Relativistic matrix elements of the energy operator in the case of complex electronic configurations

The relativistic energy operator is considered. The expressions for the matrix elements of this operator in the case of complex electronic configurations (one and two subshells of equivalent electrons) are presented. The numerical values of the coefficients of the radial integrals in these matrix elements are tabulated. The tables contain all these coefficients in the case of configurations arising from one shell l^N (l = 0?2).     

Determination of K shell absorption jump factors and jump ratios in the elements between Tm(Z = 69) and Os(Z = 76) by measuring K shell fluorescence parameters

The K shell absorption jump factors and jump ratios have been measured in the elements between Tm (Z = 69) and Os(Z = 76) without having any mass attenuation coefficient at the upper and lower energy branch of the K absorption edge. The jump factors and jump ratios for these elements have been determined by measuring K shell fluorescence parameters such as the total atomic absorption cross-sections, the K_α X-ray production cross-sections, the intensity ratio of the K_β and K_α X-rays and the K shell fluorescence yields. We have performed the measurements for the calculations of these values in attenuation and direct excitation experimental geometry. The K X-ray photons are excited in the target using 123.6 keV gamma-rays from a strong ⁵⁷Co source, and detected with an Ultra-LEGe solid state detector with a resolution 0.15 keV at 5.9 keV. The measured values have been compared with theoretical and others’ experimental values. The results have been plotted versus atomic number.     

Factorization of the j = 72 shell of neutrons and protons: Transformation coefficients to states of good particle number

Complete tabulations are given, for the $\text{j =}\text{7}\text{2}$ nuclear shell, of the transformation coefficients to states of good particle number needed in a new method of nuclear spectroscopy based on a quasiparticle factorization of the j-shell. The method leads to a classification scheme in which total angular momentum and isospin are good quantum numbers and in which the calculation of matrix elements can be carried out simply by standard techniques of Racah algebra, without the use of coefficients of fractional parentage and with the use of only a very small number of reduced matrix elements. These reduced matrix elements also are tabulated. A few sample calculations show how matrix elements of one- and two-body operators, single nucleon and pair creation operators, …, can be calculated with this technique and the present tabulations.     

Internal conversion coefficients for atomic numbers Z ≤ 30

Presented here are internal conversion coefficients (ICC) of gamma rays for 20 values of atomic number Z in the range 3 ≤ Z ≤ 30, including all Z ≥ 14. The tables provide the previously missing data for light elements. Coefficients are given for 19 values of gamma-ray transition energies up to 6 MeV for the K-electron shell and 18 values up to 2 MeV for three L-subshells. The minimum energy is 15 keV. The first five electric and magnetic nuclear transition multipolarities are covered. The calculations are relativistic, with screening and finite nuclear size effect taken into account.     

基于面元模型的河蚌剂量转换系数的计算

基于河蚌开口与闭合两种形态,利用多边形网格(PM)与NURBS曲线技术对河蚌体素模型进行光滑和变形处理,分别建立河蚌开口面元模型和闭合面元模型,并通过MCNPX模拟计算河蚌在两种模型形态下的外照射剂量转换系数(外照射DCC)与内照射剂量转换系数(内照射DCC)。通过模拟结果可得出:(1)在外照射条件下,河蚌的外照射DCC的关系为:开口面元模型>闭合面元模型;(2)在内照射条件下,河蚌的内照射DCC的关系为:闭合面元模型>开口面元模型。     

Cross sections for ionization of K, L and M shells of atoms by impact of electrons and positrons with energies up to 1 GeV: Analytical formulas

Analytical formulas are presented for the easy calculation of cross sections for ionization of K, L and M shells of neutral atoms by impact of electrons and positrons with kinetic energies up to 1 GeV. Each formula contains a number of parameters that are characteristic of the element, the active electron shell and the projectile particle. The values of these parameters were determined by fitting the cross section values in an extensive database that was calculated recently by means of a composite algorithm that combines the distorted-wave and plane-wave Born approximations. Tables of parameter values are given for all elements, from hydrogen (Z=1) to einsteinium (Z=99). The proposed analytical expressions yield ionization cross sections that agree with those in the numerical database to within about 1%, except for projectiles with near-threshold energies.     

Experimental and theoretical Gamow-Teller beta-decay observables for the sd-shell nuclei

A comprehensive comparison of experimentally observed Gamow-Teller beta decay with theory is presented for the sd-shell (A = 17–39) nuclei. Relevant experimental data on half-lives, Q values, and branching ratios are tabulated and used to deduce experimental log(ft) values and Gamow-Teller matrix elements. These are compared with theoretical values based on complete $\text{(0d}{}_{\mathrm{<mtext>5</mtext><mtext>2</mtext>}}\text{, 1s}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{, 0d}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{)-}\text{space}$ shell-model wave functions. These wave functions are obtained from diagonalizations of a model Hamiltonian formulation which reproduces observed energy-level structures throughout the sd shell. The calculations are carried out both with the “free-nucleon” normalization for the Gamow-Teller single-particle matrix elements and with effective values for these matrix elements obtained from analyses of experimental Gamow-Teller magnitudes with the shell-model wave functions.     

Non-dipole second order parameters of the photoelectron angular distribution for elements Z = 1-100 in the photoelectron energy range 1-10 keV

Presented here are the photoelectron angular distribution non-dipole parameters associated with the terms of the second order O [(kr)²] (k is the photon energy and r is the radius of the ionized atomic shell) for both unpolarized and linearly polarized radiation. The parameters are given for atomic shells with binding energies lower than 2 keV of all elements 1 ? Z ? 100 for four values of photoelectron energy in the range 1−10 keV. In this range, the second-order terms are shown to make a significant contribution (up to ∼30%) to the angular differential cross section. The inclusion of these terms becomes all the more important in calculations of the differential cross section ratio for the fixed geometry of angles which is measured experimentally in the case of linearly polarized radiation. The Dirac-Fock-Slater potential is used in the calculations. The hole left by the emitted electron is taken into account in the frozen orbital approximation.     

Vibration and stability of prestressed shells

Thin shells are widely used structural elements and are sometimes subjected to time-dependent loads after they have already acquired some prestress. The response of such a shell can be very different from that when the prestress is absent. The general methods of calculation of the response of the shell are discussed. The stability limit of the shell is viewed as a special result obtained in the free vibration analysis. An example of a typical containment shell of a nuclear power plant, housing the reactor, is worked out in detail.     

L1 and L2 sub-shell fluorescence yields for elements with 64 ? Z ? 70

The L₁ and L₂ sub-shell fluorescence yields have been deduced for elements with 64 ? Z ? 70 from the L_k(k = l, α, β_1,4, β_3,6, β_2,15,9,10,7, γ_1,5 and γ_2,3,4) X-ray production cross sections measured at 22.6 keV incident photon energy using a spectrometer involving a disc type radioisotope of Cd¹⁰⁹ as a photon source and a Peltier cooled X-ray detector. The incident photon intensity, detector efficiency and geometrical factor have been determined from the K X-ray yields emitted from elemental targets with 20 ? Z ? 42 in the same geometrical setup and from knowledge of the K shell cross sections. The present deduced ω₁(exp) values, for elements with 64 ? Z ? 70, are found to be in good agreement with those tabulated by Campbell (J.L. Campbell, Atom. Data Nucl. Data Tables 95 (2009) 115), where as these are, on an average, higher by 19% and 24% than those based on the Dirac-Hartree-Slater model (S. Puri et al., X-ray Spectrometry 22 (1993) 358) and the semi-empirical values compiled by Krause (M.O. Krause, J. Phys. Chem. Ref. Data 8 (1979) 307), respectively. The present deduced ω₂(exp) values are found to be in good agreement with those based on the Dirac-Hartree-Slater model and are higher by up to ∼13% than the semi-empirical values for the elements under investigation.     

Analysis of (n, 2n) multiplication in lead

Lead is being considered as a possible amplifier of neutrons for fusion blankets. A simple one-group model of neutron multiplications in Pb is presented. Given the 14 MeV neutron cross section on Pb, the model predicts the multiplication. Given measured multiplications, the model enables the determination of the (n, 2n) and transport cross sections. Required for the model are: P—the collision probability for source neutrons in the Pb body—and W—an average collision probability for non-virgin, non-degraded neutrons. In simple geometries, such as a source in the center of a spherical shell, P and an approximate W can be expressed analytically in terms of shell dimensions and the Pb transport cross section. The model was applied to Takahashi's measured multiplications in Pb shells in order to understand the apparent very high multiplicative power of Pb. The results of the analysis are not consistent with basic energy-balance and cross-section magnitude constraints in neutron interaction theory.     

Buckling analysis of a cylindrical shell,under neutron radiation environment

This research investigates the buckling of a cylindrical shell in the neutron radiation environment, subjected to combined static and periodic axial forces. Radiation induced porosity in elastic materials affects the thermal, electrical and mechanical properties of the materials. In this study, the data based technique was used to determine the volume fraction porosity, P, of shell material. A least-squares fit of the Young's module data yielded the estimated Young's modulus. The shell assumed made of iron irradiated in the range of 2–15e?7 dPa/s at 345–650 °C and theoretical formulations are presented based on the classical shell theory (CST). The research deals with the problem theoretically; keeping in mind that one means of generating relevant design data is to investigate prototype structures. A parametric study is followed and the stability of shell is discussed. It is concluded that both temperature and neutron induced swelling have significant effects on the buckling load.     

Particle flux in an annular gap about a sphere

We develop a method for joining diffusion theory to a void region in a consistent manner. In particular, we apply the theory to a three region problem with a central solid core, surrounded by a concentric void shell and this in turn is surrounded by a concentric shell of another solid material. The sources can be either incident on the outer surface or uniformly distributed in the inner and outer regions. The void flux is calculated from the free streaming Boltzmann equation and is linked to the diffusion equations by assuming that the angular distribution on the surfaces follows the form A + B cosϑ, where A and B are related to the diffusion theory fluxes in the inner and outer regions. This procedure allows us to calculate the flux throughout the solid and void regions without making any diffusion theory assumptions in the void. Numerical results are given to illustrate the method and comparisons with exact transport calculations are given to establish the accuracy. An extension of the method to deal with axially symmetric systems is also given in Appendix B. The theory will apply to neutrons and photons.     

Calculation of stress intensity factors of surface cracks in complex structures: Application of efficient computer program EPAS-J1

A finite element computer program EPAS-J1 was developed to calculate the stress intensity factors of three-dimensional cracks. In this program, the stress intensity factor is determined by the virtual crack extension method together with the distorted elements allocated along the crack front. The program also includes the connection elements based on the Lagrange multiplier concept to connect such different kinds of elements as the solid and shell elements, or the shell and beam elements. For the structure including the three-dimensional surface cracks, the solid elements are employed only at the neighborhood of the surface crack, while the remainder of the structure is modeled by the shell or beam elements since the crack singularity is very local. Computer storage and computational time can be highly reduced with the application of the above modeling for calculation of the stress intensity factors of the three-dimensional surface cracks, because the three-dimensional solid elements are required only around the crack front.Several numerical analyses were performed by the EPAS-J1 program. At first, the accuracies of the connection element and the virtual crack extension method were confirmed using the simple structures. Compared with other techniques of connecting different kinds of elements such as the tying method or the method using an anisotropic plate element, it is found that the present connection element provides better results than the others. It is also found that the virtual crack extension method provided the accurate stress intensity factor. Furthermore, the results are also presented for the stress intensity factor analyses of cylinders with longitudinal or circumferential surface cracks using the combination of various kinds of elements together with the connection elements.     

Shell effect on stabilization processes following multi-electron transfer in slow Arq+–Ar (q = 6–9, 11) collisions

We experimentally investigate the shell effect on the stabilization processes following the multi-electron transfer in slow collisions of Ar^q+–Ar (q = 6–9, 11). The relative cross-section ratios of multi-electron transfer and of the subsequent stabilization with respect to single-electron capture are measured meanwhile compared with the theoretical results predicted by the classical over-barrier model. Our result indicates that the multi-electron transfer is dominant when the projectile charge is large and the subsequent stabilization shows a dramatic variation if the projectile L-shell configuration becomes open. It shows that the subsequent stabilization processes of multiply excited scattering ions have a strong dependence on the projectile shell.     

Optical oscillator strengths, mean excitation energy, shell corrections and experimental values for stopping power

In a recent paper, Smith, Inokuti, Karstens and Shiles discussed optical oscillator strengths (OOS) of graphite, Al and Si and compared the mean excitation energy I obtained by integration to that from stopping power measurements. They found agreement for graphite and Al, but disagreement for Si. In this paper, we discuss the OOS of Al, Si, Cu and Au and compare the stopping powers calculated from these OOS (or from a single I-value), using program CasP40, directly to experimental stopping power values for protons between 10 and 80 MeV. We find that the choice of proper shell corrections is essential: since the shell correction built into CasP is too small, we take the correction for Al, Si and Cu from the BEST program of Berger and Bichsel. For Au, better results are obtained using Bonderup’s shell correction. With these choices, we find fair agreement between experimental and calculated stopping data, both with the I-values from ICRU Report 49 and with OOS. Even in the case of Si, the stopping curve based on OOS is not in conflict with experimental data. In all cases, the curves calculated using SRIM are in good agreement with the data.     

Calculation of K and L x rays for elements of Z = 95 to 130

Binding energies for the K, L, M, N, and O shells and the K and L x-ray energies have been determined for elements of Z = 95 to 130. The results for 15 elements are based primarily on detailed calculations made with total energies from a Dirac-Fock code of J. P. Desclaux. Special interpolation procedures are used for the remaining elements. In addition, small empirical corrections are added as a result of comparing calculations with experimentally determined binding energies for elements of Z < 95. A discussion is given on the uncertainties encountered with the calculations and on the probable errors listed with the results.     

Measurement of L X-ray fluorescence cross-sections for elements with 45 ? Z ? 50 using synchrotron radiation at 8 keV

The L shell fluorescence cross-sections of the elements in range 45 ? Z ? 50 have been determined at 8 keV using Synchrotron radiation. The individual L X-ray photons, Ll, Lα, Lβ_I, Lβ_II, Lγ_I and Lγ_II produced in the target were measured with high resolution Si(Li) detector. The experimental set-up provided a low background by using linearly polarized monoenergetic photon beam, improving the signal-to-noise ratio. The experimental cross-sections obtained in this work were compared with available experimental data from Scofield [1] and [2] Krause [3] and [4] and Scofield and Puri et al. [5] and [6].These experimental values closely agree with the theoretical values calculated using Scofield and Krause data, except for the case of Lγ, where values measured of this work are slighter higher.     

Scaling laws for spherical pinch experiments

Spherical pinch experiments are characterized by a central discharge in a spherical vessel followed by an inductive discharge in the vessel's peripheral shell gas. An analysis is carried out of the evolution of the imploding shock waves produced by the shell explosion in order to find out if the central discharge can be contained and compressed by the converging shocks, so as to maintain its temperature for a time sufficiently long for breakeven. The analytical model adopted is essentially that of the recent paper of Ahlborn and Key (Plasma Phys. 23: 435, 1981). One finds that the converging shocks are indeed capable of containing and compressing the central plasma. In addition, if the central spark reaches the critical temperatureT _L = 2.58 keV by the deposition of an energy density of 1.86×10⁸ J·g^?1, the scaling law required in order to contain such a plasma for breakeven isρ ₀ R(E_s/M_s)^1/2 ? 1.96×10⁶, whereρ ₀ is the initial fill gas density,R is the radius of the spherical vessel, andE _s is the energy deposited in the peripheral shell massM _s . The general applicability of the model to other fusion devices based on the implosion principle is discussed.     

Strain-based finite elements for the analysis of cylinders with holes and normally intersecting cylinders

A finite element solution to the problems of stress distribution for cylindrical shells with circular and elliptical holes and also for normally intersecting thin elastic cylindrical shells is given in the present paper. Quadrilateral and triangular curved finite elements are used in the analysis. The elements are of a new class, based on simple independent generalised strain functions insofar as this is allowed by the compatibility equations. The elements also satisfy exactly the requirements of strain-free-rigid body displacements and uses only the external “geometrical” nodal degrees of freedom to avoid the difficulties associated with unnecessary internal degrees of freedom.A rectangular curved element was first developed and applied to the analysis of the familiar pinched cylinder and barrel vault problems (Ashwell and Sabir [1]). The results converge rapidly for displacements as well as for stresses. Further tests were carried out to investigate the ability of this element in predicting the high stresses in the neighbourhood of applied concentrated loads, (Sabir and Ashwell [2]). The loads considered were either radial or axial forces as well as moments about tangents to the circular cross section. The results obtained were not only in agreement with those of Forsberg and Flügge [3] but when plotted for the complex parameters defining proportions of the shell and flexibility as suggested by Calladine [4], their general forms corresponded closely with theoretical predictions.In the present paper we first develop strain based quadrilateral and triangular elements and apply them to the solution of the problem of stress concentrations in the neighbourhood of small and large circular and elliptical holes when the cylinders are subjected to a uniform axial tension. These results are compared with analytical solutions based on shallow shell approximations and show that the use of these strain based elements obviates the need for using an inordinately large number of elements.Normally intersecting cylinders are common configurations in structural components for nuclear reactor systems and design information for such configurations are generally lacking. The opportunity is taken in the present paper to provide a finite element solution to this problem. A method of substructing will be introduced to enable a solution to the large number of non banded set of simultaneous equations encountered. The solutions show good agreement when compared with experimental results of Corum, Bolt, Greenstreet and Gwaltney [5].