X-RAVS AXD OR/KXTATIOX OF QUARTZ CRYSTALS 301 



Further, in llie equation ;/X = Id sin 9, the intcrplanar spacing d of the 

 atomic planes (hk-C) is 



yuif + hk + k') + 



d,,.f = / -^ (3.2) 



Where the axial ratio c/a is l.(WW7 in quartz and ao , the distance between 

 atoms along the a axis, is 4.9(KSA.^ 



The ti order reflection from the hk-f plane is sometimes spoken of as 

 the reflection from the n{hk-() plane as in Table I and P'ig. 3.8 where, for 

 example, the second order reflection from the 01 • 1 plane is listed as the 

 reflection from the 02-2 plane. The justification for this notation is that 

 the 02-2 planes would have- a d value half that of the 01-1 planes (since 

 the indices are reciprocals of the axial intercepts) and the first order reflec- 

 tion from such planes would have the same 6 value as the second order 

 reflections from the 01-1 planes i.e., if 



then 



2X = 2d sin 



X = 2 f sin ^ 



2 



Among manufacturers of piezoelectric units a rectangular coordinate 

 system is used in place of the hexagonal one described above, but the 

 indices as derived from the hexagonal axes are retained. One of the a 

 axes is chosen as the A' axis. The three a axes are identical (the vertical 

 axis of quartz is an axis of 3-fold symmetry) and therefore any one of the 

 a axes may be chosen as A'. The c axis is called the Z axis and a I' axis, 

 normal to .A and Z is so chosen as to form a right-handed coordinate system 

 for right-handed quartz or a left-handed coordinate system for left-handed 

 quartz as described in Chapter II. 



The indices of the cap faces of right and left-handed quartz crystals 

 (as viewed from above) are given in Fig. 3.6 which should be compared 

 with Fig. 2.4 and 2.6 of Chapter II. (Note that Z in Fig. 3.6 refers to the 

 Z axis whereas z in Fig. 2.6 refers to the z faces or minor cap faces). The 

 cap faces directly beneath those illustrated (i.e., on the other end of a 

 doubly terminated crystal) would have the same indices except that the 

 final digit would be negative since they cut the negative end of the Z (or 

 c) axis. Thus (Ol-l) is beneath (01 -1) and parallel to (01-1). 

 ^ Parallel crystal faces lie along the same set of atomic planes. Thus 

 (01-1) and (Oi-1) represent the same atomic plane. Further, since the 



■•Wyckoff, W. G., "The Structure of Crystals," The Chemical Catalogue Co. (19.51). 



