Group-subgroup relationship

Fig. 8. Group-subgroup relationships among phosphorus oxides (105).

More generally, the occurrence of first- or second-order phase transitions can be rationalized within the framework of Landau theory (1937) that classifies phase transitions by symmetry considerations. In this theory, the symmetry relationship of the descendant phase with the parent phase is of primary importance in the description of the possible phase transitions. Within a group-subgroup filiation, continuous- (second order) or discon-tinuous-(first order)phase transitions are possible. On the other hand, if the two phases are not related by a group-subgroup relationship, then the phase transition cannot be continuous. 

Landau also recognized that continuous phase transformations in a group subgroup relationship could be described by an order parameter, which carries the information about the broken symmetry. This order... 

Many deformations of the ideal perowskite structure are possible. In this context, one single type of deformation will be considered, namely a rotation of the octahedra along the [111] direction. Fig. 7 illustrates various ranges of to values for which many real structures have been identified. They are listed on table 2 by increasing value of (o and fig. 8 shows their group-subgroup relationships. 

The lattice gas approach is valid within certain limits for typical metallic hydrides, binaries as well as ternaries. Deviation from this idealized picture indicates that metallic hydrides are not pure host-guest systems, but real chemical compounds. An important difference between the model of hydrogen as a lattice gas, liquid, or solid and real metal hydrides lies in the nature of the phase transitions. Whereas the crystallization of a material is a first-order transition according to Landau s theory, an order-disorder transition in a hydride can be of first or second order. The structural relationships between ordered and disordered phases of metal hydrides have been proven in many cases by crystallographic group-subgroup relationships, which suggests the possibility of second-order (continuous) phase transitions. However, in many cases hints for a transition of first order were found due to a surface contamination of the sample that kinetically hinders the transition to proceed. 

In all cases we start from a simple structure which has high symmetry. Every arrow (= -) in the preceding examples marks a reduction of symmetry, i.e. a group-subgroup relation. Since these are well-defined mathematically, they are an ideal tool for revealing structural relationships in a systematic way. Changes that may be the reason for symmetry reductions include ... 

GROUP-SUBGROUP RELATIONS BETWEEN SPACE GROUPS FOR THE REPRESENTATION OF CRYSTAL-CHEMICAL RELATIONSHIPS... 

The polynomial (1.5) which I called cycle index is, if H is the symmetric group, equal to the principal character of H in representation theory. Professor Schur informed me that the cycle index of an arbitrary permutation group being really a subgroup of a symmetric group is of importance for the representation of this symmetric group. We will, however, not expand on the relationship between representation theory and our subject. 

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