 Hello and welcome to video number four of the free online version of the future research lecture We are in chapter one and you might remember in the last video We mostly talked about the nuclear fusion process itself and today we will focus about a few key parameters important for infusion research so some key parameters will be our topic today From last video you might remember that the DT fusion reaction was the most attractive one to be Pursued on earth so that is definitely the candidate for our future reactors and the DT fuses into helium nucleus with 3.52 MeV and a neutron with 14.06 mega electron volt and As I said last time only the energy of the alpha particle can be further used for heating because only this is confined in the magnetic cage So only 20% of the energy release of the fusion process is left for further heating and This means a good confinement is required to achieving a self sustained fusion reaction a fusion reaction Which is sustained by the heating of the future reaction itself So this means good confinement is required Is required for a self sustained Safe sustained. Sorry. Let's try to properly write that safe sustained fusion reaction and Quantity to measure that is the energy confinement time So that is the energy Confinement time This is defined as the plasma energy content divided by the heating power so the energy confinement time tau e is W over P where W is the plasma energy content the nominator plasma energy Content the nominator and in the denominator we have P the applied heating power applied Heating power and the plasma energy content is the sum of the energy of the electron in the ions So we have the plasma energy content being the sum of the electrons energy and The ions energy and here we take the kinetic energy. So that is three half and then electron density times electron temperature and As you might know, this is from Averaging over max valium. So this is from Averich over Max valium or Maxwell-Boltzmann distribution And we not only have the electrons but also the ions and We write it like this. Yeah And We have since we interested in the whole energy content, we have to take the integral over the whole plasma volume This allows us then to calculate the energy content of the plasma Now, let's have a look at an example. So consider let's consider the energy content of 50 50 Deterium trisium plasma because that is what we would like to have in our fusion reactor That at first since we are a plasma physics, you know, we have quasi neutrality quasi neutrality Meaning that the ion Density is equal to the electron density and we often or we are just abbreviated with n Then we have or we assume that we have no impurities No impurities meaning that the ion density is given as Just the sum of the deuterium density and the trisium density Then as we have written down above there, we have a 50 50 DT mix or mixture Which means that the deuterium density is equal to the trisium density and That would correspond in our notation here to n over 2 to n half Furthermore, we have a typical fusion plasma which has densities on the order of 10 to the 20 particles per cubic meter That means there is a strong energetic Coupling between electron and ion so there's a lot of collisions going on energetic Coupling That takes care that the energy of the electron and the ions the temperature is very similar usually So the electron and the ions temperature can be assumed to be similar and we abbreviate that with a capital T Now this allows us to write the energy content Fusion plasma again have the volume integral and then it's three half and e The electron compartment part the ion part And since as we have written it above there The electron and ion density is the same the electron and I temperature can be assumed to be very similar This can be just written as three times n T dv and This allows us to give an approximation for the energy content of the typical fusion plasma being Approximately equal to three times the volume times n bar and t bar and What are n bar and t bar? These are volume averaged quantities. So n bar t bar these are volume averages of n and t Okay, this is a formula to get an estimation on approximation of the energy content of a typical fusion plasma for typically t fusion plasma Now let's go one step further and look at the power balance The power balance Now a power balance. Let's assume we have somehow a plasma this is supposed to be some kind of plasma and Then we have some This this plasma has some energy content w. It has some Heating power going into it some auxiliary heating So we have some heating going in and we have some losses P we use an L for that L like losses And in steady state We assume that losses are the same as heating losses and sorry losses and what goes in must go off basically So this is how steady state is then defined in steady state or equilibrium losses Equal to the heating power and this means that P L Can be defined or is defined as the energy content and then over the Confinement time and that this is supposed to be the same as the heating so here we have an expression for the losses and Now let's have a look at the power input. So what about the power input? So first of all, we have the external heating power External heating P then auxiliary As I have drawn it in the sketch over there, but then we also have the alpha particle heating the alpha particle heating P alpha which I have not drawn here yet. So let's draw another arrow pointing inwards This is P alpha and This can be written as The density of deuterium times the density of trisium and we knew that this is n over 2 and Since we have both of them as n over 2 squared Yeah, well as I said and and Deuterium is equal to n trisium and that is equal to n over 2 and half as we had it written on the last slide so and over 2 squared and then times the fusion reactivity Which we had introduced in the last video the fusion reactivity times the energy of the alpha particles and Then we are again have to Take the integral over the whole volume Now this here has the unit so the first term sorry it has a unit of meter to the minus six and the second one to Meter squared meter over second So overall This is something like reactions per volume per time So this is something like reactions per Volume and per time Well and e alpha is of course the energy of the alpha particles so 3.52 mev Okay, now we have the power of the alpha particles here Serving as a heating power auxiliary heating and the losses now let's have a look at a power balance so because the power balance allows us to Define or derive conditions for a self sustained plasma. This is what we interested in so conditions for a self Sustained Plasma From the power balance Let's mark. Oops. There is a C missing I guess So we have here the important expressions here are self sustained plasma and power balance now a self sustained plasma is usually defined by The fact that the alpha heating power exceeds the the power by the losses Yes, so that there is more power provided by by the alpha heating then is transported out of the plasma by losses and this point is often defined as ignition or Way to be Yeah ignition to be more precise often you read that Ignition refers to the point where the Losses are just the same as the heating power by the alpha particles because that in principle Implies that you can turn off your auxiliary heating power and your plasma is Kept burning Okay, so since we had on the previous slide the heating power by the alpha particles We can now write this down and to go over the volume and Squared over for then the fusion reactivity then the alpha particle energy and That this is supposed to be larger then One over tau e then the energy content of the plasma for which we had also the No, I think it was two slides ago. Yes the expression three times and then you the kinetic energy and N times T so density times temperature and Now we can get rid of the volume here and we can in addition take The volume average quantities so take the average Quantities n bar and t bar and then we end up with an condition with a condition Which reads n bar squared over four times the fusion reactivity Times the energy of the alpha particles and that this has to be larger than three Volume average density times volume average temperature Divided by the energy confinement time Okay, this is the first expression for a power balance which we have to fulfill to achieve a safe sustained plasma Now there's one so far. We have only included basically losses due to Transport so the losses so far here are just due to transport because here we have the profiles of density and temperature going into it But there are also however Radiative losses which we have to mention so there are also Radiative losses and most importantly of those are at our parameter range the losses due to brems strahlen That's the brems strahlung is or can be an important loss mechanism. We have to be aware of it What is losses or what our losses via brems strahlung where if we have an electron We have an electron flying around and Encountering an ion then it is decelerated in the electric field or by the electric field of The ion and this deceleration process leads to an emission of Gamma quant or electromagnetic radiation And if the electron had the energy e1 before Encountering the ion and then being decelerated by it having the energy e2 afterwards then the energy of the gamma quant of the electromagnetic radiation corresponds to the different of Those two quantities so I e1 minus e2 Okay, so brems strahlung is basically the D. I'm sorry. This looks a bit weird the D acceleration or due to the deceleration of charged particles in In other Charges field in the electric field of another charge and this leads to the emission of electromagnetic radiation and Typically, this is in the x-ray regime and also typically these are electrons and countering Nucleus and this happens indeed quite often our plasma since we have a Lot of electrons and ions and fusion plus most typically the degree of ionization is one So there are a lot of these interactions going on So in a plasma or in general these are losses Via we can call them electron ion collisions So these are Coulomb collisions and they become non-negligible if the Electron and iron if these if the temperature of those two particles Exceeds 10 kiloelectron right now, then we can no longer just neglect the losses via brems strahlung an expression for the brems strahlung losses are given by Equation CBR so some constant times the effective charge number Times density squared and temperature Square root so first of all, let's briefly write down the constant CBR This is 1.04 times 10 to the minus 19 and then we have meter cubed electron volt over second and Z Effective here is the effective Charge number So that is the effective charge important quantity and The effective charge is defined as the sum over all ions in the plasma so all charge particles and then the number density and then the Charge number squared and that is about the charge number squared Then over the electron density So brems strahlung losses in a plasma not only occur at hydrogen Nuclei but also at impurities and balls. I mean even more impurities because due to the Squared dependence of the charge number. This can be a significant loss mechanism So brems Strahlung Losses not Only at hydrogen More Importantly, let's say at impurities oops at impurities and Only for hydrogen the charge number is equal to 1 for impurities the charge number for all impurities the charge number is larger than 1 and Also for helium so also for the helium ash the charge number is larger than 1 And as I said due to the squared dependence if we have impurities there Brems strahlung losses can be a significant loss channel of plasma This is why we have to keep impurities at a minimum level Okay, in this video we talked about a few Important parameters. This was the energy confinement time being defined as a ratio of plasma energy content by applied heating power We talked about the plasma energy content being the sum of the electron and ions kinetic energy We started with a power balance and Showed how Conditions for a safe sustained plasma is basically defined by a power balance and then just on this very last slide We talked about radiative losses And that radiative losses are mostly due to brems strahlung in our plasmas and that the effective charge number plays an important role here and That the brems strahlung losses have Squared dependence on the charge number and this basically says that we have to keep impurities at a minimum level Otherwise losses via brems strahlung might be a real problem Okay, that's it for the fourth video and I hope to see you in the next video where we will derive and define Further key parameters in future research