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Ordering by node number

In this section we perform a more elaborate investigation of the form of the different matrix elements which appear in a node number ordering of the diffusion equations (1.43). Analogously to section 1.6.1 we now define the flux vector $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}$^1.24

$$\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex} \;\hbox{$=$... ...ex}\underline{\phi}{}\hspace{0.15ex}_{K-2} \end{array} \right]$$ (1.58)

where $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}_k$ is defined by

$$\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}_k \;\hbox{$... ...}{1.1em} \phi_k^2 \\ \vdots \\ \phi_k^G \end{array} \right]$$ (1.59)

Now, we define the scattering matrix at region k^1.25

$$\hspace{0.2ex}\underline{\underline{\overline{\Sigma}{}}}{}... ...rray} \right], \hspace{5em} \forall k \in \{1,2,\ldots,(K-1)\}$$ (1.60)

the diffusion coefficient matrix at region k

$$\hspace{0.2ex}\underline{\underline{\overline{D}{}}}{}\hspa... ...rray} \right], \hspace{5em} \forall k \in \{1,2,\ldots,(K-1)\}$$ (1.61)

the absorption matrix at region k

$$\hspace{0.2ex}\underline{\underline{\overline{\Sigma}{}}}{}... ...rray} \right], \hspace{5em} \forall k \in \{1,2,\ldots,(K-1)\}$$ (1.62)

the out-scattering matrix at region k

$$\hspace{0.2ex}\underline{\underline{\overline{\Sigma}{}}}{}... ...rray} \right], \hspace{4em} \forall k \in \{1,2,\ldots,(K-1)\}$$ (1.63)

the fission spectrum matrix (which we in the mathematical description consider independent of region number)

$$\hspace{0.2ex}\underline{\underline{\chi}}{}\hspace{0.15ex}... ...2 & & \\ & & \ddots & \\ & & & \chi^G \end{array} \right]$$ (1.64)

and finally the fission source matrix at region k

$$\hspace{0.2ex}\underline{\underline{\cal F}}{}\hspace{0.15e... ...mbox{ rows, } \hspace{5em} \forall k \in \{1,2,\ldots,(K-1)\}$$ (1.65)

where

$$\hspace{0.2ex}\underline{f}{}\hspace{0.15ex}_k \;\hbox{$=$\... ...cdots &,& (\overline{\nu \Sigma_f^G}{})_k \end{array} \right]$$ (1.66)

In the above mentioned definition we work with quantities $(\overline{\nu \Sigma_g^1}{})_k$. This is due to the fact that the program used for the generation of homogenized cross sections calculates this product. In fact, even though we so far have worked with the assumption that $\nu$ is space independent this assumption does not describe the whole truth. Since $\nu$ is energy dependent and the energy spectrum changes slightly with position, $\nu$ in fact changes a (very) little with space (ie z). This comment is also valid for the fission spectrum specified by $\hspace{0.2ex}\underline{\underline{\chi}}{}\hspace{0.15ex}$--I have, however, for simplicity chosen to exclude this from the mathematical description.

With the above mentioned matrices we are able to write down expressions for the components of the block matrices $\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex}$ and $\hspace{0.2ex}\underline{\underline{C}}{}\hspace{0.15ex}$ given in equations (1.74) and (1.76) respectively.

$$\begin{eqnarray*} \hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex}_k & ... ...underline{\Sigma}}{}\hspace{0.15ex}}{}_{s,k-1} \right) \right] \end{eqnarray*}$$

$\parbox{1.50cm}{\begin{eqnarray} \end{eqnarray}}$

and

$$\hspace{0.2ex}\underline{\underline{C}}{}\hspace{0.15ex}_k ... ...{\hspace{0.2ex}\underline{\underline{D}}{}\hspace{0.15ex}}{}_k$$ (1.67)

and finally (see (1.76))

$$\hspace{0.2ex}\underline{\underline{\hat{C}}}{}\hspace{0.15... ...hspace{0.2ex}\underline{\underline{\cal F}}{}\hspace{0.15ex}_k$$ (1.68)

In terms of defining a removal matrix $\overline{\hspace{0.2ex}\underline{\underline{\Sigma}}{}\hspace{0.15ex}}{}_{r,k}$

$$\overline{\hspace{0.2ex}\underline{\underline{\Sigma}}{}\hs... ...0.2ex}\underline{\underline{\Sigma}}{}\hspace{0.15ex}}{}_{s,k}$$ (1.69)

we can write the following expression for $\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex}_k$

$$\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex}_k ... ...underline{\Sigma}}{}\hspace{0.15ex}}{}_{r,k-1} \right) \right]$$ (1.70)

The generalized eigenvalue problem may now be stated as

$$\lambda_0 \hspace{0.2ex}\underline{\underline{B}}{}\hspace{... ...hspace{0.15ex} \hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}$$ (1.71)

where $\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex}$, $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}$ and $\hspace{0.2ex}\underline{\underline{C}}{}\hspace{0.15ex}$ are defined by (1.74) (see also (1.75), (1.86) and (1.87)),(1.77) and (1.76) (see also (1.88)) respectively.

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