Cylinder, infinite

Here iC is the intrinsic anisotropy constant due to the crystalline anisotropy. After the demagnetization in the longest direction, k is the shape-dependent constant (for an infinite cylinder k = 1.38), M is the exchange constant, and R the particle radius. An infinite cylinder with only shape anisotropy gives... 

Infinite cylinder, half-circle cross section radiating to ... 

The geometrical factor, like the filling factor, shifts the position of the resonance peak. When = 0 we have the case of an infinite cylinder (see Table 1). An infinite cylinder connects one side of the crystal to the other. Therefore, the electrons travel freely through the crystal. Actually, this is not the situation of metallic particles dispersed in an insulator any more. The situation corresponds... 

Figure 9.18. Temperature as a function of axis temperature in an infinite cylinder of radius >-,.

Sphere—radiation to surface Infinite circular cylinder— radiation to curved surface Semi-infinite cylinder— radiation to base Cylinder of equal height and diameter — radiation to entire surface Infinite parallel planes — radiation to planes... 

AWBERY, J. H. 426, 563 Axial thrust in centrifugal pumps 339 Axis temperature, infinite cylinder 406 A7.1Z, K. 183,184.185,227... 

Runaway will occur when the calculated delta (8) exceeds the critical delta (8cr) which depends on the shape of the reaction mixture 0.88 for a plane slab, 2.00 for an infinite cylinder, 2.78 for a right cylinder with 1/d equal to 1, and 3.32 for a sphere. Bowes [133] provides formulas for calculation of 8cr for other geometric shapes and structures. In this model, heat is lost by conduction through the material to the edge, where the heat loss rate is infinite relative to the conduction rate. In this model, there is a maximum temperature in the center as shown in Figure 3.20 Case B. 

Deviations from the Forster decay (Eq. 9.29) arise from the geometrical restrictions. In the case of spheres, the restricted space results in a crossover from a three-dimensional Forster-type behavior to a time-independent limit. In an infinite cylinder, the cylindrical geometry leads to a crossover from a three-dimensional to a one-dimensional behavior. In both cases, the geometrical restriction induces a slower relaxation of the donor. 

Two-dimensional flow past infinite cylinders is not treated in detail since such bodies do not meet our definition of a particle (see Chapter 1). 

A calcn similar to the first case has been done by Zinn Mader (Ref 11a) for a semi-infinite slab, an infinite cylinder, and a sphere, and compared with experimental data for the cylindrical shape. In spite of the fact that the mathematical model neglects fusion (the expln temp is usually above mp of the expl), agreement betw the calcd and the exptl values is reasonably good... 

Appendix A. Homogeneous Sphere, 477 Appendix B. Coated Sphere, 483 Appendix C. Normally Illuminated Infinite Cylinder, 491 References, 499 V... 

In Chapter 8 we shall derive the field scattered by an infinite cylinder of arbitrary radius and refractive index we shall also consider scattering by a finite cylinder in the diffraction theory approximation. Although the finite cylinder scattering problem is not exactly soluble, we can obtain analytical expressions for the amplitude scattering matrix elements in the Rayleigh-Gan s approximation. 

Figure 8.2 Cylindrical polar coordinate system. The z axis lies along the axis of the infinite cylinder.

Figure 8.3 Infinite cylinder obliquely illuminated by a plane wave.

Figure 8.4 Wave front and wave normals of light scattered by an infinite cylinder.

In Chapter 3 we derived a general expression for the amplitude scattering matrix for an arbitrary particle. An unstated assumption underlying that derivation is that the particle is confined within a bounded region, a condition that is not satisfied by an infinite cylinder. Nevertheless, we can express the field scattered by such a cylinder in a concise form by resolving the incident and scattered fields into components parallel and perpendicular to planes determined by the cylinder axis (ez) and the appropriate wave normals (see Fig. 8.3). That is, we write the incident field... 

Although we have repeatedly referred to an infinite cylinder, it is clear that no such cylinder exists except as an idealization. So what we really have in mind is a cylinder long compared with its diameter. Later in this section we shall try to acquire some insight into how long a cylinder must be before it is effectively infinite by considering scattering in the diffraction theory approximation. 

In a manner similar to that in Section 3.4, where we considered cross sections of finite particles, we can calculate cross sections per unit length of an infinite cylinder by constructing an imaginary closed concentric surface A of length L and radius R (Fig. 8.6). The rate Wa at which energy is absorbed within this surface is... 

Figure 8.7 Scattering cross section per unit particle volume for normally incident light polarized parallel (---) and perpendicular (...) to the axis of an infinite cylinder in air.

We need consider only scattering directions in the plane = tt/2 (or < > = 377/2) because p vanishes outside this plane we also have 0 = 0 when = 77/2 and 0 = - 6 when = 377/2, where 0 = 0 is the forward direction. Thus, we may take the phase function for scattering by an infinite cylinder in the diffraction theory approximation to be... 

Figure 8.10 Phase function for scattering of unpolarized light by an infinite cylinder. The arrows indicate minima according to diffraction theory.

A few particles, such as spores, seem to be rather well approximated by spheroids, and there are many examples of elongated particles which may fairly well be described as infinite cylinders. Our next step toward understanding extinction by nonspherical particles is to consider calculations for these two shapes. To a limited extent this has already been done spheroids small compared with the wavelength in Chapter 5 and normally illuminated cylinders in Chapter 8. We remove these restrictions in this section measurements are presented in the following section. Because calculations for these shapes are more difficult than for spheres, we shall rely heavily on those of others. 

---