 So, what I was discussing earlier that there is a critical angle and nickel is one of the good element, possibly the best element which gives you a critical angle. This critical angle is dependent on lambda linearly, so this gives a critical angle of 6 arc minutes per angstrom. That means for a 1 angstrom neutron, it will be 6 arc minutes for a 4 angstrom neutron, for a 4 angstrom neutron, for a 4 angstrom neutron, it is 6 into 4, 24 arc minutes, this critical angle, this value theta, it will be 24 arc minutes for a 4 angstrom neutron, then it will 4. Also, as I told you that if I magnetize a nickel medium, usually it is in form of thin films, then for magnetization direction with respect to neutron spin, we will have 2 different critical angles. I stopped at this point, now I will come to neutron super mirrors. A neutron super mirror, as the name suggests that it is over much more than a mirror, thing is that we need to reflect neutrons up to a large angle, it can be theta or q whichever unit we use, how to do that? In general, we know that if I have a periodic medium, now I will straight away go into thin films. Suppose I have a thin film which has got a coating of A layer and the coating of B layer. So, this bi layer, I keep coating which has got some D spacing of D, which is equal to DA plus DB. Now, we know that for a given D spacing in crystallography, 2D sine theta equal to lambda and you have got a bright peak. Same thing can be said for this, you can say artificial crystal that you have created with A and B, artificial crystal, but this is exactly same using the same logic, we can say that if this D spacing is fixed that means what I need to do actually, I need to take a substrate on which I keep on piling up this media A and B, typically thickness about 50 to 100 angstrom of each layer. If I do that, then apart from the reflectivity curve, I should also get a bright peak at some point which is twice pi, sorry, twice pi by D dictates at which Q I will have a bright peak. Now, this is the story for a single periodicity that means I make stacks with 1D. Now, the next step is that I keep making stacks, but I keep changing the thickness continuously, that means D keeps varying, D is varying now. So, I continuously keep changing this D value, if I do that I can define this thin film stack as consisting of several D spacing and then it will have broad peaks at all these D values. So, now the critical angle, now the critical angle, I had a critical angle up to which it was one and then it was falling, but now the way I plan this D spacing that it is taken over by the bright peaks for various D spacing. So, by this technique I can extend this critical angle if it was a Theta C to a large value and this is how Theta C for super mirrors are much higher than the Theta C for a single angle. That means best example is nickel titanium mirrors. Reason being we know that the reflection takes place whenever there is a refractive index contrast. Titanium has got a negative scattering length. This is a question mark, I leave it with a question mark now, just accept the fact that few elements have got negative scattering length and then n is greater than 1. Nickel has got a positive scattering length so n is less than 1. So, a layer of nickel and another layer of titanium, they have excellent contrast because excellent difference in refractive index between the two media and there will be strong reflection and then the next step is to design a super mirror with a variable D spacing and I can get a super mirror which can reflect up to a very large angle and that means I can reflect neutrons up to a critical angle which is large compared to a single element. For a single element as I showed you earlier that it is around 6 angstrom, 6 arc minutes per angstrom for nickel, in this case this goes much larger and usual comparison is with respect to nickel always. So, if I say I have got a super mirror which has got which is m equal to 2, which has got m equal to 2, m equal to 2 then m equal to 2 super mirror will have critical angle which is 12 arc minutes per angstrom for neutrons. Similarly, m equal to 4 will have 24 arc minutes, arc minutes per angstrom for neutrons and this is not an indefinite game because you also have to think that neutrons are penetrating in a medium and getting reflected from each and every interface and how thick you can make the medium till the absorption starts taking over, but this is possible and that is how neutrons super mirrors are made. So, we have been talking about neutrons super mirrors and super mirrors are actually bilayer stacks. If I have a bilayer stack with a single periodicity then it is like one dimensional crystal which has whose periodicity dictates where will be the Bragg peak. Similarly, for the mirror if I have a single periodicity then I will have a Bragg peak at some q value, q value or at some theta value I will have a Bragg peak. Now, this Bragg peak is not an atomic Bragg peak, but by the same principle it is a it is like a virtual one dimensional crystal which gives a Bragg peak depending on its d spacing which I told you in case of nickel titanium it will be d nickel plus d titanium titanium and this is this Bragg peak position is dictated by twice pi by d for this particular single periodic step, but now the clever manipulation is that in case of neutrons super mirror we vary the d value in our thin film mirror. So, this is truly like a mirror and actually look at it it will look like a daily mirror which is we are familiar, but this is of course at a much lower angle and with a much higher precision this needs to be made the d is a variable now d is a variable for a super mirror. When d is a variable in that case the as I told you that I can have the Bragg peaks overlapping with each other because d is varying continuously and this can push the critical angle to an outer outer means to a higher angles theta and then it will fall. So, this critical angle is a critical angle for the super mirror for the super mirror. So, this can be compared with the critical angle of a single film and usually in case of in the field of neutron super mirror it is compared with nickel. I am repeating again nickel has got 6 arc minutes per angstrom of neutron wavelength that means a 4 angstrom neutron will have a critical angle of 24 arc minutes and when I talk about super mirrors and m equal to 2 super mirror will have twice this value. So, it will have 12 arc minutes per angstrom at the critical angle and for 6 angstrom it will be 36 arc minutes more than half a degree and the trick is that you have a plane substrate usually this is called float glass float glass on which you keep depositing thin films using thin film deposition techniques you have a specific bi layer and you know how you can calculate out how to change the thickness of the layers and you keep growing this step with variable thickness and ultimately you get I will show the photos later that reflectivity as a function of theta it goes out to a much larger angle compared to what you can find for a single element. So, this that is why it is called super mirror because the critical angle is large and you can reflect neutrons reflection means here is sort of Bragg law I would say snails law which follows snails law the neutron reflection follows snails law and that so far as the total scattering is concerned and it can the critical angle is much higher compared to any elements in case of neutron super mirrors. So, I just show you the so I showed you the critical angle for a silicon wafer and I discussed the super mirror now I can show you this is the source Swiss neutron is it is available from them it has got a critical angle which is almost 8 times because some super mirrors have a critical angle up to 8 times of nickel that means 8 into 648 art minutes per ounce from for this super mirror. So, these are the various super mirrors available from them written as m equal to 2 3 4 5 6 7 up to 8 you can go of course thus slow rather the reflectivity is not exactly 1 actually you can see that by the time we have 1 m equal to 8 the reflectivity has come down to almost 50 percent, but if you talk about m equal to 3 or 4 super mirrors you have got a very good reflectivity 20 percent and these are actually mirrors for neutrons and that is why I call that the neutron super mirrors, but so far I have not talked about I have I started talking about polarizers and I introduced you to neutron super mirrors now let me get back to super mirror polarizers. So, in case of super mirrors also I have got 2 different critical angles one is for b coherent plus b magnetic and the other is b coherent minus b magnetic. So, I can design now super mirrors in which a magnetic usually a magnetic like a nickel and a non magnetic material are coated in alternate layers with a variable periodicity and this super mirrors needs to be magnetized once I do that what like here is a nickel I show you the actual design data or nickel manganese super mirror and you can see that the critical angles are so different now how does it help in polarizing so now I get back to the story of polarization so now I have got this reflectivity versus theta that we just simplified so I have got a critical angle and then the fall for positive 1 and I have got the reflectivity plot for the negative polarization now you can see that if I reflect a neutron at angles beyond the critical for the negative polarization so now for reflection at angles between beyond this and less than this between these two for an unpolarized beam the one polarization is reflected and the other polarization is not reflected they are not lost they get transmitted so I have got a transmitted beam which is positively polarized and I mean sorry a reflected beam which is positively polarized and a transmitted beam which is negatively polarized so this is a positively polarized beam but this whole exercise is done mostly for cold neutrons because for thermal neutrons reason being as I said that the critical angle is dependent on lambda so for thermal neutron this lambda is of the order of one angstrom the critical angles are too small and difficult to control we talk about cold neutrons when the wavelength is four angstrom or higher or higher longer or longer when the wavelength is four angstrom or longer typically they are known as cold neutrons and for cold neutron polarization polarizing super mirrors super mirrors are used routinely nowadays they are commercially available and experiments with neutron reflectometry or small angle neutron scattering if we have to do with polarized beam it makes sense to using polarizing super mirror so I have talked to you about single crystal drag diffraction from magnetic sample for polarization of thermal neutron beams now I have explained to you how neutron polarizing super mirrors are used to polarize cold neutrons in general another class of upcoming polarizer is a helium three polarizer this is in principle it is simple in practice it is it needs very high technology so this is a iron silicon polarizing super mirror where as I told you that you can see that the up neutron reflectivity is very high up to a very large q value this q value will dictate what is the lambda or what is the theta because q is equal to 4 pi by lambda sin theta and for the down neutron it goes down to zero very often so if I consider 0.33 already the reflectivity of the down neutron is zero so any angle beyond this will give me a polarized reflected beam and also polarized transmitted beam because the neutron which is not reflected are transmitted through the polarizer so this is a iron silicon super mirror polarizer available from the sources I have taken the data from Swiss Neutronics site now coming back to helium three polarizing super mirrors these are transmission polarizer there has been long attempt to use helium three gases spin dependent when I say spin dependent is nuclear spin dependent absorption for polarizing transmitted beams large area transmitted beams so this is spin dependent neutron capture in an intermediate state where the helium three absorbs a neutron and then decays to a triton and a proton but interestingly neutrons with spin component antiparallel to the helium three nuclear spin when you talk about helium three polarizers it is a nuclear spin which absorbs so that means this has a very tall order of polarizing helium three gas by using some techniques now helium three gas polarization is done by interaction of this helium three in a cell with a laser beam and after that we get polarization of helium three so the transmission for two spins that means we have the helium three gas the nuclear polarization which has taken place and now we can write down the transmission as an exponential term sigma zero is the absorption cross section for the unpolarized neutrons but it gets boosted by one minus a one plus polarization of the helium nuclear the nuclear absorption nuclear polarization so it can be as high as 6000 burn for the antiparallel neutron so that's why one spin component goes out and since it's a transmission polarizer since it's a transmission polarizer it's a transmission polarizer the transmission polarizer in a simplistic terms you can have very large beam cross sections for some experiments which can be polarized in the transmission mode so you have you have got plus beam minus beam minus gets absorbed very strongly if the helium three nuclei helium three nuclei are polarized and this polarization is done by its interaction with the laser so I am just quoting it this is an old paper so it was the status at a in 2005 at NIST and this tells that it is done through interaction with two laser beams and how the gas is stored in a buffer cell I myself I don't want to get into the details of this details of the theory of this polarization but the basic fact is that helium three gas with nuclear polarization can be used for transmission of one component of neutron spin that is the basics of it but now it has advanced much more and today at ILL Grenoble and NIST there are helium cells which are put in line with the neutron beams and polarized beams are used so this completes my target of telling you how neutron is polarized for application next part is a brief part is flipper so here it is the data from the same paper and it shows the polarization efficiency of 91.2% but even at that time the time the polarization could be held and let us at 80% was 250 to 300 seconds it has increased a lot at the moment and that's why because if the polarization is lost quickly then the transmission polarizer also loses its efficiency so not only we need to polarize helium three but you also need to maintain the polarization for sufficiently long time for which the polarizer can be used in the neutron beam so in this paper the time was around 250 to 50 seconds I should say 250 seconds is around four minutes for which one can have a good polarized neutron beam I understand that this has gone much higher today but this is not an often used technique the often use techniques are what I discussed earlier the crystal polarizer Bragg scattering and the multi layer based super mirror polarizers now the last part of this talk is about flippers so we not only need to polarize the neutron beam but we also need to because there is a sample which is possibly magnetized in some direction and we need to have polarization along and sometimes polarization opposite to get the two different intensities of scattered Bragg scattered or mirror scattered beams and many times you also need to do a polarization analysis of the reflected beam so we need to flip the neutron spin after polarization depending on the experimental requirement so the flipping is as simple as said that up spin by flipping I should take it to down spin so there are several types of flippers I will just briefly tell you about other spin flipper here actually the neutron is polarized in the Z direction in a guide field let us call it B so if Z is the direction then I call it BZ that is the B0Z and then in radio frequency flipper if this is the if the particle is moving in Y direction and X reaction is normal to it then we can apply a radio frequency field another field which is rotating in this plane when it is rotating in this plane then you can see that this neutron sees a field which is a cone and which sees a static field of B component Y component of the B of this rotating RF frequency and then in this field the neutron will undergo precision undergo precision and if the Larmor precision frequency of this neutron I can match with the Larmor precision frequency based depending on the guide field B0 and if I can match the Larmor precision frequency with the field in the Y direction then the depending on the length of the RF cavity the neutron will go out with a flipped spin so this is called RF spin flipper being used right from the beginning of neutron scattering and later one more kind of spin flippers are introduced by Mezee known as DC flipper this is much easier to understand here actually the BG is a direction of the field guide field and neutron spins as you can see they are up over here you just take it through a guide through a field which actually there are three component of the field one is that minus BG to cancel the effect of BG and the field as you can see it is a solenoid so it is normal to the spin of the neutron and in this field depending on the velocity of the neutron and the field value the neutron undergoes a precision so if you can choose a certain wavelength and accordingly the length of the DC flipper which the neutron has to traverse then we get flipped neutron spin coming out from the other side of the flipper and the guide field you can see the field written as a guide field plus compensating fields and the field in which the neutron is undergoing precision and flipping so I have discussed with you two types of flipper one is a neutron spin flipper one is a DC flipper or a Mezee flippers most of the laboratories use these two kinds of flippers for neutron spin flipping for use in magnetic neutron diffraction or reflection or even analysis of the scattered beam so with this I come to an end for in the discussion regarding neutron polarizes and neutron flippers