in the last lecture we saw how the energy of an electron in an applied magnetic field consists of magnetic terms of different kinds one is a paramagnetic term and the other is a diamagnetic term ferromagnetic term corresponds to a positive magnetic susceptibility in the sense that the magnetic dipoles are aligned along the applied magnetic field whereas the diamagnetic term arises from a negative magnetic susceptibility and this is due to the extra centrifugal term centrifugal force due to the magnetic field on the electron in an atomic or molecular orbital so this causes the magnetic flux associated with the field to change changes and this induces inducing an EMF according to lenses law in electromagnetism so this is since this EMF is a back EMF which is opposing the influence of the field and therefore it has a negative sign that gives you the negative diamagnetic susceptibility this effect is universal the diamagnetism occurs in all substances whereas paramagnetism occurs only in cases where there is an unpaired electron giving rise to a net magnetic dipole moment which has to orient along the field so this does not occur in all situations whereas diamagnetism always occurs the total diamagnetic susceptibility is obtained by summing up the contributions from all the electrons total is sum of contributions from all electrons and then in all atoms or molecules in a given quantity of the material so the magnetization is then Chi diamagnetic type H or B where B is the in magnetic induction so if you have n atoms each with Z electrons because the atom with an atomic number Z will have Z electrons then the M is and Z times the contribution which is minus e square B by 6 M times as we saw last time so there are square is the average value of the square of the atomic radius orbital radius so thus we get KY pea yes – NZ e square by 6 M times R square so n is of course the Avogadro number if we are talking about the molar diamagnetic susceptibility and this is of the order of 10 to the power minus 8 in most substances therefore it is an extremely small value per mole so this is annex it is present everywhere in all the materials this is extremely small in comparison to the contributions from paramagnetism or ferromagnetism therefore we usually do not feel it we do not talk about it but in general for accurate work you have to take root of the diamagnetism before determining calculating the ferromagnetic or ferrimagnetic contributions next we consider paramagnetism as we said it is necessary to have an unpaired electron in the atom or molecule in order to give rise to a net orbital angular momentum nonzero orbital or spin angular momentum only when there is a non zero net orbital and our spin angular momentum then only you will have a net dipole magnetic dipole moment which gives rise to a paramagnetic effect so what kinds of systems are they which have such a possibility which are the systems in which the there is a non zero orbital or spin angular momentum given right giving right given by unpaired electron spins so this is the question so what you look at the periodic table of elements we find the so-called transition metal ions corresponding to the filling of 3d shells or even 4d shells and so on or rare earth ions corresponding to the 4f shell so these are the ions these 3d ions are also known as the iron group ions because the iron Fe 2 plus or Fe 3 plus is an important member of this group so are the other magnetic ferromagnetic ions such as cobalt nickel and so on for a nickel two plus nickel three plus nickel plus also so these are the various ions and the rare earth ions are known as the lanthanides so they correspond to the progressive filling of the 4f shell and they are the most prominent examples of paramagnetic behavior so the next table summarizes this with the giving the magnetic moment in bohr magnetons both experimental and calculated values for the different typical members of the ion ion group 3d transition metal ion as well as the lanthanide rare earth diodes in order to understand this paramagnetic behavior it is necessary for us to consider the spectroscopic notation of the ground state of an ion how do you determine the ground state and that's what will determine the corresponding paramagnetic moment for example let us consider a particularly simple example of manganese 2 plus and similarly for the length and a group let us take get a lynnium 3 plus so if you consider the manganese ion from the atomic number of manganese we can easily find that this corresponds to a configuration the electron configuration with an outermost shell in which there are five unpaired electrons in the outermost 3d shell a typical example of this is manganese oxide MN o where manganese occurs in the two plus state so they there are five electrons in this unfilled 3d shell the outer two for s2 or ionized in mn 2 plus so these go out leaving these in the unfilled outer more shell it is the behavior of these phi electrons that determines the magnetic behavior of a manganese ion in general the 3d shell can have 10 electrons so this is a half-filled shell so according to whom school all the electrons will be found in such states so as to have the maximum hoon's rule gives you maximum orbital angular momentum so as not to violate the Pauli principle consistent with the Pauli exclusion principle we have already seen what is Pauli exclusion principle according to Pauli exclusion principle no two electrons having all quantum numbers identical can occupy the same quantum state in other words applied to the manganese 2 plus for which there are different states since there are five electrons and these are D shell so you have the possibility + 2 + 1 0 minus 1 and minus 2 so this gives me these are the various ml values for the orbital angular momentum which are possible so all these are filled by electrons since there are five states each of these is filled so I a so that the total orbital angular momentum because if there if another electron is put into this it can have only spin down so we must have L equal to the total the sum of these is zero so that you can have maximum spin multiplicity in other words we want to have spin orientation this upward arrow corresponds to MS equal to plus half these are the ml and these are the MS so I have s equal to Phi by two so I have maximum spin and consistent with that I have L equal to zero which is a state L equal to 0 corresponds to an S state and the J the total angular momentum which is L plus s is Phi by 2 so the ground multi-plate has j equal to 5 by 2 so if you look at the ground state this is be the ground state for the MN 2 plus ion now in the same way we can find that gadda lynnium 3 plus which has 4 f 7 configuration in the outer shell outermost unfilled shell so since there is a possibility of 14 electrons in the F shell therefore we again have a half-filled shell and similarly we can have ml equal to + 3 + 2 + 1 0 – 1 – 2 – 3 giving L equal to 0 and M s is up spin up spin up spin up spin up spin up spin giving rise to s equal to 7 by 2 so we have an estate again with a J equal to L equal to 0 so J is 7 by 2 so that would be the spectroscopic state of the gadda lynnium 3 + ion now having determined the ground state of the Aeon BK we have to now proceed to find the G factor of this ion it is this g factor which will determine the magnetic moment so in the case of the gather idiom three plus the situation is somewhat simpler because in order to determine this we have to proceed by considering the Hamiltonian of the electron a many electron atom which is which has the following terms this is the kinetic energy term P I square by 2m summed over all the different electrons which are labeled by the denoted by this label I then you have I am leaving out the 1 by 4 pie epsilon-not etc so that I have this is the kinetic energy term this is a potential energy term I have also this is the potential energy in the nuclear nuclear field Coulomb field of the nucleus so this is minus and then I have plus Sigma I less than J this is the inter electron coupling or repulsion so beyond this we have various terms the most important is the spin-orbit coupling you charge the form lambda l dot s so if the speed in orbit coupling is sufficiently strong i am writing it in descending order of the strength of the various interactions so if this coupling is strong this gives the coupling between L and s gives you a resultant J as the total angular momentum so that it is this angular momentum which interacts with an applied magnetic field Zeeman term due to a magnetic field so in order to find this we have the hoon's rules we have applied the hoon's rules and now we can proceed to find the G factor so how do we go about doing this we have according to the g g g j the G factor associated with the total angular momentum J V is given by G L into L cos L j + g s into s cos s j so this is the projection of the contribution from the orbital angular momentum this is the contribution from this pin angular momentum now GJ is given by 1 plus or L square plus J square minus 2 s square minus s square times plus 2 into S square plus J square minus L square this 2 comes because the land a G factor for spin which we write as GS is approximately two so if you take these two together you get the net J as three J into J plus one minus L into L plus one plus s into s plus 1 by 2 J into J plus 1 that's the expression for the G factor associated with an angular momentum J here for every square of the app angular momentum operator we have replaced it by the corresponding eigen value which is J into J plus 1 for J square L into L plus 1 for l square s into S Plus 4 for L square so using this since we know the LS and J we can calculate the G value if we substitute these values we get a G value of seven point nine four for the gadolinium now GJ g J into J into J plus one which is the angular momentum which is seven point nine four four back returns so this is how we calculate the magnetic moment of a given paramagnetic area now how do we experimentally determine it we experimentally determine it by measuring the magnetization in an applied magnetic field so that DM by DB gives you the km the paramagnetic susceptibility is given by DM by D P where the magnetization is in mu where mu is the average of the magnetic moment individual magnetic moment and n is the number of ions if it is a mole of the substance this number is they have a getter number the average magnetic moment has now to be calculated in order to make a theoretical calculation of the magnetic susceptibility and then we can compare with the experimentally determined values in order to do this we have to compare this by recalling our discussion of the orientational polarization in the case of a dielectric material a para electric material where you have a number of electric dipoles distributed in different orientations in an applied electric field this was classically treated by using Boltzmann statistics the various orientations of the electric dipole with respect to the applied electric field ranges from zero to pi the angle theta goes from zero to pi all these orientations were allowed the main difference the our argument in this case proceeds in the same way we will calculate the statistical average of the dipole moment of the paramagnetic Aeon by five averaging it over all the allowed orientations in the case of a paramagnetic dipole the only thing is in the magnetic case we have spatial quantization which means that all orientations are not allowed corresponding to theta equal to zero to pi are not allowed only certain discrete orientations are allowed for example in the case of a spin half you have MS equal to plus minus half these are usually denoted by an up or down arrow that means the dipole is al either aligned parallel to the applied magnetic field or it is aligned anti parallel to the plane so you have only to such orientations possible for a spin half the spin can only be half integral in general or integral corresponding to a given s value M s has two s plus 1 value possible values and these two years plus one orientations will have different energies the applied magnetic field and we make a statistical average of the magnetic moment over these 2's plus Orion one orientation from for example plus for example if you have MJ then this goes from minus J 2 plus J which is 2 J plus 1 so this is what we have to do in order to calculate the average moment so let's calculate this norm so the average moment mu is some statistical some / GJ mu B be exponential well this can be written as MJ because you have mu B B dot J so that gives you MJ exponential GG mu b MJ b by KBT by sigma / MJ equal to minus j2 plus j / so that gives you the net average magnetic moment for shorthand let us write GJ mu b MJ by KBT B as X just for shorthand so that we can write this on doing the calculation it turns out that we arrive at GJ mu b bj of x times J so this is the average where this B J of X is known as the Bruin function the explicit form of this bellowing function is as follows what is the blowin function associated with a given J value be J of X is 2 J plus 1 by 2 J s cot H of 2 J plus 1 X by 2 minus plus 1 by 2 J cot H of X by 2 so this is minus so this is the form of the bellowing function it's rather easy to make this substitution and arrive at this form so this is the form or the Bellarmine function which enters the expression of the magnetic moment so for a given J value and a given B value and so on M can be written which is n mu can be written as M naught B J of X where m naught is a saturation value the maximum value maximum possible value when all the N spins are aligned parallel to the applied magnetic field so we can show the experiment graphically show the variation of e m J associated with a given J value for different choices of different magnetic moments so the overall variation has a form like this M versus B goes like this so that is not so this is a BJ of X BJ of X where X is GJ boobie P by Kb T so we can see several features all the way a given magnetization always approaches a saturation so this is the phenomenon of paramagnetic saturation it approaches a constant value at high enough air field so once you have the field which is sufficiently strong that all the dipoles get lined up along the field then you cannot increase the magnetic moment beyond that value so that's why you reach maximum value and close to the origin this is linear and then it slowly becomes nonlinear and tends to this saturation value which is determined by this overall behavior is determined by this argument of the Brillouin function which basically involves the applied strength to the applied field at the temperature so either you can have a very high magnetic field or you can go to a very low temperature in both cases the value of x is very high so that will give you the saturation so that is the overall behavior you have so all the graphs can be shown to be linear close to the origin this is the region corresponding to small applied magnetic fields and high temperatures this is where the susceptibility can be defined as the ratio of the magnetization to the strength of the applied magnetic field so in the limit when X tends to 0 very close to the origin we can show that BJ of X tends to the value J into J plus 1 by 3 J plus 1 into X by 3 so the slope of this is just j plus 1 by 3 so the Khayyam the magnetic susceptibility which is defined as the ratio of the magnetization to the applied field becomes in the linear region so this gives me a susceptibility paramagnetic susceptibility which is inversely proportional to the absolute temperature so the effective magnetic moment is given by n mu e effective square by 3 KBT so this is G J square J into J plus 1 mu B Square and that is mu effective is therefore GJ root J into J plus 1 Bohr Magneton so we can calculate the effective magnetic moment once you know the GJ the G factor and the J value so this can be calculated as we have already seen in the case of the gadda lynnium 3 plus ion so this is the in this form this orientation this susceptibility the paramagnetic susceptibility the resembles the dielectric orientational susceptibility in the case of a polar molecule for sufficiently high fields and our low temperatures when the spin alignment in the field is complete we can easily see that limit extended to infinity P J of X is 1 so that M will become M naught the saturation values so the chi at low temperatures goes as C by T where C is a constant known as Curie constant it has the value n mu effective square by 3 KB so if you substitute all these values this is zero point one two four one into P effective square where P effective is the effective Bohr Magneton number which we calculated as seven point nine four in the case of gate already in three plus so we have in general a diamagnetic under paramagnetic susceptibility so the next figure shows both these contributions the magnetic susceptibility which always includes a diamagnetic contribution and in the case of a system with unpaired magnetic ions and parrot electrons it also has a paramagnetic contribution and the paramagnetic contribution is positive and goes as perceptibility this is 0 so the paramagnetic contribution goes like this corresponding to this C by P variation this is paramagnetic and there is a small negative diamagnetic the total susceptibility is the sum of these two as a function of temperature this the inverse 1 by T which gives you a hyperbolic dependence for example if you take Chi inverse it is T by C so if you plot the inverse susceptibility 1 by Chi as a function of the absolute temperature you will get a straight line whose slope is 1 by C from which you can determine the effective Magneton number so this is how we compare calculate the theoretical value and compare it with experimentally measured susceptibility values so this comparison is shown in you know table yeah yeah this is the table you can see that most of the rare earth ions the comparison between the experimental and calculated values is quite satisfactory whereas this is not the case with the iron group ions there is a fundamental reason for that but before going to that let us consider the rare earth ions in the case of European 3 plus and samarium 3 plus the agreement is poor as you can see from the table this is because this was accounted for by Van Vleck the reason for this departure this disagreement between the experimental and calculated value in the case of European and samarium is because these two ions have not only ground multipliers but an excited multiplet close to the ground multiple so you have a ground multiplayer with an excited multiplayer and our assumption has been that at sufficiently low temperature all the ions are practically in the ground state and there is the excited state is sufficiently far away that there is no admits chair of the excited and state into the ground state otherwise the excited hair state will have a different magnetic moment if there is an admixture this will also mix with this ground state magnetic moment and this is precisely what happens in these cases of European samarium and because of this admixture and the excited state into the ground state because of the relatively shallow all seperation between the ground state and excited state so this can mix with this that is what changes the magnetic moment value that contributes to what is known as Van Vleck para magnetism which is temperature independent unlike the Curie para magnetism which is dependent inversely on the temperature it depends one on one by t well we have to consider the iron group ions where we know that the agreement between the experimental and calculated values in this following this procedure is rather poor and there is a very deep fundamental reason for this which has to do with the nature of the 3d ions in comparison to the four FA ohms you will see this in the next lecture you