Transcript Case Study 6c

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
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
f
Epk/Eacc
b
Bpk/Eacc
r/Q
G
704.4
3.36
0.47
5.59
173
161
MHz
mT/(MV/m)
W
W
Lacc = 5L
For a proton, the kinetic energy for b = 0.47, using relativistic
relation, reads :
 1

E0 938.3MeV
E  E0  
 1
E 124.7MeV
 1  b 2 
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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)
it yields L = b.l/2
With a wave length l = c / f
f
Epk/Eacc
beta
Bpk/Eacc
r/Q
G
704.4
3.36
0.47
5.59
173
161
0.10
L
Lacc=5*L 0.50
MHz
mT/(MV/m)
W
W
1
𝑃𝑐 = 𝑅𝑠
2
2
𝐇 2 𝑑𝑠
𝐺=
𝜔0 𝜇0
𝑆
𝑆
2
𝑟 𝑉𝑐
𝑉𝑐 𝑅𝑠
=
=
𝑄 𝑃𝑐 𝑄
𝑃𝑐 𝐺
𝑉
m
m
𝐇 2 𝑑𝑣
𝐇 2 𝑑𝑠
2
𝑟
2𝑉𝑐
=
𝑄 𝜔 𝜇
2
0 0 𝑉 𝐇 𝑑𝑣
“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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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) :
𝑅𝐵𝐶𝑆
1 𝑓
−4
= 2. 10 ∙
𝑇 1.5
2
17.67
𝑒𝑥𝑝 −
𝑇
RBCS (@ 2.0 K) 3.21E-09 W
RBCS (@ 4.3 K) 1.68E-07 W
A factor of 50!
• To choose the operating temperature, one takes into account the energy required to
cool the system. 𝑃𝑙𝑜𝑠𝑠𝑒𝑠 = 𝜂𝑚𝑎𝑐ℎ𝑖𝑛𝑒 ∙ 𝜂𝐶𝑎𝑟𝑛𝑜𝑡 ∙ 𝑃𝑐𝑜𝑜𝑙𝑖𝑛𝑔
• Assuming the same field conditions…
𝑃@2𝐾 𝑙𝑜𝑠𝑠𝑒𝑠
𝑃
@4.3𝐾
=
𝑙𝑜𝑠𝑠𝑒𝑠
𝑃@2𝐾 𝑐𝑜𝑜𝑙𝑖𝑛𝑔
𝑃@4.3𝐾 𝑐𝑜𝑜𝑙𝑖𝑛𝑔
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𝜂𝑚𝑎𝑐ℎ𝑖𝑛𝑒 ∙ 𝜂@2𝐾 𝐶𝑎𝑟𝑛𝑜𝑡 ∙ 𝑃@2𝐾 𝑐𝑜𝑜𝑙𝑖𝑛𝑔
𝜂𝑚𝑎𝑐ℎ𝑖𝑛𝑒 ∙ 𝜂@4.3𝐾 𝐶𝑎𝑟𝑛𝑜𝑡 ∙ 𝑃@4.3𝐾 𝑐𝑜𝑜𝑙𝑖𝑛𝑔
=
𝜂@4.3𝐾 𝐶𝑎𝑟𝑛𝑜𝑡 ∙ 𝑃@2𝐾 𝑙𝑜𝑠𝑠𝑒𝑠
𝜂@2𝐾 𝐶𝑎𝑟𝑛𝑜𝑡 ∙ 𝑃@4.3𝐾 𝑙𝑜𝑠𝑠𝑒𝑠
=
4.3 300 − 2 1
1
≈
2 300 − 4.3 50 25
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Operate
at 2K!
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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]
III. Superconducting Niobium cavity
Nb, 0.7 GHz
BCS model:
All contributions:
High T
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Low T
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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
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 Pc = 2π f U / Q0 = 5.7 W
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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/Q0 where Qext = ω U / Pext
For U = 65 J and Pext = 100 kW  QL = 2.9E+6
Intensity
The quality factor is associated to the bandwidth:
1/Q = Δf / f0
Q0
QL
f
for Q0, (Δf)0 = 14 mHz
for QL, (Δf)L = 245 Hz
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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VII. Limitations
If at some point of the cavity walls the magnetic field locally exceeds the critical field,
the cavity can quench.
For Nb @ 2K, Bquench = 190 mT
The limit on the maximum gradient is imposed by the magnetic field peaking point.
B pk
Eacc
 5.59 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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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.
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