Some Remarks on Superfluidity
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Lindau Nobel Laureate Meetings106 / 340
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00:00
Bloch, FelixMagnetismMusical developmentNuclear powerMeasurementKastler, AlfredModel buildingJosephson-KontaktSt. Louis CardinalsSizingAtomismCrystal habitOrder and disorder (physics)YearAngle of attackSuperconductivityPlane (tool)Telescopic sightCurrent densitySeparation processGasKnotPair productionScrew capWeightKardierenForceChemical substanceTemperatureBird vocalizationFACTS (newspaper)LiquidToolAnalog signalVisible spectrumEnergy levelHourAngeregter ZustandRailroad carFullingMetalKelvinRulerRadio jammingCartridge (firearms)ScoutingVolkswagen GolfColorfulnessCouchSelectivity (electronic)DrehmasseFocus (optics)Capital shipChannelingAlcohol proofLinear motorAtmospheric pressureIdeal gasTheodoliteHot workingCondenser (heat transfer)LightLambda baryonWireThin filmHyperbelnavigationModel buildingDirect currentVolumetric flow rateGround stationMeasuring instrumentParticleMeeting/Interview
09:19
Bloch, FelixNuclear powerKastler, AlfredModel buildingMusical developmentMagnetismMeasurementRail transport operationsPower (physics)ParticleMetreCartridge (firearms)H-alphaWind waveLiquidContrast (vision)Centre Party (Germany)Orbital periodHydrogen atomOrder and disorder (physics)Bird vocalizationZeitdiskretes SignalAtomismCosmic distance ladderRing (jewellery)TypesettingPhase (matter)TiefdruckgebietFACTS (newspaper)Spread spectrumGasFuelIPadDrehmasseLoudspeakerRulerFullingOrbitRail profileButtonSunriseAmplitudeColorfulnessTiefgangCylinder blockTurningSchwache LokalisationSizingHourAngeregter ZustandPionSpieltisch <Möbel>BackpackPair productionAngle of attackWeekAutumnEveningMultiplizitätHot isostatic pressingPagerSeparation processGentlemanMarble sculpture
18:34
Model buildingJosephson-KontaktBloch, FelixMagnetismMusical developmentNuclear powerMeasurementKastler, AlfredLiquidCapital shipCombined cycleLinear motorTypesettingVortexZeitdiskretes SignalOrder and disorder (physics)Phase (matter)Roll formingCoherence (signal processing)Color chargeAbsolute zeroAmmeterRemotely operated underwater vehicleFACTS (newspaper)DensitySpare partCartridge (firearms)Chemical substanceAtomismNoise figureCogenerationGravitational singularityContinental driftFahrgeschwindigkeitAngeregter ZustandMultiplizitätOrbital periodAlpha particleEnergiesparmodusZirkulatorYearVolumetric flow rateKelvinMassPlanck unitsFinger protocolPlant (control theory)AmplitudeVideoMoving walkwayTemperatureTelephoneMechanical fanStonewareFiling (metalworking)AutumnFoot (unit)CaliperRulerDrehmasseNorthrop Grumman B-2 SpiritFrictionHourSeparation processRelative articulationRailroad carTape recorderVisible spectrumAntenna diversityFlugbahnDirect currentAngle of attackMitsubishi A6M ZeroCardinal directionGirl (band)Ballpoint penGround stationToolMobile phone
27:49
Bloch, FelixModel buildingNuclear powerMagnetismMusical developmentMeasurementKastler, AlfredAtomismGasAngeregter ZustandTheodoliteCartridge (firearms)FACTS (newspaper)MeasurementSoundQuasiparticleParticleSteckverbinderVolumetric flow ratePhononProzessleittechnikNyquist stability criterionRoll formingLightCurrent densitySpeed of lightSeparation processSuperconductivityBookbindingGas turbineHose couplingSpeed of soundCasting defectFinishing (textiles)UniverseWavelengthAbsolute magnitudeSpare partGround stateZirkulatorAbsolute zeroOrder and disorder (physics)Composite materialMonthTurningFinger protocolStationeryHourDiesel engineIdeal gasKickstandMechanical fanEffects unitWeekDayBook designDrehmasseQuasarAngle of attackFullingNanotechnologyModel buildingMicroscopeThorns, spines, and pricklesStress (mechanics)FlightMorse codeWind farmScale (map)AutumnMeeting/Interview
37:05
Kastler, AlfredModel buildingBloch, FelixNuclear powerMagnetismMusical developmentMeasurementAlcohol proofAngeregter ZustandTemperatureSpare partZirkulatorAtomismRemotely operated underwater vehicleCasting defectSpeed of soundFACTS (newspaper)Contrast (vision)SoundRRS DiscoveryEnergy levelPartition of a setLiquidDirect currentBook designAbsolute zeroFahrgeschwindigkeitCartridge (firearms)Lambda baryonCooper (profession)TheodoliteCocktail party effectPhononNeutronScatteringH-alphaNatürliche RadioaktivitätStarMechanicFormation flyingPhononOrder and disorder (physics)Flavour (particle physics)FlugbahnMeasurementDensityGentlemanEngineDrehmassePitch (music)Vertical integrationAutumnSteckverbinderRefrigeratorBill of materialsMechanical fanElectronic mediaPlant (control theory)UniverseBusFord ExplorerWorkshopKelvinKritische MasseYearModel buildingNeutronenstreuungCabin (truck)SynthesizerBallpoint penSuperconductivityCurrent densityDayStock (firearms)TelephoneRutschungKontraktion
46:20
Bloch, FelixNuclear powerMusical developmentMagnetismMeasurementKastler, AlfredMeeting/Interview
Transcript: English(auto-generated)
00:16
I was able, at long last, to come to this Lindau conference after having heard so much
00:22
about it from my friends who had been here before. And I would like to express my thanks for the friendly reception that my wife and I have received here. Now, from my obvious accent and the fact that most of the audience is certainly more fluent in German than in English,
00:42
I might be expected to give this lecture in German, but having spent most of my life now teaching English, I think it would not be advisable for me to go back to my early habits in speaking German. Besides, Ninz Bohr said a long time ago that by now, like the Latin in
01:05
the Middle Ages, the broken English is really the international language of physics. And so I will try to speak in a sufficiently broken way so that everybody can understand me. Now, the properties of helium at low temperatures have been a personal thought of theorists for a
01:27
long time. The fact that of all the gases it requires the lowest temperature of about so difficult to understand in view of the small Van der Waals forces between atoms.
01:44
But that there was something peculiar about it was already evident from the fact that if you went to still lower temperatures, it refused, unlike any other decent substance, to solidify. Instead, at a temperature of 2.2 degree Kelvin, so-called lambda point,
02:02
it even exaggerated its nature as a liquid in becoming superfluid, which means that it was able at this point to pass even through very narrow channels without any difference pressure gradient whatsoever, rather similar to the transition of a metal into a superconductive
02:25
state. The first who attempted to explain this behavior was Fritz London, who suggested that it might somehow relate it to the Einstein-Bose condensation of an ideal gas. But it took
02:40
still many years, this was before the war, but it was several years after the war until the problem really became clarified primarily through the work of Landau. Now, I want to say that I have nothing whatsoever new to say about this. In fact, the only qualification which I have to speak about this topic at all is that I'm far from being an expert. And this
03:05
has of course a very great advantage because that means that the few things which I know about superfluidity have to be so simple that even I can understand them, and that means then that everybody else in the audience can certainly likewise understand.
03:23
So, those among you who are really sophisticated in this subject, I can only offer my apologies and hope that you will derive some amusement from hearing things which you know very well, perhaps in a somewhat different light. Now, to make things as simple as possible,
03:42
I will take the geometry of a thin circle of tube, so you have to imagine helium to flow around in a thin tube, and then I try to explain to you why it is that in analogy to a persistent current in a wire loop, there we have reasons to expect a persistent flow.
04:08
The method which I'm using is somewhat similar to one which I've used a few years ago, likewise to superconductors. It is based on basic principles rather than on microscopic models.
04:22
This of course has the disadvantage that I will not be able to give you any numerical results or derive any numerical results, and that the reasoning has to be of more qualitative nature, but it has the advantage that unlike a microscopic theory which necessarily
04:41
must be approximate, it can claim a certain amount of rigor. Unfortunately, I didn't have the clever computer that Professor Lam had at his disposal, and therefore my presentation will not be very elegant. I have only some transparencies here, which I have drawn with my own hand, and I hope that nevertheless I will be able to
05:04
show what I want to show. Now I have to learn how to operate this instrument. That looks like the right one. All right, so here I have schematically indicated a thin circular tube. These are the walls of the tube. Please imagine liquid helium being put
05:23
in here, and I will characterize the position of any atom within that tube, well, naturally by three coordinates, as to the sideways dimensions. I will, how does one focus this thing? Yeah, as to the sideways dimension,
05:45
I'm using two coordinates y and z, y this way and z that way, and the more important coordinate is the one around the ring, around this tube. Well, of course it would be reasonable then to say that one can use an angle phi, but in order to emphasize the analogy to
06:05
linear motion, rectilinear motion, I'd rather use a coordinate x counted from some arbitrary point around here. All right, now then, what we would like to know are the energy values, the whole spectrum of energy levels of this system here, and that will be given, as you know,
06:23
by the Schrodinger equation, where the Hamiltonian H shall contain not only the kinetic energy of the atoms, but also an arbitrary interaction, which of course in fact is quite strong. Now the function psi I shall use in a representation, which I make it depend on the coordinates in the
06:42
usual Schrodinger way, and so there would be an atom one and an atom two as a general, an atom s, atom n with x, y, z. Since the x and y coordinates are not terribly important, I will mention them, I will simply omit them for brevity's sake and write this only as a
07:01
function of the x coordinate, the one around the ring, and then I might occasionally go even further and write only one of them, so all these three notations are equivalent. Very good, and I indicated here this is the thickness of the ring, and for simplicity let's assume that's small compared to the radius r, although that's not a terribly important feature. Now then, the Hamiltonian
07:25
should of course in reality also contain the interaction between the atoms and the wall. This I will omit for the time being, of course I will come back to it, you may say well, so I will assume the wall to be perfectly smooth and perfectly rigid, and you might of
07:40
course say no wonder that a liquid runs around in a perfectly smooth and perfectly rigid wall forever, however I will not stick to this idealization with a bitter end but mention it of course later. So for the time being we'll assume that, and then the problem is treated in the usual way, that is to say in that case the total momentum in the x direction
08:03
is a constant of motion, this is given by the sum of the momenta of the individual atoms, and of course pr is the angular momentum which would also be a constant of motion. I introduce the coordinate of the center of gravity which is one of the total number of atoms times the sum of all of them, and now I proceed just the way you have learned it in
08:25
kindergarten about how one separates off the center of gravity, one writes psi as a function of, you might say, a plane wave regarding the x, the center of gravity corner capital x, that's the total momentum, times the function chi, or more explicitly written here,
08:44
psi of x in the notation I used before equal to this, times the function chi now, which depends only on the relative coordinates, and I wrote it simply as saying it depends on the differences between two any arbitrary coordinates x and y, and then you have a separate
09:02
equation for this quantity chi here, I will write it h prime chi, which simply means that h prime, everything is still there except the center of gravity motion that has been separated off, and then as you well know the energy is simply additive, p squared over 2m is the total kinetic energy of the system, and then an energy which I call little e, it is not little
09:25
at all, it's bigger than the other one, but it includes only the relative, the energy of relative motion, very well. Normally, as you do, for example, if you treat the hydrogen atom equation, you find that this relative energy e is totally independent of the total energy,
09:49
that is to say to any value of the momentum that belongs as an arbitrary value, e that has nothing to do with the momentum whatever, and that has of course immediate the following consequence, that if you ask now what is the minimum of this energy irrespective,
10:06
and it depends on the total momentum I mean, then irrespective of this quantity e it is clearly p equal to zero. Now I come to the difference between this normal treatment or the additional considerations which one has to make if one wants to learn something about super fluidity.
10:25
In this equation which I just removed, h prime chi equal to e chi, that is the equation which is supposed to give you the relative energy, the operator h prime is well defined as I said before, it contains everything except the center of gravity, but of course the eigenvalues,
10:46
little e which you get from that equation do not only depend upon the operator on the left side, it also depends upon the boundary value which you impose upon the function chi. Now as far as the coordinates x and y are concerned this is trivial, if you think you have a rigid wall well it would mean that for any particle as it comes
11:04
to the wall the function psi and also the function chi simply has to vanish, that's a trivial boundary condition which we need not further consider. The important boundary condition comes from the coordinate x and that's where the novelty comes, and the point that I'm making
11:20
here is a statement which is accepted I suppose as one of the axioms of quantum mechanics, is that the total function psi must be single valued, that is for each particle, for any one particle s, be it particle 1 or 2 or n, it must be true that as you take one particle
11:42
around the whole ring, that is to say you augment its x coordinate by 2 pi r which is the circumference of the tube, it ought to return to its original value. Now if you remember before the separation off of the centre of gravity I've simply written it down here once more,
12:01
the xs occurs here, xs plus 2 pi r it occurs here too and that must be equal to that, this is identically the same equation except the centre of gravity has been separated off, and now you see that this implies now a condition on that function chi, because this factor will draw out on both sides and in order that this factor here,
12:24
the exponential e to the ip over nh bar times 2 pi r, times the function chi, gives the function itself means that the function chi s as you increase it by 2 pi r must be multiplied by the inverse of that factor, I'll call it alpha, where this is given by e to the minus 2 pi
12:44
pr over nh. This has one well-known and trivial consequence, what I said here for particle, any particle s you can do for particle 1 and particle 2 and particle 3 all the way up to n, and if you do that every time you get another factor alpha, and the condition here then says
13:04
simply that the nth power of alpha must be equal to 1, well then you see the n goes out, and as you see that means simply that the angular momentum which appears here must be an integer multiple of h bar, which is the usual quantisation of the angular momentum, or for the momentum itself it means it must be hr over r, and here I would like to say
13:25
that since we consider r to be a rather large quantity, I will not say how large, but it's certainly a microscopic quantity, not metres but maybe millimetres or tens of millimetres, that you can consider p as effectively as a continuous quantity, the more so certainly
13:43
the bigger the ring is. Now then, since alpha does depend on p as you see, that means, and since it determines the boundary condition on chi, you have to expect in general that the eigenvalues of the equation for chi will likewise depend upon p,
14:04
contrary to what we said before in the normal classical, normal situation. Now, it has however a very important property, namely, if you increase now p by the amount nh over r, then you see it will return to its old value because you simply
14:25
multiply with e to the minus 2 pi i, and that means therefore that the boundary condition returns to its old value whenever p has increased by this amount, and therefore the eigenvalues will
14:41
likewise return to their old values, because again the same sort of boundary condition that I've implied before. All right, what does that mean? That means that e is a periodic function of p, in general does depend on p, but not in an arbitrary way, but in a periodic way,
15:01
in the sense that it comes back to its old value whenever you have increased the momentum by this amount. Furthermore, it doesn't matter really, it is, you might say it's only a question of a conventional sense of rotation, whether you talk about p or minus p, or if you want a little bit highbrow, you can say we believe in this case in time invariance,
15:23
and therefore it means that the function p does not change either if you reverse p into minus p, and so we come to the conclusion then that in general e p is an even periodic function of p with period n h over r. Now I said a function, now of course in reality, the energy e has a
15:45
discrete set of eigenvalues, and what it really means is that to each eigenvalue which you find for a certain momentum, you will find the same value once again if you increase the momentum by this amount n h over r, and I've tried to indicate that in this little schematic plot here.
16:02
Suppose we plot here p, and for p equal to zero you have a certain set of states, and then as p increase you have another set of states, the same for minus p, and then it repeats here, and again minus p minus p, so the blue ones, the blue things correspond to a momentum p equal to zero, or one times n h, or two times n h, and the red ones, and then of course there's
16:26
a whole lot of other sets of eigenvalues in between, with possibly quite a different arrangement. All right, now let's see what further conclusions one can draw from this.
16:42
Now let me first say something else. As I pointed out to you, this is not the normal behavior that you have in general. You will find that really this energy, e of p, the little energy, the relative energy, is independent of the momentum, and that in fact is the normal behavior. I have not said anything in so far as you may have noticed about any special property of helium,
17:06
or even less of helium at low temperatures, so certainly a normal liquid is totally false, totally within the range of my present discussion, and now I'd like to point out to you what one ought to understand, but what I said is perfectly true, and I will not revoke it
17:24
in the case of a normal substance, but I will tell you what the special property or rather general property of a normal liquid is in contrast to that of a superfluid. Well, the point here is that normally, well if you talk about the hydrogen atom,
17:41
you surely assume that the center of gravity, that you can localize the center of gravity of the atom well within the distance in which you are working, let's say within the dimensions of the gas container in which your atoms buzz around or something like that, and it is this property of localizability which characterizes a normal liquid from a superfluid liquid,
18:07
and I've tried to indicate that schematically here, I'm talking here about the function chi, well as you remember, psi and chi differ only by phase factor, so if psi is localizable, so is chi by localizable, I mean of course that you can build a wave packet,
18:22
so I've here schematically indicated the way how in the localizable case, chi will look, this will be our, this is 2 pi r, rather macroscopic this diameter, circumference of the circle, and then there is a certain delta x, the spread of that wave packet, and if delta x is very small compared to 2 pi r, which is the normal situation,
18:45
well then you can have, this is once, suppose you have a solution here, but then that means again as you go 2 pi r, you come back to the same place, you have another solution here, you have another solution here, and as you well know, all belong to the same eigenvalue,
19:01
and as you well know, any linear combination of them will likewise be a solution belonging to the same eigenvalue, and if you want to, and you can, if it pleases you, you can in fact write in this way, where this was our factor alpha, and this thing here has indeed the proper periodic dependence, because every time we go forward by 2 pi r, you change nu by nu plus 1,
19:24
so we have multiplied by alpha, everything is all right, however it turns out to be irrelevant, because in that case, no matter, this is a special linear combination, but you can choose any other linear combination, you'll still come to the same value, in other words, this is a situation where indeed the relative energy is independent of
19:42
the momentum. If you want to have a dependence, then you need a rather extraordinary behavior, after all, superfluid is a rather extraordinary substance, and the extraordinary that I refer to is indicated by this dotted curve very schematically, that really these wave packets
20:03
ought to overlap, you can express that in many ways, you can say you ought to have a phase coherence going around the whole macroscopic tube, or if you want a little bit more highbrow, you can say that you have to have off-diagonal long-range order of the reduced density matrix,
20:21
whatever that may be. Okay, fine, now all right, so let's now take a big act of faith, and believe that indeed in the case of helium, this strange substance does exist, that we have indeed a periodic dependence of the relative energy on p, and in fact perhaps
20:41
a very pronounced dependence, and that I have schematically indicated in this figure here, oh yeah, well I should say one more thing, one more simplification, I make everything as simple as I can, let's put ourselves at the absolute zero and not worry that this is rather hard to do, if I have time I'll later mention something about the modification of finite
21:05
temperatures, but let's assume, or let's put it that way, if the substance did not become superfluid at the absolute zero, well then it probably never will, and so it is a necessary but maybe not sufficient condition, so let's assume only the lowest, to each given momentum,
21:22
we consider only the lowest value of the relative energy, which I indicate by E naught, on our previous diagram, you tick for the blue and the red and so on, then you always take the lowest one, and since p is practically continuous, you can plot it as a function of p, and I plotted it indeed as a periodic function with a rather pronounced
21:45
amplitude, it need not be a sine curve, although it could be, oh I shouldn't say a cosine curve, because it has to be even, the fact that I have said it is a minimum at zero, well that sounds rather reasonable, that the absolutely lowest state of the system is the one where the
22:02
liquid is nicely addressed and doesn't move, but then of course if it has a minimum at zero, because of what I said before, it will also have a minimum at one, a minimum at two, and so forth, but that's only the relative energy, and now let's go back again to the lowest value of our total energy, four given p, I add that p square, well what that means is,
22:24
you have to put this wiggly curve here on top of the parabola p square over two m, and you see that if these minima are deep enough, you still have minima, certainly a minimum at zero, then one around one, times nh over two r, two times nh bar over r,
22:42
and so forth, that now leads us to immediately a conclusion which is rather interesting, namely the following, that in that case, as I have indicated here, you get minima here, approximately I said, integer multiples of nh bar over two r for the momentum,
23:04
let's go for the momentum to the drift velocity, which is the average velocity of the electrons, which you get of course by dividing by the total mass, I call it v, that is the definition, and that this drift velocity necessarily, if you have all you have done is divide that by
23:20
capital M, which is m, so you get a discrete set of drift velocities, that is not so interesting yet, but now you want to go to a concept which is well known to the hydrodynamics, which is called the circulation, which is the line integral of the velocity around a closed
23:42
curve, as you well know that in the case of a liquid without friction, this circulation is a constant of motion, it does not mean conservation of angular momentum, but that's what it is, well anyway, well in our simplified case, our ds, all you do is you go around your circular tube,
24:01
so and v is a constant, so you get two pi r times v, and therefore if you insert for v this value here, you will find that the circulation again has approximately a discrete spectrum, an integer multiple of n times h, that's now the real Planck's h, not the h bar,
24:22
not Dirac's h, but Planck's h, divided by the mass of a single atom. This quantization of circulation, as we may properly call it, has indeed been observed as one of the really interesting facts, one of the very nice experiments was done several years ago by Reif, who shot alpha particles into the superfluid helium, in this way produced vortices
24:47
in the form of a smoke ring, and then he accelerated them, the charge sits on the smoke ring, he accelerates them in a direct field, and from that can calculate, simply by applying classical hydrodynamics, the amount of circulation, and found it indeed very close to that value,
25:04
and the ratio h over m derived from his experiments agrees with a well-known ratios to a part in a few percent, and in fact there are some reasons why you might possibly use that sometime to make an even more accurate determination, but it's not a precision instrument, but I just
25:21
wanted to say that this is not sheer fantasy, but has something to do with reality. All right, now there's something else you can learn from this simple-minded consideration. Now I have here again drawn my my dotted p square over m curve, it is here, this time I've plotted it as in terms of velocity, because the two are very closely related instead of momentum,
25:44
and in a somewhat smaller scale, so you put these wiggles on the top of a parabola and you keep on going and going, but now something happens, because as you go higher and higher up, of course the slope of your parabola becomes steeper and steeper, and except for very
26:00
singular curves, periodic curves like that, there comes a point where there's no minimum anymore. I have called that critical velocity, I've tried to indicate it here, you see how here's still a minimum, here it becomes kind of flat, and there's no minimum anymore, and from there it stops, so that means therefore that, oh well, okay, yeah, I'll come back to that,
26:23
so therefore it means that these minima really exist only up to a certain velocity which one might well call a critical velocity, or perhaps even the critical velocity, because it is well known, and is a known fact that you cannot maintain a super flow beyond a certain velocity,
26:43
or at room temperature, at liquid helium, one degree kelvin or so, of the order of maybe 20-30 meters per second, beyond that it just won't tolerate it anymore. Now I should have said something before, and I'm sorry I apologize, about the significance of these minima. As I said before, so far I have totally neglected the walls, or rather
27:06
I have assumed the wall to be perfectly smooth and rigid, and if that were the case, anything would flow indefinitely. Now we have to ask ourselves what would the wall do. Now as I told you before, I consider the situation at absolute zero, and that means that I want to have a wall
27:22
at absolute zero, but now a certain amount of interaction between the wall and the helium atoms. Now if I sit in a minimum like that, and the wall, and somehow by interaction of the wall, the liquid would like to go to a lower momentum, as a normal liquid would do dying out,
27:43
the wall will say sorry, can't do it, I have no energy to give you, because the energy has to go up, as you see. Well you might say, but listen dear wall, I can't do a little bit more generous, suppose I sit here, and you give me enough momentum to jump directly here, then I have lowered
28:03
my energy, or from here to here. Well this is a well-known situation also in the case of persistent currents and superconductors. One does not deal with absolute minima, one deals with relative minima, and in a certain sense you may say that also superconductive current in a
28:21
loop does not run forever, but it runs several hours, or several months, or maybe a year or so. In other words, I have never calculated a certain finite probability of a jump. Oh yes, well one more thing I would like to say, this is completely analogous to the case of a superconductor. In the superconductor what corresponds to quantization of the circulation
28:44
corresponds to the quantization of the flux, and there's quite an analogous argument. Well all right, like in a superconductor, it could happen in principle, but presumably will take a rather long time, as I say I haven't calculated, maybe the age of the universe
29:00
for a rather large ring. The point is there that although you can go from here to here, it is a very improbable, a highly improbable transition. What it would mean is namely that each atom simultaneously would have to lose the exact amount h over r, or one unit of angular
29:21
momentum. Well, that all the atoms should so nicely cooperate and make the jump together, I think with a little hand waving it will be understandable. That is a rather unlikely process and accounts for the really extremely long stability of persistent flow or persistent currents. All right, and now I mentioned already the critical velocity, and you can give it an expression here,
29:46
you can say it occurs when the absolute magnitude of the derivative of dE0 by dP, well, it is defined by the maximum value of that, because if this becomes, well, the smaller
30:00
this is, the earlier this cessation of minima will occur, and no more superfluid flow. Well, yeah, I tried to make up for the sins of my predecessors and be short, but I just want, how long have I really talked? Half an hour? All right, well, I think I'll finish in the
30:25
next 15 minutes, then at least I haven't made up for their sins, but at least I have not added to them. Okay, now you will justly ask yourself now, all right, fine and good, we have assumed we have been very nice, we have did our act of faith and believed
30:42
these minima to exist, and that seems to have something to do with reality, it explains quantization of circulation, it explains cultural velocity, but now under what conditions come and reasonably expect this really to happen? Now I'm going, at this point I'm going to a
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certain extent into models, that is to say the general part is finished, what can be generally said I have said, but now it would be interesting to ask yourself what kind of more microscopic assumptions would you have to make in order to get indeed this strong, or reasonably strong, periodic dependence? And as a first case, I want to discuss with you just what London
31:26
suggested originally, namely let us forget about the interaction between the atoms, that's certainly an abominable approximation, but let's assume that we deal indeed with the Einstein-Bose gas and ask ourselves what happens to such a system at absolute zero. Of course, as you well know,
31:46
long before the absolute zero there occurs this phenomenon of Einstein-Bose condensation, but let's go even to the extreme end, to the bitter end, where in the ground state of the system all the atoms would have zero momentum, that we will not assume now, but let's see what
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happens in the case of the ideal Bose gas. Well, all right, here I've written down now the energy of a single atom, which is simply its kinetic energy, and now I will try to construct the lowest energy for a given momentum. How do you do that? Well, you take each atom and you
32:20
give it a kick in the x direction, but each only in the x direction, don't waste any momentum sideways, because that will only cost you energy without gaining you a circular momentum. So suppose you take then new atoms, and for the time being, they shall, well, better be less than n because you don't have any more new atoms, an arbitrary number of atoms, each of them,
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you give its minimum momentum, h over r, and that means that the total momentum is this much. That is the cheapest way, at the lowest energy expense, of getting a given amount of the momentum. We have new particles with momentum p equal to h over r, and the other ones we leave in their
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momentum state zero, right? Well, all right, then we can very easily calculate the total energy, simply new times their kinetic energy, p square over 2m, and that you can also write in this form if you use this form p, and now let's calculate the corresponding relative energy, as I would call
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it, which you'll get by subtracting now p square over 2m, and this is the form which you get. That holds between zero and n, and as you notice, this is a parabola, and I've plotted it here. It starts with zero, and very nicely, as it should, it comes again to zero when p is equal to nh over r. But now, of course, you can go on from here. Suppose you have brought all your
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atoms from zero to n, then you are again here, and now you start all over again, leave most of the atoms in the state one, and start the same with the other again. The other words, and in fact this is a general statement, as I said before, this little parabola will have to repeat periodically.
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So that's at least an illustration, a very simple illustration, of this case of the periodic function e naught of p. But the trouble is that when you do superimpose that function on p square over n to get the total energy, what you get is what I indicated here. You see,
34:23
of course, the energy increases simply linearly with the momentum. Yeah, here it is. And so, instead of these nice minima, which I indicated before, you simply get the connection. You take this point, that, and it's a straight line, and another straight line, another straight line. Where are the minima? Well, there's one, the one at zero, but nothing else. And this is true.
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Unfortunately, and here I'm talking about my friend Fritz Landen, it is not true that the Einstein, the ideal Einstein Bose gas, leads to superfluidity. This is a fact which has been recognized quite some time ago. In fact, Bogdan-Luboff has gone
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so far as to calculate the effect of weak binding and found that it is quite essential to have superconductivity, superfluidity, and this is what I'd like to explain now. Now, rather to refer to Bogdan-Luboff's paper, which is a mathematically very beautiful paper, has not terribly much to do with reality, because the coupling in helium is far from being weak,
35:26
it's very strong. Instead of that, I will use the terminology of Landau, I think that was, in fact, one may say the essential contribution of Landau, to postulate that rather than having to consider individual non-interacting atoms in helium, as it would be in the case of an ideal
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gas, you should think of what he called quasi-particles. Now, these quasi-particles are really the result, the final product, of a very complicated game of interactions. And again, it needs a certain amount of faith to believe in those, but it doesn't need too much
36:05
faith if you restrict yourself to particles of rather low momentum, which means rather long de Broglie wavelength. If this de Broglie wavelength is long compared to the inter-particle distance, then it is rather clear that these quasi-particles are nothing else but the quanta
36:25
of sound waves, so-called phonons. In that case, just like if this u is the sound velocity, if it were the light velocity, you would have the energy of a light quantum, velocity of light times momentum, here it's velocity of sound times the absolute magnitude of the momentum,
36:44
this would be the energy of a phonon or of a quasi-particle of Landau, at least at low momentum, and since we are interested in low energies, we might perhaps be satisfied with that. To give the, that is to say, the dependence of epsilon p is no more quadratic, but at least
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for low momenta is linear, and has in fact a discontinuity at p equals 0, discontinuity of a slope. All right, now you want to build up a certain amount of energy, again, e0, the cheapest way, how do you do it? You excite phonons only in the direction, phonons traveling
37:24
only around the thing, and don't bother of exciting side waves, sound waves, because that's a waste of energy, and that has a very simple consequence, because it means that the total energy is simply u times the total momentum, and the relative energy is then given by that.
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Well, all right, this shall be, I want to hold for relatively small momenta and this range, and I have tried to indicate it on this plot here, you see here this typical discontinuity here, p absolute, and of course these things somehow may continue, you may go on getting higher
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momenta for higher excitations, but we do know that it is periodic, so this same sharp point here occurs again, must occur again, here and here and here and so forth, superimpose that on the parabola, and now you get a nice result. In that case, namely,
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the minima are exactly at the values h, mh over r. Also, as you go higher up, of course, again, there will be a critical velocity, in that case, since the slope here is likely to
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decrease, the maximum slope is in fact the velocity of light, you would come out with a result that the critical velocity is equal to the sound velocity. This is not quite true, actually it's somewhat lower, or a factor two or three lower, this comes from the fact that this approximation or the idealization of keeping the linear law really holds only for
39:05
the smallest momenta, and if you go a bit higher, you have to make corrections. People have studied this function epsilon of p, it can be done by neutron diffraction, by inelastic scattering of neutrons in liquid helium, and it is indeed, of course, through its starts
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linearly, the way I said, but then it has some kinks, and therefore there are qualitative changes, and as I said, this statement that the critical velocity is equal to sound velocity is not true, but it is of the same order of magnitude, and more than that we cannot hope, and the circulation is exact. Consequently, it is a warning to people who want to determine
39:44
h over m, in contrast to the people who have determined e over h with very great accuracy, from flux quantization, that if they want to do with great accuracy, they better go to very, very low temperatures, because this statement holds only as long as I have stuck to the lowest
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energy. Now, I think my time is almost over, I could, would like to go only briefly then, mention to you what happens at finite temperature. Of course, this discontinuity that we have here is just an idealization, and it's the same idealization as assuming that we are at the absolute zero. If you are not at the absolute zero, then rather than the energy itself,
40:26
you have to consider the free energy, but it has rather similar properties. All right, now here f, the center of gravity still comes off, little f now corresponds to the
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free energy, to correspond to the energy levels of little e, what we had before, and is, as you know, well, well know, obtained by taking the logarithm of this partition function, this sum here, and what it does in effect, and you can calculate it easily, it means, of course, that at higher energies you are not so economical anymore, you do excite phonons
41:03
also in directions not parallel to the x direction, and what it does is it makes a rounding off here, I've indicated here for t equal to zero, f is equal to r, I drew it here only near one of the minima, equal to r, this continues, then becomes rounded off more and more and more,
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you can characterize that by a factor alpha here, that's not our old alpha, some quantity which depends on t, which is the smaller, the lower the velocity, in fact, you can calculate it and find that it goes like, here, like t over t star to the fourth, so it does decrease rather
41:43
rapidly, when you go through the whole arithmetic you'll find that there is a correction in the quantized value of the circulation, it is n times h over n, thus it would be at the absolute zero times one plus alpha, so it gets reduced, and you may in a certain sense assume that alpha goes to infinity at critical temperatures or that this value goes to zero, anyhow it's reduced,
42:06
this is a formula which holds for low temperatures t small compared to a certain temperature t star, which is given by this exact expression here, that's simple phonon calculation, and that temperature t star is about four times about 90 degree Kelvin, if you
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insert here the measured velocity of a density of helium and the sound velocity, you come out with 90 degree Kelvin, fourth, about four times the critical temperature, just why these two temperatures are of same order of magnitude may or may not be an accident, I rather think it is not an accident, anyway, this is what it is, now just very briefly, this can be used in other
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ways, because people will say, oh no, no, that's not true, the circulation is still rigidly, rigorously H over m, only you shouldn't talk about the whole helium, you should talk about the superfluid fraction, there is this often very useful model, two fluid model, where one says that
43:05
only at absolute zero is all of the helium superfluid, and that finite temperature is a fraction, and then of course at the critical temperature, at the lambda point, all is normal, you can express it that way, I never quite understood what that means, if you ask a helium
43:21
atom, are you normal, or are you superfluid, the poor fellow certainly won't know what he ought to say, it's of course only part of a total system, but it can be interpreted that way, that's not my discovery either, this is discussed in London's book on superfluid, and you can in fact say that with that definition of ratio of normal to superfluid, you can in fact say that the circulation
43:46
is that H over m multiplied in the ratio of superfluid to normal, and therefore is one at the absolute zero, this factor, and it goes to zero presumably at the lambda point, as I say, I don't want
44:00
to take your time anymore, you're probably hungry and so am I, and we want to hear Dr. Osaki, there are some interesting other points I should have liked to make about helium 3, because that's of course quite a different story, all I talked about is helium 4, because we are talking about both systems, whereas helium 3 is a family system, in that case, well it was a matter of
44:24
long debate whether helium 3 would at all become superfluid, certainly it doesn't become superfluid at 2.2 degree kelvin, but recently in the neighborhood of milli degrees, a whole lot of interesting transitions has been found, and one of them is I think pretty well established now
44:43
to be indeed superfluid, now if you want to explain that, you have of course to take the same kind of reasoning that Bardeen, Cooper and Schrieffer took in the explanation of superfluidity, I'm sorry, I mean superconductivity, namely and the essential point there is the formation of Cooper pairs, that is to say you have to assume and introduce some mechanism
45:06
by means of which roughly speaking two helium 3 atoms form a unit and then that can become effective, we are both talking in a way that that Dr. Cooper will be horrified about, but he's a nice man and will forgive me, it's a very complicated calculation, theories have
45:23
wondered for a long time, the estimates have fluctuated I think between 10 to the minus 6 and 10 to the minus 1 degree absolute, the truth is somewhere in the neighborhood of 10 to the minus 3, but it's a very much more complicated thing, one thing I would like to say however, and that I do not know where it has been measured, but surely the quantum of circulation
45:43
would have to be half of what it is in the case of liquid helium simply because you have two atoms to contribute, h over m, you must write h over 2m, that does not exclude my very general statement, although I drew you your curve pedagogically with minima only at 1 and 2 and
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integer parts of this characteristic momentum, of course a periodic function can also have periodically repeating minima in between and that is exactly what we would have to assume in the case of superfluid helium, well as I say my time is over and thank you for your attention.