 The first lecture in the course is really going to be sort of a taster of many of the things that you will go into more detail in from the other lecturers. I want to give you a general overview of neutron scattering, why we use neutrons, what types of methods are available, how we produce neutrons, and then give you a conceptual overview and essentially some initial reference material for some small angle scattering and how that actually works. And then in the further lectures you will hear in more detail about some of the topics that I bring up today. Okay, so let's talk a little bit first about neutrons and x-rays. People are most familiar with x-rays simply from the fact that I think pretty much everybody at some point has had a medical x-ray of some kind, be it a dental one or something to do with the bone breakage or damage or something like that. But neutrons are perhaps less familiar. So if we start talking about the comparison between the two, it'll perhaps give you a more intuitive field for the properties of the neutron. So neutrons themselves were discovered in the 1920s by James Chadwick. They'd been predicted by Ernest Rutherford. He had seen in some gold foil experiments that there must have been something else other than the charged particles that were known at the time. And in 1923 Chadwick set up an experiment where he used a source of alpha particles hitting beryllium and passing through paraffin to then show that there were indeed particles that could only be these neutral subatomic particles that were christened neutrons. And Chadwick received the Nobel Prize in physics for this discovery. And here we have some comparisons between the properties of neutrons and x-rays. And I think the key items to consider are the fact that firstly the neutron has mass whereas the x-ray does not. Neutrons travel at non-relativistic speeds. Their velocities are slower than the speed of light whereas x-rays travel at the speed of light. Neutrons have a spin half and this becomes important about their interactions with atoms. Neutrons have a magnetic moment. This is very important and this is the basis of everything that Elizabeth would tell you about when she talks about magnetic small angle scattering. And then I'll say a little bit more about these energy ranges in a minute. So if we look at x-rays they were discovered 20 years earlier, 30 years earlier by Wilhelm Röntgen when he was studying cathode ray tubes. And he also got the Nobel Prize in physics for this award. In fact it was the first Nobel Prize in physics was awarded for the discovery of x-rays. So in general discovering fundamental particles or new forms of radiation is a good way to get a Nobel Prize if you're angling for one. We then need to talk a little bit about how neutrons and x-rays interact with atoms. So in the case of x-rays there are electromagnetic radiation so they interact with the charge cloud around the outside of an atom. Whereas neutrons being neutral are not affected by the charge cloud and can directly interact with the nucleus of an atom. The one case where in fact they do interact with the electron system is in magnetic scattering and you'll hear more about that later in the week. The interaction between neutrons and the nucleus is a complicated one determined by the nuclear energy levels and the exact structure of the nucleus. And as a result what we see is the fact that the interaction varies somewhat haphazardly across the periodic table. If we look at this plot on the left hand side what we see is the fact that here the neutron scattering length which I'll explain a bit later varies in this haphazard manner across the periodic table. It can vary wildly from one element to the next and even vary from one isotope to another and this is an important fact in the use of neutrons for material science. X-rays on the other hand because they interact with the electron cloud their interaction simply goes as the number of electrons so it just goes as the atomic number of the atom. And what this means is that adjacent nuclei in the periodic table have very similar X-ray scattering factors whereas for neutrons they may have very different ones and this provides a different degree of chemical sensitivity between the two probes where neutrons can actually often be much more chemically sensitive. On the right hand side we have a sort of pictorial representation of what we call the cross section and this is essentially you can think of it as how big an atom looks. You know if you imagine you're throwing a tennis ball at something the bigger the thing is the more likely you are to hit it. Here this is not a representation of the physical size of the nucleus but a representation of the strength of the interaction and thus how big an inverted commas it looks when you fire neutrons at it. And what we see here is that we see this variation being different for X-rays in green and neutrons in red and in fact what we can see is the fact that for instance for X-rays for hydrogen deuterium you have a situation where because they have the same number of electrons the cross section is identical whereas it turns out that for neutrons hydrogen has a much larger cross section than that of deuterium and in fact also if we look at its scattering length hydrogen is negative whereas deuterium is positive and this allows us to play some interesting games with the neutron refractive index that will allows us to change how our sample scatters the neutrons and control it. The other useful fact is the fact that we have relatively low cross sections for some key metallic elements such as aluminium and iron this means that we can probe through materials that would not be transparent if we were using X-rays so materials that X-rays will be absorbed by neutrons can often penetrate and there are materials of course that neutrons don't penetrate but but X-rays do such as boron for instance. There we go. If we now look at the properties of the neutrons and X-rays that we actually use experimentally of course neutrons can have a very wide range of energies as can X-ray photons but here I'm focusing on the the energy ranges that we actually use for experiments. In the case of neutrons it's on the order of milli electron volts to electron volts whereas for X-rays it's on the order of tens of electron volts to kilo electron volts. This you can see here that the X-rays we use have a significantly higher energy however if we look at what that means in terms of wavelength those are very similar wavelengths so what this means is that in the case of the structures we can probe we can probe the same length scales. We can probe the atomic to macroscopic length scales using both X-rays and neutrons but the neutrons we use have a much lower energy and this plays importantly into a whole range of techniques that I'll come to later called a neutron spectroscopy methods which don't really have any analogs in the X-ray world. It also means that the X-rays have the potential to do a lot more damage to your sample by depositing a lot more energy into it and then the other aspect is the question of source brightness. So neutron sources for the experiments we do today range from 10 to the 10 to 10 to the 14 neutrons per square centimeter per second per steradian per angstrom of bandwidth whereas photons from lab and synchrotron sources range from sort of 10 to the 6 to 10 to the 20 photons per square millimeter per second per milliradian per 0.1 per cent bandwidth and at first glance you might say well those are similar but of course I cheated by using different units. So if you can see here for the neutrons I chose per square centimeter and per steradian whereas for the photons it's per square millimeter and per milliradian and also the bandwidth is much narrower for the photon measurement and what this means is that modern X-ray sources are probably about 10 orders of magnitude brighter than modern neutron sources and this folds itself into the way we design experiments and the types of experiments we can potentially do and I'll talk more about sources a little bit later. So what does this all mean in reality? Well what it means is that neutrons and X-rays see things differently and if we just look at the macroscopic question of radiography which is probably what you're most familiar with the type of things you would do with medical X-rays we can do things similarly. So on the left here you can see this is actually a boiling coffee pot with neutron radiography and you can see we see right through the aluminium of the mocha pot that small cross section whereas the water containing coffee or the water in fact going through the coffee has a lot of hydrogen in it and thus is much less transparent because of that large hydrogen cross section. Another example here is we take the same item in this case a small motor and we can look at it with both X-rays and neutrons and the X-rays and neutrons are able to highlight different parts of the structure depending upon the relative interaction of the neutrons and X-rays with the materials. And last example here is a rifle cartridge so here you see with the X-rays that the brass and lead are strongly attenuating whereas for neutrons the brass and lead are actually relatively transparent but the gunpowder is similarly not so transparent and we can actually then see inside materials that would otherwise not be transparent and this actually has a lot of scientific applications using radiography in things such as looking at the motion of water in fuel cells or the motion of lithium in lithium ion batteries. So this means that we can actually have different views of the material depending upon whether we use neutrons or X-rays. We can also have different views of the material using just neutrons. I mentioned before that hydrogen and deuterium have opposite signs of the scattering length and what this means in effect is the fact that we can make their refractive indices cancel out and so here you can see on the left that Lola she has made sure that her refractive index is the same as her surroundings whereas Harold unfortunately has not and so the neutron monster will eat Harold and interact strongly with him but will not interact so strongly with Lola and what we do what we call this is we call this contrast variation and this is something that Adrian will talk about in his lecture but we make use of selective deuteration to hide and show different parts of our sample and we can also combine X-ray and neutron measurements on the same sample to get more information. All right so that is a very brief overview of why we might use neutrons and how they compare to X-rays and so the next thing I'm going to talk about is the different types of neutron sources. Neutrons typically are bound up in the nucleus of atoms and we have to get them out in order to be able to make use of them for our experiments to make beams of neutrons and there are a number of different ways that we can obtain free neutrons. The one perhaps that you are most familiar with is nuclear fission so here you take an element that readily decays with neutron release and then in particular when you bombard it with a further neutron with a neutron it will also split and so we can generate a chain reaction whereby we start one atom splitting and then you get a chain reaction as it produces three neutrons which can then go on to split other atoms and of course we need to control this in nuclear weapons. This is an uncontrolled release of energy whereas in nuclear reactors we put some absorbing materials in amongst the uranium in order to control the rate of that chain reaction so that it removes some of the neutrons and then what we do is we in a normal power reactor those neutrons will interact with material and generate heat. In the case of a research reactor what we want to do is let those neutrons out so we can use them. Another way of producing neutrons is nuclear fusion. This is of course what happens in stars but there are lots of work ongoing to try and have fusion sources on earth primarily for power production. Here what we do is we take something like deuterium and tritium and fuse them under high temperature and or high pressure in order to cause them to react and produce a neutron and a helium atom and this high energy neutron in a power reactor then goes on to interact with lithium which then produces a tritium atom which can be reused and heat which can be used for generating power. These are not really practical well as as neutron sources they're terribly complex and they don't produce very many very good yield of neutrons and the final primary way we produce neutrons is through spallation and the European spallation source which is our work is the source we're building here in London that will make use of this method it's in the name. Here we take a high energy pulsed proton beam and we fire that onto a heavy metal target and what this does is this destabilizes the nucleus and causes it to give off a huge range of subatomic particles including a lot of neutrons. The reason we use heavy metals is because they contain a lot of neutrons and this word spallation comes from geology where it means sort of chipping away at something. There are several working spallation sources in the world at the moment the one in Europe the primary one in Europe is the ISIS facility but also there's the Paul Sincu facility at the Paul Scherer Institute in Switzerland and the last way we can produce neutrons is essentially the same way that Chadwick did we can bombard a light element like beryllium have it undergo a nuclear reaction and produce neutrons. We don't use helium or alpha particles so much these days what we do do though is we take a proton beam but a lower energy proton beam than we use in spallation sources and bombard beryllium targets which then undergoes a reaction that produces neutrons. These are what we call low energy neutron sources or compact accelerated driven neutron sources and these are actually of the scale that could be built at a university so you could more or less have a laboratory scale neutron source I mean they're still fairly big you need a low energy proton accelerator but a number of sources around the world are being built with repurposed research accelerators at universities. So in terms of the high flux sources that you actually will likely mostly use for doing experiments the two types we have are reactors and spallation sources in both cases we have a central source of neutron production around which basically we put holes in the wall let the neutrons out and can then direct them to experiments. However if we look at the energies that are involved in both of those types of reactions the neutrons that are produced are in the mega electron volts to the giga electron volt range and I said earlier that the ones we want to use for experiments are actually in the milli electron volt range and so what we have to do is we need to lower the energy and so we use what we call a moderator and essentially what this is is this is a hydrogen containing material remember that large cross section I mentioned that we keep at a specific temperature and then by the interaction of the neutrons with those hydrogen nuclei they will interact, lose some energy and eventually end up with an energy distribution that is representative of the thermal energy in the moderator and so if we want lower energy neutrons we use a lower energy lower thermal energy source so we use something cold and this is usually something like liquid hydrogen or liquid deuterium or possibly solid methane. If we want slightly higher energy neutrons we use what we call a thermal source, thermal moderator and that's usually just water at room temperature so we basically pass the neutrons through a bucket of water and they rattle around inside and their energy goes down to something similar to the the Boltzmann distribution of energies in water and if we want slightly higher energy still so these are wavelengths on a fraction of an angstrom then we use what we call a hot source which actually we take graphite and heat it up to about a thousand Kelvin and are higher and so then this then adjusts the energy of the neutrons down to a energy distribution that's represented by that temperature and so all this is to say is that we have a way of producing neutrons from the source that have varying wavelengths and equivalent energies and so depending on the type of experiment we want to do we can make use of different neutron sources so where can you go and do such experiments you can see that there are actually quite a lot of sources around the world unfortunately some of these have now started to shut down the Canadians Neutron Beam Centre has closed Los Alamos National Lab has closed to external users the LLB in France has closed its reactor but they are working on a compact accelerator source and the Helmholtz Centre in Berlin has also closed in the recent years but there are many other sources you can go to they're distributed more or less everywhere there are sources in South America and in Africa as well as in Australia Asia but you can see there's a very high density of sources in Europe and as a result neutron scattering as a community is actually very strong in Europe and there is a large research base which was made in a natural source place to build a place like ESS so why are we building new neutron sources what's wrong with the old ones well as you heard some are closing they get old and nuclear reactors are getting less popular and at some point the refurbishment costs outweigh the benefits and the other reason is that we want to try and produce higher and higher fluxes of neutrons so I mentioned before that the source brightness for neutrons was many odds of magnitude below that of x-rays and many of the techniques we use with neutrons are limited by the number of neutrons per second we can actually get onto our sample and so we always want brighter and brighter sources and this plot shows the brightness of sources all the way from Chadwick's source up through reactors here and spallation sources here and you might be wondering why can't we make the reactor sources brighter and the reason for that is simply that I mentioned before that the neutrons interact and produce heat and at some point we can't we're generating more heat than we can take away in power reactors they produce an awful lot of heat they might have giga gigawatt power levels compared to the maybe 20 to 100 megawatt power levels from a research reactor and the way they achieve this in power reactors is that they make the cores bigger because all they're interested in is the total amount of heat produced however this doesn't increase the number of neutrons per unit volume of of of reactor core and that's what we need for doing our experiments faster we need brighter sources of neutrons not just more neutrons overall we need more neutrons coming out per unit volume and we've really reached a limit with nuclear reactor design in terms of how dense we can make the reactor cores while still extracting the heat and this is where spallation sources come in these allow us to because they produce many more neutrons for each proton we can put in we can put more and more protons in and get more and more neutrons out of course we still reach some limits and ESS one of the main challenges we have is whilst we're aiming to be the brightest neutron source in the world we are having challenges with removing the heat and so we have a very unique source design in order to allow us to get rid of the heat that's produced in the spallation process but this is in general we see that the the brightness of spallation sources is on the rise and there are some developments also now in the US with the second target station at SNS where they will be able to produce even brighter beams but for much shorter durations so an example of a neutron source spallation source is the isis source this is what we call a short pulse source so the pulse is hundreds of microseconds long and you the the source operates at 50 Hertz so you get 50 pulses a second it has a variety of moderators as we discussed and over 30 instruments that make use of those sources and so you can see here we have a linear accelerator that produces the protons these get put into a synchrotron to bunch them up and store them and then they get extracted and sent to two target stations around which the new neutron instruments are placed so it really is you know the neutrons are produced here they hit the moderator they go off in all directions and then we take whatever comes out of the hole and guide it to the instruments in terms of a sort of canonical reactor source the Institut La Langevan in Grenoval in France is probably the world's leading reactor neutron source it's a 58 megawatt reactor and was a post war project between France, Germany and the UK with also it has 12 scientific members including Sweden and this you can see here it has the reactor core with a series of moderators around it and then a number of instruments both close to the reactor and further out that are all viewing these sources of neutrons right so that's been a whirlwind tour of how we produce neutrons so I'm just going to finish up now by saying a little bit about what we do with them and in particular some of aspects of small angle scattering that you'll need to consider so there are a huge range of methods that can then take advantage of the fact that the neutrons interact with nuclei and are scattered or absorbed and they can allow us to look at the structure of materials all the way from the atomic scale using diffraction methods through the nanometer scale with sands and surface structures with reflectometry all the way into the macroscopic regime with imaging but I mentioned before also that neutrons have a relatively low energy and what this means is the fact that we also can perform experiments where we look at how the energy of the neutron changes when it interacts with the sample and these are spectroscopy techniques and these allow us to measure the way that atoms in the case of time of flight and crystal spectroscopy or molecules in the case of spinneco spectroscopy and length scales in between move and you'll see there's this diagonal correlation and this is because the shorter wavelength neutrons are good at measuring atomic structures but and they end up having energies that are useful for measuring atomic motions whereas the the longer wavelength neutrons are good for measuring small nanometer scale structures and these have energies that are also then useful for measuring motion on the nanometer length scale so motion of molecules and so what this means is we say that neutrons allow us to measure the geometry of motion because the length scale and energy are well matched so that we probe both the have the right energy probe the motions at the same time as we can probe the right length scale in terms of structural measurements there are x-ray equivalents of all of these neutron techniques in terms of spectroscopy and there aren't so many x-ray techniques that are similar and this is because of the much higher energy of the x-rays and so we simply aren't in the right energy regime to look at these very small energies that are related to the motion of atoms and molecules and there are other techniques that are similar so for instance dynamic light scattering and spinneco can often overlap in their regime there are methods like NMR and various other spectroscopy and other optical spectroscopies that can also align with some of these other methods so I mentioned all these different techniques and I mentioned also that we can measure the fact that the energy changes but what do we really measure in a neutron scattering experiment and fundamentally all neutron scattering experiments apart from perhaps radiography are the same we have an incident beam of neutrons that some have some energy and some wave vector they hit the sample and then they leave the sample with some other energy and some other wave vector and what we measure then is the change in energy and or the change in direction or change in wave vector of the of the neutrons and we measure that change in wave vector as a function of this variable we call q which is essentially just literally the the difference between the incoming and outgoing wave vectors in vectorial terms and if we look at how the intensity of scattering varies as a function of q it turns out that this is related to the Fourier transform of how the atoms and molecules are arranged in our sample this gives us correlations in space on the other hand we can also look at how the energy of the neutron changes and we we we talk about the energy transfer omega and if we look at the intensity of scattering as a function of omega this then turns out to be related to the Fourier transform of how stuff is arranged in time so we can look at correlations in time if you've done any dynamic light scattering it's very similar to the the autocorrelation functions we look at in dynamic light scattering so we can look at how the direction of a neutron beam changes and how its energy changes but there are different ways in which its energy can change so it could not change at all we could have a purely elastic interaction and this is primarily what we think of when we think of atomic diffraction so here we're getting no information about the motion of the atoms all we're seeing is the information about their positions if we have a crystalline material such as you know assault crystal or something like this they the atoms are not actually stationary at any real temperature what they are is that they are all vibrating and they tend to vibrate in collective modes we refer to as phonons but it means that in fact the these vibrations are both collective and also quantized and so when we do see an energy change in the neutron beam it ends up being peaked at very specific energies and this structure of peaks tells us about those collective motions of the atoms if we have something like a liquid or a gas however what we don't have this quantized discrete set of motions we have a Boltzmann distribution of of energies as I mentioned before with the moderators right we make use of that but that is then a broad distribution of energies and what we see then is that the energy of the neutron beam isn't really produced in a quantized way but is spread out in a distribution of energies and this is what we call quasi-elastic scattering and the distribution of this the width of this distribution it shape tells us about the distribution of energies and motions in the sample all right so I mentioned at the beginning that neutrons scatter from nuclei so what does that really mean essentially if we treat our neutron beam as an instant plane wave it comes in it scatters off the nucleus the nucleus is very small compared to the wavelength of the neutron beam and so we get a point like scattering effect with a spherical scattered wave where the the the equation of the wave is given here and here we see that the scattering length comes in and it represents the interaction of the neutron with the nucleus the sign of the of this is actually arbitrary but was chosen such that most elements are positive and we've seen before that they vary across the periodic table in this haphazard way but the most useful difference is between hydrogen and deuterium where one is negative and one is positive single nuclei are interesting but what happens if we look at multiple nuclei and also what is this cross section that I mentioned at the beginning with those little pictures of disks so it we define the cross section here as being the total number of neutrons that are scattered per second divided by the incident number of neutrons per second so here we have our instant neutron beam it interacts with the sample and it's scattered in some direction theta phi and into some solid angle ds also some sort of angle d omega and so we can measure in our experiment the differential cross section so we measure the differential of the cross section with respect to solid angle and then the total cross section as I say is given by the integral of the differential cross section over all angles and so when we take again our single nucleus we can calculate out based upon the properties of the incoming beam and the properties of the scattered beam what the scattering cross section is and so the differential cross section ends out ends up as being simply the square of the scattering length and so then the total cross section where we integrate over 4 pi steradians a full sphere ends up being 4 pi times the square of the scattering length this simple case however only holds when we have one isotope or one element that has zero nuclear spin present and the presence of multiple isotopes and multiple elements or non-zero spin actually leads us to having a coherent and incoherent part of the cross section where the coherent part is this bit and is provide structural information whereas the incoherent part describes the variations around the mean scattering length and is does not provide structural information okay so if we take more than one scatterer then basically what we do is we simply we sum up all of the scattered waves from all of the atoms in the sample right and modulated by the the distance between them and the differences in their scattering vectors we do that same calculation for the differential cross sections that he did before and we end up here with what is sort of the fundamental equation of one of the fundamental equations of scattering which is that the differential cross section as a function of q is simply related to the sum of all of the scattering lengths and their relative positions I'll briefly mention here coherent and incoherent scattering this is here for your reference but essentially if we work through where there are variations in scattering length across the sample then we end up with a situation where we have a component that is dependent purely upon the average scattering n squared and this is the coherent scattering term I mentioned before but here we end up with a term that is not does not have any dependence upon scattering angle and is simply related to the average of the square of the scattering length subtracted by the square of the average so basically it is the average deviation from from the mean and that has no information in it but does produce background in our scattering experiments all right so we saw before that to calculate the scattering or we just sum up the scattering lengths of all of the atoms and know where all the atoms are that is kind of tricky if you don't have an ordered structure so it doesn't repeat and you are interested actually in length scales that are longer than atomic distances so what we want is a bulk property that we can use to describe the scattering and the one we choose is what we call scattering density and it is exactly what it sounds it's the total scattering length divided by the volume and what this means is that in small angle scattering because we're looking at structures that are large enough we can make use of this instead of having to worry about all the individual atomic scattering lengths and the way this works is that if you imagine we have water we start by sitting on oxygen and just use the volume around oxygen oxygen has a positive scattering length if we increase the radius of our sampling volume then we include some hydrogen atoms hydrogen is negative so it takes it negative we make it a bit bigger we now include more oxygen it goes positive make it a bit bigger more hydrogen and so on until eventually we're sampling enough that we lose the details of the atomic structure and we're now simply looking at the the average of the molecular structure and so as long as we're looking on the sufficiently long length scales we can use scattering density rather than having to worry about where all the individual atoms are so we take that equation from before the differential scattering cross section and we integrate it using the scattering density distribution across the whole sample normalized by that total sample volume because we've done by sample volume and we end up with what we call the macroscopic cross section with big sigma in it and this is related to the distribution of scattering density in the sample and we call this is the called the Rayleigh Gantz equation and this is the sort of fundamental message I want you to take away from and that is that small angle scattering is a result of inhomogeneities in scattering than the scattering intensity the neutron beam knows nothing about your sample other than that the scattering density varies with in space it's up to you to turn that information into something about your structure and we see here this is now as you can see this is essentially now the Fourier transform of the scattering scattering density distribution in the sample so as a last point I just want to talk about sands versus sacks which is really neutrons versus x-rays both methods of measurement provide us with similar information we get structural information on the nanometer to micrometer length scale the difference is in how the radiation interacts with the sample in the case of neutrons they scatter from nuclei case of x-rays they scatter from electrons we have a variation of neutrons scattering length which varies from element to element and isotope to isotope in a way that is haphazard across the periodic table and then the x-ray form factor which is the equivalent for x-rays this is simply a linear dependence upon the number of electrons we rarely need to worry about the absorption of neutrons there are certain elements that do absorb neutrons but we don't mostly have to worry about them and whereas the absorption of x-rays is common and has to be considered in the analysis and this is an additional term in the scattering length density our flux of neutrons we talked about the brightness of neutron sources versus x-rays in general it limits the rate of data collection and means we need to have much larger beams whereas with x-rays we can make extremely fast measurements or low concentrations or look at very small samples with neutrons there's essentially no concern about damaging our sample with the beam whereas in the case of x-rays because of the high energy and strong interact and strong absorption then potentially we will be depositing a lot of energy in and we could damage our sample so in summary that was sort of a introduction to the why we use neutrons how we produce neutrons a very quick overview of how neutrons interact with matter and some of the key equations and concepts such as scattering length and scattering density that you need to know to be able to do small-angle scattering data analysis and perform small-angle scattering experiments so any questions or if you're fallen asleep I got a thumbs up good somebody's there excellent all right well if there are no questions on that then we have a half hour break now well 20 minutes I'm afraid because I ran over my apologies and at 10 30 Adrienne will start to give you more information on those things such as scap and density and contrast variation and how we compare sands and sacks so I'll I'll talk to you later and have a good break thanks