Polymorphism, polytypism and modular aspect of compounds with the general formula A₂M₃(TO₄)₄ (A = Na, Rb, Cs, Ca; M = Mg, Mn, Fe³⁺, Cu²⁺; T = S⁶⁺, P⁵⁺): order–disorder, topological description and DFT calculations

The crystal structure of Na₂Mn₃(SO₄)₄ [unit-cell parameters a = 14.8307 (18), b = 9.9107 (18), c = 8.6845 (12) Å, space group Cmc2₁] displays order–disorder (OD) character and can be described using the OD groupoid family, more precisely a family of OD structures built up by two types of non-polar layers, with layer symmetry P(m)c2₁ (L_2n+1 type) and P(b)cm (L_2n type) (category IV). A new hypothetical MDO₂ polytype has been proposed and the geometry optimization demonstrates its reasonability as another possible stable polytype. Compounds Na₂Mn_3–xMg_x(SO₄)₄ with the unit-cell parameters a ∼ 29.2–29.7 Å, b ∼ 9.5–9.9 Å, c ∼ 8.7 Å and space group Pbca can be described in terms of modularity as a sequence of A, S₁ and S₂ modules:…|AS₁AS₂AS₁AS₂|… or (AS₁AS₂), together with MDO₁ (AS₁AS₁) and MDO₂ (AS₂AS₂). The crystal structures of itelmenite, NaCaFe³⁺₃(PO₄)₄, and Ca₂MgFe³⁺₂(PO₄)₄ are crystal-chemical isotypic to Na₂Mn_3–xMg_x(SO₄)₄ and should be considered as (A*S₁A*S₂) derivatives of the (AS₁AS₂)-type structure.