Neutron Elastic Scattering

Generally, a neutron scattering reaction occurs when a target nucleus emits a single neutron after a neutron-nucleus interaction. In an elastic scattering reaction between a neutron and a target nucleus, there is no energy transferred into nuclear excitation. The elastic scattering conserves both momentum and kinetic energy of the “system”. It may be modeled as a billiard ball collision between a neutron and a nucleus.
σs
There is usually some transfer of kinetic energy from the incident neutron to the target nucleus. The target nucleus gains the exact amount of kinetic energy that the neutron loses. This interaction can take place via compound nucleus formation, but, in case of elastic scattering, a neutron emission returns the compound nucleus to the ground state of the original nucleus. Therefore the initial and final neutrons do not need to be necessarily the same. The elastic scattering can occur by way of two interaction mechanisms:

  • Potential scattering. In potential scattering, the neutron and the nucleus interact without neutron absorption and the formation of a compound nucleus. In fact, the incident neutron does not necessarily have to “touch” the nucleus and the neutron is scattered by the short range nuclear forces when it approaches close enough to the nucleus. Potential scattering occurs with incident neutrons that have an energy of up to about 1 MeV. It may be modeled as a billiard ball collision between a neutron and a nucleus.
  • Compound-elastic scattering. In some cases, if the kinetic energy of an incident neutron just right to form a resonance, the neutron may be absorbed and a compound nucleus may be formed. This interaction is more unusual and is also known as resonance elastic scattering. Due to formation of the compound nucleus, initial and final neutron are not the same.

Conservation of momentum and kinetic energy

conservation of momentum and energy

where:

mn = mass of the neutron
mT = mass of the target nucleus T
vn,i = initial neutron speed
vn,f = final neutron speed
vT,i = initial target speed
vT,f = final target speed

Key Characteristics of Elastic Scattering

  • Elastic scattering is the most important process for slowing down neutrons.
  • Total kinetic energy of the system is conserved in elastic scattering.
  • In this process, energy lost by the neutron is transferred to the recoiling nucleus.
  • Maximum energy transfer is occurred with an head-on collision.
  • Kinetic energy of the recoiled nucleus depends on the recoiled angle φ of the nucleus.
  • Elastic scattering cross-sections for light elements are more or less independent of neutron energy up to 1 MeV.
  • For intermediate and heavy elements, the elastic cross-section is constant at low energy with some specifics at higher energy.
  • A good approximation is, σs = const, for all elements, that are of importance.
  • At low energy, σs can be described by the one-level Breit-Wigner formula.
  • Nearly all elements have scattering cross-sections in the range of 2 to 20 barns.
  • The important exception is for water and heavy water.
  • If the kinetic energy of an incident neutron is large compared with the chemical binding energy of the atoms in a molecule, the chemical bound can be ignored.
  • If the kinetic energy of an incident neutron is of the order or less than the chemical binding energy, the cross-section of the molecule is not equal to the sum of cross-sections of its individual nuclei.
  • Scattering of slow neutrons by molecules is greater than by free nuclei.
  • Therefore one nucleus microscopic cross-sections do not describe the process correctly, while the macroscopic cross-section (Σs) has a precise meaning.
Scattering of slow neutrons by molecules is greater than by free nuclei.
Scattering of slow neutrons by molecules is greater than by free nuclei.
Scattering of slow neutrons by molecules is greater than by free nuclei.
Elastic scattering of molecules.

Elastic Scattering Cross-section

To be an effective moderator, the probability of elastic reaction between neutron and the nucleus must be high. In terms of cross-sections, the elastic scattering cross section of a moderator’s nucleus must be high.
Elastic scattering cross-sections for light elements
Elastic scattering cross-sections for light elements are more or less independent of neutron energy up to 1 MeV.

Source: JANIS (Java-based Nuclear Data Information Software); The JEFF-3.1.1 Nuclear Data Library

Elastic scattering cross-sections for heavy elements
For intermediate and heavy elements, the elastic cross-section is constant at low energy with some specifics at higher energy.

Source: JANIS (Java-based Nuclear Data Information Software); The JEFF-3.1.1 Nuclear Data Library

Elastic Scattering and Neutron Moderators

As can be seen, a high elastic scattering cross-section is important, but does not describe comprehensively capabilities of moderators. In order to describe capabilities of a material to slow down neutrons, three new material variables must be defined:

  • The Average Logarithmic Energy Decrement (ξ)
  • The Macroscopic Slowing Down Power (MSDP)
  • The Moderating Ratio (MR)
Key properties of neutron moderators:

  • high cross-section for neutron scattering
  • high energy loss per collision
  • low cross-section for absorption
  • high melting and boiling point
  • high thermal conductivity
  • high specific heat capacity
  • low viscosity
  • low activity
  • low corrosive
  • cheap
During the scattering reaction, a fraction of the neutron’s kinetic energy is transferred to the nucleus. Using the laws of conservation of momentum and energy and the analogy of collisions of billiard balls for elastic scattering, it is possible to derive the following equation for the mass of target or moderator nucleus (M), energy of incident neutron (Ei) and the energy of scattered neutron (Es).

equation momentum energy

where A is the atomic mass number.

In case of the hydrogen (A = 1) as the target nucleus, the incident neutron can be completely stopped. But this works when the direction of the neutron is completely reversed (i.e. scattered at 180°). In reality, the direction of scattering ranges from 0 to 180 ° and the energy transferred also ranges from 0% to maximum. Therefore, the average energy of scattered neutron is taken as the average of energies with scattering angle 0 and 180°.

Moreover, it is useful to work with logarithmic quantities and therefore one defines the logarithmic energy decrement per collision (ξ) as a key material constant describing energy transfers during a neutron slowing down. ξ is not dependent on energy, only on A and is defined as follows:

logarithmic energy decrement - equation

For heavy target nuclei, ξ may be approximated by following formula:
the logarithmic energy decrement per collision

From these equations it is easy to determine the number of collisions required to slow down a neutron from, for example from 2 MeV to 1 eV.

Example:
Determine the number of collisions required for thermalization for the 2 MeV neutron in the carbon.
ξCARBON = 0.158
N(2MeV → 1eV) = ln 2⋅106/ξ =14.5/0.158 = 92

Table of average logarithmic energy decrement for some elements

Table of average logarithmic energy decrement for some elements.

For a mixture of isotopes:

the logarithmic energy decrement for mixtures

We have defined the probability of elastic scattering reaction, we have defined the average energy loss during the reaction. The product of these variables (the logarithmic energy decrement and the macroscopic cross section for scattering in the material) is the macroscopic slowing down power (MSDP).

MSDP = ξ . Σs

The MSDP describes the ability of a given material to slow down neutrons and indicates how rapidly a neutron will slow down in the material, but it does not fully reflect the effectiveness of the material as a moderator. In fact, the material with high MSDP can slow down neutrons with high efficiency, but it can be a poor moderator because of its high probability of absorbing neutrons. It is typical, for example, for boron, which has a high slowing down power but is absolutely inappropriate as a moderator.

The most complete measure of the effectiveness of a moderator is the Moderating Ratio (MR), where:

MR  = ξ . Σs/Σa

Table of macroscopic slowing down power MSDP for some materials.

Table of macroscopic slowing down power MSDP for some materials.

The moderating ratio or moderator quality is the most complete measure of the effectiveness of a moderator because it takes into account also the absorption effects. When absorption effects are high, most of the neutrons will be absorbed by moderator, leading to lower moderation or lower availability of thermal neutrons.

Therefore a higher ratio of MSDP to absorbtion cross sections ξ . Σs/Σa is desirable for effective moderation. This ratio is called the moderating ratio – MR and can be used as a criterion for comparison of different moderators.

Examples:

  • Light water has the highest ξ and σs among the moderators (resulting in the highest MSDP) shown in the table, but its moderating ratio is low due to its relatively higher absorption cross section.
  • On the other hand, heavy water has lower ξ and σs, but it has the highest moderating ratio owing to its lowest neutron absorption cross-section.
  • Graphite has much heavier nuclei than hydrogen in water, despite the fact graphite has much lower ξ and σs, it is better moderator than light water due to its lower absorption cross-section compared to that of light water.
Table of moderating ratios MR for some materials.

Table of moderating ratios for some materials.

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
  7. Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
  8. U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.

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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