374 



THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1951 



V. Evaluation for the Remainder of the Elastic 

 Constants over a Wide Temperature Range 



In order to evaluate the remainder of the elastic constants measure- 

 ments were made of the frequencies of a series of X-cut longitudinal crystals 

 over the same temperature range from — 100°C to +200°C.^ The longi- 

 tudinal crystals measured had their lengths at -30°, 0°, -f 30° and +60° 

 from the Y axis. For the four crystals measured the results are shown by 



70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 

 TEMPERATURE IN DEGREES CENTIGRADE 



Fig. 5 — Frequency temperature curves for fundamental and third overtone for a crystal 

 cut at 36°45' rotation. 



Table II. The analysis for fm = fbo" and the three constants ai , a^ and az 

 are also shown. 



To correct for the temperature expansion coefficients, the increase along 

 Ix is given by the last equation of (16) while / ± 30° and / ± 60° are 



/ db 30° = .25/. 4- .75/x = 7o[l + 12.9 X \^\T - 50) 



-f- 5.42 X 10-8(r - 50)2 _ 18 X \^^\T - 50)' + 



/ =t 60° = .75/, + .25/x = /o[l + 9.5 X \^\T - 50) 



-f- 3.68 X 10-9(r - 50)2 _ 16 X \0-^KT - 50)^ + 

 Since the frequency of a long thin bar is given by the equation 



1 1 „/ 1 



(19) 



/ - ^ . /—Tt 



21 V^ 



^, or 522 = ..2 ,2 

 p522 ^-'"»* P 



(20) 



introducing the length correction from (19) and the density correction 

 from (7) one can correct for the effect of temperature expansion. 



» These measurements were made by T. G. Kinsley. 



