The Burgers Vector

While in this introductory class, we mainly want to understand why dislocations are important and how we use them in engineering, actual materials engineers need to be able to quantify the stress/strain due to a dislocation. The Burgers Vector essentially tells you the magnitude and direction of the lattice distortion due to the presence of a dislocation.

Let's first derive the Burgers vector and then consider how we'll use it. Start by looking at a simple 2D array of atoms (Figure 12.8.1) and try to "quantify" the distortion due to the edge dislocation. First, let's think about what the assembly of atoms would look like without the dislocation: the three atoms belonging to the "extra half-plane" would no longer be there, recovering a perfect crystal. So, it is the extra half-plane that is yielding the distortion. Let's measure its influence by constructing the so-called Burgers Circuit:

  1. Identify the core of the dislocation (the region where the most distortion occurs, or the area of undercoordinated atoms).
  2. Identify an atom (up and to the left is conventional) that does not seem to be influenced by the dislocation. That is, it seems to be in a perfect crystalline coordination.
  3. Start drawing a circuit around the dislocation core by jumping from one atomic site to another, making equal numbers of jumps in antiparallel directions (i.e., same number right/left and up/down, but you don't need the same number in all four directions).
  4. When you've completed the jumps, draw a vector between the starting and ending point of the circuit. This is your Burgers Vector

Here is an example shown in Figure 12.8.1:

  1. The dislocation core is the open space in the middle of the atoms, indicated with a $\bot$.
  2. I start at the blue atom up and to the left of the dislocation core and take 3 steps to the right.
  3. I take 3 steps down.
  4. I take 3 steps to the left (the same number as the steps to the right).
  5. I take 3 steps up (the same number as down).
  6. I do not return to the origin, so I draw a vector from the end point to the starting point. This is my Burgers Vector, b.

A few observations:

  1. If the magnitude of the Burgers vector is 0, then there's no dislocation in the circuit! (Or, you had two dislocations in the circuit... be careful).
  2. For edge dislocations, the Burgers vector points perpendicular to the dislocation core and along the direction the dislocation will slip.
  3. The Burgers vector will tell us about the degree of strain that exists in the crystal, which is related to its energy.
Demonstration of how to find the Burgers Vector of a dislocation.

Figure 12.8.1 Demonstration of how to find the Burgers Vector of a dislocation.

Exercise 12.8.1: Slipping Around
Not Currently Assigned

Consider the dislocation in Figure 12.4.1. With respect to the Burgers Vector $\mathbf{b}$, how would you define the orientations of the other dislocation features (all of which can be described by a direction).

  1. Determine how each feature is oriented with respect to the Burgers Vector for an edge dislocation, parallel or perpendicular.

    1. Dislocation Line (aka dislocation core):
    2. Slip Direction (aka atomic motion during slip):
    3. Dislocation Motion (aka motion of the dislocation itself:)

Burgers Vector in Edge, Screw, and Mixed Dislocations

The process for deriving the Burgers vector is essentially the same for edge and screw dislocations, but for screw dislocations the circuit will be in three dimensions instead of just two as shown in Figure 12.8.2. For edge dislocations, the Burgers vector is perpendicular to the dislocation line, while for screw dislocations they are parallel.

Burgers vector examples for edge and screw dislocations. By <a href="//commons.wikimedia.org/wiki/User:MartinFleck" title="User:MartinFleck">Martin Fleck</a> - <span class="int-own-work" lang="en">Own work</span>, <a href="https://creativecommons.org/licenses/by-sa/4.0" title="Creative Commons Attribution-Share Alike 4.0">CC BY-SA 4.0</a>, <a href="https://commons.wikimedia.org/w/index.php?curid=96858277">Link</a>

Figure 12.8.2 Burgers vector examples for edge and screw dislocations.

By Martin Fleck - Own work, CC BY-SA 4.0, Link

For mixed dislocations, the angle between the Burgers vector and the dislocation line varies along the dislocation line, since the dislocation line is curved as shown in Figure 12.8.3.

For all types of dislocations, the magnitude of the Burgers vector is equal to the slip distance, i.e., how far the atoms have to move for the dislocation to slip one unit cell over. For this reason, a larger Burgers vector generally corresponds to the material being more brittle.