Journey to the centre of a Neutron Star (Lecture - 07) by G Srinivasan

so let's start today's lecture which is a journey to the center of a neutron star to see that a neutron star is not just a liquid drop of neutrons it's a very interesting object now how are we going to approach this question of water Newton scholars when a neutron star is formed its temperature will be about 10 to the power 10 or 10 to the power 11 Kelvin but please remember in the process of protons becoming neutrons an equal number of neutrinos were produced and this nuclear laws escape and carry away the heat very effectively so the temperature of the neutron star drops of about 10 to the 10 or 10 to the 11 to a modest 10 to the 9 degrees or so very quickly within a week or a few days and therefore I am entitled to treat neutron star as a ground state of matter in other words it is a lowest energy configuration and let us ask first thing is I expect that in the process of collapse here equilibrium would have been attained I mean by that strong eeeek and electromagnetic forces will adjust the nuclear composition to the most favorable 1 which is the lowest free energy configuration now I'm going to regard neutron stars as very cold objects even though the temperature may be a billion degrees as hot or cold has no meaning in physics unless you compare it with something else so I must compare KT with the characteristic energy of the system I can compare it with the Fermi energy of the system or I can compare it with the excitation energy of the system now you know that nuclei emit gamma means that the energy levels inside the nucleus are separated by a million electron volts or more so the characteristic energy scale at nuclear density is a million electron volt which is 10 to the power 10 Kelvin remember one electron volt is approximately equivalent to 10 to the power 4 Kelvin therefore a neutron star is a temperature of 100 million degrees or even a thousand million degrees is a very cold object compared to the characteristic energy the typical excitation energy is 10 to the 10 the Fermi energy will be 10 to the power 12 Kelvin so KT is a very small number therefore I am entitled to resolve a neutron star as being the loop now let's start with the surface of the neutron star there is nothing sitting on top of it therefore the pressure is zero or the surface of the star so what is the lowest energy configuration at zero pressure what do I mean by that what I what have given you or a certain number of protons an equal number of electrons and also roughly an equal number of neutrons now you have to bundle them something to create the lowest form of energy so let's start with their nucleons first you know from various things that we have discussed and also from your other studies then the nucleus which is most strongly bound is iron 56 cook we are just a hot but an input nucleon so the sensible thing would be to do would be to package neutrons and protons into iron 56 nuclear and then you attach the electrons to them to form iron 56 atoms now if they are moving around like a liquid or again consider Z there is entropy associated with this motion and you would like to mean it is that the best way to minimize that is to tell the iron atoms to stay put now they need not form a periodic lattice they could be like a glass amorphous then you can ask which is a lower energy a crystal lattice is a lower energy than an amorphous communication then you have to ask should it be a centered cubic lattice or a face centered cubic lattice and questions like that and I think my memory serves me correctly is the body centered cubic lattice of Arc fifty-six atoms lowest energy configuration that still leaves me with the degrees of freedom of the electron spins which could be a random in other words this iron could be paramagnet obviously I lower the energy if the spin find themselves either a parallel or antiparallel ferromagnet or antiferromagnet and after many years of study people came to the conclusion that the ferromagnetic will lower the spin entropy more than the anti ferromagnetic state the other thing to appreciate is that the temperature even a temperature of about 100 million Kelvin will be below the melting temperature at those densities if you look in kettles incredibly solid state physics book the melting temperature of a solid will depend upon the density also so you have every right to convince yourself the outer layers of the neutron star will be solid and that solid will consist of ferromagnetic iron-56 lattice okay this is a safe thing to say this is a view of pressure which is the surface that you will see from outside by the way we will see neutron stars are in endowed with very strong magnetic fields of the order of 10 to the power 12 to 10 to the power 14 Gauss so this iron atoms will not look like your iron atoms in the laboratory whose characteristic size is the Bohr radius either this way or this way or this way but because the magnetic field exerts force when you try to move perpendicular to it the motion of the electrons in the magnetic field will also be quantized and the characteristic radius will be the larmor radius e be over MC how does that compare with the Bohr radius which is the characteristic size of the electrons in orbit it turns out that the larger radius can be very much smaller than the Bohr radius therefore the Riggin atom will not look like a nice sphere the hydrogen atom will look like our ion atom will look like a needle its characteristic size in this dimension will be the larmor radius the characteristic size in this direction will be a bore radius okay so these are aggregate give rise to very interesting possibilities but we are not going to deal with that if you go deeper you don't have to go very much deeper convince yourself rather calculate at what depth the pressure that you feel on your head will equal to the pressure you'll feel when you go to the center of the earth in other words at what depth the gravitational pressure to the preserve the center of the earth okay to a simple calculation using Newtonian mechanics now as you go down the surface of the neutron star the density will dramatically increase and by the time you reach 10 to the power 4 grams per cubic centimeter atomic shells would have melted away from the nuclei and you will have a complete free electron gas in a of course in the laboratory it's only the valence electrons which form the free electron still bound to the nucleus here everything will be ionized now when you rate a density of 10 to the power 7 grams per cubic centimeter and you don't have to go very deep for that the electrons will become relativistic really degenerate in other words the pressure will now be then filled to the power 4 by 3 now around this density when the Fermi energy of the electrons exceeds 1 million electron volt remember the Fermi energy is equal to the density to the power 2 by 3 so at some density the Fermi energy will reach 1 million electron volt I am this is no longer the most stable nucleus that is because an electron from the top of the Fermi level which is an energy about 1 million electron volt can combine with a proton to form a neutron and a neutrino the neutrino of course will escape in other words the neutralization process will begin now when that begins iron-56 will no longer be the most stable nucleus instead it is nickel 62 then happy as you go down with a little more nickel 62 will no longer be the lowest energy nucleus it will be nickel 64 and so on so the moral of the story is as you go down the inverse beta decay which is proton plus electron combining to form Neutron and a neutrino will make the nuclei more and more Neutron rich and when you go to this magical density of 4.3 times 10 to the power 11 grams per cubic centimeter this is the number that I had mentioned in I in the context of Landau's I a neutralization of matter what you will find is exotic nuclei for example you I am 56 or I am 56 the nickel 62 and nickel 64 then selenium 84 germanium 82 zinc 80 nickel 78 and so on on all these nuclei have 50 neutrons in them then molybdenum zirconium strontium and Krypton number super Superman came from Krypton which has 82 neutrons in our periodic table you know the number of protons is roughly equal to the number of neutrons so you have a plus a 118 z is 82 another interesting thing about all these nuclei for those of you who studied nuclear physics they have closed nuclear shells so that tells you that these are the minimum energy nuclei straight away right so you will encounter these exotic nuclei as you go down from the surface oh I better remember the top as you go down I'm saying you will encounter you'll encounter these nuclear that I am having disappear just as I go down I will meet newer newer nuclei yes you are absolutely right so this is the picture of the neutron star the outer let's consider the neutron star of the temperature limiting mass because nature seems to make them with that mass the most models of the neutron star will give me a radius around 10 kilometers and all of them will agree that the outer one kilometer is a solid crust the outer layers of the crust consists of iron-56 nuclei arranged in a ferromagnetic lattice and now as you go down you find the more exotic nuclei that I have already mentioned now an important thing to understand is that if I go down a little further then you CLE the nuclei don't like to accept any more neutrons therefore the neutrons drip out of the nucleus just as the valence electrons drift out of atoms when you brought them to distances comparable to the size of atoms I'll do it free electron gas and metals now you have a free Neutron gas in the lattice and that's what I have indicated in the bottle so the lower part of the crust will consist of a crystal lattice of exotic nuclei such as the ones that I have mentioned here and sloshing around will be free neutrons so let's go back to the picture there so what I will have is a density of 4.3 times 10 to the 11 grams per cubic centimeter below that I will encounter an inner by the way this is not drawn to scale if we exaggerated I will encounter the inner crust which will consist of exotic nuclei with a Fermi liquid of free neutrons what happens if I go even further is I will encounter a density of 2.5 times 10 to the power 14 grams per cubic centimeter which is the density of atomic nuclei I convince yourself that starting from hydrogen all the way to the end of the periodic table that the typical density of atomic nuclei of terrestrial matter is 10 to the power 14 grams per cubic centimeter I say this because very often when you talk of neutron star the tendon I don't believe all that you know we have never met densities of that order so how do we know laws of physics or wearing those densities you and I are made up of matter whose density is 2.5 times 10 to the power 14 grams per cubic centimeter we float in water only because the distance between the nuclei is so large the average density is 1 all right so when you read the nuclear density what you would expect is the nuclei to melt in other words the distinction between the individual nuclei disappears and what you have is an assembly of protons and neutrons with an admixture of electrons for charge neutrality now how many protons will you have and until you have built this city of 2.5 times 10 to the 15 grams per cubic centimeter that's the answer to the question I had asked before the lecture namely why doesn't a neutron star decay back to a proton and electron stuff now the neutron decays to proton electron and an anti-neutrino and at the same time protons combine with electrons to form neutrons and neutrinos now these two chemical reactions must come to chemical equilibrium in other words for steady state the reaction going this way must be reaction rate must be the reaction rate going this way that requires that the chemical potential of the neutron which is the Fermi energy at low temperature must be equal to the sum of the chemical potential of the proton and the chemical potential of the electrons at zero the chemical potential is same as the Fermi energy so that statement is the Fermi energy of the neutron is equal to the Fermi energy of the proton and the the energy of the extra now if a neutron were to decay back going back to my question then the electron must have enough energy to climb to the top of its Fermi sea and sit there because all the states are occupied and Fermi energy several million electron volts so when a neutron decays the characteristic energy of the electron will be equal to the rest mass energy of the proton Neutron minus the rest mass energy of the proton and there is no guarantee that this electron will be able to climb the top of the skyscraper and sit there that is why a neutron star is stable against beta decay it is because of the extreme degeneracy of the electrons okay now if you work this out has this chemical equilibrium given by the equality of this chemical potential convince us we do this it will take only about two three minutes to do this convince yourself that you will get roughly 95% neutrons Cyprus protons and an equal number of electrons so let's draw the picture again so what we will have about the nuclear density of 2.5 times 10 to the power 14 grams per cubic centimeter is a liquid or a fluid core consisting essentially of neutrons a three or four or five percent of protons and electrons okay so that's what at first sight a neutron star will look like now you can ask the question but what is it going to look like right near the center now this question was asked last time so I will deal with this question in a few minutes but hence this is an overall picture to which we will come back and various things indicate to us including constraints of the equation of state deduce from observations tell us that this picture is essentially correct all right now how did we arrive at this picture I seem to be waving and telling this but is this the way people really arrived at this assertion no they actually did the hard work and since questions were asked last time let me spend a few minutes dealing this how does one calculate an equation of state which is the relation between pressure and density let us start the density is below the neutron drip namely in the outer crust the problem of determining equilibrium nucleus at any density is that is the same as minimizing the total energy per unit volume what you have is a fixed number of baryons and charge neutrality so what you have to do is to minimize this energy epsilon which is the energy per nuclear of a and Z multiplied by the number of those nuclei plus the energy of the electron or the density and a V of Iligan so the sum of nuclear energy and electronic energy and you have to minimize this well how do you calculate this nuclear energy the lowest possible nuclear energy as I said last time one uses the liquid drop model of the nucleus which deals born toddlers and use something like a semi empirical mass formula the semi empirical mass formula right the nuclear energy as the sum of surface energy which is proportional to a to the power minus 1 by 3 per nucleon there is a atomic mass and the nuclear Coulomb energy which is proportional to a to the power plus 2 by 3 and Z over a whole squared per nucleon the important thing to appreciate is that the surface energy favors larger nuclei there is a Coulomb repulsion energy prefers is smaller nuclear therefore it is this competition of this balance which will determine what is the lowest energy configuration of the nuclear so that's the principle that you use now you have to do this at all densities now how did this objective all the first person to undertake this job was John Wheeler and Princeton University with one of his students Harrison and what they did was to regard the atomic number a and that only charge Z as a continuous variable so you can do this using calculus how this expression will vary as I change a and vet but then some years later in 1961 from not mistaken saltpeter shall no no no no no you can't regard a and Z as a continuous variable because the nucleus consists of shells practice just as atom consists of electronics inspectors therefore you must regard a and Z a discrete variables and so South Peter did these calculations is proving upon Harrison wheelers earlier calculation then came along beim Pittock and Sutherland and they said look that you put the nuclei in a lattice and you have to take into account the lattice energy because you take the cohesive energy of metals it consists of the lattice energy and it consists of the electronic energy and it consists of the energy interaction between the electrons on the nuclear and the lattice energy is equal to minus 1.8 to Z squared e squared well behavior is the inter atomic distance okay now you see this is very important to have included at some stage or other because the lattice energy because it is a minus sign reduces the repulsive Coulomb energy by about 15% thereby favoring you in larger nuclei remember surface energy favored larger nuclei Coulomb repulsion favored smaller nuclei this reduces the effectiveness of Coulomb repulsion therefore so this is the way a subjective wall equation of state got more and more refined all this is a density below the Neutron drip density and here just to convince you is a plot what is plotted on the y-axis is the logarithm of the pressure what is plotted on the x-axis is a logarithm of the density I'm not going into the details of this I just want to tell you that various curves with levels correspond to the equations of State calculated by various people and i just want to convince you that some honest work has been done which led to the picture that I drew okay it will take us to far afield to discuss individual equations of scale now let us refer to the equation of state above the Neutron drip namely the lower part of the solid crust density between to the power level and the nuclear density of 2 point 5 10 to the power 14 grams per cubic centimeter now what is the essential new physics here there are free neutrons in addition to free electrons and therefore been hans bethe a and pity improved upon what had been done earlier by taking into account many body interactions of the neutrons this is what state physics big for electrons and metals and out of that came all the interesting physics of solids namely the optical properties the electronic properties magnetic properties and so on so on our hands better knew all that very well and therefore he said let's now deal with a neutron in a sophisticated fashion now what about the liquid interior above the nuclear density there the very complicated because you're dealing with densities which are even more than the density of atomic nuclei in atomic nuclei it is understood that the mean distance between the particles inside the nucleus is at the minimum of the potential energy curve now you're pushing them towards the repulsive potential energy so there you have to be guided by scattering experiments he identifications I want us to solve the many body floating near equation I want us to take into account nodal or the forces a body force in fact over there are Raja Raman who used to work here they needed to do science was now JNU he was Hans bethe a student and his thesis was on full body forces in the nuclear up so here is a question of fit calculated by Harrison and wheeler a densities above the Neutron drip namely you're inside the liquid interior and just for comparison I have shown here the bay mystic pines equation of state I just want to tell you that there is a difference because beam better and Petit included many-body fixed with Harrison and wheeler did not so the inclusion of anybody's face you clearly see and this is a logarithmic plot all right so this prince is quite significant what will this difference amount to finally it will amount to your prediction about the radius of the star the density profile and things will example yes okay now let us go to this question the Camry has last picture about whether there can be exotic States near the center of the neutron star what kind of exotic particles are having mind the primary zones came in zones and other kind of strange particles and then of course quacks so let us deal with the missiles first I am as on as you know comes in three varieties pi plus pi minus and pi zero its mass is one hundred and thirty nine point six million electron volts they obey Bose Einstein statistics and not where we do extra two sticks please remember that we are talking about the interior where there is beta equilibrium which means Neutron decaying two proton goes at the same rate as proton plus electron combining to form neutrons an upward the chemical potential of the neutron is equal to the sum of the chemical potential the proton and electrons now imagine what was happening before the neutron decayed to proton and an electron in other words the Fermi energy of the electron minus the Fermi energy of the proton gave you the Fermi energy of the electron population now a new channel opens up in other words here is the P the neutrons and there is the Fermi energy at high densities when the Fermi energy becomes high enough the neutron that should be small letters sorry the neutron can decay to proton and a PI meson instead of an electron remember faison is a mass of 140 million electron volt and an electron is a mass of only half a million electron volt therefore this cannot happen normally but it can happen at sufficiently high density and what density do you expect this to occur that depends on a level of sophistication that you bring to your analysis a simple-minded estimate will tell me that this process is unlikely to occur at a density less than twice the nuclear density will have three times the nuclear density the nuclear density is 2.5 times 10 to the power 14 grams per cubic centimeter so we are talking about a minimum of 5 times 10 to the power 14 grams per cubic centimeter so whether or not primers ohms will be spontaneously produced depends on whether as I journey to the center of the neutron star whether I will encounter a density of 5 or 6 times 10 to the power 14 grams per cubic centimeter similarly other even higher density where the Fermi the neutron becomes even higher then I can produce came a zones and other strange particles now for game essence the rough estimate is that you need at least three or four times the nuclear density before you can spontaneously produce the caiman songs for the sake of argument let us assume that such high density is obtained and primer zones and camera zones are produced in the neutron star and ask will it matter now let us remind ourselves that photons Amazon's gamma zones graviton we haven't found the net obey both Einstein statistics because their spin angular momentum is an interval of H caused by to know its growth she was questioned half and you know what I mean spin is for speed of the photon is one and yeah all right what does the bose-einstein distribution function look like what is shown below is the Fermi Dirac distribution function which we have used and that differs from the bose-einstein distribution function with a minus sign in the denominator so the bose-einstein distribution function is the probability that a state with energy ek is occupied is equal to 1 divided by exponential to the power ek minus nu divided by kt minus 1 whereas it is plus 1 in the case of Fermi Dirac distribution there is another fundamental difference and that fundamental difference concerns the chemical potential mu the chemical potential of the Fermi Dirac distribution is obviously positive it is the Fermi energy it is so many electron volts or for copper but the chemical potential of a both understand gas is negative it can most be 0 and that 0 obtains for mass party - like photons remember in Planck's law there is no mu it is e to the power H bar Omega by K T - 1 but if you have massive both Einstein particles then the chemical potential has to be negative and that is the reason why some interesting things happen the interesting that thing that happens is what is called as bose-einstein condensation it should really be called Einstein position also nothing got served to do let's recall the history Bose discovers the statistics of photons and sends the paper to Einstein with a letter saying if you agree that this derivation is valid can you please translate this to German and have it published in a German did but in the process of reading BOCES paper he realized that this distribution is far more general than this for photons and therefore Einstein realized that for all particles is integral multiple of Planck's constant for the spin angular momentum this statistics will be valid not just that and immediately you realize that the fact that the chemical potential has to hit will implies that at low enough temperature all these particles will go into the zero momentum or zero energy state and then therefore he respect which he wrote subsequent to ring most people he wrote a second paper and right there in the paper he predicted this sensation so the credit 100% of the credit should go to Einstein for this because it is an extraordinary statement to have made right then then as you know in the last five or six years having two Nobel prizes and physics because they finally found this bose-einstein condensation what is this bose-einstein condensation let us consider that degenerate both gasp that means both gasp at Absolute Zero temperature like we discussed the degenerate Fermi gas its behavior is very different from that of the Fermi gas the Fermi gas as we saw at zero temperature has an enormous amount of zero-point energy and that zero-point energy came because fermions obeyed Pauli's exclusion principle two to two you have to file them up but since both Einstein particles do not have to obey all these exclusion principle at low enough temperature all the particles we in the e equal to zero state now please remember that this fundamental difference namely that the total energy of bosons can be zero at low enough temperature arises because the Bose gas obeys has a negative chemical potential somebody is interested you ask in the end and then I will tell you a little more about why this phenomena occurs now what the Bose Einstein statistics tells you is that there is a temperature T naught below the temperature the number of particles with energy greater than zero is given by T by T naught to the power 3 by 2 therefore by inference that particles with energy e equal to 0 is given by 1 minus T by T naught to the power 3 by 2 therefore at T equal to 0 the second term vanishes and all the particles will be in the zero energy and zero momentum state please understand that this both Einstein condensation is a condensate and momentum spirits and not condensation in the real space when a gas becomes a liquid it is condensation in real space it settles down to the bottom of the beaker it settles down to the bottom of the beaker and momentum space I'm not in realist right and please remember that this temperature T dot will be a function of density I will show you the expression for T note in a minute and therefore you don't have to wait till T equal to zero at low enough temperatures you remember we said at low enough temperatures when es when KT becomes smaller than es the new statistics takes over similarly when T becomes smaller than T naught which depends on density both statistics will take over and that's going to put more on poor particles in the zero energy zero momentum state now let's get back to fire and carry on let's take PI on suppose to be specific there we're both Einstein statistics and at low temperatures they will condense into the zero momentum state what is low temperature sorry this is this P naught of the previous slide TC is the critical temperature for both condensation is proportional to density to the power 2 by 3 same as the density dependent to the Fermi energy therefore you tell me the density I will tell you what is a critical temperature below which post condensation is beginning to sit in and then you find that there is a finite number of particles and zero in the ground state therefore a low enough temperature all the bose-einstein particles will be in the zero moment and therefore the pressure will vanish just as Boyle's law pressure vanishes at T equal to zero forget about t equal to zero the moment you go below temperature TC which is a critical temperature there will be a macroscopic number of particles in the zero momentum state therefore the pressure of this gas will suddenly become much less than before or in other words the equation of state will be soft power will become suddenly highly compressible then it becomes a matter of life or death for the star therefore let us go back to our picture of the interior of a neutron star let us assume that we to reach densities of the order of 5 times 10 to the power 14 grams per cubic centimeter then below that are above the density we will encounter pi maisons and if you go ahead and go to a density of the order of 10 to the power 15 grams per cubic centimeter you will encounter keep my songs but as the star cools below the critical temperature for bose-einstein condensation the pressure will drop off and the star will collapse and it'll become a black hole regardless of whether it is below the maximum mass when you Tron star or above the maximum height or you close it there for now the question is do we think this is happening the last one has obviously not been set on any of these things but if you ask me if I were to bet today what will I bet I will say that there are no such things as pile on and carry on condensation a neutron star how do you know well I have a good reason for saying that when PI meson son came has also produced lot of neutrinos of produce the neutrino luminosity of the star will increase normal and therefore the cooling of the neutron star will be dramatic now we will see on Friday's lecture that whenever you look at the site of a supernova explosion we find the neutron star there yesterday order and Ricky are conjectured in 1934 and we see these neutron stars emitting blackbody x-rays and I know therefore the temperature of this body must be at least ten million degrees on the surface and higher inside now if these guys are going to cool very fast how come I go and see a neutron star at the site of an explosion that occurred on that 4th July 1054 ad and it's temperature is still 10 million degrees that tells me you Tron stars do not cool that dramatically if initially they cool when they form from temperature of the order of 10 to the 11 Kelvin to 10 to the 9 or so but they don't cool to temperature below blackbody exterior temperature which is 10 million degrees from Wiens displacement law I know that when the temperature of a blackbody is 10 million degrees the peak of the blackbody curve will come in extremely soft action well therefore although the last word has not been said it is unlikely that such high densities of five or ten times the nuclear density is attained in the center of neutron star but please take this as an open question now let's try to assume that even higher densities obtain in the center what do you expect at a higher density elementary particle physics tells you that when you go to a temperature of the order of 10 to the power 12 Kelvin such was was the temperature when the universe was one second all there is a phase transition that occurs from hedron to quarks in the case the early universe from quark to head-ons so what is it that we are talking about we don't know for a long time now that neutrons and protons are not fundamental particles they are made up of quarks which are bound together by gluons so inside the nucleus there are quarks and gluons now the fundamental thing about quarks is that if I separate the quarks the force between them increases are not decreases therefore conversely if I squeeze quarks together to very short distances away high energies they become free this is what elementary particle physicists call as asymptotic freedom you go to high energy you're free it is exactly the opposite of what we will normally think right therefore the question that underlies this is if I squeeze the nucleons together to so much will the quarks Deacon fine in other words will be now suddenly have a soup of quarks and blue but now I'm talking right at the center now it may happen may not happen and one reason for thinking it may not happen is that again you will encounter very high neutrino luminosity resulting in very rapid cooling of neutron star now I am talking about lots of you chaos which are met years old still radiating blackbody x-rays okay now people have also speaking very deep physicists that if quarks do happen at the center suddenly there will be a transition such that the entire star is now made up of quarks you have called strange stars and I never understood these arguments and I don't think the arguments are compelling although the people who say these things are they they are the high priests of the game but I have a serious problem with these models and one of them is the following according to these models of strange stars as they're called quark stars they won't have any crust at all maybe a very thin crust but this contradicts the fact that we have found 2,500 neutron stars functioning as pulsars as I will talk about on Friday now what are the first thing that pulsar tells us is that pulses are absolutely stable that means that the magnetic field that the neutron star is anchored solidly to the crust over millions of years if a neutron star doesn't have a crust then hydro magnetic instabilities will completely prevent you from having if fixed field configuration for millions of years and therefore this very simple experiment to fact existence the Pulsar all Jews again star which will not have a thick crust okay but let us leave it open for the moment we just don't know I just react like this when somebody tells me no no it will happen it may happen may not happen let's leave it as an open question I'm appeal always to experiments to resolve the issues okay now as I said last time and I won't have time to go into that now let me just mention it two of the fastest spinning neutron stars that we have discovered have both a period of one point five milliseconds weather magnetic fields are very different so how come two are spinning at one point five milliseconds so John Friedman and his colleagues conjecture about 30 years ago that this may be happening because you are not able to spin up a star more rapidly than a period of one point four milliseconds what is it that's going to prevent you from spinning it up gravitational radiation right now you agree that if a star is smaller in size for a given mass I can spin it faster before centrifugal force becomes dominant so large of the star larger is a limiting period to which I can spin it up so treatments point stated in simple language is that the fact that the limiting period of the neutron star is half a millisecond and you have a breach half a millisecond you have found 2,500 pulsars put none spinning faster than one point five milliseconds suggests that the equation of state of a neutron star may be stiff such that the radius never becomes smaller than about ten or eleven kilometers or a mass of one point four Solomon so there are indirect inferences to be taught concerning whether the equation of state in the interior is stiff or soft so if the equation of state is not soft then measure on States and quarks states are unhooker okay so let's leave it at that at the moment the last word has not been said and the last word will be said only by observation all right so here is a summary of what I have said up to now so let us tentatively admit that in principle there is a possibility of having a quark on plasma or soup right at the center of a neutron star now I change subject something even more exotic but this time very real and that is the super fluidity of neutron stars what is super fluid well if you remember would have studied it in BSC or MSC that cover Ling honors in Laden in Holland liquefied helium in 1911 Phenom estate and then when he liquid helium was cooled to a temperature of two point one nine Kelvin it became a superfluid in the sense that its viscosity became zero not small but zero similarly gaveling honours cool mercury and at a temperature of about eight degrees Kelvin it became a superconductor its electrical resistance became identically zero so this is the phenomena of super fluidity and super conductivity and both have some very intimate relation to this phenomena of both Einstein condensation now let us first discuss superconductivity as I said super conductivity of mercury was discovered by kamerlingh honest in 1911 itself the explanation of it was a great puzzle under tension of of the great physicist Heisenberg Pauli lambda Fineman so many other people and it was finally solved in 1957 by John Bardeen with leon cooper who's his postdoc and a student bob schrieffer now finally when they solve the problem this is what it was we normally think of interaction initial energy between two electrons being repulsive due to Coulomb interaction but under low enough temperatures this interaction can be in that positive all right sorry net negative or a collective interaction in otherwise two electrons can form a molecule these are called Cooper pairs and these molecules now obey both Einstein statistics because there are two electrons and then there can be super phenomena such as superconductivity let's try to understand this little better here I scanned an old transparency of mine because it's too lazy to draw this picture all over again so I wanted to look at that picture on the right it's an instantaneous picture of a crystal lattice the circles are atomic nuclei and you'll notice that at one particular point of the crystal lattice the atomic nuclei or closer to one another than in the average why is this possibility probable because please remember there is a crystal lattice the atoms are not in a fixed position they are vibrating now vibrating about an equilibrium position but the phase between the vibrations of different nuclei is random you can for you analyze them in terms of lattice vibrations which are called photons but instantaneously I could certainly have a situation where for photons a four nuclei are closer together than what I will have in that region of the lattice is therefore a net positive charge an electron is there already happened to be there in the region then another life passing by electron she is a region where there is a net positive charge and that electron - is attracted to the region so thereby two electrons visit that region for a very different reason and having visited there they decide okay will form a net attractive molecule stated in most scientific language it is shown on the left please remember such a static distortion is not static it is time varying therefore when you Fourier analyze it you get a spectrum of photons which are quantized sound waves and sure you have studied these things and the picture is two electrons mediated by lattice vibrations feel a net attractive force this is an ever so small attractive force the net repulsion between the electrons is of the order of electron volts whereas the net attraction of the electrons due to foreign interaction is of the order of a million Ektron walk a thousand times smaller now are they due to electrons that sort of form bound molecules role as I shown in the bottom of the slide what happens is please remember that our electrons and metals or degenerate they obey Fermi Dirac statistics all moment up to the fermi momentum or occupy and all moment routes unoccupied so what bardeen cooper and super steady state two electrons with electrically opposite on the Fermi sphere they are equal and opposite momenta their spins are oriented in opposite direction and they form a molecule there or the Cooper pair as it is called need not be small and physical size at super fluidity and super conductivity or quantum phenomenon well the coherence of the wave function can be macroscopic in science so the Cooper pair maybe is one electron going over there one electron going over here but in Fermi sea there is some coherence to that properties okay so it is these pairs which if you like sort of both condense - to put it in a very over simplified fashion at the phenomena of superconductivity arises I am stressing this because this is perhaps the most successful of all many-body theories till today this idea explained every conceivable experimental facts about superconductor the discovery was not in 1957 just as the discovery was about to be made I know this from the horse's mouth john bardeen got a phone call from stockholm saying that he had been awarded the nobel prize for the discovery of the transistor he was very irritated because he didn't want to go because he knew that they were very close to solving this problem incidentally he got the Nobel Prize for this also he's late now it is important to appreciate that the existence of the Fermi sea guarantees the formation of these bound Cooper pairs no matter how weak the interaction is so only the net interaction is attractive these bound pairs will form why is superconductivity such a low-temperature phenomena because this attractive energy is only of the order of Mille electron world one electron volt is 10 to the power 4 Kelvin therefore a milli electron volt is 10 Kelvin that's why superconductivity occurs only at liquid helium temperatures but superconductor please remember is not just a met with zero resistance it is a completely different thermodynamic state there is a difference between resistance going and something which is a different thermodynamic state on the Left I've shown an energy level diagram of a normal metal under superconductor and they've been in an ingot so there is a phase transition at a critical temperature from a normal metal to a superconductor that energy gap in a semiconductor between valence and conduction band is of the order of an electron volt here it is an order of a milli electron volt right fine now what do I do to you have to go there and click something thank you so please appreciate that a superconductor is a diamagnet whereas a normal metal is a parameter R&I that means the spins are oriented at random but if I apply an external magnetic field the spins will align and there will be a net magnetization diamagnet means the susceptibility is negative it will repel a magnetic field there it is an experimental fact of a magnet at the bottom floating above a superconductor inside a beaker of supercooled liquid helium so we look at the picture on the top if I have a metal through which a magnetic field threads and if I cool that metal to the superconducting temperature once it become a superconductor magnetic flux is expelled so this is a fundamental property of superconductors all right what is so special about superconductors and also superfluid is that it is a quantum phenomena it is a microscopically quantum phenomena in other words there is a long-range order let's consider something we are all familiar with in the second line let us consider a perfectly periodic crystal lattice as drawn in solid-state physics books with a lattice constant smaller then as you know as you destory that the wave function of the electron shy of K R is some amplitude you K of R multiplied by e to the power i ka now this amplitude has a fundamental symmetry properties namely the answer to the position R plus a is the same as the amplitude at R or n times a that is a statement of perfect periodicity of the lattice so there is long-range order there is coherence of this wave function electrical resistance arises in metal because this coherence is destroyed by scattering of impurities and various other thing in a similar way the wave function of a superfluid or a superconductor is phase coherent in other words the wave function is written in the yellow box in the form sy of R is an amplitude sy e to the power I theta bar at theta is the phase of the wave function then if I know the phase at any point R then I can uniquely determine the phase at any other point R so there is long-range order so let us expand theta of R as a Taylor series expansion you will find that theta R is Theta of our prime then you work it out then the super fluid velocity times then you'll find the Taylor series expansion you'll find a gradient of theta on it F of R is f of zero derivative of s KX minus zero and so on right so I only returned the first two terms of the Taylor series expansion so the important thing to appreciate is that the super fluid velocity is the gradient of the phase of the wave function the super fluid velocity is the gradient of the face of the wave function remember that so that's what all right now I know this from John Bardeen himself that there was a meeting in Washington state of Washington in America where the body in Cooper sleeper theory was presented for the first time and their percent of the meeting was a great question theoretical physicist Boogaloo ball those days remember there was a much traffic between Russia and the United States but Boca loophole for some reason was allowed to travel so he took back with him a copy of this paper by bardeen cooper and schrieffer and when he went to moscow gave it to the lambda gang there's a bunch of young kids working with landau in what is today called the land of institute and they immediately applied this theory to neutron stars a B McDowell one of London students immediately applied it to the protons in the interior of neutron stars and showed that the protons will become superconductors five years later Ginsburg and kids Nate's argued that the neutrons as well as the neutrons in the interior will be superfluous by applying body in Cooper sharifa theory and please remember that both these predictions were made wrong before neutron stars were discovered the Cristal 1968 when Jocelyn Bell's paper appeared in nature neutron stars were discussed seriously only as one chapter in London lifted statistical physics condensed bodies so to the rest of the world they are all forgotten about each other shaker Oppenheim Iran Volkov nobody took it seriously because there was no reason to take them seriously in terms of finding celestial body but lambdas group function very differently they treated it as a piece of theoretical physics and neutron stars exist business also happen now at this point you you will see but by doing in 2003 Ginsburg was ordered the Nobel Prize for his contribution to the super conductivity of metals now now you say about digits and super conductivity in metals arises at very low temperatures now you're talking of temperatures of 10 to the 8th Kelvin no problem because to have superfluid state there has to be an attractive interaction between the fermions so to form Cooper pairs now the strength of the attraction and the density of states at the Fermi level n of years density of states is a number of allowed energy levels in the vicinity of the Fermi energy that determines the transition temperature so if you listen at equation of the yellow box the transition temperature to super conductivity TC is the Fermi temperature TF multiplied by exponential to the - 1 divided by the density of states at the Fermi energy multiplied by the attractive potential energy V so as V increases you notice the transition temperature increases and as a density of states of the Fermi energy increases the transition temperature will increase now please remember that I don't have to do anything fancy by invoking electron phonon interactions and all that to create a net attractive interaction because the net like there the net force between 2 protons and 2 neutrons is naturally attractive due to the nuclear force and it is very strongly at like 2 therefore the transition temperature T C if you are now put in the strength of the nuclear potential attractive potential remember I drew the picture put in the strength of the attractive nuclear potentials here the minimum and the density of states the Fermi energy appropriate to a density of 10 to the power 14 grams per cubic centimeter you will get a transition temperature anywhere between 10 to the 9 Kelvin - 10 to the power 10 Kelvin compared to that the temperature of 10 to the 7 degrees or 10 to the 8 degrees is ultra low temperature therefore neutron star will become super fluid and superconducting if the body and Cooper theory is correct and we know it is correct so these are the arguments of McDonald and Ginsburg and Gibney's for why protons have a superconducting neutrons will be superfluid what the electrons will they become superconducting no they will not become because of their low as the same old problem the temperature is too high for electrons and not too high for the protons because there are 2000 times heavier remember on our expression for the Fermi energy and everything the mass of the particle comes in the denominator all right so so what is it that we are set up to now go back to the central portion of the neutron star which is the bulk of the neutron star with all a definition of there is only skimmed over there this is not electron super conductor but proton super conductor now let us turn our attention to the neutrons the neutrons of the lower crust which has spilled out of the nucleus and the neutrons in the fluid coat they will become superfluid like liquid helium becomes a superfluid at 2.19 kill now I would like to convince you that both the proton superconductor and neutral superfluid in a neutron star will be rather exotic let us first this is the neutrons now please remember that our bucket is rotating so let us discuss the problem of a superfluid in a rotating bucket here on this table now the first thing to appreciate is a super flow is a potential flow in other word a super flow cannot support circulation why do not support circulation because the curl of the velocity is 0 circulation is determined the curl of the vector field why is the curl of the velocity 0 I already explained to you that the wavefunction is of the form modular society to the power I theta R and the velocity of the superfluid is the gradient of the phase of the wave function and the curl of a gradient of a vector is zero therefore a scoop of fluid is not allowed to rotate so I have a bucket of water and if I rotate the bucket slowly the water will start rotating because gossipy which connects the water with the walls of the bucket but in a superfluid the viscosity is zero and circulation is not allowed but you know gut feeling that's of the thermodynamic point of view the free energy will be lowered if the superfluid rotates so this is a major problem in physics for a long time and the problem was solved one day by Lars Onsager in 1949 and then it was independently solved by Fineman some years later in 1956 said Lars Onsager he wrote only fourteen papers in his life if I'm not mistaken there's at least three of them you should have got nobel prize we got one Nobel Prize but he should have actually got three to five places you cannot battery out of 14 and so his papers are very seldom read and so it went unnoticed till find one find one try very hard to solve the super conductivity problem you can succeed but he came upon this deep inside if you read fragments biography written by Jack this mirror you'll see that Fineman considers this as one of his greatest discoveries something intellectually terribly challenging even some real assignment so let us understand how Lars Onsager and then they say Fineman solve this problem of super fluid in a rotating bucket which refuses to rotate okay what they said was this is very irritating if I made this slide yesterday all the symbols that supposed to be a dot product that's supposed to be why does this happen is if the version migration of Windows is different from this so does it happen if you don't use Windows I know it's not enough my friends you have Windows that I am so done that I can only use this thing anyway that meant to be a dot product solve this problem what answer and Fineman said was the following what a superfluid will do is to create a needle inside that needle it will destroy a fluidity so inside that needle you have normal fluid and a normal fluid is allowed to rotate but these are tiny ladles of microscopic sites therefore when the fluid rotates just as the angular momentum of an electron in an atom has to be in multiples of Planck's constant the circulation of this fluid must also be quantized the quantum of circulations is splines constant divided by twice the mass Y twice the mass because these two particles combine to form a boson so that is where the two comes from right so that is a unit of circulation so what we said was let us consider a bucket whose radius is R rotating with an angular velocity Omega the circulation which is the cyclic integral of the velocity dot DL DL is the length that action is two PI R multiplied by V which is omega R for rigid body rotation and what I'm sorry Ron Fineman said was that this circulation which is the left hand side must be equal to the quantum of circulation which is H divided by 2 m multiplied by the number of these vertices which are now called an Sagar Fineman quantized vortices so these are what 2 X cubes which are quantum vortex tubes each having only one quantum of circulation all right now if I now apply this to a neutron star whose radius is 10 kilometers so the bucket is 10 kilometers in radius spinning with a period of one second there will be 10 to the power 51 of these quantized vertices so the center of a neutron star is not just a superfluid of neutrons but that soup that by 10 to the power 51 quantized won't excuse what does this by you well ok fine this was also a long time ago then in 1971 a paper appeared in Physical Review Letters experiment was done in Berkeley in California the rotating beaker of superfluid helium as you rotate the vortex appears as you rotate more and faster and faster two three four and then you see they arranged themselves in a beautiful periodic lattice the number of what is this is proportional to Omega the number of vortices is proportional to the angular velocity of rotation you see that very clearly and so when Fineman read this paper he sent a postcard to that author and the postcard simply said this visions in the head that night 20 years ago coming out in black and white physical reality so the way Porcher very difficult experiment to do but here is a spectacular picture a recent picture of quantized voltages in a droplet of bose-einstein condensate they cool these atoms using lasers you heard of laser cooling of atoms cool them to nano Kelvin temperatures and then they become a bose-einstein condensate and you spin them using cymatics field and there you see quantized vortices appearing and increasing in number as they spin faster and faster and you see how they arrange themselves into a beautiful periodic lattice because it will lower the energy here is another picture we always been told rotation is analogous to magnetic field and magnetic field is analogous to rotation like current goes around in the loop produces magnetic field before you expect something like this to happen unless superconductor also ok first the superfluid so here is my bucket the superfluid and it fuses itself with just the right number of vortices corresponding to the angular velocity and what is brought below is the velocity V the vertical axis as a function of the radial distance of the center if the bucket rotated as a rigid body then V is omega R which is a straight line that is a rigid body rotation of the bucket but what you find if you really look very closely is that the velocity field diverges goes to infinity right at the center of these vortices but from a distance of a microscopic point of view it will appear to you as though the superfluid is rotating as a rigid body with the right circulation so right in that paper of 1964 Ginsburg and Krishna's mentioned this so that not only will the super neutrons will be in a superfluid state but it will be in a wort extinct now let's turn okay now let us ask a question it is all of academic interest and something so long because if super fluidity occurs then it has several very important Astrophysical consequences for example as we will see on Friday neutron stars emit radiation that radiation cap is away not only energy but angular momentum and therefore the neutron stars spin slow down so I have a bucket of superfluid here which is rotating I put my hand on it and I slow down the bucket but the superfluid won't slow down but sooner or later it has to slow down and there is a time lag and that time what causes starquakes the neutron star these talk rates are pretty spectacular there are 56 or 58 on the Richter scale remember the most powerful earthquake on earth is about eight on the Richter scale and it's a logarithmic scale so from 8 to 58 so you can imagine the energy released in small quake of the neutron star now these now as the neutron star slows down a rotation the number of vortices has to decrease proportionately due to Omega therefore these vortices will have to migrate and annihilate themselves on the wall of the bucket in suitable numbers so you can drag something with it so some years oh I came up with a theory of how the migration of these vortices outward can actually result in expulsion of the magnetic field from entron star okay now let's go to that question will become more men when we turn our attention to proton super conductivity now while the reason for the onset of proton super conductivity may be compelling the star is spaced is it embarrassing dilemma now imagine a neutron star which is born it is cooling down at some time at some time it reaches the temperature where it would like to become a superconductor because it is energetically a lower energy state but superconductors a diamagnet you just expel the magnet in this time we are not talking to a few girls you are talking of 10 to the power 12 girls how is the magnetics produce all that feel so produce by current loops so the only way you can expel the magnetic field is by expelling all the electrons and then you have come to your problem I to cost me normal electrostatic energy to expose electrons so what does the metal do it wants superconducting but it stuck with a magnetic field and this sounds crazy till it becomes a diamagnet it cannot become a superconductor well not all superconductors in the laboratory or perfect diamagnets get most metals pure metals when they become superconductors they are diamagnet but alloys also become superconductor now onward everything is a superconductors question of how low temperature you cool it now it was long time ago that alloys do not behave like this allows behavior shown in this picture what is plotted on the y-axis is the strength of the magnetic field and what is plotted on the x-axis is the temperature the transition temperature T C so you see that there are two curves if you look below it the lower critical magnetic field bc1 you find that the superconductor is a diamagnet it has expelled a magnetic field and at very high temperature or very high magnetic field superconductivity is destroyed by the magnetic field and the magnetic field stretch the metal merrily that is this picture but an intermediate region the superconductor allows magnetic field to thread it but has quantized flux tubes now this was first pointed out by another student of lambda by name abrikosov and in 1958 he constructed a successful theory of what is called as type 2 superconductors now you had a problem Landau didn't believe in this so he refused to allow him to publish this paper Landau made many mistakes like this like Einstein in many cases they were so pretty later about some issue but in the case of Landau he would actually prevent people from publishing the paper em you know Lando had this terrible car accident and then doctors from all over the world went there and operated on him they saved him he recovered but intellectually in ever recovered after that only a precursor had the courage to send the paper for publication and then he got the Nobel Prize in May 2003 for his theory of type 2 superconductors which actually goes back to 1958 now what I because of said was that a type true superconductor will allow the magnetic field to go through it but as quantized flux tubes what is the quantum of flux dimensionally it is HC by 2 e by 2 e because two electrons are involved and becoming a super content like two M's were involved in the quantum of circulation so if you look inside of what Lex tube like this the magnetic flux is a maximum at the center there is a decay length and this is a superfluid a number of superfluid superfluid density n goes to zero in the center in other words inside the tube super conductivity is destroyed thereby enabling a quantum of flux to go through exactly as in the large on solver Fineman vortech super fluidity was destroyed so that that little thing could rotate here is an experiment of such quantized flux tubes and type 2 superconductors and what you see at the bottom is as I increase the applied magnetic field 1 Tesla to spacely's 3 Tesla 5 Tesla the number of flux cubes increases as it should because the flux is the number of field lines per unit area but each plus 2 is a quantum of flux so it must increase and they also form beautiful periodic lattices and these are called a precursor of lattices and today it's a very hot subject these lattice can undergo phase transition they can become liquid glass crystal and it's a very hot subject in physics today now let's go back to our neutron star and superconductors here is our neutron star with an external dipole magnetic field but the dipole magnetic field previously was due to electron going around a current group but now and it became a superconductor that magnetism has been transferred to the proton fluid and what you have is quantized flux tubes how many flux tubes the number of flux tubes will be determined by PI R squared B which is the number of flux lines causing the equatorial plane divided by the quantum of fixed PI naught which is HC divided by 2 e so in a neutron star magnetic field is 10 to the power 12 girls there will be about 10 to the power 31 every coat of quantized flux tubes so again there are profound Astrophysical consequences of this is it occur one of the things that we know about neutron stars with over millions and billions of years their magnetic field decayed by 4 phi orders of magnitude this has been a major problem to understand because the interior was a normal fluid then the magnetic field decay timescale will be determined by ohmic dissipation like in normal metals feels this decay because ohmic current results the decay of current loops now the only timescale if you look in Griffiths book or Jackson's book you will find it's 10 to the power 13 years because all weak decay time scale depends upon the thickness of the wire and the size of the loop and the electrical conductivity of the wire these are the three things that determines decay constant now if you applied to starve the dimension of 10 kilometers you will get 10 to the power 13 years in other word magnetic field cannot behave within the age of the universe now if the interior was superconducting it even works because as Bardeen pointed or long time ago the flux expulsion timescale from the superconductor is 10 to the power 23 years so whether the interior is normal or superconducting the magnetic field cannot decay spontaneously so so some years as I said I invoke this fortuitous circumstance this is the only example we know in nature of the core existence of super fluidity and super conductivity we know on a superfluid you know superconductors you know no example so there are inside there are two brands of spaghetti there are the Fineman vortex spaghetti then there are the abrikosov voltic spaghetti one is similar to the rotation axis now there is parallel to the magnetic axis now it turns out that these two spaghetti stick to one another with it almost binding energies and therefore ask the neutron star slows down and the Fineman vortices migrate outwards they will drag the other brand of spaghetti with them so this is one way of this is the only way anybody has been able to think of how the magnetic field of a neutron star can begin it decays not despite super conductivity but because connected if you super get the magnetic field were not in this quantized spaghetti drag them up by attaching them so all I am trying to say is that these exotic states like give rise to Astrophysical consequences like stark weight flux expulsion for the interior plate tectonics or neutron stars and all these things may become terribly important when in the context of gravitational waves you deform this gravitational wave you have to deform the surface right so there it may matter whether the surface is like a nice smooth eggshell or like a cracked hard-boiled egg it's lots of plates so here is a final summary the outer layer outer crust is iron-56 mattes then as you come down further you find neutron-rich exotic nuclei arranged in a lattice but lot of neutrons are spilled out of drift out of the nucleus and there is a sea of neutrons which will be super fluid and which will have fine one unsolder what is is in them parallel to the rotation axis now if you go to the center you will find predominantly neutrons which will be in a superfluid state and some 5% or so protons which will be in a superconducting state and these will also be in vortex state so in the center you will have the two brands and spaghetti namely the final notices and the every cause of suck so much so that's a good point to end our discussion of the interior of a neutron star are there any questions yes I think Darren give us a like microphone is here can you just close it down yeah so as fast as here so you can increase them keep it is useless cause it's Cutler and going on as well yes so for this super fluid in order dating myself what is the surface of the superfluid look like I mean does it get deformed like will it have a Fisker you think about it like a like water and has is paraboloid that's what I'm saying you think about it this is a very interesting car sitting on whether it's 11 that's called the meniscus which he recorded a very interesting answer to it yes you'll have fun thinking about it let's like yeah yes go back to wild why what will have a meniscus and then you'll find on it when my when I started teaching long time ago my teacher said ancient Chinese say you give a man a fish you will have a good meal but if you teach him to fish he will eat well all his life ok so I try to live with that bottle yes and yes your agrees you said that after the iron-56 was in our NBCC right eg lattice is remember these are all things I'm out of touch for longer than your before your bond so it could be BCC it could be FCC go and shake which is the correct one but the point is people have done the hard calculations and come to the conclusion which is the lowest energy there is ferromagnetic anti ferromagnetic BCC or as you see but continue the equation I could be wrong see I don't have access to the library and so on so all these things are straight for my memory some of these things could be wrong they're not wrong intentionally access memory fills but it's a calculable things in it it's a question of calculating which is the lowest energy by the way that is why these various crystals such as a curve or various elements was given the number of electrons different crystal structures are the lowest energy configuration right okay you have been beyond but like they're assuming their spheres right the atoms are spheres all right if they but is it may not be smart on the not Restless fear so what that will do is the tensile strength of a lattice will depend upon whether they are like the atoms are like ping-pong balls are like needles in other words the forces are scalar or vector or tentorium okay so it will numerically affect all these things and people have tried to put in all these things into the calculation spaghetti like atoms needle-like atoms here the surface yes it's very important and finally they will determine and you calculate the total energy these things alone oh do you you I'm sure they appreciate that the nature of the forces per unit volume will depend right there needles as as a pasta spherical body in one case you will have tensorial person pencil means water if I you know elasticity is a tension right if I pull something in on direct information doesn't have to be necessarily in that direction I have a mocha shake-a please how would you know about how do you find out the age of us we will you ask me this on Friday when we discuss till the the cheap answer to that is in some cases I know the day on which it was born if you believe somebody said long time ago that supernovae occur because of the formation of neutron star if that is true then if you go and look at the remains of the supernova explosion is a fine neutron star and you do and because some of these supernovae were recorded by the Chinese and so on you know when they were born but even otherwise there are there is a approximate way in which deducing but the basic principle is this is a clock which is ticking very regularly the clock is slowing down so people wouldn't appreciate this because you all products of the digital era before views of clocks with you you had to wind every day grandfather clocks in the house which you found once a week and it will slowly it will slow down but on the sly strength of the slowing down you can deduce when it was last one okay so basically neutron stars are not only spinning but the rate of spinning is decreasing because they are losing energy in angular momentum so the ratio of birth divided by the period derivative of D so if I divide the period of a clock by rate at which the period lengthening then I get a characteristic is not an exact it there so we can do become observational II deduce ages of neutron stars which are for all time you may be wrong by small percentage but it's a very good measure I said you said it's permanently more favorable for the Watchers to formation why is it more favorable energy favorable for the quantum what it is to form when you rotate it why is it should dissipate in it when you rotate later so this is it's a normal fluid and when normal fill rotates to dissipate energy right and how it moves available for the verismo is now they were wrong to destroy superfluidity i must expend energy summer superfluidity is the lower energy state so it costs me it cost me so that that was a great discovery by on sagar and fine one so to do that correctly you cannot use macroscopic vortices you have to use this quantized focus if you read in fine one go to fine one volume three you will discuss superfluid go and have the pleasure of leading some fine men himself so that was the cleverness of it that's why they had to be quantized instead of you can you can otherwise regions where super fluidity is destroyed and create things right that doesn't give you the lowest energy configuration that that will also not give you the simulation of macroscopic rotation from one side okay so that's where the thing it's like that is also a difference between on solid solution and fineman's it's a very good and important question but yeah yes going there in person so you said that at very high densities the P on and K on condensates form like which leads to the equation of state being solved so if the matter becomes sort of compressive so won't it affect the stability of the star of course is the system will collapse look clear of the neutron star you're almost there is a black hole as I said a one solar mass neutron star is a ten kilometer radius the black hole radius is three kilometers so you are dangerously close to instability and then you put in general relativity and all into it it will just collapse to a black hole so soon as sure yes so people invoke this for example this supernova in which the neutrino burst was detected and nobody is still seen a neutron star there now we don't know how long before a neutron star turns on as a visible object okay we only know this is the supernova of 1054 of a thousand years ago yes only thirty years ago exactly thirty years ago right the neutron star has not turned on it some people except maybe because it's a black hole maybe because of ion condensation and there was a so this is the sort of reason why people are thinking about that secondly in a supernova explosion we assume that the rest of the star is blown apart but suppose some of the rest of the star falls back on it then you will increase the maybe under such circumstances the density of the center may go to these densities and they'll form firearms and crayons but in the end collapse okay so reason why people started thinking about that is the possibility that some supernova explosions will actually produce a black hole and not a neutron star I - I didn't get how exactly the flux explosion so let's just never expose my superconductors I chose a neutron star so it literally well the idea is that as a neutron star slows down because if everything are there is an external break put on it the work Lister's have to go out and disappear in proportionate number of it and because they are strongly into a pin is like a tennis racket these strings are very strongly coupled with these things so if I try to move one set of string now the set things will also be dragged with it so the flux can be taken from the interior to the this is it is a very important question connected with technology also you know why are people trying to build I would try to make high-temperature superconductors so then you can run trains and things like that levitation and so on right so one of the problems is and then you can have magnets it's produced very strong magnetic fields with a coil being bound with superconductors so there is no energy dissipation that so one of the problems in all this technologically is this problem of pinning of these focuses first of all when you send a current through a wire it generates magnetic field and that produces Otis's whether you like it or not okay now one fundamental property of what this is is that they pin to inhomogeneities if you take a vortex suppose what I'm saying is suppose I have a beaker of suppose let's take superfluid and it's spinning and let's consider one particular vortex that water is wants to move to the ball of the beaker and annealing but the bottom is attached to something because there is some impurity and so on or in a solid there can be dislocations so experimentally the pinning of vortices to inhomogeneities impurities dislocations is very common now your question is why do these two bands of spaghetti spin to each other that is because if the spaghetti regardless there are more exotic reasons for it for example I I said I did say it but maybe I said now I said in the case of a superconductor like tin now mercury you taught me of two electrons on the surface of the Fermi sea diametrically opposite equal and opposite momentum but is opposite spin so the two neutrons are in one s0 state 1 0 state there is no net angular momentum what you see but the neutrons the core the soup a neutron star or at much higher density at much higher density the neutrons are forming feeling a repulsive force due to nuclear interactions so how do these form bound pairs if unfair like when helium-3 becomes a superfluid came a superfluid a century ago in 1973 liquid helium 3 also became a superfluid at a temperature of one milli Kelvin and that's because helium 3 has an extreme and this helium 3 is a net spin and there is a strong repulsive energy between these two helium 3 therefore they also have a difficulty pairing and what helium 3 does is that these two helium threes go around in a sort of an orbital motion so that the net angular momentum is not zero so they take 3p to state they form an angular momentum equal to constant P state and not an estate so our traditional body Cooper S state superconductor net before that people are pointed out in the case of a neutron saw before helium 3 became a superfluid that is the interior of a neutron star the neutrons became a superfluid the only way this repulsive force can be overcome is if there is an orbital angular momentum generated between these two therefore these two neutrons are ferromagnetic aliy aligned and therefore the core of the neutron vortex is magnetic there's a magnetic field of 15 young extraordinary what is still editing or neutrons neutrons are neutral particles so this going around cannot produce a magnetic field but it does produce a magnetic field because of its spin alignment okay so there are many interesting reasons why these two spaghetti stick to each other very strongly and please remember they don't just stick in one place like in a tennis racket each string there are okay remember that's ten to the 31 and 10 to the 15 tender that is one attend to the 51 so the number of interpreting is enormous it's just that I wrote that when I wrote the paper I remember receiving a postcard from McDowell who was in Princeton at that time and Ginsburg very simple I treasured them saying how come we didn't think of it so if this is a nice feeling to know it so they already they endorse this idea that it can happen any other questions

Channel: International Centre for Theoretical Sciences

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Length: 114min 24sec (6864 seconds)

Published: Thu Jun 22 2017