Fuzzy spheres

Fuzzy spheres. Radially varying dielectric response, 79 "Point-particle" interactions, 79 Point-particle substrate interactions, 85 Particles in a dilute gas, 86 Screening of "zero-frequency" fluctuations in ionic solutions, 89 Forces created by fluctuations in local concentrations of ions, 90 Small-sphere ionic-fluctuation forces, 91... 

Fuzzy spheres. Radially varying dielectric response... 

Fuzzy spheres, radially varying dielectric response 156... 

Fixman M. Polyelectrolytes a fuzzy sphere model. J Chem Phys 1964 41 3772-3778. 

The rotational diffusion coefficient of the fuzzy cylinder can be formulated in a similar way. For the rotational diffusion process, it is convenient to imagine a hypothetical sphere which has the diameter equal to Lc, just encloses the test fuzzy cylinder, and moves with the translation of the fuzzy cylinder. If the test cylinder and the portions of surrounding fuzzy cylinders entering the sphere are projected onto the spherical surface as depicted in Fig. 15b (cf. [108]), the rotational diffusion process of the test cylinder can be treated as the translational diffusion process of a circle on the hypothetical spherical surface with ribbon-like obstacles. 

Further, ions are not hard, billiard ball like spheres. Since the wave functions that describe the electronic distribution in an atom or ion do not suddenly drop to zero amplitude at some particular radius, we must consider the surfaces of our supposedly spherical ions to be somewhat fuzzy. A more subtle complication is that the apparent radius of an ion increases (typically by some 6 pm for each increment) whenever the coordination number increases. Shannon10 has compiled a comprehensive set of ionic radii that take this into account. Selected Shannon-type ionic radii are given in Appendix F these are based on a radius for O2- of 140 pm for six coordination, which is close to the traditionally accepted value, whereas Shannon takes the reference value as 126 pm on the grounds that it gives more realistic ionic sizes. For most purposes, this distinction does not mat-... 

As a general rule greater than 70% of the particles are spherical. They may be perfect spheres or they may be distorted in some way. Three-dimensional roundness is the basis for this classification. The surface of the particles may be smooth, fuzzy, scaly, or have smaller spheres on their surface. Sometimes they are perforated, capped, broken, or stemmed. The vast majority have diameters of less than 5 pm although they range in size from <0.5 pm to >32 pm (it is not practical to detect particles less than 0.5 pm in diameter). 

Core-sphere radius Rs, fuzzy-layer thickness ARf, center-to-center distance z Small steps allowed in s s at Rs and Rs + ARf ... 

The amplitude or wave function, is the orbital. As is generally true for waves, however, it is the square of the amplitude, that has physical meaning. For electron waves, represents the probability of finding an electron at any particular place. The fuzzy balls or simple spheres we draw to show the shapes of orbitals arc crude representations of the spate within which has a particular value—the space within which the electron spends, say, 95% of its time. Whether is positive or negative, is of course positive this makes sense, since probability cannot be negative. The usual practice is to draw the lobes of a p orbital to represent if + or - signs are added, or one lobe is shaded and the other unshaded, this is to show the relative. signs of [Pg.927]

Actually, the orbits are three-dimensional with the electrons apparently traveling around the surface of an imaginary sphere or shell, and the actual paths are much more fuzzy than our sim-... 

In the commercial formulations the presence of particles at =0.20 merely increases the level of the viscosity and shifts the characteristic shear rates and frequencies. Jenkins data (48) conforms at 0.10, but at higher volume fractions the Newtonian plateau disappears. This raises the question of whether a reversible polymer network persists or the particles simply interact as fuzzy and, perhaps, sticky spheres with much slower dynamics that control the shear rate. Additional (kta exists beyond that cited here but d s not seem to resolve this issue. 

Ground-State Orbitals. The carbon-atom orbitals in the ground state can be visualized as shown graphically in Fig. 3.3. The wave-function calculations represent the s orbital as a sphere with a blurred or fuzzy edge that is characteristic of all orbital representation. As a sphere, the s orbital is non-directional. The 2p orbital can be represented as an elongated barbell which is symmetrical about its axis and consequently is directional. 

A computer then generates a contour map of the surface and the outline of individual atoms can be detected. The atoms resemble the hard spheres proposed by Dalton (Figure 2.10), but the STM images are in fact showing the electrons. The fuzziness occurs because the electrons move in a cloud and are not in fixed energy levels or orbits. Previous generations of chemists believed in atoms, but the STM provides empirical evidence for the existence of atoms. 

Pattern recognition techniques based on fuzzy objective function minimization use objective functions particular to different cluster shapes. Ways to approach the problem of correctly identifying the cluster s shape are the use of adaptive distances in a second run to change the shapes of the produced clusters so that all are unit spheres, and adaptive algorithms that dynamically change the local metrics during the iterative procedure in the original run, without the need of a second run. 

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