Material Properties

Charged Particle Ejection

Charged particle reactions are usually associated with formation of a compound nucleus, which is excited to a high energy level, that such compound nucleus can eject a new charged particle while the incident neutron remains in the nucleus. After the new particle is ejected, the remaining nucleus is completely changed, but may or may not exist in an excited state depending upon the mass-energy balance of the reaction. This type of reaction is more common for charged particles as incident particles (such as alpha particles, protons, and so on).

The case of neutron-induced charged particle reactions is not so common, but there are some neutron-induced charged particle reactions, that are of importance in the reactivity control and also in the detection of neutrons.

The most important charged particle reactions in nuclear reactor physics are reactions of thermal neutrons with boron nuclei (rather with ¹⁰B nuclei). In nuclear industry boron is commonly used as a neutron absorber and a neutron converter (in neutron radiation detectors) due to its high neutron cross-section of isotope ¹⁰B along entire neutron energy spectrum. Its (n,alpha) reaction cross-section for thermal neutrons is about 3840 barns (for 0.025 eV neutron). Examples of neutron-induced charged particle reactions are shown below:

See also: Application of boron based materials
See also: Detection of neutrons

σ

Examples of (n,alpha) reactions

10B(n,alpha)7Li

This reaction is the most important (n,alpha) reaction of isotope ¹⁰B with thermal neutrons. This reaction can be used as in the case of neutron absorbers (chemical shim, burnable absorbers or control rods), and in the case of neutron converters (neutron detectors), because its cross-section is very high and produces energetic alpha particles.

6Li(n,α)3H

This is a reaction allowing detection of neutrons. The reaction cross-section for thermal neutrons is σ = 925 barns and the natural lithium has abundance of ⁶Li 7,4%.

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

Examples of (n,p) reactions

3He(n,p)3H

This is a reaction allowing detection of neutrons. The reaction cross-section for thermal neutrons is σ = 5350 barns and the natural helium has abundance of isotope ³He 0.014%.

14N (n,p) 14C

This nuclear reaction continually take place especially in the earth’s atmosphere, forming equilibrium amounts of the radionuclide ¹⁴C. In nuclear power plants, it is important especially from radiation protection point of view. The reaction is responsible for most of the radiation dose delivered to the human body by thermal neutrons. The nitrogen atoms are contained in proteins, therefore if the human body is exposed to thermal neutrons, then these thermal neutrons may be absorbed by ¹⁴N, causing a proton emission. Protons are directly ionizing particles and deposit their energy over a very short distance in the body tissue.

Examples of (n,…) reactions

10B(n,T + 2*alpha)

This threshold reaction of fast neutron with an isotope ¹⁰B is the main way, how radioactive tritium in primary circuit of all PWRs is generated. ¹⁰B is the principal source of radioactive tritium in primary circuit of all PWRs (which use boric acid as a chemical shim). Note that, this reaction occurs very rarely in comparison with the most common (n,alpha) reaction of isotope ¹⁰B with thermal neutrons.

There are more reactions with neutrons, which can rarely lead to formation of radioactive tritium, for example:

10B(n,alpha)7Li + 7Li(n,n+alpha)3H  – threshold reaction (~3 MeV).

See also:

See also:

Neutron Reactions

See also:

Tritium

Tritium is the only naturally-occurring radioisotope of hydrogen. Its atomic number is naturally 1 which means there is 1 proton and 1 electron in the atomic structure. Unlike the hydrogen nucleus and deuterium nucleus, tritium has 2 neutrons in the nucleus. Tritium is naturally-occurring, but it is extremely rare. Tritium is produced in the atmosphere when cosmic rays collide with air molecules. Tritium is also a byproduct of the production of electricity by nuclear power plants. The name of this isotope is formed from the Greek word τρίτος (trítos) meaning “third”.

Decay of Tritium

Tritium is a radioactive isotope, bur it emits a very weak form of radiation, a low-energy beta particle that is similar to an electron. It is a pure beta emitter (i.e. beta emitter without an accompanying gamma radiation). The electron’s kinetic energy varies, with an average of 5.7 keV, while the remaining energy is carried off by the nearly undetectable electron antineutrino. Such a very low energy of electron causes, that the electron cannot penetrate the skin or even does not travel very far in air. Beta particles from tritium can penetrate only about 6.0 mm of air.

Tritium decays via negative beta decay into helium-3 with half-life of 12.3 years.

³H

Tritium in nuclear reactors

Tritium is a byproduct in nuclear reactors. Most important source (due to releases of tritiated water) of tritium in nuclear power plants stems from the boric acid, which is commonly used as a chemical shim to compensate an excess of initial reactivity. Main reactions, in which the tritium is generated from boron are below:

10B(n,T + 2*alpha)

This threshold reaction of fast neutron with an isotope ¹⁰B is the main way, how radioactive tritium in primary circuit of all PWRs is generated. ¹⁰B is the principal source of radioactive tritium in primary circuit of all PWRs (which use boric acid as a chemical shim). Note that, this reaction occurs very rarely in comparison with the most common (n,alpha) reaction of isotope ¹⁰B with thermal neutrons.

There are more reactions with neutrons, which can rarely lead to formation of radioactive tritium, for example:

10B(n,alpha)7Li + 7Li(n,n+alpha)3H  – threshold reaction (~3 MeV).

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

Tritium is also produced in reaction with ⁶Li.

6Li(n,α)3H

This is a reaction allowing detection of neutrons, but in some cases, LiOH is added to control the pH of primary coolant in some LWR. The reaction cross-section for thermal neutrons is σ = 925 barns and the natural lithium has abundance of ⁶Li 7,4%.

Tritium occurs in nuclear power plants in the form of tritiated water. Tritiated water is like normal water, but is very very weakly radioactive. Therefore it dose not pose a hazard to human health. The releases of tritiated water are closely monitored by plant operators and state supervisors.

Reference: Jacobs D.G. Sources of Tritium and Its Behaviour Upon Release to the Environment. US Atomic Energy Commission, 1968.

Tritium in Nature

Tritium is produced in the atmosphere when cosmic rays collide with air molecules. In the most important reaction for natural production, a fast neutron (which must have energy greater than 4.0 MeV) interacts with atmospheric nitrogen:

Worldwide, the production of tritium from natural sources is 148 petabecquerels per year. In result, the tritiated water produced participates in the water cycle.

• about 400 Bq/m³ in continental water
• about 100 Bq/m³ in oceans

Tritium poses a risk to health as a result of internal exposure only following ingestion in drinking water or food, or inhalation or absorption through the skin. The tritium taken into the body is uniformly distributed among all soft tissues. An average annual dose from natural tritium intake is 0.01 μSv.

In case of artificial tritium ingestion or inhalation, a biological half-time of tritium is 10 days for HTO and 40 days for OBT (organically bound tritium) formed from HTO in the body of adults. It was also shown that the biological half-time of HTO depends strongly on many variables and varies from about 4 to 18 days. During the warmer months, the average half-life is lower, which is attributed to increased water intake. As well as, drinking larger amounts of alcohol will reduce the biological half-life of water in the body.

See also: Tritium in Nature

See also:

Glossary

Liquid Drop Model of Nucleus

One of the first models which could describe very well the behavior of the nuclear binding energies and therefore of nuclear masses was the mass formula of von Weizsaecker (also called the semi-empirical mass formula – SEMF), that was published in 1935 by German physicist Carl Friedrich von Weizsäcker. This theory is based on the liquid drop model proposed by George Gamow.

According to this model, the atomic nucleus behaves like the molecules in a drop of liquid. But in this nuclear scale, the fluid is made of nucleons (protons and neutrons), which are held together by the strong nuclear force. The liquid drop model of the nucleus takes into account the fact that the nuclear forces on the nucleons on the surface are different from those on nucleons in the interior of the nucleus. The interior nucleons are completely surrounded by other attracting nucleons. Here is the analogy with the forces that form a drop of liquid.

In the ground state the nucleus is spherical. If the sufficient kinetic or binding energy is added, this spherical nucleus may be distorted into a dumbbell shape and then may be splitted into two fragments. Since these fragments are a more stable configuration, the splitting of such heavy nuclei must be accompanied by energy release. This model does not explain all the properties of the atomic nucleus, but does explain the predicted nuclear binding energies.

The nuclear binding energy as a function of the mass number A and the number ofprotons Z based on the liquid drop model can be written as:This formula is called the Weizsaecker Formula (or the semi-empirical mass formula). The physical meaning of this equation can be discussed term by term.

With the aid of the Weizsaecker formula the binding energy can be calculated very well for nearly all isotopes. This formula provides a good fit for heavier nuclei. For light nuclei, especially for ⁴He, it provides a poor fit. The main reason is the formula does not consider the internal shell structure of the nucleus.

In order to calculate the binding energy, the coefficients a_V, a_S, a_C, a_A and a_P must be known. The coefficients have units of megaelectronvolts (MeV) and are calculated by fitting to experimentally measured masses of nuclei. They usually vary depending on the fitting methodology. According to ROHLF, J. W., Modern Physics from α to Z0 , Wiley, 1994., the coefficients in the equation are following:Using the Weizsaecker formula, also the mass of an atomic nucleus can be derived and is given by:

m = Z.m_p +N.m_n -E_b/c²

where m_p and m_n are the rest mass of a proton and a neutron, respectively, and E_b is the nuclear binding energy of the nucleus.From the nuclear binding energy curve and from the table it can be seen that, in the case of splitting a ²³⁵U nucleus into two parts, the binding energy of the fragments (A ≈ 120) together is larger than that of the original ²³⁵U nucleus.

According to the Weizsaecker formula, the total energy released for such reaction will be approximately 235 x (8.5 – 7.6) ≈ 200 MeV.

See also:

Binding Energy

See also:

Nuclear Fission

See also:

Critical Energy

Weizsaecker Formula – Semi-empirical Mass Formula

This formula is called the Weizsaecker Formula (or the semi-empirical mass formula). The physical meaning of this equation can be discussed term by term.

Volume term

Volume term – a_V.A. The first two terms describe a spherical liquid drop of an incompressible fluid with a contribution from the volume scaling with A and from the surface, scaling with A^2/3. The first positive term a_V.A is known as the volume term and it is caused by the attracting strong forces between the nucleons. The strong force has a very limited range and a given nucleon may only interact with its direct neighbours. Therefore this term is proportional to A, instead of A². The coefficient a_V is usually about ~ 16 MeV.

Surface term

Surface term – a_sf.A^2/3. The surface term is also based on the strong force, it is, in fact, a correction to the volume term. The point is that particles at the surface of the nucleus are not completely surrounded by other particles. In the volume term, it is suggested that each nucleon interacts with a constant number of nucleons, independent of A. This assumption is very nearly true for nucleons deep within the nucleus, but causes an overestimation of the binding energy on the surface. By analogy with a liquid drop this effect is indicated as the surface tension effect. If the volume of the nucleus is proportional to A, then the geometrical radius should be proportional to A^1/3 and therefore the surface term must be proportional to the surface area i.e. proportional to A^2/3.

Coulomb term

Coulomb term – a_C.Z².A^-⅓. This term describes the Coulomb repulsion between the uniformly distributed protons and is proportional to the number of proton pairs Z²/R, whereby R is proportional to A^1/3. This effect lowers the binding energy because of the repulsion between charges of equal sign.

Asymmetry term

Asymmetry term – a_A.(A-2Z)²/A. This term cannot be described as ‘classically’ as the first three. This effect is not based on any of the fundamental forces, this effect is based only on the Pauli exclusion principle (no two fermions can occupy exactly the same quantum state in an atom). The heavier nuclei contain more neutrons than protons. These extra neutrons are necessary for stability of the heavier nuclei. They provide (via the attractive forces between the neutrons and protons) some compensation for the repulsion between the protons. On the other hand, if there are significantly more neutrons than protons in a nucleus, some of the neutrons will be higher in energy level in the nucleus. This is the basis for a correction factor, the so-called symmetry term.

Pairing term

Pairing term – δ(A,Z). The last term is the pairing term δ(A,Z). This term captures the effect of spin-coupling. Nuclei with an even number of protons and an even number of neutrons are (due to Pauli exclusion principle) very stable thanks to the occurrence of ‘paired spin’. On the other hand, nuclei with an odd number of protons and neutrons are mostly unstable.

Table of Calculated Binding Energies

With the aid of the Weizsaecker formula the binding energy can be calculated very well for nearly all isotopes. This formula provides a good fit for heavier nuclei. For light nuclei, especially for ⁴He, it provides a poor fit. The main reason is the formula does not consider the internal shell structure of the nucleus.

In order to calculate the binding energy, the coefficients a_V, a_S, a_C, a_A and a_P must be known. The coefficients have units of megaelectronvolts (MeV) and are calculated by fitting to experimentally measured masses of nuclei. They usually vary depending on the fitting methodology. According to ROHLF, J. W., Modern Physics from α to Z0 , Wiley, 1994., the coefficients in the equation are following:Using the Weizsaecker formula, also the mass of an atomic nucleus can be derived and is given by:m = Z.m_p +N.m_n -E_b/c²

where m_p and m_n are the rest mass of a proton and a neutron, respectively, and E_b is the nuclear binding energy of the nucleus.

From the nuclear binding energy curve and from the table it can be seen that, in the case of splitting a ²³⁵U nucleus into two parts, the binding energy of the fragments (A ≈ 120) together is larger than that of the original ²³⁵U nucleus.

According to the Weizsaecker formula, the total energy released for such reaction will be approximately 235 x (8.5 – 7.6) ≈ 200 MeV.

See also: Liquid Drop Model

See also:

Nuclear Fission

Critical Energy – Threshold Energy for Fission

In principle, any nucleus, if brought into sufficiently high excited state, can be splitted. For fission to occur, the excitation energy must be above a particular value for certain nuclide. The minimum excitation energy required for fission to occur is known as the critical energy (E_crit) or threshold energy.The critical energy depends on the nuclear structure and is quite large for light nuclei with Z < 90. For heavier nuclei with Z > 90, the critical energy is about 4 to 6 MeV for A-even nuclei, and generally is much lower for A-odd nuclei. It must be noted, some heavy nuclei (eg. ²⁴⁰Pu or ²⁵²Cf) exhibit fission even in the ground state (without externally added excitation energy). This phenomena is known as the spontaneous fission. This process occur without the addition of the critical energy by the quantum-mechanical process of quantum tunneling through the Coulomb barrier (similarly like alpha particles in the alpha decay). The spontaneous fission contributes to ensure sufficient neutron flux on source range detectors when reactor is subcritical in long term shutdown.

The amount of excitation energy required for fission to occur can be estimated from the magnitude of the electrostatic potential barrier and the dissociation energy of the fission.

In the figure, it is considered the potential energy of the fissioning nucleus as a function of the distance r between the two separating lobes. In order to deform the nucleus into a dumbbell shape, the sufficient kinetic or binding energy must be added to the system. This is due to the fact, nucleons attract one another and energy is required to increase the average distance between them. After the energy is added, the resulting nucleus is in an intermediate state with a larger potential energy than was the potential energy of original nucleus. The height of the potential barrier may be approximated given by:

E_c = Z₁.Z₂.e² / (4.π.ε₀.(R1+R2))

where R1 and R2 are the respective nuclear radii and ε₀ is the permittivity of the vacuum. If this point is reached, the two lobes of the dumbbell begin to separate.

The dissociation energy E_d is equal to the difference between the binding energy of the compound nucleus to be fissioned and the sum of the binding energies of the fission fragments. The minimum activation energy E_a that has to be added to a nucleus to cause fission reaction is thus E_c – E_d.

All fissioning nuclei do not split in the same way. Although the mass of the initial nucleus is well defined in a reaction, the masses of resulting fission fragments are not. This is the reason there is no single Q-value, but what is usually referred to as the fission Q-value is actually an average of Q-values over all ways of fission.

The excitation energy can be added to a nucleus by a neutron, but it is not the only way, how to induce fission. The excitation energy can be added also by bombardment with photons (photofission) or charged particles. In reactor engineering, the most attractive method of causing fission, however, is by forming a compound nucleus with the aid of a neutron.

In the case of neutron-induced fission reactions, an incident neutron provides additional energy to a target nucleus in the form of the kinetic energy and the nuclear binding energy. Neutrons have the principal advantage, they do not need to overcome the coulomb forces as in the case of charged particles.

It can be seen from the table that for fission of ²³⁸U or ²³²Th the neutron must have some additional kinetic energy (negative BE_n – E_crit value), while absorption of a neutron without kinetic energy can already cause fission of ²³⁵U (or ²³³U, ²³⁹Pu). For example, according to the table , the binding energy of the last neutron in ²³⁶U is 6.8 MeV (target nucleus is ²³⁵U), while the critical energy is only 6.5 MeV. Thus, when a thermal neutron is absorbed by ²³⁵U, the compound nucleus ²³⁶U is produced at about 0.3 MeV above the critical energy and the nucleus splits immediately. Nuclei such as ²³⁵U that lead to fission following the absorption of thermal neutron are called fissile nuclei.

For heavy nuclides with atomic number of higher than 90, most of fissile isotopes meet the fissile rule:

Fissile isotopes have 2 x Z – N = 43 ± 2 (example for 235U: 2 x 92 – 143 = 41)

where Z is number of protons and N is number of neutrons.
In general, the heavy nuclei with an odd number of neutrons (²³⁵U, ²³³U, ²³⁹Pu and ²⁴¹Pu) can easily be split, because the neutron that is absorbed to form a compound nucleus with these nuclei is an ‘even’ neutron, so that the binding energy due to the pairing effect is large.

On the other hand, the heavy nuclei with an even number of neutrons (²³²Th, ²³⁸U, ²⁴⁰Pu and ²⁴²Pu) have a threshold energy (the kinetic energy) for fission by neutrons, because the absorbed neutron is an ‘odd’ neutron and this neutron makes relatively little binding energy available. For these nuclides fission is thus a threshold reaction.

For example, according to the table, the binding energy of the last neutron in ²³⁹U is only 5.5 MeV (target nucleus is ²³⁸U), while the critical energy is 7.0 MeV. Thus, when a thermal neutron is absorbed by ²³⁸U, fission cannot occur. In order to cause fission of ²³⁸U, the incident neutron must have additional kinetic energy. Nuclei such as ²³⁸U that lead to fission following the absorption of fast neutron are called fissionable nuclei. It must be noted, also a ²⁰⁸Pb nucleus may be fissioned when struck by a high energy neutron of about 20MeV, but this nucleus is not ordinarily said to be fissionable.

When a nucleus is excited above the potential barrier, it is not sure that fission will occur. Most of absorption reactions result in fission reaction (σ_f = 585 barns), but a minority results in radiative capture forming ²³⁶U. The radiative capture is a reaction, in which the compound nucleus decays to its ground state by gamma emission.The cross-section for radiative capture for thermal neutrons is about 99 barns (for 0.0253 eV neutron). Therefore about 15% of all absorption reactions result in radiative capture of neutron. About 85% of all absorption reactions result in fission.

See also: Uranium 235 Fission

See also: Plutonium 239 Fission

See also:

Liquid Drop Model

See also:

Nuclear Fission

See also:

Energy Release

Photoneutrons

In nuclear reactors the gamma radiation plays a significant role also in reactor kinetics and in a subcriticality control. Especially in nuclear reactors with D₂O moderator (CANDU reactors) or with Be reflectors (some experimental reactors). Neutrons can be produced also in (γ, n) reactions and therefore they are usually referred to as photoneutrons.

A high energy photon (gamma ray) can under certain conditions eject a neutron from a nucleus. It occurs when its energy exceeds the binding energy of the neutron in the nucleus. Most nuclei have binding energies in excess of 6 MeV, which is above the energy of most gamma rays from fission.On the other hand there are few nuclei with sufficiently low binding energy to be of practical interest. These are: ²D, ⁹Be, ⁶Li, ⁷Li and ¹³C. As can be seen from the table the lowest threshold have ⁹Be with 1.666 MeV and ²D with 2.226 MeV.

In case of deuterium, neutrons can be produced by the interaction of gamma rays (with a minimum energy of 2.22 MeV) with deuterium:Because gamma rays can be emitted by fission products with certain delays, and the process is very similar to that through which a “true” delayed neutron is emitted, photoneutrons are usually treated no differently than regular delayed neutrons in the kinetic calculations. Photoneutron precursors can be also grouped by their decay constant, similarly to “real” precursors. The table below shows the relative importance of source neutrons in CANDU reactors by showing the makeup of the full power flux.Despite the fact photoneutrons are of importance especially in CANDU reactors, deuterium nuclei are always present (~0.0156%) also in the light water of LWRs. Moreover the capture of neutrons in the hydrogen nucleus of the water molecules in the moderator yields small amounts of D₂O. This enhances the heavy water concentration. Therefore also in LWRs kinetic calculations, photoneutrons from D₂O are treated as additional groups of delayed neutrons having characteristic decay constants λ_j and effective group fractions.

After a nuclear reactor has been operated at full power for some time there will be a considerable build-up of gamma rays from the fission products. This high gamma flux from short-lived fission products will decrease rapidly after shutdown. In the long term the photoneutron source decreases with the decay of long-lived fission products that produce delayed high-energy gamma rays and the photoneutron source drops slowly, decreasing a little each day. The longest-lived fission product with gamma ray energy above the threshold is ¹⁴⁰Ba with a half-life of 12.75 days.

The amount of fission products present in the fuel elements depends on how long has been the reactor operated before shut-down and at which power level has been the reactor operated before shut-down. Photoneutrons are usually major source in a reactor and ensure sufficient neutron flux on source range detectors when reactor is subcritical in long term shutdown.

In comparison with fission neutrons, that make a self-sustaining chain reaction possible, delayed neutrons make reactor control possible and photoneutrons are of importance at low power operation.

See also:

Eight Groups

See also:

Delayed Neutrons

See also:

Energy Spectra

Thermal Neutron – Reactors

Free neutrons can be classified according to their kinetic energy. This energy is usually given in electron volts (eV). The term temperature can also describe this energy representing thermal equilibrium between a neutron and a medium with a certain temperature.

Thermal Neutrons are neutrons in thermal equilibrium with a surrounding medium of temperature 290K (17 °C or 62 °F). Most probable energy at 17°C (62°F) for Maxwellian distribution is 0.025 eV (~2 km/s). This part of neutron’s energy spectrum constitutes most important part of spectrum in thermal reactors.

Thermal neutrons have a different and often much larger effective neutron absorption cross-section (fission or radiative capture) for a given nuclide than fast neutrons. Therefore the criticality of a thermal reactor can be achieved with a much lower enrichment of nuclear fuel.

Moreover, thermal neutrons are in the 1/v region and the cross-section behaves according to the 1/v Law. In this region absorption cross-section increases as the velocity (kinetic energy) of the neutron decreases. Therefore the 1/v Law can be used to determine shift in absorbtion cross-section, if the neutron is in equilibrium with a surrounding medium. This phenomenon is due to the fact the nuclear force between the target nucleus and the neutron has a longer time to interact.

See also: Compound Nucleus Reactions

Example of cross- sections in 1/v region:

The absorbtion cross-section for 238U at 20°C = 293K (~0.0253 eV) is:

.

The absorbtion cross-section for 238U at 1000°C = 1273K is equal to:

This cross-section reduction is caused only due to the shift of temperature of surrounding medium.

Thermal Neutron – Diffraction

Thermal neutrons are also widely used in neutron diffraction experiments. Neutron diffraction experiments use an elastic neutron scattering to determine the atomic (or magnetic) structure of a material. The neutron diffraction is based the fact that thermal or cold neutrons have the wavelengths similar to atomic spacings.

An examined sample (crystalline solids, gasses, liquids or amorphous materials) must be placed in a neutron beam of thermal (0.025 eV) or cold (neutrons in thermal equilibrium with very cold surroundings such as liquid deuterium) neutrons to obtain a diffraction pattern that provides information about the structure of the examined material. The neutron diffraction experiments are similar to X-ray diffraction experiments, but neutrons interact with matter differently. Photons (X-rays) interact primarily with the electrons surrounding (atomic electron cloud) a nucleus, but neutrons interact only with nuclei. Neither the electrons surrounding (atomic electron cloud) a nucleus nor the electric field caused by a positively charged nucleus affect a neutron’s flight. Due to their different properties, both methods together (neutron diffraction and X-ray diffraction) can provide complementary information about the structure of the material.

See also:

Neutron

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:

m_n = mass of the neutron
m_T = mass of the target nucleus T
v_n,i = initial neutron speed
v_n,f = final neutron speed
v_T,i = initial target speed
v_T,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, σ_s = const, for all elements, that are of importance.

• At low energy, σ_s 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.

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.

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

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

See also:

See also:

Neutron Reactions

See also:

Neutron Inelastic 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.

In inelastic neutron scattering, the incident neutron is absorbed by the target nucleus and the interaction take place via compound nucleus formation. The compound nucleus emits then a neutron of lower kinetic energy which leaves the original nucleus in an excited state. The nucleus gives up excitation energy by emitting one or more gamma rays to reach its ground state. In comparison to elastic scattering reactions, it is not easy to write an expression for the average energy loss of inelastic scattering reactions because it depends on the energy levels within the target nucleus.

Inelastic scattering occurs above a threshold energy. This threshold energy is higher than the energy the first excited state of target nucleus (due to the laws of conservation) and it is given by following formula:

E_t = ((A+1)/A)* ε₁

where E_t is known as the inelastic threshold energy and ε₁ is the energy of the first excited state.

In general, the energy of the first excited state of nuclei decreases with increasing mass number. Therefore an inelastic scattering plays an important role in slowing down neutrons especially at high energies and by heavy nuclei. In calculations of a reactor inelastic scattering by light nuclei (such as oxygen with E_t = 6.42 MeV) can ordinarily be ignored. Moreover, the hydrogen nucleus does not have excited states, so only an elastic scattering reaction can occur in that case.

On the other hand an inelastic scattering by uranium nuclei (especially by ²³⁸U) must be included. ²³⁸U is a major component of nuclear fuel in commercial power reactors, therefore its scattering data are one of the most important data in the neutron transport calculations in the reactor core.

See also: Inelastic Scattering and Capture Cross-section Data of Major Actinides in the Fast Neutron Region, INDC(NDS)-0597. IAEA, 12/2012.

Key Characteristics of Inelastic Scattering

• During an inelastic scattering the neutron is absorbed and then re-emitted.

• The reaction occurs via compound nucleus.

• While momentum is conserved in an inelastic collision, kinetic energy of the “system” is not conserved.

• Some energy of the incident neutron is absorbed to the recoiling nucleus and the nucleus remains in the excited state.

• The nucleus gives up excitation energy by emitting one or more gamma rays.

• General notation: A(n, n’)A* or A(n, 2n’)B; Example: 14O(n, n’)14O*.

• Inelastic scattering is a threshold reaction and occurs above a threshold energy.

• Inelastic scattering cross section is relatively small for light nuclei.

• For hydrogen nucleus, inelastic scattering does not occur, because it does not have excited states.

• Inelastic scattering plays an important role in slowing down neutrons especially at high energies and by heavy nuclei (e.g. ²³⁸U).

• Inelastic scattering may be significant for heterogeneous reactors with highly enriched fuel (e.g. in fast neutron reactors).

Inelastic Scattering Cross-section

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

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Neutron Reactions

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Elastic vs. Inelastic Scattering of Neutrons

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.

Besides, in an inelastic scattering reaction between a neutron and a target nucleus some energy of the incident neutron is absorbed to the recoiling nucleus and the nucleus remains in the excited state. Thus while momentum is conserved in an inelastic collision, kinetic energy of the “system” is not conserved.

Key Characteristics of Elastic Scattering

See also: Neutron 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, σ_s = const, for all elements, that are of importance.

• At low energy, σ_s 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.

Key Characteristics of Inelastic Scattering

See also: Neutron Inelastic Scattering

• During an inelastic scattering the neutron is absorbed and then re-emitted.

• The reaction occurs via compound nucleus.

• While momentum is conserved in an inelastic collision, kinetic energy of the “system” is not conserved.

• Some energy of the incident neutron is absorbed to the recoiling nucleus and the nucleus remains in the excited state.

• The nucleus gives up excitation energy by emitting one or more gamma rays.

• General notation: A(n, n’)A* or A(n, 2n’)B; Example: 14O(n, n’)14O*.

• Inelastic scattering is a threshold reaction and occurs above a threshold energy.

• Inelastic scattering cross section is relatively small for light nuclei.

• For hydrogen nucleus, inelastic scattering does not occur, because it does not have excited states.

• Inelastic scattering plays an important role in slowing down neutrons especially at high energies and by heavy nuclei (e.g. ²³⁸U).

• Inelastic scattering may be significant for heterogeneous reactors with highly enriched fuel (e.g. in fast neutron reactors).

See also:

See also:

Elastic Scattering

See also: