 From this lecture we will start our discussion on neutron reflectometry. Till now we have discussed mesoscopic structure studies using neutrons at low Q range. I have gone through the experiment that you can do using long wavelength neutrons because you will be working at lower Q range or momentum transfer range. Another lake of this mesoscopic structure studies are thin film and multi layers where we will be using the technique of neutron reflectometry. This is an extremely important tool in hands of experiment place at present specially for magnetic thin films because since the discovery of giant magneto resistance a lot of work is getting done in the field of thin films and their magnetism interface coupling, interface magnetic movement, so on and so forth. All these experiments or characterizations can be done using neutron reflectometry. Neutron reflectometry and exo reflectometry are very close. They are close cousins as experiments, very similar and can be done on the same sample provided the densities allow. I will also be discussing exo reflectometry partly with you and we will show that exo reflectometry and neutron reflectometry can be used for understanding mesoscopic structure of thin films in terms of density, thickness, interface roughness, etc. Both of these can be used. But in addition, polarized neutron reflectometry can be used for same things but with a magnetic term in front. So magnetic density, magnetic layer thickness, etc. and also magnetic roughness. In the first part of my lectures, I will be discussing specular neutron and exo whenever required reflectometry. Of specular reflectometry something which I will take towards the end. So let me quickly tell you, specular reflectometry is something in which we use Snail's law where the angle of reflection just as we have learnt from optics in our school that angle of incidence is equal to angle of reflection that is known as specular. In case of off specular, we talk about diffuse reflection that means the angle of incidence theta i, theta i is possibly not equal to angle of reflection theta r. So in case of specular, theta i is equal to theta r and most of the time we will be discussing specular neutron reflectometry and then I will take example for off specular which is also interesting study when theta i is not equal to theta r. As an example I might tell you that reflection from a mirror reflection from a mirror that we see every day is specular. Reflection from a movie screen is off specular because a movie screen the image is seen from all angles so there is nothing like an angle of incidence and angle of reflection equality whereas in case of a mirror you can see the image only when you are at a position where you are satisfying angle of incidence equal to angle of reflection. So this is the broad difference between specular and off specular and I will be discussing initially the specular reflectometry. But before I get into the discussion proper I want to point out to you this small difference between experiments in the reactor hall and experiments in the guide halls. Please know that so far we have been discussing large q experiments large q large momentum transfer experiment. So we have a powder diffraction we have discussed powder powder diffraction diffraction we have discussed single crystal diffraction and magnetic diffraction and also also liquid and amorphous liquid and amorphous these are all liquid and amorphous spectrometry on the other side of the reactor block in Truba. So all these the q ranges are large large q ranges broadly I can categorize that experiments which you do inside the reactor hall are in the large q range so 4 pi by lambda sin theta so we use shorter lambda and for the guide hall we have reserved experiments which are done at low q so 4 pi by lambda sin theta small angle and longer wavelength why so because these guides that you see they transport neutrons through total external reflection just like a mirror and the guides preferably prefer cold neutrons or neutrons which have wavelength longer than typical thermal neutrons in the higher energy and that's why like this particular guide it has got a critical angle of critical wavelength of 2.2 angstrom and there's a guide on the other side which critical angle is 3 angstrom by critical angle I mean the transmission of the guide transmission of the guide it depends on lambda if it's a curved guide and one guide I can draw like this so this transmission goes up like this and this is 2.2 angstrom for one guide and 3 angstrom for other guide so it is better to do experiments using neutrons in this range using the guide halls so in general everywhere this partitioning you will find that long wavelength wavelength experiments are done at guide hall guide halls and short wavelength wavelength at reactor hall there are exceptions to this there are thermal guides it is always preferable to go to the guide hall because of very good signal to noise ratio in case of the reactor hall the signal to noise ratio is small because they have high noise whereas in case of a guide hall you have low noise low number of background neutrons and signal to noise ratio improves so it is preferable to do experiments in the guide hall in general but because guides transfer long wavelength neutrons comparably the low Q or long wavelength experiments are preferred to be set up in the guide hall with this much I will go back to reflectometry so this is the reflectometer in Dhruva but I will discuss other instruments also all over the world but what kind of studies we can do with them so let's come back to reflection reflection is a phenomena which is our daily experience I have just shown you some photographic image of reflection of trees in water now you can see from here to here as I travel one qualitative comment that I can make that the water surface is smooth here slightly less smooth here and rough here so I can talk about roughness in a qualitative term roughness of the surface so from the reflection I can make a qualitative comment about the surface that the surface is more rough or less rough and sometimes I can say it's a very bad surface maybe there is roughness not roughness but the surface may be wavy at macroscopic length scale so but these are all what I showed you at a qualitative scale but when we do neutron reflectometry different from optical reflections that is visible to the naked eye neutrons on x-rays can also be reflected because neutrons have a de Broglie wavelength lambda and x-rays are electromagnetic so both of these they are waves and reflection is a property of waves, refraction so they can be reflected, refracted but with certain constrictions I will come to so neutron and x-ray reflectometry both can determine structure for both neutron and x-rays categorically magnetic when you use neutron polarized neutron reflectometry information of thin films and interfaces at mesoscopic length scale so I will talk about a film which may be 100 angstrom thick and magnetic moment may be say I can say per 10 angstrom thickness roughness may be 1 to 10 angstrom so these are 25 parameters at a mesoscopic length scale but both these experiments are at near raising incidence and that's why it's difficult so before I get into that let me just quickly tell you that what is the difficulty we have seen that in case of small angle we use a very narrow beam and the sample is in transmission mode transmission mode whereas in case of x-ray and neutron reflectometry we have a thin film mostly on a substrate this is a thin film I'm exaggerating and I have to put it in a reflection mode in a narrow beam so the beam is narrow because the average angle with respect to the surface is of the order of 10s of arc minutes to maybe a degree and this beam can be as narrow as few arc minutes arc minutes so the instrument needs one narrow beam often much narrower than what we use in a sans instrument and alignment of the sample alignment of the sample that is very important alignment of the sample sample in the beam so these are the difficulties because when you make a narrow beam you cut down the intensity of neutrons but lesser number of neutrons and when you have such a narrow beam let me just show it like a line like this I need to align my sample in the beam I have to bring it in a beam at a certain angle so that I can see a reflected beam and the reflection angle itself also can be questioned we have means of answering that one is the angle of reflection and second is how to bring the sample in the beam because the sample is maybe 100 nanometer thick and it needs to be placed in the beam by moving the substrate so these are the challenges with respect to a neutron reflectometer interestingly historically I feel like sharing this with you because this is a major technique today what one could say about X-ray the neutron the discoverer of X-rays Ron J. in the first Nobel Laureate you look at his comment in 1895 the question has to reflection of the X-ray maybe regarded as settled by the experiments mentioned in the preceding paragraph in favor of the view that no noticeable regular reflection of the rays take place from any of the substances examined other experiments which I omit lead to the same conclusion so he said that the X-rays cannot be reflected this is December 28, 1895 this is the year when he I think that is the year when X-rays are discovered but this technique was rediscovered in 1954 and reestablished as a technique so for as neutron is concerned historically Fermi and Zinn in 1946 were the first to present neutron reflectometry measurements for finding out coherent nuclear scattering why it is so I will tell you coherent nuclear scattering it can be found out using neutron reflectometry and approximately a decade later the first report on X-ray reflectivity actually following the comment by Ron Jen later in 1954 we get the classic paper paper where surface studies of solid by total reflection of X-rays so X-rays can be reflected and this is a classic paper which is followed by almost entire thin field community those who attempt to do X-ray reflectometry so what Barrett says in 1954 analysis of the shape of the curve of reflected X-rays so basically the case how Ron Jen missed reflection because he didn't realize that the refractive index of X-rays is almost one I will complete briefly but if you can take care of the beam that has to be very very narrow and if you can take care of the angles that you can measure then it's a this paper says analysis of the shape of the curve of reflected X-rays intensity versus glancing angle in the region of total reflection provides a new method of studying certain structural properties of the mirror surface about to several hundred angstroms D using dispersion theory extended to treat number of stratified homogenous media is used as a basis of interpretation this classic paper not only provides with experimental results but also with the formalism that we used even today to analyze or fit X-ray and neutron reflectometry data obtained from our instruments so X-rays and neutrons they give us non-destructive tools for thin film characterization in as reflectometry so here as I mentioned earlier that the refractive index for X-rays and neutrons can be written as n equal to 1 minus delta so what is the value of the delta I will come to in later part but I want to point out to you that this delta value this delta is very very small so one is that n is marginally less than 1 for neutrons and X-rays and then what it means let me compare this with what I find in general optics you are familiar that let us say water water has a refractive index around 1.33 in case of optics we use the angle with respect to the normal to the interface between the two mediums if it is water it is air when light ray comes out from water to air it undergoes total internal reflection that means as I keep increasing this angle at some angle the refractive beam goes along the boundary and beyond that it reflects back inside the medium it cannot come out from water this is known as total internal reflection just as opposite to this if n is less than 1 then every medium the medium has a lesser refractive index compared to the vacuum or air we will consider air as vacuum or air in that case when the ray comes from vacuum inside the medium just the same thing happens in the opposite direction at the interface I am not going to draw it properly let me exaggerate it it undergoes a change in direction this and up to a certain angle as we go to larger angles that means if this angle is smaller this angle is larger then it undergoes total external reflection total external external reflection so because the refractive index is less than 1 I have total external reflection and let me point out right at the onset unlike optics which you studied in our school days in this case I will consider the angle with respect to the surface and not with respect to the normal so that means as I go from smaller angle with respect to the surface actually in case of optics this would have been 90 degrees here I am starting from the surface as I go up to a certain angle the beam gets totally reflected so this is total external reflection and beyond a certain angle it starts penetrating the medium beyond the critical angle what is known as critical angle critical angle and then the ray penetrates in the medium and the reflected intensity goes down intensity goes down so please be aware of this fact that I am measuring the angle with respect to the surface and because the refractive index N is less than 1 actually it is marginally less than 1 as I told you it is 1 minus delta where delta is around 10 to the power minus 5 for x-rays even lower for neutrons so this total external reflection takes place at a very very small angle up to an angle known as theta c which is very very small and the beam gets reflected and this is the principle of neutron guides I showed you photograph of neutron guides earlier and today also the neutron guides surface it is nickel coated nickel coated, nickel coated surface it reflects the beam and allows it to travel in vacuum we have to evacuate the nickel guides like you see light traveling in an optical fiber in that case it is through total internal reflection in neutron guides it is total external reflection now let us get back to the refractive index of x-rays so I will quickly try to give you how we derive the refractive index for x-rays so this is from our master's level course in plasma physics the refractive index is given by omega p is the plasma frequency and omega square is the frequency at which we are calculating the refractive index now in case of collection because x-rays x-rays interacts interact with charged cloud electron cloud to be precise I can write down this plasma frequency omega p square I am not deriving it I request you to accept it this is all 4 pi e square rho e by m e this is the plasma frequency and then I can write n equal to 1 minus 1 upon omega square omega p square equal to 4 pi e square rho e by m e so now h cross omega is equal to e which is omega equal to h nu I can find out omega to lambda because nu is 1 by lambda so hc by lambda this is equal to by changing omega to lambda I get the expression n equal to so it is n square so I get n equal to 1 minus lambda square by 2 pi e square by mc square rho e now this is because I wrote here n square n comes as to the power half it is to the power half assuming that this value is very small I can write down n equal to 1 minus half epsilon and that gives me n equal to 1 minus lambda square by 2 pi e square by mc square rho e now e square by mc square is known as re classical electron radius and is given by 2.818 femtometers so I can write finally n equal to 1 minus lambda square by 2 pi r rho e where re is the classical electron radius rho e is the rho e is the electron density so electron density is given by rho e for a atom let us say like nickel or for a single atom like cobalt or anything you know if z i is the i th atom charge number and n i is the number per unit volume then if I sum over all the atoms so it need not be 1 it can be many this is give rho e power electrons electrons power unit volume number of electrons so this is rho e if you remember a similar thing I discussed with respect to suns I discussed sl scattering length density we will come to something similar later so for this it is 1 minus lambda square by 2 pi rho e re this is the value of refractive index for x-rays so it depends on the electron density and the classical electron radius