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