Ok, unha sábada, now Alex Trey is going to give his third lecture talking today on oscillations in matter. So thank you very much. That was last slide from the previous lecture where I have shown how things happen in a medium with constant density. Essentially the dynamics of evolution is precisely the same as in vacuum, the only difference is that now you need to take parameters namely the mixing angle and the phase of oscillations in matter. And I have shown the expressions before, so if it is just electron neutrina and two neutrina mixing then you have two wave packets and the distribution of color in each of the packets is determined by mixing angle in medium and the size of the wave packets also determined by the mixing angle in medium according to new Hamiltonian which takes into account also matter effect. So you need to substitute in formulas which we have derived mixing angle by mixing angle in medium which depends on density and on energy which is very important and also substitute the phase which also depends now on density and on energy. So I also discussed the resonance case when the mixing in matter becomes maximal and it corresponds to coincidence of eigen frequency of medium which is related to refraction length and eigen frequency of the system which is given by the oscillation length in vacuum. So this is the resonance condition and then we introduced the graphic representation some delay. So we introduced also the graphic representation of the oscillations and show the equivalence of the oscillations with spin precession of electron in magnetic field and this is the case of maximal mixing when the b vector which determines the axis of the cone is along the horizontal vertex so the double mixing angle in medium is pi over 2 and so rotation occurs rotation of the polarization vector of neutrina around the cone with axis given by vector b. So this is the case of constant density and if for instance you started from this position so you produce electron neutrina because electron neutrina corresponds to vertical direction up and if the length of matter layer is such that the phase is pi over 2 then you will end up here so you will have maximal transition from electron flavor to muon flavor. If the layer has a big length then your vector polarization vector will rotate and make many rotations. So what is the power of this graphic representation and this you can see probably we need to substitute of course. You need to see here. So suppose you have medium which consists of three layers with different densities and this actually corresponds to realistic situation when neutrinos are propagating inside the earth because in the first approximation the earth density profile looks like here this is the realistic density profile and it consists of two layers one is the mental and then the core layer and there is a sharp density change between the core and the mental. So we have roughly speaking in the first approximation such a type of profile then see how easy to see the evolution in this case. Suppose you start again with electron neutrina this is point one which corresponds to such a position and then neutrina propagates in the first layer. So that would be equivalent to precession of polarization vector at the surface of the cone with the axis which actually inclined by two theta m and mental. Remember that the b vector it has direction determined by mixing angle in matter multiplied by factor two. So if layer would have big length then you would have just this type of the precession and again let me recall that projection on this axis z minus plus one half give the probability to find electron neutrina. But suppose the layer has a finite width and neutrina arrives at the border between these two layers in the point two and so it make just this evolution part of the rotation and then in the point two it enters the second layer now in the second layer density is different and therefore the mixing angle in medium is different and therefore direction of b field is different and suppose the direction of the axis or the field b is this so that now starting from the point two the polarization vector start to precess around this new direction so it start to precess in this way. Again we have here finite size of the layer and in the point three you make an evolution that it starts here and end up in this position. Here in the point three neutrina enters the third layer and in the third layer the density is the same in the first one and therefore again the cone will have the axis in this position so that now the polarization vectors start to rotate again along the old direction and so it will make rotation here and since again the size of the layer is finite it may end up in the point four so the total evolution will be like this so you start here in this point one then it go to this way then to this way and then to this way so in such case you can get very strong transition even if mixing angle is not maximal so this is so called parametric enhancement of oscillations here you see the evolution change of the probability, transition of probability so you build up big transition probability this is in the first layer, this is in the second one and this is in the third one although in each layer the transition may not be very strong and in the end of the day you will have almost maximal transformation so this is another example which is parametric enhancement of one to mixing the evolution is also realized this is realized inside the earth but for different energy range so it may be a more complicated motion like you start here then in the first layer the evolution is like this in the second layer evolution may be even like that in the third layer like this so again you end up with almost maximal transition and actually you can cook some sophisticated density profiles you can make different transitions and one of examples is here suppose you create the density profile of this castle wall type and then the evolution can be like this like a spiral change of the direction of polarization vector questions so this is so called parametric enhancement of oscillations another effect is realized in nature and should be realized I think there is no doubt that it should at some point correctly if I get it correctly if we would be able to build an experiment with sandwich of different material density that would be optimal to get to these results so for example if we would shoot a beam through rocks and then water and then rocks again like intersecting a sea or something right so there are some in principle yes so you can even have very small mixing but due to many this type of the parts of this period quasi periodic castle wall profile you may produce big transition probability in reality I will show how it is realized in reality so the oscillation length is quite big usually and therefore you cannot use a small like you are saying ok mountain and lake so the oscillation length typically in realistic experiments is comparable say with a part of the Earth's radius and therefore you need to have big structures to realize this or you can go to very low energies however in this case it is difficult to realize such an experiment however one can think so that may be an interesting point now suppose we have a varying density smoother and slowly varying density on the way of neutrinos such a situation is realized in the case of solar neutrinos and supernova neutrinos and also in the early universe there are many applications of such a situation if the density changes with time or with distance our Hamiltonians start to be dependent on time and the situation becomes more complicated so we still have such an equation and new F are the flavor states let us again consider two neutrino mixing and we can find the eigen states of Hamiltonian in medium but now since Hamiltonian changes in time we can speak about instantaneous eigen states so to say we can diagonalize our Hamiltonian in a given moment of time and find these eigen states of the Hamiltonian in the next moment because density changes of course the state will also change flavor states are related to the eigen states of the Hamiltonian by mixing matrix in medium and the angle now becomes also the function of time because it is function of density density depends on time and therefore mixing angle changes now let us find the question of motion for these eigen states in medium so for this we take our original question and substitute relation of flavor states and eigen states and then we will get such a type of the question you see what happens in the contrast of the case with constant density Hamiltonian is not diagonal and there are off diagonal terms which depend on the change of the density and therefore mixing angle with time so because of time dependence of mixing angle Hamiltonian is not general diagonalized so if you would have constant density then here we would have zeros and the only non zero element is here which actually is realized in constant density case in in vacuum when eigen states diagonalize also evolution equation so equation of motion not only Hamiltonian instantaneously and then solution is trivial because then we just have these eigen values and we solve trivial equation so change of the density leads to appearance of off diagonal terms and therefore in this case we will have transitions between new 1m and new 2m because we have off diagonal terms of Hamiltonian which precisely describe transitions between eigen states however in many cases these off diagonal terms can be very small they can be much smaller than this diagonal term and so such an equality is realized and therefore in the first approximation which we call adiabatic approximation we can neglect these off diagonal terms and therefore we are back to our standard usual previous situation of oscillations in vacuum and in medium with constant density so if we neglect these off diagonal terms then eigen states propagate independently as we discussed before so like mass states in vacuum so I have repeated this equation again so this is condition of adiabaticity and so the essence is that if this is satisfied then I can neglect transitions between eigen states in medium this means also that shape factors do not change so remember in evolution in vacuum and in medium we have these wave packets and shape factors didn't change so the only what happened is the change of the phase this condition is actually very crucial in the resonance layer where the mixing changes fast and level splitting is minimal in this case and if vacuum mixing is not big this condition, adiabaticity condition is reduced to this one the size of the resonance layer should be bigger than the oscillation length in resonance so what is the width of the resonance layer it is given by such an expression and actually it's proportional to mixing angle remember the smaller mixing angle the narrower is resonance and so also the density change with distance plays the role so if the density changes very fast then this condition is not satisfied because if density changes fast then mixing angle changes fast and therefore you will have breaking of adiabaticity and this is oscillation length in resonance which is given by oscillation length in vacuum over sine to t time so you can extract all these results from foremost I have given last lecture so what is the adiabatic evolution in the case of adiabatic evolution new one m and new two m do not interchange so they do not transform one into another and so the sizes of the wave packets the amplitudes do not change because if they would have transition between each other then the size of the wave packets of the amplitude would be changed but what changes is actually the flavor composition of eigenstates remember that mixing determines the flavor composition of eigenstate since mixing angle changes then the flavor composition changes which means these red and blue parts in the course of propagation so if you start with high density then you may have such a configuration you see here the biggest part is red which means electric flavor and the small one is green one muon flavor but in the course of propagation mixing angle changes and therefore the flavor composition changes and you may end up with this in vacuum so suppose this is in the center of the sun and then on the surface of the sun you would have something like this and you see if in initial state red flavor dominates then here green flavor would dominate so this is the adiabatic transformation of course you would have also oscillations I haven't put here these oscillatory factors so there are two effects one is oscillatory behavior due to interference of different parts and the second and this is new degree of freedom which start to operate here is the change of flavor of individual eigenstates this change occurs in accordance to change of mixing angle which uniquely depend on the density questions see this picture if you understand this then so we are done almost you will understand all the phenomenon related to neutrino propagation essentially remember in vacuum these parts didn't change so red and green in each of the wave packs they were fixed by mixing angle so essentially this is determined again by cosine and sine but now of mixing angles in medio ok so now let me derive the formula which describes for instance evolution of solar neutrinos extremely precise so that will just be one slide and I will derive formula which is valid with accuracy 10 to the minus 8 even experimentalists are not doing just you know blind computation so they use this type of the formula and it's very simple it's valid for solar neutrino supernova so we have initial states suppose we have electron neutrino like in the sun and it is given in terms of eigenstates in initial moment so neutrinos are produced in the center part of the sun and so we need to compute the mixing angle in this initial point in the production point and then just by definition of the mixing in medio electron neutrino can be written in this way again in two neutrino case cosine theta m and this is mixing angle in matter in initial point this is nu 1 m and sine theta 0 m nu 2 m agreed so this is mixing angle in matter in initial point now what is adiabatic evolution just following what I have explained before this evolution when neutrino moves from the center parts to the surface and so potential changes from some big value to 0 what happens is that nu 1 m state just slowly transforms into nu 1 and nu 2 m slowly transforms into nu 2 because nu 1 and nu 2 are eigenstates at slow densities in vacuum and since there is no transitions between nu 1 and nu 2 then the only what happens that nu 1 eventually transforms nu 1 m into nu 1 and nu 2 m to nu 2 agreed well please ask me because it's just a few lines more and we are getting a result she wishes now it's pitted if you miss this but you can repeat this and nothing complicated ok now the final state therefore will be the following I wanted to do this on the blackboard but I don't think it's you will have any way in this slide so what will be the final state mixing angles so the mixtures of this nu 1 m and nu 2 m do not change these amplitudes do not change and they are determined by mixing in initial moment so you produce this state the mixtures of nu 1 m and nu 2 m are determined by mixing in initial state and since there is no transitions between nu 1 m and nu 2 m the amplitudes are the same and again determined by just mixing angle as in initial state and the only difference between this and that is that now I have this nu 1 and nu 2 and of course the phase, some phase appears here ok so now we can compute the probability to find electron neutrinos and for simplicity just use expressions averaged over oscillations now it starts to work better so what I will get is the following I need to compute this matrix element so I need to take this state and plug it with nu e and to compute modulized square and what I will get is just this expression so the first term just comes from this one and then I take projection of nu 1 on to nu e which is given by cosine theta and I get this and then I square this and the second term comes from here because I have this sine theta m and then I project nu to m on to nu e and this gives me this sine theta and I square this and there is of course intermediate term which is just described as oscillations and in this case I just averaged out this term which actually happens inside the sum so that's it, this is the formula and I can write this in a different way or even in some other way you can use also so this is the same formula so this formula describes solar neutrino transformations exactly with accuracy 10 to the minus 8 the only what you should remember to derive it that evolution is adiabatic and this means that there is no transitions between eigen states so we neglect transitions between eigen states but we do not neglect change of the flavor of eigen states so you neglect some derivatives in your equations roughly speaking but do not neglect the function of the pandas questions? yeah it looks like every neutrino sees the same but you neglect the function of the pandas but you neglect the function of the pandas questions? yeah it looks like every neutrino sees the same transition in the potential from V to 0 as if they were produced always in the same place you would expect neutrinos to be produced in many position in the sun and therefore transiting through very different potentials so actually neutrinos are produced not in one point they are produced in some region which is for some neutrinos it's quite small point 8 of solar radius the density profile inside the sun actually is flatten when you go to lower energy sorry to central parts it's like this so this is exponential part and neutrinos are produced somewhere here but you are right so you need to make averaging this is for neutrinos produced in a given point but eventually you need to make averaging over production region right was some other question ok good so yeah so this is for probability for neutrinos produced in a given point with a given density but of course neutrinos are produced in some region and there is some interval of densities and then you need to make this averaging good so this is how this evolution looks like when you study survival probability as a function of density so what happens is that the formula which I have shown you corresponds to this red red line when I made the averaging over oscillations and so if you start far from resonance this is resonance density you start above the resonance density then you have very small oscillations here and you end up in this way if you would start close to resonance then you will have bigger oscillations here if you start very far from the center if the density initial density is very high like in supernova case then essentially this band just degenerate in the line so the evolution becomes non oscillatory just a smooth change of the flavor of the whole state which follows the change of the density so this is the same in terms of eigen values remember we had this picture the resonance, these are eigen states as the function of density and if you start at large density somewhere here so this is the resonance you are essentially sitting on one of these eigen states so here you have electron line so this is the energy of electron neutrino effective and these are just our eigen values the fact that there is no transitions between these levels means that if you are sitting here you will continuously sit down here and evolution when you go from large density to small will be just a smooth transition along this way and you end up here so this is in terms of eigen values and let me make just one comment about adiabaticity violation actually the nature is very kind to us it seems that in all relevant cases the evolution is adiabatic but just for case, for instance if you have some sterile neutrino with very small mixing or you have shock waves propagating inside the supernova and then in the shock wave the density gradient changes is very big and so you may have violation of adiabaticity in shock waves propagating in supernova in this case of course you cannot neglect transitions between new one and new two and what will happen is the following suppose you start at this level you are sitting here but then since density changes quick you are moving with fast velocity along this level and there is a non zero probability that you will jump from this level to that it's like when you are driving the car you didn't turn with high velocity you may not turn appropriately well the same phenomenon if you want you mean so that's physics is unique stuff again so if density changes fast then you may have jump probability and jump probability is given by such an expression which is just ratio of the difference of the potential difference of the between eigenvalues over the energy associated to this motion which is kind of physically also clear and this energy associated with this motion is just inverse of gradient of density change there are more precise formals which you can see here now I will show you how these things look like using graphic representation and adiabatic conversion is this type of phenomenon so remember if you just have constant density then the state just precesses around the surface on the surface of the cone but now mixing angle changes which means that cone itself turns and this angle is 2 theta m density changes mixing angle changes and therefore the axis of the cone magnetic field changes actually the same phenomenon is also in the case of precession of electrons around magnetic field then you can switch of polarization of your electron which will follow the change of the magnetic field so you see here cones in different moments of time clear and in this way you may get a very strong transition if you start here even if the cone angle is small but because of rotation of the cone itself you can transform your between a polarization vector from this direction to this one so you may have a very strong transition flavor transition in this case the size of the cone angle doesn't change so you see the cone is here and this reflects the fact that there is no transition between eigen states so if adiabaticity is violated then the situation is like this so not only the axis of the cone changes but also the size of this rotation angle changes it increases and that reflects transitions between eigen states so you see here another degree of freedom becomes operative not only the phase change which describes rotation of the polarization vector on the cone surface but also the axis of the cone changes so this is just summary which tells you what is the difference of the oscillations and this adiabatic conversion in the case of oscillations this degree of freedom is operating the phase as the function of time and the mixing angle doesn't change but it is function of energy in the case of oscillations in matter for adiabatic conversion in the case of non uniform medium we have this and other degree of freedom mixing angle which changes with time this is dynamical wherever now and this is finally how it looks like so suppose you have the layer with constant density and you saw already this picture then in this case you will have the following distortion of energy spectrum so the ratio of the spectrum at the detection point over the original spectrum will be like this periodic or quasi periodic function it will have oscillatory behavior in the case of smoothly changing density and after averaging over oscillations you will have the following dependence of energy of probability essentially this is the probability the function of energy so it just smooth change and this is the size of these probabilities just sign square theta you saw this in one of the formulas for adiabatic transformation yeah wait wait wait so you talked about the adiabatic conversion and we also saw the density profile of the earth maybe this is related to the previous question but that does look like quite a big jump at that point does that also lead to these oscillations between the matter eigen states or is that because the oscillation length is too long not really relevant in this scenario so in the case of the earth we had kind of the system of layers with quasi constant density there is no adiabatic transformation here or transformation is adiabatic within the layers but then at the border of the layers the density changes so quickly that it is just maximal violation of adiabaticity of course and of course this means that at this point the eigen states change completely suddenly so you had one system of eigen states but then they change immediately so that's gone what happens so now you know everything about neutrino propagation almost I will skip this conclusion and there is a kind of list of the papers original Wolfenstein and so our papers and actually this this is 30 years from the my first presentation of MSW effect which is this adiabatic conversion and it was in southern lina just it was a conference from 16 to 22 of June and it was the first first presentation of this paper now phenomenology and I will discuss fluxes, detection and the results so let me tell you what I'm going to do of course it is not possible to cover all the experiments all the results so I will just try to give you the flavor about the most important experiments essentially running now and so what are what is the outcome and tomorrow I will discuss some future so what we are expecting in the coming years so I will start from solar neutrinos and here you see the flux of solar neutrinos the energy range this from zero to say 14 MEVs so solar neutrino spectrum is in this low energy range up to say actually 18 GEVs there is also half neutrino you see here several different components like proton-proton neutrinos or P-neutrino which are neutrinos first reaction inside the sun and two protons create neutron and then electron positron and electron neutrino so there are two main chains of nuclear reactions inside the sun and eventually they lead to transformation of four protons to helium four so in the sun we have burning of hydrogen and creation of helium core inside and there are several reactions the chains of reactions which produce this complicated spectrum again this P-P is the biggest one then we have beryllium neutrinos these are neutrinos which are produced due to capture of electrons and therefore you have the line then there is this famous boron neutrino which was first detected and there is also P-E-P line and here I don't see but there are some also other components which are from CNO cycle which is kind of subleading cycle of nuclear reactions inside the sun you see here also the list of different experiments and their sensitivity range so for instance super K and SNR sensitive only to this high energy part so they essentially detect boron neutrinos chlorine experiment this is historical first experiment that covers bigger range of energies and also it's sensitive to beryllium neutrinos now gallium experiments they are sensitive even to bigger energy range and boraxina also can detect very low energy neutrinos so what happens with solar neutrinos they are produced in the central part of the sun they adiabatically propagate inside the sun and they undergo conversion which I have described just a few slides before coherence here is lost already inside the sun so which means that you have no already interference effects and you can understand this in terms of the separation of the wave packets which I have discussed then this neutrinos nothing happens on the way from the sun to the earth apart from the fact that the flux just decrease like 1 over r2 and then what happens inside the earth when neutrinos and they are arriving as mass eigen states new one and new two these states start oscillating again inside the sun inside the earth because in the earth we have matter and therefore the eigen states in matter are something different new 1m new 2m and let me put here earth and this also splits and therefore we will have oscillations between new 1 and new 2 so interesting that in matter mass states oscillate not flavors, mass states oscillate and that also affect the flux which you are detecting here now these are pictures of some experiment, these are historical experiment homestake which uses this process to detect solar neutrinos and two gallium experiment sage and gallix and then it was GNO which uses this reaction of electron neutrina captured by gallium 31 these experiments do not run so this is the famous Davis experiment but still some results kind of affect us and not only on solar neutrinos gallium experiments made so called calibration of the detector so to say they put source inside the detector just to see if they understand correctly the detector if cross sections are correct and what they have found something strange so they have found that the signal is smaller than they expected so they know source because they prepared some radiative elements quite this big and it was chromium and argon put inside so they knew what is activity they could even measure by heat release so what is the power of the sources and then they saw some signal which is below what is expected something like by 20%, 15%, 20% the reference is not statistically very significant but all experiments saw this deficit and now what we call this we call this gallium anomaly I will discuss this next lecture and one of the explanation is that there are some other neutrinos species involved sterile neutrinos and this kind of small value of the flux is because electron neutrinos or electron antineutrinos produced by source partly were oscillating into sterile neutrinos and there is a number of new experiments which will start soon to check this now these are experiments which are kind of more recent supercomiacan they still running so they study scattering on the electrons and this is a huge water sharing of detector what you see here is this huge detector filled in by water and this is something like 50 kiloton of water tank and at the surface you see photomultiplier so they see what happens inside the volume and if neutrino interacts so it actually electron so electron moves quickly produces sharing coflite and this sharing coflite is detected at the surface of the detector by these photomultipliers now SNO result and this is still running experiments so I will show you some recent results from this experiment SNO, a Sudbury Neutrin Observatory uses heavy water and they managed to detect three types of reactions one is this charge current therefore this is D 2-O so that's deuterium yeah? sorry I cannot hear you how does super cave distinguish between muon and electron neutrinos? no but electron neutrinos have a bigger contribution because it's due to charge current and so muon neutrina and tau neutrina also produce effects so you need to take into account this but is there any effect in the sharing coflite that comes off from the differences? no because it's from electron if electron has the same energy then it produces the same sharing coflite so you do not detect neutrinos themselves so you detect recoil electron so the important process here was this famous neutral current process you have neutrina just which splits deuterium neutron, proton and this neutrina and this is because this reaction is actually is produced by all neutrinos species so muon and tau and it doesn't affect it's rate by oscillations neutral currents are the same for electron muon and tau neutrina ok? and therefore this reaction is not affected by oscillations this one this is due to charge current this reaction is affected by neutral current this process is also affected by neutral currents and other reactions and oscillations also comparing the rate of this charge current and neutral reaction you can immediately realize if there is some leakage of electron neutrinos and that was the first real proof which doesn't depend on solar neutrina models that we deal here with some flavor transitions now but actually decisive experiment was not even solar neutrina experiment to resolve the solar neutrina problem it was a camland and camland is this big scintillator detector you see man here staying so I just everywhere here you can see what is the size and this is scintillator something like one kiloton of scintillator also view it by photomultipliers attached at the surface of this volume so they detected this classical inverse beta decay reaction at the first reactor experiment and neutrinos were taken from atomic reactors so atomic reactors produce huge fluxes of electron anti neutrinos and they detected this neutrinos not just from the closest reactor but from many from all the reactors in Japan and even in South Korea so the idea is to have big distance from the detector to the sources and remember the process is actually characterized by oscillation lengths and the oscillation lengths is let me give you a numerical expression so that you can so this oscillation lengths can be estimated as 10 to the 3 kilometers one GEV energy over one GEV and delta m2 over 2.5 10 to the minus 3 electron volts squared so if you have neutrinos with energy one GEV and delta m2 equal 2.5 10 to the minus 3 then the oscillation lengths is 10 to the 3 kilometers so this is the biggest mass split but the smallest mass split which is actually operating in the case of solar neutrinos is something like 30 times smaller sorry split is smaller but oscillation lengths is bigger so this is for 1 2 so this is say 1 3 due to the bigger mass split and this is for 1 2 will be given by 3 10 to the 4 kilometers so this number instead of 10 to the 3 now reactor neutrinos are low energy neutrinos so this helps so you need to take here not one GEV but something which is say at least two orders of magnitude smaller and then you get in the case of solar neutrinos you need to divide this by say factor of 3 10 to the 2 and what you will get is something like 100 right so then it will give you 100 kilometers that will be the oscillation length of reactor anti neutrinos with energy say 3 MAVs and luckily Super K was in the play and also this come one because this is also in the same place in cameo commun was roughly at the distance in average 180 kilometers that was average distance to atomic reactors and so they can measure squares which are relevant for solar neutrinos I will show you in a minute what actually the present day situation so what they observe is the following so that's dependence of survival probability on the L over E so this combination enters the oscillation phase right so you can change L you can change E and they manage also to extract not only energy dependence events and measure the air construct energy of your events but also you can have some sensitivity to L because some reactors were switched off for some periods some not and so this also effective distance also changed and they saw this very nice oscillatory picture and they managed to measure with high accuracy mass square difference delta M square with worse sensitivity to mixing angle but mass square difference was measured very nice questions so unfortunately no because neutrinos have lower energies there's very little sensitivity to the direction because you need to go to higher energies comparable with proton mass to have because the energy of neutrina is comparable is much smaller than the mass of the proton so the scattering although not completely isotropic but then you have also events from different sides and so they didn't use this yeah it's a nice question there so this is the best feed point using also three neutrina oscillations you see how the data described I'm sorry but I really to understand the meaning of this oscillatory result I mean so what you are having here the land of the neutrina what is traveling or I just don't get it okay so what you see here is essentially the change of when you change let me do the following so suppose you have the same distance suppose all your reactors are the same distance 180 then you change energy so when you move from here to there so you change the energy so here you reduce energy then you reduce energy because it's indenominator the face of oscillations remember the face of oscillations let me just write probably of this picture so if I put here energy not inverse energy then what I would expect here is the following I would expect such dependence of probability survival probability this what matters here the function oh sorry better to do in this way so one over e then I would expect this this is because the face of oscillations is given to two four pi energy and over delta m2 now this is the period of oscillations and this is the depth of oscillations which is given by sin2 to theta so the formula remember it was 1- sin2 to theta sin2 f over 2 and so I'm wrong here because it should be opposite right so it's delta m2 over two energies so and L so this is how face is determined so again let me repeat this is the face and here is one half of this face and it is inversely proportional to energy if L is fixed then it's so this is just like here like this function and the depth of oscillation is given by sin2 to theta and this is precisely what you see here so taking into account also the average ink effect partial average ink effect efficiencies and so this is nothing but part of this oscillatory curve and from the depths you extract mixing angle and from the period you extract what is delta m2 yeah so how do you experimentally evaluate survival probability I mean what is the initial flux of the neutrinos that how do you know and as you said the detector is not sensitive to the direction so how can you be sure that what is the length that the neutrino has has traveled I mean whereas it produced right so what you need to do you need to compute number of absorbable effects so you know what is the probability you need to take this probability you need to make computation of the effect from each individual reactor taking into account that they are at different distances fortunately for the most power reactors and kind of those who contribute a lot they are quite close to the detector so if you would have kind of complete average ink over distance of course you would not see oscillatory picture now of course you need also to know the flux and in this experiment they just use some computed flux of electron and neutrinos in the modern experiment so in the present day experiment they have some kind of other control there are some uncertainties in this of course there are uncertainties in the flux now people say that this is maybe up to 6-10% even uncertainty in the flux you can extract the flux from the power of reactor because there is kind of good relation between power of reactor or the heat released and the flux of neutrinos ok but to extract for instance delta m2 no need to know this absolute normalization of flux so this kind of analysis doesn't depend much on normalization of the flux because you extract period from here but changing just up down you do not change period here right so delta m2 can be extracted very nicely and actually this experiment gives the best measurement of delta m2 to 1 one of this mass square differences just because of this nice description of this period less sensitivity to mixing angle which is also affected by normalization on certain days actually my main question was actually the length the length that the neutrinos travel because you have a point that say 50km per meb how do you know that I have to put the point there in 50km again what I'm saying to you I reply to your answer because this is observable curve which means that this for instance blue line it takes into count already result of summation overall of course you cannot distinguish camioca mind you cannot distinguish if this a given neutrina came from reactor at 100km or 500km or 300km so what you are doing is just doing computation of total flux of these neutrinos taking into account different distances from just summing up unfortunately this summation doesn't lead to a complete averaging on the ground of neutrina individual nutrient detection of course you cannot say anything so you detect many neutrinos taking into account all the distances all the different spectra even different reactors give slightly different spectra and then you get this green part and then you confront it with experimental observations as usual I have a very small question isn't this reaction producing a lot of energy if this reaction what isn't this reaction producing a lot of energy which one so this one you see neutron is slightly heavier than proton and so the energy which you are producing here is the energy of neutrina minus what is this two MEVs so the energy spectrum of reactor neutrinos is from say 0 up to 8, 10 MEVs so it doesn't overshine the chairing cof radiation because it's a small explosion essentially it produces chairing cof radiation of course so these detectors actually they have kind of they can detect both chairing cof light and they can detect also scintillator shining so they since they are not interested in directions so they cannot detect directions what is important for them is to detect all the energy release from a given reaction ok now solar neutrino data last year two very important events happened one is that finally super k announced say that they see a three sigma level earth matter effect a symmetry of the signal between day and night and this what I have explained this is because of these oscillations of mass states que arise at the surface of the earth inside the earth and you see here dependence of this so here is dependence on the zennet angle the biggest effect of course for neutrinos which are coming exactly from down to up right and the effect is about say 4% so 3, 4%, 3, 4% is what is asymmetry you see now we are kind of really in the phase of precision measurements also I should say that there are kind of big fluctuations of the point you see there is some point here this what is what is actually predicted this is blue and this is the best fit point and this is asymmetry when you sum up the signal over night and over day so this is the difference and this is average ok there is also energy dependence so one expects that the effect should increase with energy and it somehow increases but also some fluctuations are here and now there is some puzzle and this is one of the puzzles in Neutrinophysics and so some experiments are actually keen to check this the first of all the day night asymmetry which is observed is somehow bigger the fact is something quite too bigger than what is expected if you take delta m2 according to Camelante result which is the most precise measurements of delta m2 and you see here so these are experimental data and you see this in average something like the effect is say 3% now if you take the data from Camelante so this is the range of delta m2 determined by Camelante experiment then you should see something like less than 2% day night asymmetry so this is kind of a bit puzzling and then I will show you some other things which might probably puzzle this the second important event last year is detection of neutrinos so this is proton-proton neutrinos from the first reaction inside the sun so the proton-proton reaction and Boraxin experiment managed to do some wonderful things so they dig up the effect of this proton-proton neutrinos from big background for instance here you see some other things so this is actually to extract from the data this type of the flux so Boraxin experiment measures neutrino-electron scattering neutrino-electron scattering at very low energy so this is QEV range energy so the release is very small and so they manage this so you see here the energy in QEV something like 100-200 QEVs and so that's because of very high purity of this experiment and here you see this experimental point in the plot when I show this probability of new E to new E transition and this is what is expected according to adiabatic conversion which I have explained to you before and you see data nicely lying on what is expected now this is some more global result which we put here experimental points from different experiments this is from Boraxin this is also Boraxin major Beryllium neutrinos PEP neutrinos and there are many points these dashes are again from Boraxin but what are important here are some points which have very small error bars like this from SNO and also from super-comactant they are measuring very precisely this high energy tail of solar neutrino spectrum and here there is something first of all let me say the following here at higher energies matter effect dominates so that's essentially these lines are converging to sine square theta which was in our formula here at lower energies this is so called vacuum dominated regime this line actually converged to just averaged vacuum oscillation result because here delta m square is bigger delta m square over energy is much bigger than the potential so in fact these regimes they are when you compare delta m square over two energies so this is vacuum contribution with V potential versus V so if this guy is bigger than the potential maximum say inside the sun then we say this is vacuum dominating regime in fact the results of oscillations are close to what you get no matter effect now if this is much smaller than V and this happens when energy is big then it's matter dominated regime and you see there is kind of intermediate range which is not covered by experimental results and we expect this up turn and now there is another puzzle that this up turn hasn't been observed in no one experiment yet and you see for instance as no see even turn down of the spectrum also super K see some kind of reduction so it's not clear what happens here that's one of the important goals in neutrino physics to test what listen solar neutrino physics to test what happens in this intermediate range so we are expecting some new physics at LHC it may happen that new physics may appear here so there are several explanations possible of this and actually also the same probably this fact is related also to this day and night asymmetry bigger than is expected that may be due to so called nonstandard interactions so maybe on the top of our usual interactions there are some additional nonstandard interactions so they may produce such an effect another possibility if some new neutrino states exist I don't know if I have taken this plot no maybe no I have not taken so here you see two curves one is from best fit point of just solar neutrino data and this is okay but blue one is just going by far from this point so this blue one corresponds to delta m2 from from camland data so it's not clear maybe something is wrong with camland data and actually recently we have realized that the fluxes of anti neutrinos we used before are somehow wrong so it is realized recently that there are some certain bumps which haven't been observed taken into account before so the situation is a bit messy I don't know where is actually the reason of this type of the discrepancy actually I tried this with Maltony and the result is kind of unexpected because we tried to implement this new determinations of the flux of anti neutrinos from reactors and we have found that discrepancy even increases so this is one of the hot topics nowadays now people start to recalculate again the fluxes of anti neutrinos from atomic reactors it's old subject you know that rhinos have discovered neutrinos using atomic reactors and now we have this puzzle with atomic reactors so if you introduce non standard interactions then these lines can be changed by something like that and if additional sterile neutrinos are introduced new neutrinos species that maybe even like that something like that so it's interesting to watch what will happen in many cases in the past these anomalies appeared and disappeared but anyway it's interesting to see the developments here probably I will show you such a plot and this is the determination of 1, 2 mixing and delta m2 1, 2 and let us see just this plot so what do you see here the loud region of parameters mixing angles as you see this 1, 2 mixing is not small it's something like 0.3 and delta m2 which is around 4 or maybe 4.5 this is the best feed point from solar neutrino analysis only and you see what comelant is giving the comelant is here it's something like 2sigma from here but it seems that this new update of the spectrum may even move this up so this is the present day situation not only this you can use data to extract the potential matter potential and here what we did we are not the first that if you try to extract the potential from the data considering it has three parameters so actually Wolfenstein make these computations and it looks like it's just standard physics and this is just square root of 2 multiplied by fermi coupling constant and density this looks standard but if you extract this from data you will get something like 1.5 times bigger so another puzzle probably related to others here what you see the problems and future so what I mentioned already absence of upturn large day and night asymmetry difference of delta m2 large value of the potential and we don't know what is this another reactor another reactor normally maybe something happens with solar neutrinos what is future future we want to detect this SNO neutrinos so these are components which have not yet been detected solar neutrino components it's difficult because not because they are very small because they are in very dirty region because there are many backgrounds in this so this is one important problem because that may clarify some astrophysical issues that are related to the sun then we need to make some more precise measurements of PP and beryllum neutrino fluxes in this way to have further tests of all this type of evolution and it's interesting to make detailed study of the Earth's matter effect even tomography of the Earth can be done so let me finish and then we give a last and what are future experiments super cable continue to operate an SNO plus will start to operate soon and these are a remote future more remote so it's that June experiment can also detect solar neutrinos hopefully but it will start in 2020 and that's even more remote maybe 2025 this is the biggest version of super cable this is hyper cable experiment again I'm interested to know when they see neutrinos fun when they see signals for neutrinos inside say super k o hyper k how do they differentiate whether it's coming from a pp process of BE or SNO ok so super k doesn't see super k has quite high threshold so they just see bottom neutrinos only so there's no problem with super k and SNO they have higher threshold now a low energy range the situation is like this so boraxino experiment and there were some well let me proba go back to so that's one so you see that is the signal which is due to pp neutrinos this is due to this is due to beryllium neutrinos so that's typical spectrum of record electrons you are expecting so you see you can distinguish just using say for instance this range because pp neutrinos are not contributing here to measure beryllium neutrinos and use then this range to measure pp neutrinos so they are separated in energy so that's more there are some energies where the contribution of one of the components is dominant also is there some spectral flux that they fit to see whether in the equal energy of the electron you see a certain shape that gives you the shape of a beryllium neutrinos to some extent yes you see the shape is different here and there so you can also they are using also the shape but you see you see not the shape of the spectrum itself they see a recoil spectrum but of course if you have beryllium line which is original line still for electrons you will have a kind of white distribution for pp already original flux is kind of white it's not just line and therefore you see not like a peak here but some kind of richness and I assume that the shape would be different for atmospheric neutrinos right oh yes so here you can compute what is the contribution of atmospheric neutrinos of course they did this and for these energies it's small so let me ask you yeah by the discovery of the of matter effect what do you think about the determination of the delta m atmospheric square sorry to the determination of what delta m atmospheric square oh, that's next topic actually I started from solar neutrinos and actually sometimes this delta m square one two and quita one two are called solar mixing angle solar delta m square so and I want to ask you I can probably spend 5 minutes because there were some questions no no you are sorry you are not yeah I should so can I spend 5 minutes more we had some questions just to move probably just this and then I will finish we are moving to some then I will I'm going to discuss atmospheric neutrinos accelerator and reactor experiment so that's the plan for next slides so this what we have inside the earth we have mental we have the core, inner core and even some layers here so the earth looks like in a multilayer system and then applications and actually all these neutrino fluxes most of all we are discussing they are somehow propagating inside the earth in one way or another so that should be taken into account sometimes it produces very interesting effects applications so here we can discuss flavor to flavor transitions and this takes place for accelerator neutrinos for atmospheric neutrinos and for cosmic neutrinos so what happens is that neutrinos of certain flavor for instance muon neutrinos enter the earth and somehow oscillate and then we see the result and we also can discuss mass to flavor transition this is what happens with solar neutrinos in supernova so this is the density profile so if you have questions I can answer and this is what you expect for oscillation probability so this is the probability of new e go to new mu or new mu go to new e if you want as the function of energy in the range say from point one g e v up to these are not solar energies they have higher energies up to ten g e v and these are probabilities for different directions so this is for vertical direction when neutrinos are coming from down exactly and you see what happens here there is a big depth of oscillations which are due to enhancement of one to mixing this is one to resonance then probability is decrease and then is another same thing here and these peaks are due to this parametric enhancement of oscillations because for these vertical directions neutrinos cross both the mantle and the core of the earth these two plots correspond to different cosine theta this is my this point eight which is some kind of inclined and this is for point four so that corresponds to trajectories which do not cross the core and here you see again two resonances two regions of resonance enhancement here around point one g e v and here around six g e v remember this number we will discuss this later there are two resonances due to two delta m squares one is small solar which I have discussed and the other one is big one so this is what you are expecting now different colors correspond to different c p violating phase I will come to this now this is for new mu to new mu and you see what's going on here so again this region and these are affected by this resonance phenomenon so this one and this but in the bigger bigger range here we have just something which is very close to vacuum oscillations and to some extent we were lucky because the first analysis have been done for just two new three in a case without any matter effect and to a large extent we were right because in some big ranges of energies in fact the oscillatory pattern is close to what you expect for vacuum oscillations and this is in the three new three sorry this is in the plane energy and zenit angle so these colors give you the strengths of the size of the probability one is this black probability is one and this is for transition of new e to the sum of new mu and new tau fluxes nice pattern so this is the image of the earth in neutrino light if you want and so what you see in this picture is the following this corresponds to vertical direction when neutrinos go down from down to up this is horizontal direction and neutrino come from horizon and now you see different structures this is due to MSW resonance peak due to one to mass square difference and it corresponds to different phases this is parametric resonance this actually point corresponds to one of this geometric picture I have shown you before now what else now this is the resonance peak due to mass splitting which is the biggest one and so these are parametric regions which correspond to one three frequency to bigger mass square difference before I ask question so this is for neutrinos and anti neutrinos you have this picture in the case of so called normal mass ordering I will discuss this later and this is for anti neutrinos there is no resonance of here you see a big difference so you have one, two states down which are small splitting and one isolated but if you turn the picture then essentially you need to turn these patterns and so I will start from this next lecture so thank you


Welcome to Exploring the Solar System, our free online course that not only visits the celestial bodies of the Solar System and discusses their general properties but also presents the necessary information for their observation and exploration using modern astronomical and space technology. Together with my colleague Uli Lotzman, I designed and developed this course. You may wonder how this is possible. Uli, a dental surgeon and myself, a linguist. Well, we have both been hobby astronomers since our youth. We have been keen followers of the first manned missions to the moon. We have enjoyed the enormous progress made by several missions to the planets and together we have the equipment and the background on how to perform astronomical observation. And the digital material for this course, well, the internet has it all. So what we did, we took the texts, the images, the animations, the simulations and the videos that are freely available on the web, added our own stuff and of course our long experience with digital teaching and learning. The result is this online repository. So join us on the Virtual Linguistics Campus, our standard platform for digital teaching and learning. Take your free account and then self-enroll to exploring the solar system. In the second video, we will talk about the digital learning units of this course and about our principles of asynchronous learning using our digital material. Until then, have a nice time.


Thank you. Thank you. Thank you very much. How are you doing, all right? What are you talking, live or film? Well, the one we saw last night on the air was the dinner in Washington. The variety camp of foreigners. The variety camp. I was with you at the event last night. I was on TV. Good to see you. I'm not meeting a couple of people here. I'm here to welcome my staff to see you. And Clyde Romney. Clyde is my administrative assistant. It's good to see you. But I have a couple of things over here to sign. It's my first. You and what you did with our author. Well, at some point we'll have to report back as to how well we've done, although we'll never get considered for it. I understand, though, that it has worked very well. I think it was very well done. I know I think we'll have several million dollars that we can account for on it. And maybe a hundred. Wonderful thing to do. Thank you, Mr. President. No, this is a little enough. I appreciate it. I should have put that back in the box instead of on it. I'm used to clipping it to the bills that I forgot for a minute there. Well, thank you very, very much. Thank you so much for doing that. I remember 24th before Thanksgiving. It was the appropriate time to do it. We were just individually with you individually. Let's get out of the road here.  Well, he did. He switched on when he came with me. First time I voted for somebody. Although George Romney is a relative. I was a reprobating of the family. Let's converse. Make the best converse. Mr. President, this is a new fact from my one hand. I'm honored. My wife. Where was it? For heaven's sakes. I'll be done. I remember that day of Occidental. And I remember somebody asked me something about bedtime verbons. I'm sure it was one of many. I'm sure it was one of many. This is a nice meeting of person. I can't remember. I said something to the effect that might have been something rather... I can't remember how he framed the question. Something rather to him, but it was bread and butter to me. My wife and I will join you tomorrow night at the mall. Three days in a row will be wonderful for me. It's a pleasure. Thank you, sir. Thank you very much. Good to see you. This is my wife. And this is House. And I haven't seen you since you were here. We have some jelly hoops for you. She's in it. Well, thank you. This is Becky. And this is Brian. How are you? And John. And Patrick. And Jill. And actually, we're all here to lobby for the $2,000 full extension. I think our family will come. Well, thank you very much. Thank you.  You're going to be very proud of your father. And they're lobbying me to lobby you. To make sure that the tax reform bill gets out of the house. And over to the Senate. Well, I'm still on the underside of it. So your lobbying is very important to the neighbors. To the neighbors. To the neighbors. To the neighbors. To the neighbors. The lobbying is very important to the neighbors. They try for the Republican version. But if that doesn't work, they still have the other. Then we'll try to fix it up to the other house. And if we don't, if we can't make it acceptable, I'll scratch it when it comes to me. But we shouldn't stop the process. Well, that's a good idea. We shouldn't stop the process. I think we need a few phone calls from some of my colleagues. I'm sure he was doing that. That is very, very much needed. I'm afraid you may be in trouble on that. But I talked to some of my liberal in the steam room there. I saw some of the two liberal Democrats and they say that I would vote for them. So we need to make it public. That's right. I can't understand there's so many that don't want tax reform. I can't. How they can stand up for the present tax system we have is beyond my understanding because the Lord I have income taxes now for quite a few years. And I remember when I used to make out my own and now, when somebody else a professional makes it out for and sends it to me I don't even understand it after I've seen it made out. I don't know how the public is putting out this. Mr. President, we were at the Army Navy game and we saw your presentation up on a big screen. I thought that was very very moving. Those stuck in my mind and I was late getting the set turned on. It was a good game. I saw the game later, but I didn't see it. We saw you. It was really good. You had a few Democrats that comment about that. Very moving, I thought. What you said in the Army services throughout the world. Thank you. We appreciate your time. We appreciate it. One family photo here, Mr. President. We better get a nice family photo. That's right. I know where you are, Mr. President. Okay. Thank you, Mr. President. We appreciate your attention. Thank you very much. Mr. President, how are you? Good. Thank you. Mr. President, you made this by hand. It was a cotton ball. I was wondering what it was about. I thought it was going to be a real cotton ball. I was wondering what it was about. Well, you guys, that is a hobby. Well, Mr. President, I'm sure you've got that in your system. Well, would you please give me that? Artful things, that's very handsome. I had one just like it. I know this is going to happen this week. It might happen to be on Phoenix, but I had it on. Oh, I see. I was just... I was just telling you that officially in Arizona, these are nictics. Oh, they are. In fact, I was going to say, do you want me to pass the bill to make it better? It should make me better in Arizona. Well, that's handsome. Thank you very much. Please, thank you. Mr. President, good to see you. You're very nice to see you. You're not a rustling. This is America. This office today is better than others. Oh, my goodness. I always tell her, my whole brother needs to meet your wife. Because she's been such a wonderful woman. I wish we'd known each other. I wish we'd known each other. And she has such strength of care. I liked her when she picked out her china. And I've always said that every bride should pick out their china pattern. So, it isn't... And actually, that was so distraught with the press. Because... Well, they had it out. But she was buying this. No, people had volunteered this. And she did pick out. But they were going to marry you. Well, till like every one... Everyone had to pick out their own pattern. And that was her thought. To get to pick out. Donna brought you some things from Mississippi in seven years. Well, I was so thrilled when I heard Friday that every president needs an only china. This is my alma mater. Oh, thank you. Colonel Brad and Hotty Time. And then, my neighborhood, I live at Mill Creek in Brandon, Mississippi. And when I heard about it, two streets up, her lady makes and paints different things. So, she made this for me in a flower state, flower. And I'm actually presented to President Ms. Reagan. I didn't want to leave out Ms. Reagan. Well, thank you very much. And then, I don't know if I can afford for my husband and children to come, but they are Republicans. In fact, when the war on against Mondale, my children had achieved what everyone did for them. But I said the next best thing, and that's a picture of my children, my family. And last night, and I know you promised... Yes, these are for you. These are gifts from me, too. Last night, everyone needs a pecan pie. I think they touch it, because I know you probably can't eat it, but I just wanted you to know about our pecans in Mississippi. And I've been letting anybody open it because of security. It feels good again. Are you good? Well, anyway, it's a southern pecan pie. It's my mother's recipe, so I hope you'll enjoy it. And this morning, a quarter to seven before I left, my father-in-law called me and said, I have so many questions. How you should do it. The White House, please let us know. We'll answer what they... I said, no, I thought I could handle you by myself, but I really like to do a picture which is something for my mother-in-law and my mother-in-law. And this one's for my children because they were upset they couldn't come. What's going on, ladies and gentlemen? Oh, my goodness. Oh, I'm excited. Gosh, thank you. You've been pretty busy, I know. And when I watched the Mrs. Wolf had, I was like, how do you feel about your time as they come represent a great picture? These are postcards. I know, that's all I could find. It doesn't matter. I heard you're having trouble with Mr. McDonald over here, so did you need me to talk to him? Hello. I was just looking. I was just looking to see when you said those were the only pictures you had. Would you like to take a picture? Oh, God. Three. Oh, how exciting. It's all fun. I had to look all over for those, but it was fun. And what a great picture. Well, now this would be a little different because you've got to bring in the big ones. It doesn't matter. There's very little place to sign on this one. It's serious. What if I just put it on my name on here, but then I'll really autograph the front of the ones that are coming in. I certainly will. These are really informal. These were from the ranch. Well, I guess since we met, we can be informal. What are your children's names? Heather and Jason. I'll let you know what they're... I better not say it. Maybe we ought to have another one here because then I'm going to have to share it. Anyway, their cheer was Reagan Reagan, he's our man.  He's a student democrat. See, aren't they the greatest Republican in the United States? I've trained them right. But anyway, it is so fun to sing this at school and everywhere during the election. I thought it was very appropriate. And when I told them I was going, they said, mama, please tell them I'm a cheer and I'll tell them I'm a star. Donna's husband was a really outstanding football player at Ole Miss, Mr. President. Oh. That was about that generation. And his name was Stop Brown. He's a J-A-S-O-M. President got to see Ole Miss play Tennessee one year when he was Governor's Association. And they had a meeting in Mississippi. And we saw that in Jackson and the state of Michigan. You saw her. No, you're right. It was Tennessee and Tennessee was supposed to be one of the 30-point favors. And I remember the game because I would never forget that in a moment of quiet and up in the stands after Ole Miss had scored again, was out of the hit and it was pretty definite. I heard a voice raise. I was sitting down there with the Governor in a box and he says, man, what would they have done if John Wayne had been here? That's right. That's good. Make sure, man. There. Oh, they just want to leave this one. There's a barbershop in Maywood, Mark, in Jackson. And when I was down there right after Donna's won the Mrs. America title she came in there and sitting in the barbershop held court for all the fellas in the barbers telling them about the Miss America experience, the Mrs. America experience in the bottom.  And it was like, oh, hold on. There I was getting my hair cut. Donna was telling them all about the size of what it was going to be. I feel good. Charlie, CH, I'm such a good daughter-in-law now. So you've been giving him some problems? No. Not me. No. No records. All right. Great. I might be giving Tim some problems. This one is too. My mom and dad. Donald. This is the Oval Office. The fire is cooking. Well, on the airplane, this man said you're going to be there. I'm all in here. Are you going to have dinner with him? I said, why not? I was cooking tonight. I enjoyed it so much. You're doing a wonderful job. And what did your husband say? Linebacker. And his name was appropriate. Stomp. Yes. Trestone. Really great. I'll tell him. The school oligarch goes to Madison Regional Account. He helps coach up there. And most of the 10, 28 players go both ways. Thank you again, Mr. President. Thank you very much. You're very kind. You've made lots of people happy. You've made a lot of people. Thank you so much. Thank you very much. Mr. President, how are you? Good. We got a flop that was made and a piece of pipe and a star quill if I could show it a little bit. I hope the light is red, white, and blue. Yes, I do. It's got the red, white, and blue. It's got the symbol of the American Eagle on it. Yes. It's all handmade by our local women. Do you like it? Yes. I realize you're a very busy person, but thank you for taking the time. This is a piece of pipe. Let's get that cleaned out later on. It's got a buffalo on it. What I want you to know from way back to that size, believe it or not, I was, you know, it was way back in the Midwest, in Illinois, but it was in a place where the Black Hawk Indians had been very prominent there. As a matter of fact, the statue of them just north of there along the Black River was built, looking down river at Corn Island because they used to come through that Dixon amount up there in the Corn Island and back. Well, I just grew up with some of our local library in the little town there. They had a whole gallery of things of that kind. These are products there. I could always be found in there. These are products of stuff that's made, you know, nowadays. They hope to make an industry out of that. We're hoping to. Oh, that's great. I should think so. Yes. Well, thank you for that. Yes, thank you again. Thank you very much. Pleased for all those who had a hand in this. Thank you. President, I'm going to give you something. Yeah, man. This is Paul. Thanks to you and the script. Well, hey, good. Thank you. Thank you. How many more have we got? One. Last group. This is the business end of life. Oh. It's got more to protect it. There. There's your peace pipe. Ah. I think... I think we... Well, I'll let you put it all back together again, but I think we put that in the ranch stuff, to go on the ranch. I didn't think there was a hole in the under. There is a hole. Okay. The other things, I suppose, should go to the library. Clothing block. Yeah, I guess. You don't want that on your bedroom wall. Okay. Thank you so much. Hello, President. How are you? Nice to see you. I'm sorry that Rudy couldn't make it down here, but I guess the bed is... Well, it's nice to see you. Thank you. Mr. Schaefer has a bronze sculpture, like we said, too. Well, it's... This... is a letter of offer to test for you, which one is yours? Thank you. Thank you.