bl6: translational symmetrygoodwin.chem.ox.ac.uk/goodwin/teaching_files/bl6.pdf ·...

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BL6: Translational Symmetry

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Page 1: BL6: Translational Symmetrygoodwin.chem.ox.ac.uk/goodwin/TEACHING_files/bl6.pdf · (x+⅔,y+⅓,z+⅔) F (x,y,z) (x+½,y+½,z), x +½,yz), (x,y+½,z+½) The Bravais lattices represent

BL6: Translational Symmetry

Page 2: BL6: Translational Symmetrygoodwin.chem.ox.ac.uk/goodwin/TEACHING_files/bl6.pdf · (x+⅔,y+⅓,z+⅔) F (x,y,z) (x+½,y+½,z), x +½,yz), (x,y+½,z+½) The Bravais lattices represent

Pure translational symmetry:

Unit cell translations

Centred lattices

Page 3: BL6: Translational Symmetrygoodwin.chem.ox.ac.uk/goodwin/TEACHING_files/bl6.pdf · (x+⅔,y+⅓,z+⅔) F (x,y,z) (x+½,y+½,z), x +½,yz), (x,y+½,z+½) The Bravais lattices represent

A / B / C

(x,y,z) ⟼(x+½,y+½,z)

I

(x,y,z) ⟼(x+½,y+½,z+½)

R

(x,y,z) ⟼(x+⅓,y+⅔,z+⅓),(x+⅔,y+⅓,z+⅔)

F

(x,y,z) ⟼(x+½,y+½,z),(x+½,y,z+½),(x,y+½,z+½)

Page 4: BL6: Translational Symmetrygoodwin.chem.ox.ac.uk/goodwin/TEACHING_files/bl6.pdf · (x+⅔,y+⅓,z+⅔) F (x,y,z) (x+½,y+½,z), x +½,yz), (x,y+½,z+½) The Bravais lattices represent

The Bravais lattices represent the absolute minimum number of primitive and centred lattices

for all 7 crystal systems

Centred unit cells are used in lieu of primitive unit cells only when needed to represent the

true crystal symmetry

Page 5: BL6: Translational Symmetrygoodwin.chem.ox.ac.uk/goodwin/TEACHING_files/bl6.pdf · (x+⅔,y+⅓,z+⅔) F (x,y,z) (x+½,y+½,z), x +½,yz), (x,y+½,z+½) The Bravais lattices represent

Translational symmetry+ point symmetry:

Screw axesGlide planes

21 screw axis ∥ x

(x,y,z) [ ⟼ (x,−y,−z) ](x,y,z) [ ⟼ (x + ½,−y,−z)

Page 6: BL6: Translational Symmetrygoodwin.chem.ox.ac.uk/goodwin/TEACHING_files/bl6.pdf · (x+⅔,y+⅓,z+⅔) F (x,y,z) (x+½,y+½,z), x +½,yz), (x,y+½,z+½) The Bravais lattices represent

a-glide ⊥ b

(x,y,z) [ ⟼ (x,−y,z) ](x,y,z) [ ⟼ (x + ½,−y,z)

b

a

n

Page 7: BL6: Translational Symmetrygoodwin.chem.ox.ac.uk/goodwin/TEACHING_files/bl6.pdf · (x+⅔,y+⅓,z+⅔) F (x,y,z) (x+½,y+½,z), x +½,yz), (x,y+½,z+½) The Bravais lattices represent

M T DoveStructure and Dynamics

§ 3.5