  The structural-magnetic models of 25 AFM kagome cuprates similar to
herbertsmithite (ZnCu3(OH)6Cl2) - a perspective spin liquid - have been
calculated and analyzed. Main correlations between the structure and magnetic
properties of these compounds were revealed. It has been demonstrated that, in
all AFM kagome cuprates, including herbertsmithite, there exists the
competition between the exchange interaction and the antisymmetric anisotropic
exchange one (the Dzyaloshinskii-Moriya interaction), as magnetic ions are not
linked to the center of inversion in the kagome lattice. This competition is
strengthened in all the kagome AFM, except herbertsmithite, by one more type of
the anisotropy (duality) of the third in length J3 magnetic couplings (strong
J3(J12) next-to-nearest-neighbor couplings in linear chains along the triangle
edges and very weak FM or AFM J3(Jd) couplings along the hexagon diagonals).
The above couplings are crystallographically identical, but are divided to two
types of different in strength magnetic interactions. The existence of duality
of J3 couplings originated from the structure of the kagome lattice itself.
Only combined contributions of dual J3 couplings with anisotropic DM
interactions are capable to suppress frustration of kagome AFM. It has been
demonstrated that the possibility of elimination of such a duality in
herbertsmithite, which made it a spin liquid, constitutes a rare lucky event in
the kagome system. Three crystal chemistry criteria of the existence of spin
liquids on the kagome lattice have been identified: first, the presence of
frustrated kagome lattices with strong dominant antiferromagnetic
nearest-neighbor J1 couplings competing only with each other in small
triangles; second, magnetic isolation of these frustrated kagome lattices; and
third, the absence of duality of the third in length J3 magnetic couplings.
