Nuclear Reactions – Basic Classification

A nuclear reaction is considered to be the process in which two nuclear particles (two nuclei or a nucleus and a nucleon) interact to produce two or more nuclear particles or ˠ-rays (gamma rays). Thus, a nuclear reaction must cause a transformation of at least one nuclide to another. Sometimes if a nucleus interacts with another nucleus or particle without changing the nature of any nuclide, the process is referred to a nuclear scattering, rather than a nuclear reaction.

In order to understand the nature of nuclear reactions, the classification according to the time scale of of these reactions has to be introduced. Interaction time is critical for defining the reaction mechanism.

There are two extreme scenarios for nuclear reactions (not only neutron nuclear reactions):

  • A projectile and a target nucleus are within the range of nuclear forces for the very short time allowing for an interaction of a single nucleon only. These type of reactions are called the direct nuclear reactions.
  • A projectile and a target nucleus are within the range of nuclear forces for the time allowing for a large number of interactions between nucleons. These type of reactions are called the compound nucleus reactions.

In fact, there is always some non-direct (multiple internuclear interaction) component in all reactions, but the direct reactions have this component limited.

Compound Nucleus Reactions

There is no difference between the compound nucleus and the nuclear resonance.

The compound nucleus is the intermediate state formed in a compound nucleus reaction. It is normally one of the excited states of the nucleus formed by the combination of the incident particle and target nucleus. If a target nucleus X is bombarded with particles a, it is sometimes observed that the ensuing nuclear reaction takes place with appreciable probability only if the energy of the particle a is in the neighborhood of certain definite energy values. These energy values are referred to as resonance energies. The compound nuclei of these certain energies are reffered to as nuclear resonances. Resonances are usually found only at relatively low energies of the projectile. The widths of the resonances increase in general with increasing energies. At higher energies the widths may reach the order of the distances between resonances and then no resonances can be observed. The narrowest resonances are usually the compound states of heavy nuclei (such as fissionable nuclei) and thermal neutrons (usually in (n,γ) capture reactions). The observation of resonances is by no means restricted to neutron nuclear reactions.

The compound nucleus model (idea of compound nucleus formation) was introduced by Danish physicist Niels Bohr in 1936. This model assumes that incident particle and the target nucleus become indistinguishable after the collision and together constitute the particular excited state of nucleus – the compound nucleus. To become indistinguishable the projectile has to suffer collisions with constituent nucleons of the target nucleus until it has lost its incident energy. In fact many so these collisions lead to a complete thermal equilibrium inside the compound nucleus. The compound nucleus is excited by both the kinetic energy of the projectile and by the binding nuclear energy.

This compound system is a relatively long-lived intermediate state of particle-target composite system and from the definition, the compound nucleus must live for at least several times longer than is the time of transit of an incident particle across the nucleus (~10-22 s). The time scale of compound nucleus reactions is of the order of 10-18 s  –  10-16 s, but lifetimes as long as 10-14 s have been also observed.

Very important feature and a direct consequence of the thermal equilibrium inside a compound nucleus is the fact the mode of decay of compound nucleus does not depend on the way the compound nucleus is formed. The large number of collisions between nucleons leads to the loss of the information on the entrance channel from the system. The decay mechanism (exit channel) that dominates the decay of C* is determined by the excitation energy in C* and by the law of probability.compound nucleus reaction

These reactions can be considered as a two-stage processes.

  • The first stage is the formation of a compound nucleus expressed by σa+X➝C*
  • The second stage is the decay of a compound nucleus expressed by PC*➝b+Y
  • The result cross-section of certain reaction a+X➝[C*]➝b+Y is given by σ(a,b)= σa+X➝C* . PC*➝b+Y

Uranium absorption reaction

Absorption reaction of fissile 235U. The uncertainty of the exit channel is caused by “loss of memory” of resonance [236U].

Resonances

For the compound nucleus peaks in the cross-section are typical. Each peak is manifesting a particular compound state of nucleus. These peaks and the associated compound nuclei are usually called resonances. The behaviour of the cross-section between two resonances is usually strongly affected by the effect of nearby resonances.

Resonances (particular compound states) are mostly created in neutron nuclear reactions, but  it is by no means restricted to neutron nuclear reactions. The formation of resonances is caused by the quantum nature of nuclear forces. Each nuclear reaction is a transition between different quantum discrete states or energy levels. The discrete nature of energy transitions plays a key role. If the energy of the projectile (the sum of the Q value and the kinetic energy of the projectile) and the energy of target nucleus is equal to a compound nucleus at one of the excitation states, a resonance can be created and peak occurs in the cross section. For light nucleus, the allowable state density in this energy region is much lower and the “distance” between states is higher. For heavy nuclei, such as 238U, we can observe large resonance region in the neutron absorption cross-section.

It is obvious the compound states (resonances) are observed at low excitation energies. This is due to the fact, the energy gap between the states is large. At high excitation energy, the gap between two compound states is very small and the widths of resonances may reach the order of the distances between resonances. Therefore at high energies no resonances can be observed and the cross section in this energy region is continuous and smooth.

Compound Nucleus
The position of the energy levels during the formation of a compound nucleus. Ground state and energy states.
Energy levels of compound state. For neutron absorption reaction on 238U the first resonance E1 corresponds to the excitation energy of 6.67eV. E0 is a base state of 239U.
Region of Resonances
Region of resonances of 238U nuclei.
Source: JANIS (Java-based Nuclear Data Information Software); The ENDF/B-VII.1 Nuclear Data Library

Direct Reactions vs. Compound Nucleus Reactions

Direct Reactions

  • The direct reactions are fast and involve a single-nucleon interaction.
  • The interaction time must be very short (~10-22 s).
  • The direct reactions require incident particle energy larger than ∼ 5 MeV/Ap. (Ap is the atomic mass number of a projectile)
  • Incident particles interact on the surface of a target nucleus rather than in the volume of a target nucleus.
  • Products of the direct reactions are not distributed isotropically in angle, but they are forward focused.
  • Direct reactions are of importance in measurements of nuclear structure.

Compound Nucleus Reactions

  • The compound nucleus is a relatively long-lived intermediate state of particle-target composite system.
  • The compound nucleus reactions involve many nucleon-nucleon interactions.
  • The large number of collisions between the nucleons leads to a thermal equilibrium inside the compound nucleus.
  • The time scale of compound nucleus reactions is of the order of 10-18 s – 10-16 s.
  • The compound nucleus reactions is usually created if the projectile has low energy.
  • Incident particles interact in the volume of a target nucleus.
  • Products of the compound nucleus reactions are distributed near isotropically in angle (the nucleus loses memory of how it was created – the Bohr’s hypothesis of independence).
  • The mode of decay of compound nucleus do not depend on the way the compound nucleus is formed.
  • Resonances in the cross-section are typical for the compound nucleus reaction.
Nuclear and Reactor Physics:

  1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
  2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
  3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
  4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
  5. W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
  6. G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965

Advanced Reactor Physics:

  1. K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.
  2. K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.
  3. D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2. 
  4. E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.

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