Nuclear Cross-Section Calculations [1 meV - 5 GeV Range]

45 Pages·2002·1.092 MB·English

ORNL/TM-2001/191
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NUCLEAR CROSS-SECTION

CALCULATIONS IN THE 1 MEV

TO 5 GEV RANGE WITH COMBINED

SEMI-CLASSICAL AND QUANTUM

MECHANICAL MODELS

February 2002
Prepared by
F. B. GUIMARÃES , C. Y. FU AND L. C. LEAL
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ORNL/TM-2001/191
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Nuclear Science and Technology
NUCLEAR CROSS-SECTION
CALCULATIONS IN THE 1 MEV
TO 5 GEV RANGE WITH COMBINED
SEMI-CLASSICAL AND QUANTUM
MECHANICAL MODELS
F. B. GUIMARÃES , C. Y. FU AND L. C. LEAL
February 2002
Prepared by the
OAK RIDGE NATIONAL LABORATORY
P.O. Box 2008
Oak Ridge, Tennessee 37831-6285
managed by
UT-BATTELLE, LLC
for the
U.S. DEPARTMENT OF ENERGY
under contract DE-AC05-00OR22725
Contents
ABSTRACT ........................................................................ v
I. INTRODUCTION ................................................................. 2
II. DESCRIPTION OF THE REACTION PROCESS ........................................ 2
II.1. THE CASCADE-EXCITON MODEL CODE CEM95 ............................ 3
II.2. THE MODEL CODE TNG .................................................. 7
II.3. THE NEED FOR A QUANTUM MECHANICAL DESCRIPTION ................. 13
III. COMPARING RESULTS ......................................................... 14
III.1. AUTOMATIC GENERATION OF INPUT FILES .............................. 14
III.2. RESOLVING PHYSICAL INCONSISTENCIES ............................... 16
III.2.1. Spin Distribution ................................................ 16
IV. RESULTS AND CONCLUSION ................................................... 19
ACKNOWLEDGMENTS ............................................................ 35
REFERENCES..................................................................... 35
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List of Figures
Figure 1Production cross section for 55Co with p-induced reactions in elemental Fe, in comparison with
EXFOR data. TNG shows good agreement with the experimental data from the activation
threshold to about 45 MeV and CEM95 has a reasonable agreement for E above 30 MeV. In
inc
the 30 to 40 MeV range TNG and CEM95 calculations essentially coincide. .............. 22
Figure 2Production cross section for 56Co with p-induced reactions in elemental Fe. The TNG calcula-
tion shows good agreement with the experimental data from the activation
threshold to about 45 MeV and CEM95 has reasonable agreement for E above 50 MeV. For
inc
energies below 50MeV TNG is clearly more accurate than CEM95. .................... 23
Figure 3CETNG cross-section calculations for production reactions induced by protons in iron for
various radionuclides. ......................................................... 24
Figure 456-Fe (n,2n) cross sections calculated with CEM95 and TNG, for incident energies up to 100
MeV. One notices the smooth transition between the two calculations around 40 MeV. ..... 25
Figure 5Production cross section for 51Cr with p-induced reactions in elemental Fe. TNG and CEM95
calculations together give a reasonable account of the experimental results for all energies in the
30 to 250 MeV region. The peak around 20 MeV, due to the 54Fe(p,p 3He) reaction, cannot be
described in the present version of TNG. .......................................... 26
Figure 6Production cross section for 52Mn with p-induced reactions in elemental Fe. The TNG calcula-
tion shows good agreement with the experimental data from the activation threshold to about 45
MeV. CEM95 misses the experimental data above 80 MeV but is close to the TNG estimate in
the low energy region.......................................................... 27
Figure 7Production cross section for 54Mn with p-induced reactions in elemental Fe. The TNG calcula-
tions show reasonable agreement with the experimental data from the activation thresholds to
about 50 MeV and, for energies from 60 to 250 MeV, the agreement of CEM95 calculations with
experimental data is excellent. The general combined result of TNG plus CEM95 is reasonable
or good for all energies. ....................................................... 28
Figure 8Nonelastic cross section in elemental Fe. The data calculated with CEM95 and TNG show a
smooth transition around 25 MeV, but the results increasingly differ for increasing energies,
where CEM95 shows a better agreement with the experimental data. .................... 29
Figure 956-Fe (n,p) cross section. There is a discontinuity in the curvatures of the calculated data of
CEM95 and TNG above 30 MeV. Both calculations give similar results in the 5 to 30 MeV
region where the CEM95 results are better than the TNG results. ....................... 30
Figure 10Total cross section for n-induced reactions in elemental Fe. TNG shows a reasonable agree-
ment with the experimental data from 10 to 80 MeV. CEM95 results are much worse. ...... 31
Figure 1147-Sc production reaction in elemental Fe. TNG results strongly oscillate but show a reason-
able agreement, on the average, with the experimental data from activation threshold to about 90
MeV. CEM95 calculated cross sections are not good for energies below 150 MeV, but the result
improves for increasing energies. ................................................ 32
Figure 1248-V production reaction in elemental Fe. TNG shows a reasonable qualitative agreement with
the experimental data from activation threshold to 90 MeV but the region close to the activation
energy is not correctly described. CEM95 results are not good in almost the whole energy
interval but the estimate improves for increasing energies. ............................ 33
Figure 1346-Sc production cross section from p-induced reactions in elemental Fe. Both codes do not
give good results in this case. TNG misses the activation threshold and the points between 65
MeV to 100 MeV. CEM95 results are not good for energies below 190 MeV, but the estimate is
good in the 200 to 300 MeV range. .............................................. 34
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ABSTRACT
In this work we describe neutron and proton induced reaction cross-sections for iron produced by the
codes TNG and CEM95 in the 5 to 300 MeV energy range. TNG calculations cover the 5-90 MeV range,
while CEM95 covers the 50-300 MeV high energy range. The two codes show some disagreements in the
overlap energy range, both among themselves and with the experimental data, which are presently being
addressed. The experimental data used are from NNDC and/or from LA150 NSE references. We also
describe some developments for combining TNG and CEM95 into a new code called CETNG (Cascade
Exciton TNG).
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I. INTRODUCTION
There is an increasing need to extend the upper limit of energy of the Evaluated Nuclear Data File
(ENDF/B-VI) to high energies, e. g., to 150 MeV and above, for various technological and scientific
applications [1]. Technologies such as the Spallation Neutron Source (SNS) or the accelerator production
of tritium or the transmutation of long-lived radioactive nuclides into shorter- lived ones for the treatment
of nuclear waste need high energy nuclear data up to a few GeV. As yet, no model code covers this wide
range of energies.
In the present work we address this problem by developing a consistent nuclear cross-section code
usable from the high-keV to low-GeV range, in which the semi-classical to quantum transition is
accurately done. This goal can be achieved with two codes already in use at the Oak Ridge National
Laboratory (ORNL), CEM95 [2] and TNG [3], complemented by the development of new models which
implement a continuous transition between quantum and semi-classical regions.
CEM95 is a semi-classical high-energy nuclear cross-section model code for incident energies from
about 20 MeV to a few GeV developed by Stepan G. Mashnik et al. [2]. It has been incorporated into the
ORNL high-energy transport code, HETC [4], as an alternative to the HETC original cross-section model
that combines the intranuclear cascade model (INC) of Bertini [5] with an evaporation model (EVAP) in
accordance with the original formulation of Bethe [6]. CEM95 is a good alternative among the semi-
classical model codes because it correctly uses the pre-equilibrium (PE) model to link the INC and EVAP
models. On the other hand, the code TNG is a nuclear cross-section model code for lower energies, from
the keV region to about 90 MeV, developed at the ORNL and continuously updated as a tool for cross
section evaluation.
Due to the capabilities of CEM95 and TNG, the incident energy range of the new code, hereafter
called CETNG, ranges from about 1 MeV to 5 GeV. CETNG will allow data evaluation for a large
number of materials whose cross-section data are needed for a wide range of applications, including the
SNS design and accelerator driven systems.
In the following sections we briefly describe the available models that have been used in this work
and some developments for the new code. Results of calculations using TNG, CEM95 and CETNG are
compared with experimental results.
II. DESCRIPTION OF THE REACTION PROCESS
To achieve a consistent description of nuclear reactions one uses different models for the various
energy regions.
In the highest energy region (above 150 MeV) the reaction process can be described by the semi-
classical INC model. In this case one assumes that the components of the initial and final channels can be
treated semi-classically as “particles” instead of “waves” in the description of their motion between two
successive collisions.
For energies below a given threshold in a reaction process two PE models are needed. One PE model
suitable for energies in the range 20–40 MeV, in which the semi-classical description is still valid, and
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another one for energies below this region, in which the PE decay is described quantum-mechanically
and spins and parities are conserved. Finally, it is necessary to use a model based on the formalism of
Hauser-Feshbach (HF) for the description of the CN decay.
In the following two sections we briefly describe the codes CEM95 and TNG and show how they can
be used to make a new model code which consistently describes the whole reaction process, according to
the above description.
II.1. THE CASCADE-EXCITON MODEL CODE CEM95
In CEM95 the whole sequence of the reaction process is described semi-classically in three stages. In
the first stage (INC) the target nuclide is divided into spherical zones of the nuclear potential with
different neutron and proton number densities. The phase space of the tracked particle is uniformly
sampled using Monte Carlo techniques either for the description of its movement toward the target or
inside the intermediate composite system.
The cascade model of CEM95 considers the primary particles as belonging to a degenerated Fermi
gas in which all kinematic information is kept and the residual interaction is neglected. These primary
particles undergo many scatterings before being absorbed or escaping. The excited particles in the
different potential zones of the intermediate nucleus are able to escape unless some conditions such as
the depth of the potential of the zone or the Pauli's exclusion principle preclude escape.
The initial assumption of CEM95's INC model is the description of the diagonal elements of the
density operator for nuclear states, P(E, α, t), by means of the following master equation, [2]
∂P(E,α,t) (cid:1)[ ]
= λ(E,α,E,α')P(E,α',t)−λ(E,α',E,α)P(E,α,t) , (2.1)
∂t
α'≠0
where α represents all nuclear state quantum numbers except the energy E. The time rate of change
parameter λ(E,α,E,α') is defined by the first order approximation of the time-dependent
perturbation theory (Fermi 's Golden Rule) [3,7,8]
π
( ) 2 2
λ α α = α α ω
E, ,E, ' E, V E, ' (E) , (2.2)
(cid:1) scattering α
where ω (E) is the density of final nuclear states and the interaction matrix element and
α
( )
λ E,α,E,α' is assumed to be a smooth function of the energy of the final states. It is also assumed
that the distance between subsequent collisions is large enough so that the nuclear states Eα can be
described by plane waves in the definition of the potential matrix element.
The INC model also assumes that the many-body nuclear system is dilute with short-range interac-
tions among nucleons so that the scattering by more than two particles can be neglected. In addition, the
residual correlation among the non-excited nucleons in the target and between the cascade particles and
the excited nucleons in the intermediate nuclear system are neglected, so that their motion can be
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considered independent of each other. With these hypotheses Eq. (2.1) can be reduced to the Boltzmann
equation, which in the classical limit can be written in terms of the precise momenta and coordinates of
the scattered particles,
(cid:1)(cid:3) ∂ +pk (cid:1)∇ +F(cid:1)∇ (cid:2)(cid:4) f =(cid:1)(cid:1)dp dΩν dσ(νrel)( f f − f f ) , (2.3)
(cid:5)∂t m pk pk (cid:6) k k rel dΩ i j k l
where the function f is the particle distribution function for the single-particle state k ,
k
ν = p −p /m , is the relative velocity between two colliding particles with momenta p and p ,
rel k l k l
and the differential cross section, dσ(ν )/dΩ, is defined in the first Born approximation.
rel
Due to the assumption of quasi-free (independent) motion of the nuclear components, Eq. (2.3)
( )
becomes a linearized equation for the cascade particle distribution, f cas r, p, t ,
(cid:1) ∂ p (cid:2) ( ) ( ) ( )
(cid:3) + k (cid:1)∇ +F(cid:1)∇ (cid:4) f cas r,p,t =ρT <σν > f cas r,p,t +Q r,p,t . (2.4)
(cid:5)∂t m pk pk (cid:6) rel
( )
Here Q r,p,t is a source term which depends on f cas(r, p, t) and therefore, Eq. (2.4) defines
( )
f cas r,p,t self-consistently. The single-particle number density, ρT , is defined in terms of the
( )
equilibrium (Maxwellian) distribution function, fT r,p , for the undisturbed nucleons of the
intermediate nuclear system,
ρT =(cid:1)dpf T (r,p).
Equation (2.4) is solved using the Monte Carlo method to determine the single-particle distribution
function from which all other parameters can be obtained. For example, the number of particles in the
cascade stage is given by
Ncas(p)dp= 1 (cid:1)Rdb2 (cid:1) dr(cid:1)tcasdtf cas(r,p,t)dp , (2.5)
σ n 0 r>R 0 b
i
where the integration is performed over all possible impact parameters b, for particles emitted from a
nucleus with radius R when the time reaches the end of the cascade process t .
cas
The sequence of individual scatterings is then defined by considering the possibility of particle
emission by the intermediate system, both prior to and after each individual scattering.
When it is not possible for more particles to escape, the INC process is finished and the residual
nuclide is likely to be left in an excited state, characterized by its exciton configuration [2,9]. At this
point, begins the second stage of CEM95 in which the PE model is called upon to de-excite the nuclear
system through particle emission, n, p, d, t, 3He, α or γ-rays, and the corresponding creation of excited
residual nuclides. When the excitation energy of a residual nuclide is sufficiently low (some few MeV)
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the PE mechanism becomes unnecessary and the remaining nuclide will evolve towards more compli-
cated states, characterized by an increasing number of excitons, resulting from the residual interactions
between quasi-particles, until the formation of the compound nucleus (CN). This characterizes the third
stage of CEM95 in which the evaporation model (EVAP) is called upon to complete the de-excitation
process.
In either the PE or the EVAP stages, the excited nucleus is described by the exciton model. The
nuclear system is considered as a gas of quasi-particles (excitons) in which the residual interactions are
taken into account, but the collision kinematics and the nuclear geometry are lost. This circumstance
makes it difficult to describe the angular distributions of emitted particles using only the exciton model.
To exploit the advantages and avoid the disadvantages of the cascade and exciton models, CEM95
follows its own particular approach [10] by implementing a Cascade-Exciton Model in which it is
possible to explain the anisotropy of angular distributions of emitted particles in a large region of
incident energies, giving an improved overall description of the nuclear reaction features in comparison
with the conventional cascade-evaporation model [11].
In the modified exciton model of CEM95, the process that tends to equilibrium resulting from
Eq. (2.1) is identified with a discontinuous Markovian process in which each new step in the evolution of
the state of the system is completely determined by its state in the previous step. The random evolution
process is then simulated by the Monte Carlo technique, allowing the generalization of the exciton model
to include all nuclear transitions with ∇n = 0, ± 2 (n = number of excitons), multiple emission of
particles and depletion of nuclear states due to particle emission.
The equation of state (2.1) becomes,
dP(n)
=Λ(n,E)P(n,E,t)+λ+(n−2,E)P(n−2,E,t)
dt
+λ0(n,E)P(n,E,t)+λ−(n,2,E)P(n+2,E,t) (2.6)
(cid:1)(cid:2)dT (cid:2)dE'λj(n,E,T)P(n+n ,E't)δ(E'−(E−B −T)) ,
c j j
j
where
(cid:1)
Λ(α,E)= λ(E,α,E,α') ,
(2.7)
α'
The partial transition probabilities are given by Eq. (2.1)
2π
λ∆n = M 2ω (p,h,E) , (2.8)
(cid:1) ∆n ∆n
and the rate per unit time of emission of ν-particles Γ (n,E) is estimated with the help of the principle
ν
of detailed balance, [12]
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