Energy Density Distribution

if it is assumed that the local virial theorem is valid for the model electron densities fitted to the experimental structure factors, the kinetic, g(r), and potential, v(r), energy densities may be mapped, as well as the energy characteristics of the (3,-1) bond critical points evaluated [38]. 

In the following, this approach has been used to study the energy features of 3-NTO [29]. The kinetic, g(r), and potential, v(r), energy density maps have been calculated from the experimental electron densities with the WinXPRO program package [39] using the approach described above, as well as their difference with respect to atomic procrystals with no chemical bonds. Critical point characteristics have also been similarly analyzed. 

The potential energy density, v(r), maps reveal explicitly that the N02-group is the most potential energy-rich area in the molecule (Fig. 12b). This is confirmed by the analysis of the local energies at the bond critical points (see below). The kinetic energy densities, g(r), are also higher for these groups (Fig. 12a). 

In order to reveal the subtle changes in the energy distributions caused by the crystal/molecule formation, we have calculated the deformation kinetic and potential energy densities [34,40]  

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Chowdhury, D.Q., Barber, P.W., and Hill, S.C., 1992, Energy-density distribution inside large nonabsorbing spheres by using Mie theory and geometrical-optics, Appl. Opt. 31(18) 3518-3523. 

The interacting centres are connected by a bond path with a bond critical point p, at which the energy density distribution H(r) is stabilizing [//(p) < 0]. 

For a finite initial width lower limit of the integral is replaced by t0 = a2 j2D and we find an approximate... 

Given the complete description of the structure in terms of pseudo atoms as described above, it is possible to derive additional properties that provide insight into the chemical and physical properties of energetic materials. Two of these are described below, the molecular electrostatic potential and the energy density distribution. 

Thus, the analysis of energy density distributions should prove to be a useful tool in the study of the fundamental properties of energetic materials. 

Although X-ray crystallography has mainly been used for structure determination in the past, the diffraction data, especially if measured carefully and to high orders, contain information about the total electron density distribution in the crystal. This may be analyzed to provide essential information about the chemical properties of molecules, in particular the characterization of covalent and hydrogen bonds, both from the point of view of the valence electron density, the Laplacian of the density and derived energy density distribution. In addition, calculation of the molecular electrostatic potential indicates direction of chemical attack as well as how molecules can interact with their environment. 

Each lens produces a different energy density distribution in space that results in different plasma characteristics. A laser beam can be focused to a minimum size d given by the diffraction limit relation [139], d = 2.44/A/D, where A is the laser wavelength. 

Here, is again the surface work, S is the surface energy as previously defined and is the loss function dependent on crack speed, temperature and the strain, eo, applied to the specimen. The theory gives explicitly in terms of the energy density distribution in the specimen and the plastic or visco-elastic hysteresis of the material. 

In principle, then, the surface work which determines the fracture stress of the body can be calculated from the physical properties of the material. In practice this is not easy, since the energy density distribution can only be calculated exactly for linear elastic solids, for which 1 and Eq. (5) reverts to the Griffith theory. 

However, Eq. (5) has been found correct for a series of highly extensible materials in which the energy density distribution was measured experimentally ... 

Where G(r) is a local kinetic energy density distribution and V(r) is the local potential energy density distribution [31], If H(r) is negative at r, then the local potential energy density distribution V(r) will dominate, and accumulation of electronic charge in the internuclear region will be stabilizing. In this case, one can speak of a covalent bond. 

At what wavelength does the maximum in the energy density distribution function for a black-body occur if... 

If the energy density distribution function has a maximum at 600 nm, what is the temperature ... 

The energy density distribution of blackbody radiation as a function of wavelength. 

These two linear relationships between and and G appear to be quite simple (BSSE-free) and useful approximations, which enable the evaluation of the H-bonding energy in solid state using theoretical [66] or experimental (derived from X-ray or synchrotron diffraction experiment [67, 68]) energy densities. In the latter case the electronic kinetic energy density distribution, G, is derived from the following accurate approximation for closed-shell interactions (like H-bond is) in terms of experimental electron density, p, and its Laplacian V p at the BCP (in a.u.) [69] ... 

Fig. 12. Power and energy density distribution during excursion.

The power and energy density distributions in the cylindrical core (r, 9, z coordinate system), assumed to be separable in space and time, are also assumed to be quadratic in form ... 

Processes during a cell cycle are evidenced to be closely controlled cooperative events, including synchronisation within the ensemble. This caused us to describe relaxation within each cell by a Debye process, the relaxation time of which should increase with the size of the cell involved ( finite-size effect ). In that way ensemble structure and relaxation processes of cell ensembles are strictly interrelated. The universal energy density distribution and the universal relaxation mode distribution turn out to be copies of each other. Consequently, the spectrum depends only on the universal properties of the ensemble structure, i.e. on the value of p. Since all the cell populations studied here belong to the / = 3 class, the linear relaxation behaviour should show the same features. 

Equilibrium concentration distribution function g Internal energy density distribution function... 

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