Solution Of Certain Integral Equations Research Articles

Disordering of Solids by Neutron Radiation

A general method is outlined for determining the number of vacant lattice sites or interstitial atoms in a monatomic solid exposed to neutron radiation. The colliding atoms are assumed to be within the energy range for which the orbital picture can be applied. Following the treatment of Bohr, the scattering regions of excessive and moderate screening, Rutherford distribution, and electronic collisions are considered separately. The number of vacancies or interstitial atoms as a function of the energy of the primary knocked-out atom is given by the solution of certain integral equations that are different for various energy regions considered. It is found that if the velocity of a recoil atom resulting from neutron collision is less than $\frac{{e}^{2}}{\ensuremath{\hbar}}$ (region of elastic collisions) approximately half of its energy is used up to produce vacancies or interstitials. If the velocity of the recoil atom is above $\frac{{e}^{2}}{\ensuremath{\hbar}}$ (region of inelastic collisions) then the energy used up to produce vacancies and interstitials is approximately constant for medium and heavy elements. A simple formula has been derived expressing the average number of vacant lattice sites or interstitials produced in a collision of a neutron having energy $E$ in a monatomic solid composed of medium or heavy elements having atomic mass $M$. The formula is as follows: $G(E)\ensuremath{\sim}\frac{{(nE\ensuremath{-}\ensuremath{\alpha})}^{2}}{4\ensuremath{\alpha}\mathrm{nE}} \mathrm{for} E\ensuremath{\leqq}\frac{\ensuremath{\gamma}}{n},$ $G(E)\ensuremath{\sim}\frac{[{(nE\ensuremath{-}\ensuremath{\alpha})}^{2}\ensuremath{-}(1\ensuremath{-}\overline{R}){(nE\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\gamma})}^{2}]}{4\ensuremath{\alpha}\mathrm{nE}} \mathrm{for} E\ensuremath{\geqq}\frac{\ensuremath{\gamma}}{n},$ where $\ensuremath{\gamma}=\frac{M{e}^{4}}{2{\ensuremath{\hbar}}^{2}}$; $n=\frac{4M}{{(M+1)}^{2}}$, $\ensuremath{\alpha}$ is the binding energy of an atom in the lattice, and $\overline{R}$ is a slowly varying function of $Z$.

The Delta Wing in a Non-Uniform Supersonic Stream

SummaryA generalised conical field theory is developed and is applied to delta wings in a non-uniform stream. It is shown that a non-uniform stream may be characterised by the downwash at all points in space. The lift of a delta wing is found when the downwash in the wing plane is given as a power series in the co-ordinates in the wing plane. The basis of the conical field theory is described in some detail but the results only of the calculation of the lift distribution for various down washes are given. The solutions of certain integral equations, required in the calculations, are given in the Appendix.

Finite extension and torsion of cylinders

The theory of finite elastic deformations of an isotropic body, in which a completely general strainenergy function is used, is applied to the problem of a small twist superposed upon a finite extension of a cylinder which has a constant cross-section. The law which relates the force necessary to produce the large extension, with the torsional modulus for the small torsion superposed on that extension, is given by a simple general formula. When the material is incompressible the corresponding law is independent of the particular form of the strain-energy function which applies to the material. When the cylinder is not a circular cylinder or a circular cylindrical tube the twisting couple vanishes for a certain value of the extension ratio, this value being independent of the particular form of the strain-energy function when the material is incompressible. The problems of a small twist superposed upon a hydrostatic pressure, or upon a combined hydrostatic pressure and tension, are also solved. Attention is then confined to isotropic incompressible rubber-like materials using a strain-energy function suggested by Mooney, and the second-order effects which accompany the torsion of cylinders of constant cross-sections are examined. The problem is reduced to the determination of two functions of a complex variable which are regular in the cross-section of the cylinders and which satisfy a suitable boundary condition on the boundary of the cross-section. The solution is expressed as an integral equation and applications are made to cylinders with various cross-sections. This theory is then generalized to include the second-order effects in torsion superposed upon a finite extension of the cylinders. Complex variables are used throughout this part of the paper, and the problem is reduced to the determination of four canonical functions of a complex variable, these functions being the solutions of certain integral equations. An explicit solution is given for an elliptical cylinder but without using the integral equations.