Nomenclature for interpenetration parameters and classes of interpenetration 


Z - Total degree of interpenetration Total number of 3D interpenetrating nets. It is the product of translational (Zt), and non-translational (Zn) interpenetration.
FIV - Full interpenetration vector The shortest vector defining the direction along which all the interpenetrating nets exactly superimpose and that, applied Z-1 times to a single net, generates the whole entanglement.
TIV – Translation interpenetration vector When translational (Zt>1) and non-translational (Zn>1) operations are present we called TIV the shortest vector that relates all the Zt interpenetrating nets.
PIV – Partial interpenetration vectors When there is no FIV or TIV but more than one translational operation, different partial interpenetration vectors (PIVs) relate subgroups of Zp independent nets. PIVs could be of two kinds: Zit for integral translations and Zct for centering translations.
FISE - Full interpenetration symmetry element Single space group symmetry element that generates all Z equivalent interpenetrating nets. No translational operations relating the nets are present (no FIV, TIV or PIV). The only Z allowed are 2,3,4,6.
NISE - Non-translating interpenetration symmetry element When Zt>1 (TIV is present) a NISE symmetry element can exist that generates all Zn interpenetrating nets.
PISE - Partial interpenetration symmetry element Any space group symmetry element that generates Zs (two or more) equivalent interpenetrating nets (but not all Zn nets). If Zs=Zn PISE is equivalent to NISE (Zt>1) or to FISE (Zt=1)
PIC - Primitive interpenetration cell Any cell that contains the same number of atoms for each individual (coloured) net is an “interpenetration cell”. Among the possible minimum volume interpenetration cells we select the PIC as the one based on the vectors between atoms of the same net. Thus, PIC can be considered as the minimum primitive crystallographic cell constructed with one of the nets, i.e. the primitive cell to be assigned to the structure if it contains a single network.
PICVR - PIC volume ratio The ratio of the PIC volume (VPIC) divided by the primitive unit cell volume (V0), PICVR=VPIC/V0. PICVR is integer, greater than or equal to unity.



Class Nets relationships Z PICVR Subclass Interp. Vector Symm. Elem. Z symbol
I Only translations (integral or centering) Zt = Z Ia FIV none Zt
Ib PIV none Z(ΣZit ∗ ΣZct)
II Space group symmetry operations Zn = 1 IIa none FISE Zn
IIb none PISE Z(ΠZn)
III Translations + symmetry operations Zt × Zn = Zt IIIa TIV NISE Z(Zt ∗ Zn)
IIIb PIV NISE Z[(ΣZit ∗ ΣZct) ∗Zn]
IIIc TIV PISE Z[Zt ∗ΠZn]
IIId PIV PISE Z[(ΣZit ∗ ΣZct) ∗ΠZn]


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