1. Case study 6 Properties and test of a Superconducting RF cavity
Arnau Izquierdo, G.
Bajas, H.
Furci, H.
Geithner, O.
Zheng, S.
CERN Accelerator School – Erice 2013

2. I. Presentation of the problem
Goal: study of a SRF cavity in terms of geometry, energy losses, merit figures and limitations.
Basic parameters of cavity for proton acceleration operation in Continuous Wave p-mode: L
Lacc = 5L f
704.4
MHz
Epk/Eacc
3.36 - b
0.47
Bpk/Eacc
5.59
mT/(MV/m)
r/Q
173 W G
161
For a proton, the kinetic energy for b = 0.47, using relativistic relation, reads : E0
938.3
MeV E
124.7
19/11/2018
CAS - Erice

3. All parameters are independent of the material.
II. Size of the cavity
The distance L between two neighbouring cells is chosen to have synchronism between accelerated particles and wave oscillations.
L = b.c . ttrans
In p-mode, this is achieved when the transit time is equal to half of the period of the radiofrequency.
L = b.c / (2f) L
0.10 m
Lacc=5*L
0.50
With a wave length
l = c / f
it yields
L = b.l/2 f
704.4
MHz
Epk/Eacc
3.36 -
beta
0.47
Bpk/Eacc
5.59
mT/(MV/m)
r/Q
173 W G
161
𝑃 𝑐 = 1 2 𝑅 𝑠 𝑆 𝐇 2 𝑑𝑠
𝐺= 𝜔 0 𝜇 0 𝑉 𝐇 2 𝑑𝑣 𝑆 𝐇 2 𝑑𝑠
𝑟 𝑄 = 2 𝑉 𝑐 𝜔 0 𝜇 0 𝑉 𝐇 2 𝑑𝑣
𝑟 𝑄 = 𝑉 𝑐 𝑃 𝑐 𝑄 = 𝑉 𝑐 2 𝑅 𝑠 𝑃 𝑐 𝐺
“r” is often referred to as the geometric shunt impedance. The ratio r/Q depends only on the cavity geometry as does the geometric factor G.
All parameters are independent of the material.
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4. III. Superconducting Niobium cavity
One of the objectives is to estimate the dissipation at the surface of the cavity.
Surface resistance estimated from the BCS theory:
In case of Niobium, this becomes (f in GHz and T in K) :
𝑅 𝐵𝐶𝑆 =2. 10 −4 ∙ 1 𝑇 𝑓 𝑒𝑥𝑝 − 𝑇
RBCS 2.0 K)
3.21E-09 W
RBCS 4.3 K)
1.68E-07
A factor of 50!
To choose the operating temperature, one takes into account the energy required to cool the system.
𝑃 𝑙𝑜𝑠𝑠𝑒𝑠 =𝜂 𝑚𝑎𝑐ℎ𝑖𝑛𝑒 ∙ 𝜂 𝐶𝑎𝑟𝑛𝑜𝑡 ∙ 𝑃 𝑐𝑜𝑜𝑙𝑖𝑛𝑔
Assuming the same field conditions…
𝑃 @2𝐾 𝑙𝑜𝑠𝑠𝑒𝑠 𝑃 @4.3𝐾 𝑙𝑜𝑠𝑠𝑒𝑠 = 𝜂 𝑚𝑎𝑐ℎ𝑖𝑛𝑒 ∙𝜂 @2𝐾 𝐶𝑎𝑟𝑛𝑜𝑡 ∙ 𝑃 @2𝐾 𝑐𝑜𝑜𝑙𝑖𝑛𝑔 𝜂 𝑚𝑎𝑐ℎ𝑖𝑛𝑒 ∙𝜂 @4.3𝐾 𝐶𝑎𝑟𝑛𝑜𝑡 ∙ 𝑃 @4.3𝐾 𝑐𝑜𝑜𝑙𝑖𝑛𝑔
Operate at 2K!
𝑃 @2𝐾 𝑐𝑜𝑜𝑙𝑖𝑛𝑔 𝑃 @4.3𝐾 𝑐𝑜𝑜𝑙𝑖𝑛𝑔 = 𝜂 @4.3𝐾 𝐶𝑎𝑟𝑛𝑜𝑡 ∙ 𝑃 @2𝐾 𝑙𝑜𝑠𝑠𝑒𝑠 𝜂 @2𝐾 𝐶𝑎𝑟𝑛𝑜𝑡 ∙ 𝑃 @4.3𝐾 𝑙𝑜𝑠𝑠𝑒𝑠 = −2 300− ≈ 1 25
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5. III. Superconducting Niobium cavity
A more realistic estimation of the resistance should take into account phenomena as:
Trapped magnetic field
Normal conducting precipitates
Interface losses
Grain boundaries
Subgap states
Rs[nW]
BCS model:
Nb, 0.7 GHz
All contributions:
High T
Low T
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6. IV. Accelerating gradient
The corresponding accelerating gradient in operation reads:
Eacc= Vc / Lacc with R / Q0 = Vc2 / ω U
While doing the test, a stored energy of U = 65 J was measured, which yields:
Vc = 7 MV and Eacc = 14 MV/m
V. Dissipated power
The quality factor can be computed using the surface resistance of the material and the geometry factor:
Q0 = G / Rs ≈ 5E+10
Qo can also be expressed in terms of energy considerations, then the dissipated power in the cavity walls (in cw operation) can be calculated as:
Q0 = ω U / Pc  Pc = 2π f U / Q0 = 5.7 W
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7. VI. Quality factor and frequency response
As we have seen, if we only consider surface resistance losses, the quality factor is Q0 ≈ 5E+10.
However, the power exchanged with the coupler significantly lowers the quality factor:
1 / QL = 1/Qext + 1/Q where Qext = ω U / Pext
For U = 65 J and Pext = 100 kW  QL = 2.9E+6
The quality factor is associated to the bandwidth:
1/Q = Δf / f0
for Q0, (Δf)0 = 14 mHz
for QL, (Δf)L = 245 Hz f
Intensity Q0 QL
During operation, E and B fields can change slightly the cavity shape, as well as microphonics, pressure fluctuations. Thus the resonance frequency will shift (for fields, some Hz/(MV/m)2).
To ensure the operating frequency is at the nominal value, fine tuning is necessary.
This can be accomplished applying a compensatory mechanical deformation to the cavity.
Also, stiffening rings can be of help.
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8. VII. Limitations (Eacc)max = 34MV/m For Nb @ 2K, Bquench = 190 mT
If at some point of the cavity walls the magnetic field locally exceeds the critical field, the cavity can quench.
For 2K, Bquench = 190 mT
The limit on the maximum gradient is imposed by the magnetic field peaking point.
mT/(MV/m)
(Eacc)max = 34MV/m
The quench preferably initiates at the equator where the field is the highest.
But in real cavities the limit can be lower because of imperfections, such as:
grain boundaries, normal conducting precipitates,
hot spots (at higher T, lower Bcrit),
Surface imprefections
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9. Synthesis
During this case study we had a closer approach to the design of RF cavities.
We have seen how to choose the size of a cavity given a fixed geometry and a particle velocity requirement.
We have observed the limitations of the models to determine surface resistance and losses and how to use this notions to choose the operation temperature.
We have had an insight of how losses influence frequency response through the quality factor.
We discussed the limitations to the cavities coming from quench and real cavity imperfections.
19/11/2018
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