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BELL SYSTEM TECHNICAL JOURNAL 



Hence since the elastic constants are definitely known, the temperature 

 coefficient of any longitudinally vibrating crystal can be obtained when 

 the separate temperature coefficients are evaluated. 



The temperature coefficients appearing in equation (A. 65) can all be evalu- 

 ated from the temperature coefficient angle curves for A' cut rotated crystals 

 shown by Fig. 1.19. For an X cut crystal equation (A.65) reduces to 



Tf = 3.9 + 6.5 cos' V 



SiiTsf^ sm \p + {IsnTs,^ + ^4^^,^,) sin" \p cos" i/' 

 -f SszTs^^ cos^ \(/ — Is^iTsfi sin^ yj/ cos ^p 



Sii sin i/' + (25i3 -f- Sn) sin ^ cos \}/ 



+ 533 cos yp — 2sn sin \p cos yp. 



(A.66) 



The value of Tsf^^ is obtained directly for ^2 = or ^ = 90°, for Tj = —2 

 and hence 



T,f, = 11.8 (A.67) 



Taking three other angles and solving for the remaining constants we find 

 T,f/u = -5310; (2513^,^3 + SuT^f,) = 45,130; 

 r,3 3533 = 17,400. 



(AM) 



Inserting the values found for the elastic constants, two temperature coeffi- 

 cients are determined, and one relation is given between the others, 



Tsf, = +119; r,33 = 182; T.f, - .1112 T,f, = 228.2 (A.69) 



The values of (A.68) are sufficient to determine the temperature coefficient 

 of long thin crystals cut at any angle, for inserting these values in (A.65) the 

 temperature coefficient for any oriented crystal in longitudinal vibration 

 is given by 



T/ = 3.9 + 6.5 sin' 6 cos' i^ 



'+755 (cos- 6 cos"' xp + sin- \py + 22,565 sin- 6 cos- \p 



(cos2 d cos2 1// + sin2 xP) + 8700 sin" ^cos" \p 

 +5310 sin 6 sin \p cos \p [3(cos (p cos 6 cos \p — s\n<p sin \p)- 



— (sin (p cos 6 cos \p + cos (p sin yp)-] 



127.9 (cos2 e cos2 xP + sin2 xpy + 175.8 sin^ d cos^ xP 



(cos- 6 cos-\p + sin-i/') + 95.6 sin 6 cos \p 

 +89.2 sin 6 sin \p cos i/'[3(cos (p cos d cos \p — sin v? sin \p)- 



— (sin (p cos d cos \p + cos <^ sin xp)''] 



The only regions of low temperature coefficients are the regions for which 

 the two big middle terms are small which requires that 6 —> 0, or \p —)■ 90°. 



(A.70) 



