 Okay, thank you everyone. So, today we continue talking now about magnetic mirrors before getting there. Some of you asked about the Lawrence equation integrator. And so I added a few slides on that as well as well as a word about vacuum magnetic field solutions. So let's go through the Lawrence equation integrator fairly quickly. And as I mentioned before, this is a leapfrog implicit integration scheme of this form. And we're going to apply it to the Lawrence equation. And so it's actually simply about putting the equation in terms that you can solve for this algorithm for this equation with this algorithm, excuse me. And so it's a little bit of vector algebra. So you start by creating this normalized vectors. So in this case, Omega, excuse me, and Sigma are 3D vectors and they're normalized. As such in the Lawrence equation to get through, excuse me. This scheme. And now we start discretizing the Lawrence equation in the same way that we have here for the implicit method. And so we get this expression right here. I've just copied it up here just for reference. But then, if you create this new vector a which is for convenience if you will to just reduce the algebra a little bit and replace it here then you get something like this. If you perform. A, so if you perform the cross product and then regroup, you get something like this where now I've grouped all of this quantity on the right hand side to some new vector called C. So you've got the equation with a and also cross the equation this this original equation equation one with a as well. You get two expressions. You expand after using the usual a cross B cross C vector identity and inserted into a question to over here. Excuse me, insert a question to into it. You get this expression here so we're just doing vector algebra which after solving. You get to this point. So you've now solved for the new step in the integrator. And that's the new step of the velocity, the particle position is simply. The next I so the previous step, plus the next step times your time step. So this is a very simple algorithm that can be implemented with a few lines of code. I've used it to create the trajectories that I'm going to show here and that I showed in the previous presentation. I'm sorry, of course, available. There's a lot of resources actually on from an organization called plasma pie. So it's a Python set of Python libraries with applications to plasma physics that you may download. And I encourage you to visit it. It's all open open source freely available. I was asked about the vacuum field solver and it's also relatively simple programming. It's really mostly knowing which expressions to put in, and then a lot of summation so if you start with magnetic flux and use Stokes theorem to get to this point contour integral of the vector potential. If you have a circular loop, then the only non zero component of that vector potential is a five, which is constant along the L for a given so differential of length in your contour for a given radius. The flux through a circular contour can be given in this way in terms of the vector potential. And then there's a really nice derivation in Jackson's electrodynamics on on the vector potential for this case, which you get here in terms of elliptical integrals. And where the argument of the elliptical function is shown here, and I'll be happy to share this with you, of course, this expression so you don't have to write it down now if you don't want to but if you have questions please feel free to ask. And from the flux that you calculated then you can get the components of the magnetic field as well. In this case, I use the program Mathematica from Wolfram to calculate the flux. And I approximate the elliptical integrals with this polynomials that I found in a mathematical mathematics tables that I believe have worked really well. And that's because even though Mathematica has elliptical integrals in it. I don't need the super high precision that Mathematica has and sometimes it's difficult to dial it down so this is a convenient and faster way to do it. And I also of course calculate the inductance of the loop, or the, or multiple loops by using the basic definition of inductance, and this is assuming the loop is a fixed conductor of course, then then it becomes just a simple summation. For example, I've put here a calculation that we did some time ago. Each little dot represents a loop of elementary current with this radius where the I equals zero axis is at the bottom of this rendering. And in this case it's in meters so it's almost point 75 meters or also long machine, and you have four coils here. And so the program actually calculates the flux for each coil for each winding. So you have thousands of turns here. And it also estimates the flux everywhere for from all the coils. So you, you have just a series of summations here, where you just offset the axial coordinate by some distance, as well as the radial coordinate. And again if you have questions about this please let me know the number of lines of code for this is very short. And again this is not the only vacuum fields over but we just built around. So now, going back to equals be drifts and centrifugal confinement so last presentation I talked about drifts and how mirrors and new conservation or the first adiabatic conservation. It turns out that in mirrors you get other quantities that are conserved they're called second and third adiabatic invariance, and they have to do with the length of the bounds. As well as the drift that I talked about the grad B and curvature drift that you naturally get in mirrors, even without any electric field apply. But also I mentioned that the most important drift was the equals be drift. So, in axis symmetric mirror, if you apply a radio electric field, then you'll start to get an azimuthal meaning around the axis drift, just like it's shown here. And interestingly, again even with a single particle you start to get an idea of how it improves confinement. So the bounce length is reduced, and the loss cone is also reduced. Now if I start my simulation with a single particle at different radii. And let it run for the same time and I'm starting all particles with the same initial be parallel and be perp. And in this case, they are the same be parallel and be perp and, again, parallel and perpendicular refer to the magnetic field. And I get that the angular velocity is not the same, and it depends on where you start in radius. So this already start to hint at a shear flow that you can get in a rotating mirror which again it's advantageous to reduce the ideal losses. Now here's a little trick going from a single particle and pretending we have many particles and again and this is just recasting the Lawrence equation in cylindrical coordinates which I'm sure every everyone here has done for some in your math and physics classes. And using this expressions with the integrator that I mentioned. Now, before I continue. I want to mention that with this simple integrator that I'm going to show the simple numerical problem that I'm going to show it. Also demonstrates that even though equals be drips to not depend on charge and mass. That is only of the guiding center. If the electric field is large enough. This assumption doesn't hold perfectly, and then you start to get other pendants on charge and masses I'm going to show. So assume you have a helium ion here, and I estimate that for a certain density and temperature, the collision length, the mean free path before a collision can occur. And so I let it run for that length of time for that mean free path that particle energy, and then you get a collision. You randomized the direction of that collision. Let it run again, you get a second collision, and so on. If it gets to a certain radius which I call the, the chamber radius will call that a recycling the particle gets pushed in some distance away and then the process starts all over again. And if you do that many, many collisions, you can get a profile. And if you do that for different ions. For example, oxygen that is singly ionized so called oh too. You get a profile like this that is parabolic is not perfect because I guess I didn't let it run for long enough but you get qualitatively parabolic profile which is what you will expect us I'll show in a in a common common slides. So if you do the same for carbon three so the mass of carbon three so doubly ionized. So it has two elementary charges missing, or if you have helium to which means it's singly ionized, and you get a fairly different magnitude and profile of depending on the ion mass and charge. So the dotted lines he he referred to actual measurements in the experiment that I'm going to mention here too. Anyway, as much as Ecosby theory is useful, you always have to be careful to not take it as the ultimate word in what you're doing. We're going to go. So, going from one particle, or one particle pretending to be many particles and in the case of multiple slides, we're going to go to now a large collection of particles namely 10 to the 20 particles or more. And we're going to use single fluid m hd. And that is the simplest fluid model that we have in plasma physics. It's very useful is not the most accurate. But because it's the simplest becomes very useful. And I'm going to go through some expressions to show the applicability of single fluid m hd. First of all, you want the time scales. You take your time scale to be the cyclotron frequency, the length scale to be your armor radius, and you assume quasi neutrality by by looking at the by shielding. So, for centrifugal mirrors. It's beyond this talk to derive the expressions of single fluid m hd, but it's a absolutely worthwhile exercise of course, if you start from the blasts of equation. But here assume you've already gotten there, and you have the force balance equation. And for a magnetic mirror, you start with a grad pressure gradient. You have j cross B forces. And then you have. You assume for now that it's, it is collision now. And because this is applicable to laboratory experiments that we have performed and we're doing, but eventually collisionality goes way down, but you also have viscosity. And this viscosity is important for the rotating mirror, because shear flow that I mentioned with a single particle theory and that will appear in m hd creates heating. And that heating is important if you want to get to thermonuclear conditions. Assume that you started the rotation it's been going on for a while so it's in steady state so you can neglect the dndt components. And if you look at the radial direction components, you have that the rate is balanced by the j cross B force in the radial direction minus this losses. Radially. You also assume we can assume that the radial flows are small compared to the smithal flows. So of course you're going to have some losses radially out. The velocity of or the rate of those losses shall be much more compared to the rate of rotation that you have. And let's do this simple transformation, where the velocities just are times the angular velocity and the smithal coordinate, and you can recast this expression again in this way here. And if you replace this expression that we just derived, we see that political currents can balance the radial pressure profile, as well as the centrifugal force and radial currents balances the angular momentum loss to neutral collisions based on viscous dissipation in other words, when you're rotating, you have to keep adding power, because if you stop the power, the rotation will wind down based on collisions with neutrals, and based on viscous dissipation, as it's winding down you can build up heat. But if you want to keep it rotating steady state you have to keep adding power to it, a current through the plasma or a j cross B times a velocity. Now, if we perform volume integral in the above expression. We get the momentum confinement time, which goes as this expression, which we'll come back to later. Now if we look at the components along the magnetic field. So, until now we were concerned with the components that are radio and the radio transport. Along the magnetic field. By dotting the expression from the previous page with B. Then we drop this component of course by simple vector algebra. Excuse me. And again, assuming that axial flows are much smaller than your as a mutual flows. The expression reduces to this. And again doing the transformation for the velocity with the radius. And the angular velocity and the same transformation again I'm just copying it again here. And recall that pressure can be recast as NT or NKT if it's in SI units but T is an EV so it's just an NT. Then we have this expression here. Now, you can note that this expression has already been separated in terms of N. If now we, which is the number density. If we define the sound speed as such. And replace here, we can now integrate this survival equation and evaluate at a certain R1 and R2 along a field line so this. This is assuming that the temperature here is constant or inside your sound speed. Here is constant along a field line, which is considered a very reasonable assumption, even a few tens of EVs. And even more so at hundreds or thousands of electron volts or KVs. If you define the Mach number. That is the rotation velocity over the sound speed as such, then you can replace it here. You see that the pressure. The plane here in the bulge of the mirror is exponentially dependent on the Mach number. Compared to the pressure here which is what you want right you want most of your plasma to be confined in the center of your mirror, not at the mirror throat, where it could escape. Because it's not not always practical to measure R1 or R2, but we know we normally know what the field is, we impose the field and so we know what the minimum and maximum are. We can recast this ratio of R1 and R2 in terms of what's called the mirror ratio, which is just the ratio of the maximum over the minimum of the field and so you end up with this expression right here. You can see that even for moderate mirror ratios and by moderate I mean, say, five to 10. This is mostly depend the exponent in the exponential expression is mostly dependent on the square of the Mach number. In other words, you get a one over R dependent on the mirror ratio so a super high mirror ratio doesn't buy you much compared to a moderate mirror ratio, which is good news because making magnetic fields is very expensive. This expression also shows that with fast enough rotation, you can effectively close the mirror losses or through the loss cone. Now, simulations have been performed, you know these simulations have been around for many years now, but they essentially confirm what we just saw with single particle approximation. With the analytical expressions. Where you get most of your pressure here. The mirror is just oriented vertical. I am never sure why they do this when they perform simulations but okay, and the angular velocity. This shows that it's ice rotating in any given flux surface so a flux surface if you imagine as a shell. If you're on that flux surface, then the rotation anywhere on that surface has the same angular velocity, but if you move to the next flux surface. The angular velocity changes and indeed it has a profile that is more less parabolic. And that also gives you a pressure profile that is parabolic and that this is of course, again advantageous, because it can help destroy interchange modes like I'm going to show in the next slide. So this is another simulation where the plasma pressure in a mirror without rotation so this is a traditional mirror was let left to increase to high enough value that you get this so called flute modes or flutes so this is just a slice of your mirror. Flutes are essentially really Taylor like instabilities, where you have a heavier in really Taylor instability you have a heavy fluids on top of a lighter fluid. And the heavy fluid wants to go down the lighter fluid wants to go up, and you start to get this kind of mushrooms that are formed at least in in a 3D space in a mirror, because you have the magnetic field creating a sort of tension. Bubbles what you get are flutes essentially to the bubbles if you will, and they very quickly can remove your confined hot particles, and then to replace those particles you bring in cold particles, and essentially for a traditional mirror, you can never get to the thermonuclear conditions with this interchange. That is a that is a fundamental issue with just a traditional mirror without rotation. Now in this simulation, if you grab this last panel as the initial condition, and now you turn on the rotation. You can see that immediately because of the cross be drift. And because of sheer slow, you start to stabilize that interchange mode, until after some time. It becomes fully stable. Now, the caveat is that this interchange modes. Yes. It's like a water. Coming up, because we have only plasma. So, and you said that one and one. Yes, it's the analogy. Yes, thank you. Is the analogy that I use but in reality, there's a pressure, the magnetic field has its own pressure B squared over two mu not right that's a magnetic field pressure, and the plasma has a thermal pressure nkt. And ideally you will expect that they balance out right when the pressures are the same they balance out as it happens in plasma with plasmas. It's not really like the magnetic field is not like a solid wall is more like trying to confine, say, yellow or something that is very viscous and can flow through a spring like like a slinky say the stronger the magnetic field, the more turns per unit length you will have in this configuration right but anyway it's an imperfect confinement scheme that magnetic field is not perfectly. It's not like a solid wall. But in the analogy of the Rayleigh Taylor instability, the magnetic field will be the heavy fluid. I'm sorry the plasma will be the heavy fluid the magnetic field will be the light fluid, if you will. So the hot plasma just pushes through the magnetic field and the magnetic magnetic field has to rearrange itself to let the plasma through, as you see in the left panel. But fortunately, with the eCrosby rotation you naturally get a shear flow, which is, which is what you want. This fluids can snap out can can grow and and move plasma. Very quickly in the speed of sound essentially. So if you're rotating super sonically, then you have a chance to share those flutes out and and train any of that plasma wanted to move out. So you go into the, as you move to the direction, and now you, the plasma never made it outside. And so that's, that's what the eCrosby rotation is, is helping you with. Yes, it can. But we don't think it does experimentally and numerically. I don't have it here but there's another instability called the Kelvin Helmholtz instability. Oh, yes, I'm sorry, I thought you were talking about chaos. Yes, that's right. I don't have it in my presentation but I'll refer you to a paper by Professor Adil Hassan and one of his students where they looked at that problem and they they determined that it will not happen in this configuration. That's correct. Now, according to this theory if you scale it up to thermonuclear conditions and thermonuclear conditions really mean that you have to comply with the loss on criteria and that you've heard here, or have enough particles confined for long enough. You have to be here to start the monocular fusion right, which here it means you know very hot plasmas. Now, this confinement time for the radio losses is essentially a Brighinsky's expression and it's a assume the classical diffusion, which is has always been for all plasmas has always been very optimistic. And that is part of what why we're doing experiments that I'll mention in a moment. In the parallel direction. Electrons are a lot more mobile and they can escape until there's a potential that is formed that is a pastic of potential that helps keep them in there. But the, in order to have a measurement with a centrifugal mirror, you need to have, as I mentioned supersonic rotating velocities, and the harder your plasma, the higher the rotation velocity. And so it's estimated that you need on the order of a Mach number of six so six times the speed of sound and the plasma to impose that rotation. So let me go here. Well, how exactly do you impose a radio electric field. So a simple way to do it is just to put an electrode in the center of your configuration, bias it to a high voltage, and that gives you a radio electric field. And obviously have a magnetic field, a traditional mirror like we've discussed, and that will give you the rotation. There's something else that you have to have in order to support voltage drop between flux surfaces so let that electric field propagate through your flux surfaces cannot touch a metal, because otherwise you will short them. So the voltage in one flux surface, again, we assume that the temperature is the same. And it's reasonable to expect that the voltage in that temperature will also be the same. So long as you can have that flux surface, electrically isolated from its neighbors from it adjacent flux surfaces or really from any other flux surface. In order to do that, you have to have insulators. In terms of practical engineering of a reactor, both a center electrode immersed in a high temperature plasma and insulators are a challenge. But, you know, you, you never get anything for free so there's always some some problems you will have to solve both in physics and in engineering. But this is basically what you need to do you need to have some way to impose the electric field and some way to prevent the magnetic flux lines or flux surfaces from shorting so they can support an electric field. Oh, they can form a closed loop. Because well, you need to have a pressure, you need to have a chamber, a vacuum chamber where you sustain your plasma right because it needs to be. You want as high pressure as possible, but it won't be atmospheric pressure. And even if you had atmospheric pressure, or if you did it saying outer space, where you could inject the gas and keep it there etc. You still not need a mechanical support structure for your coils. Somehow, so that you don't want the field lines to intercept mechanical structures. And on Earth, the truth is we need, you know, densities of say 10 to 20, but air densities are what 10 to 25 10 to 26 so it's actually even when you have a plasma there. You still have a fairly high vacuum. I'm saying this in terms of engineering you know you you still need essentially a vacuum chamber. And those are typically made out of metal. It's, you know, the safest is what you can get easiest, but you don't want the field lines to intercept those metal surfaces. The field lines go outside, you know I the cartoon stops here, but these field lines do go all the way around of course you know you, you still have to meet Gauss's law, you know you still have to comply with Gauss's law so that there's few lines and of course there are few lines everywhere here right there they're going on the sides of the chamber. They're going everywhere and all the way around the magnets. That is okay, but where you want the rotating plasma. It's only reduced number of field lines and I'll talk about that more in a moment. In fact, there are of course few lines or flux surfaces here as well, but you may have plasma there but it's not rotating and so you don't expect hot plasma there. Now, a simple way to power the discharge to apply a high voltage and and let it rotate well if you have a. It's a DC electric field you don't need a oscillating electric field. It actually is more challenging DC fields are more challenging that than oscillating fields because all the power that we get from from the wall is is oscillating right. So you have to rectify it you have to make sure you know it has a low ripple etc, or you could also use capacitors and traditionally, we have used capacitors because plasma's take a lot of energy first of all just to ionize them to produce them. And then to support them, they typically get take high power, you know anywhere from hundreds of kilowatts to many megawatts, and depending on the experiment, you know not just for rotating mirrors. It could be hundreds of megawatts right to the support which is very high for any laboratory, where you will do this. But capacitors have no problem delivering that type of power, although of course we're short time, which is usually long enough. Many m HD times. So you can do experiments. So the basic circuit will be like this where you charge your capacitor with a separate circuit, you open the switch on the left hand side. So you don't destroy your charging supply. And then when you're ready to take the discharge, you close the switch. That starts the discharge and at the same time imposes the electric field that gets it to rotate. And we'll see some traces in a moment but you can model the plasma. Okay, again just looking at the electrical engineering as a capacitor and a resistor and you say well, a capacitor. How is that. How, how can that be. And if you look at the stored energy in a capacitor, you of course have the stored energy in your capacitor bank so you know how much energy you start with. So if you look at the stored energy, just assume that the plasma is a capacitor because it's in a cylindrical chamber, but because it's rotating, there's some stored energy in the rotation as well if you equal those two. You get to this expression, and assuming the rotation velocity is VP which is the plasma voltage over a distance. You know, the, it's the electric field over be that's the magnitude of the rotating the velocity rotating plasma and this is of course an average velocity, just for the sake of calculation. And you get to this expression. I don't have it in these slides but you can get to capacitance by looking at the cloud, the chamber and the center electrode as a cylindrical capacitor, calculating the capacitor from capacitance from that, and then using the dielectric of the plasma to calculate how much charge will store throughout and they're not exactly the same but they're similar, the two values that you could get from here. Now, something else that you get in rotating plasmas is what's called the magnetic currents. It turns out, again, starting from the momentum confinement equation and we're right now, this regarding the collision losses. And any viscous heating. So we're simplifying this just for the sake of argument here. If you dot the expression with be to look at the component that's aligned to the magnetic field and assume the magnetic field has this form like we've seen before actually where psi is a flux function. And you can express the component perpendicular to the field as this, which is called the grad Sheffrin of equation. And it includes as a mutual flows here. Dp in deep side so it's the pressure perpendicular to the magnetic field, or orthogonal to the magnetic field. So the expression this expression here is what I showed in the other slides except instead of using, and we use row for the density of the plasma, but it's essentially the same expression that I showed you before. It shows the exponential dependence. We can replace this by the magnum number etc but linearizing this expression. It shows you what's called the diamagnetic currents, essentially how the flux changes when you have just plasma pressure so in a mirror without rotation. The plasma pressure will tend to bow the field lines a little bit, thus creating essentially equivalent to the magnetic currents. But then when you have the plasma rotating that creates even more pressure outwards. So the diamagnetic currents are even bigger. This is a paper for a from a student that worked at Maryland, a while ago. And this shows his calculations of that magnetic currents where you start zero is your vacuum field and 1.5 Webbers is what you will get just from a certain plasma pressure. So with rotating your diamagnetic currents become significantly more and they're modified because the plasma is not nearly as distributed along the field along the magnetic mirror. So anyway, diamagnetic curves are important to take into account as well. This is the experiment that was done more than 10 years ago, which essentially was a long mirror machine with insulators here. They're here they're shown as little plates and a center conductor and post high voltage. And these are the parameters that it achieved central fields of anywhere it could be controlled, you know from zero to point two or maybe, maybe more than that. But the coils couldn't support much higher than that. The mirror field, the strongest field be max will go to typically 1.8 maybe a little bit more than that sometimes. So if you get the ratio of these two, you could get mirror ratios of 10 with this sort of fields, or more of course, if you lower the midplane field, but you didn't really want to do confinement times of, you know, 200 300 microseconds, and we demonstrated supersonic rotation with rotation velocities of more than 100 kilometers per second. The capacitor bank that was used was 1.7 milli farads using the expression that I showed a couple of slides back. You get plasma capacitances off on the order of 100 micro farads. Excuse me. Oh, I'll show I'll show that in a moment but it was also almost in equilibrium with ion with iron temperature. So experimentally, we've managed to demonstrate the hallmarks of the theory that I showed that is centrifugal confinement, which is a significant improvement from just a mirror confinement without rotation and supersonic rotation that was measured with ion Doppler spectroscopy, as well as just looking at the voltage if you measure the voltage that you have across the plasma. You know the radius extent of the plasma and you know the magnetic field. So you can calculate the average E over B, or, or E cross B velocity the magnitude of E cross B over B squared is just E over B. You can calculate the magnitude of that velocity, which again confirms that is supersonic. This results were checked for a variety of mirror ratios, which you have here in this kind of funny way. In in the Maryland centrifugal experiment, does confirming that the density that you should be getting this exponential dependence follows the theory. So, again, even though single fluid MHD is the courses, the less precise fluid theory that we have. And plasmas. This is the adequate for modeling the mirror. We used a spectroscopy as well to look at, again, Doppler spectroscopy, but also to look at what impurities we had in the system. We used high speed imaging to be able to tell at different times, what was happening. We measured the velocity in time. And we were able to put many chords. That is, have multiple views on the same image, we only had one camera, these cameras are very expensive. So we only had one camera so we we made, we commissioned a fiber. This fiber system where you had many optical fibers going into a single fiber that was attached to the spectroscopy. And in one image you could get the information from up to 10 fibers. Then you could use if you arrange your fibers in this radial way, you could get a profile for velocity and temperature. That's not to a point right. If you're, if you're using hydrogen and it's ionized and you get no atomic line emission of course, but if you see the little bit of helium, then helium doesn't, it's harder to ionize so you get signals that are weak, they're not very strong. So you can tell the Doppler broadening, you, you can get, you can get Doppler shift, etc. from those seed helium, and the mass of helium is the closest you can get to hydrogen of course. The voltage and current normally will look very quiescent, but as we started to get hotter and hotter, it started to show this high frequency and low, and I call low frequency components, which are here. And we think that's an instability, you also get, you can also decompose the rotation and Fourier modes just like you've seen in other components, and we think we had two modes beating, say imagine you had a flat tire with holes or nails in two different places and as you're running your car you hear something like that, which was confirmed using B dot probes that were all around the machine, recording the rotation. But even with this modes or disinstabilities. So you get a fast Fourier transform of this. You get that there's components, what I call the low frequency side in the, a few kilohertz to a few tens of kilohertz. Then it, you go up to 300 or so kilohertz, and then they become significant again. And I believe that the high frequency ones that hundreds of kilohertz ones are because of plasma neutral interactions. The low frequency ones are this beating mode that I that I mentioned, we managed to see how, during these modes, you get a lot of plasma expulsion. And you can look at the with the high speed image, we had to put mirrors and so on because our camera couldn't be too close to the electromagnets. You can see the cone where the plasma is rotating that's where it looks. The brightest, and you do have some plasma outside of it but again, it's fairly well defined. But the shape of this cone the width of this cone if you will, will vary with time and and we could never confirm that it was at the frequency of the burst. But it happened many times during a shot for sure. And this is just a way to know where the, the measure the width of this cones that I talked about. And you, excuse me. If you integrate over many bursts, then it gets fuzzy but also the cones are equal between more equal between times which is which indicates that the burst are fairly high frequency, compared to the length of the shop. I'm here because we're running out of time but one of the conjectures is that this mode there was some drag of the rotation at some place that creates field dragging. In other words, it starts to pull the magnetic field with it, the magnetic field lines, which in turn creates equals we drift actually does go to the center but it could also go outside. But because that's not sustainable at some point the system has to reset itself, and that was probably done by expelling plasma, and then you could build it up again and this may happen many times over so this, this was a conjecture that we published. We never had time to be absolutely sure of it. In this experiment. But there's something else called the critical ionization velocity. That critical ionization velocity is was conjectured but by Alfvane as an instability whereby, if you're rotating or you're pushing plasma across magnetic field lines that the velocity at which you can push the plasma will be limited by the essential potential of the plasma. So Alfvane was trying to explain planetary formation with this, but if applied to the rotating mirror essentially says that the kinetic energy that particles can have that ions can have in your plasma is limited. In the case of hydrogen by the ionization potential of hydrogen, which actually gives you a velocity of 50.9 kilometers per second. That is mapped throughout the plug surfaces. In other words, the limitation of 5950.9 kilometers per second is the linear velocity that you can get here at the insulators which is where you have the highest density where you have the coldest plasma, the highest density of neutrals I should say the coldest plasma etc. But we said that you need solid angular velocity rotation so the angular velocity is the same in a given flux surface. Well that gives you a velocity that is higher than 50 kilometers per second. If you map it here, but it could still be below the civ limit which, in this case was 134 kilometers per second. In other words we couldn't go past the civ limit, because the plasmas were over dense. The machine was too dirty meaning it had a lot of impurities, which gets you to radiate power away, which gets you to reduce the kinetic energy that you have. But eventually with a lot of effort, cleaning, you know, changing the culture of how we do experiments and every time we open the machine we literally cleaned everything as much as we could. Eventually we were able to overcome the civ limit in that machine, even if it was for a few for a fraction of a millisecond, and then it went back down to being below the civ limit. And again this is this is what I just explained where you equate the velocity of the particle the kinetic energy of the particle to the initial potential. So this is a summary of results. You start with, this is the angular velocity along the same flux surface, and you start with everything the same we will never tell what was going on here. The linear velocity can of course be different because you're rotating at a different radii with the same angular velocity. The temperature. This is only for ions was measured again to at least 100 eb which was fantastic but you can see the error bars indicate that some measurements were done, even 200 eb which again for us was a big deal. You can tell here that at some point it looks like the angular velocity seems to start to change and you say aha that's the measurement that you were talking about to get this field dragging. That's this is probably not what that is because of that magnetic currents so the the ion Doppler spectrometer was fixed in space, but because of that magnetic currents, the flux surfaces will expand. So we actually shift with respect to the view of the ion Doppler spectrometer, and then you could get the impression that you're rotating differentially, but in reality you may be rotating at the same velocity your, your Doppler spectrometer just didn't adapt essentially didn't change throughout the shop. 10 years ago now we have an experiment that we literally just closed a few weeks ago and we started vacuum and starting plus plasma tests and so on. This is the first superconducting rotating mirror experiment. We, you know, finally had enough budget to buy them but then we realize we didn't have the budget to buy custom made superconductors. So we ended up buying used medical magnets MRI magnets stripping them down taking all the medical stuff and the RF coils because we're not doing magnetic resonance experiments were just we just want the magnetic field from them, which would never go to a hospital and get an MRI. The tunnel is very, very slim, mainly because you have to have all these RF coils and so on and the plastic to make it look nice. If you take all that out, the board is actually almost 90 centimeters or a little bit more than 90 centimeters. And so we used as much as the board as we cool and put a simple cylindrical chamber so we learned a few lessons from the previous experiment on how to minimize plasma surface interactions which we believe were a big problem before the machine itself is longer so I have both MCX, and what's called CM effects the centrifugal centrifugal mirror fusion experiments here. The machine itself is longer, which is advantageous again to to minimize interactions from neutrals. And we have better control where the current from the imposed high voltage and the current that goes across the magnetic field where it grounds. And the fact that the mirror here the mirror throat is long is just a consequence of the medical magnets we didn't really need it to be this long. But that's what we get. Of course the field is stronger because medical magnets, typically have three Teslas. And at the meat lane field, we have right now we have 0.35 Tesla which is more than 50% of what we had before. And trust me that makes a big difference in experiments just any small game that you can get at the meat plane of the mirror is advantageous. But concomitantly with that you have to add a higher voltage so the stronger your field is better for confinement, but then you need a higher voltage, which engineering wise is more challenging. We made the machine such that we could, we could control the mirror ratio. This is not easy to do but because we have to open the machine, change the limiters which are rings essentially. So it's a it's a big disruption. Right now we're starting with this limmer limiters, because that gets you to have. This is the a parameter that I mentioned before of B over a, which is right here. That gets you to require a lower voltage to get a high velocity, you know when a small. Again, you may say well just make be small because it's in the denominator, that will also give you a high velocity of rotation. But if you go back to the single particle theory, your, the larmor radius becomes too large. And then that becomes a problem because it starts to be more non HD. And even if you say well I'll just throw all my computer resources added for the modeling, you still get this favorable conditions for confined. So anyway, we'll, we will test in the future, at least two of these conditions, but then something that we really need to test is the radial confinement essentially what, what type of transport do we really have in the past for mcx. The expressions that have been assumed in publications that we've sent out actually even before I started working in the mirror professor at the office and publish this. Again this breaking ski classical type confinement, which is of course very optimistic. It gives you curves that look like this is our lower ethnic plots. And so you can see that they vary by several orders of magnitude, depending on where you are in the applied voltage, the magnetic field, but also I need to point out that most of the power goes in the rotation. So in mcx the experiment that I mentioned before, we're able to only test. This is for a reactor let me show for cm effects in mcx in the previous experiment we're only able to test conditions way down here. So we're going to impose much higher voltages to test this curves. You know, 50 to 100 kilovolts in this case. If this works, or whatever we get will be able to inform whether we could project this to a reactor scenario. Even with a pessimistic scenario, you still get q greater than one. In the case of very optimistic scenario, you get q, q scientific which you've seen before the power that you get out of the fusion energy over the power you have to put in of more than 200 for for magnetic fields, center field that are for Tesla which are perfectly possible today with radii of two meters or so that is super optimistic, you know I created this plots. It's very hard time believing them so that's why we're doing the experiments to get a better handle of on reality of this on the reality of this. I'll skip to this. Again the machine is running. I, we're almost out of actually we're out of time. I just want to mention that. If you're doing fusion energy, you have to worry about how much the things going to be and how much it's going to be even after you learn to make it even after you are turning them out in steady state on your production plant etc. You know how much the things going to cost, even if you manage to produce beautiful confinement. If the machine is so expensive that no one's going to want to buy it knows when I want to give you the money upfront or give you a loan even to pay it over many years. It's not a very useful concept on their scenarios that we've modeled it, we received funding to pay a company actually to do the costing study for a projection to a reactor for the centrifugal mirror and assuming much more. Excuse me. I'm going to go back to 20 or 21. We get very encouraging results power electricity costs that will be very competitive and this takes into account the cost of the magnets the cost of the machine assuming a certain downtime. The cost of the leasing the land etc. And the auxiliary systems that you will have to have for a power plant. It indicates that it will be a cost effective machine. Now, there's of course right now huge range of values that we will have here. And hopefully with the experiment we're doing now and with Fisher experiments will be able to narrow down this projected Q values. So, I'll stop there. Thank you everyone. Any other question. So for ions it effectively eliminates the ion losses. Electrons can still escape. It helps but it can still escape until you build enough of a potential this past the potential to prevent them from from escaping, but for ions is effectively close and and actually that difference between electrons and ions is what helps you build up this potential very quickly and also stop parallel losses from electrons. Yes. The diagram for what. The diagram for what. You mean. You mean this course here. Oh, okay. I'll get there. Here. Yeah, we have seen that. Oh, the electrode is, you bias the center electrode to impose the voltage that gives you the radial electric field. The line here indicates that your machine is the ground. So you have to the potential differences between the center electrode and the cylindrical shell that you have in this case it's a again it's a simplified diagram. Now that this is terrible for plasma surface interactions. You don't want this type of flat field here you're just maximizing your plasma surface interactions which then take away a lot of your energy or your rotation energy and so this is a cartoon but it's not a good shape, if you will. Yeah, this is just showing the guiding center drift right like just the general direction of the plasma drift. Yes you for individual particles you will have that gyro motion that I showed before. So that doesn't stay on for long because eventually, even in a conditionless plasma eventually those that motion will change direction you'll get a few more tri-coil orbits etc. Is that what you're asking? There's this curve here is just showing the guiding center. So maybe you can pursue this. Yeah, we can we can talk offline if you want. We have to get started. Okay, thank you. Okay, thank you everyone. Thank you very much.