368 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1951 



cients for any arbitrary temperature. These calculated values have been 

 checked experimentally for the AT t)rpe crystal and the angles and proper- 

 ties are approximated by the calculations. It is shown that there is a critical 

 angle of +36°26' which results in the highest temperature of 190°C for 

 which it is possible to obtain a zero temperature coefficient AT type crystal. 



II. Evaluation or the Elastic Constants as a Function 

 OF Temperature 



A simple method for taking account of the temperature terms is to ex- 

 pand the frequencies for the known cuts in powers of the temperature 

 around some reference temperature. Since the data of Figs. 1 and 2 run 

 from -100°C to +200°C, a convenient temperature is 50°C. Then 



f = fm[i + a,{T - To) + a^iT - ToY + a,(T - ToY + • • •] (1) 



Over this temperature range the frequencies measured can be accurately 

 represented by terms including the cubic as the highest. If To is taken as 

 50°C, equation (1) can be solved for the constants ai, a2, and az and we 

 find 



— -^ [/200° — /-100°] + :r [/l25° — /-25°] 

 fll = 



300/60- 



f-lOO" "h /200° r 



2 -J^' 



(2) 



az = 



22,500/50- 



(/200° ~ /-100°) ~ 2[/i25° — /-25°] 



5,062,500/50' 



where the subscripts refer to the temperatures for which the frequencies 

 are measured. If we apply these equations to the AT crystal cut at 35° 18' 

 and the BT at — 49°16', we find, for the frequencies, the equations 



/at = 1.661 X 10^[1 + .22 X 10-«(r - 50°) 



+ 8.9 X 10-9(r - 50)2 + 82 X lO-^H^ - 50)^ + .••] 



(3) 

 /ot = 2.547 X 10^[1 - 2.2 X 10-\T - 50) 



- 55.5 X 10-9(r - 50)2 _ 73 X 10-i2(r - 50)3 + . . .] 



In order to obtain the frequency and the variation of frequency with 

 angle, use is made of the equation for a thickness shear vibration 



y __ 1 . Aef = — A /^w ^^^ ^ + ^'4 sin* — 2cu sin cos 6 



(4) 



