Eigenvalue Calculations - Mathematical Basis

The time-independent Boltzmann transport equation for neutrons is an equation for the neutron angular density. This equation is derived by applying neutron conservation to an infinitesimal element of volume, direction and energy [Bell70, Lew93]. The integro-differential form of the time-independent transport equation in a medium with volume V surrounded by a surface S, as shown in figure 2.1, can be written as,

$$\vec \Omega \cdot \vec \nabla \psi (\vec r,\vec \Omega,E )+\... ... r,E)\psi (\vec r,\vec \Omega,E )= Q(\vec r,\vec \Omega,E ) ,$$

$$\vec r\in V, \;\;\; 0 < E < \infty, \;\;\; \vert\vec \Omega\vert=1 .$$ (1)

 

Figure: Spatial Domain V for the Transport Equation with Surface S.

The boundary condition, if $\vec n$ is an outward normal to S, is given as,

$$\psi (\vec r,\vec \Omega,E ) = \psi^{b} (\vec r,\vec \Omega,E ), \;\;\; \vec \Omega \cdot \vec n < 0, \;\;\; \vec r \in S .$$ (2)

In the transport equation, $\vec r$ denotes spatial variables, E denotes energy, $\vec \Omega$ denotes angular variables, $\psi (\vec r,\vec \Omega,E )$is the angular flux, $\Sigma_t(\vec r,E)$ is the macroscopic total cross section, and $Q(\vec r,\vec \Omega,E )$ is the source emission density.

In general, the source emission density $Q(\vec r,\vec \Omega,E )$ consists of three contributors:

Q = Q_ex + Q_s + Q_f .

The above three contributors are due to the external source, scattered particles, and fission neutrons, respectively.

For nonmultiplying systems Q_f=0, and the two contributions to the source emission density are due to the external source and the scattered particles. Hence the transport equation in a nonmultiplying medium is given by,

$$\vec \Omega \cdot \vec \nabla \psi (\vec r,\vec \Omega,E )+\Sigma _t(\vec r,E)\psi (\vec r,\vec \Omega,E )=$$

$$Q_{ex}(\vec r,\vec \Omega,E )+\int dE' \int d \Omega' \Sigm... ... \Omega' \cdot \vec \Omega ) \psi (\vec r,E',\vec \Omega' ) ,$$ (4)

where $\Sigma_s(\vec r,E' \rightarrow E, \vec \Omega' \cdot \vec \Omega )$ is the macroscopic scattering cross section.

For multiplying media, the contribution Q_f due to the neutrons emitted from fission reactions must be included. We will assume that all neutrons are produced instantaneously at the time of fission and neglect the effect of delayed neutrons. This is a valid assumption, except for time dependent problems of neutron kinetics. This assumption is appropriate for steady state problems with fixed sources and for criticality calculations, where only the critical state and the flux distribution are of interest. The transport equation in a multiplying medium without delayed neutrons takes the following form:

$$\vec \Omega \cdot \vec \nabla \psi (\vec r,\vec \Omega,E )+\... ...,E)\psi (\vec r,\vec \Omega,E )= Q_{ex}(\vec r,\vec \Omega,E )+$$

$$\int dE' \int d \Omega' \Sigma _s(\vec r,E' \rightarrow E, \... ...f (\vec r, E') \int d \Omega' \psi (\vec r,E',\vec \Omega' ) .$$ (5)

Here, $\nu (E)$ is the mean number of fission neutrons produced in a fission caused by a neutron with energy E. The effect of delayed neutrons is included in the total $\bar{\nu}$. Also $\Sigma_f(\vec r,E)$ is the macroscopic fission cross section and $\chi (E)$ denotes the fission spectrum. The transport equation is a hyperbolic equation in both the time dependent and steady state forms. In a nonmultiplying medium with a steady state source, there exists a steady state flux distribution that satisfies this equation [Cas67]. In multiplying systems, the transport equation is studied by introducing the concept of criticality. Physically, a system containing fissionable material is critical if a self-sustaining time independent chain reaction in the absence of external neutron sources can be maintained. Mathematically, a system is critical if it has a time-independent, nonnegative solution to the transport equation without external source, i.e.

$$\vec \Omega \cdot \vec \nabla \psi (\vec r,\vec \Omega,E )+\Sigma _t(\vec r,E)\psi (\vec r,\vec \Omega,E )=$$

$$\int dE' \int d \Omega' \Sigma _s(\vec r,E' \rightarrow E, \... ...f (\vec r, E') \int d \Omega' \psi (\vec r,E',\vec \Omega' ) .$$ (6)

In general, it is difficult to find that combination of cross sections and geometry that will allow equation (2.6) to be satisfied. Therefore, for criticality calculations, the above equation is cast into the form of an eigenvalue problem. The eigenvalue provides a measure of the criticality of the system. Two formulations for criticality are the time-absorption or $\alpha$-eigenvalue formulation and the multiplication factor formulation with the K eigenvalue [Bell70]. The K eigenvalue approach is discussed in this thesis.

We have stated before that K can be defined as the quantity by which $\nu$must be divided to keep a system critical, i.e., $\nu$ can be adjusted to obtain a time independent solution to equation (2.6). Hence for the K eigenvalue problem $\nu$ is replaced by $\nu$/K and the transport equation becomes,

$$\vec \Omega \cdot \vec \nabla \psi (\vec r,\vec \Omega,E )+\Sigma _t(\vec r,E)\psi (\vec r,\vec \Omega,E )=$$

$$\int dE' \int d \Omega' \Sigma _s(\vec r,E' \rightarrow E, \... ...f (\vec r, E') \int d \Omega' \psi (\vec r,E',\vec \Omega' ) .$$ (7)

In an eigenvalue problem it is always possible to find the largest value of K that will give a nonnegative fundamental mode solution to equation 2.7. The system is critical if this largest value of K is equal to unity. If K < 1, then $\nu$, the actual number of neutrons per fission available, is less than $\nu$/K, the number of neutrons per fission necessary to maintain criticality, and the system is subcritical. Similarly
for K > 1, the actual number of neutrons per fission, $\nu$, is more than $\nu$/K, the number of neutrons per fission required to maintain the system critical, and hence the system is supercritical.

Equation (2.7) is the integro-differential form of the transport equation. This equation can also be expressed in integral form using the Green's function. The Green's function G($\vec r',\vec \Omega', E' \rightarrow \vec r, \vec \Omega, E$) is the neutron angular flux at $\vec r, \vec \Omega, E$ due to a unit point source at $\vec r'$ emitting 1 neutron/sec in the direction $\vec \Omega'$ with energy E'. Using the Green's function we can write down the K eigenvalue equation as follows:

$$\psi(\vec r,\vec \Omega, E ) =$$

$$\int \int \int d \vec r' d\vec \Omega' dE' G(\vec r', \vec \... ...''\nu \Sigma_f(\vec r',E'') \psi (\vec r',\vec \Omega'', E'') .$$ (8)

Both equation (2.7) and (2.8) can be expressed in terms of the fission source density, $\int \int \nu \Sigma_f(\vec r, E') \psi (\vec r, \vec \Omega', E')d \Omega'dE'$, which gives the spatial distribution of fission neutrons. From that form, both equations lead to the same matrix eigenvalue problem, to which the power iteration method can be applied to calculate K, as explained below.

For the time-independent, nonmultiplying transport equation given in equation (2.4), the transport operator is defined as [Lew93]:

$$H\psi = [\vec \Omega \cdot \vec \nabla + \Sigma _t(\vec r,E)] \psi (\vec r,\vec \Omega,E )$$

$$-\int dE' \int d \Omega' \Sigma _s(\vec r,E' \rightarrow E, \vec \Omega' \cdot \vec \Omega ) \psi (\vec r,E',\vec \Omega' ) .$$ (9)

Expressing the fission source density as,

$$F \psi = \int \int \nu \Sigma_f(\vec r, E') \psi (\vec r, \vec \Omega', E')d \Omega'dE',$$ (10)

the multiplication eigenvalue problem, equation (2.7), can be written in operator notation as

$$H \psi = \frac {1}{K} \chi F \psi .$$ (11)

Inverting H and operating on both sides by F, we obtain the following form of the eigenvalue problem:

$$F \psi = \frac {1}{K}[F H^{-1} \chi] F \psi .$$ (12)

Using the following definition for the transport operator

$$A \equiv F H^{-1} \chi ,$$ (13)

the eigenvalue equation becomes,

KS = AS ,

where S ( = F$\psi$) is the source eigenvector and A is the fission matrix. The element A(l,m) of the fission matrix represents the probability that a neutron starting in cell m will generate a fission neutron appearing in cell l. In integral form, the above equation can be expressed as,

$$KS(\vec r) = \int S(\vec r') A(\vec r,\vec r') dr',$$ (15)

where A$(\vec r,\vec r')$ is the operator that describes the probability that a neutron originating at $\vec r'$ produces a fission neutron at $\vec r$ and S$(\vec r)$ is the source distribution. The above equation can also be derived from the integral transport form of the K eigenvalue equation ( equation (2.8) ) by using the operator F and the following property of the Green's function:

$$\int \int \int d\vec r' d \vec \Omega' dE' G(\vec r', \vec \Omega',E' \rightarrow \vec r, \vec \Omega, E) = H^{-1} .$$ (16)

The multiplication eigenvalue problem of equation (2.14) is then solved by the method of power iteration [Var62, Wac66, Nak77].