**Reactor kinetics**is the study of the time-dependence of the neutron flux for postulated changes in the macroscopic cross sections. It is also referred to as reactor kinetics

**without feedbacks**.

To study the kinetic behaviour of the system, engineers usually use the **point kinetics equations**. 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**.

In this section, amongst other things we will study the **time-dependent behaviour** of nuclear reactors. An understanding of the **time-dependent behavior** of the neutron population in a nuclear reactor in response to either a **planned** change in the reactivity of the reactor or to **unplanned** and abnormal conditions is of the most importance in the nuclear reactor safety.

**Nuclear reactor kinetics** is dealing with transient **neutron flux changes** resulting from a departure from the critical state, from some reactivity insertion. Such situations arise during operational changes such as control rods motion, environmental changes such as a change in boron concentration, or due to accidental disturbances in the reactor steady-state operation.

In general:

**Reactor Kinetics.**Reactor kinetics is the study of the time-dependence of the neutron flux for postulated changes in the macroscopic cross sections. It is also referred to as reactor kinetics**without feedbacks**.**Reactor Dynamics.**Reactor dynamics is the study of the time-dependence of the neutron flux, when the macroscopic cross sections are allowed to depend in turn on the neutron flux level. It is also referred to as reactor kinetics**with****feedbacks**and with spatial effects.

Time-dependent behaviour of nuclear reactors can be also classified by the time scale as:

**Short-term kinetics**describes phenomena that occur over times shorter than a few seconds. This comprises the response of a reactor to either a**planned**change in the reactivity or to**unplanned**and abnormal conditions. In this section, we will introduce especially**point kinetics equations**.**Medium-term kinetics**describes phenomena that occur over the course of several hours to a few days. This comprises especially effects of neutron poisons on the reactivity (i.e.**Xenon poisoning**or**spatial oscillations**).**Long-term kinetics**describes phenomena that occur over months or even years. This comprises all long-term changes in fuel composition as a result of**fuel burnup**.

This chapter is concerned especially with short-term kinetics and the **point kinetics equations**. At first, we have to start with an introduction to **prompt and delayed neutrons** because they play an important role in short-term reactor kinetics. 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.

## Simple Point Kinetics Equation

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**.

### Derivation of Simple Point Kinetics Equation

Let ** n(t)** be the number of neutrons as a function of time

*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.

## Simple Point Kinetics Equation without Delayed Neutrons

An equation governing the neutron kinetics of the system without source and with the absence of delayed neutrons is **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****.**

## Simple Point Kinetics Equation with Delayed Neutrons

The simplest equation governing the neutron kinetics of the system with delayed neutrons is the simple **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.

## Point Kinetics Equations

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:

## Inhour Equation

If the reactivity is constant, the model of point kinetics equations contains a set (**1 + 6**) of linear ordinary **differential equations** with constant coefficient and can be solved analytically. Solution of six-group point kinetics equations with Laplace transformation leads to the relation between the **reactivity** and the **reactor period**. This relation is known as 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.

**General Form:**

The **point kinetics equations** may be solved for the case of an initially critical reactor without external source in which the properties are changed at t = 0 in such a way as to introduce a **step reactivity ρ _{0}** which is then constant over time. The system of coupled first-order differential equations can be solved with Laplace transformation or by trying the solution

**n(t) = A.exp(s.t)**(equation for the neutron flux) and

**C**(equations for the density of precursors).

_{i}(t) = C_{i,0}.exp(s.t)Substitution of these assumed exponential solutions in the **equation for precursors** gives the relation between the coefficients of the neutron density and the precursors.

The subsequent substitution in the equation for neutron density yields an equation for **s**, which after some manipulation can be written as:

This equation is known as the **inhour equation**, since the constants of** s _{0 – 6}** was originally determined in inverse hours. For a given value of the reactivity

**ρ**the associated values of

**s**are determined with this equation. The following figure shows the relation between

_{0 – 6}**ρ**and roots

**s**graphically. From this figure it can be seen that for a given value of ρ seven solutions exist for s. The figure indicates that for positive reactivity

**only s**. The remaining terms rapidly die away, yielding an asymptotic solution in the form:

_{0}is positivewhere **s _{0} = 1/τ_{e}** is the

**stable reactor period**or

**asymptotic period of reactor**. This root,

**s**, is

_{0}**positive for ρ > 0**and

**negative for ρ < 0**, therefore this root describes the reactor response, which is lasting after the transition phenomena have died out. The figure also shows that a negative reactivity leads to a negative period: All of the s

_{i}are negative, but the root s

_{0}will die away more slowly than the others. Thus the solution

**n(t) = A**is valid for positive as well as negative reactivity insertions.

_{0}exp(s_{0}t)To determine the reactivity required to produce a given period a plot of ρ vs. τ_{e} must be constructed using the delayed neutron data for a particular fissionable isotope or mix of isotopes, and for a given prompt generation time. To determine the stable reactor period, which results from a given reactivity insertion, it is convenient to use the following form of inhour equation.

## Special Cases of Inhour Equation

## Reactivity Pulse – Impulse Characteristics

We will now study a response of a reactor on a **reactivity pulse**, which is represented by the **Dirac delta function**, δ(t). Strictly speaking Dirac delta function is not a function, but a so-called distribution, but here the function form will be used, in which the delta function is defined as follows:

the reactivity pulse can be mathematically expressed as ** ρ(t) = ρ_{0} . δ(t)**. Using the inverse Laplace transformation and the system transfer function, G(s), it can be derived that the pulse reactivity insertion causes a transient which is characterized by following relations:

That means the **prompt neutron lifetime** plays key role in the first part of the transient, while the **delayed neutrons** play key role in the steady-state neutron level.

## Oscillation of Reactivity – Frequency Characteristics

We will now study a response of a reactor on a **reactivity oscillation**, which is represented by the following function: ** ρ(t) = ρ_{0} . cos(ωt)**. Where

*ρ*is the amplitude of the input signal (forcing function) and ω is the signal frequency expressed in radians per second.

_{0}Using the inverse Laplace transformation and the system transfer function, G(s), it can be derived that the system response is strongly dependent on the frequency, ω.

## 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

## Experimental Methods of Reactivity Determination

There are two main experimental methods for fundamental reactor physics measurements: kinetic and static.

**Static methods**are used to determine time independent core characteristics. These methods can be used to describe phenomena that occur independently of time. On the other hand they cannot be used to determine most dynamic characteristics.**Kinetic methods**are used to study parameters (parameters of delayed neutrons etc.) which determine short-term and medium-term kinetics.

There are three main kinetic methods for experimental determination of neutron kinetics parameters:

## Reactivity

In preceding chapters, the classification of states of a reactoraccording to the effective multiplication factor – k_{eff} was introduced. The effective multiplication factor – k_{eff} is a measure of the change in the fission neutron population from one neutron generation to the subsequent generation. But sometimes it is convenient to define the **change in the k _{eff}** alone, the change in the state, from the criticality point of view.

For these purposes reactor physics use a term called **reactivity** rather than k_{eff} to describe the change in the state of the reactor core. **The reactivity** (**ρ** or** ΔK/K**) is defined in terms of k_{eff} by the following equation:

From this equation it may be seen that **ρ** may be positive, zero, or negative. The reactivity 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**.

It must be noted the reactivity can be calculated also according to the another formula.

This formula is widely used in neutron diffusion or neutron transport codes. The advantage of this reactivity is obvious, it is a measure of a **reactor’s relative departure**not only from criticality (k_{eff} = 1), but it can be related to any sub or supercritical state (**ln(k _{2} / k_{1})**). Another important feature arises from the

**mathematical properties of logarithm**. The logarithm of the division of k

_{2}and k

_{1}is the difference of logarithm of k

_{2}and logarithm of k

_{1}.

**ln(k**. This feature is important in case of addition and subtraction of various reactivity changes.

_{2}/ k_{1}) = ln(k_{2}) – ln(k_{1})See more: D.E.Cullen, Ch.J.Clouse, R.Procassini, R.C.Little. Static and Dynamic Criticality: Are They Different?. Lawrence Livermore National Laboratory. UCRL-TR-201506. 11/2003.

## Inverse Reactor Kinetics – Reactimeter

The reactivity describes the measure of a **reactor’s relative departure from criticality**. During reactor operation and during reactor startup, it is important to monitor the reactivity of the system. It must be noted, **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).

On the other hand, during reload startup physics tests which are performed at the startup after refueling the commercial PWRs, it is important to monitor subcriticality continuously during criticality approach. On-line reactivity measurements are based on the inverse kinetics method. The inverse kinetics method is a reactivity measurement based on the point reactor kinetics equations. This method can used for:

**Reactivity measurement at high neutron level**–**reactimeter without source term**. Without source term a reactimeter can be constructed, but it works only at higher neutron levels, where the neutron source term in point reactor kinetics equations may be neglected.**Reactivity measurement at subcritical multiplication**–**reactimeter with source term**. For operation at low power levels or in the sub-critical domain (e.g. during criticality approach), the contribution of the neutron source must be taken into account and this implies the knowledge of a quantity proportional to the source strength, and then it should be determined. The subcritical reactimeter is based on the determination of the**source term**(source strength).

As was written, the reactivity of the system can be measured by a **reactimeter**. The reactimeter is a device (or rather a **computational algorithm**) that can continuously give real time reactivity using the **inverse kinetics method**. The reactimeter usually processes the signal from source range excore neutron detectors and calculates the reactivity of the system.

It was shown that the source term is not so easy to be determined and the problem is that the source term is of the highest importance in the subcritical domain. One of the recognize methods for source term determining is known as Least Squares Inverse Kinetics Method (**LSIKM**).

Special reference: Seiji TAMURA, “Signal Fluctuation and Neutron Source in Inverse Kinetics Method for Reactivity Measurement in the Sub-critical Domain,” J. Nucl. Sci. Technol, Vol.40, No. 3, p. 153–157 (March 2003)