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-07743 Jena, Germany
2Friedrich-Schiller-Universität Jena, Institute of Applied Physics, Albert-Einstein-Strasse 15, D-07745 Jena, Germany
April 19th 2013
ELiTES Workshop, Tokyo
Friedrich-Schiller-Universität Jena
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
Gerd Hofmann
ELiTES Workshop, Tokyo, 19th April 2013
2 / XX
MECHANICAL LOSS
Friedrich-Schiller-Universität Jena
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.
Gerd Hofmann
ELiTES Workshop, Tokyo, 19th April 2013
4 / XX
Friedrich-Schiller-Universität Jena
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.
Gerd Hofmann
mechanical loss
Mechanical loss of fused silica vs. silicon & sapphire
10
-2
10
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
-9
fused silica
silicon
sapphire
0
30
60
90
120
150
180
210
240
270
300
temperature in K
[R. Nawrodt et al.: Cryogenic Setup for Q-factor measurements on bulk materials for
future gravitational wave detectors, in Proceedings of ICEC22-ICMC2008 (2009)]
ELiTES Workshop, Tokyo, 19th April 2013
5 / XX
Friedrich-Schiller-Universität Jena
mechanical loss 
Measured mechanical loss of sapphire Ø 3“ x 24mm
2x10
-7
10
-7
10
-8
10
32.3 kHz
36.2 kHz
50.8 kHz
-9
0
30
60
90
120
150
180
210
240
270
300
330
temperature T in K
At 20 K we achieved a loss of 10−8 . Our measurements reveal a loss
peak at 35 K for all the measured modes.
Gerd Hofmann
ELiTES Workshop, Tokyo, 19th April 2013
6 / XX
Friedrich-Schiller-Universität Jena
Akhiezer damping in bulk sapphire
Loss peak at 35 K is linked to Akhiezer loss (interaction of acoustic
and thermal phonons) as follows:
10
𝜙=
𝜔𝜏𝑝
𝑇𝐶𝛾 2
𝜈2 1+ 𝜔𝜏𝑝 2
where 𝜏𝑝 =
3𝜅
.
𝐶𝜌𝜈2
[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,
𝜏𝑝 … lifetime of thermal phonons,
mechanical loss 
-6
32.3 kHz
36.2 kHz
50.8 kHz
10
-7
10
-8
10
-9
0
30
𝜅… heat conductivity, and 𝜌… density of material.
60
90
120
150
temperature T in K
 Akhiezer loss can not be overcome thus it is an intrinsic limit.
Gerd Hofmann
ELiTES Workshop, Tokyo, 19th April 2013
7 / XX
Friedrich-Schiller-Universität Jena
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
Gerd Hofmann
ELiTES Workshop, Tokyo, 19th April 2013
8 / XX
Friedrich-Schiller-Universität Jena
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
Gerd Hofmann
ELiTES Workshop, Tokyo, 19th April 2013
9 / XX
Friedrich-Schiller-Universität Jena
mechanical loss 
MolTech fiber Ø 1.8 mm x 350 mm, clamped in cone
10
-4
10
-5
10
-6
10
-7
1217 Hz
TED
10
-8
0
30
60
90
120
150
180
210
240
270
300
330
temperature T in K
• Lowest obtained loss on sapphire fiber so far: 5 × 10−8
• Thermo elastic damping (TED) above 60 K
Gerd Hofmann
ELiTES Workshop, Tokyo, 19th April 2013
10 / XX
Friedrich-Schiller-Universität Jena
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:
𝑌𝑇 𝜔𝜏 𝑇𝐸
𝜙=
𝜌𝐶 1 + 𝜔𝜏
𝜏 𝑇𝐸
2
1
𝜌𝐶𝑑
=
2.16 × 2𝜋 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
Gerd Hofmann
ELiTES Workshop, Tokyo, 19th April 2013
11 / XX
Friedrich-Schiller-Universität Jena
Impex fiber No.3, attached head clamped in cone
Mechanical loss 
10
-3
10
-4
10
-5
10
-6
10
-7
87Hz
1240Hz
3676Hz
35132Hz
0
30
60
90
120
150
180
210
240
270
300
Temperature in K
• Again: TED above 60 K seems to limit the loss
• Low temperature behaviour is not cleared and under investigation
Gerd Hofmann
ELiTES Workshop, Tokyo, 19th April 2013
12 / XX
THERMAL CONDUCTIVITY
Friedrich-Schiller-Universität Jena
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
Gerd Hofmann
ELiTES Workshop, Tokyo, 19th April 2013
14 / XX
Friedrich-Schiller-Universität Jena
Setup and measurement procedure
THeater
Distance: 10…200 mm
T1
T2
THeat Sink
Gerd Hofmann
𝐴
𝑃 = 𝜅Δ𝑇
𝐿
𝜅 … therm. conductivity
A … cross section
P … electr. power
𝐿 𝑑𝑃
𝜅=
𝐴 𝑑𝑇
L … temp-sensor distance
Δ𝑇 … temp. difference
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
ELiTES Workshop, Tokyo, 19th April 2013
15 / XX
Friedrich-Schiller-Universität Jena
Thermal conductivity of sapphire
30000
Thermal Conductivity [W/mK]
10000
1000
100
1.8mm dia, 168mm length, MolTec-Fibre (Jena)
1.55mm dia, 60mm length, unpolished (Touloukian)
2.47mm dia, 60mm length, unpolished, annealed
2.52mm dia, 60mm length, polished, annealed
recommended values for bulk sapphire
10
1
10
100
1000
Temperature [K]
• Thermal conductivity of the fiber is clearly different to that of bulk sapphire
• Surface and also heat treatment might change the thermal conductivity
Gerd Hofmann
ELiTES Workshop, Tokyo, 19th April 2013
16 / XX
Friedrich-Schiller-Universität Jena
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:
𝐴𝑤
𝑄=
𝑘 𝑇𝑇𝑀 − 𝑇𝑈𝑀 ≈ 200𝑚𝑊
𝐿
• Around 1 W of extracted heat is desirable for KAGRA, but with
fibers of Ø 1.6 mm
• Futher investigations are needed!
Gerd Hofmann
ELiTES Workshop, Tokyo, 19th April 2013
17 / XX
Friedrich-Schiller-Universität Jena
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 (𝜙𝑏𝑢𝑙𝑘 ≈ 10−8 )
• Above 60 K TED limits the loss of sapphire fibers
Losses of better than 𝜙𝑓𝑖𝑏𝑒𝑟 ≈ 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
Gerd Hofmann
ELiTES Workshop, Tokyo, 19th April 2013
18 / XX