Sphere method

Figure 14.10 Illustration of the sphere method. Energy minima on the hyperspheres are denoted by , while R indicates a (local) minimum in the full variable space...

In the case of pharmaceutical quality control, and in many other spheres, methods which give an answer in a shorter hme are being investigated, evaluated and in some cases used. Some of these quicker or rapid methods will be referred to in this sechon. 

At high-particle number densities or low coefficients of normal restitution e, the collisions will lead to a dramatical decrease in kinetic energy. This is the so-called inelastic collapse (McNamara and Young, 1992), in which regime the collision frequencies diverge as relative velocities vanish. Clearly in that case, the hard-sphere method becomes useless. 

In the following, we focus on the soft-sphere method since this really is the workhorse of the DPMs. The reason is that it can in principle handle any situation (dense regimes, multiple contacts), and also additional interaction forces—such as van der Waals or electrostatic forces—are easily incorporated. The main drawback is that it can be less efficient than the hard-sphere model. 

Cupprammonium Solution. This solvent for NC used in determination of its viscosity by the falling sphere method, contains 30 2 g of pure Cu, 165 2g of NH40H (26 to 25%... 

In [43], the carbon cage was represented as a classical infinitesimally thin spherical conducting sphere. Methods of classical physics were used to determine the dipole polarizability ad co) of the sphere. Accordingly,... 

Old observations on spheres falling through liquids are collected by Muncke.12 The falling sphere method seems to have been first used for viscous liquids (tar, glycerol) by Schottner, ho used 2-4 cm. diam. spheres in a vessel of 20 cm. diam. With. 13 cm. diam. tubes the rate of fall was slower. The values of 97 were larger than those found by Poiseuille s method. 

Compressible Sugar Confectioner s Sugar Sugar spheres Method IV NF (19, pp. 2528 and 2529)... 

We had to choose one of the earliest and least precise of the numerous methods for viscosity measurements that have been developed over the years, because the falling-sphere method was the only one that could be adapted to low-temperature measurements. The determination of viscosity was made by measuring the resistance to motion of a falling sphere in the fluid. Since the viscosities measured in our experiments on aqueous-organic mixtures at cryogenic tern-... 

Figure 2.12 The Ewald sphere method illustrates the ring pattern of diffraction from a powder specimen. The Debye ring recorded by the Elull-Debye-Scherrer method results from randomly oriented crystals in the powder specimen, in which reciprocal lattice points of (hkl) touch the Ewald sphere surface in various directions to form individual rings. It is equivalent to rotating a reciprocal lattice along an incident beam axis. (Reproduced with permission from R. Jenkins and R.L. Snyder, Introduction to X-ray Powder Diffractometry, John Wiley Sons Inc., New York. 1996 John Wiley Sons Inc.)...

Figure 3.41 Deviation from Bragg s Law diffraction illustrated by the Ewald sphere method. The deviation from exact Bragg conditions is represented by vector s in reciprocal space, ko is the primary beam vector, k is the scattering beam vector and g is a reciprocal lattice vector.

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