From the definition of a dislocation being the line of demarcation between slipped and unslipped regions, dislocations cannot end inside a crystal except at a node. The node is a point where two or more dislocations meet. The existence of dislocation nodes affirms that different types of dislocation reactions occur inside a crystal. How­ever, a dislocation reaction is totally different from a chemical reaction. So, we should not confuse ourselves looking for similarities between the two. The

dislocation reactions pertain to the addition or dissociation of initial dislocation(s) into new product dislocation(s). However, they cannot just happen arbitrarily. The dislocation reaction should follow certain geometrical and energetic rules. The energy criterion used to decide whether a dislocation addition or dissociation reaction is feasible or not is known as Frank’s rule. The underlying basis of this rule is that the total energy needs to be minimized for the reaction to be energetically feasible. To elucidate the rule, let us take an example of a dislocation reaction where two dislocations with Burgers vectors b1 and b2 meet to form a third dislocation with a Burgers vector b3. That is, the reaction is b1 + b2 ! b3. According to Frank’s rule, the reaction will be feasible when bj + b2 > b3. In case of a reaction b1! b2 + b3, the energetic feasibility condition of the reaction would be b1 > bj + b3, accord­ing to the Frank’s rule. This implies that the reaction should be vectorially correct and energetically favorable. With little introspection, we can find out that the Frank’s rule actually stems from Eq. (4.13).

& Example 4.4

Show that the following dislocation reaction is feasible:

ao [їм] ^ a0 [2ii]+a0 [112]. 2 6 6

Solution

To test the feasibility of the reaction, a two-step procedure needs to be fol­lowed. The first step is to ascertain whether the reaction is geometrically sound or not. This can be accomplished by verifying that the x, y, and z com­ponents are equal on both sides of the reaction:

ao 2ao ao

x components: — = —— + — 2 6 6

ao ao ao

y components: — (0) = 0 = — — + — 2 6 6

ao ao ao

z components: — (—1) = =

2 2 6

Therefore, the given dislocation is geometrically (or vectorially) possible. If any reaction that does not happen, there is no need to go to the second step for validation.

For the dislocation reaction to be energetically favorable, we need to show that b1 > b2 + b2.

Original dislocation: b1 = [101].

The magnitude ofthe Burgers vector is

Hence,

b1 — ^.

1 2

Product dislocation : b2 = — [211].

6 L

The magnitude of Burgers vector is b2 = | (22 + (-1)2-

Hence,

b2- a» b2 — 6 •

Product dislocation: b3 — — [112].

6

The magnitude of the Burgers vector is
b3 — a0 (12 +12 + (-2)2)1/2.

Hence,

4.2.4