 Okay, so it's time. If you want, we can start. Please, please turn on with the second lecture. Okay, thanks. So yeah, good to see you all here again. This second lecture will go into some more detail about possible ways dark matter could interact in condensed matter systems. So, talk about dark matter electron interactions and dark matter phonon interactions. So to start off, let me remind everyone of this plot I showed yesterday, just as an example to illustrate why why these condensed matter systems are so interesting for dark matter detection. So, we have the sensitivity from nuclear recoils. This, these units are this is a different axis than the usual winter recoil cross section. This is dark matter electron cross section. This is actually a particular model that I'll talk about shortly, and you can compare nuclear recoils versus looking for electronic stations and phonon excitations and various solid state materials. And so that's why there's been so much interest and development in looking at different types of materials for this KV to GV dark matter mass range. So last time I promised I would say a little bit more about what kind of models were actually looking for. So do that now. And what we covered last time was already the kinematics of why these kinds of excitations would be better. And now let's let's discuss a little bit. The models that might live in this parameter space, particularly you see this blue line. That's one example of a kind of thermo relic that that is present. So the blue line is exactly where you want to live for the couplings to get the thermo relic through a dark photon mediator. And as always, feel free to interrupt with any questions, comments. All right, so let me say a little bit about this dark photon mediator scenario that gives rise to that blue line. So the place it usually comes from ultimately is some sort of kinetic mixing between dark photon. So that's the prime with hypercharge. And there might be some mass term for the dark photon a prime. And this is something that is very easily generated. So this is a epsilon parameter very easily generated if you have some heavy particles. Just call them F. And they might be charged under both the new you one and existing you one. And so through this kind of diagram then you'll typically get small numbers for this loop diagram and so you can easily get some sort of small number for this epsilon why this kinetic mixing here. And so it's very natural to think about these dark photons. And for our purposes, we're talking about sub GV dark matter. We're really focusing on low mass mediators as well. So really focusing on sub GV a primes as well. And so the thing we'll actually focus on is you can just focus on the mixing kinetic mixing with the standard model photon. For our purposes, we won't have to talk about the Z mixing part. And so to treat this there's two different bases that are often used one you can keep keep this as it is. But I'm going to go to a different basis by rotating away this kinetic mixing. So I can do that by taking standard model photon to be photon plus a small mixture of this dark photon. And the way the reason we do that is because the dark photon already has a master. So if we were to do this rotation on the dark photon, we would get mass mixing terms. Whereas we start off knowing that the thermal photon does not have a mass and so we perform this mixing that gets rid of this kinetic mixing term. And we're instead left with coupling I mentioned yesterday which is coupling of the dark photon with all the standard model stuff and electromagnetic stuff. And so that leads to the kind of interaction that I drew before which is that this provides a very good mediator for freeze out reason direct detection. So for instance, this is a process that's possible in the early universe and couplings will consider usually G chi and epsilon E. If this happens, if this process is an equilibrium that would allow for dark matter freeze out. And there's also freeze in the line I just showed which is possibility if this diagram only happens in, whoops, in one direction which is the direction of producing dark matter out of the thermal bath. So dark matter is not an equilibrium. You just have very rare interactions that are occurring in standard model thermal bath where occasionally plus minus will annihilate to the dark matter. And you can produce the right amount of dark matter in this way through some combination of couplings is about 10 minus 10 so much smaller. And this MA prime is pretty light the dark photon is pretty light. Okay, so it's possible to have both a freeze out reason scenarios and I'll show those both on on these kinds of cross section plots. And if we have this coupling then dark matter will source a potential in materials. So this is basically just a cool or you call potential if we include the mass, and I'll write it in for a space, just get a q squared plus MA squared. Normally when we're talking about scattering events were used to thinking of like, say, computing a two to two scattering process or two to two interaction process kind of like this. But when it comes to treating the dark matter and materials. The end of diagram while it's, it's useful can can lead to, it can be misleading. So another way to think of things is by writing down the potential source by the dark matter and figuring out how this potential perturbs the material. And the reason is because when we're dealing with the material. Again, we're not necessarily dealing. It's not always easy to think of the material as just a collection of individual particles that the dark matter can scatter off of one by one. We're really thinking of the material which is a has a lot of interacting components all responding at once. And so that's why I'm treating this in the language of what's the potential of the dark matter as it's passing through this material. Okay, so mainly I'll work with one of the ones I'll work with is this dark photon example. Another case that's often considered just to cover our basis is what if there's no such electron coupling and we just have a leptophobic mediator so this would be a scalar and there are merely couples to nucleons and similar we would have a potential sourced by this dark matter as well, where now I only some over the ions in the material so I meant the ions, where I'm treating that solid state material as basically ions plus electrons. One other thing to to mention about the kind of interaction will look at is that there's usually two limits taken for this potential. I don't, of course, a prior pro or you know what's the dark photon mass is different models of different masses. In case you can do a, you know, study for a particular model but usually will take two limits, the massless mediator limit. And MA prime is much less than kev doesn't have to be massless but as long as it's much less than kev then its presence in this potential can be neglected. And we'll see that's because the typical momentum transfers are either the m chi v zero alpha me both of these. Well this is going to be good in order kev. And this one, this one can actually be as well as say few EV. So really this maybe should be stronger, much less than EV. And the other limit is the massive mediator limits. And usually this might be of order is the dark matter mass. And when that happens. Typically q. Basically the massive media limit is where q is much less than my prime. So we consider those kinds of limits for scouting potential usually so basically the first one is like a cool potential from the dark matter and the second one is like a contact interaction. So I've introduced now to possible mediators and the kinds of external potentials that they introduce. And so let me sketch out now how the scattering rate is computed in the material, and I'll draw an analogy with the calculations that we looked at yesterday for dark matter nucleus scattering. So for dark matter nucleus scattering. We found that the leading operator looks for for vector interactions had this coupled basically to the one operator which means coupling to the nucleon number density. And the same is true now we're also coupling to the number density so before it was row and coming over nucleons. And now we're coupling to row E, which is the electron number density, as well as possibly row, I, which is the ion number density. And it's appropriate to use to think about these instead of nucleons are more individual particles, especially for low q. So low momentum transfer. So to give you an idea that inter particle spacing in materials is order angstrom, which corresponds to momentum transfers of order KV, and we do one over angstrom. On the other hand, when we were looking at dark matter nucleus scattering we're looking at much larger momentum transfers that was more of order MeV or even 100 MeV. And so there it's really appropriate to deal with individual nuclei and even goes far as the nucleons within the nucleus. Because once you get to pretty high momentum transfer at 100 MeV, we saw that you're not probing the whole the nucleus as a whole coherently. There was a form factor, the Helm form factor, which led to some suppression in the coupling once we get to really high momentum transfers. And you're no longer, you no longer have this being just the sum of all the nucleon squared. But you're you're probing the structure. So similarly for the solid state systems. Once we get to momentum transfers of order KV, which is definitely relevant for the for light dark matter. That's because that's comparable to the inter particle spacing, we can't really think about probing just a single electron or a single ion in the material. We have to sum up all the electrons and ions in our electron density operator and our ion density operator. And we need to compute a different kind of form factor, which will be called the dynamic structure factor. Okay, so to remind you for the Helm form factor we just computed with the nucleon number density operator. And particular we computed with the same initial and final state because we're just recoiling against the nucleus as a whole, and we're not exciting any internal states. Now for the solid state system, we compute a similar density operator expectation value, but we, we have different initial and final states corresponding to say we might start with the ground state of the system at zero temperature, and then we might create some excitations with some energy, with some difference in energy omega. So again this is just like, this is just perturbation treating treating our external potential as some perturbation to this, this whole system, and here computing some rate to produce excitations. Okay, so I'll just give you the form of the total rate and it looks very similar to when we wrote it down for dark matter nucleus scattering, the dark matter number density here. This is the target mass density. It just replaces the number of targets before so we can still talk about the exposure in mass times time. As before we have the dark matter velocity distribution in our halo, and some additional cross section. This is the reduced mass of the dark matter either with the electron or the nucleon. We always typically put our overall form factors here. And then we want to integrate over the possible phase space of the excitations. So last time I emphasize that we treat this in energy deposition omega and momentum transfer q. There's a dark matter mediator form factor here, which I'll explain in just a moment. This is the dynamic structure factor that I just introduced which is like the form factor for the material. And this is just energy conservation saying that dark matter deposits energy omega. So that's going to be the same as that relationship between omega and q that I introduced last time. So this is relatively straightforward in that, okay, we packaged all the material properties inside this one function the dynamic structure factor. And everything else here is either the astrophysics of the dark matter density and velocity distribution, or various particle physics quantities that we can compute, namely this fiducial cross section and this mediator form factor. And so let me just define those more explicitly. So let me just do the electron case. So this combination in front, the cross section over the reduced mass is just the combination of couplings, for instance, for dark photon. And I call it a fiducial cross section because it's defined at some reference momentum transfer. So this is basically just that strength of the potential. And it's defined at a reference cross section just so we can compare and the actual momentum dependence of the potential is then absorbed into this mediator form factor. Okay, so the, the mass, the dark photon mass dependence of the potential I absorb into this mediator form factor. And this part is just pulled out in order to define the cross section. So basically it looks very similar to calculating dark matter nucleus scattering just with essentially with the replacement of these form factors, the nuclear form factors with this. So in today's lecture I wanted to focus on in on giving some intuition and sense of what this form factor looks like in various materials, and how that compares with the allowed face space for light dark matter scattering. Are there any questions so far that on the dark matter model side. Okay, well we can always return to this later. So now I want to get started into thinking about dark matter electrons gathering and computing that structure factor. So, let me recap a little bit about electrons they went fast last time at the end because we were out of time. And I've introduced this band structure plots, which, if you start looking into the literature and some of this thing you might see this a lot so I want to make sure we go over a little bit more. So, as I said one of the important elementary excitations in solid state materials are electrons, and they're not exactly electron particles but they're called electron quasi particles. So there's something that's not exactly an electron but they behave a lot like an electron in that they have the same charge. They're a fermion, but they have a really funny dispersion. So we can almost think of them as having a different kind of mass, it's called a band mass. And so the toy model of where this usually starts comes from basically looking at the electron wave function in a periodic potential. So, remember there's theorem called blocks theorem, which says that you can write the solution to Schrodinger's equation in this periodic potential, most generally as some phase factor times another periodic function. And furthermore, these solutions, the set of all solutions will be equivalent. If you kind of take this label K and change it to K plus a vector G, if G a is equal to two pi. And I'm just, I'm doing it in 1D but of course this will be in 3D you have a dot product of G dot some lattice vector. So because of this equivalency, usually the spectrum of solutions of Eigen states is shown inside the first, what's called the first Ruan zone. So for instance for just a 1D periodic lattice K would be in minus pi over a to pi over a. So you would just show the spectrum of Eigen states in a particular K range. And it's a simple, it's a, the Ruan zone is simple for this 1D periodic lattice. In a crystal, we have a three dimensional structure and it becomes a little bit more complicated. And so what's shown here is basically, if you imagine this 1D thing gets replaced with a three dimensional object. And this is a path going in that three dimensional space K space. So this is effective, this is basically showing momentum on this X axis where momentum really means the label of these solutions. But you can think of it roughly as an electron crystal momentum. And these labels are just particular labels there's a convention for them for what they mean in terms of momentum. I don't even remember them. The one that's always useful to remember is gamma gamma is always K equals zero. Okay, so there's a lot of stuff going on but let's, you can simplify a little bit. And if we look near gamma equals zero what you'll see is there is actually this quadratic dispersion. Here, which is basically the beginnings of a K squared over two and dispersion, which you would expect for free electrons. And if you were to continue that you know if you didn't do this cutting off of the momentum space you would just keep going like this. But instead it's folded over and it's mapped back into these lines here through the equivalence shown here. And so then the way we read this is you can think of this boundary here as roughly where the momentum becomes pi over the lattice spacing A, which is typically going to be around KV. So you find the gamma point and kind of just imagine that this axis now is going from zero to KV in momentum. And then you can read now the energy eigenstates going up, which will be something like a free electron, which looks like down here, but then get more and more distorted, as you can see, depending on the material. And these other points are just other finite momentum points I won't really look too much at this region. The important one point point is this dashed line here. This is the Fermi, the Fermi energy for this material which is silicon. So that means all of the eigenstates are filled up to that energy. Each one, each, you can imagine this, these bands really as having a discrete number of eigenstates, so really it's like this. And as we fill it up with all of the electrons in the material, it goes all the way up to here. So everything below is going to be the occupied states and everything above are available states we could expect to. And importantly, there's a gap here, so the highest occupied one is around here, and the lowest available one is here. So this has a gap of around one EV. So when we talk about dark matter excitations in this material, then we can think about specifically removing, for instance, an electron from here and say exciting it to some available eigenstate here. Let me draw that better because so we can remove it and so the amount, the horizontal motion we make or the horizontal part will basically be the momentum transfer and the vertical part will be the energy deposition. So this is a typical example of a semiconductor with the, like I said, electron volt gaps, and that's going to allow us to probe dark matter as light as one MEV. There are two, there are some other possibilities for where this EF, this Fermi energy might be located. So if it's down here, let's say EF were down here, that would be metal, because there's basically no gap from taking an occupied state to an excited state. Another possibility is if we had an insulator. It would be pretty similar to this band structure, but the gap would basically increase to about 10 EV. So there's no exact boundary, but if you basically moved all the bands above unoccupied bands above up by like nine EV that would be like an insulator. So if you move the Fermi energy below, that's a metal with zero gap. Okay, and so that flexibility gives a lot of options when you're considering different materials for dark matter scattering. But let's focus on the silicon example. And let me show you now just some schematic of how, how the experiment might actually go so these are schematics from real experiments. As I said, what we imagined happening when dark matter comes in is we take this occupied electron and we cite it to some states, to some, the electron occupied state to some available Eigen states. And so what happens after we excite that electron. It's a complicated process because now that electron. So we have the excited electron can decay to other electrons. So it can decay to some state there. That one could decay here. And so there's a kind of cascade process where the amount of energy you deposit will lead to one electron but then that'll cascade into a few more electrons. And so roughly speaking, you'll get out a number of electrons at the end of this cascade, which is omega over about three electron volts depending on the material, maybe higher than other insulators, but for silicon it's around three EV. And so that's the number of electrons you might observe. And so if we look on the right, how would we observe that. Now we can take this number of, you know, final electrons in the cascade. And in the Dominic and sensei experiments, what we have is a silicon CCD. And that's the this blue region. Now one of these observed electrons, you'll have an electron in a hole. And if there's a bias voltage, then it'll drift the electron to one side and the whole to another side, the whole just sorry the unoccupied and unoccupied state. Okay, but more importantly, let's just say we drift the electron to we drift the electron with this bias voltage. So that's just the operation principle of a CCD. And what Dominic and sensei experiments are doing are using particularly fixed CCDs that we have enough mass and also and a bunch of basically treating them as dark matter detectors where we're taking a picture of electron ionization events that happened. So the bottom plot is showing basically a pixel. Well, it's not a pixel. It's a bunch of pixels in a CCD. And instead of, you know, usual CCDs taking pictures if you cover it for a long time and you wait, you get a long exposure of the CCD in the dark. And for instance, this one pixel here shows has a measure charge of exactly one electron being produced. Everything else is zero. And through this, you would say I have an event with one electron which would correspond to about one to three EV. So it can be as low as the band cap that could be as high as three. Okay, so they're these are these experiments are running and I'll show you some of their results. But already from here you can see, they have some really nice techniques to basically reduce noise and see just that spot from that single, single electron type events, very low energy events. Okay, so I think before the break, I'll say a little bit more about the dynamic structure factor in in materials, and then we'll we'll continue with the electron scattering after. Okay, yeah, I'll try to finish the electron scattering before the break then. Alright, so to make us a little bit more concrete I like to use the example of the hydrogen atom just as a case where you can compute things without having to go into all the band structure. I've been talking about. So to remind you the thing that we care about this dynamic structure factor. Let's write it again. It's just a expectation value of this density operator. And the thing that can be quite complicated is identifying what the initial and final states are in a many body system. But as a warm up we can simplify things with a hydrogen atom, and just take, for instance, I hydrogen I'm simple we just have one electron and we'll take that to be separate hydrogen atoms as our target. So I will just be the ground state of hydrogen. So it'll be proportional to either minus r over for radius. And in this example I'm just going to take the outgoing states to be free electron free outgoing electron states. So k prime is the outgoing wave as a momentum the outgoing wave. And because there's just one electron and I also just, I don't have to do any sum j is equal to one and I can compute that matrix element directly. So it's shown in the plot on the right, basically computing this matrix element and summing, and including this sum over final states with the delta function, which anyway just turns into a factor. And so this is the log of the dynamic structure factor for hydrogen. And what we see is one that scattering is going to be peeked in the zero to five kv region so around five kv or less. And so that justifies what I was saying earlier that the typical momentum transfer that we're looking at in in electrons is going to be of order one over the bore radius or alpha me or five, which is about five kv. And the typical energy is 20 Ev or below. So note that this is cut off at 13.6 Ev, which is the gap for ionizations. And then the rate is kind of peaked just just about that around 20 Ev. So this is the basically this is showing us how easy it is to excite the material in a given q and omega. And then we have to compare that now with what is the q and omega available for dark matter scattering. So at that depends on the mass. But if we take a momentum of 10 kv. So if q is 10 kv, then we expect that the corresponding omega is going to be less than q times the dark matter velocity, which is 10 going to be 10 Ev or so. Well, up to, it'll be up to 20 Ev, because then it wouldn't show in the plot. So really if we looked at dark matter scattering it would actually give us a parabola that's very low down here so probably should change the axis. The axis here, but basically you're not really scattering, you're not really able to access any of the stuff up here. For dark matter scattering remember we had a parabola and it's going to be all the way down here kind of that large momentum and low, relatively low energies down there. So this kind of gives some intuition for how you might even think about comparing different materials and looking for material that would be optimal for dark matter electron scattering. Basically you want to look at this structure factor and see, you know what kinds of materials give you plenty of signal down here. Let me show you then the result for the silicon semiconductor. So that's this plot and it actually has quite some quite a few features in common with the hydrogen atom one you can explain a lot of it similarly. One is you can see that the typical momentum transfers here, where we get the most weight is similar it's going to be order 5 KV, a little less, we get similar momentum transfers and again that's that's basically coming from the that you know when we look at the electrons and material that that's a very that sets the scale for the momentum of the electrons and their wave functions and so on. That's always going to appear as a typical scale. That's generally going to appear as a typical scale won't say it for every single material. And that's nice because the band gap, the Delta E greater than one EV. The structure factor the dynamic structure factor extends to much lower energies. So it does go all the way down to one EV. And you can see a little bit of white region here, and that's basically the part that's excluded from zero to one EV due to the band gap. It's already a lot better and if we were to draw that parabola for dark matter scattering, they would look something like this. So you're still not quite reaching the most optimal part of the structure factor, but you are doing a lot better with semiconductor down here. Okay, and there are various subtleties in calculating the structure factor so these initial and final states are not. You cannot just use the block states that I mentioned as shown in our some recent work which I've cited below. We've shown that the, these are really many body states. And if you, if you do that, it introduces some additional facts, basically screening in the material, as you're exciting the material as a whole so it's a many body effect. Any case I won't go into that other than to comment, you can't just use the block states in the initial and final states when thinking about this dynamical structure factor, you have to count for various many body things. Okay, so, given that here's a comparison of different kinds of materials you'd that have been studied in the literature so far. So basically we've been looking at different structure factors and now I'll show you how that translates into a reach in the dark matter. Electron scattering cross section plane so dark matter mass and electron cross section. So, first thing to know is the left plot is the massless mediator limit I mentioned, so that was the MA prime. That's an EV. And this is the massive mediator so MA prime is in time. So each of the plots, I, I have them that kind of thick bluish line. So those are interesting thermal relic lines so those are free that's the freeze in line and I also have freeze out over on the right side. The two different lines are for scalar versus for man. And now all the different colorful lines are projections for if you had a kilogram of material and you waited a year and you had very low background. What's the best you could do in different materials. Starting on the left, if you have an insulator which my example here, if you have an insulator like diamonds, or maybe the hydrogen atom, that would give you something like this yellow curve. So that will have the highest threshold delta E of 10 electron volts. So it will only go down to order a few MeV and dark matter mass. And these next two lines, those two are silicon and germanium silicon doctors. So that corresponds to lowering your energy threshold to about an EV. So you can basically significantly improve the region around one MeV by lowering the threshold. And then you can ask going for in the future what what if we could have even lower threshold detectors. So the, the darker lines show two proposed possibilities. One is to consider a metal, and that's this line. Now, rather than a metal, what's the original proposal was to consider the metal in the superconducting phase, which is better in terms of detecting the actual excitation that you produce. But for this discussion, this discussion is not too important. I'll just, you can just think of it basically as a metal. So when you have a metal, then the gap basically goes to zero. And the experiment might not be able to go to zero. So I've put thresholds of 10 MeV and 100 MeV. So as you lower the gap, you can keep doing better and better. Particularly for this massless mediator case because remember it's like a Coulomb potential, which is highly peaked at low momentum transfers and so it's really beneficial to keep going lower. And then this one is another example, kind of exotic, more exotic material, the Dirac material, which is something which also has a really low threshold to excitations, here's about 10 MeV. So there are a bunch of existing experiments already with these kinds of energy thresholds and then in the future, we can explore more novel materials and get to this dark matter mass range. Okay, and for the massive mediator, the advantages of going to low threshold, there aren't quite as many. So all of the lines basically are cutting off at around one MeV, it's hard to go below. But the model space is also not extending to some MeV. Okay, so one quick comment about solid state materials is also that they allow for directional detection. So in this graphic, what you see is there's a particular crystal and the structure of the crystal actually depends on which direction the dark matter is coming from and how you're exciting it. And so as the Earth rotates, this is the Earth axis of rotation, T equals zero, you'll look at the crystal in one direction. Half a day later, this will have rotated around and you're basically probing the crystal in a different direction, and you can actually get a modulation in the rate, which for some materials is quite large. So in this example, it goes from below one, all the way up to three, so it can be factor of three change in rate throughout the day. So if you were to observe any signal, then you could try to use that to distinguish a dark matter origin, which is very important because with electron recoils, we don't have a way, we don't currently have ways to distinguish whether electron recoils are coming from some background or from dark matter unless we use the spectrum and maybe this effect where it's modulating throughout the day in a particular way. Okay, so I have one slide about experiments, but I think since I am running into the hour, I'm going to stop there for this first section and we can take a break. Okay, great. So we'll zoom at four or five. There's a question though, first, would you like to answer it? Sure. Okay, the first one, it's very quick. When you say direct material, you mean something that has, you know, the comb like dispersion like graphic or something. Okay. Yeah. And the second, about the plot with the reach you showed. So this one, yeah. So you said provided there is low enough background. Does this mean that, you know, this is a plot, like one of those very optimistic plots that, you know, as a theorist you like to do, but there is not much input from experimentalists or Right, this is just a, I would call this more of a materials comparison. So it's how what's the best you could do comparing these materials. But of course, it realistically, you have to consider lots of backgrounds which might be more materials dependent. But yeah, this is more of a, you know, what's the best you could do with a material just to sort of guide your thinking about different materials. So, yeah, indeed, these are optimistic with zero background assumptions very optimistic. Okay, I see no other questions. We have a break and resume in five minutes. Shall we restart? Please. Okay, so I forgot one last thing I wanted to show about dark matter electrons gathering which are actual experimental results. And so that's summarized in this plot. It's the same dark matter electron scattering cross section. And now you see that there are actually tons of experimental results in this regime with pretty low thresholds. So there's Domic and Sensei here are the CCD experiments I mentioned with silicon CCDs. And they have fairly small exposures so far order one to 200 gram days. And they're also currently dealing with some, well, this one is actually not even that far below the surface. So there's lots of potential backgrounds. But you can see, even though the exposures are still relatively small right now it's not even that far from the interesting cosmological benchmark I mentioned so this will be pretty exciting to follow in the next few years. Okay, so that's it for dark matter electron scattering then and in the remaining time I was going to briefly cover dark matter phonon excitations. So lots of extra references I can share later. I'm interested. Okay, so dark matter phonon excitations are really going to be for sub MEV dark matter. And the reason for that is related to the energy scales of phonons. So let me step back for a moment and just recap what I covered last time, maybe expand on that a little bit about what phonons are. So in a, in a solid state material will have some regular arrangement of the atoms in their lowest energy state. So some equilibrium positions. And phonons are the collective displacements from the equilibrium position. So you can talk about the position of some Adam in a lattice at a particular lattice site. So there might be, there might be a crystal like this one where you have two different types of ions at a lattice site, and you would expand about those positions so this is just a factor telling you which lattice site, the equilibrium positions plus you is the displacement. So what happens next is basically just a harmonic oscillator. There's a potential energy for where you can calculate for all the positions. And so, like any equilibrium system we can expand about it to second order in the displacements that'll be the leading correction, if the ions are displaced. So that'll just be a harmonic oscillator and if you recall from a 1D harmonic oscillator you have the position being related to creation and annihilation operators in solid state material we would have the displacements. Also being a sum over creation and annihilation operators. The difference is that we have a continuum of energy modes. So we have to sum over all omegas. Okay, so we have, these are basically the phonon dispersions, and we associate these creation operators with creating phonons, single phonon. And the E's are basically some eigenvector, some overall normalization for the phonon. It's not too important. So basically it for phonons, you basically take the eigenmodes of the displacement and when quantize it, we turn it into, we write it in terms of phonon creation operators. And similar to the electron band structures, we'll have some sort of phonon band structure. And there are different types of phonons that you could have. As I mentioned, there's the acoustic phonons which have a linear dispersion, like the goldstone mode. And there's also optical phonons which are gapped, and they typically have energy of around, you can see here this one is around 35 milliv, and they can go higher up to 100 milliv. And really because of those energies that we focus on some MEV dark matter because these are pretty low energy excitations. So they're likely to be more optimal for some MEV dark matter. Also on here, this is a log log plot, but basically this is the same parabola I've been showing in the energy transfer and momentum space for dark matter scattering. And so you can see, this is dark matter scattering for a particular mass, I think it was 35 MEV. And so you can see these optical phonons live right in the middle of this base space where you want them to. And that's why phonons are also very exciting direction to go they're right in the middle here. The acoustic ones aren't quite as good because they kind of live in the corner of the available base base. Okay, so this is basically a similar plot. There's the same parabola where you have allowed scattering. And here I'm just indicating that for different types of materials you have different possible energies for those optical phonons but they'll live somewhere in here, and the acoustic ones are down here. So this red line is actually Q squared over two MN, which you see is pretty much awful in this mass range. Schematically, how would we detect one of these phonons being produced. So let's say I didn't want to excite an optical phonon, or let's say dark matter came an exciting optical phonon. So I'll come in. So I have a phonon in some material here. This is gallium arsenide. Again, we don't detect the primary excitation, it will shower and cascade so there will be lots of other phonons being produced lower energy phonons, and they might even bounce off the surface and bounce around. And the proposal is that the surface of this cube would be instrumented with these collection fins and TES's. So what does that mean? The blue region is a region where the phonons can be collected. So that means as these lower energy phonons are bouncing around inside, some of them might hit this collection fin, this blue region, and be absorbed there. And then the purple regions are basically the calorimeter part or the kilometer part where the energy of the phonons in these collection fins is measured. So the idea is if you can cover this efficiently, then you can measure most of the energy deposited from the single optical phonon, even though you're not detecting the original phonon itself. And these, of course, have to be very low threshold, very sensitive. For instance, 10 milliv threshold. So I want to say a little bit about the dynamic structure factor for phonon excitation. It's very similar to what we just talked about, but now we're considering the coupling of the dark matter with the ions in this lattice. So here's my example of a lattice again with different ions in a periodic lattice. And the dark matter will have some different effective coupling with each of them. And again, we want to sum over all of them weighted by UVIQR. So the way the phonons come into play is through this quantized displacement field. So very schematically, if we have this matrix element, we can take our initial state to be one in which the system is in its ground state and there's no excitation. So just vacuum, no phonons. And we can take our final one to be basically one phonon, could be one phonon like this. We could also even consider two phonon states. And then this matrix element is not hard to compute if you've determined these phonon dispersions and the modes, eigen modes I mentioned, because you just have to expand this. So you'll get one, which doesn't, this is basically no scattering. But you also get a contribution of Iq.u in there. And you can see that will give us a matrix element with the one phonon state. And there's also, you can also get, well, you can also get two phonon terms, which won't worry about two phonon. So this structure factor, dynamic structure factor, then we can, once we have some methods to compute this, which will come from the condensed matter side, we can directly compute one phonon form factor, two phonon, and so on. Another one interesting thing about the phonons is that which phonons get excited depends on depends on how the dark matter couples to the various ions in the material. Sorry, I didn't quite get there yet. Actually, I'm going to skip that comment. Alright, so, given this, we can, given this one phonon structure factor, we can do a similar kind of comparison materials comparison. And this is basically the previous plot I showed, but now I've added on the phonon bits. So these are different optimal reach curves for materials which have optical phonons that can be excited. So these are all with optical phonons. And what we see is that these materials with phonon excitation are particularly interesting because they're even stronger than the ones you would get from electron scattering which are the ones shown above. And I think, again, I think that can be understood by going back to this plot I've shown several times of the available phase space for dark matter scattering, and the fact that these optical phonons kind of live more optimally in the available phase space, whereas when we study the structure factor for the electron scattering we weren't really hitting the peak of the structure factor with the kinematics of dark matter scattering. So as a result we get some few orders of magnitude here. In addition, you could also look at different types of mediators and as I said, for different types of mediators you actually excite different types of phonons. That's an advantage because you can then use that complementarity to, you can use that fact to try to distinguish whether you're looking at which type of dark matter model you are looking at if you were to detect a signal. So for a scalar mediator you would actually preferentially excite acoustic phonons, which is how these curves here were obtained by computing acoustic phonon excitations. And these lines up here are actually the nuclear recoil reach with somewhat higher threshold. So this extends it down to sub-MEV with phonons. Similarly here we're focusing on sub-MEV mass range. Again, for a solid state material we can have a very anisotropic material where the energy of the phonons and the eigenmodes depends on the direction that we're looking in. So it's the same diagram and now you can compute how much does the rate change over one day for phonon excitations and you see similar, it's a bit smaller here but depending on the dark matter mass you can get sizable daily modulations, which again helps you try to figure out if some signal was coming from dark matter. So let me actually go back and write this single phonon structure factor down. So these were the phonon eigenmodes and the energies appearing here. And so this structure factor is a little harder to show because it basically is a delta function on top of wherever the phonons are so that's why I'm not showing any plots. But you'll get different weights on different phonons depending on how the eigenmodes look in a particular material. So this will be a material dependent eigenmode which just tells you how the different ions are oscillating. And that's again powerful because you can use that information along with the coupling to figure out to try to determine if your signals from dark matter or maybe combine different materials to figure out the dark matter model or to figure out the origin of the signal. And so here's a schematic from the Herald and Spice experiments showing a plan for using in particular sapphire and gallium arsenide and looking for the phonons being produced. You can ignore this photon part that's a different signal. So these are both materials with the optical phonons and you can see the collection fins and the TS is instrumented on the surfaces. Another idea that's being proposed recently is to use liquid helium, particular super fluid helium, which also has phonon modes. This is a liquid so it's a little bit different, but the basic ideas are similar in that there are something similar to acoustic phonons and you can produce them and they will bounce around and then you can try to measure the total energy being deposited. So, these are currently being developed all based on phonon detection. Okay, and there's a lot more to say about that but I'll just leave the references later if you want. Come to my last couple of slides. So to summarize, there's a lot of potential for exploration for different types of novel materials and novel material excitations in here. And one thing I hope that you come away with is that when we think about the rate in these materials, we can kind of absorb all the material dependent effects into one quantity called the dynamic structure factor. And it's the analog of the form factor when we're talking about high energy scattering for instance or higher energy or for nuclear recoils. So it's a low energy form factor in terms of momentum and energy that we can use. If we know it then we can easily figure out how the material responds to this passing dark matter and its potential. And hopefully I've given you a sense of why so many different materials are considered. There's lots of things I wasn't able to discuss but hopefully that gives you a sense of why that's a growing and important direction for these sub GB dark matter searches. And that's it. I'm well over time. I'm well under time which is good. All right, thank you very much. Thank you very much. Thank you. So we have time for questions. Wait, please. Hi, Tony. Thanks for lecture. So I have questions regarding the diet.