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BELL SYSTEM TECHNICAL JOURNAL 



When the orientation of the atomic plane with respect to the plate edge 

 has thus been determined the next and final step is the determination of the 

 angles between the incident X-rays and the plate-face. 



3.9 Determination of Angles Between X-Rays and the Faces or a 



Finished Plate 



Procedures for determining the angles g and g + g' for certain particular 

 positions of the atomic plane with respect to the plate edges were described 

 in section 3.6 under (a), (b), and (c). For the general case (d) in which 

 the intersection between the atomic plane and the plate-face is neither 

 normal nor parallel to the plane of the instrument, the problem is best 

 solved vectorially, as follows: 



(d) Atomic plane intersecting plate-face in a line which is neither normal 

 nor parallel to the plane of the instrument (general case). (Fig. 3.16) 



Fig. 3.19 — Position 1 for a plate of general orientation (one in which no plate-edge 

 is parallel to a crystallographic axis). 



Let Ni , N2 , N3 be the components of the unit normal N to the a tomic 

 plane in terms of the plate edges Pi , Po , P3 , and A'l , .Y2 , .Y3 the components 

 of the unit vector X along the incident beam. Then 



sin Ohk.f = A'liVi + X2N2 + XgAs 



(3.22) 



(the inner product of these two vectors, which is thus equal to the cosine 

 of the angle between the incident ray and the normal to the atomic plane 

 or cos (90 - 6)). 



In matrix form this may also be written: 



Xc Ni,k-( = sin ei,k./> 

 Where Xc is the matrix A'l , A'2 , A'3 



(3.23) 



^See Bond, VV. L. "The Mathematics of the Physical Properties of Crystals," Bell 

 Sys. Tech. Jour., Vol. XXII, No. 1. 



