1. Measurement of the thermal conductivity of sapphire fibers
Takafumi Ushiba, M2
Tsubono Lab., The University of Tokyo

2. Hook the mirror on the sapphire fiber with thick edge
Suspension of KAGRA
enlarge
Hook the mirror on the sapphire fiber with thick edge

3. Sapphire Fiber with Thick Edge
Thin part:
　　about 1.8mm in diameter
Thick part:
　　10mm in diameter
These fibers are not polished.
IMPEX is now manufacturing their fibers.
Conceptional drawing of IMPEX fiber
Moltech sapphire fibers

4. Theory of the Thermal Conductivity
Thermal conductivity 𝜅= 1 3 𝐶 𝑉 𝑣𝑙 𝐶 𝑉 : volumetric specific heat → at low temperature 𝐶 𝑉 ∝ 𝑇 3 𝑣: speed of sound → in sapphire 𝑣=1.1× 10 4 m/s (constant) 𝑙: mean free path of a phonon Since 𝑙 is constant at low temperature, the thermal conductivity is proportional to 𝑇 3 at low temperature.

5. Previous Measurement They use sapphire fibers made by Photoran
Physics Letters A 301 (2002) Tomaru et al
They use sapphire fibers made by Photoran
𝜙160μm sapphire fiber
𝜅 is well fitted by 𝑇 3 .
Proportional to 𝑇 3
This result supports that the thermal conductivity is proportional to 𝑇 3 at low temperature

6. Requirement of Thermal Conductivity
For two cryocooler
(2W)
Thermal resistance of a sapphire fiber
𝑅 𝑡ℎ (𝑇)= 𝐿 4𝜅(𝑇)𝑆
𝜅(𝑇): thermal conductivity
𝐿: length of a sapphire fiber
𝑆: cross-section of a sapphire fiber
If we require 𝑑𝑇 𝑅 𝑡ℎ 𝑇 >1 W,
𝜅 should be larger than 5.6× W/m∙K
Intermediate
mass
16K
4 sapphire fibers
Laser
Test mass
absorption∼1 W
Requirement
5.6× K
(𝜅=0.7 𝑇 3 W/m∙K)
23K

7. Theory of the Thermal Conductivity
Thermal conductivity 𝜅= 1 3 𝐶 𝑉 𝑣𝑙 𝐶 𝑉 : volumetric specific heat → at low temperature 𝐶 𝑉 ∝ 𝑇 3 𝑣: speed of sound → in sapphire 𝑣=1.1× 10 4 m/s (constant) 𝑙: mean free path of a phonon
These are inherent in the material
This is the only parameter we can change
We should prepare the sapphire fibers which have the long mean free path of a phonon.

8. Thermal Conductivity of Sapphire fibers
The side of Moltech and IMPEX sapphire fibers are not polished. →The surface of these sapphire fibers are rough. It makes the thermal conductivity low. (Size effect)
Sapphire fiber
phonon
Highly polished
Completely reflected
High thermal conductivity
Not polished
Scattering
Low thermal conductivity
Even if the size effect limit the thermal conductivity of sapphire fibers, it is not a matter for KAGRA suspension.

9. Thermal Conductivity of Sapphire fibers
If the purity of the sapphire fiber is low →The sapphire fibers have defects of phonon and impurities It makes the thermal conductivity low.
Sapphire fiber
phonon
High purity
Nothing disturbs phonon
High thermal conductivity
Defects of phonon or impurities
Low purity
Scattering
Low thermal conductivity
We should check the thermal conductivity(purity) of sapphire fiber experimentally!

10. Experimental Setup
One dimensional approximation 𝜅= QL SΔ𝑇 𝑄: heat flux in the sapphire fiber 𝐿: distance between two thermometers 𝑆: cross-section of the sapphire fiber Δ𝑇: difference of values of two thermometers L
thermometer
heater Q
Heat bath
Sapphire fiber
First we measure the normal sapphire fiber
Diameter is 1.8mm.
Length is 100mm.
Polished sample.

11. Result
We could reproduce the result of Tomaru’s work around 80K →The method of this measurement might have no serious troubles. However, our result at low temperature is not proportional to 𝑇 We do not check the reproducibility of this result because of some troubles.
This value might be limited by the impurity
This value is much smaller than we expected.
We should measure the thermal conductivity of sapphire fiber again.

12. Summary Requirement of the thermal conductivity 𝜅=5.6× 10 3 W/m∙K @20K
To realize the requirement, the purity of the sapphire fiber has to be very high.
In our result, the thermal conductivity might be limited by the purity of Photoran’s sapphire fiber.
We should measure the thermal conductivity of Photoran’s fiber again.
And, we have to measure the thermal conductivity of Moltech and IMPEX sapphire fibers as soon as possible.

13. Thank you for your attention.

14. Result The thermal conductivity of photoran’s sapphire fiber
　　700
　　1100
We cannot reproduce the result of Tomaru’s work.
In this result, the thermal conductivity of sapphire fiber is not proportional to 𝑇 3 .
The purity of sapphire fiber is very low?
These results are smaller than the requirement value
窒素温度ではリーズナブルな値を得た。
We should measure the thermal conductivity of sapphire fiber again.