This content has been downloaded from IOPscience. Please scroll down to see the full text. Download details: IP Address: 128.252.67.66 This content was downloaded on 01/07/2014 at 12:30 Please note that terms and conditions apply. Evidence for thermal boundary resistance effects on superconducting radiofrequency cavity performances View the table of contents for this issue, or go to the journal homepage for more 2014 Supercond. Sci. Technol. 27 085004 (http://iopscience.iop.org/0953-2048/27/8/085004) Home Search Collections Journals About Contact us My IOPscience
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Evidence for thermal boundary resistance effects on superconducting radiofrequency cavity
performances
View the table of contents for this issue, or go to the journal homepage for more
2014 Supercond. Sci. Technol. 27 085004
(http://iopscience.iop.org/0953-2048/27/8/085004)
Home Search Collections Journals About Contact us My IOPscience
Received 13 February 2014, revised 12 April 2014Accepted for publication 13 May 2014Published 25 June 2014
AbstractThe majority of the literature on superconducting cavities for particle accelerators concentrateson the interaction of a radiofrequency (RF) electromagnetic field with a superconductor cooled inliquid helium, generally either at a fixed temperature of 4.2 K or 1.8 K, basing the analysis ofexperimental results on the assumption that the superconductor is at the same temperature as theinfinite reservoir of liquid helium. Only a limited number of papers have extended their analysisto the more complex overall system composed of an RF field, a superconductor and liquidhelium. Only a few papers have analyzed, for example, the problem of the Kapitza resistance, i.e.the thermal boundary resistance between the superconductor and the superfluid helium. Amongthem, the general conclusion is that the Kapitza resistance, one of the most controversial and lessunderstood topics in physics, is generally negligible, or not relevant for the performanceenhancement of cavities. In our work presented here, studying the performance of 6 GHzniobium (Nb) test cavities, we have discovered and studied a new effect consisting of an abruptchange in the surface resistance versus temperature at the superfluid helium lambda transition Tλ.This abrupt change (or ‘jump’) clearly appears when the RF measurement of a cavity isperformed at constant power rather than at a constant field. We have correlated this jump to achange in the thermal exchange regime across the lambda transition, and, through a simplethermal model and further reasonable assumptions, we have calculated the thermal boundaryresistance between niobium and liquid helium in the temperature range between 4.2 K and 1.8 K.We find that the absolute values of the thermal resistance both above and below the lambda pointare fully compatible with the data reported in the literature for heat transfer to pool boilinghelium I (HeI) above Tλ and for the Kapitza interface resistance (below Tλ) between a polishedmetal surface and superfluid HeII. Finally, based on the well-documented evidence that thesurface status of metal to liquid helium influences the heat exchange towards the fluid, andspecifically the Kapitza resistance below Tλ, we have tested an anodization process external tothe cavity, comparing the performances of the cavity before and after external anodization. Thetests were done without breaking the vacuum inside the cavity or modifying the innersuperconducting layer in any way, and were repeated on different samples. The results show thatwhen the cavity is externally anodized, both the Q-factor and the maximum accelerating fieldincrease. Again, when the oxide layer is removed, the Q-factor shifts towards a lower level andthe maximum accelerating field is also reduced.
(Some figures may appear in colour only in the online journal)
1. Introduction
The optimization of superconducting radiofrequency (RF)cavity performance is still a relevant issue for future particleaccelerator development. The main parameters are the abso-lute value of the cavity quality factor Q, the maximum valueof the accelerating electric field (breakdown field) and the ‘Q-slope’, i.e. the level of degradation of Q for increasing valuesof the accelerating field.
If the applied RF field can be considered homogeneousover the cavity surface, the quality factor Q is directly relatedto the superconductor surface resistance Rs by the relationQ=Γ/Rs, where Γ is a constant related to the cavity geometry(temperature- and field-independent).
The surface resistance at zero field is usually written asthe sum of two terms:
= +( ) ( )R T R T R . (1)s BCS 0
RBCS (T) is the ‘ideal’ theoretical, strongly temperature-dependent, surface resistance as given by the BCS theory andR0 is a temperature-independent residual term that accountsfor a large set of possible ‘spurious’ effects [1]. Since the BCSterm goes to zero at low temperatures, this last term generallydominates at very low temperatures and at low frequency.
The BCS term has a fairly complex expression that needsto be computed numerically [2]. In specific conditions (lowfrequency in respect to the gap frequency, dirty and locallimits, temperature T≪Tc) some approximations can bemade, leading to the simplified expression:
ω Δ= −
⎛⎝⎜
⎞⎠⎟( )R T
A
T K Texp (2)BCS
B
20
(where Δ0 is the low temperature value of the energy gap, KB
is the Boltzmann constant and A is a weak frequency-dependent constant, whose value is determined by the mate-rial parameters [3, 4].
Equation (2) is generally assumed to be valid for standardaccelerating cavities and essentially almost all papers in thefield use this expression for the BCS term to fit the data.
The power dissipated by the RF field per unit area over adepth λ from the cavity inner surface (λ is the magnetic fieldpenetration depth, generally less than 100 nm) can be writtenas:
= ( )P R T H1
2(3)d s RF
2
where HRF is the amplitude of the RF magnetic field. Thecorresponding magnetic induction μ0HRF is directly propor-tional to the accelerating field Eacc, through a constant that isequal to 4.5 mT/(MV/m) for the elliptical cavities of theTESLA design [1] accelerating relativistic electrons (v/c = 1).
If surface resistance were independent of field level, acavity characterization measurement would reveal a flatresponse of Q versus Eacc up to a certain critical field. Inpractice the Q-factor versus field tends to decay with a severeor less severe slope, depending on the case, implying a field-dependent surface resistance (increasing with the fieldamplitude level). Such field dependence can be present inprinciple both in the BCS terms (through the weak depen-dence of the energy gap Δo from HRF [5]) and in the residualterm. In the latter case (far more often considered) the specificform of the function R0 (HRF) strongly depends on the specificmechanism considered to be relevant in determining theresidual losses. Although a large number of papers in theliterature have been dedicated to trying to finally solve the so-called ‘Q-slope’ problem [6–9], no general consensus existsand the real mechanism determining this important effect hasnot been fully established.
In addition to the possible field dependence of the surfaceresistance, another effect also has to be considered whenanalyzing the Q versus Eacc cavity data. In fact, the heatgenerated by the RF power dissipation can produce a tem-perature increase of the inner cavity superconducting layer inrespect to the liquid helium bath temperature where the cavityis immersed. Using a simplified but fully effective ‘one-dimensional’ thermal model, as schematically reported infigure 1, the temperature difference ΔT between the innercavity surface T and the bath temperature T0 can be written as:
Δ = − = +⎛⎝⎜
⎞⎠⎟T T T
d
kR P . (4)
mB d0
Here d is the thickness of the cavity wall, km is thethermal conductivity of the superconducting cavity materialand RB represents the thermal resistance present at theinterface between the cavity outer surface and the He bath.
The thermal boundary resistance RB has an entirely dif-ferent nature depending on whether it is above or below thesuperfluid helium transition temperature Tλ.
HeI (above Tλ) is in fact an ordinary liquid. The heattransfer from a metal surface to a pool of boiling liquid occursvia a strong and complex convective process involving theformation of bubbles. The details of the process stronglydepend on many factors (surface chemistry and roughness,system geometry, etc) and the behavior is generally char-acterized by some degree of hysteresis [10].
In the superfluid regime, below Tλ (HeII), boiling is lar-gely suppressed and the thermal boundary resistance can beidentified with the Kapitza resistance RK. Indeed, followingKapitza’s original work [11], an extended theoretical andexperimental activity has been carried out to fully understandand quantify the effect. The first theory of the Kapitza resis-tance, proposed by Khalatnikov [12], was based on the cal-culation of the acoustic phonon transmission mismatch at a
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Supercond. Sci. Technol. 27 (2014) 085004 V Palmieri et al
perfect solid–liquid HeII interface. However, the predictedvalues for RK were far too high compared to the experimentalresults. Theoretical calculations, based on the phonon diffusescattering at a ‘rough’ interface (diffuse scattering model,[13]), gave results not far from the so-called ‘phonon radia-tion limit’ [14], and in this case the resulting RK values turnedout to be too low in respect to the experimental results. Thelatter were, in any case, not fully set, giving contradictoryresults between different papers and changing the measure-ment methodology.
It is worth mentioning that, as predicted by Landau [15],a further contribution to the boundary resistance comes fromthe fact that although in bulk superfluid 4He the heat fluxgenerates a counter-flow between the normal fluid and thesuperfluid, and no net thermal resistance is present, thecounter-flow is suppressed in the vicinity of the solid wall.This ‘singular’ contribution becomes fairly relevant only veryclose (in the μK region) to Tλ [16]; its observation requiresspecific techniques and is not relevant to the presentdiscussion.
In the presence of a thermal gradient between the innercavity surface and the helium bath, the BCS surface resistancecan be rewritten as:
ωΔ
ΔΔ
=+
−+
⎡⎣⎢⎢
⎤⎦⎥⎥( ) ( )
R TA
T T K T Texp (5)BCS
B0
2
0
0
0
with Δ ∝ ∝T P Hd rf2. This induces a dependence of the surface
resistance on the RF field, and a corresponding Q-slope effect.The relevance of the thermal effects described above on
the accelerating cavity behavior has already been discussed inthe literature. In a specific test experiment using a heater as apower source, Amrit and Francois analyzed the relativeweight of the Kapitza contribution and Nb thermal conductionas a function of different Nb surface preparations at tem-peratures well below Tλ, concluding that the Kapitza resis-tance can become predominant in the thermal behavior ofsuperconducting cavities for very clean Nb surfaces [17].
A fairly complete treatment of thermal effects has beengiven by Bauer et al [18]. In their paper the authors suggestthat in order to fully describe many Q versus Eacc data onCEBAF 1.5 GHz cavities, at temperatures between 1.6 K and
2.0 K, the inclusion of both direct RF field dependence of thesurface resistance and thermal effects have to be considered.In a later paper on 3.9 GHz nine-cell cavities at FERMILAB[19], using the thermal model only, a reasonable agreementwith the data is instead found in the same temperature range.
However, with a few exceptions (including those men-tioned above), the current assessment in the literature is thatthermal effects can be essentially neglected, at least at low ormoderate input power, and that the main mechanism causingthe Q-slope problem is the ‘direct’ dependence of the surfaceresistance on the RF field, often invoking the RF dissipationof vortices penetrating inside the superconductor.
In the present work we present new data on the Q-factoras a function of temperature and accelerating field (inputpower) on small test bulk Nb 6 GHz cavities, at a wide rangeof temperatures and input power. The extracted Rs versus Tdata show in all cases a significant anomaly around Tλ; morepronounced for increasing RF power. This anomaly is oftenseen in the literature at the same temperature range (see, forexample, the recent paper by Dhakal et al [20]).
In the literature the Rs(To) data are always reported at aconstant RF field. We noticed that if the Rs(To) data arereported at constant input power, the anomaly close to thehelium superfluid transition shows up as an abrupt change ofthe Rs value at Tλ, evidently reflecting the abrupt change ofregime in the heat transmission to the He bath that indeedoccurs at that temperature, due to the change from HeI to HeIIthermal conduction regimes. This has suggested in turn thatthe thermal effects, though often discussed in the past, havebeen underestimated in the recent literature on acceleratingcavities, and should be reconsidered.
In particular we will show that, from the Rs (T) experi-mental data, using equations (1), (4) and (5) (withoutassuming any explicit RF field dependence of Rs(T)) andthrough appropriate fitting procedures, it is possible to extractthe absolute values and the temperature dependence of thethermal boundary (Kapitza) resistance RB (RK). We will showthat the thermal boundary resistance obtained by this proce-dure exhibits values and an overall temperature dependencesubstantially consistent with measurements and theoreticalestimations typically reported in the literature. Finally we willshow that measuring cavities identical in all respects, but withdifferent external surface treatments aimed to reduce theKapitza thermal boundary resistance, we can get a significantimprovement in terms of the Q versus Eacc characteristics.
The paper is organized as follows. The next section willbe devoted to the description of the experimental procedures.In particular, the unique advantages of our 6 GHz cavities willbe described, as well as the fabrication procedure, the dif-ferent tools and techniques employed, and our cryogenic andRF measurement system.
Following that, we will present our experimental resultsand a discussion on the dependence of the surface resistanceon temperature and RF power, in line with our results.
The discussion will make clear the possible relevance ofexternal surface treatments in order to reduce the thermalboundary resistance and improve the overall cavity
Figure 1. Electrical analog (one-dimensional model) for the heattransfer in a superconducting cavity.
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Supercond. Sci. Technol. 27 (2014) 085004 V Palmieri et al
performance. This will be shown and discussed in the lastsection, devoted to analyzing the effect of the external cavitysurface.
The conclusion will finally summarize the obtainedresults and the consequent implications.
2. Experimental procedures
2.1. The 6 GHz resonators
In order to achieve higher and higher cavity performances, itis necessary to have extremely reliable manufacture protocolsvalidated by reliable statistics from RF tests on super-conducting materials. However, dealing with a large-sizeresonator is of high cost and time-consuming. Therefore theuse of superconducting samples inside suitable test super-conducting resonators is quite common. However, RF mea-sured samples will never be comparable to a real large cavity,always being an indirect measurement.
Our strategy instead has been based on the developmentof small 6 GHz cavities, which are easy to handle as a sample,but remain ‘real’ cavities [21].
The 6 GHz cavities will produce many statistics at alower cost and effort. Therefore, the main advantage of thismethod lies in it being a simple tool to open the road to newresearch, or to prove a phenomenon with a large number oftests at a limited cost. However, the extrapolation of quanti-tative information to other cavity frequencies, for instance theQ-slope versus the accelerating field, requires a deeper con-cern. The RF results cannot be straightforwardly scaled tolower frequencies with the square of the frequency ratio, sinceat 6 GHz the role of BCS losses is still very high anddominant.
Working on 6 GHz cavities has not often been consideredin the past due to major technological problems using stan-dard electron beam welding caused by the small bore of thecavity cutoff tubes.
Thanks to the spinning technology developed at INFN-LNL, our 6 GHz cavities are totally seamless [22]. Spinning isa point deformation process by which a metal disc, or acylindrical preformed hollow component, is plasticallydeformed by axial or radial motions of a tool or rollers actingon a workpiece clamped against a rotating chuck. Thispeculiar fabrication technique is a chip-less productionmethod of forming axially symmetrical hollow parts of almostany shape, and it is an ideal seamless fabrication technologyfor producing small cavities. Totally seamless cavities can bestraightforwardly spun from a circular blank by this coldforming process and no intermediate annealing is required.Flanges are also directly spun together with the cavity.Kapton joints instead of the traditional indium wires are usedto vacuum seal the cavity.
Our Nb 6 GHz cavities are 97 mm long and have a45 mm diameter cell (see figure 2), an electrical length of25 mm and the same geometrical factor Γ of a large resonator,equal to 287Ω (assuming Γ=R QS ). The area of the centralcell exposed to the RF fields is about 42 cm2.
The cutoff tubes have a length equal to three times thediameter in order to damp the RF fields.
6 GHz cavities can be produced in large numbers at lowcost from a relatively small amount of niobium and allow usto obtain many experimental statistics. The reason is thatwhile a large accelerating cavity requires expensive and time-consuming treatments, due to the small size of the cavities, allthe surface treatments (mechanical polishing, buffer chemicalpolishing, electropolishing, high-pressure rinsing, alcoholrinsing, thermal treatments and many other surface treatmentssuch as purifications and even the RF test) can be performedquickly and at low cost. Even usage of the tool for surfacetreatments (tumbler, chemical plant, high-temperature fur-nace, cryostats) is reduced and less expensive than that forlarge cavities. With a tool like this, it is possible to studytraditional and innovative surface treatments as well as newthin-film superconducting materials grown for example bysputtering or thermal diffusion techniques.
The spinning process implies material surface defects,stress and lattice dislocations. Obviously the internal surfacefinishing of a resonant structure is directly correlated to itsperformance, especially at high fields. In order to removesurface roughness and contaminations introduced during thespinning process, the cavity is mechanically polished bytumbling with silicon carbide media, electropolished up to theremoval of 300 μm in a mixture of H2SO4:HF at a ratio of 9:1with an aluminum cathode, then high-temperature annealed inUHV at 2000 °C for 60 s [23].
2.2. Cavity assembly and cryogenics
After the 100 bar deionized water rinsing, the resonators arefast mounted in a class 1000 clean room on a removableattachment as shown in figure 3. The attachment is then fixedto its own RF cryogenic vertical stand that is preliminarilypumped by a turbo-molecular pump and subsequently by anionic pump of 2 L sec−1 in order to avoid vibrations once theRF cryogenic stand is plunged into the cryostat.
Figure 2. A set of 6 GHz Nb monocell cavities.
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Supercond. Sci. Technol. 27 (2014) 085004 V Palmieri et al
This removable attachment consists of two parts: acoupler top flange, equipped with an edge-welded bellowthat allows the motion of the coupler antenna, and a pick-upbottom flange, equipped with an SMA coaxial feedthroughwelded to the antenna that picks up the RF signal. The cavityis sealed to the a.m. flanges by 100 micron Kapton joints.The cavity is pumped through a double 8 mm diametervacuum pumping line, passing through the coupler flangeand the pick-up flange.
Since the cavity is equipped with a movable coupler, theRF measurement always uses critical coupling, therebyensuring higher precision than when working either under-coupled or over-coupled in the fixed coupler configuration.
The stand is slowly inserted into a cryostat using anoverhead traveling crane and cooled at the temperature ofthe liquid helium (4.2 K) for the measurement of the Q-factor versus accelerating field. After that, the bath is readyto be pumped down in pressure to gradually reach the ulti-mate temperature of 1.8 K in order to study the variation ofcavity surface resistance as a function of the temperature.
Two Lake Shore calibrated thermometers, a silicondiode (calibrated in the range 1.4–500 K) and a negative T-coefficient germanium thermometer (calibrated in the range0.05–100 K), immersed in the liquid helium bath at a 1 cmdistance from the cavity ensure the reliability of the tem-perature measurement. For every temperature, the RF dataacquisition is done only when the temperature is stabilizedand both the thermometers measure a value within 5 hun-dredths of a Kelvin of each other.
2.3. The radiofrequency measurement system
The Q measurement and RF processing of the resonatortake advantage of computer-controlled procedures whichcommunicate with the devices, collect data and assist theoperator during the measurements. In figure 4, a schematicdiagram of the RF measurement system is shown. Thecomputer is interfaced with the signal generator, powermeters and frequency counter via an IEEE 488 bus. Theauxiliary equipment (not shown) such as the helium levelmeter, temperature, vacuum, pressure sensors and steppingmotor control are connected by an RS232 bus. The RFgenerator signal is divided into three paths: one goesthrough the programmable phase shifter to feed an LOmixer input, the second goes to the frequency counter,while the third one arrives at the power amplifier through aPIN diode switch and a programmable PIN diode attenua-tor. A bi-directional coupler allows monitoring of the for-ward and reflected powers in the feeding line that aremeasured by a two-channel power meter. The pick-upsignal passes through a computer-controlled RF relay forsensitivity selection and a low-noise amplifier, and it is alsosplit into two paths. The first reaches the mixer RF inputthrough a limiting amplifier optimizing mixer performance;the second leads to a second power meter. The phase errorsignal, coming from the intermediate frequency mixeroutput, is amplified and filtered by the dc amplifier andmonitored by USB control card ADC. The RF powerdividers/combiners are used to organize the RF signal flow.The fixed attenuators are used to optimize the signal levelin different parts of the system. We use the five-stepsemiautomatic calibration procedure for all the elements upto the cavity ports [24].
After that, we determine the cavity Q at a low fieldcavity in critical coupling. We feed the cavity, locked inphase at its resonant frequency, and adjust the couplerposition to minimize the cavity reflected power. At equi-librium (stable cavity field), the Q measurement procedureis started. It turns off the power and interpolates the pick-updecay data in order to determine the cavity decay time andthe related Q-value. From the Q-value and from the mea-sure of feeding and transmitting power, the program thencomputes the pick-up accelerating field coefficient (knownas the ratio between stored energy and the square of theaccelerating field). The accelerating field and the corre-spondent Q-value at higher accelerating fields can then bedetermined by increasing the feeding power and adjustingthe coupling in order to maintain the critical couplingcondition. In detail, what happens is that once the tem-perature value is stabilized at a fixed value, the Q-factorvalue is acquired at different values of the RF input power.Therefore, for every temperature value, the program auto-matically plots the triad of Q-factor, accelerating field andRF input power values, allowing a fast recording andanalysis of the data together with all the auxiliary para-meters [24, 25].
Figure 3. The cavity, equipped with a pick-up antenna and movablecoupler antenna, mounted onto a removable attachment that will bemounted onto the cryogenic stand.
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Supercond. Sci. Technol. 27 (2014) 085004 V Palmieri et al
3. Experimental results and discussion of untreatedcavities
As mentioned in the introduction, the starting point of ourwork was to notice that in most cases the Rs(T) valuesextracted from accelerating cavity Q-measurements reportedin the literature showed a ‘jump’ or abrupt change just abovethe HeI-HeII transition (lambda point, Tλ= 2.18K, 1/Tλ= 0.46 K−1) (as an example, refer to Dhakal et al [20]). Thiscircumstance, also systematically found in our own mea-surements, is not commented on in the literature, possibly dueto the simple consideration that in that temperature regionsome temperature instability can obviously occur. However,we noticed that if the Rs(T) data are taken keeping a constantinput power, the anomaly is far more clear and defined, andexhibits an abrupt change of the Rs value exactly at Tλ.
The conceptual difference between the data taken at aconstant RF field or constant RF power is illustrated infigure 5. According to equation (3), if we decrease thetemperature at a constant field, we will also decrease thevalue of the RF power; if we decrease the temperature atconstant power, instead, we will increase the value of themagnetic field. Therefore the acquisition of the curve of RS
versus 1/T at a constant field will be useful for the experi-mental data treatment in the framework of an electrodynamicmodel approach. The acquisition of the same curve at con-stant power will be more useful in the framework of athermal model approach.
In figure 6 the Rs(T) data obtained from the Q mea-surements at constant RF power are reported for one of our6 GHz Nb cavities following the experimental proceduresdescribed in the previous section.
The reported behavior is found with the same generalcharacteristics in all our measured samples (over 140 mea-surements of about 100 Nb 6 GHz cavities).
We can get more insight into this behavior consideringthat typically reported values in the literature for the tem-perature difference ΔT’ for heat exchange between a metallicsurface and boiling liquid helium above the lambda transi-tion are at maximum of the order of 0.1 K for input powerdensity up to 5*10−3 W cm−2 [10]. This corresponds to themaximum input power level of 200 mW that we used in ourmeasurements (this value of the power density is estimated
Figure 4. Schematic diagram of the RF measurement system for 6 GHz superconducting cavities.
Figure 5. Conceptual difference between Q data taken at constant RFfield and constant RF power.
6
Supercond. Sci. Technol. 27 (2014) 085004 V Palmieri et al
considering that the power input can be assumed to beuniformly distributed over a cavity surface of about 42 cm2
for our 6 GHz cavities). Moreover, for our cavities, given thehigh quality of our Nb (RRR≈ 300) and the small wallthickness (d = 0.2–0.3 cm), we can assume d/km⩽ 0.4 cm2 KW−1 at 4.2 K (only moderately increasingwhen lowering the temperature).
Since ΔT will be much smaller below Tλ (where thethermal boundary resistance is identified with the Kapitzaresistance) the condition ΔT≪ T will always be well satis-fied in our measurements.
In this limit we can rewrite equation (5) as:
ω Δ Δ Δ
Δ Δ
= − +⋅
≈ +⋅
⎡⎣⎢
⎤⎦⎥
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
( )
( )
R TA
T K T
T
K T
R TT
K T
exp 1
1 (6)
BCSo
B
o
B o
BCSOo
B o
0
2
0 02
0 2
where To is the bath temperature and RBCSO is the BCS sur-face resistance in the absence of thermal effects(equation (2)). From equation (4) we know that ΔT is pro-portional to the input power Pd (assuming a power-indepen-dent thermal boundary resistance, which is a reasonableassumption at low or moderate input power), so we can writefor the overall surface resistance:
Δ= +
⋅ ++
⎛⎝⎜⎜
⎞⎠⎟⎟( ) ( )
( )R T R T
P d k R
K TR1 (7)s BCSO o
o d m B
B oo0 2
(here Ro is assumed to be both temperature- and RF power-independent).
Equation (7) implies that the measured surface resistancedirectly depends on RB and this is clearly the only quantity inthe equation possibly sensitive to the HeI–HeII transition: theabrupt change in Rs at Tλ observed in figure 6 can only beascribed to a sharp change in the overall thermal boundaryresistance RB between the external cavity wall and thehelium bath.
In figure 7 we report, in a 3D plot, the evolution of theobserved ‘jump’ of Rs(T) at the helium lambda transition Tλas a function of the input power (again kept accurately con-stant) for a different sample. From the figure we can deducean approximately linear dependence of the Tλ jump as afunction of the input power and that the jump would dis-appear in the limit of zero applied RF power. Equation (7)implies indeed that the surface resistance would scale linearlywith the input power Pd at a fixed bath temperature (at a ratedepending on the overall thermal resistance). In particular, thetemperature jump at Tλ would also be expected to dependlinearly on the power, as occurs, in good approximation, withthe data reported in the figure.
It is worth pointing out now that equation (7), through
equation (3) =( )( )P R T Hd s rf1
22 , implies a square dependence
on the RF field of the BCS surface resistance term at any fixedtemperature, as already discussed by Padamsee [26] in theframe of a similar thermal mismatch model.
Moreover, equation (3) implies that if the surface resis-tance measurements were performed at a constant field (andnot at a constant power), the dissipated power at the cavityinner surface Pd would strongly decrease the lowering of thetemperature, due to the exponential decrease of Rs. Thiswould imply, in turn, a significant change in the observedtemperature dependence of Rs(To) as described byequation (7). In other words, the usual expression
= +( ) ( )R T R T Rs BCS 0, due to the finite thermal resistance, is
indeed only valid in the limit of zero power input, which, ofcourse, cannot be realized in real experiments.
Equation (7) shows that the measured surface resistanceRs(To) directly depends on the overall thermal resistance
+ Rd
k Bm. Within the assumed frame it is then possible to
extract this latter quantity by equation (8):
Δ+ =
−−
⎡⎣⎢⎢
⎤⎦⎥⎥
( )( )
d
kR
K T
P
R T R
R T1 (8)
mB
B
d
s
BCS
02
0
0 0
0 0
which is simply obtained by ‘inverting’ equation (7).In figure 8, curve 1 (squares), the temperature depen-
dence of the overall thermal resistance is reported, as deducedby equation (8) using for Rs(T0) the experimental datareported in figure 6 (the two data points just below the lambdatransition have been omitted in the fit due to the criticalthermal instability in that region). Due to the already dis-cussed small value of d/km (≪1 cm2KW−1) in our case, theoverall thermal resistance is in fact coincident with the ther-mal boundary resistance RB, in turn coincident with theKapitza resistance RK below Tλ. Given the nature of theexternal surface for the specific cavity under study (polishedbulk Nb), the circumstance that the Kapitza resistance is thedominant term in the overall thermal resistance below Tλ isnot surprising, as discussed, for example, by Amrit andFrancois [17].
The parameters used in the theoretical expression forRBCSO(To) in equation (8) to obtain the overall thermalresistance are reported in table 1.
Figure 6. Surface resistance (ln scale) versus the inverse of the bathtemperature. Measurements were taken at a constant RF power of100 mW.
7
Supercond. Sci. Technol. 27 (2014) 085004 V Palmieri et al
Δo and the parameter Aω2 were determined by extra-polating to zero power the classical BCS fitting of the Rs
curves in the high-temperature region obtained at constantdecreasing power. This is justified by equation (7), showingthat for Rs≫Ro and Pd → 0 we get RBCSO(To) =Rs(To). Thereported value of Δo implies Δo/KBTc = 1.76, very close to thetheoretical BCS value.
The value of the residual resistance Ro cannot be easilydeduced by the same procedure and has been chosen to ensurethat the low-temperature data (below Tλ) for the thermalboundary resistance follow a T−3 dependence, as theoreticallypredicted for the Kapitza resistance (continuous fitting curvein the figure).
The T−3 dependence of the Kapitza resistance [12–14] isindeed set on fairly solid grounds, simply reflecting thechange in temperature of the overall phonon number in thesolid at low temperatures. The parameters reflecting the solid/HeII interface nature determining the specific phonon inter-face scattering mechanisms between the two well-knownlimits of (1) a perfectly reflecting surface (acoustic mismatchmodel) and (2) a rough surface (diffuse mismatch model,which gives results close to the so-called ‘perfect matchinglimit’) should be essentially temperature-independent.
It is worth underlining that, although the described pro-cedure has some degree of arbitrariness, the deduced tem-perature dependence of the thermal resistance is relativelysensitive to the precise choice of the power law exponent: asan example, the assumption of a T−4.5 temperature depen-dence of the Kapitza resistance, as assumed on the basis of
Figure 7. 3D plot of the surface resistance as a function of inverse bath temperature and RF power.
Figure 8. Temperature dependence of the overall thermal boundaryresistance, as deduced by our surface resistance measurements, on atypical Nb 6 GHz cavity. Curve 1 (squares): polished Nb surface(current section). Curve 2 (diamonds): externally anodized Nbsurface (next section).
Table 1. List of parameters used in the theoretical expression ofRBCSO(To).
Δo Aω2 Ro
1.40 meV 2.18 10−3 Ω*K 20 nΩ
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Supercond. Sci. Technol. 27 (2014) 085004 V Palmieri et al
experimental evidence, for example, by Bauer et al [18],would hardly change the value of Ro or the overall shape andabsolute values of the deduced thermal resistance curve.
It is also important to notice that the absolute values ofthe thermal resistance both above and below the lambda pointare not far from the data reported in the literature for heattransfer to pool boiling HeI above Tλ [10] and for the Kapitzainterface resistance (below Tλ) between a polished metalsurface and superfluid HeII [13, 14, 17], and lie well withinthe theoretical limits of about 35 cm2KW−1 for Nb at T = 2 Kin the acoustic mismatch model and 1 cm2KW−1 in the pho-non radiation limit.
Finally, as already discussed in the introduction, thetemperature measurement sensitivity in our case is certainlynot sufficient to observe the effects predicted by Landau [15],that indeed occur in a very narrow temperature range aroundTλ [16].
4. Effect of external cavity surface treatments
In the previous section we discussed and attempted todemonstrate the relevant role of thermal effects on the surfaceresistance Rs(T,Pd) and in turn on the quality factor Q. Wealso demonstrated that, assuming a temperature- and field-independent low-temperature residual resistivity and thethermal boundary resistance reported in figure 8, curve 1, wecan perfectly describe our Rs data as a function of temperatureand power. The obvious consequence of this is, of course, toattempt to reduce the thermal boundary resistance, in order toimprove the overall cavity performance.
In fact, data in the literature clearly indicate that theKapitza boundary resistance strongly depends on surfacetreatments, being lower for etched or generally rougher sur-faces [17]. In the frame of the theory of the Kapitza resis-tance, this can be ascribed to the fact that a rougher surfaceshould tend to ‘diffuse scattering model’ behavior (in contrastto a specular surface, which should tend to ‘acoustic mis-match model’ behavior), characterized by a lower Kapitzathermal resistance. In addition, the introduction of a differentmaterial layer between Nb and the HeII bath should reduce theKapitza resistance, if the intermediate material presents an‘acoustic impedance’ Z= ρvo (ρ is the material density and vois the sound velocity) between Nb and HeII, allowing a degreeof ‘impedance matching’ for the phonon propagation at theinterface [14]. Moreover, we cannot exclude that effectsrelated to boiling processes are still active below Tλ. Thoseprocesses would certainly be strongly influenced by the sur-face roughness [27].
In figure 9 we report the surface resistance versus theinverse of the bath temperature at constant RF power for thesame cavity reported in figure 6, after an electrolytic padanodization process that grew, externally to the cavity sur-face, a blue porous Nb oxide a few tenths of a nanometerthick. The process did not influence the inner cavity surface inany case. In fact, the 6 GHz Nb cavity was initially measuredat low temperatures, then extracted from liquid helium,warmed up to room temperature and then subject to external
anodized cycling, keeping it under UHV. The anodization ofthe cavity external wall was done at room temperature by amovable electrode covered by a sponge wet in a water-dilutedsolution of 30 g l−1 citric acid at a potential of 26 Volts. Thecolor depends on the applied voltage. The anodization of thecavity exterior was performed until the niobium became anintense blue color. It is important to always keep the cavityunder UHV during the coloring. The cavity was then cooleddown to low temperature again and RF tested.
The data reported in figure 9 show a significant loweringof the surface resistance below the helium lambda transition,corresponding to an increase of the quality factor Q. This isevidently due to a lowering of the Kapitza resistance. Theexternal treatment did not influence the Rs values above Tλ asexpected due to the nature of the thermal boundary resistancein that temperature region (due, as already mentioned, to astrong convective process involving the formation ofbubbles).
The data points reported in figure 8, curve 2, refer to thesame 6 GHz cavity whose behavior was reported in the pre-vious section, and have been obtained by equation (8) usingthe surface resistance data reported in figure 9, i.e. after theexternal surface anodization process. The parameters used toobtain the data from equation (8) are the same used to extractcurve 1, and reported in table 1. Indeed, the external anodi-zation treatment, as discussed above, has no possible influ-ence on the superconducting properties of the inner surface.
The observed reduction of the Kapitza resistance belowTλ due to the different states of the external surface (from apolished surface to a ‘rough’ one) is consistent with manyreports from the literature [17, 27]. However, it is worthwhileto note that all of our experimental measurements show aparallel shift of the Q versus Eacc curve and do not seem toaffect the Q-slope in the way described by manyauthors [27–30].
Figure 9. Surface resistance at constant RF power versus the inverseof the bath temperature around the HeII lambda transition before(squares) and after (diamonds) the external anodization process (seetext). The data before anodization are the same as already discussedin figure 6.
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As reported in figure 10 (which refers to the same cavityas figure 9), the external blue anodization process increasesboth the Q-factor and the accelerating field.
When the anodized layer is further removed we see thatQ is again lowered to a level even lower than it was beforeanodization. We suppose that the reason for this is that theblue oxide layer is removed chemically in a mixture of HF,H2NO3 and H3PO4 at a 1:1:4 ratio. After the chemistry, thecavity external surface is shining, and we have observed infurther experiments that the specularity of the external surfacenegatively affects the cavity performance. As already dis-cussed, a mirror-like surface acts as a mirror for phonons,producing a higher acoustic mismatch and, in turn, a higherKapitza resistance.
Analogous results were found in several tests on differentcavities. It is important to note that all operations were donewithout breaking the vacuum and that the movable couplerallowed us to perform high-precision measurements.
5. Conclusions
From this paper we can extract some important conclusions,rather new in the panorama of RF superconductivity.
We have discovered the presence of a steep jump in thesurface resistance versus temperature curve at the superfluidtransition temperature Tλ of liquid helium, when measuring atfixed input power rather than at a fixed accelerating field (inthe latter case the jump is seen as a much smoother anomaly).This jump clearly indicates that the losses abruptly decrease atthe superfluid transition, and that the external environment isfundamental for the thermal exchange. This conclusion canappear obvious, but in practice in the literature, attention hasalways been focused on the effect of the RF fields on theinternal surface of the Nb cavity, considering the approx-imation in which the latter is in contact with an infinitereservoir of liquid helium with infinite thermal conductivity.
We have seen that when injecting more RF power in thecavity, the jump increases, indicating that the thermalexchange with the He bath becomes more and more impor-tant, and, in turn, that thermal boundary resistance effectsbecome more and more important.
From the experimental data acquired, we have found anew way to extract the thermal boundary resistance (i.e. theKapitza resistance below Tλ). We have shown that a super-conducting cavity can be a valid alternative for measuringKapitza resistance, since it is a quasi-spherical system con-fining the RF heat load almost uniformly inside itself, andabove all without the edge effects that affect thermal devicesgenerally used to measure Kapitza resistance. The extractedvalues are compatible (on the higher side) with current lit-erature data for the thermal boundary resistance both aboveand below Tλ. We then tested different external surfacetreatments on the cavity aimed at reducing the Kapitzaresistance, and proved that electrolytic pad anodization, pro-ducing porous Nb oxide a few tenths of a nanometer thick onthe external cavity wall (without affecting the internal cavitysurface in any way), systematically and significantly lowersthe Kapitza resistance and, in turn, the measured surfaceresistance below Tλ, improving the overall Q versus Eacc
cavity performance at the operating temperature.
Acknowledgements
The authors are indebted to F Stivanello for the chemical andelectrochemical treatments, M Checchin and M Martinello fortheir help in RF tests, R K Thakur, G Yu and A Battistello forthe mechanical finishing of the 6 GHz cavities, and C Pira forassistance in editing the paper. The research was developed inthe framework of the NTA-SHAMASH Experiment fundedby the 5th National Board of INFN.
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Figure 10. RF test of the cavity of figure 9. The squares refer to thecavity prepared in a standard way. The diamonds refer to the samecavity after anodization of the external surface. The circles refer tothe same cavity after the stripping of the external anodization.
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