TN method for the critical thickness of one-speed neutrons in a slab with forward and backward scattering

The critical slab problem in the case of combination of forward and backward scattering with usual isotropic scattering is studied for one-speed neutrons in a uniform finite slab by using T_N method based on Chebyshev polynomial approximation and Marshak boundary conditions. It is shown that T_N method gives accurate results in one-dimensional geometry and is very efficient both in derivation of equations and rapid convergence. Numerical results obtained by T_N method are compared against the P_N method in tabular form, which agreed quite well.     

TN approximation to neutron transport equation and application to critical slab problem

The critical slab problem has been studied in one-speed neutron transport equation with isotropic scattering by using the T_N method. T_N moment criticality solutions are obtained for the uniform finite slab using Mark and Marshak type vacuum boundary conditions. Results obtained by T_N method, using the two type boundary conditions mentioned above, were presented in the Tables and also the Tables included the results obtained by P_N method for the comparisons.     

Mössbauer and magnetization studies of the U1−xNpxO2 fluorites

²³⁷Np Mössbauer effect and magnetic susceptibility measurements of the U_1?Np_xO₂ fluorite solid solution have been performed in the composition range 0.15 ? x ? 0.75. For x = 0.15 and 0.25, the Np ions order magnetically at a lower temperature T₀ than the bulk material (T_N) (T₀ ～ 19 K, T_N ～ 27 K for x = 0.15). For x = 0.50, T₀ ～ 10 K (T_N ～ 12 K from recent neutron diffraction measurements). For x = 0.75, T₀ ～ T_N ～ 9 K. The Np (induced) ordered moment is ～ 0.5 μ_B. The ²³⁷Np Mössbauer isomer shift shows that the Np ions are in a IV charge state.     

Specific heat of NaNO2 near its transition points

Using an a.c. technique, the specific heat of NaNO₂ was measured as a function of temperature near its antiferroelectric-to-paraelectric phase transition point (T_N). The transition was found to be of the second order. The critical exponents are; α = 0·38 for ? = 2 × 10^?4 ～ 1 × 10^?1, and α′ = 0·18 for ? = ?2 × 10^?4 ～ ?3 × 10^?3. The critical exponents deduced from the scaling-law relations are roughly close to the values obtained from a random phase approximation for a system with an isotropic interaction. However, a difference was recognized between the observed exponent for the specific heat and the values theoretically given for T > T_N by the random phase approximation for a system with a short-range interaction or for a system with a long-range dipolar interaction. A thermodynamical analysis was made by using the generalized Pippard relation, and the present result was found to be consistent with the pressure dependence of the antiferroelectric transition point.     

Eigenvalue spectrum with chebyshev polynomial approximation of the transport equation in slab geometry

Using certain well-known properties of chebyshev polynomials, a simple and highly efficient approach to evaluate eigenvalue in radiation transport is presented. The spectrum of eigenvalues has been studied for slabs with isotropic scattering of different magnitudes of the cross section parameter c (i.e., the mean number of neutrons emitted per collision). It is shown that in the presence of the chebyshev polynomial approximation (T_N) there are both discrete and continuum of eigenvalues. It is found that the T_N method gives very good agreement with conventional spherical harmonics approximation (P_N).     

Crystal field and magnetization of canted antiferromagnet CoCO<Subscript>3</Subscript>

The magnetization of the canted antiferromagnet CoCO₃ (T _N = 18.1 K) is calculated in the Weiss molecular field approximation taking into account the microscopic state of the Co²⁺ ion in the entire range of temperatures and magnetic fields. The values of T _N, magnetic susceptibility in the basal plane, and ferromagnetic moment were used as parameters. It is shown that the anisotropy of the g factor and of the exchange interaction at low temperatures (T < 30 K) including the magnetic ordering temperature is correctly described in the Abragam-Pryce approximation. At high temperatures, the g factor increases and becomes isotropic, but it cannot be described using the Abragam-Pryce approximation. The reasons for g factor variation and the magnitude of the magnetic moment are discussed.     

Kondo behaviour of the solid solution (Ce1−xLax)PtGa

The solid solution (Ce_1−xLa_x)PtGa has been studied through X-ray diffraction, magnetization (σ(B)), magnetic susceptibility (χ(T)), electrical resistivity (ρ(T)), magnetoresistivity (MR) and heat capacity (C_P(T)) measurements. The Néel temperature (T_N=3.3 K) for CePtGa is lowered upon La substitution as observed from χ(T) and ρ(T) measurements. The Kondo temperature T_K as calculated from MR measurements is comparable to T_N and also decreases with La substitution. The volume dependence of T_K is in accordance with the compressible Kondo lattice model and a Doniach diagram of the results is presented. C_P(T) measurements are presented for CePtGa, Ce_0.2La_0.8PtGa and LaPtGa and the results are discussed in terms of the electronic and magnetic properties. Other features of interest are anomalies in ρ(T) and C_P(T) due to crystalline electric field effects and metamagnetism as observed in σ(B) studies for samples with 0≤x≤ 0.3.     

Relaxation function of a Heisenberg paramagnet at large wavevectors

An approach based on the Mori's theory to calculate the Kubo relaxation function is presented in order to explain the structure of neutron scattering cross section at large wavevectors for Heisenberg paramagnets. The limits of validity of the previous approximate theories are discussed. A low-frequency expansion for the third order memory function is performed: this corresponds to considering a more consistent long-time approximation for the second order memory function. The relaxation function has been computed for RbMnF₃ at T?T_N and T = 1.125T_N.     

Two-dimensional and layered structures in the discrete φ4 model

The properties of phase transitions in two-dimensional and layered systems are investigated on the basis of a discrete φ⁴ model by numerical and analytical methods. The only parameter a of the discrete φ⁴ model determines the behavior of the system and makes it possible to investigate phase transitions ranging from transitions of the displacement type (a → +0) to order-disorder type (a → +∞). The behavior of a two-dimensional system is investigated in a wide range of values of the parameter a. The temperature dependences of the squared order parameter η²(T) and the phase transition temperature T _c as a function of the thickness N of the system are obtained for three characteristic values of the parameter a using the Monte Carlo method. The properties of phase transitions in the discrete φ⁴ model are investigated on the basis of the mean-field approximation and the independent-mode approximation. The results obtained in the numerical experiments are compared with the analytical approximations. It is shown that the mean-field approximation qualitatively describes the behavior of the phase-transition temperature T _c as a function of the thickness N of the system for a wide range of values of the parameter a, and the independent-mode approximation describes quantitatively, to within 5%, the results of the numerical simulation for small values of a.     

Magnetic phase diagram of GdAg1−xZnx

The magnetic phase diagram of GdAg_1?xZn_x, an intermetallic solid solution of an antiferromagnet (GdAg: T_N = 136 K) and a ferromagnet (GdZn: T_C = 269 K), has been elaborated from magnetization measurements. The antiferromagnetic phase boundary T_N(x) first passes a broad maximum meeting the ferromagnetic phase boundary T_C(x) at x₁ = 0.575 and T₁ = 72 K, where four phases coexist. On approaching (x₁, T₁) along T_N(x) the magnetization phenomena vanish. At x₁ the phase transition still has ferromagnetic appearance but proceeds into a state without spontaneous magnetic moment. Two different ferromagnetic phases (F₁, F₂) and one ferrimagnetic phase (F₃) occur in the composition intervals 0.69 < x₁ <1, and 0.61 <x₂ < 0.69 and 0.51 < x₃ < 0.61. All phase transitions seem to be of second order except th e F₁?F₂ one at x = 0.69 which is of first order. This phase line meets the paramagnetic to ferromagnetic phase boundary T_C(x) in a multicritical point with the coordinates x_m = 0.69, T_m = 123 K.Six further mixed magnetic phases, M_{1, 2, ..., 6}, are observed between x = 0.33 and 0.6 below the antiferromagnetic branch and exhibit irreversible thermodynamic properties, such as hysteresis, below about 40 K.Assuming local magnetic interactions only between nearest and next-nearest Gd neighbours, the T_N(x) and T_C(x) phase boundaries can be described fairly well by a simple model calculation using different exchange parameters for a few relevant distributions of Ag and Zn atoms.     

Strings in the CP2N−1 model

The CP₂^N?1 model is discussed using strings as collective variables in the hamiltonian formulation. The large N limit is obtained as a semiclassical approximation. The mass gap and β-function are computed. In the limit N → ∞, it is shown that the singlet spectrum contains both bound states and scattering states whose energies and wave functions are calculated.     

EPR of Cr2O3 in the phase transition region

The temperature dependence of the Cr₂O₃ EPR spectra is obtained. The linewidth of the EPR is defined by the phonon's modulation of the exchange interactions at T >T_N + 4–5 K though at T_N < T < T_N + 4 K the critical broadening of the EPR in Cr₂O₃ is determined by fluctuation effects.     

Extension of the weak-line approximation and application to correlated-k methods

Global climate models require accurate and rapid computation of the radiative transfer through the atmosphere. Correlated-k methods are often used. One of the approximations used in correlated-k models is the weak-line approximation. We introduce an approximation T_γ which reduces to the weak-line limit when optical depths are small, and captures the deviation from the weak-line limit as the extinction deviates from the weak-line limit. This approximation is constructed to match the first two moments of the gamma distribution to the k-distribution of the transmission. We compare the errors of the weak-line approximation with T_γ in the context of a water vapor spectrum. The extension T_γ is more accurate and converges more rapidly than the weak-line approximation.     

Effect of surface orientation and thickness on the magnetization of anisotropic FCC ferromagnetic films

The dependence on temperature of the layer magnetization of a Heisenberg ferromagnetic ultrathin film in presence of magnetocrystalline single-ion anisotropy was theoretically investigated in the framework of a Green's function approach using the random phase approximation (RPA). The effect of surface orientation and of film thickness N on the Curie temperature T_C was carefully investigated in the case of face centered cubic (FCC) films: the steepest increase of T_C(N) was found in the case of the FCC(1 1 1) orientation and the smoothest in the FCC(1 1 0) one. Our results for T_C(N) were successfully fitted by a finite-size scaling relation [T_C(∞)−T_C(N)]/T_C(N)=(N/N₀)^−λ, giving a shift exponent λ≃1.5, irrespectively of the surface orientation. Finally, the temperature evolution of the magnetization profile was analyzed, as well as its limiting shape at T_C.     

T = 1 excitation mechanism in pion induced reactions on 12C

The cross sections for the reactions π^± → π^±, π⁰ on ¹²C with excitation of the T = 1 states are calculated for the N₃₃¹ energy region. The Chew-Low amplitude for pion-nucleon scattering is used in the framework of impulse approximation which has been supported by recent calculation of pion scattering with excitation of definite nuclear states.     

Modification of the magnetic structure of UAs — Appearance of an incommensurate phase below TN in UAs0.97Se0.03, studied by neutron diffraction from a single crystal

The magnetic structure of UAs_0.97Se_0.03 has been studied by neutron diffraction from a single crystal in zero applied magnetic field. It is found to be antiferromagnetic, of type-IA (++--) below T_o = 75.6 K and of type-I (+-+-) above T_o. The type-I persists till T_IC = 113.5 K, while above it and up to T_N = 122 K an incommensurate phase appears, thereby modifying the magnetic structure in pure UAs. The k-value of the wavevector $\text{K}$ (along cubic axes) is changing from 0.642 at T_N to 0.652 at T_IC. The transitions at T_o and T_IC are first-order while the transition at T_N is second-order. The ordered magnetic moment is 2.15 μ_B at T = 4.2 K, it varies smoothly to 1.95 μ_B at T = 75.4 K and drops drastically to 1.47 μ_B at T = 76 K.     

Effect of Al doping on the magnetic and electrical properties of layered perovskite La1.3Sr1.7Mn2−xAlxO7

The effect of Al substitution for Mn site in layered manganese oxides La_1.3Sr_1.7Mn_2−xAl_xO₇ on the magnetic and electrical properties has been investigated. It is interesting that all the samples undergo a similar and complex transition with lowing temperature; they transform from the two-dimensional short-range ferromagnetic order at T*, then enter the three-dimensional long-range ferromagnetic state at T_C, at last they display the canted antiferromagnetic state below T_N. T*, T_C and T_N are all reduced with Al content. Resistivity increases sharply with increasing Al concentration, and the metal-insulator transition disappears when x reaches 10%. Additionally, magnetoresistance (MR) effect is weakened. Al substitution dilutes the magnetic active Mn-O-Mn network and weakens the double exchange interaction, and further suppresses FM ordering and metallic conduction. Owing to the anisotropic interaction in the layered perovskite, the magnetic and electrical properties are more sensitive to Al doping level than those in ABO₃-type perovskite.     

Effect of antiferromagnetic ordering on the optical transitions in MnF2

Two optical absorption bands (viz. C and F) in MnF₂, whose line positions are nearly temperature-independent above T_N = 67.3 K, are investigated for the effect of magnetic ordering. Both bands are blue-shifted by ΔE (≌90 cm^?1 for C and 120 cm^?1 for F at 10 K) and the lines show additional narrowing on cooling through T_N. It is found that ΔE varies as the sublattice lattice magnetization below T_N and nearly as the magnetic energy above T_N.     

Elliptic PDE formulation and boundary conditions of the spherical harmonics method of arbitrary order for general three-dimensional geometries

The inherent complexity of the radiative transfer equation makes the exact treatment of radiative heat transfer impossible even for idealized situations and simple boundary conditions. Therefore, a wide variety of efficient solution methods have been developed for the RTE. Among these solution methods the spherical harmonics method, the moment method, and the discrete ordinates method provide means to obtain higher-order approximate solutions to the equation of radiative transfer. Although the assembly of the governing equations for the spherical harmonics method requires tedious algebra, their final form promises great accuracy for any given order, since it is a spectral method (rather than finite difference/finite volume in the case of discrete ordinates). In this study, a new methodology outlined in a previous paper on the spherical harmonics method (P_N) is further developed. The new methodology employs successive elimination of spherical harmonic tensors, thus reducing the number of first-order partial differential equations needed to be solved simultaneously by previous P_N approximations (=(N+1)²). The result is a relatively small set (=N(N+1)/2) of second-order, elliptic partial differential equations, which can be solved with standard PDE solution packages. General boundary conditions and supplementary conditions using rotation of spherical harmonics in terms of local coordinates are formulated for the general P_N approximation for arbitrary three-dimensional geometries. Accuracy of the P_N approximation can be further improved by applying the “modified differential approximation” approach first developed for the P₁-approximation. Numerical computations are carried out with the P₃ approximation for several new two-dimensional problems with emitting, absorbing, and scattering media. Results are compared to Monte Carlo solutions and discrete ordinates simulations and a discussion of ray effects and false scattering is provided.