I 



Wyckoff — Crystal Structure of Alahandite. 241 



groups, TS T-, and T^, likewise have special cases with 

 four equivalent positions. These special cases are as 

 follows : 



Tetartohedral symmetry: 



/Space Group T\' 



4a uiiu/ uuu/ uim/ uuu. 

 Space Group T^: 



U 000; UO; iOi; OU- 



4c iH; OOi; 0^0; iOO. 



4cl 4^4 ; 444; 444; 4T4 ♦ 



4e 4-4^; 4T¥J 44 4) T44' 



/Space Group T%' 



4f uuic; u-{-^^^,—u^u; u^u-\-^^\—u; ^—u^u^u-\~^. 



Tetrahedral symmetry (hemimorphic hemihedry) : 



Space Group T/: 4:a. 



Space Group T^: 45, 4c, 4c?, 4e. 



From these special cases all of the possible ways of 

 arranging the atoms in a unit of alabandite which has 

 four chemical molecules within it are as follows : 



(I) Mn: uuw, uuu; uuu; uuu. 



S: u'u'u' ; u'uu'; uuu'; u'u'u'. 



On the basis of the symmetry of the arrangement of its 

 atoms this structure would possess tetrahedral symmetry. 



(II) Mn: 000; i^O; iO^; Oii. 



S: iH; ooi; OiO; iOO. 

 The symmetry of this arrangement is the complete 

 symmetry (the holohedry) of the cubic system. If a 

 crystal having this arrangement of its atoms exhibits a 

 lower symmetry than the holohedral, as alabandite does, 

 this low degree of symmetry must be accounted for by 

 some such effect as a dissymmetry in the shape of the 

 fields of force surrounding the atoms. 



(III) Mn: 000; i^O; iOi; 0^- 



S • iii • 13.3. . 3 13 . 3 3 1 

 *->• 444) 444) 444) '¥¥4' 



The symmetry of this arrangement is tetrahedral. The 

 other arrangements deducible from the space groups T^ 

 and Td 2 are capable of reduction to this same arrange- 

 ment. 



(IV) Mn: uuu; u-\-\^\ — u^u; u^u-\-^,^—u; ^—u^u^u-{-\. 



S: u'u'u' ; u'-\-^,^ — u'^u'; u',u'-{-^,^—u' ; ^—u',u',u'-{-^' 



