Superconducting Radio Frequency - Physics of SRF Cavities

Physics of SRF Cavities

The physics of Superconducting RF can be complex and lengthy. A few simple approximations derived from the complex theories, though, can serve to provide some of the important parameters of SRF cavities.

By way of background, some of the pertinent parameters of RF cavities are itemized as follows. A resonator's quality factor is defined by

,

where:

ω is the resonant frequency in ,

U is the energy stored in, and

P_d is the power dissipated in in the cavity to maintain the energy U.

The energy stored in the cavity is given by the integral of field energy density over its volume,

,

where:

H is the magnetic field in the cavity and

μ₀ is the permeability of free space.

The power dissipated is given by the integral of resistive wall losses over its surface,

,

where:

R_s is the surface resistance which will be discussed below.

The integrals of the electromagnetic field in the above expressions are generally not solved analytically, since the cavity boundaries rarely lie along axes of common coordinate systems. Instead, the calculations are performed by any of a variety of computer programs that solve for the fields for non-simple cavity shapes, and then numerically integrate the above expressions.

An RF cavity parameter known as the Geometry Factor ranks the cavity's effectiveness of providing accelerating electric field due to the influence of its shape alone, which excludes specific material wall loss. The Geometry Factor is given by

,

and then

The geometry factor is quoted for cavity designs to allow comparison to other designs independent of wall loss, since wall loss for SRF cavities can vary substantially depending on material preparation, cryogenic bath temperature, electromagnetic field level, and other highly variable parameters. The Geometry Factor is also independent of cavity size, it is constant as a cavity shape is scaled to change its frequency.

As an example of the above parameters, a typical 9-cell SRF cavity for the International Linear Collider (a.k.a. a TESLA cavity) would have G=270 Ω and R_s= 10 nΩ, giving Q_o=2.7×1010.

The critical parameter for SRF cavities in the above equations is the surface resistance R_s, and is where the complex physics comes into play. For normal-conducting copper cavities operating near room temperature, R_s is simply determined by the empirically measured bulk electrical conductivity σ by

.

For copper at 300 K, σ=5.8×107 (Ω·m)−1 and at 1.3 GHz, R_{s copper}= 9.4 mΩ.

For Type II superconductors in RF fields, R_s can be viewed as the sum of the superconducting BCS resistance and temperature-independent "residual resistances",

.

The BCS resistance derives from BCS theory. One way to view the nature of the BCS RF resistance is that the superconducting Cooper pairs, which have zero resistance for DC current, have finite mass and momentum which has to alternate sinusoidally for the AC currents of RF fields, thus giving rise to a small energy loss. The BCS resistance for niobium can be approximated when the temperature is less than half of niobium's superconducting critical temperature, T<T_c/2, by

,

where:

f is the frequency in ,

T is the temperature in, and

T_c=9.3 K for niobium, so this approximation is valid for T<4.65 K.

Note that for superconductors, the BCS resistance increases quadratically with frequency, ~f 2, whereas for normal conductors the surface resistance increases as the root of frequency, ~√f. For this reason, the majority of superconducting cavity applications favor lower frequencies, <3 GHz, and normal-conducting cavity applications favor higher frequencies, >0.5 GHz, there being some overlap depending on the application.

The superconductor's residual resistance arises from several sources, such as random material defects, hydrides that can form on the surface due to hot chemistry and slow cool-down, and others that are yet to be identified. One of the quantifiable residual resistance contributions is due to an external magnetic field pinning magnetic fluxons in a Type II superconductor. The pinned fluxon cores create small normal-conducting regions in the niobium that can be summed to estimate their net resistance. For niobium, the magnetic field contribution to R_s can be approximated by

,

where:

H_ext is any external magnetic field in ,

H_c2 is the Type II superconductor magnetic quench field, which is 2400 Oe (190 kA/m) for niobium, and

R_n is the normal-conducting resistance of niobium in ohms.

The Earth's nominal magnetic flux of 0.5 gauss (50 µT) translates to a magnetic field of 0.5 Oe (40 A/m) and would produce a residual surface resistance in a superconductor that is orders of magnitude greater than the BCS resistance, rendering the superconductor too lossy for practical use. For this reason, superconducting cavities are surrounded by magnetic shielding to reduce the field permeating the cavity to typically <10 mOe (0.8 A/m).

Using the above approximations for a niobium a SRF cavity at 1.8 K, 1.3 GHz, and assuming a magnetic field of 10 mOe (0.8 A/m), the surface resistance components would be

R_BCS = 4.55 nΩ and

R_res = R_H = 3.42 nΩ, giving a net surface resistance

R_s = 7.97 nΩ. If for this cavity

G = 270 Ω then the ideal quality factor would be

Q_o = 3.4×1010.

The Q_o just described can be further improved by up to a factor of 2 by performing a mild vacuum bake of the cavity. Empirically, the bake seems to reduce the BCS resistance by 50%, but increases the residual resistance by 30%. The plot below shows the ideal Q_o values for a range of residual magnetic field for a baked and unbaked cavity.

In general, much care and attention to detail must be exercised in the experimental setup of SRF cavities so that there is not Q_o degradation due to RF losses in ancillary components, such as stainless steel vacuum flanges that are too close to the cavity's evanescent fields. However, careful SRF cavity preparation and experimental configuration have achieved the ideal Q_o not only for low field amplitudes, but up to cavity fields that are typically 75% of the magnetic field quench limit. Few cavities make it to the magnetic field quench limit since residual losses and vanishingly small defects heat up localized spots, which eventually exceed the superconducting critical temperature and lead to a thermal quench.