The Deformed Lattice Potential

In the theoretical developments that follow, two of the standard assumptions of conventional plastic flow are retained. It will be assumed that plastic flow occurs by the creation and motion of dislocations and that to first order the length of the crystal slip planes does not change during plastic deformation. The development of a deformed lattice potential will be undertaken first and then incorporated into a quantum mechanical approach to plastic deformation required to describe plastic deformation due to shock or impact. 

The deformed lattice potential, U, can be written as an expansion about the undeformed potential, Uo in terms of the number of dislocations, N, on an active slip plane [12]. 

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Section (2) develops a theoretical account of plastic deformation and energy dissipation at the atomic or molecular level. The AFM observations show that plastic deformation of shocked or impacted crystals can significantly deform both the crystal lattice and its molecular components. These molecular and sub-molecular scale processes require a quantum mechanical description and necessarily involve the lattice and molecular potentials of the deforming crystals. A deformed lattice potential is developed which when combined with a quantum mechanical account of plastic flow in crystalline solids will be shown to give reasonably complete and accurate descriptions of the plastic flow and initiation properties of damaged and deformed explosive crystals. The deformed lattice potential allows, for the first time, the damaged state of the crystal lattice to be taken into account when determining crystal response to shock or impact. 

The remaining sections employ the deformed lattice and quantum picture of plastic flow to account for shear band formation as a means of achieving fte energy localization and hot spot temperatures necessary for initiation of crystalline explosives by shock or impact. Also briefly examined will be the role of the deformed lattice potential in causing particle size effects and its effect on the plastic deformation and energy dissipation rates. Finally, the dependence of the energy dissipation rate on shear stress will be shown to imply that reaction initiation will be dependent on the shape of the shock wave or impact stimulus. These predictions will be compared with experiment. 

A better alternative is to use the difference structure factor AF in the summations. The electrostatic properties of the procrystal are rapidly convergent and can therefore be easily evaluated in direct space. Stewart (1991) describes a series of model calculations on the diatomic molecules N2, CO, and SiO, placed in cubic crystal lattices and assigned realistic mean-square amplitudes of vibration. He reports that for an error tolerance level of 1%, (sin 0/2)max = 1-1.1 A-1 is adequate for the deformation electrostatic potential, 1.5 A-1 for the electric field, and 2.0 A 1 for the deformation density and the deformation electric field gradient (which both have Fourier coefficients proportional to H°). 

Much of this data has been or will be published elsewhere [5-10]. The data shows that the lattice and molecules of plastically deformed crystals experience significant and semi-permanent deformation. From this, insights are obtained that permit the development of an approximate deformed lattice potential for shocked or impacted crystals. Shear bands have been observed in shocked or impacted crystals. Some of shear bands show that molten material had been extruded from deep within the bands. These are possibly the source of the hot spots thought to be responsible for initiation during shock or impact. On the basis of these and other experimental observations it is concluded that energy dissipation and localization during plastic deformation is likely to be responsible for initiation of chemical reaction. 

In summary, the RDX molecular and lattice structures were significantly distorted by shock induced deformation, especially on the surface of the shear bands. The amplitude of the lattice potential as measured by the AFM was most reduced in the highly deformed shear band regions. The presence of molten RDX extruded from deep within the interior of the shear bands indicates that the deformed lattice extended deep into the crystal. 

The recent AFM experimental data concerning plastic flow place severe restrictions on possible theoretical accounts of plastic deformation in crystalline solids due to shock or impact. The high spatial resolution of the AFM, = 2 x lO " m, reveals substantial plastic deformation in shocked or impacted crystal lattices and molecules. Understanding how this occurs and its effect on plastic flow requires a quantum mechanical description. The semi-permanent lattice deformation has necessitated the development of a deformed lattice potential which, when combined with a quantum mechanical theory of plastic deformation, makes it possible to describe many of the features found in the AFM records. Both theory and the AFM observations indicate that shock and impact are similar shear driven processes that occur at different shear stress levels and time durations. The role of pressure is to provide an applied shear stress sufficient to cause initiation. 

FIG. 9. Configuration of a two-dimensional lattice filament (Fig. 1) under shock compression at time step J=3220. Each periodic block contained 12x12 bcc unit cells, one unit cell thick. Each block contained a cluster of ten vacancies. Morse potential (equation (1)) with ai=JKn2/0.10, =1.7. Other conditions as described in the text. Points (rectangles and crosses) mark the positions of the atoms in the two lattice planes of the unit cell. Solid lines joining the atoms show the misalignments (dislocations) in the deformed lattice under shock loading. 

Figure 1 2 10. The reduced Lifshitz parameter"z" - (ET - EF)/(EA- ET), where (EA- Er) is the full energy band dispersion in the c-axis direction, as a function of the number of holes in the G subband in A1 doped MgB2. The quantum uncertainty in the z value is indicated by the error bars that are given by D (

The liquid metal mercury-solution interface presents the advantage that it approaches closest to an ideal polarizable interface and, therefore, it adopts the potential difference applied between it and a non-polarizable interface. For this reason, the mercury-solution interface has been extensively selected to carry out measurements of the surface tension dependence on the applied potential. In the case of other metal-solution interfaces, the thermodynamic study is much more complex since the changes in the interfacial area are determined by the increase of the number of surface atoms (plastic deformation) or by the increase of the interatomic lattice spacing (elastic deformation) [2, 4]. 

Lattice parameters (LPs) of a semiconductor depend on the following factors [1] (i) chemical composition (including deviation from stoichiometry), (ii) presence of free-charge acting via the deformation potential of the energy-band extremum occupied by this charge, (iii) presence of foreign atoms and defects, (iv) external stresses (for example, exerted on a heteroepitaxial layer by its substrate), and (v) temperature. These factors are not independent [1], For nitrides, studies of such factors are in a state of infancy. 

More difficult to calculate are the properties which depend on the response of the solid to an outside influence (stress, electric field, magnetic field, radiation). Elastic constants are obtained by considering the response of the crystal to deformation. Interatomic potential methods often provide good values for these and indeed experimental elastic constants are often used in fitting the potential parameters. Force constants for lattice vibrations (phonons) can be calculated from the energy as a function of atomic coordinates. In the frozen phonon approach, the energy is obtained explicitly as a function of the atom coordinates. Alternatively the deriva-tive, 5 - can be calculated at the equilibrium geometry. 

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