3.1.1 Neutron spectrum

For neutrons with energies of the order of eV or lower, the target nucleus cannot be considered stationary because the neutron energy is comparable to the kinetic energy of the thermal motion of the scattering target nucleus. When an elastic collision occurs between the neutron and the scattering target nucleus, the neutron may obtain energy from the vibration of the scattering nucleus. Hence, the energy of the outgoing neutron may be larger than that of the incident neutron, which is the phenomenon of neutron upward scattering in the thermal energy region [23]. With an increase in the graphite-moderator temperature, the vibration of the graphite nucleus becomes violent, the neutron obtains energy from the graphite nucleus more easily, and the upward scattering effect becomes more obvious. The Maxwellian component then shifts to a higher energy region, the neutron spectrum becomes harder, and the graphite moderator’s ability to slow the neutrons down weakens.

Figure 2 gives the neutron spectra for two different graphite-moderator temperatures (880 K and 980 K). The spectra are normalized to their respective total number of counts in the energy range from 1E-09 MeV to 20 MeV. It can be seen that the neutron energy shifts with a slight increase in the normalized flux values. More importantly, the neutron spectrum hardening mostly occurs below 1 eV, and there is an intersection point of the neutron spectrum curves for different temperatures around 0.16 eV. The stronger the slowing ability of the graphite, the more obvious the Maxwellian component, the more apparent the hardening of the neutron spectrum, and the greater the magnitude of the MTC. The sign ("+" or "-") of the MTC is primarily dependent on the distance between two adjacent fuel cells. Through research, this distance should be less than about 15 cm to obtain a negative MTC. Such an investigation involves the influence of the geometry of the graphite fuel cell on the MTC, which is out of the scope of this article and not discussed here. Hence, for the graphite fuel cell used in this work, the distance was 3.1 cm corresponding to a negative MTC.

Fig. 2.Full-screen View

Neutron energy shift with changing moderator temperature.

To describe the neutron spectrum quantitatively, a spectrum factor defined as the Energy of Average Lethargy of Fission (EALF) is introduced. A larger EALF represents a harder neutron spectrum. EALF is defined using the following equations [24] and calculated by MCNP5:

where E₀ is the maximum energy considered (10 MeV), m is the index for material regions, g is the index for energy groups, ${\overline{u}}_g$ is the average lethargy for energy group g, Φ_g,m is the neutron flux in the material region m and the energy group g, and ∑_f,g,m is the macroscopic fission cross section in the region m and the energy group g.

Figure 3 shows the EALF as a function of ²³⁵U enrichment and HM proportion. It can be seen that a higher ²³⁵U enrichment and/or a higher HM proportion has a larger EALF corresponding to a harder neutron spectrum. Furthermore, the EALF is more sensitive to the change of the ²³⁵U enrichment for a higher HM proportion. The lower the HM proportion, the smaller the increment of the EALF owing to the increasing ²³⁵U enrichment. For instance, as the ²³⁵U enrichment increases from 1 wt% to 20 wt%, the EALF increases by 7.9 times and 1.6 times for HM=12 mole% and HM= 2 mole%, respectively. Similarly, the lower ²³⁵U enrichment presents a smaller increment in the EALF on account of the increasing HM proportion. For example, when the HM proportion increases from 2 mole% to 12 mole%, the EALF increases by 1.4 times and 6.8 times for a ²³⁵U enrichment equal to 1 wt% and 20 wt%, respectively. In other words, it is conducive to alter the neutron spectrum by adjusting the ²³⁵U enrichment for a higher HM proportion or changing the HM proportion for a higher ²³⁵U enrichment.

Fig. 3.Full-screen View

EALF as a function of ²³⁵U enrichment and HM proportion.

There are two highlights. First, for a fixed graphite fuel element, the sign of the MTC is related to the type of nuclear fuel. As the graphite-moderator temperature increases, the Maxwellian component shifts to higher energies, the neutron spectrum becomes harder, the average thermal absorption (capture and fission) cross section of nuclides, such as ²³⁵U, ²³⁸U, graphite, and FLiBe, decreases and makes a positive contribution to the MTC. The term "FLiBe" is used here to include ¹⁹F, ⁶Li, ⁷Li, and ⁹Be. Different nuclides have different decrements in their respective thermal absorption cross section, which are mainly related to the distance between two adjacent fuel cells. A negative or a positive MTC is then presented. Second, the magnitude of the MTC is mainly dependent on the neutron spectrum shape, such as the degree of thermalization or hardness. As the neutron spectrum hardens owing to the increasing ²³⁵U enrichment and/or HM proportion, the percentage of thermal neutrons decreases, compounded by the fact that the increasing graphite-moderator temperature mainly affects the thermal region, the contributions of thermal absorption and total absorption of the nuclides to the MTC weaken. The following discussions are based on the above two points. It should also be noted that the neutron energy in the thermal region is less than 0.625 eV.

3.1.2 Contributions of α_TM(ε) and α_TM(p) to the MTC

The temperature coefficient for ε (α_TM(ε)) and that for the resonance escape probability (α_TM(p)) are strongly affected by the neutron spectrum. As the graphite-moderator temperature increases, the Maxwellian component shifts to higher energies and the neutron spectrum becomes harder, corresponding to a larger EALF (Fig. 2), which then enhances the probability of fast fissions. Thus, a positive α_TM(ε) is presented. As mentioned above, the ratio of the moderating atoms (molecules of graphite) owing to the increasing graphite-moderator temperature is neglected. However, an increased graphite-moderator temperature leads to a harder neutron spectrum and results in a decreasing macroscopic scattering cross section for the moderator. The macroscopic slowing-down power of the graphite moderator then weakens, the transport mean free path becomes larger, and more neutrons are absorbed by the nuclear fuel or the moderator in the resonance region. As such, the probability of absorption by the nuclear fuel or the moderator in the thermal region decreases, and a reduction of the resonance escape probability is introduced. Therefore, a negative α_TM(p) is revealed.

To evaluate the actual contributions of α_TM(ε) and α_TM(p) to the MTC, Fig. 4 presents the variations of α_TM(ε) (top) and α_TM(p) (bottom) with ²³⁵U enrichment for different HM proportions. We will discuss each of them in turn.

Fig. 4.Full-screen View

Variations in α_TM(ε) (top) and α_TM(p) (bottom) with ²³⁵U enrichment for different HM proportions.

First, the magnitude of α_TM(ε) (Fig. 4 (top)) is mainly dependent on the neutron spectrum shape. The main contributions to α_TM(ε) are the thermal fission (neutron energy E lt;0.625 eV) and the total fission (1d-9 eV lt;E lt;20 MeV) of heavy metal (mainly ²³⁵U). As discussed before, the increasing graphite-moderator temperature leads to a reduction in both thermal fission and total fission. This means that both thermal fission and total fission have positive effects on α_TM(ε). As the ²³⁵U enrichment or the HM proportion increases, the neutron spectrum becomes harder, and the contribution of thermal fission and total fission decrease gradually. The decrement of the contribution of total fission is slower than that of thermal fission owing to the decreasing percentage of thermal fission of ²³⁵U. An increasing α_TM(ε) is then revealed. That is, the magnitude of the positive α_TM(ε) increases gradually with the increasing ²³⁵U enrichment and/or HM proportion.

Second, differences in the magnitude of α_TM(p) (Fig. 4 (bottom)) are owing to accumulation effects of neutron spectrum shape as well as the ²³⁵U enrichment and HM proportion. As the Maxwellian component shifts to higher energies and the neutron spectrum becomes harder owing to the increasing graphite-moderator temperature, the thermal absorption for all materials in the core, including the heavy metal (i.e. ²³⁵U and ²³⁸U), graphite, and FLiBe, decreases in addition to their respective total absorption. They all make a positive contribution to α_TM(p). As the neutron spectrum hardens owing to the increasing ²³⁵U enrichment and/or the HM proportion, the positive contribution of thermal absorption and total absorption of the nuclides to α_TM(p) decreases. For ²³⁵U and FLiBe, the contribution to α_TM(p) decreases more slowly in thermal absorption than in total absorption. This is mainly owing to the decrease of the thermal region (²³⁵U and FLiBe) or the enhancement of the resonance absorption effect (especially ²³⁵U). Thus, the contribution of ²³⁵U and FLiBe to α_TM(p) increases with the hardening neutron spectrum. However, for ²³⁸U, the increasing fast fission effect makes up for the decrease of the contribution of the total absorption. This results in a faster decrease in the contribution of thermal absorption than in that of the total absorption. A gradually weakening contribution of ²³⁸U to α_TM(p) is then introduced. In this way, the heavy metal, including ²³⁵U and ²³⁸U, has a positive contribution to α_TM(p), and its contribution initially decreases mainly owing to the weakening contribution of ²³⁸U, and then increases mainly owing to the increasing contribution of ²³⁵U as the neutron spectrum hardens. Taken together, the variation of the magnitude of α_TM(p) with ²³⁵U enrichment is primarily dependent on the contribution of heavy metal. That is, at a fixed HM proportion, the magnitude of α_TM(p) decreases initially and then increases with the increase in the ²³⁵U enrichment. Meanwhile, the difference in the magnitude of α_TM(p) between different HM proportions mainly depends on the contribution of FLiBe. With the increase in the HM proportion, the magnitude of α_TM(p) monotonically increases.

Moreover, it is worth mentioning that α_TM(ε) and α_TM(p) show an almost similar magnitude but with opposite signs. This also means that the contributions of ε and p to the MTC compensate for each other.

3.1.3 Contribution of α_TM(η) to the MTC

The temperature coefficient for η (α_TM(η)) is closely related with the ratio of the thermal production (Fiss=∑f⋅ν) to the thermal absorption (Abs=∑f+∑γ) of the fuel material. Here, ∑f and ∑γ are the macroscopic fission and absorption cross sections of fuel material, while ν is the average number of neutrons produced per fission. As the graphite-moderator temperature increases, the variation of η is mainly owing to the following two aspects. First, because the changes in the thermal production/absorption of ²³⁸U are much less than in those of ²³⁵U, the variation of η mainly stems from ²³⁵U. In addition, ∑f is much larger than ∑γ for ²³⁵U, and the value of ν is usually higher than 2. Hence, the change in $Fis{s}_{235}{}_U$ (the thermal production of ²³⁵U) is much larger than that in $Ab{s}_{235}{}_U$ (thermal absorption of ²³⁵U). The hardening neutron spectrum weakens $Fis{s}_{235}{}_U$ and $Ab{s}_{235}{}_U$, and thus, a reduction of η is introduced. Second, the Maxwellian component shifts to higher energies, the neutron spectrum becomes harder, and then the average thermal neutron energy increases. Since the fissile material is all ²³⁵U is this work, η will decrease as the neutron energy increases, especially in the thermal neutron ranges (lt;1eV) [25]. Both of the above two aspects lead to a negative α_TM(η).

Figure 5 presents the variation of α_TM(η) with ²³⁵U enrichment for different HM proportions. The magnitude of α_TM (η) is mainly related to the neutron spectrum shape and its main contributions, including its thermal production and thermal absorption, are from ²³⁵U. As mentioned previously, both the thermal production and thermal absorption have positive contributions to α_TM(η), and their contributions decrease as the neutron spectrum hardens. However, because the thermal fission cross section is nearly an order of magnitude lower than the thermal capture cross section, while the average fission neutron number (ν) is larger than 2, the contribution of thermal production decreases faster than that of thermal absorption as the neutron spectrum hardens. A decrease in the magnitude of α_TM(η)) is then presented. That is, a lower ²³⁵U enrichment and/or a lower HM proportion results in a stronger α_TM(η)).

Fig. 5.Full-screen View

Variation in α_TM(η) with ²³⁵U enrichment for different HM proportions.

3.1.4 Contribution of α_TM(f) to the MTC

For a graphite-moderated MSR, the thermal utilization factor (f) is calculated by:

where "Abs" represents the thermal absorption, while the subscripts HM, FLiBe, and Grap correspond to heavy metal, light elements in fuel salt (including fluorine, lithium, and beryllium), and graphite, respectively. The graphite here includes the moderator and reflector. As the graphite-moderator temperature increases, the hardening neutron spectrum weakens the thermal absorption of the fuel salt and the graphite moderator. That is, Abs_HM, Abs_FLiBe, and Abs_Mod (the thermal absorption of the graphite moderator) show decreasing trends. The Evaluated Nuclear Data File on the website (https://www-nds.iaea.org/exfor/endf.htm) shows that the microscopic absorption cross section of ²³⁵U decreases faster with increasing neutron energy than the other elements, including ²³⁸U, graphite (i.e. ¹²C), fluorine, lithium, and beryllium, especially in the thermal regions (lt; 1 eV). The cross section of several important nuclides are presented in Fig. 6. Thus, the decrement of Abs_HM is larger than that of Abs_FLiBe. The ratio of Abs_FLiBe to Abs_HM will increase. Moreover, because the thermal absorption of the graphite reflector (Abs_Refl) strengthens owing to more fast neutrons leaking from the active core, and the calculations indicate that the increment of Abs_Refl is faster than the decrement of Abs_Mod, Abs_Grap will increase. Combining the decreased Abs_HM, the ratio of Abs_Grap to Abs_HM increases. Above all, a reduction of f is introduced, and a negative α_TM(f) is shown. It should be especially explained that the possibility of the sign of α_TM(f) being positive is significant for a large graphite assembly size, which results in a positive α_TM(f). The effects of geometric factors, such as fuel salt fraction and graphite assembly cell size on the temperature coefficient are outside the scope of this article and are not discussed here.

Fig. 6.Full-screen View

Absorption cross section for several important nuclides.

Figure 7 presents the variation of α_TM(f) with ²³⁵U enrichment for different HM proportions. The magnitude of α_TM(f) is mainly related to the neutron spectrum shape. When the graphite-moderator temperature increases, the neutron spectrum hardening occurs mainly in the region where the neutron energy is less than 1 eV. In this region, the fastest decrease in the absorption cross section of the nuclides is ²³⁵U, followed by ⁷Li, and then ⁹Be, ²³⁸U, ¹²C, and ¹⁹F (Fig. 6). In the reactor core, the thermal absorption change of heavy metals mainly comes from ²³⁵U, while that of non-heavy metals (including FLiBe, the moderator and the reflector) mainly comes from FLiBe (but especially ⁷Li). As mentioned in section 3.1.2, with the increase in ²³⁵U enrichment and HM proportion, the neutron spectrum hardens, and the contribution of ²³⁵U and FLiBe to α_TM(f) decreases. It can be seen from the cross section (Fig. 6) that when the neutron energy is between 1E-8 and 1E-6 MeV, the decrease in the rate of absorption cross section of ²³⁵U with neutron energy is obviously faster than that of any nuclide in FLiBe. The positive contribution of ²³⁵U to α_TM(f), therefore, decreases faster with the hardening neutron spectrum than FLiBe. In short, the harder the neutron spectrum, the smaller the magnitude of α_TM(f). That is, a lower HM proportion and/or a lower ²³⁵U enrichment results in a higher magnitude of the negative α_TM(f).

Fig. 7.Full-screen View

Variation in α_TM(f) with ²³⁵U enrichment for different HM proportions.

3.1.5 Contribution of α_TM(Λ) to the MTC

The distance traveled by fast neutrons during moderation (the slowing-down length) and the distance traveled by thermal neutrons during diffusion (the neutron diffusion length, L) in a reactor are important in reactor design because of their effects on the neutron leakage. The first directly affects the fast non-leakage probability (P_FNL), while the second directly influences the thermal non-leakage probability (P_TNL). For the graphite-moderated MSR used in this work, the neutron diffusion length was much larger than the slowing-down length. Therefore, the temperature coefficient for Λ, (α_TM(Λ)), was linked closely with the neutron diffusion length rather than the slowing-down length. A larger neutron diffusion length results in a reduction of P_TNL, and then a decreasing Λ is presented. Because the neutron diffusion length is dependent on the macroscopic cross section for elastic scattering (∑s) and the macroscopic cross section for neutron absorption (∑a), i.e. L²∝ $\frac{1}{{\sum }_s{\sum }_a}$, we will study the impact of the change in graphite-moderator temperature on these two processes. As the graphite-moderator temperature increases, ∑a decreases owing to the hardening neutron spectrum. Meanwhile, because the microscopic scattering cross section is almost independent of neutron energies, especially for low-energy neutrons, and the atomic density of solid graphite is nearly invariable, the value of ∑s shows little change. That is, the above two processes make L² greater. Therefore, as the graphite-moderator temperature increases, the neutron diffusion length increases, and the thermal non-leakage probability then decreases corresponding to a reduction of the total non-leakage probability (Λ). Thus, a negative α_TM(Λ) is introduced.

Figure 8 presents the variation of α_TM(Λ) with ²³⁵U enrichment for different HM proportions. Other than the neutron spectrum shape, the magnitude of α_TM(Λ) is mainly related to heavy metals, including the HM proportion and ²³⁵U enrichment. At a fixed ²³⁵U enrichment, a decreasing HM proportion leads to a decrease in the neutron absorption of heavy metals. At a fixed HM proportion, because the absorption cross section of ²³⁵U is much higher than that of ²³⁸U (Fig. 6), a decreasing ²³⁵U enrichment also corresponds to a reduction in the neutron absorption of heavy metal. Therefore, as the HM proportion and/or the ²³⁵U enrichment decreases, the neutron leakage increases owing to the reducing neutron absorption of the heavy metal. A large variation of neutron leakage will then be induced, and even the graphite-moderator temperature will change slightly It, therefore, means that the magnitude of α_TM(Λ) increases as ²³⁵U enrichment and/or HM proportion decrease.

Fig. 8.Full-screen View

Variation in α_TM(Λ) with ²³⁵U enrichment for different HM proportions.

3.1.6 The MTC as a function of ²³⁵U enrichment for different HM proportions

Based on the above discussions, one knows that the MTC is presented as a combination of the temperature effects of five factors. The sign of α_TM(ε) is positive, while the other four are negative. To ensure a negative MTC, one usually suppresses the positive α_TM(ε) and the possible positive α_TM(f) in the design of the MSR core. It is also significant that the sign of the MTC is closely related with the nuclear fuel type, while the magnitude of the MTC is primarily dependent on the neutron spectrum shape.

The variation of the MTC with ²³⁵U enrichment for different HM proportions is shown in Fig. 9. It can be found that the MTC can remain negative for all of the selected ²³⁵U enrichments and HM proportions for the MSR core chosen in this work. As mentioned above, the signs of α_TM(ε) and α_TM(p) are opposite while their magnitudes are very similar, and so, these two temperature coefficients approximately compensate for each other. Thus, the trends of the MTC with ²³⁵U enrichment depend primarily on α_TM(η), α_TM(Λ), and α_TM(f). The magnitudes of the MTC for different HM proportions decrease monotonously as the ²³⁵U enrichment increases. Meanwhile, at a fixed ²³⁵U enrichment, a lower HM proportion presents a more negative MTC. That is, a soft neutron spectrum is beneficial to obtain a more negative MTC.

Fig. 9.Full-screen View

Variations in the MTC with ²³⁵U enrichment for different HM proportions.