Real crystals

Equation (Bl.8.6) assumes that all unit cells really are identical and that the atoms are fixed hi their equilibrium positions. In real crystals at finite temperatures, however, atoms oscillate about their mean positions and also may be displaced from their average positions because of, for example, chemical inlioniogeneity. The effect of this is, to a first approximation, to modify the atomic scattering factor by a convolution of p(r) with a trivariate Gaussian density function, resulting in the multiplication ofy ([Pg.1366]

A first step towards a systematic improvement over DFT in a local region is the method of Aberenkov et al [189]. who calculated a correlated wavefiinction embedded in a DFT host. However, this is achieved using an analytic embedding potential fiinction fitted to DFT results on an indented crystal. One must be cautious using a bare indented crystal to represent the surroundings, since the density at the surface of the indented crystal will have inappropriate Friedel oscillations inside and decay behaviour at the indented surface not present in the real crystal. 

In ordinary diamond (2inc-blende stmcture) the wrinkled sheets He in the (111) or octahedral face planes of the crystal and are stacked in an ABCABC sequence. In real crystals, this ABCABC sequence continues indefinitely, but deviations do occur. For example, two crystals may grow face-to-face as mirror images the mirror is called a twinning plane and the sequence of sheets crossing the mirror mns ABCABCCBACBA. Many unusual sequences may exist in real crystals, but they are not easy to study. 

It is emphasized that the delta L law does not apply when similar crystals are given preferential treatment based on size. It fails also when surface defects or dislocations significantly alter the growth rate of a crystal face. Nevertheless, it is a reasonably accurate generahzation for a surprising number of industrial cases. When it is, it is important because it simphfies the mathematical treatment in modeling real crystallizers and is useful in predicting crystal-size distribution in many types of industrial crystallization equipment. 

In the last chapter we examined data for the yield strengths exhibited by materials. But what would we expect From our understanding of the structure of solids and the stiffness of the bonds between the atoms, can we estimate what the yield strength should be A simple calculation (given in the next section) overestimates it grossly. This is because real crystals contain defects, dislocations, which move easily. When they move, the crystal deforms the stress needed to move them is the yield strength. Dislocations are the carriers of deformation, much as electrons are the carriers of charge. 

The exchange reactions (6.20) and (6.21) have been among the basic objects of chemical-reaction theory for half a century. Clearly further investigation is needed, incorporating real crystal dynamics. It is worth noting that the adiabatic model, upon which the cited results are based, can prove to be insufficient because of the low frequency of the promoting vibrations. 

The key here was the theory. The pioneers familiarity with both the kinematic and the dynamic theory of diffraction and with the real structure of real crystals (the subject-matter of Lai s review cited in Section 4.2.4) enabled them to work out, by degrees, how to get good contrast for dislocations of various kinds and, later, other defects such as stacking-faults. Several other physicists who have since become well known, such as A. Kelly and J. Menter, were also involved Hirsch goes to considerable pains in his 1986 paper to attribute credit to all those who played a major part. 

The orientational relationships between the martensite and austenite lattice which we observe are partially in accordance with experimental results In experiments a Nishiyama-Wasserman relationship is found for those systems which we have simulated. We think that the additional rotation of the (lll)f< c planes in the simulations is an effect of boundary conditions. Experimentally bcc and fee structure coexist and the plane of contact, the habit plane, is undistorted. In our simulations we have no coexistence of these structures. But the periodic boundary conditions play a similar role like the habit plane in the real crystals. Under these considerations the fact that we find the same invariant direction as it is observed experimentally shows, that our calculations simulate the same transition process as it takes place in experiments. The same is true for the inhomogeneous shear system which we see in our simulations. 

M.A. Krivogiaz, Theory of X-ray and Thermal Neutron Scattering by Real Crystals , (Plenum, New York,... 

The reason for this can be seen as follows. In a perfect crystal with the ions held fixed, a positive hole would move about like a free particle with a mass m depending on the nature of the crystal. In an applied electric field, the hole would be uniformly accelerated, and a mobility could not be defined. The existence of a mobility in a real crystal derives from the fact that the uniform acceleration is continually disturbed by deviations from a perfect lattice structure. Among such deviations, the thermal motions of the ions, and in particular, the longitudinal polarisation vibrations, are most important in obstructing the uniform acceleration of the hole. Since the amplitude of the lattice vibrations increases with temperature, we see how the mobility of a... 

Ionic transport in solids originates from the atomic disorder in real crystals compared with ideal crystal lattices. The most important defects of this kind are ... 

Obviously this model is very simplified, compared with the real crystal which contains many defects, dislocations and entanglements. In particular, it neglects many aspects of the true three-dimensional nature of the lamella which one may have thought to be important the influence of the stacking of folds, which is... 

Types of imperfection which represent deviations from the ideal lattice in real crystals... 

ANALYSIS OF A REAL CRYSTAL USING A THERMODYNAMIC METHOD... 

All real crystals have deviation from the ideal crystallographic structure due to atomic displacements from their ideal positions in the unit cell. These displacements may... 

Crystals are distinguished by the regular, periodic order of their components. In the following we will focus much attention on this order. However, this should not lead to the impression of a perfect order. Real crystals contain numerous faults, their number increasing with temperature. Atoms can be missing or misplaced, and dislocations and other imperfections can occur. These faults can have an enormous influence on the properties of a material. 

Strictly speaking, a symmetry-translation is only possible for an infinitely extended object. An ideal crystal is infinitely large and has translational symmetry in three dimensions. To characterize its translational symmetry, three non-coplanar translation vectors a, b and c are required. A real crystal can be regarded as a finite section of an ideal crystal this is an excellent way to describe the actual conditions. 

To a first order approximation, the scattering potential of a crystal may be represented as a sum of contributions from isolated atoms, having charge distributions of spherical symmetry around their nuclei. In a real crystal the charge distribution deviates from the spherical symmetry around the nucleus and the difference reflects the charge redistribution or bonding in the crystal. The problem of experimental measurement of crystal bonding is therefore a problem of structure factor refinement, i.e. accurate determination of the difference between the true crystal structure factors... 

B GO Solid Amorphous solid Real crystal Perfect crystal... 

Talc is a hydrated magnesium silicate, Mg6(Si802o)(OH)4. It is a layerd compound like mica. One layer of its crystal structure is shown schematically in Figure 11.3. Such layers are stacked up like playing cards in real crystals. Notice that the top and bottom of the layer consist of slicate tetrahedra with oxygen... 

L. Zuppiroli, Irradiation Effects Perfect Crystals and Real Crystals... 

The heat capacity models described so far were all based on a harmonic oscillator approximation. This implies that the volume of the simple crystals considered does not vary with temperature and Cy m is derived as a function of temperature for a crystal having a fixed volume. Anharmonic lattice vibrations give rise to a finite isobaric thermal expansivity. These vibrations contribute both directly and indirectly to the total heat capacity directly since the anharmonic vibrations themselves contribute, and indirectly since the volume of a real crystal increases with increasing temperature, changing all frequencies. The constant volume heat capacity derived from experimental heat capacity data is different from that for a fixed volume. The difference in heat capacity at constant volume for a crystal that is allowed to relax at each temperature and the heat capacity at constant volume for a crystal where the volume is fixed to correspond to that at the Debye temperature represents a considerable part of Cp m - Cv m. This is shown for Mo and W [6] in Figure 8.15. 

From the above paragraph it is clear that very many different models may be subjected to calculation. The numerical results, naturally, vary from model to model (for a given real crystal), and at present it would be too risky to claim correctness for any of them. Consequently, only a few (mainly recent) examples of the approaches to the computation of 7 are reviewed in this section, chiefly to indicate the unsettled state of affairs at present. Older theories and their numerical results may be consulted, for instance, in Ref.6. ... 

The excess energy of a real crystal fragment depends on how this was obtained usually, the fracture surface is imagined to be perfectly smooth, but real surfaces are rough, have steps, and so on. The surfaces assumed in theoretical calculations are not in contact with any other substance. If a gas, however, is admitted, then the unsaturated valencies will be saturated with adsorbed gas molecules, and the asymmetry of the field will be reduced. Thus the structure calculated for the external layer may be in equilibrium only in excellent vacuum, and the duration of this equilibrium would depend on how rapidly gases and vapors leak into the evacuated vessel. This remark shows, by the way, how illogical are the attempts to correlate the experimental estimates of ys or 7 with those calculated from the theories of Chapter II. 

The essence of the problem, as pointed out in Section II, is that conventional X-ray diffraction does not provide information as to whether the molecule W-Y points in the +h or in the —b direction. In other words, it does not allow one to distinguish between the real crystal (Scheme 9a) and the hypothetical enantiomeric structure in which the orientation of W-Y with respect to the b axis is opposite (Scheme 9b). -... 

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