The structure of lovozerite is derived from perovskite. For 24 members of the lovozerite family an aristotype is postulated. The method of quantitative comparison using the concept of mappings is applied to the lovozerite family using the aristotype as a 'structural unit'. The method is extended to relationships of symmetry-type II, i.e. the derived structure and the aristotype have only a common subgroup, the remaining non-common symmetry of the derived structure is used as 'distribution' symmetry for the structural unit. The numerical results are discussed in detail.
AKIJI YAMAMOTO 483 it into the first one. (hi is the identity operator.) The groupoid T is therefore the set of operations which superposes each substructure onto itself (diagonal terms) or onto the other substructure (off-diagonal terms). The superspace group G is called the kernel and H the hull of the superspace groupoid. The structure analysis can be made within the framework of the present theory if we use the operators in G and H.It is possible to classify the structure into two groups of substructures, one of which consists of the second substructure of Hg and part of the first substructure and the other consists of the third substructure of Hg and the remaining part of the first substructure. The two groups are transformed into each other by the glide plane normal to a *l-b .1 which transforms a *~ into b *l. In the first substructure, each part has a tetragonal lattice but with monoclinic symmetry. These two parts have no common atoms because they are related by the glide plane. We can apply the groupoid theory to these two groups by recognizing the groups as the substructures in the above discussion. The two structure groups have a fivedimensional superspace group with monoclinic symmetry. When the structure factor of the first group Fo(h e) and h2 = {R[r} transforming the first part into the second one are considered, the structure factor of the total structure is given by F(h e)=
F0(he)+exp (27rher)Fo(R-~h~).In the present case, h2 is the glide plane so that R-1 = R. Then the diffraction pattern shows the rotational symmetry due to h2" F(Rhe)=exp(2rrRher)F(h e) because {Rlr} 2= {Elr+Rr}={E]O} and therefore exp(27rRh~r) = exp (-27rh~r), where E is the identity operator. This ensures orthorhombic diffraction symmetry. Thus, instead of applying the superspace group for the merged Hg substructure, we can use the superspace groupoid as given in a previous paper (Yamamoto & lshihara, 1988
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