## Point Kinetics Equations

As we have seen in previous chapters, the number of neutrons is multiplied by a factor k_{eff} from one neutron generation to the next, therefore the multiplication environment (nuclear reactor) behaves like the exponential system, that means the power increase is not linear, but it is **exponential**.

It is obvious** the effective 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**_{eff}**< 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 self-sustaining. This condition is known as**the subcritical state**.

**k**_{eff}**= 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**.

**k**_{eff}**> 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**.

But we have not yet discussed the **duration of a neutron generation**, that means,** how many times in a one second we have to multiply the neutron population by a factor k _{eff}**. This time determines the

**speed of the exponential growth**. But as was written, there are different types of neutrons: prompt neutrons and delayed neutrons, which completely change the kinetic behaviour of the system. Therefore such a discussion will be not trivial.

To study the kinetic behaviour of the system, engineers usually use the **point kinetics equations**. The name **point kinetics** is used because, in this simplified formalism, the **shape** of the neutron flux and the neutron density **distribution** are **ignored**. The reactor is therefore **reduced to a point**. In the following section we will introduce the point kinetics and we start with point kinetics in its** simplest form**.

**be the number of neutrons as a function of time**

*n(t)**t*and

*l*the

**prompt neutron lifetime, which**is the

**average time from a prompt neutron emission**to either

**its absorption**(fission or radiative capture) or to

**its escape**from the system. The average number of neutrons that disappear during a unit time interval

*dt*is

**But each disappearance of a neutron contributes an average of**

*n.dt/l.**k*new neutrons.

Finally, the change in number of neutrons during a unit time interval *dt *is:

**where:**

**n(t) = transient reactor power**

**n(0) = initial reactor power**

**τ**_{e}** = reactor period**

**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. The

**smaller**the value of

**τ**

**, the**

_{e}**more rapid**the change in reactor power. The reactor period may be positive or negative.

**the point kinetics equation**(in certain form). 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 prompt neutron lifetime**. The role of prompt neutron lifetime is evident. Shorter lifetimes give simply faster responses of 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 form of point kinetics equation (without source and without delayed neutrons).

This simple point kinetics equation is often expressed is terms of reactivity and prompt generation time, **Λ**, as:

where

**ρ**= (k-1)/k is the reactivity, which describes the**deviation of an effective multiplication factor from unity**.**Λ = l/k**_{eff}**= prompt neutron generation time,**which is the average time from a prompt neutron emission to an absorption that results only in fission.

Both forms of the point kinetics equation are valid. The equation using **Λ, prompt neutron generation time, **is usually better for calculations. This is because most reactivity transients are induced by changes in the absorption cross-section rather than in the fission cross-section. During this transients the prompt neutron lifetime is not constant whereas the prompt generation time remains constant.

Example:

Let us consider that **the prompt neutron lifetime is ~2 x 10**** ^{-5}** 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 / (k**_{∞ }**– 1) = 2 x 10**^{-5 }**/ (1.0001 – 1) = 0.2s**

**This is a very short period.** In one second the neutron flux (and power) in the reactor would increase by a factor of e^{5} = 2.718^{5}, in 10 seconds the reactor would pass through 50 periods and the power would increase by e^{50} = ……

Furthermore in case of fast reactors in which prompt neutron lifetimes are **of the order of 10**^{-7}** second**, the response of such a small reactivity insertion will be even more unimaginable. In case of 10^{-7} the period will be:

**T = l / (k**_{∞ }**– 1) = 10-7 / (1.0001 – 1) = 0.001s**

**Reactors with such a kinetics would be very difficult to control.** **Fortunately this behaviour is not observed** in any multiplying system. Actual reactor periods are observed to be considerably longer than computed above and therefore the nuclear chain reaction can be **controlled more easily**. The longer periods are observed due to the presence of **the delayed neutrons****.**

**point kinetics equation with delayed neutrons**. 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**

**(l**

_{d}**)**.

**l**_{d}** = (1 – β).l**_{p}** + ∑l**_{i}** . β**_{i}** => l**_{d}** = (1 – β).l**_{p}** + ∑τ**_{i}** . β**_{i}

where

**(1 – β)**is the fraction of all neutrons emitted as prompt neutrons**l**is the prompt neutron lifetime_{p}**τ**is the mean precursor lifetime, the inverse value of the decay constant_{i }**τ**_{i}**= 1/λ**_{i}- The weighted delayed generation time is given by
**τ = ∑τ**_{i}**. β**_{i}**/ β = 13.05 s** - Therefore the weighted decay constant
**λ = 1 / τ ≈ 0.08 s**^{-1}

The number, **0.08 s**** ^{-1}**, is relatively high and have

**a dominating effect of reactor time response**, although delayed neutrons are a small fraction of all neutrons in the core. This is best illustrated by calculating a weighted mean generation time with delayed neutrons:

**l**_{d}** = (1 – β).l**_{p}** + ∑τ**_{i}** . β**_{i}** = (1 – 0.0065). 2 x 10**^{-5}** + 0.085 = 0.00001987 + 0.085 ≈ 0.085**

In short, **the mean generation time with delayed neutrons** is about **~0.1 s**, rather than ~**10**** ^{-5}** as in section Prompt Neutron Lifetime, where the delayed neutrons were omitted.

The role of **l**** _{d}** is evident. Longer lifetimes give simply slower responses of multiplying systems. The role of reactivity (k

_{eff}– 1) is also evident. Higher reactivity gives simply larger response of multiplying system.

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****):**

Example:

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 (e.g. one step by control rods). 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 / (1.0001-1) = 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.

Both previous simple point kinetics equations are only an approximation, because they use many simplifications. The simple **point kinetics equation with delayed neutrons **completely fails for higher reactivity insertions, where is significant difference between the production of prompt and delayed neutrons. Therefore a more accurate model is required. The **exact point kinetics equations**, that can be derived from the general neutron balance equations without making any approximations are:

In the **equation for neutrons**, the first term on the right hand side is the production of prompt neutrons in the present generation, ** k(1-β)n/l**, minus the total number of neutrons in the preceding generation,

**. The second term is the production of delayed neutrons in the present generation. As can be seen, the rate of absorption of neutrons is the same as in the simple model (**

*-n/l***). But a distinction is between the direct channel for prompt neutrons**

*-n/l***production and the delayed channel resulting from radioactive decay of precursor nuclei (λ**

*(1-β)*_{i}C

_{i}).

In the **equation for precursors**, there is the balance between the production of the precursors of i-th group and their decay after the decay constant λ_{i}. As can be seen, the rate of the decay of precursors is the radioactivity rate (λ_{i}C_{i}) and the rate of production is proportional to the number of neutrons times **β**_{i}**, which **is defined as the fraction of the neutrons which appear as **delayed neutrons in the i th group**.

As can be seen, the point kinetics equations include two differential equations, one for the neutron density *n(t)* and the other for precursors concentration *C(t)*.

Again, the point kinetics equations are often expressed is terms of reactivity **(ρ = (k-1)/k)** and prompt generation time, **Λ**, as:

Both forms of the point kinetics equation are valid. The equation using **Λ, prompt neutron generation time, **is usually better for calculations. This is because most reactivity transients are induced by changes in the absorption cross-section rather than in the fission cross-section. During this transients the prompt neutron lifetime is not constant whereas the prompt generation time remains constant.

The previous equation defines the reactivity of a reactor, which describes the **deviation of an effective multiplication factor from unity**. For critical conditions the reactivity is equal to zero. The larger the absolute value of **reactivity** in the reactor core, the further the reactor is from **criticality**. In fact the reactivity may be used as a measure of a **reactor’s relative departure from criticality**. According to the reactivity, we can classify the different reactor states and the related consequences as follows:

**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**

^{-5}**s**).

**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**.

## Approximate Solution of Point Kinetics Equations

Sometimes, it is convenient to predict qualitatively the behaviour of a reactor. The exact solution can be obtained relatively easy using computers. Especially for illustration, the following approximations are discussed in the following sections:

- Prompt Jump Approximation
- Prompt Jump Approximation with One Group of Delayed Neutrons
- Constant Delayed Neutron Source Approximation

**ρ**<

**) causes at first a sharp change in prompt neutrons population and then the neutron response is slowed as a result of the more slowly changing number of delayed neutrons. The rapid response is a result of the small value of prompt neutron generation time in the denominator of point kinetics equation.**

*β*If we are interested in **long-term behaviour** (asymptotic period) and not interested in the details of the prompt jump, we can simplify the point kinetics equations by assuming that the **prompt jump takes place instantaneously** in response to any reactivity change. This approximation is known as the **Prompt Jump Approximation (PJA)** in which the rapid power change due to prompt neutrons is neglected, corresponding to taking ** dn/dt |_{0} = 0** in the point kinetics equations. That means the point kinetics equations are as follows:

From the equation for neutron flux and the assumption, that the delayed neutron precursor population does not respond instantaneously to a change in reactivity (i.e. C_{i,1} = C_{i,2}), it can be derived that the ratio of the neutron population just after and before the reactivity change is equal to:

The prompt-jump approximation is usually valid for smaller reactivity insertion, for example, for **ρ < 0.5 β. **It is usually used with another simplification, the

**one delayed precursor group approximation**.

**prompt jump approximation (PJA)**. This eliminated the fast time scale due to prompt neutrons. In this section we consider that delayed neutrons are produced only by

**one group of precursors**with the same decay constant (averaged) and delayed neutron fraction. Point kinetics equation using PJA and one group of delayed neutrons becomes:

This simplification then leads to:

Assuming that the reactivity is constant and n_{1}/n_{0} can be determined from prompt jump formula this equation leads to very simple formula:

**ρ**<

**) causes at first a sharp change in prompt neutrons population and then the neutron response is slowed as a result of the more slowly changing number of delayed neutrons. The rapid response is a result of the small value of prompt neutron generation time in the denominator of point kinetics equation.**

*β*If we are interested in short-term behaviour and not interested in the details of the asymptotic behaviour, we can simplify the point kinetics equations by assuming that the production of the delayed neutrons is constant and equal to the production at the beginning of the transient. This approximation is known as the **Constant Delayed Neutron Source Approximation** (CDS) in which changes in the amount of delayed neutrons is neglected, corresponding to taking dC_{i}(t)/dt = 0 and C_{i}(t) = C_{i,0} in the point kinetics equations. That means the point kinetics equations are as follows:

The above equation can be solved analytically and assuming that the reactivity is constant the solution is given as:

## Point Kinetics Equations – Notes

**Prompt neutron lifetime, l**, is the

**average time from a prompt neutron emission**to either

**its absorbtion**(fission or radiative capture) or to

**its escape**from the system. This parameter is defined in multiplying or also in nonmultiplying systems. In both systems the prompt neutron lifetimes depend strongly on:

- material composition of the system
- multiplying – nonmultiplying system
- system with or without thermalization
- isotopic composition of the system

- geometric configuration of the system
- homogeneous or heterogeneous system
- shape of entire system

- size of the system

In an infinite reactor (without escape) prompt neutron lifetime is the sum of **the slowing down time and the diffusion time**.

**l=t _{s} + t_{d}**

In an infinite thermal reactor **t _{s} << t_{d}** and therefore

**l ≈ t**. The typical prompt neutron lifetime

_{d}**in thermal reactors**is on the order of

**10**. Generally, the longer neutron lifetimes take place in systems in which the neutrons must be thermalized in order to be absorbed.

^{−4}secondSystems in which most of the neutrons are absorbed at higher energies and the neutron thermalization is suppressed (e.g. in fast reactors), have much shorter prompt neutron lifetimes . The typical prompt neutron lifetime **in fast reactors** is on the order of **10 ^{−7} second**.

In multiplying systems, in which the absorption of a prompt fission neutron can initiate a fission reaction, l is equal to the average time between two generations of prompt neutrons (at k_{eff}=1). This time is known as the **prompt neutron generation time**.

**Prompt Neutron Generation Time** (or **Mean Generation Time**), **Λ**, is the average time **from a prompt neutron emission** **to a capture that results only in fission**. The prompt neutron generation time is designated as:

**Λ = l/k _{eff}**

In power reactors **the prompt generation time** changes with the fuel burnup. In LWRs increases with the fuel burnup. It is simple, fresh uranium fuel contains much fissile material (in case of uranium fuel about 4% of ^{235}U). This causes significant excess of reactivity and this **excess must be compensated** via chemical shim (in case of PWRs) or via burnable absorbers.

Owing to these factors (high probability of absorption in fuel and high probability of absorption in moderator) the prompt neutron lives much shorter and prompt neutron lifetime is low. With fuel burnup the amount of fissile material as well as the absorption in moderator decreases and therefore the prompt neutron is able to “live”much longer.

**key neutron-physical characteristics**of reactor core. Its value depends especially on the type of the moderator and on the energy of the neutrons causing fission. Its importance for nuclear reactor safety is well known for a long time.

The longer prompt neutron lifetimes can substantially improve kinetic response of reactor (**the longer prompt neutron lifetime gives simply slower power increase**). For example under RIA conditions (**Reactivity-Initiated Accidents**) reactors should withstand a jump-like insertion of relatively large (~1 $ or even more) positive reactivity and the PNL (prompt neutron lifetime) plays here the key role. Therefore the PNL should be verified in a reload safety evaluation process.

In some cases (especially in some fast reactors) reactor cores or can be modified in order to increase the PNL and in order to improve nuclear safety.

Reactor-kinetic calculations with considering of such a number of initial conditions would be correct, but it also would be very complicated. Therefore G. R. Keepin and his co-workers suggested **to group together the precursors** based on their half-lives.Therefore delayed neutrons are traditionally represented by **six delayed neutron groups**, whose yields and decay constants (λ) are obtained from nonlinear least-squares fits to experimental measurements. This model has following disadvantages:

- All constants for each group of precursors are empirical fits to the data.

- They cannot be matched with decay constants of specific precursors.

- These constants are different for each fissionable nuclide.

- These constants change also with the neutron energy spectrum.

It was recognised that the half-lives in six-group structure **do not accurately reproduce** the asymptotic die-away time constants associated with the three longest-lived dominant precursors: ^{87}Br, ^{137}I and ^{88}Br.

**This model may be insufficient** especially in case of epithermal reactors, because virtually all delayed neutron activity measurements have been performed for fast or thermal-neutron-induced fission. In case of fast reactors, in which the nuclear fission of six fissionable isotopes of uranium and plutonium is important, the accuracy and energy resolution may play an important role.

**reactor kinetics**and in

**a subcriticality control**. Especially in nuclear reactors with D

_{2}O moderator (CANDU reactors) or with Be reflectors (some experimental reactors). Neutrons can be produced also in

**(γ, n) reactions**and therefore they are usually referred to as

**photoneutrons**.

A high energy photon (gamma ray) can under certain conditions **eject** a neutron from a nucleus. It occurs when **its energy exceeds** the binding energy of the neutron in the nucleus. Most nuclei have binding energies in excess of **6 MeV**, which is above the energy of most gamma rays from fission. On the other hand **there are few nuclei** with sufficiently low binding energy to be of **practical interest**. These are: ** ^{2}D, ^{9}Be**,

^{6}Li,

^{7}Li and

^{13}C. As can be seen from the table

**the lowest threshold**have

**and**

^{9}Be with 1.666 MeV**.**

^{2}D with 2.226 MeVIn case of deuterium, neutrons can be produced by the interaction of gamma rays (with a minimum energy of 2.22 MeV) with deuterium:

Because gamma rays can be emitted by fission products with certain delays, and **the process is very similar** to that through which a** “true” delayed neutron** is emitted, **photoneutrons** are usually treated no differently than regular delayed neutrons in the kinetic calculations. Photoneutron precursors can be also grouped by their decay constant, similarly to “real” precursors. The table below shows the relative importance of source neutrons in CANDU reactors by showing the makeup of the full power flux.

Despite the fact photoneutrons are of importance especially in CANDU reactors, deuterium nuclei are always present **(~0.0156%**) **also in the light water of LWRs**. Moreover the capture of neutrons in the hydrogen nucleus of the water molecules in the moderator yields small amounts of D_{2}O. This enhances the heavy water concentration. Therefore also in LWRs kinetic calculations, photoneutrons from D_{2}O are treated as additional groups of delayed neutrons having characteristic decay constants λ_{j} and effective group fractions.

After a nuclear reactor has been operated at full power for some time there will be a considerable build-up of gamma rays from the fission products. This **high gamma flux** from short-lived fission products will **decrease rapidly after shutdown**. **In the long term** the photoneutron source decreases with the** decay of long-lived fission products** that produce delayed high-energy gamma rays and the photoneutron source drops slowly, decreasing a little each day. The longest-lived fission product with gamma ray energy above the threshold is ** ^{140}Ba** with a half-life of

**12.75 days.**

The amount of fission products present in the fuel elements depends on **how long** has been the reactor operated before shut-down and **at which power** level has been the reactor operated before shut-down. **Photoneutrons** are usually major source in a reactor and ensure **sufficient neutron flux** **on source range detectors** when reactor is **subcritical** in long term shutdown.

In comparison with **fission neutrons**, that make a **self-sustaining chain reaction possible**, **delayed neutrons** make reactor **control possible** and **photoneutrons** are of importance **at low power operation**.

See also: Effective Delayed Neutron Fraction – βeff

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%.

**Mean Generation Time with Delayed Neutrons**,

**l**, is the weighted average of the prompt generation times and a delayed neutron generation time. The delayed neutron generation time,

_{d}**τ**, is the weighted average of mean precursor lifetimes of the six groups (or more groups) of delayed neutron precursors.

It must be noted, the true lifetime of delayed neutrons (the slowing down time and the diffusion time) is very short compared with the mean lifetime of their precursors (t_{s} + t_{d} << τ_{i}). Therefore τ_{i} is also equal to the mean lifetime of a neutron from the ith group, that is, **τ _{i} = l_{i}** and the equation for mean generation time with delayed neutrons is the following:

**l _{d} = (1 – β).l_{p} + ∑l_{i} . β_{i} => l_{d} = (1 – β).l_{p} + ∑τ_{i} . β_{i}**

where

**(1 – β)**is the fraction of all neutrons emitted as prompt neutrons**l**is the prompt neutron lifetime_{p}**τ**is the mean precursor lifetime, the inverse value of the decay constant_{i }**τ**_{i}= 1/λ_{i}- The weighted delayed generation time is given by
**τ = ∑τ**_{i}. β_{i}/ β = 13.05 s - Therefore the weighted decay constant
**λ = 1 / τ ≈ 0.08 s**^{-1}

The number, **0.08 s ^{-1}**, is relatively high and have

**a dominating effect of reactor time response**, although delayed neutrons are a small fraction of all neutrons in the core. This is best illustrated by calculating a weighted mean generation time with delayed neutrons:

**l _{d} = (1 – β).l_{p} + ∑τ_{i} . β_{i} = (1 – 0.0065). 2 x 10^{-5} + 0.085 = 0.00001987 + 0.085 ≈ 0.085**

In short, **the mean generation time with delayed neutrons** is about **~0.1 s**, rather than ~**10-5** as in section Prompt Neutron Lifetime, where the delayed neutrons were omitted.

**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**. In power reactors such a reactivity insertion is

_{eff}≈ 1.006 or ρ = +600 pcm)**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.

**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).

**Nuclear and Reactor Physics:**

- J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
- J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
- W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
- Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
- W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
- G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley 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: 0-894-48033-2.
- K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.
- D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2.
- E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.

### See above:

Contents

- Point Kinetics Equations
- Approximate Solution of Point Kinetics Equations
- Point Kinetics Equations – Notes
- Prompt Neutron Lifetime
- Prompt Generation Time - Mean Generation Time
- Effect of Prompt Neutron Lifetime on Nuclear Safety
- Six Groups of Delayed Neutrons
- Photoneutrons
- Effective Delayed Neutron Fraction – βeff
- Mean Generation Time with Delayed Neutrons
- Effect of Delayed Neutrons on Reactor Control
- Interactive chart – Infinite Multiplying System Without Source and Delayed Neutrons
- References:

- See above:

- Reactor Dynamics