Lecture 11: Radiation

Key Takeaways

so today we are starting a new topic and we're going to be looking at radiation or self emission from the plasmas in a way this is like the natural language of the plasma previously we have surrounded it with Magnetic probes we prodded it with lar probes we fired various beams of lasers microwaves through it but now we're looking at what plasma produces itself as a way of trying to diagnose what's going and it's important to not of course plasmas are not special in this sent so all objects will emit electromagnetic radiation humans of course mostly in the IR Pocus when they get red hot famously glow different colors uh and of course the universe in the form of things like stars and black holes but even in the completely empty parts of the universe we still have the cosmic microwave background left over from The Big Bang rattling around and so even something that is as cold as out of space still has some radiation associated with it in general if we have an object which is hotter we get higher energy photons out so we could say higher frequency or energy which is just of course H Bar times the frequency photons out of it and so we might expect in the case of plasmas which are usually pretty hot that we would be able to get some pretty high energy photons and indeed plasmas t a Spam the gamma all the way from the radio frequency up to xrays and even gamma R so we're going to be trying to talk about radiation that spans different energy bands varying by many many orders of magnitude and just like when we talked about radiation sorry when we talked about the different plasmas we worked on at the start list class we saw there was a huge diversity here we're also dealing with a huge diversity so we're going to try and use a framework that treats all of these in a similar way but of course there will be some subtle details uh and at the end of the day some things are going to look more like radio waves where maybe you need to think more about the wave equation because the wavelength of the radiation is on the order of the size of your plasma and some things like X-rays and gamma rays are going to be up where the wavelength is so small compared to the size of our plasma that we can use like a Ray treatment instead so just keep in mind that we have different conceptual Frameworks dealing with em radiation and we may need to modify it a little bit depending on what we're dealing with now the fact that plasmas are hot and emit all of these uh different types of radiation has two consequences first of all this can be a significant cooling term so for example the radiation May cool down our plasma you saw this particularly if you took Fusion Energy class and we remember that our brmr and cooling there was a significant loss mechanism in our zero power that balance but as well as being maybe pesky in the sense that they cool our nice hot plers down of course they also give us information so if we can study the Spectrum and the location and the temporal variation of this electromagnetic radiation we can get information on our plasma and this of course is our focus in this course because this is a diagnostic of course right so in this course we like radiation maybe in the previous course decided you didn't like radiation because he wants to do Fusion but here we like the radiation now in general the way that radiation moves around inside the plasma or indeed inside any fluid is not trivial right and we want to think about how that radiation is transported this is a whole Topic in itself that we'll only cover briefly here the topic of radiation transport the basic idea in r radiation transport is that you have some sort of plasma there'll be radiation emitted in one place and that will locally cool the plasma and that radiation may be em absorbed in another place which will locally heat the plasma but then that bit of plasma will also Reit and so the cycle continues and you can see straight away that in three dimensions in a in homog system this is going to be very very complicated but this effectively the transport of energy and this transport can be highly non-local so you may be used to thinking about heat transport in uh you know a system where we've got some sort of uh heat uh thermal conductivity uh and we look at the profusion of the Heat through the material that's a very local process the amount of heat that's traveling just depends on the local temperature gradients but here we could have a region which is emitting and it gets absorbed a very very long way away and because that radiation is traveling close to the speed of light this can be a very very fast process so this non-locality makes solving the full radiation transport problem extremely difficult um in reality if we want to solve radiation transport we often assume that this is a diffusive process and there are reasons why that assumption might be valid but there are also good reasons why it may not be valid in general so just the give you an overview that radiation transfer is complicated we're just be doing a simplified version of it here but radiation transport is extremely important for understanding uh how how plasmas work um for example the significant cooling but it's also very important for understanding how we get information from the plasmas because if you just have a camera sitting out here looking at the plasma and you just see some light coming out you really want to have some model that tells you where that light came from inside your plasma if you just think it's all coming from the surface you'll get one result if you think it's coming from Mod within the plasma you'll get a different result so it's very important to understand this radiation transport conure so we're going to start by having a look at radiation transport before we even start to talk about what the radiation is or how it's produced so even without knowing any of the details what's making all these radio waves and x-rays and gam rays and things like that we're going to look at a radiation transport framework which will then apply for all of the different wavelengths that we're working okay that's very high level any questions on that so far um Professor yes um when you say assume diffuse what what does that exactly mean sorry I was talking at the same time so the idea was that um most of the time when we're trying to solve radiation transport uh we can't uh use this non-local model because it's very very complicated uh and so we often make assumptions that our transport is diffusive I'm not really going into this in great deal of detail here but there are I can give some references later if you want for yeah great thank you the other thing I will say is that a lot of this radiation transport stuff as we'll find out uh is important when you have some sort of uh opacity in your system for sufficiently high energy x-rays in a sufficiently sparse plasma a toac then radiation transport is not necessarily very important and so you might think oh I'm a toac person I don't need to know this but we'll find out very very quickly that radiation transport is still incredibly important uh for the lower frequency waves like the electron cyron emission in a toomax so people still need to pay attention even if they think they're too good for radiation transport so let's have a look at what's going on here again we're going to have some sort of plasma and we're going to have some radiation coming into the plasma maybe this is radiation from another part of the plasma maybe we have generated a beam of x-rays or lasers or microwaves that we're using to shine through the plas doesn't really matter here this radiation is going to have an intensity I and we're going to parameterize the path of the radiation with this parameter s so we're going to call the point where the radiation enters the plasma S1 and then there is some path s through the plasma and we're interested in the properties of the radiation at Point S2 that's I at S2 when the radiation has exited the plasma here so what we want to know is how does I change along this path s now in general this path s could be curved because we've talked extensively about the fact that when you have changes in the refractive index you're going to have refraction of your red now although I have drawn it curved here a lot of the time I'm going to assume it's a straight line um so just watch out to that but if you want to solve the full thing you need to take this curvature into account okay and this quantity I that we're dealing with here this is a quantity which we can formly Define as the spectral Radiance I've used the symbol I which is often used for intensity because this is the symbol that's commonly used but we although people often refer to the spectral Radiance as the intensity they also use the word intensity to mean lots of different things and so if you go on Wikipedia on the radiosy Wikipedia page there's an incredible table that has very very Niche words for all thoughts of different quantities and this is the one that the table I always go to when I want to work out exactly what I'm talking about and the reason is that this word spectral Radiance here corresponds to one exact set of units which is the Watts the Jews per unit time being admitted being emitted through a area meter squared here through a solid angle per unit um per some spectral unit here so the spectral units could be something like HS it could be something like fules could be electron volts it could be meters the difference in this being depends how you're resolving your radiation spectrally so in terms of Herz that's where we're using like angular frequency and EV that's where we're using energy and meters that's where we're measuring it in weight so if we're talking about the spectral Radiance in terms of per I don't know gigahertz or per EV or per meters here of course if you put meters in this one then it starts to get complicated because this becomes a cubed and stuff like that and so people will actually write this as like watts per met squ per radian per meter just to remind themselves that they have an accident prob this it so what this means is that there is some amount of radiation that's going through uh some unit area that's meter squar and we've got whats going in this direction and that watts is subtended by some solid angle that's measured in stair Radiance and we're also resolving it in terms of like electron volts you this is a complicated quantity but it will correspond to your intuition of what intensity of light is like if you think about it for a little bit okay so let's just have an example here maybe our spectral Radiance and I may start calling this intensity every now and again initially has a spectrum and I do this in hertz here because this is what pson uses in his book maybe our Spectrum initially you know has some sort of complicated set of features spectral lines in the background stuff like that like this and by the time that it has traveled through the plasma there will be some emission inside the plasma the plasma will make more light which adds to our beam of light and there'll be some absorption inside the plasma gets rid of these so for example we might have absorption at these low frequencies they've been completely wiped out and then we will have some additional emission from inside the so here we would have a region where there was absorption and here we have a region where there was a mission so in general we want to come up with some formulas that will tell us how for a given amount absorption a given amount of emission in different parts of the plasma how the Spectra will change from one point to the other and this is the radiation transport equation which is the very fundamental equation in this field the radiation transport equation says that the change in spectral Radiance along our path that we parameterized by S is simply equal to this quantity J which is the emissivity minus the intensity or the spectral radius times by another quantity Alpha which is the opacity the important thing about the equation as we've written it here is that each of these quantities are functions of frequency and effectively this equation is linear in a frequency in the sense that we can solve the equations separately to each frequency so whatever is happening down the gahz range has nothing to do what's happening up in the X-ray range this is not necessarily true in order to make this assumption we have to neglect interesting processes like uh fluoresence where we might excite paral plasma of one frequency and get back lines of different frequencies but in order to be able to solve this equation easily we're going to neglect that so this is an assumption that we've made and I'll just tell you what the units of all these things are you can work them out yourself but the units of opacity very clearly it has units of inverse length scale and the units of emissivity are still watts per scar radian per Hertz but now it's not per meter squar but per meter cub and this reflects the fact that the emissivity is a measure of the light coming out from a small volume in every direction like this the per staran means that if you integrate over four Pi stents you get the total radiated power per cubic meter our MS and opacity are also functions of s right yes I've I've suppressed the s in here but you're quite right like as we go through the plasma these will in general change yeah these are properties of the material could be a plasma it could be a lump of Cloudy glass we haven't actually decided or said anything about them and we won't learn how to calculate these for some time you just have to take my word for it that you can calculate the Amity and the opacity you've come across some of this stuff before if you've done like black body radiation so um but we're just going to leave it like that at the moment okay any questions yes SE does this equation if you include temperature dependence of the inity andity in this equation be used to understand eating of plasma if you're thinking about maybe you're injecting waves or something so the question was can this equation be used to understand heating of the plasma um yeah effectively this term is the amount of energy lost from the beam uh per unit length and so you could add that energy into your plasma so you could couple this to an energy equation in your plasma but this itself is not an energy equation for the plasma this is an energy equation effectively for the energy in the radiation I see okay yeah yeah but if you wanted to do full radiation transport you would couple this to your energy equation yeah other questions okay it's very convenient to introduce a slightly mysterious but important quantity called the optical depth and the optical depth which is still a function of frequency and I'm just going to keep trying to write these as long as possible to remind you functions of frequency this is the quantity which is defined as the integral of the opacity along some section of path and just written like this this is an indefinite integral we will come up with some limits to it later on so at the moment it doesn't have a huge meaning but the important thing about it is that it depends on the actual path s and frequency so the optical depth can be very different than x-rays than for microwaves so again very important to realize that the reason we introduced this is that if you stare at this definition and this equation you might be able to convince yourselves that we can then rewrite the equation as d i is equal to J upon Alpha minus I and this is rather a nice equation because we're used to equations in form D to equals minus I these are sort of exponential growth exponential decay equations and we're also used to adding an extra term in here which is like a Source ter so what this equation says is that the intensity is going to drop off along the path parameterized by S which is now sitting inside C but the intensity is going to increase with a parameter that looks like J upon Alpha and I'll show you how we're going to use that in a second just label explicitly this this looks like a function so now we can go back to this problem and we can solve this equation so we can solve for S1 to S2 like that and we're going to solve it by integrating up with with respect to tow here and we're going to find that the intensity at S2 on the other side of the plasma is going to be the intensity that we started out with at S1 Times by a factor of to one minus to 2 so this is an attenuation that takes into account any absorption along the path and then along with the initial radiation we started with we're going to have a second term which corresponds to the radiation we've picked up through the plasma du to the enity and that's going to be the integral from S1 to S2 our emissivity now explicitly parameterized as a function of s as someone asked already all of the JS and alas our functions of s but now I'm making explicit here e to the minus L2 DS what this exponential here is doing is saying yeah okay you're adding radiation but that radiation may also be absorbed if that radiation is emitted very early on near S1 it's going to be absorbed a lot as it goes through the plasma if the radiation is emitted late along the path close to S2 it's not going to be absorbed very much so it matters where along the path the radiation is actually emitted so let me just label these terms here so this is the original uh I'm going to call it spectral Radiance right so I don't accidentally say intensity so the original spectral Radiance this is the optical depth from S1 to S2 this is the emission along the path s and this is the depth from s to S2 where s is wherever along the path this radiation has been emitted this is a complicated equation and I'm now going to give you two simple examples to try and build up your intuition for what this equation is saying if you don't think it's complicated well L um I think it's quite a complication equ especially what these tow are doing here any questions on it though before we keep going yeah um so for the emission along s term um if we're wondering about what's you know appearing at the at the point S2 I'm guess some confused why we don't care about the the angle that that ised from right if it's emitted back along the path then that right it's not heading towards S2 you know what I mean like I guess where is that dependence there right so the question was where is like the direction of the emission uh featuring into this equation and the question is because when we're talking about the emissivity we said that it's into some uh solid angle so we could imagine that this emissivity has like an angle with respect to the magnetic field right and that would be a perfectly reasonable thing to think about and indeed this is where I've started slipping into a sort of straight line kind of picture and also like an isotropic emission kind of picture if you want to do this properly all of these things that are scaler start to become vector or you know even worse some sort of horrific tensor and things like that and you have to account for all of that properly so just to try and simplify this and can do that but you're right if you're dealing with a mission that is an isotropic like electron cyron emission that might be very important yeah another question from shop an integral over the yeah a little confused how like how inside the exponential inside the integral is this a sub integral to be evaluated before you evaluate true integral uh I think there needs to be some f primes and things like that in order to do this you're certainly going to evaluate this Tow and it's going to be evaluated based on whereabouts in this integral s you are so as you're incrementing s here this tow is going to be evaluated from for example whatever is halfway between s one.5 to S of two where the end point of this Tower is fixed already um I don't think I gave very good explanation of that I think probably if I went back and did this more rigorously I put some more primes on this we have integral okay other questions yeah I see Nicol the that the radiation takes through the plasma is going to depend on the radiation itself right so it's going to depend on what kind of radiation it is uh yeah could be so if you're a different frequency you might have a different refractive index so you'd have a different bending yes exactly but in this equation the bth kind of predetermined so it doesn't like we're saying that if we know that the path is this this is how we would Cate right but the pa the the path of the radiation although it does depend on a frequency doesn't depend on the intensity necessarily and so you can for a certain frequency you can trace out a ray through your plasma you'll know what trajectory it is and then you can go back and solve the radiation transport equation along that path and all the emissivity and opacity figures for that particular frequency that we yes at the moment we're splitting it up so you could solve this for all any different frequencies you want and they don't interact in this simple model here yeah okay um when you're answering Sean's question I'm a little confused about how the like s 1.5 Works into the balance of the to integral um you know like what is if you're providing with some s integrate up what's the starting s or if you're providing it starting s what's your ending s in that to integral so in this integral it's an indefinite integral so we not provided any limits what all at all on this the limits only come when you start specifying the end points so for example here we completely specified both the start and the ends to one and to two and that integral is going to look like this is going to just look like e to the integral of alpha DS from S1 2 S2 this is this is equivalent these two are equivalent here in this case for Town two at the moment what we've got here we don't really have the lower bound so we have e to the integral of s to S2 of alpha DS like that where the actual s that we put in the bottom here depends on whereabouts and evaluating the syal we actually are maybe that also might answer Sean's question I don't know okay good thank you for teasing that out so yeah okay U just any questions online okay n is it possible that radiation emitted from outside of that line would end up kind of joining that same rate yes it could do and we not in in this case no what we're going to do in an awful lot of this is assume that the radiation is moving in straight lines in which case it won't cross um the reason is because this is already complicated enough to solve without that if you need to solve it with that then you do but if you're dealing with like if you're dealing with something like an x-ray going through a toac plasma the we are so far away from the critical density with an x-ray that the refractive index is unity and it doesn't change and so therefore they do go in straight lines but you're absolutely right you know that you could imagine all sorts of radiation coming through Crossing through this point going back to an earlier question that radiation could heat up a bit of the plasma and then that plasma could then emit in a way that you didn't expect before by changing youru so radiation transport is uh is extremely complicated even without magnetic fields and then when you put the magnetic fields in the whole thing becomes much worse so okay any other questions I will give some examples of this equation so hopefully it will begin to make more sense but okay let's keep going on to that good so let's consider a really simple system where we have some radiation coming in here we have a plasma which I'm just going to split up into two regions here these regions are actually going to be identical but I want to show you how the intensity changes in each of these regions and then we've got some radiation coming back out here so uh in this plasma we're going to have and opacity which is just simply some amount of opacity at a single frequency new1 and we're going to have some emissivity which is simply some amount of emission at a different frequency new2 like this and my initial radiation that I'm going to put through here is just going to have some amount of a mission at new one here so I've really simplified this we effectively we are solving the radiation transport for all frequencies it's just that I come up with a problem where there are two frequencies right so this is much easier you can come up with something arbit more complicated than this if you want to I just want to point out you're going to find out soon that these two uh emissivity and opacity are not compatible in fact there's a really strong thermodynamic link between the two so don't shout at me there this is just to make life easier for you right now but it turns out that you couldn't physically have a pattern like this we'll talk about that a little bit okay so what does this look like as we go through the plasma so at this first step here we're going to absorb some amount of the initial radiation which was that new one right so our radiation is going to go down but we're going to pick up some amount of radiation at new2 because the plasma is emitted at this next step here this process continues new one goes down new two goes up and then finally as we exit the plasma we might end up with a system where new1 is very very small and new2 is very very large so effectively as we've gone through the plasma we have been absorbing that new1 the size of that line has gone down and we've been admitting that new2 the size of that line has gone up here so that's a cartoon example it turns out that for this approximation that the plasma is homogeneous then we can solve this equation analytically as well um professor yeah so just to clarify um the absorption is throughout both sections of the plasma yeah I I split it into I split into two sections so I could just draw the intensity in two places but the plasma has uniform properties throughout so yeah so these this applies to both sections and this applies to both sections like that yeah good question I thank you okay any other questions okay um these times I wish had more blls really want that equation yeah have we um assumed anything about energy conservation in this equation or could your emission like have more energy out than what the I guess so the question was have we are we conserving energy in this equation so explicitly we are not a conserving energy in this equation the plasma is the source and the sink of the energy any radiation we lose goes to heating the plasma any radiation we gain goes to cooling the plasma that's why you would need to couple an energy equation into this to do it properly okay yeah okay so put that equation there so if we have a system which is now homogeneous so that along our path s this quantity J upon Alpha doesn't change so again this is just the homogenous condition we can solve this analytically and we get that the intensity at Point S2 is equal to the intensity of Point S1 attenuated by a factor e minus Cal to one which is as we said before the integral from L to one is equal to the integral from S1 to S2 of alpha DS so this is just an exponential damping Factor on the intensity this is what you would expect um for some sort of constant opacity through your system the further you go uh the more that initial signal is going to be down and then we're also going to have a term J upon Alpha 1us eus T 2 one same again here okay and when we look at this equation we can see that there's a strong dependence on T and so we want to identify two limit which have names one of which is this T to one the optical depth is much much less than one and the other one is to 2 one is much much more than one and these are called optically thin and optically thin in these limits this equation redu es to either is2 is equal to I S1 or I S2 is equal to J on Alpha here optically thin corresponds to the radiation streaming through with an being absorbed optically thick corresponds to the radiation being very very strongly absorbed so in the optically thick case we have no information about the initial intensity in the optically thin case uh we only have information about the initial intensity we don't have any added emission here so people may also call these transparent and opaque and remember this is a strong function of frequency and so you can be optically thin or transparent to X-rays and optically thick or opaque to the first and second harmonics of the electron cyron emission in top of right and these are not contradictory we have solved these equations separately for every different frequency that we're interested in in our system now the interesting thing about this optically thick case is that this corresponds to a black body which is a thermodynamic system that I'm sure you've stud in which again the radiation is in equilibrium with uh the temperature inside the plasma perfect semic equilibrium and for a black body we know a second expression for the intensity here for black body I black body or yeah is a function of frequency is equal to a function that's often called B I'm guessing the black m and this is equal to ^2 c^2 Conant H new over the exponential of H new upon T minus one or in many cases it suffices to use the approximation mu^ 2 T upon c^ 2 and this is valid for H much much less than T and this is the clal limit so this was the limit we had in classical pH for some time and it was the violation of this the ultraviolet catastrophe implied by the fact that as new keeps getting larger you keep emitting more radiation that catastrophe is well me to in some part the development of quantum mechanics and this Quantum mechanically correct correction it just turns out that this uh is pretty good a rally genes law it's a pretty good approximation uh for low frequencies where the frequency is low compared the temperature what is interesting then is we now have two expressions for the optically thick case we have an expression for the black body radiation and we have an expression that's the as J upon Alpha and these Expressions must be equal to each other and so this means that the emissivity of the opacity is equal to I'm just going to use the clage new squ upon c s times the temperature and this is called C along with all the other things called C kof law is very profound although we derived it in the limit of an optically thick body it still has to apply into an optically thin ball and effectively what this says is if you know the emissivity then you automatically know the opacity these two quantities are intimately related this is why I said that this wasn't actually valid because this does not obey K's law so the nice thing about that is you only have to calculate the emissivity for a system and then you automatically get the opacity so if you help L J you get Alpha to three that's rather convenient that's nice but maybe the more profound thing about kop law is the fact that where you have high emissivity we also have high capacity what this means is that regions which strong L emit also strongly absorb and that also means that regions regions which weakly emit do not ah sorry regions which weakly absorb do not strongly emit they also weakly emit oh that was confusing this is where this comes in if some of you been staring at this being like wait is this obvious the point is if you have very very low absorption you are also going to have very very low emission and so you're not going to see significant self emission from the plasma being added to your initial beam radiation and so you might think oh perhaps I would get the initial beam plus some extra term well you would get an extra term but that term would be proportional to T to one and we've already said to derive this it's much less than one so that extra radiation wouldn't be very significant compared to the initial beam put through okay questions bu in theramic Assumption or something like you always have to the same temperature everywhere along your pack or something like that or is it just no I think K power of K's law is although you derive it using all these black body assumptions it then works everywhere else as well because these two have to obey this principle for a black body but it means it also pins it even when you're not in camic equilibria non loal th equ um yeah so I think that's why it's a powerful result it doesn't just apply properly yeah another question say that you can calculate the um aity and get absorption that also repace you know the temperature though right well if you're going to calculate the emissivity you need to know the temperature of your plasma we haven't got there yet but the emissivity is always going to be a very strong function of temperature okay yeah so so you to calculate the emissivity you're going to need to know the density and temperature of your plasma and if you want to do it an isotropically you'll need to know like magnetic field Direction all sorts of fun things like that but at the very Bas basic we're going to do breem strum in a little bit brm stong density squared temperature to half right so you need to know those two things in order to be able get the brst so yeah soe that we also have AL C or oh maybe this is more powerful than just simply a homogeneous plasma yeah it looks like if they both go up and down together uh yeah I I I don't think we need to assume that they're actually homogeneous in order to get this result so maybe you're right yeah maybe with the as long as J and Alpha increase and decrease together I'm trying to work out this is is that effec to be a statement that the temperature is constant I think it is so the temperature is constant because of this result but I think that means the density could change so it's homogeneous in temperature only okay any questions online yeah but this differential equation alal Z how is that why do we need law conclude that I J and I Alpha are the same you know that they increase and decrease in lock step isn't that what that differential equation says this is an assumption this equation here is an assumption that we've made in order to derve this equation which is particularly simple to understand the optically Thin and optically Thick limits because to is just the linearly increases with distance rather than like changing rapidly over different bits of plasma once we've done that we then make we find out for the optical bit case that this is true and then I don't know I don't think it's obvious from the start that we were going to end up with with this result and this result is still true for in homogeneous plasmas as well it just happens to be true for we' proved it in the case of a homogeneous plasma okay yeah I I think I see your question but I don't think we've baked in our result in our assumptions if if that's what you're asking yeah I don't think so but yeah okay okay other questions anything online anything in the room all right so now what we probably want to do is calculate J for a variety of different cases so we can calculate J the emissivity from three electrons what sort of radiation do we get from three electrons RS okay anything else synon synchron we're not going to do synchron compon scattering compson scattering oh no um Compton Compton scattering compson scattering is just relativistic Thompson scattering um so we're not going to consider that because that is scattering of radiation from external radiation by the plasma uh so we're talking about radiation being produced by the plasma here but it's it's a good point um we will talk about scattering extensively later on um but not in this case yeah anything else recombination recombination yeah starts with a free electron Beast so it counts anything else uh Lam more radiation or does that just kind of yeah cyron radiation to be honest if that's the same a syncron that explains synchron that explains a lot so I've always been confused about that but we're definitely doing cyron so the same think is it one the relativistic version of the other they're definitely related any any free positrons think what would what would we get if we had free positrons you know yeah okay so we're not going to do that but that would be cool yeah no no no no no like that's that that would be neat we're not going to drive that it's a little bit terminal for the plaz isn't it yeah uh that that's all I've got here um and then we'll also do this for bound electrons and really for bound electrons this is the whole zoo of line emission I think I've be very inconsistent whether I've put two M's or one m in Mission throughout this lecture so you can tell how good my spelling okay now in a previous version of this course we went through all of these without talking at any point about what you'd actually do with it and I found this uh very difficult to teach so what we're going to do is going to cover each of these topics sort of in turn and then interject with like here's how a Bomer works or here's how a pinhole camera works or here's how a spectrometer works but the point is that all of the techniques I'll discuss are applicable to all of these different types of radiation I'm just trying to break it up so it's not like learn every type of radiation and then learn every type of diagnosing radiation but we will start as Hutchinson does with cyron but before that I have a little question for you about something that I thought about in the shower the other day and thought I wonder if this is obvious or not and you can tell me whether or not it's obvious and my question for you is we've encounted two places in which radiation ceases to go through a plasma right we have found a limit where the optical depth is much much greater than one so that's particularly thick and we've also talked about a cut off that's where the reflective index is less than zero right in the context of plasma remember we might have n is 1 minus NE over 2 C and so when we get close to N critical the wave is cut off and that also appeared to stop the radiation going so what I want to ask you is are these two things the same and if not why they not St there's not the same reason being that the O case includes absorption but not reflection whereas less zero case can be reflection okay so let's start ring down to differences I agree with you I don't think they are the same so here we have reflection there's no energy absorbed at least in our sort of wkb picture of this there might be reality here we just have absorption what happens to the wave in these two cases the electric what happens to the electric field in these two cases okay so in both cases okay what happens to the electric field I guess um we have our critical surface here our wave is coming in what happens to the electric field past the critical surface in the case of a cutter it can't propagate it can't propagate so what does it do instead reflect is there no electric field esses it esses so what does it do Decay exponentially decays exponentially an function somewhere okay so this and this is because our refractive index is less than zero our refractive index is defined as k 2 over Omega S c^ 2 like this Omega and C are positive which means that k s is less than zero which means we can Define some qu which I'm going to write is like very exaggerated Kappa to try make it look different Mye which is equal to i k like this and then our electric field now goes as exponential to the minus Kappa X like this the electric field itself decays it does not oscillate what about in the case where optically thick does the wave oscillate or not do you just attenuate the amplitude yeah so in the cases where optically thick the amplitude just goes down so we still have a wave which is oscillating so e is going as exponential of I KX minus Omega T it's just that this envelope I which is proportional to etimes its complex conjugate not proportional just is e times conjugate that is dropping like exponential of minus to right but the wave is still oscillating any other differences about this it's a clarifying question I mean these cases are are actually the same right just that in one case you've entirely gone to an imaginary K and in the other case you would you in principle could have a complex day or something that actually feeds into my final point so yeah keep going with that thought I just I guess just to say one case could become the other assuming you lost all of your real part but yeah so people on line that the question was you know may actually be the same in some sense this is a slow Decay this is a very rapid Decay the difference I see here is that you can have this slow Decay for many different values of the optical depth this happens suddenly and only when we get to the critical density before that the wave knows nothing about it in reality if the density is ramping up the wavelength will get longer but the amplitude of the wave packet in our wkp approximation that we use to derive equations that amplitude doesn't change so I guess the way I summarize this in my notes is that this is a gradual process and this is a sudden process in a sense this is an overdamped oscillator the the the amplitude just drops without oscillating and this is an underdamped or maybe critically damped oscillator where the wave keeps oscillating but it goes down slowly so and you know this will happen there will always be some absorption in some plasma so you always have very gradual decrease whereas you don't actually have to have this happen in any plas at all there doesn't have to be a critical test but there will always be some finite capacity even if it's very very small anyway I don't know if that's profound or not I just thought it was interesting that these two phenomena look quite similar but they're actually very good okay more questions yes weird we talk a little bit not being over able to like see over the build or some reflectometry yes yeah so if you have a case where your Decay is longer than the length scale over which your critical deny is maintained do you get weird things where it becomes wave on the other side I like so the question was if you have a region where the density drops below the critical density for example here so any less than critical density what happens to wave has anyone done this experiment I did it in underground it was incredible life changing yeah it will start oscillating again yeah so you will couple an evanescent wave you can couple an oscillating wave through an evanescent Gap where the wave itself does not propagate and energy will start coming out the other side we did it with wax blocks and microwaves in undergr lb these big blocks of wax and a microwave generator and as you move the wax blocks apart you can like generate a wave with increasingly small amplitude but the wave is still there and it bridged a gap and then the remaining energy because now your wave is just oscillating on the other side of much smaller that remaining energy is reflected instead but this is this is a really cool example of electron yeah I assume the same thing happens in a toat or an neoplasm does it people who do waves for I in lower hybrid right okay so you can get it evanescently crossing a bit of the plasma and then coming back to life on the other side yeah that's why you want to put your laer as close to the as you can right obviously that gives you you know CL service problem okay just for the people online Grant is saying that you want to put your launcher very very close to last close block surface because otherwise it's evanescently decaying in free space so you want it very close to plasma where it's actually in oscillating mode instead okay any other questions on this and then we will go on to deriving radiation from recharges oh no screams none of you have done Jackson the good news is we're not going to do the P Jackson treatment of this it's extremely boring and you hopefully done it before and if you haven't there's no way I'm going to teach you it in a couple of lectures we're going to quote some of the main results if they look completely perplexing then might be worth going to have a look at something that deals with radiation from pre chapters but we're not going to do the whole thing if you want to see it in very rigorous detail you should look at hinson's book it really does go into this um with a lot of rigor but what I want to start with is I want to start with a very simple picture of why charges radiate and I have not been able to find this in any textbook but it was thought to me in underground and I thought it was a rather nice physical picture so I will teach it to you and it may or may not be helpful to you radiation is measured by the pointing flux the pointing flux is equal to the electric field crossed with the magnetic field over a factor of M or probably C if you're using CGS I don't know um the point is that radiation moves in a direction which is perpendicular to the electric and the magnetic fields in the system and we're going to use this simple formula and sketch a few different moving charges and we're going to use that to see whether those charges radiate or not so let us start with a very simple system an electron at rest what are the electric and magnetic fields in this system yes a radial electric field no there's a radial electric field like this this electric field drops off as one upon r s G's law in the r hat Direction the magnetic field is zero because there are no moving charges so there's no currents and so therefore the pointing Vector is zero our stationary charge is not R yet okay next one now we got a particle and it's traveling at a constant velocity for example it's traveling in this direction at a snapshot in time I am looking at this particle from my lab frame what are the electric and the magnetic fields here not you again we'll have someone else but I'm glad you know uh the magnetic field you can do by like the uh right hand rule so it's a moving charge yeah so there'll be some sort of there's current in this direction so there'll be a magnetic field surrounding it I've grown this sort of tilted kind of out of the page otherwise I just have to draw it straight up and down which would be hard to do so okay this is B Theta what about the electric Fields know question yes there are still the same electric field s RA in ask any question we're observing from close enough that able to see it's moved and all that good stuff yes yes this is this is definitely not rigorous but it does get the right answer so please bear with me even if you're like okay so then we've got electric fields which go as one upon r s r hat magnetic fields which are going to go as one upon R Theta hat this is assuming that we've sort of got this electron as a current carrying wire like clearly it's not a current carrying wire that's the result for a current carrying wire um but it's sort of dropping off in this kind of fashion and so that means what is s how does it scale and what directions it pointing pointing in the th direction or whatever depends on your units yeah I system's a bit screwy here um let's say I'm using cylindrical coordinates where I've got Z Direction here a radial Direction here and I've got some Theta angle like this I know that doesn't make sense with the definition of r that we had earlier but again yeah maybe yeah I'm going to drop the treat myself um and what directions it pointed in the Z Direction yes what is this pointing flux doing pointing flux is the transport of electromagnetic energy what's it doing why is it pointing in the Z direction is it radiating is following the particle this pointing flux is simply moving the electric and magnetic fields that the particle has with the particle okay and it drops off as one / R Cub it drops off very very quickly away from the particle right if I draw a surface with surface area r s around it and I ask how much total power do I have going through that surface that amount of total power will drop off as I make my service bigger this is not a propagating wave you can't observe this I mean again you need to be close enough to see what these Electric magnetic fields but this is not a radiation you can see from far away okay so this moves [Music] energy with the electron like that this is the point of it I can add sorry now we decided it's in the Z Direction okay now finally we're going to look at a system where the velocity is not constant and I'm going to look at a very very specific velocity profile here and you'll see why I've chosen this in a moment this is a profile in which the velocity of our particle in this sort of Z direction is initially at some value and then suddenly instantaneously drops to zero a function of time which means that the x coordinate is going to go up and then flatten on the particle will then be at rest and we'll Define this to be b equals zero here am I going to get this change something just before the elction to make it clearer now I'm not sure it's consistent give me a doing it the other way around let's see the particle is initially at rest and then it suddenly accelerates the some velocity I think this will make it work so the particle is initially at rest let's say it's at rest here and then all of a sudden it's over moving in this direction like this now information about the electric and magnetic fields of this particle can only propagate at the speed of light and so that means if I an observer some distance away and let's say that that distance is this circle like that the information I have about the electrical magnetic fields is the same information I had oh from my point of view the particle is still at rest which means that outside of this circle all I can say is that the electric fields are poin