## Abstract

The purpose in discussing this topic is to show, by example, the types of efforts that are involved in order to fulfill the potential for scientific advances offered by some initial mathematical analysis of a problem. For example, there may exist a mathematical analysis of a problem which offers a solution under the quite ideal circumstances of having an infinite amount of exactly correct data. How is it possible to start with a theory in the real world in which experimental data are far from infinite and have a certain amount of error? How is it possible to proceed when it is readily shown mathematically that a solution to a problem exists, but there is no indication of how to proceed? These specific topics will be discussed by examples derived from diffraction techniques for determining structures of molecules in the gaseous and crystalline states.

Both my wife and I carried out our research for the Ph.D. with Professor Lawrence Brockway of the University of Michigan chemistry department. One of the reasons that I mention my wife here is because, although much of our research over the years has been independent, her participation with me at times was very beneficial, in fact, crucial to the outcome of my work. Lawrence Brockway was one of the first graduate students of Linus Pauling and he had established under Pauling’s tutelage, electron diffraction of gases for structure determination in the United States.

In 1941, P.W. Debye published a paper on gas electron diffraction that had a great influence on the future course of our scientific lives. Debye developed a mathematical description, from a probabilistic point of view, of the nature of an electron diffraction experiment, with the objective of including in the results, not only the mean positions of the atoms of a molecule, but also the root-mean-square vibrational amplitudes between pairs of atoms. This was the theory, and it was quite obvious that there was a real challenge to work out the practice.

Data were limited in amount and accuracy. The molecular scattering, the part of the scattering of interest, was recorded on an inherently steeply falling background. An unknown line had to be drawn through these data, compensation for limited data had to be made and a damping function had to be applied to the data which would be accounted for toward the end of the analysis.

The solution to these problems led to an accurate procedure for studying the arangements of atoms in molecules in the vapor state and also information concerning their vibrational motions.

In addition, it will be seen how discoveries in the study of gas electron diffraction influenced the path toward a general solution of the phase problem in crystal structure analysis, resulting in a method applicable to all types of crystals. Various aspects of this will be covered in my presentation.

Both my wife and I carried out our research for the Ph.D. with Professor Lawrence Brockway of the University of Michigan chemistry department. One of the reasons that I mention my wife here is because, although much of our research over the years has been independent, her participation with me at times was very beneficial, in fact, crucial to the outcome of my work. Lawrence Brockway was one of the first graduate students of Linus Pauling and he had established under Pauling’s tutelage, electron diffraction of gases for structure determination in the United States.

In 1941, P.W. Debye published a paper on gas electron diffraction that had a great influence on the future course of our scientific lives. Debye developed a mathematical description, from a probabilistic point of view, of the nature of an electron diffraction experiment, with the objective of including in the results, not only the mean positions of the atoms of a molecule, but also the root-mean-square vibrational amplitudes between pairs of atoms. This was the theory, and it was quite obvious that there was a real challenge to work out the practice.

Data were limited in amount and accuracy. The molecular scattering, the part of the scattering of interest, was recorded on an inherently steeply falling background. An unknown line had to be drawn through these data, compensation for limited data had to be made and a damping function had to be applied to the data which would be accounted for toward the end of the analysis.

The solution to these problems led to an accurate procedure for studying the arangements of atoms in molecules in the vapor state and also information concerning their vibrational motions.

In addition, it will be seen how discoveries in the study of gas electron diffraction influenced the path toward a general solution of the phase problem in crystal structure analysis, resulting in a method applicable to all types of crystals. Various aspects of this will be covered in my presentation.