**The analytical theory of point systems**

by J. D. Bernal, 1923.

In 1923, J. D. Bernal, when still an undergraduate, wrote a brilliant Prize Essay for Emmanuel College, Cambridge, in which he applied quaternions to the structure of crystal lattices. It remained unpublished until 1981, when it was published by Birkbeck College under the title ‘The analytical theory of point systems’, *Occasional Paper No.* 1, 1981, Department of Crystallography, Birkbeck College, London. Thanks to the efforts of Alan Mackay, the IUCr has acquired a copy which we are happy to make available to a wider audience through this web site. We are grateful to Birkbeck College for allowing us to do so.

*Massimo Nespolo, Chair, Commission on Mathematical and Theoretical Crystallography*

“… This leads to the most striking aspects of the manuscript: the influence of modern algebra in the discussion of quadratic forms (in Chapters II and III the symbol *SXY* is the scalarproduct of *X* and *Y*) and in the use of quaternions throughout. In crystallography the linear part of a symmetry is, of course, a 3 × 3 matrix determined by coefficients. The fact that ρ preserves lengths and angles imposes 6 conditions so only 3 degrees of freedom are left for ρ. Clearly a more precise analytic approach would benefit from a more economic method of presentation of 3 parameters. It was fairly well known among theoretical physicists and mathematicians that all such linear parts arise from transformations of the form

*X* → ± *qXq*^{-1}

where *q* is a unit quaternion (the correspondence between unit quaternions and linear parts is actually not one-one, because –*q* determines the same transformation as *q* ; this does not cause any confusion in practice). Most readers of this note will be familiar with this fact but, for those who are not, a brief note on quaternions is appended [see pages 10 and 11!] which covers these facts assumed without comment by Bernal in Chapters II and III. Note that the quaternion notation – unlike matrix notation – allows angles of rotation, axis of rotation, plane of rotation to be read off immediately. This is a great advantage when trying to decide whether two groups are or are not equivalent. …”

*From introductory note by R. L. E. Schwarzenberger, Science Education Department, University of Warwick*

**Download/HTML links**

- Bernal_monograph.pdf (The analytical theory of point systems)
- Editor’s Preface (Alan Mackay)
- Introductory note (R. L. E. Schwarzenberger)

*Source: *Emails by M. Nespolo (2009/10/06 23:55, mathcrystATuhp-nancy.fr), H. Wondratschek (2011/02/22 3:35, wondraATphysik.uni-karlsruhe.de), http://www.iucr.org/education/teaching-resources/bernal-essay