Rayleigh limitations

Since the intercept corresponds to the Rayleigh limit also (i.e., 0 = 0°), Eq. (10.57) demonstrates this to be the weight average value of M. 

The rival theory of Iribame and Thomson54 assumes that gas-phase ions solvated by a few solvent molecules are emitted into the gas phase by charged droplets with radii R < 10 nm, which are somewhat below the Rayleigh limit. This ion evapora-... 

On the contrary, it may be argued that the electric field strength locally necessary to evaporate ions from a droplet cannot be attained because of the prior fission of the droplet due to crossing the Rayleigh limit. [23,90]... 

Here (7, is the surface tension, and Sq is the permittivity of free space. The mode n = 0 corresponds to the equilibrium sphere, and n = 1 is a purely translational mode. The first unstable mode is n = 2. The critical charge for this mode is given by setting Eq. (37) to zero, which yields the Rayleigh limit of charge,... 

A number of investigators attempted to verify the Rayleigh limit using Millikan condensers and quadrupoles (Doyle et ai, 1964 Abbas and Latham, 1967 Berg et al., 1970), but these studies were not sufficiently precise because... 

Richardson et al (1989) performed similar measurements for droplets of sulfuric acid and dioctyl phthalate (DOP) in a quadrupole. Sulfuric acid droplets exploded prior to the Rayleigh limit (at 84 20% of the Rayleigh limit), and the DOP droplets fissioned approximately at the Rayleigh limit... 

The low-frequency limit of c" (9.16) correctly describes the far-infrared (1 /X less than about 100 cm-1) behavior of many crystalline solids because their strong vibrational absorption bands are at higher frequencies. This limiting value for the bulk absorption, coupled with the absorption efficiency in the Rayleigh limit (Section 5.1), gives an to2 dependence for absorption by small particles this is expected to be valid for many particles at far-infrared wavelengths. 

The origin of the misconception that the absorption spectrum of particles in the Rayleigh limit is not appreciably different from that of the bulk parent material is easy to trace. Again, for convenience, let us take the particles to be in free space. In Chapter 3 we defined the volume attenuation coefficient av as the extinction cross section per unit particle volume if absorption dominates extinction, then av for a sphere is 3Qabs/4a, where a is the radius. If we assume that n k, which is true for most insulating solids at visible wavelengths, then... 

Measurements of extinction by small particles are easier to interpret and to compare with theory if the particles are segregated somehow into a population with sufficiently small sizes. The reason for this will become clear, we hope, from inspection of Fig. 12.12, where normalized cross sections using Mie theory and bulk optical constants of MgO, Si02, and SiC are shown as functions of radius the normahzation factor is the cross section in the Rayleigh limit. It is the maximum infrared cross section, the position of which can shift appreciably with radius, that is shown. The most important conclusion to be drawn from these curves is that the mass attenuation coefficient (cross section per unit particle mass) is independent of size below a radius that depends on the material (between about 0.5 and 1.0 fim for the materials considered here). This provides a strong incentive for deahng only with small particles provided that the total particle mass is accurately measured, comparison between theory and experiment can be made without worrying about size distributions or arbitrary normalization. 

Figure 12.12 Maximum infrared extinction cross sections of spheres normalized by the value in the Rayleigh limit.

If Vs/Is is to be nonzero, either S34/Sn and Ums/Ims, or S44/S] i and Vms/Ims, or both, must not be zero. S34 is zero for nonabsorbing particles in the Rayleigh limit and for arbitrary particles in the Rayleigh-Gans approximation. Even for particles—spherical and nonspherical—of size comparable with the wavelength, however, 34 tends to be small, particularly in the forward direction (see Figs. 13.13 and 13.14). 

An additional question arises concerning the way the Rayleigh criterion terminology has traditionally been used. There is a vague perception that the Rayleigh limit is a barrier to be penetrated only by supernatural means— that it is a limit of the most fundamental kind. And so it is, but only in a sense... 

This chapter has attempted to give some flavor of the historical development of nonlinear methods. Early investigators of these methods expended great effort in overcoming the popular notion that bandwidth extrapolation was not possible or practical. It was, for example, believed that the Rayleigh limit of resolution was a limit of the most fundamental kind—unassailable by mathematical means. To be sure, the popular notion was reinforced by a long history of misfortune with linear techniques and hypersensitivity to noise. Anyone who still needs to be convinced of the virtues of the nonlinear methods would benefit from reading the paper by Wells (1980) the nonlinear point of view is nowhere else more clearly stated. 

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