1. Introduction
Advances in particle physics are inextricably linked to advances in particle accelerators. Many parameters affect the performance of the accelerators, and the physics they make accessible. A very important factor is the availability of higher energy particle beams, for higher energy beams allow for increasingly finer measurement of the fundamental nature of particles.
The technology for accelerating particles is dependent on the type of particle being accelerated. Heavy particle (proton) accelerators are generally circular machines, with the technological challenges lying in high field magnet development. The Tevatron, at
Fermilab, and the the LHC, proposed for construction at
CERN are examples of this type of machine.
Electron positron colliders have been almost exclusively circular machines up to the present. The synchrotron radiation which results from accelerating charged particles in a circle makes circular machines impractical for further increase in energy. For this reason, the next generation of electron positron colliders will have to be linear. A linear collider with center of mass energy in the 1 TeV range would be a complimentary machine to the previously mentioned LHC. Several proposals have been put forward for the construction of such a machine,
[1]-[3] including
TESLA, which would be an SRF-based machine.
N.B. : This primer will concern itself
only with the type of accelerating cavities used in electron-positron machines. This is somewhat unfair, as much of the work done in SRF technology has been performed with cavities designed for heavy ion machines, such as those in the
ATLAS facility at
Argonne National Laboratory. However, since the author of this document knows primarily electron-positron machines, these cavities are all that will be covered here.
1.1 RF Cavities
An RF cavity is the device through which power is coupled into the particle beam of an accelerator. In electron-positron colliders, RF accelerating cavities are microwave resonators which generally derive from a "pillbox" shape (right circular cylinder), with connecting tubes to allow particle beams to pass through for acceleration. Figure 1 shows a typical cylindrically symmetric cavity. The fundamental, or lowest RF frequency, mode (TM 010) of the cavity has fields as shown. The electric field is roughly parallel to the beam axis, and decays to zero radially upon approach to the cavity walls. Boundary conditions demand that the electric surface be normal to the metal surface. The peak surface electric field is located near the iris, or region where the beam tube joins the cavity. The magnetic field is azimuthal, with the highest magnetic field located near the cavity equator. The magnetic field is zero on the cavity axis.
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Figure 1. A typical accelerating cavity geometry, showing particle beam and fundamental fields of the RF cavity.
The particle beam traverses the cavity as shown, experiencing an accelerating force along the axis of the cavity due to the electic field. Since the RF fields alternate in time, the particle beam must, of course, be in the proper phase with respect to the fields in order that the force be accelerating rather than decelerating. In addition, since the particles take a finite time to cross the cavity, the accelerating field is the time average of the electric field along the particles flight. The average gradient is defined in equation 1:
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TRF is the RF period, and
E(z,t) is the electric field at the time and position of the particle.
The
Q0 of an accelerating cavity is defined as the RF angular frequency (
w) times the ratio of the stored energy in the electromagnetic fields (
U) to the dissipated power (
Pdiss), as shown in equation 2.
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The relationships between the stored energy and the magnitudes of the electric fields are obtained by numerical solution of Maxwell's Equations for the cavity geometry. Several computer program packages are available to solve the equations for typical cavity geometries, e.g.
SUPERFISH[4] ,
URMEL[5] and
URMEL-T[6] , or
MAFIA[7] . Of particular interest in these solutions are the ratios of peak surface electric and magnetic fields to the square root of the stored energy, and the ratio of peak surface electric electric field to the average accelerating field in the cavity, given by:
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In the situation where all cavity losses are due to surface currents, the
Q0 can alternately be defined as the ratio of the geometry factor
G to the microwave surface resistance
Rs, as shown in equation 4.
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The geometry factor is defined in equation 5. Geometry factors have units of resistance, and generally have values between 200 and 300 Ohms.
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The microwave surface resistance in a normal conductor is given by equation 6, and is approximately 15 milliOhms for copper at 3 GHz.
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where delta is the RF skin depth in a normal conductor.
The surface resistance causes power dissipation by the surface currents which arise in order to support the magnetic fields at the RF surface. Wall losses are the primary reason for investigating superconducting cavities, as the RF surface resistance is five to six orders of magnitude lower than that of a normal conduncting surface. Superconducting surface resistance will be discussed in the next section.
1.2: Basics of RF Superconductivity
The intention of this section is to provide the basics of RF superconductivity which are necessary to understand the work performed in the HPP project. A more in depth discussion can be found in several of the listed references.
[8]-[13] The famous BCS theory of superconductivity was originally worked out for DC conditions by Bardeen, Cooper, and Schrieffer.
[14] Extension of BCS theory to RF conditions was made by Mattis and Bardeen
[15] and Abrikosov, Gor'kov, and Khalatnikov.
[16] In addition, the empirical models of London
[17] prove to be useful in understanding the basic nature of RF superconductivity.
Niobium is currently the material of choice for superconducting cavities. The primary reason for this choice is that niobium has the highest critical temperature of all pure metals (*Tc*= 9.25 K), and in addition is relatively simple to use in terms of fabrication. Many compounds (including the new high-*Tc*ceramic materials) have shown higher critical temperatures than niobium. None of these materials can match niobium, however, either in terms of its ease of use, or in terms of its performance with increasing RF fields.
It is well known that many materials, known as superconductors, lose all DC electrical resistance when the temperature drops below the critical temperature
Tc. The BCS theory has quite successfully described this phenomenon. According to BCS theory, below
Tc the electrons of a conductor gain a small net attraction through their interaction with the surrounding lattice. The electrons then condense into "Cooper pairs," which move without resistance through the conductor. The Cooper pairs have binding energy Delta, which is dependent on temperature.
Unlike DC resistance, RF surface resistance is zero only at*T = 0 K*(absolute zero). At temperatures above absolute zero, but below the critical temperature, the surface resistance is greatly reduced, yet non-zero. This can be most easily understood through the "London two-fluid model." The two fluids are paired (superconducting) and unpaired (normal conducting) electrons. The binding energy of the Cooper pairs is comparable to thermal energies, therefore we can express the fraction of unpaired electrons by a Boltzman distribution:
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where
kB is the Boltzman constant.
Cooper pairs move without resistance, and thus dissipate no power. The Cooper pairs do nonetheless have an inertial mass, and thus the electromagnetic fields must extend into the surface of the conductor in order to provide the forces to accelerate the pairs back and forth to sustain the RF surface currents. The EM fields will act on the unpaired electrons as well, therefore causing power dissipation.
For temperatures less than
Tc / 2, the superconducting surface resistance can be well represented as:
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The first term on the right hand side of equation 8 is the BCS resistance. For temperatures less than
Tc / 2, the binding energy is nearly unchanged from its value at absolute zero,
D (0). The coefficient
A is a complex function of material parameters such as the superconducting coherence length, the penetration depth, the electron mean free path, and the Fermi velocity. A can be evaluated computationally via programs by Turneaure
[18] or Halbritter.
[19] The BCS resistance in a typical 3 GHz cavity varies from 3 microOhms at
T = 4.2 K to less than 1 nanoOhm at
T = 1.4 K.
The second term on the right hand side of equation 8 is the residual, or temperature independent, resistance
R0. Mechanisms for
R0 are not well understood, though several possibilities have been proposed and investigated.
[20],[21] Residual resistance values are generally found to be between 5 and 100 nanoOhms, though values as low as 1 nanoOhm have been measured.
[22]
1.3: Advantages of SRF Technology
As previously mentioned, the chief advantage in the use of SRF cavities is the reduced dissipation due to wall losses. The wall loss power dissipation is proportional to the surface resistance, which is reduced by a factor of one million in superconducting cavities. The total power usage does not reflect all of this gain, however, due to the need to refrigerate the cavities to liquid helium temperatures. Even including refrigerator power, using typical refrigerator efficiencies, the net power usage drops by a factor of several hundred to a thousand in superconducting cavities. In CW operation this means greatly reduced power and higher accelerating gradients. In pulsed operation, SRF cavities offer long pulse lengths and high duty cycles compared to NC cavities.
Many proposed accelerator projects (e.g. B-factories) require significant improvements in the luminosity, which is a measure of the rate at which particles in counter rotating beams will collide. One factor in the luminosity is the average beam current. The maximum current can be limited by beam-cavity interactions. The beam can be significantly disrupted anytime the surrounding environment is changed, for example cavities, vacuum connections, etc. The typical normal conducting cavity is an extreme change in the surrounding environement. The extreme shape of normal conducting cavities is necessary in order to minimize the power dissipation for a given electric field. SRF cavities avoid the disruptive cavity shapes by instead reducing the power dissipation with superconductivity. Given their smoother shape and larger beam holes, superconducting cavities present less of a disruption to the beam than their normal conducting counterparts.
In addition, in continuous wave operating conditions, the higher accelerating gradient which SC cavities can sustain compared to NC cavities minimizes the required length of accelerating sections, therefore minimizing the overall disruption to the beam quality.
The reduced interaction of an SRF cavity is to be exploited in the
CESR Phase 3 high luminosity upgrade. Figure 2 shows the cross sections of the normal conducting cavity currently in use, and the SRF cavity which will be installed in the new machine.
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Figure 2 . Comparison of 500 MHz cavities: On the left is the normal conducting cavity currently used in CESR. On the right is the SRF B-cell cavity, which will be used in the
CESR Phase 3 high luminosity upgrade.
1.4: Theoretical Potential of SRF Cavities
It is well established that even at
T = 0 K, a sufficiently high surface magnetic field can destroy superconductivity. The limiting field is referred to as the critical field. There are two types of classical superconductors, which have the same fundamental mechanism for superconductivity, but differ in their behavior with increasing magnetic fields. The difference in behavior can be traced to differences in the free energy associated with NC/SC boundaries on the RF surface, which are controlled by such parameters as the coherence length and the penetration depth. For a more complete description of these phenomena, see reference 8.
In Type I superconductors, the magnetic field is completely shielded from the superconductor interior for fields up to the critical field
Hc. Above this field, the magnetic field penetrates completely, destroying the superconductivity.
In Type II superconductors, the magnetic field is completely expelled up to a first critical field,
Hc1. Above
Hc1, the magnetic field penetrates partially, with normal conducting regions isolated on the surface of the superconductor. This behavior persists up to a second critical field,
Hc2. Above
Hc2, the field penetrates completely, destroying the superconductivity.
These descriptions are for DC or steady fields at
T = 0 K. Above absolute zero, the critical fields drop approximately according to equation 9.
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In RF conditions, the requirements are relaxed somewhat, as the penetration of the magnetic field into the RF surface requires nucleation of a flux line, which requires a finite amount of time. The nucleation time has been determined
[23] to be such that the complete shielding of magnetic fields can persist to fields higher than the critical field, up to a limit termed the superheating critical field,
Hsh. In niobium, the superheating critical field is estimated to be approximately
Hsh = 2300 Oe.
[8]
Experimentation with specially designed SRF non-accelerating cavities
[24],[25] has clearly shown that there are no fundamental limits to the peak electric field on a niobium surface up to
Epeak = 200 MV/m. The theoretical limit on accelerating cavity performance is therefore dependent on the cavity magnetic fields.
Recent experiments in the [[SrfHpp]HPP]] experimental program have begun to investigate further the behavior of SRF cavities at increasing magnetic fields.
In typical SRF accelerating cavities,
Hpeak = 2300 Oe corresponds to accelerating gradients of 50 to 60 MV/m. Given that accelerators with niobium cavities generally operate at
Eacc = 5-10 MV/m, the need for further work is clear. Accelerating gradients of 20 to 30 MV/m are necessary to make linear colliders with SRF technology economically attractive.
[1]
