Dislocations density

Germanium single crystals intended for electronic apphcations are usuaHy specified according to conductivity type, dopant, resistivity, orientation, and maximum dislocation density. They may be specified to be lineage-free unless the specified resistivity is below about 0.05 H-cm. Minority carrier lifetime and majority carrier mobHity are occasionaHy specified. 

An analogy to sHp dislocation is the movement of a caterpillar where a hump started at one end moves toward the other end until the entire caterpillar moves forward. Another analogy is the displacement of a mg by forming a hump at one end and moving it toward the other end. Strain hardening occurs because the dislocation density increases from about 10 dislocations/cm to as high as 10 /cm. This makes dislocation motion more difficult because dislocations interact with each other and become entangled. SHp tends to occur on more closely packed planes in close-packed directions. 

This phenomena has been attributed to the very high strain rates associated with shock loading and the subsonic restriction on dislocation velocity requiring the generation and storage of a larger dislocation density during the shock process than for quasi-static processes [1], [2], [12],... 

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. 

The shock-induced micromechanical response of <100>-loaded single crystal copper is investigated [18] for values of (WohL) from 0 to 10. The latter value results in W 10 Wg at y = 0.01. No distinction is made between total and mobile dislocation densities. These calculations show that rapid dislocation multiplication behind the elastic shock front results in a decrease in longitudinal stress, which is communicated to the shock front by nonlinear elastic effects [pc,/po > V, (7.20)]. While this is an important result, later recovery experiments by Vorthman and Duvall [19] show that shock compression does not result in a significant increase in residual dislocation density in LiF. Hence, the micromechanical interpretation of precursor decay provided by Herrmann et al. [18] remains unresolved with existing recovery experiments. 

Figure 7.3. Dislocation densities required to fit the precursor curves as a function of the initial quasi-static yield stress.

Figure 7.4. Dislocation density at the shock front as a function of shear stress on primary slip planes.

Kumar and Clifton [31] have shock loaded <100)-oriented LiF single crystals of high purity. The peak longitudinal stress is approximately 0.3 GPa. Estimates of dislocation velocity are in agreement with those of Flinn et al. [30] when extrapolated to the appropriate shear stress. From measurement of precursor decay, inferred dislocation densities are found to be two to three times larger than the dislocation densities in the recovered samples. 

To answer questions regarding dislocation multiplication in Mg-doped LiF single crystals, Vorthman and Duvall [19] describe soft-recovery experiments on <100)-oriented crystals shock loaded above the critical shear stress necessary for rapid precursor decay. Postshock analysis of the samples indicate that the dislocation density in recovered samples is not significantly greater than the preshock value. The predicted dislocation density (using precursor-decay analysis) is not observed. It is found, however, that the critical shear stress, above which the precursor amplitude decays rapidly, corresponds to the shear stress required to disturb grown-in dislocations which make up subgrain boundaries. 

Champion and Rohde [42] investigate the effects of shock-wave amplitude and duration on the Rockwell C hardness [41] and microstructure of Hadfield steel over the pressure range of 0.4-48 GPa (pulse duration of 0.065 s, 0.230 ls, and 2.2 ps). The results are shown in Fig. 7.8. In addition to the very pronounced effeet of pulse duration on hardness shown in Fig. 7.8, postshoek electron microscope observations indicate that it is the final dislocation density and not the specific microstructure that is important in determining the hardness. 

The shock-hardening response (total dislocation density) of Cu-8.7 Ge [43] is found to vary as a function of pulse duration as shown in Fig. 7.9. 

Figure 7.9. Dislocation density in recovered Cu-8.7 Ge subject to shock compression of various peak pressures and pulse durations.

How many atoms must be included in a three-dimensional molecular dynamics (MD) calculation for a simple cubic lattice (lattice spacing a = 3 x 10 ° m) such that ten edge dislocations emerge from one face of the cubic sample Assume a dislocation density of N = 10 m . ... 

Assume the edge dislocation density to be divided into positive and negative populations, N+ and N, moving only on slip planes at 45° (maximum shear stress) to the planar shock front. For a dislocation multiplication (annihilation) rate M, show that conservation of dislocations requires that... 

When metals are deformed plastically at room temperature the dislocation density goes up enormously (to =10 m ). Each dislocation has a strain energy of about Gb /2 per unit length and the total dislocation strain energy in a cubic metre of deformed metal is about 2 MJ, equiva-lent to 15 J mol k When cold worked metals are heated to about 0.6T new strain-free grains nucleate and grow to consume all the cold-worked metal. This is called - for obvious reasons - recrystallisation. Metals are much softer when they have been recrystallised (or "annealed"). And provided metals are annealed often enough they can be deformed almost indefinitely. 

In one of the most significant observations, small amounts of recrystallized material were observed in rutile at shock pressure of 16 GPa and 500 °C. Earlier studies in which shock-modified rutile were annealed showed that recovery was preferred to recrystallization. Such recrystallization is characteristic of heavily deformed ceramics. There has been speculation that, as the dislocation density increases, amorphous materials would be produced by shock deformation. Apparently, the behavior actually observed is that of recrystallization there is no evidence in any of the work for the formation of amorphous materials due to shock modification. Similar recrystallization behavior has also been observed in shock-modified zirconia. 

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