GENERATION, CONTROL, AND MEASUREMENT 205 



"replica gratings" and are available in both transmission and reflection 

 gratings at *i fraction of the cost of the originals. Modern replica reflec- 

 tion gratings are practically as good as the originals for monochromator 

 use. 



The angular position of the various wave lengths diffracted by a grating 

 are given by the equation 



mX ^ d{sin i + sin 0), (3-20) 



where m is the order of the spectrum on the right or left side of the 

 central, or zero-order, image, in integers of 1, 2, 3, etc., and d is the 

 grating spacing or the distance from the center of one line to the center 

 of the next. For a 600-line-per-millimeter grating, d equals 3'^oo mm 

 per Hne. The angles i and 6 are the angles of incidence and diffraction, 

 respectively, measured from a normal to the grating surface. This equa- 

 tion i? valid for both transmission and reflection gratings and for all 

 values of i and 6. In the reflection grating, sin i and sin 6 will have 

 opposite signs if they are on opposite sides of a normal to the grating 

 surface; they will have the same sign if on the same side of the normal. 

 It will be noted that, in contrast to a prism, the shortest wave lengths 

 are deviated the least by a grating. 



By diff'erentiating the grating formula and keeping i constant, an 

 expression for the angular dispersion is obtained: 



— = ^ C3-21) 



d\ d cos d \ - ) 



The dispersion is proportional to the number of lines per millimeter (l/d) 

 and the order of the spectrum m. It is a minimum when ^ = 0, i.e., 

 when the spectrum is observed normal to the grating. For the so-called 

 "normal spectrum" the dispersion is approximately constant for small 

 changes in wave length, and dd/d\ = m/d. For a grating with 600 lines 

 per millimeter, the first-order normal dispersion is 600 X 10"^ radian/m^u, 

 since there are 10^ m^u/mm. This amounts to 



1 X 57.3 X 600 X 10-« = 3.4 X 10"- Vm^. 



The linear dispersion dl/d\ for the normal spectrum of a grating is 



where r is the distance from the grating to the focal curve of the spectrum. 

 At 2-m distance the linear dispersion of the first order for this grating 

 would be 1.2 mm/m/u. Thus two lines differing in wave length by 1 m/x 

 would appear as two lines 1.2 m/i apart. 



With early gratings much of the energy was dissipated into undesired 

 orders, since only the spectrum of one order and on one side could be 



