 Hello everybody and welcome to video number 5 of the free online version of the Fusion Research Lecture. We are in chapter 1 entitled Fusion Research and you might remember that in the last video we introduced a power balance for a fusion reactor, for a fusion plasma. And this is where we will continue this time. We will continue to derive an ignition condition from the power balance. So we will start to talk about the ignition condition derived from the power balance which we discussed last time from power balance. So the ignition condition and the ignition condition is given by the fact that the power produced by the fusion process should be larger than the sum of the losses. So that we have a self sustained plasma, meaning that on the left hand side of the inequality we have the fusion heating processes. So the fusion heating power is given by the density of the fusion ingredients, which is the tertiary mantricium. We assume both have the same density, so in half times in half is n squared over 4. Then we have the fusion reactivity, sigma fusion times u. Then we have the energy of the alpha particles and that this is supposed to be larger than the losses. We have two contributions here. One, the losses due to transport, as we talked about last time, which is three times volume average density and temperature over the energy confinement time tau e. The other part is due to radiative losses due to bremsstrahlung, where we had the coefficient or a constant rather C Br for bremsstrahlung Z effective for the effective charge number then n squared and then the temperature of the temperature. So on the left hand side, this is the heating by the alpha, so this is fusion heating and then as I said, this is the contribution due to the losses. Let's say, sorry, losses is obvious, I guess, right? What I wanted to write is transport losses. Transport losses and this one is the losses due to bremsstrahlung, so bremsstrahlung losses. And from this we can derive by just rearranging the quantities and inequality, which reads that on the left hand side we have the density times the energy confinement time. This has to be larger than 12 times the volume average temperature and then a large fraction where we have the fusion reactivity times the alpha, so the fusion power minus the bremsstrahlung losses C Br Z effective square root of the temperature. And this is obviously a function of the temperature. Now let's give it a box as this is such an important equation, rather inequality of course. And this is called the Lawson criterion. This is the Lawson criterion, which gives us the values of density and energy confinement time that we need to achieve to get a burning plasma. Now Lawson derived that in 1957 and Lawson was an engineer, he shared his office with a fusion physicist and he sort of failed the responsibility as he said in an interview to pin down unrealistic expectations of his enthusiastic physics fellows. However, his physics fellows look at the inequality and said, oh, well, that's very handy. Now we precisely know which parameters we need to achieve. So now let's have a look at the Lawson criterion when we plot that. Here you see n times tau e as a function of the iron temperature. You see both. Sorry, you see two curves. One is for deuterium trisium fusion, including bremsstrahlung losses. This is the blue curve, the orange, or maybe it's brown. I'm slightly colorblind, I guess it's brown. DT fusion neglecting the bremsstrahlung losses and obviously, as expected, the one where we neglect bremsstrahlung losses, the values we need to overcome in terms of density and energy confinement time are a bit lower. As compared to the curve including bremsstrahlung losses, however, in both cases, there's a minimum around that area here and that minimum corresponds to a temperature at roughly 27 kilo electron volt. So there's a minimum at approximately 27 kilo electron volt. This is the value which we are aiming at to get a burning plasma. Now, in nowadays, it is more common to use the triple product instead of the original Lawson criterion. So the triple product is the more common quantity or measure. Nowadays, the triple product is more common to measure the success, if you want, of a fusion device, triple product. Quite a common expression and often used quantity. This is simply, again, it's n times, now in addition to the energy confinement times, also the temperature on the left hand side. We have now all three quantities which are hard to achieve on one side of the inequality. This has to be larger than 12 and now it's t squared because the equation has just been multiplied with t. The same large fraction as we had in the previous one, it's the fusion reactivity, the alpha energy, so the fusion power basically minus than the bremsstrahlung losses, CBR, Z effective, square root of the temperature. And this expression is often abbreviated with a capital F. So you find that quite often in literature. So this is the triple product. Okay, as I said on the left hand side, or the advantage of the triple product is on the left hand side, we have the relevant global quantities that are hard to achieve to get fusion. And on the right hand side, we have more or less the same fraction as in the origin loss criterion, except for the factor, for an additional factor of T, the temperature. And just as a reminder, here you can see, since we have Z effective here, low Z effective is important. So as I said last time, this is the effective charge number. And it is important to keep that at a low level, meaning we want to avoid impurities, which would increase the Z effective, the effective charge number, thus increase losses due to bremsstrahlung. The expression on the right hand side has a minimum. So this has a minimum roughly at 15 keV, minimum at T approximately, at an electron temperature of approximately 15 keV or higher temperature. Here we assume both to be the same. This is why, oops, here we assume both to be the same. And now let's have a look at some numbers. What does that mean? If we look at a typical fusion experiment, it's always good to have an idea what numbers we actually need to achieve. So some numbers. If we are on the range from 10 to 20 keV, and we want to be in that range, because we have a minimum at 15 keV, as I said on the right hand side. And in this range, the fusion reactivity can be expressed by a simple parabola. So here we have a constant C, then times T squared, where this constant C sigma U in the index has a value of 1.1 times 10 to the minus 30 meter cube over electron volt squared in seconds. Now if we neglect bremsstrahlung for now, neglect bremsstrahlung, then it means our inequality reads N times T times tau E has to be larger than an inserting that parabola expression. We get a value of 0.5 megajoule second per cubic meter, or in more plasma physics common units, 3 times 10 to the 21 kilo electron volts over meter cube times a second. Okay, this is the number we need to overcome. And what does that mean for a fusion plasma? A typical fusion plasma has densities of 2 times 10 to the 20 particles per cubic meter. Temperature as I said is supposed to be 15 keV. And then the third parameter, the energy confinement time tau E, need to be approximately three seconds. So this is the minimum requirement for a burning plasma. And note that the density and temperature, these are values we have achieved in experiments already. The energy confinement time, the value of three seconds is something we have not yet achieved, but is something Eta is aiming on. Talking about values which have been achieved so far, let's have a brief overview and look at Nt tau, so the triple product as a function of year. So here I've plotted the triple product as a function of time and each dot represents an experiment. These are all tokamaks and you can see there is a strong scaling, a strong increase over time until the year 1997. And this note, there's a logarithmic scalar and there's a 1.7 year doubling rate until the year 1997. And just as a reminder, Moore's law has a doubling rate of two years. Moore's law which says that the number of transistors in an integrated circuit doubles every two years. So just as a reminder, Moore's law says that the number of transistors in an integrated circuit doubles every two years. Every two years is supposed to read years. Now you can see in the plot that at 1997 the doubling rate basically stopped and then there's a long break until the dot to Eta which gives us another, again an increase. But it takes some time, so the slope is strongly reduced. Now there are a few reasons for that and would require probably an own lecture, sorry, an own talk to discuss that. One reason is of course politics. It took a bit to decide where Eta is going to be built, then how Eta is organized, that it is a multinational project, a lot of countries contributing to it. And of course Eta has a number of new physics or challenges which were not tackled in other experiments before. For example, the major significant alpha particle heating. So there are a few reasons for this long break so far. Okay, this is anti-Tau as a function of time. Let's now look at the triple product as a function of the temperature and looking at different experiments, not just Tokamaks. So here you can see the filled squares, these are Tokamaks, the names indicated next to the symbols. And then we have the filled circles and these are stellarators. And then the green filled diamonds and these are spherical Tokamaks. And you can clearly see that Tokamaks are leading in this plot. So the blue dots are way ahead. Tokamaks are just performing better. Tokamaks are performing better. And are performing better and you can see that they are close to the ignition region which I have indicated by the blue, by this blue area here at the top. This is when ignition is achieved and they are getting close to it. Eta will get very close to it. And the reason why Tokamaks are performing better is basically they are one machine generation ahead. They are one generation ahead. W7X has codally the record, so W7X the symbol is there. This is currently the record on the stellarator for a triple product achieved. As you can see here, but it is still lagging behind the Tokamak record value. Reason being as I said that the machine generation is one generation behind. Let's now have a look at another important quantity for measuring the achievement of ignition and that is the Q value. The Q value is another important measure for the achievement of ignition. It's a quantity to measure how close we are to ignition. That is the Q value. The Q value is basically the fusion power produced in an experiment on a plasma. The fusion power produced and then divided by the external heating power. Divided by the external heating power like this. And in a DT experiment, this means that the Q value is 5 times the alpha particle power divided by the external heating power. And to give you an idea, of course, if we would achieve complete ignition and turning off the external heating power, Q would go to infinity, right? Eta aims on a Q on the order of 10. So Q equals to 10 is what Eta is aiming on. And a power plant or a reactor fusion power plant would probably be have something like Q equals to 30. Because you would still like to have some kind of control, local control about the plasma there. So what have been achieved so far? These are time traces of the record shot. So this is the record shot in jet performed in 1997. You can see a few time traces there. So let's have a look at the top time trace. So here we can see the injected heating power. There are some variations in it and it's turned off again. And then we can see the heating power produced by the fusion process itself, so by the alphas. And putting those two quantities into relation, we get the Q value. So the fusion power over the external heating power and a value of 0.67 has been achieved there. 16 megawatt of fusion power. So jet in 1997 achieved 16 megawatt fusion power. And the resulting Q value was equal to 0.67, which is the record Q value which we have achieved so far. Okay, time to introduce another important quantity and that is the plasma beta. Now you know by now that fusion magnetic confined fusion happens at high pressures, at high plasma pressure. It happens at high plasma pressure. Plasma pressure is P equal to density times temperature. Now this is only possible if the plasma pressure or this is only okay if the plasma pressure is not getting too high. There's a counter acting or stabilizing magnetic field. And if this is too low as compared to the plasma pressure, instabilities occur. So instabilities occur, instabilities, instabilities occur. If the counter acting or stabilizing, counter acting or stabilizing magnetic field pressure is not high enough. Not high enough. And whoops, there's probably a letter missing, not high enough. And that is the topic of MHD, magnetic hydrodynamic instabilities, which we will tackle in the next chapter. Now quantity to measure the plasma pressure as a function of magnetic field pressure or how close we are to that region where the plasma becomes unstable. This is the normalized pressure, also called simply beta. The normalized plasma pressure, simply beta. And beta is the plasma pressure, so P over the magnetic field pressure, and that is B squared over 2 mu naught. That is the plasma beta. And since the plasma pressure is the sum of the electron and ion components and assuming we have same density and temperature for electron and ions. It is 2 times N times T, and then over B squared over 2 mu naught. And this is the plasma beta. Now, sometimes it is also useful to separate the polar and magnetic, sorry, in the toroidal magnetic field direction. So for the sake of completeness, we should also write that here, so sometimes useful to be aware of the fact that the magnetic field can be separated into a polar and a toroidal magnetic field. Likewise, for the beta, then the overall of the beta, 1 over beta would be the sum of 1 over beta polaroidal plus 1 over beta toroidal. And you see in the denominator of the beta, you have the magnetic field, and this is being the sum of toroidal and polar magnetic field. Now, to give a few numbers in the solar corona, for example, in the solar corona, we have a beta on the order of 10 to the minus 5 to 10 to the minus 4. In a tokamak or a stellarator, the beta values are the maximum beta values which are achieved. So usually it's 0.05 and sometimes in tokamaks, 0.1 is achieved. And if the beta values become larger, then instabilities occur terminating your plasma discharge, basically. Instabilities, there's something wrong, I guess. Instabilities occur and then terminating your plasma discharge. Another example is the solar wind, where beta can be much higher. Especially when you are far away from the sun, the beta can definitely be larger than one. Now, beta can be used to rearrange the triple product, writing it into a slightly different form. So let's rearrange the triple product, and then it reads B squared over 4 u0 times beta times the energy confinement time, and that this has to be larger than the value of the number F, which we already had. And as I said, beta is the number which is basically fixed, it cannot be arbitrarily large. The magnetic field B cannot also not be arbitrarily large. And this is why it's important for the energy confinement time, tau e, to be large. So let's have a look at the energy confinement time. The energy confinement time is something we know from scaling loss, basically. So, which is determined from scaling loss. The energy confinement time, especially when we do extrapolations to new experiments, we know that from empirical scaling loss. And if you think, ooh, this is a bit scary that we just use empirical scaling loss to define our next generation experiments, well, these scaling laws are really good. So on the right hand side, you see the measured confinement time has a function of the predicted confinement time. And you can see that the values are nicely agreeing with each other. So the extrapolation to eta is based on a very solid database. And writing down the scaling loss, instead of writing down the parameter dependence of every parameter, we try to just write down the major, the most important ones. So tau e is proportional to the FH, which is the measure for the confinement quality. Then the plasma volume, the magnetic field to the power of 0.8. And the heating power, the applied heating power to the power of 0.6. This gives you an idea of the scaling of the energy confinement time. And FH, this is a measure for the confinement quality. Now that sounds maybe like a bit of an abstract quantity. And we will talk about that later on, just to be short, there exist different confinement modes in a plasma. For example, the H mode and the L mode, low and high confinement mode. By going from the low confinement to the high confinement mode, the confinement quality factor changes from a factor 1 to a factor 2. This is why the H mode is so important if you have heard of the H mode so far, because it basically doubles the energy confinement time by going from a factor 1 to a factor 2. Then the plasma volume is important, magnetic field is important, and you can see that putting more and more heating power in is not very helpful. An important thing here, apart from the confinement quality, is as I said that the size is obviously important. This is why Eta is also so large. So the size is important. Something we definitely should write down, as it is an important consequence. So size is important. And you can see this in the right-hand side. So there are four experiments listed. Compas D, then Aztecs, Upgrade, Jet and Eta. These are poloial cross-sections. And you can see how much that Eta is much larger than Jet in cross-section. And this is due to the energy confinement time, which we want to have sufficiently large to fulfill the loss criteria and add the triple product to achieve a significant outward fusion power there. Now, the energy confinement time scaling, which we had discussed on the previous slide, allows us to rewrite the ignition condition into a slightly different form. Ignition condition. And it then reads FH, the confinement quality, one for the L-mode, two for the H-mode. And then the magnetic field squared times the plasma volume over 4 mu naught, and then times beta, and then times the magnetic field 0.8. Then the heating power minus 0.6. And this has to be larger than F and Y. Is this a useful quantity? Because, as I said, the plasma beta is limited due to physics reasons. The magnetic field might also be limited. And the confinement quality is crucial to get there a factor of 2. And this expression here, this factor, this basically defines the cost of your experiments because magnetic field volume is expensive. If you want to have a high magnetic field over a large volume, this is really expensive. And this inequality, therefore, allows you, in a way, also to get an estimation for your costs of your fusion experiments which you are designing. Okay, that's it for the fifth video of the free online version of the Fusion Research Lecture. And in this video, we talked about an ignition condition, the Lawson criterion, and the triple product, which tells us the numbers we need to achieve to get ignition in our plasma, so a self-burning plasma where the alpha particle produced in the fusion process deliver enough power to the plasma to have a self-sustained reaction. We have looked at the energy confinement, sorry, the triple product values which we have been achieved, which have been achieved so far, and also at the Q value. And keep in mind, we have already achieved experiments at the jet tokamak with 16 megawatt of fusion power output where the Q value was 0.67, which we have achieved there so far. This is the record from 1997, and we are looking forward to ETA to get higher values than this. Finally, we talked about the normalized plasma pressure, the beta, which is an important quantity, usually on the order of 0.05, maximum 0.1 in large-scale tokamaks, and often it is a percentage value which is used there. And then we talked about the empirical scaling laws for the energy confinement time because the energy confinement time is a very important quantity, something which we have to increase and something which is increased in ETA to achieve the, to fulfill the ignition condition at the triple product. And an important consequence of all this is that the size is important and size means also expensive, and this is why ETA is so expensive. Okay, that's it, and I hope to see you in the next video.