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.
- 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
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.
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.
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 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
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)
- 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
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:
For heavy target nuclei, ξ may be approximated by following formula:
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
For a mixture of isotopes:
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
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.
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