Fluorite, , has space group , whose general position has multiplicity 192 per…

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

Solid State Physics, Ch Symmetry and the classification of crystals

Problem

Fluorite,

\mathrm{CaF_2}

, has space group

Fm\bar3m

, whose general position has multiplicity 192 per conventional cell. The

\mathrm{Ca^{2+}}

ions occupy Wyckoff position

4a

(0,0,0)

, site symmetry

m\bar3m

of order 48; the

\mathrm{F^-}

ions occupy Wyckoff position

8c

(\tfrac14,\tfrac14,\tfrac14)

, site symmetry

\bar43m

of order 24. ((a)) Find the multiplicity of each site and the total number of atoms per conventional cell. ((b)) Check this total against

Z=4

formula units of 3 atoms each. ((c)) A proposed alternative puts fluorine on

4b

, at

(\tfrac12,\tfrac12,\tfrac12)

, also of site symmetry

m\bar3m

. Show that it cannot describe

\mathrm{CaF_2}

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