The impact of thermal agitation on Doppler coefficient for Gdbearing fuel was analyzed. It was found through the analysis that the impact increases when a small amount of Gd_{2}O_{3} is added to pure UO_{2} fuel although the impact decreases for a large amount of Gd_{2}O_{3}. This tendency was discussed with the usage of simplified expression for the difference of Doppler coefficient. The simplified expression was used to consider the tendency, and it was revealed that the tendency mainly comes from the rapid decrement of multiplication factor and the relatively slow decrement of the magnitude of sensitivity coefficient of U238 capture cross section at low Gd_{2}O_{3} concentration. Similar tendency which shows a maximum impact on Doppler coefficient at interior concentration is expected for other UO_{2} fuel with a slight content of strong absorber. This indicates that Doppler coefficient of UO_{2} fuel system with low content of strong absorber should be analyzed carefully by considering thermal agitation in epithermal range.
Scattering kernel used in epithermal range is usually treated as the asymptotic model in solving the slowing down equation, which does not take thermal agitation into account. If the targets are scatterings of light isotopes, the asymptotic model is appropriate since the change of scattering kernel by thermal agitation is trivial. But for heavy isotopes, the asymptotic model is not appropriate for scattering kernel especially in epithermal range [
Many evaluations were performed by many researchers but the target composition to evaluate Doppler coefficient is mainly UO_{2} without any poison. As easily expected, the addition of poison such as Gd_{2}O_{3} to UO_{2} fuel causes the change in multiplication factor and also causes the change in Doppler coefficient by the increment of capture rate of the poison. Thus, it is expected that the impact of thermal agitation on Doppler coefficient varies as a function of poison content in UO_{2} fuel.
The purpose of this paper is to evaluate the impact of thermal agitation on the Doppler coefficient for Gdbearing fuel and to reveal the mechanism of the impact of thermal agitation on Doppler coefficient in Gdbearing fuel. Section
The calculation procedure and the calculation results are described in this chapter.
The calculation geometry is modeled from conventional PWR cell for 4.8 wt% enriched UO_{2} fuel with Gd_{2}O_{3} as shown in Figure
Calculation conditions.
Fuel  
Material  UO_{2} with Gd_{2}O_{3} 
U235 enrichment  4.8 wt% 
Gd_{2}O_{3} concentration  0~20 wt% using following Gd vector 
Gd152: 0.2 wt%  
Gd154: 2.2 wt%  
Gd155: 14.8 wt%  
Gd156: 20.5 wt%  
Gd157: 15.6 wt%  
Gd158: 24.8 wt%  
Gd160: 21.9 wt%  
Temperature  600 K, 900 K 


Cladding  
Material  Zircalloy4 
Temperature  600 K 


Coolant  
Material  H_{2}O 
Temperature  600 K 


Boundary condition  White reflection 
Pin cell model.
The Doppler reactivity is calculated from two multiplication factors of different fuel temperatures as
The calculation is performed by continuous energy Monte Carlo code MVP which can handle thermal agitation in epithermal range [
Impact evaluated by MVP for various Gd_{2}O_{3} concentrations.
Gd_{2}O_{3} concentration 
Impact on Doppler 
Error [%1  

0  8.1  ±  0.3 
0.2  10.6  ±  0.5 
2  9.9  ±  0.7 
5  8.9  ±  0.7 
10  7.7  ±  0.7 
20  5.6  ±  0.7 
The calculation process and the calculation results for detailed evaluation of the impact are described in this chapter.
Calculation geometry and calculation conditions are the same as previous chapter. Deterministic evaluations of Doppler coefficient with considering thermal agitation were performed by the following process. The multiplication factor with considering thermal agitation is evaluated with the usage of sensitivity coefficients for the multiplication factor and the relative difference of cross sections between asymptotic model and exact model as expressed in
The technique shown in this section is useful to obtain breakdown in isotopes, reaction types, and energy group because each of the parameters is evaluated separately.
The impact on Doppler coefficient is summarized in Table
Impact evaluated by UA method for various Gd_{2}O_{3} concentrations.
Gd_{2}O_{3} concentration [wt%]  Impact on Doppler 

0  9.4 
0.2  12.2 
2  12.0 
5  10.6 
10  8.5 
20  5.6 
Doppler coefficients calculated by conventional asymptotic kernel model and UA method are shown in Figure
Comparison of Doppler coefficient.
Table
Nuclide and reaction breakdown of impact on the Doppler coefficient of Gdbearing fuel.
Nuclide reaction  Gd_{2}O_{3} concentration [wt%]  

0  0.2  2  5  10  20  
U238 capture  9.4%  12.2%  12.0%  10.6%  8.4%  5.4% 
U235 capture  0.0%  0.0%  0.0%  0.0%  0.0%  0.0% 
Gd155 capture  —  0.0%  0.0%  0.0%  0.0%  0.0% 
Gd156 capture  —  0.0%  0.0%  0.0%  0.0%  0.1% 
Gd157 capture  —  0.0%  0.0%  0.0%  0.1%  0.1% 
Gd158 capture  —  0.0%  0.0%  0.0%  0.0%  0.0% 


All  9.4%  12.2%  12.0%  10.6%  8.5%  5.6% 
Energy breakdown of impact on the Doppler coefficient of Gdbearing fuel.
Energy range [eV]  Gd_{2}O_{3} concentration [wt%]  

0  0.2  2  5  10  20  
3.93~5.04  0.0%  0.0%  0.0%  0.0%  0.0%  0.0% 
5.04~6.48  −0.1%  −0.2%  −0.1%  −0.1%  −0.1%  0.0% 
6.48~8.32  0.9%  1.1%  0.8%  0.7%  0.4%  0.2% 
8.32~10.68  0.0%  0.0%  0.0%  0.0%  0.0%  0.0% 
10.68~13.71  0.0%  0.0%  0.0%  0.0%  0.0%  0.0% 
13.71~17.60  0.0%  0.0%  0.0%  0.0%  0.1%  0.1% 
17.60~22.60  2.1%  2.7%  2.6%  2.2%  1.7%  1.0% 
22.60~29.02  0.0%  0.0%  0.0%  0.0%  0.0%  0.0% 
29.02~37.27  4.6%  6.1%  6.2%  5.6%  4.6%  3.1% 
37.27~47.85  −0.2%  −0.2%  −0.2%  −0.2%  −0.2%  −0.1% 
47.85~61.44  0.0%  0.0%  0.0%  0.0%  0.0%  0.0% 
61.44~78.89  1.4%  1.8%  1.8%  1.6%  1.3%  0.8% 
78.89~101.3  0.0%  −0.1%  0.0%  0.0%  0.0%  0.0% 
101.3~130.1  0.7%  1.0%  0.9%  0.8%  0.7%  0.5% 


All  9.4%  12.2%  12.0%  10.6%  8.5%  5.6% 
The impact of thermal agitation on Doppler coefficient increases up to about 12% when Gd_{2}O_{3} concentration is around 0.2 wt% and 2 wt% and the impact decreases according to the increment of Gd_{2}O_{3} concentration although Gd isotopes do not have the impact of thermal agitation on Doppler coefficient as shown in Table
In order to discuss this tendency, a simplified expression of the difference in Doppler coefficient is derived in Section
In this section, the impact on Doppler coefficient (
The impact of Doppler coefficient by thermal agitation at energy group
For further simplification of (
The capture cross section of Gd nuclide is remarkably large at thermal range, but the magnitude is not so large compared to that of U238 at the energy group where the agitation effect is sensitive to Doppler coefficient. Thus,
Relative difference of U238 capture cross section.
Sensitivity coefficient of U238 capture cross section to multiplication factor.
In this section, the impact of thermal agitation on Doppler coefficient in epithermal range is discussed by using (
Multiplication factor and inverse of multiplication factor.
Difference in Doppler coefficient.
As already described, there is a maximum impact of thermal agitation on Doppler coefficient at low Gd_{2}O_{3} concentration and this tendency comes from the rapid increase of
Similar tendency which shows a maximum impact is expected for the fuel with the slight content of strong absorber. Therefore, Doppler coefficient of the system which contains low content of strong absorber should be analyzed carefully with the consideration of thermal agitation.
The impact of thermal agitation on Doppler coefficient of Gdbearing fuel was analyzed. The results show that the impact increases by adding a small amount of Gd_{2}O_{3} (~2 wt%) to UO_{2} fuel, although the impact decreases by adding a large amount of Gd_{2}O_{3} (~10 wt%) to UO_{2} fuel. This tendency was analyzed with the usage of simplified expression of the difference in Doppler coefficient. The difference in Doppler coefficient varies as a function of Gd_{2}O_{3} concentration, and this is mainly caused by the two factors: sensitivity coefficient of U238 capture cross section to multiplication factor and multiplication factor. The magnitude of both factors decreases according to the increment of Gd_{2}O_{3} concentration, but the decrement rate of multiplication factor is remarkable compared to that of the sensitivity coefficient at low Gd_{2}O_{3} concentration, where the selfshielding of Gd isotopes is not so important and the content of U238 is roughly the same to the fuel without Gd_{2}O_{3}. The decrement rates of both factors are almost the same at high Gd_{2}O_{3} concentration, which brings almost the same difference in Doppler coefficient between with and without considering thermal agitation. On the other hand, the magnitude of Doppler coefficient increases monotonically and linearly according to the increment of Gd_{2}O_{3} mainly caused by the decrement of multiplication factor. Therefore, there is a maximum point of the relative difference in Doppler coefficient caused by thermal agitation.
Similar tendency is expected for other strongabsorberbearing fuel. Thus, it should be noted that the Doppler coefficient of the fuel with a slight amount of strong absorber should be analyzed carefully with the consideration of thermal agitation.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank Dr. Nagaya for his kind support for the evaluation by continuous energy Monte Carlo code MVP.