Dislocation Emission : The Gamma Surface.

a study of iron

INTRODUCTION

In a macroscopic sense, it seems easy to imagine how a crack propagates through metal. At the atomic scale, however, when one sees that metal is a lattice arrangement of atoms, the idea of a "crack" even becomes fuzzy. Similarly, when a metal deforms in a plastic manner (i.e. it doesn't return to its original shape), it doesn't make sense to think that each small element of the metal's lattice has assumed a similarly deformed shape. And in fact, this doesn't happen. To allow for gross deformation of a specimen, defects are created in the lattice in nature's attempt to keep the potential energy in the metal as low as possible. A "shifting" of atoms takes place in such a way that, if you could sit inside and watch it happen, you might see dislocations shooting away from crack tips, where stress is generally highest. But how does nature determine which direction to shoot out dislocations? Nature creates dislocations, or planar faults, in a manner that takes the least amount of energy as possible.

For this reason, to help in predicting which direction a dislocation will be emitted, it is helpful to calculate the potential energy (Gamma) involved in "forcing" a dislocation and compare its energy to that of other "forced" planar faults.

• The lattice parameter for BCC Fe is 2.87 Å and the cohesive energy is 4.28 eV.

• The plane that was chosen for study was the (112) plane.

Energies were obtained and a matlab file was used to create the following output:

THESE TWO PLOTS REPRESENT THE SAME DATA. The first is a colorized equivalent of a countour plot and the second is a surface plot. The values being represent in the "y-direction" are the energy it takes to displace a (1,1,2) plane from the lowest potential energy (perfect lattice) in the x and z-directions as labeled in the plots. The plot covers one "period" of translation. That is, displacement of any integer multiple of the maximum x and z displacements plotted would result in the exact same energy as the reference (starting) point.

It should be noted that originally only 100 values were obtained (10 in the x by 10 in the z). Cubic splines were fit to the data in order to interpolate the values. Slight anomalies might have been introduced not only due to the interpolation of such sparse data (75 points are represented in each directoin of these plots), but also the order in which the interpolation was performed. Cubic splines were fit first to the rows of constant z and then to rows of constant x.

DISCUSSION

From the data, one can see what appear to be discrete points, represented by rounded peaks on the plots, where the energy is higher than the surroundings. In fact, the dropoff of energy immediately surrounding all peak values is steep. This would indicate that within each flat "trough," displacement is relatively unhindered. Around the lattice point (0,0) there is approximately a 2Å by 2Å square (remember periodicity) in which movement will occur rather readily in comparison to large displacement in the x-direction (over the large ridge). Having mentioned the difficulty in large displacement in the x-direction, it follows that displacement in the z-direction will occur easily (two small obstacles before returning to a locally perfect lattice). Of course, the above statements about the ease with which certain planar displacements can be made must be taken wiht a grain of salt. This data represent purely one plane in three-dimensional space. One must consider that displacement in another plane might present less of an hinderance (as measured by the component of the gradient of the above plotted surface opposite the direction of displacement).