Neutron diffusion Diffusion neutrons, the propagation of neutrons in matter, accompanied by multiple changes in the direction and speed of movement as a result of their collisions with atomic nuclei. The radiation of neutrons is similar to the radiation in gases and obeys the same laws (see. Diffusion). Fast neutrons, i.e. neutrons with energy many times greater than average energy thermal movement of particles of the environment, during D. they give up energy to the environment and slow down. In weakly absorbing media, neutrons arrive in thermal equilibrium with the medium (thermal neutrons). In an unbounded environment, a thermal neutron diffuses until it is absorbed by one of the atomic nuclei. The dispersion of thermal neutrons is characterized by the diffusion coefficient D and the mean square of the distance from the point of formation of a thermal neutron to the point of its absorption, equal to L 2
T = 6Dt, where t is the average lifetime of a thermal neutron in the medium.
To characterize the dynamics of fast neutrons, use the mean square of the distance L 2
B between the point of formation of a fast neutron (in a nuclear reaction, for example a fission reaction) and the point of its deceleration to thermal energy. In table L values are given for some media 2
T for thermal neutrons and L 2
B is for neutrons emitted during the fission of uranium.
L values 2
T and L 2
B for some substances
L 2
T, cm 2
L 2
B, cm 2
D2 0 ..... Beryllium Be .... Graphite C...
1.5 105
When D. in a limited environment, a neutron with high probability flies beyond its limits if the half-size (radius) of the system is small compared to the size
on the contrary, a neutron is likely to be absorbed in a medium if its radius is large compared to this value.
D. neutrons play a significant role in the work nuclear reactors. In this regard, the development of nuclear reactors was accompanied by intensive development of the theory of neutron radiation and methods for its experimental study.
Lit.: Bekurts K., Wirtz K., Neutron physics, trans. from English, M., 1968.
Big Soviet encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .
See what “neutron diffusion” is in other dictionaries:
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Let us give another example, giving an equation of the same type, but this time related to diffusion. In ch. 43 (issue 4) we examined the diffusion of ions in a homogeneous gas and the diffusion of one gas through another. Now let's take another example - neutron diffusion in a material like graphite. We chose graphite (a type of pure carbon) because carbon does not absorb slow neutrons. Neutrons travel freely in it. They travel in a straight line for an average of several centimeters before they are dispersed by the core and deviated to the side. So if we have a large piece of graphite several meters thick, then the neutrons that were initially in one place will move to other places. We will describe their average behavior, i.e. their average flow.
Let N(x, y,z)
ΔV
— number of neutrons in a volume element Δ
V
V point (x, y,z).
The movement of neutrons leads to the fact that some leave Δ
V,
and others fall into it. If there are more neutrons in one region than in the neighboring one, then from there more neutrons will move to the second region than vice versa; the result will be a flow. Repeating the proofs given in Chap. 43 (issue 4), the flow can be described by the flow vector J. Its component J x
is the resulting number of neutrons passing per unit time through a unit area perpendicular to the axis X. We'll get it then
where is the diffusion coefficient D
is given in terms of the average speed ν
And medium length free path l between collisions:
The speed at which neutrons pass through some surface element da,
equal to Jnda
(where n, as usual, is unit vector normals). Resulting stream from element of volume then equals (using the usual Gaussian proof) v J dV.
This flux would result in a decrease in the number of neutrons in ΔV unless neutrons are generated within ΔV (by some nuclear reaction). If the volume contains sources producing S
neutrons per unit time per unit volume, then the resulting flux from ΔV
will be equal to [ S—(∂Nl∂t)]
ΔV. Then we get
Combining (12.21) and (12.20), we obtain neutron diffusion equation
In the static case, when ∂N/
∂t =0,
we have equation (12.4) again! We can use our knowledge of electrostatics to solve neutron diffusion problems. Let's solve some problem. (You may be wondering: For what decide new task, if we have already solved all the problems in electrostatics? This time we can decide faster precisely because electrostatic tasks deyindeed already decided!)
Let there be a block of material in which neutrons (say, due to the fission of uranium) are produced uniformly in a spherical region of radius A(Fig. 12.7). We would like to know what is the neutron density everywhere? How uniform is the neutron density in the region where they are born? What is the ratio of the neutron density in the center to the neutron density on the surface of the production region? The answers are easy to find. Neutron density in the source S o
stands instead of the charge density ρ, so our problem is the same as the problem of a uniformly charged sphere. Find N- is the same as finding the potential φ. We have already found fields inside and outside a uniformly charged sphere; to obtain potential we can integrate them. Outside the sphere, the potential is equal to Q/4πε 0 r, where the total charge Q
is given by the ratio 4πа 3 ρ/3. Hence,
For internal points only charges contribute to the field Q(r),
located inside a sphere with radius r;Q(r)
=4πr 3 ρ/3, therefore,
The field increases linearly with r. Integrating E, we get φ:
At a radius distance and φ externally must match φ
internal, therefore the constant should be equal to ρa 2 /2ε 0. (We assume that the potential φ equal to zero on long distances from the source, and this for neutrons will correspond to the circulation N to zero.) Therefore,
Now we will immediately find the neutron density in our diffusion problem
Figure 12.7 shows the dependence N from r.
What is now the ratio of the density in the center to the density at the edge? In the center (r=0) it is proportional to For 2/2, and at the edge (r=a) proportional to 2a 2 /2; therefore the density ratio is 3/2. A uniform source does not produce a uniform neutron density. As you can see, our knowledge of electrostatics provides a good basis for studying the physics of nuclear reactors.
Diffusion plays a large role in many physical circumstances. The movement of ions through a liquid or electrons through a semiconductor obeys the same equation. We end up with the same equations over and over again.
To describe some important regularities of the diffusion process in reactors, we introduce and clarify some definitions. Let's define neutron flux density F, more often called “flux” as the number of neutrons crossing a spherical surface of 1 cm 2 per second, so the dimension of the flux will be 1/(cm 2 *s). We have already defined earlier microscopic section reactions of type “” of isotope “i” i as the area of interaction of one nucleus in barns. Now let's define the so-called macroscopic section reactions of type “” of isotope “i” as the cross section of the interaction of all nuclei “i” located in 1 cm 3 of substance i.
These two sections are interconnected by the so-called value. “nuclear density” or nuclear density , which characterizes the number of molecules (or nuclei) in 1 cm 3 of a substance.
= N A * /
N A – Avogadro’s number (equal to 0.6023*10 24 molecules/gmol);
- physical density of any complex substance (g/cm 3);
- molecular weight of the substance (g/gmol).
Then the relationship between the microscopic and macroscopic sections can be written as:
i = i * i
At the same time, the density of nuclei of a given isotope i will be related to the density of molecules through the number of atoms of a given type “i” in a molecule of a substance.
Finally, the only quantity that can actually be measured in nuclear reactions (including in dosimetric instruments, fission chambers, and realized inside the reactor) is reaction speed of a given type “” for the selected isotope “i” A i:
A i = Ф* i
This value is measured in units of the number of reactions in 1 cm 3 per second (1/(cm 3 *s)). Moreover, for the fission process there is an important connection between the number of fissions and the power released during this process: 1W = 3.3 * 10 10 div/s.
Thermal neutron diffusion. When the neutron energy decreases to energies characteristic of the energies of thermal motion of the atoms of the medium, the neutrons come into equilibrium with these atoms. Now, when colliding with an atom of the medium, a neutron can not only transfer part of its energy to it, but also receive a portion of energy. As a result, the neutron continues to move in the medium, but now its energy from collision to collision can not only decrease, but also increase, fluctuating around a certain average value, depending on the temperature of the medium. For room temperature, this average energy value is approximately 0.04 eV. A neutron that has come into thermal equilibrium with its medium is called thermal neutron, and the movement of thermal neutrons with a constant average speed is thermal neutron diffusion. Similar to the slowing down process, the diffusion process is characterized by diffusion lengthL d, which is equal to the average distance from the point where the neutron became thermal to the point where it ceased to exist freely as a result of absorption by some oncoming nucleus (see Table 1.8).
Table 1.8. Neutron moderation and diffusion lengths in various substances
The processes of neutron moderation and diffusion are illustrated in Fig. 1.4
Rice. 1.4. Illustration of the processes of neutron moderation and diffusion in matter.
Neutron diffusion, as well as the diffusion of other substances in liquid and gaseous media, is described by the universal Fick law, which relates the diffusion current J D to the particle density N or flow through a proportionality coefficient called diffusion coefficient D:
J D = -D*grad(N) = -D* (N)
The propagation of neutrons in the diffusion model (although subject to a number of assumptions) is well described by mathematical functions. For non-breeding media with a source (which corresponds to a subcritical reactor), in the simplest case these are exponentials:
Ф(z)= С 1 exp(+z/ L d)+ C 1 * exp(-z/ L d)
What the functions for breeding media will be will be shown in the next chapter.
Let us consider the balance of neutrons per unit volume dV for given Ф( r), S s.
Neutron balance
Changes in the number of neutrons lead to absorption, leakage, and birth. Then
birth – leakage – absorption.
The birth of neutrons is caused by a source : S( r) -the number of neutrons produced per unit time in a unit volume near r. Neutron absorption is determined by the number of reactions per unit time per unit volume. We need to find the yield of the reaction in a volume element
Let's find the neutron leak, knowing the density vector J from Fick's law
If known vector J at each point on the surface of the elementary volume dV, then the leakage is equal to div J - the number of neutrons crossing the surface of a unit volume per unit time. Moreover
div /D= const/= – D D F
Thus we have the equation
In stationary case
Notes:
When deriving these equations, we used Fick's law, which is valid if the flow distribution along the coordinates is linear at a distance of several. This means that these equations do not work well near the source boundary. Coefficient D here it already takes into account the possible non-sphericity of scattering (see earlier).
Boundary conditions:
1) the neutron flux F is finite and non-negative in the region where the diffusion equation is applicable;
2) at the boundary of two media that differ in at least one characteristic of the interaction of neutrons with nuclei.
Interaction of neutrons with nuclei
It is clear that this boundary condition cannot be written down knowing only the dependence of Ф on r . We use the following technique: draw F (r) in a flat reactor. Obviously, the flux at the boundary is less than at the center of the core, but is not equal to 0, i.e. . The equation is most easily solved under zero boundary conditions.
Flow on the border
| X |
| F(x) |
| Ф max |
| F |
| α |
Solving the diffusion equation is especially simple when the flux is 0 at some boundary. We will assume that the flux is formed at 0 not at the physical boundary, but at some extrapolated boundary of the reactor (linear extrapolation).
Extrapolation length d– an uncertain quantity, but it makes a small correction to the diffusion equation. Grade d was done both theoretically and experimentally. It turned out that when d = 0,71λ tr the best agreement between theory and experiment is observed.
End of work -
This topic belongs to the section:
Physical theory of reactors
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The simplest nuclear reactor
The content of the theory of nuclear reactors is most easily understood using the example of the simplest reactor - a sphere made of the fissile isotope 235U. The diameter of this sphere in which non-existence can be carried out
Nuclear reactor fuel
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Reproduction rate
The ratio of the number of fissile nuclei formed in a reactor during the absorption of neutrons to the number of burned-out fissile nuclei is called the breeding factor (BR).
Mechanism of nuclear reactions
Nucleon energy in the nucleus En r Fig. 2.1.1. For interaction
Nuclear energy levels
Just like in the atom, complete internal energy Evn core has certain discrete levels. Evn is understood as the sum of kinetic energy and potential energy
Resonance absorption
Let a stationary flow of neutrons fall on a layer of matter. We will assume that we can smoothly change the energy of incident neutrons. Then you can notice that for certain values of kinetic energy
Neutron scattering
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Neutron scattering and moderation
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Neutron cross sections
Let us consider the flow of neutrons penetrating the flow of matter with nuclei. We will assume that the flow is so thin that the nuclei do not obscure each other, that is (d<< λ).
Поперечным
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The yield of neutron reactions is the number of reactions occurring per unit time in a unit volume. Let us calculate the yield of neutron reactions under the assumption that all neutrons have the same energy, then
Neutron emission
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Mechanism of nuclear fission
The properties of heavy nuclei are in many ways similar to the properties of a drop of liquid. Nuclear forces tend to give the nucleus a spherical shape. An analogue of nuclear forces are molecular forces in a liquid, which also
Balance of released energy
The reason for the release of energy during fission is the higher binding energy per coulomb for lighter nuclei. The total energy released in one act of decay of uranium is about 204 MeV, including: kinetic
Fission chain reaction
Each fission reaction of U235 produces 2 or more neutrons. A necessary condition for a chain reaction is that more particles are born than are absorbed by the reaction initiators (neutrons
Multiplication factor of a reactor of infinite size
For a reactor of infinite size, the multiplication factor must be greater than 1 to start it. For thermal reactors, the problem of finding the multiplication factor can be solved. Let us have ra
Amount of enrichment required to maintain a steady-state chain reaction
Is enrichment necessary for nuclear reactors? To answer the question, let's consider. Obviously necessary for a stationary chain reaction ³1. In the expression for the product epf»1, therefore
Neutron leak
For a reactor of finite dimensions, the expression Keff = K∞P is valid, where P is the probability of avoiding a leak. Then the condition is critical
Action of delayed neutrons
Let's consider the influence of delayed neutrons on the control of a nuclear reactor. Previously, we used the average lifetime of the neutron generation, taking into account the delay, equal to 0.1 sec. (lifetime is instant
Neutron distribution in the reactor
In a reactor, neutrons are produced at all points of the active zone, that is, neutron sources are evenly distributed throughout space. The energy of the produced neutrons is ~2 MeV, they have different directions
Neutron moderation in infinite media
Let us have an infinite homogeneous active medium. Then the dependence n(E) remains. Let's consider the main processes that occur when neutrons are slowed down: 1. elastic
Elastic neutron scattering
Elastic dissipation is the main process in thermal reactors. Its consideration makes it possible to find the energy spectrum of moderated neutrons. Let neutrons be scattered on stationary free nuclei (p
Slowdown in hydrogen without absorption
Slowdown on hydrogen is considered due to the particular simplicity of its spectrum, because a neutron can slow down to zero energy. Slowing down a neutron on hydrogen to zero energy
Slowdown Density
The moderation density q(E) is the number of neutrons that, in a unit volume per unit time, cross the energy value E. This quantity is convenient when considering
Slowdown without absorption in non-hydrogen environments
Let A>>1 (A>10), then the change in energy per collision is small, small is the average logarithmic decrement energy, and the solution becomes simpler. Fermi proposed a model in which
Slowdown in infinite media in the presence of absorption
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Probability of avoiding resonant capture in media with a mass number greater than one
Let Σa<<Σs, а также пусть спектр с учетом резонансного захвата мало отличается от спектра Ферми. В отсутствии поглощения плотность замедления постоянн
Effective resonant integral
In nuclear reactors using thermal neutrons Sa<
Doppler effect
The Doppler effect is the dependence of the interaction macrosection on the velocity of nuclei and, consequently, on the temperature T of the medium, i.e. with increasing T, resonant peaks of the interaction macrosection, if any
Neutron current density. Fick's law
Let there be a medium with a given distribution of neutrons over space (given F(r)) and a scattering cross section Ss (with Sa=0). Let's find the current density through a unit area dS, l
Diffusion length
This concept is introduced in order to characterize the distance by which neutrons are displaced during diffusion from the point of creation to the point of absorption. Consider a point source of neutrons
Albedo
This is the reflection coefficient. And the surrounding zone reflects (neutron return to the active zone). Each environment has systems ΣS and Σа. Reflection properties cf
Continuous deceleration model
A neutron slows down as it diffuses. it is necessary to look for the distribution of neutrons of a given energy in space, i.e. energy spectrum of neutrons at any point in space. Age theory created by E. Fehr
Diffusion equation taking into account retardation
Let us denote Ф(r, u) - the sum of the paths traversed by neutrons with lethargy in a unit interval near lethargy u and in a unit volume near r per unit in
Assumptions and limitations of age theory
Age is associated with lethargy. We obtained the distribution of neutrons of a given age, and therefore of a given energy, in space, i.e. spectrum of neutrons at each given point. When deriving the diffusion equation we
Let =0 be given in an infinite medium, and all neutrons have energy E=2MeV. Let's find the neutron moderation density. for a spherically symmetric problem, i.e. . Solution of equations
Physical meaning of age
Age was introduced as a convenience variable, [t]=cm2, related to the nature of the environment. Let's find the average distance rdflhfn from the point of birth to the point where it intersects the values
Diffusion time and deceleration time
It is necessary to know how the time of slowing down a neutron to thermal energy and the time of diffusion of a neutron as thermal energy relate. According to the elastic dispersion model.
Criticality condition. Geometric and material parameter
If the composition in the core is given, then certain characteristics, such as the age of thermal neutrons, the square of the diffusion length, and the multiplication factor are given. The criticality condition gives only
Probability of avoiding leakage
We have Keff = KR1P2 where P1 is the probability of avoiding leakage during deceleration, where P2 is the probability of avoiding leakage during diff
Geometric parameters for reactors having dimensions and shapes in the form of a sphere and a cylinder
The most common shape of the core is cylindrical. The geometric parameter is the minimum eigenvalue of the wave equation: . A solution needs to be found to satisfy
Experimental determination of the critical reactor size
How to build a critical size reactor? If we start building a reactor, then as a result of the absence of neutrons in a subcritical reactor, we will not be able to consider the degree of approximation to the critical
Reflector properties
The critical mass of the reactor can be reduced by surrounding the core with a scattering substance. Will there be an effect if the core is surrounded with a well-absorbing substance? It can't get any worse. The worst thing is the vacuum. There is no dispersion in it
Neutron distribution and critical dimensions of a reactor with a reflector
The easiest way to build a reactor is using a single-speed (single-group) model. Neutrons are born, diffuse and are absorbed at the same energy. One can consider the energy spectrum
Effective reflector additive
A decrease in the critical size of the reactor due to the presence of a reflector is characterized by the effective addition of a reflector: , where H0 are the critical dimensions (thickness of the core
Reactor period
Knowledge of this section is necessary for practical work at the reactor as an operator, because you need to be able to predict the behavior of the neutron flux and heat release over time and at any point in the world
Large reactivity
Let T be so small that, i.e. Then Again is a straight line, the slope of which is characterized by the average lifetime of prompt neutrons
Thermal explosion
The reactor period may become short, the operator will not react, and a thermal explosion will occur. A reactor consists not only of fuel; any reactor also contains a moderator and a coolant. In a uranium-water reactor
Neutron balance disturbance
In order for the reactor to operate for a long time at a given power, it is necessary that during this time Keff = 1. However, in a power reactor there are reasons that lead to a decrease in Keff:
Control rods
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Reactor poisoning by fission products
Poisoning is caused by almost one radioactive isotope Xe135 (sa=2.7×106barn). This cross section is very large, because it corresponds to a linear size of 1.7 × 10-9 cm, i.e. order of size
Slagging
Slagging is the absorption of neutrons by stable or long-lived isotopes. This process is similar to poisoning, only here radioactive decay occurs slowly and at a speed
Sequential neutron absorption
There are such chains nuclear reactions, when each successive absorption of neutrons does not lead to the destruction of the slag nucleus, i.e., nuclei with a sufficiently large absorption cross section are formed.
Changes in reactivity during fuel burnout and reproduction
Basic nuclear reactions in fissile matter Let us assume that the decay rate of long-lived isotopes can be
Fuel burn-up
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About the atomic bomb
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Measuring fuel reserves as fuel burns out
To start a reactor and reach power, you need to have a reactivity reserve, i.e. Keff ~ 1.3. As the reactor operates, it becomes poisoned. In 20 hours the reactivity reserve of 0.05 will be used up, then
Perturbation theory in the one-group effective approximation
; Let us have an unperturbed reactor. The neutron flux in it obeys the diffusion equation (wave equation): ; Let it be in a small volume
Features of a heterogeneous reactor
It is convenient to divide the consideration of the theory of nuclear reactors into 2 parts: 1. Microscopic theory, which deals with the calculation of K and M2. These quantities are essentially internal x
Main effects of placing uranium in the form of blocks
1. The internal blocking effect for the probability of avoiding resonant capture is due to external peaks of resonant absorption on uranium 238. The presence of strong resonant absorption ensures
Calculation of the multiplication factor for heterogeneous systems
The thermal utilization factor f is the ratio of the number of thermal neutrons absorbed by the fuel to the total number of thermal neutrons. Fuel and moderator in a heterogeneous reactor are completely
Fast neutron multiplication factor
In a homogeneous reactor, ε differs slightly from unity. For heterogeneous 1.03 ¸ 1.06. Every hundredth is worth its weight in gold, since the maximum possible kef = 1.08 for a bang
Lecture 4. Neutron moderation and diffusion The process of reducing the average kinetic energy of neutrons when scattered by nuclei is called moderation. Neutron scattering by nuclei can be elastic or inelastic. Elastic scattering occurs with conservation of the total kinetic energy of the neutron and nucleus. The energy loss of a neutron E 1-E 2 during one elastic scattering is usually characterized by the average logarithmic energy loss (slowdown parameter) ξ = ‹In (E 1/E 2)› ≈ 2/(A + 2/3) Using ξ, you can calculate the average the number of collisions nzam of a neutron with nuclei, which leads to its slowdown from the initial energy to the thermal region (Et): nzam = ln(E 0/Et)/ ξ. 1
To select substances that can be used as moderators, the concept of moderating ability is introduced, which shows not only the average energy loss per collision, but also takes into account the number of such collisions in a unit volume of the substance. The product ξ Σs, where Σs is the macroscopic scattering cross section, takes into account both of the above factors, therefore its value characterizes the moderating ability of the substance. The higher the value of ξ Σs, the faster the neutrons slow down and the less volume of matter is needed to slow down the neutrons. 2
A MODERATOR must have a minimum absorption capacity in the region of thermal energies, and the absorption capacity of a substance is characterized by the value Σa, t. Therefore, the main characteristic of substances used as a moderator is the moderation coefficient kzam, which shows the ability of a substance not only to slow down neutrons, but also to preserve them after moderation: kzam = ξ Σs / Σa, t. The greater kzam, the more intense the accumulation of thermal neutrons in the moderator due to the high moderating ability of the substance and the weak absorption of neutrons in it. Substances with high values of kzam are the most effective moderators (see Table 2. 2). The best moderator is heavy water, but the high cost of heavy water limits its use. Therefore, ordinary (light) water and graphite are widely used as moderators. 3
In the process of slowing down to the thermal region, the neutron experiences a large number of collisions, and its average displacement (in a straight line) occurs at a distance from the place of generation (see Fig. 2. 8.). The value Ls = 1/2 is called the slowdown length, and the square of the slowdown length is called the neutron age τ. Neutrons, after being slowed down to the thermal region, move chaotically in the medium for a relatively long time, exchanging kinetic energy in collisions with surrounding nuclei. This movement of neutrons in a medium, when their energy remains constant on average, is called diffusion. The diffusion motion of a thermal neutron continues until it is absorbed. During the process of diffusion, a thermal neutron moves from the place of its birth to the place of absorption by an average distance “diff”. The value L = 1/2 is called the diffusion length of thermal neutrons. The average distance by which a neutron moves from the place of its birth (fast) to the place of its absorption (thermal) is characterized by the migration length M: M 2 = τ + L 2. 4
5
3. 3. Separation of the neutron energy range in a nuclear reactor Of the variety of processes that occur during the interaction of neutrons with nuclei, three are important for the operation of a nuclear reactor: fission, radiation capture and scattering. The cross sections of these interactions and the relationships between them depend significantly on the neutron energy. Energy intervals of fast (10 Me. V-1 kE. V), intermediate or resonant (1 kE. V-0.625 e. V) and thermal neutrons (-e. V) are usually distinguished. Neutrons produced during nuclear fission in reactors have energies above several kiloelectron volts, i.e. they are all fast neutrons. Thermal neutrons are so called because they are in thermal equilibrium with the reactor material (mainly the moderator), i.e., the average energy of their motion approximately corresponds to the average energy of thermal motion of the atoms and molecules of the moderator. 6
As can be seen, for all moderators the diffusion time is significantly longer than the deceleration time, with the largest difference occurring for heavy water. This means that in a large volume of the moderator, the number of neutrons with thermal energy is approximately 100 times greater than the number of all other neutrons with higher energy. 9
Structural materials and fuel weakly slow down neutrons compared to heavy or light water. In graphite reactors, the volume of the moderator in the cell significantly exceeds the volume of the fuel assembly, and the age of neutrons in the reactor is close to the age of neutrons in graphite 10
Multiplication factor To analyze the fission chain reaction, a multiplication factor is introduced, showing the ratio of the number of neutrons ni of any generation to their number ni-1 in the previous generation: k = ni/ ni -1 11
PHASES OF A CLOSED NEUTRON CYCLE The value of k∞ in a breeding medium containing nuclear fuel and a moderator is determined by the participation of neutrons in the following four processes, representing different phases of a closed neutron cycle: 1) fission by thermal neutrons, 2) fission by fast neutrons, 3) moderation of fast neutrons neutrons to the thermal region, 4) diffusion of thermal neutrons to absorption in nuclear fuel 12
1. Fission by thermal neutrons (10 -14 s). 1) Fission by thermal neutrons is characterized by the fission coefficient by thermal neutrons η, which shows the number of secondary neutrons produced per one absorbed thermal neutron. The value of η depends on the properties of the fissile substance and its content in nuclear fuel: η = νσf 5/(σf 5 + σγ 8 N 8/N 5). The decrease in η compared to the number ν of secondary neutrons produced during fission) is due to the radiative capture of neutrons by nuclei 235 U and 238 U, having concentrations N 5 and N 8, respectively (for brevity, we will indicate the last digit of the nuclide mass number in the subscript). 13
For the nuclide 235 U (σf 5 = 583.5 b, σγ 5 = 97.4 b, N 8 = 0) the value is η = 2.071. For natural uranium (N 8/N 5 = 140) we have η = 1, 33.14
2. Fission with fast neutrons (10 -14 s.). Some of the secondary neutrons produced during fission have an energy greater than the energy of the fission threshold of 238 U. This causes the fission of 238 U nuclei. However, after several collisions with moderator nuclei, the neutron energy becomes below this threshold and the fission of 238 U nuclei stops. Therefore, neutron multiplication due to the fission of 238 U is observed only during the first collisions of generated fast neutrons with 238 U nuclei. The number of secondary neutrons produced per absorbed fast neutron is characterized by the fast neutron fission coefficient μ. 16
3. Slowing down fast neutrons to the thermal region (10 -4 s) In the resonant energy region, the main absorber of moderating neutrons are 238 U nuclei. The probability of avoiding resonant absorption (coefficient φ) is related to the density N 8 of 238 U nuclei and the moderating ability of the medium ξΣs by the ratio φ = exp[ – N 8 Iа, eff/(ξΣs)]. The quantity Ia, eff, which characterizes the absorption of neutrons by an individual 238 U nucleus in the resonant energy region, is called the effective resonance integral. 17
The greater the concentration of 238 U nuclei (or nuclear fuel Nyat) compared to the concentration Nreplacement of moderator nuclei (ξΣs = ξσs. Nreplacement), the lower the value of φ 18
Diffusion of thermal neutrons before absorption in nuclear fuel (10 -3 s). Neutrons that reach the thermal region are absorbed either by fuel nuclei or moderator nuclei. The probability of thermal neutrons being captured by fuel nuclei is called the thermal neutron utilization factor θ. θget = Σа, yatΦyat/(Σа, yatΦyat + Σа, zamΦzam) = Σа, yat/(Σа, yat + Σа, zamΦzam/Φyat). 19
The four processes considered determine the neutron balance in the breeding system (see Fig. 3. 3). As a result of the absorption of one thermal neutron of any generation, ημφθ neutrons appear in the next generation. Thus, the multiplication coefficient in an infinite medium is quantitatively expressed by the formula of four factors: k∞ = n ημφθ/n = ημφθ. 20
Rice. 3. 3 Neutron cycle of a fission chain reaction using thermal neutrons in a critical state (k∞ = ημφθ = 1). 21
The first two coefficients depend on the properties of the nuclear fuel used and characterize the production of neutrons during the fission chain reaction. The coefficients φ and θ characterize the useful use of neutrons, but their values depend on the concentrations of moderator nuclei and fuel in the opposite way. Therefore, the product φθ and, consequently, k∞, have maximum values at the optimal ratio Nzam/Nyat. 22
a fission chain reaction can be carried out using different types of nuclear fuel and moderator: 1) natural uranium with a heavy water or graphite moderator; 2) weakly enriched uranium with any moderator; 3) highly enriched uranium or artificial nuclear fuel (plutonium) without a moderator (chain fission reaction with fast neutrons). 23
