Past exam of the materials science course of the University of Cambridge / 2022 / Paper 1 / 5 / b / iii / Solution

Burgers vector conservation gives two candidate reactions, depending on the relative signs of the approaching segments:Initially $b_1^2+b_2^2=3a^2/2$. The sum product has $b^2=a^2$, while the difference product has $b^2=2a^2$. Thus the $a[001]$ product is favoured by Frank's rule; the $a[1\bar{1}0]$ product is disfavoured. The result is not annihilation, because the two supplied vectors are not opposite. The same ordering survives a simple character correction: with $\nu =0.30$, line energy proportional to $G{b}^2[{\cos }^2\chi +{\sin }^2\chi /(1-\nu )]$ gives the two parent segments a combined coefficient $1.786a^2$, whereas any character of the sum product has coefficient between $a^2$ and $1.429a^2$ and any character of the difference product at least $2a^2$, comparing equal lengths and common logarithmic factors. A favourable dislocation junction can obstruct further glide and harden the crystal. Its actual mobility requires the junction line and allowed glide plane: the second screw can occupy several $\{110\}$ planes containing $[\bar{1}11]$, so geometry is not unique. For instance, its $(101)$ plane intersects $(10\bar{1})$ along $[010]$; an $a[001]$ junction on that line would glide geometrically on $(100)$, not either original plane. Dislocation cross-slip, junction unzipping or bypass can compete under load. An energy inequality alone is not a universal proof of a permanently sessile product.