Test of nuclear level density inputs for Hauser-Feshbach model
  calculations
Test of nuclear level density inputs for Hauser-Feshbach model calculations
1A.V. Voinov∗, 1S.M. Grimes, 1C.R. Brune, 1M.J. Hornish, 1T.N. Massey, 1,2A. Salas
Department of Physics and Astronomy, Ohio University, Athens, OH 45701, USA and
Los-Alamos National Laboratory, P-25 MS H846, Los Alamos, New Mexico 87545, USA
The energy spectra of neutrons, protons, and α-particles have been measured from the d+59Co
and 3He+58Fe reactions leading to the same compound nucleus, 61Ni. The experimental cross
sections have been compared to Hauser-Feshbach model calculations using different input level
density models. None of them have been found to agree with experiment. It manifests the serious
problem with available level density parameterizations especially those based on neutron resonance
spacings and density of discrete levels. New level densities and corresponding Fermi-gas parameters
have been obtained for reaction product nuclei such as 60Ni,60Co, and 57Fe.
I. INTRODUCTION
The nuclear level density (NLD) is an important in-
put for the calculation of reaction cross sections in the
framework of Hauser-Feshbach (HF) theory of compound
nuclear reactions. Compound reaction cross sections are
needed in many applications including astrophysics and
nuclear data for science and technology. In astrophysics
the knowledge of reaction rates is crucial for understand-
ing nucleosynthesis and energy generation in stars and
stellar explosions. In many astrophysical scenarios, e.g.
the r-process, the cross sections required to compute the
reaction rates are in the regime where the statistical ap-
proach is appropriate [1]. In these cases HF calculations
are an essential tool for determining reaction rates, par-
ticularly for reactions involving radioactive nuclei which
are presently inaccessible to experiment. HF calculations
are likewise very important for other applications, e.g.,
the advanced reactor fuel cycle program [2].
The statistical approach utilized in HF theory [3] re-
quires knowledge of the two quantities for participating
species (see details below). These are the transmission
coefficients of incoming and outgoing particles and level
densities of residual nuclei. Transmission coefficients can
be obtained from optical model potentials established on
the basis of experimental data of elastic and total cross
sections. Because of experimental constraints, the dif-
ference between various sources of transmission coeffi-
cients usually does not exceed 10 − 15%. Level densi-
ties are more uncertain. The reason is that it is diffi-
cult to obtain them experimentally above the region of
well-resolved discrete low-lying levels known from nuclear
spectroscopy. At present, the level density for practi-
cal applications is calculated mainly on the basis of the
Fermi-gas [4] and Gilbert-Cameron [5] formulas with ad-
justable parameters which are found from experimental
data on neutron resonance spacing and the density of low-
lying discrete levels. Parameters recommended for use in
HF calculations are tabulated in Ref. [6]. The global pa-
rameter systematics for both the Fermi-gas and Gilbert-
Electronic address: voinov@ohio.edu
Cameron formulas have been developed in Ref. [7]. How-
ever, it is still unclear how well these parameters repro-
duce compound reaction cross sections. No systematic
investigations have been performed yet. Experimental
data on level density above discrete levels are scarce.
Some information is available from particle evaporation
spectra (i.e. from compound nuclear reactions). The lat-
est data obtained from (p,n) reaction on Sn isotopes [8]
claims that the available level density parameters do not
reproduce neutron cross sections thereby indicating the
problem with level density parameterizations. It becomes
obvious that more experimental data on level density are
needed in the energy region above discrete levels.
In this work, we study compound nuclear reactions
to obtain information about level densities of the resid-
ual nuclei from particle evaporation spectra. Two dif-
ferent reactions, d+59Co and 3He+58Fe, which produce
the same 61Ni compound nucleus, have been investigated.
This approach helps to eliminate uncertainties connected
to a specific reaction mechanism. As opposed to most of
the similar experiments where only one type of outgo-
ing particles has been measured, we have measured cross
sections of all main outgoing particles including neu-
trons, protons, and α-particles, populating 60Ni, 60Co,
and 57Fe, respectively.
We will begin with a discussion of the present status of
level density estimates used as inputs for HF calculations.
II. METHODS OF LEVEL DENSITY
ESTIMATES FOR HF CODES
The simple level counting method to determine the
level density of a nucleus works only up to a certain exci-
tation energy below which levels are well separated and
can be determined from nuclear spectroscopy. This re-
gion is typically up to 2 MeV for heavy nuclei and up
to 6-9 MeV for light ones. Above these energies, more
sophisticated methods have to be applied.
http://arxiv.org/abs/0704.0916v2
A. Level density based on neutron resonance
spacings
In the region of neutron resonances which are located
just above the neutron binding energy (Bn), the level
density can again be determined by counting. In this case
neutron resonances are counted; one must also take into
account the assumed spin cut-off factor σ. Traditionally,
because of the the absence of reliable data below Bn, the
level density is determined by an interpolation procedure
between densities of low-lying discrete levels and den-
sity obtained from neutron resonance spacing.The Bethe
Fermi-gas model [4] with adjustable parameters a and δ
is often used as an interpolation formula:
ρ(E) =
exp[2
a(E − δ)]
2σa1/4(E − δ)5/4
, (1)
where the σ is the spin cut-off factor determining the level
spin distribution. There are a few drawbacks to this ap-
proach. One shortcoming is that it uses an assumption
that the selected model is valid in the entire excitation
energy region including low-lying discrete states and neu-
tron resonances. Undoubtedly this is correct for some of
the nuclei. A nice example is the level density of 26Al
which exhibits Fermi-gas behavior up to 8 MeV of exci-
tation energy [7]. On the other hand the level densities
of 56,57Fe measured with Oslo method [9], for example,
show complicated behavior which cannot be described
by simple Fermi-gas formula. The reason for this might
be the influence of pairing correlations leading to step
structures in vicinity of proton and neutron paring en-
ergies and above. In such cases the model function fit
to discrete levels may undergo considerable deviations in
the higher excitation energy region leading to incorrect
determination of level density parameters.
Another consideration is associated with the spin cut-
off parameter which is important in determination of the
total level density from density of neutron resonances at
Bn. In Fermi-gas model the spin cut-off parameter is
determined according to:
σ2 = m2gt =
t, (2)
wherem2 is the average of the square of the single particle
spin projections, t =
(E − δ)/a is the temperature,
g = 6a/π2 is the single particle level density, I is the rigid
body moment of inertia expressed as I = (2/5)µAR2,
where µ is the nucleon mass, A is the mass number and
R = 1.25A1/3 is the nuclear radius. The spin cut off
parameter in rigid body model is :
σ21 = 0.0146A
5/3t = 0.0146A5/3
((E − δ)/a)). (3)
On the other hand the Gilbert and Cameron [5] used
m2 = 0.146A2/3. The corresponding formula for σ is:
σ22 = 0.089A
((E − δ)/a). (4)
Eqs. (3) and (4) have the same energy and A dependence
(σ2 ∼ A7/6(E − δ)1/2) but differ by a factor of ≈ 2. It
should be mentioned also that the recent model calcula-
tions [10] show the suppression of the moment of inertia
at low temperatures compared to its rigid body value.
Thus uncertainties in spin cut off parameter transform
to corresponding uncertainties of total level densities de-
rived from neutron resonance spacings.
Experimentally, the spin cutoff parameter can be ob-
tained only from spin distribution of low-lying discrete
levels. However, because of the small number of known
spins, the uncertainty of such procedure is large. It turns
out that reported systematics based on such investiga-
tion σ = (0.98 ± 0.23)A(0.29±0.06) [7] is different from
above expressions for which σ ∼
A7/6 = A0.58. At
higher excitation energies determining the cutoff param-
eter becomes problematic due to the high level density
and the absence of the reliable observables sensitive to
this parameter. One can mention Ref. [11] where the
spin cutoff parameter has been determined from the an-
gular distribution of evaporation neutrons with α and
proton projectiles. The deviation from the expected A
dependence has also been reported. The absolute values
of the parameter agree with Eq. (3).
The parity dependence of level densities is also not
established experimentally beyond the discrete level re-
gion. At the neutron binding energy the assumption is
usually made about the equality of negative and positive
parity states. This is supported by some experimental
results [12]. However, recent calculations, performed for
Fe, Ni, and Zn isotopes, show that for some of them the
assumption of equally distributed states is not fulfilled
even far beyond the neutron binding energy, up to exci-
tation energies 15-20 MeV [13].
As is seen from the above considerations, the cal-
culation of the total level density from neutron res-
onance spacing might contain uncertainties associated
with many factors such as the possible deviation from
Fermi-gas dependence in interpolation region, uncertain-
ties in spin cutoff parameter and inequality of states with
different parity. Thus the question of how large these
uncertainties are or to what extent the level density ex-
tracted in such a way can be applicable to calculations
of reaction cross sections still remains important and not
completely resolved.
B. Level density from evaporation particles
The cross section of evaporated particles from the first
stage of a compound-nuclear reaction (i.e. when the out-
going particle is the first particle resulting from com-
pound nucleus decay ) can be calculated in the framework
of the Hauser-Feshbach theory:
(εa, εb) = (5)
σCN(εa)
Iπ Γb(U, J, π, E, I, π)ρb(E, I, π)
Γ(U, J, π)
Γ(U, J, π) =
Γb′(U, J, π, Ek, Ik, πk)+ (6)
∫ U−B
dE′ Γb′(U, J, π, E
′, I ′, π′) ρb′(E
′, I ′, π′)
Here σCN (εa) is the fusion cross section, εa and εb
are energies of relative motion for incoming and outgo-
ing channels (εb = U − Ek − Bb, where Bb is the sepa-
ration energy of particle b from the compound nucleus),
the Γb are the transmission coefficients of the outgoing
particle, and the quantities (U, J, π) and (E, I, π) are the
energy, angular momentum, and parity of the compound
and residual nuclei, respectively. The energy Ec is the
continuum edge, above which levels are modeled using a
level density parameterization. For energies below Ec the
known excitation energies, spins, and parities of discrete
levels are used. In practice Ec is determined by the avail-
able spectroscopic data in the literature. It follows from
Eq. (6) that the cross section is determined by both trans-
mission coefficients of outgoing particles and the NLD of
the residual nucleus ρb(E, I, π). It is believed that trans-
mission coefficients are known with sufficient accuracy
near the line of stability because they can be obtained
from optical model potentials usually based on experi-
mental data for elastic scattering and total cross sections
in the corresponding outgoing channel. Transmission co-
efficients obtained from different systematics of optical
model parameters do not differ by more that 15-20 %
from each other in our region of interest (1− 15 MeV of
outgoing particles). The uncertainties in level densities
are much larger. Therefore the Hauser Feshbach model
can be used to improve level densities by comparing ex-
perimental and calculated particle evaporation spectra.
Details and assumptions of this procedure are described
in Refs [14, 15].
The advantage of this method is that because of the
wide range of spin population in both the compound and
final nuclei, evaporation spectra are determined by the
total level density (integrated over all level spins) as op-
posed to the neutron resonance technique where reso-
nances are known for one or two spins and one parity.
The drawback stems from possible direct or multistep
compound reaction contributions distorting the evapora-
tion spectra, especially in the region of low-lying discrete
levels needed for the absolute normalization of obtained
level densities.
According to Hodgson [16], the interaction process can
usefully be considered to take place in a series of stages
corresponding to the successive nucleon-nucleon interac-
tion until complete equilibrium is reached. At each stage
it is possible for particles to be emitted from the nucleus.
The direct reactions refer to the fast, first stages of this
process giving forward peaked angular distribution. The
term multistep direct reaction implies that that such pro-
cess may take place in a number of states. Compound
nuclear reactions refers to all processes giving angular
distributions symmetric about 900; they are subdivided
into multistep compound reactions that take place before
the compound system has attained final statistical equi-
librium and statistical compound reactions that corre-
spond to the evaporation of particle from an equilibrium
system.
The use of evaporation spectra to infer level densities
requires that the reaction goes through to complete equi-
librium. Significant contributions from either multistep
direct or multistep compound reactions could cause in-
correct level density parameters to be deduced. Multi-
step direct reactions would usually be forward peaked
and also concentrated in peaks. If the reaction has a lim-
ited number of stages, the two-body force cannot cause
transitions to states which involve a large number of rear-
rangements from the original state. Multistep compound
reactions would be expected to lead to angular distribu-
tions which are symmetric about 900. They would, if
complete equilibration has not occurred, also preferen-
tially reach states which are similar to the target plus pro-
jectile. The shape of spectra from a multistep compound
reaction would be different for a deuteron-induced as op-
posed to a 3He-induced reaction. Hodgson has reviewed
[16] the evidence for multistep compound reactions. He
finds the most convincing evidence for such contributions
comes from fluctuation measurements for the 27Al(3He,p)
reaction. In this case, certain low-lying states show level
widths in the compound system which are larger than
expected. These states are low-lying and are the ones
which would be most likely to show such effects. It ap-
pears that measurements of continuum spectra do not
show evidence of such contributions. The uncertainties
connected to contributions of pre-equilibrium reactions
are generally difficult to estimate experimentally. The
measurement of angular distribution does not solve the
problem in the case of multistep compound mechanism.
We believe that the use of different reactions to form the
same compound nucleus is the most reliable way to esti-
mate and eliminate such contributions.
In this work we investigate reactions with deuteron and
3He projectiles on 59Co and 58Fe, respectively. These two
reactions form the same compound nucleus, 61Ni. The
purpose was to investigate if the cross section of outgo-
ing particles from both reactions can be described in the
framework of Hauser-Feshbach model with same set of
level density parameters. This is possible only when pro-
duction cross section is due to compound reaction mecha-
nism in both reactions. Neutron, protons, and α-particles
have been measured. These outgoing particles exhaust
the majority of the fusion cross section. The ratio be-
tween cross sections of different particles is determined
by the ratio of level densities of corresponding residual
nuclei. It puts constraints on relative level density val-
ues obtained from an experiment. In our experiment, the
level densities of 60Ni, 60Co, and 57Fe residual nuclei have
been determined from the region of the energy spectra of
neutron, proton, and α-particles where only first state
emission is possible.
III. EXPERIMENT AND METHOD
The tandem accelerator at Ohio University’s Edwards
Accelerator Laboratory provided 3He and deuteron
beams with energies of 10 and 7.5 MeV, respectively.
Self-supporting foils of 0.625-mg/cm2 58Fe (82% en-
riched) and 0.89-mg/cm2 59Co (100% natural abun-
dance) have been used as targets. The outgoing charged
particles were registered by charged-particle spectrome-
ters as shown in Fig. 2. The setup has ten 2-m time-
of-flight legs ending with Si detectors (see Fig. 1). Legs
are set up at different angles ranging from 22.5◦ up to
157.5◦. The mass of the charged particles is determined
by measuring both the energy deposited in Si detectors
and the time of flight. Additionally, a neutron detector
was placed at the distance of 140 cm from the target to
measure the neutron energy spectrum. The mass resolu-
tion was sufficient to resolve protons, deuterons, 3H/3He,
and α-particles.
He-3 beamTarget
2m flight path
FIG. 1: Charge particle spectrometer utilized for the mea-
surements.
Additionally, the neutron spectra from both the
58Fe(3He, Xn) and the 59Co(d,Xn) reactions have been
measured by the time-of-flight method with the Swinger
facility of Edwards Laboratory [17]. Here a flight path
of 7 m has been used to obtain better energy resolution
for outgoing neutrons, allowing us to measure the shape
of neutron evaporation spectrum more accurately. The
energy of the outgoing neutrons is determined by time-of-
flight method. The 3-ns pulse width provided an energy
resolution of about 100 keV and 800 keV at 1 and 14 MeV
of neutrons, respectively. The neutron detector efficiency
was measured with neutrons from the 27Al(d, n) reaction
on a stopping Al target at Ed = 7.44 MeV [18]. This
measurement allowed us to determine the detector ef-
ficiency from 0.2 to 14.5 MeV neutron energy with an
accuracy of ∼ 6%. The neutron spectra have been mea-
sured at backward angles from 110◦ to 150◦. Additional
measurements with a blank target have been performed
at each angle to determine background contribution. The
absolute cross section has been calculated by taking into
account the target thickness, the accumulated charge of
incoming deuteron or 3He beam, and the neutron detec-
tor efficiency. The overall systematic error for the abso-
lute cross sections is estimated to be 15%. The errors in
ratios of proton and α cross sections are only a few per-
cents because they are determined by counting statistics
alone.
IV. EXPERIMENTAL PARTICLE SPECTRA
AND LEVEL DENSITY OF PRODUCT NUCLEI
Energy spectra of neutron, protons, and α-particles
have been measured at backward angles (from 112◦ to
157◦) to eliminate contributions from direct reaction
mechanisms. Fig. 2 show energy spectra of outgoing
particles for both the 3He+58Fe and d+59Co reactions.
The calculations of particle energy spectra have been
performed with Hauser-Feshbach (HF) program devel-
oped at Edwards Accelerator Lab of Ohio University
[19]. Particle transmission coefficients have been calcu-
lated with optical model potentials taken from the RIPL
data base [6]. Different potentials have been tested and
found to be the same within 15%. Alpha-particle po-
tentials are more uncertain. Differences between corre-
sponding α-transmission coefficients depends on the α-
energy and varies from ∼ 40% for lower α-energies to
< 1% for higher α-energies in our region of interest (8-
18 MeV). In order to reduce these uncertainties the RIPL
α-potentials have been tested against the experimental
data on low energy α−elastic scattering on 58Ni [20]. The
data have been reproduced best by the potential from
Ref.[21] which has been adopted for our HF calculations.
Four level density models have been chosen for testing:
• The M1 model uses the Bethe formula (1) with pa-
rameters adjusted to fit both discrete level density
and neutrons s-wave resonance spacings.
• The M2 model uses the Gilbert-Cameron [5] for-
mula with parameters adjusted to fit both discrete
level density and neutrons s-wave resonance spac-
ings.
• The M3 model uses Bethe formula but δ parame-
ters are obtained from pairing energies according
to Ref. [1]. The a parameter has been adjusted
to match s-wave neutron resonance spacing. This
model does not fit discrete levels.
• The M4 model is based on microscopic Hartree-
Fock-BCS calculations [22] which are available from
RIPL data base [6]. According to Ref. [22], this
model has also been renormalized to fit discrete lev-
els and neutron resonance spacings.
0 2 4 6 8 10 12 14
4 6 8 10 12 14 16 18 6 8 10 12 14 16 18 20
0 2 4 6 8 10 12 14
 Neutron energy  (MeV)
2 4 6 8 10 12
 Proton energy (MeV)
6 8 10 12 14 16
 Alpha energy (MeV)
FIG. 2: Particle energy spectra for the 3He+58Fe (upper panel)and d+59Co (lower panel) reactions. The experimental data
are shown by points. Solid lines are HF calculations with level density parameters extracted from the experiment. Calculations
have been multiplied by reduction factor K = 0.52 due to direct reaction contributions. Arrows show energies above which
spectra contain only contributions from the first stage of the reaction.
The value of total level density derived from neutron res-
onance spacings depends on spin cutoff parameter used.
Therefore two prescriptions (3) and (4) for this parame-
ter have been tested for M1-M3 models. The M4 model
uses its one spin distribution which it is close to the pre-
scription (3).
The measured particle energy spectra include particles
from all possible stages of the reaction. However, by lim-
iting our consideration to particles with energies above
a particular threshold, we can ensure that only particles
from the first stage of the reaction contribute. These
thresholds depend on the particular reaction and are in-
dicated by the arrows in Fig. 2. In this energy interval
cross sections are determined exclusively by the level den-
sity of those residual nuclei. Another aspect which should
be taken into consideration when comparing calculations
and experiment is the contribution of direct processes.
Direct processes take away the incoming flux resulting in
reduction of compound reaction contribution. Assuming
that the total reaction cross section (σR) can be decom-
posed into the sum of direct (σdr ) and compound reaction
mechanisms (σcR), we have σR = σ
R + σ
R. In this case,
the HF calculations should be multiplied by the constant
factorK = σ
c exp
R /σR to correct for the absorbed incident
flux which does not lead to compound nucleus formation.
In our experiment the K has been estimated from the ra-
tioKexp = (σexpn +σ
α )/(σ
α ) ≈ K
where the experimental cross sections have been mea-
sured at backward angles. If level densities used in calcu-
lations are correct, Kexp = K. However, the calculations
show that this parameter is not very sensitive to input
level densities and can be estimated with ∼20% accuracy
with any reasonable level density models.
Table I shows the ratio of theoretical and experimen-
tal cross sections for different level density models used
in calculations. Calculations have been multiplied by re-
duction factor K which for both reactions varied within
0.48-0.54 for different level density models. Results show
that all of the models reproduce neutron cross sections
within ∼20%. However, they overestimate α-particle
cross sections by ∼30% in average and underestimate
protons by 5-80%. None of the models reproduce the
ratio of p/α cross section; for example all models sys-
tematically overestimate this ratio by a factor of ∼2 for
the d+59Co reaction. Assuming that particle transmis-
sion coefficients are known with sufficient accuracy, we
conclude that the level density of residual nuclei is re-
sponsible for such disagreement. In particular, the level
density ratio ρ[57Fe]/ρ[60Co] is overestimated by model
calculations.
In order to obtain correct level densities, the following
procedure has been used as described in Ref. [23]. The
NLD model is chosen to calculate the differential cross
section of Eq. (6). The parameters of the model were
adjusted to reproduce the experimental spectra as closely
as possible. The input NLD was improved by binwise
renormalization according to the expression:
ρb(E, I, π) = ρb(E, I, π)input
(dσ/dεb)meas
(dσ/dεb)calc
. (7)
The absolute normalization of the improved level den-
sities (later referred to as experimental level densities)
has been obtained by using discrete level densities of
60Ni populated by neutrons from the 59Co(d,n) reac-
tion. Protons and α-particles populating discrete lev-
els behave differently for different reactions. The Fig. 2
shows that the ratio between experiment and calculations
in discrete energy region is greater for 59Co(d,p) com-
pared to 58Fe(3He,p) and for 58Fe(3He,α) compared to
59Co(d,α). These enhancements are apparently reaction
specific and connected to contribution of direct or/and
multistep compound reaction mechanism. We are not
able to make the same comparison for neutron spectra
because the counting statistics in the region of discrete
levels for 58Fe(3He,n) reaction are rather poor. However,
our recent result from 55Mn(d,n) [23] indicates that the
neutron spectrum measured at backward angles is purely
evaporated even for high energy neutrons populating dis-
crete levels. Therefore we used the neutron spectrum
from the 59Co(d,n) reaction to determine the absolute
normalization of the level density for the residual nu-
cleus 60Ni. The absolute level densities of both 60Co and
57Fe nuclei have been adjusted in such a way as to repro-
duce ratios of both neutron/proton and neutron/alpha
cross sections. Uncertainties of obtained level densities
have been estimated to be about 20% which include un-
certainties of absolute cross section measurements and
uncertainties of particle transmission coefficients.
Both experimental and calculated level densities are
displayed in Fig. 3. The level density for 60Ni has been
extracted from (d,n) spectra because of better counting
statistics but (3He,p) and (3He,α) reactions have been
used to obtain level density for 60Co and 57Fe, respec-
tively, because of larger Q value. This approach allows
one to obtain level densities in a larger excitation energy
interval. Calculations have been performed with models
M1-M4 with spin cutoff parameters σ1 and σ2 for M1-M3
models. The M4 model uses its own spin distribution
which is close to σ1 for these nuclei. The χ
2 values for
calculated and experimental level densities are shown in
the Table III. Results show that the M1 model with σ1
gives worse agreement with experimental data. The use
of σ2 improve the agreement for all of the models. The
M2 and M3 models using σ2 give best agreement with
experiment on average, however level density for 60Co
agrees better when using σ1 and the best agreement is
reached with M4 model. It appears that the spin cut-
off parameter is very important when deriving the total
level density from neutron resonance spacings. However,
none of the models give a perfect description of the ex-
perimental data.
In order to improve level density parameters, the
experimental level densities have been fitted with the
Fermi-gas function (1) for two different spin cutoff fac-
tors σ1 and σ2. Best fit parameters are presented in the
Table II. They allow one to reproduce both shapes of
particle spectra (fig.2) as well as ratios of neutron, pro-
ton and α cross sections for both 3He+58Fe and d+59Co
reactions (Table I). Level density parameters have been
adjusted independently for both spin cutoff parameters
resulting in the approximately same final ratio of exper-
imental/calculated cross sections and χ2 values. There-
fore the only one entry M1exp is presented in tables. The
fact that a single set of level density parameters allows
one to reproduce all particle cross sections from both re-
actions supports our conclusion that the compound nu-
clear mechanism is dominant in these reactions. Finally
we note that the HF calculations do not perfectly repro-
duce the low-energy regions of the proton spectra where
the second stage of outgoing protons dominate. Here
the calculations also depends on additional level densi-
ties of corresponding residual nuclei as well as on the
γ-strength functions. We leave this problem for further
investigations.
The level density of 57Fe below the particle separation
threshold has also been obtained [9] by Oslo technique
using particle-γ coincidences from 57(3He,3He′)57Fe re-
action. We performed a similar comparison for the 56Fe
nucleus where we confirmed consistency of both the Oslo
technique and the technique based on particle evapo-
ration spectra. Figure 4 shows the comparison for the
57Fe nucleus. Here we also see good agreement between
level densities obtained from two different experiments.
It supports the obtained level densities. The Fermi-
gas parameters for 60Ni have been obtained in Ref. [24]
from 63Cu(p,α)60Ni reaction at Ep=12 MeV . The val-
ues a=6.4 and δ=1.3 are in a good agreement with our
parameters presented in the Table II.
V. SPIN CUTOFF PARAMETER
As it has been mentioned in the previous section the
spin cutoff parameter σ2 obtained according to Eq. (4)
gives slightly better agreement with the experiment com-
pared to σ1 obtained from Eq. (3). On the other hand,
the spin cut off parameters at the neutron binding energy
can be directly obtained from the experimental total level
density and the density of levels for one or several spin
states which are known from the analysis of neutron res-
TABLE I: Ratio of experimental and calculated cross sections obtained with four prior level density models M1-M4 and one
posterior M1exp which uses parameters fit to experimental level densities (see Table II). The spin cutoff parameters σ1 and σ2
are defined according to Eqs. (3) and (4).
M1 M2 M3 M4 M1exp Kexp
σ1 σ2 σ1 σ2 σ1 σ2
58Fe(3He,n) 0.79(12) 1.04(16) 1.03(16) 1.22(19) 1.03(16) 1.05(16) 0.90(14) 1.03(16) 0.52
58Fe(3He,p) 1.23(20) 1.01(15) 1.05(16) 0.93(14) 1.03(15) 1.01(15) 1.11(17) 0.98(15)
58Fe(3He,α) 0.66(10) 0.81(12) 0.66(10) 0.86(13) 0.73(11) 0.81(12) 0.72(11) 1.01(15)
59Co(d,n) 0.81(12) 0.90(14) 0.84(13) 0.91(14) 0.92(14) 0.93(14) 0.89(13) 0.97(15) 0.53
59Co(d,p) 1.82(27) 1.42(21) 1.70(26) 1.40(21) 1.32(20) 1.24(19) 1.41(21) 1.07(16)
59Co(d,α) 0.69(11) 0.59(10) 0.64(10) 0.57(10) 0.59(9) 0.70(11) 0.64(10) 0.97(15)
0 2 4 6 8 10 12
0 2 4 6 8 10 0 2 4 6 8 10 12
 Excitation energy (MeV)
57Fe60Co
FIG. 3: Our experimental level density are shown as points. Curves indicate level densities from the four model prescriptions
M1-M4. The upper and lower curves for M1-M3 relate to two spin cutoff parameters σ1 and σ2 used to determine total level
densities from neutron resonance spacings. The histogram is the density of discrete levels.
onances [6]. We used the spin distribution formula from
Ref. [5]:
G(J) =
(2J + 1)
[−(J + 0.5)2
with normalization condition:
G(J) = 1 (9)
TABLE II: Fermi-gas parameters obtained from experimental
level densities
Nucleus 60Ni 60Co 57Fe
a, δ for Eq.(3) 6.16;1.43 6.91;-1.89 5.92;-0.13
a, δ for Eq.(4) 6.39;0.80 7.17;-2.6 6.14;-0.78
TABLE III: χ2 of experimental and calculated total level den-
sities for different level density models and spin cutoff factors
Nucleus M1 M2 M3 M4 M1exp
σ1 σ2 σ1 σ2 σ1 σ2
60Ni 15.3 3.8 1.5 0.2 0.9 1.4 2.5 0.6
60Co 1.3 1.9 2.9 3.1 1.8 2.0 0.8 0.6
57Fe 20.2 2.5 18.9 2.0 10.8 1.8 5.8 0.6
All nuclei 11.5 2.7 7.5 1.8 4.3 1.8 3.1 0.6
0 2 4 6 8 10 12
 Excitation energy (MeV)
FIG. 4: The experimental level densities of 57Fe nucleus.
Filled points are present experimental values. Open points
are data from Oslo experiment [9]. Histogram is density of
discrete levels.
The total level density ρ(U) can be connected to the neu-
tron resonance spacing by using the expression:
= ρ(Bn + 0.5∆E)
|I0+0.5+L|
J=|I0−0.5−L|
G(J), (10)
where DL is the neutron resonance spacing for neutrons
with orbital momentum L, ∆E is the energy interval con-
taining neutron resonances. The assumption of equality
of level numbers with positive and negative parity is used.
Because the total level density ρ(Bn+0.5∆E) around the
neutron separation energy is known from our experiment,
the parameter σ can be obtained from Eqs.(8)-(10).
The data on neutron resonance spacings for nuclei un-
der study are taken from Ref. [25]. The estimated spin
cut off parameters from both s-wave (L = 0) and p-
wave (L = 1) resonance spacings are presented in the
Table IV. The uncertainties include a 20% normaliza-
tion uncertainty in total level densities and uncertainties
in the resonance spacings. For 57Fe, we have obtained
good agreement between two values of σ derived from s-
and p-wave neutron resonances. It indicates the parity
equilibrium of neutron resonances. For 60Co, the uncer-
tainties are too large to draw a definite conclusion. For
60Ni, the onlyD0 is known and one value of σ is obtained.
It agrees better with σ2 but σ1 cannot be excluded.
The calculations of spin cutoff parameter have been
performed with Eqs. (3) and (4) with Fermi-gas param-
eters from Table II. The experiment shows better agree-
ment with σ2 for
57Fe. Spin cut off parameters for 60Ni
and 60Co agree better with σ2 and σ1 respectively, how-
ever because of the large uncertainties, it is impossible to
draw an unambiguous conclusion.
TABLE IV: Spin cut off parameter obtained from s-wave σexps
and p-wave σexpp resonances with using the total level density
from the experiment. σcal1 and σ
2 have been calculated ac-
cording to Eqs. (3) and (4), respectively, with parameters
from Table II.
Nucleus 60Ni 60Co 57Fe
s 3.3(8) 3.6(15) 2.80(30)
p 5.2(12) 2.88(35)
1 4.13 3.95 3.76
2 3.22 3.26 3.0
VI. DISCUSSION
The consistency between results from two different re-
actions supports our conclusion that these reactions are
dominated by the compound-nuclear reaction mechanism
at backward angles. Our results show that level densi-
ties estimated on the basis of interpolation procedure be-
tween neutron resonance and discrete energy regions do
not reproduce experimental cross sections of all outgo-
ing particles simultaneously. The reason is that for some
of the nuclei the level density between discrete and con-
tinuum regions has a complicated behavior which can-
not be described by simple formulas based on Fermi-gas
or Gilbert-Cameron models. It is seen for 57Fe (Fig. 4)
where the level density exhibits some step structure at en-
ergy around 3.7 MeV. Nevertheless, the Fermi-gas model
can still be used to describe the level density at higher ex-
citation energies where density fluctuations vanish. The
M3 model, which does not use discrete levels, gives best
agreement. However, a problem apparently connected to
the spin cutoff parameters is still present. These results
indicate that it is necessary to use level density systemat-
ics obtained from compound-nuclear particle evaporation
spectra. Obviously, the region of discrete levels should be
excluded from such an analysis.
Spin cut off parameters obtained from this experiment
are in general agreement with model prediction of Eqs.(3)
and (4). However it is difficult to reduce uncertainties
to make more specific conclusions about the origin of
this parameter. Most probably, this parameter fluctuates
from nucleus to nucleus and is determined by the internal
properties of nuclei such as the specific population of shell
orbits.
As it has been discussed in the introduction, level den-
sities affect reaction rates which are important in astro-
physics and other applications. The magnitude of this
affect depends mainly on level densities and contribution
of the channel of interest to the total reaction cross sec-
tion. According to the Table I, the neutron outgoing
channel is less sensitive to variations of level densities
while changes in proton and α cross sections can reach a
factor of 2 from corresponding changes in level densities.
Changes in predicted cross sections will also occur at this
level.
VII. CONCLUSION
The neutron, proton, and α-particle cross sections have
been measured at backward angles from 3He+58Fe and
d+59Co reactions. The calculations using HF model have
been performed with three level density models adjusted
to match discrete levels and neutron resonance spacings
and one model adjusted to match neutron resonances
only. None of the model reproduces cross sections of all
outgoing particles simultaneously from both reactions.
However, the model M3 suggested in Ref.[1] gives the
best agreement with experiment.
Level densities of residual nuclei 60Ni, 60Co, and 57Fe
have been obtained from particle evaporation spectra.
Experimental level densities have been fit by Fermi-gas
function and new level density parameters have been ob-
tained. The new level densities allow us to reproduce
all particle energy spectra from both reactions that in-
dicate the dominance of compound nuclear mechanism
in particle spectra measured at backward angles. The
contribution of compound mechanism to the total cross
section is estimated about 50% for both reactions.
The total level density obtained from particle spectra
and neutron resonance spacings have been used to extract
the spin cut off parameter at the neutron separation ener-
gies. The extracted parameters agree with predictions of
Eq. (4) for 57Fe but no definite conclusions can be made
for 60Ni and 60Co. A better understanding of parity ra-
tio systematics would help to make this technique more
reliable.
VIII. ACKNOWLEDGMENTS
We are grateful to J.E. O’Donnell and D. Carter for
computer and electronic support during the experiment,
A. Adekola, C. Matei, B. Oginni and Z. Heinen for
taking shifts, D.C. Ingram for target thickness calcula-
tions done for us. We also acknowledge financial sup-
port from Department of Energy, grant No. DE-FG52-
06NA26187/A000.
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