Quantisation of magnetic flux in a superconductor
Magnets and electrical currents produce magnetic fields, regions of space where other magnets and electrical wires experience a mechanical force. For a circle of wire in the presence of a magnetic field the magnetic flux is defined as the strength of the magnetic field passing through the circle multiplied by the area of the circle. A law of electromagnetism states that if the field varies with time, then a voltage is produced in the wire with a magnitude that is proportional to the rate at which the magnetic flux through the circle changes. This is the physics behind all electrical motors and electrical generators. In the everyday world magnetic flux can have any value and can be varied continuously by changing the strength of magnetic field. In the quantum world that is not the case. Magnetic flux is quantised.
In 1961, two experimental groups independently reported the first observation of a macroscopic quantum effect, the quantisation of the magnetic flux passing through a superconducting cylinder (Figure 30). One team was Bascom Deaver and William Fairbank and the other Robert Doll and Martin Nabauer. A tall thin cylinder made of tin was placed in a magnetic field and cooled down to a low enough temperature that it entered the superconducting state. The magnetic flux passing through the cylinder was then measured as the magnetic field was varied. The resulting graph has four noteworthy features. First, there are clear steps, showing that the magnetic flux has discrete values. In contrast, in the normal metallic state the graph was a straight line. Secondly, the magnitude of the steps was the same, to within about one per cent, suggesting quantisation of a single unit of magnetic flux. Thirdly, the value of this quantum of magnetic flux was equal to the value of h/2e. Thus, it was completely determined by the two fundamental constants, h and e, Planck’s constant and the charge on an electron, respectively. And fourthly, graphs with the same three features noted above were later observed in other superconducting materials and cylinders. This showed that flux quantisation is independent of details such as the chemical composition and dimensions of the cylinder. This flux quantisation is a macroscopic quantum effect. It is macroscopic because the system is macroscopic, and the magnetic flux is a macroscopic property. It is quantum as and the magnitude of the quantisation is determined by Planck’s constant.
(b)
Figure 30. Quantisation of magnetic flux in a superconducting cylinder. (a) A tall thin cylinder of tin was placed in a magnetic field. (b) The graph shows the value of the magnetic flux passing through the cylinder as the magnetic field was varied. Note the step like structure, showing quantisation of the flux.
The quantum of magnetic flux is denoted Φ0 (= h/2e) and has the value 2.067833848...×10−15 tesla (metre)2. This number also determines the scale of quantum interference effects between two superconductors, as we will see shortly. The flux quantum is also relevant to vortices that form when some superconductors are placed in a magnetic field (Figure 20). A persistent electrical current flows around the vortex and the magnetic field penetrates the core of the vortex. It can also be shown, both theoretically and experimentally, that the magnetic flux associated with each vortex is exactly equal to one quantum of flux. Something similar happens in superfluids.
Macroscopic quantum effects in superfluids
When a cylinder containing a fluid is rotated about an axis passing down the centre of the cylinder the fluid will also rotate. The faster the cylinder is rotated the faster the fluid rotates. A physical quantity known as the circulation is proportional to the speed of rotation and the diameter of the cylinder. With a variable speed motor, the rotation speed can be continuously varied and in normal fluids the circulation has continuous values. But not in a superfluid, as shown in a beautiful experiment done by W.F. Vinen in 1961 using liquid 4He. He observed that when the liquid was cooled below the superfluid transition temperature that the circulation could only take on discrete values. Furthermore, these discrete values are multiples of h/M where h is Planck’s constant and M is the mass of one atom of helium. This value was predicted by Lars Onsager in 1949 who identified h/M with the circulation of a single vortex in the superfluid. This is another macroscopic quantum effect.
The quantisation of magnetic flux in superconductors and of circulation in superfluids showed that both superconductors and superfluids can be classified as quantum states of matter. The close similarity of these quantum phenomena, even though superconductivity occurs in solids and superfluidity in liquids. This indicates a deep underlying unity, demonstrated through the study of condensed matter physics.
This is an extract from Chapter 7, Quantum Matter, in Condensed Matter Physics: A Very Short Introduction.
Hi Ross, I am a PhD student studying superconductivity in Australia and I have commented on a couple of your other posts regarding the superconductivity and just wanted to bring something semi-relevant to this post to your attention I discovered the other day that you may find interesting, however I apologise if you are already familiar with the matter and the comment is long and should be taken with a grain of salt in case I got something wrong I was going to potentially write something more formal up later for the student Physics Society at my University a bit later also I have broken up the comment as it was too long for one comment.
ReplyDeleteHave you heard of Max Schafroth, John Blatt or Stuart Butler? I have been doing my PhD at the University of Sydney where they are from and have never heard of them up until stumbling on them last week. But reading more on them I honestly am in shock, as these guys were some of the world leaders in the theory of superconductivity in the 1950s where prior to BCS theory they had a theory quasichemical equilibrium theory where they alleged superconductivity was explained as Bose-Einstein Condensation (BEC) of electron pairs (they were unaware of Richard Ogg's work initially when doing this). In 1954 they proved an ideal BEC would exhibit the Meissner effect https://doi.org/10.1103%2FPhysRev.96.1149 and in an additional one Schafroth suggested electrons in a metal could form bound pairs that would behave as Bosons. https://journals.aps.org/pr/abstract/10.1103/PhysRev.96.1442 and in 1955 they visited a number of US universities including Bardeen in 1955 and were influential on Cooper. Meissner even wrote a letter to them saying he read the papers. Here is a link to the University of Sydney archive of letters of Schafroth https://archives-search.sydney.edu.au/nodes/view/22472 you could probably also find it yourself with a search engine.
Their theory for superconductivity Quasichemical Equilibrium Theory was submitted to Physical Review in 1956 where the editor Samuel Goudsmit personally wrote to them in a letter in the archives saying the unsigned report considered the statistical method publishable after condensation but called the superconductivity application highly speculative and recommended removing the most speculative sections in effect delaying the publication so they published in Helvetica Physica Acta instead https://www.e-periodica.ch/digbib/view?pid=hpa-001%3A1957%3A30%3A%3A95 . Their PhD student Lord Robert (Bob) May later stated Bardeen was the referee and held the manuscript for a while before it was rejected, however I can not verify that. Goudsmit was the Physical Review editor who also handled the short February 1957 BCS announcement where Bardeen was given special treatment a they thought it was a major breakthrough. I would be interested to know the referees in both cases to be honest. Also Blatt had wrote a letter to Bardeen in 1955 saying Bardeen didn't have a theory for superconductivity either and in some of the letters amongst the Sydney group members there seemed to be some frustration with Bardeen where Frohlich, Gorter and Luttinger were somewhat supportive in their letters and said Bardeen was the cause of opposition in the US however in some later letters from Blatt it appears there was significant respect for Schreiffer and Cooper. Cooper even apparently told Blatt their result was independently developed but he would cite there papers and Blatt even said he was a really nice guy.
However the biggest problem for Sydney researchers theory was that they assumed pairs would be compact molecular objects which didn't seem to for work for rudimentary metals as the coherence length for starters is very large in comparison BCS theory seemed to work very well for explaining the properties in the metals but they were actually cited in the original BCS papers. Schafroth and co did also point out a Gauge invariance objection https://journals.aps.org/pr/abstract/10.1103/PhysRev.111.72 that was not trivial in the original BCS papers and the BCS people knew there was an apparent gauge/particle-number problem. Anderson, Bogoliubov, Nambu, Pines, Schrieffer and others showed that collective response restores gauge invariance an example is given here https://journals.aps.org/pr/abstract/10.1103/PhysRev.110.827 https://link.springer.com/article/10.1007/BF02859833 . Schafroth according to the internal letters didn't seem to be treated the best by the University and was going to take a position in 1959 as the chair of theoretical physics at the University of Geneva in Switzerland where he was from. Unfortunately Schafroth and his wife died in a plane crash with the pilot near Townsville and Ingham in North Queensland whilst on holiday before he could take up the position however their son who wasn't on the plane Mark who was only born in 1955 was not on the plane and from what I have read he was taken back to Schafroth's family in Switzerland (I would be interested to know if he knew of the impact of his fathers work).
ReplyDeleteBlatt with Robert May and Takeo Matsubara (known for Matsubara frequencies) recast the original Schafroth–Butler–Blatt theory in second-quantized many-body language, introduced off-diagonal pair correlations, corrected errors in the 1957 treatment and made the formalism capable of describing more general correlated pairs https://academic.oup.com/ptp/article/20/4/553/1877767 then developed a gauge-invariant calculation of magnetic response https://academic.oup.com/ptp/article/21/5/745/1820335. This was the same Robert May who later became famous for ecology and chaos theory and explained the central result that the Bogoliubov–BCS wavefunction is a special case of the extended quasi-chemical pair-condensation wavefunction, corresponding to perfect condensation of the electron pairs into one collective pair state https://academic.oup.com/ptp/article/23/3/447/1839556 . In a couple of papers they developed the mathematics of condensating composite fermion pairs while accounting for antisymmetry and the Pauli principle, showed that the extended theory could recover the characteristic magnetic response of superconductors and then in the restricted case of one condensed pair state and “simple pairing,” the entropy and variational equations become identical to the BCS/Bogoliubov equations at nonzero temperature https://academic.oup.com/ptp/article/29/4/494/1872312 . In his 1979-1980 work Tony Leggett used the same BCS variational state over the entire range of interaction strengths and instead of assuming the chemical potential remained near the Fermi energy, he solved the gap and particle-number equations self-consistently while varying 1/k_F a_s and showed a weak interaction when μ≃E_F where pairs are very large and overlap—BCS, a strong interaction when 𝜇<0 so tightly bound fermion molecules form and Bose-condense into a BEC where between them for an s-wave continuum gas, the ground state changes smoothly rather than undergoing a separate phase transition https://link.springer.com/chapter/10.1007/BFb0120125. Pretty much showing that BCS and the Sydney picture are not mutually exclusive and they were opposite ends of one continuos family of paired states. David Eagles did perform an explicit BCS to BEC crossover calculation in 1969 though which Leggett was unaware of. Leggett’s treatment became particularly influential because it gave a clean, general zero-temperature formulation and Nozières and Schmitt-Rink extended the crossover treatment to finite temperature in 1985. Leggett and Shoucheng Zhang wrote a nice history of it here https://link.springer.com/chapter/10.1007/978-3-642-21978-8_2 .
ReplyDelete
I should also point out in the archive Berndt Matthias also wrote letters to Schafroth and seemed to take his side and wrote this really nice paper with Anderson in 1964 discuss these two models https://www.jstor.org/stable/1713425 and I really want to stress here that Anderson says "I think it is fair to say that this equivelance, and the importance of the ideas of Blatt, Butler and Schafroth have been overlooked to some extent." and then discusses and compares the contributions of each side. Anderson wrote that Blatt and others had shown that the BCS results could be obtained from the pair-condensation picture and that the importance of Schafroth, Butler and Blatt had been “overlooked to some extent.” But Anderson still gave BCS the main credit because BCS produced the experimentally successful equations first, whereas the Sydney formalism became equivalent only after substantial post-BCS modification and was much harder to use. Also Matthias interestingly points out how the theories could explain the known properties but could not predict which materials would superconduct and adds that magnetic interactions might sometimes produce superconductivity.
I will also state there are a lot of Physicists that are interacted with in the letters to list a few Pauli and Oppenheimer who Schafroth used to work for and he was initially from Switzerland, David Pines who he worked with, Pippard, Dyson, Wentzel, Buckingham and Seitz also all correspond. There is even some remarks between Schafroth and Blatt about Feynman not having much of a clue or along those lines on superconductivity.
ReplyDeleteDue to the death of Schafroth it kind of broke the team, where Blatt up leaving the university due to difficulties with management to go to UNSW and became a foundation professor in mathematics and Butler changed fields into more plasma Physics (I did not say a lot about him but he also did his PhD under Peierls). Now the PhD student Bob May was supposed to go to Geneva with Schafroth but as this no longer happened he changed field and became pivotal in the development of chaos theory and was the leading theoretical ecologist at least of his generation https://science.org.au/our-focus/history-australian-science/conversations-australian-scientists/lord-robert-may-physicist-ecologist . One other speculative other piece of information to finish it all off is that the person who took the Schafroth chair position at Geneva was Josef-Maria Jauch who made an important no go hidden variables proof there, but he also met John Bell there in 1963 and had some very intense discussions on the matter and led to the development in 1964 of Bell's Theorem.
Honestly I am completely bemused by all of this as someone who has studied superconductivity for 6 years now as a PhD student at their university I have never heard of them and I am not sure if many of the other condensed matter staff have either which given the contributions of them is a massive shame and perhaps they should be more celebrated in the Australian Condensed Matter Community.
Thanks for the comments. I agree this story needs to be told more.
ReplyDeleteAt the 50 years celebration of BCS, Jorg Schmalian gave a talk about "failed theories of superconductivity"
https://arxiv.org/abs/1008.0447
He did not mention the Australians.
I was familiar with some of the details, but not at the level that you describe. Having access to the archival letters is helpful and fascinating.
When I did my PhD at Princeton in the 1980s I remember my adviser, Jim Sauls, saying that he thought that Blatt, Butler and Schafroth did not get the credit they deserved.
I agree this work should be celebrated more in Australia. I am hard pressed to think of other cases where work being done in Australia was at the absolute forefront of theoretical physics? The problem with the sociology of science and how scientific history is written is that everyone quickly forgets the "near misses" that may have been foundational for the ultimate success.
Given you are drawing on the archival material you might consider whether it is worth trying to publish a paper in a History of physics journal. However, they may be too much of a diversion from your main PhD work.
First, you should look at books on the history of superconductivity
See also
https://en.wikipedia.org/wiki/Max_Robert_Schafroth
and
https://share.google/aimode/gQqDuWkRVF2ag3Okd
Thanks again for bringing this to everyones attention.
Hi Ross,
DeleteThank you for the extra sources I will endeavor to incorporate them. Regarding publishing an article on it, that's not a bad idea if I have some time after the PhD I could try to do so then, and also potentially ask the University archivists for a bit of help in case they have some more documents, and I would also love to see in person some of these handwritten letters from the legends in the field particularly people like Meissner, Bardeen and Berndt Matthias. I might even be better off writing the article in the AIP magazine as more Australians would be likely to read about the matter. As I also struggle to think of other instances at least in a semi-recent time where there were Australian's who were at the absolute forefront of a major mystery in theoretical Physics that perplexed many of the worlds all time greatest Physicists, and in a way aided with a foundational theory in the field of Condensed Matter Physics just had there contribution forgotten about. The closest maybe would be Rodnet Baxter and his Ising work or John Cowley and Alan Moodie who came up with the multislice formulation in TEMs which is used in all simulation software in modern TEMs but I dont know much about it as when I use the TEMs it is more chuck the sample in and image it and both cases are in highly specialised areas of Physics rather than something you get taught about in the last year of high school. I am rather suprised Schmalian missed them to be honest, but I guess as you say everyone forgets the "near misses" even if it very important to the field, and Australia has not exactly remembered them much either. Especially in this case with Schafroth dying very young and the others changing fields, along with the fact that other than a few like Matthias in the 60's many people were convinced the whole problem of superconductivity was solved and there was nothing more to it.