中子输运的随机理论

Stochastic Theory of Neutron Transport

运用概率理论,考虑t时刻n个相空间点(r,uiΩi)单位体元中分别出现Ni(i=1,2,…,n)个中子的概率PN(r,t,uΩ),提出一个新的中子输运的随机理论,导出概率母函数Fn的非线性积分微分方程组。在某些近似下,n=1概率分布一阶矩方程恰好是中子平均数玻尔兹曼方程。将各向同性散射的单速中子随机理论应用于点堆模型。在一个超临界系统中,当t→∞时,出现有限个中子的概率为零,PN=0(0<N<∞),即系统内或没有中子,或有无限多中子。给出了母函数的近似解,导出了母函数概率分布各阶矩的近似方程及解式。标准差公式表明,当初始中子数起伏ξ0较大,初始中子平均数N0不够多,或中子源强Q很弱时,对于0<λ1的增殖系统,中子数的起伏很大,应予以重视。

A stochastic neutron transport theory, in which we consider the probability PN（r, t,uΩ） that the neutron densities Ni （i= 1, 2, ..., n） emerge in the phase space point （r, ui Ωi） at time t respectively, was given by means of the probability theory, and a set of non-linear integral-differential equations for the probability generating functions Fn （r, t, uΩ, S ） was derived. The equation for one-order moment δF1/δS1 under some approximation is just the Boltzman equation for the average neutron number. Onevelocity neutron stochastic theory with isotropic scatting was applied to a point model. An approximate solution for the generating function and the equations for moments of the probability distribution and their solutions were derived. It is shown that in a supercritical system, at t→∞, the probability appearing finite neutrons is zero, PN = 0 （0〈 N 〈∞）, in other words, the system has no or infinite neutrons. A formula for standard deviation shows that the fluctuation of neutron number in the near critical （0 〈λ 〈〈 1） system should be paid our attention when the fluctuation of initial neutron number ξ0 is larger and the initial neutron average number N0 is not large enough, or neutron source Q is weaker.