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

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

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

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.

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 0.0625 𝑇 3 . Proportional to 𝑇 3 This result supports that the thermal conductivity is proportional to 𝑇 3 at low temperature

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 16 23 𝑑𝑇 𝑅 𝑡ℎ 𝑇 >1 W, 𝜅 should be larger than 5.6× 10 3 W/m∙K Intermediate mass 16K 4 sapphire fibers Laser Test mass absorption∼1 W Requirement 5.6× 10 3 W/m∙K @ 20K (𝜅=0.7 𝑇 3 W/m∙K) 23K

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.

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.

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!

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.

Result We could reproduce the result of Tomaru’s work around 80K. 1300 W/m∙K @82K 700 W/m∙K @90K →The method of this measurement might have no serious troubles. However, our result at low temperature is not proportional to 𝑇 3 . 700 W/m∙K @12K 1100 W/m∙K @17K 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.

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.

Thank you for your attention.

Result The thermal conductivity of photoran’s sapphire fiber 　　700 W/m∙K @12K 　　1100 W/m∙K @17K 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.