Chord of a circle

A chord of a circle (or sphere) is a line segment whose end points lie on the circle (or sphere). A line which intersects the circle (or sphere) in two points is a secant of the circle (or sphere). A diameter of a circle (or sphere) is a chord containing the center and a radius is a line segment from the center to a point on the circle (or sphere). 

The majority of samples used in NMR are cylindrical. The echo decay of a cylindrical sample in the presence of a uniform field gradient perpendicular to the cylinder axis represents a Fourier transform of a distribution of chords of a circle which, in turn, is a semi-ellipse. Thus, the echo can be written in terms of a Bessel function as J (t)/t, where t is the time coordinate, because this represents a Fourier transform of a semicircle which differs from a semi-ellipse only by a multiplicative constant. (See, for example, Chapter 19 of Bracewell, Appendix A.) Because of the way in which the two domains are related (as discussed in sections I.D.l. and I.D.2.), there is a reciprocal relationship between the spread of the Larmor frequencies across the cylindrical sample and the extent of the signal with the shape J (t)/t in the time domain. Thus the information on the field gradient experienced by the sample is contained in the echo if the sample dimension is known. 

A variety of other configurations and modifications of the basic design shown in Figures 3-6 and 3zZ are possible. Valve trays (see Chapter lOi are popular. Downcomers can be chords of a circle as shown or circular pipes. Both partial and total condensers and a variety of reboilers are used. The column may have multiple feeds, sidestream withdrawals, intermediate reboilers or condensers, and so forth. The column also usually has a host of temperature, pressure, flow rate, and level measurement and control devices. Despite this variety, the operating principles are the same as for the sinple distillation column shown in Figure 3-6. 

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