QUARTZ CR VSTA L A PPI.ICA TIONS 201 



very low and nearly independent of the width length ratio. The NT low 

 coefficient flexurally vibrating crystal is similar to the MT but requires a 

 somewhat higher rotation about the V axis to produce a low coefficient. 



The MT crystal has been used in narrow band filters such as pilot channel 

 filters of the cable carrier system and in oscillators having frequencies be- 

 tween 50 kilocycles and 100 kilocycles. The NT flexure crystal is capable 

 of producing frequencies as low as 4 kilocycles, and can be used to produce 

 filters and control oscillators in the frequency range from 4 kilocycles to 50 

 kilocycles. Crystals of this type have been used with the Western Electric 

 frequency modulation broadcast transmitter.^^ Operating in the region 

 of 5 kilocycles, they maintain the frequency of the transmitter to ±.0025 

 per cent without temperature regulation. These two crystals will be 

 described in more detail in a subsequent paper. 



APPENDIX A 



\'OIGT'S ELASTIC AND PIEZOELECTRIC RELATIONS AND 



THEIR APPLICATION TO THE DETERMINATION OF LOW 



TEMPERATURE COEFFICIENT CRYSTALS 



A.l Mathematical Expressions for Piezoelectric Relations 



As mentioned in the historical introduction, Voigt formulated a mathe- 

 matical relation between the stresses, strains, polarizations, and electric 

 fields existing in a crystal. For a general crystal devoid of symmetry these 

 relations take the form 



— -Vj- = 5ii Ai -\- 5i2 Yy -\- S\zZz -j- ^14 Yz -\- Si^Zx 



+ sf^Xy — duEi — d-iiEy — dzxEz 



— y'v = Sii Xj, -f 5^2 Y,j + slzZ, -f 5^4 Yz + 5^5 ^x 



-\- s^^Xy — dnEx — d-22Ey — d-i«Ez 



— Zz = 5^1 Xx + sli Yy + S33 Zz + 5^4 Yz + ^fs Zx 



+ 536 Xy — di3 Ex — dis Ey — dss Ez 

 — y'z = su Xx + sli Yy -f- suZz + S44 Yz + SiiZx 



+ Si^Xy — d\iEx — d^iEy — dz\Ez 



(A.l) 



'•'' "A New Broadcast Transmitter Circuit Design for Frequency Modulation," J. F. 

 Morrison, Proc. I. R. E., Vol. 28, No. 10, Oct. 1940, i)p. 414-449. 



