Study On Pure Mode Axes Of Hexagonal And Trigonal System Crystals

Pure mode axis of a crystal is the direction in which energy velocity and phase velocity are in the same direction when an elastic wave propagates in the crystal.Propagating along a pure mode axis,the elastic wave in crystals will show special properties which has theoretical guidance function for the design of transducer and the improvement of the receive effect.In addition,piezoelectric crystals adopted by piezoelectric transducers are mostly the crystals of hexagonal system or trigonal system.Piezoelectric crystal is an important part of piezoelectric devices.The chip randomly cut from a piezoelectric crystal cannot serve as a piezoelectric element.The cutting directions are decided according to the piezoelectric coefficients and energy transmission characteristic of crystals.In this paper,the slowness surface method is used to study the pure mode axes of the crystals of hexagonal and trigonal system.The influence of piezoelectric effect on pure mode axes of crystals will be discussed.Using the MATLAB software,three-dimensional slowness surfaces which are three-dimensional grid figures are plotted.By using slowness surfaces,it can not only avoid lots of complex calculation but also can solve the problem of not factoring in some directions for Christoffel equations.The direction of a radius vector on a slowness surface is the direction of a phase velocity.A normal direction on the slowness surface represents the direction of the energy velocity.Thus,the direction of a radius vector which is consistent with the normal direction on a three dimensional slowness surface is a pure mode axis.It is found that thel pure mode axes of hexagonal system crystals are the positive and negative direction of Z axis respectively and all directions in the XY plane by data analysis and comparison.The pure mode axes of trigonal system crystals with symmetry 32,3m,3 have six directions which are(1,0,0),(-1,0,0),(1/2,(?)/2,0),(1/2,-(?)/2,0),(-1/2,(?)/2,0),(-1/2,-(?)/2,0).And the pure mode axes of crystals of 3,3 class are also six directions in XY plane.These directions doesn’t contain the positive and negative direction of X axis and the angles between the other four directions and X axis are no longer integer multiple of 60 degrees.They are equivalent to pure mode axes of 32,3m,(?)m class integrally rotated a certain angle around Z axis,but the rotation angles vary with different kinds of crystals.If considering piezoelectric effect,only the crystals belonging to 6,(?),622,6mm,(?)2m classes of hexagonal system have piezoelectric effect and the independent components of elastic stiffness coefficients increase dramatically.Slowness surfaces of zinc oxide,cadmium sulfide,zinc sulfide crystals with symmetry of 6mm class change obviously,but pure mode axes don’t change.For trigonal system,only the crystals belonging to 32,3m,3 classes have piezoelectric effect and the independent components of elastic stiffness coefficient also increase dramatically.By comparing the pure mode axes data,it is found that piezoelectric effect have no influence on the pure mode axes of quartz crystal with symmetry 32 and that of lithium niobate crystal with symmetry 3m,respectively.