 So far, I have discussed with you various components of a spectrometer starting from beam tailoring at the reactor to detectors but neutron being a spin half magnetic nuclear particle The few more important parameters are the neutron polarization for some experiments and the capability to flip the neutron spin because often in Magnetic neutron scattering we need to polarize the beam of neutrons and Then use it for scattering experiments and sometimes even analyze the polarization of the reflected beam So in this lecture, I will introduce you to neutron polarization and Very briefly about spin flippers because these designs can be intricate and elaborate and if you get In details, then we might not be able to reach the target of neutron diffraction So I will be briefly discussing spin flippers and neutron polarizers Now we know that neutron is a spin half particle a Magnet with minus one point nine one nuclear magneton Which is of course the nuclear moment is much lesser than what we have for electron nearly two thousand times weaker but it's a tiny magnet which can penetrate deep and Sometimes for many magnetic studies We need to polarize the neutron beam That means with respect to some direction the neutron beam will either have plus of spin or minus of spin and Also for such experiments often we need to Flip the polarization of the neutron beam from plus half to minus half or vice versa. So so far The point that I've discussed direction. Yes we get the Direction defined using various kinds of collimeters in pile collimeters as well as solar collimeters We get the energy of the beam defined using monochromators If we want a broad beam then using velocity selectors or even sometimes using filtered neutron beams So direction energy how they have to be defined we have discussed now comes the polarization polarization can be done By various means using drag diffraction from a magnetized crystal using super mirror reflection and also presently a very novel Polarizer called helium-3 transmission polarizer are used in some few Advanced neutron sources. So I will give brief introduction to these techniques Now it is I will prefer to start with an expression for Bragg diffraction from magnetic crystal the derivation of this expression is long So I'll skip that but I will Give you the expression for intensity of the Bragg reflection which has got nuclear part magnetic part and Various terms as I defined So if there is a K is a unit vector along the Momentum transfer and if mu Is the unit vector along the magnetic moment in general? I can consider this Happening from a plane sample then Q is a vector Q is a vector which basically if I take out the component of Mu That is along the K direction then I get this u vector once I define the vector then the Bragg intensity is given as Fn square Plus Q square if m square plus 2 Fn Fn P dot P dot Q P is the polarization of the neutron now this expression. I will explain you term by term So I am talking about a sample. Let me just Try to give us so which is a crystalline sample crystalline sample Which also has got magnetic moments and these magnetic moments are aligned And I am Carrying out a Bragg diffraction from this sample now here the term Fn is the nuclear scattering amplitude Which I had defined earlier to you which was actually It was a coherent scattering length to do by IQ dot rj if you remember The counter part of it is a magnetic scattering amplitude. So that also looks so here you can see this is the Nuclear scattering amplitude which is BJ times to the power IQ dot rj that Q I have replaced it with a G which is a which is a reciprocal lattice vector and then twice pi hx plus ky per cell z is Fn square and There is a thermal term. There's a Debye-Waller factor which I discussed earlier that because of the atom Oscillating around its mean position you have a reduction in the Bragg intensity and That reduction is given by a to the power minus To W where it is W is basically earlier I had discussed it is sigma square q square by sigma is a root mean square displacement similarly Apart from this constant terms which you can just accept for the timing point five G is the London splitting factor J is the angular moment of the nucleus total J value of the nucleus it is same Except for a F which is the magnetic form factor and rest of it is same. So we have very similar terms So there are terms which are square of this and there's an interference term. So this has got This is nuclear term This is the magnetic term and this is the interference term Where P is the polarization of the neutron and q is the vector which are defined just now so now this is the expression of Bragg intensity coming from a certain plane of a crystal with magnetic moment at the sides when we write this We can see now Q is this in case the Q is if the magnetic moment is in plane Then you can see that mu dot K will be zero Then Q is equal to minus mu which is the I mean the unit vector. So Q square Q square is equal to one and Q is equal to plus or minus one When I have in plane The magnetization vector is in plane in that case you see I have got several cases P dot Q can be plus one or minus one and then I Is equal to If n square Plus if m square Q square equal to one I have put plus 2 fn if m fn fm and P dot Q P is the polarization of the Neutron beam. So this is what I wrote I have put Q square equal to one because I have assumed that the magnetization is in plane in that case please note that if polarization Please note that this P is not the P vector. Let me make it capital P. I'm sorry Just make it capital P. So that he did not get confused This is capital P capital P This is not the vector here and this is also capital P This is also capital P so now You can see when Q is one when the the Neutron If it is plus half this P will be one. Let us say P will be one and When it is opposite To the magnetization of the sample it will be minus one in that case when P is plus one I is equal to intensity is equal to fn plus fn Square so this is when the nuclear form factor or nuclear scattering amplitude interferes constructively With the magnetic scattering amplitude when P is minus one That same is equal to fn Minus fm So then they are in opposite acting opposite to each other and They are destroy destroying each other or interfering destructively. I must mention here one thing when I am talking about Magnetic thing there is one F came form factor. Now if you remember Earlier when I discussed Nuclear scattering at that time I said that the nuclear scattering potential is a delta function and the delta function Does not have any drop as a function of Q because it is at R equal to 0 a delta function and whose Fourier transform is one all over Q space Now for magnetism. I cannot say the same thing to understand this fact that Tiny nucleus of femtometa size is here, but the magnetism comes from unpaired electrons Unpaired electrons Electrons so you have nickel Hobart Iron these are 3d elements or you have rare earth switcher 4f or 6f that means the these shells Are basically not filled and that's why I can consider it as a shell classically in which you have a Magnetic moment, but now Delta function is Fourier transform as a function of Q is One everywhere acceptable, but this shell But this has got a finite subscending of the order of angstrom and This falls even faster than the Form factor I calculated for x-rays For an atom because then you consider the whole sphere here. It's a shell. So it will fall fast It will fall fast. So that's why the magnetic peaks or magnetic contribution goes at lower Q's Lower Q's because At high Q at high Q Hi Q the Magnetic form factor form factor Factor Falls Rapidly, please remember this You will get back to this point when he come to actual magnetic neutron diffraction So now after explaining this thing that sample magnetized in plane and Q is equal to minus mu Q square equal to 1 that gives me If the polarization is parallel to Q p dot Q is 1 I have Fn plus fm square constants apart They boost each other and they add up the form factor the magnetic scattering amplitudes when the neutron has opposite polarization in that case they Don't add up But the magnetic part gets subtracted from the nuclear part. So the picture is somewhat like this. This is the Scattering vector perpendicular to the plane. This is the planes which are reflecting the neutrons Not just the external planes of a crystal, but they are the reflecting planes Now if magnetization is in plane, then you have two cases either plus or minus half and they are Intensities are different because one is fn plus fm square other is fn minus fm square Now I have to look for a reflection like we know that in crystallographic Perlens we talk about 1 1 0 1 0 0 2 2 0 reflections So if I get a reflection for which Fn is almost equal to fm then the beauty is that fn minus fm is 0 and Fn plus fm is 2 fn. So we have only one polarization. So in that case This are the planes of The crystal The neutron comes they are magnetizing plane With plus or minus polarization with respect to the magnetization But because fn is equal to fm So the down part goes to 0 and then the back reflected beam has it is polarized This is a Crystallographic polar crystal polarizer the black diffracted neutron is polarized and we can use this polarized beam for further experiment so So this is what I want to say that the high polarization is possible in black reflected beam and the polarization efficiency is defined as i plus minus i minus plus is the Reflected intensity of the up neutron minus is the reflected intensity of the down neutrons and this is given by 2 fn fm Fn square plus fm square or when fn is equal to fm the polarization efficiency will be 1 often it is not exactly same, but they are close and Historically Fe3O4 2 2 0 was the first reflection used for polarization The polarization efficiency was not too high. It was 95 percent FCC cobalt gives very good polarization But I must mention that this is cobalt is COO should be small. This is not Carbon carbon net. This is CO Sorry for this. It's a cobalt cobalt It's cobalt. This cobalt 92 percent and iron for 8 percent Alloy it gives 2 to 2 0 0 plane gives very high efficiency But the issue is that cobalt is a strong neutron absorber. So we lose intensity in absorption At present the Heusler along which is Cu2MNAL and Iron silicon Fe3Si are two materials They have a very similar characteristics as polarizers in both of them the 1 1 1 reflections are matched means Fn is nearly equal to fm and these are used as Neutron polarizing monochromatase. So these neutron Here we not only choose the polarization depending on the Bragg angle We also choose the lambda. So it monochromatizes So these monochromatizes plus polarizer So Heusler alloy is used in one of the magnetic neutron diffractometers at Dhruva So this is about Crystal polarizers where the monochromatization as well as polarization takes place Next I will discuss with you about Mirror-based polarizers. But before that I want to bring to notice one thing Suppose we are doing diffraction from a magnetized sample that means apart from crystallographic long range order There is magnetic long range order. We shall come to later when I deal with specific examples Then using this expression These expressions Fn plus fm square and fn minus fm square when you add them up because the beam is Unpolarized so half the neutrons are coming with up Spin with respect to the sample magnetization half of them are coming with minus of Orientation with respect to the sample and the intensity is half of i plus plus i minus From those expressions and their fn square plus fm square. So even for an unpolarized beam we can find out fm square through fitting processes and we can find out the magnetic structure with an unpolarized beam for a magnetically aligned samples and This is used I would say most of the time this has been used and I will discuss it under the Heading of diffraction from crystallographic samples just diffraction Plus diffraction to find out magnetic structures. Now I talk about mirror based Some neutron polarizers. So one can define refractive it takes of a medium for neutrons. So far we have been talking about the atomic structure of the medium when you talked about Neutron diffraction, but at a much lower Q If I do think of Q Which is of the ordinary say 0.1 angstrom inverse Then twice pi by Q max, let us say twice pi by Q max is about 6 by 0.1 60 angstrom. So for such a low Q scattering experiment the medium does not come as a medium comprising atoms and molecules, but as a uniform medium the way you see the medium in case of light So then in this case We can define the refractive index of a medium for neutron in terms of me if it is a non magnetic sample and I'm talking about unpolarized neutron beam then n is given as 1 minus lambda square by pi Rho into B coherent where lambda is the wavelength of the neutron Rho is the number density of the scatterer and B coherent is the coherent scattering length density in this case you can see for Rho some values of Rho B coherent n is less than 1 n is less than 1 you can write n equal to 1 minus delta Where delta is of the order of 10 to the power minus 6 per most materials materials in case of neutrons that means You can say it is 0.999999 It's that close. This means that since n is less than 1 Comparing it with our experience with light rays traveling from a denser medium to a rarer medium here the any medium having refractive index less than 1 behaves like a Lighter medium with respect to air or vacuum because its refractive index is less than 1 So in case of optics, we know that total internal reflection takes place In case of neutron for most materials total external reflection will take place and since n is given by So for large B coherent large density, this n will be more different from 1 this delta value will increase and that means Our total external reflection will take place up to a larger angle So in general if I say a refractivity profile as a function of q or Theta because q is equal to 4 pi by lambda sin theta Just like life. So reflected intensity up to certain point It will be 1 because total external reflection Takes place and then beyond that critical angle the intensity starts falling So this is a typical reflectivity curve of a medium. So in brief that means neutron Allows the lower especially the long wavelength neutrons or rather low q Scattering experiments or that means at grazing incidence the medium behaves like an isotopic medium of a given row B coherent and Total external reflection takes place so I Have just shown you the Experimental data from a silicon wafer and then In the silicon wafer you can see up to the critical angle the reflectivity is 1 and then it falls actually When I go to large q value that falls as 1 by q to the power 4 known as Fresnel Reflectivity, but that come that comes later, but the fact is that I have a critical angle up to which the Medium reflects neutron totally now Reflectivity is less than 1 if there is a magnetized medium then Similar to what I did in case of Bragg scattering now Apart from be coherent For a magnetic medium. We also have a magnetic scattering length, which is be magnetic and now this be magnetic depending on the magnetization direction with respect to the Neutrons pin can either be plus be coherent or be Minus it can be either be coherent plus be magnetic or be coherent minus be magnetic So refractive indices are different for two different spins of neutron So that means now this angle they draw which I drew here So now I can have two different critical angles For plus for minus this is very interesting and you can see that if I go Beyond this angle one Spin component gets reflected the other spin component doesn't get reflected interesting So I will stop now and Then I will continue with the same thing Well introduced to the now it is for a single medium. I'll talk to about neutron super mirror polarizers What are super mirrors? These are beyond mirrors. I will discuss in the next module