I have drawn the cubic system and have marked the tetrahedral and octahedral voids myself(see attachment).However I do not understand why all of them will be having index (1/4,1/4,1/4).Someone suggested me to imagine each of them as mere vectorial translation along three directions. Someone else told me that moving equal distances from each of the vertices in the cube will be equivalent owing to its symmetry.However I do not understand why should I interpret symmetry that way .Is it not about mirror images ?Secondly is it not that vectorial translation will land me into any site/location that I feel like. Sir please explain me this in detail and with close reference to my doubts.
Unfortunately you do not show your image which would be helpful in order to see what do you understand as fcc and tetrahedral.
fcc means you have 8 (actually 1 belongs to a unit cell) atom at each corners and 6 (actually 3 belong to the unit cell) at the center of each plane. Using the single atom at the origin (0,0,0) and the three centered atoms at (1/2, 1/2,0), (1/2,0,1/2), (0, 1/2, 1/2) form already the tetrahedron. The tetrahedral site is in the center of this tetrahedron.
Crystal symmetry if valid for everything, atom, physical or chemical properties, vacancies (like your tetrahedral sites). You can try to find other tetrahedral positions than those 8 which are either all symmetry-equivalent or pairwise symmetric, so that they are covered by (1/4, 1/4, 1/4) + symmetry or (1/4, 1/4, 1/4) and (-1/4, -1/4, -1/4) + symmetry. The last two symmetries which require 2 positions are 23 and -43m. For m-3, 432 and m-3m only one position is required to generate all 8 symmetry-equivalent positions.
For octahedral sites you ac do the same.
BTW: There are hundreds of descriptions in fundamental books about crystallography and materials science or the internet. It is basic knowledge so that I can recommend you to read and try to understand it. Reserachgate cannot compensate fundamental knowledge and I doubt that Researchgate is the suitable platform for such equations. If you do not understand such basics I doubt that you can follow the majority of the asked questions.
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