Contents

- Prompt and Delayed Neutrons
- Delayed Neutrons
- Precursors of Delayed Neutrons
- Six Groups of Delayed Neutrons – Parameters
- Eight Groups of Delayed Neutrons – Parameters
- Photoneutrons
- Delayed Neutrons Fraction
- Delayed Neutrons Energy Spectra
- Effective Delayed Neutron Fraction – βeff
- Mean Generation Time with Delayed Neutrons
- Example – Infinite Multiplying System Without Source and with Delayed Neutrons
- Interactive chart – Infinite Multiplying System Without Source and Delayed Neutrons
- Effective Precursor Decay Constant – Lambda-Effective
- Effect of Delayed Neutrons on Reactor Control

## Prompt and Delayed Neutrons

It is known the fission neutrons are of importance in any chain-reacting system. Neutrons trigger the nuclear fission of some nuclei (^{235}U, ^{238}U or even ^{232}Th). What is crucial the fission of such nuclei produces **2, 3 or more** free neutrons.

But not all neutrons are released **at the same time following fission**. Even the nature of creation of these neutrons is different. From this point of view we usually divide the fission neutrons into two following groups:

**Prompt Neutrons.**Prompt neutrons are emitted**directly from fission**and they are emitted within**very short time of about 10**.^{-14}second**Delayed Neutrons.**Delayed neutrons are emitted by**neutron rich fission fragments**that are called**the delayed neutron precursors**. These precursors usually undergo beta decay but a small fraction of them are excited enough to undergo**neutron emission**. The fact the neutron is produced via this type of decay and this happens**orders of magnitude later**compared to the emission of the prompt neutrons, plays an extremely important role in the control of the reactor.

Source: JANIS (Java-based Nuclear Data Information Software); The JEFF-3.1.1 Nuclear Data Library

## Delayed Neutrons

While the most of the neutrons produced in fission are prompt neutrons, **the delayed neutrons are of importance in the reactor control.** In fact the presence of delayed neutrons is perhaps **most important aspect of the fission process** from the viewpoint of reactor control.

The term **“delayed”** in this context means, that the neutron is emitted with half-lifes, ranging **from few milliseconds up to 55 s** for the longest-lived precursor ^{87}Br. These neutrons have to be distinguished from the prompt neutrons which are emitted immediately (on the order of 10^{-14} s) after a fission event from a neutron-rich nucleus. Despite the fact the **amount of delayed neutrons** is only on the order of **tenths of percent** of the total amount, **the timescale in seconds plays the extremely important role.**

## Precursors of Delayed Neutrons

**Delayed neutrons** originate from the radioactive decay of nuclei produced in fission and hence they are different for each fissile material. They are emitted by excited neutron rich fission fragments (so called **the delayed neutron precursors**) some appreciable time after the fission. How long afterward, is dependent **on the half-life of the precursor**, since the neutron emission itself occurs in a very short time. The precursors usually undergo beta decay without any neutron emission but a small fraction of them (highly excited nuclei) can undergo **the neutron emission** instead of the gamma emission.

In addition current nuclear physics facilities can produce more neutron-rich isotopes that can emit **multiple neutrons**. Currently about **18 2n-emitters** are experimentally confirmed [IAEA – INDC(NDS)-0599], but only two of them are also fission products.

As an example,** the isotope ^{87}Br** is the major component of the first group of precursor nuclei. This isotope has half-life of

**55.6 seconds**. It undergoes negative beta decay through its two main branches with emission of 2.6 MeV and 8 MeV beta particles. This decay leads to the formation of

^{87}Kr* and

^{87}Kr (ground state) and the

^{87}Kr nucleus subsequently decays via two successive beta decays into the stable isotope

^{87}Sr. But there is also one possible way for the

^{87}Br nucleus to beta decay. The

^{87}Br nucleus can beta decay into an excited state of the

^{87}Kr nucleus at an energy of 5.5 MeV, which is larger than the binding energy of a neutron in the

^{87}Kr nucleus. In this case, the

^{87}Kr nucleus can undergo (with probability of 2.5%)

**a neutron emission**leading to the formation of stable

^{87}Kr isotope.

According to the JEFF 3.1 database, about **240 n-emitters** are known between ^{8}He and ^{210}Tl, about 75 of them are in the non-fission region. Furthermore **18 2n-emitter**, and only **four 3n-emitters** ( ^{11}Li, ^{14}Be, ^{17}B, ^{31}Na) are experimentally confirmed. These numbers are not certainly final. Since new IAEA Co-ordinated Research Project (CRP) on Beta-delayed neutron emission evaluation has been started in 2013, it is expected these numbers will change significantly.

See also: IAEA, Beta-delayed neutron emission evaluation, INDC(NDS)-0599

As can be seen it was identified many precursor nuclei. Not all of them are fission products (about 75 of them in the non-fission region A<70), but there are also many precursor nuclei, that are in the fission region between A=70-150. Their half-lives range between tenths of second (0.12 s) and tens of seconds (55.6 s), therefore their delayed neutrons appear with considerably differing delay times.

## Six Groups of Delayed Neutrons – Parameters

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.

## Eight Groups of Delayed Neutrons – Parameters

In order** to reduce discrepancies** between measured and calculated values of the reactivity scale based on reactor kinetics (this discrepancy results in excessive conservatism in the design, because it must be covered by uncertainties during core design and safety analyses calculations), the NEA (Nuclear Energy Agency) and its NEANSC/WPEC Subgroup 6 recommends **a new eight-group representation**. It seems reasonable to increase the number of delayed neutron groups, because many studies have shown that **most of delayed neutrons** are produced approximately **by twelve precursors** that are common to all fissioning isotopes.

The eight-group representation uses a set of eight-group half-lives for all fissioning systems, with the half-lives adopted for the** three longest-lived groups** corresponding to the **three dominant long-lived precursors**: ** ^{87}Br, ^{137}I and ^{88}Br** (these precursors was separated into single groups).

See also: Delayed Neutron Data for the Major Actinidies, NEA/WPEC–6. OECD 2002.

## Photoneutrons

In nuclear reactors the gamma radiation plays a significant role also in **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**.

## Delayed Neutrons Fraction

**The total yield of delayed neutrons per fission**, v_{d}, depends on:

**Isotope, that is fissioned**(see table).

**Energy of a neutron that induces fission**(see chart).

**delayed neutron fraction (DNF)**. At the steady state condition of criticality, with k

_{eff}= 1,

**the delayed neutron fraction is equal to the precursor yield fraction (β)**.

where **β _{i} **is defined as the fraction of the neutrons which appear as

**delayed neutrons in the i**. In contrast to the prompt neutrons, which are emitted with a continuous energy spectrum, the delayed neutrons in each group appear with a more or less well defined energy.

*th*group**In general the delayed neutrons are emitted with much less energy than the most prompt neutrons.**

Distinction between these two parameters is obvious. The delayed neutron fraction is dependent on certain reactivity of multiplying system, on the other hand

**β**is not dependent on the reactivity. These two factors,

**DNF and β**, are not the same thing in case of a rapid change in the number of neutrons in the reactor.

In LWRs the delayed neutron fraction **decreases with fuel burnup**. This is due to isotopic changes in the fuel. It is simple, 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**, the resultant **core delayed neutron fraction** of a multiplying system **decreases** (it is the weighted average of the constituent delayed neutron fractions). This is also the reason why the neutron spectrum in the core **become harder with fuel burnup**.

**β _{core}= ∑ P_{i}.β_{i}**

where P_{i} is fraction of power generated by isotope i.

**Example:**

Let say the reactor is at the beginning of the cycle and approximately

**98%**of reactor power is generated by

^{235}U fission and

**2%**by

^{238}U fission as a result of fast fission. Calculate the core delayed neutron fraction.

β_{core}= ∑ P_{i}.β_{i} = 0.98 x β_{235} + 0.02 x β_{238}

= 0.98 x 0.0065 + 0.02 x 0.0157

= 0.0064 + 0.0003

= 0.0067

## Delayed Neutrons Energy Spectra

The key properties of delayed neutrons, which are **very important for the nuclear reactor design**, belong also **delayed neutron energy spectra**. The energy spectra of the delayed neutrons are the poorest known of all input data required, because it **very difficult to measure it**.

Depending on the type of the reactor, and their spectrum, the delayed neutrons may be more (in thermal reactors) or less effective than prompt neutrons (in fast reactors). In order to include this effect into the reactor kinetic calculations **the effective delayed neutron fraction – β _{eff} must be defined**.

## Effective Delayed Neutron Fraction – β_{eff}

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%.

Table of main kinetic parameters.

## Mean Generation Time with Delayed Neutrons

**Mean Generation Time with Delayed Neutrons**, **l _{d}**, is the weighted average of the prompt generation times and a delayed neutron generation time. The delayed neutron generation time,

**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.

## Example – Infinite Multiplying System Without Source and with Delayed Neutrons

**An 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.**

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. 10 pcm 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 _{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.

Reactors with such a kinetics would be quite **easy to control**. From this point of view it may seem that reactor control will be a quite boring affair. It will not! The presence of delayed neutrons entails many many specific phenomena, that will be described in later chapters.

## Interactive chart – Infinite Multiplying System Without Source and Delayed Neutrons

Press the “**clear and run**” button and try to increase the power of the reactor.

Compare the response of the reactor with the case of Infinite Multiplying System Without Source and without Delayed Neutrons (or set the β = 0).

## Effective Precursor Decay Constant – Lambda-Effective

**The effective delayed neutron precursor decay constant** (pronounced **lambda effective**) is a new term, which has to be introduced in the reactor period equation in case of **a single precursor group model**. For the purpose of creating a simple kinetic model conducive to understanding reactor behavior, it is useful to further **reduce the precursors to a single group**. But if we do this, the convention is to employ a **constant precursor yield fraction** and **a variable precursor decay rate**, as defined by **lambda effective (λ _{eff})**. In the single precursor group model the

**lambda effective is not a constant**, but rather a dynamic property that depends on the mix of precursor atoms resulting from the reactivity.

The reason the constant decay constant cannot be used, is as follows. **During power transients**, there is a difference in the decay and the creation of **short-lived** and **long-lived** precursors.

**During a power increase** (positive reactivity), the short-lived precursors decaying at any given instant were born at a higher power level than the longer-lived precursors decaying at the same instant. **The short-lived precursors become more significant**. As the magnitude of the positive reactivity increases, the value of lambda effective increases closer to that of the short-lived precursors (let say 0.1 s-1 for +100pcm).

**During a power decrease** (negative reactivity), the long-lived precursors decaying at a given instant were born at a higher power level than the short-lived precursors decaying at that instant. **The long-lived precursors become more significant.** As the magnitude of the negative reactivity increases, the value of lambda effective decreases closer to that of the long-lived precursors (let say 0.05 s-1 for -100pcm).

If the reactor is operating at steady-state operation, all the precursor groups reach an equilibrium value and the **λ _{eff}** value is approximately 0.08 s

^{-1}.

## Effect of Delayed Neutrons on Reactor Control

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**. 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.