Video: “Space Filling in Molecular Solids” Jack Dunitz
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Ken Hedberg: ...he was born, received his education in Scotland with a degree in 1947 from the University of Glasgow. After that time, he spent several visiting years, fellowship years, in the United States at the NIH, at Caltech, at the Royal Institution in London, and in Oxford.
In Oxford, he worked with Dorothy Hodgkin, who was a distinguished crystallographer and working on what at the time were incredibly complicated and difficult molecules. She later won the Nobel Prize for her work on Vitamin B-12. I happened, as Jack was, to be at the lecture she gave on Vitamin B-12 at the International Union of Crystallography meeting, the conclusion of which was greeted by an assembly of some several thousand people of a standing ovation.
I tell this story because it reflects directly on Jack's subsequent career. Dorothy Hodgkin talked to some people in Zurich who were enthralled by her account of this work and a very complicated organic molecule and asked her for suggestions of somebody that they might bring to their institute to carry on a similar kind of work. She suggested Jack's name. He went there in 1956 and has been there ever since until his retirement here in 1990.
Well, those of us who follow his career know that he has distinguished himself in all aspects of his research. He has published some 300 articles. He is a member of our National Academy. He's a fellow of the Royal Society. He's written a couple of books, which I will read to make sure that I have the titles right here. X-ray Analysis of the Structure of Organic Compounds - I believe, Jack, that grew out of the Baker Lectures, didn't it? At Cornell University, which he gave. Which incidentally is the same set of lectures that gave birth to the Nature of the Chemical Bond when Linus Pauling gave the lectures there in 1936. He also, Professor Dunitz, has written a book with a friend, co-authored by a friend of his, Edgar Heilbronner, called Reflections on Symmetry in Chemistry and Elsewhere.
Well, I have a couple of personal notes that I would just like to say. I've known him since he arrived at Caltech in 1948, where we became acquainted and became good friends. I've spent many pleasant afternoons talking science with Jack and many pleasant social evenings, as well, as the time went on. He and I were talking the other evening and we agreed that those years from 1948, and perhaps a year or two earlier, and on were what he and I would term "the golden years" at Caltech. They were the years that saw the birth of the alpha helix and a number of other things related to structural chemistry. So please help me wel-, join me in welcoming our speaker, Professor Dunitz. [3:43]
Jack Dunitz: Thank you.
Ken Hedberg: ...turn it on.
Jack Dunitz: [Whistles] Booming. Well it's nice to know everything is booming. Mmhmm. I think it's enough if I have this one on. Is that right? I can turn this one off just now because I think it's too much. Too much of a good thing. Yeah.
Thank you, Ken, for your introduction. You referred to the years in Caltech, late-'40s, and also to Pauling's book, The Nature of the Chemical Bond, and like many chemists of my generation, I first encountered the great man, Linus Magnus, as I liked to call him, through his writings. Sometime during my undergraduate career at Glasgow, I discovered The Nature of the Chemical Bond. Now, at that time, I had already chosen chemistry as my main subject, but I was having my doubts. I have to confess that I was wondering whether I hadn't perhaps made the wrong choice. There were, there was so much that had to be learned, so many reactions, so many experimental details, so few principles, and the subject seemed to be built from vast collections of facts, many of them weak facts.
Pauling's book came as a revelation to me. He set out to author an introduction to modern structural chemistry and he explained how the structures and energies of molecules could be discussed and understood in terms of a few simple principles. The emphasis was on structure. The essential first step in understanding chemical phenomena was to establish the atomic arrangements in substances of interest. And to try to understand chemical reactivity without this knowledge was a waste of time. That was what I needed to know. And that helped me to make up my mind that my future was to be in structural chemistry and for that, alone, I am grateful to Linus Pauling. Later on, I met Pauling when he came to Oxford as visiting professor and as I result of that I came to Caltech as a post-doc and had a marvelous time there.
The twentieth century, the one that has passed, has been the century of the chemical bond, as I think Ahmed told us this morning. There was Pauling's book, which influenced my generation, and even before that there was G.N. Lewis with his insights into electron pairing, and Heitler and London with their theory of the hydrogen molecule. And the emphasis was on the chemical bond. It's fair to say that by the end of the century, given the molecular formula, it was possible to explain, to obtain the shape, dimensions, and many properties of that molecule with the help of quantum chemistry. All you needed was a big, a big enough computer. This changed the perspective in chemistry because what one, it's an exaggeration to say that one didn't need to underst-, that one didn't need to study chemical bonding anymore, but many aspects of chemical bonding were understood and those which were not could often be derived by computer calculations.
The next level down from covalent bonds are weaker interactions between molecules, such as the hydrogen bond, which Pauling also played an important, pioneering role in understanding. Now the energy of a hydrogen bond is about order of magnitude five percent the energy of a strong covalent bond. And weaker still, now, are all the other weak, individually weak, intermolecular forces which hold the world together in condensed phases. I mean, what holds us together? Our skin is not held together by covalent or even by hydrogen bonds. It's held together - the membranes which holds us together, which holds sails together, which holds trees together - are built from weak interactions, weaker than hydrogen bonds often, but very, very many of them. And they are the basis for what has been called supramolecular chemistry and the crystal is the supra- supramolecule, par excellence. [10:27]
Now I promise now to talk about, since Ahmed talked about time, I decided to talk about space. How molecules occur in space. And, of course, you can ask what is space? What is time? What is life? And these questions cannot be answered in a way that's going to suit everybody. The answers depend on a point of view or an attitude, a focus of interest. And when we talk about space, there are also many, many spaces, it depends on what we are talking about. Whether we're talking about the space measured in light years or the space of the small particles which determine the structure of atoms or the space, our space, everyday space, the space of what Bohr called "the middle dimensions" or psychological space. Goodness knows, all sorts of spaces. And, I'm going to talk about the space in one of the intermediate regions, the space of the size of molecules. Now, for this purpose, it is true that a proper deep theory of molecular structure and supramolecular forces rest on a quantum mechanical basis but many aspects of it can be understood in quite simple terms and I shall try to cover some of these.
Now I am going to put this on and walk away from there... Turn this one on, it seems to work, put it in my pocket, and look at my slides. Now the first person who had the idea that you could explain the shapes and properties of microscopic objects in terms of small, fundamental particles in a detailed way was Johannes Kepler and he wrote a book, beginning of the seventeenth century, nearly four hundred years ago, called De Nive Sexangula, concerning hexagonal snow. And he had noticed that snow crystals, whatever their shape, there's all sorts of different shapes, but they always show six-fold symmetry, nearly exactly, and he was fascinated by this observation. And I think he was the first person to try to explain it. And he explained it by assuming that ice molecules were little spheres - they're not, the assumption is wrong, but it doesn't matter. He assumed that the molecules of ice are little spheres and that they pack so as to fill space as closely as possible as shown - everybody who has tried to stack, to put a lot of beer bottles or saucers or cups into a small space knows that that's how you pack them. Anybody who has observed how people pack oranges at the fruit store knows that the oranges are packed in these hexagonal arrays. Now, there are here two kinds, notice there are two kinds of triangles here. There are little, these kinds of triangles that point downwards and these kinds of triangles that point upwards in this picture. There are equal numbers of both kinds of triangles. [15:06]
And now to fill space - this is just two-dimensional layer - and to fill space, Kepler thought that the next layer of spheres will go in one of the sets of triangular holes. Say this set. And then this one, then this one. And the third layer can go then either on top of the first layer or on top of the second kind of layer. So there are two possible ways, two possible regular ways, of stacking oranges, ice molecules, spheres and these are called cubic closest packing and hexagonal closest packing.
Now it's an interesting thing. Two things, first of all, the idea that cubic closest packing represents the densest packing of spheres had been communicated, actually, in a letter to Kepler about ten years before he wrote this book by an English mathematician called Thomas Harriot. Thomas Harriot was given the job by Sir Walter Raleigh, who was the admiral in charge of Elizabeth's fleet and wanted to equip his ships to sail to the Spanish main to whatever they, to pirate, to fight. And to fight, he needed cannon balls. And on a ship there's, on the ships of those days, there was a limited space for cannon balls and he wanted to put as many cannon balls as possible into the limited space. So he asked Harriot two questions, if I have so much space, how many cannon balls can I put in there and what is the best way to pack cannon balls?
So Harriot came up with exactly the same solution and was very pleased with this solution and communicated it to Kepler. But see, that was in a letter, and Kepler published first. So, the discovery of this is generally attributed to Kepler. Until very recently, Kepler postulated that this was, that there is no, that this was the densest packing of spheres but he couldn't prove it. And in fact, proof was only obtained two years ago by a man called Thomas Hails and there's an internet, there's a website, which one can give you where it describes the proof of this postulate of Kepler but I warn you the proof is very, very long and difficult and depends on the use of computers.
Now, the one thing about Kepler's pattern is that it's a regular pattern and he really was suggesting that ice crystals are composed of a regularly repeating array of fundamental particles. And this was a great idea. You see Harriot didn't, Harriot was thinking about cannon balls and Kepler was thinking about what we would now call molecules. And since then, in the four hundred years, of course, between then and now, we know a great deal about molecules and crystals and it's become, it was established two hundred years ago that crystals, not only of ice but any sort of crystalline mineral, is obtained by having a regular repeating pattern of objects.
And here is just a way crystals and wallpaper patterns. Crystals are three dimensional wallpaper patterns. And here are some simple things and it shows that wallpaper patterns of this kind can only have certain kinds of symmetry. See? And for example, with pentagons, with five-fold symmetry, you can't make a wallpaper pattern of this kind, at least not one that will fill space. [19:53]
At the beginning, end of the last century, a Russian mathematician called Fyodorov did the problem in three dimensions and showed that there are only five shapes with which you can fill three-dimensional space completely by simple repetition and this is an obvious one. And this one, the hexagonal prism. This one here, the rhombic dodecahedron, interestingly enough, was discovered by Kepler again and this is just the object you get which, if you take that cubic closest packing and between every pair of centers of spheres you draw a plane and where, this is the intersection of all those planes. And finally, these two were new ones, which were discovered by Fyodorov. And Fyodorov showed that these are the only possibilities. There aren't any other ones.
Now apart from the scientists, of course, space filling, particularly in two dimensions, has been an object of great interest to artists for hundreds, hundreds of years. Here is a picture from the Alhambra in Granada. And you see that the Moorish artists of the fifteenth, fourteenth and fifteenth centuries were adept in filling up space in a beautifully symmetrical pattern. The Alhambra is full of hundreds of these patterns and anybody who has never had the experience of going there, I recommend, as one of the most exciting artistic experiences that it is possible to have, to go there.
In more recent times, we have the Dutch artist, Maurits Escher, who played much more playfully in all sorts of patterns of filling two-dimensional space by regular repetition of patterns, no longer simply geometric patterns but all sorts of fantastic shapes.
Now there's an interesting question arises, here is the unit of one of, of that Escher pattern that I just showed you, now just look at it. Is it obvious that with this pattern, that one can fill, that can one can tile the wall with this pattern. And, that is to say this little bird here has a property which it shares with a parallelogram or a regular hexagon. There's no apparent, I've looked for the word, there's no word for this property and it's not so obvious as to whether if I take a given figure like this, it's not so obvious to say "Aha! With this figure I can tile the plane." An even more difficult problem comes if I take another figure in which this is slightly deformed and I say, what percentage of space can I fill? To what degree can I cover the plane? 50%? 60%? 70%? Or what? [23:57]
Now I mentioned before that you can't tile the plane with five-fold symmetry and therefore people thought for many years that you can't make crystals with five-fold symmetry and it was only in the 1980's that Dan Schectman - Ahmed showed a picture on the frontispiece of the 1991, the 90th birthday Pauling book, which showed also a pattern like this. And this pattern here, if you look at it closely, has got ten-fold symmetry and that means it's also got five-fold symmetry and that is forbidden by the regular rules of crystallography because you can't make regular repeating patterns with a five-fold axis. And these patterns called, now called quasicrystals, Pauling, in his latter years, argued that this was not a new form of matter, that these were simply twins of types of structures that he could describe as normal crystals, but I think at the end of his life he considered that this was indeed something else and not a twin.
And the interesting thing, however, is that whereas the cubic closest packing and all these other packings I've made are made with one type of object, say the bird or one type of molecule, an ice crystal, an ice molecule, you can only make these quasicrystal patterns with two types of objects. You can't do it with a unique object. And this pattern here is called, this a type of such a pattern called a Penrose pattern and you see there are two kinds, there are two kinds of objects here. There are these rhombs shaped like that and there's a second type of rhomb and you've got to mix the two up in a very carefully arranged way to get this pattern, in which you have local five-fold symmetry but the pattern as a whole hasn't got any symmetry whatsoever. You can go out, take this little bit here and go out to forever and you'll never find exactly that environment repeating itself.
Now this raises problems in mathematics, deep problems. I mentioned that if you have an object which you cannot tile the plane with or fill three-dimensional space exactly, how much space does it fill? And this is one of the famous problems of the mathematics of the last century, it's still unsolved.
This is one of, in 1900 Hilbert, the great German mathematician, put forward twenty unsolved problems of mathematics which he upheld as problems for the twentieth century. And some of them have been solved, some of them have been solved for special cases, and some not. And he wants to know, he thinks that there's a problem important to number theory and possibly useful to physics and chemistry: How can one arrange most densely in space an infinite number of equal solids of given forms, spheres with given radii? We know the answer to that now. Or regular tetrahedral with given edges - how can one fit them together so that the ratio of the filled to unfilled space may be as great as possible? And that's what I'm going to be talking about really; not a mathematical solution to this problem, because it's never been obtained, but some facts about how well do molecules do. When you pack molecules in crystal, they tend to fill space as much as possible, how do they do it? [28:28]
Now, chemists portray molecules in all sorts of, all sorts of ways. Here's the typical, here's how mole-, here's how a chemist usually writes a molecule and in fact, if it's a hydrocarbon molecule, he even misses out these, these things, you see, the skeleton of this object here are supposed to be carbon atoms, a chemist understands that mostly, and the little spikes going outward are carbon-hydrogen bonds. And many chemists simply forget about the spikes sticking out, it's enough if you just show them that little polyhedral sketch in the middle, that's enough, they know that is a molecule, it's C12, it's a possible isomer of C12H12. Now you can also draw the same molecule here in which you represent the points by little spheres which somehow represents some property of the atom, so this is a carbon atom and this is a hydrogen atom. Or another way to do it, which is perhaps more useful for our purpose, is to draw, compute a space filling model in which characteristic radii, the atoms are now represented by little spheres and caps and it looks like that, it looks like a Michelin tire. [30:02]
Now, of course, this isn't correct. The electron density in this molecule, strictly speaking, goes to infinity, but for many purposes and for the purposes in which we are going to be interested in, you don't make too much of a mistake. If you bound the atoms by surfaces, in fact, Pauling was one of the first to do this, and he, in The Nature of the Chemical Bond, there is a table, page such and such, of so-called van der Waal's radii of atoms and this is this molecule drawn with Pauling's van der Waal's radii. Now if you accept that picture, you can calculate the volume of a molecule. It's the volume inside this set of boundary spheres.
Now I give you the following thought: Molecules are not polyhedra and they have curved boundary surfaces so they cannot pack to fill space completely. I didn't mention, but I should have mentioned, that Kepler's close packing of spheres, of course there are holes between the spheres, and in fact the ratio of the filled, the volume of the spheres to the total space is 74%. 0.74 is the densest sphere packing and it is found that in crystals the packing coefficient, the same ratio for molecules of arbitrary shape varies between about 0.65 and 0.8. It's roughly the same as for densest sphere packings.
If this ratio is less than about 0.6, you don't make a crystal at all, you get a liquid. Molecules have more space to move around. And if that packing coefficient becomes less than about 0.5, the liquid vaporizes, we get a gas. The cohesive forces between the molecules are now not strong enough to hold the molecules together in a condensed state. Now the problem is that there seem to be, see if molecules filled space with 100% efficiency and you have a given molecule, there would only be one way to fill space. For the Escher bird, there is only one way to pack those birds so as to fill space but once you have a shape which fills space to 70%, there's not a unique way to do it. There are hundreds, possibly hundreds, of ways to do it with very nearly the same density of packing and that's Hilbert's problem: How do you find out which is the densest packing? And it's also something that makes crystal structure prediction so difficult.
Now we found that when we took, went through the Cam-, there's a, in Cambridge there's a database called the Crystal, the Cambridge Structural Database, and it contains structural information for about 200,000 crystal structures which have been determined mostly in the last few years but going back solely to 1912. And we took 164 structures of hydrocarbons and found empirically that the highest packing coefficient was 0.78, the lowest was 0.64, the average was 0.72, so again very close to closest packing of spheres. This by the way was the molecule with the highest density. See? It's got two little cubes joined by a triple bond. And here is the one with the lowest one, that's this one here. And you see, it's low, it's got a lot of these things that chemists recognize these as these spikes as methyl groups. And the methyl groups on molecules are rather like umbrellas, and it's rather as if you have a lot of people in a room and everybody is holding in front of them an umbrella, not this way but this way. So obviously it's not easy to pack such molecules together. [35:05]
Now one of the rules which Pauling recognized about how to obtain between molecules the best fitting to fill space is contained in a little, small article, a one page article, 1940. [L. Pauling, M Delbrück. "The nature of the intermolecular forces operative in biological processes." Science, 1940, 92, 77-79.] And he says, "in order to achieve the maximum stability," and stability and density generally go together, the "molecules must have complementary surfaces, like die and coin." So it's just that a bump in one molecule fits into a gap in another molecule and he mentions here, this very interesting sentences where he is looking forward to the complementariness of the two strands of DNA, even though they hadn't been discovered yet. So here he is, he and Delbrück are explaining something which hasn't yet been discovered, which is rather good.
But you don't need to go to Pauling, exactly the same idea had been put forward 2000 years earlier because in the, the Roman scientist-poet wrote, I took this, I made this in all languages, you see, so that I could show it in all sorts of, but I'll do the English one here. It's rather fancy English. "Those things, whose textures fall so aptly contrary to one another that hollows fit solids, each in the one and the other, make the best joining." That's the Pauling-Delbrück complementariness 2000 years earlier.
Now as an example of this, here are two, see I've taken this arbitrary shape here, see, same shape, and if you fit them together in this pattern it corresponds to one of the crystallographic plan groups. And you see in this one, bump fits into hollow and in this one, bump fits against bump. So in this pattern, imagine this as a wallpaper pattern going forever. This one has regions where things are crowded together and empty spaces. This in exactly the same volume has bumps into hollows so the space is evenly distributed. Now when I looked through again for crystals which show this pattern, it turned out there were about 300 crystal structures with this pattern and one with this pattern. So this also in nature is a far more efficient way of packing objects than this one here. [38:27]
Nowadays, of course, when we sit in laboratory we don't just play with shapes or models or something like that, we try to imitate the physics of packing molecules together. And we imitate the physics by doing calculations with so-called pair-pair potential functions. We assume that between every pair of atoms in the solid, there is an interaction potential of a shape something like this where all of these letters are, can be adjusted to which particular kind of atoms we're talking about. And, you see if we have two atoms, of this particular kind, we have a postulated structure. We calculate all the distances and for each distance, here we look up what the energy is and simply in the computer add up all those energies together. And when we've added them all up we have an estimate of the packing energy of a crystal. We don't look at shapes anymore, now - does it fit nicely or not nicely. We can do the whole thing with our eyes closed and just look at the final answer and decide out of two patterns which one has the best packing. This [microphone] is really booming but perhaps it's a good effect behind to boom.
Well this energy, you see, there isn't just, there are very many groups of scientists in the world who are involved with this problem and there are all sorts of different atom-atom potentials which have been developed by different groups of workers and aimed at for different purposes but we ask at least that when you apply a set of such potentials to a known crystal structure that the forces resulting from the calculation, the calculated forces on all the atoms should be zero or nearly zero because if it weren't so the structure would be unstable and would move.
There are some problems, which I won't go into in great detail, about the fact that the calculations here refer to a collection of stationary atoms - this is atoms, this refers to a hypothetical structure which you would have at the absolute zero of temperature without zero point motion. And in real, in real crystals, actual crystals, the atoms are vibrating about their equilibrium positions. And in the simple version of using those potentials this motion part is not included. So we have the problem that the computed energies refer to atoms at rest but in the real crystal the atoms are vibrating. To allow vibrations, free space is allowed. That's why the structure of an actual crystal is not always the one in which the molecules are packed as densely as possible because if they were packed as densely as possible there would be less space for the atoms to vibrate. So there's a sort of balance between two things: Compress as much as possible to get as big a lattice energy as possible but on the other hand relax this condition a little bit to allow the molecules to vibrate. And we have recipes for deciding what that balance should be.
And I'll just mention for the chemists and physicists here how that is done. We calculate, in the computer, we calculate the vibrational entropy of the molecules inside the solid and this is done by using this formula, which is actually a century old nearly and due to Albert Einstein. And the calculations provide the necessary information to calculate the frequencies of motion for the molecules doing this, and doing that, about the three axes.
This is just the same as spectroscopists use to calculate force constants and frequencies but there is a little complication, more than a little complication in crystals, because if you look at a molecule and do spectroscopy, you get a set of force constants, a set of vibrations, and each, all of these vibrations are, as it were, going on at once. But each one has got a characteristic frequency at least as long as there's not too much anharmonisticity in, so-called, in the motion. But in a crystal it's different because now we've got a lot of vibrators but they're all coupled throughout the whole crystal and since there are, in each direction, in a crystal possibly something like ten to the power fifteen to twenty molecules, they're all coupled together. So this is an enormous number of coupled vibrations. [44:51]
And you won't understand this in the short time that this thing is shown but in the physics is, you have a lot of springs here, you see, a lot of atoms connected by springs but the springs are all now connected. And because of this you have a set of, continuous set, not just one frequency but of waves going along here with all sorts of frequencies. And we have recipes for how to calculate the frequency of these waves against their wavelengths and this is done by a particular piece of sophistry called sampling over the Brouillon zone. And if you do that properly you get a set of frequencies which correspond very well to observed frequencies which are seen in the spectrum of the crystal.
Now I'm nearly finished. I just want to mention one little exercise in this domain about different molecular shapes. Molecules containing the same atoms but the molecules have different shapes - here is a set of all these molecules, they are all C12H12. Here's this fellow which we saw before. And from these are, the crystal structure of all these compounds are known and since the crystal structure is known, the density of the crystal is known. And since the crystals all contain the same atoms, the higher the density, the higher, the more the molecules are, the more tightly the atoms are packed together. And you see the difference in density is quite considerable. This fellow here has a density, they're in grams per cubic centimeter. This, a gram of this weighs, sorry, one c.c. of this weighs 1.34 grams whereas of this it only weighs 1.1 grams. So a 20% difference but in a sense the same atoms.
Now can we understand why that is? Well, I don't know whether one understands it, one can take all these molecules and nowadays with a computer program compute for each of these molecules as many crystal structures as you can and look at their energies. And that's what we've been doing with Angelo Gavezzotti in Milano. These letters here just refer to the codes in the Cambridge database, you can just forget about them. These are all known crystal structures. Here are two molecules which we invented because we tried to imagine a molecule which would have the worst possible packing density, full of spikes and this is the one we decided, it's got to still be C12H12, and this was the imaginary C12H12 isomer that we thought would pack as badly as possible. And this is another one which we thought would be as badly as possible. And this molecule, as opposed to all the others, you may notice, is not superimposable on its mirror image and its called chiral. Now here are the six you see and here for each property we have six. So this number corresponds to that.
Now as far as the density is concerned, the density of the four known crystal structures, the calculated densities agreed quite well with the observed ones. And indeed the two molecules that we invented as being poor packers have very low densities. Partly, this is due to the fact that this molecule one is a very compact molecule - see its molecular volume is only 146 cubic Angstroms. Whereas this molecule one has got a much larger molecular volume. This number here refers to the packing coefficients. Once again you see all the packing coefficients are nearly the packing coefficients of spheres, but not quite. And indeed these two molecules here pack a little better than spheres and these two down here pack a little worse. [49:46]
And finally, the energies, the energies of the best crystal structure we could find and the interesting thing is here that although this molecule here, see 84.5, has a density much less than this one, its packing density is nearly as good. This one here has a very poor packing density. So that's how we amuse ourselves nowadays, we imagine molecules, take some of them, real molecules, some real crystal structures, and we also can take imaginary molecules, calculate their properties and we have a nice time comparing the results. I won't bother about the entropic slide.
And the, as you know, Pauling started off as a crystallographer and always throughout his whole life he was interested in crystallography, which during the course of his active scientific career changed enormously because at the beginning, crystal structures which were very, very difficult or impossible to analyze by the end of the century had become commonplace and every issue nowadays of Science or Nature will contain two or three crystal structures of proteins or enzymes or collections of proteins and collections of enzymes, collections of nucleic acids and proteins, and it's now become so that only the people who actually work on the problem or are interested in the problem can possibly read the papers which are published.
Nevertheless, crystallography, I'll give, this talk has had a few quotes and here is the last one I'm going to give you, it's not from Pauling, it's by Goethe, the translation is by mys- I translated, yeah? "Crystallography, considered as a science, gives rise to quite specific viewpoints. It is not productive, it exists only for itself, and has no consequences, especially now that one has encountered so many isomorphic substances." That's incredible. This was written in 1822 and the concept of isomorphous crystals was announced in 1820 so Goethe must have been reading the scientific literature of the time. Anyway, "as it is really nowhere useful, it has developed largely into itself," we're a community who talk to ourself. But "it does provide the intellect a certain limited satisfaction, and its details are so diverse that one can describe it as inexhaustible;" and "that is why it binds even first-rate people so firmly and so long."
Thank you very much. [53:16]
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