# Subcritical Reactor

## Subcritical – Critical – Supercritical Reactor

The basic classification of states of a reactor is according to the multiplication factor as eigenvalue, which measures the change in the fission neutron population from one neutron generation to the subsequent generation.

• keff < 1. Suppose the multiplication factor for a multiplying system is less than 1.0. In that case, the number of neutrons decreases in time (with the mean generation time), and the chain reaction will never be self-sustaining. This condition is known as the subcritical state.
• keff = 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 self-sustaining. This condition is known as the critical state.
• keff > 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 self-sustaining. The number of neutrons is exponentially increasing in time (with the mean generation time). This condition is known as the supercritical state.

The definitions described above are fully applicable to a reactor at zero power level, that is, at such a power level. All thermal considerations are not important to the chain reaction (from zero power to 1% of rated power).

Six Factor Formula
The effective multiplication factor (keff) 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 non-leakage probability) and neutron leakage during neutron diffusion (thermal non-leakage probability) by following equation, usually known as the six factor formula:

keff = k . Pf . Pt

Effective Multiplication Factor in Reactor Kinetics
The simplest equation that governs the neutron kinetics of the system with delayed neutrons is the point kinetics equation. This equation states that the time change of the neutron population is equal to the excess of neutron production (by fission) minus neutron loss by absorption in one mean generation time with delayed neutrons (ld). The role of ld is evident. Longer lifetimes give simply slower responses to multiplying systems. 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 the 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 = ld / (k-1) = 0.085 / (1.0001-1) = 850sThis is a very long period. In ~14 minutes, the reactor’s neutron flux (and power) would increase by a factor of e = 2.718. This is a completely different dimension of the response on reactivity insertion compared to the case without delayed neutrons, where the reactor period was 1 second. Reactors with such kinetics would be quite easy to control. From this point of view, it may seem that reactor control will be quite a boring affair. It will not! The presence of delayed neutrons entails many specific phenomena that will be described in later chapters.
Interactive Chart - Reactor Kinetics
Press the “clear and run” button and try to increase the power of the reactor.Compare the response of the reactor with the case of Infinite Multiplying System Without Source and without Delayed Neutrons (or set the β = 0).
Example - Neutron Generation and Neutron Population
The number of neutrons (the neutron population) in the core at time zero is 1000 and k = 1.001 (~100 pcm).

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:

Nn=N0. (k)n
N1=N0.1.001 = 1001 neutrons after one generation
N2=N0.1.001.1.001 = 1002 neutrons after two generations
N3=N0.1.001.1.001.1.001 = 1003 neutrons after three generations
.
.

N50=N0. (k)50 = 1051 neutrons after fifty generations.

.

.

N100=N0. (k)100 = 1105 neutrons after hundred generations.

If we consider the mean generation time to be ~0.1s, the increase from 1000 neutrons to 1105 neutrons occurs within 10 seconds.

## Prompt Criticality

The basic classification of states of a reactor may be insufficient, and a finer classification is needed. The finer classification takes into account the two groups of neutrons that are produced in fission.

The prompt critical state is defined as:

The prompt subcritical and delayed supercritical state is defined as:

The prompt subcritical and delayed critical state is defined as:

• keff = 1; ρ = 0, where the reactivity of a reactor is equal to zero. In this case, the production of prompt neutrons alone is insufficient to balance neutron losses, and the delayed neutrons are needed to sustain the chain reaction. There is no change in neutron population in time, and the chain reaction will be self-sustaining. This state is the same state as the critical state from basic classification.

The prompt subcritical and delayed subcritical state is defined as:

• keff < 1; ρ < 0, where the reactivity of a reactor is lower than zero. In this case, the production of all neutrons is insufficient to balance neutron losses, and the chain reaction is not self-sustaining. If the reactor core contains external or internal neutron sources, the reactor is in the state that is usually referred to as the subcritical multiplication.

References:
Nuclear and Reactor Physics:
1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
5. W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
6. G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
7. Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
8. U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.