2.1 Neutron slowing-down equation

For an isolated fuel rod, the neutron slowing-down equation is:

where, α=[(a -1)/(a+1)] 2, a is mass number of the nuclide; ∑_t,f(E) is total cross-section in fuel region; ∑_s,f(E) and ∑_s,m(E) are scattering cross-section in fuel and moderator region, respectively; ϕ_f(E) and ϕ_m(E) are flux in fuel and moderator region, respectively; P_ff(E) is first-flight collision probability in the fuel region; P_m0(E) is first-flight escape probability in the moderator region; V_f is volume of fuel region and V_m is volume of moderator region.

In moderator region, assuming that the 1/E spectrum is suitable and ∑_s,f(E) is constant in the integral, the second term on the right part in Eq. (1) can be simplified. And assuming that the absorption cross-section in moderator is negligible, then total cross-section is equal to the scattering cross-section. Substituting the reciprocity relationship of Eq. (2) into Eq. (1) and using some approximation, the neutron spectrum can be solved as Eq. (3).

where λ is the Goldstein-Cohen factor, with λ =1, λ =0 and 0<λ <1 being the narrow, wide and intermediate resonance approximation, respectively.

If the P_ff(E) in isolated fuel rod is replaced with the first-flight collision probability in a lattice model, P_FF(E), the neutron spectrum in the lattice model can be written as:

2.2 Stamm’ler’s method for evaluating the effective cross-section

In an infinite lattice system, according to Stamm’ler’s method, the P_FF(E) and P_ff(E) can be correlated as:

where, $A=S_{\text{b}}{\gamma }_{\text{b}}^0/(S_{\text{f}}{t}_{\text{fb}}^2)$, S_b is surface area of cell boundary, S_f is surface area of fuel lump, ${\gamma }_{\text{b}}^0$ is blackness of the cell at _t,f = 0, and t_fb is transmission probability that neutrons leaving a fuel surface with cosine angular distribution will reach the cell boundary; x=∑_t,f/∑_e, ∑_e=S_f/(4V_f), V_f is volume of fuel lump.

The expression of P_ff in Carlvik’s two-term rational approximation is as follows:

Substituting Eq. (6) into Eq. (5), we have the expression of P_FF with two-term rational approximation:

where, α_1,2 = (5A+6) (A² +36A +36)/[2(A+1)], β₁=[(4A +6)/(A+1) - α₁]/(α₂ - α₁), and β₂= 1 -β₁.

Substituting Eq. (7) into Eq. (4), and assuming that (1 -λ)_s,F is far smaller than _t,F, we have:

Equation (9) defines effective cross-section of a single resonance nuclide in group g. Substitute Eq. (8) into Eq. (9) and assumeing λ _rs,f(E) is far smaller than _α,f(E), too, the effective cross-section can be defined by Eq. (10):

where, RI_n,x = βnFx(_bn), _bn = λσ_p.f +αnσ_e, $F_x({\sigma }_{\text{b}n})={\displaystyle {\int }_{\Delta E_g}}\frac{{\sigma }_{\text{b}n}{\sigma }_{\text{x,f}}(E)}{{\sigma }_{\alpha \mathrm{,}\text{f}}(E)+{\sigma }_{\text{b}n}}g\frac{\text{d}E}{E}$, and $\Delta u_g={\displaystyle {\int }_{\Delta E_g}}\frac{\text{d}E}{E}$.

In Eq. (10), Δu is usually adjusted to unity by the library. The resonance integral Fx(_bn) will be determined by the interpolation in the resonance integral table which will be supplied by the nuclear data library. Therefore, to evaluate the effective cross-section, parameter A in Stamm’ler’s method is calculated in advance. In the original Stamm’ler derivation, A is a characteristic constant for a given infinite lattice problem and is evaluated by using collision probability method. Instead of using the collision probability method to calculate A, we will relate A to Dancoff factor and then use Yamamoto’s neutron current method to calculate the Dancoff factor in a simple way, which is valid even for the non-lattice assembly geometry.

2.3 The neutron current method for calculating Dancoff factor

According to Eq. (4), the total reaction rate can be written as:

Taking the black-limit of very large total cross-section, the limit of Eq. (11) can be expressed as:

In the black-limit Wigner’s one-term rational approximation to P_FF is valid:

Substituting Eq. (12) into Eq. (13), we have:

If the total cross-section is large enough, the total reaction rate is very close to Eq. (14). So when the total reaction rate in the resonance region is obtained by Yamamoto’s fixed source transport calculation in the limit of large total cross-section, the Dancoff factor can be calculated by:

In using the Dancoff method for effective resonance cross-section calculation, the Dancoff factor from Eq. (15) is applied directly to the equivalent escape cross-section of an isolated rod to calculate the background cross-section for resonance integral table lookup. Instead, we propose to apply the Dancoff factor from Eq. (15) to the Stamm’ler method equations.

2.4 Application of Dancoff factor to Stamm’ler method

An alternative equivalent definition of Dancoff factor is given as [2, 3]:

Also note that the first flight collision probability satisfies the following relation:

Substituting Eqs. (5) and (17) into Eq. (16), the relationship between the Dancoff factor and the parameter A in Stamm’ler method is obtained:

Therefore the neutron current method can be used to calculate D and thus obtain A without having to use the collision probabilities in Eq. (5). Furthermore the neutron current method is applicable to the general non-lattice assembly geometry. Once A is known, the two-term coefficients in Eq. (7) can be calculated. The Stamm’ler method can then be used without any change to calculate the effective cross-section by Eq. (10).

2.5 Simplification of the resonance interference factor table

Traditionally the background cross-section iteration method is used for the resonance interference correction, but the iteration method is not yet effective. A method using simplified resonance interference factor (RIF) table is given below.

In a typical LWR core, the number density of ²³⁸U is usually much larger than that of other resonance nuclides. The spectrum in the resonance region is dominated by resonance absorption from ²³⁸U. To simplify the resonance interference treatment, only the interference effects between ²³⁸U and other resonance nuclides are considered. To correct the resonance overlap (interference) effects, the resonance interference factors (RIFs) prepared with Eq. (19) are used.

where, ${\sigma }_{x\mathrm{,}i}^{\text{ef}\text{f}}({\sigma }_{\text{b}\mathrm{,}i}\mathrm{,0})$ is effective cross-section for nuclide i and reaction type x, at zero number density of ²³⁸U, and background cross-section _b,i; ${\sigma }_{x\mathrm{,}i}^{\text{ef}\text{f}}({\sigma }_{\text{b}\mathrm{,}i}\mathrm{,}{N}_i/N_{238})$ is effective cross-section for nuclide i and reaction type x, at a ratio of number density of Ni/N₂₃₈, and background cross-section _b,i; $RI{F}_x^{238\to i}({\sigma }_{\text{b}\mathrm{,}i}\mathrm{,}{N}_i/N_{238})$ is resonance interference factor from ²³⁸U to nuclide i and reaction type x, at number density ratio Ni/N₂₃₈, and background cross-section _b,i.

Corresponding to the overlap effects from ²³⁸U to other resonance nuclides, the interference effects from other resonance nuclides to ²³⁸U at the same number density ratio (Ni/N₂₃₈) are calculated by Eq. (20):

Taking ²³⁸U as an example, the whole process of evaluating a resonance interference factor is described below.

For ${\sigma }_{x\mathrm{,238}}^{\text{eff}}({\sigma }_{\text{b}\mathrm{,238}}\mathrm{,0})$, a calculation through solving the slowing-down equation is done by RMET21 [9], in which only ²³⁸U and hydrogen are present. The background cross-section is defined as:

The number density of ²³⁸U will be set to unity for convenience. To get different background cross-sections, we just need to adjust the number density of Hydrogen.

For ${\sigma }_{x\mathrm{,}i}^{\text{eff}}({\sigma }_{\text{b}\mathrm{,}i}\mathrm{,}{N}_i/N_{238})$, the calculation is done by RMET21 in presence of ²³⁸U, hydrogen and nuclide i. The background cross-section is defined as:

where, _b,238 and ${{\sigma }^{\prime }}_{\text{b}\mathrm{,238}}$ are effective cross-section for nuclide i, group g, and resonance reaction type x, without and with resonance overlap correction, respectively. In Eq. (22), the value of Ni/N₂₃₈ will cover the possible range in a typical LWR. To make ${{\sigma }^{\prime }}_{\text{b}\mathrm{,238}}$ equal _b,238, we just need to adjust the number density of hydrogen.

When the effective cross-sections with and without the interference effect is obtained, RIF from resonance nuclide i to ²³⁸U is calculated by Eq. (20). For a typical lattice, the flux spectrum varies with the background cross-section, caused by different number densities of ²³⁸U or different shielding conditions. So, a series of calculations with different background cross-sections and different number density ratios are carried out.

For resonance nuclides other than ²³⁸U, the effective cross-section for a specific resonance nuclide without the resonance interference correction is calculated by interpolating the nuclear data library, before interpolating the RIFs table to obtain RIF. Finally the RIF is multiplied to the uncorrected effective cross-section as:

where, $RI{F}_{x\mathrm{,}g}^{238\to i}$ is resonance interference factor for nuclide i, group g, and resonance reaction type x.

For ²³⁸U, the resonance interference effects from other resonance nuclides to ²³⁸U are counted into the correction as:

where ${\prod }_{i^{\prime }}RI{F}_{x\mathrm{,}g}^{i^{\prime }\to 238}$ is cumulative product of resonance interference factors from all resonance nuclides to ²³⁸U.

2.6 Calculation procedure

The calculation procedures for implementing the improvement are described as follows.

i) Prepare table of RIFs as described in Sect. 2.5.

ii) Perform fixed source transport calculation for each resonance group. The macroscopic total cross-section in resonance region is set to 100000 (or any number that is large enough), and set to λ_p for regions without resonance nuclide. The neutron source is set to λ_p for all regions. The transport equation, Eq. (25), is solved with the MOC (Method of Characteristics):

where ∑_t,g=10⁵ in resonance region, and _t,g= λ_p,g in non-resonance region. With the solution for the fixed source transport solution, the total reaction rate for each resonance region is calculated.

iii) Dancoff factor for each resonance region in each resonance group is calculated through Eq. (15). And parameter A is calculated through Eq. (18).

iv) In resonance regions and for resonance nuclides, calculate the effective cross-section through Eq. (10).

v) For resonance nuclide other than ²³⁸U the effective cross-section is corrected by Eq. (23). For ²³⁸U, the effective cross-section is corrected by Eq. (24).

3.1 Infinite lattice cases

An infinite lattice of hexagonal fuel cells, as shown in Fig. 1, was used in the lattice calculation. Each side of the hexagonal cell is 7.83 mm. In the cell, the fuel rod is of 7.862-mm diameter, the clad made of Zr-Nat is 0.649 mm thick, and the moderator is light water at 300 K in both resonance and non-resonance regions.

Fig. 1.Full-screen View

(Color online) Cell geometry for infinite lattice calculation.

3.1.1 Infinite lattice case with two resonance absorbers

Let the fuel rod contain only UO₂, with ²³⁵U enrichment of 0.8%–3.6% in steps of 0.2%. Figure 2 shows errors of the eigenvalue as function of the ²³⁵U enrichment. One sees that Method A differs little from Method C. This indicates that our goal to find an alternative way to use Stamm’ler equations without possibility calculation is achieved. As Methods A and C have almost the same result, the following discussions will focus on Methods A and B.

Fig. 2.Full-screen View

(Color online) Error to reference eigenvalue as function of ²³⁵U enrichment for the two absorbers case.

There is a significant improvement in eigenvalue for Method B over Method A. This shows that the RIF table works much better than the background cross-section iteration in capturing resonance interference. The error of using Method B is around 100 pcm, which is quite acceptable in practice.

The verification results of 1.8% enrichment, as a typical case named as Case-A for the convenience of discussion, are given in Table 2. Table 3 shows the nuclide composition in Case-A. The reference eigenvalue at 1.8% enrichment is 1.22667± 0.00009 and the errors to the reference eigenvalue are -290 and -115 pcm (Fig. 2), for Methods A and B, respectively. From the microscopic cross-section for absorption and neutron production by each resonance nuclide (Table 2), one sees that the interference effect is mostly on ²³⁵U, not on ²³⁸U, as the Method B results are almost the same as the Method A results for ²³⁸U. This is expected because of the large ²³⁸U number density versus the much smaller number density of ²³⁵U.

TABLE 2.Full-screen View

Micro cross-section and relative error in absorption and neutron production for ²³⁵U of 1.8% inrichment and ²³⁸U in Case-A

Energy groups	Absorption of 235U			Neutron production of 235U			Absorption of 238U
	∑a	δ (%)b		∑a	δ (%)b		∑a	δ (%)b
		Method A	Method B		Method A	Method B		Method A	Method B
15	4.4904	0.02	0.02	7.9300	0.05	0.04	0.69442	-0.19	-0.17
16	5.6824	-0.12	-0.10	10.420	-0.12	-0.09	0.73277	-0.81	-0.78
17	6.9880	-0.23	-0.17	12.586	-0.22	-0.22	0.94726	-1.65	-1.60
18	9.5619	-0.04	0.17	14.984	-0.34	0.11	0.87767	-1.62	-1.56
19	12.184	-0.69	-0.11	19.408	-1.07	-0.08	1.1712	-1.75	-1.71
20	16.946	-0.2	0.25	28.906	-0.13	0.43	1.3449	-2.29	-2.26
21	27.897	-0.36	-1.07	45.743	-1.06	-0.76	1.5361	-0.03	0.05
22	34.768	-0.37	-0.18	53.527	-0.12	-0.51	2.4168	0.06	0.08
23	58.251	-2.36	0.55	99.149	-2.75	0.58	2.1192	-0.71	0.12
24	69.201	-3.37	-0.01	102.82	-2.80	-0.05	4.6044	2.25	2.27
25	74.991	6.58	-1.15	114.81	2.23	-0.56	6.2095	0.40	0.36
26	74.892	-1.03	-0.73	88.854	-0.75	-0.53	0.47409	-0.54	-0.27
27	67.381	-1.04	-1.46	98.015	-8.06	0.25	7.8266	0.79	0.42

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^a cross-section of reference;

^b relative error;

TABLE 3.Full-screen View

Material and number density in Case-A

Materials	Fuel rod			Clad	Moderator
	235U	238U	16O	Zr-Nat	H	16O
Number density (1024/cm3)	4.21236E-04	2.26905E-02	4.62234E-02	3.89087E-02	4.42326E-02	2.21163E-02

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3.1.2 Infinite lattice case with multiple resonance absorbers

As the fuel burns, different resonance nuclides build up. A fresh UO₂ fuel with 3.1% enrichment of ²³⁵U is taken as an example to verify the proposed method at different burnup steps. Figure 3 shows the error in eigenvalue versus burnup of 10–40 GW d/t using Method A or Method B to model the case. It can be seen that the RIF table captures the interference effect much better than the background cross-section iteration method, and the Method B results are quite acceptable.

Fig. 3.Full-screen View

(Color online) Error in eigenvalue versus burnup.

The verification results is given in Tables 4 and 5 for 30 GW d/t burnup, as a particular case named as Case-B for the convenience of discussion. Table 4 shows the material and number density of nuclides, and Table 5 shows the effective cross-sections for major absorbers (²³⁵U, ²³⁸U and ²³⁸Pu). Again, Method B gives better performance than Method A in most cases for all nuclides except for ²³⁸U. ²³⁸U is almost not affected by resonance interference due to its very large number density. Therefore the ²³⁸U effective cross-section is not sensitive to using Method A or Method B. These results are consistent with those in Sec. 3.1.1.

TABLE 4.Full-screen View

Material and number density in Case-B at Burnup depth of 30 GW d/t

Materials	Fuel rod								Clad	Moderator
	235U	236U	238U	239Pu	240Pu	241Pu	242Pu	16O	Zr-Nat	H	16O
Na	2.115d-2	8.449d-3	2.166	1.505d-2	4.441d-3	2.833d-3	7.811d-4	4.581	3.891	4.423	2.212

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^a Number density in 10²²/cm³.

TABLE 5.Full-screen View

Micro cross-section and relative error in absorption and neutron production for U and Pu in Case-B at burnup of 30 GW d/t

Energy groups	Absorption of 235U			Neutron production of 235U			Absorption of 236U
	∑a	δ (%)b		∑a	δ (%)b		∑a	δ (%)b
		Method A	Method B		Method A	Method B		Method A	Method B
15	4.4903	0.04	0.03	7.9297	0.06	0.05	1.0488	-1.51	-1.53
16	5.6829	-0.11	-0.09	10.421	-0.12	-0.09	1.2468	-1.65	-1.66
17	6.9896	-0.22	-0.16	12.588	-0.21	-0.21	1.5081	-1.75	-1.77
18	9.5685	-0.07	0.13	14.996	-0.37	0.06	1.9279	-2.88	-2.39
19	12.188	-0.65	-0.08	19.419	-1.06	-0.07	2.7944	-2.97	-1.95
20	16.959	-0.13	0.30	28.924	-0.06	0.49	5.0874	-4.49	-2.60
21	27.983	-0.28	-1.03	45.859	-0.97	-0.71	6.0751	-5.57	-9.20
22	34.903	-0.30	-0.17	53.623	0.05	-0.39	17.340	-7.43	-3.49
23	59.000	-2.68	0.11	100.37	-3.04	0.15	17.684	-11. 6	-5.61
24	70.387	-3.19	0.06	104.40	-2.49	0.16	38.844	-14.2	-5.53
25	77.462	5.97	-1.77	118.40	1.62	-1.22	0.0511	-2.98	-1.28
26	78.153	-2.13	-1.86	91.932	-1.95	-1.77	0.1365	-1.10	-1.46
27	70.004	-1.65	-1.81	103.76	-10.13	-1.64	208.25	-4.97	-2.09
Energy groups	Absorption of 239Pu			Neutron production of 239Pu			Absorption of 238U
	∑	δ (%)		∑	δ (%)		∑	δ (%)
		Method A	Method B		Method A	Method B		Method A	Method B
15	3.6384	-0.01	-0.02	6.1712	-0.04	-0.04	69.268	-0.23	-0.20
16	4.5680	0.08	0.04	7.1936	0.11	0.08	0.7303	-0.87	-0.83
17	6.1122	-0.18	0.11	9.8722	0.14	0.41	0.9418	-1.70	-1.61
18	7.2395	-0.01	-0.46	10.503	0.55	-0.56	0.8698	-1.61	-1.53
19	10.240	3.16	-0.51	17.738	3.53	-0.59	1.1610	-1.82	-1.68
20	14.618	-2.45	0.34	24.590	-1.91	0.66	1.3305	-2.41	-2.19
21	29.322	-1.82	-3.08	44.010	-1.14	-1.64	1.5170	-0.02	0.06
22	49.759	-1.95	-1.14	85.083	-2.36	-1.34	2.3971	-0.26	0.18
23	100.02	4.41	-2.72	177.93	5.57	-2.95	2.0807	0.10	1.15
24	37.159	-9.15	0.90	29.986	-8.77	0.31	4.5449	2.19	2.71
25	79.071	-8.77	-0.89	122.97	-8.99	-1.21	6.1062	0.60	0.76
26	217.05	-1.83	-1.56	396.23	-2.03	-1.83	0.4759	-0.94	-0.10
27	54.238	-14.30	-2.16	89.712	-13.35	-2.55	7.8411	-0.11	0.95

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^a cross-section of reference;

^b relative error;