## Reactivity

**The prompt critical state is defined as:**

**k**, 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_{eff}> 1; ρ ≥ β_{eff}**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**, where the reactivity of a reactor is_{eff}> 1; 0 < ρ < β_{eff}**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**, where the reactivity of a reactor is_{eff}= 1; ρ = 0**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**, 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 s_{eff}< 1; ρ < 0**ubcritical multiplication**.

In preceding chapters, the classification of states of a reactor according 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.

**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}=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**.

N_{50}=N_{0}. (k_{∞})^{50} = **1051 neutrons after fifty generations**.

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

## Units of Reactivity

Mathematically, reactivity is a **dimensionless number**, but it can be expressed by various units. The most common units for **research reactors** are units normalized to the **delayed neutron fraction (e.g. cents and dollars)**, because they exactly express a departure from prompt criticality conditions.

The most common units for **power reactors** are units of **pcm** or **%ΔK/K**. The reason is simple. Units of **dollars are difficult to use**, because the normalization factor, **the effective delayed neutron fraction**, significantly **changes with the fuel burnup**. In LWRs the delayed neutron fraction decreases with fuel burnup (e.g. from **β _{eff} = 0.007** at the beginning of the cycle up to

**β**at the end of the cycle). This is due to isotopic changes in the fuel. It is simple,

_{eff}= 0.005**fresh uranium fuel**contains only

^{235}U as the fissile material, meanwhile during fuel burnup the importance of fission of

^{239}Pu increases (in some cases up to 50%). Since

^{239}Pu produces significantly less delayed neutrons (

**0.0021**for thermal fission), the resultant core delayed neutron fraction of a multiplying system decreases (it is the weighted average of the constituent delayed neutron fractions).

β_{core}= ∑ P_{i}.β_{i}

### dollar ($)

The unit of reactivity which has been **normalized** to the delayed neutron fraction. Reactivity in **dollars = ρ / βeff**. The cent is 1/100 of a dollar. This is very useful unit, because the reactivity in dollars (rather in cents) determines exactly the response of the reactor on the reactivity insertion. Conversion of dollars to pcm depends on βeff. For reactor core with β_{eff} = 0.006 (0.6%) **one dollar** is equal to about **600 pcm**. It is very important amount of reactivity, because if the reactivity of the core is one dollar, the reactor is prompt critical.

**BOC and β _{eff} = 0.006**

**k _{eff} = 0.99** ρ = (k

_{eff}– 1) / k

_{eff}= -0.01 ρ = -0.01 / 0.006 =

**-1.67 $**=

**-167 cents**

**EOC and β _{eff} = 0.005**

**k _{eff} = 0.99** ρ = (k

_{eff}– 1) / k

_{eff}= -0.01 ρ = -0.01 / 0.005 =

**-2.00 $ = -200 cents**

### %ΔK/K

The unit of reactivity in **percents** of the effective multiplication factor. For example, the subcriticality of **k _{eff} = 0,98** is equal to

**-2%**in units of

**%ΔK/K**. Since this is

**very large amount of reactivity**, these units are usually used to express significant quantities of reactivity like power defects,

**xenon worth**,

**integral worth of control rods**or

**shutdown margin**. For operational changes that affect the effective multiplication factor this unit is inappropriate, because these changes are of the lower order.

**k _{eff} = 0.99** ρ = (keff – 1) / keff = -0.01 ρ = -0.01 * 100% =

**-1 %**

### percent mille (pcm)

The unit of reactivity which is **one-thousandth** of a percent **%ΔK/K** (equal to 10^{-2}x10^{-3} = **10 ^{-5}** of k

_{eff}). The unit of

**pcm**is used at many

**LWRs**because reactivity insertion values are generally quite small and units of pcm allows reactivity to be written in

**whole numbers**. The operational changes such as control rods movement causes usually reactivity insertion of the order of units of pcm per one step. The fact that the effective delayed neutron fraction changes with the

**fuel burnup**have an important consequence. Due to the difference in

**β**a response of a reactor on the same reactivity insertion (in units of pcm) is different at the beginning (

_{eff}**BOC**) and at the end (

**EOC**) of the cycle.

For example,** one step** of control rods causes **greater response** at EOC than at BOC. Despite the fact that we assume in both cases, that one step causes the same reactivity insertion (e.g. +10pcm). Moreover, this assumption is not always correct, because the control rods worth increases with fuel burnup.

(10 pcm = 1.43 cents for **β _{eff} = 0.007**; 10 pcm = 2.00 cents for

**β**)

_{eff}= 0.005**keff = 0.99** ρ = (keff – 1) / keff = -0.01 ρ = -0.01 * 105 = **-1000 pcm**

**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**, is the same fraction at thermal energies.

_{eff}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%.

**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.

## Reactivity in Reactor Kinetics

In preceding chapters, the classification of states of a reactor according to **the effective multiplication factor – k _{eff}** was introduced. The effective multiplication factor –

**k**is a measure of the change in the fission

_{eff}**neutron population**from one neutron generation to the subsequent generation. Also the reactivity as a measure of a reactor’s relative departure from criticality was defined.

In this section, amongst other things it will be briefly described how **the neutron flux** (i.e. the reactor power) changes if **reactivity** of a multiplying system is not equal to zero. 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. This subject is usually called **reactor kinetics** (without reactivity feedbacks) or **reactor dynamics** (with reactivity feedbacks and with spatial effects).

**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.

## Point Kinetics Equation – One Delayed Neutron Group Approximation

**The simplest equation governing 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 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):**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.

**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**is based on many assumptions and it is only

_{d}/ (k-1)**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**= effective delayed neutron precursor decay constant

_{eff}**τ**= reactor period

_{e}**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**. For this state calculate the reactor period –

_{d}= 0.01s**τ**, doubling time –

_{e}**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**. For this state calculate the reactor period –

_{d}= 0.01s**τ**, doubling time –

_{e}**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**. For this state calculate the reactor period –

_{d}= 0.01s**τ**, doubling time –

_{e}**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**

## Effect of Presence of Delayed Neutrons

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