welcome to lecture number 3. so, what arewe going to do is, we will start with where we finished off in lecture number 2. so, letus recall what we did in at the end of lecture number 2 was we had said that if you knowthat you have n electrons per unit volume which is this n write here. is the n electron per unit volume, then asyou start filling, now you have a different case state, we derived quantized case stateavailable and now we said that if you want to take this n electrons and start puttingthem from lowest energy up slowly, slowly start putting them up. we will start lookingat this graphically also in little bit as you start putting them up, then you will fillup the last electron up to the energy level
e f, which is this energy.this relationship between them this n and this e f was derived has to be this pi thereas shown here. similarly, we had said if you want momentum, then if you want to do momentumthen this momentum was equal to h k f and even this we can now in terms of this quantityn in terms of n we can also determine momentum. once we determine momentum, then in termsof this n, we can also determine what is the fermi velocity? and as a momentum, we meanmomentum of the electrons which are at the fermi energy.that means, the highest highest energy which electrons are having from lowest energy onwardsis up to that energy where electrons are filling up. the velocity of those electrons then isalso given by this quantity which is where
we finished in the previous lecture. now,let us look at this little bit further that we also said that, now that you know sinceyou know how to calculate this quantity n. so, substitute n in to this equation of vf and say v f we will find is about 10 to power 8 centimeters per second.that is the kind of velocity which you have which means that you are in about one percentof velocity of light. remember in drude’s theory, this velocity turned out the velocityof the electrons that where conducting turned out to be 10 to the power 7 centimeters perseconds, that is the difference we have. gain in free electron is what the free electronicstheory is saying so far. now, we want to move forward in this lecture three what we willdo is that we will look at four topics today
will look at the density of electron stage. this is a side topic, this topics is goingto be useful when we start doing semiconductors, then i am going to use the result from this.also after that we will start looking at electrical conductivity which we derived in context ofdrude theory. now, we will start looking at the looking at that point of free electronversion, once we have done that i will show you the difference between drude theory andfree electron theory. finally, today i will show you that even freeelectron theory has series limitation and we have to see what next. so, that is whatthis lecture number 3 would be today. so, let us get going let us start with this densityof electrons state let us start with this
topic. so, what we derive was the less turnaround this equation for fermi energy we have derive if we take this equation right hereif you take this equation and turn this around then i would write this equation. then, i would write it as n as equal to iwill write n as equal to 1 by 3 pi square two m e divided by h bar square power 3 by2 e f power 3 by 2. so, what we mean by density of states that was defined meaning of densityof states. this is the symbol i am going to use. so, we will say that g of e d e is thenumber of electron states in energy interval of e and e plus d e. this is the number ofelectrons that is number of electrons states which are in energy level energy e and e plusd e.
so, that is what the meaning of this densityof states says, now i like to pause for a minute and clear up whole idea which may becomeconfusing later. so, let us clear it up right now. first thing is, we could also definesomething called as density of k states remember what we have is the way you should think islike this that there are allowed k energies, the allowed k states, not k energy the allowedk states. so, if you have different k states which allowedthen you can ask the question what is the density of these k states. so, therefore icould have defined another quantity called density of k states then since each of thisk states can take two electrons. so, if i multiply by 2 these densities of k states,then i would get what i what we call as density
of electron states. so, either you work withdensity of electrons states or you can rather density of k states as long as you rememberwhether you multiply with this factor of 2 then you would be ok.in this course what we will do is, we will never deal with density of k states, we willalways deal with density of electron states which means that there is a difference offactor 2 only. that means, i have certain days to k states, since each k state couldhave taken 2 electrons it has been up, it has been down electrons. therefore, i willalways call that density multiplied by 2 as the density of electron states; remember theseare density of electron state. it does not mean that electron indeed is necessarily presenton those states is that the states are there.
then that means, electron can occupy thatit is not necessary that electron does occupy those states.in fact, if you want to know whether electron is there or not we will have to do somethingelse which i will do later, but notice i have been silent so far about this aspect. i havederived this relationship right there saying that if i have n electrons up to what energythey will fill. in that sense it is a hypothetical question that if i have n electron then howfar they will fill that is the question. that is the question i have answered i have notsaid whether i have indeed put those electrons in or not.so, you can think it think of it like that or you can think it think differently thatindeed all these states are indeed filled
with the electrons you remember that all thiscalculation are revealed for 0 k. it is possible we will see later that if we go for highertemperature above 0 k, then some of these electrons could go to even higher energiesleaving some of state vacant. in that case, these formulas directly will not be valid.so, that is why i am saying that you think as if i have all this k states and therefore,electron states multiplied by 2. if i had n electrons, then how far they will fill isthe questions we have answered so far, and indeed if they do occupy if that the probabilityof occupying those states is 1; that means, indeed all those states are occupied.then precisely, these would be the energy, these would be the momentum, these would bethe velocity, fermi velocity etcetera of those
electrons. so, with this background, now isvery simple to derive what the, remember now i can instead of writing this fermi energy,let me write another quantity called n prime and i will write this same expression as likethis n prime. i am using only for explanation in future; i will drop this and keep callingback again and itself. so, there is no confusion sorry let me remove this f here i just wante, what i am trying to show is by exactly same logic instead of saying earlier i saidi have n electrons, if i have n electrons how far they fill and defined that is a fermienergy. but, you could think slightly differentlythe way which will also think is that up to energy e. i can fill total of n prime electronsand if i go slightly d e energy higher then
i would have filled this n prime plus then prime electrons extra and we are seeking what is that number of electrons. so, if youcan calculate that number of electron, then you can immediately define our g e clearly,then this quantity g e would be equal to simply d n prime by d e why because d n prime wouldbe equal to g e d e. this is the quantity we are trying to whatis that mean this is the number of electron in energy, this is the number of electronwe read the definition, here is the number of electron states in energy of this intervals.so, i will explain n prime electrons are filling up to energy e and if i go slightly higherenergy, the d e then the number increase the electrons is d n prime. so, we are basicallytrying to calculate d n prime by d for the
density of states. so, now once we have thisdefinition it is straight forward enough to write this. density of states has been equal to just takingthe derivative of this quantity n d n d e call it prime or do not drop this prime doesnot matter really 1 over 2 pi square 2 m e. at presume you can carry out this derivationi have just written it out here and to save time, 3 by 2 energy to power now half, whichis what density of state is. so, what density of state goes as square root of energy? ifthis is it this energy then g of e goes as square root, it goes something like this.given this, now you stop at this topic, these density electrons, density of electron statesjust remember this expression i am going to
use as about 4, 5, 6 lecture down the line.i will start using this when i start doing semiconductors, but in context of free electrontheory, we derived this we given this definition and from gives given this certain definitionwhich will use subsequently. so, now let us move on to the next topic which is electricalconductivity free electron version is now what you wantto do. so, what do we have we know that h bar k ismomentum. so, if i divide this by mass of electron i get velocity of any electron, iget velocity of any electron in this in this way now let look at this the way i am sayingis that if you have all this k states. if you have all this k states that is a k x ky k z and in this in there is a sphere up
to which electrons are all filling up. allthese electrons are filling up to these this k f and this is what k f is up to here allthese electrons are filling up. now, if we notice that for every k in for fermi sphere,there is a minus, there is a minus k vector also. what is that mean for every velocity of electron, there is exactlyopposite velocity of electrons also. so, net velocity net velocity is equal to0, so that means, clearly if nothing then this electron cannot cut material is thisas this no conductivity, we expected there should be no net velocity if there was netvelocity then we would have correct. but, clearly without application of electric field,we can’t be having current. and therefore, as we expected the net velocity must be 0,but what happens if i apply high electric
field. if we now apply electric field then what happens, we know that d p its rateof change of momentum, now should be equal to minus q times their electric field whichis a minus sign because a electron q is a absolute charge 1, but positive number 1.6into 10 to the power minus 19. so, erase that number any way. so, this is the rate of changeof momentum which implies, i can write this us h d k d t as equal to minus q times thiselectric field e, what are that this mean. so, in time t this fermi sphere starts moving,this start moving how does it move? if i integrate this expression right here, equation whatdo i get, i get that k vector at any time
t then will be equal to k vector at time equalto 0. that at time equal to 0 is when we apply the electric field say t equal to 0 electricfield is applied in time, this k vector began to move and this becomes equals to q minusq e by h bar t. this amount by which this k vector moves intime t, now that in time as time increases this k vector can keep increasing is likeshifting of origin of the fermi sphere. we can think of this as follows, we can imaginewe have draw in a minute, but before that you can think this is just a center the fermisphere changing from a different k position moving from one k position other k position,but the problem is now that t is changing, this fermi’s starts keeps shifting.so, this causes of problem same problem which
we had when we started that if i applied electricfield if there is nothing to dump the motion then the current will continuously increasesfor a given electric field which is contrive to all observation. same thing will happenhere if k continuously increases then therefore, the same token this we will continuously increase.and therefore, current will continuously increase the current will also continuously increase.so, we face with exactly same problem as we started in the lecture number one who is wherewe started drude’s theory and we invoked collusion which was really giving as stamping.hence, the velocity we found to be average velocity was a fix velocity when we couldit collisions. now, in this free electron theory we will make more such statement asto what the mechanism of mechanism of this
damping is, but we know that from experiencethey must be some damping. so, let us assume, let us invoke the sametime tau let say that by some mechanism there is a relaxation time called tau the same relaxationtau, we started using in drude’s theory may be because of pollution with ions. but,same idea that in time tau that this k state changes to that value to k state changes upto this time tau, and then it is held at that fermi, then no more changes happened. youthink of tau of relaxation time, means if you want to think same way as drude, thenmean time between collisions. and therefore, after tarp time tau on average is it a valueof k average value of k would have. then, in that case become as you write thenthe same expression by saying k, then they
will become after applying electric fieldk, then will become equal to k. whatever it was a 0 minus or plus, i should say minusq times plus e, what is the expression was minus q e by h bar minus q e by h bar downhere. so, if tau is the relaxation time, then k average value k states where is a stateapplying when apply electric field is then k of tau, this quantity.so, effectively therefore i can show in two dimensions is easy to show you can imaginethis to be c. you can imagine this as a 3 dimensional figure, also what i will do islet us say this is k x. let us say this is k y and let us say that i am going to drawit at a circle or everything not as sphere for a circle in two dimensions. you can keepimagine, this is a 3 dimensional figure as
a sphere. so, what i do is i will draw a circlehere. let us chose a different color, we will chose a circle here like this. so, here isa circle i have drawn here and this is at t this is a t equal to 0 and what you havedone is what you have done is we applied a electric field e.let us say, we applied a electric field in x direction initially t equal to 0. we haveelectric field equal to 0 and then at t equal to 0, turn on electric field and what happensthen when we applied an electric field what happens is a the picture will draw here righthere. this is k x, this is k y and let me draw the same sphere first, the same spherewhich i had for i will draw, show it is a dotted line. so, this is the sphere at t equalto 0 which originally i draw in the previous
figure.now, i will show with another color pen, let us say blue, let us say now i have a situationwhere this is t is equal to tau and e field which i have lied is let us say minus e magnituden x. i have direction we applied this field, let us say if we applied this field then thisnow see where the k will be every k point, whatever the k point was a time equal to 0in time tau. its value would be that number plus this number plus this number at i havetaken e to be minus field with every where plus number here.so, it is going to shift in the x direction. so, let us draw that is speed also the circlenow in this case and you can imagine sphere, then this after shifting will become somethinglike this right this whole thing as shifted
this center is shifted to here. now, thiswhole thing moves something like this now something like this. so, this was at t equalto 0 and this is at and this is at t equal to tau. so, what you notice. so, what do yousee here. so, what has happened you my figures are not nice you can imagine all of theseto be circles, but anyway. so, now what you see now notice that there as in this particularcase. for every vector k for every vector k i hadexactly equal minus k vector also and therefore, what happened was for every vector k any vectork i had a equal and opposite minus vector. therefore, net velocity was 0 where i whenif we go for 0 at that time the net velocity will it trans for zero and therefore, no current.but, now its notice in this blue circle which
is what it will be at t equal to tau wheni have applied electric field. when i applied electric field now the netvelocity will not cancel, now notice again a velocity, like this will have exactly componentlike this. and all this velocity will cancel out, but the electrons which have velocitycorresponding to k states which are near the surface of this blue circle rose velocity.now, will not cancel out what is that mean this very important fact this very important. therefore, since now the electrons the electronvelocity near fermi surface will not cancel out meaning net velocity of electrons willbe due to those electrons, which are near fermi surface. what is that mean? let me repeatwhat i said again i said that most of these
velocity like this k component will cancelout this minus k component, but the k now there will be those case, which will not cancelout. and on those case will be the once which lie on the surface which are not going tocancel out any more. so, what do we see that the electrons thatconduct implies the electrons that conduct are once near the fermi surface and have velocities.approximately as v f fermi velocity which we recall was 10 to power 8 centimeters perseconds not 10 to power 7 centimeter per second at root s we assume in drude’s theory. therefore,you can see the first error which was in drude’s theory. now, we can we have we have a betterestimates of what the velocity of electrons that conduct should be now i will not derivethe expression, but only thing is that if
you go through a process of calculating thisaverage that velocity through this process exactly, what the way it has been.describe if you carry out summation over l if you average over all the velocity thencalculate what the net velocity is and you also calculate, what will be the number ofelectrons that will be what the number of electrons, which will be is. therefore, conductingthen you will find if you go through this whole process, what i will do is write thefinal expression that. in this case we find that the conductivity will become equal to1 by 3 q square the same, which is going to has to appear and this becomes equal to thentau i am going to chose the punting test g itself.this is the density of electrons near the
fermi energy for this expression for conductivitybecomes this which should make sense, it is should have a dependence on the number ofelectrons that are near the fermi energy. it should depend on the velocity of the electronswhich is of the velocity of those electrons, which conduct, which is the fermi velocitythis gives to say estimates of what the conductivity of a material will be any way this was onlyside topic i just since, we derive the same expression for drude’s theory. so, therefore,i am done that for free electron theory, now let us there is a time to take stop of whatis happened lets analyze what we have gain. so, far and where the problem continues tobe. so, let us first start looking at what have been gain drude’s velocity verses whiletalk about drude’s per velocity drude’s
theory verses free electron theory. let usdo this first what is happened what are we gain by doing this free electrons theory clearly find this case our estimates of thingswill become better where ever velocity is going involve where velocity is going involve.since, we know that the velocity is order without between drude’s theory and freeelectron theory we found the disciplines of order of magnitude in velocity.velocity being higher in case of free electron theory, then we know that root theory wherevervelocity of electrons works will be involve independently for that. it does not cancelwith something else then that case free electron theory will make it better prediction. thendrude’s theory in order to really show you
advantage of brief free electron theory iwill take you through some other topic couple of topics.this means sound out of context in context is electronic properties of material, whati want to talk about this thermal conductivity and thermal electric power we just show youhow free electron gives free electron theory gives you better estimates then drude’stheory. so, let us let us look at two topics in this case one is thermal conductivity in metals in particular i am going to talkabout this what is called as wiedemenn-franz law. i will try to show you what have we gained,now if we without any remember the conduction thermal conduction in metals occurs by sameprocess by which electrical conduction occurs. that means, the mechanism by which the heattransport remember we said electrons are arrive
local thermal equilibrium.so, now if electrons are moving now due to electric field for example, if you wish thatgives you correct, but then this motion of motion of electrons also carries heat. so,the thermal conduction in metals also happen through this m not same exact same motionof electrons in that sense we are not talking out of context some. when you talk of thermalconductivity in metals we are at microscopic level at atomic level, we are talking aboutthe same process which we talk for electrical conductions.so, if is you therefore, you are free to apply the same theory to develop the idea of thermalconduction also in which case by classic in roots approach or classical approach without proof. i will write this thermal conductivityor capital k to be equal to 1 by velocity
square tau times c v in quantity of drude’stheory. if we just derive then this quantity is half derived as half b square tau timesspecific heat c v this is the specific heats of a material. this also can be written as1 by 3 being the l being v times tau of what being that l by 3 should here 1 by 3 v l cv see the weight room number v times tau as been equal to l therefore, we using here.this quantity is derived like this second thing also, i say there by classical approachby same approach its specific heats specific heat is derived as 3 by 2 n times boltzmann constantk b. there is a boltzmann constant and estimate of velocity is estimate of velocity is by half m m e v square is equal to 3 by 2k b times t which you have seen before which
is same thing. this we have use before alsoin when we are doing drude’s theory i am giving it two expressions right here.this i am giving without proof that in root context of root theory you can derive specificheat of metals to be 3 by 2 n k time k boltzmann constant. you can derive thermal conductivityby same transport mechanism to be equal to 1 by 3 v square tau c v or 1 by 3 v timesl time c v. for this you can derive by classical theory drude’s theory now there is a lawwhich is called wiedemann-franz law, which says for metal k divided by sigma t will bea constant and this is this law. so, this is what this wiededmann-franz lawis, lets derive this expression got it now. so, what is this quantity v equal to? so,this we going to derive, so let us write down
this quantity k sigma by t will be equal tothe substitute k 1 by 3 v square tau c v divided by conductivity. remember what is conductivity,conductivity we had derived as equal to n q square tau by m e .that is what we derive,this as therefore, you can write substitute that in head. so, n q square tau by m e timest. therefore, then substitute here also for, now we gone substitute, make two substitutionwhich is what we are going to substitute. first we substitute for velocity this squareof velocity we should estimate from right from here.so, will say these square as being equal to 3 k b t by m e, that is what will substitutein here and we will also substitute c v what equal to of course, we have written therec v is already there given there. so, let
us substitute this in here also, so 1 by 3,we going to write and then for v square we going to write 3 k b t by m e.and then there is a tau here, then c v we substitute in here as 3 by 2 n k b and thiswhole thing divided by n q square tau m e by t. now, notice this tau and tau cancelsthis t and this t cancels this m e and m e cancels and this n and this n cancels. so,what have we left with and lets allow the 3 to cancel this three. so, we have left withessentially 3 by 2 k b by q whole square, indeed is a constant boltzmann constant dividedby this happens for charge electronic charge 1.6 into 10 power minus 19, indeed this isconstant which is what is observe. in this wiedemann-franz law, which is x remittalobservation even in roots theory. this value
came out this constant which and this valuewent plugged in. you can plugged in this value what you get is the number which is like 1.11into 10 to power minus 8 watts ohms or kelvin square. now, if you look at this number, thisnumber is about only half of what actually is observed. so, it is pretty remarkable thateven drude’s theory basically predicted right result anybody thought. at this, drude’stheory is predicting this wiedemann-franz law will because only half of what is experimentallyobserved. so, the thing of it is this that what happen was that electronic contribution. the electronic contribution to specific heat is at least hundred times more than observedand what is the electronic contribution remember
3 by 2 n k b. this is the classical theoryand this is the electronic contribution, the specific heat and this number is at leasthundred times more than ever done any electrons. then what the electronic contribution to specificheat is, so what happens, how it still work does.now, let us go back if you go back notice then also v square term in here there is thev square term also in here and this v square is what is helped this. and remember drude’stheory v also under estimates we under estimates velocity by 10 times. we have v square andover estimates of c v 100 times, therefore 2 error canceled out, and you got somethingvery close to what reality is in drude’s theory 10 times. let me just repeat this part,so this is in the drude’s theory this is
in the drude’s theory.now, let us see what happens in free electrons, free electron theory in free electron theory,the estimate of c v becomes equal to pi square without proof. again i just simply give youby free electron theory what happens k b boltzmann constant, this fermi energy now 10 times kb another post two in drude’s theory. where it was c v equal to 3 by 2 n k b this factorright here, then k b remember is same in both of them classical theory, the free electrontheory. this factor if you look at even at room temperature, this t at room temperatureif you look at this factor this pi square by 2 k b t by e f, this factor at least hundredtimes smaller than this 3 by 2. compare it 3 by 2 this factor is about hundredtimes smaller which is why you are saying
that the electronic contribution to the specificheat is at least 100 times more than observed. this is what electronic contribution reallyis which is being predicted well by this free electron theory. now, if you put this simultaneouslyestimates this velocity if you estimates by the velocity by 2 times e f by m e not whereis according to half m v square m e v square mass of electron as equal to the fermi energy.if we going that to fermi energy then you estimate the velocity to be like this if youestimate the velocity to be like this. then remember in classical theory we have estimatethis by as equal to 3 by 2 k b t and we have under estimated this velocity 10 times. so,now this velocity estimate is 10 for 8 centimeter per second, as we have derived in free electrontheory earlier if you now plug in this number.
then in this case, k by sigma t becomes equalto pi square by 3 k b by q whole squared again constant number even by free electron theorywe were getting k by sigma t with the wiedemann-franz law. i have we say that this constant indeed,we find derive by free electron theory also that k divided by sigma t is constant. now,this substitute in the numbers this number comes out as 2.44 into 10 to power minus 8watts ohms per kelvin square which is a even better estimates of which is now correct estimates.of what we have what observed value, because remember i said this was 1.11 in case of roottheory and it is more half of what it is actually observe.now, this is beginning to predict even better of course, it is a minor gain if you lookat another example which is thermo electric
power. now, you will see what we are goingto do is, what we going to do is in case of thermal conductivity thermal conductivity.the two errors velocity and specific heat canceled out each other, now we were goingto take a example where there is no velocity term, but only c v term.this means, now drude’s theory will completely go here where i will show to you the freeelectron theory continues to predict. now, predict much better and that incase of thermoelectric power let us look at this, which means if i take a hot bar, first on material,then the number what will happen, if we keep one end at t 1 temperature other one at t2 temperature where t 1 is greater than t 2. then what will happen, remember electronscome to thermal equilibrium we had assume
that local thermal equilibrium. so, the oncewhich are near t 1 will have high energy higher kinetic energy then the once which have otherend at t 2 end, which have lower kinetic energy. so, there will be net flux of electrons goingfrom t 1 side to t 2 side, but that cannot continue in differently because it can’tbe a current flow in this system. so, what will happen as more and more electrons movefrom t 1 sides to t 2 temperature sides. then a electric field will set up, this electricfield will oppose the motion of motion of electrons from t 1 side to t 2, that meandue to electric field you will have electron going from t 2 to t 1. due to thermal energy,the net flow will be from t 1 to t 2 and in equilibrium these two will balance out. so,that there no current and this is well known
effect called see-back effect. remember thatfewer the electric field that will develop if you apply temperature gradient across thebar. then, the field that will develop inside forthat the system comes in a steady state is e equal to few times gradient in gradientin temperature. that is the electric field that is develops and this is called the thermoelectric power and this is well known see back effect which you have seen in a varioussituations thought in schools also. so, this the thermo electric power, now if you go throughthis again to have net 0 velocities, no current flow, when there is temperature gradient electricfield also developing and both balancing. so, there is no net velocity under those conditionsthe expression which derives for q is the
q should be equal to minus c v divided by3 n q that what this q should be thermo power. now, notice in thermo power we have only havec v if we incorrectly estimated, we going to get wrong thermo power and if you put itright number then only we get thermo power which is today, if you use classical estimatesof pacific heats. so, roots theory classical estimates if you use estimate we get q whichis equal to minus 4.3 into 10 to power minus 4 volts per kelvin, which is hundred timesmore than observed value clearly. then, since estimates of free electron theoryof c v is hundred times less we will start getting right numbers. so, that is reallythe success free electron theory. this is where the free electron theory has there isan improvement from roots theory for classical
theory going to quantum mechanical free electrontheory, this is where we start seeing improvement. now, let us look at also failures of freeelectronic and there are many, now notice what about hall coefficient they call r hwas equal to 1 by n q. now, notice that drude’s theory we found that does not work for metalsuch as aluminum. now, when you are done, free electron theory what we have gained nothing,we cannot say anything about hall coefficient i mean no improvement, there is nothing isonly n appearing here that estimates have not change. therefore, we can’t and we foundaluminum the halls sign of hall coefficient was positive the hall efficient was sorrythe hall coefficient, it appeared hall for conducting.so, therefore, in free electron theory also
we make no improvement in this regard samefollowing magneto resistance in this regard also. the field depended of magneto resistance is again not predictedgive nothing done nothing. in this theory which says that let me ask many questionsto you, now as we start discussing this there too many thing which are which are problem.how would you explain that carbon comes in many form diamond, it comes in graphite. whyis it carbon which is insulator why its graphite which is conductor why that happen wiedemann-franzlaw which is obeyed well at in room temperature. but, we go to low temperature that even wiedemann-franzlaw its cooperated well by this what is observed this discrepancy between free electron theoryand free electron theory and what is berated by what is observed. similarly, this directedconstant of a material could be very, very
complex which is not by pirated by free electrontheory. now, already talked you talked about, i give you already hint that when you talkabout thermal when you talk about conductivity in graphite or in diamond.now, both are carbon, this nothing we said about differences between graphite and diamondand therefore, the difference in conductivity we just can’t bring it. bring it in whyis a aluminum good conductor, but bismuth and antimony which are also metal which arenot why boron is insulator. why boron is insulator, now these are the things which these are thethings, which are not coming, which are the other observation. however, this free electrontheory we are not gaining anything out of it why is it. so, you should already haveit to that our materials have special structure.
then that materials are, they have a structurealso they underline atom that periodicity in what is that mean, remember the true formgraphite is conductor diamond is not what is the difference between them both are carbon-carbonbased materials. now, all we have done is in copper we have taken copper atom and considervalence electron and went ahead and did both this case.but, you see there are underline latex, underline periodicity, in these materials these periodicitythe electron wave which is coming. now, you notice this electron wave is traveling. now,there is interference of this electron wave if the latex is perfect, then there couldbe constructive interference, and this electron wave could just simply go through, where asif we break the periodicity of the latex somewhere.
then you can have distractive interferenceand therefore, you could have the electron wave dying out and; that means, conductivitybeing low. now, all these there, therefore in all the explain conductivity, we must takeinto home what is the underline structure of the material. so, what in the next lecturei will do is i will cover two topic, one just to build up the case, i will cover two topicsone is that will start with crystal structure of material then after that i will take youto something take you through, what is called reciprocal lattice. some of you would allready be familiar with it in context of x-ray diffractions or electron diffraction.so, you see will go through these two topic first and then you will see the relevancewhere we are heading what ultimately, i want
to do is now i am going to write down herethat what have we done. so, far we have said that energy is equal to h square by 2 m ksquare, but ultimately i want to show you is that there is something called e k diagram.so, thought this e k diagram e k verses energy, then clearly it look something like this it clear looks it look something like this,this is this, this expression being ported here.what i am doing of course, k appears only a distinct levels allowed values of k r. onlythese are the allowed values of k and corresponding energy is ported here, but now these k alsototally pack that we draw then continues line that it the line that the lack line i havedrawn is sufficient enough because this k also. so, in this picture i show you by reddot which a far apart, but in practice there,
so close there is no need to show those dotthey look like the black line which is continuous line.now, what happens when what we have done so far is that when we start filling of electron,when we start filling of these electron we start putting them two electrons in each ofthese k states. in each of these k states we start putting 2 electrons and when youstart putting them in we fill up to let us say these energy and this energy we call usee f. this energy we call e f that is the something which we have done, i shown you the questionis what this quantity. you also remember i have pointed out thatthis is e k diagram, meaning this energy was a momentum diagram k precociously representmomentum as well. h bar multiplied by k represents
momentum. so, if i get a proper e k diagramfor each material, i should energy and momentum there is 2 quantity of sufficient to do allthe dynamics we want to do whatever we want to do.with the dynamics of a electron you can do if you have a e k diagram essentially fora real material, i want to do this e k diagram. in order to understand this e k diagram iwas understand k this k remembered in out of plot like this, i have drawn at as a k.but, the k really is a vector k is a big vector, which is a in which direction should i drawit how to draw with e k diagram this issue we will start looking at almost 2, 3 lectureson the line slowly will build the case in order to use. so, we have to understand reciprocallattice because k belongs to reciprocal space
and in order to understand reciprocal lattice,we must do real lattice, which we will start next lecture.thank you.
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