 So, in the previous part, I introduced the diffraction pattern as we expect from a neutron. The specific diffraction patterns are caused by coherent scattering length that I explained. I thought that since neutron and extra diffractions, especially extra diffraction is the most commonly used technique, it is important that I provide a comparison of the two to the students so that the subject becomes clearer. I am talking about neutron and extra diffraction on the same footing and I will show you that they are almost same except for a few differences like coherent and incorrent scattering length part. So, at the onset I must mention clearly that in any diffraction experiment, we are not taking photographs. So, I just showed you atoms in a in an FCC crystal that is how they are arranged, but to get that what I do is do a diffraction experiment as a function of angle or we can write it as a function of the wave vector transfer Q, which is k minus k prime given by 4 pi by lambda sin theta where there is no energy transfer and we are working Fourier space and all your diffraction experiments you can see on the right bottom, I show a typical diffraction pattern of some samples at various temperatures, you get this kind of patterns which are actually caused by the diffraction and the diffraction peaks signify various arrangement, crystallographic arrangement like one I shown as a photograph. So, we are working the Q space known as scattering law so SQ, I am working SQ space and the information I am getting is the real part real space distribution and their Fourier transfer of each other and in these experiments our task is to rebuild N of r in a way. So, N of r is the starting point and with this I will go to our master's degree, but before that I just quickly give a comparison of the properties between exers and neutrons which are repeatedly said that their wavelengths are typically 1.54 angstrom you know for x-rays for copper k alpha neutrons also have same, but in a tablet of experiment this 1.54 angstrom comes from a copper target. So, some characteristics wavelength of a specific target molybdenum if you use it will be even lower 0.6 or 0.7 angstrom in case of neutron I have a continuous Maxwellian distribution and we select the wavelength which are typically in this range interaction of x-rays this is something one needs to digest that depends on electron density in that rather electron density distribution. So, this is not the free electron it is the bound electron density distribution as caused by Thomson scattering in case of neutron it is neutron nuclear interaction and later I will bring in also with the atomic magnetic movement so far I have not discussed it it will be introduced a little later because x-rays in electromagnetic wave it strongly attenuates in a medium in an actually we get information typically from few microns to tens of microns neutrons can penetrate very deeply even tens of centimeters it can go in because it is a charge less particle. Another important difference is that in case of x-ray the scattering length increases monotonically with z we know it follows most less law and it increases monotonically with z whereas neutron nucleus interaction I at some point I mentioned that it is fluctuating across the z values in a periodic table and it provides good contrast between isotopes many times and that I am sorry I mentioned some point that is some point that the x-rays it follows a linear increase I am sorry it is most less law is a power law so in case of x-rays the scattering intensity or scattering length varies z minus mu to the power q for x-rays so primarily what I am trying to evaluate is this I have got an incident wave function e to the power ikr and I have an outgoing wave function which is e to the power ikr and this k to k prime change in direction is caused by the crystal scattering potential so if we go back to our Kittel or any one of the master's degree books basically the scattering amplitude a is given as a Fourier transform of the electron density in this case over the entire crystal which I have written down as an integration over is a Fourier transform of the density I can write d3r which is a 3 hour dv which I wrote there this is how I expressed my scattering amplitude a you can see that this is very similar to what I obtained from the Fermi golden rule in case of neutrons I will go ahead and I will show you the same day the expression for a scattering this is scattering amplitude and nr depends on electron density so this nr is the electron density in case of any condensed matter by electron density I mean there are atoms at sides with electron clouds which causes Thompson scattering of the x-rays for diffraction and in case of neutrons it will be coherent scattering length density so when I write that it is a Fourier transform over the entire crystal the crystal consists of lattice so it's a combination of unit cells combination of unit cells this is one one unit cell where I have just drawn a square lattice I can easily extend it to three dimensions this depends on the unit cells now this integration I can represent it as sum over unit cells cells and the integration over a single unit cell of it's a three-dimensional so I integrate over one unit cell one unit cell and then sum over all the unit cells then I get the scattering amplitude because the crystal is built with the repetition of the unit cells so now this is what the expression is sum over unit cells I wrote d3 are there dv are the same volume integral n of r equal now for extra scattering we have been taught that all values of the reciprocal lattice vector are not allowed you have specific peaks in the Bragg pattern Bragg scattering and actually when q is equal to g then the Bragg scattering occurs so we don't have all values of reciprocal lattice vectors but when q is equal to g a reciprocal lattice vector then we get Bragg scattering that's what the world construction tells us and if I consider one of the lattice sides one of the lattice sides one of the lattice sides then there is electron charge cloud around it so if this is the jth lattice side so the unit cell consists of these lattice sides where I have the electron charge cloud and in this case then I can write n of r equal to so this is rj let us say I have some arbitrary origin so this is rj and any arbitrary point is r so I am the density of charge cloud is given by r minus rj and summed over all the points in the lattice and that gives me one unit cell then I sum up over all the unit cells so this is the expression for the density so now I can write the scattering amplitude is integration over unit cell sum over all the unit cells sum over all the unit cells then density which I wrote n is actually r minus rj and then e to the power minus g dot r because whenever reciprocal amplitude q is equal to I mean sorry the wave vector transfer q is equal to a reciprocal vector in the reciprocal lattice vector then then and then only Bragg scattering takes place I understand I that you are familiar of reciprocal lattice of a crystal lattice this is given by the expressions in case of cubic lattice if a is the real lattice then reciprocal lattice will be twice pi by a but in general reciprocal lattice a a is given by b cross c you would a a dot b cross c this is the definition of reciprocal lattice so basically because it depends inversely if in one direction it is a in the that direction will contract to 1 by a and so if I have a real lattice which is like this two-dimension the reciprocal lattice will have a dimension which is slightly like this so the reciprocal lattice and let me get back to my point that whenever q is equal to g I have a reciprocal lattice vector equal to reciprocal lattice vector q then I will have Bragg scattering and I have to add up over the all the charge clouds which are centered around j point I have for the unit cell I have to sum over all the j or all the lattice points and the general wave vector in a charge cloud is given by nj of r minus rj so now in a simple way I write r minus rj define another vector then r equal to rho plus rj and in that integration d3r n of r minus rj so instead of r I put rho plus rj r minus rj becomes rho and then the variable becomes d3 rho so then I have integration over a single charge cloud summation over j so now you can see that here my integration breaks up into a summation over all the lattice points and integration over a single charge cloud so what I am left with actually I can show you I am left with an integration over a single charge cloud which gives me if it is a j charge cloud then it is fj which is nj rho it is over i g dot rho and there is a summation over all these charge clouds in a single single unit cell so what I mean is that let me take one unit cell let me take one unit cell so I have these charge clouds I have these charge clouds around each point so that integration over one unit cell goes to integration over a charge cloud a single charge cloud and summation over all the charge clouds so I will write this actually now that previous expression so summation over all the charge clouds f of j where the f of j f of j is equal to use q anymore it's a triple integral triple integral so this integral is basically if you see there is a charge cloud around an atom this is nj rho and there is a Fourier transfer over the charge cloud this is known as form factor many of you are familiar with this term form factor and now what we have for the scattering amplitude is this form factor multiplied by e to the power minus i g dot rj let me just take a typical example so now you see here in real space any point rj is given by axj plus byj plus cj in terms of abc and in the reciprocal space it is the hkl the Bragg coefficients is given by h a plus ab plus lc so one is the Fourier transform over the charge cloud around one point which is fj and the other is g dot rj which is here g dot rj is equal to axj plus byj plus cj dot h a plus ab plus lc which gives me twice pi xj h because a dot a is 1 b dot b is 1 a dot b is 0 so you just get axj xj h yjk and zjn so now I have the form factor fj and I have e to the power minus i g dot rj in real lattice real lattice is given by axj plus byj plus cj which is a real lattice vectors and the reciprocal lattice vector g is given a is equal to which is a real lattice points and the reciprocal lattice point is given by a h or h a the way normally it is written h a hkl plane is kb plus lc lc in the reciprocal space and g dot rj for a crystalline lattice is basically dot product of these two a dot a is 1 b dot b is 1 c dot c is 1 because the reciprocal of each other size-wise especially in a cubic lattice this c is nothing but 1 by c this b is nothing but 1 by b and this a is nothing but 1 by a then 1 but otherwise in general also this is true so it is nothing but xj h plus yj l plus zj k hkl so if I talk about an hkl point then g dot rj for that point is given by xj h yj l plus zj k for the real lattice now it is interesting this gives me the selection groups but before that what is fj let me discuss the fj now you see in the expression for scattering amplitude I wrote earlier if you remember the scattering amplitude in case of neutron it was sum over j bj e to the power i q dot rj this becomes g when I come to a crystal lattice otherwise q remains q because I will also discuss with you when q is not equal to g in case of liquid in our system but that later excuse me so this q is g and now what I have in case of scattering amplitude for x rays is fj e to the power i q or g dot rj you see they are same for the selection rule for a crystal and lattice for neutrons so for neutron scattering length was equal to which is f k k prime for a crystal and lattice it comes down to exactly same as x rays and in case of x rays I have scattering amplitude is equal to except for these two terms bj and fj they are identical now bj I know which is the coherent scattering length we wrote as bj and what is fj fj is the Fourier transform of a charge cloud at the side j Fourier transform over this but this is an important difference between x rays and neutrons if you do this Fourier transform actually in Kittel it is shown for a spherical charge cloud you can write n r equal to some constant when r is less than radius I can assume a constant charge cloud and then you can do this d3r will be r square d theta dr d phi and q dot r is q r cos theta you can try this integral but this is not my aim to evaluate this integral over here but the fact is it's an extended charge cloud because it is an extended charge cloud so the Fourier transform will look somewhat like this so if I draw them draw it the Fourier transform will fall in space but if you think in terms of the scattering length average scattering length for neutron it does not have any angle dependence so again I am writing q here it's constant and it's so obvious because your scattering potential for neutrons is a delta function and if you take a delta function it's Fourier transform is constant all over q a delta function in real space will give a constant value in all over q whereas in case of x rays because it is an extended charge cloud by extended I mean the extension is of the order of the wavelength of x rays you have this scattering amplitude this form factor falling in q and larger your atom the faster it will fall that means the form factor for uranium because uranium has a much larger charge cloud will fall much faster than the charge cloud for an aluminum where aluminum are smaller charge so this charge cloud this form factor falls depending on the atom and its size of charge cloud whereas in case of neutron the form factor's equivalent is b which is a current scattering length it does not have any angular dependence so these are very interesting difference so in case of x rays you may not see large angle peaks so so this is one part your f part the other part is your the part which depends on the structure of the crystal xj plus h plus yj k plus zj this is the part which is the structure factor so the structure factor gives this given by the form factor multiplied by part which depends on the structure for example just as an example if you have a bcc crystal then a bcc crystal is one in which it's a cubic crystal you have atoms at the corner at the corner corner at the corner and then you have an atom at the body center at half half half so that means if i consider the coordinates there is for unit cell there is one at zero zero zero one is at half half half so now you see for a bcc crystal the atoms are at zero zero zero and half half half in the unit cell and now you can write this s of h k equal to i consider only one atom so that fj will become f for all of them and the summation gives me one plus e to the power minus it was twice pi i xj h yj k zj this is what i showed you and this is the structure factor of hkl reflection for x-rays this is the structure factor for neutrons as i again repeat there is a form factor there is a coherent scattering length there is the you can consider it as a Fourier transform of a delta function which is constant at all q space whereas fj falls so with this you can see that for a bcc crystal i can write down this expression as f into one plus e to the power minus i pi h plus k plus l my apologies that i here i miss a 2 pi term here there should be because there is a one as i wrote here this is there will be a 2 pi term so second here when i wrote down this part xj h y there will be a 2 pi part because i mean when i define reciprocal lattice i missed one 2 pi point so here i have to put a 2 pi term 2 pi term so with that here is i say 2 pi h plus k plus l because xj yj zj you have to put 000 in this summation so that is one and when i put half half half it comes pi i h plus k plus l so now you see the structure factor is f into one plus h plus k plus l now you can see if h plus k plus l there are certain selections for them this s h k l if h plus k plus l is even then it will give me two f you can see that from this expression so this gives me the selection rule and if i use the values for neutron it will be b but this part remains same so selection rule will remain same for x rays and neutrons so if we do extra diffraction the position of the peaks for let us say bcc crystal will remain same as that what you get in case of neutrons so neutrons and x rays the selection rules remain same but the form factors are different and that's why your high angle peaks may be less intense because this f is multiplying this factor so your structure factor has a prefactor f which is falling in q space or falling in theta but b does not fall in theta so apart from that this part of the expression remains same 1 plus e to the power minus i pi h plus k plus l remains same and depending on the hkl values certain peaks will be allowed certain peaks will be forbidden like as i told you because e to the power minus i pi h plus k plus l is incidental cost pi h plus k plus l plus i sine pi h plus k plus l whenever cost term is 1 you will have this hkl present when it is minus 1 then it will not be present that particular reflection will not be present so you get the selection rules which are same for x-ray the neutron form facts are different so this way i have established an equivalence between neutron diffraction and the way many of you have learned x-ray diffraction they are identical i started from slightly different points and reached at the same place next i need to discuss the thermal effect before i go forward because so far i have considered a lattice which is at zero degree kelvin there are no fluctuations or no thermal effects in this lattice i will introduce the thermal effect and then i will go ahead thank you