Superconducting radio frequency

Superconducting radio frequency science and technology involves the application of electrical superconductors to radio frequency devices. The ultra-low electrical resistivity of a superconducting material allows an RF resonator to obtain an extremely high quality factor, Q. For example, it is commonplace for a 1.3 GHz niobium SRF resonant cavity at 1.8 kelvins to obtain a quality factor of Q=5×10¹⁰. Such a very high Q resonator stores energy with very low loss and narrow bandwidth. These properties can be exploited for a variety of applications, including the construction of high-performance particle accelerator structures.

Introduction

The amount of loss in an SRF resonant cavity is so minute that it is often explained with the following comparison: Galileo Galilei was one of the first investigators of pendulous motion, a simple form of mechanical resonance. Had Galileo experimented with a 1 Hz resonator with a quality factor Q typical of today's SRF cavities and left it swinging in an entombed lab since the early 17th century, that pendulum would still be swinging today with about half of its original amplitude.
The most common application of superconducting RF is in particle accelerators. Accelerators typically use resonant RF cavities formed from or coated with superconducting materials. Electromagnetic fields are excited in the cavity by coupling in an RF source with an antenna. When the RF fed by the antenna is the same as that of a cavity mode, the resonant fields build to high amplitudes. Charged particles passing through apertures in the cavity are then accelerated by the electric fields and deflected by the magnetic fields. The resonant frequency driven in SRF cavities typically ranges from 200 MHz to 3 GHz, depending on the particle species to be accelerated.
The most common fabrication technology for such SRF cavities is to form thin walled shell components from high purity niobium sheets by stamping. These shell components are then welded together to form cavities.
A simplified diagram of the key elements of an SRF cavity setup is shown below. The cavity is immersed in a saturated liquid helium bath. Pumping removes helium vapor boil-off and controls the bath temperature. The helium vessel is often pumped to a pressure below helium's superfluid lambda point to take advantage of the superfluid's thermal properties. Because superfluid has very high thermal conductivity, it makes an excellent coolant. In addition, superfluids boil only at free surfaces, preventing the formation of bubbles on the surface of the cavity, which would cause mechanical perturbations. An antenna is needed in the setup to couple RF power to the cavity fields and, in turn, any passing particle beam. The cold portions of the setup need to be extremely well insulated, which is best accomplished by a vacuum vessel surrounding the helium vessel and all ancillary cold components. The full SRF cavity containment system, including the vacuum vessel and many details not discussed here, is a cryomodule.
Entry into superconducting RF technology can incur more complexity, expense, and time than normal-conducting RF cavity strategies. SRF requires chemical facilities for harsh cavity treatments, a low-particulate cleanroom for high-pressure water rinsing and assembly of components, and complex engineering for the cryomodule vessel and cryogenics. A vexing aspect of SRF is the as-yet elusive ability to consistently produce high Q cavities in high volume production, which would be required for a large linear collider. Nevertheless, for many applications the capabilities of SRF cavities provide the only solution for a host of demanding performance requirements.
Several extensive treatments of SRF physics and technology are available, many of them free of charge and online. There are the proceedings of CERN accelerator schools, a scientific paper giving a thorough presentation of the many aspects of an SRF cavity to be used in the International Linear Collider, bi-annual International Conferences on RF Superconductivity held at varying global locations in odd numbered years, and tutorials presented at the conferences.

SRF cavity application in particle accelerators

A large variety of RF cavities are used in particle accelerators. Historically most have been made of copper – a good electrical conductor – and operated near room temperature with exterior water cooling to remove the heat generated by the electrical loss in the cavity. In the past two decades, however, accelerator facilities have increasingly found superconducting cavities to be more suitable for their accelerators than normal-conducting copper versions. The motivation for using superconductors in RF cavities is not to achieve a net power saving, but rather to increase the quality of the particle beam being accelerated. Though superconductors have little AC electrical resistance, the little power they do dissipate is radiated at very low temperatures, typically in a liquid helium bath at 1.6 K to 4.5 K, and maintaining such low temperatures takes a lot of energy. The refrigeration power required to maintain the cryogenic bath at low temperature in the presence of heat from small RF power dissipation is dictated by the Carnot efficiency, and can easily be comparable to the normal-conductor power dissipation of a room-temperature copper cavity. The principle motivations for using superconducting RF cavities, are:

High duty cycle or cw operation. SRF cavities allow the excitation of high electromagnetic fields at high duty cycle, or even cw, in such regimes that a copper cavity's electrical loss could melt the copper, even with robust water cooling.
Low beam impedance. The low electrical loss in an SRF cavity allows their geometry to have large beampipe apertures while still maintaining a high accelerating field along the beam axis. Normal-conducting cavities need small beam apertures to concentrate the electric field as compensation for power losses in wall currents. However, the small apertures can be deleterious to a particle beam due to their spawning of larger wakefields, which are quantified by the accelerator parameters termed "beam impedance" and "loss parameter".
Nearly all RF power goes to the beam. The RF source driving the cavity need only provide the RF power that is absorbed by the particle beam being accelerated, since the RF power dissipated in the SRF cavity walls is negligible. This is in contrast to normal-conducting cavities where the wall power loss can easily equal or exceed the beam power consumption. The RF power budget is important since the RF source technologies, such as a Klystron, Inductive output tube, or solid state amplifier, have costs that increase dramatically with increasing power.

When future advances in superconducting material science allow higher superconducting critical temperatures T_c and consequently higher SRF bath temperatures, then the reduced thermocline between the cavity and the surrounding environment could yield a significant net power savings by SRF over the normal conducting approach to RF cavities. Other issues will need to be considered with a higher bath temperature, though, such as the fact that superfluidity would not be present with liquid nitrogen. At present, none of the "high T_c" superconducting materials are suitable for RF applications. Shortcomings of these materials arise due to their underlying physics as well as their bulk mechanical properties not being amenable to fabricating accelerator cavities. However, depositing films of promising materials onto other mechanically amenable cavity materials may provide a viable option for exotic materials serving SRF applications. At present, the de facto choice for SRF material is still pure niobium, which has a critical temperature of 9.3 K and functions as a superconductor nicely in a liquid helium bath of 4.2 K or lower, and has excellent mechanical properties.

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:
The energy stored in the cavity is given by the integral of field energy density over its volume,
where:
The power dissipated is given by the integral of resistive wall losses over its surface,
where:
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 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 ⁻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:
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:
The Earth's nominal magnetic flux of 0.5 gauss translates to a magnetic field of 0.5 Oe 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.
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, the surface resistance components would be
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.
Image:SRF Cavity Max Qo vs H 2.jpg|frame|Plot of SRF cavity ideal Q_o vs external DC magnetic field for the same cavity frequency, temperature, and geometry factor as used in the text.
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.