This section reviews the theory on the higher-order modes in the Feynman-a method in an ADS based on the work of Yamamoto [15]. Neglecting the energy — and spatial dependence of neutrons in a subcritical system driven by a neutron source with Poisson character, we obtain the Feynman Y function (the variance-to — mean ratio of neutron counts minus unity) as

where A = counting gate width, C1(4) = neutron counts in A, a0 = fundamental mode prompt neutron time-decay constant. When considering the energy and spatial dependence in an ADS, however, the Feynman Y function is more involved, as shown next.

The formula for the Feynman Y function in an ADS where q spallation neutrons are emitted from the beam target at a constant period T is given by this expression

[15]

:

the angle brackets denote the ensemble-averaging operator, Cr = count rate, and am = time-decay constant of the mth-order mode. (Refer to Yamamoto [14, 15] for other nomenclature.) Equation (12.3) represents the correlated component of the Y function, which also appears in a subcritical system with Poisson source. The correlation in Eq. (12.3) results from the multiple neutron emissions per fission

S: spallation neutron source Dj: detector 1 D2: detector 2

reaction. Equation (12.4) represents another correlated component caused by peri­odically pulsed multiple neutrons. Equation (12.5) represents the uncorrelated component caused by the periodically pulsed spallation neutron source.

A numerical example is considered for a one-dimensional slab with infinite height. The thickness of the slab is H = 55 cm. The vacuum boundary conditions are imposed on both ends of the slab. The spallation neutron source and neutron detectors are allocated as shown in Fig. 12.1. This chapter considers a one-energy — group problem. The constants used for the numerical example are Xt = 0.28 cm-1, Xf = 0.049 cm-1, Xc = 0.05 cm-1, и = 2,200 m/s, v = 2, and q = 60, T = 0.01 s (100 Hz). This system is sufficiently subcritical and keff = 0.95865 ± 0.00002, which is obtained by a Monte Carlo criticality calculation (it is referred to as “large subcritical system” hereinafter). The Feynman Y function versus counting gate width Д at the position of the detector 1 in Fig. 12.1 is calculated with a Monte Carlo simulation of the Feynman-a method. The simulation result at detector 1 is shown in Fig. 12.2 as “Monte Carlo.” In Fig. 12.2, “Theory (correlated)” shows a
theoretical value of the sum of YC(A) and YCS(A) calculated with Eqs. (12.3) and (12.4). “Theory total” shows “Theory (correlated)” plus the theoretical value of the uncorrelated component YuNA), calculated with Eq. (12.5). The neutron flux and am, which are needed to calculate the theoretical values of Eqs. (12.3), (12.4), and (12.5), are calculated with the Monte Carlo method up to the third-order mode

[16] . Beyond the third order, those are approximated with the diffusion theory:

am = v{Xf + Xc + DB2m — vXf), m > 4 ¥m(x) = {~^2dsinBmO + d),m > 4

Bm

where d = extrapolated length(=0.7104/Xt). The summation in Eqs. (12.3), (12.4), and (12.5) is taken up to the 250th mode. As shown in Fig. 12.2, there is good agreement between the Monte Carlo simulation and the theory, which shows verification of the theoretical formula of Eq. (12.2). Using Eqs. (12.3), (12.4), and (12.5), the Feynman Y function is decomposed into mode components. Figures 12.3 and 12.4 show the mode components of the correlated component and of the uncorrelated component, respectively. In these figures, each mode component includes the cross terms with the lower-order mode components. For example, “1st higher” includes the cross terms between the fundamental mode and the first higher-order mode as well as the first higher-order mode itself. Figure 12.1 shows that detector 1 is located at the bottom of the first higher-order mode. Thus, the first higher-order mode has a significant effect on the Feynman Y function of detector

1. Especially, the higher-order mode is more remarkable in the uncorrelated component, as shown in Fig. 12.4.

Fig. 12.5 Feynman Y function in the nearly critical system

>4

For a “nearly critical system” (keff = 0.99242 ± 0.00002), the Feynman

Y function is calculated using Eq. (12.2). The constants used for the nearly critical

system are Zt = 0.2834 cm-1, = 0.0524 cm-1, Zc = 0.05 cm-1, и = 2,200 m/s,

v = 2, q = 60, and T = 0.01 s (100 Hz). The Feynman Y function versus the counting gate width is shown in Fig. 12.5. “Total” in Fig. 12.5 shows the sum of the correlated and uncorrelated components. As shown in Fig. 12.5, the uncorrelated component is very minor in the nearly critical system. Thus, the Feynman

Y function is almost the same as the correlated component. The higher modes are negligibly small in the correlated component in the nearly critical system. Thus, the accurately approximated fundamental mode a can be obtained by fitting the Feyn­man Y function to the conventional formula, Eq. (12.1). On the other hand, if the subcriticality is not small enough, the uncorrelated component and higher-order modes have significant effects on the Feynman Y function. Therefore, obtaining a fundamental mode a would become difficult by simply fitting the Feynman

Y function to Eq. (12.1). The Feynman-a method is not necessarily a suitable method as a subcriticality measurement technique.