Scattering, azimuthally symmetric

Isotropic scattering indicates that the radiant energy incident on a volume element is uniformly distributed to all directions. For an isotropically scattering medium, all a, coefficients of the phase function are zero, except a0. If only a0 and the first coefficient a, are considered, then one obtains the linearly anisotropic phase function, which means that the phase function is a linear function of cos 0 (or, in the case of an azimuthally symmetric medium, a linear function of p = cos 9). 

Equation (5.36) gives the current through unit area normal to the x axis (located at the origin) due to all last scatterings in a cone of width dp about the x axis. This result will be very useful for computing the current in systems in which the neutron density is azimuthally symmetric. 

The velocity vector v is used here to represent the condition on the probability, namely, that the initial coordinates of the neutron be y, /i, and For the present calculation, scattering collisions are assumed to be azimuthally symmetric about That is, it is equally probable that the final vector O will lie in any position on the conical surface which has... 

It shall be assumed in this chapter that molecular arrangement in the bulk of solid explosives, and all amorphous and liquid explosives, has no preferred orientation direction. The diffraction patterns in this case are isotropic around the primary X-ray beam, and the vector quantity, x, can be replaced by its scalar magnitude. It is customary to speak of diffraction profiles, rather than patterns, when isotropy obtains and the diffraction profiles are derived by integration of the (circularly-symmetric) diffraction pattern over the azimuthal component of the scattering angle. 

In a scattering experiment a beam of electrons of momentum k hits a target. We consider the target to be represented by a potential V(r). Electrons are observed by a detector placed at polar and azimuthal angles 9,(f) measured from the direction of the incident beam, which is the z direction in a system of spherical polar coordinates (fig. 4.2). For a central potential the problem is axially symmetric. Relevant quantities do not depend on (f). The detector subtends a solid angle... 

The simplest example in which the singular solutions of (4) form a continuous spectrum, but can be easily overlooked, concerns one of the simplest problems of transport theory the diffusion of monoenergetic neutrons in plane geometry. In this case, t, as well as y, z, cease to be variables because the flux is assumed to be independent of them. Furthermore, since the flux is zero except for a single value of E, this is not a significant variable either. Finally, the double variable 2 can be replaced by the direction cosine /jl = 0.x of the velocity with respect to the x axis because the flux does not depend on the azimuthal angle either. If the further simplification is made that the scattering is spherically symmetric and if the total cross section is measured in appropriate units, the transport equation assumes the form... 

For a spherically symmetrical potential there is no azimuthal dependence of the scattered intensity per unit solid angle. Thus, one can work with the differential (polar) cross-section dcr/d9, which represents the fractional contribution to q-R from any polar angle 9, by integrations over the azimuthal angle... 

Again, for a spherically symmetrical potential with no azimuthal dependence of the scattered intensity, one can write... 

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