Similarity dimension

X-ray diffraction experiments revealed a psendo-cubic orthorombic unit cell with cell dimensions similar to the expected cubic F centered arrangement of the predesigned diamond-like crystal. 

Note that partial differentials are used in the 2nd Law (see 4.6.4. given above) since they relate only to the "x" dimension- similar equations may... 

Liquid transport is achieved by hydrostatic action, pumping or electroosmotic flow (EOF). So far, chip reactors have been employed at low to very low flow rates, e.g. from 1 ml min to 1 pi min. Applications consequently were restricted to the laboratory-scale or even solely to analytics. However, this is not intrinsic. By choosing larger internal dimensions, similar throughputs as for the other classes of liquid or liquid/liquid micro reactors are in principle achievable. 

So far, we have talked about phasing 1-D spectra but this is also valid for some 2-D experiments. Phase-sensitive 2-D experiments also require phasing in one or both dimensions. Similar approaches are used as described here. Note that this is not necessarily the case for all 2-D experiments as some of them are collected in magnitude mode where we look at only the intensity of the signals, not their sign. 

Frequency of energy deposition in volumes of dimensions similar to DNA and other biological molecules provides a method of interpretation of mechanism of radiation effects. In this method, the assumption is that radiation quality effects are mostly because of the spatial properties of the radiation on a microscopic scale. The database of absolute frequencies of energy depositions in cylindrical targets of dimensions 1-100 nm include ... 

The results of the hole-opening measurements are presented in the series of oscilloscope traces shown in Figure 16. Traces 16a and b are the results obtained on a sample fabricated with polystyrene of molecular weight 4 K. From trace a it can be seen that under the action of the 8 nsec dye laser pulse, the holes or pits develop fully in —30 nsec. Furthermore, from the measurements taken at 100 nsec/div (b), it can be seen that the pit, once fully developed, maintains its dimensions. Similar conclusions can be drawn from oscilloscope traces c, d and e which show measurements taken on samples... 

For diffusion in one and two dimensions, similar calculations show that (R2 (r)) = 2Dt and 4Dt, respectively. An analogous expression for (R2 (r)) when the diffu-sivity is anisotropic is explored in Exercise 7.4. 

Another important feature is the relative size of the diffusion layer with respect to the double layer. The diffusion-layer dimensions are proportional to the size of the electrode, and can approach the size of the double layer with electrodes of molecular dimensions [95]. This can also occur in other situations explored with microelectrodes. For example, in solutions of very dilute electrolyte, the diffuse double layer extends several nanometers into solution [62]. Alternatively, very fast cyclic voltammetry results in a very small diffusion layer, which may be of dimensions similar to the double layer [46]. In all these situ-... 

The first really successful artificial material used in the manufacture of synthetic blood vessels was Dacron , a polyester fiber made from polyethylene terephthalate (PET). The material is woven or knitted into thin tubes with dimensions similar to those of a natural blood vessel. The tubes are then treated with coagulated blood or with albumin, an important blood protein, to block the tiny holes in the fabric of which they are made. Over time, cells migrate into the blood or albumin trapped within the Dacron matrix and deposit collagen. As the blood or albumin degrades, it is replaced by the collagen, producing a vessel with some properties similar to those of natural blood vessels. 

The ceramic based on lead zirconate (PZ) has intermediate figure of merit values. Its final form as a thin plate is obtained by sawing up a hot-pressed block and lapping and polishing to the required dimensions. Similar methods must be applied to single crystals and so the costs of manufacture are not very different. 

Berkelium(III) orthophosphate has been characterized both by X-ray powder diffraction and by solid state absorption and Raman spectroscopies (111, 115). The X-ray data confirm the hypothesis that BkP04 is isostructural (monazite-type monoclinic) with the lighter lanthanide orthophosphates (177) and exhibits unit-cell dimensions similar to those of SmP04 and EuP04 (111). 

Three-Dimensional Model. On the other hand, if the interfacial layer is thick enough compared to the molecular size of SRIOI and if SRIOI molecules adsorbed on the interface are weakly oriented, the rotational motions of SRIOI take place in three dimensions, similar to those in a bulk phase. If this is the case, the contribution of the fluorescence with the excited dipole moment of SR 101 directed along the z-axis cannot be neglected, so that the time profile of the total fluorescence intensity must be proportional to / (0 + 2/i(t). Thus, fluorescence dynamic anisotropy is given by Equation (15), as is well known for that in a macroscopically isotropic system [10,13] ... 

From specimens cut in the x-z and y-z shearing planes, mean square chain dimensions similar to Rg j and Rg q can also be determined in the principal shear directions x, y and z. Typical isointensity patterns in the three shearing planes are shown in Fig. 28 for sample B. It can be seen that the shape of these curves in the x-z and y-z planes is nearly elliptical like in the x-y plane. However, due to the symmetry of the shear flow, the z direction is a principal direction for molecular orientation at all scales and the major and minor axes of the isointensity curves are indeed found to be parallel to the x,z and y,z directions. Therefore, the analysis of the scattering data in the x-z and y-z planes has been carried out by fixing the direction of the axes of the ellipse (to 0° and 90°) in the least-squares fit program. The major and minor axes of these ellipses then define the scattering vectors q, q and q in the principal shear directions and allow us to calculate the mean square chain dimensions Rg, Rg y and Rg 2-... 

The presence of a delta function in this expression indicates the absence of damping for paramagnons in a Luttinger liquid - in contrast to a Fermi liquid (see Eq. 9) - this also indicates that spin collective excitations are exact eigenstates of the system in one dimension (similar conclusions also hold for the charge part). Now close to q = 2k% one can use the following expression [118] ... 

Preliminary trials in complex aerospace type structures suggest that selectively placed blocks of Z-pins, of dimensions similar to those used in the laboratory specimens, have a significant capacity to resist or even stop the propagation of an inter-laminar crack in the structure. The success of even relatively simple FE models in predicting the behaviour of any specific block of Z-pins in the laboratory specimens would seem to indicate that the same approach will be successful in more sophisticated local-global models of structures containing such reinforcement. 

We can also use the 1-D Green s theorem, equation (15.51), to prove the reciprocity theorem in one dimension, similar to the general reciprocity theorem 27 ... 

The inhaler should have dimensions similar to those of the pMDIs, fitting the pocket or the handbag and permitting discrete use. 

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