N.G. Sjöstrand

The discrete ordinates method compared to Carlvik's and Syros's methods for anisotropic neutron scattering in slabs

The discrete ordinates method has been used to determine the criticality factor of infinite slabs with monoenergetic neutrons scattering anisotropically. The slab thickness was 0.2, 1, 2 or 20 mean free paths and the average cosine of the scattering angle 0, 0.1 or 0.2. The calculations were extended up to the S32 approximation for the thinnest slab. It was found that the Sn results extrapolate towards the values obtained by Dahl and Sjostrand using Carlviks method. The reason why Syros and Theocharopoulos obtain deviating results is not known.

Eigenvalues for reflecting boundary conditions in one-speed neutron transport theory

Abstract The spectrum of criticality eigenvalues for the one-speed neutron transport equation has been studied for an infinite slab with reflexion coefficients R 1 and R 2 at the surfaces. For R 1 = R 2 = −1 or +1 the problem can be solved exactly. In another case, R 1 = −1 and R 2 = 0, the problem is simplified considerably. When both reflexion coefficients are close to unity the criticality eigenvalues follow a very accurate approximation formula. For the high-order eigenvalues a semi-empirical formula is given. Some results have also been obtained for the corresponding time-dependent problems.

One-speed neutron transport in reflected spheres

Abstract The transport of one-speed neutrons has been studied in isotropically scattering spheres with reflective boundaries. An integral equation for the total flux has been derived and solved numerically for various radii and reflexion coefficients. Calculations have also been done for spheres in vacuum with varying degree of strongly peaked forward and backward scattering, so called Inonu scattering. Some special cases have been studied in detail, such as that when the forward-backward scattering dominates over the isotropic scattering.

Time eigenvalue spectrum for one-speed neutron transport in spheres with strong forward-backward scattering

Abstract The spectrum of time eigenvalues has been studied for one-speed neutrons in spheres with strong forward and backward scattering together with isotropic scattering. It is shown that in the presence of strong backward scattering there are both discrete eigenvalues and a continuum of eigenvalues. The lower limit of the continuum has been determined. The results are supported by numerical calculations.

Time-eigenvalue spectra for one-speed neutrons in systems with vacuum boundary conditions

Abstract Detailed calculations have been performed to establish the real time-eigen-values of one-speed neutrons. The systems studied are spheres and infinite slabs and cylinders. Vacuum boundary conditions and linearly anisotropic scattering are assumed. The eigenvalue curves in the three cases show great similarities, but the odd mode eigenvalues in slabs require a closer investigation. A comparison is made with eigenvalue curves obtained assuming periodic boundary conditions. The resemblances and differences are discussed.

Extrapolation distance of spheres and of infinite slabs and cylinders for monoenergetic neutrons scattering anisotropically

Abstract In an earlier paper the critical size of spheres and infinite slabs was calculated with a method by Carlvik for monoenergetic neutrons scattering anisotropically. From the results and from the work by Sanchez on the critical size of infinite cylinders, accurate extrapolation lengths have been derived for various dimensions and for an average cosine of the scattering angle up to 0.3. For not too thin slabs there is good agreement with an approximation formula derived by Davison. However, the formula is not applicable to spheres or cylinders in the presence of anisotropic scattering.

Criticality of reflected spherical reactors for neutrons of one speed

Abstract One-group transport theory is applied to two-medium spherical systems in the case when the mean-free-path is the same in both media. Two coupled integral equations are derived for the neutron flux in the respective medium. Accurate numerical values are calculated for critical, reflected reactors. Results are also given for pulsed-neutron systems used in neutron cross-section measurements.

On an alternative way to treat highly anisotropic scattering

It is shown that for several scattering functions, which describe highly anisotropic scattering, an integration over the polar angle can be performed analytically. This makes it easier to use such functions in calculations using, e.g. the Sn method and to avoid long series in Legendre polynomials.

One-speed neutron transport eigenvalues for an infinite, two-medium slab lattice

Abstract The monoenergetic transport equation with isotropic scattering is applied to an infinite, two-medium slab lattice. One of the media is assumed to have a multiplication factor larger than unity and the other smaller than unity. The two coupled integral equations that are derived are numerically solved using the spatial Legendre expansion method (Carlviks method). Tables of seven eigenvalues for various dimensions of the bodies are given and the first four flux modes for some cases are plotted. In addition, simple formulae for very high-order eigenvalues are given.

On a problem with strong forward and backward neutron scattering in spheres

Abstract The one-speed neutron transport equation with strong forward and backward scattering (characterized by the parameters α and β) may be transformed, according to Inonu (1973), into an “ordinary” transport equation. However, for spheres this transformation gives an artificial limit for α + β, beyond which it is not useful. In the present work it is shown that numerical results can be obtained in the “forbidden” region by solving the original transport equation by the Sn method, not using the Inonu transformation.