A rectangular plane lattice has primitive vectors and with and (lengths on a scale…

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

Invariance, Ch Point groups and crystallographic symmetry

Problem

A rectangular plane lattice has primitive vectors

\mathbf a=(a,0)

and

\mathbf b=(0,b)

with

a=4.00\,\text{cm}

and

b=3.00\,\text{cm}

(lengths on a scale drawing; only their ratio matters). A second net is built by adding one extra lattice point at the centre of every rectangle, at

(a/2,b/2)

relative to each corner point. ((a)) Show that the vectors

\mathbf a_1=(a/2,b/2)

and

\mathbf a_2=(a/2,-b/2)

generate every point of the centred net, corners and added centres alike, and find the area of the primitive cell they define. ((b)) Compute

|\mathbf a_1|

|\mathbf a_2|

and the angle between them for the given

a

and

b

, and show that this cell is a rhombus, not a rectangle. ((c)) Compare

|\mathbf a_1|,|\mathbf a_2|

with

a

and

b

to argue that the rhombus, not the original rectangle, is the true primitive cell of the centred net. State whether the centred net drawn here is the same plane lattice as a primitive rectangular net, and say what changes if instead

a=b

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