| The following article was originally prepared by M. B. Chadwick, Los Alamos National Laboratory, as an introduction for a chapter on nuclear models for the International Commission on Radiation Units (ICRU). Its emphasis is on models suitable for fast neutron and proton medical radiotherapy applications, which implies an energy range of interest to about 250 MeV, and materials of interest including carbon, oxygen, and calcium. However, it provides a nice summary of the current status of theoretical models for computing reaction cross sections for many applications. |
| A variety of theoretical models are in general use at this time for calculating nuclear reaction cross sections. To put these models in perspective, an overview is presented here of the physical features that they emphasize. Since the nature of the nuclear force (including its origin in quark and gluon interactions) is not fully understood, and since nuclei consist of nucleons interacting through complicated many-body interactions, a comprehensive theory of nuclear reactions and nuclear structure derived from fundamental principles, with good predictive abilities, does not yet exist. Instead, nuclear physics researchers often develop models which typically emphasize one or more physical features over others, depending on the context of the investigation. Examples include compound nucleus, direct, and various preequilibrium nuclear reaction theories, including exciton and intranuclear cascade semiclassical models and quantum mechanical multistep approaches. The drawbacks inherent in emphasizing certain physical aspects over others are partly compensated by a corresponding insight, and mathematical simplicity, exhibited by a model. |
| Many different interaction mechanisms can occur when a nucleon of a few hundred MeV and below strikes a target nucleus. At low incident energies (a few MeV, say), nuclear reactions take place by the compound nucleus process, in which the incident particle is captured by the target nucleus, and its energy is shared statistically among all the nucleons of the compound system. After a time much greater than the interaction time, the compound nucleus emits one or more particles and generally attains its ground state by gamma-ray emission. As the incident energy increases, it becomes more likely that particle emission will take place in the first stage of the reaction, when the incident particle interacts with the target nucleus as a whole (for example, a collective excitation) or a nucleon within it. Many theories have been developed to enable the cross sections of these direct reactions to be calculated, and they facilitate an understanding of elastic and inelastic scattering and particle transfer reactions. However, experimental and theoretical research in the last few decades has shown that particle emission can take place with a time scale longer than the very rapid direct reactions (about 10-22 sec) but much much shorter than the slower compound nucleus reactions (about 10-16 to 10-18 sec). These emission processes are known as preequilibrium or multistep reactions, and they are characterized by particles emitted with relatively high energies and with angular distributions that are peaked in the forward direction. One of the problems in nuclear reaction theory is a proper treatment of the scattering at energies where two or more reaction mechanisms apply. |
| There are two main approaches for calculating the decay of excited equilibrated nuclei that have been used for modeling nuclear reactions on light biological elements: the compound nucleus theory (as used for our calculations) ; and the Fermi breakup theory (as used, for instance, by Brenner and Prael (1989)). Both use statistical arguments in their derivation, and both can use experimental information on the excited nuclear level structure of the residual nuclei (when the Hauser-Feshbach theory is used for the compound nucleus calculation). The compound nucleus theory invokes detailed balance to predict particle and gamma-ray decay rates in terms of transmission coefficients for penetrating the nuclear potential barrier (obtained from the inverse process of a particle incident on a nucleus), and nuclear level densities of the residual and decaying nucleus (the phase space for the reaction). Many-body breakup processes are assumed to proceed through sequential two-body breakups. Fermi breakup models, on the other hand, determine the decay probabilities from the statistical weights for different breakup channels, typically without following all possible intermediate excited states. In fact, both approaches tend to work reasonably well in nuclear reaction calculations and comparisons with measurements have not been able to indicate a preference (Subramanian et. al. (1983, 1886), and Chadwick et al. (1996)). The formal relation between these two models has been described by Epherre et al. (1969). |
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Numerous models have been developed to account for the high-energy
preequilibrium particles that make up the "continuum" region of the
secondary particle emission spectra above the evaporation peak (for
incident energies above approximately 10 MeV). Beginning in the
1940's, intranuclear cascade (INC) models were developed that use
Monte Carlo techniques to simulate nucleon-nucleus reactions in terms
of individual successive nucleon-nucleon collisions. The INC theory
makes use of free nucleon-nucleon experimental cross sections, and
it accounts for Fermi motion and Pauli blocking in a semiclassical
manner, following the trajectories of the nucleons in coordinate and
momentum space. Perhaps the most questionable assumption within the
INC model is the utilization of free-space nucleon-nucleon cross
sections for scatterings that take place within a nuclear medium,
which is most accurate only at high incident energies (above
approximately 100-200 MeV). However, some authors, such as Brenner and
Prael (1989), have had some success with INC calculations even at
energies as low as 20 MeV.
Other widely used semiclassical preequilibrium reaction models are the exciton (Griffin, 1966) and hybrid (Blann, 1971) models. Griffin's work provided an explanation of the high energy emitted particles first observed in the measurements of Wood et al. (1965). These models also picture the evolution of the nuclear reaction in terms of successive nucleon-nucleon collisions, but do so within a particle-hole, or "exciton", formalism (nucleons excited from within the Fermi sea leave a hole). The exciton and hybrid models differ in terms of the statistical postulates made to treat the evolution of the reaction (Hodgson and Gadioli, 1992). In these models the reaction is usually followed in energy space, which simplifies the calculations and leads to closed-form deterministic equations. Emission probabilities from particle-hole excitations are obtained by applying detailed balance, making use of inverse reaction cross sections and the phase space of particle-hole excitations. The bulk of the our calculations were performed using the exciton model for the preequilibrium phase of the reaction; its predictive capability for reactions up to a few hundred MeV is rather good, and it requires a relatively small amount of computer time. We note, however, that these models deal with reaction probabilites instead of amplitudes coming from solution of the wave equation, and therefore observables that are sensitive to interference effects in the scattering, such as angular distributions, are often not calculated accurately. In recent years, quantum mechanical theories have been developed to describe preequilibrium processes. Various theories have been proposed which differ in the underlying quantum statistical assumptions made. The most widely studied theory is that of Feshbach, Kerman, and Koonin (FKK) (1980), due to its relatively straightforward computational structure. This theory partitions the reaction process into two types of scattering: multistep compound (important at the lowest incident energies), and multistep direct (important at higher incident energies). The multistep direct reaction mechanism, which is generally most important, predicts multistep scattering cross sections in terms of a convolution of the 1-step scattering cross section, an attractive result that follows from the statistical assumptions (particularly the random phases of matrix elements) made by FKK. The 1-step cross section is obtained by extending direct reaction distorted wave Born approximation (DWBA) theory into the continuum. One of the main advantages of quantum mechanical theories is that they accurately predict the angular distributions of preequilibrium ejectiles, which are governed by inherently quantum effects, such as diffraction and refraction. In most cases, semiclassical exciton models can only compete in predictive power with the FKK theory because they use angular distributions from phenomenological systematics based on measurements (Kalbach, 1989). For some of the nuclei that we calculated, the FKK theory was used. However, the long computer time required for its computation makes it somewhat impractical for large-scale calculations. IThe FKK theory is, however, useful for comparisons with exciton model results to help validate the exciton model predictions. |
| The optical model is another nuclear reaction model that is widely used in our calculations, particularly for determining elastic scattering and total cross sections, and transmission coefficients (needed for Hauser-Feshbach calculations). The nucleus is considered to provide a nuclear potential in which the projectile nucleon moves that is complex, having real and imaginary terms. The scattering solutions to the Schroedinger equation are then determined. Elastic scattering off the nucleus can occur, and diffractive effects in the elastic angular distribution are obtained. The imaginary term in the potential accounts for all nonelastic processes by leading to a damping of the projectile wavefunction. While this model, therefore, predicts the total reaction cross section, it does not provide information on the subsequent nonelastic partial cross sections for various decay channels - for this, the aforementioned compound nucleus and preequilibrium reaction models are needed, and they depend crucially upon the optical model potential. |
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Certain properties of nuclear structure are utilized in nuclear
reaction model calculations. At low excitation energies the properties
of discrete nuclear levels are known for most of the stable nuclei
from measurements (e.g. their
energy, spin, parity, deformation, and decay properties). These are
used in calculations of equilibrium particle and gamma-ray decay where
angular momentum and parity conservation are included (the
Hauser-Feshbach theory). These properties are also important in direct
reaction calculations, such as inelastic scattering and particle
transfer reactions to low lying
nuclear levels, where the angular momentum, and the collectivity of
the excited nuclear level, strongly influence the magnitude of such
processes.
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 (1975), account for the dependence of the level density on shell effects (near closed shells the nuclear single-particle levels are widely spaced at the Fermi level so that the level density is reduced, though this effect is damped out at higher excitation energies), 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 is known from the measured discrete levels, and the density at approximately 7-8 MeV is known for many nuclei from neutron resonance spacing measurements, but at other energies the experimental information is sparse. Thus, even the most current theoretical models of the level density are often inaccurate in some regions of excitation energy. Level densities are used in Hauser-Feshbach calculations since particle emission probabilities are proportional to the accessible phase space (i.e. level density), and because they make the computations tractable. |
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