THERMAL MODELING OF RING-TYPE DEFECTS

Yi Xie, Matthias Liepe, David Meidlinger, Hasan PadamseeCornell Laboratory for Accelerator-Based Sciences and Education,

Cornell University (CLASSE), Ithaca, NY 14853, USA

AbstractThermometry and optical inspection results suggest that

pit-like structures in the high magnetic field region of anSRF cavity are possbile defect candidates that cause cavityquench. Previous thermal modeling of pit-like structuresbased on a disk-type defect model suggests that the ob-served pit is much larger than the actual normal conductingregion responsible for initiating the quench. Guided by themagnetic enhancement calculations of pits, a ring-type de-fect could be a better model for quench caused by the sharpboundary segment of a pure niobium pit. The relationshipbetween quench field and inner radius of a ring-type defectis presented based on calculations of an improved ring-typedefect model.

INTRODUCTION

The maximum field achieved by superconducting rf cav-ities is limited by the thermal breakdown initiated fromso-called ”defect” areas in the high magnetic field regionaround the equatorial welding zone. Optical inspectiontechniques found that many of these defect regions canbe categorized as pit-like structures [1]. Although mostof the pits found did not cause quench at the magneticfield achieved, some of them are responsible for initiaingquench. This indicates that sizes, depth and edge sharpnessof the pits may determine the quench onset field. Previousthermal modeling of pit-like structures based on a disk-typedefect model suggest that the observed pit is much largerthan the actual normal conducting region causing quench[2]. Figure 1 shows an actuall pit which causes a cavityquench at maximum cavity surface magnetic field 1200Oewith its diameter of nearly 1mm. Calculations of the mag-netic field enhancement at pits and bumps on the surface ofsuperconducting cavities show that the field enhancementfactor for pits can reach the value of 4 depending on theslope of the edge [3]. Figure 2 shows the relative high cur-rent density distribution calculated by Microwave Studio@

at the edge of a pit structure modeled as a hole on the axisof a pill-box cavity. This indicates that the edge of a pitor a part of it can become normal conducting due to thestrong magnetic field enhancement. In order to study thequench initiated from such a ring-shaped normal conduct-ing region at the edge of a niobium pit, a ring-type defectthermal model was developed. Due to the axi- symmetricmesh configurations in the thermal model, arc-type defectmodel can not be simulated currently.

Figure 1: SEM image of a quench causing pit-like defectfound in the cavity LE1-HOR.

Figure 2: Current density distribution on a pit presented asa hole on the axis of a pill-box cavity. [3]

THERMAL MODELING AND CODEBENCHMARK

The 2-d ring-defect thermal program was developedfrom a disk-defect thermal program. Heat balance equa-tions and boundary conditions are the same as they wereused in a disk-defect model [2]. Figure 3 shows the meshconfiguration difference between a ring-type defect and adisk-type defect. For a ring-type defect, the program splitsa cylindrical section of the niobium wall into many circularring-shaped mesh elements. The normal conducting de-fect is located in a ring section located at a certain distanceto the center of the entire modeled niobium disk. Giventhe temperature dependent thermal conductivity of niobiumand Kapitza conductance between niobium and helium, r.f.power produced at the surface is compared with poweremitted into the helium bath at a given iteration number.The over-relaxation method is used to estimate the (n+1)-th iteration from n-th iteration. Once the two heat transfer

Proceedings of SRF2009, Berlin, Germany TUPPO048

05 Cavity performance limiting mechanisms

331

numbers are sufficiently equal (e.g. their difference is lessthan 0.01%), thermal equilibrium is reached. For initialstudies, the defect resistance was taken as 10mΩ which isassumed as the normal resistance of niobium. The real nor-mal conducting surface resistance of a given niobium pitedge does depend on temperature, and therefore the defectregion may have a large range of resistances (1∼10mΩ).To speed up simulations, the mesh density is high nearthedefect element and lower away from it where temperaturegradient are smaller. The mesh spacing in the radial di-rection was chosen to increase exponentially (the distancebetween the i-th element to the ring-defect is proportionalto exp(i)). The z direction is also meshed using an expo-nential function. Figure 4 is shown to verify that the calcu-lated results are independent of the different mesh densityconfigurations. It shows that the results obtained from thethermal model become independent of the mesh density forsufficiently dense meshes.

(a) Mesh configuration for a disk-type defect

(b) Mesh configuration for a ring-type defect

Figure 3: Different mesh distributions of ring-type anddisk-type defect models with normal conducting (red) andsuperconducting (blue) mesh elements.

To test the ring-shape defect thermal modeling program,we have calculated the heating due to a ring-type defectand a disk defect. The ring defect inner radius is 1μm witha ring width of 49μm and the disk defect radius 50μm. Thetemperature distribution of those two cases should approx-imately be the same because the rf deposited power in bothcases was is nearly equal. Figure 5 shows that the calcu-lated temperature distribution agrees very well indeed forthe two cases. In a second test of the program, a large ring-type defect with an inner radius 5mm and ring width 1μm.The result of this simulation is shown in Figure 6. The ringinduced heat distribution is nearly symmetric as expected.

2

2.5

3

3.5

4

4.5

5

5.5

6

1e-06 1e-05 1e-04 0.001 0.01 0.1

RF

sur

face

tem

pera

ture

T(K

)

Radial distance from the center of Nb slab r(m)

f=1.5GHz,RRR=300,PMFP=1mm,Tb=2K

Hpeak=800 Oe, Ring defect radius R=20μm, width ΔR=5μm

Mesh configuration 1 (insufficient)Mesh configuration 2Mesh configuration 3Mesh configuration 4

Figure 4: rf surface temperature distribution along radialdirection for a given shape ring-type defect with differentmesh densities

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

1e-07 1e-06 1e-05 1e-04 0.001 0.01 0.1

RF

sur

face

tem

pera

ture

T (

K)

Radial distance from the center of Nb slab r(m)

f=1.5GHz, RRR=300, PMFP=1mm, Tb=2KHpeak=500 Oe, Defect resistance Rd=0.01Ω

Point defect model R=50μmRing defect model, R=1μm, ΔR=49μm

Figure 5: rf surface temperature distribution of a disk de-fect (radius 50 = μm) and of a ring defect (outer radius =50μm, inner radius = 1μm). The rf deposited power is thesame in both cases

2.5

3

3.5

4

4.5

5

5.5

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

RF

sur

face

tem

pera

ture

T(K

)

Radial distance from the center of the Nb slab(m)

f=1.3GHz,RRR=300,PMFP=1mm,Tb=2K

Rdefect=0.01Ω,Ring defect width Δr=1μm

rf side temperature

Figure 6: rf surface temperature distribution of a 5mm ring-type defect with 1μm width. The modeled niobium platehas a radius of 10mm.

TUPPO048 Proceedings of SRF2009, Berlin, Germany

05 Cavity performance limiting mechanisms

332

CALCULATION RESULTS

Simulations on ring-shaped defects of different radiusand width were performed to explore the relationship be-tween the pre-quench (a steady state with cavity maximumsurface magnetic fields just below the quench field) tem-perature distributions and ring geometries. Figure 7 showsa typical pre-quench temperature distribution along the ra-dial direction both in the helium side and in the rf surfacefor different inner radius ring-type defect with the samering width of 1μm. The defect resistance was assumedas 10mΩ. The results show a clear dependence of heliumside pre-quench temperature on the inner radius of the ring-shaped defect which was similarly discovered in the previ-ous study of the disk-type defect model [2].

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

1e-06 1e-05 1e-04 0.001 0.01 0.1

Hel

ium

inte

rfac

e te

mpe

ratu

re T

(K)

Radial distance from the center of Nb slab r(m)

Inner radius r0=500μmInner radius r0=400μmInner radius r0=300μmInner radius r0=200μmInner radius r0=100μmInner radius r0=50μmInner radius r0=20μmInner radius r0=10μm

(a) Helium side surface temperature distribution.

2

2.5

3

3.5

4

4.5

5

5.5

6

1e-06 1e-05 1e-04 0.001 0.01 0.1

RF

sur

face

tem

pera

ture

T(K

)

Radial distance from the center of Nb slab r(m)

f=1.3GHz,RRR=300,PMFP=1mm,Tb=2K

Rdefect=0.01Ω,Ring defect width Δr=1μm

Inner radius r0=500μmInner radius r0=400μmInner radius r0=300μmInner radius r0=200μmInner radius r0=100μmInner radius r0=50μmInner radius r0=20μmInner radius r0=10μm

(b) RF surface temperature distributions.

Figure 7: Temperature distributions for different sizes ofring-type defects with the same ring width of 1μm.

Figure 8 shows the calculated quench field distributionas function of the ring-type defect inner radius for two ringwidth 1μm and 2μm. For a pit quenched at 1200Oe asobserved in figure 1, a ring width around 1.8μm can beconcluded. Direct measurements by contact profilometryconfirmed that the edge of the pit shown in figure 1 was1∼2μm. Considering the unincluded magnetic field en-

hancement factor at the edge of the pit, the agreement ofpit width information between simulated and measured re-sults is acceptable. Combined with thermometry data ofquench field Hquench and pre-quench helium side temper-ature reading ΔT, the simulation results of quench field ofa ring defect size with the measured pit radius R by pro-filometry or optical inspections can provide estimates ofthe resistance of the normal conducting region of the pitas well as of the width of the edge of the pit with strongmagnetic field enhancement.

1100

1150

1200

1250

1300

1350

1400

1450

1500

0 100 200 300 400 500 600

Bre

akdo

wn

field

H(O

e)

Inner radius of ring-type defects ri(μm)

f=1.3GHz,RRR=300,PMFP=1mm,Tb=2Kf=1.3GHz,RRR=300,PMFP=1mm,Tb=2K

Rdefect=0.01Ω

1μm2μm

Figure 8: Quench fields v.s. difference ring-type defectsizes for two difference ring width.

CONCLUSION AND FUTURE WORK

A 2-d ring-type defect thermal model has been devel-oped and is benchmarked. This model assumes an idealring-shape defect at the edge of pit structures and canprovide information on normal conducting resistance andwidth of normal conducting pit edge. Since both opticaland profilometry studies suggests an irregular ring widthstructure, only a fraction of the ring may go normal con-ducting. Therefore the calculation results may need a ge-ometrical correction factor depending on the actual pitsgeometry. Also the magnetic field enhancement factor βwhich is determined by slopes of ring edges should be con-sidered in the future improved version of thermal ring de-fect model.

REFERENCES

[1] Y. Iwashita et.al., “Developement of high resolution cam-era and observations of superconducting cavities”, EPAC08,Genoa, Italy.

[2] Y. Xie et.al., “Possible relationship between defects pre-heating and defects size”, SRF09, Berlin, Germany.

[3] V. Shemelin, H. Padamsee, “Magnetic eld enhancement atpits and bumps on the surface of superconducting cavities”,SRF Report 080903-04, Cornell University.

Proceedings of SRF2009, Berlin, Germany TUPPO048

05 Cavity performance limiting mechanisms

333