278 BELL SYSTEM TECHNICAL JOURNAL 



The critical lengths are defined by the series of numbers m — in -\- \) tt 

 where n takes the values 1, 2, 3, etc. and apply to successive modes of a 

 bar clamped at both ends. Beyond the first mode, the critical wire lengths 

 are spaced at equal intervals, corresponding to increments of m each equal 



to TT. 



There is also a series of wire lengths which will present minimum impedance 

 to the motion of the plate. These may be considered as corresponding to a 

 resonance of the wire. These minima of impedance are obtained for lengths 

 of wire defined by the series of numbers m — {n — \) ir. They apply to a 

 bar which is clamped at one end and, while free to vibrate at the other end, 

 is constrained to a slope perpendicular to the plate. 



In selecting a desirable length for the supporting wire, it is not essential 

 that this length be such as to cause the wire to present minimum impedance 

 to motion of the plate. As a matter of fact, since the wire is of relatively 

 low characteristic impedance a small departure from the critical length is 

 sufficient to avoid trouble from wire resonance. In order to allow for as 

 wide a manufacturing tolerance as possible the supporting wire is usually 

 designed to have a length half-way between two successive critical lengths. 

 For a 6.3-mil phosphor-bronze wire, the spacing between successive critical 

 lengths ranges from about 58 mils at 100 kc to about 15 mils at 1000 kc. 

 Hence, even at 100 kc the length of the supporting wire must be controlled 

 within a tolerance of about 20 mils. 



These supporting wires are formed to have definite bends along their 

 length and the location of these bends varies slightly from one wire to 

 another. In addition, the wires are terminated by solder at both ends. 

 Because of these complications it is impractical to meet such close tolerances 

 on the effective length of the wires. Furthermore, a wire that does have a 

 suitable effective length at room temperatures may exhibit sufficient change 

 of properties with variations in temperature so that it becomes of critical 

 length at some other operating temperature. 



Much of the difficulty due to wire resonance is avoided by use of a solder 

 ball on the supporting wire, as described in Chapters VIII and XIII. The 

 solder ball is located near the quartz plate. Since it serves as a clamp at that 

 point, it makes the supporting wire short. By locating and forming the 

 solder ball accurately, the length of the supporting wire is controlled within 

 a close tolerance. Further, since the wire is shortened by use of the ball 

 it is less affected by changes in temperature. Experience at about 500 kc 

 indicates that a tolerance of about 10 mils in locating the solder ball is prac- 

 tical and has provided satisfactory operation between — 40 C and -f 85 C. 



14.4 Need For Cleanliness and Low Relative Humidity 



One of the most serious difficulties encountered in manufacturing quartz 

 crystal plates is that of assuring sufficient cleanliness. Even minute par- 



