Spherical polar components

Here, 9 and are the spherical polar components, used to define the angular dependency of direction of propagation of radiation intensity. 

The five second-moment spherical tensor components can also be calculated and are defined as the quadrupolar polarization terms. They can be seen as the ELF basin equivalents to the atomic quadrupole moments introduced by Popelier [32] in the case of an AIM analysis ... 

To describe the velocity profile in laminar flow, let us consider a hemisphere of radius a, which is mounted on a cylindrical support as shown in Fig. 2 and is rotating in an otherwise undisturbed fluid about its symmetric axis. The fluid domain around the hemisphere may be specified by a set of spherical polar coordinates, r, 8, , where r is the radial distance from the center of the hemisphere, 0 is the meridional angle measured from the axis of rotation, and (j> is the azimuthal angle. The velocity components along the r, 8, and (j> directions, are designated by Vr, V9, and V. It is assumed that the fluid is incompressible with constant properties and the Reynolds number is sufficiently high to permit the application of boundary layer approximation [54], Under these conditions, the laminar boundary layer equations describing the steady-state axisymmetric fluid motion near the spherical surface may be written as ... 

The components of the direction vector are related in the usual way to the azimuthal () and polar (9) angles of a spherical polar coordinate system,... 

From the definition of a particle used in this book, it follows that the motion of the surrounding continuous phase is inherently three-dimensional. An important class of particle flows possesses axial symmetry. For axisymmetric flows of incompressible fluids, we define a stream function, ij/, called Stokes s stream function. The value of Imj/ at any point is the volumetric flow rate of fluid crossing any continuous surface whose outer boundary is a circle centered on the axis of symmetry and passing through the point in question. Clearly ij/ = 0 on the axis of symmetry. Stream surfaces are surfaces of constant ij/ and are parallel to the velocity vector, u, at every point. The intersection of a stream surface with a plane containing the axis of symmetry may be referred to as a streamline. The velocity components, and Uq, are related to ij/ in spherical-polar coordinates by... 

The velocity components ur and ue in the spherical polar coordinates r and 9 are in this case given by the Stokes expressions... 

Hence there exists a complete set of common eigenfunctions for L2 and any one of its components. The eigenvalue equations for L2 and Lz are found to be separable in spherical polar coordinates (but not in Cartesian coordinates). Using the chain rule to transform the derivatives, we can find... 

Equation (11) is written in the form of Newton s second law and states that the mass times acceleration of a fluid particle is equal to the sum of the forces causing that acceleration. In flow problems that are accelerationless (Dx/Dt = 0) it is sometimes possible to solve Eq. (11) for the stress distribution independently of any knowledge of the velocity field in the system. One special case where this useful feature of these equations occurs is the case of rectilinear pipe flow. In this special case the solution of complex fluid flow problems is greatly simplified because the stress distribution can be discovered before the constitutive relation must be introduced. This means that only a first-order differential equation must be solved rather than a second-order (and often nonlinear) one. The following are the components of Eq. (11) in rectangular Cartesian, cylindrical polar, and spherical polar coordinates ... 

Figure 2.12 Definition of the components of angular momentum in cartesian and in spherical polar coordinates.

In order to evaluate the matrix elements of the dipole moment operator in Eq. (24), it is convenient to separate out the geometrical aspects of the problem from the dynamical parameters. To that end, it is convenient to decompose the LF scalar product of the transition dipole moment d with the polarization vector of the probe laser field e in terms of the spherical tensor components as [40]... 

The LF spherical tensor components of the electric-field polarization are defined... 

The electric field polarization is conveniently described in the LF. The MF spherical tensor components of the electric field polarization tensor are related to the components in the LF through a rotation... 

Consider the two-spin system, with energy levels shown in Fig. 7.1. We can derive a simple and useful expression for a pair of spins by expanding the scalar product of Eq. 7.2 in terms of the x, y, and 2 spin components and using spherical polar coordinates (r, 0, and ) for the spatial coordinates. This gives the expression... 

The motion of a free particle on the surface of a sphere will involve components of angular momentum in three-dimensional space. Spherical polar coordinates provide the most convenient description for this and related problems with spherical symmetry. The position of an arbitrary point r is described by three coordinates r, 0, 0, as shown in Fig. 6.2. 

Equations (D.6) and (D.7) are Gilbert s equation in spherical polar coordinates. To obtain the Gilbert-Langevin equation in such coordinates we augment the field components //, and with random field terms and h. By graphical comparison of the Cartesian and spherical, polar coordinate systems, we find these to be... 

Optimizing correlating and polarization functions presents issues that are similar to those of the SCF functions. If the spin-orbit components of a subshell are sufficiently different, it might be necessary to use j- or k-optimization for the correlating functions. The use of j-optimization raises a fundamental question what do we now mean by a polarization function A j = 1/2 shell contains both an si/2 function and a pi/2 function, but the charge distributions for both of these are spherical. Polarization by a uniform electric field introduces functions with the angular momentum incremented by one unit. Thus for a, j = 1/2 shell, the polarization functions would have to come from a j = 3/2 shell, which contains both a P3/2 and dy2 function. If the basis set already contains a j = 3/2 shell with appropriate radial distributions, it is not necessary to add another function, just as in the non-relativistic case it is not necessary to add p functions for polarization of an s set if the functions already exist in the basis set. 

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