Here is another example giving an equation of the same kind, but this time referring to diffusion. In ch. 43 (Issue 4) we considered the diffusion of ions in a homogeneous gas and the diffusion of one gas through another. Now let's take another example - the diffusion of neutrons in a material like graphite. We chose graphite (a form of pure carbon) because carbon does not absorb slow neutrons. Neutrons travel freely in it. They travel in a straight line for a few centimeters on average before being scattered by the nucleus and deviated to the side. So if we have a large piece of graphite several meters thick, then the neutrons that were first in one place will go to other places. We will describe their average behavior, i.e. their average flow.

Let N(x, y,z) ΔV is the number of neutrons in the volume element Δ V in 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 region, then more of them will pass into the second region than vice versa; the result is a stream. Repeating the proofs given in Chap. 43 (issue 4), one can describe the flow 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 then

where is the diffusion coefficient D is given in terms of the average velocity ν and mean free path l between collisions:

The speed at which neutrons pass through some element of the surface da, is equal to Jnda (where n is, as usual, the unit normal vector). Result stream from elevolume ment then equals (using the usual Gaussian proof) v J dV. This flow would result in a decrease in the number of neutrons in ΔV, if neutrons are not generated inside ΔV (by some nuclear reaction). If the volume contains sources that produce 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 get neutron diffusion equation

In the static case, when ∂N/ ∂t=0, we again have equation (12.4)! We can use our knowledge of electrostatics to solve neutron diffusion problems. Let's solve some problem. (Perhaps you are wondering: why solve a new problem if we have already solved all the problems in electrostatics? This time we can decide faster precisely because electrostatic problems deyalready resolved!)

Let there be a block of material in which neutrons (say, due to the fission of uranium) are produced uniformly in a spherical region with a radius a(Fig. 12.7). We would like to know what is the density of neutrons everywhere? How uniform is the density of neutrons in the region where they are born? What is the ratio of the neutron density at the center to the neutron density on the surface of the birth region? The answers are easy to find. Neutron density in the source So 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; we can integrate them to get the potential. 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. Consequently,

For interior points, only the charges contribute to the field Q(r), inside a sphere with a radius r;Q(r) =4πr 3 ρ/3, therefore,

The field grows linearly with r. Integrating E, we get φ:

At a distance of radius and φ external must match φ internal, so the constant must be equal to ρа 2 /2ε 0 . (We assume that the potential φ is equal to zero at large distances from the source, and this for neutrons will correspond to the reversal 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 2/2, and on the edge (r=a) in proportion to 2a 2 /2; so the density ratio is 3/2. A homogeneous source does not produce a uniform neutron density. As you can see, our knowledge of electrostatics provides a good starting point 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 come to the same equations over and over again.

Moderation and diffusion of neutrons.

During the existence of a neutron from the moment of emission at pressure until the moment of absorption, 2 processes take place:

1). The process of slowing down a fast neutron from fission energy (~2 MeV) to thermal energy (<0,2эв)(0,025эв);

2) the process of diffusion of a thermal neutron.

The lifetime of a neutron is ~0.001 sec and depends on the composition of the active zone.

Neutrons, like gases, diffuse from a region of higher density to a region of lower density.

Between collisions is a straight section. A typical trajectory is a zigzag view of straight line segments of different lengths.

If there were no neutron capture, the trajectory would be infinite. After a dissipative impact, it moves in a direction that forms an angle ψ with the initial direction of motion.

Angle ψ-y scattering. Important for the study of diffusion and retardation, what is the probability of scattering in any direction. It has been experimentally established that it tends to scatter in the direction of its initial movement.

If scattering occurred with the same probability in all directions (isotropic scattering), then the value of const averaged over all collisions would be =0.

In reality, the average cos ψ >0 (zero) and is determined by the equality cos ψ= ,

where A is the mass number of the scattering nucleus.

Starting with beryllium, the deflection is almost isotropic. For isotropic scattering, the average distance traveled between scattering collisions is

In reality, the effective distance is greater than the mean free path λ s due to predominant forward scattering. This distance is called the transport free path:

By analogy with e, the concept of the transport section is also introduced

Because light elements are used as a moderator in nuclear reactors, then the process of slowing down fast neutrons occurs mainly as a result of elastic scattering.

The energy loss upon impact depends on ψ. When ψ=0 E 2 /E 1 =1. The greatest loss of E during collisions occurs at ψ= 0-π. Other things being equal, the moderator is the more effective, the more energy will be lost by fast fission

upon collision with moderator nuclei.

As a measure of the change in the energy of a neutron in an elastic collision, the average logarithmic energy decrement per 1 collision (or the average logarithmic energy loss) is emitted:

ξ \u003d (ln E 2 / E 1) cf,

E 1 - before the collision

E 2 - after the collision

The value ξ averaged over all possible scattering angles depends only on the atomic weight of the element A:

i.e. ξ does not depend on the initial energy .

This means that, on average, it loses the same fraction of its initial energy, regardless of the initial energy of the neutron at which the collision occurred.

The height of the steps indicates a change in ln E per 1 collision, i.e. determines ξ., since ξ does not depend on E, then on average the height of the steps is the same throughout the entire deceleration time.

The average number of collisions with the atoms of a substance required to reduce the energy from E 1 to E 2 is determined by the relation

Physically, with increasing ξ. The loss of E increases by 1 atom, which means that the average number of collisions required to reduce E = 2 MeV to 0.025 eV decreases.

C increases with an increase in the mass number of moderator nuclei (19 collisions are required on water, and 114 on graphite). The smaller C, the better the moderator. However, both C and ξ do not adequately reflect the moderating properties of matter. They are determined by the average loss of energy per 1 collision, but do not reflect how likely a scattering collision of a neutron with the nuclei of a given moderator is. The latter is determined by the macroscopic scattering cross section.

Σs = σs ∙N,

where σ s - microscopic section;

N-density of moderator nuclei

Therefore, as a more appropriate characteristic of the slowing down properties, the product is introduced:

ξΣ s , called the retarding ability, because it is characterized by both the loss of E(ξ) and the probability that a collision will occur. When choosing a moderator, one has to take into account the important requirement that it should absorb neutrons as little as possible. Therefore, a retarder is introduced:

For the moderator of nuclear reactors, only such substances can be used that simultaneously have high values ​​of kz and slowing ability ξΣ s. Such materials are ordinary water, heavy water, graphite, beryllium, beryllium oxide and some organic liquids. The best is heavy water. In ordinary water, kz is the smallest due to the increased capture of thermal neutrons in hydrogen.

substance	ξ.	FROM		to z	σ a	σs
Water	0,918		1,53		0,66		0,0218	1,45	2,7
Heavy water	0,51		0,37		2,6∙10 -3		0,86∙10 -4	0,50
Beryllium	0,207		0,176		9∙10 -3		10,8∙10 -4	0,84
beryllium oxide	0,174		0,129		9∙10 -3	11,2	6,5∙10 -4	0,81
Diphenia	0,892		1,5		4∙10 -3	4,8	3,32∙10 -4	0,998
Differential mixture	0,886		1,61	117,5
Graphite	0,158		0,064		4∙10 -3	4,8	3,32∙10 -4	0,998
Helium in good condition	0,525		1,6∙10 -5
Lithium	0,268		0,0172	negligible
Bor	0,171		0,0875

In the process of deceleration, in addition to a change in energy, there is a displacement of the neutron in space from the point of its emission to the point where it becomes thermal. The displacement in space continues in the process of diffusion, which has reached the thermal level.

Lecture 4 Scattering of neutrons by nuclei can be elastic or inelastic. Elastic scattering occurs with the conservation of the total kinetic energy of the neutron and the nucleus. The energy loss of a neutron E 1-E 2 with one elastic scattering is usually characterized by the average logarithmic energy loss (deceleration parameter) ξ = ‹In (E 1/E 2)› ≈ 2/(A + 2/3) the number of collisions ndeputy of a neutron with nuclei, which leads to its slowing down from the initial energy to the thermal region (Et): ndet = ln(E 0/Et)/ ξ. one

To select substances that can be used as moderators, the concept of retarding ability is introduced, which shows not only the value of the average energy loss in one collision, but also takes into account the number of such collisions in a unit volume of a substance. The product ξ Σs, where Σs is the macroscopic scattering cross section, takes into account both of the above factors, so its value characterizes the moderating ability of a substance. The higher the value of ξ Σs, the faster the neutrons slow down and the smaller the volume of matter is needed to slow down the neutrons. 2

The moderator must have a minimum absorbing capacity in the range of thermal energies, and the absorbing 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 kde, which shows the ability of a substance not only to slow down neutrons, but also to keep them after deceleration: kdet = ξ Σs / Σа, t. The more kdet, the more intensively thermal neutrons accumulate in the moderator due to the large moderating ability of the substance and 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, while its average displacement (along a straight line) occurs at a distance ‹rreplacement› from the place of generation (see Fig. 2. 8.). The value Ls= 1/2 is called the deceleration length, and the square of the deceleration length is called the neutron age τ. After slowing down to the thermal region, neutrons move randomly 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. In the process of diffusion, a thermal neutron is displaced from the place of its birth to the place of absorption by an average distance ‹rdiff›. The value L = 1/2 is called the diffusion length of thermal neutrons. The average distance that a neutron moves from its place of birth (fast) to its place of 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 whole 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 essentially on the neutron energy. Usually, energy intervals are distinguished for fast (10 Me. V-1 ke. V), intermediate or resonant (1 ke. V-0.625 e. V) and thermal neutrons (-e. V). The neutrons produced during the fission of nuclei in reactors have energies above a few kiloelectron volts, i.e., they all belong to fast neutrons. Thermal neutrons are so called because they are in thermal equilibrium with the material of the reactor (mainly the moderator), i.e., the average energy of their movement approximately corresponds to the average energy of the thermal movement of atoms and molecules of the moderator. 6

As can be seen, for all moderators, the diffusion time is much longer than the deceleration time, and the greatest difference occurs 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 moderate neutrons weakly compared to heavy or light water. In graphite reactors, the volume of the moderator in the cell significantly exceeds the volume of fuel assemblies, and the age of neutrons in the reactor is close to the age of neutrons in graphite 10

Multiplication factor To analyze a fission chain reaction, the 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 on thermal neutrons (10 -14 s). 1) Thermal neutron fission is characterized by the thermal neutron fission coefficient η, which indicates the number of secondary neutrons produced per absorbed thermal neutron. The value of η depends on the properties of the fissile material 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 235 U and 238 U nuclei, which have concentrations N 5 and N 8, respectively (for brevity, we will indicate the last digit of the mass number of the nuclide in the subscript). 13

For the nuclide 235 U (σf 5 = 583.5 b, σγ 5 = 97.4 b, N 8 = 0) the value η = 2.071. For natural uranium (N 8/N 5 = 140) we have η = 1, 33.14

2. Fission on fast neutrons (10 -14 s.). Some of the secondary neutrons produced during fission have an energy greater than the energy of the 238 U fission threshold. This causes the fission of 238 U nuclei. However, after several collisions with the moderator nuclei, the neutron energy falls below this threshold and the fission of 238 U nuclei stops. Therefore, neutron multiplication due to 238 U fission is observed only in the first collisions of produced fast neutrons with 238 U nuclei. The number of produced secondary neutrons per absorbed fast neutron is characterized by the fast neutron fission coefficient μ. 16

3. Moderation of fast neutrons to the thermal region (10 -4 s) In the resonant energy region, 238 U nuclei are the main absorber of moderating neutrons. 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 relation 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 resonant integral. 17

The greater the concentration of 238 U nuclei (or nuclear fuel Nfl) compared to the concentration Ndm of the moderator nuclei (ξΣs = ξσs. Ndm), the lower the value of φ 18

Diffusion of thermal neutrons before absorption in nuclear fuel (10 -3 s). Neutrons that have reached the thermal region are absorbed either by the fuel nuclei or by the moderator nuclei. The probability of capture of thermal neutrons by fuel nuclei is called the thermal neutron utilization factor θ. θhet = Σa, yatΦat/(Σa, yatΦyat + Σa, zamΦzam) = Σa, yat/(Σa, yat + Σa, zamΦzam/Φyat). 19

The four processes considered determine the balance of neutrons in the multiplying 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 factor in an infinite medium is quantitatively expressed by the formula of four factors: k∞ = n ημφθ/n = ημφθ. twenty

Rice. 3. 3 Neutron cycle of fission chain reaction on thermal neutrons in the critical state (k∞ = ημφθ = 1). 21

The first two coefficients depend on the properties of the nuclear fuel used and characterize the birth of neutrons in the course of a fission chain reaction. The coefficients φ and θ characterize the useful use of neutrons, but their values ​​depend on the concentrations of the moderator nuclei and fuel in the opposite way. Therefore, the product φθ and, consequently, k∞, have maximum values ​​at the optimal ratio Nsub/Nat. 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) low enriched uranium with any moderator; 3) highly enriched uranium or artificial nuclear fuel (plutonium) without a moderator (fast neutron fission chain reaction). 23

To describe some important regularities of the diffusion process in reactors, we introduce and refine some definitions. Let's define neutron flux density F, more commonly called "flux" as the number of neutrons crossing a spherical surface of 1 cm 2 per second, so the flux dimension will be 1/(cm 2 *s). We have previously defined microscopic section reactions like "" isotope "i"   i as the area of ​​interaction of one nucleus in barns. Now let's define the so-called. macroscopic section reactions of the “” type of the isotope “i” as the cross section for the interaction of all nuclei “i” located in 1 cm 3 of the substance   i .

These two sections are interconnected by the value of the so-called. "nuclear density" or the density of nuclei , which characterizes the number of molecules (or nuclei) in 1 cm 3 of a substance.

 = N A * / 

N A is the Avogadro number (equal to 0.6023*10 24 molecules/gmol);

 - physical density of any complex substance (g / cm 3);

 is the molecular weight of the substance (g/gmol).

Then the relationship between microscopic and macroscopic cross sections can be written as:

  i =  i *  i

In this case, the density of nuclei of a given isotope  i will be related to the density of molecules  through the number of atoms of this type "i" in the molecule of the substance.

Finally, the only quantity that can actually be measured in nuclear reactions (including dosimetric instruments, fission chambers, and is realized inside the reactor) is speed reaction of the 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)). At the same time, for the fission process, there is an important relationship between the number of fissions and the power allocated in this case 1W = 3.3 * 10 10 divisions / s.

Diffusion of thermal neutrons. When the energy of the neutrons decreases to the energies characteristic of the energies of the 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 is about 0.04 eV. A neutron in thermal equilibrium with a medium is called thermal neutron, and the movement of thermal neutrons with a constant average speed - diffusion of thermal neutrons. Similar to the deceleration 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 its free existence as a result of absorption by some counter nucleus (see Table 1.8).

Table 1.8. Neutron deceleration and diffusion lengths in various substances

The processes of slowing down and diffusion of neutrons are illustrated in fig. 1.4

Rice. 1.4. Illustration of the processes of slowing down and diffusion of neutrons in matter.

The diffusion of neutrons, as well as the diffusion of other substances in liquid and gaseous media, is described by the universal Fick's law, which relates the diffusion current J D to the particle density N or flux through a proportionality coefficient called diffusion coefficient D:

J D = -D*grad(N) = -D* (N)

The propagation of neutrons in the diffusion model (however, under 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 the exponents:

Ф(z)= С 1 exp(+z/ L d)+ C 1 * exp(-z/ L d)

What will be the functions for breeding environments will be shown in the next chapter.

Diffusion of neutrons

Neutrons slowed down to thermal energies begin to diffuse, propagating through the substance in all directions from the source. This process is already approximately described by the usual diffusion equation with the obligatory allowance for absorption, which is always large for thermal neutrons (in practice, they are made thermal in order for the desired reaction to proceed intensively). This possibility follows from the fact that in a good moderator (in which the scattering cross section ys is much larger than the absorption cross section ya), a thermal neutron can experience a lot of collisions with nuclei before being captured:

N= us/ua=la/ls, (3.10)

in this case, due to the smallness of the mean free path ls, for thermal neutrons, the condition for the applicability of the diffusion approximation is satisfied - the smallness of the change in the neutron density over ls. Finally, the speed of thermal neutrons can be considered constant: .

The diffusion equation has the following form:

where c( r, t) is the density of thermal neutrons at the point r at time t; D is the Laplace operator; D is the diffusion coefficient; tcap is the average lifetime of thermal neutrons before capture; q is the density of thermal neutron sources. Equation (3.11) expresses the balance of the change in the neutron density over time due to three processes: the influx of neutrons from neighboring regions (DD s), the absorption of neutrons (- s /tzap) and the production of neutrons (q). In the general case (taking into account scattering anisotropy), the diffusion coefficient is:

however, for thermal neutrons it can be written with a good degree of accuracy in the simplest form:

This is due to the fact that the energy of thermal neutrons is less than the energy of the chemical bond of atoms in a molecule, due to which the scattering of thermal neutrons occurs not on free atoms, but on heavy bound molecules (or even on crystalline grains of the medium).

The main characteristic of the medium that describes the diffusion process is the diffusion length L, which is determined by the relation

where is the average square of the distance traveled by a thermal neutron in matter from the place of birth to absorption. The diffusion length is approximately of the same order as the deceleration length. Both of these quantities determine the distances from the source at which there will be an appreciable amount of thermal neutrons in the substance. Table 3.1 shows the values ​​of f and L for the most commonly used moderators. From this table, it can be seen that ordinary water has >>L, which indicates strong absorption. In heavy water, on the contrary, L>>. Therefore, it is the best retarder. The value of L depends not only on intrinsic diffusion, but also on the absorbing properties of the medium. Therefore, L does not fully characterize the diffusion process. An additional independent characteristic of diffusion is the lifetime of a diffusing neutron.

Table 3.1

Values ​​and L for the most commonly used moderators

Diffuse reflection of neutrons

An interesting property of neutrons is their ability to be reflected from various substances. This reflection is not coherent, but diffuse. Its mechanism is this. A neutron entering the medium experiences random collisions with nuclei and, after a series of collisions, can fly back. The probability of such an emission is called the neutron albedo of the given medium. Obviously, the higher the albedo, the larger the scattering cross section and the smaller the absorption cross section of neutrons by the nuclei of the medium. Good reflectors reflect up to 90% of the neutrons falling into them, that is, they have an albedo of up to 0.9. In particular, for ordinary water, the albedo is 0.8. It is not surprising, therefore, that neutron reflectors are widely used in nuclear reactors and other neutron installations. The possibility of such an intense reflection of neutrons is explained as follows. A neutron entering the reflector can be scattered in any direction during each collision with the nucleus. If the neutron is scattered back near the surface, then it flies back, i.e., it is reflected. If the neutron is scattered in another direction, then it can be scattered in such a way that it leaves the medium in subsequent collisions.

The same process leads to the fact that the concentration of neutrons decreases sharply near the boundary of the medium in which they are born, since the probability for a neutron to escape is high.