Sec. 9.7] MASS SPECTROGRAPHS 287 



deflected through a smaller or greater angle to be focused at a distance 

 \/l R m along the central path from the edge of the magnetic field. The 

 displacement of the focus from the apex of the magnetic field along the line 

 AO for such ions is D = IRmo. Actually, only those ions of mass m for 

 which R e = R m and which enter and leave the magnetic field normal to its 

 edge are focused exactly on this line. The error in the focus for other masses 

 however is small for a considerable range in mass about m . The width of 

 the final focus of mass m for which the conditions above hold is 



8 = R m a* 



where a = one-half angular divergence of beam entering electric field from 

 the source, or 



sum of slit widths 



2 X slit separation 



The maximum geometrical resolving power for ions on the central path, 

 defined as the mass separation equivalent to the width of the focal image 

 (complete separation), can be calculated from 



m R m 



Am AR m 



The resolving power falls off slowly on either side of m owing to the increasing 

 image width from imperfect focusing. 



9.7. Trochoidal-trajectory Mass Spectrograph. A combination of crossed 

 uniform electric and magnetic fields has been proposed which accomplishes 

 complete focusing of ion beams with large initial angular divergence and 

 energy spread [8]. Ions injected into a region containing a uniform electric 

 and magnetic field arranged at right angles are known to follow trochoidal 

 trajectories that converge to a single point for any one value of m/e (see 

 Fig. 66). The shapes of the ion trajectories in terms of field strengths and 

 initial conditions is given in rectangular coordinates by the equations 



x = a(\p — yp ) + p(sin ^„ — sin \p) 

 y = L — p cos yp = p(cos <^ — cos \J/) 

 Ec- m 



, _ lirEc-m 



= lira = — jjr, — 



Since a and b are proportional to m/e and not the initial angle or energy, 

 all ions of the same m/e converge to a point or a line image located at a 

 distance of x = b from the source. It also follows from this that the mass 

 scale is strictly linear over the whole mass range. The secondary generating 



