Chaplin - Nuclear Interactions

NUCLEAR ENERGY MATERIALS AND REACTORS – Nuclear Interactions - R.A. Chaplin

NUCLEAR INTERACTIONS R.A. Chaplin University of New Brunswick, Canada Keywords: Nuclear Interactions, Nuclear Cross Sections, Neutron Energies, Fission and Fusion Contents

S S L R O E E T P A O H C C S E E N L U P M A S

1. Neutron Interactions 2. Nuclear Cross Sections 3. Neutron Scattering and Capture 4. Neutron Moderation 5. Fission and Fusion Glossary Bibliography Biographical Sketch To cite this chapter Summary

When a heavy element, such as uranium, fissions into two mid range elements, binding energy is released. released. Furthermore since the neutron to proton ratio is about about 1.5 for the heaviest elements but in the range of 1.2 to 1.3 for mid range elements there is a surplus of neutrons after a fissioning process. Some heavy elements elements fission spontaneously at a very very slow rate due to inherent instability. However fissioning can be induced induced by adding energy to the nucleus of some elements. This can be done by allowing the nucleus nucleus to capture a free neutron which then adds sufficient binding energy, as it combines with the nucleus, to cause the nucleus to become highly unstable and to split into two parts with additional free neutrons. These components components fly apart with high kinetic kinetic energy which is subsequently degraded to produce heat. Free neutrons interact with the t he nuclei of other materials in various ways, the m most ost common common being absorption and scattering. Scattering results in the transfer of some energy energy and the neutron continues to move through the medium but at a lower energy and hence lower velocity. velocit y. Neutrons being uncharged do not interact with the electron cloud surrounding the nucleus and, since the nucleus occupies such a tiny space within the atom, the probability of interaction is quite low. This probability is not not necessarily related to the size of the nucleus but is measured as as a cross section in units of area. The cross sections of different nuclei vary widely and may be greater or smaller than the projected area of the nucleus itself. To maintain an ongoing chain chai n reaction of nuclear fissions to release energy at least one free neutron from a previous fission must go on to induce fission in another fissile element element such as uranium. The probability of this occurring can be enhanced enhanced by reducing the velocity velocity of the neutron so that, when encountering a fissile nucleus, it spends more time in the

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immediate vicinity of the nucleus. Thus surplus neutrons produced produced by fission are made to pass through a suitable medium, known as a moderator, where their velocity is reduced by multiple scattering collisions with moderator nuclei. They then re-enter the fissile fuel to produce at at least one one further fission. Some neutrons neutrons are absorbed by various nuclei. This process is carefully balanced to ensure the steady and continuous release release of energy. Since only about 200 MeV or 32 pJ is released by each fission, many parallel processes as described above must occur simultaneously. 1. Neutron Interactions 1.1. Neutron Production


Neutrons can be created by the integration of an electron and a proton. proton. Furthermore a free neutron will in time disintegrate into a proton and an electron. Neutrons interact with the nuclei of atoms in various ways and may also be produced by the nuclei of certain atoms. The most common source of neutrons is the fissioning process where a heavy nucleus splits into two lighter nuclei. This fissioning of nuclei and the subsequent subsequent interaction of the resultant neutrons with other nuclei are the fundamental processes governing the production of power from nuclear energy. energy. Knowledge of these processes processes is important in the study of nuclear engineering. A heavy nucleus such as Uranium-235 will occasionally fission spontaneously into two lighter nuclei. A heavy nucleus such as this has about one and a half as many neutrons as protons in the nucleus. A mid-range nucleus however has only about one and a third as many neutrons as protons in its nucleus. Thus, when a heavy nucleus fissions into two lighter nuclei, not as many neutrons are required to maintain a stable configuration in the nucleus and some some neutrons are rejected immediately immediately the fission occurs. occurs. Generally two to three neutrons are emitted during the fission process. In a nuclear reactor, fissile nuclei such as Uranium-235 and Plutonium-239 are induced to fission by having having their nuclei excited excited beyond the level of of stability. This is done done by subjecting them to the influence of free neutrons. Free neutrons interact with various nuclei in different ways causing a range of different reactions of which fission is just one. Most interactions involve scattering (non-absorption) or capture (absorption) of the neutrons and a transfer of energy. energy. These reactions reactions are important important in maintaining and controlling the fission reactions in nuclear reactors. 1.2. Elastic Scattering (Elastic Collision)

Elastic scattering occurs when a neutron strikes a nucleus and rebounds elastically. In such a collision kinetic energy is transmitted elastically in accordance with the basic laws of motion. If the nucleus is of the same mass mass as the neutron then a large amount of kinetic energy is transferred to the nucleus. If the nucleus is of a much greater mass than the neutron then most most of the kinetic energy is retained by the neutron as it rebounds. rebounds. The amount of kinetic energy transferred also depends upon the angle of impact and hence the direction of motion of the neutron and nucleus after the impact.



1.3. Inelastic Scattering (Inelastic Collision) Inelastic scattering occurs when a neutron strikes and and enters a nucleus. nucleus. The nucleus is excited into an unstable condition and a neutron is immediately emitted but with a lower energy than that of the entering neutron. neutr on. The surplus energy is transferred to the nucleus as kinetic energy and and excitation energy. energy. The excited nucleus nucleus subsequently subsequently returns to the ground state by the emission of a γ -ray. Such collisions are inelastic since since all the initial

kinetic energy does not reappear as kinetic energy. energy. Some is absorbed by the nucleus nucleus and subsequently emitted in a different form ( γ -ray). The emitted neutron neutron may or may not be the one that initially struck the nucleus. nucleus. In simplistic terms the neutron neutron can be considered considered simply to be bouncing off an energy absorbing nucleus.


1.4. Radiative Capture

Radiative capture can be considered to be similar to the initial process leading to inelastic scattering. A neutron strikes and enters a nucleus. The nucleus is excited but the level level of excitation is insufficient to eject a neutron. Instead all the energy energy is transferred to the nucleus as kinetic energy and excitation energy. The excited nucleus subsequently subsequently returns returns to the ground state by the emission of a γ -ray. The incoming incoming neutron remains in the

nucleus and the nuclide increases increases its number of neutrons by one. one. This is a very common common type of reaction. It leads to the creation of heavier heavier isotopes of of the original element. element. Many of these may be radioactive and decay over time in different ways. 1.5. Nuclear Transmutation (Charged Particle Reaction)

Nuclear transmutation tran smutation is similar to radiative capture capture and inelastic scattering. A neutron strikes strike s and enters a nucleus. The nucleus is excited excited into an unstable condition but a particle other than a neutron is emitted. The emitted particles are either protons or α -particles. This leaves the nucleus still in an excited state and it subsequently returns to the ground state by the emission emission of a γ -ray. In this process the total number of protons in the nucleus

is reduced reduced by one for proton proton emission and by two for α -particle emission. The original element is thus changed or transmuted into a different element. 1.6. Neutron Producing Reaction

Neutron producing reactions occur when one or two additional neutrons neut rons are produced from a single neutron. As before a neutron neutron strikes and and enters a nucleus. nucleus. The nucleus is excited into an unstable condition as with inelastic scattering but two or three neutrons instead of only one neutron are emitted. emitted. The still excited nucleus subsequently subsequently returns to its ground state by the emission of a γ -ray. This is an uncommon reaction occurring in only a few

isotopes. 1.7. Fission

Although spontaneous fission f ission occasionally occasionall y occurs, fission is generally induced by neutrons. A neutron strikes and enters a heavy nucleus. The nucleus is excited into an unstable unstable condition as with most of the foregoing interactions. In this unstable condition the nucleus nucleus



splits into two new mid-range nuclei usually of unequal mass. Since these new nuclei do not need as many neutrons for stability some neutrons are emitted immediately. The surplus binding energy drives the new nuclei (fission fragments) and neutrons away from one another with high velocity. The new nuclei subsequently lose their kinetic energy by ionizing reactions with the surrounding nuclei through which they pass and return to their ground states by emission of γ -rays. They are invariably still unstable with too many neutrons and subsequently decay usually by β -particle and γ -ray emission. The high energy neutrons lose energy by scattering collisions with nuclei of the surrounding medium and are subsequently generally captured by other nuclei to produce one of the reactions described in this section.


1.8. Neutron Flux

Neutrons created by fission pass freely through solid material since atoms consist mainly of empty space. They have no charge and so are not affected by the charged electron cloud surrounding the nucleus. Furthermore the nucleus is so small compared with the size of the atom that the chance of the neutron colliding with it is extremely small. In a uniform material the neutrons travel randomly in all directions and some measure of their number or influence is required. A convenient parameter is neutron flux. Neutron flux ϕ is defined as the number of neutrons per unit volume n multiplied by their velocity v . ϕ = nv

(1)

Neutron flux so defined has units of number per unit area per unit time. This can be considered as the number of neutrons passing through a particular cross sectional area in any direction per second. If the neutrons travel in a parallel beam the area through which the neutrons pass may be considered to be at right angles to the beam and the given area will then be equal to the cross sectional area of the beam. This is the case in irradiation experiments where a beam of neutrons is directed out of a nuclear reactor through special ports which trap neutrons moving in other directions. Such a beam is known as a collimated beam. Within the reactor the neutrons travel in all directions and the neutrons will pass through a given area in all directions and from both sides. This area is more difficult to define hence the definition of neutron flux as number multiplied by velocity. 1.9. Neutron Energy

During the fission process, in which a heavy nucleus splits into two fission fragments and some residual neutrons, some 200 MeV of binding energy is released. This appears as kinetic energy as the fragments and neutrons separate at high velocity. Most energy is carried by the fission fragments and is deposited as heat in the surrounding material as the fragments come to rest. The two or three residual or prompt neutrons carry away about 5 MeV as kinetic energy so on average a neutron produced by fission has an energy of about



-12

-27

2 MeV or 0.32 x 10 J. Considering that the mass of a neutron is 1.67495 x 10 kg its velocity can be calculated from the basic equation for kinetic energy E KE where m is mass and V velocity: EKE

= ½ mV 2

(2) 6

This gives an average velocity of about 20 x 10 m/s. This is the average based on an average energy of 2 MeV. The actual range of energies however can range from near zero 6 to about 8 MeV as shown in Figure 12 giving velocities anywhere up to about 55 x 10 m/s.


These high energy neutrons interact with the nuclei of the medium through which they pass. In the process some are captured but most are scattered by elastic or inelastic collisions with the nuclei. Such scattering collisions result in a transfer of energy from the neutrons to the nuclei until the neutrons reach an equilibrium condition with t he medium. In this condition the nuclei, being in a state of vibratory motion by virtue of their temperature, give as much energy to the neutrons as they receive from the neutrons. The neutrons are thus in thermal equilibrium with the medium and are said to be thermalized. Even though the medium may be at a uniform temperature, subsequent scattering collisions occurring in random directions relative to the motion of the nuclei, result in thermal neutrons having a range of energies above and below the thermalization energy as shown in Figure 1. This figure also shows the corresponding velocity distribution of the neutrons.

Figure 1. Energy and velocity distribution of thermalized neutrons



This is a Maxwellian distribution with the energy E given in terms of the Boltzmann constant k and temperature T as well as in electron-volts while the velocity V is given in meters per second. The Boltzmann constant is as follows: k = 13.8 × 10 −24 J/K 6 k = 86.2 × 10 − eV/K

The average energy E ave and the most probable energy E mp of the neutrons are given by: Eave

= (3/2) kT = ½ kT


Emp

In neutron studies however the most probable velocity V mp is considered. This is given by: Vmp

= [2kT / m]1/ 2

Hence the corresponding neutron energy E is given by: E = kT

(3)

All thermal neutrons in a system are considered to have this velocity which is then given by: ½ mV 2

= kT

At an ambient temperature of 20°C or 293K this velocity is 2200 m/s and the corresponding energy is 0.025 eV. These are the values traditionally used in neutron scattering calculations involving thermal neutrons. 2. Nuclear Cross Sections

2.1. Microscopic Cross Sections

A solid material may be considered as being made up of tiny nuclei suspended in empty space (the electron cloud of negligible mass). Each nucleus has an imaginary projected area which may interfere with the passage of a neutron. A neutron entering the solid will see these projected areas scattered everywhere but they are so small and so far apart (as seen by the neutron) that the chances of hitting one is practically nil. Eventually a neutron may hit a nucleus and will then interact with it in any of a number of possible ways. Other neutrons will simply pass it without any interaction. It is interesting to note that the imaginary projected area or target area of a nucleus, as shown in Figure 2, may be larger or smaller than the actual projected area as determined from the physical size of the nucleus. It may be larger because the nucleus has a sphere of influence surrounding it and any neutron passing within this sphere of influence may be



attracted to interact with it. It may be smaller because some nuclei may allow neutrons to pass right through themselves without any interaction taking place. The imaginary projected area may thus be considered as being related to the probability of a reaction occurring-the larger the area, the greater the probability of interaction. It is also interesting to note that for different reactions with the nucleus there are different degrees of probability of interaction and therefore effectively different imaginary projected areas. Uranium-238 for example has a larger imaginary target area for elastic scattering than for radiative capture illustrating that there is a greater probability of elastic scattering occurring. It is convenient for illustrative purposes to draw a pie diagram with the total area signifying the probability of all interactions occurring and each slice representing the probability of individual interactions taking place.

S S L R O E E T P A O H C C S E E N L U P M A S Figure 2. Target areas of nuclei for different reactions

These imaginary projected areas are known as nuclear cross sections and indicate the probability of any interaction occurring. The cross sections of the nuclei of individual atoms are measured in square centimeters, square meters or barns where: 1 barn = 1 x 10 -24 cm2 -28 2 1 barn = 1 x 10 m



If the actual projected area of a nucleus is calculated it is found that for mid-range elements with an atomic mass number of about 90 this area is equal to 1 barn. Lighter elements have smaller projected areas and heavier elements larger projected areas. A cross section of 1 barn indicates immediately that the imaginary target area i s roughly equal to the actual projected area of the nucleus. This allows cross sections to be visualized. A cross section of several hundred barn indicates that the nucleus has a large sphere of influence around it while a cross section very much smaller than a barn indicates that the nucleus allows neutrons to pass through it with practically no chance of an interaction occurring.


The cross section of Uranium-235 for example is 687 barn whereas its physical cross section is 1.87 barn. Therefore, with an effective area for neutron interaction so much bigger than that its actual area, it is "as big as a barn" from a nuclear point of view and hence the term "barn". The term "barn" was proposed in 1942 by physicists M.G. Holloway and C.P. Baker as a result of such humorous association of ideas. There are different types of cross sections, in fact there is one type of cross section for each type of neutron interaction with the nucleus except for the relatively rare nuclear producing and nuclear transmutation reactions. These different cross sections may be added to give a total cross section or probability of reaction as shown in Figure 3. The magnitude of each slice of the "pie" represents the probability of that type of reaction. The nomenclature for different cross sections is given below with the different types of interactions: σ s = Elastic scattering cross section

σ i = Inelastic scattering cross section σ n,γ = Radiative capture cross section

σ a = Absorption cross section σ f = Fission cross section

Values for these are tabulated but are often combined into two main types of interactions: σ s = Scattering cross section ( σ s and σ i )

σ a = Absorption cross section ( σ n,γ and σ f )

When these are combined they are added together so that the scattering cross-section includes both elastic and inelastic scattering and the absorption cross-section includes both radiative capture and fission.



S S L R O E E T P A O H C C S E E N L U P M A S Figure 3. Cross sections of U-235 for various nuclear reactions

For a particular isotope all the individual microscopic cross sections can be added to give the total microscopic cross section. σ total =σ s +σ i +σ a + . . . . .

Generally however any particular calculation requires the application of a specific microscopic cross section only. 2.2. Macroscopic Cross Sections

The macroscopic cross section Σ is the cross section density in a material. It is defined as the number of nuclei per unit volume N multiplied by the microscopic cross section σ . -1 -1 The units are the inverse of length (cm or m )

Σ = N σ

(4)

This provides a basis for the comparison of different materials. A dense material with nuclei of small cross section would be seen by neutrons to be effectively t he same as a rare material with nuclei of large cross section. For a single isotope the macroscopic cross section can be determined from the above equation. This gives the effective cross section density in a pure material and indicates the probability of a neutron interaction within that material. If there is a homogeneous mixture of different isotopes the cross section density can be calculated separately for each and then added to give the total macroscopic cross section.



Σ

= N aσ a + N bσ b + N cσ c + . . . . .

Note that N is the number of nuclei or atoms of each element per unit volume in the material. 2.3. Number of Nuclei

The number of nuclei per unit volume N in a sample is given by the following equation where N A is Avogadro's number, M the atomic weight, and ρ the density. N

S S L R O E E T P A O H C C S E E N L U P M A S = ( N A / M ) ρ

(5)

For a material such as water H 2O or uranium dioxide UO 2 where the elements are bound together as molecules the number of nuclei of each element needs to be properly accounted for. For example when calculating the number of nuclei of each element in H2O or UO2 using Avogadro's Number N A and the molecular weight M m (approximately 18 for H2O and 270 for UO 2) the number of molecules is obtained as follows: N

= ( N A / M m ) ρ

In the case of H 2O the number of oxygen atoms is equal to the number of molecules while the number of hydrogen atoms is double the number of molecules. In many cases it is required only to know the number of nuclei of a specific isotope in a mixture of elements. In the case of natural uranium with a mass fraction γ of U-235 the number of U-235 nuclei will be: N 235

= γ 235 ( N A / M 235 ) ρ 235

Two assumptions may be made in evaluating the number of nuclei without excessive error:

• The molecular or atomic weight may be taken as an integer corresponding with the •

atomic mass number of each constituent. Mass ratio (enrichment) and volume ratios (isotopic abundance) may be considered equal for any single element.

In the case of uranium dioxide UO 2 enriched to 3% in U-235 and having a density ρ of 3 10.5 g/cm the number of U-235 nuclei per unit volume is: N 235

= γ 235 ( N A / M 235 ) ρ 235 =γ 235 ( N A / M 235 ) f ρ fuel

f

= [ γ M 235 + (1- γ )M 238 ] /[γ M 235 + (1- γ ) M 238 + M oxygen ] = 0.881



N 235

= 0.03 (0.6022 ×10 24 / 235) 0.881 ×10.5 = 0.711 ×10 21 nuclei/cm 3 = 0.711 ×1027 nuclei/m 3

2.4. Reaction Rate

Since the macroscopic cross-section Σ is effectively the material parameter seen by the neutrons and since neutron flux ϕ is effectively the number of neutrons passing through a given place per unit time it follows then that the reaction rate R between neutrons and nuclei is given by:


R = Σϕ

(6)

This may also be written as: R = Nσ nv

(7)

This is perfectly logical since the reaction rate R would likely be proportional to the number of nuclei N , the cross-section σ , the number of neutrons n and the velocity of the neutrons v -the greater the number of nuclei and neutrons, the greater the chances of a reaction. The bigger the effective area (cross section) the more likely a nucleus will intercept a neutron. The higher the velocity of a neutron the sooner it will meet a nucleus. 2.5. Summary

The following relationships with units summarize the key factors given above. 2.5.1. Macroscopic cross-section

N = nuclei per unit volume σ = microscopic cross-section Σ = N σ

(nuclei/m3) 2 (m ) -1 (m )

2.5.2. Neutron Flux

n = neutrons per unit volume v = neutron velocity ϕ = nv

3

(neutrons/m ) (m/s) 2 (neutrons/m s)

2.5.3. Reaction Rate ϕ = neutron flux

Σ = macroscopic cross-section R = ϕ Σ


2

(neutrons/m s) (m -1) 3 (reactions/m s)


3. Neutron Scattering and Capture 3.1. Neutron Attenuation

When a beam of neutrons impinges upon a solid body the neutrons interact with nuclei within the body. Those not interacting continue through the body. As the beam progresses through the body more and more interactions occur and less and less neutrons continue on through the material. The beam of neutrons diminishes in intensity and is attenuated by the material as shown in Figure 4. The decrease in intensity dI over any section of material is proportional to the neutron beam intensity I , microscopic cross-section of the material σ , number density of nuclei N and the thickness of the material dx : dI

S S L R O E E T P A O H C C S E E N L U P M A S = − Iσ Ndx

If the macroscopic cross section Σ is used this becomes: dI

= − I Σ dx

The solution to this differential equation is: I

= I 0 e − Σ x

(8)

This is the equation for the attenuation of a neutron beam. The attenuation of a γ -ray beam is similarly: I

= I 0e− μ x

(9)

Here μ is the attenuation coefficient of the γ -ray beam.

Figure 4. Neutron attenuation in a material



3.2. Mean Free Path

It is convenient to express the attenuation of neutrons in terms of the average distance traveled by a neutron before interacting with a nucleus. This is known as the neutron mean free path λ . If the value for intensity I from the solution of the differential equation is substituted into the differential equation the following is obtained. dI

= − I 0e -Σ x Σ dx

S S ∫ ∫ L R O E E ∫ T P A O H C C S E E N L U P M A S

Neutrons in this beam have traveled a distance x without interacting with any nucleus. For an infinite slab the total distance traveled by all neutrons is: x =∞

x = 0

x dI

= I 0 Σ

∞

0

x e−Σ x dx

The mean free path λ is this total interaction divided by the original beam intensity λ = I 0 Σ

λ

∞

0

xe

−Σ x

dx / I 0

= 1/ Σ

(10)

The neutron mean free path λ may also be deduced from the probability of an interaction and the distance traveled before that interaction as illustrated in Figure 5.

Figure 5. Neutron mean free path



The reaction rate R is equal to the macroscopic cross section Σ multiplied by the neutron flux ϕ . R = Σϕ

R = Σ nv The reaction rate R can also be written in terms of the number of neutrons n multiplied by their velocity v and divided by their mean free path λ . R = nv / λ


This in effect states that more reactions will occur when the velocity is higher and the mean free path lower. If these two equations for reaction rate are combined then the following is obtained:

Σ nv = nv / λ λ

= 1/ Σ

(11)

Thus the mean free path λ is the inverse of the macroscopic cross-section Σ . 3.3. Scattering Characteristics

It was seen previously that, with elastic scattering, the neutron rebounded from a nucleus with kinetic energy conserved and no excitation of the nucleus. Furthermore, with inelastic scattering, the neutron interacted with the nucleus leaving it in an excited state. Both of these scattering effects may occur in a single nuclide and it is found that the probability of these reactions is, to a large degree, dependent upon the energy of the incoming neutron. At very low energies, the neutron does not excite the nucleus and is scattered as if influenced by the physical size of the nucleus which is given in terms of atomic mass number A by the following: −

r = 1.25 × 10 15 A1/ 3 (m)

(12)

This gives the physical cross sectional areas for most nuclei of about 1 barn. The apparent area of the nucleus for elastic scattering is the neutron scattering cross section σ s which generally ranges from about 4 barn to 12 barn for most elements. This discrepancy indicates that the neutron itself has a certain physical size and that the nucleons of certain elements are not necessary closely packed in the nucleus. This scattering at low neutron energy is called potential scattering and is constant over a range of low neutron energies. At intermediate energies some neutrons have an energy that raises the nucleus to a discrete excitation level. Under these conditions absorption and subsequent emission of a neutron



occurs more easily. Since the nucleus is left in an excited state the emitted neutron is at a lower energy. This results in inelastic scattering. If there is no match in vibrational characteristics of the neutron and the nucleus, absorption does not occur easily. However, if there is a match between the vibrational characteristics, the neutron is readily absorbed resulting in a lesser chance of scattering. This results in widely varying scattering probabilities over a certain range of neutron energies. This is called the resonance region. At very high energies there is no longer a match in vibrational characteristics and the probability of scattering falls with increasing energy as those neutrons passing close to the nucleus are less affected by it. This is known as the smooth region.


These regions and other characteristics are shown in Figure 7. 3.4. Absorption Characteristics

It was seen previously that, with both inelastic scattering and radiative capture, the neutron interacted with the nucleus leaving it in an excited state. Both of these interactions may occur in a single nuclide and it is found that the probability of these reactions is to a large degree dependent upon the energy of the incoming neutron. For many nuclides there is a threshold neutron energy above which inelastic scattering occurs and below which radiative capture occurs. This is due to the fact that the neutron brings with it a certain amount of energy which is transferred to the nucleus when it enters the nucleus. If the neutron energy is sufficient to raise the energy of the nucleus above the threshold value then the excited nucleus can emit a neutron along with a γ -ray. If the energy of the excited nucleus remains below the threshold value no neutron wil l appear and only a γ -ray will be emitted. The threshold energy corresponds with the binding energy of the additional neutron while the γ -ray corresponds with the amount of energy remaining above the ground state of the nucleus. High velocity (high energy) neutrons are thus likely to be elastically scattered while low velocity (low energy) neutrons are likely to suffer radiative capture. 3.5. Radiative Capture Model

From the above it is evident that radiative capture is likely to occur with neutrons below the threshold energy value, that is, with lower velocity neutrons. As the velocity is decreased further it is found, for many nuclides, that the probability of radiative capture increases. This probability is in fact inversely proportional to the velocity (square root of energy). This can be visualized by imagining that the nucleus has a sphere of infl uence around it as illustrated in Figure 6. A neutron passing through this sphere of influence will spend a certain period of time within that sphere of influence. For a given path, the higher its velocity the shorter the time spent within the sphere of influence. If the probability of capture is proportional to the time spent within the sphere of influence, then the probability of capture (absorption crosssection σ a ) will be inversely proportional to velocity v .



σ a

∝ 1/ v

(13)

S S L R O E E T P A O H C C S E E N L U P M A S Figure 6. Interaction probability with respect to neutron velocity.

3.6. Cross Sections

The above may be summarized and illustrated by plotting on a composite diagram as in Figure 7.

Figure 7. Variation of typical cross sections with neutron energy The elastic scattering cross-section σ s is constant in the low energy potential region,



fluctuates in the resonance region and falls slowly with increasing energy in the smooth region. The inelastic cross-section σ i is only apparent above a certain threshold energy. The radiative capture cross-section σ γ is inversely proportional to velocity in the 1/ v region, fluctuates in the resonance region and drops to a low value or disappears at high energies. The total cross-section σ t is a summation of all the individual cross-sections including fission. Note that both the cross-section and neutron energy are plotted on logarithmic scales. 4. Neutron Moderation


4.1. Neutron Energy Changes

When neutrons interact with nuclei by elastic or inelastic scattering their energy is degraded. Generally neutrons produced from fission have an energy of about 2 MeV while neutrons after thermalization have an energy of about 0.025 eV. The number of interactions to bring about this degradation depends upon several factors including the initial energy of the neutrons and the type of scattering (elastic or inelastic). Inelastic scattering generally requires that the incoming (captured) neutron have sufficient energy to excite the nucleus to a level that will result in the ejection of a neutron. Hence inelastic scattering occurs only at high neutron energies and the resulting neutrons will have very much lower energies. Elastic scattering, on the other hand, occurs at all neutron energies and may not necessarily degrade the neutron energy very much. Hence, any neutrons produced from fission that suffer inelastic collisions initiall y will subsequently be subject to a series of elastic collisions. Those that suffer elastic collisions initially will also likely continue to degrade their energy by elastic collisions. Hence most collisions are elastic. 4.2. Logarithmic Mean Energy Decrement

When neutrons interact with nuclei in elastic scattering collisions they lose energy. The amount of energy lost depends upon the mass of the nucleus and the angle of incidence of the neutron. More energy is lost when a neutron strikes a light nucleus than when it strikes a heavy one. Also more energy is lost in a head-on collision than in a glancing collision. The minimum energy E min after one collision is: Emin

= α E 0

where: α = [( A -1) /( A + 1)]2 Here A is the atomic mass number of the nucleus and it is evident that the higher this number the closer α will be to unity and the smaller the maximum loss in energy. Considering the results of various angles of incidence of the incoming neutron, it is found that the average energy E ave after one collision is:



Eave

= (1/ 2)(1 + α ) E 0

The average energy loss after one collision is thus given by:

Δ E = E0 - E ave Δ E = (1 / 2)(1 -α ) E 0 where:


α = [( A -1) /( A + 1)]2

Since the loss or change in energy depends upon the incoming neutron energy and since this is lower in each subsequent collision in an exponential manner, it is convenient to express the change in energy in logarithmic terms. Furthermore, since the change in energy is different for each collision, the average of the logarithmic values of the initial energy E 0 and resultant energy E is used. The logarithmic mean energy decrement ξ is the average of the difference of these logarithmic energy values: ξ = [ln E0 - ln E ]average

ξ = [- ln( E / E 0 )]average

The value of the logarithmic mean energy decrement for any isotope of atomic mass number A is given as follows: ξ = 1 + [( A -1)2 / 2 A]ln[( A -1) /( A + 1)]

(14)

An approximate value for the logarithmic mean energy decrement is given by the following empirical equation: ξ = 2 /[ A + (2 / 3)]

(15)

This equation has negligible error for all but the very lowest atomic mass numbers hence it is widely adopted in place of the theoretical equation. The number of elastic collisions N required for the neutron energy to drop from an initial energy E i to a final energy E f is then given by: N ξ = ln( Ei / E f )

(16)

The value of N for a high energy neutron from fission (2 MeV) to become thermalized at ambient conditions (0.025 eV) is 18 for Hydrogen, 43 for Helium and 115 for Carbon. Lighter elements are efficient at reducing neutron energy because they are light and absorb a lot of energy when struck by a nucleus. Collision parameters for a few other common



materials are given in Table 1.

Nucleus

Mass Number

Mass Number Ratio

A

α

Mean Logarithmic Energy Decrement ξ

Number of Collisions to Thermalize

N

S S L R O E E T P A O H C C S E E N L U P M A S Hydrogen H2O Deuterium D2O Helium Beryllium BeO Carbon Oxygen Sodium Iron Uranium

1

0

2

0.111

4 9

0.360 0.640

12 16 23 56 238

0.716 0.779 0.840 0.931 0.983

1.000 0.920* 0.725 0.509* 0.425 0.209 0.174* 0.158 0.120 0.0825 0.0357 0.00838

18 20 25 36 43 83 105 115 152 221 510 2172

α = [( A -1) /( A + 1)]2 * An appropriate average value. Data obtained from Lamarsh and Baratta, Introduction to Nuclear Engineering, Prentice Hall, 2001 Table 1. Scattering collision parameters of some common materials

4.3. Definitions

It has already been shown that the probability of radiative capture of neutrons by many nuclides increases as the energy (and hence velocity) of the neutrons is decreased. The probability of capture (absorption) is inversely proportional to the neutron velocity over a range of neutron energies. The fissioning of certain fissile materials such as U-235 is the result of the absorption of a neutron so it follows that the probability of fission in these materials will also increase with a reduction in neutron velocity. To enhance the fission process therefore it is advantageous to reduce the energy of the neutrons to a l ower value by passing them through a suitable material or moderator . This material however should not absorb neutrons (by radiative capture) too strongly as this would reduce the number of neutrons available for causing fission. Materials suitable for slowing down or moderating neutrons without excessive absorption of them may be assessed by using the following equations and definitions:



4.3.1. Mean Logarithmic Energy Decrement ξ N ξ = ln( Ei / E f )

N = number of collisions E i = initial energy (2 MeV after fission) E f = final energy (0.025 eV when thermalized)

4.3.2. Macroscopic Scattering Cross Section Σ s

Σ s

= N σ s

(m-1)

S S L R O E E T P A O H C C S E E N L U P M A S 3

N = nuclei per unit volume σ s = microscopic scattering cross section

(nuclei/m ) 2 (m )

4.3.3. Slowing Down Power

-1

Slowing down power = ξ Σ s

(m )

4.3.4. Moderating Ratio

Moderating ratio = ξ Σ s / Σ a

Table 2 gives the above parameters for some materials suitable as moderators.

Moderator

Mean Logarithmic Energy Decrement

Macroscopic Scattering Cross Section(a)

Slowing Down Power

Macroscopic Absorption Cross Section

Moderating Ratio

Σ s

ξ Σ s

Σ a

ξ Σ s / Σ a

ξ

(b)

He Be C(c) BeO H2O D2O D2O D2O

0.425 0.206 0.158 0.174 0.927 0.510 0.510 0.510

-1

-1

(cm )

2 x 10 0.74 0.38 0.69 1.47 0.35 0.35 0.35

(cm )

-6

-6

9 x 10 0.15 0.06 0.12 1.36 0.18 0.18 0.18

very small -3 1.17 x 10 0.38 x 10-3 -3 0.68 x 10 -3 22 x 10 -4(d) 0.33 x 10 -4(e) 0.88 x 10 -4(f) 2.53 x 10

large 130 160 180 60 (d) 5500 (e) 2047 (f) 712

Data obtained from NB Power Nuclear Training Course 22007 (a) (b) (c) (d)

Average value for epithermal neutrons (energies between 1 eV and 1000 eV) At standard temperature and pressure Reactor-grade graphite 100% pure D2O



(e) (f)

Reactor-grade D2O (99.75 pure) 99% pure D2O Table 2. Slowing down and moderating properties of moderators

5. Fission and Fusion 5.1. Energy Release

It has been shown that both the fusion of light elements and the fission of heavy elements will produce energy. This is due to the fact that the binding energy per nucleon is less for light and heavy elements than for mid-range elements. The amount of energy released can be calculated from the mass defect if the final products are known. For fusion a range of different reactions is possible as hydrogen fuses into helium. For fission only one reaction is possible for any particular fuel but a range of fission products is produced. On average about 200 MeV is produced from a fission reaction. Typical fusion and fission reactions are shown in Figure 8.

S S L R O E E T P A O H C C S E E N L U P M A S Figure 8. Typical fusion and fission reactions



5.2. Fission

During the fission process a number of neutrons is released since otherwise the resulting fission products would have too many neutrons and be too far off the stability range. Even so they have an excess of neutrons and decay towards a more stable condition. These neutrons are free to enter other fissile nuclei and so cause further fissions to maintain a chain reaction. If the same number of neutrons continues into the next generation the chain reaction is stable. To achieve this some neutrons must be captured without producing fission since, for every neutron causing fission, on average two or three are produced. Figure 9 shows the number of neutrons emitted from the fission of U-235 for different fission reactions (different fission products).

S S L R O E E T P A O H C C S E E N L U P M A S Figure 9. Prompt neutron emission from U-235 per 100 fissions.

Fission occurs spontaneously in some heavy nuclides but is rare. This contributes to the gradual decay of the nuclide and creates a few free neutrons within the fuel. This is an important factor when loading new fuel into a reactor as the resulting low level nuclear chain reactions could inadvertently grow out of control. Fission induced by neutrons is due to the fact that the incoming neutron adds sufficient energy to the nucleus to raise its energy level enough for it to become unstable. Nuclides that fission when unstable are known as fissile materials. There are four such fissile isotopes: Uranium-233 Uranium-235 Plutonium-239 Plutonium-241 Thermal neutron interaction parameters for these fissile materials are given in Table 3.



Nuclide

Microscopic Absorption Cross Section

Microscopic Capture Cross Section

σ a

U-233 U-235 Pu-239 Pu-241 Natural U

σ γ

578.8 680.8 1011.3 1377 7.59

47.7 98.6 268.8 368 3.40

Microscopic Fission Cross Section

Capture Fission Ratio

σ f

α

531.1 582.2 742.5 1009 4.19

0.090 0.169 0.362 0.365 0.811

Neutrons Emitted per Absorption

η

2.287 2.068 2.108 2.145 2.24

Neutrons Emitted per Fission

ν

2.492 2.418 2.871 2.927 3.06


σ a =σ γ +σ f

α

= σ γ /σ f = (σ a - σ f ) / σ f

η = ν (σ f /σ a )

Data obtained from Lamarsh and Baratta, Introduction to Nuclear Engineering, Prentice Hall, 2001 Table 3. Thermal neutron (0.025 eV) data for fissile nuclides

A number of other nuclides will fission if the incident neutron has a high kinetic energy. This kinetic energy together with the binding energy can raise the energy level of the nucleus sufficiently for it to become unstable and to fission. Such nuclides are known as fissionable materials. Fissionable isotopes thus require energetic neutrons to cause fission and as such are nonfissile. 5.3. Fission Characteristics

Uranium-235 and Uranium-238 have scattering and absorption cross-sections similar to other materials. Refer to Figure 10. In U-235 absorption usually leads to fission and in the low neutron energy region the absorption cross-section is very high but decreases with increasing neutron energy since it is inversely proportional to the neutron velocity. There is then a resonance region where there are peaks with a high probability of absorption. At high energies there is a low probability of absorption and hence fission and the cross-section is low. In U-238 absorption does not lead to fission except at very high neutron energies. At low neutron energies there is a low probability of absorption and this is also inversely proportional to neutron velocity. In the resonance region however there are very high peaks of absorption. The absorption cross-section then falls again to low values in the high energy region. At very high energies absorption leads to fission.



S S L R O E E T P A O H C C S E E N L U P M A S Figure 10. Fission and absorption characteristics of uranium.

5.4. Fission Products

During fission two fission fragments usually of unequal mass are produced. These generally have atomic mass numbers of between 100 and 140 though a range of possibilities exists from an atomic mass number of about 70 to about 160 as shown in Figure 11. The amount of a particular fission product occurring is known as the fission yield . Fission yields vary for different fissile materials and for fission with higher energy neutrons. The fission yields of Plutonium-239, for example, show that somewhat more fission products of intermediate mass number are produced than is the case with Uranium235. For high energy neutrons the fission yield curve is much flatter still with even more fission products of intermediate mass being produced.



S S L R O E E T P A O H C C S E E N L U P M A S Figure 11. Fission yields for U-235 and Pu-239

5.5. Neutron Energy Spectrum

Neutrons produced at the time of fission are known as prompt neutrons. Some neutrons appear a short time later and these are known as delayed neutrons. The prompt neutrons are produced with a range of different energies. Most energy from fission appears as kinetic energy of the heavy fission products but some is carried away by the neutrons also as kinetic energy.

Figure 12. Prompt neutron energy distribution per 100 neutrons



The energy of prompt neutrons varies from about zero to about 8 MeV. If a sample of 100 prompt neutrons is analyzed, as in Figure 12, it is found that some 35 have an energy of about 1 MeV, the most probable energy, while the average energy is about 2 MeV. The results are usually plotted as a smooth curve of fraction emitted versus neutron energy. 5.6. Delayed Neutrons

Delayed neutrons are emitted from some fission products a short while after fission has occurred. Most fission products are unstable and decay towards a more stable state by emitting particles, usually β -particles to convert a neutron into a proton. Some however are sufficiently unstable to emit neutrons directly or subsequently (after β -particle emission) to reduce the neutron number. An example is the fission product Bromine-87. This decays to Krypton-87 by the emission of a β -particle and then to Krypton-86 by the emission of a neutron. The half-lives for these reactions are so short that the neutrons appear almost immediately but the time lag is sufficiently important to have a very marked influence on the control of nuclear reactors. The delay is long enough to be detected by control systems which can respond rapidly enough to changes in delayed neutron production. No control system can respond rapidly enough to changes in prompt neutron production.


Delayed neutrons come from some twenty fission products or delayed neutron precursors. Each precursor produces a neutron following decay or decays of different half-lives. For convenience these are grouped into six groups of precursors, as shown in Table 4, such that each group produces neutrons following decay according to a particular half-life. The first group has a half-life of 55 seconds while the last group has a half-life of only 0.2 second. Each group has a different yield of neutrons per fission with the fourth group producing nearly 40% while the first and last groups produce only about 3% and 4% respectively. Overall the total yield of delayed neutrons is only 0.65% of all neutrons produced in fission. This small amount however is very important in the control of nuclear reactors and the control system must be able to detect small enough changes in the neutron flux to maintain control on delayed neutrons.

Group

Half life t 1/2

(s)

1 2 3 4 5 6

Decay constant λ (s-1)

55.72 22.72 6.22 2.30 0.610 0.230


0.0124 0.0305 0.111 0.301 1.14 3.01

Relative Yield

(%)

3 22 20 39 12 4

Yield (neutrons per fission) ν

0.00052 0.00346 0.00310 0.00624 0.00182 0.00066

Fraction β

0.000215 0.001424 0.001274 0.002568 0.000748 0.000273


Total

100

0.01580

0.006502

Data obtained from Lamarsh and Baratta, Introduction to Nuclear Engineering, Prentice Hall, 2001 Table 4. Delayed neutron data for thermal fission of U-235 5.7. Fission Process Summary


For fission to occur the incoming neutron must add sufficient energy to the fissile nucleus to raise its energy above the critical value for fissioning. For the four fissile materials, thermal neutrons add sufficient binding energy to achieve this. Low energy neutrons interact more readily with Uranium-235 to cause fission than do high energy neutrons. Uranium-238 on the other hand will only undergo fission with high energy neutrons. The shape of the neutron-proton ratio curve results in additional neutrons being produced in fission. These additional neutrons allow for a chain reaction to be established with subsequent fissions with each new generation of neutrons. Neutrons produced in fission have a range of energies with an average of about 2 MeV. These high energy neutrons must be slowed down or moderated to reduce their energy so as to be able to interact easily with further Uranium-235 nuclei to start a new cycle. The energy produced in one fission process is about 200 MeV made up as tabulated in Table 5. By arranging for multiple parallel fissions in a continuing controlled chain reaction a steady production of energy can be achieved. 5.8. Charged Particles

Fission products are produced as a light fragment and a heavy fragment from each fission. The lighter fragments have kinetic energies of about 100 MeV while the heavier fragments have energies of about 70 MeV. This division of energies arises from the conservation of momentum as two initially stationary parts of different mass recoil from one another. These fission fragments leave behind some twenty electrons and immediately become positively charged. They lose kinetic energy rapidly in the surrounding material producing heat and ionization along their path. Their range is very short being in the order of 1.4 x -3 10 cm (0.014 mm) in uranium dioxide fuel (UO 2)

Energy Source


Recoverable energy (MeV)


Fission fragments lighter fragment (kinetic energy) heavier fragment (kinetic energy) Fission product decay β-rays γ-rays Prompt γ-rays Fission neutrons (kinetic energy) Capture γ-rays

100 68 8 6 7 5 6

S S L R O E E T P A O H C C S E E N L U P M A S Total

200

Table 5. Energy produced by the fission of U-235

Alpha particles emitted from heavy nuclides also interact with other atoms causing ionization. They travel in a short straight path with a range dependent upon their energy according to the following formulae where ρ is the density and M the molecular weight Rangeair

=

Rangemedium

function( Energy)

= Rangeair ( ρ air / ρ medium )( M medium / M air )1/ 2

Beta particles travel in a zigzag path and are not very penetrating since they are very light. Their range is also a function of their energy Rangemax

=

function( Energy) / ρ medium

Glossary Fission: Fusion:

Macroscopic Cross Section: Microscopic Cross Section: Neutron Flux:

Nuclear reaction involving the splitting of a heavy atom into two lighter atoms. Nuclear reaction involving the joining of two light atoms into a single heavier atom Cross section density in material for a specified nuclear reaction Effective cross section of nucleus for a specified nuclear reaction Flow of neutrons per unit area per unit time

Nomenclature A b

Atomic mass number -28 2 Barn (1 x 10 m )



E E ave

Energy Average energy

E f

Final energy

E i

Initial energy

E KE

Kinetic energy

E min

Minimum energy

E mp

Most probable energy

E 0

Energy before interaction

f

Mass fraction of fuel


I k m M M m n N N N A r R T v V V mp x

α γ λ μ ξ ρ σ σ a

Neutron beam intensity -24 Boltzmann constant (13.8 x 10 J/°K) Mass Atomic weight Molecular weight

Number of neutrons Number of collisions

Number of nuclei per unit volume Avogadro's number

Radius Reaction rate Absolute temperature Neutron velocity Particle velocity

Most probable particle velocity Material thickness Collision parameter Mass fraction of isotope Neutron mean free path

Attenuation coefficient for γ-rays Logarithmic energy decrement

Density Microscopic cross section Microscopic absorption cross section

σ s

Microscopic scattering cross section

σ f Σ ϕ

Microscopic fission cross section Macroscopic cross section Neutron flux

Bibliography

El-Wakil, M.M. (1993), Nuclear Heat Transport , The American Nuclear Society, Illinois, United States. [This text gives a clear and concise summary of nuclear and reactor physics before addressing the core material namely heat generation and heat transfer in fuel elements and coolants]. Foster, A.R., and Wright, R.L. (1983), Basic Nuclear Engineering, Prentice-Hall, Englewood Cliffs, New Jersey, United States. [This text covers the key aspects of nuclear reactors and associated technologies. It gives a good fundamental and mathematical basis for the theory and includes equation derivations and worked


Chaplin - Nuclear Interactions

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