Corresponding author: Vitaliy V. Dulin ( vitdulin@yandex.ru ) Academic editor: Georgy Tikhomirov
© 2021 Vladimir A. Grabezhnoy, Viktor A. Dulin, Vitaliy V. Dulin, Gennady M. Mikhailov.
This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation:
Grabezhnoy VA, Dulin VA, Dulin VV, Mikhailov GM (2021) On the determination of neutron multiplication by the Rossialpha method. Nuclear Energy and Technology 7(3): 253257. https://doi.org/10.3897/nucet.7.74156

Introduction. This work contains the results of determining the prompt neutron multiplication factor in the subcritical state of a onecore BFS facility, obtained by the neutron coincidence method, for which the influence of the error in the β_{eff} in determining the multiplication factor turned out to be insignificant. The core of the facility consisted of rods filled with pellets of metallic depleted uranium, 37% enriched uranium dioxide and 95% enriched plutonium, sodium, stainless steel and Al_{2}O_{3}. Stainless steel served as a reflector.
Methods. In contrast to the inverse kinetics equation solving (IKES) method, which is convenient for determining reactor subcritical states, the neutron coincidence method practically does not depend on the error in the value of the effective fraction of delayed neutrons β_{eff}. If in the IKES method the reactivity value is obtained in fractions of β_{eff}, i.e., from the measurement of delayed neutrons, the neutron coincidence method is based on the direct measurement of the value (1 – k_{σ}_{p})^{2}, where is the effective multiplication factor by prompt neutrons. The total multiplication factor is defined as k_{eff} = k_{σ}_{p} + β_{eff}. If, for example, k_{eff} ≈ 0.9 (which is typical for determining the fuel burnup campaign), then it is the error in determining k_{σ}_{p} that is the main one in comparison with the error in β_{eff}. Thus, a 10% error in β_{eff} of 0.003–0.004 (typical for plutonium breeders) will make a contribution to the error 1 – k_{eff} equal to 1 – k_{σ}_{p} + β_{eff} ≈ 0.00035, i.e., approximately 0.35%, but not 10%, as in the IKES method.
Rossialpha measurements were carried out using two ^{3}He counters and a time analyzer. The measurement channel width Δt was 1.0 μs. From these measurements, the value of the prompt neutron multiplication factor was obtained. In this case, the spaceisotope correlation factor for the medium with a source was calculated using the following values: Φ(x) – solutions of the inhomogeneous equation for the neutron flux and Φ^{+}(x) – solutions of the ajoint inhomogeneous equation.
Results. The authors also present a comparison of the results of the Rossialpha experiment and measurements of the BFS73 subcritical facility by the standard IKES method in determining the multiplication factor value. The data of the IKES method differ insignificantly from the results of the Rossialpha method over the entire range of changes in the subcriticality with an increase in the subcriticality of the BFS73 onecore facility.
Conclusion. It was impossible to apply the neutron coincidence method to fast reactors; however, the method turned out to be quite workable on their models created at the BFS facility, which was successfully demonstrated in this study.
Neutron coincidence method, Rossiα method, spatialisotope correlation factor, prompt neutron multiplication factor
The multiplication factor, estimated by the standard methods adopted at the BFS facility, i.e., the IKES method (
The work contains the results of determining the multiplication factor in the subcritical state, obtained by the neutron coincidence method (
The effective neutron multiplication factor in the subcritical state of a singlecore BFS facility was determined using both of the above methods. The core of the facility consisted of rods filled with pellets of metallic depleted uranium, 37% enriched uranium dioxide and 95% enriched plutonium, sodium, stainless steel and Al_{2}O_{3}. Stainless steel served as a reflector. The height of the core was 45 cm, the equivalent diameter ≈ 65 cm. The subcritical state was achieved by removing from the core 11 fuel rods, of which the controls and CPS were composed.
Measurements by the neutron coincidence method were carried out using two SNM18 ^{3}He counters placed in the central channel (instead of a fuel rod) and a time analyzer (TA). The helium pressure in the counters was about two atmospheres. One of the counters was located 1 cm above the central plane of the critical facility core, and the other was below it. The length of the counter was approximately 25 cm. The first was used as a starting one for the TA, and the second was used as a counting one. The width of the VA channel Δt = 1 μs. The delay T in the counting channel was measured and turned out to be 40.1 μs.
As an example, Figure
In (
[N/(2J_{right}ΔtQ_{sp})]_{exp} D = (1 – k_{σ}_{p} _{exp})^{2}, (1)
where D is the calculated spatialisotope correlation factor for the medium with the source:
$D=\left.\right)"\; close=">">{Q}_{cf}\left(x\right)$;
Θ = [(4.91∑_{f}_{235}(x) + 7.06∑_{f}_{239}(x) + 5.28∑_{f}_{238}(x)) Φ(x) + 3.825Q_{sp} (x)](Φ^{+}(x))^{2};
Ξ = [(2.50∑_{f}_{235}(x) + 2.98∑_{f}_{239}(x) + 2.59∑_{f}_{238}(x)) Φ(x) + 2.156Q_{sp} (x)]Φ^{+}(x)^{2},
where Φ(x) is the solution to the inhomogeneous equation for the neutron flux; Φ^{+}(x) is the fission spectrumaveraged solution to the ajoint inhomog eneous equation, which is the probability for the detector to register a neutron that appears at the point x (
The processing of Rossiα temporal distributions was carried out taking into account the asymmetry of the temporal distribution of the background relative to the center of the correlated signal, which was at 40.1 µs of the TA scale. For this, the signal of a periodic pulse generator with a duration of 0.05 µs was applied simultaneously to the starting and counting (through the delay line T) TA channels with a channel width of 0.1 µs, and the count was recorded in the 401^{st} channel, which corresponded to the delay value T = 40.1 µs.
This type of background asymmetry was found experimentally on a fast critical assembly (
${N}_{left}\left(t\right)=N\left\{1A\frac{C\alpha}{\alpha +C}\left[\frac{2\alpha}{\alpha C}\xb7{\mathrm{e}}^{C(tT)}\frac{\alpha +C}{\alpha C}\xb7{\mathrm{e}}^{\alpha (tT)}\right]\right\}$; t < T, (2)
${N}_{right}\left(t\right)=N\left\{1A\frac{C\alpha}{\alpha +C}\xb7{\mathrm{e}}^{\alpha (tT)}\right\}$; t ≥ T. (3)
Note that the temporal distribution of the background is continuous, i.e., N_{left}(T) = N_{right}(T).
The correlated part of the distribution at t ≥ T obtained after the constant mean background is subtracted, equal to [ACα/(α + C)]e^{−α(}^{t} – ^{T}^{)} (the right side visible in the experiment), according to (
R _{cor}(t) = AC_{count}Ce^{−α(}^{t} – ^{T}^{)}; t ≥ T. (4)
Correspondingly, the area under the right correlated component (integral over t ≥ T) is
J _{right} = AC_{count}C/α. (5)
To obtain the true measured correlated areas J_{cor} from those ‘visible’ above the constant background level N, it is necessary to additionally measure the speed of incoming counts C at the start and then find the α value from the experiment. According to (7), it is on the right side of the equality R_{cor}^{exp}(t) (see Figs
Knowing C and α, we can calculate the value of the addition to the area AC/α to the right of t ≥ T:
ΔJ_{right} = (AC/α)×[C/(α + C)]. (6)
When calculating ΔJ_{left}(t), the integration, according to (
$\Delta {J}_{left}=\frac{AC}{\alpha}\frac{C}{(\alpha +C)}\left[\frac{2{\alpha}^{2}}{(\alpha C)\stackrel{~}{N}}\frac{\alpha +C}{(\alpha C)\alpha}\right]$ (7)
Table
Files  Starts C, s –1  Channel background on the right  α×10^{5}, s^{–1}  J _{left}  J _{right} 

RT1  1544 ± 44  8666  1.191  36811  73436 
RT21  1816 ± 46  13836  1.002  56401  121326 
RT3  1796 ± 15  14993  1.119  62041  130442 
Files  J _{left} + ΔJ_{left}  J _{right} + ΔJ_{right}  N/(2J_{right}ΔtQ)  N/[(J_{right}+J_{left})ΔtQ] 

RT1  73150  74376  0.0859  0.0858 
RT21  111799  123485  0.0820  0.0869 
RT3  123102  132503  0.0831  0.0859 
0.0837 ± 0.0017  0.0862 ± 0.0005  
k _{eff} (Rossiα) – k_{eff} (IKES)  0.9624 ± 0.0008 –  0.9630 ± 0.0003 –  
– 0.9595 ± 0.0030 =  – 0.9595 ± 0.0030 =  
= 0.0029  = 0.0035 
The source of spontaneous fissions Q_{sp} was the ^{240}Pu source, which was included in the plutonium fuel. Using the calculated factor D = 0.0472 and the effective source of spontaneous fissions ^{240}Pu Q_{sp} = (7.64 ± 0.21)×10^{6} n/s (1σ), we obtain the prompt neutron multiplication factor 1/(1 – k_{σ}_{p} _{exp}) and the values k_{eff}(φ^{+}_{σ}) = k_{σ}_{p} _{exp} + β_{eff}. In this method the absolute error β_{eff} but not the relative error, as in the ρ_{$}subcriticality (in units of β_{eff}) of the assembly. The difference between the k_{eff} values obtained by different methods is shown in the last row of the Table
Table
The results of using the experiment of Rossiα and measurements of the subcritical assembly by the IKES method
File  α×10^{4}  S _{right}, in range channels (ch)  S _{left}, in range channels (ch)  J _{left}×10^{4}  J _{right}×10^{4}  N/(2J_{right}ΔtQ)  N/[(J_{right}+J_{left})ΔtQ] 

S2404A  11.16  6026  11299  1.205  1.130  0.117  0.113 
(30–50 ch)  (51–98 ch)  
S2404В  13.93  2733  11299  1.205  1.130  0.111  0.108 
(1–49 ch)  (50–110 ch)  
S2804A  8.43  5463  11511  1.091  1.155  0.110  0.113 
(1–26 ch)  (27–68 ch)  
S2704А  10.66  5755  9761  1.150  0.978  0.119  0.109 
(1–26 ch)  (27–110 ch)  
S2804В  10.78  6823  13959  1.362  1.402  0.113  0.114 
(1–26 ch)  (27–110 ch)  
S2904А  12.41  5057  10626  1.009  1.067  0.111  0.114 
(1–26 ch)  (27–68 ch)  
S2904В  9.07  5926  12325  1.182  1.239  0.112  0.115 
(1–26 ch)  (27–68 ch)  
R809A  10.52  20914  41141  4.154  4.170  0.0960  0.0962 
(20–79 ch)  (80–130 ch)  
R0909A  7.12  15998  31581  3.178  3.202  0.0984  0.0987 
(20–83 ch)  (84–130 ch)  
R1109A  8.76  35021  70874  6.947  7.203  0.0907  0.0924 
(20–79 ch)  (80–130 ch)  
R1109B  8.76  14238  27711  2.821  2.822  0.103  0.103 
(20–84 ch)  (85–130 ch)  
R1409A  6.71  17119  32147  3.387  3.283  0.0982  0.0967 
(54–84 ch)  (85–145 ch)  
R1609A  9.79  9718  18913  1.943  1.892  0.102  0.101 
(62–82 ch)  (83–114 ch)  
R1609B  9.92  20714  41624  4.128  4.191  0.0831  0.0837 
(20–81 ch)  (82–150 ch)  
Average,  0.113  0.112  
S files  ± 0.003  ± 0.002  
Average,  0.0959  0.0960  
R files  ± 0.0064  ± 0.0059  
Average,  0.090  0.090  
without СН_{2}  ± 0.009  ± 0.009 
A similar method was used to determine the subcritical state of a singlecore BFS73 facility, consisting of metallic uranium (18% enrichment), steel and sodium, in proportions close to the composition of a fast reactor (Dulin VA and Dulin VV 1998).
The multiplication factor, estimated by the standard methods adopted at the BFS facilities (by the IKES method (
The processing of these experiments carried out in (
The results of using the Rossiα experiment and measurements of the subcritical assembly by the IKES method presented in Table
The data of the IKES method differ insignificantly from the results of the Rossiα method with an increase in the subcriticality of the assembly (the difference did not exceed the errors). Perhaps, this was due to the simplicity of the geometry of the subcritical singlecore BFS73 facility (