▁used ▁in ▁cry stal log raph y ▁to ▁indicate ▁the ▁magn itudes ▁and ▁directions ▁of ▁the ▁thermal ▁v ibration ▁of ▁atoms ▁in ▁cry stal ▁structures . ▁Since ▁the ▁v ibr ations ▁are ▁usually ▁an is otrop ic ▁( d ifferent ▁magn itudes ▁in ▁different ▁directions ▁in ▁space ), ▁an ▁elli ps oid ▁is ▁a ▁convenient ▁way ▁of ▁visual ising ▁the ▁v ibration ▁and ▁therefore ▁the ▁symmetry ▁and ▁time ▁aver aged ▁position ▁of ▁an ▁atom ▁in ▁a ▁cry stal . ▁ ▁Th erm al ▁elli ps oid s ▁can ▁be ▁defined ▁by ▁a ▁tensor , ▁a ▁mathematical ▁object ▁which ▁allows ▁the ▁definition ▁of ▁magnitude ▁and ▁orientation ▁of ▁v ibration ▁with ▁respect ▁to ▁three ▁mut ually ▁per pend icular ▁axes . ▁The ▁three ▁principal ▁axes ▁of ▁the ▁thermal ▁v ibration ▁of ▁an ▁atom ▁are ▁denoted ▁, ▁, ▁and ▁, ▁and ▁the ▁corresponding ▁thermal ▁elli ps oid ▁is ▁based ▁on ▁these ▁axes . ▁The ▁size ▁of ▁the ▁elli ps oid ▁is ▁scale d ▁so ▁that ▁it ▁occup ies ▁the ▁space ▁in ▁which ▁there ▁is ▁a ▁particular ▁probability ▁of ▁finding ▁the ▁electron ▁density ▁of ▁the ▁atom . ▁The ▁particular ▁probability ▁is ▁usually ▁ 5 0 %. ▁ ▁See ▁also ▁ ▁Deb ye – W aller ▁factor ▁ ▁References ▁ ▁Category : C ry stal log raph y <0x0A> </s> ▁B irk ▁Man ufact uring ▁is ▁a ▁mult im ill ion - d ollar ▁priv ately ▁owned ▁company ▁which ▁was ▁recently ▁recognized ▁as ▁one ▁of ▁the ▁countries ▁fast est ▁growing ▁companies ▁by ▁the ▁Inc ▁ 5 0 0