Peierls force

Provided that the Peierls force is not too large (see Section 11.3.1). 

However, dislocations will still move by thermally activated processes below the Peierls force. 8For more about the dispersion relation, see a reference on solid-state physics, such as Kittel [6]. 

Peierls Force Continuous vs. Discontinuous Motion. In some crystals (e.g., covalent crystals) the Peierls force may be so large that the driving force due to the applied stress will not be able to drive the dislocation forward. In such a case the dislocation will be rendered immobile. However, at elevated temperatures, the dislocation may be able to surmount the Peierls energy barrier by means of stress-aided thermal activation, as in Fig. 11.5. 

After the dislocation has moved by half a Burgers vector, the Peierls force pushes it forwards and moves it to the position of the next energy minimum. The stored energy is usually dissipated as heat (i.e., as random crystal vibration) in the crystal. The Peierls force thus acts as a kind of frictional force and reduces the effective stress that can be used to drive the dislocation to overcome other obstacles. 

Dislocations can be retarded by different kinds of obstacles. We already know one of these, the Peierls force. Other types, such as precipitates of a second phase, grain boundaries, or impurity atoms, will be discussed below in section 6.4 when we look at strengthening mechanisms. Here we want to understand in what ways a dislocation can overcome an obstacle. As we will see,... 

If we consider the Peierls force from section 6.2.9 as obstacle, it can also be overcome by thermal activation. This is especially relevant if the Peierls force is large i. e., when slip is along planes that are not close-packed, for example in body-centred cubic lattices. For this reason, the yield strength of body-centred cubic lattices is strongly dependent on the temperature, different from face-centred cubic metals (figure 6.29). The Peierls stress can reach values of up to several hundred megapascal. 

Let ns recall that for metallic monocrystals, the creep speed in a permanent regimen is expressed as a power of the applied load function, with an exponent n = 3 - 5, characteristic of a deformation by dislocation mobihty. We will show some of the characteristics of the deformation of iono-covalent sohds. The latter are generally less plastic and the dislocations are aligned in a stable manner in potential valleys because of the existence of high Peierls forces. The propagation and multiphcation of dislocations are therefore not very easy and the density of dislocations is generally low. 

Figure 11.6 illustrates the energy that must be supplied by thermal activation. The curve of ab vs. A shows the force that must be applied to the dislocation (per unit length) if it were forced to surmount the Peierls barrier in the manner just described in the absence of thermal activation. The quantity A is the area swept out by the double kink as it surmounts the barrier and is a measure of the forward motion of the double kink. A = 0 corresponds to the dislocation lying along an energy trough (minimum) as in Fig. 11.5a. A2 is the area swept out when maximum force must be supplied to drive the double kink. A4 is the area swept out when the saddle point has been reached and the barrier has been effectively surmounted. The area under the curve is then the total work that must be done by the applied stress to surmount the barrier in the absence of thermal activation. When the applied stress is a a (and too small to force the barrier), the swept-out area is A, and the energy that must be supplied by thermal activation is then the shaded area shown in Fig. 11.6. The activation energy is then...

A solution to this problem requires a knowledge of the atom positions in the dislocation core. Using the atom positions in the Peierls dislocation, Pacheco and Mura (1969) estimated the force on a dislocation due to just a single interface. They obtained the increase in shear flow stress, Ate, due to an elastic modulus change across a sharp interface as... 

As one approaches the Peierls temperature from above, the restoring force for a distortion which has the symmetry that would create a gap at the Fermi surface gets smaller and smaller, until at the Peierls temperature it goes to zero and the lattice distorts spontaneously to the new structure. Thus phonons with the symmetry of this distortion become soft as one approaches this temperature and the amplitude of thermal excitation of the mode grows enormous. These effects show up in x-ray studies of such compounds and allow the identification of the instability. 

One can describe this as the presence of a charge-density wave in the electronic system. In this case the charge-density wave follows from displacement of the atoms. One can ask the question whether in a rigid lattice the electron system itself can distort spontaneously to lower its symmetry, producing an effect that would then attempt to force the nuclei to follow suit. The driving force in this case would not be the movement of the atoms as in the Peierls instability but rather the inherent instability of the electron system itself. The answer to this question is yes. The study of these types of instabilities and associated instabilities in the spin system of the electrons has become an important part of the physics of limited dimensionality. 

Note In the spin-Peierls transitions on the Ni, Pt, the driving forces seem to be electron-phonon interactions and not spin-phonon, since the field dependence of the transition temperatures are of Peierls type and not spin-Peierls. 

Our treatment in this section will cover three primary thrusts in modeling dislocation core phenomena. Our first calculations will consider the simplest elastic models of dislocation dissociation. This will be followed by our first foray into mixed atomistic/continuum models in the form of the Peierls-Nabarro cohesive zone model. This hybrid model divides the dislocation into two parts, one of which is treated using linear elasticity and the other of which is considered in light of a continuum model of the atomic-level forces acting across the slip plane of the dislocation. Our analysis will finish with an assessment of the gains made in direct atomistic simulation of dislocation cores. 

Fig. 8.26. Schematic of key elements in Peierls-Nabarro dislocation model (courtesy of R. MiUer). The key idea is the use of a nonlinear constitutive model on the relevant slip plane which is intended to mimic the atomic-level forces resulting from slip.

Our preliminary analysis centers around the geometry shown in fig. 11.18 in which an atomically sharp crack is subjected to mode II loading and the slip distribution is assumed to occur along the prolongation of the crack plane. Borrowing from the Peierls-Nabarro analysis described in section 8.6.2, it is assumed that the atomic-level forces across the slip plane may be characterized in terms of an interplanar potential (5). This description asserts that the tractions which arise on account of the sliding discontinuity are given by r = —d

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From a mechanistic perspective, what transpires in the context of all of these strengthening mechanisms when viewed from the microstructural level is the creation of obstacles to dislocation motion. These obstacles provide an additional resisting force above and beyond the intrinsic lattice friction (i.e. Peierls stress) and are revealed macroscopically through a larger flow stress than would be observed in the absence of such mechanisms. Our aim in this section is to examine how such disorder offers obstacles to the motion of dislocations, to review the phenomenology of particular mechanisms, and then to uncover the ways in which they can be understood on the basis of dislocation theory. 

Th.e refinements of the theory, which have been worked out in particular by Houston, Bloch, Peierls, Nordheim, Fowler and Brillouin, have two main objects. In the first place, the picture of perfectly free electrons at a constant potential is certainly far too rough. There will be binding forces between the residual ions and the conduction electrons we must elaborate the theory sufficiently to make it possible to deduce the number of electrons taking part in the process of conduction, and the change in this number with temperature, from the properties of the atoms of the substance. In principle this involves a very complicated problem in quantum mechanics, since an electron is not in this case bound to a definite atom, but to the totality of the atomic residues, which form a regular crystal lattice. The potential of these residues is a space-periodic function (fig. 10), and the problem comes to this— to solve Schrodinger s wave equation for a periodic poten-tial field of this kind. That can be done by various approximate methods. One thing is clear if an electron... 

Drag force, Bv, where B is the drag coefficient and v is the dislocation velocity. Peierls stress Fpeierb. 

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