Reactor Physics Experiments

Abstract

A correct comprehension of fission chain reactions in a reactor core is a significant clue for carrying out reactor physics experiments, including approach to criticality, control rod calibration and subcriticality experiments as described in this chapter. Based on an easy understanding of neutron characteristics on a reactor core, such as fission chain reactions, approach to criticality experiment is the start of the first touch of reactor physics itself. Before explaining about the approach to criticality experiment, neutron multiplication is briefly described by using two theoretical concepts with and without a neutron source. In terms of control rod calibration experiment, measurement methodologies are provided for excess reactivity and control rod worth by the positive period and the rod drop methods, respectively. Noteworthy is that subcriticality measurement methodologies are uniquely introduced, including the neutron source multiplication method, the source jerk method, the inverse kinetics method and the reactor noise analysis method. Also, the methodologies could be helpful to understand the magnitude of negative reactivity in the core: to offer a kindly response to “how much margin (i.e., subcriticality) is left in the core from the critical state?”.

2.1 Approach to Criticality Experiment

2.1.1 Purpose

In fission chain reactions, fission reactions are induced by the U-235 absorption reactions with neutrons, and two or three neutrons generated at this time are moderated to be thermal neutrons, and one or more of the neutrons could contribute to fission reactions in a next generation. To reduce the ratio of wasted neutrons, it is necessary to reduce the number of the neutrons that are absorbed into U-238 and other structural materials of the reactor or leaked out of the core. Thus, we would like you to learn how to appropriately control the absorption and leakage of neutrons through the approach to criticality experiment, in addition to fission reactions in the system.

When a new reactor core is constructed in a reactor building, fuel is loaded into the core little by little. Then, to confirm the birth of the new reactor, a first experiment is always conducted by measuring the amount of the fuel that is just enough to induce fission chain reactions, i.e., the amount of the fuel that is sufficient to reach a critical state. The experiment is called the approach to criticality experiment. The main purpose of the experiment is to understand how the critical state is reached in UTR-KINKI.

2.1.2 Principle of Measurement

2.1.2.1 Neutron Multiplication

• (1) Balance of neutrons

Balance of neutrons in a nuclear reactor was explained in Sect. 1.2.5, as can be summarized as follows: “Neutrons are generated by fission reactions, move around the reactor repeatedly by scattering reactions, and eventually leak out of the reactor or are absorbed by structure materials in the reactor.”

The balance of neutrons is assumed to be in a nuclear reactor, and then, let the balance in a system with a certain volume, and focus on the change in the number of neutrons in unit volume and unit time. When the increase or decrease of the number of neutrons is considered as a positive or negative effect, the rate of change can be finally expressed by using the effects of increase and decrease as follows:

$$\begin{aligned} {``}{\text{Rate of change}}\text{''} & = \left( {\text{Effect of increase}} \right){-}\left( {\text{Effect of decrease}} \right) \\ & = \left( {{``}{\text{Production rate}}\text{''} + {``}{\text{Inflow rate}}\text{''}} \right) \\ & \quad {-}\left( {{``}{\text{Absorption rate}}\text{''} + {``}{\text{Outflow rate}}\text{''}} \right) \\ \end{aligned}$$

We transform “Rate of change” mentioned above as follows:

$$\begin{aligned} {\text{``Rate of change{''}}} & = ``{\text{Production rate{''}}}{-}``{\text{Absorption rate{''}}} \\ & \quad {-}\left( {{\text{``Outflow rate{''}}}{-}``{\text{Inflow rate{''}}}} \right) \\ \end{aligned}$$

If the third term on the right-hand side in the above “equation” is “net outflow rate (leakage rate),” the “Rate of change” can be rewritten as follows (We call this the “equation of continuity,” but we skip the details):

$${\text{``Rate of change{''}}} = ``{\text{Production rate{''}}}{-}``{\text{Absorption rate{''}}}{-}``{\text{Leakage rate{''}}}$$

Let’s imagine which reaction in the behavior of neutrons in a nuclear reactor corresponds to three terms that constitute the above “Rate of change.” Here, we consider that the “rate (probability) is proportional to the number of neutrons. The “Production rate” is the number of neutrons generated by fission reactions, the “Absorption rate” is that by absorption reactions, and the “Leakage rate” is that by leaked outside. Furthermore, when the three terms are divided into positive and negative groups again, the overall balance between two groups, that is, the ratio of “increase due to positive factors” and “decrease due to negative factors” is not always unity. When the coefficient that adjusts this balance (adjustment factor) is k, k can be expressed as follows:

$$\begin{aligned} k & = ``{\text{Increase due to positive factors{''}}}/``{\text{Decrease due to negative factors{''}}} \\ & = ``{\text{Production reactions{''}}}/\left( {{\text{``Capture reactions{''}}} + ``{\text{Leakage{''}}}} \right) \\ \end{aligned}$$

The above idea is a basic approach to treat the behavior of neutrons in the neutron diffusion theory, which is called the “effective multiplication factor” with the notation “$k_{eff}$” for the adjustment factor k.

Here, in the neutron diffusion theory with one-energy group, the behavior of neutrons is expressed by using the effective multiplication factor $k_{eff}$ as follows (Here, scattering reactions are neglected. For an explanation of neutron leakage, we refer the reader to other textbooks.):

$$- D\nabla^{2} \phi + \Sigma_{a} \phi = \frac{1}{{k_{eff} }}\nu \Sigma_{f} \phi ,$$

(2.1)

where D is the diffusion coefficient, $\phi$ the neutron flux, $\Sigma_{a}$ the absorption cross section, $\nu$ the average number of neutrons triggered per fission reaction, and $\Sigma_{f}$ the fission cross section. It is easy to understand that the left-hand side in Eq. (2.1) is the “decrease of number of neutrons,” the right-hand side is the “increase of number of neutrons,” and their adjusting factor is then $1\,/\,k_{eff}$.

Once again, interpreting the balance of neutrons in the system as the ratio of increase and decrease of number of neutrons, the balance of neutrons is a straightforward concept for understanding neutron multiplication. The $k_{eff}$ in Eq. (2.1) is then written as follows:

$$k_{eff} = \frac{{\nu \Sigma_{f} \phi }}{{ - D\nabla^{2} \phi + \Sigma_{a} \phi }}.$$

(2.2)

The change in the value of $k_{eff}$ depends on the balance of denominator and numerator factors shown in Eq. (2.2) and can be interpreted as follows:

• When the number of neutrons generated by fission reactions is larger than those by absorption reactions and leakage, $k_{eff} > 1$.

• When the number of neutrons generated by fission reactions is equal to those by absorption reactions and leakage, $k_{eff} = 1$.

• When the number of neutrons generated by fission reactions is smaller than those by absorption reactions and leakage, $k_{eff} < 1$.

From the above interpretation, we think it is easy to judge that $k_{eff}$ is closely related to the reactor condition. In other words, when the effective multiplication factor $k_{eff}$ is unity, the reactor reaches a critical state. Moreover, when $k_{eff}$ is smaller than unity, the reactor condition is called a subcritical state. If $k_{eff}$ is larger than unity, the reactor condition is called a supercritical state.

• (2) Generation of neutrons

The effective multiplication factor explained by the concept of the balance of neutrons was based on the assumption that the neutrons generated by fission reactions already existed in the reactor. Noteworthy is that the behavior of neutrons in this section is, however, different from that in the previous section ((1) Balance of neutrons), because the balance of neutrons is assumed to observe the system back to the origin of fission reactions and to recognize the existence of the neutron source that is an initial trigger of fission reactions.

In a system with the U-235 fuel and light-water moderators, fission reactions are impossible to be triggered with only the existence of fuels and moderators in the system. Here, seeds, as is called “source neutrons,” are required for inducing an actual trigger of fission reactions.

Let us suppose that number of source neutrons S is emitted from a plutonium–beryllium (Pu-Be) neutron source. Also, let us focus on S emitted from the neutron source and observe how these neutrons move around in the core. Here, we introduce the ratio of change of neutrons (balance between increase and decrease in the number of neutrons) in the system introduced in the balance of neutrons.

In addition, let us denote the ratio of change as $k_{s}$(note that this is not the same as for the balance of neutrons, since a neutron source is existed in the core), number of source neutrons S is then inserted into the reactor, and neutrons cause repeatedly scattering, capture and fission reactions, and leak out of the reactor. When a series of events is considered an initial generation, one of two or three of the neutrons produced in the initial generation of fission reactions will contribute to fission reactions in the next generation. In short, before the initial trigger of fission reactions (zero generation), the number of source neutrons S is provided from the neutron source. After the first generation, the number of neutrons $k_{s} \,S$ will be produced.

Neutrons in a successive generation are multiplied by $k_{s}$ again after the first generation, and the number of neutrons $k_{s}^{2} \,S$ is produced after the second generation. Thus, a total number of neutrons are $S + k_{s} S + k_{s}^{2} S + k_{s}^{3} S + k_{s}^{4} S + \cdots$ for the third, fourth generations, and so on up to infinity, including the initial number of source neutrons S. Then, if the total number of neutrons generated by fission reactions in all generations is F, that is, all generations are expanded to infinity, F can be expressed by using infinite geometric series as follows:

$$\begin{aligned} F & = S + k_{s} S + k_{s}^{2} S + k_{s}^{3} S + k_{s}^{4} S + \cdots = S\left( {1 + k_{s} + k_{s}^{2} + k_{s}^{3} + k_{s}^{4} + \cdots } \right) \\ & = \frac{S}{{1 - k_{s} }}. \\ \end{aligned}$$

(2.3)

Here, the multiplication of neutrons in the system is determined by the ratio of the number of source neutrons S and the total number of neutrons generated by fission reactions F. If the multiplication of neutrons is denoted as M, M can be expressed by using Eq. (2.3) as follows:

$$M = \frac{F}{S} = \frac{S}{{1 - k_{s} }} \cdot \frac{1}{S} = \frac{1}{{1 - k_{s} }}.$$

(2.4)

In Eq. (2.4), the multiplication M is diverging to infinity when $k_{s}$ is approaching 1 (unity: a critical state); i.e., when $k_{s}$ is less than 1 (a subcritical state), M converges to a certain value (saturation). In the approach to criticality experiment, to make an easy understanding of the multiplication M by increasing $k_{s}$, we transform Eq. (2.4) as follows:

$$\frac{1}{M} = 1 - k_{s} .$$

(2.5)

If Eq. (2.5) is defined as the inverse multiplication $1\,/\,M$, we observe that the inverse multiplication approaches zero as $k_{s}$ approaches 1. Using the property, in the approach to criticality experiment, the parameter relating to $k_{s}$ is set on the horizontal axis and the inverse multiplication on the vertical axis. The criticality of the core can be then determined by gradually bringing the reactor closer (larger) to the criticality (horizontal axis) and finding the point where the inverse multiplication (vertical axis) is zero (extrapolation).

2.1.2.2 Role of Neutron Source

The behavior of neutrons in the core was explained by introducing two concepts of “balance” and “generation” of neutrons, to understand easily the concept of neutron multiplication. The absence of neutron source was confirmed in the concept of the balance of neutrons, and conversely, in that of the generation of neutrons, the neutron source was present. Since a series of reactor physics experiments, except for control rod calibration experiment, could be conducted with the use of neutron source, an actual role of neutron source should be understood in the procedures of the experiments.

The neutron source used in UTR-KINKI is Pu-Be, and neutrons are generated by the ⁹Be($\alpha$, n)¹²C reactions using particles emitted from Pu-239, which is one of the radioactive sources. The energy of the neutrons generated in the ⁹Be($\alpha$, n)¹²C reactions is 4.5 MeV, and we can easily image that the neutrons are sufficiently fast because of their considerable energy. Considering that the half-life of Pu-239 is 24,110 years, Pu-Be can be used as a neutron source semipermanently.

Based on the neutron diffusion theory with one-energy group, how can we describe the behavior of neutrons in the core under the existence of neutron source? Neutrons have repeatedly scattering, capture and fission reactions, and leak out of the core. At the same time, source neutrons are continuously supplied from the neutron source. If the total number of neutrons supplied from the neutron source is S, the balance of neutrons can be then expressed as follows:

$$- D\nabla^{2} \phi + \Sigma_{a} \phi = \nu \Sigma_{f} \phi + S.$$

(2.6)

It can be observed that the ratio of change of neutrons in the presence of a neutron source is always constant; i.e., no significant change in the behavior of neutrons occurs in the reactor while neutrons are constantly supplied. As an aside, from Eq. (2.4), the right-hand side in Eq. (2.5) is positive at a subcritical state, $1 - k_{s} > 0$, resulting in $k_{s} < 1$. The multiplication of neutrons is also then, $M > 0$. The interpretation implies that the ratio of change of neutrons is less than 1 (unity) in the system where the neutron source is used; i.e., the steady-state reactor is always at a subcritical state under the existence of the neutron source in the core.

Here, we should always confirm whether a neutron source is in the system or not at the time of the experiment. This confirmation is a significant clue to determine whether the reactor is at a subcritical or not.

[Column] Extension of neutron diffusion equation to two-energy groups

The behavior of neutrons is conveniently described by the neutron diffusion theory with one-energy group as shown in Sect. 2.1.2.1, and here, let us consider what it looks like when the one-energy group is extended to the two-energy groups. Although Eq. (2.1) showed the neutron diffusion equation with one-energy group, please think about the reason why it is necessary to extend the equation from the one-energy group to the two-energy groups.

When we imagine the behavior of neutrons in a nuclear reactor, it is convenient to model the behavior in one-dimensional plate or two-dimensional cylinder to understand the physical phenomena qualitatively. Suppose the behavior is modeled in three-dimensional geometry. Then, we could model the behavior of neutrons in three-dimensional geometry more precisely than that in one-dimensional or two-dimensional geometry.

How the neutron energy is then? If the energy distribution (neutron spectrum) of neutrons in the reactor is dominated by fast neutrons, as in fast reactors, there is no serious obstacle to understanding the behavior of neutrons even if we assume the energy distribution of neutrons as one-energy group.

In UTR-KINKI, U-235 is used as a fuel, light water as a moderator and graphite as a reflector. When we imagine the behavior of neutrons in the reactor from this composition, the neutrons generated by U-235 fission reactions are almost fast neutrons, which are eventually changed into thermal neutrons through repeated reactions with light water as moderators, and graphite as reflectors playing a similar role as moderators in the reactor. Then, the energy of neutrons has a wide distribution of fast and thermal neutrons. If a wide range of energy is treated as one-energy group, the behavior of neutrons in a nuclear reactor cannot be accurately understood.

Therefore, we roughly divide the energy distribution of neutrons into two-energy groups, fast and thermal neutrons, and summarize the physical phenomena and interactions (reactions) of each group as well as the one-energy group diffusion equation. Here, the subscripted numbers represent the energy group (“1” is the fast group and “2” is the thermal group), $\Sigma_{a}$ is the capture reaction, $\Sigma_{s}$ the scattering reaction, and $\Sigma_{f}$ the fission reaction. Let us consider the behavior of fast and thermal neutrons in terms of the effect of increasing or decreasing the number of neutrons in the core, as used in the concept of neutron multiplication, as follows:

• (1) Fast neutron

(Effect of removal) Leakage outside the reactor, capture reaction, downward scattering reaction from group 1 to group 2.

(Effect of production) Fission reaction (the only fast group is neutron source by fission).

The effects can be described by the diffusion equation using the neutron fluxes $\phi_{1}$ and $\phi_{2}$, the effective multiplication factor $k_{eff}$ as follows:

$$- D_{1} \nabla^{2} \phi_{1} + \left( {\Sigma_{a,1} + \Sigma_{s,1 \to 2} } \right)\phi_{1} = \frac{1}{{k_{eff} }}\left( {\nu_{1} \Sigma_{f,1} \phi_{1} + \nu_{2} \Sigma_{f,2} \phi_{2} } \right),$$

(2.7)

• (2) Thermal neutron

(Effect of removal) Leakage out of the reactor and capture reaction. Here, upward scattering reactions from group 2 to group 1 are assumed to be absent.

(Effect of production) Neutron source by fission is not considered to be thermal neutron group.

The behavior of thermal neutrons, as well as that of fast neutrons, can be described by the diffusion equation as follows:

$$- \,D_{2} \,\nabla^{2} \,\phi_{2} \, + \,\Sigma_{a,\,2} \,\phi_{2} \, - \,\Sigma_{s\,1 \to \,2} \,\phi_{1} = \,0.$$

(2.8)

Using Eqs. (2.7) and (2.8) as governing equations, for example, and giving the diffusion coefficient of each group and the cross section of each reaction, the neutron fluxes $\phi_{1}$ and $\phi_{2}$ of the fast and thermal groups, respectively, can be easily determined. The details of the two-energy-group diffusion equation are given in Refs. [1, 2].

2.1.3 Method of Measurement

Four possible ways are generally available to reach a critical state as follows:

• (1) Increasing gradually the amount of fuel and reaching a critical state

• (2) Increasing gradually the fuel concentration while keeping the core volume almost constant

• (3) Loading the fuel as much as possible to reach a critical state in advance, and increasing gradually the water level to reach a critical state

• (4) Loading the fuel as much as possible to reach a critical state in advance, and withdrawing gradually control rods to reach a critical state.

Among the four methods, the most common method is method (1), and, in this experiment, approach to criticality is performed by two methods: method (1) by increasing number of fuel plate; method (4) by operating control rods (to change the position). One of the two methods is, however, chosen depending on the experimental week, so you could observe the reactor carefully to confirm which method is used in the experiment.

2.1.3.1 Inverse Multiplication

The multiplication factor and its inverse value are obtained from neutron counts measured by the neutron detectors. Here, for the convenience of the experiment, we assume that the core is loaded with some fuel (hereinafter referred to as the initial core).

Let $k_{s,0}$ and $k_{s,i}$ the ratio of change of neutron multiplication in the initial core (subscript is “0.”) and the core with the ith state of loading fuel (subscript is “i.”), respectively. Since the neutrons generated by fission reactions correspond to the neutron fluxes $\phi_{s,0}$ and $\phi_{s,i}$ multiplied by the neutron source in the initial and i-th states, respectively, as shown in Eq. (2.3), we approximate them as follows:

$$\phi_{s,0} \approx \frac{S}{{1 - k_{s,0} }},$$

(2.9)

$$\phi_{s,i} \approx \frac{S}{{1 - k_{s,i} }}.$$

(2.10)

When the state of the core varies from the initial core into i-th state core, the value of $\phi_{s,i} /\phi_{s,0}$ can be expressed by using Eqs. (2.9) and (2.10), as follows:

$$\frac{{\phi_{s,i} }}{{\phi_{s,0} }} \approx \frac{{1 - k_{s,0} }}{{1 - k_{s,i} }}.$$

(2.11)

Here, for any state i, the state of the initial core is the same; i.e., the numerator $1 - k_{s,0}$ of the right-hand side in Eq. (2.11) can be then regarded as being constant (this is an important assumption). Therefore, we can approximate Eq. (2.11) as follows:

$$\frac{{\phi_{s,i} }}{{\phi_{s,0} }} \approx \frac{1}{{1 - k_{s,i} }}.$$

(2.12)

The inverse value of Eq. (2.12) can be expressed as follows:

$$\frac{{\phi_{s,0} }}{{\phi_{s,i} }} \approx 1 - k_{s,i} .$$

(2.13)

Equation (2.13) has the same form as the inverse value of multiplication (inverse multiplication) $1\,/\,M$ in Eq. (2.5), and the method for obtaining inverse multiplication $1\,/\,M$ can be directly applied by attaining the inverse of the neutron flux ratio $\phi_{s,0} /\phi_{s,i}$ in Eq. (2.13).

Let us assume that the response (counting rate) per unit time obtained from the neutron detector is proportional to the neutron flux (this is a vital assumption). When obtaining the counts $C_{0}$ and $C_{i}$ in states 0 and i, respectively, the ratio of the counting rates, $C_{0} \,/\,C_{i}$, can be expressed from Eq. (2.13) as follows:

$$\frac{{C_{0} }}{{C_{\,i} }} \approx \frac{{\phi_{k,0} }}{{\phi_{k,i} }} \approx 1 - k_{s,i} \left( { = \frac{1}{{M_{i} }}} \right).$$

(2.14)

The obtained $C_{0} /C_{i}$ is the inverse counting rate (inverse multiplication $1\,/\,M_{i}$ in state i). In short, experimentally, the inverse counting rate of Eq. (2.14) corresponds to the inverse multiplication of Eq. (2.5) itself.

2.1.3.2 Settings of Detectors

To reach safely a critical state, detectors are set at several appropriate locations in the reactor. Here, the outline of approach to criticality by method (1) shown in Sect. 2.1.3 is described as follows:

As shown in Fig. 2.1, the decrease in the inverse of multiplication (vertical axis) against the increase in the amount of fuel to be loaded (horizontal axis) is plotted on a graph, and a straight line is obtained from the two most recent points and extrapolated to find the intersection point with the horizontal axis. The curve of this plot may be convex upward or convex downward, depending on the position of the detector. In the approach to criticality experiment, the detectors are arranged so that both curves can be obtained, and especially at a near-critical state, it is desirable to obtain a curve with a downward convex trend from the viewpoint of criticality safety. (see Fig. 2.1).

Fig. 2.1

Relationship between inverse multiplication 1/M and critical mass of U-235 fuel

2.1.3.3 Fuel Loading

Fuel loading starts from the region of core center unless fuel is a liquid state. This is because the number of neutrons is highest near the center of the core, no matter what the core configuration is. If fuel is loaded here after the core is close to a critical state, the core will reach a critical state immediately.

The amount of the first fuel loaded is much less than the predicted critical mass. Let $C_{0}$ the total number of neutrons before the first fuel is loaded. After the first fuel is loaded, we wait for a while until the reactor power is constant (stable state), and measure the neutron counts in each condition (full insertion or full withdrawal of control rods) at a stable state. Let $C_{1}$ the total number of neutrons obtained in this way for the first fuel loading, and the inverse counting rate for the first fuel loading $\left( {C_{0} /C_{1} } \right)$ is obtained. Since the multiplication factor of neutrons is proportional to the counting value, the inverse counting rate $C_{0} /C_{1}$ is equivalent to the inverse multiplication factor $k_{0} /k_{1}$, where $k_{0}$ and $k_{1}$ are the multiplication factors of neutrons before and after the first fuel loading, respectively. As shown in Fig. 2.1, extrapolating the straight line obtained after repeated fuel loading (3 or 4 times), there is an intersection point between the straight line and the horizontal axis: the predicted critical mass. The additional fuel estimated from the predicted critical mass should not be loaded at once, and less than half of the predicted critical mass should be used as a guide for the next loading. Therefore, the amount of additional fuel should be reduced as the reactor is reaching a critical state, and the additional fuel loading should be handled with sufficient care.

2.1.3.4 Control Rod Operation

Approach to criticality experiment by operating control rods is, in principle, exactly the same as that by loading fuel. When all the control rods are withdrawn, the reactor is assumed to be at a supercritical state; i.e., there are enough fuel plates loaded in the core to make it critical. The control rod in UTR-KINKI is a plate absorber made of cadmium (Cd), so the control rod is sometimes referred to as a control plate or Cd plate in this text. (For details, please refer to Sect. 1.1.4.2).

The initial condition is when the control rod is at the lower limit, and the total value at that time is $C_{0}$. Here, the inverse counting rate of neutrons is the same as in the case of fuel loading, whereas the control rod position is on the horizontal axis, differing from the number of fuel plates (critical mass) shown in Sect. 2.1.3.3. The control rod position in the horizontal axis corresponds to the amount of fuel loaded in the case of fuel loading. The control rods move along the axial direction of the core, and the control rods are gradually withdrawn from the core, as in the case of fuel loading. For example, if $C_{1}$ is the value of neutron counts when the control rod is withdrawn by 10 cm from the lower limit (0 cm: center position in axial direction), the inverse counting rate is $C_{0} /C_{1}$, given by the initial inverse multiplication when the control rod is withdrawn. By repeating this process several times, the inverse multiplication curve resulted in the control rod operation is obtained. Also, “position of the control rod at a critical state” can be determined by extrapolating the straight line obtained from the last two points, since the horizontal axis is the position of the control rod. Although the method of the control rod operation is different from that of the fuel loading, the basic principle of measurement is the same in the two methods. In short, when the control rod position is gradually changed toward the upper limit, the predicted control rod position at a critical state can be estimated by measuring quantitatively the neutron multiplication.

2.1.4 Procedure of Measurement

The specific procedure for the approach to criticality experiment is described in this section. The schematic of the measurement system of the neutron detector used in the approach to criticality experiment is shown in Fig. 2.2, and data sheet in the approach to criticality experiment is shown in Fig. 2.3.

Fig. 2.2

Neutron measurement system used in approach to criticality experiment

Fig. 2.3

Data sheet in approach to criticality experiment

2.1.4.1 Approach to Criticality by Loading Fuel Plates

• (1) Set the detector.

• (2) Check the detector.

The detector has already been set, and its location should be checked. We will consider the relationship between the detector location and the concavity of the $1/M$ curve later.

The UTR-KINKI core that is a coupled core is divided into two parts (cores), but in principle, the core is possible to be considered as a single core.

• (3) Check the conditions of the control rod and light-water moderators, and then, insert the neutron source.

A quantity of fuel (U-235) has already been loaded in the core, reaching a deep subcritical state.

In the state (denoted $i = 0$) before the first fuel is loaded, let $C_{0}$ the count under the existence of the neutron source.

• (4) Measurements of the counting rate of neutrons are made as follows:

• (a) When two control rods (Shim Safety Rod: SSR and Regulating Rod: RR) are inserted, withdrawing two Safety Rods (SRs #1 and #2) (denoted $j\, = \,1$).

• (b) When two control rods (SSR and RR) have been withdrawn, withdrawing two safety rods (SRs #1 and #2) (denoted $j\, = \,2$).

In other words, two counting rates, $C_{0,1}$ and $C_{0,2}$, are obtained as the state before the fuel is loaded. (The measurements of $C_{0,1}$ and $C_{0,2}$ are also meant to check the effectiveness of the control rods at the same time in the process of the approach to criticality experiment.)

• (5) All control rods are inserted when the measurement in (b) of step (4) is completed.

• (6) The first fuel is loaded (fuel elements are carefully inserted one by one, paying attention to the increase of the counting rate (defined $i = 1$).

• (7) The counting is conducted in the same way as in step (4). (Two neutron counting rates, $C_{1,1}$ and $C_{1,2}$, are obtained.)

• (8) All control rods are inserted when the measurement in (b) of step (4) is completed.

• (9) Take the fuel loading (mass of U-235) on the horizontal axis and the inverse counting rate (inverse multiplication) on the vertical axis of the graph, and plot the value of $C_{0,1} /C_{1,1}$ (and $C_{0,2} /C_{1,2}$).

• (10) Assuming that the first plot point $C_{0,1} /C_{1,1}$ is normalized as unity (= 1), a straight line is obtained with the combined use of the next point obtained in step (9) and the first point, and the critical mass can be then predicted by extrapolating the line. Based on the results, the amount of fuel to be loaded next is determined. It is evident that the counting rate of neutrons increases as additional fuel is loaded, and let us observe that it takes time for the counting rate to converge (saturate).

• (11) Repeat steps (6) to (10). (i = 2, 3, 4, etc., correspond to the number of fuel loadings.)

• (12) At a near-critical state (e.g., $i = 4$), SRs #1 and #2 are withdrawn, and as a next step, following the withdrawal of SSR and RR. Here, the reactor is not yet at a critical state when the neutron source is withdrawn and the indicated value by the recorder shows a decreasing tendency. Conversely, when the indicated value reveals an increasing tendency with the fluctuation of reactor power, the neutron source is withdrawn. It is then observed whether the value is a constant value or not. The reactor reaches a critical state when the value of the recorder reveals a constant value, adjusting the positions of SSR and RR.

2.1.4.2 Approach to Criticality by Operating Control Rods

This section describes the procedure (step) of the approach to criticality experiment with the control rod operation.

• (1) Set the detector.

• (2) Check the detector.

• (3) Insert the neutron source.

We know that fission chain reactions never occur at the positions where the SRs #1 and #2, the RR are at the lower limit, the SSR is at the upper limit. First, make sure that the positions of SRs #1 and #2 are at the upper limit, the RR is at the lower limit, and the SSR is at the lower limit (center of the Cd plate: 0%; 0 cm; denoted $i = 0$). The counting rate in the state $i = 0$ is $C_{0}$.

• (4) The counting rate of neutrons is measured with SSR at the lower limit.

• (5) Withdraw the SSR by 30% from the lower limit 0%. (Note the increase in the counting rate of neutrons; this is assumed to be $i = 1$).

• (6) Take the position of SSR on the horizontal axis of the graph and the inverse counting rate (inverse multiplication) on the vertical axis, and plot the value of $C_{0} /C_{1}$.

• (7) Assuming that the first plot point $C_{0} /C_{1}$ is normalized as unity (= 1), a straight line can be obtained with the combined use of the next point obtained in step (6) and the first point, and the SSR position that the reactor could reach a critical state is predicted by the linear extrapolation. Based on the result, the next step is available to determine a position where SSR should be withdrawn. It is evident that the counting rate of neutrons will increase as the SSR is withdrawn, and let us observe that it will take time for the counting rate to converge (saturate).

• (8) Repeat steps (5) to (7) in the proportions of 50%, 70%, 80%, etc. ($i = 2,3,4,$ etc., corresponding to the number of times where the control rod is withdrawn.)

• (9) When the neutron source is withdrawn at a near-critical state (e.g., $i = 4$) and the counting rate shows a decreasing tendency of reactor power, the reactor is not yet at a critical state. Conversely, the value indicated by the recorder shows an increasing tendency with the fluctuation of reactor power, when the neutron source is withdrawn from the core at this time. Then, we observe whether the indicated value is a constant value or not. When the indicated value of the recorder shows a constant value, the reactor is at a critical state.

2.1.5 Discussion

• (1) Does the inverse multiplication curve show an upward convexity, a downward convexity, or a shape close to a straight line? In particular, let us observe the position of the detector and the positional relationship with the neutron source, and consider why different shapes are obtained depending on the position of the detector, including physical reasons (Only the case of “Approach to Criticality by Loading Fuel Plates” in Sect. 2.1.4.1 will be considered.).

• (2) Let us determine whether the critical mass obtained from each curve can be regarded as appropriate values.

• (3) Assuming that there is only one detector to be used in the experiment, let us examine from the viewpoint of criticality safety which detector should be chosen to predict the critical mass (positions of control rods at a critical state) of the core appropriately, i.e., which of the above inverse multiplication curves is suitable for predicting the critical mass.

2.2 Control Rod Calibration Experiment

2.2.1 Purpose

The reactivity of a reactor depends on various factors, including the amount of fuel, the temperature of moderators, the positions of control rods, etc. In a low-power reactor like UTR-KINKI, the reactivity that the reactor has when all the control rods are fully withdrawn is called the “excess reactivity.” When all control and safety rods are withdrawn fully from a criticality state (for k_eff = 1, reactivity in Eq. (1.9) is $\rho = 0$), the reactor is reaching at a supercritical state and the reactor power increases. If the reactivity of the reactor at this time is defined as $\rho_{excess}$, the change in the position of the control rod causes the reactivity change of the reactor, $0 \to \rho_{excess}$. Therefore, the change in the position of the control rod is said to have “an equivalent reactivity value of $\rho_{excess}$,” which is an index of the effect of the control rod on the reactor. In the control rod calibration experiment, the “equivalent reactivity” of each control rod is obtained in this way.

The control rod calibration experiment is carried out just after the reactor reaches a critical state for the first time, to confirm the essential characteristics of control rods that play a significant role in the safety of the reactor. The reactivity worth of the control rod obtained by the control rod calibration experiment is used as a reference for various reactivity measurements, such as the reactivity of irradiated materials inserted in the core: sample reactivity worth. In the control rod calibration experiment, excess reactivity and control rod reactivity worth in the UTR-KINKI core can be determined using the positive period method and the rod drop method, respectively, which are typical reactivity measurement methods. The experiment aims to deepen the understanding of the role of control rods and nuclear safety of nuclear reactors by comparing and discussing the measured reactivity values.

2.2.2 Structure of Control Rod

The control rods used in UTR-KINKI (strictly speaking, they are control plates made of Cd) are described in terms of their positions in the core (see Sect. 1.1.4). Figure 2.4 shows a longitudinal cross section of the core in which the control rod is inserted. Looking at Fig. 2.4 carefully, you will see that the control rods in UTR-KINKI are not located next to most of the fuel region, as is usually the case in research reactors.

Fig. 2.4

Axial view of positional relationship between control rod (safety rod) and fuel plate in UTR-KINKI

As shown in Fig. 2.4, when the control rod is fully inserted into the core (left-side control rod), the position of the center of the absorber (control plate) corresponds to that of the center of the fuel region (fuel plate). The control rod is then at the lower limit. On the other hand, when the control rod is fully withdrawn (right-side control rod), the center of the absorber is about 10 cm higher than the upper height of the fuel plate, and here, the control rod is at the position of the upper limit. The axial relationship between control rods and fuel is shown in Fig. 2.4.

2.2.3 Method of Measurement

2.2.3.1 Reactivity Measurement by Positive Period Method

Of several methods of measuring the reactivity, the most standard “positive period method” is used to measure the relatively small positive reactivity.

As shown in Fig. 2.5, the reactor power increases exponentially when a positive reactivity is added to the reactor at a critical state. State a is an example of the case where the reactivity $\rho_{a}$ is large, and state b is that of the case where the reactivity $\rho_{b}$ is small $\left( {\rho_{a} > \rho_{b} } \right)$. The doubling time $T_{2a}$ for state a is the same at any time, and the doubling time $T_{2b}$ for state b is also the same. Here, the doubling time is the time it takes for a reactor power to double. As shown in Fig. 2.5, the larger the reactivity, the larger the rate of increase of the reactor power and the shorter the doubling time: $T_{2a} < T_{2b}$.

Fig. 2.5

Relationship between reactivity and reactor power

The increase of the reactor power p can be then expressed as follows:

$$p\left( t \right) = p_{0} e^{\frac{t}{T}} ,$$

(2.15)

where p₀ is the reactor power at the time when the measurement is started (t = 0), t the time from the start of measurement and T the reactor period. Here, the reactor power is notably proportional to the number of neutrons generated by fission reactions.

The relationship between the reactor period T (Sect. 1.2.8.3) and the reactivity $\rho$ (Sect. 1.2.7) is determined by the inhour equation as follows (The details are omitted. Please refer to Ref. [3]):

$$\rho = \frac{{\ell_{0} }}{{T + \ell_{0} }} + \frac{T}{{T + \ell_{0} }}\sum\limits_{i = 1}^{6} {\frac{{\beta_{eff,i} }}{{1 + \lambda_{i} T}},}$$

(2.16)

$\ell_{0}$:

Prompt neutron lifetime of prompt neutrons (1.605 × 10^–4 s by MVP3.0 [4] with JENDL-4.0 [5] for UTR-KINKI),

$\beta_{eff,i}$:

Effective delayed neutron fraction of the i-th delayed neutron precursor,

$\lambda_{i}$:

Decay constant of the i-th delayed neutron precursor,

where the delayed neutron parameters of UTR-KINKI are given in Table 1.4.

The reactivity $\rho$ can be obtained by substituting the reactor period T into Eq. (2.16), together with the parameters in Table 1.4.

Since it is difficult to directly measure the reactor period T (time for the reactor power to increase by $e$ times) in actual measurement, when the doubling time $T_{d}$ is obtained at the experiment, the reactor period T is determined by the following equation:

$$T = \frac{{T_{d} }}{\ln 2} = \frac{{T_{d} }}{0.693}.$$

(2.17)

The procedure for measuring the reactivity by the positive period method is as follows, referring to the data sheet shown in Fig. 2.6:

Fig. 2.6

Data sheet in control rod calibration experiment

• (1) The reactor is kept at a critical state with low power.

• (2) The control rod is withdrawn, and a relatively small positive reactivity is added.

• (3) Since the reactor power increases exponentially, the doubling time $T_{d}$ at which the reactor power doubles is measured by using a stopwatch.

• (4) Using Eq. (2.17), calculate the reactor period T from the doubling time $T_{d}$. Substitute the reactor period T and the delayed neutron parameters in Table 1.4 into the inhour equation in Eq. (2.16) to obtain the reactivity $\rho$.

[Column] Delayed criticality and prompt criticality

Let us consider the magnitude of the reactor period using the following discussion: Suppose that the reactor condition varies from a critical state $\left( {k_{eff} = 1.000} \right)$ to a supercritical state $\left( {k_{eff} = 1.001} \right)$ while UTR-KINKI is in operation. Assuming that the prompt neutron lifetime is approximately 0.0001 s (10^–4), under no existence of delayed neutrons, as shown in Sect. 2.2.3, $\ell_{0} = 1.605 \times 10^{ - 4} {\text{s}}$, the reactor period T is about $T = \ell_{0} /(k_{eff} - 1) = 10^{ - 4} /10^{ - 3} = 0.1\,{\text{s}}$ from Eq. (1.14). Using Eq. (2.15), the number of neutrons increases by $e^{1/0.1} = 22,000$ times in one second. Then, it is evident that the neutron multiplication is so fast that the reactor power cannot be controlled very well.

Conversely, the situation is very different when delayed neutrons are present. The evaluation equation is slightly different (see Ref. [3] for details). If we substitute $k_{eff} = 1.001$ and the delayed neutron parameters in Table 1.4 into Eq. (1.14), we obtain $T = \left\{ {k_{eff} /(k_{eff} - 1)} \right\} \cdot \left( {\beta /\lambda } \right)$, {1.001/(1.001–1.000)} ⋅ (0.007342/0.0746) = 98.52 s. The reactor power after one second is only $e^{1/98.52} = 1.0102$ times, and, as a result, the reactor power is possible to be controlled with a sufficient margin.

In this way, the state in which a reactor is at a critical state with both prompt and delayed neutrons is called delayed critical, whereas the state in which a reactor is critical with only prompt neutrons is called prompt critical.

2.2.3.2 Reactivity Measurement by Rod Drop Method

One of the methods for obtaining a relatively large negative reactivity is the control rod drop method (rod drop method). In the rod drop method, when a control rod is inserted into the reactor at once, the reactivity of the control rod is obtained soon. As shown in Fig. 2.7, when a control rod is inserted into the reactor at the time $t_{0}$, the degree of decrease in the counting rate of neutrons after the insertion depends on the reactivity of the inserted control rod, even though the counting rate (power) of the neutron detector before the insertion is the same as that after the insertion. When the reactivity of the control rod to be inserted is large, the power “a” decreases rapidly, as shown in Fig. 2.7.

Fig. 2.7

Relationship between reactivity and output change by rod drop method

The ratio $\frac{{n_{0} }}{{\int_{0}^{\infty } {n(t)dt} }}$ of the pre-insertion counting rate $n_{0}$ and the post-insertion total counting rate of the control rod $\int_{0}^{\infty } {n(t)dt}$ is proportional to a negative reactivity $\left( { - \rho } \right)$ as follows:

$$- \rho = \frac{{n_{0} }}{{\int_{0}^{\infty } {n(t)dt} }}\sum\limits_{i = 1}^{6} {\frac{{\beta_{eff,i} }}{{\lambda_{i} }}} ,$$

(2.18)

where $n(t)$ is the counting rate of neutrons at the time t after the start time of measurement t₀ when the control rod is dropped.

Equation (2.18) was given by Hogan [6] (integral method) and has been often used to determine the integral reactivity of control rods. The reactivity value of a dropped control rod can be obtained by substituting the delayed neutron parameters in Table 1.4, the measured results of $n_{0}$ and $\int_{0}^{\infty } {n(t)dt}$ into Eq. (2.18).

The procedure for measuring the reactivity by the rod drop method is as follows, referring to the data sheet in Fig. 2.6:

• (1) The reactor is kept at a critical state and a reactor power ($n_{0}$) of about 0.1 W (steady state).

• (2) The counting rate of neutrons $n_{0}$ (s⁻¹; cps: counts per second) is actually obtained from the measurement values of m (cps) that is corrected for the dead time of the measuring instrument (see [Column] “Dead time correction”).

• (3) The control rod is dropped rapidly to add a relatively large negative reactivity in the core.

• (4) The value of the time integral of the neutron counts $\int_{\,0}^{\infty } {n\,(t)\,dt}$ is measured after the control rod is dropped.

• (5) The negative reactivity $\left( { - \rho } \right)$ is obtained by substituting the measured $n_{0}$ and $\int_{0}^{\infty } {n(t)dt}$, and the delayed neutron parameters in Table 1.4 into Eq. (2.18).

2.2.4 Discussion

• (1) Let’s check whether the excess reactivity obtained by the positive period method was an appropriate value within the critical limits (or reactor limits) of UTR-KINKI (Table 1.1 in Chap. 1).

• (2) Do the results of reactivity of the control rods obtained by the rod drop method satisfy the limiting conditions of reactivity (critical limits or reactor limits) of UTR-KINKI, as shown in Table 1.1? If the reactivity of each of several control rods obtained by the rod drop method seems to be different from each other, let us consider the reason why the results are different even though the same method was used.

[Column] Reactivity calibration curve of control rods

The main role of control rods is to control fission chain reactions for keeping the reactor at a critical state and adjusting the reactor power. Here, if a control rod is inserted into the reactor, neutrons are absorbed, while fission chain reactions are suppressed. Conversely, fission chain reactions are activated when the control rod is withdrawn from the reactor. Before the operation of the reactor, it is crucial to know in advance how much (positive or negative) reactivity is added to the reactor by the control rod operation.

In the control rod calibration experiment in UTR-KINKI, when the control rods are moved for the whole stroke from the upper limit (full withdrawal) to the lower limit (full insertion) (or from the lower limit to the upper limit), the reactivity of the control rod can be obtained by using the positive period method and/or the rod drop method. The control rod calibration experiment is then conducted to know the reactivity to shut down the reactor safely.

By using the rod drop method, the response of the control rod to a full stroke of the control rod was obtained. Meanwhile, what about the response to the reactor power when the control rod is partially moved from a certain position to another? Is the amount of reactor power change always constant at any position for the stroke of control rod movement? Surprisingly, the answer is that the reactor power change is not constant. As shown from the reactor structure described in Chap. 1 (cross-sectional view of UTR-KINKI in Fig. 1.4), the control rod moves in the region adjacent to the fuel. From the positional relationship between the control rod and the fuel (Fig. 2.4), you can easily imagine that the effectiveness of the control rod depends strongly on neutron density of the fuel region along an axial direction. Without investigating the effect of control rods on the direction of movement, the reactor operation cannot be conducted safely. Therefore, the control rod calibration experiment is considered very important in reactor physics experiments, in accordance with the approach to criticality experiment.

[Column] Dead time correction

When radiation is incident on a measuring instrument, there is a short time when the instrument cannot measure the next incident radiation due to the recovery of the instrument and data processing. This time is called the “dead time.” If the measured counting rate is m (s⁻¹: cps) and the dead time is τ (s), the correct measurement time of the instrument in one second is $(1 - m\tau )$. Therefore, the actual counting rate $N_{0}$ (cps), which takes into account the counting off due to the dead time in the measured value, is expressed by the following equation:

$$N_{0} = \frac{m}{1 - m\tau }.$$

(2.19)

Typically, the dead time of the fission counter used for the rod drop method measurement is about 1.4 $\mu {\text{s}}$.

2.3 Subcriticality Measurement Experiment

The subcriticality is the index that quantifies “how much margin is left in the system from the critical state $k_{eff} = 1$?” and is the quantity defined by the negative reactivity $\left( { - \rho } \right) = \left( {1 - k_{eff} } \right)/k_{eff}$.

The criticality state of a nuclear reactor is defined as a state in which neutron production by fission reactions is balanced by neutron annihilation due to absorption reactions in the system or leakage outside the system. When the reactor core is maintained at a steady state for a long period without an external neutron source (Pu-Be start-up neutron source) inserted into the core, the core is judged to be at a critical state.

Here, if the reactor core is at a subcritical state $\left( {k_{eff} < 1} \right)$, the number of neutrons in the target system will be zero under no existence of external neutron source in the core. In this case, if the counting rate of neutrons measured by a neutron detector is zero, the core may be judged to be at “a subcritical state.” It is, however, impossible to determine the magnitude of subcriticality from the information that the counting rate of neutrons is zero. Therefore, to investigate the magnitude of subcriticality $\left( { - \rho } \right)$ in a subcritical core, the number of neutrons is measured with the help of the external neutron source that induces the trigger of fission chain reactions, and the magnitude of subcriticality $\left( { - \rho } \right)$ is then analyzed.

Although several methods have been proposed to measure the subcriticality [6,7,8,9,10], in this section, we focus on measurement methods that can be used in UTR-KINKI and introduce several subcriticality measurement methods.

2.3.1 Neutron Source Multiplication Method

2.3.1.1 Principle of Measurement

Consider the case where the reactor is at a subcritical state and the reactor power is constant (no change with time) due to an external neutron source. In this case, if the number of neutrons and the number of delayed neutron precursors are $n_{0}$ and $C_{0}$, respectively, the point-reactor kinetics equations with the addition of the neutron source are shown in Eqs. (1.10) and (1.11), considering that the reactor is at a steady state. Then, Eqs. (1.10) and (1.11) can be expressed as follows, respectively, neglecting the change in time t:

$$\frac{{dn_{0} }}{dt} = \frac{{\rho_{0} - \beta }}{\Lambda }n_{0} + \lambda C_{0} + S = 0,$$

(2.20)

$$\frac{{dC_{0} }}{dt} = \frac{\beta }{\Lambda }n_{0} - \lambda C_{0} = 0,$$

(2.21)

where $\rho_{0}$ is a negative value of reactivity and S neutron source intensity.

From Eq. (2.21), the following equation can be obtained:

$$\lambda C_{0} = \frac{\beta }{\Lambda }n_{0} .$$

(2.22)

Here, Eq. (2.22) can be rewritten using $\Lambda = \ell /k_{eff}$ and $\rho_{0} = 1 - \frac{1}{{k_{eff} }}$ in Eq. (1.9) as follows:

$$\lambda C_{0} = \frac{{\beta k_{eff} }}{\ell }n_{0} = \frac{\beta }{{\ell \left( {1 - \rho_{0} } \right)}}n_{0} .$$

(2.23)

Inserting Eq. (2.23) into Eq. (2.20), we obtain the following equation for $n_{0}$, and noteworthy is that $\rho_{0}$ has a negative value:

$$n_{0} = - \ell \left( {1 - \rho_{0} } \right)\frac{S}{{\rho_{0} }} = \ell \left( {1 - \frac{1}{{\rho_{0} }}} \right)S.$$

(2.24)

Assuming that the reactor is at a near-critical state, and using $1 - \rho_{0} \approx 1$ and $\Lambda \approx \ell$, the $n_{0}$ in Eq. (2.24) can be approximated as follows:

$$n_{0} \approx \Lambda \frac{S}{{\left( { - \rho_{0} } \right)}}.$$

(2.25)

Equation (2.25) shows that the power of a subcritical reactor is proportional to the neutron source intensity $S$ and inversely proportional to $\left( { - \rho_{0} } \right)$. The method for measuring the reactivity in the subcritical state above is called the “neutron source multiplication method.”

How can we measure the subcriticality $\left( { - \rho_{0} } \right)$ in an actual reactor using this principle? We will discuss this question by modifying Eq. (2.24).

The reactor is at a known subcritical state where the multiplication factor is $k_{0}$, and the detection efficiency of the detector set in the core is $\varepsilon$. Equation (2.24) is transformed into the following expression for the multiplication factor $k_{0}$:

$$n_{0} = \frac{\ell \varepsilon S}{{1 - k_{0} }}.$$

(2.26)

If the multiplication factor $k_{0}$ and the counting rate $n_{0}$ measured by the detector are the known values, the inverse value of the unknown $\ell \varepsilon S$ in Eq. (2.26) can be obtained as follows:

$$\frac{1}{\ell \varepsilon S} = \frac{1}{{n_{0} \left( {1 - k_{0} } \right)}}.$$

(2.27)

Next, when the reactor condition varies from the known subcritical state (where the multiplication factor is $k_{0}$) to the unknown subcritical state (where the multiplication factor is $k_{1}$), the multiplication factor $k_{1}$ is expressed as the same formation as Eq. (2.26). If the counting rate from the detector at this time is $n_{1}$, the multiplication factor $k_{1}$ can be obtained by using Eq. (2.27) as follows:

$$k_{1} = 1 - \frac{{n_{0} }}{{n_{1} }}\left( {1 - k_{0} } \right).$$

(2.28)

From Eq. (2.28), the unknown multiplication factor $k_{1}$ obtained by measuring the counting rate $n_{1}$.

Then, using Eq. (1.9), the known subcriticality $\left( { - \rho_{0} } \right)$ can be expressed as $\left( { - \rho_{0} } \right) = \frac{{1 - k_{0} }}{{k_{0} }}$, and the unknown multiplication factor $k_{1}$ in Eq. (2.28) can be expressed by using the known subcriticality $\left( { - \rho_{0} } \right)$ as follows:

$$k_{1} = 1 - \frac{{n_{0} }}{{n_{1} }}\left( {\frac{{ - \rho_{0} }}{{1 - \rho_{0} }}} \right).$$

(2.29)

Since the unknown subcriticality $\left( { - \rho_{1} } \right)$ can be expressed by using Eq. (1.9) as $\left( { - \rho_{1} } \right) = \frac{{1 - k_{1} }}{{k_{1} }}$, and the multiplication factor $k_{1}$ in Eq. (2.29), $\left( { - \rho_{1} } \right)$ can be finally expressed as follows:

$$\left( { - \rho_{1} } \right) = \frac{{1 - k_{1} }}{{k_{1} }} = \frac{1}{{k_{1} }} - 1 = \frac{1}{{\frac{{n_{1} }}{{n_{0} }}\left( {\frac{{1 - \rho_{0} }}{{ - \rho_{0} }}} \right) - 1}},$$

(2.30)

where the unit of $\left( { - \rho_{0} } \right)$ is needed to be converted into $\Delta k/k$, when obtaining $\left( { - \rho_{0} } \right)$ in the units of pcm (10^–5$\Delta k/k$) or %$\Delta k/k$ (10^–2$\Delta k/k$). Then, the unit of $\left( { - \rho_{1} } \right)$ indicates $\Delta k/k$.

2.3.1.2 Method of Measurement

Suppose that the excess reactivity $\rho_{excess}$ and the reactivity worth of the control rod $\left( { - \rho_{rod} } \right)$ obtained by the positive period method and the rod drop method, respectively, two values of $\rho_{excess}$ and $\left( { - \rho_{rod} } \right)$ are given in advance by the control rod calibration experiment. Let the subcritical state obtained by the drop method known (subscript is “0”), and consider the unknown state when another control rod is dropped from this state (subscript is “1”).

The subcriticality of the unknown state is obtained by the neutron source multiplication method using the following procedures:

• (1) The $\rho_{0}$ was obtained in advance, using the excess reactivity $\rho_{excess}$ and the reactivity of the control rod $\left( { - \rho_{rod} } \right)$ $\left( {\rho_{0} = \rho_{excess} - \left( { - \rho_{rod} } \right)} \right)$.

• (2) Setting the neutron source in the reactor, we wait for a while until the reactor power is constant (stable) in a known subcritical state with one control rod inserted.

• (3) Using a detector in the reactor, such as a fission chamber (FC) in the startup system, the number of neutrons is counted for 100 s and repeated, for example, five times. The average value $n_{0}$ of the counting rate is obtained.

• (4) Keep the neutron source and insert another control rod at the known subcritical state. When the reactor is at an unknown subcritical state, we wait for a while until the reactor power is at a stable state.

• (5) Using the same method for the known subcritical state, obtain the average value $n_{1}$ of the counting rate in the unknown subcritical state.

• (6) The unknown subcriticality $\left( { - \rho_{1} } \right)$ can be attained by substituting the counting rates $n_{0}$ and $n_{1}$, and the known subcriticality $\left( { - \rho_{0} } \right)$.

2.3.1.3 Discussion

Let’s discuss the results of the reactivity (subcriticality) in an unknown subcritical state $\left( { - \rho_{1} } \right)$ according to the following point:

The reactivity of another control rod inserted into the known subcritical state is assumed to be attained in advance. Next, the subcriticality of the unknown state can be easily estimated from the sum of the subcriticality of the known state and the reactivity of another inserted control rod. Finally, we shall use this summation as a reference value for the experiment.

• (1) How about the measurement accuracy (relative difference between the reference and measured values, or relative error) of the $\left( { - \rho_{1} } \right)$ by the neutron source multiplication method when comparing with the reference value? If a significant difference is found, what is the cause of the difference?

• (2) Let us consider whether there is any difference between the neutron source multiplication method and the other methods introduced in Sect. 2.3, including the source jerk method and the inverse kinetics method.

• (3) Let us judge whether the measured results are subcritical enough to be regarded as a near-critical state, which is an assumption of using the neutron source multiplication method.

• (4) Are there any significant differences in the measurement results when multiple detectors are used? In short, let’s discuss the difference between the measurement results at several detector positions. In addition to the positions of the detectors, we should also discuss spatial relationships between the detector and the neutron source; the detector and the fuel region; the structural materials around the detector.

2.3.2 Source Jerk Method

2.3.2.1 Principle of Measurement

Let us assume that the external neutron source is withdrawn instantaneously in a steady state of a subcritical core with an external neutron source. By measuring neutron counts until delayed neutron precursors in the subcritical core are completely decayed to be zero, the subcriticality $\left( { - \rho } \right)$ can be experimentally analyzed. The above-mentioned subcriticality measurement technique is called the “source jerk method” [6].

First, let us suppose that an external neutron source with the intensity S is inserted into a subcritical core with the subcriticality $\left( { - \rho } \right)$. Based on a simplified point kinetics equation with one neutron energy group and one delayed neutron precursor group, the numbers of neutrons and delayed neutron precursors, n(t) and C(t), can be, respectively, described as follows:

$$\frac{dn\left( t \right)}{{dt}} = \frac{\rho - \beta }{{\Lambda }}n\left( t \right) + \lambda C\left( t \right) + S,$$

(2.31)

$$\frac{dC\left( t \right)}{{dt}} = \frac{\beta }{{\Lambda }}n\left( t \right) - \lambda C\left( t \right).$$

(2.32)

Compared with Eq. (1.10), the major modification is that the external source term S is added to the right-hand side in the case of Eq. (2.31) because the S neutrons per second are emitted due to the decay of the external source. When a sufficient time is passed after the insertion of the external neutron source, the numbers of neutrons and delayed neutron precursors become constant values, n₀ and C₀, respectively. The analytical solutions of n₀ and C₀ in the steady state can be obtained by considering the time derivatives in Eqs. (2.31) and (2.32) are zero, respectively:

$$n_{0} = \Lambda \frac{S}{{\left( { - \rho } \right)}},$$

(2.33)

$$C_{0} = \frac{\beta }{\lambda \Lambda }n_{0} = \frac{\beta }{\lambda }\frac{S}{{\left( { - \rho } \right)}}.$$

(2.34)

Namely, these constant values of n₀ and C₀ before withdrawing the external neutron source are proportional to the magnitude of the neutron source strength S and inversely proportional to the subcriticality $\left( { - \rho } \right)$.

Next, let us suppose that the external neutron source is instantaneously withdrawn from the subcritical core at time t = 0; i.e., the source intensity S can be rapidly reduced to zero. In this case, as shown in Fig. 2.8, the prompt neutron component of the fission chain reaction due to the external neutron source disappears instantaneously. Note, however, that there are still delayed neutron precursors which are accumulated in the subcritical system before withdrawing the external neutron source. Therefore, the neutron count rate after withdrawing the external neutron source does not instantaneously become to be zero, although the count rate promptly decreases to a certain count rate n_d due to the decay of delayed neutron precursors. Such a physical phenomenon of the neutron count rate in a neutron multiplication system after an instantaneous change (e.g., withdrawal of external neutron source) is called “prompt jump.” After the prompt jump, the number of neutrons in the target subcritical system decreases exponentially with time until the delayed neutron precursors in the core completely disappear due to their decays.

Fig. 2.8

Time variation of neutron count rate in the source jerk method

Here, the numbers of neutrons and delayed neutron precursors, n(t) and C(t), after withdrawing the external neutron source decrease according to the point kinetics equation with S = 0, i.e., Eqs. (1.10) and (1.11). Furthermore, n(t) and C(t) become to be zero after a sufficiently long time passes, i.e., $\mathop {\lim }\limits_{t \to \infty } n\left( t \right) = \mathop {\lim }\limits_{t \to \infty } C\left( t \right) = 0$. To obtain the subcriticality estimation formula in the source jerk method, both sides of Eqs. (1.10) and (1.11) are integrated within the range of $0 \le t \le \infty$. Then, both time-integrated equations are summed up and further transformed by substituting Eq. (2.34) into the term C_0. Finally, the following formula is obtained to estimate the subcriticality $\left( { - \rho } \right)$ from the measured neutron count rates:

$$- \rho = \frac{{n_{0} }}{{\mathop \smallint \nolimits_{0}^{\infty } n\left( {t^{\prime}} \right)dt^{\prime}}}\frac{\beta }{\lambda }.$$

(2.35)

If delayed neutrons with six groups are considered, the more rigorous formula can be obtained to estimate $\left( { - \rho } \right)$ as follows:

$$- \rho = \frac{{n_{0} }}{{\mathop \smallint \nolimits_{0}^{\infty } n\left( {t^{\prime}} \right)dt^{\prime}}}\mathop \sum \limits_{i = 1}^{6} \frac{{\beta_{{{\text{eff}},i}} }}{{\lambda_{i} }}.$$

(2.36)

Equation (2.36) is just the same formula as the integral method in the rod drop experiment [6].

[Column] Inherent neutron sources in nuclear fuel

Even if the external neutron source is completely withdrawn in the source jerk method, the neutron count rate may not be just zero due to another contribution of the “inherent neutron” in the nuclear fuel. For example, in the case of UTR-KINKI, the uranium–aluminum (U-Al) alloy is used as a nuclear fuel material. Because U-235 is a radioactive nuclide, secondary neutrons are emitted by the ($\alpha$, n) reaction which rarely occurs when the Al-27 nuclide collides with an $\alpha$ particle emitted by the $\alpha$-decay of uranium. As another example, in the case of low-enriched uranium, neutrons are rarely emitted by the spontaneous fission reaction of U-238. These reactions in the nuclear fuel can be regarded as very weak neutron sources that cause fission chain reactions [7]. Therefore, even if a long time passed after completely withdrawing the external neutron source, an extremely low neutron count rate ${n}_{\infty }$ is maintained by the inherent neutron source. If ${n}_{\infty }$ is nearly equal to zero, the impact of the inherent neutron source on the source jerk method is negligible. Otherwise, the subcriticality $\left( { - \rho } \right)$ can be more accurately estimated by considering the background neutron count rate $n_{\infty }$ due to the inherent neutron source, i.e., by correcting the contribution of $n_{\infty }$ as follows [8]:

$$- \rho = \frac{{n_{0} - n_{\infty } }}{{\mathop \smallint \nolimits_{0}^{\infty } \left( {n\left( {t^{\prime}} \right) - n_{\infty } } \right)dt^{\prime}}}\mathop \sum \limits_{i = 1}^{6} \frac{{\beta_{{{\text{eff}},i}} }}{{\lambda_{i} }}.$$

(2.37)

2.3.2.2 Method of Measurement

To measure the subcriticality using the source jerk method for a target subcritical core, the experimental procedures are described as follows:

• (1) A neutron detector (e.g., BF-3 detector) is installed in a hole of the graphite reflector to measure the neutron count rate using a typical nuclear instrumentation system as shown in Fig. 2.2.

• (2) A target subcritical core is configured by an arbitrary pattern of inserting control rods (two safety rods: SRs, a shim safety rod: SSR, and a regulating rod: RR).

• (3) A startup Pu-Be neutron source is inserted into the target subcritical core. Note that the effective source intensity (or the inserted axial position of the Pu-Be source) should be appropriately adjusted to reduce the dead time effect on the neutron count rate n₀ in the steady state after inserting the Pu-Be source.

• (4) The target core is maintained without any change until the neutron count rate becomes a constant value.

• (5) For example, the measurements of the neutron count during 100 s are repeated five times. From the five measured counts, the sample mean of neutron count rate n₀ (count per second, cps) before withdrawing the Pu-Be source is calculated.

• (6) The neutron source is withdrawn instantaneously, and the time integral of the neutron count rate after the withdrawal, $\mathop \smallint \limits_{0}^{T} n\left( {t^{\prime}} \right)dt^{\prime}$, is cumulatively measured. Here, the total measurement time T for the time integral depends on the magnitude of the subcriticality $\left( { - \rho } \right)$. A typical value of T is approximately 300 to 500 s.

• (7) If possible, the background neutron count rate $n_{\infty }$ after withdrawing the Pu-Be source is measured in a similar way as step (5) to check whether the approximation of $n_{\infty } \approx 0$ is applicable.

• (8) The target subcriticality $\left( { - \rho } \right)$ is estimated by substituting measured n₀ and $\mathop \smallint \limits_{0}^{\infty } n\left( {t^{\prime}} \right)dt^{\prime}$ into Eq. (2.36).

2.3.2.3 Discussion

Let us discuss the experimental results of subcriticality $\left( { - \rho } \right)$ using the source jerk method from the following viewpoints:

• (1) As can be seen from the experimental result, the time variation of the count rate after withdrawing the external neutron source decreases exponentially according to the decay of the delayed neutron precursors. Based on the half-life of Br-87 (i.e., 55.6 s shown in Table 1.2), let us discuss whether the total measurement time T =  ~ 300 s is sufficiently long.

• (2) Let us consider the following two subcritical cores: (1) a shallow subcritical (or near-critical) core where the subcriticality $\left( { - \rho } \right)$ is close to zero, and (2) a deep subcritical case where $\left(-\rho \right)$ is large. Based on Eq. (2.36), which of them is more accumulated delayed neutron precursors before withdrawing the external neutron source?

• (3) Before the measurement using the source jerk method, let us measure the excess reactivity $\rho_{{{\text{excess}}}}$ and the control rod worth $\rho_{{{\text{rod}},i}} < 0$ using the positive period method and the control rod drop method, respectively, in advance. Then, based on the control rod pattern for the target subcritical core, let us evaluate the reference value of the subcriticality $\left( { - \rho_{{{\text{ref}}}} } \right)$. By comparing with the reference value $\left( { - \rho_{{{\text{ref}}}} } \right)$, is there a significant difference in $\left( { - \rho } \right)$ between the source jerk method and the reference? If any, what is the major reason?

• (4) In the case of the source jerk method, the external neutron source is instantaneously withdrawn. Instead of withdrawing the external neutron source, let us discuss whether the subcriticality can be estimated by instantaneously dropping (or inserting) the external neutron source. Based on the point kinetics equation, derive the theoretical formula to estimate the subcriticality using the “source drop method”.

2.3.3 Inverse Kinetics Method

2.3.3.1 Principle of Measurement

Let us change the reactivity $\rho \left( t \right)$ by inserting or withdrawing the control rod position in a critical or subcritical core. According to the positive or negative reactivity change, the number of neutrons, n(t), increases or decreases with time. In other words, the total number of neutrons n(t) can be modeled by some kind of “function” where the input variable is reactivity $\rho \left( t \right)$. Then, by considering the inverse function as the input variable of n(t), can we deduce the reactivity from the measured neutron count rates? The “inverse kinetics method” is a technique to estimate the time variation of the reactivity $\rho \left( t \right)$ using the measured time variation of n(t).

First, let us suppose that an external neutron source of intensity S is inserted into a core where the reactivity is $\rho \left( t \right)$. Even if the reactivity $\rho \left( t \right)$ changes with time due to the control rod operation, the numbers of neutrons and delayed neutron precursors, n(t) and C(t), respectively, can be described based on the point kinetics equation in a similar way as the source jerk method:

$$\frac{dn\left( t \right)}{{dt}} = \frac{\rho \left( t \right) - \beta }{{\Lambda }}n\left( t \right) + \lambda C\left( t \right) + S,$$

(2.38)

$$\frac{dC\left( t \right)}{{dt}} = \frac{\beta }{{\Lambda }}n\left( t \right) - \lambda C\left( t \right).$$

(2.39)

If the time variation of reactivity $\rho \left( t \right)$ can be given as the input variable for Eqs. (2.38) and (2.39), the time variation of the neutron number n(t) can be numerically predicted using the numerical integration of the first-order differential equation, e.g., the Euler or Runge–Kutta method.

Reversing this idea, we can numerically estimate the time variation (increase or decrease) of $\rho \left( t \right)$ as shown in Fig. 2.9, by giving n(t) as the input variable. This method is called the “inverse kinetics method,” and the measurement principle for the “reactivity meter” that is widely utilized for the real-time monitoring of reactivity in nuclear reactors [9].

Fig. 2.9

Example of reactivity estimation using inverse kinetics method

As a precondition, let us assume that the neutron count rate n(t) is proportional to the total number of neutrons in the core and the time-series data are successively measured with a certain time width Δt using a neutron detector located in the core. For simplicity, by considering delayed neutrons group with one group, the number of delayed neutron precursors C(t) can be numerically estimated by the following equation using the input value of the measured neutron count rate n(t):

$$C\left( t \right) \approx C\left( {t - {\Delta }t} \right){\text{e}}^{{ - \lambda {\Delta }t}} + \frac{{1 - {\text{e}}^{{ - \lambda {\Delta }t}} }}{\lambda }\frac{\beta }{{\Lambda }}n\left( t \right).$$

(2.40)

Equation (2.40) can be derived based on Eq. (2.39), and $C\left( {t - {\Delta }t} \right)$ denotes the number of delayed neutron precursors at the previous time $t - {\Delta }t$. Note that the initial value of C(0) = C₀ can be estimated by Eq. (2.34) if the core is a steady state at time t = 0. By giving C₀ as the initial value for Eq. (2.40), C(t) can be successively estimated from the continuous measurements of n(t). Finally, the reactivity $\rho \left( t \right)$ can be inversely estimated by substituting the estimated values of C(t) into the following equation that is obtained by transforming the one-point kinetics equation of Eq. (2.38):

$$\begin{aligned} \rho \left( t \right) & = \frac{\Lambda }{n\left( t \right)}\left( {\frac{dn\left( t \right)}{{dt}} + \frac{dC\left( t \right)}{{dt}} - S} \right) \\ & \quad \approx \frac{\Lambda }{n\left( t \right)}\left( {\frac{{n\left( t \right) - n\left( {t - {\Delta }t} \right)}}{{{\Delta }t}} + \frac{{C\left( t \right) - C\left( {t - {\Delta }t} \right)}}{{{\Delta }t}} - S} \right). \\ \end{aligned}$$

(2.41)

Equation (2.41) is simplified by considering only delayed neutrons with one group. If delayed neutrons with six groups are considered, the reactivity $\rho \left( t \right)$ can be more rigorously estimated using the modified equation as follows:

$$\begin{aligned} \rho \left( t \right) & = \frac{\Lambda }{n\left( t \right)}\left( {\frac{dn\left( t \right)}{{dt}} + \mathop \sum \limits_{i = 1}^{6} \frac{{dC_{i} \left( t \right)}}{dt} - S} \right) \\ & \quad \approx \frac{\Lambda }{n\left( t \right)}\left\{ {\frac{{n\left( t \right) - n\left( {t - {\Delta }t} \right)}}{{{\Delta }t}} + \left( {\mathop \sum \limits_{i = 1}^{6} \frac{{C_{i} \left( t \right) - C_{i} \left( {t - {\Delta }t} \right)}}{{{\Delta }t}}} \right) - S} \right\}, \\ \end{aligned}$$

(2.42)

where the numerical value of the delayed neutron precursor, C_i(t), can be obtained by Eq. (2.40) with the ith delayed neutron fraction $\beta_{{{\text{eff}},i}}$ and decay constant $\lambda_{i}$ summarized in Table 1.4.

In order to inversely estimate the reactivity $\rho \left( t \right)$ using the inverse kinetics method of Eq. (2.42), it is necessary to provide not only the point kinetics parameters but also the value of the source intensity S (neutrons/s). If the core is in a critical state without an external neutron source, the source intensity is given by S = 0. On the other hand, in the case of a source-driven subcritical core, if the initial subcriticality $\left( { - \rho_{0} } \right)$ is known in advance and the initial neutron count rate n₀ in the steady state can be measured, the effective intensity S can be calibrated as follows:

$$S = - \rho_{0} \frac{{n_{0} }}{\Lambda }.$$

(2.43)

2.3.3.2 Method of Measurement

To measure the reactivity using the inverse kinetics method for a critical core or a source-driven subcritical core, the experimental procedures are described as follows:

• (1) In advance, the excess reactivity and the control rod worth are measured by the positive period method and the rod drop method, respectively.

• (2) If the initial condition is a critical state without an external neutron source, the control rod positions are adjusted to maintain an appropriate constant value of neutron count rate. In this case, the following procedures (3) to (5) are skipped.

• (3) If the initial condition is a steady state of a subcritical core with an external neutron source, four control rods (two safety rods: SRs, a shim safety rod: SSR, and a regulating rod: RR) are arbitrarily inserted to configure a certain subcritical system. Then, let us wait until the neutron count rate becomes a constant value.

• (4) For example, the measurements of the neutron count during 100 s are repeated five times. From the five measured counts, the sample mean value of neutron count rate n₀ is calculated.

• (5) Using the excess reactivity and control rod worth, the initial subcriticality $\left( { - \rho_{0} } \right)$ is obtained. Then, the effective source intensity S is determined by Eq. (2.43).

• (6) A neutron measurement system is constructed to successively measure the neutron count rate n(t) at a certain time interval Δt. Typically, Δt = 0.5 s.

• (7) While the time-series data of neutron count rate are successively measured, the reactivity is arbitrarily changed by adjusting the control rod position. In this operation, the time variation of the control rod position should be recorded.

• (8) Based on the inverse kinetics method, Eqs. (2.40) and (2.42), with the measured neutron count rate n(t), the time variation of reactivity $\rho \left(t\right)$ is estimated.

2.3.3.3 Discussion

Let us discuss the experimental results of reactivity $\rho \left(t\right)$ using the inverse kinetics method from the following viewpoints:

• (1) Let us investigate the relationship between the time variations of the neutron count rate n(t) (e.g., magnitude and sign for the slope) and the estimated reactivity $\rho \left( t \right)$ by the inverse kinetics method. For example, how the reactivity $\rho \left( t \right)$ is inversely estimated for the following three cases of n(t)?

• (a) n(t) is constant

• (b) n(t) increases with time

• (c) n(t) decreases with time.

• (2) As can be seen from the experimental results of $\rho \left( t \right)$ using the inverse kinetics method, the estimated $\rho \left( t \right)$ has statistical uncertainty. Why does the statistical uncertainty of $\rho \left( t \right)$ arise? Let us consider how the statistical uncertainty of $\rho \left( t \right)$ can be reduced in the inverse kinetics method.

• (3) Let us evaluate the reference value of reactivity $\rho_{{{\text{ref}}}}$ at a specific time t, based on the records of control rod positions, the excess reactivity $\rho_{{{\text{excess}}}}$, and the control rod worth $\rho_{{{\text{rod}},i}}$. By comparing the reactivity estimated by the inverse kinetics method with the reference value $\rho_{{{\text{ref}}}}$, is there a significant difference in $\rho$ between them? If any, what is the major reason?

• (4) Are there any differences in the reactivity $\rho \left( t \right)$ between the neutron source multiplication and inverse kinetics methods? Let us compare and discuss these results by both methods for the following cases: (a) during the control rod operation, and (b) after sufficient time has passed from the end of control rod operation.

2.3.4 Reactor Noise Analysis Method

2.3.4.1 Principle of Measurement

Let us suppose that the time-series data of neutron counts are successively measured in a steady state for a source-driven subcritical core or a critical core with an external neutron source. Then, as shown in Fig. 2.10, the measured neutron counts fluctuate around a certain mean value. As explained later, there is a unique phenomenon that the “reactor noise” (fluctuation of the neutron count from the mean value) becomes larger as the target core approaches the criticality or the subcriticality becomes shallower. Based on this phenomenon, the subcriticality measurement technique by analyzing the measured reactor noise is called the “reactor noise analysis method.”

Fig. 2.10

Examples of temporal fluctuations in neutron counts (reactor noises)

First, let us consider the phenomenon of the decays of radioactive nuclides. The probability of the radioactive decay per unit time is constant, and these decays are independent and random events each other. Therefore, it is well known that the number of decays of radioactive nuclides (or the total number of detected primary radiation emitted by the decay), C, follows the Poisson distribution. According to the statistical property of the Poisson distribution, the standard deviation (i.e., a measure for the amount of dispersion from the mean value) can be approximated by $\sqrt C$.

Next, let us consider the number of neutrons in a nuclear reactor where fission chain reactions happen. In this case, neutrons emitted by an external neutron source or delayed neutron precursor have the role of “seed” to yield the fission chain of “neutron family tree.” Because of the fission chain reaction, the neutron-descendants belonging to the same fission chain family form a group like “fish shoal.” Let us suppose that, like the fishing for the shoal, some neutrons in the group are detected using a neutron detector. If the neutron multiplication factor is k_eff = 0 (i.e., the neutron family disappears immediately after the first generation), we have to wait for the next neutron emitted by the neutron source after one neutron is detected, because there are no neutron family within the neighborhood of the first detected neutron. On the other hand, if the neutron multiplication factor k_eff is close to 1 to induce the longer fission chain reaction, we expect a higher possibility that other neutrons belonging to the same fission chain can be detected within the neighborhood of the first detected neutron. In other words, if a neutron were a fish, we have a better chance to catch other descendant neutrons by dropping a fishing rod immediately after catching one neutron. Therefore, as shown in Fig. 2.10, there are sparse or dense time domains in the reactor noise like “fish shoal”.

Various reactor noise analysis methods have been proposed in the field of reactor physics experiments [11]. Among them, the “Feynman-$\alpha$ method (variance-to-mean ratio method) [10]” is explained below in this section. Let us suppose that the neutron counts C(T) are detected during a counting gate of width T in a steady state for a target subcritical core, as shown in Fig. 2.11. By successively measuring the time-series data of neutron counts $\left( {C_{1} \left( T \right),C_{2} \left( T \right), \cdots ,C_{N} \left( T \right)} \right)$, where N is the total number of count data, the sample mean $\mu \left( T \right)$ and the unbiased variance $\sigma^{2} \left( T \right)$ are calculated as follows:

$$\mu \left( T \right) = \frac{1}{N}\mathop \sum \limits_{i = 1}^{N} C_{i} \left( T \right),$$

(2.44)

$$\sigma^{2} \left( T \right) = \frac{1}{N - 1}\mathop \sum \limits_{i = 1}^{N} \left( {C_{i} \left( T \right) - \mu \left( T \right)} \right)^{2} .$$

(2.45)

Fig. 2.11

Example of reactor noise analysis using the Feynman-$\alpha$ method

If no fission chain reaction occurs in the target system, neutrons are randomly detected. Then, the frequency distribution of C(T) follows the Poisson distribution, thus the mean $\mu \left( T \right)$ is equal to the variance $\sigma^{2} \left( T \right)$. On the other hand, as more fission chain reactions occur due to the shallower subcriticality, the variance $\sigma^{2} \left( T \right)$ becomes larger than the mean $\mu \left( T \right)$ because of larger differences between the sparse and dense time domains in the measured C(T). By focusing on this feature, the magnitude of “neutron correlation” caused by the fission chain reaction can be investigated by analyzing the relative deviation of variance in neutron counts compared with the variance in the case of the Poisson distribution. Namely, the neutron correlation $Y$ value is defined as follows:

$$Y\left( T \right) = \frac{{\sigma^{2} \left( T \right)}}{\mu \left( T \right)} - 1.$$

(2.46)

Now, let us analyze the variation of $Y\left( T \right)$ defined by Eq. (2.46) for various counting gate widths T. For example, as shown in Fig. 2.12, if the time-series data of C(T) are continuously measured, different $Y$ values (e.g., $Y\left( {2T} \right)$ and $Y\left( {4T} \right)$ corresponding to the gate widths 2 T and 4 T) by bunching the adjacent data of neutron counts (e.g., $C_{i} \left( T \right)$ and $C_{i + 1} \left( T \right)$). As shown in Fig. 2.11, $Y\left( T \right)$ is almost equal to zero at T = 0. Namely, when the counting gate width T is very small, the measured neutron counts can be reasonably regarded as the random detection process which follows the Poisson distribution. On the other hand, as T becomes larger, $Y\left( T \right)$ converges to a certain saturation value. Although the detailed theoretical derivation of the Feynman-$\alpha$ method is omitted in this section, the variation of $Y\left( T \right)$ can be expressed by the following analytical formula [12]:

$$Y\left( T \right) = Y_{\infty } \left( {1 - \frac{{1 - {\text{e}}^{ - \alpha T} }}{\alpha T}} \right) + c_{1} + c_{2} T \approx Y_{\infty } \left( {1 - \frac{{1 - {\text{e}}^{ - \alpha T} }}{\alpha T}} \right),$$

(2.47)

$$Y_{\infty } = \varepsilon \frac{\left\langle {\nu \left( {\nu - 1} \right)} \right\rangle}{\left\langle {\nu} \right\rangle^{2}}\frac{1}{{\left( {\beta_{{{\text{eff}}}} - \rho } \right)^{2} }}.$$

(2.48)

Fig. 2.12

Bunching process for time-series data of neutron counts

In Eq. (2.47), $Y_{\infty }$ and $\alpha$ represent the saturation value of $Y$ and the prompt neutron decay constant (s⁻¹), respectively. Furthermore, c₁ and c₂ are additional terms to correct effects on the $Y$ value mainly due to the dead time and the delayed neutrons, respectively. If these effects are negligible, we can approximate Eq. (2.47) by c₁ = c₂ = 0. In Eq. (2.48), $\varepsilon$ is the detection efficiency, $\nu$ is the number of fission neutrons emitted per fission reaction, the bracket 〈 〉 is the expected value, and $\left\langle {\nu \left( {\nu - 1} \right)} \right\rangle /\left\langle {\nu} \right\rangle^{2}$ is the nuclear data called the “Diven factor” [13].

If the position of the neutron detector and the external neutron source in the target core is the same and only the subcriticality $\left( { - \rho } \right)$ changes, Eq. (2.48) indicates that the saturation value $Y_{\infty }$ increases inversely proportional to $\left( {\beta_{{{\text{eff}}}} - \rho } \right)^{2}$ (i.e., the square of the negative reactivity considering only prompt neutrons) as the subcriticality becomes shallower. However, the subcriticality is not easily estimated only from the saturation value $Y_{\infty }$, because $Y_{\infty }$ is also proportional to the detection efficiency $\varepsilon$ and some kind of measurement is necessary to determine $\varepsilon$. Therefore, in the Feynman-$\alpha$ method, the prompt neutron decay constant $\alpha$ is often estimated using the nonlinear least-squares fitting method using Eq. (2.47) for the measured $Y\left( T \right)$ values as shown in Fig. 2.11. Here, the prompt neutron decay constant $\alpha$ is a time constant (s⁻¹) that expresses how fast the neutron family decreases exponentially. The relationship between $\alpha$ and the subcriticality $\left( { - \rho } \right)$ can be well approximated as follows:

$$\alpha = \frac{{\beta_{{{\text{eff}}}} - \rho }}{\Lambda } = \frac{{1 - \left( {1 - \beta_{{{\text{eff}}}} } \right)k_{{{\text{eff}}}} }}{\ell }.$$

(2.49)

Using Eq. (2.49) with the point kinetics parameters $\Lambda$ and $\beta_{{{\text{eff}}}}$ for the UTR-KINKI, the subcriticality $\left( { - \rho } \right)$ can be indirectly estimated from the measurement results of prompt neutron decay constant $\alpha$.

2.3.4.2 Method of Measurement

To measure the prompt neutron decay constant $\alpha$ using the Feynman-$\alpha$ method for a target subcritical core, the experimental procedures are described as follows:

• (1) To measure the reactor noise, a neutron detector (e.g., BF₃ detector) is placed in a hole in the graphite reflector, and a neutron measurement system is constructed as shown in Fig. 2.2.

• (2) Four control rods (two SRs, SSR and RR) are arbitrarily inserted to configure a certain subcritical system. Or, a shutdown state (i.e., all control rods fully inserted) may be useful to carry out a longer reactor noise measurement using the inherent neutron source in the nuclear fuel.

• (3) If an external neutron source is inserted into the core, or even if the inherent neutron source due to the U($\alpha$, n)²⁷Al reaction is utilized, let us wait until the neutron count rate becomes a constant value.

• (4) According to the subcriticality, the counting gate of width T should be appropriately set to analyze the prompt neutron decay constant $\alpha$. Typically, T = 0.1 ms = 10^–4 s. The time-series data of neutron count $\left( {C_{1} \left( T \right),C_{2} \left( T \right), \cdots ,C_{N} \left( T \right)} \right)$ are successively measured using the basic gate width T.

• (5) From the measured reactor noise, the Feynman-$\alpha$ histogram is analyzed using the bunching method to evaluate the prompt neutron decay constant $\alpha$.

• (6) Based on Eq. (2.49), the subcriticality $\left( { - \rho } \right)$ is finally converted from the measured $\alpha$ value.

2.3.4.3 Discussion

Let us discuss the experimental results of Feynman-$\alpha$ method from the following viewpoints:

• (1) Let us evaluate the reference value of subcriticality $\left( { - \rho_{{{\text{ref}}}} } \right)$ based on the control rod pattern using the excess reactivity $\rho_{{{\text{excess}}}}$ and the control rod worth $\rho_{{{\text{rod}},i}}$ in advance. By comparing the subcriticality estimated by the Feynman-$\alpha$ method with the reference value $\left( { - \rho_{{{\text{ref}}}} } \right)$, is there a significant difference in $\left( { - \rho } \right)$ between them? If any, what is the major reason?

• (2) Let us suppose that the subcriticality of a target core can be varied by changing the control rod pattern. According to the magnitude of subcriticality $\left( { - \rho } \right)$, how do the saturation value $Y_{\infty }$ and the prompt neutron decay constant $\alpha$ change? Let us discuss the physical meanings of the above-mentioned changes.

• (3) In the Feynman-$\alpha$ method, because of the statistical uncertainty of neutron counts, the $Y$ and $\alpha$ values also have statistical uncertainties. Let us discuss how we can reduce the statistical uncertainties of the $Y$ and $\alpha$ values.

[Column] Statistical uncertainty in the Feynman- ${\varvec{\alpha}}$ method

Because the total measurement time is limited in the actual reactor noise measurement, it is a complicated question to quantify the statistical uncertainty $\sigma_{Y}$ in the measured $Y$ value (i.e., the uncertainty of fluctuation in neutron counts). Various estimation methods for the statistical uncertainty $\sigma_{Y}$ have been studied, e.g., (1) the resampling technique using the moving block bootstrap method and (2) the analytical formula using the uncertainty propagation (or sandwich formula). The latter method (2) utilizes the statistical property that the probability distribution of neutron counts in a neutron multiplication system follows a special distribution known as the Pál-Mogil’ner-Zolotukhin-Bell-Babala (PMZBB) distribution [14, 15]. For example, if the subcriticality for the target core is $\left( { - \rho } \right) < 10,000$ pcm (= 10%Δk/k = 0.1 Δk/k), the statistical uncertainty $\sigma_{Y}$ can be easily estimated by the following formula using the total number of count data N, the sample mean $\mu$, and the measured $Y$ value [16]:

$$\sigma_{Y} \approx \left( {Y + 1} \right)\sqrt {\frac{{Y\left( {2Y + 1} \right)\left( {5Y + 2} \right)}}{{N\left( {Y + 1} \right)^{2} \mu }} + \frac{2}{N - 1}} .$$

(2.50)