A Machine for Synthesizing Two-Dimensional Fourier Series in the Determination of Crystal Structures

An electromechanical analog machine is described which performs a two‐dimensional Fourier synthesis for use primarily in x‐ray crystal structure calculations, $$\mathrm{\rho (xy)=\Sigma }{\displaystyle {lim}_h}\Sigma {\displaystyle {lim}_k}{F}_{\mathrm{hko}}\cos \text{2π(hx+ky).}$$ The analog to a term in this summation is obtained through the use of a pair of selsyns as a synchro‐variable transformer somewhat similar to its use in servomechanisms. ρ(xy)_hk is represented by the output voltage, F_hko by the input voltage, and 2πhx and 2πky are the angles of rotation of the primary and secondary rotors respectively. A system of crossed shaft and cascading gear trains turns the rotors to the appropriate angles for each selection of point (x, y) in the unit cell. The present machine handles h and k values up through 8, but provision is made for future extension through 16. Computations are shown for two known structures to indicate the results obtainable with the machine.