Spherical coordinates angle

Since the vector g is represented above in terms of the g-coordinate system (i i is) having — g as the i3 axis, it is necessary to determine the transformation to the (iI,iJ/,i2) coordinate system in which the particle velocities are written, in order to evaluate certain integrals. If we let be the spherical coordinate angles of the vector v2 — vlt in the v-coordinate system, then ... 

Since the applied field is fixed in the laboratory, the components in the crystal coordinate system will be Hz = Ho cos 0, H = Ho sin 0 cos <(>, and H, = Ho sin 0 sin , where 0 and are the usual spherical coordinate angles. Thus, for NMR in a single crystal of a metallic solid for which the Knight shift tensor has axial symmetry, the dependence of the resonance frequency v on crystal orientation with respect to the laboratory field Ho will be given by [Abragam (I960)] ... 

Consider the case of two neutral, linear, dipolar molecules, such as HCN and KCl, in a coordinate system with its origin at the CM of molecule A and the z-axis aligned with the intemiolecular vector r pointing from the CM of A to the CM of B. The relative orientation of the two molecules is uniquely specified by their spherical polar angles 0, 03 and the difierence <]) = - <])3 between their azimuthal angles. The leading temi in the... 

In addition, the volume element of interest is not the box dx dy dz shown in Fig. 1.6a but, rather, a spherical shell of radius r and thickness dr as shown in Fig. 1.6b. The result of expressing the volume element in spherical coordinates and integrating over all angles is the replacement... 

Consider the specific example of a spherical electrode having the radius a. We shall assume that diffusion to the spherical surface occurs uniformly from all sides (spherical symmetry). Under these conditions it will be convenient to use a spherical coordinate system having its origin in the center of the sphere. Because of this synunetry, then, aU parameters have distributions that are independent of the angle in space and can be described in terms of the single coordinate r (i.e., the distance from the center of the sphere). In this coordinate system. Pick s second diffusion law becomes... 

They often appear as products of the function l/v/2jre/m. The angles 6 and p are just the two angles defined in spherical coordinates, as shown in Fig. 6-5. The function sin 0 appearing in the integral arises from the appropriate volume element. The functions... 

We have gone from Cartesian to spherical coordinates in the last member, as a convenience in imposing the limit on the integrations, and carried out the integration over angles, giving the factor of 4ir the division by 8 is because only positive values of nx, ny, and nz are to be considered. Now we have... 

As seen from Chapter 2, adsorbed molecules often form monolayers with chain orientational structures in which the chains with identically oriented molecules alternate (Fig. 2.4). Consider the Davydov splitting of vibrational spectral lines in such systems. Let molecular orientations be specified by the angles 6> and spherical coordinate system with the z-axis perpendicular to the lattice plane ... 

We now need to convert dco into terms involving the spherical coordinates 9 and cf>. As shown in Fig. 3.19a, a given solid angle co traces an area a on the surface of a sphere of radius r. When a = r2, the solid angle co is by definition 1 sr (sr = steradian). For the more general case of a surface area a subtended by the solid angle co,... 

FIGURE 3.19 Conversion of solid angle u> to spherical coordinates. 

This expression gives the probability that the loose end of the molecule is in a volume element located at some particular values of x, y, and z, as shown in Figure 2.13a. Our interest is not in any specific x, y, z coordinates, but in all combinations of x, y, z coordinates that result in the end of the chain being a distance L from the origin. This can be evaluated by changing the volume element in Equation (69) to spherical coordinates and then integrating over all angles. 

As is the case with all differential equations, the boundary conditions of the problem are an important consideration since they determine the fit of the solution. Many problems are set up to have a high level of symmetry and thereby simplify their boundary descriptions. This is the situation in the viscometers that we discussed above and that could be described by cylindrical symmetry. Note that the cone-and-plate viscometer —in which the angle from the axis of rotation had to be considered —is a case for which we skipped the analysis and went straight for the final result, a complicated result at that. Because it is often solved for problems with symmetrical geometry, the equation of motion is frequently encountered in cylindrical and spherical coordinates, which complicates its appearance but simplifies its solution. We base the following discussion on rectangular coordinates, which may not be particularly convenient for problems of interest but are easily visualized. 

Following the physicists convention, we use for longitude and d for colatitude, i.e., the angle of formed hy a point, the center of the sphere and the north pole. We can express Cartesian coordinates in terms of spherical coordinates on the two-sphere as follows ... 

The angles a, d, are spherical coordinates on the sphere clearly d and are the usual spherical coordinates on momentum space. The surface element on the unit sphere... 

State a set of plausible boundary conditions that would be needed to solve the equation for i)0. Remember that in spherical coordinates the angle 6 is measured from the normal to the stationary plate. (It will turn out that an explicit boundary condition at the outer radius, r = R will not be required.)... 

This gives the probability for the absorption of a photon with a frequency oo traveling along a particular angle pair in spherical coordinates. This must then be integrated over by the solid angle <70 and evaluated. 

Coordinates of molecules may be represented in a global or in an internal coordinate system. In a global coordinate system each atom is defined with a triplet of numbers. These might be the three distances x,, y,-, z, in a crystal coordinate system defined by the three vectors a, b, c and the three angles a, / , y or by a, b, c, a, P, y with dimensions of 1,1,1,90°, 90°, 90° in a cartesian, i. e. an orthonormalized coordinate system. Other common global coordinate systems are cylindrical coordinates (Fig. 3.1) with the coordinate triples r, 6, z and spherical coordinates (Fig. 3.2) with the triples p, 9, . 

The overall emission rate of photons with energy tku at time t reads (note the use of spherical coordinates for k and the abbreviation of the solid angle integration by f do)... 

Figure 4.11 Definition of the solid angle fi0 of a point source Q accepted by the entrance slit S of a sector CMA. Two cross-cuts are shown (a) for a plane containing the symmetry axis of the analyser, (b) for a plane perpendicular to this axis. The principal ray starting at Q is shown together with the angular spreads from the finite acceptances in A and Acp. If expressed in spherical coordinates, the slit S has a width b = r2 A and a length i = i ,2 + A

Orientation angle, or polar spherical coordinate. Brownian noise function, (7.96). 

Kinetics of establishing of orientational equilibrium of a magnetic moment of a single-domain particle in the presence of thermal fluctuations is described by FPE (4.27). We express it in spherical coordinates at the surface of a unit sphere. Assuming that all the functions depend only on the meridional angle H, we obtain, [47]... 

In each case the probability density may be represented in spherical coordinates in the form of a three-dimensional diagram (see Fig. 2.2) in which the angles 0, ip correspond to the arguments of the function pb(0,ip), whilst the value of the function is plotted along the radius-vector r(0, ip). 

A system of N spherical particles in an electrolyte solution with permittivity em is considered. Particle radii are denoted as ak, and their permittivities are denoted as ek (k = 1, 2,TV). We link the local polar spherical coordinates (rk, 0k, (pk) with the particle centers (rk is a polar radius, 6k is an azimuth angle, q>k is a polar angle). The arrangement of two arbitrarily chosen particles from the ensemble is shown in Figure 1 with corresponding coordinates indicated. Global coordinates (x,y,z) of an observation point P(x,y,z) are determined by vectors rk, r. in the local coordinates, and a distance between centers of the spheres is Rkj (Figure 1). 

---