Algebraic Basis Of Symmetric Group Of Crystallography And Its Applications

AbstractEach field of modem sciences and technology involves the stUdyand aPPlication of all the kinds of solid materials inevitably A crystal isthe moSt imPortan and most widely used solid material. The proPertiesof the crystal, including mechanical, physical, chendcal and geometricalones are determined by itS atomic smicture uniquely The unit cell andsyInmetry are the basic parameters of the crysta. The valid use ofsymmetry operaion will simPlify calculation of varous physicalproblems maximally; therefore, the stUdy of symmetry grouP theory isone of the imPortan subjects in the fields of physics, chemiStry andbiology that is the reason we choose this subject as the research project.From the poiflt of view of mathematics, the SynUnetry of the crystalcan be described by grouP theory The grouP consisting of syITUn6tricaloPerations is defined as crystallOgraPic syInmetry grOuP. TheWetrical theory became an integraed classical theory at the end ofl9th century and promoted the progress of space-grouP theory On theother hand, sPace grou theory exercises a significam effect on theProgress of physics. With the develoPment of sciences and techaologymany subects, such as the strUctUre analysis of the crystal, eleCtronenergy bond theory lattice dynamics, and frequency spectrUms etC,involve the knowledge of crystal space-grouP theoryAt the same time, crystallograPhers realized that the geometricmethod of deriving space grouP was failed to work fdr higherdiInensional sPace, so that the crystalograpers and mathemdticianswere trying to develoP an abStract algebraic method to solve the problemof symrnetry of high dimensional space.The potential aPplication of grouP theory in physics, derivingmethod and the status ofgron theory are reviewed in the Preface at first.On the basis of review the research method of higher dimensionalsHetrical grouP theory is proposed.In order to prove the accuray of this new method, the aPPlicatonin 3-dimensional sPace is analyzed in the main part, and a comParisonbetWeen algebraic method and geometric ones is stated.The article emPhasized on the foundaton of the algebraic methodand the process of deriving crystallograPhic symmetry grouP. Anuniversal formula:NTN = T.Which can find all the elements of a symmetry grouP for n dimensionalspace was derived first. Based on this formula, by using the methodgiven by Dade to classify the maximal finite grouPs of 3x3 integramatrices, we proved that the mchic tensor T falls into four arithInaticallynonequivalent classes r,Tz,Tl,T4 and twO geometrically noneqUhalentclasses r and T4. Then substituting r,T. into above whversalformla, we found twO maximal finite grouPs of Which the ranks are 48and 24 respectively Finally, we derived tWo subgrouP networscorresPonding to the tWo mbomal finite grous using the generalprinciples of grouP theory and find that the crystallograPhic point grouPsof 3 dimensional SPace are really 32. We have develoPed a new method,so called generator method, to derive subgrouP nberks that work atleast for 3- dimensional space. The generators of 230 sPace grous havebeen calculated one by one. Using these generators as the data base, wehave writtn a program to calculate all the equivaent points of 230sPace grouPs Which can be available for any one Who are interested incrystallograPhy...