Neutron excess method for performance assessment of thorium-based fuel in a breed-and-burn reactor with various coolants

Neutron excess method for performance assessment of thorium-based fuel in a breed-and-burn reactor with various coolants 1 2 3 first-author 1 2 3 1 2 corresp chenjg@sinap.ac.cn chenjg@sinap.ac.cn chenjg@sinap.ac.cn 1 2 corresp caixiangzhou@sinap.ac.cn caixiangzhou@sinap.ac.cn caixiangzhou@sinap.ac.cn Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China CAS Center for Excellence in TMSR Energy System, Chinese Academy of Sciences, Shanghai 201800, China CAS Center for Excellence in TMSR Energy System, Chinese Academy of Sciences, Shanghai 201800, China University of Chinese Academy of Sciences, Beijing 100049, China University of Chinese Academy of Sciences, Beijing 100049, China Fertile fuel, such as thorium or depleted uranium, can be bred into fissile fuel and burnt in a breed-and-burn (B&B) reactor. Modeling a full core with fertile fuel can assess the performance of a B&B reactor with exact quantitative estimates, but costs too much computation time. For simplicity, performing the recently developed neutron balance method with a zero-dimensional (0-D) model can also provide a reasonable result. Based on the 0-D model, the feasibility of the B&B mode for thorium fuel in a fast reactor cooled by sodium was investigated by considering the (n, 2n) and (n, 3n) reaction rates of fuel and coolant in this work, and compared with that of depleted uranium fuel. Afterwards, the performance of the same thorium-based fuel core, but cooled by helium, lead-bismuth, and FLiBe, respectively, is discussed. It is found that the (n, 2n)&(n, 3n) reactions should not be neglected for the neutron balance calculation for thorium-based fuel to sustain the B&B mode of operation. Neutron excess Thorium-breed-and-burn Monte Carlo method 1 Introduction 1 Breed-and-burn (B & B) is a strategy used to breed fertile material into usable fissile fuel and then to burn it in situ in a fast reactor. The concept of a B & B reactor using only natural uranium or depleted uranium as fuel was initially proposed and investigated in 1958 by Saveli Feinberg [ 1 1 ]. Then, the travelling nuclear burning wave in infinite U-Pu medium sustaining the B & B mode of fuel was proven [ 2 2 ]. Sekimoto proposed a CANDLE nuclear reactor requiring only natural or depleted uranium for the nuclear non-ignition region of core, in which the spatial distribution of the nuclide density, neutron flux and power density would move stably in the core axial direction without changing their shapes [ 3 3 , 4 4 ]. Terra Power LCC proposed a TWR to produce fissile plutonium from fertile uranium, and the nuclear breeding/burning wave of the TWR to move gradually from the inside to the outside of the core [ 5 5 ] . Compared with the uranium fuel cycle, the thorium fuel cycle has several interesting advantages, which perhaps enable thorium-based fuel to become an alternative approach in nuclear fission energy utilization [ 6 6 - 9 9 6-9 ]. Thorium has a greater abundance on earth than uranium and the use of thorium fuel can reduce the radiotoxicity due to its much lower quantity of plutonium and long-lived Minor Actinides (MA) accumulation than uranium fuel. The average number of neutrons produced in each fission reaction per neutron absorbed in 233 U could be greater than 2 over a wide range of the neutron spectrum, which is of primary importance for the nuclear fuel B & B mode. It, therefore, raises the possibility of a nuclear breeding/burning wave by using thorium in a fast reactor [ 10 10 ]. The full core model of a B & B reactor provides accurate results but costs too much computation time. Florent Heidet et al., proposed a neutron balance method based on a simplified 0-D core model, which can save much computational time and provide reasonable estimates of the minimum required burnup (BU) and the maximum attainable BU, despite the neutron spectrum evolution during operation [ 11 11 , 12 12 ]. The (n, 2n) and (n, 3n) reaction contributions, however, were not considered in the method, which would bring an underestimation to the neutron balance performance. This paper investigates the use of thorium in a B&B reactor with the similar neutron balance method with consideration of the effects of the (n, 2n)&(n, 3n) reactions. We will also compare the result of thorium with that of uranium in a sodium cooled fast reactor. Then the feasibilities of a thorium B&B mode in the reference reactor with other coolants, including lead-bismuth, helium, and FLiBe, are assessed by performing the modified neutron balance method. The methodology modified by considering the (n, 2n)&(n, 3n) reaction channels is described in Sec. 2. The description of the reactor core is provided in Sec. 3. The simulated results and a detailed discussion are presented in Sec. 4. The conclusion is given in Sec. 5. 2 Neutron balance method 1 The neutron excess concept used to determine the required minimum BU to sustain B & B reactors was proposed and developed recently [ 11 11 - 15 15 11-15 ]. The basic principle of B & B is expected to provide a possible approach to operate a reactor with only fertile fuel feeding and without any fuel reprocessing, provided that the core is ignited successfully by fissile fuel loaded in the starter zone of the core. It requires that fertile fuel must produce more neutrons than its capture in the process of transmutation into fissile fuel. The excess neutrons are defined as the net number of neutrons generated or absorbed in a unit volume by a given material [ 14 14 ]. The total number of excess neutrons during operation can be obtained as a function of BU [ 11 11 ]: <math display="block" id="M1"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>Δ</mi><mi>N</mi><mi>E</mi><mo>=</mo></mrow></mtd><mtd columnalign="left"><mrow><msub><mi>N</mi><mrow><mi>H</mi><mi>M</mi></mrow></msub><mo>×</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><mi>B</mi><mi>U</mi></mrow></msubsup></mrow></mstyle><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><mrow><msub><mi>k</mi><mi>∞</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mi>U</mi><mo stretchy="false">)</mo><mo>×</mo><msub><mi>P</mi><mrow><mi>N</mi><mi>L</mi></mrow></msub><mo>×</mo><msub><mi>P</mi><mrow><mi>C</mi><mi>R</mi></mrow></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>×</mo><mover accent="true"><mi>ν</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>B</mi><mi>U</mi><mo stretchy="false">)</mo><mo>×</mo><mtext>d</mtext><mi>B</mi><mi>U</mi><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where N HM is the initial heavy metal atomic density; <math id="M2"><mrow><mover accent="true"><mi>ν</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>B</mi><mi>U</mi><mo stretchy="false">)</mo></mrow></math> is the average number of neutrons produced per fission; P NL is the neutron nonleakage probability and P CR is the fraction of neutrons not lost in the reactivity control rods. The value of P NL is deduced from the full core Mont-Carlo calculation. It is unnecessary to have control rods for the adjustment of burnup reactivity at equilibrium if the effective multiplication factor of the B & B reactor is equal to unity, namely, P CR also equals unity. The infinite multiplication factor of the core at a level of burnup can be expressed as a ratio of produced neutron numbers from the fuel to the absorbed one by all the materials in the core [ 11 11 ]: <math display="block" id="M3"><mrow><msub><mi>k</mi><mi>∞</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mi>U</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>ν</mi><mo>×</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover></mstyle><mrow><mo>[</mo><mrow><msub><mi>φ</mi><mi>i</mi></msub><mo>×</mo><msubsup><mi>Σ</mi><mrow><mtext>f</mtext><mn>,</mn><mtext>Fuel</mtext></mrow><mi>i</mi></msubsup></mrow><mo>]</mo></mrow></mrow><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover></mstyle><mrow><mo>[</mo><mrow><msub><mi>φ</mi><mi>i</mi></msub><mo>×</mo><mrow><mo>(</mo><mrow><msubsup><mi>Σ</mi><mrow><mi>a</mi><mn>,</mn><mtext>Fuel</mtext></mrow><mi>i</mi></msubsup><mo>+</mo><msubsup><mi>Σ</mi><mrow><mi>a</mi><mn>,</mn><mtext>FP</mtext></mrow><mi>i</mi></msubsup><mo>+</mo><msubsup><mi>Σ</mi><mrow><mtext>a</mtext><mn>,</mn><mtext>Coolant</mtext></mrow><mi>i</mi></msubsup><mo>+</mo><msubsup><mi>Σ</mi><mrow><mi>a</mi><mn>,</mn><mtext>Struct</mtext></mrow><mi>i</mi></msubsup></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mrow></mfrac><mn>,</mn></mrow></math> where φ i is the neutron flux of the fuel zone i ; <math id="M4"><mrow><msubsup><mo>∑</mo><mrow><mtext>f,Fuel</mtext></mrow><mi>i</mi></msubsup></mrow></math> , <math id="M5"><mrow><msubsup><mo>∑</mo><mrow><mtext>a,Fuel</mtext></mrow><mi>i</mi></msubsup></mrow></math> , <math id="M6"><mrow><msubsup><mo>∑</mo><mrow><mtext>a,FP</mtext></mrow><mi>i</mi></msubsup></mrow></math> , <math id="M7"><mrow><msubsup><mo>∑</mo><mrow><mtext>a,Coolant</mtext></mrow><mi>i</mi></msubsup></mrow></math> , and <math id="M8"><mrow><msubsup><mo>∑</mo><mrow><mtext>a,Struct</mtext></mrow><mi>i</mi></msubsup></mrow></math> are the effective one-group macroscopic fission for fuel and absorption cross sections for fuel, fission products, coolant, and structural material, respectively. It can be seen that the (n, 2n)&(n, 3n) reactions are neglected in the above neutron balance model. Assuming that the two reactions additively contribute to the neutron excess, the modified neutron excess and the infinite multiplication factor can be expressed as: <math display="block" id="M9"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>Δ</mi><mi>N</mi><mi>E</mi><mo>=</mo></mrow></mtd><mtd columnalign="left"><mrow><msub><mi>N</mi><mrow><mi>H</mi><mi>M</mi></mrow></msub><mo>×</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><mi>B</mi><mi>U</mi></mrow></msubsup></mrow></mstyle><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><mrow><mi>k</mi><msub><mo>'</mo><mi>∞</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mi>U</mi><mo stretchy="false">)</mo><mo>×</mo><msub><mi>P</mi><mrow><mtext>NL</mtext></mrow></msub><mo stretchy="false">(</mo><mi>B</mi><mi>U</mi><mo stretchy="false">)</mo><mo>×</mo><msub><mi>P</mi><mrow><mtext>CR</mtext></mrow></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>×</mo><mrow><mo>(</mo><mrow><mover accent="true"><mi>ν</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>B</mi><mi>U</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mn>2</mn><mo>×</mo><mover accent="true"><mrow><msub><mi>Σ</mi><mrow><mtext>n2n</mtext></mrow></msub></mrow><mo stretchy="true">¯</mo></mover><mo>+</mo><mn>3</mn><mo>×</mo><mover accent="true"><mrow><msub><mi>Σ</mi><mrow><mtext>n3n</mtext></mrow></msub></mrow><mo stretchy="true">¯</mo></mover></mrow><mrow><mover accent="true"><mrow><msub><mi>Σ</mi><mtext>f</mtext></msub></mrow><mo stretchy="true">¯</mo></mover></mrow></mfrac></mrow><mo>)</mo></mrow><mo>×</mo><mtext>d</mtext><mi>B</mi><mi>U</mi><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> <math display="block" id="M10"><mrow><mi>k</mi><msub><mo>'</mo><mi>∞</mi></msub><mo>=</mo><msub><mi>k</mi><mi>∞</mi></msub><mo>+</mo><mfrac><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover></mstyle><mrow><mo>[</mo><mrow><mn>2</mn><mo>×</mo><msub><mi>φ</mi><mi>i</mi></msub><msubsup><mi>Σ</mi><mrow><mtext>n2n</mtext></mrow><mi>i</mi></msubsup><mo>+</mo><mn>3</mn><mo>×</mo><msub><mi>φ</mi><mi>i</mi></msub><msubsup><mi>Σ</mi><mrow><mtext>n3n</mtext></mrow><mi>i</mi></msubsup></mrow><mo>]</mo></mrow></mrow><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover></mstyle><mrow><mo>[</mo><mrow><msub><mi>φ</mi><mi>i</mi></msub><mo>×</mo><mrow><mo>(</mo><mrow><msubsup><mi>Σ</mi><mrow><mtext>a,Fuel</mtext></mrow><mi>i</mi></msubsup><mo>+</mo><msubsup><mi>Σ</mi><mrow><mtext>a,FP</mtext></mrow><mi>i</mi></msubsup><mo>+</mo><msubsup><mi>Σ</mi><mrow><mtext>a,Coolant</mtext></mrow><mi>i</mi></msubsup><mo>+</mo><msubsup><mi>Σ</mi><mrow><mtext>a,Struct</mtext></mrow><mi>i</mi></msubsup></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mrow></mfrac><mn>,</mn></mrow></math> where <math id="M11"><mrow><mover accent="true"><mrow><msub><mo>∑</mo><mrow><mtext>n2n</mtext></mrow></msub></mrow><mo stretchy="true">¯</mo></mover></mrow></math> , <math id="M12"><mrow><mover accent="true"><mrow><msub><mo>∑</mo><mrow><mtext>n3n</mtext></mrow></msub></mrow><mo stretchy="true">¯</mo></mover></mrow></math> , and <math id="M13"><mrow><mover accent="true"><mrow><msub><mo>∑</mo><mtext>f</mtext></msub></mrow><mo stretchy="true">¯</mo></mover></mrow></math> are the average effective one-group macroscopic cross sections of (n, 2n), (n, 3n) and fission reactions, respectively. Comparing Eq. (3) with Eq. (1), and Eq. (4) with Eq. (2), one can find that the neutron excess and the infinite multiplication factor considering the (n, 2n) and (n, 3n) reactions would become greater. It perhaps plays an important role in the nuclear fuel B & B mode, especially for the Th-U fuel cycle since the average effective fission neutron number of 233 U is smaller than that of 239 Pu in a fast neutron reactor. As a neutron absorber, fresh fertile fuel absorbs more neutrons than it produces at BOL, and k ’ (BU)× P NL × P CR < 1 makes its neutron excess negative (see Eq. (3)). When the fertile fuel builds up enough fissile elements to become a neutron producer, its k ’ (BU)× P NL × P CR reaches unity, which implies that the net neutron excess equals zero, namely the fuel has given back as many neutrons as it has absorbed. Thus, the minimum burnup required to sustain the B & B mode of the fuel is satisfied. As the fission products accumulate, the fuel absorbs neutrons at a higher rate than it produces neutrons. Therefore the fuel becomes an absorber of neutrons again. In other words, when the neutron excess crosses the zero line for the second time, the maximum achievable BU is obtained, at which the k eff of the core is equal to unity [ 11 11 ]. 3 Zero-dimensional B&B core 1 The influences of (n, 2n) & (n, 3n) are investigated by performing the neutron balance calculation in the B & B core with thorium- and uranium- based fuels, respectively, for comparison. A higher fuel volume fraction can lower the minimum required BU and increase the maximum attainable BU, which is assumed to be 65% in this work, near the maximum value in a Triangular lattice adopted by Nagata et al. [ 16 16 ]. The fuel cell and pin parameters are also taken from Ref. [ 16 16 ]. The core radius is set to be 250 cm to obtain a lower neutron leakage probability. The other geometrical parameters are taken from a CANDLE design [ 17 17 ]. This work is focused on the neutron performances of the reference core without considering the thermal-hydraulics limits. All the important parameters required for simulation are listed in Table 1 Table 1 . 2 1 Parameters 1 1 Values 2 1 Thermal power (MWth) 1 1 3000 2 1 Core height (cm) 1 1 800 2 1 Core radius (cm) 1 1 250 1 1   1 1 Graphite reflector thickness 1 1   1 1   1 1 Top (cm) 1 1 100 1 1   1 1 Side (cm) 1 1 100 1 1   1 1 Bottom (cm) 1 1 100 2 1 Fuel cell parameters 1 1   1 1   1 1 Fuel/coolant volume fractions (%) 1 1 65/23.8 1 1   1 1 Fuel pin o.d. (mm) 1 1 13.2 1 1   1 1 Fuel pin P/D ratio 1 1 1.091 1 1   1 1 Cladding thickness (mm) 1 1 0.5 The B & B core uses thorium or depleted uranium as feed fuel. The thorium fuel is made of pure 232 Th metal, since its crystal structure is stable [ 11 11 ], while the depleted uranium fuel is assumed to be a U-Zr binary metallic fuel. The initial density is 11.65 g/cm 3 for thorium fuel and 15.85 g/cm 3 for uranium fuel [ 11 11 ]. The cladding material is composed of HT-9 alloy [ 12 12 ]. Sodium is chosen as a coolant, which has been widely used in fast reactor designs due to its relatively low vapor pressure during operation. The B & B core is studied using a 0-D core model, which is a homogenized unit cell with a reflective boundary. The unit cell consists of fuel, coolant, and cladding material with their respective fraction or geometrical size, as listed in Table 1 Table 1 . The minimum required BU obtained using the 0-D simple model is approximately 4% higher than the result for the full core [ 11 11 ]. The 0-D fuel cell is exposed under a constant power density of 20 W/cm 3 , which is near the average value in the full core. The burnup analysis is performd by MCNPX 2.6 [ 18 18 ] with the ENDF/B-VII library [ 19 19 ]. 4 Results and discussion 1 4.1 Neutron balance of thorium- and uranium-based fuel core 2 The neutron excess quantity for the reference B & B core fueled with uranium and thorium is calculated by Eq. (1) and Eq. (3), respectively, and shown in Fig. 1 Fig. 1 . For the uranium-based fuel core, it is found that the minimum required BU and the maximum achievable BU without considering (n, 2n) & (n, 3n) reactions are predicted to be about 15.9% FIMA and 68.1% FIMA, respectively. They can be decreased by 4.4% and increased by 1.3%, respectively, if the above two reactions are included in the calculation. Therefore, the (n, 2n) & (n, 3n) reactions bring a very limited influence on the neutron excess of uranium-based fuel. The minimum amount of neutrons required for converting the depleted uranium to a net neutron producer is about 1.26×10 -3  mol/cm 3 , which corresponds to the minimum value in the neutron balance plot around 5% FIMA when considering (n, 2n) & (n, 3n) reactions, as shown in Fig. 1(a) Fig. 1(a) . When the fertile fuel provides enough excess neutrons to make up its debt, the neutron excess increases with burnup and the minimum BU required to sustain the B & B mode in the reference sodium cooled core is about 15.2% FIMA (the first cross point on the zero line). The uranium fertile fuel can maintain its k ’ (BU)× P NL × P CR > 1 up to 47.1% FIMA, and it can provide about a 3.61×10 -3  mol/cm 3 net neutron excess. In addition, it is sufficient to extend the fuel BU up to 69.0%, the maximum achievable BU, at which the equilibrium k eff value is equal to unity. If the remaining fissile fuel in the reference core at the minimum required BU is discharged, it is capable of starting more than 2 new B & B cores. For the thorium-based fuel core, it shows that about 1.55×10 -3  mol/cm 3 neutrons should be provided from the igniting fuel to convert the fertile fuel into the driver fuel if the (n, 2n) & (n, 3n) reactions are considered. The minimum required BU is about 36.5% FIMA. The feed fuel can maintain the B & B mode of operation up to the maximum achievable BU, ∼ 51% FIMA. If (n, 2n) & (n, 3n) reactions are ignored, however, it needs about 1.88×10 -3  mol/cm 3 net neutrons to convert the thorium fuel from a neutron absorber to a neutron producer, which is about 0.28×10 -3  mol/cm 3 higher than the neutron excess number of the neutron producer. It implies that ignoring the (n, 2n) & (n, 3n) reactions cannot maintain the B & B mode for thorium-based fuel, even in a fast reactor, which is significantly different from uranium-based fuel. To better understand the above distinct difference between uranium- and thorium-based fuels, we present the neutron spectra, fission cross section, and (n, 2n) & (n, 3n) cross sections [ 19 19 ] for the two fuels in Fig. 2 Fig. 2 . Based on the distinct differences in neutron excess for the two fuels from BOL, one learns that 238 U or 232 Th play a predominant role in the neutron balance evolution as a starter fuel in the core. The difference in the results with and without considering the (n, 2n) & (n, 3n) reactions for the sodium cooled thorium core is 0.31×10 -3  mol/cm 3 , which is about 4 times greater than that for the sodium cooled uranium core. This is because the fission cross section of 232 Th is smaller than 238 U, as shown in Fig. 2(a) Fig. 2(a) , which implies the term 1/ <math id="M14"><mrow><mover accent="true"><mrow><msub><mo>∑</mo><mtext>f</mtext></msub></mrow><mo stretchy="true">¯</mo></mover></mrow></math> in Eq. (3) for thorium fuel is higher than for uranium fuel, while the (n, 2n) cross sections of 232 Th is slightly greater than that of 238 U. The (n, 3n) cross section also displays this trend, but its effect is very weak due to its smaller cross section and higher energy threshold. The difference between the results with and without considering (n, 2n) & (n, 3n) reactions for the thorium-based core is greater than that of the uranium core, but the former displays a slower increase than the latter with BU. The reason is that, with the breeding of 233 U from 232 Th or of 239 Pu from 238 U, the fission of fissile fuel plays a more important role gradually. The mean fission cross section of 233 U is higher than that of 239 Pu, as shown in Fig. 2(b) Fig. 2(b) , which implies that the term 1/ <math id="M15"><mrow><mover accent="true"><mrow><msub><mo>∑</mo><mtext>f</mtext></msub></mrow><mo stretchy="true">¯</mo></mover></mrow></math> in Eq. (3) for thorium fuel is smaller than that for uranium fuel. The (n, 2n) & (n, 3n) reaction effects of 233 U and 239 Pu can be neglected due to their significantly smaller cross sections compared to those of fission reactions. 4.2 Breed and burn feasibility for alternative coolants 2 With the exception of the sodium coolant, it is interesting to analyze the neutron balance of thorium-based fuel in the reference reactor with other coolants, including lead-bismuth eutectic (LBE), helium, and FLiBe, since they have been adopted in various advanced reactor designs. The LBE coolant can eliminate the need for a secondary coolant loop, unlike sodium [ 12 12 , 20 20 ]. Helium can efficiently improve the neutron economy due to its excellent characters of low neutron capture and slowing-down [ 11 11 ]. The FLiBe molten salt features a smaller neutron leakage probability. To investigate an upper possible bound of a thorium-based fuel B & B core, the sodium coolant is replaced by lead-bismuth coolant, helium coolant, and FLiBe coolant, respectively, using the same volume fractions and geometry parameters as those for the sodium cooled core. The neutron excess quantity for the reference thorium-based B & B core cooled by LBE, helium, and FLiBe is calculated and shown in Fig. 3(a) Fig. 3(a) , Fig. 3(b) Fig. 3(b) , and Fig. 3(c) Fig. 3(c) , respectively. The results with (n, 2n) & (n, 3n) consideration are presented in Fig. 3(d) Fig. 3(d) for comparison. It can be seen that the neutron balance of the helium cooled core features the best neutron economy. The minimum required BU and the theoretical maximum attainable BU for the helium cooled core are about 32.1% FIMA and 55.6% FIMA, respectively, corresponding to the smallest and greatest value for the reference core with the above three coolants. This is because the helium coolant has a significantly lower atomic density and can hardly interact with neutrons, which leads to a harder spectrum, as shown in Fig. 4 Fig. 4 . Consequently, the harder spectrum of the helium cooled core would play an important role in the neutron balance evolution. The LBE cooled core features a harder neutron spectrum than the sodium cooled core due to the significantly heavier atomic mass and the higher inelastic scattering threshold of LBE, which is helpful for the B & B operation. However, the neutron leakage probability with LBE coolant is slightly larger than sodium coolant, implying the neutron nonleakage probability, P NL , in Eq. (3) for the LBE cooled core is smaller than the sodium cooled core. The greater parasitic neutron capture cross section for LBE coolant also worsens the neutron economy. As a consequence, the minimum required BU of an LBE cooled core is ∼ 38.5% FIMA, which is higher than a sodium cooled core, ∼ 36.3% FIMA. The theoretical maximum attainable BU of an LBE cooled core is ∼ 48.3% FIMA, also smaller than the sodium cooled core, ∼ 51.1% FIMA. One can find that the FLiBe cooled reference core cannot sustain the B & B mode. This is because the reference core with FLiBe coolant has a significantly softer spectrum, which results in a stronger minus neutron excess from the parasitic neutrons captured by fuel and FLiBe than that of the core with the other three coolants. The number of excess neutrons that can be generated by the fuel in the FLiBe cooled core is about 1.01×10 -3  mol/cm 3 , which is not sufficient to make up the 2.00×10 -3  mol/cm 3 neutrons required for B & B. It is also found that the influence of the (n, 2n) & (n, 3n) reactions for the FLiBe cooled core is the greatest. This is because the (n, 2n) & (n, 3n) cross sections of FLiBe coolant are higher than those of other coolants, especially the (n, 2n) reaction of 9 Be isotope that makes a predominant contribution. The difference for the neutron excesses between with and without (n, 2n) & (n, 3n) reactions for the lead-bismuth cooled core is somewhat higher than that for helium cooled core due to the obvious (n, 2n) & (n, 3n) effects of 209 Bi vs the negligible (n, 2n) & (n, 3n) cross sections of helium. The evolutions of the 233 U mass for the reference core with the four coolants are presented in Fig. 5 Fig. 5 for comparison. When the feed fuel becomes a net neutron producer corresponding to the minimum value in Fig. 3(d) Fig. 3(d) around 5%-10% FIMA, the mass of 233 U per cubic meter is about 473.6, 488.3, 497.8 and 602.3 kg for the reference helium, lead-bismuth, sodium, and FLiBe cooled cores, respectively. The weight of fissile material 233 U is nearly the same for helium, lead-bismuth, and sodium cooled cores due to the similar spectrum and parasitic neutron capture, however, harder spectrum and lower parasitic neutron capture make a smaller initial mass of fissile isotopes required for establishing the criticality. The critical mass for the FLiBe cooled core is ∼ 23% higher than that for other three cores. 5 Conclusion 1 Based on a 0-D fuel cell model, we have performed the neutron balance calculation for thorium-based fuel in a fast reactor cooled by sodium by considering the (n, 2n)&(n, 3n) reactions to analyze the feasibility of B&B mode. The analysis for uranium-based fuel in the same reactor was also performed for comparison. It was found that the (n, 2n)&(n, 3n) reaction rates should be considered to achieve the B&B operation for the thorium-based fuel core. Then, we analyzed the B&B mode in the same reference core, but with helium, LBE, and FLiBe coolants, respectively. 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(Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2016 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-016-0096-4 http://dx.doi.org/10.1007/s41365-016-0096-4 . 2016 Nuclear Science and Techniques 04 27