Annual Emeriti Colloquium

Friday, April 16, 2004
3:00 PM, room 3206

THE CRYSTALLOGRAPHIC PHASE PROBLEM

 

Herbert A. Hauptman

Ph.D. in Applied Mathematics, University of Maryland, 1955

Nobel Prize in Chemistry, 1985

 

            We begin with a collection

                                                                                                                      (1)

of N three-dimensional vectors rj, each having three components lying in the interval (0,1).  For each vector H having three integer components define the (complex-valued) normalized structure factor EH by means of

                                                                       (2)

where

                                                                                                                              (3)

and the N real numbers fj (atomic scattering factors) are specified.  The crystallographic phase problem is the problem of determining the values of the so-called phases fH when only the values of the magnitudes |EH| are presumed to be known.

 

            The central importance of the phase problem stems from the fact that the vectors r for which the function r(r), defined by

                                                                                          (4)

takes on its maximum values, coincide with the elements of S.  The magnitudes |EH| are presumed to be known since they can be measured experimentally.  Furthermore the phase problem is a solvable one, in fact an over-determined one, provided only that a sufficiently large number of magnitudes |EH| have been measured.  Thus the function r(r) (Eq. 4), which requires a knowledge of the phases fH, not obtainable experimentally, yields the values of the vectors rj, not known a priori.  In this way the problem of crystal structure determination, equivalent to the problem of determining the vectors r which maximize r(r), is here formulated as a problem in pure mathematics.