A two-dimensional crystal has primitive lattice vectors and

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Tensors and differential geometry

Conditions of Validity, Ch A coordinate system is a map: the metric, the scale factors and where a chart fails

Problem

A two-dimensional crystal has primitive lattice vectors e1=(2.0,0) nm\mathbf{e}_1=(2.0,0)\ \text{nm} and e2=(1.0,2.0) nm\mathbf{e}_2=(1.0,2.0)\ \text{nm}, oblique axes with no orthogonality. ((a)) Construct the reciprocal basis ei\mathbf{e}^i satisfying ei⋅ej=δji\mathbf{e}^i\cdot\mathbf{e}_j=\delta^i_j (no sum implied by the jj on the right, the summation convention applies only to repeated indices in an expression). ((b)) A point defect sits at r=(3.0,4.0) nm\mathbf{r}=(3.0,4.0)\ \text{nm} from the origin. Write r=ciei\mathbf{r}=c^i\mathbf{e}_i (summation convention) and find the expansion coefficients c1,c2c^1,c^2.

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