Reactor Dynamics

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_eff ≈ 1.006 or ρ = +600 pcm). In power reactors such a reactivity insertion is 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.

• material composition of the system
  • multiplying – nonmultiplying system
  • system with or without thermalization
  • isotopic composition of the system
• geometric configuration of the system
  • homogeneous or heterogeneous system
  • shape of entire system
• size of the system

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

l=t_s + t_d

In an infinite thermal reactor t_s << t_d and therefore l ≈ t_d. The typical prompt neutron lifetime in thermal reactors is on the order of 10⁻⁴ second. Generally, the longer neutron lifetimes take place in systems in which the neutrons must be thermalized in order to be absorbed.

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

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

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

Λ = l/k_eff

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

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

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

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

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

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

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

• These constants are different for each fissionable nuclide.

• These constants change also with the neutron energy spectrum.

Although this six group parameterization still satisfies the requirements of commercial organizations, a higher accuracy of the delayed neutron yields and a better energy resolution in the delayed neutron spectra is desired.

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: ⁸⁷Br, ¹³⁷I and ⁸⁸Br.

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.

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: ²D, ⁹Be, ⁶Li, ⁷Li and ¹³C. As can be seen from the table the lowest threshold have ⁹Be with 1.666 MeV and ²D with 2.226 MeV.

In 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 D₂O. This enhances the heavy water concentration. Therefore also in LWRs kinetic calculations, photoneutrons from D₂O 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 ¹⁴⁰Ba with a half-life of 12.75 days.

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

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

See also: Effective Delayed Neutron Fraction – βeff

The delayed neutron fraction, β, is the fraction of delayed neutrons in the core at creation, that is, at high energies. But in case of thermal reactors the fission can be initiated mainly by thermal neutron. Thermal neutrons are of practical interest in study of thermal reactor behaviour. The effective delayed neutron fraction, usually referred to as β_eff, is the same fraction at thermal energies.

The effective delayed neutron fraction reflects the ability of the reactor to thermalize and utilize each neutron produced. The β is not the same as the β_eff due to the fact delayed neutrons do not have the same properties as prompt neutrons released directly from fission. In general, delayed neutrons have lower energies than prompt neutrons. Prompt neutrons have initial energy between 1 MeV and 10 MeV, with an average energy of 2 MeV. Delayed neutrons have initial energy between 0.3 and 0.9 MeV with an average energy of 0.4 MeV.

Therefore in thermal reactors a delayed neutron traverses a smaller energy range to become thermal and it is also less likely to be lost by leakage or by parasitic absorption than is the 2 MeV prompt neutron. On the other hand, delayed neutrons are also less likely to cause fast fission, because their average energy is less than the minimum required for fast fission to occur.

These two effects (lower fast fission factor and higher fast non-leakage probability for delayed neutrons) tend to counteract each other and forms a term called the importance factor (I). The importance factor relates the average delayed neutron fraction to the effective delayed neutron fraction. As a result, the effective delayed neutron fraction is the product of the average delayed neutron fraction and the importance factor.

β_eff = β . I

The delayed and prompt neutrons have a difference in their effectiveness in producing a subsequent fission event. Since the energy distribution of the delayed neutrons differs also from group to group, the different groups of delayed neutrons will also have a different effectiveness. Moreover, a nuclear reactor contains a mixture of fissionable isotopes. Therefore, in some cases, the importance factor is insufficient and an importance function must be defined.

For example:

In a small thermal reactor with highly enriched fuel, the increase in fast non-leakage probability will dominate the decrease in the fast fission factor, and the importance factor will be greater than one.

In a large thermal reactor with low enriched fuel, the decrease in the fast fission factor will dominate the increase in the fast non-leakage probability and the importance factor will be less than one (about 0.97 for a commercial PWR).

In large fast reactors, the decrease in the fast fission factor will also dominate the increase in the fast non-leakage probability and the β_eff is less than β by about 10%.

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_p is the prompt neutron lifetime
• τ_i is the mean precursor lifetime, the inverse value of the decay constant 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.

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_eff ≈ 1.006 or ρ = +600 pcm). In power reactors such a reactivity insertion is 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.

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

k_eff > 1; ρ ≥ β_eff, where the reactivity of a reactor is higher than the effective delayed neutron fraction. In this case, the production of prompt neutrons alone is enough to balance neutron losses and increase the neutron population. The number of neutrons is exponentially increasing in time (as rapidly as the prompt neutron generation lifetime ~10^-5s).

k_eff > 1; 0 < ρ < β_eff, where the reactivity of a reactor is higher than zero and lower than the effective delayed neutron fraction. In this case, the production of prompt neutrons alone is insufficient to balance neutron losses and the delayed neutrons are needed in order to sustain the chain reaction. The neutron population increases, but much more slowly (as the mean generation lifetime with delayed neutrons ~0.1 s).

k_eff = 1; ρ = 0, where the reactivity of a reactor is equal to zero. In this case, the production of prompt neutrons alone is insufficient to balance neutron losses and the delayed neutrons are needed in order to sustain the chain reaction. There is no change in neutron population in time and the chain reaction will be self-sustaining. This state is the same state as the critical state from basic classification.

k_eff < 1; ρ < 0, where the reactivity of a reactor is lower than zero. In this case, the production of all neutrons is insufficient to balance neutron losses and the chain reaction is not self-sustaining. If the reactor core contains external or internal neutron sources, the reactor is in the state that is usually referred to as the subcritical multiplication.

where:

l = prompt neutron lifetime

β_eff = effective delayed neutron fraction

λ_eff = effective delayed neutron precursor decay constant

τ_e = reactor period

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 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_d = 0.01s. For this state calculate the reactor period – τ_e, doubling time – 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

As can be seen, this special case results in the same period as in the case of simple point kinetics equation, which also uses the mean generation time with delayed neutrons (l_d):

Thus for small reactivities—positive or negative—the reactor period is governed almost completely by the delayed neutron properties. 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 (τ_i) plays the extremely important role.

The assumption is s₀ ≪ λ_i. The largest value of τ_i = 1/λ_i = 80s, therefore the this formula is valid for periods, τ_e, higher than 80s. On the other hand, this formula is useful and accurate enough for most purposes for reactivities up to about ρ = 0.0005 = 50pcm.

Note that:

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_p is the prompt neutron lifetime
• τ_i is the mean precursor lifetime, the inverse value of the decay constant 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:

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.

This causes the first root, s₀, of the reactivity becomes large compared to each λ_i. In this case, λ_i ≪ s₀ and therefore, we can ignore the term λ_i in the denominator of the inhour equation. The inhour equation then takes form:

As can be seen, this special case results in the same period as in the case of simple point kinetics equation without delayed neutrons.

Obviously, reaching the reactivity of β (e.g. 650pcm) completely changes the response of a reactor, therefore this situation can only be accidental. As prompt criticality is reached, the distinction between the prompt jump and the reactor period vanishes, for now the prompt neutron lifetime rather than the delayed neutron half-lives largely determines the rate of exponential increase. Reactors with such a kinetics would be very difficult to control by mechanical means such as the movement of control rods. In normal operation, the reactivity of a reactor must remain far below the prompt criticality threshold with sufficient margin.

In design basis accidents (DBA), such a kinetics can occur. For example under RIA conditions (Reactivity-Initiated Accidents) reactors should withstand a jump-like insertion of relatively large (~1 $ or even more) positive reactivity (e.g. in case of control rod ejection) and the PNL (prompt neutron lifetime) plays here the key role. The longer prompt neutron lifetimes can substantially improve kinetic response of reactor (the longer prompt neutron lifetime gives simply slower power increase). Therefore the PNL should be verified in a reload safety evaluation (RSE) process. Management of this accident is based on reactivity feedbacks, especially on the doppler temperature coefficient (DTC). This coefficient is of the highest importance in the reactor stability. The doppler temperature coefficient is generally considered to be even more important than the moderator temperature coefficient (MTC). Especially in case of a control rod ejection the doppler temperature coefficient will be the first and the most important feedback, that will compensate the inserted positive reactivity. The time for heat to be transferred to the moderator is usually measured in seconds, while the fuel temperature coefficient is effective almost instantaneously. Therefore this coefficient is also called the prompt temperature coefficient because it causes an immediate response on changes in fuel temperature.

But even if it were possible to insert an infinite negative reactivity, the neutron flux would not immediately fall to zero. Prompt neutrons will be absorbed almost immediately. It is consistent with the prompt drop formula. Therefore the resulting neutron flux will be:

It is obvious, the neutron flux cannot drop below the value βn₁.  The real values are much higher. The integral worth of all control and emergency rods (PWRs) is for example -9000pcm. It is equal to ρ = -9000/600 = -15β = -0.09 (β= 600pcm = 0.006)

For this negative reactivity the prompt drop is equal to:

n₂/n₁ = 0.006/(0.006+0.09)=0.063

which is about ten times higher than in case of an infinite negative reactivity insertion.

The neutron flux then continues to fall according to stable period. The first root of reactivity equation occurs at s₀ = – λ₁. the decay constant of the long-lived precursors group. The shortest negative stable period is then τ_e = – 1/λ₁ = -80s. The neutron flux cannot be reduced more rapidly than this period. On the other hand, the prompt drop causes an immediate drop to about 6% of rated power and within few tens of seconds the thermal power which originates from nuclear fission is below the thermal power which originates from decay heat.

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

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

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

This simplification then leads to:

Assuming that the reactivity is constant and n₁/n₀ can be determined from prompt jump formula this equation leads to very simple formula:

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

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

As was written in previous chapters, we can expect that the solution of point kinetics equation can be n(t) = A.exp(s.t) and C_i(t) = C_i,0.exp(s.t). In the asymptotic term (i.e. after the transition phenomena have died out) the asymptotic solution is in the form:

where s₀ = 1/_e is the stable reactor period or asymptotic period of reactor. This root, s₀, is 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 measurement of asymptotic period,  can be used to determine (directly from inhour equation) the reactivity inserted to the system.

As was described in the Prompt Jump Approximation, the response of a neutron detector (n₁ ➝ n₂) immediately after a control rod is dropped into a critical reactor (ρ₁ = 0) is related by:

which allows determination of the reactivity worth of the rod. The rod drop method is advantageous because it is very quick to perform and it requires no extra equipment. Moreover it can easily and safely measure large amounts of reactivity. On the other hand the rod drop method is associated usually with reactor shutdown and subsequent reactor startup. Also, the rod drop time is not instantaneous as is theoretically assumed, therefore limiting the accuracy of the method. In PWRs, the drop time of all control rods is usually about 2 – 4 seconds.

This method is widely used to determine the worth of all control rods (i.e. of an emergency shutdown system). This method can be also used to determine the parameters of delayed neutrons. These data can be obtained by decomposition of neutron flux coastdown.

A more accurate method is, for example, in:

Moore, K. V. Shutdown Reactivity by the Modified Rod Drop Method, USAEC Report ID0-16948, 1964.

In this case the subcriticality of the reactor (subcritical multiplication) can be determined from the measurement of the prompt drop. Also here the prompt drop can be measured because the delayed neutron precursor population will not change immediately. Following sudden source removal, the neutron population undergoes a sharp negative jump.

As was described in the Prompt Jump Approximation, the response of a neutron detector (n₁ ➝ n₂) immediately after a source removal from a subcritical reactor (ρ₁ < 0) is related by:

where n₀ is the neutron level with the source in place and n₁ is the neutron level immediately after the source jerk. Since a source is much smaller and lighter than a control rod, it is much easier to quickly remove from the core. On the other hand, in commercial reactors, this method cannot be used, because in commercial reactors neutron sources (when used) cannot be simply removed from the core. Moreover commercial reactors contains high burnup fuel which is very important source of neutrons.

See also: Source Neutrons

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_eff = 0.005 at the end of the cycle). This is due to isotopic changes in the fuel. It is simple, fresh uranium fuel contains only ²³⁵U as the fissile material, meanwhile during fuel burnup the importance of fission of ²³⁹Pu increases (in some cases up to 50%). Since ²³⁹Pu 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

where:

• n(t) is the neutron density in the core (which is proportional to the detector count rate)
• C_i(t) is the precursors density of delayed neutrons of group i
• Λ is the prompt generation time
• ρ is the reactivity
• the constants β_i and λ_i are the fraction and decay constant of delayed neutron precursor of group i
• 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.

According to Seiji TAMURA, the Inverse Kinetics equations (flux ⟶ reactivity) for discrete time series data can be derived from the ordinary point kinetics equations (reactivity ⟶ flux) assuming that the reactor power change for the interval of Δt is as n(t)=n_j-1exp(μ_jt), where μ_j = log(n_j/n_j-1)/Δt:

When the reactor power is at a steady state n_o, there are seven initial conditions that can be obtained as:

The equation for ρ_j provides a real time reactivity calculation for successive reactor power data. When the reactor is operating at sufficiently high power level the last term of the right side, the source term, may be neglected, because it becomes negligible.

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. Otherwise zero reactivity will be obtained for any subcritical reactor condition (after the transition phenomena have died out). The source term can be determined using data from a transient state after the introduction of a given reactivity. The source term can be determined from data from rod drop experiment using the Least Squares Inverse Kinetics Method (LSIKM).

Special reference: M. Itagaki, A. Kitano, “Revised source strength estimation for inverse neutron kinetics,” Preprints 1999 Fall Mtg., At. Energy Soc. Jpn., Hiroshima, Japan, G20, (1999)

Special reference: Renato Yoichi Ribeiro Kuramoto, Anselmo Ferreira Miranda. SUBCRITICAL REACTIVITY MEASUREMENTS AT ANGRA 1 NUCLEAR POWER PLANT. 2011 International Nuclear Atlantic Conference – INAC 2011. ISBN: 978-85-99141-04-5.

↑T_M ⇒ ↓k_eff = η.ε.  ↓p  .f.  ↓P_f  .  ↓P_t  (EOC)

Resonance escape probability. It is known, the resonance escape probability is dependent also on the moderator-to-fuel ratio. As the moderator temperature increases the ratio of the moderating atoms (molecules of water) decreases as a result of the thermal expansion of water. Its density decreases. This, in turn, causes a hardening of neutron spectrum in the reactor core resulting in higher resonance absorption (lower p). Decreasing the density of the moderator causes that neutrons stay at a higher energy for a longer period, which increases the probability of non-fission capture of these neutrons. This process is one of two processes (or three if chemical shim is used), which determine the moderator temperature coefficient.

↑T_M ⇒ ↓k_eff = η.ε.  ↓p  .f.  ↓P_f  .  ↓P_t  (EOC)

Thermal utilization factor. The impact on the thermal utilization factor depends strongly on the amount of boron which is diluted in the primary coolent (chemical shim). As the moderator temperature increases the density of water decreases due to the thermal expansion of water. But along with the moderator also boric acid is expanded out of the core. Since boric acid is a neutron poison, and it is expanding out of the core, positive reactivity is added. The positive reactivity addition due to the expansion of boron out of the core offsets the negative reactivity addition due to the expansion of the moderator out of the core. It is obvious this effect is significant at the beginning of the cycle (BOC) and gradually loses its significance as the boron concentration decreases.

↑T_M ⇒ ↓k_eff = η.ε.  ↓p  .f.  ↓P_f  .  ↓P_t  (EOC)

Change of the neutron leakage. Since both (P_f and P_t) are affected by a change in moderator temperature in a heterogeneous water-moderated reactor and the directions of the feedbacks for both negative, the resulting total non-leakage probability is also sensitive on the change in the moderator temperature. In result, an increase in the moderator temperature causes that the probability of leakage increases. In case of the fast neutron leakage, the moderator temperature influences macroscopic cross-sections for elastic scattering reaction (Σ_s=σ_s.N_H2O) due to the thermal expansion of water, which results in an increase in the moderation length. This, in turn, causes an increase of the leakage of fast neutrons.

• For the thermal neutron leakage there are two effects. Both processes have the same direction and together causes the increase in the thermal neutron leakage. This physical process is a part of the moderator temperature coefficient (MTC).
  • Macroscopic cross-sections for elastic scattering reaction Σ_s=σ_s.N_H2O, which significantly changes due to the thermal expansion of water. As the temperature of the core increases, the diffusion coefficient (D = 1/3.Σ_tr) increases.
  • Microscopic cross-section (σ_a) for neutron absorption changes with core temperature.  As the temperature of the core increases, the absorption cross-section decreases.

Let assume that the reactor is critical at 75% of rated power and that the plant operator wants to increase power to 100% of rated power. The reactor operator must first bring the reactor supercritical by insertion of a positive reactivity (e.g. by control rod withdrawal or boron dilution). As the thermal power increases, moderator temperature and fuel temperature increase, causing a negative reactivity effect (from the power coefficient) and the reactor returns to the critical condition. In order to keep the power to be increasing, positive reactivity must be continuously inserted (via control rods or chemical shim). After each reactivity insertion, the reactor power stabilize itself proportionately to the reactivity inserted. The total amount of feedback reactivity that must be offset by control rod withdrawal or boron dilution during the power increase (from ~1% – 100%) is known as the power defect.

Let assume:

• the power coefficient:                 Δρ/Δ% = -20pcm/% of rated power
• differential worth of control rods:    Δρ/Δstep = 10pcm/step
• worth of boric acid:                                      -11pcm/ppm
• desired trend of power decrease:              1% per minute

75% → ↑ 20 steps or ↓ 18 ppm of boric acid within 10 minutes → 85% → next ↑ 20 steps or ↓ 18 ppm within 10 minutes → 95% → final ↑ 10 steps or ↓ 9 ppm within 5 minutes → 100%

Let assume that the reactor is critical at 100% of rated power and that the plant operator wants to decrease power to 75% of rated power. The reactor operator must first bring the reactor subcritical by insertion of a negative reactivity (e.g. by control rod insertion or boric acid addition). As the thermal power decreases, moderator temperature and fuel temperature decrease as well, causing a positive reactivity effect (from the power coefficient) and the reactor returns to the critical condition. In order to keep the power to be decreasing, negative reactivity must be continuously inserted (via control rods or chemical shim). After each reactivity insertion, the reactor power stabilize itself proportionately to the reactivity inserted.

Let assume:

• the power coefficient: Δρ/Δ% = -20pcm/% of rated power
• differential worth of control rods: Δρ/Δstep = 10pcm/step
• worth of boric acid: -11pcm/ppm
• desired trend of power decrease: 1% per minute

100% → ↓ 20 steps or ↑ 18 ppm of boric acid within 10 minutes → 90%→ next ↓ 20 steps or ↑ 18 ppm within 10 minutes → 80% → final ↓ 10 steps or ↑ 9 ppm within 5 minutes→ 75%

α_f = ^dρ⁄_dTf

The magnitude and sign (+ or -) of the fuel temperature coefficient is primarily a function of the fuel composition, especially the fuel enrichment. In power reactors, in which low enriched fuel (e.g.  PWRs and BWRs require 3% – 5% of 235U) is used, the Doppler coefficient is always negative. In PWRs, the Doppler coefficient can range, for example, from -5 pcm/°C to -2 pcm/°C. This coefficient is of the highest importance in the reactor stability. The fuel temperature coefficient is generally considered to be even more important than the moderator temperature coefficient (MTC). Especially in case of all reactivity initiated accidents (RIA), the fuel temperature coefficient will be the first and the most important feedback, that will compensate the inserted positive reactivity. The time constant for heating fuel is almost zero, therefore the fuel temperature coefficient is effective almost instantaneously. Therefore this coefficient is also called the prompt temperature coefficient because it causes an immediate response on changes in fuel temperature.

Fuel Temperature Coefficient and Fuel Design

In general, the fuel temperature coefficient is primarily a function of the fuel enrichment. There are two phenomena associated with Doppler effect. It results in increased neutron capture by fuel nuclei (both fissile and fissionable) in the resonance region, on the other hand it results in increased neutron production from fissile nuclei. Therefore DTC may be also positive. A net positive Doppler coefficient requires higher enrichments of fissile nuclei. Some calculations shows, that enrichments higher than 30% are associated with slightly positive DTC, but this enrichment results in a harder neutron spectrum. Low enriched uranium oxide fuels  usually provide a large negative DTC, because of the elastic scattering reaction of oxygen nuclei (soften spectrum). Another way to produce more negative DTC is to soften the neutron spectrum by addition of moderator nuclei directly into the fuel matrix. For example, the TRIGA reactor uses uranium zirconium hydride (UZrH) fuel, which has a large negative fuel temperature coefficient. The rise in temperature of the hydride increases the probability that a thermal neutron in the fuel element will gain energy from an excited state of an oscillating hydrogen atom in the lattice. As the neutrons gain energy from the ZrH, the thermal neutron spectrum in the fuel element shifts to a higher average energy (i.e., the spectrum is hardened). This spectrum hardening is used differently to produce the negative temperature coefficient.

In the United States, the Nuclear Regulatory Commission will not license a reactor unless α_Prompt (FTC) is negative. All licensed reactors are thereby assured of being inherently stable. This requirement is followed by many other countries.

The moderator-to-fuel ratio strongly influences especially:

• Resonance escape probability. An increase in moderator-to-fuel ratio causes an increase in resonance escape probability. As more moderator molecules are added relative to the amount of fuel molecules, than it becomes easy for neutrons to slow down to thermal energies without encountering a resonance absorption at the resonance energies.
• Thermal utilization factor. An increase in moderator-to-fuel ratio causes a decrease in thermal utilization factor. The value of the thermal utilization factor is given by the ratio of the number of thermal neutrons absorbed in the fuel (all nuclides) to the number of thermal neutrons absorbed in all the material that makes up the core.
• Thermal and fast non-leakage probability. An increase in moderator-to-fuel ratio causes a decrease in migration length, which in turn causes an increase in non-leakage probability.

As can be seen from the figure, at low moderator-to-fuel ratios the product of all the six factors (k_eff) is small because the resonance escape probability is small. At optimal value of moderator-to-fuel ratio, k_eff reaches its maximum value. This is the case of so called “optimal moderation”. At large ratios, k_eff is again small because  the thermal utilization factor is small.

Under-moderated vs. Over-moderated Reactor

From the moderator-to-fuel ratio point of view, any multiplying system can be designed as:

• Under-moderated. Under-moderation means that there is less than optimum amount of moderator between fuel plates or fuel rods. An increase in moderator temperature and voids decreases k_eff of the system and inserts negative reactivity. An under-moderated core would create a negative temperature and void feedback required for a stable system.

• Over-moderated. Over-moderation means that there is higher than optimum amount of moderator between fuel plates or fuel rods. An increase in moderator temperature and voids increases k_eff of the system and inserts positive reactivity. An over-moderated core would create a positive temperature and void feedback. It will result in an unstable system, unless another negative feedback mechanism (e.g. the Doppler broadening) overrides the positive effect.

Reactor engineers must balance the composite effects of moderator density, fuel temperature, and other phenomena to ensure system stability under all operating conditions. Most of light water reactors are therefore designed as so called under-moderated and the neutron flux spectrum is slightly harder (the moderation is slightly insufficient) than in an optimum case. But this design provides important safety feature. An increase in the moderator temperature results in negative reactivity which tends to make the reactor self-regulating. It must be added, the overall feedback must be negative, but local positive coefficients exists in areas with large water gaps that are over-moderated such as near control rods guide tubes.

Another phenomenon associated with under-moderated core is called the neutron flux trap effect. This effect causes an increase in local power generation due better thermalisation of neutrons in areas with large water gaps (between fuel assemblies or when fuel assembly bow phenomenon is present). Note that “flux traps” are a standard feature of most modern test reactors because of the desire to obtain high thermal neutron fluxes for the irradiation of materials, but basically it can occur also in PWRs.

On the other hand, also under-moderation has its limits. In general, it causes a decrease in overall k_eff, therefore more fissile material is needed to ensure criticality of the core. Moreover, there is also a limit on the minimal value of MTC (most negative). It is due to the fact the negative temperature feedback acts also against decrease in the moderator temperature. Consider what happens when moderator temperature is decreased quickly, as in the case of the main steamline break (MSLB – standard initiating event for PWRs). The steamline break causes the steam pressure, the saturation temperature in the steam generators to fall rapidly. As a result of falling saturation temperature in the steam generators the moderator temperature will rapidly decrease. The rapid moderator temperature drop causes a positive reactivity insertion. The amount of reactivity inserted depends also on a magnitude of the MTC and therefore it must be limited. Typical values for lower limit is MTC = -80 pcm/°C, but it is a plant specific value limited in technical specifications.

Let assume that the reactor is critical at 75% of rated power and that the plant operator wants to increase power to 100% of rated power. The reactor operator must first bring the reactor supercritical by insertion of a positive reactivity (e.g. by control rod withdrawal or boron dilution). As the thermal power increases, moderator temperature and fuel temperature increase, causing a negative reactivity effect (from the power coefficient) and the reactor returns to the critical condition. In order to keep the power to be increasing, positive reactivity must be continuously inserted (via control rods or chemical shim). After each reactivity insertion, the reactor power stabilize itself proportionately to the reactivity inserted. The total amount of feedback reactivity that must be offset by control rod withdrawal or boron dilution during the power increase (from ~1% – 100%) is known as the power defect.

Let assume:

• the power coefficient:                 Δρ/Δ% = -20pcm/% of rated power
• differential worth of control rods:    Δρ/Δstep = 10pcm/step
• worth of boric acid:                                      -11pcm/ppm
• desired trend of power decrease:              1% per minute

75% → ↑ 20 steps or ↓ 18 ppm of boric acid within 10 minutes → 85% → next ↑ 20 steps or ↓ 18 ppm within 10 minutes → 95% → final ↑ 10 steps or ↓ 9 ppm within 5 minutes → 100%

Let assume that the reactor is critical at 100% of rated power and that the plant operator wants to decrease power to 75% of rated power. The reactor operator must first bring the reactor subcritical by insertion of a negative reactivity (e.g. by control rod insertion or boric acid addition). As the thermal power decreases, moderator temperature and fuel temperature decrease as well, causing a positive reactivity effect (from the power coefficient) and the reactor returns to the critical condition. In order to keep the power to be decreasing, negative reactivity must be continuously inserted (via control rods or chemical shim). After each reactivity insertion, the reactor power stabilize itself proportionately to the reactivity inserted.

Let assume:

• the power coefficient: Δρ/Δ% = -20pcm/% of rated power
• differential worth of control rods: Δρ/Δstep = 10pcm/step
• worth of boric acid: -11pcm/ppm
• desired trend of power decrease: 1% per minute

100% → ↓ 20 steps or ↑ 18 ppm of boric acid within 10 minutes → 90%→ next ↓ 20 steps or ↑ 18 ppm within 10 minutes → 80% → final ↓ 10 steps or ↑ 9 ppm within 5 minutes→ 75%

1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
5. W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
6. G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
7. Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
8. U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.

Advanced Reactor Physics:

1. K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.
2. K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.
3. D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2. 
4. E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.