Crystal structure, dislocation

An example of research in the micromechanics of shock compression of solids is the study of rate-dependent plasticity and its relationship to crystal structure, crystal orientation, and the fundamental unit of plasticity, the dislocation. The majority of data on high-rate plastic flow in shock-compressed solids is in the form of ... 

Calculations of this type are carried out for fee, bcc, rock salt, and hep crystal structures and applied to precursor decay in single-crystal copper, tungsten, NaCl, and LiF [17]. The calculations show that the initial mobile dislocation densities necessary to obtain the measured rapid precursor decay in all cases are two or three orders of magnitude greater than initially present in the crystals. Herrmann et al. [18] show how dislocation multiplication combined with nonlinear elastic response can give some explanation for this effect. 

When crystals yield, dislocations move through them. Most crystals have several slip planes the f.c.c. structure, which slips on 111) planes (Chapter 5), has four, for example. Dislocations on these intersecting planes interact, and obstruct each other, and accumulate in the material. 

The surface of a solid electrode is not homogeneous even an apparently smooth surface as observed in an optical microscope contains corners and edges of the crystal structure of the metal and dislocations, i.e. sites where the regular crystal structure is disordered (see Section 5.5.5). 

When the two vectors are parallel, the crystal planes perpendicular to the line form a helix, and the dislocation is said to be of the screw type. In a nearly isotropic crystal structure, the dislocation is no longer associated with a distinct glide plane. It has nearly cylindrical symmetry, so in the case of the figure it can move either vertically or horizontally with equal ease. 

Figure 4.1 Schematic dislocation line a simple cubic crystal structure. The line enters the crystal at the center of the left-front face. It emerges at the center of the right-front face. The shortest translation vector of the structure is the Burgers Vector, b. The line bounds the glided area of the glide plane (100) from the unglided area.

Figure 4.2 Quasi-hexagonal dislocation loop lying on the (111) glide plane of the diamond crystal structure. The <110> Burgers vector is indicated. A segment, displaced by one atomic plane, with a pair of kinks, is shown a the right-hand screw orientation of the loop. As the kinks move apart along the screw dislocation, more of it moves to the right.

If a dislocation line lies parallel to the x-axis of an xy-plane, and is kinked, the kink lies parallel to the y-axis. Therefore, if the line is of edge character, the kink is of screw character. If the line is of screw character, the kink is of edge character. In either case, the displacement gradient is indefinite at the center of the kink. This means that whatever symmetry exists in the undislocated crystal, structure is destroyed at a kink. 

Since covalent bonding is localized, and forms open crystal structures (diamond, zincblende, wurtzite, and the like) dislocation mobility is very different than in pure metals. In these crystals, discrete electron-pair bonds must be disrupted in order for dislocations to move. 

When the stress (compressive) rises to a value approaching G/10 near the Debye temperature, motion of gliding dislocations tends to be replaced by the formation of phase transformation dislocations. The crystal structure then transforms to a new one of greater density. This occurs when the compressive stress (the hardness number) equals the energy band gap density (gap/molecular volume). 

Figure 5.1 shows a schematic elevation through a kink on a screw dislocation in the diamond crystal structure. The black circles lie in the plane of the figure. The white ones lie in a plane in front of the figure, and the gray ones in a plane behind the figure. The straight lines represent electron pair bonds... 

The next two figures show that crystal structure type and ionicity also play a role in determining dislocation mobility, and therefore hardness. First, if data for the III-N compounds are plotted on Figure 5.2 they do not fall on the regression line. The reason is that they have hexagonal rather than cubic crystal structures. However, when plotted by themselves as in Figure 5.3 their hardnesses are proportional to their bond moduli. 

At roughly the same time, Peierls (1940), and Nabarro (1947) developed a two-dimensional model of a dislocation in a simple square crystal structure. This model indicated that a small, but finite, amount of energy is needed to... 

When normal sites in a crystal structure are replaced by impurity atoms, or vacancies, or interstitial atoms, the local electronic structure is disturbed and local electronic states are introduced. Now when a dislocation kink moves into such a site, its energy changes, not by a minute amount but by some significant amount. The resistance to further motion is best described as an increase in the local viscosity coefficient, remembering that plastic deformation is time dependent. A viscosity coefficient, q relates a rate d8/dt with a stress, x ... 

The crystal structure of NiAl is the CsCl, or (B2) structure. This is bcc cubic with Ni, or A1 in the center of the unit cell and Al, or Ni at the eight comers. The lattice parameter is 2.88 A, and this is also the Burgers displacement. The unit cell volume is 23.9 A3 and the heat of formation is AHf = -71.6kJ/mole. When a kink on a dislocation line moves forward one-half burgers displacement, = b/2 = 1.44 A, the compound must dissociate locally, so AHf might be the barrier to motion. To overcome this barrier, the applied stress must do an amount of work equal to the barrier energy. If x is the applied stress, the work it does is approximately xb3 so x = 8.2 GPa. Then, if the conventional ratio of hardness to yield stress is used (i.e., 2x3 = 6) the hardness should be about 50 GPa. But according to Weaver, Stevenson and Bradt (2003) it is 2.2 GPa. Therefore, it is concluded that the hardness of NiAl is not intrinsic. Rather it is determined by an extrinsic factor namely, deformation hardening. 

Dislocation lines do not move concertedly, that is, all at once. They move, by forming kinks along their lengths, and when the kinks move, the lines move. The open crystal structure of quartz (crystobalite) results in a relatively large amount of volume being associated with a kink on a dislocation line. This relatively large volume lowers the value of quartz s bond modulus, making its hardness consistent with those of other covalently bonded substances. 

In order for a kink on a dislocation line to move it must shear (destroy) AI2O3 subunits of the crystal structure. This requires approximately the heat of formation, AHf of A1203 which is 402kcal/mol = 17eV/molecule (Roth et al., 1940). The work done by the applied shear stress must supply this energy. This is about xb3 so the shear stress required is about 13.7 GPa, and the hardness, H, is about twice this, or 27.4 GPa, which is close to the observed hardness of 27 GPa. 

In relatively recent years, it has been found that that indentations made in covalent crystals at temperatures below their Debye temperatures often result from crystal structure changes, as well as from plastic deformation via dislocation activity. Thus, indentation hardness numbers may provide better critical parameters for structural stability than pressure cell studies because indentation involves a combination of shear and hydrostatic compression and a phase transformation involves both of these quantities. 

Although several types of lattices have been described for ionic crystals and metals, it should be remembered that no crystal is perfect. The irregularities or defects in crystal structures are of two general types. The first type consists of defects that occur at specific sites in the lattice, and they are known as point defects. The second type of defect is a more general type that affects larger regions of the crystal. These are the extended defects or dislocations. Point defects will be discussed first. 

Fig. 12. Derivative curves of EPR in a highly dislocated As-doped germanium crystal grown in a H2 atmosphere. The magnetic field is oriented along the [100] direction. T= 2 K, /= 25.16 GHz. Note the sign reversal of the new lines as compared to the As-donor hyperfine structure. Dislocation density 2 x 104 cm 2. (Courtesy Pakulis and Jeffries, reprinted with permission from the American Physical Society, Pakulis, E.J., Jeffries, C D. Phys. Rev. Lett. (1981). 47, 1859.)...

A unit, or perfect, dislocation is defined by a Burgers vector which regenerates the structure perfectly after passage along the slip plane. The dislocations defined above with respect to a simple cubic structure are perfect dislocations. Clearly, then, a unit dislocation is defined in terms of the crystal structure of the host crystal. Thus, there is no definition of a unit dislocation that applies across all structures, unlike the definitions of point defects, which generally can be given in terms of any structure. 

The aggregation of vacancies or interstitials into dislocation loops will depend critically upon the nature of the crystal structure. Thus, ionic crystals such as sodium chloride, NaCl, or moderately ionic crystals such as corundum, AI2O3, or rutile, TiC>2, will show different propensities to form dislocation loops, and the most favorable planes will depend upon chemical bonding considerations. 

---