1. Mechanical Loss and Thermal Conductivity of Materials for KAGRA and ET
Gerd Hofmann1, Julius Komma1, Christian Schwarz1, Daniel Heinert1, Paul Seidel1, Andreas Tünnermann2, and Ronny Nawrodt1
1Friedrich-Schiller-Universität Jena, Institute for Solid State Physics, Helmholtzweg 5, D Jena, Germany
2Friedrich-Schiller-Universität Jena, Institute of Applied Physics, Albert-Einstein-Strasse 15, D Jena, Germany
April 19th 2013
ELiTES Workshop, Tokyo

2. Outline Test mass materials for future GWDs
Fused silica – state of the art, certainly at RT (optics & suspension)
Silicon  ET
Sapphire  KAGRA
Bulk loss of silicon & sapphire vs. fused silica
Mechanical loss of sapphire fibers for suspension
Different lengths, single vs. double head
Thermal conductivity of sapphire fibers
Summary & Outlook

3. Mechanical loss

4. Basic layout of an interferometric GWD
Extremely sensitive Michelson interferometer, several noise sources
Main topic: Brownian thermal noise arising from the mechanical loss of the materials (currently fused silica) for the optics, the test masses, and their suspensions.

5. Mechanical loss of fused silica vs. silicon & sapphire
The mechanical loss of fused silica strongly increases when being cooled down. Much more suitable are single crystalline materials like silicon or sapphire. For KAGRA, IMs and EMs will be made of sapphire. In ET-LF they will be made of silicon.
[R. Nawrodt et al.: Cryogenic Setup for Q-factor measurements on bulk materials for future gravitational wave detectors, in Proceedings of ICEC22-ICMC2008 (2009)]

6. Measured mechanical loss of sapphire Ø 3“ x 24mm
At 20 K we achieved a loss of 10 −8 . Our measurements reveal a loss peak at 35 K for all the measured modes.

7. Akhiezer damping in bulk sapphire
Loss peak at 35 K is linked to Akhiezer loss (interaction of acoustic and thermal phonons) as follows:
𝜙= 𝑇𝐶 𝛾 2 𝜈 2 𝜔 𝜏 𝑝 1+ 𝜔 𝜏 𝑝 where 𝜏 𝑝 = 3 𝜅 𝐶𝜌 𝜈 2 .
[A. Akhieser: On the absorption of sound in solids. Journal of Physics (1939)]
[V. B. Braginskyet al.: Systems with Small Dissipation.The University of
Chicago Press, Chicago and London (1985)]
𝐶… heat capacity, 𝛾… Grüneisen‘s constant,
𝜈… solid‘s speed of sound,
𝜏 𝑝 … lifetime of thermal phonons,
𝜅… heat conductivity, and 𝜌… density of material.
 Akhiezer loss can not be overcome thus it is an intrinsic limit.
𝛾 … Grüneisen‘s constant used to fit: 1.8, 2.1, 3.3

8. Sapphire fibers measured in Jena
MolTech fibers (4 in total)
single nail head with flat Ø 10 mm x 5 mm
fiber Ø 1.8 mm
1 unbroken (350 mm)
1 broken (86 mm & 264 mm)
Impex fibers (5 in total)
double nail head Ø 10 mm x 5 mm
fiber Ø 1.6 mm
total lenght 100 mm

9. Measurement setup Use of massive cooper supports and clamps:
Flat drill hole vs. Cone drill hole
Electrostatic driving plates for excitation
Optical readout by use of shaddow sensor
Ring down technique
Liquid helium cryostat 𝑇=5…300 K
Cone seems to be better suited for the measurement as we achieve lower losses.

10. MolTech fiber Ø 1.8 mm x 350 mm, clamped in cone
Lowest obtained loss on sapphire fiber so far: 5× 10 −8
Thermo elastic damping (TED) above 60 K

11. Thermo elastic damping in sapphire fibers
Thermo elastic damping (TED) occurs from irreversible heat flow between compressed and strechted areas of the fiber. The loss is given by:
𝜙= 𝑌𝑇 𝜌𝐶 𝜔 𝜏 𝑇𝐸 1+ 𝜔𝜏 2
𝜏 𝑇𝐸 = ×2𝜋 𝜌𝐶𝑑 𝜅
[C. Zener : Internal Friction in Solids: I. Theory of Internal Friction in Reeds. Physical Review 52 (1937)] [C. Zener : Internal Friction of Solids: II.General Theory of Thermoelastic Internal Friction. Physical Review 53 (1938)] 𝑌… Young‘s Modulus,
𝜏 𝑇𝐸 …characteristic time,
𝑑… diamter of the fiber

12. Impex fiber No.3, attached head clamped in cone
At low temperature we observe different behaviour for different modes. It is not clear why this happens. We take into account some week interaction with the support (recoil loss) or the influence of the surface (unlikely). This is under investigation right now and has to be taken as preliminary result.
Again: TED above 60 K seems to limit the loss
Low temperature behaviour is not cleared and under investigation

13. Thermal conductivity

14. Thermal conductivity measurement
Measured with the broken piece of MolTech fiber:
Ø 1.8 mm
264 mm in length
Copper clamps to attach
the heater
the sensors
the heat sink

15. Setup and measurement procedure
THeater
THeat Sink
Distance: 10…200 mm
𝑃= 𝐴 𝐿 𝜅Δ𝑇
𝜅= 𝐿 𝐴 𝑑𝑃 𝑑𝑇
𝜅 … therm. conductivity L … temp-sensor distance
A … cross section Δ𝑇 … temp. difference
P … electr. power
Measurement Procedure:
#1 – Wait until all sensors are in thermal equilibrium
#2 – Set a given Heater Power and wait until all
sensors reach thermal equilibrium again
#3 – Repeat #2 until a maximum given temperature
difference between T1 and T2 is reached
#4 – Plot T1-T2 vs. PHeater + linear fit of the data

16. Thermal conductivity of sapphire
Thermal conductivity of the fiber is clearly different to that of bulk sapphire
Surface and also heat treatment might change the thermal conductivity

17. Heat extraction from fibers
If we asume
L = 30 cm, Ø 1.8 mm
Test mass TM 20K
Upper mass UM 16K
Thermal conductivity of k 2 x 10^3 W/m/K
Heat extraction of one fiber:
𝑄 = 𝐴 𝑤 𝐿 𝑘 𝑇 𝑇𝑀 − 𝑇 𝑈𝑀 ≈200𝑚𝑊
Around 1 W of extracted heat is desirable for KAGRA, but with fibers of Ø 1.6 mm
Futher investigations are needed!

18. Nevertheless: Sapphire will fulfill the requirements for KAGRA
Summary
Cooling of the test masses and suspensions will reduce brownian thermal noise in future GWDs using silicon or sapphire
Bulk sapphire is limited by phonon-phonon-interaction at the desired temperature of 20 K ( 𝜙 𝑏𝑢𝑙𝑘 ≈ 10 −8 )
Above 60 K TED limits the loss of sapphire fibers Losses of better than 𝜙 𝑓𝑖𝑏𝑒𝑟 ≈ 10 −7 are achieved below 10 K
Heat extraction by suspension fibers needs to be slightly improved
Nevertheless: Sapphire will fulfill the requirements for KAGRA