Question

Magnitude of Burgers vector of the dislocation resulting from reaction of dislocations with Burgers vectors \( \frac{a}{2}[101] \) and \( \frac{a}{2}[0\bar{1}\bar{1}] \) is

• A. \( \frac{a}{\sqrt{2}} \)
• B. \( \sqrt{2}a \)
• C. \( \frac{a}{2} \)
• D. \( 2a \)

Hint:

Remember that dislocation reactions conserve the Burgers vector. Treat the Miller indices as vector components and simply add them up. A bar over a number just means it's negative.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
When two dislocations meet and react, their Burgers vectors combine. The Burgers vector of the resulting dislocation is the vector sum of the Burgers vectors of the reacting dislocations. This is based on the principle of conservation of the Burgers vector.
Step 2: Key Formula or Approach:
1. Add the reacting Burgers vectors component-wise to find the resultant Burgers vector, \(b_{r}\). \[ b_{r} = b_1 + b_2 \] 2. Calculate the magnitude of the resultant Burgers vector. The magnitude \(|b|\) of a Burgers vector \( b = k[uvw] \) in a cubic system is given by: \[ |b| = k \sqrt{u^2 + v^2 + w^2} \] Step 3: Detailed Calculation:
The two reacting Burgers vectors are: \[ b_1 = \frac{a}{2}[101] \] \[ b_2 = \frac{a}{2}[0\bar{1}\bar{1}] = \frac{a}{2}[0, -1, -1] \] First, we find the resultant Burgers vector \(b_r\) by vector addition: \[ b_r = b_1 + b_2 = \frac{a}{2}[101] + \frac{a}{2}[0\bar{1}\bar{1}] \] \[ b_r = \frac{a}{2} \left( [1, 0, 1] + [0, -1, -1] \right) \] \[ b_r = \frac{a}{2} [1+0, 0-1, 1-1] \] \[ b_r = \frac{a}{2} [1, -1, 0] = \frac{a}{2}[1\bar{1}0] \] Next, we calculate the magnitude of the resultant Burgers vector \(b_r\): \[ |b_r| = \frac{a}{2} \sqrt{1^2 + (-1)^2 + 0^2} \] \[ |b_r| = \frac{a}{2} \sqrt{1 + 1 + 0} \] \[ |b_r| = \frac{a}{2} \sqrt{2} \] \[ |b_r| = \frac{a}{\sqrt{2}} \] Step 4: Final Answer:
The magnitude of the resulting Burgers vector is \( \frac{a}{\sqrt{2}} \).
Step 5: Why This is Correct:
The solution correctly applies the principle of vector addition for dislocation reactions and then uses the standard formula for calculating the magnitude of a vector in Miller indices notation for a cubic lattice. The calculation is straightforward and yields the result in option (A).