A triclinic crystal has primitive lattice vectors

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Condensed Matter, Materials & Soft Matter · Crystal structure, bonding and diffraction

Invariance, Ch General bases, the metric and tensors in spacetime

Problem

A triclinic crystal has primitive lattice vectors, given by their Cartesian components, a1=(3.00,0,0) A˚,a2=(1.00,2.00,0) A˚,a3=(0.50,0.50,2.00) A˚.\mathbf a_1=(3.00,0,0)\,\text{\AA},\quad \mathbf a_2=(1.00,2.00,0)\,\text{\AA},\quad \mathbf a_3=(0.50,0.50,2.00)\,\text{\AA}. ((a)) Compute the cell volume V=a1⋅(a2×a3)V=\mathbf a_1\cdot(\mathbf a_2\times\mathbf a_3). ((b)) Construct the dual basis vector e1\mathbf e^1, defined by e1⋅aj=δ1j\mathbf e^1\cdot\mathbf a_j=\delta^1{}_j, giving its Cartesian components. ((c)) The reciprocal lattice vector is b1=2πe1\mathbf b_1=2\pi\mathbf e^1. Find its magnitude ∣b1∣|\mathbf b_1|.

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