Rationalization of the Substructures Derived from the Three Fluorite-Related [Li6(MVLi)N4] Polymorphs: An Analysis in Terms of the “Barnighausen Trees” and of the “Extended Zintl–Klemm Concept” - Inorganic 3D Structures - page 94

simple to explore in detail such pathways, and this we have done with a view to
facilitating a search for substructures contained within each of these three poly-
morphs. This major publication lists all the relationships between the Wyckoff
positions of a group and its immediate subgroups.
We focus attention initially on the

g -phase of Li 7 VN 4 . It is cubic, S.G. P 43 n ,

9.606 ˚

with a 8[ 6 ] and consider first the parent anti-fluorite structure

of Li 64 O 32 . We then explore wh at happ en s to the various Li and O atoms if the
symmetry were reduced from

¼ and Z ¼

43 n when we convert all 32 O atoms to N
ato m s and 8 of the 64 Li atoms to V atoms. We believe that, whatever subgroup of
F

F m 3 m to P

m 3 m we mig h t consider, the possible atom sites must have their counterparts in
the parent m 3 m space group itself. Expressed in another way, any particular

framework of atoms in the subgroup is inherently present in the supergroup. We
can then attempt to explain why it is that par ti cular atoms of g -Li 7 V V N 4 lodge in
particular sites of the observed space group

F

43 n to yield various substructures.
Table 1 is a summary of several p ossible pathways from the anti-fluorite-type
unit cell of L i 2 O in space group

P

-Li 7 V V N 4 with a doubled unit cell in

F m 3 m to g

space group arnighausen Tree

(not the same as that of Niewa et al. [ 6 ]) which links the atom sites of

P 43 n . Table 2 provides just one example of a single B €

-Li 7 V V N 4 to
those of the parent anti-fluorite compound Li 2 O. However, not all the pathways
shown in Table 1 lead to the actual atom sites occupied by the atoms in

g

-Li 7 V V N 4 . g

Table 1 Space group pathways from Fm 3 m to P 43 n . Transition from red to black indicates
doubling of unit cell edge
Cas e 1 Cas e 2 Cas e 3 Cas e 4 Cas e 5 Cas e 6 Cas e 7

Fm 3 mFm 3 mFm 3 m Fm 3 m Fm 3 m Fm 3 m Fm 3 m

22 5 22 5 22 5 2 25 2 25 225 225

Pn3m Pn3m Pn3m F43m F43m Pm3m Pm3m

22 4 2 24 2 24 2 16 2 1 6 2 2 1 2 21

Fd3c P43m P43m P43m P43m P43m P43m

2 28 2 15 2 1 5 2 15 2 1 5 2 15 2 1 5

F43c F43c I43m F43c I43m F43c I43m

2 19 2 19 2 17 2 19 2 1 7 2 1 9 2 17

P43n P43n P43n P43n P43n P43n P43n

218 218 218 218 218 218 218

Case 8 Case 9 Case 10 Case 11 Case 12 Case 13

Fm 3 mFm 3 mFm 3 m Fm 3 m Fm 3 m Fm 3 m

225 225 225 225 225 225

Pm 3 mPm 3 mPm 3 m Pm 3 m Pm 3 m Pm 3 m

22 1 22 1 22 1 221 221 221

Im3m Im3m Im3m Fm3c Fm3c Fm3c

2 2 9 22 9 229 22 6 226 2 26

I43m Pn3n Pm3n Pn3n Pm3n F43c

2 17 2 22 2 23 2 22 2 2 3 2 1 9

P43n P43n P43n P43n P43n P43n

218 218 218 218 218 218