A computational study has been performed for various options of the thorium reactor core loading. Neutronic studies of fuel have been conducted, its isotopic composition has been calculated, and the alpha emitters and the sources of neutron and photon radiation in the microencapsulated nuclear fuel have been analyzed. The studies had the purpose of developing the methodology used to estimate the radiation characteristics of nuclear fuel with a complex inner structure. Emphasis is placed on calculating the quantitative and spectral composition of the neutrons formed as the result of (a, n) reactions on small- and average-mass nuclei. The ratio of the quantity of the neutrons resulting from the (a, n) reactions to the quantity of the neutrons formed as the result of spontaneous fission has been calculated for fuel with heterogeneous and homogeneous arrangements of fissionable and structural elements. The developed tools will make it possible to estimate the neutron radiation dose, to revise the traditional fresh and spent fuel handling procedures, and to estimate, using the Rossi alpha method, the neutron multiplication factor in deeply subcritical systems. The neutron yield and spectrum were calculated using an analytical model and verified codes such as WIMS-D5B, ORIGEN-APP, SOURCES-4C and SRIM-2013.

Thorium reactor, isotopic composition, alpha particle transport, neutron source, spectrum of radiation sources

In (Shamanin et al. 2015, Shamanin et al. 2016, Shamanin et al. 2018), the author studied the physics of a small high-temperature gas-cooled thorium reactor plant (НGTRU, Russia), and selected the best reactor core configuration and the best nuclear fuel material composition. The reactor considered in (Shamanin et al. 2018) is capable to operate for not less than 3000 effective days at a power level of 60 MW. Peculiarities of the reactor core design, the material composition of dispersed microencapsulated nuclear fuel (microfuel), and operating conditions affect the radiation characteristics of fuel and require traditional nuclear fuel handling procedures to be revised at the reactor design stage.

Emphasis in the study is placed on calculating the neutron component of the dispersed nuclear fuel’s radiation characteristics. Neutrons are formed in nuclear fuel due to spontaneous and induced fissions, and as a result of (a, n) reactions on small- and average-mass nuclei (Vlaskin et al. 2015, Bulanenko 1979, Murata and Shibata 2002, West and Sherwood 1982). In such studies, the yield q_sf and the spectral distribution (spectrum) of spontaneous fission neutrons S_sf (E) are comparatively easy to determine. For this reason, of greater interest is determination of the ratio of the quantity of the neutrons q_an formed as a result of (a, n) reactions to the quantity of the neutrons resulting from spontaneous (q_sf) and induced (q_f) fission (in the event of large-size spent nuclear fuel (SNF) blocks) which is of interest both for calculating the neutron radiation dose rate during handling of such fuel (Dulin and Zabrodskaya 2005, Dulin and Matveyenko 2002) and in using the Rossi alpha method to determine the multiplication factor (k_eff) in subcritical systems (Dulin and Matveyenko 2002), such as the НGTRU fuel block (Shamanin et al. 2018). This problem was examined rather thoroughly and solved in (Vlaskin et al. 2015, Bulanenko 1979, Dulin and Zabrodskaya 2005, Dulin and Matveyenko 2002, Shamanin et al. 2017, Bogatov et al. 2015) and as part of other studies for fuel in the form of homogeneous dioxide. In the event when dispersed fuel and media containing heterogeneous inclusions of different configurations and sizes are considered, this problem is more difficult to solve and the method to calculate the quantitative and spectral compositions of the neutron source is of interest as such.

The purpose of the study is to develop the methodology to calculate the quantitative and spectral compositions of the neutron radiation source with a heterogeneous arrangement of fissionable and structural elements in nuclear fuel. Such studies are conducted with the use of verified codes and dedicated programs. It often turns out during numerical modeling of radiation fields and in development of new procedures and regulations for the SNF handling in a nuclear fuel cycle of a new generation (Shamanin et al. 2018, Bogatov et al. 2015, Glukhov et al. 2016, Spirin et al. 2015) that no standard codes can be used without respective scientific rationale and upgrades. For instance, the ORIGEN-S code enables determination of the nuclear fuel nuclide composition and the spectral composition of the SNF neutron and photon radiation sources depending on burn-up and decay time. This code is one of the advanced tools used internationally to calculate radiation sources. It is normally employed to calculate the burn-up and radiation sources of standard oxide fuel (UO₂, (U,Pu)O₂) for LWR-type reactors, while the use of ORIGEN-S may lead to incorrect results in the event of other reactor types and for fuel other than of the standard type (Bogatov et al. 2015). It should be noted that, apart from ORIGEN-S, there are other neutron source codes, e.g., SOURCES-4С (Wilson et al. 2009), NEDIS-2m (Vlaskin and Khomyakov 2017), CHARS (Leniau and Wilson 2014), and others (Spirin et al. 2015, Vlaskin and Khomyakov 2017). These codes fully focus on the neutron yield and spectrum calculation; the most commonly used code is SOURCES-4С, as the NEDIS-2m code (Vlaskin et al. 2015, Spirin et al. 2015, Vlaskin and Khomyakov 2017) is the one best adapted to problems of present-day nuclear power.

The nuclear reactor considered in (Shamanin et al. 2018) is a reactor plant using prompt neutrons (~63.03 % of neutrons with an energy of 4 eV to 183 keV) and fast neutrons (~24.5 % of neutrons with an energy of 183 keV to 10 MeV) with the (Th,Pu)O₂ composition used as nuclear fuel. The (a, n) reaction can run in the nuclear fuel of such reactor not only on ^17,18O isotopes but also on ¹³С and ^29,30Si contained in the microfuel (Ti₃SiC₂) and fuel pellet (SiC) coatings. The rated neutron spectrum c_an(E) on ¹³С and ^29,30Si is much higher than the (a, n) neutron spectrum on ^17,18O (Bulanenko 1979, Jacobs and Liskien 1983), and the content of ¹³С carbon in the pellet matrix may reach 62.5 %.

For unirradiated dispersed nuclear fuel, c_an(E) and S_an(E) can be calculated using available high-accuracy experimental data (Experimental Nuclear Reaction Data, see https://www-nds.iaea.org/exfor/exfor.htm) and special-purpose codes (NEDIS-2.0, SOURCES-4C, SRIM-2013 (Ziegler et al. 2010)). To obtain the correct result, though, it is enough to take into account the energy distribution of alpha particles f_a(E) within the fuel kernel and in the coatings (Bedenko et al. 2018). The distribution f_a(E) can be obtained provided the sources of alpha particles (²³²Th,^239,240,241Pu) are distributed uniformly within the kernel and create spherically symmetrical radiation.

The irradiated fuel kernel represents a mixture of unburnt Pu and Th isotopes, minor actinides, oxygen, and fission fragments. The initial functions f_a(E), c_an(E) and S_an(E) in such kernel will depend on the distribution P (E, r) of heavy isotopes and fission fragments within it, which complicates considerably the conditions of the problem, as the use of (Fomushkin 2010, Bak et al. 1965) and other similar models requires a scientific rationale.

The energy distribution of neutrons c_an(E), S_an(E) sought after depends in a general case on the geometry of the fuel pellet and the material composition of nuclear fuel, the reactor type and operating modes, and the distribution of radiation sources (Pu, Th, minor actinides) f_НM (r), fission products f_FP (r), and alpha particles f_a(E, r) within the fuel kernel, on the fuel surface and in the coatings.

Three types of fuel pellets designated 0817, 1017 and 1200 were used in (Shamanin et al. 2018); the authors have shown that an increase in the fuel pellet diameter leads to a smaller fraction of graphite and a reduced reactivity margin r_inf (t), and the reactor life depends on the initial quantity of ²³⁹Pu and ²³²Th and on the quantity of the ²⁴¹Pu and ²³³U formed. When a fuel pellet of the type 0817 is used, ²³⁹Pu burns out rapidly and ²³³U and ²⁴¹Pu do not have enough time to accumulate in such quantity as required for maintaining the steady-state fission reaction. Therefore, after 1500 days of operation, the quantity of ²³⁹Pu and ²³⁹Pu left in the reactor is too small and r_inf (t) decreases abruptly. The use of pellets of the types 1017 and 1200 leads to the accumulation of ²³³U and ²⁴¹Pu and to longer operation of the reactor. We shall note that the burn-up of ²³⁹Pu is practically the same for the pellets of the types 0817 and 1017 amounting to 96 and 97% respectively. Therefore, the option with a fuel pellet of the type 1200 is the best one in terms of lifetime and the involvement of ²³²Th in the fuel cycle, but an increase in the dispersed phase fraction in such pellet (w_fuel) will lead to a smaller fraction of the burnt-out ²³⁹Pu. The analysis of results in (Shamanin et al. 2015, Shamanin et al. 2016, Shamanin et al. 2018) shows that the best burn-up of ²³⁹Pu requires the selection of a fuel pellet of the type 1017 (Fig. 1).

Figure 1. 

Fuel pellet of the type 1017 (a) and a 2D computational model of the equivalent cell (b).

Further neutronic experiments were conducted in WIMS-D5B, a versatile code for calculating the cell of different reactor types. The WIMS code uses a 69-group system of constants based on the evaluated nuclear database, ENDF/B-VII.0, which makes it possible to calculate fast and thermal neutron reactors.

The geometry module of the WIMS code is not capable to create cells of a hexagonal shape, so the hexagonal НGTRU cell (Shamanin et al. 2018) was transformed into a 2D cylindrical cell. The cylindrical cell is an equivalent Wigner-Seitz cell with preset boundary conditions on the calculated area surface and with a translational symmetry on the upper and lower faces. To calculate k_eff, axial and radial geometrical parameters are introduced as calculated with regard for the transition from the actual core size to the equivalent Wigner-Seitz system (see Fig. 1b).

Computational studies were performed for 30 reactor core loading options differing in composition. The content of heavy metal in the pellet kernels (%) for all of the calculated options is as follows: Pu – 50, ²³²Th – 50. The isotopic composition of Pu (%): 238 – 0, 239 – 94, 240 – 5.4; 241 – 0.6; 242 – 0. The results of the studies are shown in Fig. 2 where it can be seen that an over 17-18% increase of ω_fuel is not reasonable since the fuel life increases insignificantly, and the share of the burnt-out ²³⁹Pu will decrease.

Figure 2. 

Duration of operation as a function of fraction of dispersed phase.

We shall note that the overall nature of the dependence t (w_fuel) (see Fig. 2) obtained for the full-scale heterogeneous reactor model will be the same. The results w_fuel and t will be not differ significantly, despite the fact that the homogenization of the cell’s fuel portion leads to a 3 to 7% reduction of k_eff in relation to more accurate modeling (Chukbar 2015).

Thus, we have selected a fuel pellet of the type 1017 with the dispersed phase content of 17 % (see Fig. 2). For a fuel pellet of the type 1017 with w_fuel = 17 % and the reactor thermal power of P = 60 MW, the reactor life will be ~3000 days (see Fig. 2). In a cell of such reactor, ²³⁹Pu burns up to 84.5%, ²⁴⁰Pu to 36.9%, and ²³²Th to 9.68%.

Irradiated microfuel represents a complex mixture of isotopes, so the functions f_a(E), c_an(E) and S_an(E) sought after will depend on the density distribution of the probability of interactions P (E, r) for the considered radiation within the kernel. The density of the probability of neutron interactions P (l₀, r) within the kernel at the distance r from its center (Fomushkin 2010) can be described by the function of the form

P (l₀, r) = ∫v (x, r)×Y(x)dx, (1)

where v (x, r)∙is the probability for experiencing the interaction within the kernel at the distance r from its center; Y(x) is the exponential law of the neutron radiation attenuation in the kernel material; l₀ = l_t /R is the rated neutron path length; l_t is the average-weighted neutron path; and R is the kernel radius.

As of the time of the irradiation start, l_t1 = 1/S_t1 = 1/0.309 = 3.24 cm, l₀₁ = 93, and l_t2 = 1/0.313 = 3.19 cm, l₀₂ = 91 as of the irradiation cycle end. The average-weighted value of the neutron free path l_t calculated with regard for the spectrum of the neutron flux j_n(E) (Shamanin et al. 2018) was used to estimate l₀, that is, all materials of the fuel pellet (graphite matrix, kernel and coatings) can see one and the same spectrum. The diagram of the dependence of P (l₀, r) for several values of l₀ is presented in Fig. 3. It can be seen (Fig. 3) that the distribution of P (l₀, r) for l_{01 and}l₀₂ (l_t >> R) is practically uniform (P (l_01.02, 0)/P (l_01.02, 175) = 1.26) and differs greatly from P (l₀₄, r) for the case when l_t ≈ R (Fomushkin 2010). We shall note that, with l_t << R, the probability P (l_03,04,r) is greater in the kernel’s peripheral portion (Fomushkin 2010), that is, the fragments formed from fission will shield the inner part of the kernel from the incident neutron flux. The “grain self-shielding” effect is taken into account practically by all neutron transport codes. Model (Fomushkin 2010) makes it possible to estimate the grain self-shielding effect (l_t < R), so will give consequently a correct result for the case when the neutron free path length is much greater than the kernel’s cross-section dimension (l_t >> 2R).

Figure 3. 

Probability density of the neutron interactions within the kernel: 1 − l₀₁ = 91; 2 – l₀₂ = 93.

We shall assume that heavy isotopes in the kernel burn out uniformly (P (l₀, r) = 1.26 ≈ 1), so, then, the sources of alpha particles and fission fragments are distributed uniformly and homogeneously within the kernel as of the end of irradiation and the spatial distribution f_FP (r) of fission fragments, as shown in (Fomushkin 2010), can be determined from the relation

where r = x/R; l = l/R; and l is the fission fragment path in the kernel.

The average energy of excitation for a light-weight fragment (A₁ = 90) is equal to about 10 MeV, and that of a heavy-weight fragment (A₂ = 140) is equal to about 8 MeV. The path length of light-weight fragments in the kernel material as of the end of irradiation will be 4 to 7 μm (l₁ = l/R = 4 μm / 175 μm = 0.023), and their spatial distribution (Fig. 4, line 1), similarly to the case of heavy metals (except the peripheral layer), does not depend on r.

Figure 4. 

Spatial distribution of fission fragments within the kernel: 1 – l₁ = 0.023; 2 – l₂ = 1.

The computational studies performed have helped to formulate the following assumptions.

1. The irradiated kernel is a homogeneous mixture of heavy isotopes, fission fragments, and oxygen.

2. The sources of alpha particles, alpha particles as such and fission fragments are distributed uniformly and homogeneously in the kernel.

3. Each source creates isotropic and spherically symmetrical radiation.

4. The functions χ_an(E) and S_an(E) depend only on the differential energy spectrum of alpha particles f_a(E) = dN_a(E)/dE inside of the kernel and on its surface.

The subject of the study is the HGTRU microfuel and fuel pellet. The configuration of the kernel, the coatings and the fuel pellet is shown in Fig. 1. To find f_a(E), one needs to solve the problem of the kernel surface radiation. A similar problem was solved, e.g., in (Fomushkin 2010, Bak et al. 1965) and by other researchers, and the theoretical basis can be found in training books on theory of grids.

The fraction of the radiation P (probability) from the inside of the kernel in (Bak et al. 1965) was obtained for a source with self-absorption:

(2)

where m = 1/l₀ is the absorption coefficient; and l₀ is the average particle path.

Relation (2) is valid with any mR, but f (x)dx = exp(−x/l₀)dx/ l₀ for the particles (neutrons, gamma-quanta) the path of which is described by an exponential distribution. In the event of μR >>1 (λ₀<<R), the key role in (2) is played by the first term and then P ≈ 3 l₀/(4R). The probability P sought after in (Bak et al. 1965) was obtained as a triple integral, the subintegral function was expanded into series, and a term-wise integration was performed, after which the infinite series was summed up. All these operations are quite cumbersome for a general problem but they are simplified to a great extent with l₀ << R, as the obtained solution is valid also for particles with a fixed free path (alpha particles, fission fragments, protons). In (Fomushkin 2010), the solution to this problem was obtained thanks to using principles of geometrical probabilities. Such approach makes it possible to reduce the finding of a general solution to elementary computations of volumes and to the subsequent integration of fairly simple functions. In (Fomushkin 2010), the author considered different types of radiation, including radiation with a fixed free path. For radiation with a fixed and relatively short path of (l ≤ R), the probability P is given by the formula

P = 0.75lR^–1 – 0.0625(l/R)³, l << R, P ≈ 3l/(4R).

If the full number of the alpha particles formed in the kernel at the time of the decay is equal to N_a0, and P_a(E) is the fraction of the radiation escaping from the kernel, then the number of the alpha particles remaining in and those leaving the kernel is equal respectively to N_a0(1 − P_a(E)) and N_a(E) = N_a0∙P_a(E).

The differential energy spectrum of the alpha particles escaping from the kernel surface for a unit of time at a solid angle of 4p is connected with P_a(E) through the relation (Fig. 5)

Figure 5. 

Spectrum of alpha particles escaping from the kernel surface at a solid angle of 4p (1 – irradiated fuel; 2 – unirradiated fuel).

(3)

where l_a(E) = ∫dE/(−dE/dx) is the alpha particle path in the kernel; and e_a(E) = (−dE/dx) is the stopping power of the alpha particles.

The path 1/ l_a(Е) = S(w_i/ l_ai) and the stopping power e_a(Е) = S(w_i∙ e_ai) in the kernel material and in the coating were calculated using the Bragg-Kleeman additivity rule. The path R_ai (E) and the stopping power e_ai (E) of the i-th nuclide were calculated using the SRIM code.

Fig. 5 presents spectra of alpha particles for irradiated and unirradiated fuel. The intensity of the alpha particle generation N_a0 in the kernel in an unirradiated fuel pellet (w_fuel = 17 %) is equal to ~2.39∙10⁵ a/s/kernel; 99.9% of the alpha particles result from the decay of the ^239,240Pu isotopes; the average energy of the alpha particle spectrum is <E_a> = Sd_i∙E_i = 5.15 MeV; and the probability of the alpha particle escape from the kernel surface is P_a(5.15) = 7.11%.

For an irradiated fuel pellet, N_a0 = 3.08∙10⁷ a/s/kernel; 99.06 % of the alpha particles result from the decay of the ^242,244Сm isotopes; <E_a> = 5.93 MeV; and P_a(5.93) = 7.94%. The calculation has shown that the alpha particles formed in unirradiated and irradiated fuel pellet kernels (see Fig. 5) remain in the first coating since their path lengths do not exceed 35 μm. For further calculations, the continuous spectrum f_a(E) was transformed into a group form for the SOURCES.

The subject of the study is a hexagonal fuel block of the HGTRU reactor plant (Shamanin et al. 2015, Shamanin et al. 2016, Shamanin et al. 2018) representing a block of dense high-quality graphite machined at a temperature of 3000 to 3300 K and having 76 small-diameter channels for fuel pellets of the type 1017 (∅10.17×10^–3 m) and seven large-diameter channels (∅24×10^–3 m) for the coolant. The flat-to-flat dimension of the fuel block is 0.207 m and its height is 0.80 m.

Fig. 6 presents the results of the neutron source calculations in terms of one fuel channel (basis) the diameter of which is 10.17×10^–3 m and the height is 0.8 m. The fuel channel is filled with pellets of the type 1017.

Figure 6. 

Energy spectrum of the neutron source: 1 – integral yield of neutrons; 2 – yield of neutrons from spontaneous fission and the (α, n) reaction on ^17,18O [(Th,Pu)O₂]; 3 – yield of neutrons from to the (α, n) reaction on ^17,18O; 4 – yield of neutrons from the (α, n) reaction on¹³С (RuS).

The integral yield of neutrons q_S = (q_an(^17,18O,¹³С) +q_sf) for w_fuel = 17 % (Fig. 6, line 1) is equal to ~4.58∙10³ neutr./s/basis. The contribution of neutrons from the (a, n) reactions on ^17,18O of the oxide fuel ceramics is q_an(^17,18O)/q_S = 41.1% (Fig. 6, line 3); the contribution of neutrons from the (a,n) reactions on ¹³С is q_an(¹³С)/q_S = 1.18% (Fig. 6, line 4); q_an(^17,18O,¹³С)/q_sf = 0.7325, and q_an(^17,18O,¹³С) = 2.98∙10¹ neutr./s/cm³. For a homogeneous fuel pellet of a VVER-1000 reactor (fresh UO₂ with an enrichment of 4.4%), the similar ratio is equal to q_an(^17,18O)/q_sf = 0.0084, and the yield is q_an(^17,18O) = 1.011∙10^–3 neutr./s/cm³.

The yield of neutrons q_an from the (a, n) reactions (see Fig. 6, line 3) for an unirradiated pellet defines the shape of the total distribution of S_an(E) (see Fig. 6, line 1), which has the maximum in the energy region of 2.25 to 2.5 MeV. The maximum is defined by the (a, n) reaction on the ^18,17O nuclei of the oxide fuel ceramics during the interaction of the alpha particles resulting from the decay of the ^239,240,241Pu and ²³²Th isotopes. For a homogeneous pellet, q_an also defines the shape of S_an(E). To describe S_an(E), heterogeneous and homogeneous problems can use the function of the form c_an(Е) = [a (2p)^1/2]⁻¹∙exp[−(E−b)²/2a²]. For an irradiated heterogeneous pellet, the total yield of neutrons can be approximated by the spectral Watt function c_sf (Е) = с×exp(−E/a)×sinh(bE)^1/2, since q_an/q_sf = 0.0050 and q_an = 8.26∙10¹ neutr./s/cm³ (w_fuel = 17 %, Z = 400 GW·day/tTh-Pu).

The spectrum of the source of photons resulting from the decay of the ^239,240,241Pu and ²³²Th isotopes was prepared in the Origen-Arp code in a group form and is presented in Fig. 7.

Figure 7. 

Energy spectrum of the photon source.

The integral yield of photons for w_fuel = 17 % is equal to ~8.2∙10¹⁰ gamma-quantum/s/basis. Over 99.01 % of photons are formed in the energy region of 10 to 30 keV. The energy spectra of the radiation sources (see Figs 6, 7) were prepared in a group form for further calculations using the MCNPX2.6.0 code.

The fuel block investigated in this paper is a subcritical multiplying system with a complex inner structure. The (a,n) reaction in the fuel block of such reactor needs to be considered not only on the oxygen nuclei of the (Th,Pu)O₂ composition but also on the carbon nuclei contained in the first coating of dispersed microencapsulated fuel. Procedures have been developed to calculate the quantitative and spectral compositions of the neutron radiation from this fuel which made it possible to determine the ratio of the quantity of the neutrons q_an from the (a, n) reactions to the similar quantity q_sf from spontaneous fission.

The tools developed as part of the study will make it possible to estimate the neutron radiation dose from the fuel under investigation, to revise the traditional procedures for handling fresh and irradiated fuel, and to estimate the neutron multiplication factor in the НGTRU fuel block using the Rossi alpha method.

The work has been supported by the Russian Scientific Fund. Grant No. 18-19-00136, dated 18.04.2018.