Critical mass

Nuclear fission chain reaction

The Nuclear reaction fission with suitable examples and sketches are discuss here advanced. Nuclear reaction fission is needed to read as engineer student.

Nuclear chain reactions are nuclear reactions in which the particles causing them are formed as products of these reactions. They are realized if more than 1 neutron flies out during nuclear fission. Comment. In nuclear physics, chain processes are not always associated with fission. In 1934 (i.e., long before the discovery of fission, but shortly after the discovery of artificial radioactivity and neutron), the Hungarian scientist Leo Szilard took the English secret patent for the first atomic bomb, based on any nuclear reaction in which two or more particles arise in exchange for an spent particle. Specifically, he meant the explosion as a result of a chain reaction of neutron multiplication in beryllium:

n + 9Be → 8Be + 2n. In practice, this reaction could not be realized,

since energy is not released in it but absorbed.  As already mentioned, 2 or 3 neutrons are released in the process of nuclear fission under the influence of neutrons. Under favorable conditions, these neutrons can enter other nuclei and cause their fission. At this stage, from 4 to 9 neutrons will appear that can cause new decays of uranium nuclei, etc. Such an avalanche-like process is called a chain reaction. Due to the high speed of the fission process, the number of fissile nuclei in a short time reaches a huge value, as a result of which enormous intranuclear energy will be released.

The appearance of secondary neutrons in the process of fission of heavy nuclei by neutrons allows the chain fission reaction to be carried out. The chain process is characterized by the fact that it is based on an exoenergic reaction excited by a neutron, which generates secondary neutrons. In this case, the appearance of a neutron in a fissile medium causes a chain of fission reactions following one after another, which continues until the termination due to the loss of the neutron carrier of the process.

There are two main reasons for the loss: neutron absorption by the nucleus without the emission of secondary neutrons (for example, radiation capture) or neutron departure beyond the limits of the substance volume (called the active zone), in which the chain fission process takes place. If the reaction results in more than one neutron, which in turn cause fission, then such a reaction is a branched reaction.

The development scheme of the fission chain reaction
The development scheme of the fission chain reaction

 The average path length of a neutron from the point of birth to the point at which the neutron produces fission is a macroscopic quantity. Therefore chain

fission reaction is a macroscopic process. Each neutron participating in the chain process undergoes a circulation cycle: it is born in the fission reaction, for some time it exists in a free state, then it is either lost or gives rise to a new fission event and gives neutrons of the next generation. The neutron needs, albeit a small, but finite time to pass through the circulation cycle. The average time τ obtained by averaging over a large number of neutron fission cycles is called the neutron cycle time or the average neutron lifetime. Thus, the chain process of division can be represented as a sequence of successive avalanches or generations separated by a time interval τ:

 N0 àN1 → N2 → … → Ni → Ni + 1 → …, (1

where N is the number of neutrons in a given generation. The ratio of the number of neutrons of the next generation to their number in the previous generation in the entire volume of the active zone is called the neutron multiplication factor:

k = Ni + 1 / Ni. (2)

The values ​​of τ and k completely determine the development of the chain process in time. Indeed, the number of neutrons in the next generation is Ni + 1 = kNi, then, after a period of time τ, the number of neutrons N i + 2 = k N i + 1 = k2Ni, after a time of 2 τ, the number of neutrons will be Ni + 3 = kNi + 2 = k2Ni +1 = k3Ni, etc. The number of neutrons in generation m (the number of neutron cycles) will be

Nm = N0km, (3)

 if the number of neutrons at the initial instant of observation time is N 0. In this case, the observation time will be t = mk, which allows us to write dependence (3) in explicit form of time:

N (t) = N0kt / τ. (4)

However, the expressions obtained are only approximately true, since the cases of the birth and disappearance of neutrons occur randomly, and at any time in the core there are neutrons from different generations, i.e. the process of changing the number of neutrons in the core occurs continuously. The increment of the number of neutrons in the chain process for an infinitely small period of time dt will be:

However, the obtained expressions are true only approximately, since the cases of the birth and disappearance of neutrons occur randomly, and at any time in the core there are neutrons from different generations, i.e. the process of changing the number of neutrons in the core occurs continuously. The increment of the number of neutrons in the chain process over an infinitely small period of time dt will be:

dN = Nk – N = N (k – 1), (5)

and the rate of change of the number of neutrons will be equal to

dN/dt = N(k-1)/ τ   (6)

Equation (6) is called the point kinetics equation without delayed neutrons. Separating the variables in (6), we obtain a solution to this equation:

N(t) = N0 exp((k-1)/ τ) (7)

where N0 = N (t = 0) is the number of neutrons at the initial moment of observation. If k> 1, then the number of neutrons in the core will continuously increase and the chain reaction process, once it occurs, will develop in time by itself.The process with k> 1 is called the supercritical mode. For k = 1, the number of neutrons in the core and the number of ones per unit The fission times do not change with time and remain constant. This mode is called the critical mode.

Finally, if k <1, the neutron multiplication process decays and is called a subcritical mode, respectively. For a self-sustaining fission chain reaction to occur, k ≥ 1. the possibilities of a chain reaction are usually considered multiplication coefficient k ∞ in a medium with an infinite volume, when neutron leakage through the surface of the active zone can be neglected. s of finite dimensions

k = κk∞, (8)

where κ is the probability of a neutron to avoid leakage from the core of a finite volume. If there is some finite volume, then the configuration, composition, and mass of the core under which the condition

k = κk∞ ≥ 1 is satisfied, (9)

are called critical parameters. The value of k depends on many parameters: the nuclide composition of the core, its shape and size, and the energy spectrum of neutrons that cause fission. The calculation of k is a complex engineering and physical task and requires knowledge of a huge number of constants that determine the course of the chain process.

Let us consider a chain process in which the neutron cycle time is τ≈10-3 s. This value of τ is characteristic of thermal neutron reactors. If the multiplication coefficient is k = 1.005, then in one second the number of neutrons, according to (7). will increase by times

N(t)/N0 = exp((k-1) t/ τ) = e^5 = 150 pas  (10)

The number of divisions per unit time and, consequently, the power of the installation will increase by the same number of times. In the installation, an excess of k over unity by only 0.5% is unacceptable. The above estimate does not take into account delayed neutrons and is therefore overestimated. Indeed, the number of neutrons in the core at a given time can be represented as follows:

Ni + 1 = Np + Nd,  ( 11)

where: Np is the number of instantaneous neutrons (p – prompt) that occurred immediately at the moment of nuclear fission; Nd is the number of delayed neutrons (d – delay) arising from fission fragments as a result of delayed energy release. by the number of neutrons N i in the previous cycle and taking into account definition (2), we find that the multiplication coefficient can be represented as the sum of two terms:

k = kp + kd, (12)

of which the first is a coefficient multiplication on instant neutrons, and the second – multiplication factor on delayed ones. Then in the chain process going to 235U under the influence of thermal neutrons

kp = k (1 – β) = 1.005 (1 – 0.0065) = 0.9985, (13)

kd = kβ = 1.005 · 0.0065 = 0.0653 . (13)

where β is the fraction of delayed neutrons in the total number of secondary fission neutrons. The chain process on instantaneous neutrons alone is subcritical, and the process is controlled by changing the number of delayed neutrons. If kp becomes equal to or greater than unity, which corresponds to k≥ (1 + β), then the chain process becomes uncontrollable. Let us find the average time τ0 of the neutron cycle taking into account delayed neutrons. By the rule of finding the average,

τ0 = (1 – β) τp + βτd, (14)

where τp is the average lifetime of instantaneous neutrons, and τd are delayed ones. For example, the average lifetime of delayed neutrons for 235U is 13 s and for τp≈ 10-3 s we get

τ0 = 10-3 + 0.085≈ 0.085 s. (15)

From the given example, an important conclusion follows that the average neutron cycle time of the chain process is determined by the average lifetime of delayed neutrons and does not depend on the lifetime of fast neutrons, but under the condition k <(1 + β). Using the time τ = τ0 = 0.085 s in example (10), we find that in one second the power of the chain process will increase by only 6%, which does not pose a problem for regulation. In the theory of control of a chain process, the value of T is usually used, called the period of the reactor, which is the time during which the number of neutrons in the core increases by “e” times. From (7) we have

T = τ/(k-1)  (16)

If, again, k = 1.05, and τ = 0.085 s, then the period of the reactor is T = 17 s. As k → 1, T → ∞, which follows directly from (16). Let us consider the kinetics of the chain process taking into account delayed neutrons. The increment rate of instant neutrons, by analogy with (6), will be equal to

dNp/dt = (kN(1- β) )/ τ  (17)

and the increment rate of delayed fragments in accordance with the law of radioactive decay of fragments relative to the emission of delayed neutrons (λ is the decay constant of this decay) is

dNd/dt = λc, (18)

where с is the number of accumulated in previous generations of retarded neutron precursor nuclei The total rate of change in the number of neutrons

dN/dt = dNp/dt + dNd/dt = (kN(1- β) )/ τ  + λc  (. (19)

This equation must be supplemented with the equation for the rate of formation of the nuclei of the precursors:

dc/dt = k β N / τ  = λc (20)

Equations (19) and (20) form a system of coupled linear differential equations of point kinetics taking into account delayed neutrons. At a more accurate consideration, six time groups of delayed neutrons are taken into account and a system of seven equations is obtained. Let us show that, using thermal neutrons, it is possible to organize a chain process of fission of natural composition on uranium: the relative atomic content of 238U is 99.28%, 235U is 0.714% and 234U is 0.006%. Only 235U and 234U nuclides are fused with thermal neutrons. Due to the insignificant content of 234U, we will not take into account his participation in the chain process. The average number η of secondary neutrons per act of absorption of a thermal neutron by natural uranium will be equal to (the fraction determines the probability of a neutron to produce fission):

where: – the average number of secondary neutrons per act of fission; n is the concentration of 235U or 238U nuclide nuclei (with corresponding superscripts); σа is the neutron capture cross section for 235U or 238U nuclei; 5σf is the cross section for fission of 235U nuclei by neutrons. For thermal neutrons, these values ​​are equal: = 2.44; σа = 694 barn for 235U nuclei; σa = 2.8 bar for 238U nuclei; 5σf = 582 barn for 235U cores. For natural uranium, 8n / 5n = 99.28 / 0.714 = 139. Substituting these values ​​in

where: – the average number of secondary neutrons per act of fission; n is the concentration of 235U or 238U nuclide nuclei (with corresponding superscripts); σа is the neutron capture cross section for 235U or 238U nuclei; 5σf is the cross section for fission of 235U nuclei by neutrons. For thermal neutrons, these values ​​are equal: = 2.44; σа = 694 barn for 235U nuclei; σa = 2.8 bar for 238U nuclei; 5σf = 582 barn for 235U cores. For natural uranium, 8n / 5n = 99.28 / 0.714 = 139. Substituting these values ​​in formula (21), we obtain η = 1.31. Thus, the chain process on 235U nuclei in the composition of natural uranium can be realized if, upon deceleration of the secondary fission neutrons to thermal energies, an average of no more than 0.3 neutrons is lost.

However, a spontaneous chain process of fission in natural uranium cannot occur and that is why. In nuclear fission, the average energy of secondary neutrons is ~ 2 MeV. For neutrons with such energy, the quantities included in formula (21) are equal to: = 2.65; σа = 2.1 bar for 235U nuclei; σa ≈ 0.1 barn for 238U nuclei; 5σf = 2 bar for 235U cores. Substituting these values ​​into formula (21), we obtain η (235U) ≈0.33. Now it is necessary to take into account the fission of 238U nuclei by fast neutrons. The 8σf fission cross section of 238U nuclei at an energy of 2–6 MeV is ~ 0.5 bar and actually has a threshold of 1.4 MeV). The fraction of neutrons in the fission spectrum, whose energy exceeds 1.4 MeV, is 60%. The maximum possible cross section for the interaction of neutrons with nuclei in the energy range 2–6 MeV does not exceed the geometrical cross section for the nucleus σΣ = πR2 = π (1.4 · 10-13 238 1/3) 2 ≈ 2.4 bar. Thus

The total number of neutrons per trapped will be η = η (235U) + η (238U) = 0.3 +0.3 = 0.6 <1. There is a possibility of a self-sustaining chain reaction of fission of 235U nuclei excited by fast neutrons. If we substitute in formula (21) the values ​​for neutrons with an energy of ~ 2 MeV: = 2.65; σа = 2.1 bar for 235U nuclei; σa ≈ 0.1 barn for 238U nuclei; 5σf = 2 bar for 235U cores; we find that at 8n / 5n <30 (corresponding to 235U enrichment of up to 3% or more), the total number of secondary neutrons per captured primary will exceed unity even without taking into account the fission of 238U nuclei.

The chain reaction to 235U is actively developing under the influence of thermal neutrons. However, when fused by thermal neutrons, fast neutrons are produced, which, before slowing down to thermal energy, can be absorbed. The radiation capture cross section of 238U is resonant in nature, i.e., it reaches very large values ​​in certain narrow energy ranges. In a homogeneous mixture, the probability of resonant absorption is too high for a chain reaction on thermal neutrons to occur. This difficulty is circumvented by locating uranium in the moderator discretely, in the form of blocks forming the correct lattice.

The resonance absorption of neutrons in such a heterogeneous system sharply decreases for two reasons: 1) the cross section of resonance absorption is so large that neutrons entering the block are absorbed in the surface layer, so the inside of the block is screened and a significant part of the uranium atoms does not participate in resonance absorption: 2) the neutrons of resonant energy generated in the moderator may not get into uranium, but, slowing down when scattering on the nuclei of the moderator, “escape” from the dangerous energy range. Slowing down, neutrons that escaped absorption diffuse from the graphite to the uranium block, where they cause 235U fission.

Since the number of fissions and, consequently, the number of secondary neutrons in a breeding medium is proportional to its volume, and their outflow (leakage) is proportional to the surface of the environment, a nuclear chain reaction is possible only in a medium of sufficiently large sizes. For example, for a ball of radius, the ratio of volume to surface is R / 3, and therefore, the larger the radius of the ball, the smaller

neutron leak. If the radius of the propagating medium becomes large enough so that a stationary chain reaction takes place in the system, i.e., R-1 = 0, then such a system is called critical, and its radius is the critical radius. A nuclear chain reaction is carried out on 235U enriched uranium and in pure 235U. In these cases, it goes on fast neutrons. Upon absorption of neutrons in 238U, 239Np is formed, and after two β decays, 239Pu is formed, which is fused by thermal neutrons, with n = 2.9. Upon irradiation with 232Th neutrons, 233U fissile on thermal neutrons is formed.

In addition, a chain reaction is possible in 231Pu and Cm and Cf isotopes with an odd mass number. Of the u neutrons generated in 1 fission event, one goes on to continue the chain, and if losses are reduced, more than one neutron can be saved for the reproduction of nuclear fuel, which can lead to expanded reproduction of the fuel.

The Nuclear reaction fission with suitable examples and sketches are discuss here advanced. Nuclear reaction fission is needed to read as engineer student.

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