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Notes on Superconducting Intermediate Sates, and Shubnikov De Haas Oscillations

This is just a series of rather scattered notes on things that I need to keep in mind for the h-ray experiment as well as things that are going on in class this week and how they're not that disconnected.

Shubnikov, who I've mentioned before[1], (picture 1), in reference to the intermediate state of superconductors, came up in quantum mechanics class this week.  The topic of discussion was Shubnikov-DeHaas oscillations.  These are oscillations of the resistance of a material with respect to the strength of the magnetic filed it is exposed to.  It occurred to me the the graphs of the oscillating resistance[2], (picture 2 below), looked a bit like magnetron operation because at low magnetic fields nothing much happened due to the field being too low to bend the electrons into a complete orbit.


A little more searching and reading revealed I wasn't necessarily the only person who ever thought so.  I came across an article about the intermediate state of superconducting tin by Schawlow and Devlin that mentioned the Corbino disc of magnetoresistance experiments in comparison to magnetron operation[4].  The paper was interesting for another reason though.  The researches were using visual means to quantify the extent to which flux was trapped in tin, (a type II superconductor), in it's intermediate state.  I had heard that you could see flux vortices in type II superconductors using magnetic or diamagnetic powders in a manner similar to looking at the field lines from a magnet, and in this paper that's exactly what they did.  The picture below, taken from the article shows a tin crystal that has been sprinkled with niobium powder.


Last year while working on research involving levitating superconductors with AC magnetic fields[8], I came across the Bean model[6] of type II superconductors.  Bean frequently mentioned flux creep, a phenomenon where lines of trapped flux in a superconductor would slowly move.  I hadn't realized that this could actually be observed, but that's exactly what Schawlow and Devlin did.  They comment that when the external magnetic field was removed from their sample in it's intermediate state, they could observe most of the trapped flux lines moving towards the edge of the crystal over the course of thirty seconds.

Mendelssohn[7] comes up in this article as well.  He had commented to the authors that If they reduced the temperature of a sample therefore increasing the critical filed, (see the graph below), that they should see the trapped flux vortices shrink in surface area.  Basically, the same amount of flux is trapped, but since the critical field is increased, the flux will be trapped in a proportionately smaller area.  That is in fact what the researchers wound up seeing.  Picture 4 below shows the flux spots before further cooling, and picture 5 shows the spots afterwards.




One last note.  It turns out that the critical magnetic field Hc for quenching is temperature dependent as well.  The temperature dependence was measured by Daunt and Mendelssohn[5] and is included below, (picture 6).



Refeences:
1.  On Shubnikov on the inetermediate state of superconductors here
http://copaseticflow.blogspot.com/2013/03/intermediate-state-of-superconductors.html

2.  Shubnikov and DeHaas article on the Shubnikov-DeHaas oscillation
http://www.dwc.knaw.nl/DL/publications/PU00015868.pdf

3.  The second Shubnikov-DeHaas article
http://www.dwc.knaw.nl/DL/publications/PU00015868.pdf

4.  Brillouin magnetron paper
http://dx.doi.org/10.1103%2FPhysRev.110.1011
Schawlow A. & Devlin G. (1958). Intermediate State of Superconductors: Influence of Crystal Structure, Physical Review, 110 (5) 1011-1016. DOI:

5.  Daunt and Mendelssohn on critical field vs. temperature
http://dx.doi.org/10.1098%2Frspa.1937.0099
Daunt J.G. & Mendelssohn K. (1937). Equilibrium Curve and Entropy Difference between the Supraconductive and the Normal State in Pb, Hg, Sn, Ta, and Nb, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 160 (900) 127-136. DOI:

6.  Bean model of type II superconductors
http://dx.doi.org/10.1103%2FRevModPhys.36.31
BEAN C.P. (1964). Magnetization of High-Field Superconductors, Reviews of Modern Physics, 36 (1) 31-39. DOI:

7.  Notes on Mendelssohn
http://copaseticflow.blogspot.com/2013/03/intermediate-state-of-superconductors.html

8.  My paper on superconductor levitation
http://dx.doi.org/10.1016%2Fj.physc.2012.11.005
Carter H., Pate S. & Goedecke G. (2013). Dependence of levitation force on frequency of an oscillating magnetic levitation field in a bulk YBCO superconductor, Physica C: Superconductivity, 485 92-94. DOI:

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