The finer classification takes into account the two groups of neutrons that are produced in fission.

The prompt critical state is defined as:

• k_eff > 1; ρ ≥ β_eff, where the reactivity of a reactor is higher than the effective delayed neutron fraction. In this case, the production of prompt neutrons alone is enough to balance neutron losses and increase the neutron population. The number of neutrons is exponentially increasing in time (as rapidly as the prompt neutron generation lifetime ~10^-5s).

The prompt subcritical and delayed supercritical state is defined as:

• k_eff > 1; 0 < ρ < β_eff, where the reactivity of a reactor is higher than zero and lower than the effective delayed neutron fraction. In this case, the production of prompt neutrons alone is insufficient to balance neutron losses and the delayed neutrons are needed in order to sustain the chain reaction. The neutron population increases, but much more slowly (as the mean generation lifetime with delayed neutrons ~0.1 s).

The prompt subcritical and delayed critical state is defined as:

• k_eff = 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 in order 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:

• k_eff < 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.

Calculate the reactivity in the reactor core when k_eff is equal to 1.005.

The reactivity is given by following formulas.

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:

N_n=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, so the increase from 1000 neutrons the 1105 neutrons occurs within 10 seconds.

See also: Neutron Generation – Neutron Population

The delayed neutron fraction, β, is the fraction of delayed neutrons in the core at creation, that is, at high energies. But in case of thermal reactors the fission can be initiated mainly by thermal neutron. Thermal neutrons are of practical interest in study of thermal reactor behaviour. The effective delayed neutron fraction, usually referred to as β_eff, is the same fraction at thermal energies.

The effective delayed neutron fraction reflects the ability of the reactor to thermalize and utilize each neutron produced. The β is not the same as the β_eff due to the fact delayed neutrons do not have the same properties as prompt neutrons released directly from fission. In general, delayed neutrons have lower energies than prompt neutrons. Prompt neutrons have initial energy between 1 MeV and 10 MeV, with an average energy of 2 MeV. Delayed neutrons have initial energy between 0.3 and 0.9 MeV with an average energy of 0.4 MeV.

Therefore in thermal reactors a delayed neutron traverses a smaller energy range to become thermal and it is also less likely to be lost by leakage or by parasitic absorption than is the 2 MeV prompt neutron. On the other hand, delayed neutrons are also less likely to cause fast fission, because their average energy is less than the minimum required for fast fission to occur.

These two effects (lower fast fission factor and higher fast non-leakage probability for delayed neutrons) tend to counteract each other and forms a term called the importance factor (I). The importance factor relates the average delayed neutron fraction to the effective delayed neutron fraction. As a result, the effective delayed neutron fraction is the product of the average delayed neutron fraction and the importance factor.

β_eff = β . I

The delayed and prompt neutrons have a difference in their effectiveness in producing a subsequent fission event. Since the energy distribution of the delayed neutrons differs also from group to group, the different groups of delayed neutrons will also have a different effectiveness. Moreover, a nuclear reactor contains a mixture of fissionable isotopes. Therefore, in some cases, the importance factor is insufficient and an importance function must be defined.

For example:

In a small thermal reactor with highly enriched fuel, the increase in fast non-leakage probability will dominate the decrease in the fast fission factor, and the importance factor will be greater than one.

In a large thermal reactor with low enriched fuel, the decrease in the fast fission factor will dominate the increase in the fast non-leakage probability and the importance factor will be less than one (about 0.97 for a commercial PWR).

In large fast reactors, the decrease in the fast fission factor will also dominate the increase in the fast non-leakage probability and the β_eff is less than β by about 10%.

Despite the fact the number of delayed neutrons per fission neutron is quite small (typically below 1%) and thus does not contribute significantly to the power generation, they play a crucial role in the reactor control and are essential from the point of view of reactor kinetics and reactor safety. Their presence completely changes the dynamic time response of a reactor to some reactivity change, making it controllable by control systems such as the control rods.

Delayed neutrons allow to operate a reactor in a prompt subcritical, delayed critical condition. All power reactors are designed to operate in a delayed critical conditions and are provided with safety systems to prevent them from ever achieving prompt criticality.

For typical PWRs, the prompt criticality occurs after positive reactivity insertion of β_eff (i.e. k_eff ≈ 1.006 or ρ = +600 pcm). In power reactors such a reactivity insertion is practically impossible to insert (in case of normal and abnormal operation), especially when a reactor is in power operation mode and a reactivity insertion causes a heating of a reactor core. Due to the presence of reactivity feedbacks the positive reactivity insertion is counterbalanced by the negative reactivity from moderator and fuel temperature coefficients. The presence of delayed neutrons is of importance also from this point of view, because they provide time also to reactivity feedbacks to react on undesirable reactivity insertion.

Reactivity is not directly measurable and therefore most power reactors procedures do not refer to it and most technical specifications do not limit it. Instead, they specify a limiting rate of neutron power rise (measured by excore detectors), commonly called a reactor period (especially in case of BWRs).

The reactor period, τ_e, or e-folding time, is defined as the time required for the neutron density to change by a factor e = 2.718. The reactor period is usually expressed in units of seconds or minutes.

At this time:

where:
n(t) = transient reactor power
n(0) = initial reactor power
τ_e = reactor period

The smaller the value of τ_e, the more rapid the change in reactor power. The reactor period may be positive or negative. If the reactor period is positive, reactor power is increasing. If the reactor period is negative, reactor power is decreasing. If the reactor period is constant with time, as associated with exponential power change, the rate is referred to as a stable reactor period. If the reactor period is not constant but is changing with time, as for non-exponential power change, the period is referred to as a transient reactor period.

Derivation of the formula τ_e = l_d / (k-1) is based on many assumptions and it is only simplest approximation of the reactor period. A much more exact formula reactor period is based on solutions of six-group point kinetics equations. From these equation an equation called the inhour equation (which comes from inverse hour, when it was used as a unit of reactivity that corresponded to e-fold neutron density change during one hour) may be derived.

where:
l = prompt neutron lifetime
β_eff = effective delayed neutron fraction
λ_eff = effective delayed neutron precursor decay constant
τ_e = reactor period
ρ = reactivity

The first term in this formula is the prompt term and it causes that the
positive reactivity insertion is followed immediately by a immediate power increase called the prompt jump. This power increase occurs because the rate of production of prompt neutrons
changes immediately as the reactivity is inserted. After the prompt jump, the rate of change of power cannot increase any more rapidly than the built-in time delay the precursor half-lives allow. Therefore the second term in this formula is called the delayed term. The presence of delayed neutrons causes the power rise to be controllable and the reactor can be controlled by control rods or another reactivity control mechanism.

The relationship between reactor period and startup rate is given by following equations:

Example:

Suppose k_eff = 1.0005 in a reactor with a generation time l_d = 0.01s. For this state calculate the reactor period – τ_e, doubling time – DT and the startup rate (SUR).

ρ = 1.0005 – 1 / 1.0005 = 50 pcm

τ_e = l_d / k-1 = 0.1 / 0.0005 = 200 s

DT = τ_e . ln2 = 139 s

SUR = 26.06 / 200 = 0.13 dpm

Doubling time is unit similar as in radioactive decay calculations. Doubling is defined as the amount of time it takes reactor power to double the initial power level. The reactor period is usually expressed in units of seconds or minutes. If the reactor period is known, doubling time can be determined as follows.

Doubling time = τ_e . ln2

where:
τ_e = reactor period
ln2 = natural logarithm of 2

The smaller the value of DT, the more rapid the change in reactor power. The doubling time may be positive or negative. If the doubling time is positive, reactor power is increasing. If the value is negative, we talk about the halving time and reactor power is decreasing.

Example:

Suppose k_eff = 1.0005 in a reactor with a generation time l_d = 0.01s. For this state calculate the reactor period – τ_e, doubling time – DT and the startup rate (SUR).

ρ = 1.0005 – 1 / 1.0005 = 50 pcm

τ_e = l_d / k-1 = 0.1 / 0.0005 = 200 s

DT = τ_e . ln2 = 139 s

SUR = 26.06 / 200 = 0.13 dpm

Reactivity is not directly measurable and therefore most power reactors procedures do not refer to it and most technical specifications do not limit it. Instead, they specify a limiting rate of neutron power rise (measured by excore detectors), commonly called a startup rate (especially in case of PWRs).

The reactor startup rate is defined as the number of factors of ten that power changes in one minute. Therefore the units of SUR are powers of ten per minute, or decades per minute (dpm). The relationship between reactor power and startup rate is given by following equation:

n(t) = n(0).10^SUR.t

where:
SUR = reactor startup rate [dpm – decades per minute]
t = time during reactor transient [minute]

The higher the value of SUR, the more rapid the change in reactor power. The startup rate may be positive or negative. If SUR is positive, reactor power is increasing. If SUR is negative, reactor power is decreasing. The relationship between reactor period and startup rate is given by following equations:

Example:

Suppose k_eff = 1.0005 in a reactor with a generation time l_d = 0.01s. For this state calculate the reactor period – τ_e, doubling time – DT and the startup rate (SUR).

ρ = 1.0005 – 1 / 1.0005 = 50 pcm

τ_e = l_d / k-1 = 0.1 / 0.0005 = 200 s

DT = τ_e . ln2 = 139 s

SUR = 26.06 / 200 = 0.13 dpm

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.

Advanced Reactor Physics:

1. K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.
2. K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.
3. D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2. 
4. E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.