Field Limitations

In order to reduce occurrences of RF breakdown the surface fields on the cavity walls are limited, these limits are largely informed by experimental data. The electric field was originally believed to be responsible for the initiation of RF breakdown through it’s influence on field emission. However, the magnetic field is now also believed to play an important role in the BDR, thus it must also be limited for reliable performance.

2.3 RF Breakdown 43

Kilpatrick Criterion

W.D.Kilpatrick was the first person to provide a quantitative limitation to the peak surface electric field that can exist on a cavity wall without arcing. TheKilpatrick Criterion describes a maximum field threshold beyond which the safe operation is

not guaranteed. Equation(2.41) was reformulated by T.J.Boyd [88], and was based on experimental data [89].

f[M Hz] = 1.64(Ek[M V /m])2e−8.5/(Ek[M V /m]) (2.41)

where f is the RF frequency and Ek is the Kilpatrick limit for the electric field.

The expression shows that higher fields can be reached with higher frequencies. It does not however depend on pulse length or acknowledge that breakdown is somewhat statistical in nature and breakdowns can indeed occur in the ‘safe operation’ region it is just unlikely. This threshold became obsolete with advances in vacuum technology and improved surface treatments. Now the limit is used as a figure of merit, a cavity may operate at 2 or 3 times Kilpatricks.

Modified Poynting Vector

Although the Kilpatrick criterion can still be used to inform the design of high gradient cavities, a quantitative theory that can explain and predict RF breakdown is still lacking. The CLIC study at CERN collected high gradient test data with the aim of deriving the high gradient limit due to RF breakdown.

The electric field was originally believed to be solely responsible for the onset of RF breakdown, but the data collected showed a large variation in achievable surface electric field before breakdown, which prompted the idea that it was in fact the power flow rather than the electric field alone limiting high gradient operation. The power flow model proposed that the ratio of the input power to the

iris circumference was the parameter limiting gradient in travelling wave structures and developed the scaling law [90]:

Pin·t

1_/ 3

p

C < constant (2.42)

wherePin is the input power to the structure, C is the minimum circumference of

the structure or the beam aperture, andtp is the pulse length. This scaling law fit

the data more closely than the peak surface electric field but still had limitations, one being that it does not account for standing wave structures which have no power flow through the beam aperture.

In 2009 a new local field quantity was proposed, themodified Poynting vector [82]

44 2. RF Particle Acceleration Sc= R{S} + 1 6 I{S} (2.43)

The real part of the conventional Poynting vector describes the active power flow or the net flux of power that passes through a travelling wave structure. The imaginary part describes the reactive power flow or the cyclic energy transfer between the electric and magnetic field in both travelling and standing wave structures. The

1_/

6 weighting factor was largely determined by fitting to experimental data and

accounts for the phase shift between the active and reactive power flow. Figure 2.19 shows that the reactive power flow is zero when field emission is maximum whereas the active power flow maximum is in phase with the field emission. The weighting factor accounts for the reactive power flow being less efficient in feeding the field emission mechanism.

Fig. 2.19 Time dependences of electric field (dashed black line), active power flow (blue), reactive power flow (red), and field emission power flow (green) [82]

Figure 2.20 shows the square root of the scaled modified Poynting vector agrees well with the high-gradient structure data accumulated in the CLIC study. The dependence of the breakdown rate on the accelerating gradient and the pulse length was given in Equation 2.40, and the BDR also scales withSC:

2.3 RF Breakdown 45

Fig. 2.20 Square root of the scaled modified Poynting vector calculated for the high-gradient performances of several 12 GHz travelling-wave structures (Black), 12 GHz standing-wave structures (Red), and 30 GHz travelling wave structures (Blue) [? ]. Breakdown rate measurements for the 3 GHz TERA single-cell cavity

are found in the yellow band [91].

Pulsed Surface Heating

In an ideal perfect conductor, the fields and currents inside the conductor are zero. As perfect conductors do not exist, RF cavities tend to be made from copper which has a room temperature resistivity ofρr=1/σc= 1.7×10−8Ωm. Copper is a very

good, but not perfect conductor so the fields and currents decay exponentially with distance from the surface, this is called theskin effect. When RF fields are applied

at the surface of a conductor, a current is induced at the surface of the conductor, which then shields the inside of the conductor from any fields or currents. The thin layer containing non-zero fields is called theskin depth δ

δ=

s

2

σcµ0ω

. (2.45)

The surface resistance of an RF structure isRsurf=1/σδ this is not the same as DC

due to the skin effect. Substituting Equation (2.45) into the expression forRsurf,

we find that Rsurf=

q

µ₀ω/2σc, and the AC surface resistance is proportional to

the square root of frequency. The average power dissipated per cycle is

Pd=

Rsurf

2

Z

H2dS (2.46)

where dS is an area on the surface of the cavity.

Cavities are typically kept cool with water cooling tubes along the outside surface of the cavity. As the magnetic field is pulsed, a layer δ on the inside of

46 2. RF Particle Acceleration

cavity body which is being water cooled. The pulsed fields cause cyclic thermal expansion of the skin depth layer of the cavity, which experiences fatigue as a result. If the temperature differences reaches around 50 K the yield strength of copper is exceeded and micro cracking occurs. Any surface imperfections caused by this cyclic heating can result in field enhancement and field emission. This effect can be minimised by limiting the temperature increase by limiting the peak surface magnetic field. The surface temperature rise ∆T is given by [92]

∆T = Pd

√

tpulse

2√πρkcϵ