 Hello everyone, welcome back to this material characterization course. In the last class, we just reviewed the electron optical systems and its governing principles and electron lens design and it is analogy with the light optical system. And I mentioned that we will discuss the aberrations in this class. And as I mentioned in the fundamentals of the optical system, we have gone through all the types of aberrations which the glass lens will exhibit. Other type of aberrations will also the electron optical system will encounter and then since we have already seen them in much more detailed manner about what is the each aberration and its definition, I will just mention how this is taken care in this electron optical system and then we will take up some few examples and some of the numerical significance of a spherical and chromatic aberrations. As we all know, this spherical aberration is very important and inherent to these lenses in light optical glass lenses as well as in the electro magnetic lenses. And we also appreciate that this is one particular aberration which directly influence the resolution of the microscope and we will see them in little more detail. So now we will go to this lens aberration and optical resolution with regard to this electron optical system. You look at this schematic, what you are just seeing is a lens plane and where you have the range of alpha that is aperture angle and then you see that look at this ray tracing path and then each ray is focusing at different different direction and basically and this is a region we say that the disc of a least confusion and then and if you look at this all this pair of rays intersecting the image plane at a different point and then eventually you see this aberration disc in the image plane. So the point you have to remember here is it is a general schematic which is shown here and whatever the aberration we talk about whether it could be a simple astigmatism to chromatic to spherical aberration all what all these aberrations they do to this light ray are electron beam they have they are directing this electron beam into a different focal point whether it is on axis or off axis that we have seen. So if you just think of all the aberrations which impact the resolution of the optical system or electromagnetic lens system it is the total combination of all this aberrations put together. So you can consider this schematic a general schematic where you see this the distance delta not is the de-focusing condition and this is also considered as the disc of least confusion and then you see the aberration disc which we have already seen in the beginning and then you see that delta optimum with associated with the electron lens in general. So the resolution becomes the total aberration dependent to a large degree on the half angle subtended from the image by the lens aperture as designated alpha this we have already seen. So I want to emphasis again please have make sure that you understand this. So you have the it is not a completely focused condition this could be because of any aberration but this is the smallest distance that is delta not is the least confusion and then and if you look at this delta optimum which is much larger in the image plane. So you may see that you are at the de-focusing condition your image resolution is better is that so it may be the case we will see in the coming slide. So let me read out some few introductory remarks for this aberrations of the electron optical or electromagnetic lens systems the ultimate resolution of the object signal is influenced by mechanical flaws in the lens design which produce imperfect lens field pole piece fabrication and also by mutual repulsion of electrons at constricted points that is a focal points lens apertures etc along the optical axis particularly the focal points. The variation in the electron energy at various points in the beam give rise to image distortion and contribute generally to loss of contrast and sharpness. So this is the fundamental point which you have to keep in mind whatever happens in the electromagnetic lens system it is the variation in the electron energy at various points in a beam give rise to image distortion and causes the loss of contrast and sharpness. The lens aberrations primarily responsible for deviations in electron ray intersections and concomitant loss in image clarity may be classified as geometrical aberrations chromatic aberrations and field effect aberrations including a space charge of the electrons. We will see one by one and the space charge of electrons we did not discuss in the light optical system we will see in this system how it is affecting the resolution. So it is just a recap of what we have seen the type of aberrations the this the schematic I have just put it everything in one image because we have already seen them in much more detail and when we discussed in the light optical systems. So the first schematic shows the coma effect and the second schematic describes the curvature of field and third schematic C is astigmatism and the fourth one is lens distortion and the fifth one is spherical aberration I will not describe them in detail because you have already seen it if you have a doubt you can go back to that lecture and then look at all this individual defects and then make yourself clear about this and it is the same thing I will only discuss about how these defects are taken care in this electron optical system in terms of coma it can be eliminated almost entirely in an electromagnetic lens by the establishment of field conditions giving rise to unity magnification and in terms of curvature field it is reduced by properly shaping the electromagnetic lens field the astigmatism on the other hand is correctable by inserting stigmators in the appropriate lens system to compensate the non-circularity of the image beam profile on the image plane. So what is stigmator the stigmator containing symmetrical arrangements of tiny ferro magnets are suitable permanently magnetized components acts to circularize the image and the lens distortion the correction of coma in an electromagnetic lens and currently eliminates the lens distortions as well. So what you should appreciate here is in electro optical systems the most of your aberrations is controlled by the field strength and the field distribution in an optical system we just all these aberrations were compensated with the an additional glass lens here since all the your focal length everything is controlled by the field strength and your aberrations also controlled by the appropriate field strength and its distribution in the appropriate lens system. So we will see the other aberrations spherical aberrations the correction of the spherical aberrations rest in the design of lens lenses with special field distributions for allowing smaller aperture angles to be attained with the simultaneous reduction in CS possibly by a design aperture aimed at producing less symmetrical lens fields. So as I mentioned this particular aberration is very important and how much we can reduce this will finally determine the resolution of the optical system and then we will see them and its numerical significance in a few minutes and chromatic aberration which is caused by the fluctuations occurring in the lens coils becomes simply a problem of electronic regulation as do fluctuations in the cathode and anode potentials to this extent this defect is correctable. However the energy losses resulting from the inelastic scattering in the object cannot be dealt with to the same extent and it is overcome by operation the system at higher voltages. The another important aspect of this electron optical system is a space charge effect what is this space charge effect at the focal points along the electron optic axis the concentration of electrons into small volumes produces a strong mutual repulsive action and a concomitant tendency of the beam to spread from the point of constriction that is from the point of focus this produces an effective reduction in the associated accelerating potential of the electrons and they lose velocity and this problem is somewhat less at very high voltage and where lower beam currents are employed with an associated low beam intensity. So this particular effect is specially belong to this electron optical system and you have to remember the aberrations which we talk about and its effect on resolution we simply assume that or we simply do not consider the specimen condition. For example whatever the aberration we talk about we assume that the specimen is pure and it does not have any contaminating I mean constituents in it or it does not react with the beam and then produce its own new product that will impair the resolution. So all this the treatment which we are talking about or the compensating effect we talk about assuming that the specimen is in the ideal condition. So in the mathematical treatment which we are going to look at is also in this similar manner that we are not taking the specimen effect that means we assume that specimen is ideally prepared and it does not have any contamination or any other reacting constituents with the electron beam. So now we will just take up this two spherical I mean two aberrations first we will talk about spherical aberration so what I am trying to write here is the image forming lens or the critical beam forming lens in an electron microscope or micro probe system in the objective lens we always talk about objective lens whether it could be a any image forming lens or it could be an electron forming lens I mean you know electron microscope critical beam forming lens or it could be electron micro probe analyzer we always concerned about the aberrations of objective lens we can describe the disc of confusion caused by the spherical aberration as that is delta SP delta not as general notation for disc of confusion here delta SP is it is exclusively caused by the spherical aberration can be represented as two times CS alpha cube where CS the spherical aberration coefficient which is also given by gamma naught CS equal to gamma naught times B naught divided by Ni whole square bracket square. So this expression you are familiar with already this is potential this is number of coil this is current which is which is observed to be proportional to the square of the focal length so this can be written as CS equal to gamma naught focal length of objective lengths and its potential divided by Ni whole square where gamma is a constant ranging from 100 for strong lens and 150 for weak lens gamma is a constant ranging from 100 for a strong lens 150 for weak lens so similarly we will see this chromatic aberration as we discussed earlier it depends on electron energy loss lens current fluctuation delta I is expressed as delta chromatic disc of confusion created by chromatic aberration can be related to 2 times chromatic CC alpha delta CR equal to 2 times CC that is chromatic aberration coefficient alpha times delta V by V naught whole square plus delta I by I whole square 2 power half that is square root of the whole expression we write where CC chromatic aberration coefficient which is also given by CC equals gamma naught prime focal length of objective where gamma naught prime is a constant varying from 0.5 to lens action so you have this chromatic aberration constant is equal to gamma naught prime times focal length of objective and gamma naught prime is the constant varying from 0.5 to 1 for a strong or weak lens action respectively so now what we will do is we will see that how all this aberrations we will now try to write some expressions for object resolution and then image quality involving all this aberrations which we talk about so we see that so what I have written as a consequence of the uncertainty principle the exact image displacement of electrons diffracted from the object area will be subjected to uncertainty of discrete line displacement in the object of the order delta X which is equal to L L that is list of based conclusion line to line which can be written as lambda divided by 2 sin alpha you are familiar with this expression and it can be assumed like this so I read out again because of the uncertainty principle the electrons diffracted from the object area will be subject to uncertainty of discrete line displacements in the object so if alpha is 0.01 to 0.001 gradient then we can write delta L L is 0.5 lambda divided by alpha for example you can write in a typical electron microscope alpha is 0.003 at 100 kV your delta L L could be roughly about 7 angstrom this could give you an idea a typical case where you see that delta L how to appreciate this so now we will include the lens aberrations and see how this expression is modified where so what I have done is where the lens aberration included in the real electron optical system the ultimate resolution is given by considering in addition to the diffraction uncertainty chromatic and spherical aberrations the combination of the error disc radiate that is delta optimum in the image plane is found from delta optimum equal to square root of delta L L square plus delta sp by 2 whole square plus delta chromatic divided by 2 square we consider limit of two points see we always talk about point resolution as well as line resolution you can consider these two if you consider two points in the image plane optimum in the resolution limit delta pp that is point to point disc of conclusion this also you are familiar with where again include so if you include this the spherical and chromatic aberration expression into this the point to point disc of this conclusion then you obtain delta optimum point equal to square root of delta pp square plus delta sp by 2 whole square plus delta chromatic divided by 2 whole square so these basic expressions are further modified by several researchers and then we can write one more general expression for disc of least confusion I will we will talk about all this much more detail how this is really going to affect the practical resolution when we look at the actual microscopic operation but you should appreciate that the importance of this two spherical and chromatic aberrations of it really influence the resolution limit of the optical system so what before we just look at this expression if you recall this the ray diagram which I showed in the beginning of this class where you see that delta naught was defined as disc of least confusion in a deep focus plane and then if you look at the image plane where delta optimal when on the image plane which is much larger than the D naught delta optimum was much larger than in the array disc we describe which is larger than the D naught so that clearly implies that if you reduce the field strength then you will automatically get the better resolution so to emphasis this point these to get an expression for delta naught itself that is what we are trying to show here the minimum constriction of the beam described as the disc of least confusion on the deep focus plane on the optical axis the diameter of the disc of least confusion is given by delta naught equal to square root of four times delta p p square plus c s alpha cube divided by two whole square plus c c alpha delta v by v naught whole square where delta p p is equal to 0.61 times lambda by alpha and where c s and c c are the spherical and chromatic aberration proficiency delta v is voltage change for a acceleration potential v naught and alpha is an objective aperture angle so in this class I hope you have at least have some basic idea about how this aberrations in an electron optical system is considered and its influence on the resolution of the image and the microscope so now we will now move on to the actual electron optical system especially we will start with a scanning work on microscopy and its working principle and its application from the next class thank you