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# Lecture 2 phenomenology

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### Lecture 2 phenomenology

1. 1. Lecture 2. Phenomenological Theories of Superconductivity
2. 2. Superfluids and their properties • Electrodynamics and the magnetic penetration depth • The London Equations and magnetic effects • Fluxoids
3. 3. WHAT IS SUPERCONDUCTIVITY?? For some materials, the resistivity vanishes at some low temperature: they become superconducting. Superconductivity is the ability of certain materials to conduct electrical current with no resistance. Thus, superconductors can carry large amounts of current with little or no loss of energy. Type I superconductors: pure metals, have low critical field Type II superconductors: primarily of alloys or intermetallic compounds
4. 4. MEISSNER EFFECT B T >Tc T < Tc B When you place a superconductor in a magnetic field, the field is expelled below TC. Magnet Superconductor Currents i appear, to cancel B. i x B on the superconductor produces repulsion.
5. 5. In a normal conductor, consider a particle of mass ‘m*’ and charge ‘q’ in motion:  v m qE dt dv  * Normal relaxation term due to scattering
6. 6. Here ‘v’ is the average velocity = In a superconductor, there is no scattering Now , nq J  J m Enq dt dJ  * 2 * 2 m Eqn dt dJ ss  dt dB Ex   (leave off vector signs, we’ll ultimately solve a 1-dimensional case)
7. 7. dt dB m qn xE m qn dt dJ x sss * 2 * 2   * 2 m Bqn xJ s s or B m qn or xJxBxJxB s ss * 2 0 00 ................................    Now B m qn B BBxBx s * 2 02 2 )(   and
8. 8. 2 1 2 0 *        qn m s  In 1-D, this has solution  x eBxB   )0()( If the dimensions of SC >> λ B=0 in the interior (Meissner Effect) If the dimensions are comparable to λ, get exponentially decreasing flux penetration. where 2 1 * 2 01        m qns  , a magnetic field penetration depth
9. 9. We will not cover the 2-fluid model , but it can be shown* that in the 2-fluid model of a superconductor, * from Gibb’s free energy considerations and 4 1        c s T T n n   2 1 4 )/(1)0()(   TcTT 
10. 10. London Theory In 1935, Fritz and Heintz London postulated 2 equations: I (1) II (2) These are the 2 London Equations Additionally, we’ll write the Maxwell Equations as: (3) (4) (5) (D=ϵE) (6) Ej dt d Bjx s s   )( )(   OB D t B xE t D JxH         
11. 11. Additionally, And, Take Now differentiate (7) with respect to t, and use (2) Take of each side => Whose solution is:          BB B xHx andusex continuity t J EJJJJ SNS     )6(),4(),1(),3( )8.......(.......... )7.......(  )exp()exp( 0. 1 21 tBtA      
12. 12. Where and are roots of: One can show that: One can estimate 1 2 2 1 4 , 0 1 2 2 21 2 e                  112 2 119 1 sec10~ sec10~     21,
13. 13. Rate of change in SC is controlled by slower relaxation Hence for use only Frequencies Supercurrents It can be shown also that     0,10 sec10~ 1 12 12 2    Hz   JJJj 00 02    Superconductor Current density Normal current density Displacement current density
14. 14. If path contains no hole, use Stoke’s Theorem for Deep in SC, and flux is excluded (part of Meissner effect) i.e. fluxoid vanishes for any surface entirely in the SC (assuming there is no hole). Sj     L S S SSS daBdajxdj ...   0Sj    S L SSC djdaB 0.. 
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