Question 1 of 7: BCC Vanadium Lattice Constant and Density; Grain-Boundary Types

Question 1: BCC Vanadium Lattice Constant and Density; Grain-Boundary Types (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Vanadium (V) is body-centred cubic (BCC); $d_{110}=0.215$ nm. Atomic mass $M_V=50.94$ g/mol (page-1 table); Avogadro's number $N_A=0.602\times10^{24}$ mol$^{-1}=6.02\times10^{23}$ mol$^{-1}$.

Find. (a) Lattice constant $a$ and theoretical density $\rho$ of BCC vanadium. (b) Distinguish twin, tilt, and twist boundaries.

Approach

The $(110)$ plane in BCC has $h+k+l=2$ (even), so it is an allowed reflection and its interplanar spacing follows the standard cubic relation $d_{(hkl)}=a_0/\sqrt{h^2+k^2+l^2}$ directly — no correction for a systematic absence is needed. Once $a$ is known, density follows from the BCC unit cell holding $Z=2$ atoms.

1. (a) Lattice constant from $d_{110}$. Using the page-1 formula $d_{(hkl)}=a_0/\sqrt{h^2+k^2+l^2}$ with $(h,k,l)=(1,1,0)$: $$a_0=d_{110}\sqrt{1^2+1^2+0^2}=(0.215)\sqrt2$$ $$\boxed{a_0\approx0.3041\ \text{nm}}$$
2. (a) Theoretical density. The BCC cell holds $Z=2$ atoms (one at each corner, shared $\tfrac18$ each $=1$, plus one full body-centre atom). Converting $a_0=3.041\times10^{-8}$ cm, so $a_0^3=2.812\times10^{-23}$ cm$^3$: $$\rho=\frac{ZM_V}{N_Aa_0^3}=\frac{2(50.94)}{(6.02\times10^{23})(2.812\times10^{-23}\,\text{cm}^3)}$$ $$\boxed{\rho\approx6.02\ \text{g/cm}^3}$$ This matches the accepted handbook density of vanadium ($6.11$ g/cm$^3$) closely, confirming the BCC assignment and the given $d_{110}$ value.
3. (b) Twin boundary. A twin boundary is a planar defect across which the lattice on one side is the mirror image (or an equivalent $180^\circ$ rotation) of the lattice on the other side, about the twin plane. It is a highly coherent, special-orientation interface (e.g. a $\Sigma3$ coincidence-site-lattice boundary at $60^\circ$ about $\langle111\rangle$ in FCC metals): atoms on the boundary itself are shared between both orientations with essentially no dangling bonds or misfit, so the interfacial energy is unusually low compared with an "ordinary" grain boundary of similar misorientation. Twins form either by mechanical twinning (a shear produced by rapid/low-temperature deformation, common in BCC and HCP metals) or by annealing/growth twinning (a stacking error during recrystallization or crystal growth, common in low-stacking-fault-energy FCC metals like brass and copper alloys).
4. (b) Tilt boundary. A tilt boundary is a low-angle grain boundary produced when two adjacent crystal regions are misoriented by a small rotation angle $\theta$ about an axis lying in the boundary plane. It can be modelled exactly as a regular, parallel array of edge dislocations of the same sign, stacked one above another; Frank's formula gives the dislocation spacing $D=b/\theta$ (Burgers vector $b$), so a smaller misorientation angle corresponds to more widely spaced dislocations.
5. (b) Twist boundary. A twist boundary is also a low-angle boundary, but the misorientation rotation axis is perpendicular (normal) to the boundary plane instead of lying within it. It is modelled as a crossed grid (two intersecting arrays) of screw dislocations rather than the single parallel array of edge dislocations used for a tilt boundary. A general low-angle boundary of arbitrary rotation axis is simply a mixture of tilt and twist character (a mixed dislocation array).
6. (b) Comparison. Tilt and twist boundaries are both low-angle ($\theta\lesssim10$–$15^\circ$) grain boundaries built from a regular array of dislocations whose type (edge vs. screw) is set purely by whether the misorientation axis lies in or normal to the boundary plane; their energy rises roughly with $\theta\ln\theta$ up to the low-angle limit. A twin boundary, by contrast, is a distinct, often large-but-special-angle planar defect that is not built from a dislocation array at all — it is a coherent mirror interface — and consequently has a much lower interfacial energy than a tilt or twist boundary of comparable misorientation.