Effective boundary conditions in the theory of neutron diffusion

In this review, a method of determining effective boundary conditions (EBC) is described which guarantees the asymptotic agreement of the solution of the neutron diffusion equation with the solution of the corresponding kinetic equation. For monoenergetic neutrons, the EBC are considered for plane and cylindrical surfaces of black and gray bodies. Results are given for the EBC for the case of cylinders of arbitrary section.The authors discuss the simplest problem in the determination of EBC for neutrons that are slowed-down in a medium with heavy atoms and with an energy-independent cross section. The review gives also results obtained by various authors in the USSR and in other countries.     

Time-dependent boundary source term for advanced nodal diffusion theory

An extension to nodal diffusion theory was developed to deal with time-dependent boundary source terms. It is shown that this extension is easily introduced in advanced, full-functional diffusion theory by means of Dirac's delta spatially-dependent functions at the nodal boundaries, and allows the evaluation of reactor transients showing the propagation of neutron waves or the effects of pulsed neutron sources. A model problem with exact solutions for both the diffusion and the P₁ (telegrapher's) equations was developed to test the capabilities of the theoretical extension. The larger discrepancies occur at the earliest times computed showing, at t=50 μs, mean-square deviations between the exact diffusion solution and the numerical approximations of 2.95, 1.72, 0.32, and 0.16% for the fourth, sixth, eighth, and tenth polynomial expansion order, respectively. The availability of the exact telegrapher's solution, however, demonstrates that improved accuracy is meaningless since the mean-square deviation between the exact diffusion and exact telegrapher's solution is very similar to the mean-square deviation between the exact diffusion and the poorer (fourth-order) diffusion approximation.     

The transport and diffusion theory of a line source in an infinite half-space with internal reflection

A closed form analytical solution is obtained for a three-dimensional transport theory problem, namely that of a line source in a half-space in one-speed transport theory with an internally reflecting surface. The theory is developed by using Fourier transforms in the transverse directions and a Laplace transform in the axial direction, together with the Wiener–Hopf technique. Both specular and diffuse internal reflection are considered and it is shown that a closed form solution is not available for the specular case but is available for the diffuse case. The surface particle scalar intensity and current are obtained and their sensitivity to absorption and the reflection coefficient are assessed. We also obtain a number of analytic asymptotic estimates for the intensity valid at large distances from the source. Diffusion theory is also employed and compared numerically with the transport solution. In general, diffusion theory gives very satisfactory results within its regions of limitation. We also offer this paper as an exercise in analytical transport theory; an aspect of reactor physics which is not often seen in recent times.     

Transport consistent diffusion coefficient for CMFD acceleration and comparison of convergence properties

A diffusion coefficient for the coarse mesh finite difference (CMFD) acceleration is derived from the semi-analytic solution of one-group, one-dimensional, even-parity transport equation. The derived diffusion coefficient, i.e., the transport consistent diffusion coefficient (TCD), depends on the optical length of a mesh and shows similar behavior with the artificial grid diffusion (AGD) and the effective diffusion (EffD) coefficients for an optically thick mesh. Convergence properties of typical diffusion coefficients are evaluated using the linearized Fourier analysis. Analyses of the C5G7 3D benchmark problems with and without voided region are carried out to compare the convergence properties. The number of transport sweeps to reach convergence using TCD is smaller than that using EffD or AGD.     

A multi-region boundary element method for multigroup neutron diffusion calculations

For the analysis of a two-dimensional nuclear system consisting of a number of homogeneous regions (termed cells), first the cell matrices which depend solely on the material composition and geometrical dimension of the cell (hence on the cell type) are constructed using a boundary element formulation based on the multigroup boundary integral equation. For a particular nuclear system, the cell matrices are utilized in the assembly of the global system matrix in block-banded form using the newly introduced concept of virtual side. For criticality calculations, the classical fission source iteration is employed and linear system solutions are by the block Gaussian-elimination algorithm. The numerical applications show the validity of the proposed formulation both through comparison with analytical solutions and assessment of benchmark problem results against alternative methods.     

First-principles theory for helium and xenon diffusion in uranium dioxide

The diffusion properties of He and Xe in UO₂ have been investigated, using density-functional calculations employing the projector-augmented-wave (PAW) method and the generalized gradient approximation (GGA). The migration energies corresponding to both interstitial and vacancy-assisted mechanisms have been calculated and the results for the two noble gas atoms are compared with each other. We suggest that He likely diffuses by hopping through a single vacancy with computed low migration energies smaller than 0.79 eV and its diffusivity is much higher than that of Xe. Xe has a quite large migration energy compared to He; the strain energy plays a key role in Xe diffusion in UO₂.     

One- and two-group diffusion theory for eccentric control rods—I

We have solved one- and two-group diffusion equations for a cylindrical homogeneous reactor with an eccentric cylindrical control rod. In the limit of zero eccentricity, solutions in both cases reduce to those corresponding to a cylindrical reactor with a coaxial cylindrical control rod. From this study we find that the control-rod worth decreases as eccentricity increases and, further, that the one-group theory gives higher values compared to the two-group theory.     

Methods of calculating radial boundary conditions for flux trap control assemblies

Some methods of calculating a two- or four-group albedo matrix for the control assemblies of VV'ER-440 reactors are discussed. All of these methods use the S_N program DTF-IV to calculate partial neutron currents, but the group constants for the DTF-IV calculations were obtained by different methods. The influence of the boron steel absorber on the thermalization cross section of the water in the control assembly turned out to affect the albedo matrix significantly. The control assembly itself does not, however, have any significant effect on the group constants of adjacent fuel assemblies.Published in Atomnaya Energiya, Vol. 52, No. 1, pp. 39–44, January, 1982.     

Riemann boundary conditions for the Boltzmann transport equation using arbitrary angular approximations

In this paper a method for resolving the various boundary conditions (BCs) for the first order Boltzmann transport equation (BTE) is described. The approach has been formulated to resolve general BCs using an arbitrary angular approximation method within any weighted residual finite element formulation. The method is based on a Riemann decomposition which is used to decompose the particles’ angular dependence into in-coming and out-going information through a surface. This operation recasts the flux into a Riemann space which is used directly to remove any incoming information, and thus satisfy void boundary conditions. The method is then extended by its coupling with a set of mapping operators that redirect the outgoing flux to form incoming images resembling other specified boundary conditions. These operators are based on Galerkin projections and are defined to enable reflective and diffusive (white) BCs to be resolved. A small number of numerical examples are then presented to demonstrate the method’s ability in resolving void, reflective and white BCs. These examples have been chosen in order to show the method working for arbitrary angled surfaces. Furthermore, as the method has been designed for an arbitrary angular approximation, both _SN$S_N$ and _PN$P_N$ calculations are presented.     

A mixed P1–DP0 diffusion theory for planar geometry

A variational analysis is used to derive a mixed P₁–DP₀ (P₁ spherical harmonics–double P₀ spherical harmonics) angular approximation to the time-independent monoenergetic neutron transport equation in one-dimensional planar geometry. This mixed angular approximation contains a space-dependent weight factor α(x) that controls the local angular approximation used at a spatial point x: α(x) = 1 yields the standard P₁ (diffusion) approximation, α(x) = 0 gives the standard DP₀ approximation, and 0 < α(x) < 1 produces a mixed P₁–DP₀ angular approximation. The diffusion equation obtained differs from the standard P₁ diffusion equation only in the definition of the diffusion coefficient. Standard Marshak incident angular flux boundary conditions are also obtained via the variational analysis. We examine the use of this mixed angular approximation coupled with the standard P₁ approximation to more accurately treat material interfaces and vacuum boundaries. We propose a simple but effective functional form for the weight factor α(x) that removes the need for the user to specify the value. Numerical results from several test problems are presented to demonstrate that significant improvements in accuracy can be obtained using this method with essentially no computational penalty.     

The Borrmann effect in parametric X-radiation under asymmetric reflection conditions

Parametric X-radiation (PXR) of a relativistic electron traversing a single crystal plate is considered in Laue geometry. The expressions describing spectral-angular distributions of PXR formed on the atomic planes situated under arbitrary angle δ to surface of the plate (asymmetric reflection) obtained on basis of two-wave approximation of dynamic diffraction theory are used for definition of the conditions of the most pronounced manifestation of the Borrmann effect (optimal value of angle δ) are clarified. This effect leads to considerable increase of the intensity of the quasi-monochromatic tuning source of coherent X-radiation built on basis of PXR.     

Ab initio-based diffusion theory and tracer diffusion in Ni-Cr and Ni-Fe alloys

In this paper, ab initio modeling is used to predict diffusion relevant thermodynamic and kinetic information for dilute Ni-Cr and Ni-Fe alloys. The modeling results are then used to determine the phenomenological coefficient matrices and the tracer diffusion coefficients for both vacancy and interstitial mediated diffusion. In addition to predicting diffusion coefficients, this ab initio-based approach provides information typically inaccessible to experiments, including the different contributions to diffusion (e.g., electronic excitation effects), the species dependence of interstitial diffusion, and the deviations from Arrhenius-type relations, which are often used to describe and extrapolate experimental diffusion data. It is found that: (1) Cr is the fastest diffusing species in Ni by both vacancy and interstitial diffusion, followed by Fe and then Ni. The enhanced diffusivity of Cr is primarily due to differences in migration barriers and binding energies, not pre-exponential factors. (2) Fe and Cr solutes in Ni have weak interactions with vacancies but Cr solutes bind strongly to interstitial defects. (3) Cr exhibits non-Arrhenius behavior in both vacancy and interstitial mediated diffusion. (4) Temperature dependent electronic contributions have a significant impact on the diffusion in some cases. (5) The vacancy diffusion mechanism in Ni-Cr changes as a function of temperature resulting in vacancy-solute drag below 460 K.