon Cooper pairs from
the core. The resulting two nucleons may thereby be thermally
excited rather independently to the available single-particle
levels around the Fermi surface. We therefore interpret, e.g.,
the steplike increases in entropy around 2–3 MeV excitation
energy in Fig. 7 as because of the breaking of the first nucleon
Cooper pair.
The entropies of odd-mass nuclei are higher than those
of their even-even neighbors, even when their mass numbers
are lower. It is interesting to compare entropies between
neighboring nuclei. The difference in entropy
S(E) = Sodd−mass − Seven−even
(10)
is assumed to be a measure for the single-particle entropy. The
entropies of the almost spherical 93 Mo and 94 Mo (upper panel
56
of Fig. 7) follow each other closely above E ∼ 2.5 MeV. Here,
the odd valence nucleon behaves almost as a passive spectator.
For 93,94 Mo, we find S >
∼ 0 for E > 2.5 MeV. The deformed
case, (lower panel of Fig. 7) exhibits an entropy difference of
S >
∼ 1. This is less than the value of S ∼ 2 found for
rare-earth nuclei [44,45].
These observations can be explained qualitatively by the
fact that the single-particle entropy depends on the number of
single-particle orbitals that are available for excitations at a
certain temperature. For 93,94 Mo at low energies, the single
neutron outside the closed shell can only occupy the two
d5/2 and g7/2 orbitals giving an entropy of ln 2 ∼ 0.7. For
the case of deformed nucleus 97,98 Mo, symmetry breaking
results in a splitting of these two single-particle orbitals into
seven Nilsson orbitals, giving a total entropy of ln 7 ∼ 1.9, i.e.,
about one unit more than for the 93,94 Mo case. In the rare-earth
region strong deformation and intruder orbitals from other
shells result in an even higher single-particle level density,
giving rise to an even larger single-particle entropy. Although
our simple argument somewhat overestimates the observed
single-particle entropies, it provides a satisfactory explanation
for the differences between the single-particle entropies in the
different cases.
The entropy in atomic nuclei at low energies does not simply
scale with the total number of nucleons. In the presence of
pairing correlations, i.e., away from closed shells, the entropy
scales rather with the number of unpaired nucleons at a
certain excitation energy. When pairing correlations cannot
form because of the large single-particle level spacings around
closed shells, an unpaired nucleon will behave almost as
a passive spectator without contributing significantly to the
entropy of the system.
At excitations energies around 5.5 MeV, a bump is observed
in the entropy curves for the lighter 93,94 Mo nuclei. In light
of the discussion above, it is unlikely that such a bump can
be interpreted as the breaking of a nucleon Cooper pair.
We propose that this bump is related to the sudden onset of
neutron excitations across the N = 50 shell gap. Because of
the generally higher level density and the onset of deformation
in the heavier Mo isotopes, the opening of the g9/2 shell might
be less significant, leading to the effect being spread out in
energy. However, such an effect might become visible in the
lighter 93,94 Mo nuclei. This interpretation is supported by the
fact that the large transfer peak at 5.5 MeV excitation energy
in the particle spectrum of the 97 Mo(3 He,αγ )96 Mo reaction
at a beam energy of 45 MeV (see Fig. 1) has been shown
in an experiment at the Yale University Enge splitpole to be
dominated by high ℓ transfer, most likely ℓ = 4h̄ [46].
IV. PHASE TRANSITIONS
A. Model
In this section we utilize a recently developed thermodynamic model [41,47,48] that allows the investigation and
classification of the pairing phase transition. The model is
based on the canonical ensemble theory where equilibrium is
obtained at a certain given temperature T. It can describe level
densities for midshell nuclei in the mass regions A = 58, 106,
162, and 234.
The basic idea of the model is the assumption of a reservoir
of nucleon pairs. These nucleon pairs can be broken and the
unpaired nucleons are thermally scattered into an infinite,
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R. CHANKOVA et al.
PHYSICAL REVIEW C 73, 034311 (2006)
equidistant, doubly degenerated single-particle level scheme.
The nucleon pairs in the reservoir do not interact with each
other and are thought to occupy an infinitely degenerated
ground state. The nucleons in the single-particle level scheme
do not interact with each other either, but they have to obey
the Pauli principle.
The essential parameters of the model are the number of
pairs n in the reservoir at zero temperature, the spacing of
the single-particle level scheme ǫ = 30 MeV/nucleon, and√the
energy necessary to break a nucleon pair 2 = 24 MeV/ A.
Quenching of pairing correlations is introduced in this model
by reducing the energy required to break a nucleon pair in
the presence of unpaired nucleons. We assume that for every
already broken nucleon pair, the energy to break a further
nucleon pair is reduced by a factor of r = 0.56, which is
suggested by theoretical calculations [49]. To simulate the
effect of the N = 50 shell closure on the A < 98 isotopes,
we depart from the global systematics for ǫ and replace it
with ǫ ′ = ǫa(A = 98)/a(A < 98) using the experimentally
deduced a values of Ref. [40]. We use the same parameters for
both protons and neutrons, keeping the proton pairs fixed to
seven because there are 14 more protons outside the Z = 28
shell closure.
The total partition function is written as a product of
proton (Zπ ), neutron (Zν ), rotation (Zrot ), and vibration (Zvib )
partition functions where the parameters for the collective
excitations are the rigid moment of inertia Arig = 34 MeV
A−5/3 and the energy of one-phonon vibrations h̄ωvib =
1.5 MeV taken from spectroscopic data [39]. Thermodynamical quantities of interest can be deduced from the Helmholtz
free energy defined as
F (T ) = −T ln (Zπ Zν Zrot Zvib ) .
(11)
This equation connects statistical mechanics and thermodynamics. Quantities such as entropy, average excitation energy,
and heat capacity can be calculated by
∂F
S(T ) = −
(12)
∂T V
E(T ) = F + ST
∂E
,
CV (T ) =
∂T V
(13)
(14)
respectively.
In Fig. 8, the Helmholtz free energy F, entropy S, average
excitation energy E, and heat capacity CV are shown as
functions of temperature for even-even, odd, and odd-odd
systems in the 96 Mo mass region. The free energy F and
the average excitation energy E are rather structureless as
functions of temperature. The odd-even effects are small: The
even-even, odd, and odd-odd systems have different excitation
energies at the same temperature, where the even-even system
requires the highest E to be heated to some given temperature
T. Around Tc ∼ 0.9–1.1 MeV the nuclei are excited to their
respective nucleon-separation energies.
The entropy S and heat capacity CV are more sensitive
to thermal changes. The entropy difference S between
systems with A and A ± 1 is a useful quantity. At moderate
temperatures, it is approximately extensive (additive) and
represents the single-particle entropy associated with the
FIG. 8. Model calculation for nuclei around
Mo. The four panels show the free energy
F, the entropy S, the thermal excitation energy
E, and the heat capacity CV as a function
of temperature T. The arrow at Tm ∼ 0.9 MeV
indicates the local maximum of CV where the
pair-breaking process takes place in the eveneven system. The same parameter set is used for
even-even (solid lines), odd (dashed lines), and
odd-odd systems (dash-dotted lines).
96
57
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LEVEL DENSITIES AND THERMODYNAMICAL . . .
valence particle (or hole) [41]. In the upper right panel, we
find, e.g., that the nucleon carries a single-particle entropy of
S ∼ 2.0 at T ∼ 0.4 MeV.
The shape of the heat-capacity curve is related to the
level density. Traditionally, level-density curves have been
described by the two-component model as proposed by Gilbert
and Cameron [34]. Within this model, the low energetic
part is a constant-temperature level density and the high
energetic part is a Fermi-gas model. It has been shown in
Ref. [12] that the inclusion of a constant-temperature part
in the level-density formula creates a heat-capacity curve as
function of temperature with a pronounced S shape similar
to that shown in Fig. 8. With our model parameters, the
maximum of the local increase in the CV curve takes place
at about T ∼ 0.9 MeV. This temperature compares well with
the temperature determined in the microcanonical ensemble
from Eq. (9), giving a temperature of T ∼ 0.9 MeV for 96 Mo
(see also Fig. 5).
B. Comparison with experimental data
Our model is described within the canonical ensemble,
whereas experimental data refer to the microcanonical ensemble. There are two ways to compare our model with
experiments. Details are given in Ref. [41]. In this work we
will make use of the saddle-point approximation [50]
ρ(E) =
T
exp(S)
,
√
2π CV
(15)
which gives satisfactory results for the nuclear level density
[41,48].
Figure 9 shows the theoretical level densities calculated
using Eq. (15). The agreement with the anchor points and
the experimental level densities for 97,98 Mo isotopes is satisfactory. Some of the model parameters could be adjusted
more precisely, however, in this work we have chosen to use
parameters taken from systematics.
To investigate the behavior of the pairing correlations when
approaching a major shell gap, we compare the canonical
CV curves that are based on the Laplace transforms of the
experimental level densities. The curves are plotted in Fig. 10
for even 94,96,98 Mo (upper panel) and odd 93,95,97 Mo (lower
panel) nuclei. The CV curves resemble washed-out step
structures and show an S shape as a function of temperature
quite similar to the model calculation on the lower right
panel of Fig. 8. Because of the averaging performed by
the Laplace transformation discrete transitions between the
different quasiparticle regimes, as observed within the microcanonical ensemble, are hidden. Only the phase transition
related to the quenching of the pair correlations as a whole can
be seen. Details are given in Ref. [17].
The canonical heat-capacity curves show local enhancements around T ∼ 0.5–1.0 MeV. Such enhancements were
predicted in the calculations of Fig. 8, and they are expected
58
to be larger in the even-mass nuclei compared to the odd-mass
neighbors. The experimental heat capacities show this feature
for the 97,98 Mo pair, and up to a certain extend for the
93,94
Mo pair, but it is not very obvious for the 95,96 Mo pair,
FIG. 9. Calculated level density of 98 Mo (solid line) and 97 Mo
(dashed line) as function of average excitation energy E. The big
open circles and squares are experimental level-density anchor points
from Ref. [41]. The small filled and open circles are experimental
data points measured with the (3 He,α) and (3 He,3 He′ ) reactions,
respectively for the two isotopes.
where 95 Mo shows a more pronounced enhancement than
expected. Approaching the N = 50 closed shell, the local
enhancements become less and less pronounced. The general
behavior of pairing correlations is that at shell closure there
are almost no pairing correlations and, as particles are added,
the pairing correlations increase. Therefore the signature of
a transition from a “paired phase” to an “unpaired phase”
when approaching a major shell gap becomes less and less
pronounced. We should note that very recently an alternative
interpretation has been given [51]. These authors find that the S
shape can be accounted for as an effect of the particle-number
conservation, and it occurs even when assuming a constant gap
in the BCS theory.
From the CV curves, we have extracted the critical temperature for the quenching of pair correlations. The critical
temperatures have been obtained by a fit of the canonical heat
capacity of a constant-temperature level-density model to the
data for the first 600 keV in temperature. The algorithm and
its sensitivity are discussed in Ref. [12]. The values obtained
are plotted in Fig. 11; there is a tendency for the critical
temperature to be slightly higher for odd 93,95,97 Mo than for
even 92,94,96 Mo nuclei, similar to the local enhancement of the
heat-capacity curve in the model calculation (see the lower
right panel of Fig. 8) that is observed at higher temperatures
for odd-mass Mo isotopes. The higher critical temperature for
odd-mass nuclei is because of the Pauli blocking effect of
the unpaired quasiparticle that increases the distance to the
Fermi surface for low-lying orbitals with coupled pairs and
thus increases the cost in energy to break pairs into more
034311-9
PHYSICAL REVIEW C 73, 034311 (2006)
R. CHANKOVA et al.
from a phase with strong pairing correlations to a phase
where the pairing correlations are quenched [12]. Shell-model
Monte-Carlo calculations [7] have shown that the pairing phase
transition is strongly correlated with the suppression of neutron
pairs with increasing temperature. It has also been observed
that the reduction of the neutron-pair content of the wave
function is much stronger in the even-even than in the odd-mass
isotopes, giving rise to the more pronounced S shape in the
canonical heat-capacity curves in the even-even nuclei. The
same odd-even difference in the heat capacity is also observed
experimentally between 161 Dy and 162 Dy, as well as 171 Yb and
172
Yb [12].
C. Entropy as function of neutron number
FIG. 10. Observed heat capacity as functions of temperature in
the canonical ensemble for the even 94,96,98 Mo (upper panel) and odd
93,95,97
Mo (lower panel) nuclei.
quasiparticles. Incidentally, the critical temperature for the
quenching of pairing correlations coincides (by construction)
quite well with the temperature-like parameter τ of Fig. 5.
A discontinuity of the heat capacity in a macroscopic
system indicates a second-order phase transition according to
the Ehrenfest classification; this is observed in the transition
of a paired Fermion system such as a low-temperature
superconductor or superfluid 3 He to their normal phases. Thus,
the experimentally observed local enhancement of the heat
capacity is interpreted as a fingerprint of a phase transition
To study entropy as a function of neutron number, we
compare the microcanonical entropy obtained by the saddlepoint approximation of Eq. (15) to our experimental data.
In Fig. 12 the data are plotted as a function of the neutron
number N (left panel) and as a function of the number of
available neutrons in the reservoir (right panel). Although only
qualitative agreement is achieved, some simple conclusions
can be drawn.
For the isotopes under investigation in this work, we see
that the entropy at 1 MeV in both panels increases moderately
as a function of the number of particles. The entropy at
7 MeV increases more rapidly and this is correlated to the
evolution of the temperature-like parameter τ (see Fig. 5).
Both theoretically and experimentally, the odd systems show
S = 1.0kB higher entropy than their neighboring even-even
systems.
59
FIG. 11. Critical temperature for the quenching of pair correlations for 93−98 Mo isotopes.
FIG. 12. Entropy extracted at excitation energies of 1 and 7 MeV
as a function of neutron number N (left panel) and number of available
neutrons in the model (right panel) for odd-even (open circles) and
even-even (filled circles) molybdenum isotopes.
034311-10
LEVEL DENSITIES AND THERMODYNAMICAL . . .
PHYSICAL REVIEW C 73, 034311 (2006)
The slopes at 7 MeV in the left panel of Fig. 12 give
dS/dN = 0.5kB . Thus, going from 98 Mo to 93 Mo the level
density drops when approaching the N = 50 shell gap by a
factor of ∼0.03. This mechanism is also reflected in the η
parameter of Eq. (5), which drops from 0.87 for 98 Mo to 0.08
for 93 Mo.
As we already mentioned, the less pronounced S shape
shows that the pairing correlations decrease when approaching
the N = 50 shell gap. At the same time, the critical temperature
for the quenching of pair correlations increases, which is
the opposite of what one might expect. This effect can be
explained by the increase in single particle level spacing when
approaching the N = 50 shell gap. We have already seen in the
discussion in the previous section, that this increase, together
with the weakening pairing correlations, which fail to push
the nuclear ground state sufficiently down in energy, lead to a
decrease in single-particle entropy, see Figs. 7 and 12.
Therefore, the increase in critical temperature for the
quenching of pairing correlations when approaching the N =
50 shell gap is because of the competition between the weakening pairing correlations and the increasing single-particle
level spacing. Just as the weakened pairing correlations in odd
nuclei cannot compensate for the effect of Pauli blocking on
Tc , they cannot compensate for the effect of an increase in
single-particle level spacing on Tc when approaching a major
shell gap.
spectra. Within the canonical ensemble, thermodynamical
observables were deduced from the level density; they display
features consistent with signatures of a phase transition from a
strongly pair-correlated phase to a phase without strong pairing
correlations. This conclusion is supported by recent theoretical
calculations within shell-model Monte Carlo simulations by
Alhassid et al. [7,8,50], where it is shown that the expectation
value of the pair operator decreases strongly around the
critical temperature. However, we would like to point out that
other interpretations are not excluded. Different mechanisms
governing the thermodynamic properties of odd and even
systems were studied. A simple, recently developed model
for the investigation and classification of the pairing phase
transition in hot nuclei has been employed and qualitative
agreement with experimental data achieved. Using the saddlepoint approximation the experimental level densities of eveneven and odd-even systems are reproduced. Estimates for
the critical temperature of the pairing phase transition yield
Tc ∼ 0.7–1.0 MeV.
ACKNOWLEDGMENTS
Levels in
Mo in the excitation-energy region up to
the neutron-separation energy were populated using (3 He,αγ )
and (3 He,3 He′ γ ) reactions. The level densities of 93−98 Mo
were determined from their corresponding primary γ -ray
Part of this work was performed under the auspices of the
U.S. Department of Energy by the University of California,
Lawrence Livermore National Laboratory under contract W7405-ENG-48. U.A., E.A., G.E.M., and A.V. acknowledge
support from U.S. Department of Energy grant DE-FG02-97ER41042 and from the National Nuclear Security Administration under the Stockpile Stewardship Science Academic Alliances program through Department of Energy Research grant
DE-FG03-03-NA00074 and DE-FG03-03-NA00076. M.G.,
F.I., S.M., J.R., S.S., A.C.L., and S.Ø acknowledge financial
support from the Norwegian Research Council (NFR).
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[33] Handbook for Calculations of Nuclear Reaction Data (IAEA,
Vienna, 1998).
[34] A. Gilbert, A. G. W. Cameron, Can. J. Phys. 43, 1446 (1965).
[35] T. von Egidy, H. H. Schmidt, and A. N. Behkami, Nucl. Phys.
A481, 189 (1988).
[36] G. Audi and A. H. Wapstra, Nucl. Phys. A729, 337 (2003).
[37] J. Dobaczewski, P. Magierski, W. Nazarewicz, W. Satuła, and
Z. Szymański, Phys. Rev. C 63, 024308 (2001).
[38] D. Bucurescu, T. von Egidy, J. Phys. G: Nucl. Part. Phys. 31,
S1675 (2005).
[39] R. B. Firestone, and V. S. Shirley, Table of Isotopes, 8th ed.
(John Wiley & Sons, New York, 1996), Vol. I.
[40] A. S. Iljinov, M. V. Mebel, N. Bianchi, E. De Sanctis,
C. Guaraldo, V. Lucherini, V. Muccifora, E. Polli, A. R. Reolon,
and P. Rossi, Nucl. Phys. A543, 517 (1992), and references
therein.
[41] M. Guttormsen, M. Hjorth-Jensen, E. Melby, J. Rekstad,
A. Schiller, and S. Siem, Phys. Rev. C 63, 044301 (2001).
[42] A. Schiller et al., AIP Conf. Proc. 777, 216 (2005).
[43] A. Schiller, M. Guttormsen, M. Hjorth-Jensen, J. Rekstad, and
S. Siem, nucl-th/0306082.
[44] M. Guttormsen, A. Bagheri, R. Chankova, J. Rekstad, S. Siem,
A. Schiller, and A. Voinov, Phys. Rev. C 68, 064306 (2003).
[45] U. Agvaanluvsan et al., Phys. Rev. C 70, 054611 (2004).
[46] T. F. Wang (private communication).
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A. Schiller, and S. Siem, Phys. Rev. C 64, 034319 (2002).
[48] A. Schiller, M. Guttormsen, M. Hjorth-Jensen, J. Rekstad, and
S. Siem, Phys. Rev. C 66, 024322 (2002).
[49] T. Døssing et al., Phys. Rev. Lett. 75, 1276 (1995).
[50] H. Nakada and Y. Alhassid, Phys. Rev. Lett. 79, 2939 (1997).
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(2005).
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Microcanonical entropies and radiative strength functions of 50,51 V
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Microcanonical entropies and radiative strength functions of 50,51 V
A. C. Larsen,∗ R. Chankova, M. Guttormsen, F. Ingebretsen, S. Messelt, J. Rekstad, S. Siem, N. U. H. Syed, and S. W. Ødegård
Department of Physics, University of Oslo, P.O. Box 1048, Blindern, N-0316 Oslo, Norway
T. Lönnroth
Department of Physics, Åbo Akademi University, FIN-20500 Åbo, Finland
A. Schiller
National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824, USA
A. Voinov
Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA
(Received 23 November 2005; published xxxxx)
The level densities and radiative strength functions (RSFs) of 50,51 V have been extracted using the (3 He,αγ )
and (3 He,3 He′ γ ) reactions, respectively. From the level densities, microcanonical entropies are deduced. The
high γ -energy part of the measured RSF fits well with the tail of the giant electric dipole resonance. A significant
enhancement over the predicted strength in the region of Eγ 3 MeV is seen, which at present has no theoretical
explanation.
DOI: 10.1103/PhysRevC.00.004300
PACS number(s): 21.10.Ma, 24.10.Pa, 25.55.Hp, 27.40.+z
I. INTRODUCTION
The structures of vanadium isotopes are based on simple
shell-model configurations at low excitation energies. The
valence protons and neutrons occupy the single-particle πf7/2
and νf7/2 orbitals, respectively. These shells are isolated from
other orbitals by the N, Z = 20 and 28 shell gaps, making
the vanadium isotopes interesting objects for studying various
nuclear shell effects. In particular, it is well known that
the number of available singe-particle levels is significantly
reduced for nuclei at closed shells.
The density of states or, equivalently, the entropy in these
systems depends on the number of broken Cooper pairs
and single-particle orbitals made available by crossing the
shell gaps. The 50,51 V nuclei are of special interest because
the neutrons are strongly blocked in the process of creating
entropy; 50 V and 51 V have seven and eight neutrons in the
νf7/2 orbital, respectively. Thus, the configuration space of the
three protons in the πf7/2 shell is of great importance.
These particular shell-model configurations are also expected to govern the γ -decay routes. Specifically, as within
every major shell, the presence of only one parity for singleparticle orbitals in the low-spin domain means that transitions
of E1 type will be suppressed. The low mass of the investigated
nuclei causes the centroid of the giant electric dipole resonance
(GEDR) to be relatively high, while the integrated strength
according to the Thomas-Reiche-Kuhn sum rule is low; both
observations work together to produce a relatively weak
low-energy tail when compared to heavier nuclei. Hence,
possible nonstatistical effects in the radiative strength function
(RSF) might stand out more in the present investigation.
∗
Electronic address: a.c.larsen@fys.uio.no
0556-2813/2006/00(0)/004300(8)/23.00
The Oslo Cyclotron group has developed a method to
extract first-generation (primary) γ -ray spectra at various
initial excitation energies. From such a set of primary spectra,
the nuclear level density and the RSF can be extracted
simultaneously [1,2]. These two quantities reveal essential
information on nuclear structure such as pair correlations and
thermal and electromagnetic properties. In the last five years,
the Oslo group has demonstrated several fruitful applications
of the method [3–7].
In Sec. II an outline of the experimental procedure is given.
The level densities and microcanonical entropies are discussed
in Sec. III, and in Sec. IV the RSFs are presented. Finally,
concluding remarks are given in Sec. V.
II. EXPERIMENTAL METHOD
The experiment was carried out at the Oslo Cyclotron
Laboratory using a beam of 30-MeV 3 He ions. The selfsupporting natural V target had a purity of 99.8% and a
thickness of 2.3 mg/cm2 . Particle-γ coincidences for 50,51 V
were measured with the CACTUS multidetector array [8].
The charged ejectiles were detected using eight Si particle
telescopes placed at an angle of 45◦ relative to the beam
direction. Each telescope consists of a front E detector and
a back E detector with thicknesses of 140 and 1500 µm,
respectively. An array of 28 collimated NaI γ -ray detectors
with a total efficiency of ∼15% surrounded the target and
the particle detectors. The reactions of interest were the
pick-up reaction 51 V(3 He, αγ )50 V, and the inelastic scattering
51
V(3 He,3 He′ γ )51 V. The typical spin range is expected to be
I ∼ 2−4 h̄. The experiment ran for about one week, with beam
currents of ∼1 nA.
The experimental extraction procedure and the assumptions
made are described in Refs. [1,2]. The data analysis is based
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FIG. 1. γ spectra of 50 V for excitation energy E = 6–8 MeV.
FIG. 2. Unfolded γ spectra of
6–8 MeV.
on three main steps: (1) preparing the particle-γ coincidence
matrix, (2) unfolding the γ -ray spectra, and (3) constructing
the first-generation matrix.
In the first step, for each particle-energy bin, total spectra
of the γ -ray cascades are obtained from the coincidence measurement. The particle energy measured in the telescopes is
transformed to excitation energy of the residual nucleus, using
the reaction kinematics. Then each row of the coincidence
matrix corresponds to a certain excitation energy E, while
each column corresponds to a certain γ energy Eγ .
In the next step, the γ -ray spectra are unfolded using the
known response functions of the CACTUS array [9]. The
Compton-subtraction method described in Ref. [9] preserves
the fluctuations in the original spectra without introducing
further spurious fluctuations. A typical raw γ spectrum is
shown in the top panel of Fig. 1, taken from the 50 V
coincidence matrix gating on the excitation energies between
E = 6–8 MeV. The middle panel shows the unfolded spectrum,
and in the bottom panel this spectrum has been folded with the
response functions. The top and bottom panels are in excellent
agreement, indicating that the unfolding method works very
well.
The third step is to extract the γ -ray spectra containing
only the first γ rays in a cascade. These spectra are obtained
for each excitation-energy bin through a subtraction procedure
as described in Ref. [10]. The main assumption of this method
is that the γ -decay spectrum from any excitation-energy bin
is independent of the method of formation, either directly by
the nuclear reaction or populated by γ decay from higherlying states following the initial reaction. This assumption
is automatically fulfilled when the same states are equally
populated by the two processes, since γ branching ratios are
properties of the levels themselves. Even if different states are
populated, the assumption is still valid for statistical γ decay,
V for excitation energy E =
which only depends on the γ -ray energy and the number of
accessible final states. In Fig. 2, the total unfolded γ spectrum,
the γ spectrum of second and higher generations, and the
first-generation spectrum of 50 V with excitation-energy gates
E = 6−8 MeV are shown. The first-generation spectrum is
obtained by subtracting the higher-generation γ rays from the
total γ spectrum.
When the first-generation matrix is properly normalized [2],
the entries of it are the probabilities P (E, Eγ ) that a γ ray
of energy Eγ is emitted from an excitation energy E. The
probability of γ decay is proportional to the product of the
level density ρ(E − Eγ ) at the final energy Ef = E − Eγ and
the γ -ray transmission coefficient T (Eγ ), that is,
P (E, Eγ ) ∝ ρ(E − Eγ )T (Eγ ).
(1)
This factorization is the generalized form of the Brink-Axel
hypothesis [11,12], which states that any excitation modes
built on excited states have the same properties as those built
on the ground state. This means that the γ -ray transmission
coefficient is independent of excitation energy and thus of the
nuclear temperature of the excited states. There is evidence
that the width of the giant dipole resonance varies with the
nuclear temperature of the state on which it is built [13,14].
However, the temperature corresponding to the excitationenergy range covered in this workis rather low and changes
slowly with excitation energy (T ∼ Ef ); thus, we assume that
the temperature is constant and that the γ -ray transmission
coefficient does not depend on the excitation energy in the
energy interval under consideration.
The ρ and T functions can be determined by an iterative
procedure [2], with which each data point of these two functions is simultaneously adjusted until a global χ 2 minimum
with the experimental P (E, Eγ ) matrix is reached. No a
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FIG. 3. Experimental
first-generation
γ spectra (data points with error bars) at six
different initial excitation energies compared
to the least-χ 2 fit (solid lines) for 50 V.
The fit is performed simultaneously on the
entire first-generation matrix of which the
six displayed spectra are a fraction. The
first-generation spectra are normalized to unity
for each excitation-energy bin.
priori assumptions about the functional form of either the
level density or the γ -ray transmission coefficient are used.
An example to illustrate the quality of the fit is shown in
Fig. 3, where we compare for the 51 V(3 He, αγ )50 V reaction the
experimental first-generation spectra to the least-χ 2 solution
for six different initial excitation energies.
The globalized fitting to the data points determines the
functional form of ρ and T ; however, it has been shown [2]
that if one solution for the multiplicative functions ρ and
T is known, one may construct an infinite number of other
functions, which give identical fits to the P matrix by
ρ̃(E − Eγ ) = A exp[α(E − Eγ )]ρ(E − Eγ ),
T (Eγ ) = B exp(αEγ )T (Eγ ).
(2)
(3)
Thus, the transformation parameters α, A, and B, which
correspond to the physical solution, remain to be determined.
III. LEVEL DENSITY AND MICROCANONICAL ENTROPY
The parameters A and α can be obtained by normalizing
the level density to the number of known discrete levels at low
excitation energy [15] and to the level density estimated from
neutron-resonance spacing data at the neutron binding energy
E = Bn [16]. The procedure for extracting the total level
density ρ from the resonance energy spacing D is described
in Ref. [2]. Since our experimental level-density data points
only reach up to an excitation energy of E ∼ Bn − 1 MeV,
we extrapolate with the back-shifted Fermi-gas model with a
global parametrization [17,18]
ρBS (E) = η
(4)
where a constant attenuation coefficient η is introduced to
adjust ρBS to the experimental level density at Bn . The intrinsic
excitation energy is estimated by U = E − C1 − Epair , where
C1 = −6.6A−0.32 MeV is the back-shift parameter and A
is the mass number. The pairing energy Epair is based on
pairing gap parameters p and n evaluated from even-odd
mass differences [19] according to [20]. The level-density
parameter a and the spin-cutoff parameter σI are given by
a = 0.21A0.87 MeV−1 and σI2 = 0.0888T A2/3 , respectively.
√
The nuclear temperature T is described by T = U/a. The
parameters used for 50,51 V in Eq. (4) are listed in Table I.
Unfortunately, 49 V is unstable, and no information exists
on the level density at E = Bn for 50 V. Therefore, we estimate
the values from the systematics of other nuclei in the same
mass region. In order to put these data on the same footing,
we plot the level densities as a function of intrinsic energy
U . Due to the strongly scattered data of Fig. 4, the estimate
is rather uncertain. We chose a rough estimate of ρ(Bn ) =
5400 ± 2700 MeV−1 for 50 V. This value gives an attenuation
η = 0.46, which is in good agreement with the obtained value
of η = 0.51 for the 51 V nucleus. Figure 5 demonstrates the
level-density normalization procedure for the 50 V case, i.e.,
how parameters A and α of Eq. (3) are determined to obtain
a level-density function consistent with known experimental
data.
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exp(2 aU )
,
√
12 2a 1/4 U 5/4 σI
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TABLE I. Parameters used for the back-shifted Fermi-gas level density.
Nucleus
50
51
a
V
V
Epair
(MeV)
a
(MeV−1 )
C1
(MeV)
Bn
(MeV)
D
(keV)
ρ(Bn )
(103 MeV−1 )
η
0
1.36
6.31
6.42
−1.89
−1.88
9.33
11.05
2.3(6)
5.4(16)a
8.4(26)
0.46
0.51
Estimated from systematics.
The level densities of 50,51 V are also compared to the
constant-temperature formula
The experimentally extracted and normalized level densities of 50 V and 51 V are shown in Fig. 6 for excitation
energies up to ∼8 and 9 MeV, respectively. The level density
of 50 V is relatively high and has a rather smooth behavior due
to the effect of the unpaired proton and neutron, while the
level density of 51 V displays distinct structures for excitation
energies up to ∼4.5 MeV. This effect is probably caused by
the closed f7/2 neutron shell in this nucleus.
The level densities of 50,51 V obtained with the Oslo method
are compared to the number of levels from spectroscopic
experiments [21]. The 51 V nucleus has relatively few levels
per energy bin because of its closed neutron shell, so using
spectroscopic methods to count the levels seems to be reliable
up to ∼4 MeV excitation energy in this case. For higher
excitations, the spectroscopic data are significantly lower
compared to the level density obtained with the Oslo method.
This means that many levels are not accounted for in this
excitation region by using standard methods. The same can
be concluded for 50 V, and in this case the spectroscopic level
density drops off at an excitation energy of about 2.5 MeV.
which is drawn as a solid line in Fig. 6. Here the parameters
C and T are the level density at about zero excitation energy
and the average temperature, respectively; both are estimated
from the fit of the exponential to the region of the experimental
level density indicated by arrows. From this model, a constant
temperature of about 1.3 MeV is found for both nuclei.
The level density of a system can give detailed insight into
its thermal properties. The multiplicity of states s (E), which
is the number of physically obtainable realizations available
at a given energy, is directly proportional to the level density
and a spin-dependent factor (2J (E) + 1), thus
FIG. 4. Level densities estimated from neutron resonance level
spacings at Bn and plotted as a function of intrinsic excitation energy
Un = Bn − C1 − Epair . The unknown level density for 50 V (open
circle) is estimated from the line determined by a least-χ 2 fit to the
data points.
FIG. 5. Normalization procedure of the experimental level density (data points) of 50 V. The data points between the arrows are
normalized to known levels at low excitation energy (histograms)
and to the level density at the neutron-separation energy (open circle)
using a Fermi-gas level-density extrapolation (solid line).
ρfit = Cexp(E/T ),
s (E) ∝ ρ(E)(2J (E) + 1),
(6)
where J (E) is the average spin at excitation energy E.
Unfortunately, the experimentally measured level density in
this work does not correspond to the true multiplicity of
states, since the (2J + 1) degeneracy of magnetic substates
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FIG. 6. Normalized level density of 50,51 V compared to known
discrete levels (jagged line) and a constant temperature model
(straight line). The fits are performed in the region between the arrows.
FIG. 7. Entropies of 50,51 V (upper panel), and entropy difference
between the two vanadium isotopes (lower panel).
is not included. If the average spin of levels J at any
excitation energy were known, this problem could be solved by
multiplying an energy-dependent factor (2J (E) + 1) by the
experimental level density. However, little experimental data
exist on the spin distribution. Therefore, we choose in this
work to use a multiplicity l (E) based on the experimental
level density alone:
protons, a ρ0 which is typical for even-even nuclei in this
mass region is used for both the 50 V and the 51 V nucleus.
The normalization factor ρ0 used is 0.7 MeV−1 , found from
averaging data on 50 Ti and 52 Cr.
The entropies of 50,51 V extracted from the experimental
level densities are shown in the upper panel of Fig. 7. Naturally,
they show the same features as the level-density plot, with the
odd-odd 50 V displaying higher entropy than the odd-even 51 V.
Since the neutrons are almost (50 V) or totally (51 V) blocked at
low excitation energy, the multiplicity and thus the entropy is
made primarily by the protons in this region.
At 4 MeV of excitation energy, a relatively large increase
in entropy is found in the case of 51 V. This is probably because
the excitation energy is large enough to excite a nucleon across
the N, Z = 28 shell gap to other orbitals.
In the excitation region above ∼4.5 MeV, the entropies
show similar behavior, which is also expressed by the entropy
difference S displayed in the lower panel of Fig. 7. We
assume here that the two systems have an approximately
statistical behavior and that the neutron hole in 50 V acts as
a spectator to the 51 V core. The entropy of the hole can be
estimated from the entropy difference S = S(50 V) − S(51 V).
From the lower panel of Fig. 7, we find S ∼ 1.2kB for E >
4.5 MeV. This is slightly less than the quasiparticle entropy
found in rare-earth nuclei, which is estimated to be S ∼
1.7kB [5]. This is not unexpected since the single-particle
levels are more closely spaced for these nuclei; they have
therefore more entropy.
The naive configurations of 50,51 V at low excitations are
3
3
8
7
πf7/2
νf7/2
and πf7/2
νf7/2
, respectively. Thus, by counting
the possible configurations within the framework of the BCS
model [22] in the nearly degenerate f7/2 shell, one can estimate
the multiplicity of levels and thus the entropy when no Cooper
l (E) ∝ ρ(E).
(7)
The entropy S(E) is a measure of the degree of disorder of
a system at a specific energy. The microcanonical ensemble
in which the system is completely isolated from any exchange
with its surroundings, is often considered as the appropriate
one for the atomic nucleus since the strong force has such a
short range, and because the nucleus normally does not share
its excitation energy with the external environment.
According to our definition of the multiplicity of levels
l (E) obtained from the experimental level density, we define
a “pseudo” entropy
S(E) = kB ln
l (E),
(8)
which is utilized in the following discussion. For convenience,
Boltzmann’s constant kB can be set to unity.
In order to normalize the entropy, the multiplicity is written
as l (E) = ρ(E)/ρ0 . The normalization denominator ρ0 is to
be adjusted such that the entropy approaches a constant value
when the temperature approaches zero in order to fullfill the
third law of thermodynamics: S(T → 0) = S0 . In the case
of even-even nuclei, the ground state represents a completely
ordered system with only one possible configuration. This
means that the entropy in the ground state is S = ln1 = 0,
and the normalization factor 1/ρ0 is chosen such that this is
the case. Since the vanadium nuclei have an odd number of
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pairs are broken in the nucleus, one pair is broken, and so
on. We assume a small deformation that gives four energy
levels with Nilsson quantum numbers = 1/2, 3/2, 5/2, 7/2.
Furthermore, we neglect the proton-neutron coupling and
hence assume that the protons and neutrons can be considered
as two separate systems; the total entropy based on the number
of energy levels can then be written as S = Sp + Sn . This gives
S = 2.8kB for the nucleus 50 V, and S = 1.4kB for 51 V when
two protons are coupled in a Cooper pair. These values are in
fair agreement with the data of Fig. 7 at an excitation energy
below ∼2 MeV. It is gratifying that these crude estimates give
an entropy of the neutron hole in 50 V of S = 1.4kB , in
good agreement with the experimental value for the entropy
difference of 1.2kB found from Fig. 7.
With the three f7/2 protons unpaired, we obtain a total
entropy of S = 3.5 and 2.1kB for 50,51 V, respectively. This
means that the process of just breaking a proton pair within
the same shell does not contribute much to the total entropy,
but when a nucleon has enough energy to cross the shell gap
a significant increase of the entropy is expected. As already
mentioned, at excitation energies above ∼4 MeV, it is very
likely that configurations from other shells will participate in
building the total entropy.
IV. RADIATIVE STRENGTH FUNCTIONS
The γ -ray transmission coefficient T in Eq. (1) is expressed
as a sum of all the RSFs fXL of electromagnetic character X
and multipolarity L:
T (Eγ ) = 2π
Eγ2L+1 fXL (Eγ ).
(9)
XL
The slope of the experimental γ -ray transmission coefficient
T has been determined through the normalization of the level
densities, as described in Sec. III. The remaining constant
B in Eq. (3) is determined using information from neutron
resonance decay, which gives the absolute normalization of
T . For this purpose, we utilize experimental data [16] on the
average total radiative width Ŵγ at E = Bn .
We assume here that the γ decay taking place in the
quasicontinuum is dominated by E1 and M1 transitions and
that the number of positive and negative parity states is equal.
For initial spin I and parity π at E = Bn , the expression of
the width [23] reduces to
Bn
1
Ŵγ =
dEγ BT (Eγ )
4πρ(Bn , I, π ) I 0
FIG. 8. Unnormalized γ -ray transmission coefficient for 51 V.
Lines are extrapolations needed to calculate the normalization integral
of Eq. (10). Arrows indicate the lower and upper fitting regions for
the extrapolations.
MeV. In addition, the data at the highest γ energies, above
Eγ ∼ Bn − 1 MeV, suffer from poor statistics. We therefore
extrapolate T with an exponential form, as demonstrated for
51
V in Fig. 8. The contribution of the extrapolation to the total
radiative width given by Eq. (10) does not exceed 15%, thus
the errors due to a possibly poor extrapolation are expected to
be of minor importance [6].
Again, difficulties occur when normalizing the γ -ray
transmission coefficient in the case of 50 V because of the lack
of neutron resonance data. Since the average total radiative
width Ŵγ at E = Bn does not seem to show any clear
systematics for nuclei in this mass region, we choose the same
absolute value of the GEDR tail for 50 V as the one found for
51
V from photoabsorption experiments. The argument for this
choice is that the GEDR should be similar for equal number
of protons provided that the two nuclei have the same shapes.
Since it is assumed that the radiative strength is dominated
by dipole transitions, the RSF can be calculated from the
normalized transmission coefficient by
f
× ρ(Bn − Eγ , If ),
f (Eγ ) =
(10)
where Di = 1/ρ(Bn , I, π ) is the average spacing of s-wave
neutron resonances. The summation and integration run over
all final levels with spin If , which are accessible by dipole
(L = 1) γ radiation with energy Eγ . From this expression,
the normalization constant B can be determined as described
in Ref. [6]. However, some considerations have to be made
before normalizing according to Eq. (10).
Methodical difficulties in the primary γ -ray extraction prevent determination of the function T (Eγ ) in the interval Eγ <1
(11)
We would now like to decompose the RSF into its components
from different multipolarities to investigate how the E1 and
M1 radiation contribute to the total strength.
The Kadmenski{\u{\i}}, Markushev, and Furman (KMF)
model [13] is employed for the E1 strength. In this model,
the Lorentzian GEDR is modified in order to reproduce the
nonzero limit of the GEDR for Eγ → 0 by means of a
temperature-dependent width of the GEDR. The E1 strength
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.
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TABLE II. Parameters used for the radiative strength functions.
Nucleus
50
51
V
V
EE1,1
(MeV)
σE1,1
(mb)
ŴE1,1
(MeV)
EE1,2
(MeV)
σE1,2
(mb)
Ŵ;E1,2
(MeV)
EM1
(MeV)
σM1
(mb)
ŴM1
(MeV)
Ŵγ (meV)
T
(MeV)
κ
17.93
17.93
53.3
53.3
3.62
3.62
20.95
20.95
40.7
40.7
7.15
7.15
11.1
11.1
0.532
0.563
4.0
4.0
–
600(80)
1.34
1.31
0.75
0.74
in the KMF model is given by
2
2
0.7σE1 ŴE1
Eγ + 4π 2 T 2
1
fE1 (Eγ ) =
,
2 2
3π 2h̄2 c2
EE1 Eγ2 − EE1
(12)
where σE1 is the cross section, ŴE1 is the width, and EE1 is
the centroid of the GEDR determined from photoabsorption
experiments.
We adopt the KMF model with temperature T taken as a
constant to be consistent with our assumption that the RSF
is independent of excitation energy. The possible systematic
uncertainty caused by this assumption is estimated to have a
maximum effect of 20% on the RSF [24]. The values used for
T are the ones extracted from the constant-temperature model
in Eq. (5).
The GEDR is split into two parts for deformed nuclei. Data
of 51 V from photoabsorption experiments show that the GEDR
is best fitted with two Lorentzians, indicating a splitting of the
resonance and a non-zero ground-state deformation of this
nucleus. Indeed, B(E2) values [16] suggest a deformation
of β ∼ 0.1 for 50,51 V. Therefore, a sum of two modified
Lorentzians each described by Eq. (12) is used (see Table II).
For fM1 , which is supposed to be governed by the spinflip M1 resonance [6], the Lorentzian giant magnetic dipole
resonance (GMDR)
fM1 (Eγ ) =
2
σM1 Eγ ŴM1
1
2
2
2
3π 2h̄2 c2 Eγ2 − EM1
+ Eγ2 ŴM1
present in the whole excitation-energy region. In the case of
the 57 Fe RSF, the feature has been confirmed by an (n, 2γ )
experiment [25]. However, it has not appeared in the RSFs of
the rare-earth nuclei investigated earlier by the Oslo group.
The physical origin of the enhancement has not, at present,
any satisfying explanation, as none of the known theoretical
models can account for this behavior.
So far, we have not been able to detect any technical problems with the Oslo method. The unfolding procedure with the
NaI response functions gives reliable results, as demonstrated
in Fig. 1. Also, Fig. 2 indicates that the low-energy γ intensity
(13)
is adopted.
The GEDR and GMDR parameters are taken from the
systematics of Ref. [16] and are listed in Table II. Thus, we fit
the total RSF given by
f = κ(fE1,1 + fE1,2 + fM1 )
(14)
to the experimental data using the normalization constant κ as
a free parameter. The value of κ generally deviates from unity
because of theoretical uncertainties in the KMF model and
the evaluation of the absolute normalization in Eq. (10). The
resulting RSFs extracted from the two reactions are displayed
in Fig. 9, where the data have been normalized with parameters
from Tables I and II.
The γ -decay probability is governed by the number and
character of available final states and by the RSF. A rough
inspection of the experimental data of Fig. 9 indicates that the
RSFs are increasing functions of γ energy, generally following
the tails of the GEDR and GMDR resonances in this region.
At low γ energies (Eγ 3 MeV), an enhancement
of a factor of ∼5 over the KMF estimate of the strength
appears in the RSFs. This increase has also been seen in some
Fe [25] and Mo [24] isotopes, where it has been shown to be
FIG. 9. Normalized RSFs of 50,51 V. Dashed and dash-dotted lines
show the extrapolated tails of the giant electric and giant magnetic
dipole resonances, respectively. Solid line is the summed strength for
the giant dipole resonances.
004300-7
69
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May 29, 2006
22:8
A. C. LARSEN et al.
PHYSICAL REVIEW C 00, 004300 (2006)
is subtracted correctly; if not, one would find less intensity in
the higher-generation spectrum at these γ energies. Figure 3
shows the final test, where the result from the least-χ 2 fit nicely
reproduces the experimental data. In addition, investigations
in 27,28 Si [26] showed that our method produced γ -transition
coefficients in excellent agreement with average decay widths
of known, discrete transitions. Hence, we do not believe that
the enhancement is caused by some technical or methodical
problems. Still, independent confirmation of the increasing
RSF from, e.g., (n, 2γ ) experiments on the V and Mo isotopes,
is highly desirable.
to be ∼1.2 kB , which is less than the quasiparticle entropy of
∼1.7 kB found in rare-earth nuclei.
The experimental RSFs are generally increasing functions
of γ energy. The main contribution to the RSFs is the GEDR;
also the GMDR is present. At low γ energies, an increase
in the strength functions is apparent. A similar enhancement
has also been seen in iron and molybdenum isotopes. There is
still no explanation for the physics behind this very interesting
behavior.
ACKNOWLEDGMENTS
The Oslo method has been applied to extract level densities
and RSFs of the vanadium isotopes 50,51 V. From the measured
level densities, microcanonical entropies have been derived.
The entropy carried by the neutron hole in 50 V is estimated
A.V. acknowledges support from a NATO Science Fellowship under Project Number 150027/432 given by the
Norwegian-Research Council (NFR) and from the Stewardship Science Academic Alliances, Grant Number DE-FG0303-NA0074. Financial support from the NFR is gratefully
acknowledged.
[1] L. Henden, L. Bergholt, M. Guttormsen, J. Rekstad, and
T. S. Tveter, Nucl. Phys. A589, 249 (1995).
[2] A. Schiller, L. Bergholt, M. Guttormsen, E. Melby, J. Rekstad,
and S. Siem, Nucl. Instrum. Methods Phys. Res. A 447, 498
(2000).
[3] E. Melby, L. Bergholt, M. Guttormsen, M. Hjorth-Jensen,
F. Ingebretsen, S. Messelt, J. Rekstad, A. Schiller, S. Siem,
and S. W. Ødegård, Phys. Rev. Lett. 83, 3150 (1999).
[4] A. Schiller, A. Bjerve, M. Guttormsen, M. Hjorth-Jensen,
F. Ingebretsen, E. Melby, S. Messelt, J. Rekstad, S. Siem, and
S. W. Ødegård, Phys. Rev. C 63, 021306(R) (2001).
[5] M. Guttormsen, M. Hjorth-Jensen, E. Melby, J. Rekstad,
A. Schiller, and S. Siem, Phys. Rev. C 63, 044301 (2001).
[6] A. Voinov, M. Guttormsen, E. Melby, J. Rekstad, A. Schiller,
and S. Siem, Phys. Rev. C 63, 044313 (2001).
[7] S. Siem, M. Guttormsen, K. Ingeberg, E. Melby, J. Rekstad,
A. Schiller, and A. Voinov, Phys. Rev. C 65, 044318
(2002).
[8] M. Guttormsen, A. Atac, G. Løvhøiden, S. Messelt, T. Ramsøy,
J. Rekstad, T. F. Thorsteinsen, T. S. Tveter, and Z. Zelazny, Phys.
Script. T 32, 54 (1990).
[9] M. Guttormsen, T. S. Tveter, L. Bergholt, F. Ingebretsen,
and J. Rekstad, Nucl. Instrum. Methods Phys. Res. A 374, 371
(1996).
[10] M. Guttormsen, T. Ramsøy, and J. Rekstad, Nucl. Instrum.
Methods Phys. Res. A 255, 518 (1987).
[11] D. M. Brink, Ph.D. thesis, Oxford University, 1955.
[12] P. Axel, Phys. Rev. 126, 671 (1962).
[13] S. G. Kadmenski{\u{\i}}, V. P. Markushev, and V. I. Furman,
Yad. Fiz. 37, 277 (1983) [Sov. J. Nucl. Phys. 37, 165 (1983)].
[14] G. Gervais, M. Thoennessen, and W. E. Ormand, Phys. Rev. C
58, R1377 (1998).
[15] Data extracted using the NNDC On-Line Data Service from the
ENSDF database, www.nndc.bnl.gov/ensdf.
[16] Data extracted using the Reference Input Parameter Library,
http://www-nds.iaea.org/RIPL-2/
[17] A. Gilbert and A. G. W. Cameron, Can. J. Phys. 43, 1446 (1965).
[18] T. von Egidy, H. H. Schmidt, and A. N. Behkami, Nucl. Phys.
A481, 189 (1988).
[19] G. Audi and A. H. Wapstra, Nucl. Phys. A595, 409 (1995).
[20] A. Bohr and B. Mottelson, Nuclear Structure (Benjamin,
New York, 1969), Vol. I, p. 169.
[21] R. Firestone and V. S. Shirley, Table of Isotopes, 8th ed. (Wiley,
New York, 1996), Vol. II.
[22] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108,
1175 (1957).
[23] J. Kopecky and M. Uhl, Phys. Rev. C 41, 1941 (1990).
[24] M. Guttormsen, R. Chankova, U. Agvaanluvsan, E. Algin,
L. A. Bernstein, F. Ingebretsen, T. Lönnroth, S. Messelt,
G. E. Mitchell, J. Rekstad, A. Schiller, S. Siem, A. C. Sunde,
A. Voinov, and S. Ødegård, Phys. Rev. C 71, 044307 (2005).
[25] A. Voinov, E. Algin, U. Agvaanluvsan, T. Belgya, R. Chankova,
M. Guttormsen, G. E. Mitchell, J. Rekstad, A. Schiller, and
S. Siem, Phys. Rev. Lett. 93, 142504 (2004).
[26] M. Guttormsen, E. Melby, J. Rekstad, A. Schiller, S. Siem,
T. Lönnroth, and A. Voinov, J. Phys. G 29, 263 (2003).
V. SUMMARY AND CONCLUSIONS
004300-8
70
3
Summary and future perspectives
3.1
Fine structures in the level density and phase transitions
The interpolations of the level density between ∼ 2 and ∼ 8 MeV without
assuming any functional form enables us to observe fine structure in the
level density. The difference in entropy ∆S(E) between the odd-mass and
even-even nuclei is assumed to be a measure of the single particle entropy.
In Paper II, the entropy of the 161 Dy nucleus displays an almost constant
entropy excess compared to 160 Dy. This difference is nearly independent
of excitation energy, thus showing an entropy of ∆S ∼ 2 assigned to each
quasi particle. The probability that a system at fixed temperature T has an
excitation energy E has been evaluated. The most interesting temperature
region has been found around T = 0.5 − 0.6 MeV, where the Cooper pair
breaking process is strongest. At this point, the even-even and odd-even
nuclei behave differently; 160 Dy shows a broader distribution than 161 Dy.
This is due to the explosive behavior of ρ for E > Epair = 1.5 − 2 MeV in
even-even nuclei. Roughly, the number of levels for the breaking of neutron
or proton pairs increases by a factor of exp(∆S) ∼ 50 giving totally ∼ 100
times more levels.
In Paper V, for the almost spherical 93,94 Mo, we find ∆S & 0 for E >
2.5 MeV. The deformed case 97,98 Mo exhibits an entropy difference of ∆S &
1. These observations can be explained qualitatively by the fact that the
single particle entropy depends on the number of single particle orbitals that
are available for excitations at a certain temperature. For 93,94 Mo at low
energies, the single neutron outside the closed shell can only occupy the two
d5/2 and g7/2 orbitals giving an entropy of ln 2 ∼ 0.7. For the deformed
nucleus 97,98 Mo, symmetry breaking results in a splitting of these two single
particle orbitals into seven Nilsson orbitals, giving a total entropy of ln 7 ∼
1.9, i.e., about one unit more than for the 93,94 Mo case.
In conclusion, the entropy in atomic nuclei at low energies does not simply
scale with the total number of nucleons. This is a direct consequence of the
strong pairing interaction between the nucleons. In the presence of pairing
correlations, i.e., away from closed shells, the entropy scales instead with the
number of unpaired nucleons at a certain excitation energy. When pairing
correlations cannot form due to the large single particle level spacings around
closed shells, an unpaired nucleon will behave almost as a passive spectator
without contributing significantly to the entropy of the system.
In Paper V, a model based on the canonical ensemble theory [46, 47, 48],
which allows the investigation and classification of the pairing phase transition, has been utilized. The total partition function is written as a product
of proton (Zπ ), neutron (Zν ), rotation (Zrot ), and vibration (Zvib ) partition
72
functions. Thermodynamic quantities of interest such as entropy, average
excitation energy, and heat capacity can be deduced from the Helmholtz free
energy defined as
F (T ) = −T ln (Zπ Zν Zrot Zvib ) .
(23)
The free energy F and the average excitation energy hEi are rather structureless as functions of temperature. The entropy S and heat capacity CV
are more sensitive to thermal changes. The extracted canonical heat-capacity
curves as a function of temperature show local enhancements both theoretically and experimentally. This is interpreted as a fingerprint of a phase
transition from a phase with strong pairing correlations to a phase where the
pairing correlations are quenched [20].
The qualitative agreement between the model and the experiments shown
in Paper II, Fig. 7 indicates that the model describes the essential thermodynamic properties of the heated systems. The heat capacity curves show
clearly a local increase in the T = 0.5 − 0.6 MeV region, hinting at the
collective and massive breaking of nucleon Cooper pairs.
This feature is discussed in Paper I, where experimental level densities
for 171,172 Yb, 166,167 Er, 161,162 Dy, and 148,149 Sm are analyzed within the microcanonical ensemble. The two different critical temperatures have been
discovered using the method of Lee and Kosterlitz [38, 39]: (i) The lowest
critical temperature is due to the zero to two quasi-particle transition, and
(ii) the second transition is due to the continuous melting of Cooper pairs at
higher excitation energies. The first contribution is strongest for the eveneven system (160 Dy), since the first broken pair represents a large and abrupt
step in level density and thus a large contribution to the heat capacity. In
161
Dy, the extra valence neutron washes out this step. The second contribution to CV is present in both nuclei signaling the continuous melting of
nucleon pairs at higher excitation energies. This second critical temperature
appears at a ∼ 0.1 MeV higher value.
In Paper V, the behavior of the pairing correlations when approaching a
major shell gap has been investigated. It was found that approaching the
N = 50 closed shell, the local enhancements become less pronounced. The
general trend is that at shell closure there are almost no pairing correlations
and, as particles are added, the pairing correlations increase. Therefore the
signature of a transition from a ’paired phase’ to an ’unpaired phase’ when
approaching a major shell gap becomes less pronounced.
Shell-model Monte-Carlo calculations [12] have shown that the pairingphase transition is strongly correlated with the suppression of neutron pairs
with increasing temperature. It has also been observed that the reduction of
the neutron-pair content of the wavefunction is much stronger in the even73
even than in the odd-mass isotopes, giving rise to the more pronounced S
shape in the canonical heat-capacity curves in the even-even nuclei. The
same odd-even difference in the heat capacity has also been observed experimentally between 161 Dy and 162 Dy, and 171 Yb and 172 Yb [20].
74
Figure 5: Experimental radiative strength function for some rare-earth nuclei
compared to a model description. Left: RSFs for 160,162 Dy (data points) using
a temperature dependent GEDR (solid line). Right: A pygmy resonance in
171
Yb observed in (3 He,3 He′ ) and (3 He,α) reactions. The solid line in the
upper graphs is a fit to the data including all contributions, the dashed lines
are with the contribution from the pygmy resonance removed.
3.2
Radiative strength function and resonance structures
The radiative strength functions in all nuclei studied show a characteristic
increase with increasing γ-ray energy, generally following the tails of the
giant electric (GEDR) and magnetic (GMDR) resonances. However, the
detailed structures in the radiative strength function show different behavior
in various mass regions.
3.2.1
Local enhancement of the RSF at low γ-ray energies
For nuclei in the rare-earth region, an anomalous resonance structure is observed in the radiative strength function, the so-called pygmy resonance.
These observations have been previously verified for several well-deformed
rare-earth nuclei [35, 37].
Figure 5 (left two panels) shows fits to the experimental RSFs obtained
from the (3 He,α)160 Dy and (3 He,3 He′ )162 Dy reactions. The approaches using
75
a varying temperature, hfE1 i, and a fixed temperature, fE1 (T = 0.3 MeV),
are displayed as solid and dash-dotted lines, respectively. Since these nuclei have axially deformed shapes, the GEDR is split into two components:
GEDR1 and GEDR2. Thus, the two RSFs with different resonance parameters, taken from the systematics of Ref. [18] are added. The pygmy resonance
is described with a Lorentzian function fpy as described in Eq. (21). The total
RSF given by
f = κ(fE1 + fM1 ) + fpy ,
(24)
is fitted to the experimental data using the pygmy-resonance parameters σpy ,
Γpy and Epy and the normalization constant κ as free parameters.
The situation is similar for the 171,172 Yb nuclei, as seen from Figure 5,
(right four panels). The RSF is composed of five parts
I,II
f (Eγ ) = κ(fE1
+ fM 1 ) + Eγ2 fE2 + fpy ,
(25)
I,II
where fE1
is the sum of the two components of the GEDR given by the
KMF model Eq. (19), fM 1 and fE2 are the giant magnetic dipole and electric
quadrupole resonances given by Eqs. (21) and (22), respectively. The upper
graphs contain the total radiative strength function (RSF) and the lower
graphs show the contribution from the pygmy resonance. The solid lines in
the upper graphs represent a fit to the data using Eq. (25). The dashed
lines are fit functions when the contribution from the pygmy resonances is
excluded. After subtracting the fit function without the pygmy resonances
(dashed lines) from the data points of the upper graph, the pygmy resonance
is clearly identified. The fit using only the pygmy resonances is shown in
solid lines in the lower graphs.
76
Strength function, arb.units
-4
10
-5
10
-6
10
0
2
4
6
8
10
energy, MeV
Figure 6: Left panel: total RSF of 57,56 Fe (filled and open circles, respectively), Lorentzian (dashed line) and KMF model (dash-dotted line) descriptions of the GEDR. Center panel: fit (solid line) to 57 Fe data and decomposition into the renormalized E1 KMF model, Lorentzian M 1 and E2 models
(all dashed lines), and a power law to model the large enhancement for low
energies (dash-dotted line). Right panel: The RSF obtained from Oslo firstgeneration matrix P (Ex , Eγ ) with level density from (d, n) reaction (filled
circles). The RSF obtained solely from P (Ex , Eγ ) (open circles).
3.2.2
Large enhancement of the RSF at low γ-ray energies
Recently, more than a factor of ten enhancement of soft transition strengths
(a soft pole) in the total RSF has been observed using the
57
Fe(3 He,αγ)56 Fe and 57 Fe(3 He,3 He′ γ)57 Fe reactions [22].
The normalized RSFs in 56,57 Fe are displayed in Fig. 6, left panel. The total RSF has been decomposed into a KMF model for E1 radiation, Lorentzian
models for M 1 and E2 radiation, and a power law to model the soft pole,
center panel of the same figure.
To ensure that the observed enhancement is not connected to peculiarities
of the nuclear reaction or analysis method, a TSC measurements at the dualuse cold-neutron beam facility of the Budapest Research Reactor has been
performed. The TSC technique for thermal neutron capture is described
in Ref. [49]. The TSC intensities from the 56 Fe(n, 2γ)57 Fe reaction have
been measured. Details are given in Paper III. Model calculations based on
separated RSFs from the decomposition of the experimental total RSF and
on experimental level densities from the Oslo experiment can reproduce the
experimental TSC intensities with soft primary γ rays only in the presence
of the soft pole in the total RSF.
The unusual low-energy enhancement has been confirmed very recently
by an experiment performed with a 7 MeV deuteron beam from the John
77
Figure 7: Experimental radiative strength function of lighter nuclei compared
to a model description. Left panel: For 98 Mo including GEDR, GMDR
and the isoscalar E2 resonance. The empirical soft-pole component is used
to describe the low energy part of the RSF. Right panel: RSFs of 50,51 V.
The dashed and dash-dotted line show the extrapolated tails of GEDR and
GMDR, respectively. The solid line is the summed strength for the giant
dipole resonances.
Edwards Accelerator Laboratory tandem at Ohio University. Details are
given in Ref. [36]. The RSF for the 56 Fe isotope obtained in Ref. [22] has
been extracted by using the level density function from the neutron evaporation spectrum. The new RSF agrees well with the previous one within
experimental errors, as shown in Fig. 6, right panel.
Subsequently, the enhancement has been observed for other lighter nuclei
(A < 100). The structure is called a soft pole in the RSF and parametrization
is given by
fsoftpole =
1
3π 2 ~2 c2
AEγ−b ,
(26)
where A and b are fit parameters, and Eγ is given in MeV.
In Fig. 7, left panel, the various contributions to the total RSF of 98 Mo
are shown. The main components are the GEDR resonance and the unknown
low-energy structure. The composition of the total RSF is
f = κ(fE1 + fM1 + fsoftpole ) + Eγ2 fE2 ,
78
(27)
where κ is a normalization constant.
At low γ energies (Eγ . 3 MeV), a similar enhancement of a factor of ∼
5 over the KMF estimate appears in the RSFs for the case of 51 V (right panel
of Fig. 7). In the 50,51 V, the GEDR is fitted with two Lorentzians, indicating
a splitting of the resonance and a non-zero ground-state deformation of this
nucleus.
79
3.3
Future plans
The Oslo group has obtained funding for building a new particle telescope
system, called SIRI. The array of detectors will be placed 5 cm from the target
and in a forward angle of θ = 45◦ . Totally, 8 × 8 silicon telescopes will give
typically five times more efficiency and two times better energy resolution
than the previous system. With this set-up there is a strong hope that
several questions concerning the physics of warm nuclei can be answered.
More fine structures in the level density may be seen, and there could be
sufficient events to perform p − γ − γ coincidence measurements. Also the
angular distributions as function of the forward angle θ can provide more
information about reaction spin transfer.
Presently, in the process of investigation are the pairing properties in a
series of odd- and even-nuclei 93−98 Mo using the SPA+RPA in the monopole
pairing model. The calculations reproduce very well the experimental level
densities, and explain the unusual feature found for 93−98 Mo (Paper V), that
the heat capacities do not show clear odd-even staggering. Further studies
in this direction are in progress.
An important topic for further theoretical investigations is the physical
origin of the low-energy enhancement (the so called ’soft pole’), which does
not as yet have any satisfactory explanation. The soft pole is a general phenomenon across large areas of the nuclear chart, and its impact on nuclear
reaction network calculations is still unknown, since none of the known theoretical models can account for this behavior. More experimental data on
Mo and V isotopes from (n,2γ) reaction can give independent confirmation
of the low-energy increase of the RSF.
Finally, physics with radioactive ion beams is an area of active research
at present, since it offers unique opportunities to explore the rich nuclear
"landscape". The astrophysical implications of such experiments on the r-, sor possibly even p-process, and the possible implications to applied physics,
such as transmutation of radioactive waste and the possible resurgence of
nuclear power if the waste problem can be managed, is of great interest. To
transfer the Oslo method to radioactive beams, in order to obtain information
on level density and RSF for unstable beams, will be very challenging.
80
3.4
Conclusion
Levels for different nuclei in the excitation-energy region up to the neutronseparation energy have been populated using the (3 He,αγ) and (3 He,3 He’γ)
reactions. The level densities have been determined from their corresponding
primary γ-ray spectra. Thermodynamical studies have been explored within
both microcanonical and canonical statistical ensembles, and thermodynamic
observables were deduced from the level density. Essential information such
as pair correlations and phase transitions was revealed. The fine structures
observed in the level density have been used to obtain experimental values for
the critical temperature of the pair-breaking process. Different mechanisms
governing the thermodynamic properties of odd and even systems have been
studied.
Fine structures in the RSF have been investigated. In the rare-earth
region a local enhancement of the RSF at low γ-ray energies, denoted as
pygmy resonance has been found. For the lighter nuclei an unexpected large
enhancement in the RSF has been observed at low γ-ray energies, denoted
as a soft-pole, whose physical origin does not have any satisfactory explanation at present. More experimental and theoretical studies are necessary to
explain this unusual structures.
81
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