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Thermal Conductivity of Ingot Niobium – Estimatingwith Processing History
S.K. Chandrasekaran1, T.R. Bieler2, C.C. Compton3,W. Hartung3 and N.T. Wright1
1Department of Mechanical EngineeringMichigan State University
2Department of Chemical Engineering and Materials ScienceMichigan State University
3National Superconducting Cyclotron LaboratoryMichigan State University
1st International Symposium on Superconducting Science andTechnology of Ingot Niobium
Chandrasekaran et al. (Michigan State Univ.) Conductivity of Nb SSTIN 1 / 17
Thermal Conductivity of Superconducting Nb
Conduction in Nb at2 – 4.2 K is a function of
purityimperfection densitygrain sizegrain orientation?
Chandrasekaran et al. (Michigan State Univ.) Conductivity of Nb SSTIN 2 / 17
Motivation for this study
Need to relate thermal conductivity k with
metallurgy
processing history
Doing so,
Allows prediction of k in final cavity
k can be used as a diagnostic tool
Chandrasekaran et al. (Michigan State Univ.) Conductivity of Nb SSTIN 3 / 17
Measurement Apparatus
Steady state temperatureprofile measured
Apparatus can accommodate4 samples
Ge and C resistors used tomeasure temperature
3 – 4 sensors on each sample
Apparatus tested in LHeDewar at 1.5 – 1.9 K
Chandrasekaran et al. (Michigan State Univ.) Conductivity of Nb SSTIN 4 / 17
Temperature data
Points representthermistor temperaturemeasurements
Dotted lines assumeuniform conductivitybetween the sensors
Thermal conductivityestimated by k = − q′′
dT/dx
Chandrasekaran et al. (Michigan State Univ.) Conductivity of Nb SSTIN 5 / 17
Model for k
k(T ) = R(y)[ ρ295
LRRRT+ aT 2
]−1+
[1
De−yT 2 +1
BλT 3
]−1
ρ295 – electrical resistivity at 295 KL – Lorentz constantRRR – ratio of electrical resistivity at 295 K to that at 4 Ka – coefficient of momentum exchange within latticeD – quantifies phonon scattering by electronsB – value from Casimir for scattering at crystal boundariesλ – phonon wavelengthy ≈ αTc/T
Ref.: F. Koechlin and B. Bonin, Supercond. Sci. Technol. 9 (1996)
Chandrasekaran et al. (Michigan State Univ.) Conductivity of Nb SSTIN 6 / 17
Method often used
T and q′′ measurements
↓Estimate k using k = − q′′
dT/dx
↓Estimate parameters using model
Current method
T and q′′ measurements
↓Estimate parameter groups directly using model
Ref.: S.K. Chandrasekaran et al., Therm. Cond. 30 (2009)
Chandrasekaran et al. (Michigan State Univ.) Conductivity of Nb SSTIN 7 / 17
Parameters to be estimated
k(T ) = R(y)[ ρ295
LRRRT+ aT 2
]−1+
[1
De−yT 2 +1
BλT 3
]−1
⇓
k(T ) = R(y)
[β1
T+ β2T 2
]−1
+
[β3
e−yT 2 +β4
T 3
]−1
and
y ≈ αTc
T
⇓
y ≈ β5Tc
TChandrasekaran et al. (Michigan State Univ.) Conductivity of Nb SSTIN 8 / 17
Scaled Sensitivity Coefficients
Sensitivity coef.:
γi = βi∂T∂βi
β1 = ρ295LRRR
β2 = aβ3 = 1
D
β4 = 1Bλ
β5 = α
Chandrasekaran et al. (Michigan State Univ.) Conductivity of Nb SSTIN 9 / 17
Specimens Analyzed
Parameters estimated from
15 ingot specimens with 70 ≤ RRR ≤ 450
As rec’d condition
2 specimens heat treated at 600 C, 6 hrs
4 specimens heat treated at 750 C, 2 hrs
2 specimens heat treated at 800 C, 2 hrs
Not presented here
2 specimens heat treated at 140 C, 48 hrs
2 specimens from zone melted tube
Chandrasekaran et al. (Michigan State Univ.) Conductivity of Nb SSTIN 10 / 17
Example: Sample A
K & B method
β1 = 0.058 ΩK 2mW−1
β3 = 234.0 mK 3W−1
β4 = 0.076 mK 4W−1
β5 = 1.530
New estimates
β1 = 0.033 ΩK 2mW−1
β3 = 404.083 mK 3W−1
β4 = 0.447 mK 4W−1
β5 = 2.045Chandrasekaran et al. (Michigan State Univ.) Conductivity of Nb SSTIN 11 / 17
β1 and RRR
0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 00 . 0 0
0 . 0 1
0 . 0 2
0 . 0 3
0 . 0 4
0 . 0 5
0 . 0 6
0 . 0 7β 1 (Ω
K 2 mW -1
)
E s t i m a t e d R R R
β1 =ρ295
LRRR
β1 floats from initial guessEstimated relationship gives confidence in reduction
Chandrasekaran et al. (Michigan State Univ.) Conductivity of Nb SSTIN 12 / 17
β4 and Heat Treatments
β4 =1
Bλ
0 2 0 0 4 0 0 6 0 0 8 0 00 . 0
0 . 2
0 . 4
0 . 6β 4 (m
K 4 W -1)
Chandrasekaran et al. (Michigan State Univ.) Conductivity of Nb SSTIN 13 / 17
β4 and Heat Treatments
0 2 0 0 4 0 0 6 0 0 8 0 00 . 0
0 . 2
0 . 4
0 . 6Av
g. β 4 (m
K 4 W -1)
β4 =1
Bλ
0 C→ As rec’d600 C→ 6 hrs.750 C→ 2 hrs.800 C→ 2 hrs.
Lower β4 with increasing heat treatment temperatureAs rec’d specimens exhibit greater thermal resistance
Chandrasekaran et al. (Michigan State Univ.) Conductivity of Nb SSTIN 14 / 17
Conclusions
k of superconducting Nb is a function of material processing
Koechlin and Bonin good starting point – but more needed
New parameter estimation technique skips intermediate step
β1 - RRR relation gives confidence
Data shows strong relationship between β4 and heat treatments
Greater β4 values for as rec’d – Residual stresses from ingotproduction?
Chandrasekaran et al. (Michigan State Univ.) Conductivity of Nb SSTIN 15 / 17
Conclusions
Refine specimen characterizations
grain orientations
mis-orientation map and angle
Other heat treatment protocols are being characterized140 C, 48 hrs
600 C, 10 hrs
More data at high temperatureCapture the kinetics of heat treatment
Correlation with mechanical properties sought
Chandrasekaran et al. (Michigan State Univ.) Conductivity of Nb SSTIN 16 / 17
Acknowledgements
Ingot Nb discs
Dr. G.R. Myneni, Jefferson Laboratory
Funding
U.S. Department of Energy
Fermi National Accelerator Laboratory
National Superconducting Cyclotron Laboratory
Collaborators
Di Kang, Michigan State University
Lindsay Dubbs and Kyle Elliott, NSCL
Travel Support
Dr. G.R. Myneni and ISOHIMChandrasekaran et al. (Michigan State Univ.) Conductivity of Nb SSTIN 17 / 17
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