14 



BELL SYSTEM TECHNICAL JOURNAL 



The I.R.E. Orientation Angles and the I.R.E. Matrix 



The Institute of Radio Engineers has proposed that, for quartz crystals, 

 all orientations be given in terms of three rotations </>, d, \f/ about .T3, .T2 

 and xz respectively, starting with the plate length along Xi width along 

 X2 and thickness along X3. (Here .T3 is the z or optic axis, Xi is the electric 

 axis.) 

 Whence, here: 



cos (j) sin 0\ 



— sin (j) cos I 



1/ 



and carrying out the two matrix multiplications: 



Xi Xi X3 



cos cos cos\f/ sin cos cos ^ —sin cos i/' 

 ■ sin sin)/' +cos0sini/' 



- cos (/) cos 6 sin ^ — sin (p cos 6 sin xf/ 



- sin cos 1/' 4-COS0COS1/' sin sin 1/' 



cos sin 6 



sin <f) sin 6 



cos 6 



•Tl 



.^2 



.-vs 



(5.6) 



If we denote the unit vectors along the length, width and thickness as P1P2 

 and P3 respectively we have as a matrix defining the plate: 



P = Rx 



(5.7) 



The I.R.E. orientation system is useful to the designer of crystal plates 

 because his problem is to choose such values of 0, 0, \p as to give the plate 

 certain physical properties along its length, width and thickness. The man 

 who cuts the plate has a different problem, that of moving the crystal (and 

 hence the .Ti .to .T3 axes) about a fixed saw so that the plate cut parallel to the 

 saw blade is what the designer ordered. 



Let us consider such a system as shown in Figs. 37, 38 and 39. In Fig. 

 38 the crystal stands with its optic axis along P3, its + electric axis (for 

 right hand quartz) along P. Since the shop man considers clockwise rota- 

 tion as positive we now rotate the crystal through angle C/3 about Pz clock- 

 wise, we then turn the crystal through angle U2 clockwise about Pi, and 

 finally, after cutting out a slab of required thickness, we turn it clockwise 

 through angle Ui about Pz to cut its length and width. 



On the plate axes Pi the crystallographic axes .Ti .T2 :V3 are now given by 



X = rP 



(5.8) 



