## Subcritical Multiplication

In previous chapters, especially in the Reactor Criticality chapter, **the basic classification** of states of a reactor according to the **multiplication factor** was defined. One of these states is a subcritical state defined as:

**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**.

But this statement is not completely correct. To clarify this issue, we use a finer classification of subcritical states. The subcritical state can be subdivided into:

**transitional subcritical state**. Typical feature of this state is a**decrease in neutron flux**. The above definition is, in fact, a definition of the transitional subcritical state, in which the reactivity of a reactor is lower than zero (**k**_{eff}**< 1; ρ < 0**). In this case, the production of all neutrons by fission is insufficient to balance neutron losses and the chain reaction is not self-sustaining.- steady-state subcritical state –
**subcritical multiplication**. Typical feature of subcritical multiplication is a presence of**source neutrons**(supplied by external or internal neutron source). The source neutrons balances neutron losses and the neutron flux is constant. If an external neutron source is present, then any transitional subcritical state inevitably pass in subcritical multiplication. If not, the neutron flux will approach zero.

As can be seen, the neutron flux in a subcritical reactor **with source** **neutrons** stabilizes itself at a corresponding level, which is determined by **source strength, S**, and by the multiplication factor, **k**** _{eff}**. On the other hand, the amount of time it takes to reach the steady-state neutron level is dependent only on

**k**

**.**

_{eff}**Definition:**

In general, **subcritical multiplication** is the phenomenon that accounts for the changes in neutron flux that takes place in a **subcritical reactor** with source neutrons due to reactivity changes.

An equation governing the kinetics of a subcritical reactor can be written as:

where:

- n(t) is transient reactor power
- n(0) is initial reactor power
- k
_{eff}is the effective multiplication factor - l
_{d}is the mean generation time with delayed neutrons - S(t) is the source term, which characterizes number of neutrons (source neutrons) added to the system from an external source. As can be seen, the
**source term does not influence the dynamics**of the system, since it does not influence the reactivity of the system.

**“source neutrons”**refers to neutrons other than prompt or delayed fission neutrons. In general, they come from sources other than neutron-induced fission. These neutrons are very important during reactor startup and shutdown operations when the reactor is subcritical, because they allow to monitor the subcriticality of a reactor usually via source range excore neutron detectors.

Do not confuse terms source neutrons and external source of neutrons. The source neutrons can be classified by the place of origin:

- Internal Source of Neutrons. A very important source of neutrons arises from the nature of the nuclear fuel. Nuclear fuel, especially when irradiated, produces source neutrons, that significantly contribute to monitoring of the subcriticality of a reactor. Irradiated nuclear fuel contains almost entire periodic table of elements and there is very high radiation background inside fuel elements, therefore there is very much channels, how these neutrons are produced. 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. These neutrons can be classified by type of the reaction as:**Spontaneous fission**of certain nuclides. There are nuclides, that undergo a process of spontaneous fission. Spontaneous fission is, in fact, a form of radioactive decay that is found only in very heavy chemical elements (especially transuranic elements). For example,^{240}Pu has very high rate of spontaneous fission. For high burnup fuel, the neutrons are provided predominantly by spontaneous fission of curium nuclei (Cm-242 and Cm-244)**(α,n) reactions.**In certain light isotopes the ‘last’ neutron in the nucleus is weakly bound and is released when the compound nucleus formed following α-particle bombardment decays. For example, the bombardment of beryllium by α-particles leads to the production of neutrons by the following exothermic reaction:^{4}**He +**^{9}**Be→**^{12}**C + n + 5.7 MeV.**This reaction yields a weak source of neutrons with an energy spectrum resembling that from a fission source. Another very important source is**(α,n) reaction**with oxygen-18. This element is present in the uranium oxide fuel. The neutron producing reaction is:**18O(α,n)21Ne.**Alpha particles are commonly emitted by all of the heavy radioactive nuclei occuring in fuel (uranium), as well as the transuranic elements (neptunium, plutonium or americium).**(γ,n) reactions – photoneutrons.**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.**The lowest threshold**have^{9}**Be with 1.666 MeV**and^{2}**D with 2.226 MeV**. Therefore photoneutrons are very important especially in nuclear reactors with D_{2}O moderator (CANDU reactors) or with Be reflectors (some experimental reactors).

- External Source of Neutrons. Sometimes source neutrons must be artificially added to the system. External source of neutrons contains a material, that emits neutrons and cladding to provide a barrier between reactor coolant and the material. External sources are usually loaded directly into the reactor core (e.g. into guide thimble tubes).
- Primary Source of Neutrons. Primary source of neutrons does not need to be irradiated to produce neutrons. These sources can be used especially in case of a first core (i.e. a core that consist of fresh fuel only). Primary source of neutrons are based on the spontaneous fission reaction. The most commonly used spontaneous fission source is the radioactive isotope
**californium-252**. Cf-252 and all other spontaneous fission neutron sources are produced by irradiating uranium or another transuranic element in a nuclear reactor, where neutrons are absorbed in the starting material and its subsequent reaction products, transmuting the starting material into the SF isotope.**(α,n) reactions**can be also used to produce neutrons. The bombardment of beryllium by α-particles leads to the production of neutrons by the following exothermic reaction:^{4}**He +**^{9}**Be→**^{12}**C + n + 5.7 MeV.**This reaction yields a weak source of neutrons with an energy spectrum resembling that from a fission source and is used nowadays in**portable neutron sources.**Radium, plutonium or americium can be used as an α-emitter. - Secondary Source of Neutrons. Secondary source of neutrons must be irradiated to produce neutrons. These sources usually contains two components. The second component is a neutron emitter element, for example,
^{9}**Be,**which has the lowest threshold (1.666 MeV)**for neutron emission.**The first component is a material, which can be irradiated by fission neutrons (e.g. antimony) and then emits particle capable to knock out a neutron from the second component. The (γ,n) source that uses antimony-124 as the gamma emitter is characterized in the following endothermic reaction. The antimony-berylium source produces nearly monoenergetic neutrons with dominant peak at 24keV.

- Primary Source of Neutrons. Primary source of neutrons does not need to be irradiated to produce neutrons. These sources can be used especially in case of a first core (i.e. a core that consist of fresh fuel only). Primary source of neutrons are based on the spontaneous fission reaction. The most commonly used spontaneous fission source is the radioactive isotope

^{124}**Sb→**^{124}**Te + β− + γ**

**γ + **^{9}**Be→**^{8}**Be + n – 1.66 MeV**

The primary and secondary sources are similar to a **control rod** in mechanical construction. Both types of source rods are clad in **stainless steel**.

**Geometry of Source Neutrons Assemblies**

Since at PWRs the source range neutron detectors are usually placed outside the reactor (excore). A source neutrons assemblies should be placed at least a few migration lengths from core periphery. The main reason is that source range detectors should not register primarily the source neutrons.

**Sb-Be neutron source**is typical external neutron source used in commercial nuclear reactors. It is usually loaded into

**fuel assemblies**near the periphery of the core, because the source-range excore detectors are installed outside the pressure vessel. Sb-Be source is a two component source and prior to irradiation contains natural antimony (57.4% of

^{121}Sb and 42.6% of

^{123}Sb) and natural beryllium (100% of

^{9}Be). The first component serves as a source of strong gamma rays, while the second component is a neutron emitter element.

Thus the **Sb-Be source** is based on **(γ,n) reaction** (i.e. it emits photoneutrons). The neutron flux inside the core activates (σ_{a} = 0.02 barn) the isotope ^{123}Sb resulting in ** ^{124}Sb**.

**decays (with half-life of 60.2 days) via beta decay to**

^{124}Sb^{124}Te. The decay scheme of

^{124}Sb shows two relevant groups of γ-ray energies, namely 1691 keV and 2091 keV with absolute intensities of 0.484 and 0.057 per decay respectively.

These γ-rays have sufficient energy to knock out a neutron from the second component, ^{9}**Be,** which has the lowest** threshold** (**1.66 MeV**)** for neutron emission. **The (γ,n) source that uses antimony-124 as the gamma emitter is characterized in the following endothermic reaction.

^{124}**Sb→**^{124}**Te + β− + γ**

**γ + **^{9}**Be→**^{8}**Be + n – 1.66 MeV**

The** antimony-beryllium source** produces nearly **monoenergetic neutrons** with dominant peak at **23keV**. Using the laws of energy and momentum conservation one can derive that the 1691 keV and 2091 keV gamma rays produce two groups of neutrons:

- 23 keV (~97%)
- 378 keV (~3%)

Sb-Be sources have three main disadvantages:

- They have an extremely high photon to neutron ratio which complicates the work with such sources.
- Yield of neutrons is significantly lower than for (α,n) sources. On the other hand (α,n) sources contains transuranic elements as americium which can be converted into fissile isotope
^{242m}Am. These source are not appropriate for commercial reactors. - When loaded into a reactor, Sb-Be source may contribute to production of tritium. Most important source (due to releases of tritiated water) of tritium in nuclear power plants stems from the boric acid, but it can be also produced from beryllium via following sequence of reactions:
^{9}Be(n,α)^{6}He^{6}He**→**^{6}Li + e^{–}^{–};^{ }T_{1/2}=0.8s)^{6}Li(n,α)^{3}H

## Kinetics of Subcritical Multiplication

Source neutrons play an important role in **reactor safety**, especially during **shutdown state** and **reactor startup**. Without source neutrons, there would be no subcritical multiplication and the neutron population in the subcritical system would gradually approach to zero. That means, each neutron generation would have fewer neutrons than the previous one because k_{eff} is less than 1.0.

With source neutrons, the population remains at levels that can be measured by the source range excore neutron detectors, so that operators can always monitor **how fast the neutron population is changing **(can always monitor the reactivity of subcritical reactor). Note that, if neutrons and fissionable material are present in the subcritical reactor, fission will take place (even a deep subcritical reactor will always be producing a small number of fissions).

The source neutrons enter the life cycle and experience the same environment that fission neutrons experience. It must be noted, source neutrons are produced at different energies (e.g. 24keV for Sb-Be source), usually below energies of fission neutrons.

When the reactor is made subcritical after operating at a critical state, the neutron population at first undergoes a prompt drop as a result of rapid decrease in prompt neutrons. After a short time begins to decrease exponentially with a period corresponding to decay of the longest-lived delayed neutron precursors (i.e. ~80s). With source neutrons, the neutron flux **stabilizes itself** at a corresponding level, which is determined by** source strength, S**, and by the **multiplication factor, k _{eff}**.

For a defined source strength of *n***_{0}** neutrons per neutron generation, the neutron population (i.e. the neutron flux) is given by:

where, *n** _{0}* are source neutrons (0. generation) and

*n*

_{0}*k*

_{eff}*are neutrons from i-th generation. As*

^{i}*i*goes to infinity, the sum of this

**geometric series**is (for k

_{eff}< 1):

As can be seen, the neutron population of a subcritical reactor with source neutrons does not drop to zero, the neutron population stabilizes itself at level *n*, which is equal to source multiplied by factor M.

## Example: Subcritical Multiplication

*k*_{eff}* = 0.6*

*k*

_{eff}*= 0.6*

Consider a reactor in which:

*k*_{eff}*= 0.6*- the mean neutron generation to be
*l*_{d}*= 0.1s* - the external source provides
*n*_{0 }*= 100 neutrons/s.*

That means, when 100 neutrons are suddenly introduced into the reactor via source n_{0}. These neutrons will produce 60 neutrons (100 x 0.6) from fission to start the next generation. Duration of each neutron generation is equal to *l*_{d}** = 0.1s**. From 60 neutrons of the second generation the system will produce 36 neutrons (60 x 0.6) to start the third generation (during

*l*

_{d}**). The number of neutrons produced by fission in subsequent generations due to the introduction of 100 source neutrons into the reactor is shown in the figure.**

*= 0.1s*As can be seen, the neutron population stabilizes itself at level *n = 250*, which is equal to source multiplied by factor M. The addition of source neutrons to the reactor containing fissionable material has the effect of maintaining a much higher stable neutron level due to the fissions occurring than the neutron level that would result from the source neutrons alone. As can be seen, the amount of time it takes to reach the steady-state neutron level is approximately equal to time of 10 neutron generations (i.e. 10 x 0.1s = 1s).

Now, we will illustrate, what happens to the neutron population when the value of k_{eff} is changed. During a reactor startup, for example, an initially subcritical reactor is driven to criticality via a series of discrete control rod withdrawals. During these withdrawals a reactivity increases. After each control rod withdrawal, the operator must wait to allow the reactor to attain the steady-state neutron level. The amount of time it takes to reach the steady-state neutron level after a reactivity change increases as k_{eff} approaches the critical state.

*k*_{eff}* = 0.9*

*k*

_{eff}*= 0.9*

Consider a reactor in which:

*k*_{eff}*= 0.9*- the mean neutron generation to be
*l*_{d}*= 0.1s* - The external source provides
*n*_{0 }*= 100 neutrons/s.*

That means, when 100 neutrons are suddenly introduced into the reactor via source n_{0}. These neutrons will produce 90 neutrons (100 x 0.9) from fission to start the next generation. Duration of each neutron generation is equal to *l*_{d}** = 0.1s**. From 90 neutrons of the second generation the system will produce 81 neutrons (90 x 0.9) to start the third generation (during

*l*

_{d}**). The number of neutrons produced by fission in subsequent generations due to the introduction of 100 source neutrons into the reactor is shown in the figure.**

*= 0.1s*As can be seen, the neutron population stabilizes itself at level *n = 1000*, which is equal to source multiplied by factor M. As can be seen, the amount of time it takes to reach the steady-state neutron level is significantly higher, since it is approximately equal to time of 50 neutron generations (i.e. 50 x 0.1s = 5s). The reason is evident in the series expression in the figure for k_{eff} = 0.9. As k_{eff }increases, more terms in the series are significant, meaning that more previous generations of source neutrons contribute to the present population.

### Summary

Therefore, during a reactor startup, the value of k_{eff} increases and:

- the steady-state neutron level increases (by factor M)
- time of stabilization increases (let say 1/100 = k
_{eff}^{i}→ t_{stab}= ln(0.01) / ln (k_{eff}) x l_{d})

It must be noted, the real PWR cores reach reactivities around k_{eff} = 0.92 – 0.95 at the beginning of the reactor startup. Values around 0.6 are not common.

## 1/M Plot

It is obvious the subcritical multiplication factor significantly rises:

Because the **subcritical multiplication factor** is **related to** the value of k_{eff} (**core subcriticality**), it is possible to monitor the approach to criticality through the use of the subcritical multiplication factor. The closer the reactor is to criticality, the faster M will increase for equal step insertions of positive reactivity. When the reactor becomes critical, M will be infinitely large. Monitoring and **plotting M** during a criticality approach is impractical because there is no value of M at which the reactor clearly becomes critical.

On the other hand, the neutron flux is monitored via **source-range detectors**, where **count rate (CR)** gets infinitely large as the core approaches k_{eff} = 1.0. Note that, the count rate of source-range detectors is proportional to the neutron population (n ∝ CR). which can be determined by source-range count rate.

Therefore, instead of plotting M directly, its inverse (1/M or 1/CR) is plotted on a graph of:

- 1/CR versus rod elevation (in case of criticality approach by control rod withdrawal)
- 1/CR versus boron concentration (in case of criticality approach by boron dilution)
- 1/CR versus number of fuel assemblies (in case of loading of fuel into the core)

Note that, a true 1/M plot requires knowledge of the neutron source strength. Because the actual source strength is usually unknown, a r**eference count rate** is substituted. As criticality is approached, 1/CR approaches zero. Therefore in startup procedure the value of 1/CR provides engineers with an effective tool for monitoring the approach to criticality.

**Example of 1/M Plot**

Assume a criticality approach by control rod withdrawal. For this procedure operators check especially the control rods position (usually in steps) and the count rate (CR) from source-range detectors. The initial count rate on the detectors prior to rod withdrawal is 100 cps. There are main parameters in the following table. From these parameters the 1/M Plot is constructed. As can be seen, the estimated critical parameters can be predicted with extrapolation to the point, where CR_{0}/CR → 0.

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