Presentation on theme: "Two Major Open Physics Issues in RF Superconductivity H. Padamsee & J. Sethna What is the RF critical magnetic field? Is it –Hc1, Hc, Hsh? –How does it."— Presentation transcript:
Two Major Open Physics Issues in RF Superconductivity H. Padamsee & J. Sethna What is the RF critical magnetic field? Is it –Hc1, Hc, Hsh? –How does it depend on temperature? –How does it depend on Ginzburg-Landau parameter ( )? High-Field Q-slope (next session) –Why does the RF surface resistance of niobium increase sharply at high RF magnetic field?
Hc Quick Review of DC Critical Magnetic fields Meissner State Normal State External field Internal magnetic field
Ginzburg and Landau treated the surface energy associated with a normal/SC phase boundary. Qualitatively, the free energy per unit volume increases by µ 0 Ha 2 l /2 over the penetration depth ( L due to the diamagnetism; work is done to exclude the magnetic flux and falls by µ 0 Hc 2 0 /2 over the coherence length due to the increase of the super-electron density. If the coherence length is smaller than the penetration depth, there is a negative surface energy and it is energetically favorable at equilibrium to have normal/superconducting boundaries… Type II. Type I and Type II SC If 0 > L, there is a positive surface energy and the formation of normal/SC regions is not favorable, Type I
Magnetization vs External Field H c1, H c2, H c Boundary between Type I and Type II defined by G-L parameter G-L theory relates Hc1, Hc2 and Hc to , over a restricted range of . = H c2 /√2H c Hsh is the maximum permissible value of the applied field, which satisfies GL equations. Metastability allows Hsh > Hc1 (or Hc)
Measuring Hc1 and Hc of Nb, 4.2 K from Magnetization Curves Gives questionable results for Hc1 and Hc due to flux pinning, which depends on state of sample DESY
H c1 and H c are also difficult to measure because of hysteresis (flux pinning)… Rarely (if ever) do you get a reversible curve
Attempted Magnetization Measurements on NON-Reversible Nb (Saito)
Attempted Determination of = H c2 /√2H c
Hsh is defined as the maximum permissible value of the applied field, which satisfies GL equations. Matricon and Saint-James solved GL equations numerically for the one- dimensional case where half of the space is occupied by a superconductor. Superheating Field
Hsh in RF Fields In RF, fields change rapidly, within nanoseconds. If the time it takes to nucleate fluxoids is long compared to the rf period ( s) There is a tendency for the meta-stable superconducting state to persist up to Hsh > Hc1. T. Yogi Measured Hsh > Hc1 for Alloys Sn-In and In-Bi over a range of kappa values to show no discontinuity across Type I and Type II
T. Yogi’s Results Nb
T. Hays Measured the RF Critical Field for Superconductors: Nb and Nb3Sn Using High Pulse Power We plan to repeat these measurements with large grain and single crystal cavities (if funded)
In the process of phase transition, a boundary must be nucleated. In a Type I superconductor, the positive surface energy suggests that, in dc fields, the Meissner state can persist metastably beyond the thermodynamic critical field, up to the superheating field, H sh. At this field, the surface energy per unit area vanishes: For Type II superconductors, it is also possible for the Meissner state to persist meta-stably above H c1, How far above H c1 ? is the open question Heuristic Arguments to determine Hsh
Saito extended the energy balance argument to other dimensional forms of nucleation such as a line nucleation (vortex nucleation). The diamagnetic energy is given by and the condensation energy is Balancing the two contributions, the superheating field is Is line nucleation the proper model for the RF critical field, i.e. the superheating field?
Issues with this Energy Balance Approach for H > Hc1 As an energy-balance argument, the vortex nucleation model gives an upper bound on the equilibrium critical field for vortex penetration, which is related to Hc1. Nothing in the energy balance argument discusses meta-stability, which is the key aspect for Hsh The line nucleation model is useful in the context of nucleation on in-homogeneities on the scale of the coherence length, but not as a fundamental limit for uniform, flat, pure superconductors.
Temperature Dependence Saito determines = Hc2 (T) /√2Hc (T) and Hc(T) from DC magnetic field data This is questionable for two reasons Superheating is a prediction from the Abrikosov, G-L theory, which is a perturbation theory. Therefore it is valid for T ~ Tc or ~ 0 Non-local effects important at lower temperatures have been shown to introduce large qualitative changes in vortex behavior. Hysteresis (pinning) in magnetization gives unreliable answers for Hc1 and Hc Final problem, Saito introduces Hsh-rf = √2 Hsh-dc ?? This is incorrect for phase transition field
1700 – 1750 Oe Best
Problem If Hrf (0) = 1800 Oe, According to Saito, Hsh (dc) at zero temperature = 1270 Oe –which is << Hc 1 (known to be 1740 – 1900 Oe) !!
Experimentally Peak RF Magnetic Fields Are Rising ! Cavities material and surfaces are getting better !
How to correctly calculate Hsh? Field where barrier vanishes Linear stability analysis will also determine vortex array At large and T~Tc, 1-D analysis gives Hsh = Hc (as discussed) At lower T, we need the Eilenberger equations –(Non-local, Green’s functions, …)
Why a superheating field? Fundamental limits to H sh in Niobium What can theory tell us? James P. Sethna, Gianluigi Catelani How to calculate H sh ? Costly core enters first; gain from field later “Line nucleation” wrong Yogi, Saito H sh ~H c / discouraging Via “energy balance”, no barrier calculation Does not work for large H sh < H c1 = H c ln /(√2 ) Correct balance theory gives H c1 not H sh Field where barrier vanishes Linear stability analysis determines nucleation mechanism: vortex array At large and T~T c, 1-D analysis H sh = H c Barrier
Which Theory? James P. Sethna, Gianluigi Catelani If we trusted Ginsburg-Landau far below T c, things looks good: Niobium: H sh ~ 2400 Oe → 63 MV/m Nb 3 Sn: H sh ~ 4000 Oe → 110 MV/m But Ginsburg-Landau is only valid near T c ; RF cavities at 2K << T c for Nb At lower T, we need the Eilenberger equations (Non-local, Green’s functions, …) Known corrections from G-L: small effects on H c1, H c2 huge effect on vortex core (collapse from to 1/k F ) Unknown so far: Effect on H sh Work proposed & in progress…
Eilenberger/Gorkov for H sh in Nb James P. Sethna, Gianluigi Catelani Equations of motion for the (anomalous) Green’s functions f( ,n,x) and g( ,n,x) Self-consistent equation for gap Maxwell equation for H from current Constraint on the Green’s function Matsubara frequencies and Fermi wavevector direction n Solve 1D ODE for uniform, superconducting state Find 2D instability threshold H sh (functional eigenvalue crosses zero)