88 BELL SYSTEM TECHNICAL JOURNAL 



temperature coefficient crystals whose plot as a function of the two 

 rotations would be a line in which the A T and BT cuts would be in the 

 region of two points on the line. A few of these crystals whose angles 

 were in the region of the A T crystal were measured and were found to 

 have zero coefficients but also had a much more complicated frequency 

 spectrum than the AT or BT crystals when cut to have their major 

 faces more nearly parallel to the x axis. 



Recently Bechmann ^^ has made calculations and experiments in 

 respect to double orientation crystals which have zero temperature 

 coefficients. The calculations were made by means of the Christofel 

 formula of equations (1) and (2). Although this gives the same result 

 as that calculated from the constant Cee' for rotations around the x axis, 

 it differs from it somewhat for more general rotations. If we expand 

 equation (1) we obtain the cubic equation for the frequency of 

 oscillation. 



- Ja'/b'/cK^ - Kab' -Kac-- Kbc' + IKabKacKbc) = 0, (16) 

 where 



r _ C_ _ VXll/p ^ VX22/P . r ^ VX33/P 



^ " 2t' ^^ ~ It '' ^^~ 2t ' ^^ 2t 



j^ X12 j^ Xi3 „ X23 



J^AB — I ; J^AC — r ', J^BC 



'VX11X22 vXiiXas '\X22X33 



Ja, /jB. fc can be interpreted as the three primary frequencies and would 

 correspond to the three solutions of (16) if the couplings Kab, etc., 

 were zero. The three solutions of (16) then will be these three primary 

 modes modified by the coupling between them. If we let 



P = LifA'-^fB'+fc') - SUAja'Ci- - Kab') 



+ ///c^(l - Kac') +fB'fc' (1- Kbc')']U9, (17) 



Q = [2(/^^ + Jb' + Icy - 9(fA' + Ib' + fc') 



X [///^2(1 - Kab') +fA'fc'{i - Kac') -^-fs'fc'a - Kbc')1 



+ 27fA'fB'fc'{l - Kab' - Kac' - Kbc' + 2KabKacKbc)']/S4, 



^^ "Researches on Natural Elastic Vibrations of Piezo-Electrically Excited Quartz 

 Plates," R. Bechmann, Zeit.f. Technisch Physik, Vol. 16, No. 12, 1935, pp. 525-528. 

 This multiple orientation of high-frequency shear crystals is also the basis of the 

 V cut crystal of Bokovoy and Baldwin discussed for example in British Patent 

 No. 457,342 issued May 27, 1936. 



