372 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1951 



To obtain the angles for zero coefficient, we set the multipUer of AT 

 equal to zero, giving 



807 cos^ 6 - 1236 sin^ - 154 sin d cos 6 



- 17 sin^ e cos^ ^ - 53 sin ^ cos (9 = 



Substituting in for the values of d, we find that this equation is satisfied by 



= + 36°30' and d = - 41°20' (17) 



Substituting in the values of 6 into equations (15) and extracting the 

 square root, the variations of frequency with temperatures are as shown by 

 equation (18) 



/3,.3,, = l:666>O0_^ [1 + 35 X 10-(Ar)^ 



+ 77X io~''(Ar)'+ •••] 



5 



- 2.503 X 10 r, ^^ vx 4rw— 9/A/n\2 



/_4i°-2o/ = j [1 - 64 X 10 \ATy 



(18) 



-62 X 10"''(Ar)^+ •..] 



IV. Experimental Results 



Since the AT type crystal appeared to be more constant with tempera 

 ture and to have a higher electromechanical coupHng factor than the BT 

 type, some measurements were made for a number of crystals ranging from 

 36°26' to 36°45'. These were units made by A. W. Warner, all having 

 dimensions of diameter 12.5 mm, thickness 0.166 mm. With a ratio of 

 diameter to thickness of about 75 the dimensioning was not critical, and 

 the correction for the length thickness ratio was smal). These had special 

 solders and holders that will be described in a paper by A. W. Warner. 



These units were measured over a temperature range from 60°C to 250°C 

 by T. G. Kinsley. The results for a crystal cut at +36°26' are shown by 

 Fig. 3, 36°30' by Fig. 4, and 36°45' by Fig. 5. Figure 5 shows the fun- 

 damental and third overtone frequency plotted as a function of temper- 

 ature. The 36°30' crystal had a fundamental frequency constant of 



1.660 X 10** 



— which agrees well with that given in equation (18) since a small 



allowance has to be made for the weight of the plating. However, the tempera- 

 ture of zero coefficient is 182°C instead of the value of 200°C calculated and 

 the curvature constants 02 and az are +73 X 10~^ and +254 X 10~^2. 

 These deviations probably occur because of the fact that only three terms in 

 the power series were included; whereas, to agree with experiment, further 

 terms should be included. On the other hand, the change in orientation for a 

 zero temperature coefficient at a given temperature is fairly closely pre- 



