On the 30th of April 1852 thomas henry huxley one of my scientific heroes walked through that door to give his first fight Friday evening discourse his heart was beating like a sledgehammer as his mind he wrote to his wife later I now know what it feels like to be going to be hanged his nervousness I think was partly the fault of the audience he continued in his letter the audience is a most peculiar one at once the best and Worst in London such as tonight the best because you have all the first scientific men the
worst because you have a great number of fashionable ladies the only plan is to take a preferring subject and play at battle and shuttlecock with it so as to suit both well Huxley's outrageous division of the sexes is of course of its time we know now thanks to quantum mechanics that ladies can be fashionable and intelligent at the same time As can men but we're I agree with Huxley and I agree with him on many other topics as well is in the choice of a profound subject and I hope I have chosen a suitably deep
subject for tonight what I want to tell you about is the technique of x-ray crystallography a technique that has a very famous and intimate relationship and with this place and it's a technique we are celebrating the 100th anniversary of the invention of this method which has shown Us the world in unprecedented detail and we're celebrating the centenary this year now unfortunately it doesn't really have the pool or the appeal in the public domain and that I would like it's hidden from us and it's partly because the technique relies on a phenomenon that is not available
to us in our everyday lives we don't come across x-ray diffraction which is the physical principle on which the method relies but it does speak to a broader subject which Is humankind's with session with seeing things that are normally too small to see and that's something that actually has grabbed the human imagination for as long as we have been putting lenses together to make telescopes and microscopes and one of the most famous publications from the 1600s from 1665 and we have an original copy here is micrographia written by Robert Hooke who was a pioneer of
microscopy and this was a very popular book in its day it was a Best-seller peeps tweeted about it it is the most ingenious book I have ever read and the secret of Hook's success I think was that he included images of things that were familiar such as his famous flea even if they were largely unseen to most people and these astonishing detailed pictures of the horrific flea had delighted and enthralled the people who bought his book but the trouble with microscopy is that it has got limits it's fantastic in its Own way but there are
limits in a number of ways one is that you Jen with when you're limiting material with visible light you generally only see the surface you see the light that is reflected off the surface of the willit as with the flea here you can't see inside and also you are limited in the size of the things that you can see what if you want to see the molecules that the flea is made of and what I want to tell you tonight is about the use of x-rays in Order to solve both of those problems and I
hope in doing so to sort of set the ball rolling in a popularization of the technique because neither we're celebrating the centenary it is a it is a method that has really transformed all of the sciences it relies on a rather beautiful piece of physics and mathematics but it has informed not only those subjects but mineralogy chemistry and biology it has really and brought us an acquaintance with the world at the Molecular and an atomic level which was simply not appreciated and prior to the turn of the 20th century so I'm going to make you
work a little bit hard a couple of times doing this as I go through the theory but I am NOT going to delve into all the gory details I will spare you that ladies and gentlemen but I do want to tell you about how you use x-rays and how you use them in a very particular way now from the very beginning it was obvious that x-rays was a special kind Of light because it could see through matter and here you can see one the very first it wasn't quite a medical x-ray I think it was
more an experiment that run jinhoo discovered x-rays in 1895 did not on his own hand smartly but on that of his fashionable wife and what you can see is that the skin and soft-tissue is largely transparent and because matter is mostly transparent or semi-transparent to x-rays but the bone is slightly denser it's got calcium Atoms in it and so it cancer shadow and the ring let's give rent in a bit of credit and assume that it's gold which with an atomic number of 79 is very electron dense and so it's a scatterers x-rays and Casson
a darker shadow so what's happening here is that the electrons in the atoms and molecules of Anna Bertha's hand are scattering x-rays out of the beam and so it attenuates the transmitted beam and so you see the Shadow but rather than being a solid shadow we see that it's semi-transparent because the x-rays are very penetrating so let's think about the scattering that happened that's happening and that explains that x-ray pattern that x-ray shadow so x-rays we now know is a special kind of light it just has a very short wavelength and it effectively consists of
an oscillating electric and magnetic field here we're just showing the electric field and it's about to hit An electron and when an electric field hits an electron a charged particle it causes that electron to oscillate up and down and an oscillating electron will itself emit radiation of the same wavelength and so it emits x-rays and in fact almost in all directions are x-rays emitted and so the transmitted beam is attenuated somewhat because of that most of the x-ray is unaffected but a small part of the energy is reradiating scattered we say in all directions and
If we look at a more realistic representation of an atom we can see that what happens is the straight through Bream is attenuated as the x-ray is partially scattered by the electrons in the atom so what the x-ray is doing is penetrating and sampling all of that structure that's inside the atom now when we take medical x-rays we're only really looking at the transmitted beam and we ignore the scattered radiation but the technique of x-ray Crystallography is actually relies on the scattered radiation because it in itself contains information about the object that's doing the scattering
and that is the basis of the technique that I want to tell you about tonight and what this technique can do because of the penetrating power of the x-rays it shows us the interior structure of things and it does so not just at a fairly anatomical level we can actually begin to learn about the structures of Molecules and of atoms and that happens because of a very peculiar property of light of any kind of radiation when it interact with matter which is of the similar size to its own wavelength and we can get an understanding
of that that process that phenomenon of diffraction so what happens is when light scatters from an object that is about the same size of the wavelength it scatters in in many directions and we can get an idea of that by repeating classic optics Experiments that were done in the early part of the 19th century by the polymath at Thomas Young and I don't know how young managed to do these because I think he only had candles at his disposal but fortunately we have a laser so I would like to I would like to just show
you the principle of diffraction that occurs in order to give you an idea of the strange property of light when it interacts with matter so if I just blank the slides and we could have the lights Down a little bit we'll be able to see what happens so this is a very cheap laser and the beauty of laser light is that it gives us a very fine very bright and a very parallel beam and that beam is traveling in a dead straight line from the laser bouncing off the mirror course to the screen so we
see a fine spot but if we then take an object very simple object and this is just a single slit so there's a small gap here we of approximately the same size as the Wavelength of light we can see NEC let me get that right okay so you have to look a bit closely but what you see is we no longer have a simple spot and we don't have an image of the slit either we should just the slit is just a rectangular aperture but you see what has happened is that let the light is
spreading out at an angle okay not necessarily very much here but enough that you can see it and you can see there's a bright band in the middle and Then flanking it on either side if my laser holds up well there we go then you can see that there are fainter bands and on either side as well and what that tells us is two things one is that the the the light is bent it's scattered at an angle by the small object and that there's a pattern here that appears to be related to the particular
structure that's doing the scattering and so there's a relationship that if you understand the rules of optics we could Work backward from that pattern to figure out that we had a single slit here now let's see what happens when we put two slits into the beam so these are just two slits both the same width as the original slit in the the first slide and now we can see that there is a different pattern a little is similar to the first one so again there's a central broadband and it's flanked by fainter bands on the
other side but now each of those bands is actually split into a Series of dots so again that tells us that we've changed the structure here that's doing the scattering and that changes the diffraction pattern that we see on the wall but because the structure here is simply a duplicate of the first structure the overall pattern the envelope as aware of the intensity variation changes is just that we see different points of light and if we go from two slits nine to six then we see that again we get a similar overall Pattern a sort
of central bright band flanked by fainter bands on either side and again now they are divided into dots and the pattern of dots is exactly the same as with the two slits because the spacing between the six slits is the same as the spacing between the two slits and that's a point I want you to keep in mind because we will come back to talk about that when we talk about crystals and what crystals do because a Crystal is simply a repeating form of matter you have a structure that is repeated and we will be
using that to probe the structure of matter but the optics shows us at high that principle works so that's my diffraction but so I've mentioned crystals there and I showed you that with a repeating pattern of the slits was important for getting an interesting diffraction pattern from the on the screen that we could then start to think about in interpreting Although I haven't explained to you yet how you do that but before I do that we want to then actually grow some crystals of our own so if J Shan my assistant is going to come
here and so JN he's a novice crystallographer and so what J Shan is doing here is we have a a solution of Ed protein a purified protein called lysozyme which some of you may be familiar with and J Shan we have prefilled these small Wells with a solution that's got a high concentration of sodium chloride and an organic solvent called polyethylene glycol or peg and Jason is currently mixing some of that solution in with a drop of protein in this well and he's then going to seal the chamber and what's going to happen over the
course of lecture we hope is that because we have a high concentration here you have a lower vapor pressure above the reservoir and so water is Gradually moves through the vapor phase of the drop and into the reservoir and what that will do over a course of minutes that that will reduce the volume of the drop and it will increase the concentration of protein inside the drop to a point where we hope it starts to precipitate out of solution and when a protein precipitates out of solution what that means is that the molecules are starting
to stick together they interact with one Another rather than interact with the water molecules and what we hope is that they will aggregate together in a very regular array to give us a protein crystal so we are almost done with that I did say he was a novice I was going to do this myself I thought I'd be too nervous you know so but my hand looks okay the trouble is that's the hand I pipette with I believe that is a Tommy Cooper joke just to give proper attribution so We're going to put that on
one side we want to take it out of the bright lights in case of temperature and perturbs it but we will come back to that experiment later in the lecture but getting a little bit ahead of myself with crystals I've told you about x-rays and diffraction let's think about and how the two we're finally brought together so as I said x-rays were discovered in it in at 95 by rinchen and they were quite a puzzle at the beginning although We now know that they are a type of electromagnetic radiation and so there are waves at
the time there was a great debate are they where's or are they particles that really was wasn't known and many people spent a lot of time trying to investigate their properties in order to work it out and in Germany then a scientist called Max von Laue he thought that he was able would be able to crack the problem because he reasoned that inside crystals and in a crystal of Copper sulfate this is a beautiful blue crystal here it was well understood because of the regular phases that one sees in crystalline solids it was already presumed
that the atoms or molecules within them would be lined up in regular arrays and am now II figured that the wavelength of x rays was probably about the same size as the spacing between the atoms in crystals like copper sulfate and so he persuaded walter friedrich and paul nipping to do The experiment for him so I guess they were the gems of their day and what he did was he took a tube which generates x-rays and put it through fine slits in order to get a pencil beam as they called it then placed a crystal
here in their experiment and then pressed a photographic plate at the far side and they expose that for several hours and then developed it on tenterhooks in the darkroom and then produced one of the Most remarkable images of the 20th century now I know it doesn't look like much but it is a truly significant scientific result it does look like a horrible smudge maybe it's a Rorschach diagram I'm not quite sure but what you can see are two things one there is a diffraction pattern there it's not a very regular diffraction pattern not immediately apparent
but you can see that the beam has been split into different rays and so it's scattering so There's a point array here an array here and these are spread out from the middle so you are so the audience here is where the x-ray beam came from and this is the pattern so lamby's hypothesis was supported so it did suggest that x-rays are waves and that crystals could be used as a diffraction as a diffraction grating for them but as well as showing that there were waves because you have a pattern here what that tells you
is that the structure within the crystal because The beam is passed straight through and been scattered off at angles the structure of the crystal has imprinted itself some high on the pattern of scattering and so what you have in this pattern and when they move the the photographic plate back a bit they got a slightly better resolved separation of the spots but there's information in here about the structure of the crystal that they were analyzing now this wasn't the most regular pattern it turns out There were good reasons initially for choosing copper sulfate but although
it is a nice regular array the atoms are not arranged in copper sulfate in a very symmetric pattern so they then chose a crystal of zinc blende which is zinc sulphide and from the appearance of these beautiful crystals you can tell that it does look like the atoms are lined up in a cubic type of formation and that certainly turns out to be the case and if you take one of these Crystals and you oriented it properly in the beam then you get a much nicer pattern and you can see actually that the pattern of
scattering of the x-rays has a four-fold symmetry you can see this makes a sort of square that is hinting at fourfold symmetry which you would expect for a cube inside the crystal but now II tried to analyze this pattern and he was mostly able to solve it but he wasn't quite able to get the whole way and this is in 1912 so it was A major breakthrough it was big news in Europe but now he himself wasn't able to solve the structure wasn't able to figure out exactly how it is that you use x-ray diffraction
information in order to work out structures that fail and to the father-and-son team of William and Lawrence Bragg I they had William was already at that time a renowned physicist and had been working on the problem of x-rays and up to that point for the past 10 or 15 years and Was a world expert on that and so a norwegian colleague of his Lars regarde wrote to him because he'd been to Munich and he'd talked to Lowry and he sent a letter saying recently however certain new curious properties of x-rays have been discovered by dr.
Lowry in Munich and the letter was a long and detailed a kind of the experiments and Larry was even kind enough to give regard a photograph which he would then knew he was going to send to the Bragg so this Was a very nice example of scientific international cooperation and let this happened in the summer of 1912 and Lawrence Williams son had just completed his second degree at degree in physics in Cambridge he derly er and got a degree in mathematics when they were living in Adelaide in Australia and it was Lawrence really who was
able ultimately to crack the problem of figuring out how there is a relationship between the scattering pattern of the X-rays and the and the structure of the inside the crystal and it was or chant for him in a way there was a sort of coming together of events he just finished his physics degree so you had been learning about a electromagnetic radiation he'd been learning about the symmetry of crystals and he also was blessed with a father who was a world export world expert in x-rays and who had then received this letter they haven't actually
seen Lowery's and paper and Lawrence realized that the key to the problem was realizing that what's happening is that the x-rays are reflecting off planes of atoms in the crystal and so I'm going to make you work hard here by explaining to you a little bit of the theory so we are all familiar at least those of us who wash regularly with the laws of reflection so you know that the angle of incidence equals the angle of reflection for light impinging on a silver or Polished surface now that law of reflection applies equally although I
won't explain the details even if that surface is discontinuous say for example it was made up of a set of atoms but remember if we're now thinking about what x-rays do to atoms matter is mostly transparent to x-rays and so a large fraction of the energy will actually pass through certainly the top layer of atoms only a small fraction of it will be reflected most of the Energy more than 99% goes straight through and so that will interact them with the layers of atoms below that in the crystal and we have regular arrays of atoms
many many of them and each layer will then itself reflect a tiny fraction of the incident x-rays and what Lawrence realized was that if what he needed to do was to work out how much and the beam is scattered in this direction he needed to add up all of these rays now this diagram is getting a Little bit complicated but Lorentz had a gift I think for and simplifying things so let's just think about one of the Rays going through and you get a partial reflection of every single layer within the crystal from this law
of reflection now we also have to remember that what's happening here is a electromagnetic phenomenon so the x-rays are waves and so we have to think about what the waves are doing and to make it even simpler mathematically we can boil That down to just two is although let me just tell you so here the the scattered rays are what we call in phase and what that means is that the peaks are lined up and so all the peaks are lined up and all the troughs are lined up and when that happens when you add
those waves together you get a very big wave a bit away with a very big amplitude it oscillates massively so that's called constructive interference and so that's the best way that waves can add up Although there are other relationships too as we'll see but let's ply on and see how these relationships arise so the if we just think about what's happening in the reflections of adjacent rays so the top one and the one below so we have the incident ray comes here and goes away and then the lower ray hits the bottom layer and is
reflected off the the lower layer now what am so let me just get that so so what you see is that the lower ray has to travel a bit Further okay so it sort of falls out of step with the upper ray because it has a shorter path to go around it's like running in the outside lane of a on an athletics track as it where but as long as this extra distance traveled which indicated by the two black arrows on my diagram is equal to one wavelength or two wavelengths or three wavelengths or any
whole number of wavelengths then the waves will be back in step and so they will add up and you will get appreciable Scattering in this direction so Lawrence Bragg worked out by simple trigonometry that this extra distance from the spacing D and when you're reflecting at an angle theta the path difference is equal to 2 D sine theta and if that path difference 2 D sine theta is equal to a whole number of wavelengths so the wavelength is simply the distance between two adjacent Peaks then you will get an appreciable scattering in this direction and
that is Bragg's law now if You look at a slightly different angle and here we've just changed the angle of incidence and in this particular case the way that I've chosen it the the path difference is such that the lower wavelength is out of step by half a wavelength or a whole a whole odd number of half wavelengths so in this case the peak of the top ray lines up with the trough of the bottom ray and that means in this case that these two rays cancel one another out so you get no scattering Now
if you think about it you'll have the same relationship for the next pair of layers down and then the next pair after that and so you get no scattering in that direction and so what Bragg's law tells you is which directions you're going to get scattering in depending on the angle that you're looking at and the spacing which is an indication of the structure within the crystal now you might think well okay he's just showing us two particular special directions There but actually Bragg's law is a very severe law and it only had only allows
scattering if it is true so let me show you how that works out by a slightly more sophisticated example and I'm going to come to the front so I can see this with you as well so let's think about a case where it's coming in as an angle and the difference between adjacent layers is such that the path difference is only one point zero one wavelength it's a percent out so this ray this wave Is almost exactly in phase with the lower one near is damn it and if you then think about the next ray
up that is twice as far away from the first ray and so the path difference is doubled and in this case the path difference is two point zero two lambda which is again almost in phase with the the lower one so you kind of think well this is going to add up there's going to be quite an appreciable and a mind and the same for the third one it's only three percent is Not really too bad but by the time you get up to the 51st layer and remember that in any given crystal there will
be thousands upon thousands of layers if not millions upon millions when you get to the fifty first the path difference is 50.5 wavelengths and so this ray is half a wavelength out of step with the one from layer one and so those two will cancel out the diagram is a little bit misleading but they are close enough in space that you do get Interference here and so that the fifty first layer will cancel with the first one now what that means also then is that the scattering from 52 is going to be half a step
out with the scattering from layer two and so on up the stack of layers in the crystal so unless Bragg's law is satisfied and that is bang-on one two three or four wavelengths then you do not get any scattering and so Bragg's law places a severe restriction on the scattering of The plants but it also tells you interesting information about the spacing between those plants and also we've looked just looked at one horizontal set of plans in the crystal but there are many other ways of looking at crystals these are the horizontal plans but you
can equally imagine that the atoms line up in a completely different way and so we have an angled set of plans here but you can look at the same thing again and you have Another set of angled plans each with different spacings and what Lawrence realized was what's the spots in a diffraction pattern are simply all the various different reflections that are allowed by his law coming off the interior of the crystal and we can maybe get an idea of that just if we look in three dimensions so here is a atomic lattice but if
we sort of tilt it in three dimensions you can see that we see a horizontal set of plans there from the Way that the atoms line up but we can rotate it again and here we have a set of another set of plans which are sloping down from left to right and then one final one rotate in another direction and here you can see there's another set of plain sloping from right to left and so all of those sets of planes are inside the crystal and when an X rays come in to the crystal they
are bouncing off those sets of planes in all different directions and so when he Looked at a diffraction pattern let me get back to the slide when he looked at a diffraction pattern Lorenz realized that what he was seeing was each of these rays was one of the reflections of one set of plans and so it was telling him and the angle of the Ray which you could measure was telling him about the angle and the separation of the plans in the crystal and he realized that by if he could figure out where all the
angled plans where then the atoms would be Lying at the intersections of all those plans and so he would then be able to work out once he'd sold that puzzle he would be able to work out where the atoms where and so he was the first in that case then to solve the structure of a the atomic arrangement with inside a crystal and this is from zinc blende and this is Lawrence's interpretation of the pattern or his prediction of what the diffraction would look like once he had figured out the structure and as you can
See there's a very good correspondence and he did this just in the few months after the summer of 1912 so this was by December of that year and he'd gone back to Cambridge because he was working with JJ Thomson at the time but he was in regular correspondence with his father dear dad he writes and is sort of describing the experiment and it's a you know it's a nice sort of chatty letter sending him some photographs but he says are not you know some very bad prints From the photos and they're not very good but
you can see there's a very sort of nice statement about the his excitement at the achievement so if you can read the handwriting it's better than mine I have to say Larry's thing was equivalent to reflection but of course he didn't see it and it's great it's great fun getting it it's great fun getting it straight off isn't it you know so it's the young Bragg was he was 20 to 23 at this time and it was just You just get a real sense of his excitement and it was a real puzzle solving exercise for
him and he was jolly pleased with the with a result and quite right too so they went on certain the Bragg's father-and-son team were working together analyzing lots of crystal structures this is one of their own early pictures this is from us crystal of soda of potassium chloride beg your pardon which is a close Relation of sodium chloride and again at Lawrence was able to interpret the diffraction pattern and this is his own prediction from his from having worked out the structure you can see that he's annotated it and here sorry so you can see
that he has annotated it and then they also analyzed em sodium chloride which actually has a similar atomic arrangement but you can see that there are small differences in the pattern you can see you get many of the Spots in the same places but some of them change in intensity in darkness and and this was an early hint that it's not just the positions of the spots that are important but the intensities are as well now initially there had mostly just been working on the positions of the spots which told them about the angles of
the plans and it was by working out the angles of the plans that they could figure out where the atoms were but this was an early indication of a more Sophisticated analysis that was to follow it so this was the structure of sodium chloride that they worked out they published it in 1913 and it's even then although they worked harder and he wasn't entirely sure of himself he was on this rather slender and indirect evidence that I assigned the structure in a paper editor or society in 1913 fortunately few further investigation established its correctness now
it was a major breakthrough this was the first Time that people had seen the interior of matter had seen the atomic arrangements inside a piece of crystalline matter however it didn't please everybody and it took the chemists actually in particular a long time to get on board with crystallography or some of them at least so Bragg was writing as late as 1927 in sodium chloride there appears to be no molecules represented by sodium Tory the chemists had expected to see molecules Of an atom of sodium and an adverb flow rate and stuck together the Equality
of number of sodium and chloride atoms is arrived at by a chessboard pattern of these atoms it is a result of geometry and not a pairing of the atoms now a British chemist by the name of Henry Armstrong took exception to this he got rather cross and he wrote in pages of nature this statement is absurd to the nth degree not chemical Cricket chemistry is neither just nor geometry Of whatever x-ray physics may be you can just hear the disdain in his voice it's time that chemists took charge of chemistry and once more so he
was a little bit out of step perhaps with many of his colleagues because many chemists did adopt crystallography but it didn't faze the Bragg's they carried on working with crystallography an early structure that they solved the following year was the structure of diamond and that was followed and again ten years later by The structure of graphite now both of these are forms of carbon and this shows high understanding the atomic structure helps to us to understand the material properties of these so in diamond carbon is bonded in a three-dimensional pattern we have a tetrahedral arrangement
and you have strong bonds in all three directions whereas in graphite again a pure form of carbon the atoms are arrayed in two-dimensional hexagonal Nets and these Nets can slide pass one Another and that is why graphite is used as the lead in your pencil and is very soft and leaves a smudge and on a paper so understanding the intrinsic structure helps us to understand exactly what the material properties of these are so we're getting new insights into material science from this and the brides kept going further and further into this I solve more and
more structures mainly initially working on types of material that only naturally occur as crystal so I am pyrite calcite and quartz and I agree with him when he wrote and again this is early June 1914 we are scarcely guilty of overstatement if we say that lousy experiment and it was good of him to credit Larry has led to the development of a new science and I think that's absolutely true now until that point they had only been looking at relatively simple structures of types of matter that naturally occur as crystals such as the beautiful crystal
of rock Salt that we have here but and they have been largely relying simply on measuring the positions of the at the spots in the diffraction pattern in order to then solve the puzzle of how to work out the internal structure but again early on 1915 this is William Bragg gave a lecture at the Royal Society and identified the opportunities for advancing the technique for improving it and applying it to more complicated structures so in The simple cases we be considering the consideration of the crystal symmetry though unable themselves to determine the crystal structure comes
so near to doing so that a few plain hints given by the new methods that is the positions of the spots in the diffraction pattern have been sufficient for the completion of the task the exact positions the atoms are then now so they could work it out this is not the case with more complicated crystals and he realized That a more sophisticated mathematical approach he realized even early in 1915 that Fourier methods needed to be applied in order to bring in the information that was in the intensities of the spots in the diffraction pattern so
that they could start to analyze more complicated things now I'm not going to go through Fourier theory with you here tonight in gory detail but I do want to give you a type of graphical explanation for it so let's think about a more Complicated molecule so this is a complicated molecule I hope you'll agree it's a little bit more complicated than the two atoms of sodium chloride this is in fact a protein doesn't really matter and what it is but you can see that it's a complex set of bonded atoms and what I'm showing here
in this blue mesh is the electron density so we see exactly where the electrons are so this is a representation so let's think about how x-rays interact with a molecule and like This so an x-ray comes in from the side and here it comes and as we saw before with the electron and with a single atom the x-ray illuminates the whole of the molecule and we get scattering in every direction from every part of the molecule ok and what you'll notice here is that some of these scattered rays are beefier than others so this is
quite a strong one and it's coming from this particular direction we have quite a strong Oscillation here whereas this one is quite weedy and what that means is that this is scattering from a point for this quite high electron density strong electron density whereas here is a part of the structure which has weak electron density so although the scattering is a bit of a mess because it's in all directions what each scattered ray is doing its carrying off a little bit of information about The electron density at the point where it was scattered from now
we can't do mathematical analysis on things like this we like to break down the problem and take it step by step so let's simplify it now just by considering all the scattered rays in one particular direction there will be thousands of them I have only had time to draw four so again the the amplitude so the size of the oscillation varies according to the position and that will vary in Different positions are indicated by a vector in the structure now each of these waves is heading off in a particular direction and can be represented mathematically
by this function now this function looks horrendous okay I agree with you but it's actually not very difficult so as we said the amplitude depends on the electron density and this function here Rho of R tells you what is the electron density at position R in the molecule so That's a measure of the electron density if that's big that number then you'll get a big amplitude and this dxdyd said is just that the little volume of electron density that we have at this point now this exponential function looks a bit odd but that's just encoding
the fact that this is a wave it tells us about the wavelength it also depends on s which is a number that just varies with the angle at which we are thinking about so we're just thinking arbitrarily About this one angle but we generate the mathematical methods just for that direction but it also tells us about the phase and the phase of the wave depends on R and so the phase has got information about the position of the electrons doing the scattering as well and the phase is simply where is the position of the first
peak relative to some arbitrary origin so to calculate the total scattering in this direction we have to add these up and when we do Addition of funny terms like this we have to integrate so again looks horrible but all we're doing is just adding up the waves okay and when we add up the ways to get one particular wave that travels in that direction we can think about the whole contribution from the whole molecule emerging from some arbitrarily chosen point in the middle and then we have a wave which we describe as this function f
of s so it's just but this is just an Expression this means a way of scattered in the direction associated with F which we also know as theta so that's all the addition so that's that particular direction but we would also then to calculate we could sum up then think about all the waves scattered by all parts of the molecule in another direction but of course we've got to think about all possible directions because that will give us the most possible information so if we then were To sort of turn the detector around to face
us and see what happens then you would you might get something that looked a bit like this so there's it's not doesn't look very regular looks a bit messy but there's a pattern of sort of dark and light areas we've got strong scattering here medium strength scattering here quite weak scattering here quite weak scattering there but this is a very definite pattern and this is the diffraction pattern it's just Like the optical pattern that we saw from the laser at the beginning and it's got information in it about the structure of the molecule because it's
the x-rays that have come scatter from all points within the molecule in its three-dimensional interior and this mathematical formula that we've worked out is actually known as the Fourier transform after the French mathematician jean-baptiste Joseph Fourier what's interesting about three is that he lived And died before the x-ray was even discovered but his mathematical methods have turned out to have very general use in physics he won a prize in it from the French Academy of Sciences but they were a bit sniffy about it they give him the prize but they did note in the citation
the manner at which the author arrives his equations is not without difficulties and that his analysis for integrating them still leaves something to be desired but there you go that's The French for you Lord Kelvin and Ulsterman will say right on the money we said it is one of the most beautiful results of modern analysis now the beauty of freeze analysis when it's applied to x-ray crystallography is that we can work out what the scattering should look like from the electron density but if you can calculate the Fourier transform mathematically you can actually then also
simply do the inverse Fourier Transform so we've basically rearranged the equation you're allowed to do this according to the mathematicians so if you measure all the scattering you can work out from the maths what the electron density is and this is basically a mathematical description of the shape of the molecule and so this is the entire mathematical basis of x-ray crystallography that we use and still use in the modern era it's just the maths is all night done inside a Computer thank goodness so this allows us this mathematical technique would allow us to work out
the structure of a single molecule however we cannot work with single molecules they are really small so they're very hard to pick up and even if you could pick it up and put it in x-ray beam it would scatter so few photons so view x-rays that you wouldn't be able to measure it so the trick to get around that is to try and crystallize your molecule and so you Take your protein or whatever compound you're in says now analyzing and you try to grow a crystal of it it may not occur naturally but chemistry has
shown us that the final purification step is often crystallization so many compounds can crystallize and we now know that even many proteins can crystallize as well and so if you get a crystal you'll see that it behaves exactly like a crystal of sodium chloride or a salt all you have is a regular array not of atoms This time but of molecules but you can identify planes just in the same way as we did for sodium chloride and zinc blende and so Bragg's law still applies and so there are only certain directions in which you will
get diffraction but at least as we saw with the the slits when we put in six slits rather than two we got much brighter diffraction because we rely more light through and so the crystal is basically gives us an amplified signal That we can interpret so I wonder now can we have a look and see how our crystals of lysozyme are doing by the magic of ooh and it looks quite nice so this was a clear drop earlier but you can see here it's there's a sort of grittiness to it but this looks like a
whole cluster of little jewels and so these are crystals of lysozyme a protein that have grown in the last 20 minutes or so that I've been talking they're a little bit small but by modern standards They are perfectly adequate perfectly serviceable for doing x-ray diffraction so as long as you can crystallize it you can solve the structure of it and that was what the Bragg's realized back as early as 1915 so the application the three methods was allowed crystallographers allowed the Bragg's initially to use the positional information and the fact that the intensity of the
spots varied in different directions in order to Calculate the electron density map so they applied Fourier methods and this is one of the very first published by Lawrence Bragg this is from 1929 from diopside still a fairly simple structure but much more complicated than and sodium chloride and so this is a section through the electron density map you can see the contour so here's strong electron density here this is kind of medium and here is the structure that Bragg built into that electron density Map and solved for diopside and so through the 20s and 30s
they showed that they could apply the technique to more and more complex molecules and the chemists didn't all listen to Armstrong and this is beautiful work done by one of Britain's sort of most celebrated female scientist Dorothy Hodgkin or crowfoot as she was before her marriage this is the structure of penicillin and that was sold during World War Two and published and shortly afterwards and This she published in nineteen in the late 1950s is the structure of vitamin b12 a massive and molecule and was the biggest molecule to have been sold at that time so
a single molecule has over a hundred atoms and so the technique moved on in power thanks to the application of the a method which had been pioneered by the Bragg's and crystallography is now simply an embedded part of chemistry so as well as telling us about material Science it's now a regular routine to love chemistry the database has over half a million crystal structures in it and 40,000 new structures are added every single year so much for chemistry which isn't really my subject what about biology so again this started relatively early so it was in
the 20s and 30s that people started to think about how you could apply x-ray methods to look at biological problems biological proteins are much bigger in general and so it's Much more challenging and demanding but again brag brag father and son actually we're both very instrumental in inspiring and guiding people to tackle these problems one of the first to get involved was a chap called Bill Astbury who worked here at the Royal Institution as part of William Bragg's and group in the 20s and then moved to Leeds and here we see the work of one
of his PhD students from 1937 or 38 Florence Bell and this is actually one of the very First x-ray diffraction patterns of DNA of nucleic acid now you don't have a pattern of spots because this is not exactly a crystal that they're analyzing here this is a fiber but DNA has a very regular structure so it is crystal like and you can see even in this early pattern this is 1938 the typical sort of X pattern that is characteristic of the double helix that was eventually to emerge the whole story of the work on the
structure of DNA itself is Convoluted and tortuous and deserves a whole lecture in itself what I want to focus on tonight is the crystal the crystal side of biological crystallography and that was kind of kicked off by this chap JD Bernal an Irishman from Tipperary who worked at one of his first students was Dorothy Hodgkin or Dorothy crow food and burn now he started off as a theorist he wasn't very good with his hands initially but under William Bragg's Tutelage here at the Royal Institution he eventually mastered the technique and became a crystallographer of some
renown and so people would just send Krystal's and so a friend of his was traveling in Sweden in Swedberg slab he they had accidentally grown crystals of pepsin and their friend said I know someone who would give his eyes his eyes for those crystals and so he was allowed to take them away carry them in his coat pocket back to London or sort of back to Cambridge wear em for now was working at the time and they put them into the beam and they produced the very first x-ray diffraction pattern from a protein crystal it
must have been quite a moment unfortunately the photograph that they took is lost it was probably destroyed in a bomb during World War two however it probably looked something like this so this is a diffraction pattern from a crystal of hemoglobin which was taken by Max Perutz Who was also a student of burnell's just a few years later but you see a fairly similar must have been very similar to the pattern from pets inand you can see many many spots in here this is you get many more spots because the molecules are much bigger now
at that point they couldn't really analyze the structure they couldn't work it out because it was too complicated was even beyond the Fourier methods that they had at the time but they realized and this is 1934 That it was initially published by crowfoot and Burnell so Dorothy and that you know while sage as he was known because he was such a an intelligent and well-read person everybody called him sage and they there was two things they realized from this one was that because they had regular diffraction it meant that the protein molecule is of a
perfectly definite kind that meant that the protein molecule had a definite structure before that until that point There had been a lot of debate on this point it was thought that it was like a colloid or rather sort of loose Association of peptides but here they should because it diffracts then it is a definite structure and they further realized that they now had through this x-ray method the means of really getting to grips with what proteins look like and what they could do and although it was going to take quite some time before they could
realize this They they knew and certainly a spree and Burnell knew that they were on the cusp of I think a major breakthrough and there's a very nice quotation in a letter from a spree tuber now they were pals whereas free says if you know I do not make the most of biological crystallography we should have our respective bottoms kicked well I don't think they deserve to have their bottoms kicked they are unfortunately not as well known in the scientific world as They deserve to be there are known their names are well known inside the
crystallographic community that's partly I think because they didn't themselves solve any landmark structures but they both did in early work and inspired other people to go on to work on structures that wear themselves and landmark achievements asprey's initial work helped to inspire Morris Wilkins to get involved in analyzing DNA and Wilkins was very instrumental in and Pursuing the project to fruition for now he had mentored a Hodgkin and also people like Max Perutz but it took until 1959 before the very first protein structure was solved and what a moment that must have been when this
model appeared I won't offend the sensibilities of the fine ladies and gentleman of London by telling you what I think it looks like but the disappointment in peruses voice is palpable could the search for ultimate Truth really have revealed so hideous and visceral object party at this point was because the crystals they initially had weren't very good and they didn't scattered a high angle and so that limited the information that they could get and so the model really just shows the fold of the polypeptide chain that reveals the structure but you don't really get any
biological insight from a model like this and it's certainly this came six years after the structure of DNA had been solved but actually DNA had been solved on the basis of much sparser information and but it produced this beautiful double helical molecule it was just elegant in its simplicity and it immediately suggested the mechanism for that how genetic information is transmitted and copy to copy however within a couple of years the crystals improved and their analysis improved and soon they had a model of the protein structure that Really did have all the atoms in it
and was starting to give us real biological and mechanistic insights so this is myoglobin this is a protein that comes from a sperm whale it is an oxygen storage molecule and so it is the molecule in the muscles of the sperm wheel that allows this incredible beast to hold its breath for a very long time Perutz and so the initial myelin structure was sold by John Kendrew that was done in the Cavendish lab at Cambridge again under the watchful eye of Lawrence Bragg who was director of it at the time and Perutz himself working in
the same lab produced the structure of hemoglobin a few years later this is in the early 60s and this is a protein that's found in the blood of all mammals it's the oxygen transporter so it carries oxygen your blood from the lungs all the way through your tissues and and back into the lungs and immediately this was just the second molecule structure To be solved but what was remarkable about this was that when they overlaid the structure of myoglobin on and they saw that it was almost the same as one of the chains of hemoglobin
and myoglobin from a sperm whale the hemoglobin here was from a horse but it showed an evolutionary connection and this was the first time that people had seen evolution working at a structural level at a molecular level and this is a type of insight that we get from Structure biology all the timeline previously it had really just been looking at morphological similarities in the external appearance appearance of different animals and different species of plants and so on now we can look at and study evolution at the molecular level so this is another benefit of the
technique of crystallography so a later structure that lies as I'm was solved here in the Royal Institution in a group here involving DC David Phillips and Again this was not at the time when Lawrence Bragg was director and this sketch of the protein molecule is actually done by Lawrence's own hand he was quite a talented artist and what a pleasure it must have been for him and the is 1965 there abouts 50 years after he and his father had first sort of worked on this technique and solved the structure of just two atoms here was
the structure of lysozyme this is an enzyme This one is from chicken eggs it actually helps the chicken egg to fight off bacteria because it chews up the bacterial cell wall this is a modern representation of it and we can now see in atomic detail the orange molecule is a small segment that looks like a bacterial cell wall and so we can see how the enzyme works how it catalyzes pulling apart and this molecule so crystallography gives us incredible molecular atomic level insights into the Workings of biology and it has just gone from strength to
strength that result was from 1919 60s these days we now go to we have much better kit but we're basically doing the same thing we are still growing crystals we are still collecting x-ray diffraction patterns and we are still solving structures the kits a bit better so rather than a Crookes tube which is what the brand started out with we've got one of these this is a particle accelerator it Nestle's in the Oxfordshire countryside and you did cut this is the Diamonds light sources is the most expensive scientific facility that Britain has built in the
last 10 or 20 years and it is something that we should be very proud of it's an excellent world-class facility so now we just grow our crystals and we take them to Diamond and we fire x-rays at them and instead of the sort of early patterns like haemoglobin these this is now a Modern-day diffraction pattern and as we illuminate the crystal we can rotate it and instead of collecting debt on photographs we now have electronic detectors that capture the diffraction almost in real time and from all of these different images we can then work out
structure after structure so the technique really has matured really has grown up from sodium chloride which was just two atoms in 1912-1913 to lies as I'm which is probably a couple of Thousand atoms in the 60s we now have structures like this this is actually a sodium potassium ATPase so this is clear this is salt and this is a molecule sporter that sits in your cell membranes and regulates the influx and efflux of the sodium and potassium ions to maintain a healthy salt balance within the living cell so this is a gigantic structure however it's
not very big compared to this this is the ribosome and the ribosome is a monster so this is The ribosome from the yeast which is a eukaryotic cell it's a sophisticated type of cell which is the same type of organism that we are is just we are many celled organism and yeast is only a single celled autism but this is a truly gigantic structure Bragg's first structure had two atoms the ribosome has four hundred and four thousand seven hundred and fourteen atoms in this crystal structure it's truly a monster it might even be said to
be akin to the Monstrosity of hooks flea I wish that it would capture the public imagination in quite the same way but it's worth bearing in mind that this flea contains many millions of copies of a molecule just like this the ribosome is a enormous and powerful machine which helps us to decode the the genetic code and synthesizes proteins so from all of those techniques from x-ray crystallography applied to salt we have moved to brand-new territory and so we Are learning all the time about life and we're learning about it in a ways that are
brand-new and so if we now just finally look at the ribosome in the wild so to speak you can see where it takes its place inside the cell and this is a beautiful painting from a book by David Goodsell if I showed up here to give him the credit for it this shows the molecular interior of a cell this is bringing together all the knowledge we have about what molecules Are present and what structures they're likely to have we haven't yet sold all these structures but thanks to crystallography and we very soon will be able
to do so finally we are solving structures on earth but crystallography has moved beyond this planet and on the Curiosity rover that NASA sent to Mars last year there is an instrument that does x-ray diffraction and the robot arm can toss a sample of soil into this device and it Can take an x-ray diffraction pattern on Mars and so x-ray crystallography has moved off world so to speak and is now telling us that the solar Mars is a bit like Hawaii I don't think the climate is quite as nice and this again is just a
structure from this week's nature published on Wednesday another monster this is a polymer is that actually makes the RNA that is used to build the ribosome and so we are making connection after connection at the atomic level so I hope you will now agree with me that x-ray crystallography is one of the most powerful methods that science has produced in the 20th century and has truly allowed us to see the world in a completely different light you you
