CN106202868A - A kind of method of the intermediate resonance factor obtained in reactor multigroup nuclear data depositary - Google Patents

Abstract

A kind of method of the intermediate resonance factor obtained in reactor multigroup nuclear data depositary, based on free gas model, by the derivation of equation drawn resonance absorption nucleic about the Continuous Energy neutron scattering matrix expression σ at temperature TK_r,s,T(E → E '), and solve moderation of neutrons equation based on the Continuous Energy collision matrix at temperature TK and obtain the netron-flux density at temperature TK, and then by the multigroup absorption cross-section under also group calculates temperature TK of resonance absorption nucleic, calculate the intermediate resonance factor eventually through the intermediate resonance factor, and obtain the intermediate resonance factor multinomial coefficient as independent variable with temperature and background cross section by least-square fitting approach；The intermediate resonance factor after the present invention makes the intermediate resonance factor calculate is more accurate, improves computational accuracy and the computational efficiency of the intermediate resonance factor, and finally provides the accuracy and speed calculated that resonates in reactor physics calculating.

Description

Method for obtaining intermediate resonance factors in reactor multi-group nuclear database

Technical Field

The invention relates to the field of nuclear reactor multi-group nuclear databases and reactor physical computation, in particular to a method for acquiring intermediate resonance factors in a reactor multi-group nuclear database.

Background

In order to meet the requirement of high-fidelity resonance calculation of a numerical reactor, it is important to provide accurate intermediate resonance factors in a multi-group database.

Currently, for the intermediate resonance factor, an intermediate resonance factor calculation method proposed in an internationally popular evaluation nuclear database processing program NJOY (hereinafter referred to as NJOY) is widely used. With the increasing requirements for reactor resonance calculation, many models of the method in calculating the intermediate resonance factor and the applicability of the method have not been satisfactory.

The intermediate resonance factor calculation method proposed by NJOY comprises four steps:

1. appointing a temperature, a resonance absorption nuclide and a moderation nuclide, uniformly mixing the resonance absorption nuclide and the moderation nuclide, and calculating a corresponding background cross section value sigma according to the nuclear density proportion of a main resonance absorption nuclide and a main moderation nuclide_b,case1Solving the corresponding neutron moderation equation to obtain the continuous energy neutron flux density, and finally obtaining a plurality of groups of absorption cross sections of the resonance absorption nuclide through group combination calculation, wherein the group cross section of a certain group of the main resonance absorption nuclide is abbreviated as sigma at the temperature_case1,g。

2. Uniformly mixing the primary resonance absorption nuclide and the primary moderation nuclide according to the temperature specified in the step 1, wherein the nuclear density ratio of the primary resonance absorption nuclide and the primary moderation nuclide is 90% of the ratio in the step 1, and calculating the corresponding background cross section value sigma_b,case2Solving the corresponding neutron moderation equation to obtain the continuous energy neutron flux density, and finally obtaining a plurality of groups of absorption cross sections of the resonance absorption nuclide through group combination calculation, wherein the group cross section of a certain group of the main resonance absorption nuclide is abbreviated as sigma at the temperature_case2,g。

3. Assigning a new primary moderating species to replace the primary moderating species assigned in step 1, uniformly mixing the primary and new moderating species, and adjusting the primary resonance absorbing species in accordance with the temperature and primary resonance absorbing species assigned in step 1The ratio of the nuclear of the new main moderating nuclide is calculated to obtain the corresponding background section value sigma_b,case3Equal to the background cross-section value (σ) in step 2_b,case2＝σ_b,case3) Solving the corresponding neutron moderation equation to obtain the continuous energy neutron flux density, and finally obtaining the multi-group absorption cross section of the resonance absorption nuclide through group combination calculation, wherein the group cross section of a certain group of the main resonance absorption nuclide is abbreviated as sigma at the temperature_case3,g。

4. Obtaining the designated energy group g, the designated temperature T and the designated background section sigma of the new nuclide in the step 3 by the following formulas_b,spe＝σ_b,case1Lower intermediate resonance factor

${\mathrm{\λ}}_{g,T,{\mathrm{\σ}}_{b,spe}}=\frac{{\mathrm{\σ}}_{case3,g}-{\mathrm{\σ}}_{case1,g}}{{\mathrm{\σ}}_{case2,g}-{\mathrm{\σ}}_{case1,g}}---\left(1\right)$

The important concepts in each step are as follows:

1) cross section σ with respect to background_bThe calculation formula is

${\mathrm{\σ}}_b=\frac{N_m\·{\mathrm{\σ}}_{m,p}}{N_r}---\left(2\right)$

Wherein,

N_mnuclear density of moderating nuclides

σ_m,pPotential elastic scattering cross-section of a moderating nuclide

N_rNuclear density of resonance-absorbing nuclides

2) And calculating the group cross section sigma of a certain group of main resonance absorption nuclides under the temperature TK_g,TThe calculation method comprises the following steps:

${\mathrm{\σ}}_{g,T}=\frac{{\∫}_{{\mathrm{\ΔE}}_g}{\mathrm{\σ}}_{a,T}\left(E\right){\mathrm{\φ}}_T\left(E\right)dE}{{\∫}_{{\mathrm{\ΔE}}_g}{\mathrm{\φ}}_T\left(E\right)dE}---\left(3\right)$

wherein,

e-neutron incident energy

E' — neutron extraction energy

g-neutron incident energy group

g' — neutron emergent energy group

ΔE_gOf neutron incident energy groupsEnergy interval

ΔE_g′Energy separation of neutron emergent energy groups

φ_T(E) Continuous energy neutron flux density at temperature TK

σ_a,T(E) Continuous energy absorption Cross section at temperature TK

According to the steps, the method can involve solving a neutron moderation equation, while in the traditional method, when the neutron moderation equation is solved, a model describing elastic scattering of neutrons is a progressive scattering nuclear model, and the model ignores increase of resonance absorption caused by scattering on resonance of a resonance nuclide, so that the calculated fine energy spectrum is inaccurate, the multi-group cross section is further inaccurate, and finally the intermediate resonance factor is inaccurate; on the other hand, as is known from the above, the intermediate resonance factor is a function of the energy group, the background section and the temperature, and the background section and the temperature are constantly changed along with the increase of the burnup of the reactor in the full-reactor resonance calculation. And if scattering on resonance is considered, an iterative method is adopted to solve the slowing equation, so that the calculation time is also increased rapidly, and the full-stack resonance calculation is not facilitated.

Therefore, in order to solve the above problems, it is necessary to invent an accurate, feasible and fast intermediate resonance factor calculation method.

Disclosure of Invention

In order to solve the problems in the prior art, the invention aims to provide a method for obtaining an intermediate resonance factor in a reactor multi-group nuclear database.

In order to achieve the purpose, the invention adopts the following technical scheme:

a method of obtaining intermediate resonance factors in a nuclear database of a plurality of groups of reactors, comprising the steps of:

step 1: appointing a temperature, a resonance absorption nuclide and a moderation nuclide, uniformly mixing the resonance absorption nuclide and the moderation nuclide, and calculating a corresponding background cross section value sigma according to the nuclear density ratio of the resonance absorption nuclide and the moderation nuclide_b,case1Solving the corresponding neutron moderation equation to obtain the continuous energy neutron flux density, and finally obtaining a plurality of groups of absorption cross sections of the resonance absorption nuclide through group combination calculation, wherein the group cross section of a certain group of the resonance absorption nuclide is abbreviated as sigma at the temperature_case1,g；

Step 2: uniformly mixing the resonance absorption nuclide and the moderating nuclide according to the temperature, the resonance absorption nuclide and the moderating nuclide specified in the step 1, wherein the nuclear density ratio of the resonance absorption nuclide and the moderating nuclide is 95% of the ratio in the step 1, and calculating the corresponding background cross section value sigma_b,case2Solving the corresponding neutron moderation equation to obtain the continuous energy neutron flux density, and finally obtaining a plurality of groups of absorption cross sections of the resonance absorption nuclide through group combination calculation, wherein the group cross section of a certain group of the resonance absorption nuclide is abbreviated as sigma at the temperature_case2,g；

And step 3: according to the temperature and the resonance absorption nuclear species specified in the step 1, a new moderating nuclear species is specified to replace the main moderating nuclear species specified in the step 1, the resonance absorption nuclear species and the new moderating nuclear species are uniformly mixed, the ratio of the resonance absorption nuclear species to the new moderating nuclear species is adjusted, and the corresponding background cross section value sigma is calculated_b,case3Equal to the background cross-section value in step 2, i.e. sigma_b,case3＝σ_b,case2Solving the corresponding neutron moderation equation to obtain the continuous energy neutron flux density, and finally obtaining the neutron flux density through group-combining calculationMultiple group absorption cross-sections of a resonance-absorbing nuclear species, the group cross-section of a certain group of resonance-absorbing nuclear species at that temperature being abbreviated as σ_case3,g；

And 4, step 4: obtaining the designated energy group g, the designated temperature T and the designated background section sigma of the new nuclide by the formula (1)_b,spe＝σ_b,case1Lower intermediate resonance factor

${\mathrm{\λ}}_{g,T,{\mathrm{\σ}}_{b,spe}}=\frac{{\mathrm{\σ}}_{case3,g}-{\mathrm{\σ}}_{case1,g}}{{\mathrm{\σ}}_{case2,g}-{\mathrm{\σ}}_{case1,g}}---\left(1\right)$

And 5: for the background cross-section values σ in steps 1-3_bThe calculation is performed using equation (2):

${\mathrm{\σ}}_b=\frac{N_m\·{\mathrm{\σ}}_{m,p}}{N_r}---\left(2\right)$

wherein,

N_mnuclear density of moderating nuclides

σ_m,pPotential elastic scattering cross-section of a moderating nuclide

N_rNuclear density of resonance-absorbing nuclides

For the neutron moderation equation in steps 1-3, the form is as follows:

$\begin{array}{c}\[{\mathrm{\σ}}_{r,t,T}\left(E\right)\·N_r+{\mathrm{\σ}}_{m,t,T}\left(E\right)\·N_m\]{\mathrm{\φ}}_T\left(E\right)={\∫}_0^{\mathrm{\∞}}{\mathrm{\σ}}_{r,s,T}(E^{\′}\→E)\·N_r\·{\mathrm{\φ}}_T\left(E^{\′}\right){\mathrm{dE}}^{\′}\\ +{\∫}_0^{\mathrm{\∞}}{\mathrm{\σ}}_{m,s,T}(E^{\′}\→E)\·N_m\·{\mathrm{\φ}}_T\left(E^{\′}\right){\mathrm{dE}}^{\′}\end{array}---\left(4\right)$

wherein,

e-neutron incident energy

E' — neutron extraction energy

T-Kelvin temperature

N_rNuclear density of resonance absorption nuclides

N_m-nuclear density of moderating nuclides

φ_T(E) Continuous energy neutron flux density at temperature TK

σ_r,s,T(E' → E) -a resonance-absorbing nuclide continuous energy neutron scattering matrix at temperature TK

σ_t,T(E) Continuous energy total cross section of resonance absorption nuclide at TK

σ_m,s,T(E' → E) -a moderator continuous energy neutron scattering matrix at temperature TK

σ_m,t,T(E) -the total cross section of the continuous energy of the moderating nuclide at the temperature of TK;

1) adopting a progressive scattering model for the moderated nuclide, adopting a free gas model for the resonance absorption nuclide, and adopting the progressive scattering model:

${\mathrm{\σ}}_{m,s,T}(E^{\′}\→E)=\frac{{\mathrm{\σ}}_m\left(E^{\′}\right)}{(1-{\mathrm{\α}}_m)E^{\′}},{\mathrm{\α}}_m={\left(\frac{A_m-1}{A_m+1}\right)}^2---\left(5\right)$

wherein,

e-neutron incident energy

E' — neutron extraction energy

A_mMass ratio of target nuclei of moderated nuclides to neutrons

Free gas model:

$\begin{array}{c}{\mathrm{\σ}}_{r,s,T}\left(E\→E^{\′}\right)=\frac{{\mathrm{\β}}^{5/2}}{4E}\exp \left(E/kT\right){\∫}_0^{\mathrm{\∞}}{\mathrm{t\σ}}_{r,s,0}\left(\frac{kT}{A_r}{t}^2\right)\\ \begin{array}{cc}\×\exp \left(-t^2/A_r\right)\mathrm{\ψ}\left(t\right)dt,& \mathrm{\β}=\left(A_r+1\right)/A_r\end{array}\end{array}---\left(6\right)$

$\begin{array}{c}{\mathrm{\ψ}}_n\left(t\right)=H\left(t_{+}-t\right)H\left(t-t_{-}\right)\\ \×{\∫}_{{\mathrm{\ϵ}}_{\max }-t}^{t+{\mathrm{\ϵ}}_{\min }}\exp \left(-x^2\right)Q_n\left(x,t\right)dx+H\left(t-t_{+}\right)\\ \×{\∫}_{t-{\mathrm{\ϵ}}_{\min }}^{t+{\mathrm{\ϵ}}_{\min }}\exp \left(-x^2\right)Q_n\left(x,t\right)dx\end{array}---\left(7\right)$

$\begin{array}{c}{\mathrm{\ϵ}}_{\max }=\sqrt{(A+1)\max (E,E^{\′})/kT}\\ {\mathrm{\ϵ}}_{\min }=\sqrt{(A+1)min(E,E^{\′})/kT}\end{array}---\left(8\right)$

$t_{\±}=\frac{{\mathrm{\ϵ}}_{max}\±{\mathrm{\ϵ}}_{\min }}{2}---\left(9\right)$

$\begin{array}{c}{Q}_n\left(x,t\right)=\frac{4}{\sqrt{\mathrm{\π}}}{\∫}_0^{2\mathrm{\π}}{P}_n\left({\mathrm{\μ}}_{lab}\right)P\left(\mathrm{\μ}\right)d\mathrm{\φ}\\ \mathrm{\μ}=\frac{1}{4x^2{t}^2}\left(A+B\cos \mathrm{\φ}\right)\\ {\mathrm{\μ}}_{lab}=\frac{1}{4x^2{\mathrm{\ϵ}}_{\max }{\mathrm{\ϵ}}_{\min }}\left(C+B\cos \mathrm{\φ}\right)\end{array}---\left(10\right)$

$\begin{array}{c}A=\left({\mathrm{\ϵ}}_{\max }^2-x^2-t^2\right)\left({\mathrm{\ϵ}}_{\min }^2-x^2-t^2\right)\\ C=\left({\mathrm{\ϵ}}_{\max }^2+x^2-t^2\right)\left({\mathrm{\ϵ}}_{\min }^2+x^2-t^2\right)\\ B=\[{\left(t+x\right)}^2-{\mathrm{\ϵ}}_{\max }^2\]\[{\left(t+x\right)}^2-{\mathrm{\ϵ}}_{\min }^2\]\\ \×\[{\mathrm{\ϵ}}_{\max }^2-{\left(t-x\right)}^2\]\[{\mathrm{\ϵ}}_{\min }^2-{\left(t-x\right)}^2\]\end{array}---\left(11\right)$

wherein,

e-neutron incident energy

E' — neutron extraction energy

A_rMass ratio of target nuclei of resonance-absorbing nuclides to neutrons

T-Kelvin temperature

k-Boltzmann constant

σ_r,s,T(E' → E) -a resonance-absorbing nuclide continuous energy neutron scattering matrix at temperature TK

σ_t,T(E) Continuous energy total cross section of resonance absorption nuclide at TK

σ_m,s,T(E' → E) -a moderator continuous energy neutron scattering matrix at temperature TK

σ_m,t,T(E) Continuous energy total cross section of moderated nuclide at TK

Wherein H is Heaviside step function, P_n(μ_lab) Is a Legendre polynomial of order n, mu_labFor the scattering angle under the experimental system, P (mu) is the scattering with the scattering azimuth angle mu under the centroid systemProbability distribution;

2) solving the neutron moderation equation by using an ultrafine group method, wherein in the ultrafine group method, a resonance energy region is divided into very fine energy intervals, each energy group is called an ultrafine group, the width of each ultrafine group is considered to be far smaller than the maximum logarithmic energy drop obtained by collision of neutrons and final nuclein, namely, the ultrafine group is considered to be impossible to perform self-scattering, so that the fine flux can be obtained by sequentially solving from high energy to low energy group by group as long as a scattering source of the highest energy group is given; under 200eV, a free gas model is considered, the model can cause the up-scattering effect of neutrons, the flux cannot be obtained by solving from high energy to low energy group by group once when the energy spectrum is calculated, and the flux is calculated until phi is obtained through iteration_T(E) Converging;

for the group combination calculation in step 1-3, the group cross section σ of a certain group of resonance absorbing nuclides at TK_g,TThe calculation method comprises the following steps:

${\mathrm{\σ}}_{g,T}=\frac{{\∫}_{{\mathrm{\ΔE}}_g}{\mathrm{\σ}}_{a,T}\left(E\right){\mathrm{\φ}}_T\left(E\right)dE}{{\∫}_{{\mathrm{\ΔE}}_g}{\mathrm{\φ}}_T\left(E\right)dE}---\left(3\right)$

wherein,

e-neutron incident energy

E' — neutron extraction energy

g-neutron incident energy group

g' — neutron emergent energy group

ΔE_gEnergy separation of neutron incident energy packets

ΔE_g′Energy separation of neutron emergent energy groups

φ_T(E) Continuous energy neutron flux density at temperature TK

σ_a,T(E) Continuous energy absorption Cross section at temperature TK

Step 6: repeating the steps 1-4, wherein 10 background sections are taken, namely the nuclear proportion of the resonance nuclide and the moderation nuclide is adjusted to enable 1E1barn, 1E2barn, 1E3barn, 1E4barn, 1E5barn, 1E6barn, 1E7barn, 1E8barn, 1E9barn and 1E10 barn; ten temperature points are taken, namely 300K, 400K, 500K, 600K, 700K, 800K, 900K, 1000K, 1100K and 1200K; thereby obtaining a plurality of intermediate resonance factors of each energy group under different temperatures and background sections;

and 7: and 6, obtaining polynomial coefficients by a least square fitting method by taking a large number of intermediate resonance factors as fitting points and the background section and the temperature as independent variables in each energy group according to the plurality of intermediate resonance factors of the energy groups obtained in the step 6 under different temperatures and the background section.

Compared with the prior art, the invention has the following outstanding advantages:

1. when the slowing equation is solved, a free gas model is introduced, the scattering effect on resonance can be accurately considered, more accurate multi-group absorption cross sections can be obtained, more accurate intermediate resonance factors can be finally obtained, and the accuracy of resonance calculation is improved.

2. The intermediate resonance factor is preset by a least square fitting method, and the problem of extremely long time consumption caused by continuously solving a neutron moderation equation introducing free gas is avoided to the maximum extent in the full-stack resonance calculation.

Detailed Description

The present invention will be described in further detail with reference to specific embodiments below:

the invention discloses a method for acquiring intermediate resonance factors in a reactor multi-group nuclear database, which comprises the following steps:

1. appointing a temperature, a resonance absorption nuclide and a moderation nuclide, uniformly mixing the resonance absorption nuclide and the moderation nuclide, and calculating a corresponding background cross section value sigma according to the nuclear density ratio of the resonance absorption nuclide and the moderation nuclide_b,case1Solving the corresponding neutron moderation equation to obtain the continuous energy neutron flux density, and finally obtaining a plurality of groups of absorption cross sections of the resonance absorption nuclide through group combination calculation, wherein the group cross section of a certain group of the resonance absorption nuclide is abbreviated as sigma at the temperature_case1,g。

2. Uniformly mixing the resonance absorption nuclide and the moderating nuclide according to the temperature, the resonance absorption nuclide and the moderating nuclide specified in the step 1, wherein the nuclear density ratio of the resonance absorption nuclide and the moderating nuclide is 95% of the ratio in the step 1, and calculating the corresponding background cross section value sigma_b,case2Solving the corresponding neutron moderation equation and obtaining the continuous energy neutron flux densityFinally, the multi-group absorption cross section of the resonance absorption nuclide is obtained by group combination calculation, and the group cross section of a certain group of the resonance absorption nuclide is abbreviated as sigma at the temperature_case2,g。

3. According to the temperature and the resonance absorption nuclear species specified in the step 1, a new moderating nuclear species is specified to replace the main moderating nuclear species specified in the step 1, the resonance absorption nuclear species and the new moderating nuclear species are uniformly mixed, the ratio of the resonance absorption nuclear species to the new moderating nuclear species is adjusted, and the corresponding background cross section value sigma is calculated_b,case3Equal to the background cross-section value (σ) in step 2_b,case3＝σ_b,case2) Solving the corresponding neutron moderation equation to obtain the continuous energy neutron flux density, and finally obtaining the multi-group absorption cross section of the resonance absorption nuclide through group combination calculation, wherein the group cross section of a certain group of the resonance absorption nuclide is abbreviated as sigma at the temperature_case3,g。

4. Obtaining the designated energy group g, the designated temperature T and the designated background section sigma of the new nuclide in the step 4 by the formula (1)_b,spe＝σ_b,case1Lower intermediate resonance factor

${\mathrm{\λ}}_{g,T,{\mathrm{\σ}}_{b,spe}}=\frac{{\mathrm{\σ}}_{case3,g}-{\mathrm{\σ}}_{case1,g}}{{\mathrm{\σ}}_{case2,g}-{\mathrm{\σ}}_{case1,g}}---\left(1\right)$

5. For the background cross-section values σ in steps 1-3_bThe calculation is performed using equation (2):

${\mathrm{\σ}}_b=\frac{N_m\·{\mathrm{\σ}}_{m,p}}{N_r}---\left(2\right)$

wherein,

N_mnuclear density of moderating nuclides

σ_m,pPotential elastic scattering cross-section of a moderating nuclide

N_rNuclear density of resonance-absorbing nuclides

For the neutron moderation equation in steps 1-3, the form is as follows:

$\begin{array}{c}\[{\mathrm{\σ}}_{r,t,T}\left(E\right)\·N_r+{\mathrm{\σ}}_{m,t,T}\left(E\right)\·N_m\]{\mathrm{\φ}}_T\left(E\right)={\∫}_0^{\mathrm{\∞}}{\mathrm{\σ}}_{r,s,T}(E^{\′}\→E)\·N_r\·{\mathrm{\φ}}_T\left(E^{\′}\right){\mathrm{dE}}^{\′}\\ +{\∫}_0^{\mathrm{\∞}}{\mathrm{\σ}}_{m,s,T}(E^{\′}\→E)\·N_m\·{\mathrm{\φ}}_T\left(E^{\′}\right){\mathrm{dE}}^{\′}\end{array}---\left(4\right)$

wherein,

e-neutron incident energy

E' — neutron extraction energy

T-Kelvin temperature

N_rNuclear density of resonance absorption nuclides

N_m-nuclear density of moderating nuclides

φ_T(E) Continuous energy neutron flux density at temperature TK

σ_r,s,T(E' → E) -a resonance-absorbing nuclide continuous energy neutron scattering matrix at temperature TK

σ_t,T(E) Continuous energy total cross section of resonance absorption nuclide at TK

σ_m,s,T(E' → E) -a moderator continuous energy neutron scattering matrix at temperature TK

σ_m,t,T(E) Continuous energy total cross section of moderated nuclide at TK

1) Adopting a progressive scattering model for the moderated nuclide, adopting a free gas model for the resonance absorption nuclide, and adopting the progressive scattering model:

${\mathrm{\σ}}_{m,s,T}(E^{\′}\→E)=\frac{{\mathrm{\σ}}_m\left(E^{\′}\right)}{(1-{\mathrm{\α}}_m)E^{\′}},{\mathrm{\α}}_m={\left(\frac{A_m-1}{A_m+1}\right)}^2---\left(5\right)$

wherein,

e-neutron incident energy

E' — neutron extraction energy

A_mMass ratio of target nuclei of moderated nuclides to neutrons

Free gas model:

$\begin{array}{c}{\mathrm{\σ}}_{r,s,T}\left(E\→E^{\′}\right)=\frac{{\mathrm{\β}}^{5/2}}{4E}\exp \left(E/kT\right){\∫}_0^{\mathrm{\∞}}{\mathrm{t\σ}}_{r,s,0}\left(\frac{kT}{A_r}{t}^2\right)\\ \begin{array}{cc}\×\exp \left(-t^2/A_r\right)\mathrm{\ψ}\left(t\right)dt,& \mathrm{\β}=\left(A_r+1\right)/A_r\end{array}\end{array}---\left(6\right)$

$\begin{array}{c}{\mathrm{\ψ}}_n\left(t\right)=H\left(t_{+}-t\right)H\left(t-t_{-}\right)\\ \×{\∫}_{{\mathrm{\ϵ}}_{\max }-t}^{t+{\mathrm{\ϵ}}_{\min }}\exp \left(-x^2\right)Q_n\left(x,t\right)dx+H\left(t-t_{+}\right)\\ \×{\∫}_{t-{\mathrm{\ϵ}}_{\min }}^{t+{\mathrm{\ϵ}}_{\min }}\exp \left(-x^2\right)Q_n\left(x,t\right)dx\end{array}---\left(7\right)$

$\begin{array}{c}{\mathrm{\ϵ}}_{\max }=\sqrt{(A+1)\max (E,E^{\′})/kT}\\ {\mathrm{\ϵ}}_{\min }=\sqrt{(A+1)min(E,E^{\′})/kT}\end{array}---\left(8\right)$

$t_{\±}=\frac{{\mathrm{\ϵ}}_{max}\±{\mathrm{\ϵ}}_{\min }}{2}---\left(9\right)$

$\begin{array}{c}{Q}_n\left(x,t\right)=\frac{4}{\sqrt{\mathrm{\π}}}{\∫}_0^{2\mathrm{\π}}{P}_n\left({\mathrm{\μ}}_{lab}\right)P\left(\mathrm{\μ}\right)d\mathrm{\φ}\\ \mathrm{\μ}=\frac{1}{4x^2{t}^2}\left(A+B\cos \mathrm{\φ}\right)\\ {\mathrm{\μ}}_{lab}=\frac{1}{4x^2{\mathrm{\ϵ}}_{\max }{\mathrm{\ϵ}}_{\min }}\left(C+B\cos \mathrm{\φ}\right)\end{array}---\left(10\right)$

$\begin{array}{c}A=\left({\mathrm{\ϵ}}_{\max }^2-x^2-t^2\right)\left({\mathrm{\ϵ}}_{\min }^2-x^2-t^2\right)\\ C=\left({\mathrm{\ϵ}}_{\max }^2+x^2-t^2\right)\left({\mathrm{\ϵ}}_{\min }^2+x^2-t^2\right)\\ B=\[{\left(t+x\right)}^2-{\mathrm{\ϵ}}_{\max }^2\]\[{\left(t+x\right)}^2-{\mathrm{\ϵ}}_{\min }^2\]\\ \×\[{\mathrm{\ϵ}}_{\max }^2-{\left(t-x\right)}^2\]\[{\mathrm{\ϵ}}_{\min }^2-{\left(t-x\right)}^2\]\end{array}---\left(11\right)$

wherein,

e-neutron incident energy

E' — neutron extraction energy

A_rMass ratio of target nuclei of resonance-absorbing nuclides to neutrons

T-Kelvin temperature

k-Boltzmann constant

σ_r,s,T(E' → E) — resonance at temperature TKNuclide-absorbing continuous energy neutron scattering matrix

σ_t,T(E) Continuous energy total cross section of resonance absorption nuclide at TK

σ_m,s,T(E' → E) -a moderator continuous energy neutron scattering matrix at temperature TK

σ_m,t,T(E) Continuous energy total cross section of moderated nuclide at TK

Wherein H is Heaviside step function, P_n(μ_lab) Is a Legendre polynomial of order n, mu_labFor the experimental system scattering angle, P (μ) is the scattering probability distribution for the azimuthal scattering angle μ in the centroid system.

2) The neutron moderation equation is solved using a super-fine group method in which the resonance energy region is divided into very fine energy intervals, each such energy group is called a super-fine group, and the width of each super-fine group is considered to be much smaller than the maximum logarithmic energy drop obtained by the collision of neutrons with the final nuclei, i.e. it is considered that no self-scattering of the super-fine group occurs, so that the fine flux can be solved from high energy to low energy group by group in sequence as long as the scattering source of the highest energy group is given. Under 200eV, a free gas model is considered, the model can cause the up-scattering effect of neutrons, the flux cannot be obtained by solving from high energy to low energy group by group once when the energy spectrum is calculated, and the flux is calculated until phi is obtained through iteration_T(E) And (6) converging.

For the group combination calculation in step 1-3, the group cross section σ of a certain group of resonance absorbing nuclides at TK_g,TThe calculation method comprises the following steps:

${\mathrm{\σ}}_{g,T}=\frac{{\∫}_{{\mathrm{\ΔE}}_g}{\mathrm{\σ}}_{a,T}\left(E\right){\mathrm{\φ}}_T\left(E\right)dE}{{\∫}_{{\mathrm{\ΔE}}_g}{\mathrm{\φ}}_T\left(E\right)dE}---\left(3\right)$

wherein,

e-neutron incident energy

E' — neutron extraction energy

g-neutron incident energy group

g' — neutron emergent energy group

ΔE_gEnergy separation of neutron incident energy packets

ΔE_g′Energy separation of neutron emergent energy groups

φ_T(E) Continuous energy neutron flux density at temperature TK

σ_a,T(E) Continuous energy absorption Cross section at temperature TK

6. And (5) repeating the steps 1-4. The background section is 10, namely the nuclear proportion of the resonance nuclide and the moderation nuclide is adjusted to be 1E1barn, 1E2barn, 1E3barn, 1E4barn, 1E5barn, 1E6barn, 1E7barn, 1E8barn, 1E9barn and 1E10 barn. Ten temperature points were taken, i.e. 300K, 400K, 500K, 600K, 700K, 800K, 900K, 1000K, 1100K, 1200K. Thereby obtaining the resonance factors among the energy groups under a plurality of different temperatures and background sections.

7. According to the plurality of different temperatures obtained in step 6 and the intermediate resonance factors of the energy groups under the background section. And obtaining polynomial coefficients by a least square fitting method by taking a large number of intermediate resonance factors as fitting points and background sections and temperature as independent variables in each energy group.

Claims (1)

1. A method of obtaining intermediate resonance factors in a nuclear database of a plurality of groups of reactors, comprising: the method comprises the following steps:

step 1: appointing a temperature, a resonance absorption nuclide and a moderation nuclide, uniformly mixing the resonance absorption nuclide and the moderation nuclide, and calculating a corresponding background cross section value sigma according to the nuclear density ratio of the resonance absorption nuclide and the moderation nuclide_b,case1Solving the corresponding neutron moderation equation to obtain the continuous energy neutron flux density, and finally obtaining multiple groups of the resonance absorption nuclide by group combination calculationAbsorption cross section, the group cross section of a group of resonance absorption species at this temperature being abbreviated as σ_case1,g；

Step 2: uniformly mixing the resonance absorption nuclide and the moderating nuclide according to the temperature, the resonance absorption nuclide and the moderating nuclide specified in the step 1, wherein the nuclear density ratio of the resonance absorption nuclide and the moderating nuclide is 95% of the ratio in the step 1, and calculating the corresponding background cross section value sigma_b,case2Solving the corresponding neutron moderation equation to obtain the continuous energy neutron flux density, and finally obtaining a plurality of groups of absorption cross sections of the resonance absorption nuclide through group combination calculation, wherein the group cross section of a certain group of the resonance absorption nuclide is abbreviated as sigma at the temperature_case2,g；

And step 3: according to the temperature and the resonance absorption nuclear species specified in the step 1, a new moderating nuclear species is specified to replace the main moderating nuclear species specified in the step 1, the resonance absorption nuclear species and the new moderating nuclear species are uniformly mixed, the ratio of the resonance absorption nuclear species to the new moderating nuclear species is adjusted, and the corresponding background cross section value sigma is calculated_b,case3Equal to the background cross-section value in step 2, i.e. sigma_b,case3＝σ_b,case2Solving the corresponding neutron moderation equation to obtain the continuous energy neutron flux density, and finally obtaining the multi-group absorption cross section of the resonance absorption nuclide through group combination calculation, wherein the group cross section of a certain group of the resonance absorption nuclide is abbreviated as sigma at the temperature_case3,g；

And 4, step 4: obtaining the designated energy group g, the designated temperature T and the designated background section sigma of the new nuclide by the formula (1)_b,spe＝σ_b,case1Lower intermediate resonance factor

${\mathrm{\λ}}_{g,T,{\mathrm{\σ}}_{b,spe}}=\frac{{\mathrm{\σ}}_{case3,g}-{\mathrm{\σ}}_{case1,g}}{{\mathrm{\σ}}_{case2,g}-{\mathrm{\σ}}_{case1,g}}---\left(1\right)$

And 5: for the background cross-section values σ in steps 1-3_bThe calculation is performed using equation (2):

${\mathrm{\σ}}_b=\frac{N_m\·{\mathrm{\σ}}_{m,p}}{N_r}---\left(2\right)$

wherein,

N_mnuclear density of moderating nuclides

σ_m,pPotential elastic scattering cross-section of a moderating nuclide

N_rNuclear density of resonance-absorbing nuclides

For the neutron moderation equation in steps 1-3, the form is as follows:

$\begin{array}{c}\[{\mathrm{\σ}}_{r,t,T}\left(E\right)\·N_r+{\mathrm{\σ}}_{m,t,T}\left(E\right)\·N_m\]{\mathrm{\φ}}_T\left(E\right)={\∫}_0^{\mathrm{\∞}}{\mathrm{\σ}}_{r,s,T}\left(E^{\′}\→E\right)\·N_r\·{\mathrm{\φ}}_T\left(E^{\′}\right){\mathrm{dE}}^{\′}\\ +{\∫}_0^{\mathrm{\∞}}{\mathrm{\σ}}_{m,s,T}\left(E^{\′}\→E\right)\·N_m\·{\mathrm{\φ}}_T\left(E^{\′}\right){\mathrm{dE}}^{\′}\end{array}---\left(4\right)$

wherein,

e-neutron incident energy

E' — neutron extraction energy

T-Kelvin temperature

N_rNuclear density of resonance absorption nuclides

N_m-nuclear density of moderating nuclides

φ_T(E) Continuous energy neutron flux density at temperature TK

σ_r,s,T(E' → E) -a resonance-absorbing nuclide continuous energy neutron scattering matrix at temperature TK

σ_t,T(E) Continuous energy total cross section of resonance absorption nuclide at TK

σ_m,s,T(E' → E) -a moderator continuous energy neutron scattering matrix at temperature TK

σ_m,t,T(E) -the total cross section of the continuous energy of the moderating nuclide at the temperature of TK;

1) adopting a progressive scattering model for the moderated nuclide, adopting a free gas model for the resonance absorption nuclide,

progressive scattering model:

${\mathrm{\σ}}_{m,s,T}(E^{\′}\→E)=\frac{{\mathrm{\σ}}_m\left(E^{\′}\right)}{(1-{\mathrm{\α}}_m)E^{\′}},{\mathrm{\α}}_m={\left(\frac{A_m-1}{A_m+1}\right)}^2---\left(5\right)$

wherein,

e-neutron incident energy

E' — neutron extraction energy

A_mMass ratio of target nuclei of moderated nuclides to neutrons

Free gas model:

$\begin{array}{c}{\mathrm{\σ}}_{r,s,T}\left(E\→E^{\′}\right)=\frac{{\mathrm{\β}}^{5/2}}{4E}\exp \left(E/kT\right){\∫}_0^{\mathrm{\∞}}{\mathrm{t\σ}}_{r,s,0}\left(\frac{kT}{A_r}{t}^2\right)\\ \×\exp \left(-t^2/A_r\right)\mathrm{\ψ}\left(t\right)dt,\mathrm{\β}=\left(A_r+1\right)/A_r\end{array}---\left(6\right)$

$\begin{array}{c}{\mathrm{\ψ}}_n\left(t\right)=H\left(t_{+}-t\right)H\left(t-t_{-}\right)\\ \×{\∫}_{{\mathrm{\ϵ}}_{\max }-t}^{t+{\mathrm{\ϵ}}_{\min }}\exp \left(-x^2\right)Q_n\left(x,t\right)dx+H\left(t-t_{+}\right)\\ \×{\∫}_{t-{\mathrm{\ϵ}}_{\min }}^{t+{\mathrm{\ϵ}}_{\min }}\exp \left(-x^2\right)Q_n\left(x,t\right)dx\end{array}---\left(7\right)$

$\begin{array}{c}{\mathrm{\ϵ}}_{\max }=\sqrt{(A+1)\max (E,E^{\′})/kT}\\ {\mathrm{\ϵ}}_{\min }=\sqrt{(A+1)\min (E,E^{\′})/kT}\end{array}---\left(8\right)$

$t_{\±}=\frac{{\mathrm{\ϵ}}_{max}\±{\mathrm{\ϵ}}_{\min }}{2}---\left(9\right)$

$\begin{array}{c}{Q}_n\left(x,t\right)=\frac{4}{\sqrt{\mathrm{\π}}}{\∫}_0^{2\mathrm{\π}}{P}_n\left({\mathrm{\μ}}_{lab}\right)P\left(\mathrm{\μ}\right)d\mathrm{\φ}\\ \mathrm{\μ}=\frac{1}{4x^2{t}^2}\left(A+B\cos \mathrm{\φ}\right)\\ {\mathrm{\μ}}_{lab}=\frac{1}{4x^2{\mathrm{\ϵ}}_{\max }{\mathrm{\ϵ}}_{\min }}\left(C+B\cos \mathrm{\φ}\right)\end{array}---\left(10\right)$

$\begin{array}{c}A=\left({\mathrm{\ϵ}}_{\max }^2-x^2-t^2\right)\left({\mathrm{\ϵ}}_{\min }^2-x^2-t^2\right)\\ C=\left({\mathrm{\ϵ}}_{\max }^2+x^2-t^2\right)\left({\mathrm{\ϵ}}_{\min }^2+x^2-t^2\right)\\ B=\[{\left(t+x\right)}^2-{\mathrm{\ϵ}}_{\max }^2\]\[{\left(t+x\right)}^2-{\mathrm{\ϵ}}_{\min }^2\]\\ \×\[{\mathrm{\ϵ}}_{\max }^2-{\left(t-x\right)}^2\]\[{\mathrm{\ϵ}}_{\min }^2-{\left(t-x\right)}^2\]\end{array}---\left(11\right)$

wherein,

e-neutron incident energy

E' — neutron extraction energy

A_rMass ratio of target nuclei of resonance-absorbing nuclides to neutrons

T-Kelvin temperature

k-Boltzmann constant

σ_r,s,T(E' → E) -a resonance-absorbing nuclide continuous energy neutron scattering matrix at temperature TK

σ_t,T(E) Continuous energy total cross section of resonance absorption nuclide at TK

σ_m,s,T(E' → E) -a moderator continuous energy neutron scattering matrix at temperature TK

σ_m,t,T(E) Continuous energy total cross section of moderated nuclide at TK

Wherein H is Heaviside step function, P_n(μ_lab) Is a Legendre polynomial of order n, mu_labThe scattering angle under the experimental system is shown, and P (mu) is the scattering probability distribution with the scattering azimuth angle mu under the mass center system;

2) the neutron moderation equation is solved using a super-population method in which the resonance energy region is divided into very fine energy intervals, each such energy population is called a super-population, and the width of each super-population is considered to be much less than the maximum logarithmic energy drop obtained by the collision of a neutron with the final nucleus, i.e., it is considered that no self-scattering of the super-population is possible, so that given the scattering source of the highest energy population, it is possible to pass the energy back through the high-energy population in turnSolving the low energy group by group to obtain fine flux; under 200eV, a free gas model is considered, the model can cause the up-scattering effect of neutrons, the flux cannot be obtained by solving from high energy to low energy group by group once when the energy spectrum is calculated, and the flux is calculated until phi is obtained through iteration_T(E) Converging;

for the group combination calculation in step 1-3, the group cross section σ of a certain group of resonance absorbing nuclides at TK_g,TThe calculation method comprises the following steps:

${\mathrm{\σ}}_{g,T}=\frac{{\∫}_{{\mathrm{\ΔE}}_g}{\mathrm{\σ}}_{a,T}\left(E\right){\mathrm{\φ}}_T\left(E\right)dE}{{\∫}_{{\mathrm{\ΔE}}_g}{\mathrm{\φ}}_T\left(E\right)dE}---\left(3\right)$

wherein,

e-neutron incident energy

E' — neutron extraction energy

g-neutron incident energy group

g' — neutron emergent energy group

ΔE_gEnergy separation of neutron incident energy packets

ΔE_g′Energy separation of neutron emergent energy groups

φ_T(E) Continuous energy neutron flux density at temperature TK

σ_a,T(E) Continuous energy absorption Cross section at temperature TK

Step 6: repeating the steps 1-4, wherein 10 background sections are taken, namely the nuclear proportion of the resonance nuclide and the moderation nuclide is adjusted to enable 1E1barn, 1E2barn, 1E3barn, 1E4barn, 1E5barn, 1E6barn, 1E7barn, 1E8barn, 1E9barn and 1E10 barn; ten temperature points are taken, namely 300K, 400K, 500K, 600K, 700K, 800K, 900K, 1000K, 1100K and 1200K; thereby obtaining a plurality of intermediate resonance factors of each energy group under different temperatures and background sections;

and 7: and 6, obtaining polynomial coefficients by a least square fitting method by taking a large number of intermediate resonance factors as fitting points and the background section and the temperature as independent variables in each energy group according to the plurality of intermediate resonance factors of the energy groups obtained in the step 6 under different temperatures and the background section.