VI. Chain Reaction VI.1. Basic of Chain Reaction Two basic requirements must be filled in order to produce power in a reactor: The fission rate should be high. This rate must be continuously maintained. The fission process may be repeated due to the fact that each fission releases neutrons. The condition for the maintaining of a chain reaction is that the neutron produced in a fission be capable of producing another fission. This process can be seen in the next figure. Figure 1 Such a chain reaction can be described qualitatively by the multiplication factor, which is indicated by the symbol k and is defined as the ratio of the number of fission neutrons in one generation divided by the number of fissions neutrons of the preceding generation. K= number of fission neutrons in one generation number of fission neutrons in preceding generation If k is equal to one the chain reaction progresses at constant rate and the system is critical If k is greater than one the chain reaction progress increases the fission neutrons and the system is supercritical If k is less than one the chain reaction progress decreases the fission neutrons and the system is subcritical In a nuclear reactor the fission chain reaction can proceed in a controlled manner, this control is done by varying the value of k through manipulation of the control rods. For example to increase the power, the operator will increase k to a value greater than 1 by removal of the control rods and the reactor becomes supercritical; when the desired power has been reached the operator returns the reactor to critical condition (k=1) by insertion of the control rods. To reduce the power the operator decreases k to a value less than 1 and the reactor becomes subcritical, and the power decreases. The definition of multiplication factor can also be understood as a neutron balance between the rate at which they are produced and the rate at which they disappear. Neutrons can disappear in two ways: they can be absorbed or they can escape from the surface of the reactor. 1
k= production rate (absorption + leakage) rates If the production rate is greater than the sum of absorption and leakage rates the reactor is supercritical and if it is smaller the reactor is subcritical. VI.2. Example of chain reaction with natural uranium Since more than one neutron is generally produced per fission (n = 2.43 for U 235 with thermal neutrons), it would appear to be a simple matter to utilize one of them to produce another fission. We will see what happens in practice when we analyze the case of natural uranium. Suppose we take 100 fission neutrons in a block of natural uranium, of which, 70 are above the energy threshold for U 238, which is equal to 1.2 MeV, and the remaining 30 neutrons are below this energy level. Figure 2 shows the microscopic cross sections for U 235. Figure 3 shows the microscopic cross sections for U 238. Figure 4 shows the microscopic cross sections for natural uranium. Figure 2. The microscopic cross sections for U 235. 2
Figure 3. The microscopic cross sections for U 238 Figure 4. The microscopic cross sections for natural uranium. 3
In analyzing the 30 neutrons mentioned above (see graph), we see that the majority of them are scattered as a result of elastic collisions with U 238 (n,n)u 238, producing a moderation of these neutrons, and with no considerable loss of energy, due to the great difference in masses. In (n,n')u 238 reactions, the neutrons lose considerable energy, leaving the nucleus of the uranium excited. Following many collisions, the neutrons will drop to an energy level of under 100 kev. At these energy levels, the greatest probability of reaction is for (n,γ)u 238, which exceeds the probability of fission by 40 times, so that less than one of the 30 neutrons in question will produce a fission, with the rest being captured by U 238. The 70 neutrons whose energy level is above 1.2 MeV will undergo different reactions in direct proportion to their neutron cross sections. In the first collisions we have the following: 38 neutrons react as (n,n) U 238 27 neutrons react as (n,n') U 238 About 4 neutrons react as (n,f) U 238, and approximately 1 neutron reacts as (n,γ) U 238, with a slight probability of interaction with U 235. The 38 elastically scattered neutrons take part in a second collision with the uranium, whose possible reactions are approximately 2 neutrons (n,f)u 238, 15 neutrons (n,n')u 238 with energies of under 1.2 MeV, and 21 neutrons (n,n)u 238 of 1.2 MeV. These 21 neutrons produce a third collision and those obtained from that collision produce another one and so on, following which, of the 100 initial neutrons, 8 produce fission with U 238 and 2 with U 235, with energy levels of about 0.14 ev. The values assigned in the distribution of the number of neutrons out of the 70 that surpass the threshold of U 238 were obtained via the quantification of the neutron cross sections for each type of reaction, as seen in the previous Figure 2, Figure 3 and Figure 4. 0.3 = value of reaction (n,n) U 238 0.2 = value of reaction (n,n ) U 238 0.03 = value of reaction (n,f) U 238 or, that is to say, 57% of the initial 70 neutrons produce an (n,n) U 238 reaction. In the case of the 27 (n,n') U 238 reaction neutrons: 70 57% = 38 (n,n') U 238 neutrons 4
In the case of the 4 (n,f) U 238 reaction neutrons: 70 38% = 27 (n,n') U 238 neutrons 70 6% = 4.2 (n,f) U 238 neutrons CONCLUSION From the foregoing analysis (10 fissions) we obtain the number of new fission neutrons. 10 fissions 2.5 = 25 fission neutrons On analyzing the multiplication factor, that is to say, the relationship between available neutrons in the 1st generation and those obtained in the 2nd generation: This is a value lower than k=1, which is a condition for the maintaining of the chain reaction. Thus, the system is described as being sub-critical. SYSTEMS THAT PRODUCE CHAIN REACTIONS We have already seen that it is impossible to maintain the chain reaction in a system using natural uranium alone, but there are ways of achieving this: Fast Reactors One way is to increase the quantity of U 235 in relation to U 238, so as to augment the probability of U 235 fission at energies of under 1.2 MeV. If, for example, of the neutrons that enter this region, 32 produce fission in U 235, the chain reaction can be maintained, or, that is to say: 100 neutrons 8 neutrons fission with U 238 32 neutrons fission with U 235 If we assume that n = 2.5 we have 40 fissions x 2.5 neutrons/fission = 100 neutrons 5
in the second generation and k = 1. This process is known as fuel enrichment, a process which has given birth to the development of fast reactors. Thermal Reactors Another way of maintaining the chain reaction is by adding material with a low atomic number to natural uranium. The main objective here is to slow down neutrons to a low energy level (0.025 ev), with a material that produces elastic scattering and does no possess appreciable absorption. This process of slowing down neutrons is known as moderation, and it takes neutrons to thermal energy levels at which the probability of fission with U 235 is very high (about 580 barns, compared with 1.2 barns at 1 MeV), whereas at this same energy level, the probability of radioactive capture in U 238 is 2.7 barns. Even considering the low U 235 content in natural uranium, fission is more probable than capture. The materials that achieve this effect in neutrons are called moderators. The reactor which possesses a large mass of moderator and where the majority of the neutrons reach thermal energy levels (thermal equilibrium with the atoms of the moderator) are called thermal reactors. VI.3. TYPES OF REACTORS The nuclear reactors can be classified according to different criteria. The next paragraphs present two conceptual classification: 1) Utilization of the reactor. The fission process produces neutrons and energy. Based on this, there are conceptually two types of reactors: Research Reactors, which use the neutrons generated from the fission, and Power Reactors, which use the energy generated from the fission. 2) Energy of the neutrons which cause the fission in the system. Thermal Reactors, in which fissions are produced by neutrons that are approximately in thermal equilibrium with atoms in the system and have an energy below 0.3 ev. Fast Reactors, in which fissions are induced by neutrons with an energy above 100 kev. Intermediate Reactors, in which most of the fissions are produced by neutrons with an energy above thermal to about 10 kev. 6