Dissertation in Draft: Appendix II

Date:: 1953

Reference:: PP/CRI/F/1/16

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Licence: In copyright

Credit: Dissertation in Draft: Appendix II. Source: Wellcome Collection.

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APPENDIX 33 THE SURFACE AREA OF BODIES OP UNIFORM DENSITY The formula given on page of the text was originally derived from the following simple consideration. Consider a periodic one- dimensional distribution of electron density, which only takes the values 0 and 1, as shown in Figure . Bow imagine that we slur over the sharp edges of this distribution. This is necessary to ensure the convergence of the mathematical formula we shall derive. This slurring can conveniently be done by folding Figure with some simple smoothing function, such as a gaussian function. We shall then obtain Figure . Now let us differentiate this curve, and so obtain Figure . Finally we take the square of this last curve, as shown in Figure . Now if the smoothing function used has been fairly narrow, and if none of the edges in Figure are very close together, the peaks in our final figure will not overlap. Moreover each peak will be the same size and shape, which will depend only on the smoothing function we have used. Thus if we could add up the area under the curve of Figure for one repeat, it would be proportional to the number of edges per repeat in the original distribution. The interesting fact emerges from the mathematics that if we have the Fourier component of our original distribution we can obtain the area under the curve of Figure , and thus the number of edges by manipulating the Fourier intensities alone. It is not necessary to know the phases. This can easily be seen by carrying through the mathematics. Let the Fourier components be denoted by F^ f where h. is the -1-