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

A kind of method of the intermediate resonance factor obtained in reactor multigroup nuclear data depositary

Technical field

The present invention relates to nuclear reactor multigroup nuclear data depositary and reactor physics calculates field, be specifically related to a kind of acquisition instead The method answering the intermediate resonance factor in heap multigroup nuclear data depositary.

Background technology

In order to meet the demand that the resonance of numerical response heap high-fidelity calculates, Multi-group data storehouse provides and is total in the middle of accurately The factor of shaking is most important.

Currently in the intermediate resonance factor, widely used is the most popular evaluation nuclear data depositary processing routine Intermediate resonance factor computational methods proposed in NJOY (hereinafter referred to as NJOY).Require gradually along with reactor resonance calculates Improve, the method in the calculation between resonate because many models of the period of the day from 11 p.m. to 1 a.m and the suitability of the method can not meet requirement.

The intermediate resonance factor computational methods that NJOY is proposed are divided into four steps:

1. one temperature of appointment, a resonance absorption nucleic and a slowing down nucleic, this resonance absorption nucleic is slow with this Change nucleic uniformly to mix, absorb nucleic and the nucleon density ratio of main slowing down nucleic according to dominant resonant, calculate corresponding Background cross section value σ_b,case1, solve the moderation of neutrons equation of correspondence and obtain Continuous Energy netron-flux density, finally by And group is calculated the multigroup absorption cross-section of this resonance absorption nucleic, at a temperature of being somebody's turn to do, certain of dominant resonant absorption nucleic is group of Group cross-section is referred to as σ_case1,g。

2., according to the temperature specified by step 1, dominant resonant absorbs nucleic and main slowing down nucleic, by this resonance absorption Nucleic and this slowing down nucleic uniformly mix, and it is step 1 that dominant resonant absorbs the nucleon density ratio of nucleic and main slowing down nucleic The 90% of middle ratio, calculates corresponding background cross section value σ_b,case2, solve the moderation of neutrons equation of correspondence and obtain continuously Energy neutron flux density, is calculated the multigroup absorption cross-section of this resonance absorption nucleic finally by also group, at a temperature of being somebody's turn to do, main Certain the group of group cross-section wanting resonance absorption nucleic is referred to as σ_case2,g。

3. absorb nucleic according to temperature specified in step 1 and dominant resonant, it is intended that new main slowing down nucleic replaces Main slowing down nucleic specified in step 1, uniformly mixes this resonance absorption nucleic and new slowing down nucleic, adjusts main being total to Shake and absorb nucleic and new main slowing down nucleic nucleon ratio example, make to calculate corresponding background cross section value σ_b,case3Equal to step Value (σ in background cross section in 2_b,case2=σ_b,case3), solve corresponding moderation of neutrons equation and obtain Continuous Energy netron-flux density, Be calculated the multigroup absorption cross-section of this resonance absorption nucleic finally by also group, at a temperature of being somebody's turn to do, dominant resonant absorbs nucleic Certain group of group cross-section is referred to as σ_case3,g。

4. it is obtained by the following formula the appointment energy group g of nucleic new in step 3, assigned temperature T, specific context cross section σ_b,spe =σ_b,case1Under the intermediate resonance factor

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

Wherein, the key concept in each step has:

1) about background cross section σ_b, its computing formula is

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

Wherein,

N_mThe nucleon density of slowing down nucleic

σ_m,pThe elastic potential scattering cross section of slowing down nucleic

N_rThe nucleon density of resonance absorption nucleic

2) and group calculates, temperature is certain group of group cross-section σ that dominant resonant under TK absorbs nucleic_g,TComputational methods are:

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

Wherein,

E neutron projectile energy

E ' neutron emanated energy

G neutron incident energy group

G ' neutron outgoing energy group

ΔE_gThe energy bite of neutron incident energy group

ΔE_g′The energy bite of neutron outgoing energy group

φ_T(E) the Continuous Energy netron-flux density at temperature TK

σ_a,T(E) the Continuous Energy absorption cross-section at temperature TK

From above-mentioned steps, the method can relate to solve moderation of neutrons equation, and present traditional method is solving During moderation of neutrons equation, the model describing elastic neutron scattering is progressive scattering nucleus model, and this model have ignored resonance nucleic In resonance, scattering is for the increase of resonance absorption, causes the fine power spectrum calculated inaccurate, further results in multigroup cross section Inaccurate, finally make the intermediate resonance factor inaccurate；On the other hand, by above-mentioned known, the intermediate resonance factor be about can group, Background cross section, the function of temperature, in the resonance of full heap calculates, along with the intensification of reactor burnup, background cross section and temperature be not Disconnected change, it is possible to use the method according to the continuous repeat the above steps of practical situation in reactor line computation intermediate resonance because of Son, but owing to the slow calculating time is dragged in the meeting in large scale of full heap.And if consider the upper scattering of resonance, can take solving slowing-down equation Alternative manner solves, and the calculating time also can be made to sharply increase, and pole is unfavorable for that the resonance of full heap calculates.

Therefore, for above existing problem, need a kind of intermediate resonance factor meter accurate, feasible, quick of invention Calculation method.

Summary of the invention

For the problem overcoming above-mentioned prior art to exist, it is an object of the invention to provide a kind of acquisition reactor multigroup The method of the intermediate resonance factor in nuclear data depositary, in order to obtain the intermediate resonance factor accurately, the inventive method is in NJOY institute On the basis of proposition method, slowing-down equation will be solved based on free gas model scattering model, be simultaneously based on least square fitting Method obtains the intermediate resonance factor accurately, and calculating for the resonance of full heap provides reliable data.

To achieve these goals, this invention takes techniques below scheme:

A kind of method of intermediate resonance factor obtained in reactor multigroup nuclear data depositary, comprises the steps:

Step 1: specify a temperature, a resonance absorption nucleic and a slowing down nucleic, by this resonance absorption nucleic and This slowing down nucleic uniformly mixes, and according to the nucleon density ratio of resonance absorption nucleic and slowing down nucleic, calculates the corresponding back of the body Scape cross section value σ_b,case1, solve the moderation of neutrons equation of correspondence and obtain Continuous Energy netron-flux density, finally by also group It is calculated the multigroup absorption cross-section of this resonance absorption nucleic, at a temperature of being somebody's turn to do, certain group of group cross-section letter of resonance absorption nucleic It is referred to as σ_case1,g；

Step 2: according to the temperature specified by step 1, resonance absorption nucleic and slowing down nucleic, by this resonance absorption nucleic Uniformly mixing with this slowing down nucleic, the nucleon density ratio of resonance absorption nucleic and slowing down nucleic is in step 1 the 95% of ratio, Calculate corresponding background cross section value σ_b,case2, solve the moderation of neutrons equation of correspondence and to obtain Continuous Energy neutron flux close Degree, is calculated the multigroup absorption cross-section of this resonance absorption nucleic, at a temperature of being somebody's turn to do, certain of resonance absorption nucleic finally by also group Group of group cross-section is referred to as σ_case2,g；

Step 3: according to temperature specified in step 1 and resonance absorption nucleic, it is intended that new slowing down nucleic step of replacing 1 Main slowing down nucleic specified by, uniformly mixes this resonance absorption nucleic and new slowing down nucleic, adjusts resonance absorption core Plain Yu new slowing down nucleic nucleon ratio example, makes to calculate corresponding background cross section value σ_b,case3Equal to background cross section in step 2 Value, i.e. σ_b,case3=σ_b,case2, solve corresponding moderation of neutrons equation and obtain Continuous Energy netron-flux density, finally by also Group is calculated the multigroup absorption cross-section of this resonance absorption nucleic, at a temperature of being somebody's turn to do, and certain group of group cross-section of resonance absorption nucleic Referred to as σ_case3,g；

Step 4: obtained the appointment energy group g of new nucleic, assigned temperature T, specific context cross section σ by formula (1)_b,spe= σ_b,case1Under the intermediate resonance factor

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

Step 5: σ is worth for the background cross section in step 1-3_b, use formula (2) to calculate:

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

Wherein,

N_mThe nucleon density of slowing down nucleic

σ_m,pThe elastic potential scattering cross section of slowing down nucleic

N_rThe nucleon density of resonance absorption nucleic

For the moderation of neutrons equation in step 1-3, 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 projectile energy

E ' neutron emanated energy

T Kelvin

N_rResonance absorption nucleic nucleon density

N_mSlowing down nucleic nucleon density

φ_T(E) the Continuous Energy netron-flux density at temperature TK

σ_r,s,TResonance absorption nucleic Continuous Energy neutron scattering matrix at (E ' → E) temperature TK

σ_t,T(E) the resonance absorption nucleic Continuous Energy total cross section under temperature is TK

σ_m,s,T(E ' → E) temperature is the slowing down nucleic Continuous Energy neutron scattering matrix under TK

σ_m,t,T(E) the slowing down nucleic Continuous Energy total cross section under temperature is TK；

1) progressive scattering model is used for slowing down nucleic, free gas model is used for resonance absorption nucleic, 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 projectile energy

E ' neutron emanated energy

A_mSlowing down nucleic target nucleus and the mass ratio of neutron

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 projectile energy

E ' neutron emanated energy

A_rResonance absorption nucleic target nucleus and the mass ratio of neutron

T Kelvin

K Boltzmann constant

σ_r,s,TResonance absorption nucleic Continuous Energy neutron scattering matrix at (E ' → E) temperature TK

σ_t,T(E) the resonance absorption nucleic Continuous Energy total cross section under temperature is TK

σ_m,s,T(E ' → E) temperature is the slowing down nucleic Continuous Energy neutron scattering matrix under TK

σ_m,t,T(E) the slowing down nucleic Continuous Energy total cross section under temperature is TK

Wherein, H is Heaviside jump function, P_n(μ_lab) it is n rank Legnedre polynomials, μ_labFor the lower scattering of experiment system Angle, P (μ) is to scatter the probability of scattering distribution that azimuth is μ under center-of-mass angle；

2) use ultra-fine group's method to solve this moderation of neutrons equation, in ultra-fine group's method, resonance energy district is partitioned into The finest energy bite, each such energy group be referred to as a ultra-fine group, it is believed that each ultra-fine group's width is much smaller than The max log that the collision of neutron and final nucleic is obtained can drop, and i.e. thinks and the self-scattering of ultra-fine group can not occur, so Just acquisition fine flux can be solved to mental retardation by group by high energy successively after the scattering source of group as long as given；At 200eV Below, it is considered to free gas model, this model can cause the upper scattering effect of neutron, can not be disposably by height calculating energy time spectrum Can solve by group to mental retardation and obtain flux, by iterative computation until φ_T(E) convergence；

Calculating in step 1-3 and group, temperature is certain group of group cross-section σ of the resonance absorption nucleic under TK_g,TMeter Calculation method is:

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

Wherein,

E neutron projectile energy

E ' neutron emanated energy

G neutron incident energy group

G ' neutron outgoing energy group

ΔE_gThe energy bite of neutron incident energy group

ΔE_g′The energy bite of neutron outgoing energy group

φ_T(E) the Continuous Energy netron-flux density at temperature TK

σ_a,T(E) the Continuous Energy absorption cross-section at temperature TK

Step 6: repeating step 1-4, background cross section takes 10, the nucleon ratio i.e. adjusting resonance nucleic and slowing down nucleic makes Obtain 1E1barn, 1E2barn, 1E3barn, 1E4barn, 1E5barn, 1E6barn, 1E7barn, 1E8barn, 1E9barn, 1E10barn；Temperature spot takes ten, i.e. 300K, 400K, 500K, 600K, 700K, 800K, 900K, 1000K, 1100K, 1200K； Each energy group's intermediate resonance factor under a large amount of different temperatures and background cross section is obtained with this；

Step 7: according in step 6 obtain a large amount of different temperatures and background cross section under each can group's intermediate resonance because of Son, each can group with a large amount of intermediate resonance factors as match point, with background cross section and temperature as independent variable, pass through least square Approximating method obtains multinomial coefficient.

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

1. when solving slowing-down equation, introduce free gas model, the upper scattering effect that resonates can accurately be considered, it is possible to obtain Obtain multigroup absorption cross-section more accurately, finally give the intermediate resonance factor more accurately, improve the precision that resonance calculates.

2., by the least-square fitting approach preset intermediate resonance factor, in the resonance of full heap calculates, keep away largely The moderation of neutrons equation having exempted from constantly to solve introducing free gas is brought the most time-consuming problem.

Detailed description of the invention

Below in conjunction with detailed description of the invention, the present invention is described in further detail:

The method of a kind of intermediate resonance factor obtained in reactor multigroup nuclear data depositary of the present invention, comprises the steps:

1. one temperature of appointment, a resonance absorption nucleic and a slowing down nucleic, this resonance absorption nucleic is slow with this Change nucleic uniformly to mix, according to the nucleon density ratio of resonance absorption nucleic and slowing down nucleic, calculate corresponding background and cut Face amount σ_b,case1, solve the moderation of neutrons equation of correspondence and obtain Continuous Energy netron-flux density, calculating finally by also group Obtaining the multigroup absorption cross-section of this resonance absorption nucleic, at a temperature of being somebody's turn to do, certain group of group cross-section of resonance absorption nucleic is referred to as σ_case1,g。

2. according to the temperature specified by step 1, resonance absorption nucleic and slowing down nucleic, by this resonance absorption nucleic and should Slowing down nucleic uniformly mixes, and the nucleon density ratio of resonance absorption nucleic and slowing down nucleic is in step 1 the 95% of ratio, calculates Go out corresponding background cross section value σ_b,case2, solve the moderation of neutrons equation of correspondence and obtain Continuous Energy netron-flux density, Finally by and group be calculated the multigroup absorption cross-section of this resonance absorption nucleic, should at a temperature of, resonance absorption nucleic a certain The group cross-section of group is referred to as σ_case2,g。

3. according to temperature specified in step 1 and resonance absorption nucleic, it is intended that institute in new slowing down nucleic step of replacing 1 The main slowing down nucleic specified, uniformly mixes this resonance absorption nucleic and new slowing down nucleic, adjust resonance absorption nucleic with New slowing down nucleic nucleon ratio example, makes to calculate corresponding background cross section value σ_b,case3It is worth equal to background cross section in step 2 (σ_b,case3=σ_b,case2), solve corresponding moderation of neutrons equation and obtain Continuous Energy netron-flux density, finally by also group's meter Calculating the multigroup absorption cross-section obtaining this resonance absorption nucleic, at a temperature of being somebody's turn to do, certain group of group cross-section of resonance absorption nucleic is called for short For σ_case3,g。

4. obtained the appointment energy group g of nucleic new in step 4, assigned temperature T, specific context cross section σ by formula (1)_b,spe =σ_b,case1Under the 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. σ is worth for the background cross section in step 1-3_b, use formula (2) to calculate:

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

Wherein,

N_mThe nucleon density of slowing down nucleic

σ_m,pThe elastic potential scattering cross section of slowing down nucleic

N_rThe nucleon density of resonance absorption nucleic

For the moderation of neutrons equation in step 1-3, 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 projectile energy

E ' neutron emanated energy

T Kelvin

N_rResonance absorption nucleic nucleon density

N_mSlowing down nucleic nucleon density

φ_T(E) the Continuous Energy netron-flux density at temperature TK

σ_r,s,TResonance absorption nucleic Continuous Energy neutron scattering matrix at (E ' → E) temperature TK

σ_t,T(E) the resonance absorption nucleic Continuous Energy total cross section under temperature is TK

σ_m,s,T(E ' → E) temperature is the slowing down nucleic Continuous Energy neutron scattering matrix under TK

σ_m,t,T(E) the slowing down nucleic Continuous Energy total cross section under temperature is TK

1) progressive scattering model is used for slowing down nucleic, free gas model is used for resonance absorption nucleic, 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 projectile energy

E ' neutron emanated energy

A_mSlowing down nucleic target nucleus and the mass ratio of neutron

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 projectile energy

E ' neutron emanated energy

A_rResonance absorption nucleic target nucleus and the mass ratio of neutron

T Kelvin

K Boltzmann constant

σ_r,s,TResonance absorption nucleic Continuous Energy neutron scattering matrix at (E ' → E) temperature TK

σ_t,T(E) the resonance absorption nucleic Continuous Energy total cross section under temperature is TK

σ_m,s,T(E ' → E) temperature is the slowing down nucleic Continuous Energy neutron scattering matrix under TK

σ_m,t,T(E) the slowing down nucleic Continuous Energy total cross section under temperature is TK

Wherein, H is Heaviside jump function, P_n(μ_lab) it is n rank Legnedre polynomials, μ_labFor the lower scattering of experiment system Angle, P (μ) is to scatter the probability of scattering distribution that azimuth is μ under center-of-mass angle.

2) use ultra-fine group's method to solve this moderation of neutrons equation, in ultra-fine group's method, resonance energy district is partitioned into The finest energy bite, each such energy group be referred to as a ultra-fine group, it is believed that each ultra-fine group's width is much smaller than The max log that the collision of neutron and final nucleic is obtained can drop, and i.e. thinks and the self-scattering of ultra-fine group can not occur, so Just acquisition fine flux can be solved to mental retardation by group by high energy successively after the scattering source of group as long as given.At 200eV Below, it is considered to free gas model, this model can cause the upper scattering effect of neutron, can not be disposably by height calculating energy time spectrum Can solve by group to mental retardation and obtain flux, by iterative computation until φ_T(E) convergence.

Calculating in step 1-3 and group, temperature is certain group of group cross-section σ of the resonance absorption nucleic under TK_g,TMeter Calculation method is:

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

Wherein,

E neutron projectile energy

E ' neutron emanated energy

G neutron incident energy group

G ' neutron outgoing energy group

ΔE_gThe energy bite of neutron incident energy group

ΔE_g′The energy bite of neutron outgoing energy group

φ_T(E) the Continuous Energy netron-flux density at temperature TK

σ_a,T(E) the Continuous Energy absorption cross-section at temperature TK

6. repeat step 1-4.Background cross section takes 10, and the nucleon ratio i.e. adjusting resonance nucleic and slowing down nucleic makes 1E1barn, 1E2barn, 1E3barn, 1E4barn, 1E5barn, 1E6barn, 1E7barn, 1E8barn, 1E9barn, 1E10barn.Temperature spot takes ten, i.e. 300K, 400K, 500K, 600K, 700K, 800K, 900K, 1000K, 1100K, 1200K. Each energy group's intermediate resonance factor under a large amount of different temperatures and background cross section is obtained with this.

7. can group's intermediate resonance factor according to each under a large amount of different temperatures obtained in step 6 and background cross section. Each can group with a large amount of intermediate resonance factors as match point, with background cross section and temperature as independent variable, intended by least square Conjunction method obtains multinomial coefficient.

Claims (1)

1. the method for the intermediate resonance factor obtained in reactor multigroup nuclear data depositary, it is characterised in that: include walking as follows Rapid:

Step 1: specify a temperature, a resonance absorption nucleic and a slowing down nucleic, this resonance absorption nucleic is slow with this Change nucleic uniformly to mix, according to the nucleon density ratio of resonance absorption nucleic and slowing down nucleic, calculate corresponding background and cut Face amount σ_b,case1, solve the moderation of neutrons equation of correspondence and obtain Continuous Energy netron-flux density, calculating finally by also group Obtaining the multigroup absorption cross-section of this resonance absorption nucleic, at a temperature of being somebody's turn to do, certain group of group cross-section of resonance absorption nucleic is referred to as σ_case1,g；

Step 2: according to the temperature specified by step 1, resonance absorption nucleic and slowing down nucleic, by this resonance absorption nucleic and should Slowing down nucleic uniformly mixes, and the nucleon density ratio of resonance absorption nucleic and slowing down nucleic is in step 1 the 95% of ratio, calculates Go out corresponding background cross section value σ_b,case2, solve the moderation of neutrons equation of correspondence and obtain Continuous Energy netron-flux density, Finally by and group be calculated the multigroup absorption cross-section of this resonance absorption nucleic, should at a temperature of, resonance absorption nucleic a certain The group cross-section of group is referred to as σ_case2,g；

Step 3: according to temperature specified in step 1 and resonance absorption nucleic, it is intended that institute in new slowing down nucleic step of replacing 1 The main slowing down nucleic specified, uniformly mixes this resonance absorption nucleic and new slowing down nucleic, adjust resonance absorption nucleic with New slowing down nucleic nucleon ratio example, makes to calculate corresponding background cross section value σ_b,case3It is worth equal to background cross section in step 2, i.e. σ_b,case3=σ_b,case2, solve corresponding moderation of neutrons equation and obtain Continuous Energy netron-flux density, calculating finally by also group Obtaining the multigroup absorption cross-section of this resonance absorption nucleic, at a temperature of being somebody's turn to do, certain group of group cross-section of resonance absorption nucleic is referred to as σ_case3,g；

Step 4: obtained the appointment energy group g of new nucleic, assigned temperature T, specific context cross section σ by formula (1)_b,spe=σ_b,case1 Under the intermediate resonance factor

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

Step 5: σ is worth for the background cross section in step 1-3_b, use formula (2) to calculate:

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

Wherein,

N_mThe nucleon density of slowing down nucleic

σ_m,pThe elastic potential scattering cross section of slowing down nucleic

N_rThe nucleon density of resonance absorption nucleic

For the moderation of neutrons equation in step 1-3, 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 projectile energy

E ' neutron emanated energy

T Kelvin

N_rResonance absorption nucleic nucleon density

N_mSlowing down nucleic nucleon density

φ_T(E) the Continuous Energy netron-flux density at temperature TK

σ_r,s,TResonance absorption nucleic Continuous Energy neutron scattering matrix at (E ' → E) temperature TK

σ_t,T(E) the resonance absorption nucleic Continuous Energy total cross section under temperature is TK

σ_m,s,T(E ' → E) temperature is the slowing down nucleic Continuous Energy neutron scattering matrix under TK

σ_m,t,T(E) the slowing down nucleic Continuous Energy total cross section under temperature is TK；

1) progressive scattering model is used for slowing down nucleic, free gas model is used for resonance absorption nucleic,

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 projectile energy

E ' neutron emanated energy

A_mSlowing down nucleic target nucleus and the mass ratio of neutron

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 projectile energy

E ' neutron emanated energy

A_rResonance absorption nucleic target nucleus and the mass ratio of neutron

T Kelvin

K Boltzmann constant

σ_r,s,TResonance absorption nucleic Continuous Energy neutron scattering matrix at (E ' → E) temperature TK

σ_t,T(E) the resonance absorption nucleic Continuous Energy total cross section under temperature is TK

σ_m,s,T(E ' → E) temperature is the slowing down nucleic Continuous Energy neutron scattering matrix under TK

σ_m,t,T(E) the slowing down nucleic Continuous Energy total cross section under temperature is TK

Wherein, H is Heaviside jump function, P_n(μ_lab) it is n rank Legnedre polynomials, μ_labFor the lower angle of scattering of experiment system, P (μ) for scattering the probability of scattering distribution that azimuth is μ under center-of-mass angle；

2) use ultra-fine group's method to solve this moderation of neutrons equation, in ultra-fine group's method, resonance energy district is partitioned into very Fine energy bite, each such energy group be referred to as a ultra-fine group, it is believed that each ultra-fine group's width is much smaller than neutron The max log obtained with final nucleic collision can drop, and i.e. thinks and the self-scattering of ultra-fine group can not occur, as long as so Give and just can be solved acquisition fine flux to mental retardation by group by high energy successively after the scattering source of group；At below 200eV, Consider free gas model, this model can cause the upper scattering effect of neutron, calculate can time spectrum can not disposably by high energy to Mental retardation solves by group and obtains flux, by iterative computation until φ_T(E) convergence；

Calculating in step 1-3 and group, temperature is certain group of group cross-section σ of the resonance absorption nucleic under TK_g,TCalculating side Method is:

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

Wherein,

E neutron projectile energy

E ' neutron emanated energy

G neutron incident energy group

G ' neutron outgoing energy group

ΔE_gThe energy bite of neutron incident energy group

ΔE_g′The energy bite of neutron outgoing energy group

φ_T(E) the Continuous Energy netron-flux density at temperature TK

σ_a,T(E) the Continuous Energy absorption cross-section at temperature TK

Step 6: repeating step 1-4, background cross section takes 10, the nucleon ratio i.e. adjusting resonance nucleic and slowing down nucleic makes 1E1barn, 1E2barn, 1E3barn, 1E4barn, 1E5barn, 1E6barn, 1E7barn, 1E8barn, 1E9barn, 1E10barn；Temperature spot takes ten, i.e. 300K, 400K, 500K, 600K, 700K, 800K, 900K, 1000K, 1100K, 1200K； Each energy group's intermediate resonance factor under a large amount of different temperatures and background cross section is obtained with this；

Step 7: according to each energy group's intermediate resonance factor under a large amount of different temperatures obtained in step 6 and background cross section, Each can group with a large amount of intermediate resonance factors as match point, with background cross section and temperature as independent variable, intended by least square Conjunction method obtains multinomial coefficient.