ELECTRO-MECHANICAL REPRESENTATION OF CRYSTAL 723 



resistances 



Rr = 4\Uo. (15) 



If all other sources of dissipation were eliminated, the radiation re- 

 sistance would produce a limiting value for the decrement of a crystal 

 which may be calculated as follows. From equation (11), the value 

 of q for the mechanical system is 



on inserting the values of Rr, Ce, and /a = Xjllm^'ps. Since p = 2.65 

 and 5 = 1.27 X lO^^^ for a perpendicularly cut quartz crystal 



g = 2.24 X 10". (17) 



The decrement of a crystal in terms of the circuit of Fig. 1 is 



Inserting the values of footnote (4) we find 



5 = ^ = 1.14 X 10-". (19) 



Trg 



Van Dyke ^ has measured the limiting value of the decrement of a 

 perpendicularly cut quartz crystal vibrating in air and finds it to be 

 1.26 X lO"". Since the residual losses were about 5 per cent of the 

 radiation losses, this agrees well with the value found in equation (19). 

 The equivalent circuits of Figs. 2 and 3 may also be used as the 

 basis of design for mechanical systems, such as loud speakers, micro- 

 phones and supersonic radiators. 



^Proc. I. R.E., April, 1935. 



