130 Acta Cryst. (1990). A46, 130-133 Second-Order Piezomagnetism in Polychromatic Crystals BY K. RAMA MOHANA RAO* Department of Applied Mathematics, AUPG Extension Centre, Nuzvid 521201, Andhra Pradesh, India (Received 3 April 1989; accepted 13 September 1989) Abstract The group-theoretical method established for obtain- ing the non-vanishing independent number of con- stants required to describe a magnetic/physical property in respect of the 18 polychromatic crystal classes [Rama Mohana Rao (1987). J. Phys. A, 20, 47-57] has been explored to enumerate the second- order piezomagnetic coefficients (n~) for the same classes. The advantage of Jahn's method [Jahn (1949). Acta Cryst. 2, 30-33] is appreciated in obtaining these n~ through the reduction of a representation. The different group-theoretical methods are illustrated with the help of the point group 4. The results obtained for all 18 classes are tabulated and briefly discussed. 1. Introduction The non-vanishing independent first-order piezomag- netic constants (ni) for the 90 magnetic classes [32 conventional point groups (Gk) and 58 double colour point groups (G~,)] were studied in detail by Bhagavantam (1966) and Bhagavantam & Suryanarayana (1949) using the character method (Bhagavantam, 1942) based on the computation of characters for deriving the number of independent constants for the description of various mag- netic/physical properties. Jahn (1949), employing a general method also based on group theory, provided an alternative procedure for deriving these constants using the reduction of a representation corresponding to the physical property considered. The representa- tion in each case was obtained in terms of the rep- resentation Va or Vp of an axial or polar vector, depending on the nature of the physical property under question. Subsequently, Krishnamurty & Gopala Krishna Murty (1969) extended Jahn's method to find the second-order piezomagnetic coefficients for the 90 magnetic classes. The non-vanishing first-order piezomagnetic con- stants (n~) for the applicationally important polychro- matic crystal classes G~k p), p = 3, 4 or 6 (Indenbom, Belov & Neronova, 1960; Rama Mohana Rao, 1985), have already been derived in an earlier paper by the present author (Rama Mohana Rao, 1987). With the * Junior Associate Member at the International Centre for Theoretical Physics, Trieste, Italy. 0108-7673/90/020130-04503.00 Table 1. Second-order piezomagnetic coefficients needed by the 18 polychromatic crystal classes Second-order piezomagnetic Polychromatic coefficients class needed 1 6 (6) 12 2 5(6) 0 3 3(3)/m ' 12 4 6(3) 9 5 3 (3) 21 6 3(3)/m 9 7 6(6)/m 0 8 6(3)/m 9 9 6(6)/m ' 12 10 6(3)/m' 0 11 3 (3) 21 12 4 (4) 17 13 ~(4) 17 14 4(4)/m 0 15 4(4)/m ' 17 16 3(3)/2 4 17 ()(3)/2 4 18 ()(6)/2 0 group-theoretical method (Rama Mohana Rao, 1987), the second-order piezomagnetic coefficients (n'i) are enumerated for these classes in this paper by computing the character for the 10 crystallographic point groups Gk that generate the 18 polychromatic ¢ (P) classe~. Gk , p = 3, 4 or 6. To appreciate the advantage of Jahn's method, the non-vanishing second-order piezomagnetic coefficients are rederived in § 3 by the method of reduction of a representation and the results obtained through the former (character) method are compared. The different group-theoretical procedures are illustrated here, with the help of the point group 4 that induces the polychromatic class 4 (4). The results obtained for the rest of the 17 classes are tabulated in Table 1 and a brief discussion of the results is provided in § 4. The nomenclature adopted in this paper for the point groups is that of Hermann- Mauguin (International) and the notation for the polychromatic classes is that of Indenbom, Belov & Neronova (1960). 2. Second-order piezomagneti~ coefficients of the polychromatic classes Piezomagnetism is the appearance of a magnetic moment M (Mi, i = 1, 2, 3) on the application of stress o-. The occurrence of this phenomenon has already © 1990 International Union of Crystallography