Cosine Volume

Cosine smearing. Because instrument volume and experiment time must both be minimized for a planetary Mossbauer spectrometer, it is desirable in backscatter geometry to illuminate as much of the sample as possible with source radiation. However, this requirement at some point compromises the quality of the Mossbauer spectrum because of an effect known as cosine smearing [327, 348, 349] (see also Sects. 3.1.8 and 3.3). The effect on the Mossbauer spectrum is to increase the linewidth of Mossbauer peaks (which lowers the resolution) and shift their centers outward (affects the values of Mossbauer parameters). Therefore, the diameter of the source y-ray beam incident on the sample, which is determined by a... 

Since mercury has a contact angle with most solids of about 140°, it follows that its cosine is negative (i.e., it takes applied pressure to introduce mercury into a pore). In a mercury porosimeter, a solids sample is evacuated in a cell, mercury is then intruded, and the volume, V, is noted (it actually reads out), and the pressure, P, is then increased stepwise. In this fashion it is possible to deduce the pore volume of a particular radius [corresponding to P by Eq. (21)]. A pore size distribution will give the total internal pore area as well, which can be of importance in dissolution. 

Where there is more than one cell of the same volume (and shape appropriate to the crystal system) which will account for all the reflections, as in monoclinic and triclinic orystals (see Fig. 76), the most nearly rectangular cell will usually prove the most convenient to accept. There has been much discussion of conventions in the triclinic system the convention recommended bj" International Tables (1952) is that all three angles of the cell should be obtuse, and the direction cosines of the axes with the fill] zone axi should all be positive. 

Determine the eigenvectors associated with each of the eigenvalues. With the eigenvectors being the direction cosines of the rotation to the principal axes, make a sketch showing the orientation of an r-z control volume in the principal directions. 

Any departure introduced into an assumed steady state of a system. The magnitude of the departure is often assumed to be small so that product terms in the dependent variables may be neglected the term perturbation is therefore sometimes used as synonymous with small perturbation. The perturbation may be concentrated at a point or in a finite volume of space or it may be a wave (sine or cosine function) or, in the case of a rotating system, it may be symmetric about the axis of rotation. 

In the above, ijk denote the Cartesian coordinates of a molecule, IJK those of a crystal (a unit cell), 7Vt is the number of molecules in a unit volume occupying each particular inequivalent site in the unit cell,/ is the number of inequivalent positions of a molecule in a unit cell, and Ng is the number of equivalent positions in a unit cell. The directional cosines are used to transform each of the molecular 0 components to those of the new coordinate system (bUK) and the contributions are summed. 

For liquid water, the relative dielectric constant at 25 °C is 78. Calculate its deformation polarizability at this temperature if there are 10 moles of deformable molecules per unit volume. Use the average of the cosines of the angles... 

Figure 26.2 The effect of contact time on the cosine of static contact angles of many different size droplets equilibrating on Teflon and nylon polymer surfaces the arrows indicate the increase of time the cosine of the static contact angle decreases with droplet volume.

When the angles are all 90° (so that their cosines are 0), this formula reduces to the simple result V = abc for the volume of a rectangular box. If the mass of the unit cell contents is known, the theoretical cell density can be computed. This density must come close to the measured density of the crystal, a quantity that can be... 

Two main types of models for tubular lamps (the most widely used) will be described. There are lamps that produce an arc that emits radiation and, consequently, photons come out directly from such an arc. Emission is made by the whole lamp volume. We call this process Voluminal Emission. There are other types of lamps in which the discharged arc between electrodes induces an emission produced by some particular substance that has been coated on the lamp surface. We call this process Superficial Emission. Voluminal emission may be safely modeled as an isotropic emission in this case the specific intensity associated with each bundle of radiation originated in some element of volume of the lamp is independent of direction, and the associated emitted energy (per unit time and unit area) is also isotropic (Figure 6.6). On the other hand, it seems that superficial emission can be better modeled by a diffuse type of emission that is also known as one that follows the Lambert s cosine law of emission in this case the emitted intensity is independent of direction but the emitted energy depends on the surface orientation and follows the cosine law equation (Figure 6.7). The following assumptions are made (Irazoqui etal., 1973) ... 

The formula (5.1) is true for a scalar, vector or tensor function Fat a fixed index i the summation over i is also allowed. The function F is supposed to be continuously differentiable in volume V enclosed by a sufficiently smooth surface 2. Symbol rii designates the i-th component of the normal n external to V (the direction cosine). If F = u are the velocity components, then performing in (5.1) the summation over i, one obtains... 

The sine of the loss angle (sin d) or the cosine of the phase angle (cos 6) is termed the power factor. In electrical applications the power loss (PL) is defined as the rate of energy loss per unit volume and is derived to be... 

An infinitely long cylindrical Newtonian liquid jet, is disturbed with a spatially harmonic surface displacement of a cosine shape R = a — Cocoskz, where k = Ina/X, and a is determined such that the volume of the jet is kept constant when the initial amplitude is changed. Therefore, a = (1 — Co/2) - The dynamics of this jet due to capillary forces was investigated for various values of initial disturbance wave number k, and initial amplitude i o> and of the jet Ohnesorge number, Oh. 

We consider first the nucleation under an applied simple shear stress a without the presence of an accompanying mean normal stress We consider the nucleation as a reversible experiment in which the transformation strain in the volume element Qf can be built up by an imposed external shear strain y. As is clear, and as discussed by Orowan (1954), this results in a rise in the Helmholtz free energy AF(y) that initially is given by a quadratic function and for the whole process can be taken as being given by a cosine potential (Kocks et al. 1975) (see also Johnson and Samwer (2005)),... 

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