Circles-on-a-sphere model

The experimental designs that enable us to study this type of model arc all established according to the same principles They are composed of points that arc regularly distributed on a circle / = 2). a. sphere (A = 3). or a hyperspherc (it > 3). 

The 3D representation of the test object can be rotated by means of an ARCBALL interface. Clicking on the main client area will produce a circle which is actually the silhouette of a sphere. Dragging the mouse rotates the sphere, and the model moves aceordingly. An arc on the surface of the sphere is drawn for visual feedback of orientation additionally a set of coordinate axes in the bottom left comer provides further feedback. 

Fig. la. Atomic structure ofa two-dimensional nano-structured material. For the sake of clarity, the atoms in the centers of the crystals are indicated in black. The ones in the boundary core regions are represented by open circles. Both types of atoms are assumed to be chemically identical b Atomic arrangement in a two-dimensional glass (hard sphere model), c Atomic structure of a two-dimensional nanostructured material consisting Of elastically distorted crystallites. The distortion results from the incorporation of large solute atoms. In the vicinity of the large solute atoms, the lattice planes are curved as indicated in the crystallite on the lower left side. This is not so if all atoms have the same size as indicated in Fig. la [13]... 

Figure 8.9 Variation of the entropy convergence temperature with increasing hard-sphere radius. The thin solid line is the convergence temperature determined under the assumption that the heat capacity is independent of temperature, and the thick solid line is the exact entropy convergence temperature for spheres smaller than R < (Tww/2 (Ashbaugh and Pratt, 2004). The dashed line smoothly interpolates between the exact and constant heat capacity curves at 1.25 A and 3.3 A, respectively. The filled circle indicates the entropy convergence temperature of a methane-sized solute (7), = 382K). The open circle indicates the entropy convergence temperature based on the information model = 420 K) (Ashbaugh and Pratt, 2004).

Fig. 18. Average fractional energy transfer of diretly scattered oxygen atoms as a function of deflection angle, x. for i i) = 47 kJ mol and 6i = 60° (circles). The dashed line is the hard-sphere model prediction based on the effective surface mass, ms, shown. The solid line is the revised prediction after the hard-sphere model is corrected for the internal excitation of the interacting surface fragment. The correction is derived from a kinematic analysis of scattering in the c.m. reference frame.

Certain old assumptions die hard, particularly those having to do with circularity of motion. The circle is a sacrosanct figure for many cultures and, though it may be refuted as the actual description of a natural phenomenon, it is never completely abandoned as the ideal for which nature somehow strives. Bohr s circular orbits would give way to other models but the role of the circle itself continued to enlarge. Here we are going to explore, not quantum theory itself but the assumptions on which it is based. These include the circle, or more accurately the sphere, in a prominent role. 

However, the motions are small amplitude. Small-amplitude wiggling can equally well be modeled by aromatic CH motion on a circle or sphere, or by C2 rotations. Thus, any of these... 

Good measurements of contact are difficult to carry out and even more difficult to interpret because of the individualistic character of a given surface. Therefore the tendency has been to fit experimental results to behavior inferred from models. In one of the early simple models, a surface is viewed as an assembly of spherical asperities, and one of the basic schemes of contact is the mechanical interaction between a deformable plane surface and a spherical asperity. If the deformation of the sphere is elastic the deformed area on the sphere is a circle, and the relation between the load pressing the flat against the sphere and the radius of the circle is given by the familiar formula of elastic theory... 

Fig. 53. Hard sphere model of three interchangeable configurations for the atom pairs of Mo on the W 21l plane. Small open circles on each surface channel indicate the adsorption sites for adatoms, separated, respectively, by 2.7 A. (From ref. 414.)...

A major complication with this cosmic model was the Pythagorean demand of perfect circular orbits for all celestial bodies. The more accm-ate astronomical observations became, the more complicated the required corrections to restore the circular pattern. The authority of Pythagoras and Thales must have been enormous, especially as Plato also endorsed the geometrical perfection of the cosmic composition. The planets, including the srm and the moon, like the fixed stars, circled the earth on concentric spheres with numerical regularity, each contributing its harmonious pitch to the music of the spheres. 

Figure 19-8. Measured second virial coefficients ofSTA (soUd squares) in dffferent background salt concentrations compared with data on a number of proteins (Lysotyme, BPTI open circles) in different buffer solutions. The second virial coefficients are nondimensionalized with the hard sphere value and plotted against the solubility (volume fraction 0at) of the respective species. The solid lines are calculations of the attractive Yukawa potential with two different ranges of attractions (2ak) of 7 and 15. The values of 7 and 15 indicate that attractions between the particles are short ranged. The experimental datafor STA (at high salt concentrations) and proteins collapse within the narrow range of attractions which are only a fraction of the particle diameter. The collapse also indicates thatproteins and STA are thermodynamically similar, iftwo suspensions have the same B2 then they have the same solubility. This plot also provides an opportunity to extract interaction potential parameters for a given experimental system in a model independent manner. For detailed discussions, please refer to (Ramakrishnan, 2000).

For the entropic repulsion calculations, the model system considered is that of a spherical particle, radius a, separated by a distance H from a plate, with both surfaces coated with adsorbed polymer chains of root-mean-square (rms) height /r at a surface coverage 0. If pairs of chains of height /i on opposing surfaces just touch when those on the sphere lie on a circle which subtends, at the centre of the sphere, a semi-angle

entropic repulsion Kr for the system is given in kT units by Equation 9.9, where is the... 

Figure 17 Demonstration of molecule manipulation with a virtual trackball. Depressing the left mouse button activates the virtual trackball manipulation mode. The white circle represents the two-dimensional projection of a sphere surrounding the molecular model. Depressing the mouse while inside the circle grabs the sphere and allows the user to rotate the sphere and enclosed molecule. Mouse motion outside the circle rotates the molecule about the Z axis, llie white circle is only displayed while the mouse button is depressed. The mouse cursor, shown near the perimeter of the circle, changes shape depending on whether the mouse is positioned inside or outside the circumference of the circle and thus provides additional feedback to the user...

It seems quite natural to describe the extended part of a quantum particle not by wavepackets composed of infinite harmonic plane waves but instead by finite waves of a well-defined frequency. To a person used to the Fourier analysis, this assumption—that it is possible to have a finite wave with a well-defined frequency—may seem absurd. We are so familiar with the Fourier analysis that when we think about a finite pulse, we immediately try to decompose, to analyze it into the so-called pure frequencies of the harmonic plane waves. Still, in nature no one has ever seen a device able to produce harmonic plane waves. Indeed, this concept would imply real physical devices existing forever with no beginning or end. In this case it would be necessary to have a perfect circle with an endless constant motion whose projection of a point on the centered axis gives origin to the sine or cosine harmonic function. This would mean that we should return to the Ptolemaic model for the Havens, where the heavenly bodies localized on the perfect crystal balls turning in constant circular motion existed from continuously playing the eternal and ethereal harmonic music of the spheres. 

In the real world, however, the objects we see in nature and the traditional geometric shapes do not bear much resemblance to one another. Mandelbrot [2] was the first to model this irregularity mathematically clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. Mandelbrot coined the word fractal for structures in space and processes in time that cannot be characterized by a single spatial or temporal scale. In fact, the fractal objects and processes in time have multiscale properties, i.e., they continue to exhibit detailed structure over a large range of scales. Consequently, the value of a property of a fractal object or process depends on the spatial or temporal characteristic scale measurement (ruler size) used. 

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