I. INTRODUCTION
Neutron source multiplication (NSM) method, a simple measurement technique that uses an external neutron source and an ordinary neutron detector, is suitable for real-time measurement. With the recent development of accelerator-driven system (ADS) [1], an increasing attention has been paid to NSM experiments associated with various subcritical systems [2, 3]. Implementing this kind of experiments can acquire subcritical multiplication factor ks and external neutron source efficiency Φ∗, which can be used to design the ADS systems, and its neutron multiplication [4, 5]. The subcritical multiplication factor ks is related to the neutron multiplication M [6] by ks =1-1/M, but in practice one needs to know the effective multiplication factor keff.
In this paper, in order to overcome the difficulties of getting keff directly from the conventional neutron source multiplication method, we propose a numerically associated experiment method. In Sec.II, the NSM method and the relationship between effective multiplication factor keff, subcritical multiplication factor ks and neutron source efficiency Φ∗ are reviewed briefly. The experiment apparatus is described in Sec. III. In Sec.IV, for verifying the proposed NSM method, the numerical analysis and a series of NSM experiments were carried out at the Chinese Fast Burst Reactor-II (CFBR-II) at various subcritical states with 252Cf neutron source located at different positions in each subcritical states (from center of the system to outside).
II. THEORETICAL BASIS
For the critical systems composed of the fissile materials and other materials, the neutron balance equation in fundamental mode is expressed as:
where, φ0(r, E) is the neutron flux distribution at position r(x,y,z) at energy E, L is the destruction operator, and P is the production operator. By taking the inner product <⋅> of Eq. (1), which indicates the integration over the space and energy variables, keff in Eq. (1) can be expressed as:
For the subcritical system with an external source at steady state, the neutron balance equation is:
Similarly, the subcritical multiplication factor ks can be expressed as:
where, φs(r, E) is the neutron flux in the subcritical system with the external neutron source s(r, E).
The external source efficiency, Φ∗, is the ratio of average importance between the source neutron and fission neutron. The Φ∗ implies that a neutron emitted from source, with a certain position and energy distribution, will produce certain "important" neutrons that can lead to higher or lower neutron multiplication for a given proliferation system.
Deduced from properties of the operator and Eqs. (1)– (4), we have Eq. (5):
Then, the multiplication factors ks and keff can be theoretically correlated with the neutron source efficiency Φ∗.
Obtaining keff of the system through Eq. (5) needs to know the ks and Φ∗. As mentioned above, the ks can be measured by neutron multiplication experiment, but the quantity of neutron source efficiency Φ∗ cannot be obtained from experiments directly. In the following sections, we acquired Φ∗ from numerical analysis and evaluated keff successfully.
III. EXPERIMENT APPARATUS REVIEW
Experiments were carried out at the Chinese Fast Burst Reactor-II (CFBR-II) [7], an ellipsoidal assembly, which consists of, from inside to outside, the enriched-uranium spherical shells, inner brass reflector, depleted uranium reflector and outer brass reflector. The reactor can be divided into the upper, middle and lower parts (Fig. 1). The middle part is a 5.2 cm thick stainless steel plate, which has three horizontal holes for three control rods to pass through, i.e. an auto- adjustment rod, a composition rod and a pulse rod. The three rods are made of casted enriched-uranium. A 10 cm×4.2 cm×14.5 cm (w × h × d) irradiation chamber and two horizontal experimental holes are located in the plate. The neutron source can be put at any position in the horizontal holes by a neutron source transmission device. Reactivity of the assembly can be controlled by adjusting insert depth of the three control rods. The assembly works at subcritical, delayed critical or super prompt critical states.
A 252Cf spontaneous fission neutron source (in activity of 1.41×10-5 s-1) sealed in an aluminum box of Φ7.8 mm×11 mm can be sent into a horizontal hole of the stainless steel plate by the transmission device when the system is in deep subcritical state. Then, the control rods are inserted to different depths to operate the reactor at different subcritical power states.
Two BF3 proportional counters, of the SZJ-1 type, produced by Beijing Nuclear Instrument Factory, are used to record the neutrons (which is proportional to the neutron density in the system). They consist of the paraffin barrel (Hanson long counter), preamplifier, HV power supply, main amplifier and multi-scalar. The detectors are 180 cm away from the system center.
IV. NUMERICAL ANALYSIS AND EXPERIMENTS
A. Numerical calculations
Before a series of experiments are carried out, Monte-Carlo numerical method [8] was used to build the subcritical system model that is identical to CFBR-II in Fig. 1. Four subcritical states of keff =0.996;0.994;0.991;0.986 were selected, which are conditions for our multiplication experiments. The source positions were changed along the r-axis at each subcritical state (i.e. 0 mm, 20 mm, 40 mm, 60 mm, 80 mm, 100 mm, 120 mm and 150 mm). The spontaneous 252Cf neutron source, described by Watt fission spectrum [9], is positioned the same way as that in the planned experiments. The ks of all the experimental conditions were calculated by MCNP5 with statistic difference of <0.7%. The neutron source efficiency Φ∗ is closely connected with neutron importance, which is proportional to the detector response to a unit source. In other words, neutron importance is a measure of the "importance" of a neutron in contributing to the detector response, or the expected counts per neutron in certain position, direction and energy. The detector counting rates to the system with the 252Cf neutron source in the core could be obtained, and the counting rate to the eigen neutron source distributed in the system could be recorded, too, by using MCNP code. The ratio of the two counting rates should be the external neutron source-related average efficiency Φ∗ [10, 11]. In terms of eigen-distribution neutron source acquisition, we used the KCODE and SSW cards in MCNP5 critical calculation to achieve the goal. Thus, we tallied Φ∗ in MCNP5 calculation following certain regulation according to MCNP rules.
By calculating the ks, Φ∗ and keff inside and outside of the neutron source, we were able to investigate the effect of system subcriticality and source position.
For the subcritical states (keff =0.994), the effect of the source positions along the r-axis are demonstrated by the calculation results in Fig. 2. The source efficiency Φ∗ and subcritical multiplication factor ks at different source positions indicate that the smallest discrepancy between ks and keff is achieved when the source is at the center of the core.
Table 1 shows the subcriticality effect on ks and Φ∗ at keff =0.996; 0.994; 0.991; 0.986. One sees that for the four subcritical states, when the source is at the core center, ks is very close to keff, with a relative difference of <0.8%, and the Φ∗ was greater than 0.91. The results indicate that the closer the system reaches to the critical state, the more likeliness the relative values of φs and φ0 become equal to each other, and so do the values of ks and keff. Inversely, placing the source outside the core, the discrepancy between ks and keff becomes much larger, with the maximum relative difference being 22.1340% at keff = 0.986 and the source being 150 mm away from the core center.
| case | keff | Source position (mm) | (ks/keff -1) (%) | Φ∗ |
|---|---|---|---|---|
| 1 | 0.9960 | 0 | -0.6436 | 0.9599 |
| 150 | -12.2234 | 0.0251 | ||
| 2 | 0.9940 | 0 | -0.5851 | 0.9272 |
| 150 | -15.2628 | 0.0887 | ||
| 3 | 0.9910 | 0 | -0.7513 | 0.9471 |
| 150 | -18.6502 | 0.0939 | ||
| 4 | 0.9860 | 0 | -0.7852 | 0.9176 |
| 150 | -22.1340 | 0.1036 |
In order to check the source position effects on the neutron leakage rate, we did another numerical analysis to investigate the detector response to non-collided neutrons emitted from the source, and to total neutrons. The results in Fig. 3 show that the relative difference is below 0.25%. Placing the neutron source closer to the system edge, the counting rate for non-collided neutrons from the source becomes greater, while the counts of total neutrons decrease. In other words, the detector counting rate decreases for the eigen-distribution neutron (fission neutron inside the system). This explains why the discrepancy between ks and keff becomes large while moving the 252Cf neutron source away from the system center.
B. Experiments Results and Discussion
On the basis of the numerical investigation in Sec.IV.A, the NSM experiments were carried out at CFBR-II to measure ks and evaluate Φ∗. The subcritical states were experimentally achieved by adjusting insert depth of the control rods. All the experiment apparatus and conditions were completely identical to the numerical ones. The results are shown in Fig. 4. The ks decreased with increasing distance of the source from the core center and with deepening subcriticality. This is identical to the calculation results in Fig. 2.
In order to test the feasibility of this improved NSM method, we considered four subcritical states (keff= 0.996; 0.994; 0.991; 0.986) and eight neutron source positions according to each subcritical states. For the sake of the calculated source efficiency Φ∗ and the measured ks, we obtained the effective multiplication factor keff according to Eq. (5) at 32 studied conditions. For comparison, we used the positive periodic method to measure the reactivity as the reference method in the work. The specific process is that we measured reactivity in super-criticality, and extrapolated to subcriticality with calibrated reactivity curve of control rods.
As shown in Table 2, a comparison between the reference method and the results in this work at the four subcritical conditions with the source located at the core center revealed fairly good agreement with a maximum relative difference of 0.18%. For a certain subcriticality (keff=0.991), the discrepancy between keff measured by positive period method and the evaluated ones in this work is the smallest of all. When keff > 0.9910, the evaluated keff are a bit larger than experimental measured ones. However, at a very deep subcriticality, e.g. keff =0.9860, the evaluated keff is a little less than the measured one. This is because the source neutrons could not propagate efficiently in deep subcriticality. The results demonstrated the good precision of this improved method and the validation of the Monte-Carlo calculation model.
| Case | keffa | keffb | ksc | Φ∗d | ( keffa /keffb-1) / % |
|---|---|---|---|---|---|
| 1 | 0.9960 | 0.9977 | 0.9936 | 0.9599 | 0.1707 |
| 2 | 0.9940 | 0.9948 | 0.9899 | 0.9272 | 0.0805 |
| 3 | 0.9910 | 0.9915 | 0.9847 | 0.9471 | 0.0505 |
| 4 | 0.9860 | 0.9844 | 0.9757 | 0.9176 | -0.1623 |
In general, keff of the system for all studied cases were well approximated as shown in Fig. 5.
The comparison between the keff results in references (positive period method, marked as keff-exp) and our method (keff-eva) is done by Δϵ=[(keff-eva - keff-exp)/keff-exp]×100%. As shown in Fig. 6, the space effect on keff can be seen in a large range of the neutron position. Generally, when the neutron source is inside the subcritical system, the relative difference is small, comparing to that when the source is outside the system; and the relative difference increase rapidly with distance of the neutron source to the system center, being irregularly large at the distance of >85 cm, because this is where the brass and depleted uranium reflectors locate, causing uncertainty factor related to complicated neutron behavior.
Therefore, to reduce the relative difference in experiments implemented, the neutron source should be put inside the subcritical system.
V. CONCLUSION
A methodology to evaluate the effective multiplication factor keff from measured ks and numerical analyzed Φ∗ was investigated via NSM experiments on CFBR-II. The evaluated results were approximately equal to the reference method (positive period method).
On basis of the theoretical and numerical preparations, ks was measured through a series of NSM experiments with a 252Cf spontaneous fission neutron source located at different positions in four subcritical states of CFBR-II. The effective neutron multiplication factor keff obtained in all the subcritical states when neutron source located inside of the subcritical system, within a relative difference of <0.2% between the values obtained by the improved NSM method and positive period method. So, to reduce the relative difference in experiments implemented, the neutron source should be put inside the system. The numerical analysis-associated NSM we proposed is feasible for the simple subcritical apparatus, such as CFBR-II.
