Nuclear Chain Reaction
The chain reaction can take place only in the proper multiplication environment and only under proper conditions. It is obvious, if one neutron causes two further fissions, the number of neutrons in the multiplication system will increase in time and the reactor power (reaction rate) will also increase in time. In order to stabilize such multiplication environment, it is necessary to increase the nonfission neutron absorption in the system (e.g. to insert control rods). Moreover, this multiplication environment (the nuclear reactor) behaves like the exponential system, that means the power increase is not linear, but it is exponential.
On the other hand, if one neutron causes less than one further fission, the number of neutrons in the multiplication system will decrease in time and the reactor power (reaction rate) will also decrease in time. In order to sustain the chain reaction, it is necessary to decrease the nonfission neutron absorption in the system (e.g. to withdraw control rods).
In fact, there is always a competition for the fission neutrons in the multiplication environment, some neutrons will cause further fission reaction, some will be captured by fuel materials or nonfuel materials and some will leak out of the system.
In order to describe the multiplication system, it is necessary to define the infinite and finite multiplication factor of a reactor. The method of calculations of multiplication factors has been developed in the early years of nuclear energy and is only applicable to thermal reactors, where the bulk of fission reactions occurs at thermal energies. This method well puts into the context all the processes, that are associated with the thermal reactors (e.g. the neutron thermalisation, the neutron diffusion or the fast fission), because the most important neutronphysical processes occur in energy regions that can be clearly separated from each other. In short, the calculation of multiplication factor gives a good insight in the processes that occur in each thermal multiplying system.
Infinite Multiplication Factor – Four Factor Formula
The required condition for a stable, selfsustained fission chain reaction in a multiplying system (in a nuclear reactor) is that exactly every fission initiate another fission. The minimum condition is for each nucleus undergoing fission to produce, on the average, at least one neutron that causes fission of another nucleus. Also the number of fissions occurring per unit time (the reaction rate) within the system must be constant.
This condition can be expressed conveniently in terms of the multiplication factor. The infinite multiplication factor is the ratio of the neutrons produced by fission in one neutron generation to the number of neutrons lost through absorption in the preceding neutron generation. This can be expressed mathematically as shown below.
It is obvious the infinite multiplication factor in a multiplying system is a measure of the change in the fission neutron population from one neutron generation to the subsequent generation.
 k_{∞} < 1. If the multiplication factor for a multiplying system is less than 1.0, then the number of neutrons is decreasing in time (with the mean generation time) and the chain reaction will never be selfsustaining. This condition is known as the subcritical state.
 k_{∞} = 1. If the multiplication factor for a multiplying system is equal to 1.0, then there is no change in neutron population in time and the chain reaction will be selfsustaining. This condition is known as the critical state.
 k_{∞} > 1. If the multiplication factor for a multiplying system is greater than 1.0, then the multiplying system produces more neutrons than are needed to be selfsustaining. The number of neutrons is exponentially increasing in time (with the mean generation time). This condition is known as the supercritical state.
Calculate the number of neutrons after 100 generations. Let say, the mean generation time is ~0.1s.
Solution:
To calculate the neutron population after 100 neutron generations, we use following equation:
N_{n}=N_{0}. (k_{∞})^{n}
N_{1}=N_{0}.1.001 = 1001 neutrons after one generation
N_{2}=N_{0}.1.001.1.001 = 1002 neutrons after two generations
N_{3}=N_{0}.1.001.1.001.1.001 = 1003 neutrons after three generations
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N_{50}=N_{0}. (k_{∞})^{50} = 1051 neutrons after fifty generations.
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N_{100}=N_{0}. (k_{∞})^{100} = 1105 neutrons after hundred generations.
If we consider the mean generation time to be ~0.1s, so the increase from 1000 neutrons the 1105 neutrons occurs within 10 seconds.
See also: Neutron Generation – Neutron Population
If there are neutrons in the system at t=0, that is, if n(0) > 0, the solution of this equation gives the simplest point kinetics equation with delayed neutrons (similarly to the case without delayed neutrons):Let us consider that the mean generation time with delayed neutrons is ~0.085 and k (k_{∞} – neutron multiplication factor) will be step increased by only 0.01% (i.e. 10pcm or ~1.5 cents), that is k_{∞}=1.0000 will increase to k_{∞}=1.0001.
It must be noted such reactivity insertion (10pcm) is very small in case of LWRs. The reactivity insertions of the order of one pcm are for LWRs practically unrealizable. In this case the reactor period will be:
T = l_{d} / (k_{∞}1) = 0.085 / (1.00011) = 850s
This is a very long period. In ~14 minutes the neutron flux (and power) in the reactor would increase by a factor of e = 2.718. This is completely different dimension of the response on reactivity insertion in comparison with the case without presence of delayed neutrons, where the reactor period was 1 second.
Reactors with such a kinetics would be quite easy to control. From this point of view it may seem that reactor control will be a quite boring affair. It will not! The presence of delayed neutrons entails many many specific phenomena, that will be described in later chapters.
Compare the response of the reactor with the case of Infinite Multiplying System Without Source and without Delayed Neutrons (or set the β = 0).
See also: Fast Fission Factor
See also: Resonance Escape Probability
describes how effectively (how well are utilized) are thermal neutrons absorbed in the fuel.
The value of the thermal utilization factor is given by the ratio of the number of thermal neutrons absorbed in the fuel (all nuclides) to the number of thermal neutrons absorbed in all the material that makes up the core.
See also: Thermal Utilisation Factor
of thermal neutrons absorbed in the fuel.
See also: Reproduction Factor
k_{∞} = η.ε.p.f
In reactor physics, k_{∞} or its finite form k_{eff} is the most significant parameter with regard to reactor control. At any specific power level or condition of the reactor, k_{eff} is kept as near
to the value of 1.0 as possible. At this point in operation, the neutron balance is kept to exactly one neutron completing the life cycle for each original neutron absorbed in the fuel.
From infinite to finite multiplication factor
The effective multiplication factor (k_{eff}) may be expressed mathematically in terms of the infinite multiplication factor (k_{∞}) and two additional factors which account for neutron leakage during neutron thermalisation (fast nonleakage probability) and neutron leakage during neutron diffusion (thermal nonleakage probability) by following equation, usually known as the six factor formula:
k_{eff} = k_{∞} . P_{f} . P_{t}
Operational factors that affect the fission chain reaction in PWRs.
If any operational factor changes one of the contributing factors to k_{eff} (k_{eff} = η.ε.p.f.P_{f}.P_{t}), the ratio of 1.0 is not maintained and this change in k_{eff} makes the reactor either subcritical or supercritical. Some examples of these operational changes, that may take place in PWRs, are below and are described below:
Control rods (insertion/withdrawal) influences the thermal utilization factor. For example, control rods insertion causes an addition of new absorbing material into the core and this causes a decrease in thermal utilization factor.
The thermal utilization factor for heterogeneous reactor cores must be calculated in terms of reaction rates and volumes, for example, by the following equation:
where Σ_{a} is the macroscopic absorption cross section, which is the sum of the capture cross section and the fission cross section, Σ_{a} = Σ_{c} + Σ_{f}. The superscripts U, M, P, CR, B, BA and O, refer to uranium fuel, moderator, poisons, control rods, boric acid, burnable absorbers and others, respectively. It is obvious, that the presence control rods, boric acid or poisons causes a decrease in the neutron utilization, which, in turn, causes a decrease of multiplication factor.
In comparison with the chemical shim, which offset positive reactivity excess in entire core, with control rods the unevenness of neutronflux density in the reactor core may arise, because they act locally.
↑boron ⇒ ↓k_{eff} = η.ε.p. ↓f .P_{f}.P_{t}
The concentration of boric acid diluted in the primary coolant influences the thermal utilization factor. For example, an increase in the concentration of boric acid (chemical shim) causes an addition of new absorbing material into the core and this causes a decrease in thermal utilization factor.
The thermal utilization factor for heterogeneous reactor cores must be calculated in terms of reaction rates and volumes, for example, by the following equation:
where Σ_{a} is the macroscopic absorption cross section, which is the sum of the capture cross section and the fission cross section, Σ_{a} = Σ_{c} + Σ_{f}. The superscripts U, M, P, CR, B, BA and O, refer to uranium fuel, moderator, poisons, control rods, boric acid, burnable absorbers and others, respectively. It is obvious, that the presence control rods, boric acid or poisons causes a decrease in the neutron utilization, which, in turn, causes a decrease of multiplication factor.
When compared with burnable absorbers (long term reactivity control) or with control rods (rapid reactivity control) the boric acid avoids the unevenness of neutronflux density in the reactor core, because it is dissolved homogeneously in the coolant in entire reactor core. On the other hand high concentrations of boric acid may lead to positive moderator temperature coefficient and that is undesirable. In this case more burnable absorbers must be used.
Moreover this method is slow in controlling reactivity. Normally, it takes several minutes to change the concentration (dilute or borate) of the boric acid in the primary loop. For rapid changes of reactivity control rods must be used.
↑T_{M} ⇒ ↓k_{eff} = η.ε. ↓p .f. ↓P_{f }. ↓P_{t } (EOC)
This operational change is very difficult to describe, because changes in moderator temperature lead to the change of almost all the coefficients. Major impacts on multiplication of the system arise from the change of the resonance escape probability and the change of total neutron leakage (see thermal nonleakage probability and fast nonleakage probability).
 Change of the resononce escape probability. It is known, the resonance escape probability is dependent also on the moderatortofuel ratio. All PWRs are designed as undermoderated reactors. As the moderator temperature increases the ratio of the moderating atoms (molecules of water) decreases as a result of the thermal expansion of water. Its density simply decreases. This, in turn, causes a hardening of neutron spectrum in the reactor core resulting in higher resonance absorption (lower p). Decreasing density of the moderator causes that neutrons stay at a higher energy for a longer period, which increases the probability of nonfission capture of these neutrons. This process is one of two processes, which determine the moderator temperature coefficient (MTC). The second process is connected with the leakage probability of the neutrons. The moderator temperature coefficient must be for most PWRs negative, which improves the reactor stability, because a reactor core heating causes a negative reactivity insertion.

Change of the thermal utilization factor. The impact on the thermal utilization factor depends strongly on the amount of boron which is diluted in the primary coolent (chemical shim). As the moderator temperature increases the density of water decreases due to the thermal expansion of water. But along with the moderator also boric acid is expanded out of the core. Since boric acid is a neutron poison, and it is expanding out of the core, positive reactivity is added. The positive reactivity addition due to the expansion of boron out of the core offsets the negative reactivity addition due to the expansion of the moderator out of the core. It is obvious this effect is significant at the beginning of the cycle (BOC) and gradually loses its significance as the boron concentration decreases.
 Change of the neutron leakage. Since both (P_{f} and P_{t}) are affected by a change in moderator temperature in a heterogeneous watermoderated reactor and the directions of the feedbacks is the same, the resulting total nonleakage probability is also sensitive on the change in the moderator temperature. In result, an increase in the moderator temperature causes that the probability of leakage increases. In case of the fast neutron leakage, the moderator temperature influences macroscopic crosssections for elastic scattering reaction (Σ_{s}=σ_{s}.N_{H2O}) due to the thermal expansion of water, which results in an increase in the moderation length. This, in turn, causes an increase of the leakage of fast neutrons.
 For the thermal neutron leakage there are two effects. Both processes have the same direction and together causes the increase in the thermal neutron leakage. This physical process is a part of the moderator temperature coefficient (MTC).
 Macroscopic crosssections for elastic scattering reaction Σ_{s}=σ_{s}.N_{H2O,} which significantly changes due to the thermal expansion of water. As the temperature of the core increases, the diffusion coefficient (D = 1/3.Σ_{tr}) increases.
 Microscopic crosssection (σ_{a}) for neutron absorption changes with core temperature. As the temperature of the core increases, the absorption crosssection decreases.
 For the thermal neutron leakage there are two effects. Both processes have the same direction and together causes the increase in the thermal neutron leakage. This physical process is a part of the moderator temperature coefficient (MTC).
Change in the fuel temperature affects primarily the resonance escape probability, which is connected with the phenomenon usually known as the Doppler broadening. The Doppler effect is generally considered to be most important effect, which improves the reactor stability. Especially in case of reactivity initiated accidents (RIA), the Doppler coefficient of reactivity would be the first in the compensation of the inserted positive reactivity. The time for heat to be transferred to the moderator is usually measured in seconds, while the Doppler coefficient is effective almost instantaneously. The Doppler broadening with the process of selfshielding causes, that the Doppler coefficient (or the fuel temperature coefficient) is for all power reactors always negative. Therefore an increase in the fuel temperature promptly causes an increase in the resonance integral (I_{eff}), which, in turn, causes a negative reactivity insertion. It is of the highest importance in the reactor safety.
See also: Doppler Broadening
See also: Selfshielding
Although water is considered to be incompressible, in reality, it is slightly compressible (especially at 325°C (617°F)). It is obvious, the effect of pressure in the primary circuit have similar consequences as the moderator temperature. In comparison with effects of moderator temperature changes, changes in pressure have of lower order impact on reactivity and the causes are only in the density of moderator, not in the change of microscopic crosssections.
The pressure coefficient of reactivity has a slightly positive effect on reactivity as the pressure of the system is increased if. At high boron concentrations, the pressure coefficient may reach negative values, but for many PWRs it is prohibited to operate under such conditions. Therefore burnable absorbers are usually added into the fuel, they lower initial boron concentration.
Note: Effects of the nuclate boiling of the primary coolant are not discussed here.
The effect of change in the flow rate through the primary circuit have identical consequences as the effects of the moderator temperature. In reality, when there is an abrupt change (e.g. as a result of a disconnection of the reactor coolant pump) in the flow rate and the reactor power remains the same, the difference between inlet and outlet temperatures must increase. It follows from basic energy equation of reactor coolant, which is below:
P=↓ṁ.c.↑∆t
The inlet temperature is determined by the pressure in the steam generators, therefore the inlet temperature changes minimally during the transient. It follows the outlet temperature must change significantly as the flow rate changes. When the inlet temperature remains almost the same and the outlet changes significantly, it stands to reason, the average temperature of coolant (moderator) will change also significantly. Therefore the effect of change in the flow rate through the primary circuit have identical consequences as the effects of the moderator temperature.
The decrease in flow rate is associated with negative reacivity insertion. Special attention is needed in case of an abrupt increase in the flow rate. At normal operation such increase in the flow rate can not occur, except the controlled reactor coolant pump connection, which can be connected only under specific conditions.
In pressurized water reactors, nucleate boiling may occur even during operational conditions. Nucleate boiling occurs when any surface of fuel cladding reach the saturation temperature (e.g. 350°C), which is determined by the pressure in the pressurizer (e.g. 16MPa). Such local nucleate boiling does not pose any problem for the reactor operation.
On the other hand during abnormal condition, boiling in the reactor core is one of the most important phenomena, that may take place in the core. From the reactivity point of view, nucleate boiling have very important consequences on the reactivity of the reactor core. Boiling affects reactivity in the same manner as the presence of voids and therefore it is characterized by the void coefficient.
The formation of voids in the core has the same effect as the change in the moderator temperature (change in the density of the moderator). In comparison with the change in the moderator temperature, boiling minimally affects the neutron leakage, because it is unlikely that local boiling occurs at the periphery of the reactor core, where the local power drops significantly.
↑burnable absorbers ⇒ ↓k_{eff} = η.ε.p. ↓f .P_{f}.P_{t}
The amount of burnable absorbers in the nuclear fuel influences the thermal utilization factor. In some cases (espacially in the case of gadolinium absorbers) the presence of burnable absorbers influences all the factors in the four factor formula due to very high selfshielding effects. But at this place we consider only the change in the thermal utilization factor. An increase in the amount of burnable absorbers causes an addition of new absorbing material into the core and this simply causes a decrease in thermal utilization factor.
The thermal utilization factor for heterogeneous reactor cores must be calculated in terms of reaction rates and volumes, for example, by the following equation:
where Σ_{a} is the macroscopic absorption cross section, which is the sum of the capture cross section and the fission cross section, Σ_{a} = Σ_{c} + Σ_{f}. The superscripts U, M, P, CR, B, BA and O, refer to uranium fuel, moderator, poisons, control rods, boric acid, burnable absorbers and others, respectively. It is obvious, that the presence control rods, boric acid or burnable absorbers causes a decrease in the neutron utilization, which, in turn, causes a decrease of multiplication factor.
It is hard to describe the effects of fuel burnup on the six factor formula. It must be noted, the criticality must be maintained for long period and therefore all the negative effects must be compensated by the increase in the thermal utilization factor (boron dilution or compensating rods withdrawal).
Thermal Utilization Factor
The thermal utilization factor slightly changes with the fuel burnup. The fresh fuel at the beginning of the cycle comprises only the absorption by the ^{235}U. As the amount of ^{239}Pu and other higher transuranic elements increases because of the radiative capture of neutron by the ^{238}U in the core, it is necessary to consider the change of fuel composition in determining the value of f at different times of the fuel cycle.
In general, the thermal utilization factor decreases in time as the total content of fissile isotopes decreases and the total content of neutron poisons (fission products with high absorption crosssections) increases. But in the power reactors, in which the criticality must be maintained for long period (e.g. 12month or up to 24month) without refueling, the thermal utilization factor may not decrease. Such the decrease would imply inevitable reactor shutdown. The continuous decrease in Σ_{a}^{U} must be offset by the continuous decrease in Σ_{a}^{B}, which means the concentration of boric acid (in case of PWRs) must be continuously decreased as the fuel loses its reactivity (k_{inf}). For reactors, in which the chemical shim can not be used, the excess of reactivity is compensated by compensating rods.
Reproduction Factor
There is essentially small change in η over the lifetime of the reactor core (decreases).This is due to the fact there is a continuous decrease in Σ_{f}^{U}, but on the other hand this decrease is partially offset by the increase in Σ_{f}^{Pu}. As the fuel burnup increases, the ^{239}Pu begins to contribute to the neutron economy of the core.
See also: Nuclear Breeding
Neutron Leakage
In power reactors, the total nonleakage probability also significantly changes with fuel burnup. This dependency is not associated with any of the parameters like the diffusion coefficient or the geometrical buckling. In power reactors, the total nonleakage probability strongly depends on the certain fuel loading pattern and also on the reload strategy. The neutron leakage is one of key parameters in the neutron and fuel economy.
In order to enhance the neutron and fuel economy, core designers designs the low leakage loading patterns, in which fresh fuel assemblies are not situated in the peripheral positions of the reactor core. The peripheral positions are loaded with the fuel with highest fuel burnup. These “high” burnup assemblies have inherently lower relative power (due to the lower k_{inf} and due to the fact they feel the presence of nonmultiplying environment) in comparison with the average assemblies. In short, this parameter is significantly dependent on the certain loading pattern. During fuel burnup, the neutron leakage usually increases, especially in low leakage loading patterns. This process is caused by reducing the differences in k_{inf} between fresh fuel assemblies and peripheral highburnup assemblies.
 J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., AddisonWesley, Reading, MA (1983).
 J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., PrenticeHall, 2001, ISBN: 0201824981.
 W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0 471391271.
 Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 9780412985317
 W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 9780198520467
 G.R.Keepin. Physics of Nuclear Kinetics. AddisonWesley Pub. Co; 1st edition, 1965
 Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
 U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.
Advanced Reactor Physics:
 K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0894480332.
 K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0894480294.
 D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0894484532.
 E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0894484524.
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