Fermi-Dirac and Bose-Einstein statistics - basic introduction

hello today we're going to take a very simple and basic introduction to Fermi Dirac and Bose Einstein statistics and by way of comparison we're going to compare them with the Maxwell Boltzmann statistics or distributions as far as physics is concerned all basic particles come in two kinds fermions and bosons fermions are things like electrons neutrons protons and bosons are things like photons or the famous Higgs boson now as we shall see Maxwell Boltzmann statistics deal with molecules or particles which are what are called distinguishable and I'll explain what I mean by that in just a moment Fermi Dirac statistics deal with fermions for example electrons and they are indistinguishable and they are subject to the power E exclusion principle I will just review what both of those mean in a moment Bose Einstein statistics deal with bosons for example photons which are also regarded as indistinguishable but do not obey the Pauli exclusion principle and yes before you ask that is my washing machine making a row in the background now what do I mean by distinguishable or indistinguishable what we mean is can two particles be identified as separate well in the case of molecules yes you can see that one molecule is here one molecule is there but in the case of electrons that's not the case when we think about these things quantum mechanically you will recall that when we say we have an atom which has a nucleus and an electron orbiting we said that if you think of an electron as a particle then you would expected to spiral into the nucleus and for the whole thing to disintegrate in ten to the minus fourteen seconds but that clearly doesn't happen and the reason it doesn't happen is because quantum mechanically we don't think of an electron as a particle we think of it as a wave so here is a wave now suppose we take another electron if the wave distribution is very close to the first then we say that those two particles are indistinguishable because you cannot resolve the distance between them and the basic rule of thumb is that the wavelength of the wave associated with the particle has to be very much less than the separation between the two particles in the case of molecules that's perfectly true the molecular wave function has a very small wavelength compared with the distance between molecules on average whereas the wavelength for electrons and their separation is broadly comparable so they are indistinguishable now what do i mean by saying that electrons or fermions obey the Pauli exclusion principle but photons don't well in my videos on atomic physics you will see that the consequence of regarding electrons as having a wave nature as opposed to a particle nature is that they satisfy what's called the Schrodinger wave equation and the Schrodinger wave equation has solutions which essentially say that the energy if this is the energy of an electron must occupy certain states and it cannot occupy any states other than those so there's the one state the two state three state the four state in fact there's an infinite number of states but they are all discrete and separate so no electron can hover in the gap between the two it has to be in one of those states and we also showed that the number of electrons that can occupy these states is governed by the power exclusion principle essentially it says no two electrons can have the same four quantum numbers so for example in the lowest state you can have one electron spin up one electron spin down that is actually called the 1s state and you can have two electrons in it when you get to the N equals two in energy level you can have one electron spin up one electron spin down in the S state so this is the second energy level it's the S state and you can have two electrons in it but you can also have one two three up and one two three down electrons in what is called the p orbital so there's a total of six in that level plus the 2's orbital electrons makes a total of eight in that energy level there's two in this one here so essentially there are two states in the first energy level and eight states in the second level and here's the point power exclusion principle says once those levels are full if another electron comes along and wants to occupy an energy level it cannot occupy level 1 because that's full it cannot occupy level 2 because that's 4 it has to go up into the next energy level so even though electrons like all of nature want to occupy the lowest possible energy level in fact they are constrained to occupy a higher energy level because all the spaces in the lower levels are already full and that's what we mean by the power Lea exclusion Prince by contrast if you are a photon and you don't obey the Pauli exclusion principle then that means that all photon all photons can occupy the lowest energy level and you can have billions of them in that energy level there is no constraint they think that separates bosons from fermions is that we say that fermions have half-integer spin bosons have an integer spin but that's a technicality they are different one of them is subject to power Li the other isn't now to get us going I think we'll start with the Maxwell Boltzmann distribution because we can then compare the other two to it so let's take this room for there so there's the room and it's got air in it which will be basically molecules of oxygen and molecules of nitrogen and we know that those molecules will be moving around the room occasionally they will be bumping into one another occasionally they will be bumping into the walls of the room and as they do so they exchange both momentum and energy and we know that the each molecule will therefore have a velocity when it hits another molecule it may change that velocity but for the time being the energy of each molecule is given by 1/2 MV squared that's the kinetic energy where V is the velocity we can also say that whenever a molecule hits the wall and rebounds off of the wall there will be a change in momentum and we know that force is the rate of change of momentum we also know that pressure is force over area so the pressure on the walls of the room will be determined by the force which is the rate of change of momentum as the molecules hit the wall divided by the area of the walls of the room so that's what actually constitutes the pressure of the gas this is a gas here pressure of the gas on the walls the ceiling and the floor of the room is governed by the fact that the molecules are every now and again hitting that wall rebounding off of it the change of momentum or more importantly the rate of change of momentum determines the force on that wall and if you divide that force by the area you get the pressure now there is a pressure law which I've covered again in one of my in one of my other videos which says that if you keep the volume constant then the pressure this is the pressure and we're plotting against temperature if you keep the volume constant then the pressure is directly proportional to the temperature as the temperature increases so does the pressure and as the temperature decreases so does the pressure so that essentially if you start in your room with a pressure and a temperature of this kind as you reduce the temperature the pressure Falls and it does so linearly until some point here you will see that there is no pressure in the room at all we're down to pressure equals zero and this happens at a point which we call zero degrees Kelvin which is also minus 273 degrees Celsius or centigrade and what does it mean if the pressure is zero well if the pressure is zero that means that the force must be zero so the rate of change of momentum must be zero which means that the molecules are not moving the molecules have zero velocity that way they don't hit the walls that way there is no change in momentum no pressure so at Absolute Zero all the energy is or at least all the kinetic energy has been taken out of the molecules and they all essentially are stationary now if you were to plot the number of molecules in the room against their energy so this is the number of molecules against the energy of each molecule recognizing that some molecules will be moving fast some molecules will be moving slow so 1/2 MV squared the kinetic energy of each molecule will be different you will find that there is a distribution and the distribution looks something like this what that means is that there are some but not very many molecules which have a very low energy some but not very many that have a very high energy most are in this region here which is round about the average and we showed that actually the average energy in the room the average energy of all the molecules in a room is it is essentially the temperature that the temperature is just a measure of the average energy and in fact it's related in that the energy is equal to K times T where K is Boltzmann's constant and T is the temperature and Boltzmann's constant is very roughly 10 to the minus 23 joules per degree Kelvin or very roughly 10 to the minus 4 electron volts per degree Kelvin now if you increase the temperature of the room you would expect the average energy which is a measure of the temperature to increase and what you get is a distribution that looks like this as you increase the temperature what happens is you get more molecules now at higher energy and the average has moved up as well but the area underneath the curves must be the same because essentially the area under the curve is the total number of molecules in the room and total number of molecules in the room is not going to change on the other hand if you cool the room the distribution does this it flattens out once again the air under the curve will remain the same because it's the total number of molecules in the room but essentially as the temperature in the room increases so this curve will flatten out more at very high temperatures it will look like that and there will be far more molecules with higher energy at high temperatures than there are at very low temperatures and of course the implication is that if you go to zero degrees Kelvin and the curve will essentially look like this all will have not necessarily zero energy because in quantum mechanics you don't always get zero energy but the lowest possible energy state that the molecules can occupy they'll have no kinetic energy they might have some internal energy that's there that's quantum mechanically there but they have no kinetic energy now if we take this co ordinate here which at the moment is the total number of molecules having a particular energy if we divide that by the total number of molecules in the room then these graphs remain the same but now they show you the proportion of molecules having a particular energy so a very small proportion have a low energy a much higher proportion have that energy and this is also of course a measure of the probability of the molecule having a particular energy so if you take this curve here it's highly probable that a molecule will have this energy it's very much lower probability that it will have this energy and so the idea of probabilities comes into the ideas and that's what Mike Maxwell Boltzmann Fermi Dirac and Bose Einstein statistics are basically all about they are telling you what is the probability that any one of the molecules or fermions or photo or boson we'll have or we'll be occupying a particular energy State now the maxwell-boltzmann statistics and I'm not going to derive these it takes about 45 minutes to do these and they're mathematics it's it's mathematics not physics but the probability is often written as f of e that means the probability that a particular molecule in this case because we're going to do maxwell-boltzmann at the moment the probability that any molecule will have the the energy state e or will occupy energy state e is given by 1 over e to the e over K T so this is the exponential times e that's the energy that we're talking about divided by K Boltzmann constant times T the temperature so that's the probability that an energy state will be occupied is given by this formula and by way of illustration what happens when T equals nought in other words when we were Absolute Zero well before I proceed I better just deal with one issue which comes up in the comments on the YouTube channel from time to time and that is my assertion that 1 over 0 is infinity or 1 over infinity 0 and people point out that mathematically those are undefined but what I think I can reasonably say is that if you say Y is 1 over X then it is certainly true that as X tends to 0 Y will tend to infinity and as X tends to infinity Y will tend to 0 in other words as X gets smaller Y gets bigger as X gets bigger Y gets smaller we can all agree with that and therefore in the limit we can say that 1 over 0 is infinity and 1 over infinity is 0 for the purposes of physics and so if in this formula here T is 0 if we're if we are at Absolute Zero we get that the problem leti that an energy state will be filled is one over e to the e over KT well ka ki is zero so KT is going to be zero which of course is one over e to the infinity because anything divided by zero I've said is infinity e to the infinity is infinity and one over infinity is zero and so what you've shown is that there is no probability that any of the molecules will occupy energy States at Absolute Zero because they will all be in the lowest possible energy set and the graph of this equation here it's simply a 1 over exponential graph so it looks like this here is the probability of a molecule occupying an energy state and this is the energy itself and the graph is just an exponential which shows that for a given temperature there is a much higher probability that a lower energy State will be occupied than a higher energy State but as the temperature goes up the probabilities go up too so that there's a much higher probability that in a high energy state will be occupied at higher temperature then at lower temperature so that's the probability that that energy State will be occupied at low temperature this is the probability much higher probability that that energy State will be occupied at a higher temperature now we move on to Fermi Dirac statistics and I explained earlier in this video about Polly's exclusion principle and how for exam electrons occupy certain levels so let's just take by where example hydrogen and the hydrogen atom has one electron in it and in its ground state that electron will be in the 1s level and there'll only be one of them so we call it 1s one now let's take another hydrogen atom again within electron in its ground state 1s 1 what happens if we bring those two atoms together to form a hydrogen molecule well Pauli's exclusion principle kicks in and says hang on a minute these two electrons appear to have the same four quantum numbers they occupy exactly the same state they're not allowed to do so and so what actually happens is that the two states are very slightly pulled apart one goes very slightly above the other and then the two electrons occupy those states but where do they go well they will both try to get in the lowest possible energy level and they can as long as one is spin up and the other is spin down they can both occupy this lower state and the upper state is empty and this gives rise to the idea that the lower state is called the valence band and the upper state is called the conduction band I've dealt with this in the videos on atomic physics but what happens if you take a substance like copper well in a copper wire for example you will have billions and billions of electrons billions and billions of atoms copper and all copper atoms have the first three energy levels full and one electron in the fourth energy level which is a 4s 1 electron and every atom and there are billions of them so I thought draw them all we'll have one electron in the fourth energy shell so what happens when they will come together this principle applies all the energy levels have to be slightly different and so what you end up with is a essentially a band of levels which occupy of the order of two to three electron volts in separation move this up a band of all these levels have to be slightly different otherwise Pauli's exclusion principle would be violated and then what happens is that the electrons start to fill those levels in the same way as we did up here by going to the lowest levels first and so up down up down up down up down up down all the way up but because each of these levels has got one electron in it when it has the capacity to hold two you could in theory have another electron in here down the spin electron since you don't what will happen is when you fill from the bottom you will filter the halfway point everything below this will be full everything above it will be empty just as up here the lower one was full the upper one was empty and this is sometimes called the Fermi sea and this is called the Fermi level it is the highest level at which electrons occupy at the energy level strictly at temperature T equals zero and I'll show you why that is in a moment but what we know is that an electron that is just below the Fermi level might be promoted into the upper band this is called the this is called the conduction band this is called the valence band an electron that's hovering just a little below the Fermi level might be promoted up into the conduction band and then if you put a battery across that wire plus and minus the electrons in the conduction band flow towards the positive terminal and that's an electric current flowing and that's how electricity flows that's the quantum mechanical explanation of it how do electrons get from just below the Fermi level into the higher band well they absorb energy where do they get energy from well possibly from the room temperature remember that we said that e equals kt K is 10 to the minus 4 electron volts per degree Kelvin so if you've got room temperature of say 300 which is about what 20-year 27 degrees Celsius then you've got 10 to the minus 4 times 300 which equals naught point naught 3 electron volts well naught point naught 3 electron volts when you consider this whole band occupies 2 to 3 electron volts naught point naught 3 is not very much but if an electron is very close to the EF level the Fermi level then it can be it's got just enough energy this and all point not three electron volts just enough energy to push it up into the conduction band now let's have a look at the actual statistics that govern all this the probability that an energy state will be occupied as is exactly the same principle as we had for Maxwell Boltzmann he is in this case given by 1 over e to the e minus EF divided by kt + 1 where e is the energy state that we are asking is it occupied or not EF is the Fermi level the maximum level that is occupied when T is 0 divided by cave Boltzmann constant times T the temperature plus one now what happens when T equals zero well if the energy level you're considering is greater than EF in other words if you're talking about energy levels in this area here if E is greater than EF then e minus EF will be positive a positive term divided by K T where T is zero a positive term divided by zero will be infinity and e to the Infinity will be infinity and infinity plus one will be infinity so one divided by infinity is zero and so the probability act when temperature is zero the probability of the energy levels above the Fermi level being occupied is zero and that's exactly what we said at the temperature T is zero all the lower levels are full all the upper levels are unoccupied and there is no energy here to promote any electrons because T is zero and therefore the energy available to promote electrons into the conduction band is zero so when T is zero these are all full these are all empty and there's no energy around to promote any electrons up and that's why you get this solution here the probability of an energy level above the Fermi level which is what this means being occupied is zero well now let's keep T equals zero but this time we're going to say what happens if the energy is less than the Fermi level that means a level in this range here well in that case if we look at this formula which tells us the probability if E is less than EF then e minus EF will be a negative tell'em and that means that you effectively are talking about one over e to the e minus EF you will still get an infinity term here but essentially what you're looking at is f e equals one over this term is an e to the minus term so it's essentially another one over e to the e minus EF now this term will become infinite because it's divided by K T T is 0 so this becomes infinity 1 over infinity is naught so now we've got 1 over 1 over infinity which is naught but we must also remember that plus 1 term there that should be a plus 1 term there this is a plus 1 term here and that gives us that the probability of the energy level being full is 1 over 1 over infinity which is naught plus 1 which is 1 over 1 which is 1 so the probability of the energy state which is less than the Fermi level being occupied is 1 and that's exactly what we said when T equals 0 all of these energy levels below the end of the Fermi level are full all the energy levels above the Fermi level are empty so if we plot the probability that an energy level will be occupied against that energy and we'll say that this is not 0.5 and this is 1 then what we say is if this is let's make this the Fermi level here what we say is that when T is 0 then there is a hundred percent probability that all the energies below the Fermi level will be occupied and there is a zero probability that the energy levels above the Fermi level will be occupied so we know that the Fermi level is occupied because that's the definition of the Fermi level it is it is the highest level that is occupied when T is zero so hundred percent chance one the probability is one that's a hundred percent chance that the energy levels below will be full and a hundred cent chance that they will be empty or in other words a nor percent chance that they will be full above the Fermi level now what happens if we raise the temperature and apply it to our Fermi Dirac equation what you will get is a graph that looks a little bit like that and if you raise the temperature a little more you will get a graph that looks like that what that is saying is that as the temperature increases the closer you get to the Fermi level there is a chance that or put it another way there is less than a hundred percent chance that that energy level will be full it's now any 90 percent or 80 percent likely to be fall and by contrast there's a 10 or 20 percent chance that there is an energy level above the Fermi level that is occupied and again you can see why that is if I redraw all the energy states we said that that's the fermi level and that at T equals zero everything below is full and everything above is empty if you have an electron very close to the Fermi level and the temperature starts to rise it may have enough energy to be promoted into the conduction band this is the conduction band this is the valence band and that's exactly what this is showing that when you get close to the Fermi level it's possible that an electron will be moved or promoted from just below the level to just above which means that there will be a vacancy here so the probability is not 100% that the electron will be here because it might be here but of course it's still 100% likely that the electrons will be occupied in the lower levels because at lower levels an electron down here if all its going to get from let's suppose that T equals one well if T was one degree Kelvin then the energy available to the electrons will be KT which is 10 to the minus 4 electron volts well we said that this gap here is something like 2 to 3 electron volts if all you've got available to you is 10 to the minus 4 electron volts and you're sitting right down here you can't jump all the way up there you haven't got enough energy and you can't jump to the next available energy level because that's already full and power Li says sorry keep out we're full there's no more room so electrons down at the very lower levels cannot move because they don't have enough energy and that's why as the temperature increases there will still be a probability of a hundred percent probability it was one that the electron that the energy levels at the lower level down here will continue to be full because there's only enough energy to promote the electrons that are very close to the Fermi level into the conduction band but as the temperature increases so there is scope for electrons lower down in the band to be promoted so that if temperature goes up enough you might get something like this where now an electron quite low down in the band can that's that's the new curve an electron quite low down in the band has got enough energy to get up into the conduction band and now let's look and ask what happens if the temperature is greater than zero but you're asking what happens if the energy level we're concerned about is the fermi-level well then the Bo's sorry the Fermi Dirac statistics formula becomes one over e to the e minus EF over KT plus one whoops that's e to the e to e minus KF over KT plus one that was the formula that's the Fermi direct formula and that of course becomes one over e to be well if e equals EF then e minus EF is zero so it becomes zero over KT plus 1 and e to the zero over KT will e to the 0 is 1 so this becomes 1 over 1 plus 1 which is 1/2 so what we're showing there is that when the temperature is greater than 0 the probability that the Fermi level itself will be occupied is exactly 1/2 which is why when I drew this chart up here I made sure that no matter what the curve does at the Fermi level it always has a value of 1/2 so effectively these lines become symmetric as the temperature goes up you might get something that looks like this but at the Fermi level the probability will always be 1/2 and finally let's consider Bose Einstein statistics and the formula associated with the Bose Einstein statistics is actually very similar to Maxwell Boltzmann and Fermi Dirac it's that Fe the probability that the state E is full is e to the e over KT minus 1 and that minus 1 simply reflects the fact you can derive these equations as I say it takes about 45 minutes in each case from just basic statistics theory but what that minus 1 does is to tell you that the bosons if we're considering for example photons are not subject to power these exclusion principle so as so just to remind you that whereas with the power Li exclusion principle you've got a band and those bands would be occupied up to the Fermi level that's electrons or fermions when it comes to bosons they can all occupy the lowest energy state and there can be billions of them an unlimited number can all occupy the lowest state now they don't walk you pi the lowest state because if there is temperature if there is energy then those photons can be promoted to much higher states but the question is if you take bosons photons down to T equals 0 what happens the answer is they all condense into a single state and that is actually called a condensate so when you hear the term the bose-einstein condensate what that really means is that in relation to bosons when the temperature is zero they all condense into a single lowest possible energy State now Bose and Einstein predicted this condensate in 1925 it wasn't actually achieved until 1995 when using rubidium atoms a temperature of about one billionth of a degree Kelvin was achieved and the bosons were observed to go into a condensate and that work was done by Cornell vmon and kettle for which they won the Nobel Prize in 2001 well that is a very simple basic overview of the maxwell-boltzmann the Fermi Dirac and the Bose Einstein statistics so you have a rough idea of what they are and what they're used for in each case they tell you the probability that the particle that you're looking at whether it be a molecule a fermion like an electron or a boson like a photon is occupying a particular energy level

Channel: DrPhysicsA

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Keywords: basic, introduction, Fermi-Dirac, Bose-Einstein, statistics, Maxwell, Boltzmann, Quantum Mechanics, Probability, Energy, Distribution, physics, atomic, states

Id: 2wF_CVuWyEg

Length: 39min 59sec (2399 seconds)

Published: Thu Nov 08 2012