A parametric study of BCS RF surface impedance with magnetic field using the Xiao Code Page: 3 of 5
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A PARAMETRIC STUDY OF BCS RF SURFACE IMPEDANCE WITH
MAGNETIC FIELD USING THE XIAO CODE *
Charles E. Reecel and Binping Xiao1,2
1 Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606
2 Brookhaven National Laboratory, Upton, New York 11973
A recent new analysis of field-dependent BCS rf
surface impedance based on moving Cooper pairs has
been presented. Using this analysis coded in
MathematicaTM, survey calculations have been completed
which examine the sensitivities of this surface impedance
to variation of the BCS material parameters and
temperature. The results present a refined description of
the "best theoretical" performance available to potential
applications with corresponding materials.
The radiofrequency (RF) surface impedance of a
superconductor may be considered a consequence of the
inertia of the Cooper pairs in the superconductor. The
resulting incomplete shielding of RF field allows the
superconductor to store RF energy inside its surface,
which may be described as surface reactance. The RF
field that enters the superconductor interacts with quasi-
particles, causing power dissipation, represented by
surface resistance. Based on the BCS theory  and
anomalous skin effect theory , a derivation of a
superconductor's surface impedance was developed by
Mattis and Bardeen [3, 4]. Mattis-Bardeen theory,
however, does not consider the field dependence of
surface impedance. In particular, its real part, surface
resistance, which is of great interest in superconducting
radiofrequency (SRF) applications, is unaddressed.
Several models have been previously proposed to address
this issue. [5, 6]
Recently a new model has been put forward by Xiao et
al., starting from the BCS theory with a net current in a
superconductor, the electron states distribution at 0 K
were calculated, together with the probability of electron
occupation with finite temperature and applied to
anomalous skin effect theory, to obtain a new form of RF
field dependence of the surface impedance of a
superconductor. A Mathematicam program has been
developed by Xiao to accomplish the calculation of the
resulting challenging quadruple integral. It is applicable
to any standard superconductor described by BCS theory.
The code reproduces the standard Mattis-Bardeen theory
result at zero field as calculated, for example, by the
commonly used Halbritter code, SRIMP. 
* Work supported by DOE. Authored by Jefferson Science Associates,
LLC under U.S. DOE Contract No. DE-ACO5-060R23177. The U.S.
Government retains a non-exclusive, paid-up, irrevocable, world-wide
license to publish or reproduce this manuscript for U.S. Government
A rather surprising result of the calculation with
potential importance to SRF applications is the prediction
of non-linear, decreasing surface resistance in an RF field
regime that is prime domain for accelerator applications.
The corresponding prediction of increasing Qo with field
matches remarkably well recent reports of record-
breaking low losses [8, 9] and raises the prospect that the
common expectation of "best theoretical" cryogenic
performance from Nb, and in principle other BCS
superconductors, may be dramatically revised for the
We have used this code to perform a parametric
sensitivity survey with each the characteristic BCS
material parameters of the field-dependent RF surface
impedance in hopes of supporting increased insight into a
performance optimization strategy.
The present analysis focused on niobium, using the
following characteristic parameters as standard
conditions: 4A/kTc(0) = 1.85, Tc(0) = 9.25 K, o = 40 nm,
)L(0) = 32 nm, and mean free path a = 50 nm 
exploring the predicted surface impedance with
departures from these values.
The calculated standard condition surface impedance of
niobium at 1.5 GHz and 2.0 K is shown as a function of
Cooper pair velocity in Figure 1.
Magnetic flux density (mT)
0 50 100 150
0 200 400 600 800 1000
FIG. 1. Surface resistance, Rs, (red line) and reactance, Xs,
(blue dashed line) versus Cooper pair velocity for Nb at
2 K and 1.5 GHz.
The surface resistance Rs, with a value of 10.9 nSi at
0 m/s cooper pair velocity vs, first decreases with
increasing vs, then increases with increasing vs, with a
minimum RS of 2.0 nSi at 230 m/s vs. Since the
supercurrent density varies both with depth into the
surface and time, the local surface impedance does as
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E., Reece C. & B., Xiao. A parametric study of BCS RF surface impedance with magnetic field using the Xiao Code, article, September 23, 2013; United States. (digital.library.unt.edu/ark:/67531/metadc864910/m1/3/: accessed December 14, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.