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.

8.2.1 Outline of TEF-T

For the JAEA-proposed ADS, Pb-Bi is a primary candidate of coolant and spall­ation target. To solve technical difficulties for Pb-Bi utilization, construction of TEF-T is planned to complete the data sets that are required for the design of ADS.

The experiments to obtain the material irradiation data for the beam window are the most important mission of TEF-T.

A high-power spallation target, which will be mainly used for material irradia­tion of candidate materials for a beam window of full-scale ADS, is an essential issue to realize a TEF-T. To set up the beam parameters, future ADS concepts are taken into account. In the reference case of the target, proton beam current density of 20 ^A/cm2, which equals the maximum beam current density of the JAEA — proposed 800 MWth ADS, was assumed.

For practical application, the scale of an NRD facility should be minimized. Figure 2.1 shows a rough draft of the NRD facility. An electron linear accelerator with a power of 1 kW and acceleration voltage of 30 MeV is assumed [11]. High-energy neutrons are generated in the order of 1012 n/s by photonuclear reactions following Bremsstrahlung at the electron target. The generated neutrons are slowed down to epithermal energy by collisions in a moderator surrounding the target. Neutrons from the moderator are collimated to supply for NRTA and for NRCA/PGA.

The length of the flight path is important to design a TOF system, because the longer flight path reduces the neutron flux whereas it increases the energy resolution of the system. It may require at least a 5-m flight path to achieve a good enough resolution to resolve resonances of NMs below 50 eV in NRTA [9, 10]. A shorter neutron flight path is feasible for NRCA/PGA because the nuclei in Table 2.1 are identified by the prompt y-ray energies. We consider that a 2-m flight path is sufficient for NRCA/PGA. The beam line lengths mainly determine the scale of

the facility. One beam line for NRTA and three beam lines for NRCA/PGA are placed as shown in Fig. 2.1. The sample size for NRTA is assumed to be 10-30 cm in diameter and 1-2 cm in thickness. In comparison, the sample size for NRCA/ PGA is smaller; the diameter is 1-2 cm, and the thickness is 1-2 cm. A collimator is placed between the NRCA/PGA sample and the y-ray detector to reduce the background y-rays from the sample. Because optimal sample thickness for NRTA strongly depends on the amount of impurities or matrix material, the quantity of the interfering nuclei in debris has to be measured roughly by NRCA/PGA preceding NRTA measurements [12].

The statistical uncertainties of NMs quantified by NRTA were estimated [12]. The size of a MF sample is assumed to be 1 cm in thickness and 30 cm in diameter. The weight of the sample becomes about 4 kg: it consisted of nuclear fuel (64 vol.%), natFe (8 vol.%), natB (8 vol.%), and 20 vol.% of vacancy. The compo­sition of the nuclear fuel was taken from Ando and Takano [13] [a fuel of 40 GWd/t burn-up in a boiling water reactor (BWR)]. The measurement was assumed to be carried out for 40 min, in which 20 min was for sample and 20 min for background. Table 2.2 shows the estimated statistical uncertainties of quantified Pu and U isotopes in the sample. The achieved statistical uncertainties are less than 1 %.

With the measurement cycle given here, about 0.15 ton of debris can be handled in a day; this enables us to measure 30 tons of debris in a year (200 working-days are assumed). This amount can be increased with the number of NRTA beam lines.

The wettability of the sample before and after irradiation is evaluated by measuring the contact angle of a water droplet on a sample surface. The measurement system (Fig. 10.2) consists of a digital video camera, a stage (with a biaxial stage and a goniometer), a backlight, and a PC. Pure water of 2 ці is dropped onto the horizontal surface of the sample using a micropipette. The water droplet is imaged by the camera and the images are processed to obtain the contact angle. In the image processing, it is assumed that the droplet is a part of a sphere (Fig. 10.3), and the contact angle is estimated by the following equation:

Fig. 10.2 Contact angle measurement system

Fig. 10.3 Estimation of contact angle

where r and h are obtained by using an image processing software (ImageJ).