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Infinite crystal

Figure B3.2.12. Schematic illustration of geometries used in the simulation of the chemisorption of a diatomic molecule on a surface (the third dimension is suppressed). The molecule is shown on a surface simulated by (A) a semi-infinite crystal, (B) a slab and an embedding region, (C) a slab with two-dimensional periodicity, (D) a slab in a siipercell geometry and (E) a cluster. Figure B3.2.12. Schematic illustration of geometries used in the simulation of the chemisorption of a diatomic molecule on a surface (the third dimension is suppressed). The molecule is shown on a surface simulated by (A) a semi-infinite crystal, (B) a slab and an embedding region, (C) a slab with two-dimensional periodicity, (D) a slab in a siipercell geometry and (E) a cluster.
Ewald summation was invented in 1921 [7] to permit the efl5.cient computation of lattice sums arising in solid state physics. PBCs applied to the unit cell of a crystal yield an infinite crystal of the appropriate. symmetry performing... [Pg.462]

The electronic structure of an infinite crystal is defined by a band structure plot, which gives the energies of electron orbitals for each point in /c-space, called the Brillouin zone. This corresponds to the result of an angle-resolved photo electron spectroscopy experiment. [Pg.266]

As examples for our investigation we have chosen Fe/Cu/Fe bcc (001) and Co/Cu/Co fee (001) trilayers. A trilayer consists of two semi infinite crystals of Fe separated by a paramagnetic Cu spacer. The entire trilayer has the same crystal structure which means that all effects from lattice relaxations are excluded. The experimental lattice constant of bcc Fe was chosen for the Fe/Cu/Fe bcc trilayers, whereas for the Co/Cu/Co fee trilayers we used the lattice constant of fee Cu. [Pg.240]

At a temperature such that the solid phase is stable AF is necessarily positive and the stable crystal has infinite thickness. However, any crystal which has AG < 0 will be stable compared with the liquid, so that a crystal of finite thickness may be metastable if there is a free energy barrier to the formation of an infinite crystal. AG < 0 if ... [Pg.229]

A different melting point, and hence supercooling, is predicted for the strained sector. This is the basis for a different interpretation of the (200) growth rates a regime //// transition occurs on (110) but not on (200). This is despite the fact that the raw data [113] show a similar change in slope when plotted with respect to the equilibrium dissolution temperature (Fig. 3.15). It is questionable whether it is correct to extrapolate the melting point depression equation for finite crystals which is due to lattice strain caused by folds, to infinite crystal size while keeping the strain factor constant. [Pg.279]

The valence band structure of very small metal crystallites is expected to differ from that of an infinite crystal for a number of reasons (a) with a ratio of surface to bulk atoms approaching unity (ca. 2 nm diameter), the potential seen by the nearly free valence electrons will be very different from the periodic potential of an infinite crystal (b) surface states, if they exist, would be expected to dominate the electronic density of states (DOS) (c) the electronic DOS of very small metal crystallites on a support surface will be affected by the metal-support interactions. It is essential to determine at what crystallite size (or number of atoms per crystallite) the electronic density of sates begins to depart from that of the infinite crystal, as the material state of the catalyst particle can affect changes in the surface thermodynamics which may control the catalysis and electro-catalysis of heterogeneous reactions as well as the physical properties of the catalyst particle [26]. [Pg.78]

In order to deduce Scherrer s equation first an infinite crystal is considered that is, second, restricted (i.e multiplied) by a shape function (cf. p. 17). Thus from the Fourier convolution theorem (Sect. 2.7.8) it follows that in reciprocal space each reflection is convolved by the Fourier transform of the square of the shape function - and Scherrer s equation is readily established. [Pg.42]

The number of netplanes in the crystal (that are related to an observed peak) is not infinite (crystal size). [Pg.119]

The physics problem that needs to be addressed is that of an isolated impurity in an infinite crystal. This problem is clearly too complex to treat exactly specific geometrical arrangements have to be chosen that closely represent the physical situation while being computationally tractable. [Pg.603]

Having briefly noted the historical highlights of the WSL effect, we now examine the basic mathematical argument for its existence, namely, the idea that the energy spectrum of an infinite crystal is discretized by an applied held. For the present discussions, we assume that the applied field is linear, with its strength given by its gradient 7. [Pg.118]

A special group of particles that are often produced are the icosahedral (I5) and decahedral (D5) structures shown in Fig. 9. These particles have a fivefold symetry axis which is forbidden for infinite crystals. Yang (1 0) has described these particles using a non-Fcc model. The particles are composed by five (D5) and twenty (I5) tetrahedral units in twin relationship. However the units have a non-Fcc structure. The decahedral is composed by body-centered orthorhombic units and the icosahedral by rhombohedral... [Pg.335]

This section deals with the dynamics of collective surface vibrational excitations, i.e. with surface phonons. A surface phonon is defined as a localized vibrational excitation of a semi-infinite crystal, with an amplitude which has wavelike characteristics parallel to the surface and decays exponentially into the bulk, perpendicular to the surface. This behavior is directly linked to the broken translational invariance at a surface, the translational symmetry being confined here to the directions parallel to the surface. [Pg.221]

The dispersion curves of surface phonons of short wavelength are calculated by lattice dynamical methods. First, the equations of motion of the lattice atoms are set up in terms of the potential energy of the lattice. We assume that thejxitential energy (p can be expressed as a function of the atomic positions 5( I y in the semi-infinite crystal. The location of the nth atom can be... [Pg.224]

Thus, to the best of our knowledge, there is a lack of embedding cluster studies on the yttrium ceramics where with a sufficient precision both aspects of the ECM were taken in account. In the study [44], we attempted to fill such a gap and carried out the electronic structure calculations of the YBa2Cu307 ceramics at the Moller-Plesset level with a self-consistent account of the infinite crystal surrounding to the quantum cluster. The Gaussian basis set employed (6-31IG) was larger than those used in previous cluster calculations [16,20,22,29]. [Pg.145]

Table 1 Madelung potential on the sites of the YBaf u Oj unite cell obtained with Gupta-Gupta [14] charges, in a.u. (infinite crystal)... Table 1 Madelung potential on the sites of the YBaf u Oj unite cell obtained with Gupta-Gupta [14] charges, in a.u. (infinite crystal)...

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See also in sourсe #XX -- [ Pg.13 ]




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