Ewald's sphere transcript

we will discuss evolves sphere construction this is a beautiful construction to answer an important question the question is what are the directions of diffracted beams from a crystal for a given direction of incident being let's look at it suppose this is a crystal specimen and an x-ray is shining in this particular direction as shown by the arrow here so the question is for this particular direction of the incident beam and this beam is of a particular wavelength and the crystal is shown here with its periodicity what are the directions of the diffracted beam of course to answer such question we have the diffraction condition in terms of bragg's law lambda is equal to 2d sine theta here lambda is the wavelength but d is the interplanar spacing and theta is the angle of incidence of the x-ray on that plane as yet i have not shown any plane there are many possible planes which can be drawn or which exist in a given crystal so one particular set of plane i have now shown for this plane the spacing d is equal to d1 as shown here and the angle of incidence of the beam with the plane is theta 1 so we have to check whether this interplanetary spacing d1 and this angle theta 1 satisfies the bragg's law lambda is equal to 2d sine theta so we have to check whether lambda is equal to 2 d1 sine theta 1 if it is equal to this if this relationship is right then we will have a diffracted beam and the diffracted beam will appear to be reflected from this set of planes this is what is bragg's law if we don't satisfy this relationship then there will be no diffracted beam so we have answered the question but we have answered the question partially partially because we have answered it only for one set of planes a crystal has many in fact infinitely many sets of planes so for another set of plane we have to again apply bragg's law here i am showing another set of plane in green now the interplanetary spacing changes to d2 and the angle of incidence also changes to theta 2 so with this changed value of d and theta i'll again have to apply bragg's law to see if there is a diffracted beam or not so bragg's law can answer the question which we posed but then you can see it will be a tedious process of checking plane after plane this is where a wall came in and he solved the problem by taking it from the direct space from the real space shown here to the reciprocal space so let us look at his solution so in the reciprocal space there is an incident wave vector ki a wave vector is a vector which is in the direction of the incident wave and whose length is 1 by the wavelength of the wave now let us say that the tail of this vector is at c and the head is at o with the same tail c let us try to draw the diffracted wave vector kd so cp is the diffracted wave vector kd so ki and kd are sharing the same tail c but k i has its head at o and kd has its head at p of course we actually don't know the direction of kd but still we know something about kd the fact that because the scattering is elastic the wavelength of the scattered wave wavelength of the diffracted wave is same as the wavelength of the incident wave this is the assumption of elastic scattering under these condition kd and ki will have the same length equal to 1 by lambda since they have the same length one thing we know about kd is that if i draw a sphere if i draw a sphere with center at c and radius at 1 by lambda kd will end somewhere on this sphere the head of kd the point p has to lie somewhere on this sphere of course i have drawn a circle here and in the plane of the screen you have to complete the sphere in your imagination by adding a hemisphere below the screen and hemisphere above above the screen now we have to find out actually the direction currently we have drawn this vector cp we have drawn the vector kd arbitrarily we are not claiming that cp actually is a direction a diffracted diffracted beam direction how do i find that to find that of course we again have to use bragg's law but we will use bragg's law in reciprocal space this has been discussed in another video the link for which i have shared in the description below here we just summarize the result for looking at bragg's law in reciprocal space we need to construct the difference of kdn ki the scattering wave vector delta k which is kd minus ki shown in blue here from o to p now bragg's law in reciprocal space states that this vector delta k has to be a reciprocal lattice vector g star h k l for some crystal lattice plane h k l for some integers h k l which means g star h k l is h a 1 star plus k a 2 star plus l a 3 star where a 1 star a2 star and a3 star are reciprocal basis vectors so this is interesting that this delta k is also a reciprocal lattice vector g star hkl which means i have not yet drawn the reciprocal lattice of my crystal so i am free to choose the origin and if i choose the origin at o the red dot shown here so if i choose the origin at o then and fill the space with reciprocal lattice points then p will also be one of the reciprocal lattice point if cp is a diffracted wave vector so let me fill the space with the reciprocal lattice points and we see that p is one of the reciprocal lattice point so the reverse is also true that whenever a reciprocal lattice point lies on the evolved sphere we will get a diffracted wave in that direction so for example another point q q shown here is also lying on the evolved sphere so i'll get another diffracted beam for the same incident wave vector so cq will also be a diffracted wave of course we are again reminding you that this is a two-dimensional section which we are drawing the real sphere the real evolved sphere is in 3d and there can be diffracted beams out of this plane shown here also now let me show you the equivalence of this construction to the standard bragg's law this is not difficult to show all i have to do is to drop a perpendicular cn on op if i do that notice that op is the reciprocal lattice vector j star h k l so its length op is simply 1 by dhkl where dhkl is the interplanar spacing of the plane hkl so op is 1 by dhkl and on by geometry is half of op and so it is 1 by 2 dhkl we can now see that cn is perpendicular to op and we know that the reciprocal lattice vector is perpendicular to the corresponding direct lattice plane so since n is perpendicular to g star h k l c n is parallel to the direct lattice plane h k l which means the angle between c n and c o the incident wave vector is the incident brag angle theta now i can use the triangle c o n to find sine theta as on by co which is equal to 1 by 2 d h k l by 1 by lambda if i simplify this you get the brach's law lambda is equal to 2 dhkl sine theta so what we are showing here that the condition that the reciprocal lattice point p lies on the evol sphere is equivalent to the condition that the corresponding plane hkl satisfies bragg's law in direct space so the evol sphere construction in reciprocal space is exactly and entirely equivalent to bragg's law in direct space so let us summarize the evolved sphere construction now the steps are we first begin with the reciprocal lattice space with its origin o and draw the incident wave vector k i as co with it with its head at the origin of the reciprocal space we then use the tail of ki c as a center of a sphere and we draw a sphere of radius 1 by lambda this is the evol sphere and then we check whether any reciprocal lattice point lie on the evol sphere for any reciprocal lattice point p lying on the evol sphere we get a diffracted beam in the direction cp from the center of the evolved sphere to the reciprocal lattice point so this is a beautiful construction this is one of the most beautiful results in the diffraction theory and we thank evolve for this and i thank you for listening thank you very much