Two methods are in general use for the solution of the nuclide rate equations:one is the numerical integration method and the other is the analytic relations method.
In the numerical integration method, three principal difficulties are encountered in employing this method to solve a large system of equations. First, a large amount of memory is required to store the transition matrix and the matrix exponential functions, and second, computational problems are encountered in applying matrix exponential method to a system of equations with widely separated eigenvalues, and third, the results contain truncation errors due to discrete time steps. A typical example that is based on this method is the ORIGEN2 code.
The analytic relations method directly integrates the linear differential equations governing radioactive transformation and mass transfer using analytic relations and recursion relations. Thus, the computation is efficient and fast and involves no truncation errors. The code KIGEN was developed on the basis of this method with the assumption of no backward decay reactions in the case of fission products and with the assumption of purely forward or backward decay reactions in the case of actinides. Under the above assumptions, the analytic relations are obtained for each nuclide in terms of decay chain paths, which make it possible to compute the concentration of each nuclide in a relatively short computer time with an acceptable accuracy.
However the whole decay paths cannot be considered in KIGEN due to the above assumptions, and, in particular, the errors in the concentrations of actinides become significant after long cooling periods. The accurate calculation of the concentrations of the actinides is of importance because they are major contributors to the health effects to the human body.
To calculate the concentrations of actinides accurately, the computer program KIGEN2 was developed in this study by improving KIGEN. It allows both forward and backward decay reactions simultaneously, thus removing the major limitation in KIGEN. The transition matrix for the whole actinides was constructed using the ENDF/B-IV decay library. The whole decay-chain-path file was found for all actinides by matching every daughter nuclide to its all parent nuclides. For a particular reactor model this file is found only once. Using this decay-chain-path file, KIGEN2 calculates the concentrations of the whole nuclides by evaluating analytic relations. Once the concentrations of the nuclides are found, the calculation of the following quantity is almost trivial: decay heat, radioactive toxicity index, cancer doses of toxic nuclides whose toxicities are evaluated on the basis of the maximum permissible concentrations, and fatal cancer risk per radiation absorbed dose to various body organs. Thus, KIGEN2 developed in this study allows us to obtain accurate concentrations of the fission products and actinides and other important related quantities, using approximately half of the computer time required by ORIGEN2. Major assumptions employed in other analytic relations methods are removed except that there is no neutron, which is generally assumed in decay calculations.
방사능 물질의 천이와 붕괴를 나타내는 선형 미분 방정식들을 푸는 많은 방법들이 개발되어 왔으며 알려져 있다. 그 중에서 수치적분을 이용하는 방법과 해석적 관계식을 이용하는 두가지 방법이 널리 쓰이고 있는데 수치적분 방법은 많은 전산시간을 필요로하는 단점을 가지고 있으며 한편으로 절단오차를 포함한다. 이러한 단점들을 제거하기 위해 해석적 관계식들을 이용하는 방법이 개발되었으며 이 방법은 짧은 전산 시간으로써 짧은 냉각주기 동안에 각 핵종들의 붕괴열을 아주 정확하게 계산 한다.
그러나 장 냉각주기동안에는 방사능 물질의 안전해석을 위해서는 액티나이드의 핵종농도가 정확히 계산되어야 하나 이 방법에 의하면 액티나이드에 대한 상대오차가 급속히 증가된다.
따라서 이 연구에서는 액티나이드에 대한 정확한 붕괴경로를 구하여 액티나이드의 핵종계산을 수행하였다. 이 작업을 위해서 ENDF/B-IV 라이브러리에 의거하여 천이 매트릭스 (matrix)를 구성하고 이 천이 매트릭스(matrix)에 의해 액티나이드 핵종의 붕괴경로를 구하고 이 붕괴경로에 의한 해석적 관계식을 계산하기 위해 기존의 프로그램 KIGEN을 KIGEN2로 수정하였다. 이 계산의 결과는 붕괴열과 핵종농도에 의거해 계산되는 많은 값들에 대해 ORIGEN2 코드의 결과와 비교되어졌다. 이 결과에 의하면 붕괴열은 $10^7$ 년까지 최대 11% 의 상대 오차를 보였으며 전산시간은 ORIGEN2의 약 1/2 정도였다.
따라서 이 방법에 의해 초기 핵종농도를 알고 있는 사용후 핵연료에 대해 짧은 전산시간으로써 방사능 물질의 안전해석을 위한 정확한 핵종농도와 이에 관련된 여러가지 상수들을 계산할 수 있게 되었다.
