Azimuthal symmetry

To simulate the PECVD process, a design team creates a PDE model involving momentum and mass balances, as summarized below. It is sufficient to assume the plasma to be a continuum, with physical properties of the gas constant (independent of position and time), negligible volume change of the reacting gases, and velocity and concentration fields symmetric about the reactor centerline (azimuthal symmetry). 

The projection of T,p on each of the radial unit vectors can be evaluated in terms of the basic angular functions which make up the vector spherical harmonics.(27) Although these functions are associated Legendre polynomials for an arbitrarily oriented donor dipole, for the case of full azimuthal symmetry shown in Figure 8.19 the angular functions are ordinary Legendre functions, P (i.e., w = 0). Under these circumstances,... 

Angular light scattering measurements are sometimes classified as either absolute or relative. In an absolute measurement 7s//(, which is directly related to the differential scattering cross section (13.2), is determined in a relative measurement the irradiance is referred to some arbitrary scattering angle, say 10°, so that (assuming azimuthal symmetry)... 

While direct diagonalization of the Hamiltonian matrix works well for situations in which there is a finite number of states, as in Fig. 9.1, it is clearly hopeless to try it in this case. A useful WKB approach was proposed by Edmonds12 and refined by Starace.13 Using the fact that azimuthal symmetry exists, Starace writes the wavefunction of the spinless Rydberg electron in cylindrical coordinates as13... 

When an isolated sphere is held stationary in a creeping flow field containing suspended particles (see Figure 2), the equation of continuity in spherical coordinates, allowing azimuthal symmetry, is given by... 

We are considering collision processes with azimuthal symmetry only hence dtp - = 2n ... 

Variables are defined in the nomenclature section. Differing from the case of the flat plate reactor where the radiation field in the employed part of the surface was almost uniform (differences were never larger than 9%), in this reactor the whole system exhibit only azimuthal symmetry and significant differences were observed in both directions (r, z). However, even more... 

For the mass balance, the following assumptions and operating conditions are considered (i) steady state, (ii) unidirectional, incompressible, continuous flow of a Newtonian fluid under laminar flow regime, (iii) only ordinary diffusion is significant for a mixture where the main component is water, (iv) azimuthal symmetry, (v) axial diffusion neglected as... 

In other words, assuming azimuthal symmetry (no dependence of the radiance on [Pg.194]

An energy balance can be made in the wafer to determine the temperature distribution along the wafer radius radial uniformity of temperature is important for temperature sensitive processes. Since the thickness of the wafer is much smaller than its radius, and assuming azimuthal symmetry, a one-dimensional analysis suffices. The wafer energy balance reads,... 

Calculating procedure for the LVRPA. In order to apply equation 6.95 we need to solve the RTE (equation 6.32) for this particular reactor set-up. As shown by Alfano etal. (1995) and Cabrera etal. (1994) the radiation field of this reactor can be modeled with a ID, one-directional radiation model and rather simple boundary conditions (Figure 6.10). Hence, with azimuthal symmetry derived from the diffuse emission at x = 0 ... 

The potentials (25), often referred to as mexican hat , represent the prototype of a conical intersection. The azimuthal symmetry of the adiabatic potentials reflects the existence of a constant of motion of the linear Jahn-Teller problem, the so-called vibronic angular momentumh ... 

In an electric field the spherical symmetry of the atom is broken, but the azimuthal symmetry remains thus if we ignore the spin of the valence... 

It can be seen that this problem has an azimuthal symmetry and the general solution is of the form [1] ... 

---