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Dynamical Theory of X-Ray Diffraction$

André Authier

Print publication date: 2003

Print ISBN-13: 9780198528920

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198528920.001.0001

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(p.577) Appendix 2 The early days of dynamical theory

(p.577) Appendix 2 The early days of dynamical theory

Source:
Dynamical Theory of X-Ray Diffraction
Author(s):

Paul P. Ewald

Publisher:
Oxford University Press

(p.577) Appendix 2

The early days of dynamical theory

I was asked to say something about the origins of dynamical theory, and what I would like to do is to stress that dynamical theory is part of General Optics. It started out from Optics and the formulae for the Dynamical Theory were really ready in 1910 or 1911 before Laue’s discovery. So, let us find out what this means. I was a student in Munich and I decided to do Theoretical Physics for Professor Sommerfeld. When I came to ask him for a thesis, he had a long list of papers and at that time he worked mainly on boundary problems, on questions of self-inductance, on the propagation of wireless waves which was new at the time and, quite at the end of the list, there was to explain double refraction in crystals by an anisotropic arrangement of isotropic dipoles. Isotropic dipoles means that there is no preferential direction in the dipole for its vibrations and an anisotropic arrangement, the simplest you can think of, is that they are arranged in an orthorhombic simple lattice like match boxes packed in a package. Well, I told Sommerfeld that I chose that subject in spite of his warning that he did not know how to tackle that problem and that he could not be of as much help, while the boundary problems were all more or less routine and he could tell me exactly what to do. So, I insisted on the problem which he did not know the solution of. How did Sommerfeld come to take this interest?

In Munich there were several famous professors at the time. Röntgen led a laboratory of his own and actually did not take much part in the discovery of X-ray interference; that is to say, when Laue, Friedrich and Knipping had done the experiments, he came over and looked at the diagrams they had, shook his head and said: well this is very beautiful but certainly it is not diffraction! He had tried himself, many years ago, to obtain diffraction, without success of course, for various reasons. There was also a mineralogist, Groth. Sommerfeld was a friend both of Schoenflies and Groth. Groth was his colleague in Munich and the two, Schoenflies and Groth, were very interested in crystal structure theory but they could not come to terms, they could not understand one another. Schönfliess did it from a mathematical point (p.578) of view—in fact he had been a student and a Privatdozent of mathematics and had worked out all the space groups and brought the manuscript to Felix Klein. Felix Klein looked at the manuscript and told him: what you are doing is crystallography. Felix Klein was a famous mathematician in Göttingen at the time, with very wide interests and an enormous survey of all branches of Mathematics, Physics and Technology. Schoenflies really looked at the space groups as something as operators which operate on something and that duplicated and multiplied, whereas Groth had a much more physical approach. He thought of the Bravais space lattices and he said a crystal consists of a Bravais lattice or rather of the groups which his friend and colleague Sohncke had established—the 65 or 66 groups which are not as general as the Schoenflies or Fedorov ones—and that each kind of atom forms one of what they called space groups. You see, in German, it is all Gitter, Kristall Gitter and there is not the difference between lattice and structure which has been introduced by the English. These atomic space groups are intercalated, they penetrate one another.

Laue made his discovery in1912, early 1912, at about Easter time. Itwas published. I was in Göttingen at the time and I heard of the experiment only when Sommerfeld went to Göttingen and gave a lecture to the Physical Society there at Whitsuntide, and then I sat down and discussed the formulae which I had developed in my thesis and that gave the sphere—the usual construction of which I am not really very proud of because it was obvious whereas the dynamical theory was not quite obvious! Well, I have to come back on my thesis, you see. If you have dipoles which are in this anisotropic array and if they swing this way, they will interact in a different way when they swing along the long distance of separation than if they swing normal to it, that is to say in the direction of the small separation. And therefore, you can expect that by the coupling of the dipoles a difference in refractive index will be produced, that is to say double refraction. To establish the theory, it was first of all necessary to find out how to sum the dipoles which issue from the various points of the lattice.

Each dipole emits a spherical wavelet and you have to sum them. This took about a year and when one day I had found a solution, I wanted to tell it to Sommerfeld. He said ‘wait a moment before you start, I have something to tell you’ and he showed me that he had found exactly the same mode, how to transform the sum of spherical waves into a sum of plane waves. The interesting point in this is that neither Sommerfeld nor I ever thought of Fourier transforms. In fact what was usual at the time was to speak of Fourier developments. Sommerfeld’s own thesis was on the development of arbitrary functions according to certain types of other functions to orthogonal sets of functions. But the idea of Fourier transforms came much later.

I remember when I first read the words Fourier transform. I was so horrified because I thought it was such bad English. I remember very well that this was quite new to me but I cannot remember the name of the author of the paper. I have asked several mathematicians and people interested in the history of mathematics to find out who invented the point of view of a transformation. Of course Hilbert had it, but that was in very abstract and highbrow mathematics and it hadn’t penetrated to us. This summation is now done in a few lines; if you know Fourier transform theory, there is nothing to it. At that time, I had to do it first by inventing complex integration; it (p.579) was quite a job and it took me quite long. But it didn’t solve the problem because if you want to find out the double refraction, you have to find out the field that is active on one particular dipole, that is to say, from the transformation of the sum of all the spherical waves you have to take out one spherical wave and see what remains, that is the so-called field of excitation. While for X-rays this does not play any role, for visible optics and for any static problem like for instance the anisotropy of the dielectric constant, it is most important to consider the field of excitation. The whole theory resulted in a formula expressing the double refraction in terms of the parameters of the crystal, the axial ratio etc. and the polarizability of the dipole of course.

When I was that far, I went to Groth and asked him: well, I had done this work, could you tell me of a crystal which is built up according to this simple orthorhombic lattice which I took as a model? And so Groth was silent for a while and, suddenly, his face lit up and he said: yes, there is one crystal where I can guarantee that it is built according to this simple lattice, that is anhydrite, a form of gypsum, without water. Anhydrite! I said, why? Well, it has three cleavage planes which are orthogonal to one another and which are fairly good. One is excellent and the others are still fairly good. So I calculated the double refraction for the anhydrite model and I can only say, it came out quite wrong, compared to the experiment! The only thing I could conclude is that the order of magnitude was correct, so it was correct to assume that by the influence of the coupling through radiation of the dipoles there would result double refraction of an acceptable order, but the sign was wrong. It was positive when it should be negative and it was 10 times bigger or twice smaller, so there was no correlation and I concluded that this was evidently because the structure was not the correct one!

Well, the structure of anhydrite was determined many years after and it has a face-centred lattice or I have forgotten what, but it is entirely different, has nothing to do with Groth’s prediction. That was the state of structure theory at the time before Laue. Nobody could tell what was the object of the repetition and how it came about. Well now, in doing my work, I came across a difficulty. I could find the radiation coupling of the dipoles and obtained, let us say, a determinant which contained the refractive index and that could be calculated from the determinant having to be zero. But in all this calculation, I never used any incident radiation and in the usual theory of dispersion, as it was called at the time, which had been started by Drude and other people, there always played a role the incident beam, the incident radiation. I could really not do anything with it. The whole system was a linear system so you could superimpose solutions and if you wanted the incident ray to continue in the crystal, this upset everything so I left out the incident ray and I did not like to do that, you see, being an inexperienced young graduate student. I thought, all these big venerable people have spoken of an incident radiation, so how could I leave it out?

The man to whom I went was Laue because he had a real feeling for fundamental things, but in this case he was not very attentive to what I told him. I began telling him when we left the University, and he had invited me for supper at his house, and so we went through the Änglische Garten and I began telling him of my work of (p.580) which he did not seem to know anything. He always asked what happens if you have very small waves, what are the distances of the atoms? I said the distances of the atoms I can’t tell, it depends on the structure you assume, whatever you put in, but what I know is that the atomic distances must be about 10 000 times smaller than the wavelength of the light I am considering and that is quite sufficient because it tells me that my approximations are correct. So Laue repeated his question and I got no answer from him on the problem on which I wanted to consult him, but I left out the incident wave and it was always a sore point in my mind how this could be justified. Actually, I think I have told Laue: well here is the transformation of these spherical wavelets into plane waves; this is a strict transformation, you can discuss it yourself if you just make the wavevector k long instead of very small as in the optical case. But he of course never did it. It would have been a geometrical theory, at all events a kinematical theory of the diffraction effects.

So, after a while, the war broke out—First World War by the way—and the X-ray installation which we had in Sommerfeld’s Institute was taken to an improvised hos-pital,2 and I went with the equipment and learned to do medical work. Then after a while I thought the war would end victoriously and I would come late, not seeing anything of it. So I was eager and besides I might have been drafted, so I was quite eager to see something and I applied for a position as Feld-Röntgen Mechaniker, field X-ray mechanic. I was ordered to Russia;3 at first to assist the building of the whole equipment in Berlin and then it was resent to Russia. They wouldn’t divulge where the Headquarters in Russia were but I was told, well, take a ticket to Königsberg, that’s good enough. So I took a ticket to Königsberg and, being of very non-military nature, I never reported, of course! I took a room in the best hotel, by making very long steps in front of several generals and other high brass. I got the last room and there I was. I went every day to the station to see whether my wagon, my X-ray wagon, had appeared, which came there by train, but I always got the answer ‘no’.

So, I had plenty of time and enjoyed Königsberg and the good marzipan and worked out the theory of reflection and refraction on a half crystal, that is to say a crystal which has a boundary, because no boundary had appeared in my theory. And, you see, this is the point which I think the optical people have never acknowledged, that the problem which usually was called theory of dispersion really consists of two entirely different parts. It is first of all the propagation of a set of waves or a single wave in the interior of the crystal to determine under which conditions such a wavefield can propagate unchanged through the crystal, and then a set of problems is how such an allowed field is generated in the crystal by the incident wave. So these were clearly separated I think for the first time in my thesis and it led in my thesis already to the statement that at the surface there must be generated waves which annihilate the incident wave, which is quite true. In the work I did in Königsberg, I proved this to be correct for the half crystal. At nearly the same time, Oseen in Stockholm also deduced the same (p.581) result from Maxwell’s equations but he could not really describe the field he wanted annihilated—it was more an assumption that this field would be annihilated. So this extinction, so-called Ewald–Oseen extinction theorem, has been very much discussed recently because you have now much more complicated theories of dispersion, with polarons and other things moving along and transformation of radiation into other forms of energy, and so it has become quite a subject of discussion again.

Finally, my money ran out and I phoned headquarters not knowing where they were. But then came the connection and I asked for some more money to be sent, and then I got a telephone back and I nearly fell flat on my back because I had never heard such cursing and such strong language. I should take the next train, it would be at 7 o’clock in the evening, I could take a train to Tilsitt and report the next morning at 7 o’clock at the Sanität Depot to which I belonged by formation. I was there punctually, again taking long steps in order to secure a room and, the next morning, I got up and went there and saw the sergeant who had telephoned me. It so happened that the chef of the formation was a physicist and we became very good friends. He took me out riding and we had a very nice time. It took another three weeks, I believe, in Tilsitt, until my wagon finally appeared. Well, then I was taken through Russian landscape which was beautiful—the first snow falling—and in a beautiful train, in a hospital train that had been established by the town of Bremen and was full of good wines and good cigars and a cook who was so fat that he could only go outside the train when it stopped in order to report what meals he would prepare for the next day. And in the little place where I finally ended, I always had a room for myself. My X-ray equipment was there and I had a table on which to lay the patients but you could also write on that table. There was not very much to be done because the fighting had stopped, and the only casualties were the casualties of the Russian winter and the old people who were transferred from our western front to the eastern front in order to recover.

So I had a quiet time, and I could go on with the theory of dispersion and that is where it originated. It was not quite easy and I would like to tell you some of the problems. At first I tried, quite in vain of course, to find out, with an incident ray, how many interferences would be produced. This is a problem of the theory of numbers and I think it is insoluble and so it took some time until I finally said: well, if I can’t find that answer, let us assume that there are a certain number of waves generated and from there on an important role was played by the reciprocity theorem. That is to say, if you have a ray falling in this direction and generating a secondary ray in another direction, then, if you let the incident ray take the direction of the secondary one, the primary ray will also be generated again. So this showed that you cannot consider a single wave but you have to consider a whole bundle of waves or, as we now say, a wave field propagating in the crystal and then, of course, the problem was what is the velocity of one of the rays, because if you fix that, then the others are automatically fixed. The whole thing is based on Lorentz’ theory of electrons which was then the outstanding theory. It was not very old, Lorentz’ book appeared I think in about 1908 or 1910.

This is just to illustrate the reciprocity theorem: if you have the origin there and you get interference things there, if you change the direction of the primary ray to (p.582) one of the ‘h’ rays, then of course the whole picture will repeat because the vectors all end in lattice points so that shows that you have to consider a bundle of waves instead of a single wave which is sufficient in Optics.

It took quite some hard work to get the ideas correct, I mean to treat the problem of what you assume to find the idea of the dispersion surface if you vary the direction of incidence without really getting very far away from the case where there are only a certain number of secondary rays split up. But all that can be solved and also I solved the problem of what happens at a boundary because again this problem of propagation of X-rays is divided up into one internal problem and of how its internal field is related to the incident outside field. So when all this work was ready, I wrote it up and sent it to Sommerfeld as a thesis for becoming a lecturer in Munich. Sommerfeld looked at it and said it seemed very nice work, but of course entirely unpractical, it would never find any application! But in spite of that, he agreed that it be handled for my Habilitation so, in 1917, I became Privatdozent and I had to prepare my theses. Now they were very disappointed that I did not come in uniform! but I preferred to be in mastic4 for the lecture I had to give—a trial lecture—and I also had to prepare the theses, which in the old days were to be defended against somebody who opposed them, the opponent. In my father’s day this was in Latin, but in my days it was in German. I would like to show you this excerpt: If the absorption of X-rays is similar to that of optical waves, and is caused by loss of energy in the vibrations of the dipoles, then, under certain conditions, X-rays should find no absorption in an absorbing crystal. Well, that was the origin. I had forgotten all about it until I found—40 years later—a copy of this thesis. By that time Borrmann had made his experiments, but it was obvious from my work that if there was a full resonance between the wavefield and the dipoles, then the dipoles were not required to vibrate at all and the wavefield could pass through unmolested as it were.

This dynamical theory was rigged up only for a simple lattice—no structure factor— but from the beginning I stressed to have n-beams. This is the difference with Darwin’s theory of which I must have known because I remember that I had given a report of Darwin’s theory in a colloquium but I had forgotten everything about it. For the simple reason that we considered the phenomenon as a phenomenon of diffraction (and we knew that, if you have a plane filled regularly with atoms, besides the specularly reflected ray there are a number of other rays), I did not believe in Darwin’s theory at all. Darwin had the argument that these other rays are not in phase and so need not be regarded but, of course, this has to be proved and it is not true in various cases where you have multiple reflections, so I always treated the case of n rays but I could not do it including the structure factor. This was done later and I gave a lecture at the Institut Henri Poincaré on that in 19325 where the full structure factor is there.

So I developed the theory during my lifetime from 1910 when I started work on the optical problem and I always took it up again… I am still interested in it!

(p.583)

Question (M. Kuriyama): I would like to know from you, personally, what was the interaction between you and Professor Bloch?

Answer (P. P. Ewald): Professor Bloch at the time was probably in the cradle.6 Actually, I think it was the Solid State people who invented the name Bloch wave. If anybody of the Solid State people had known about my work, they would have known that this was not new.

Notes:

(2) In 1914 (mentioned in Ewald Special Lecture, Kyoto, 1961, communicated by D. W. J. Cruickshank).

(3) In 1915 (mentioned in Ewald Special Lecture, Kyoto, 1961, communicated by D. W. J. Cruickshank).

(4) Civilian clothes.

(5) Published in 1938: Ewald, P. P. (1938). Optique cristalline (lumière et rayons X), interaction des atomes par rayonnement. Annales de l’Institut Henri Poincaré, VIII, 79–110 (in French).

(6) The actual reference to Bloch's work is: Bloch, F. (1928).Z Physik 52, 555 (Note by the present author).