Elasticity electron density

Elastic properties, ultrasonic evaluation Electronic properties radial electron density... 

Since these structures are formed by filling the open spaces in the diamond and wurtzite structures, they have high atomic densities. This implies high valence electron densities and therefore considerable stability which is manifested by high melting points and elastic stiffnesses. They behave more like metal-metalloid compounds than like pure metals. That is, like covalent compounds embedded in metals. 

The structures of the prototype borides, carbides, and nitrides yield high values for the valence electron densities of these compounds. This accounts for their high elastic stiffnesses, and hardnesses. As a first approximation, they may be considered to be metals with extra valence electrons (from the metalloids) that increase their average valence electron densities. The evidence for this is that their bulk modili fall on the same correlation line (B versus VED) as the simple metals. This correlation line is given in Gilman (2003). 

Entropy versus temperature data give values for 0S, so values for g can be obtained from Equation 10.3. These values depend on valence electron densities just as the elastic stiffnesses do. 

The mechanical behavior of TiB2 is characterized by its lattice parameters, valence electron density, elasticity tensor, plasmon tensor, and its heat of... 

The formal definition of the electronic chemical hardness is that it is the derivative of the electronic chemical potential (i.e., the internal energy) with respect to the number of valence electrons (Atkins, 1991). The electronic chemical potential itself is the change in total energy of a molecule with a change of the number of valence electrons. Since the elastic moduli depend on valence electron densities, it might be expected that they would also depend on chemical hardness densities (energy/volume). This is indeed the case. 

According to Eq. (1.16), the elastic coherent X-ray scattering amplitude is the Fourier transform of the electron density in the crystal. The crystal is a three-dimensional periodic function described by the convolution of the unit cell density and the periodic translation lattice. For an infinitely extended lattice,... 

The wave function P contains all information of the joint probability distribution of the electrons. For example, the two-electron density is obtained from the wave function by integration over the spin and space coordinates of all but two electrons. It describes the joint probability of finding electron 1 at r, and electron 2 at r2. The two-electron density cannot be obtained from elastic Bragg scattering. 

An impurity atom in a solid induces a variation in the potential acting on the host conduction electrons, which they screen by oscillations in their density. Friedel introduced such oscillations with wave vector 2kp to calculate the conductivity of dilute metallic alloys [10]. In addition to the pronounced effect on the relaxation time of conduction electrons, Friedel oscillations may also be a source of mutual interactions between impurity atoms through the fact that the binding energy of one such atom in the solid depends on the electron density into which it is embedded, and this quantity oscillates around another impurity atom. Lau and Kohn predicted such interactions to depend on distance as cos(2A pr)/r5 [11]. We note that for isotropic Fermi surfaces there is a single kp-value, whereas in the general case one has to insert the Fermi vector pointing into the direction of the interaction [12,13]. The electronic interactions are oscillatory, and their 1 /r5-decay is steeper than the monotonic 1 /r3-decay of elastic interactions [14]. Therefore elastic interactions between bulk impurities dominate the electronic ones from relatively short distances on. 

Figure 4 The modified stalk mechanism of membrane fusion and inverted phase formation, (a) planar lamellar (La) phase bilayers (b) the stalk intermediate the stalk is cylindrically-symmetrical about the dashed vertical axis (c) the TMC (trans monolayer contact) or hemifusion structure the TMC can rupture to form a fusion pore, referred to as interlamellar attachment, ILA (d) (e) If ILAs accumulate in large numbers, they can rearrange to form Qn phases, (f) For systems close to the La/H phase boundary, TMCs can also aggregate to form H precursors and assemble Into H domains. The balance between Qn and H phase formation Is dictated by the value of the Gaussian curvature elastic modulus of the bIlayer (reproduced from (25) with permission of the Biophysical Society) The stalk in (b) is structural unit of the rhombohedral phase (b ) electron density distribution for the stalk fragment of the rhombohedral phase, along with a cartoon of a stalk with two lipid monolayers merged to form a hourglass structure (reproduced from (26) with permission of the Biophysical Society).

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