Gamma multiplicity and gamma and neutron mixing moment computational formula derivation method based on probability generation function

Abstract

The invention discloses a probability generation function-based gamma multiplicity and gamma-neutron mixing moment calculation formula derivation method, relates to the technical field of nuclear physical calculation, and has the technical key points that the invention provides a gamma multiplicity and gamma-neutron mixing moment algorithm, which provides a theoretical basis for improving the quality detection efficiency of nuclear material samples. And the method has very important significance for improving the management and control capability of the nuclear material.

Description

Technical Field

[0001] The invention relates to the field of nuclear physics calculation technology, and in particular to a method for deriving a calculation formula for gamma multiplicity and gamma and neutron mixing moment based on a probability generating function. Background Art

[0002] With the continuous advancement of nuclear material identification technology, the attribute authentication of nuclear materials such as uranium and plutonium has received more and more attention. Attribute authentication can be performed by measuring the neutron multiplicity and gamma multiplicity of nuclear material fission. At present, the theoretical equations and measurement methods for the neutron multiplicity of nuclear materials have been relatively mature after years of research. Although the calculation formula of the gamma multiplicity of nuclear materials is similar to the derivation method of the neutron multiplicity calculation formula, it considers many factors, has strong nonlinearity, and is more complicated. Its analysis model and measurement equation are still under development.

[0003] To this end, the present invention aims to provide a method for deriving a formula for calculating the gamma multiplicity and the gamma and neutron mixing moment based on a probability generating function to solve the above problems. Summary of the invention

[0004] The purpose of the present invention is to solve the above problems, provide a method for deriving the calculation formula of gamma multiplicity and gamma and neutron mixing moment based on probability generating function, and propose an algorithm for gamma multiplicity and gamma and neutron mixing moment, which provides a theoretical basis for improving the quality detection efficiency of nuclear material samples and is of great significance for improving the control capability of nuclear materials.

[0005] In order to achieve the above object, the technical solution of the present invention is as follows:

[0006] The present invention provides a method for deriving a calculation formula of γ multiplicity and γ and neutron mixing moment based on a probability generating function, and the method comprises the following steps:

[0007] S1, Probability production function:

[0008]

[0009] S2. Calculation of γ multiplicity: Derive the probability production function of f1(n) and the first three factorials of γ:

[0010] The probability production function of f1(n) is

[0011]

[0012] The first three factorial distances derived from γ are

[0013] G"'(z)=r s "'(z)q s [g(z)]+3r s "(z)q s'[g(z)]g'(z)+3r s '(z)q s "[g(z)](g'(z)) 2

[0014] +3r s '(z)q s '[g(z)]g”(z)+r s (z)g s "'[g(z)](g'(z)) 3

[0015] +3r s (z)q s "[g(z)]g'(z)g" (z)+r s (z)g' s [g(z)]g"'(z).

[0016] S3. Derivation of the formula for calculating the mixing moment of γ and neutron: triggered by a neutron, n1 neutrons and n2 γ are emitted from a sample, and there is no absorption. The joint distribution probability is:

[0017]

[0018] The joint distribution probability of emitting n1 neutrons and n2 gamma rays from a sample triggered by a source is

[0019]

[0020] Its probability generating function is:

[0021]

[0022] In step S1, P s (n) is the probability of producing n neutrons for each initial source, and P(n) is the probability of producing n neutrons for each initial source;

[0023] f s (n) is the probability of an initial event producing n gammas, f(n) is the probability of an induced event producing n gammas, p is the probability of a neutron generated in the sample inducing fission before leaving the sample, f1(n) is the probability of a single neutron producing n gammas in all fissions, and F(n) is the probability distribution of the total number of gammas produced by a source event.

[0024] In step S2, multiply both sides of the expression f1(n) by z n , and then sum from n = 0 to n = ∞ to get:

[0025]

[0026] The left side of the above formula is g(z), and the first term on the right side is (1-p), so we can get:

[0027]

[0028] Similar results are available:

[0029] Let z = 1, and we can derive the first three factorial distances of γ:

[0030] G"'(z)=r s "'(z)q s [g(z)]+3r s "(z)q s '[g(z)]g'(z)+3r s '(z)q s "[g(z)](g'(z)) 2

[0031] +3r s '(z)q s '[g(z)]g”(z)+r s (z)g s "'[g(z)](g'(z)) 3

[0032] +3r s (z)q s "[g(z)]g'(z)g" (z)+r s (z)g' s [g(z)]g"'(z).

[0033] Compared with the prior art, this solution has the following beneficial effects:

[0034] The present invention has higher efficiency in measuring gamma multiplicity than measuring neutron multiplicity in discovering, identifying and quantifying special nuclear materials, provides a theoretical basis for improving the efficiency of nuclear material sample quality detection, and is of great significance for improving the management and control capabilities of nuclear materials. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 is a flow chart of a method in an embodiment of the present invention. DETAILED DESCRIPTION

[0036] In order to enable those skilled in the art to better understand the solution of the present invention, the technical solution of the present invention will be further described in detail below in conjunction with the embodiments of the present invention and the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work should fall within the scope of protection of the present invention.

[0037] It should be noted that, in the absence of conflict, the embodiments and features in the embodiments of the present invention can be combined with each other. The present invention will be described in detail below in conjunction with the embodiments.

[0038] Example:

[0039] 1. Probability Production Function

[0040] 1.1 Probability Generating Function of Seed Distribution

[0041] (1) Let P s (n) is the probability that each initial source produces n neutrons, then YesP s The probability generating function of (n) is denoted by its factorial moment v s , <v s (v s -1)〉,…

[0042] (2) Let P(n) be the probability that each initial source produces n neutrons, then is the probability generating function of P(n), whose factorial moment is v,<v(v-1)> ,…

[0043] 1.2 Distribution of γ

[0044] (1) Let f s (n) is the probability that the initial event generates n γ, then Yes s The probability generating function of (n), the factorial moment of f(n) is μ, <μ(μ-1)>,…

[0045] (2) Let f(n) be the probability that the triggering event generates n γs, then is the probability generating function of f(n), f s The factorial moment of (n) is μ s ,<μ s (μ s -1)>,…

[0046] (3) p is the probability that a neutron generated in the sample will induce fission before leaving the sample, f1(n) is the probability that a single neutron will produce n gammas in all fissions, is the generating function of f1(n), and the values ​​of the first, second, and third derivatives of g(z) at z = 1 are denoted by g1, g2, and g3. F(n) is the probability distribution of the total number of source events. is the probability generating function of F(n), and the factorial moment of F(n) is

[0047] The expressions of f1(n) and F(n) are as follows:

[0048]

[0049] 2. Calculation of γ multiplicity

[0050] 2.1 Derivation of the probability generating function of f1(n)

[0051] Multiply both sides of the expression f1(n) by z n , and then sum from n = 0 to n = ∞ to get:

[0052]

[0053] The left side of the above formula is g(z), and the first term on the right side is (1-p). The key is to analyze and study the second term on the right side:

[0054] When n=0, the first formula of the second term on the right side is:

[0055] pf(0)(p(1)f1(0)+2p(2)f1(0)f1(1)+3p(3)f1 2 (0)f1(1)+…)z 1

[0056] When n=1n0=0, pf(0)(p(1)f1(0)+2p(2)f1(0)f1(1)+3p(3)f1 2 (0)f1(1)+…)z 1 ;

[0057] When n=1n0=1, pf(1)(p(1)f1(0)+p(2)f1 2 (0)+p(3)f1 3 (0)+…)z 1 ;

[0058] When n=2n0=0,

[0059] pf(0)(p(1)f1(2)+p(2)(2f1(0)f1(2)+f1 2 (1))+p(3)(3f1 2 (0)f1(2)+3f1 2 (1)f1(0))+…)z2 ;

[0060] When n=2n0=1, pf(1)(p(1)f1(1)+2p(2)f1(0)f1(1)+3p(3)f1 2 (0)f1(1)+…)z 2 ;

[0061] When n=2n0=1, pf(2)(p(1)f1(0)+p(2)f1 2 (0)+p(3)f1 3 (0)+…)z 2 …

[0062] Sum each term, then pf(0), pf(1)z, pf(2)z 2 , the multiple of ... is the same, and this multiple is:

[0063] p(1)(f1(0)+f1(1)z+f1(2)z 2 +…+f1(n)z n …)+p(2)(f1 2 (0)+2f1(0)f1(1)z+(2f1(0)f1(2)+f1 2 (1))z 2 +…)+…

[0064] The second item is:

[0065] Let p(0) = 0, then then

[0066]

[0067] Similar to

[0068] 2.2 Derivation of the first three factorial distances of γ

[0069] By taking the derivative of g(z) we get

[0070] Let z = 1, g1 = g′(1) = pμ + pvg′(1), then

[0071] From G(z) we can get G′(z)=r s ′(z)q s [g(z)]+r s (z)q s ′[g(z)]g′(z),

[0072] Let z = 1, then

[0073] Taking the derivative of g′(z) we get

[0074]

[0075] Let z = 1,

[0076]

[0077] Taking the derivative of G′(z) again, we get:

[0078] G″(z)=r s ″(z)q s [g(z)]+r s ′(z)q s ′[g(z)]g'(z)+r s ′(z)q s ′[g(z)]g'(z)

[0079] +r s (z)q s ″[g(z)](g'(z)) 2 +r s (z)q s ′[g(z)]g”(z)

[0080] Let z = 1, By further derivation of g″(z), we can get:

[0081]

[0082] Let z = 1:

[0083]

[0084] Taking the derivative of G″(z) again, we get:

[0085] G"'(z)=r s "'(z)q s [g(z)]+3r s "(z)q s '[g(z)]g'(z)+3r s '(z)q s "[g(z)](g'(z)) 2

[0086] +3r s '(z)q s '[g(z)]g”(z)+r s (z)g s "'[g(z)](g'(z)) 3

[0087] +3r s (z)q s "[g(z)]g'(z)g" (z)+r s (z)g' s [g(z)]g"'(z)

[0088] Let z = 1:

[0089]

[0090] 3. Derivation of the formula for calculating the γ-neutron mixing moment

[0091] Triggered by a neutron, n1 neutrons and n2 gamma rays are emitted from a sample, with no absorption, and the joint distribution probability is:

[0092]

[0093] Its probability generating function is:

[0094]

[0095] The joint distribution probability of emitting n1 neutrons and n2 gamma rays from a sample triggered by a source is:

[0096]

[0097] Its probability generating function is:

[0098]

[0099] By simplifying the above two joint probability generating functions u(z1,z2) and U(z1,z2), we can get:

[0100]

[0101] Method for obtaining

[0102]

[0103]

[0104] Similarly:

[0105] remember:

[0106]

[0107]

[0108] The above specific embodiments are merely explanations of the present invention and are not limitations of the present invention. After reading this specification, those skilled in the art may make modifications to the embodiments without any creative contribution as needed. However, such modifications are protected by the patent law as long as they are within the scope of the claims of the present invention.

Claims

1. A method for deriving the formula for calculating the γ multiplicity and γ-neutron mixing moment based on the probability generating function, characterized by: The method comprises the following steps: S1, Probability production function: S2. Calculation of γ multiplicity: Derive the probability production function of f1(n) and the first three factorials of γ: The probability production function of f1(n) is The first three factorial distances derived from γ are G”'(z)=r s "'(z)q s [g(z)]+3r s "(z)q s '[g(z)]g'(z)+3r s '(z)q s "[g(z)](g'(z)) 2 +3r s '(z)q s '[g(z)]g”(z)+r s (z)g s "'[g(z)](g'(z)) 3 +3r s (z)q s "[g(z)]g'(z)g”(z)+r s (z)g' s [g(z)]g"'(z)。 S3. Derivation of the formula for calculating the mixing moment of γ and neutron: triggered by a neutron, n1 neutrons and n2 γ are emitted from a sample, and there is no absorption. The joint distribution probability is: The joint distribution probability of emitting n1 neutrons and n2 gamma rays from a sample triggered by a source is Its probability generating function is:

2. The method for deriving the formula for calculating the gamma multiplicity and the gamma and neutron mixing moment based on the probability generating function according to claim 1, characterized in that: In step S1, P s (n) is the probability of producing n neutrons for each initial source, and P(n) is the probability of producing n neutrons for each initial source; f s (n) is the probability of an initial event producing n gammas, f(n) is the probability of an induced event producing n gammas, p is the probability of a neutron generated in the sample inducing fission before leaving the sample, f1(n) is the probability of a single neutron producing n gammas in all fissions, and F(n) is the probability distribution of the total number of gammas produced by a source event.

3. The method for deriving the formula for calculating the gamma multiplicity and the gamma and neutron mixing moment based on the probability generating function according to claim 1 is characterized in that: In step S2, multiply both sides of the expression f1(n) by z n , and then sum from n = 0 to n = ∞ to get: The left side of the above formula is g(z), and the first term on the right side is (1-p), so we can get: Similar results are available:

4. The method for deriving the formula for calculating the gamma multiplicity and the gamma and neutron mixing moment based on the probability generating function according to claim 3 is characterized by: Let z = 1, and we can derive the first three factorial distances of γ: G”'(z)=r s "'(z)q s [g(z)]+3r s "(z)q s '[g(z)]g'(z)+3r s '(z)q s "[g(z)](g'(z)) 2 +3r s '(z)q s '[g(z)]g”(z)+r s (z)g s "'[g(z)](g'(z)) 3 +3r s (z)q s "[g(z)]g'(z)g”(z)+r s (z)g' s [g(z)]g"'(z)。

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