A two-dimensional crystal lattice has basis vectors and

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

Conditions of Validity, Ch Vector spaces, linear maps and the dual: the Riesz theorem

Problem

A two-dimensional crystal lattice has basis vectors

\mathbf{e}_1=(1,0)\,\mathrm{nm}

and

\mathbf{e}_2=(1,3)\,\mathrm{nm}

, non-orthogonal, with the ordinary Euclidean dot product. A uniform force field does work

\omega(\mathbf{v})=\mathbf{F}\cdot\mathbf{v}

on a displacement

\mathbf{v}

lying in the lattice plane. It is measured that the work done for a displacement equal to

\mathbf{e}_1

2\,\mathrm{eV}

and for a displacement equal to

\mathbf{e}_2

5\,\mathrm{eV}

. ((a)) Write down the metric

g_{ij}=\mathbf{e}_i\cdot\mathbf{e}_j

of the lattice basis and invert it. ((b)) Use

u^i=g^{ij}\omega_j

to find the components of

\mathbf{F}

in the lattice basis, then convert to Cartesian (nm) components. ((c)) Check the result directly against the two given work values.

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