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scintillation counters. The wedges are adjusted by a feedback servotype 

 system. A record of wedge positions across the limb gives a measure 

 of both soft tissue and bone mass. Inherently, such a null system is less 

 convenient but far more accurate than any of the film techniques. 



4. Curve Fitters 



The analog computer described in the last section carried out an actual 

 calculation; starting with the experimentally determined data (the 

 X-ray photograph of bone and wedge), it computed the bone density. 

 Another type of problem for which analog computers are widely used 

 is to determine the parameters of an equation to give the best fit of the 

 theoretically computed curves to the experimental data. 



For example, it is known that the electron density p in a crystal 

 must have the form in any plane perpendicular to the w axis 1 



p(u, v) = ^ |^fc|[ cos <Phk cos {27Thu + 2-rrkv) 

 h,k 



+ sin (p hk sin {2-rrhu + 27rkv)] (1) 



where (u, v) are fractions of the unit cell length, \F hk \ are amplitudes 

 determined by X-ray diffraction, <p hk are parameters depending on z only, 

 and h, k are positive integers or zero. To determine the electron density, 

 one must know the parameters <p hk . In general, there is no experimental 

 method to measure them. To find the electron density, one must guess 

 the values of cp hki compute p, and then ascertain if it leads to a reason- 

 able atomic arrangement. For a complex crystal, each such computa- 

 tion without the aid of electronic computers would take many human 

 lifetimes. The analog computer X-RAC can compute p(u, v) for all u, 

 v and for 400 <p hk parameters in 1 second. The result is displayed on 

 an oscilloscope screen as equal-density contour lines ; this result can then 

 be compared with the investigator's notions of the particular atoms 

 present and the probable bonds between them. 



The initial choice of the values for the parameters <p hk is a very difficult 

 one. Once one is reasonably close, it is possible to adjust these values to 

 bring the electrons into sharper and sharper contours about atomic 

 locations. This is very much like focusing a microscope. The initial 

 choice of the <p hk s for crystals of complex molecules is usually calculated 

 with a digital computer. 



1 This can be derived from Equation 3 of Chapter 15. 



