 Hello everybody and welcome to video number 21 of the online version of the fusion research lecture We are in chapter 4 and in the last video we talked about Particles who are referred to as passing particles. Those are particles who are not trapped We have a strong parallel velocity component and who are traveling around the torus This video we will talk about particles who have a significant perpendicular velocity and this might lead to trapping as we will see So if we have particles whose parallel velocity is small compared to the or smaller than the a perpendicular velocity Then the particles can actually be reflected. So the particles can be reflected when they are traveling in the torus because they can be reflected on their trajectory from the out Board side to the inboard side and What causes this reflection simply the gradient of the magnetic field because you know on the outboard side We have a lower magnetic field as compared to the inboard side So this is basically the same as a magnetic mirror where reflection can happen And this has of course the or this has the result that no full Pull loyal turn Sorry is possible So particles Who are reflected they can no longer make a full pull loyal turn now has a let's have a look at an example So here you see I will start a video now where you can see color coded the magnetic field strength in a typical talk amok And you will see now the simulation of a particle. So this is just the Equation of motion solved and you can see how the particle bounces around if it reaches regions of higher magnetic field now to get lower magnetic field strength higher magnetic field strength Lower magnetic field strength higher magnetic field strength, and you see how the particle bounces around the talk amok so Here even you see the gyro Orbit is included in the solution something which we will usually not do But this is definitely possible and this is a rather old simulation which has been done years ago Okay, this was an example of a Standard talk amok with the particle bounces around. Let's have a look at another video So once again color coded the magnetic field strength blue means low magnetic field strength red higher magnetic field strength You see the particle bouncing around between regions of higher magnetic field strength it is reflected When the field strength is too high Assuming it has a significant perpendicular velocity And now you see a cut Through the experiment To realize the trajectory as it might look from this perspective Okay Now Here you can see snapshots of the two simulations which I have just shown as a video and You might have seen that the duration of The particles or the duration of their stay rather is longest at the reflection points as longest around the reflection points and that is simply because the Parallel velocity decreases when they reach the deflection points more and more energy is Transferred into the perpendicular velocity and since the duration of this day is thus the longest there The displacement is the strongest there. So they have the strongest Displacement at the regions close to the reflection strongest displacement there Okay, so now let's have a look at the explanation for this Deflection or for this displacement. This is basically the same. So this is an equivalent not the same Let's rather say an equivalent explanation as Compared to the passing particles as equivalent like the passing particles Now, what does that mean? So let's make a drawing where we have First we start with a coordinate system So this is X and here we have the Toroidal angle Phi and Here we have R Then let's try to draw a flux surface Maybe something like this So a pole loyal cross section here. We have our not Magnetic field points of course into the board What else is missing? We start with the Positive charge located here. So there's a positive charge here and Since we have a positive charge you remember the Drift velocity for the ions is pointing upwards Look at the video from the last video to to get a reminder on that Now let's start with an iron which moves along the magnetic field line direction then here the Sorry, the theta the pole oil velocity will oops. Sorry will point downwards So this is the pole oil velocity for a particle moving downwards And if we follow that Same explanation as for the passing particles the radio getting smaller due to the drift velocity pointing upwards But then as compared on it different which is now different to the passing particles since we do not have enough Parallel velocity, but a stronger particular velocity the particle will get reflected here then it moves upwards again Like this then it moves upwards again, and then it will again get reflected And due to the drift velocity pointing upwards in the end it will close its orbit, and we have such an orbit which is called due to its shape in the projection here a banana orbit, so this is a banana orbit and Due to the so here we have a poloidal projection, of course the poloidal projection As you have seen in the videos the particle also moves in the toroidal direction But due to the shape of the trajectory in the poloidal projection We call these particles also banana particles So we have banana orbits banana particles oops and This is what is also referred to as a trapped particle. This is a trapped particle and Usually a definition for a trapped particle is that the parallel velocity is 0 somewhere on along its trajectory somewhere along its Trajectory now an important quantity is the width of this orbit, so this From here to here. Well roughly this is referred to the width of the banana orbit data B a for banana and Since this is an important quantity for transport because again this allows the particle to deviate from its original flux surface We will have a closer look at that so important for Transport processes is the width of the banana orbit This one is an important quantity To estimate to get an estimation of this width what one has to do is compare the flux surface radii so compare the flux surface radii at the outboard midplane Where we let our particle start at the outboard midplane and in the reflection point and in the reflection point Inserting this into the equation of motion, which we do not do here and then One ends up with an expression for the banana width, which is approximately Data be a which can be approximated by the Larmor radius which we had introduced yesterday times epsilon or This is maybe more interesting here. We are the Displacement of a particle for passing particle and then times one over epsilon And the square root This is maybe more and I said this may be more interesting because it tells us that the banana orbit is by a factor of One over the square root of epsilon larger then the drift plane displacement The drift plane displacement This is important to realize that the banana orbit is larger by a factor of one over the square root of epsilon Then the drift plane displacement so another Important quantity which we will use later is the reflection time The reflection time sometimes also referred to as bounce time. So it's an expression which you might read somewhere bounce time which is the time it takes between two reflections and That is expressed or can be expressed by usually we use tau and then be a for banana Then this is first we have to pie Because tau is defined as one over omega usually so we have to pie and then the width So the distance over the drift velocity and then Assuming an average angle since the particle is above the mid-plane and below the mid-plane moves around Assuming an average angle then we get something like first inserting the expression for the banana width Now also inserting the expression for the polar llama radius so fully inserting everything then we have MV parallel square root of epsilon over qb theta Then the drift velocity for an average angle as I said then we have qr b Then over MV parallel squared and now having a close look at the equation We can see that this one cancels out This one cancels out Square here goes away. This one cancels out and we end up with an expression where we can insert now what we had used yesterday the transit time for passing particles times one over the square root of epsilon and again You have an expression of the transit time for the banana particles Sorry of the trapping time of the banana particles, which is larger by a factor of one over the square root of epsilon as compared to the passing Time for the trapping particles So let's write that down. So the banana particles take Factor of one over the square root of epsilon longer To complete their orbit to complete their orbit Then the passing particles then the passing particles Again two important factors here, which I have written down here Okay, finally This is all interesting, but maybe a few words about the fraction of trap particles So it is an important effect or not the fraction of trap particles So as I said the mechanism for reflection, this is basically a magnetic mirror and a magnetic mirror In a magnetic mirror particles are trapped if The sign this is something which you had learned a plasma physics one the sign of alpha or the squared of the sign of alpha Which is V parallel over V if this is larger than some critical angle often referred to as alpha not Which is the minimum magnetic field strengths over the maximum magnetic field strengths and Using now the expressions for the minimum magnetic field strengths, which we had introduced the beginning of this chapter So be not R not over R not plus R on the output side the magnetic field is the smallest and the maximum magnetic field strengths Be not R not Over and now this is the magnetic field springs at the input side the high-field side inserting this now years for the sign of alpha not then Be not an R not cancel out so it's R not a minus R R not plus R and this can be written as we can get R not Out of the expression meaning that we have here minus one minus small R over R not We do the same in the denominator one minus R over R not and Well, of course these cancel out right so this can we can get rid of this and Then we can write this as one minus epsilon over one plus epsilon and Now one can integrate one can perform an integration Over a Maxwell-Boltzmann velocity distribution so integrate over a Maxwell-Boltzmann Velocity distribution in spherical coordinates in Spherical coordinates using alpha not as a boundary because this defines the trapping condition and then we get a relative fraction of the banana particles The banana particles and just to be sure that the sticks to your mind banana parties here means trapped particles We get a relative fraction of banana particles with respect to the velocity distribution the respect to the velocity distribution We have Ft the fraction of trapping particles of trapped particles. Let's write it. Yeah, that's right a small t FT trapped particles This can be approximated by two times Epsilon and the square root of it Okay That's it for this video where we talked about trapping particles So those are particles who have which have somewhere the parallel equal to zero along their orbit This means like in a magnetic mirror They can be reflected and since in a toka muck we have low magnetic field springs on the outboard side high magnetic field springs on the Inboard side it simply can get reflection due to that gradient in the magnetic field The shape of the particles in the pool oil projection looks like a banana this is why they are called the banana particles and There the width of this orbit is Larger than the corresponding Displacement of passing particles and also the bouncing time the reflection time is longer by a factor of one over square root of Epsilon then for the passing particles and finally we look at the Estimation for the fraction of trap particles which can be estimated by the square root of two times Epsilon yes, that's it for this video. Hope to see you in the next video