q basic lecture series - granite.phys.s.u-tokyo.ac.jp
TRANSCRIPT
QBasic Lecture series
Ooi Ching Pin
Contents
โข The classical dissipative oscillator
โข The fluctuation dissipation theorem
โข Thermal noise spectrum
โข Q
โข Surface loss
The classical dissipative oscillator/viscous dampingโข It is traditional to consider a dissipative oscillator with the following
equation of motion
โข ๐ แท๐ฅ + ๐ แถ๐ฅ + ๐๐ฅ = 0
โข i.e. a simple harmonic oscillator with a dissipative term that is proportional to the velocity
โข Solution
โข ๐ฅ ๐ก = ๐ฅ0 ๐โ
๐
2๐๐ก cos๐0๐ก + ๐
โข We shall focus on small dissipation time
๐0 =๐
๐
Loss of energy in one cycle
โข Now let us take a look at the energy loss per cycle, because we care about dissipative energy loss.
โข Note that the energy is proportional to the square of the amplitude
โขฮE
๐ธ=
12โ ๐โ
๐2๐
1๐0
2
12=
1โ๐โ
๐๐๐0
1โ
1โ1+๐
๐๐0
1=
b
mf0
time
The fluctuation dissipation theorem
โข First formulated by Harry Nyquist in 1928โข in the context of Johnson noise (electric noise in circuits)
โข Generalised to all dissipative systems in (at least local) thermal equilibriumโข Brownian noise
โข Plank radiation law
โข Via the same mechanism that movement turns into heat (dissipation), heat leads to fluctuations (in movement)
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Fluctuation dissipation theorem
โข Intuitively, any useful energy in a system (from mechanical or electrical, for example) turning into heat is dissipation
โข Dissipation is microscopic random motion of the molecules in question
โข If mechanical/electric energy can turn to heat, heat can turn into mechanical/electric energy, but as fluctuations (noise)
โข This is thermal noise, and is a real limitations to high precision experiments today
The fluctuation dissipation theorem
โข Given by
And valid only when i.e. โhighโ temperatures
โข Displacement version:
Generalised force, e.g. voltage Real part of a generalised impedance, e.g. resistance
displacement Admittance, ๐ =1
๐
generalised impedance definition:
๐ โซโ๐
๐๐ต= 4.8 ร 10โ11
๐
1 ๐ป๐ง๐พ
Generalised Johnson-Nyquist noise
Thermal noise
โข Thermal noise is dependent on the generalised admittance
โข How to calculate?
โข ๐ ๐ =1
๐(๐)=
แถ๐ฅ
๐น
โข For the classical dissipative oscillator, using ๐๐๐ฅ = แถ๐ฅ and ๐๐ แถ๐ฅ = แท๐ฅ
โขแถ๐ฅ
๐น=
1
๐๐๐+๐+๐
๐๐
โข ๐ฅ2 =4๐๐ต๐
๐2
๐
๐2+ ๐๐โ๐
๐
2
displacement Admittance, ๐ =1
๐
generalised impedance definition:
Actual thermal noise spectrum
๐น = ๐ แท๐ฅ + ๐ แถ๐ฅ + ๐๐ฅ
Thermal noise spectrum
โข ๐ฅ2 =4๐๐ต๐
๐2
๐
๐2+ ๐๐โ๐
๐
2
โข Graphs with differentdissipative values
โข ๐ โ๐0๐
๐
Q
โข Defined as the resonance frequency ๐0 divided by the full width half maximum (of energy) ฮ๐
โข Dimensionless
โข ๐ =๐0
ฮ๐
โข For the classical dissipative oscillator
โข ๐ โ๐0๐
๐
Thermal noise spectrum
โข ๐ฅ2 =4๐๐ต๐
๐2
๐
๐2+ ๐๐โ๐
๐
2
โข ๐ โ๐0๐
๐
โข For ๐ โช ๐0
โข ๐ฅ2 โ4๐๐ต๐๐
๐2=
4๐๐ต๐๐0๐
๐2๐
โข For ๐ โ ๐0
โข ๐ฅ2 โ4๐๐ต๐
๐02๐=
4๐๐ต๐๐
๐03๐
โข For ๐ โซ ๐0
โข ๐ฅ2 โ4๐๐ต๐๐
๐4๐2 =4๐๐ต๐๐0
๐4๐๐
Another definition of Q
โข Energy lost in a cycle was calculated to be
โขฮE
๐ธโ
b
mf0
โข ๐ โ๐0๐
๐
โข โด ๐ = 2๐E
ฮ๐ธ
โข Ringdown method: ๐ =๐0
2๐, where ๐ is the time constant of the
ringdown envelope
time
Viscous damping and structural damping
โข While classically viscous damping has been well studied, it doesnโt apply to all cases in real life, from experiment.
โข Especially how Q doesnโt change with frequency.
โข ๐ โ๐0๐
๐for viscous damping
โข Structural damping is another common form of damping that is amplitude based instead of velocity based
โข ๐น = ๐ แท๐ฅ + ๐ 1 + ๐๐ ๐ฅ
โข ๐ is known as the loss factor
โข Intrinsic material losses generally follow this model
Viscous damping and structural damping
โข Repeating the calculations for structural damping, we get
โข ๐ฅ2 =4๐๐ต๐
๐2
๐๐
๐
๐๐
๐
2+ ๐๐โ
๐
๐
2
โข ๐ โ1
๐
โข Note the ๐ โ๐๐
๐replacement
Viscous damping: ๐ฅ2 =4๐๐ต๐
๐2
๐
๐2+ ๐๐โ๐
๐
2
๐ โ๐0๐
๐
๐0 =๐
๐
Thermal noise spectrum (structural damping)
โข For ๐ โช ๐0
โข ๐ฅ2 โ4๐๐ต๐๐
๐๐=
4๐๐ต๐
๐๐๐
โข For ๐ โ ๐0
โข ๐ฅ2 โ4๐๐ต๐
๐0๐๐=
4๐๐ต๐๐
๐0๐
โข For ๐ โซ ๐0
โข ๐ฅ2 โ4๐๐ต๐๐๐
๐5๐2 =4๐๐ต๐๐
๐5๐2๐
Background: Quest for High Q
โข Resonant-mass detectors (Weber bars)โข Gravitational wave detectors
โข The higher the Q, the easier it will resonate, and thus the more sensitive the detectors will be
โข Increase accuracy of G measurementsโข Anelastic properties of the fibre will bias the results of the popular time of swing
method
โข Reduce thermal noise in any precision experimentโข Usage of fused silica for both the mirror and suspensions of aLIGO and AdVirgo
โข Needed for high precision timekeeping โข High Q means that the resonant frequency is more consistent
16
Various Q
โข Q is very versatile and appears in many systems
GREEN, ESTILL I. "THE STORY OF Q." American
Scientist 43, no. 4 (1955): 584-94.
http://www.jstor.org/stable/27826701. 17
Reported values for bulk Q (material properties)
*Represents fibre measurements and not bulk
โข Crystals have higher Q in general (lower loss)
108
1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 1.E+10
Aluminium 5056
Copper Beryllium
Niobium
Molybdenum
Tungsten
Fused Silica*
Quartz
Sapphire (Al2O3)
Silicon
Literature values of bulk Q
50 mK 4 K 300K
Crystals
Metals
Technically amorphous
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Effective Q
โข Anything that dissipates energy contributes to the effective Q
โข ๐ = 2๐๐ธ
ฮ๐ธ= 2๐
๐ธ
ฮ๐ธ1+ฮ๐ธ2+ฮ๐ธ3+โฏ
โข ๐โ1 =1
2๐
ฮ๐ธ1+ฮ๐ธ2+ฮ๐ธ3+โฏ
๐ธ=
1
2๐
ฮ๐ธ1
๐ธ+
ฮ๐ธ2
๐ธ+
ฮ๐ธ3
๐ธ+โฏ
= ๐1โ1 + ๐2
โ1 + ๐3โ1 +โฏ
โข Note that here the various Qs are defined here with respect to the total (relevant) energy of the system
Dilution factor
โข You can increase the effective Q by diluting it with a lossless potential energy
โข Classical case is gravitational energy in a pendulum
โข Qeffโ1 =
1
2๐
ฮ๐ธ๐ ๐ก๐๐๐๐
๐ธ+
ฮ๐ธ๐๐๐๐ฃ๐๐ก๐๐ก๐๐๐๐๐
๐ธ=
1
2๐
ฮ๐ธ๐ ๐ก๐๐๐๐
๐ธ=
1
2๐
ฮ๐ธ๐ ๐ก๐๐๐๐
๐ธ๐ ๐ก๐๐๐๐+๐ธ๐๐๐๐ฃ๐๐ก๐๐๐๐๐๐
โข ๐๐๐๐ = 2๐๐ธ๐ ๐ก๐๐๐๐+๐ธ๐๐๐๐ฃ๐๐ก๐๐๐๐๐๐
ฮ๐ธ๐ ๐ก๐๐๐๐= ๐๐๐๐ก๐๐๐๐๐ + 2๐
๐ธ๐๐๐๐ฃ๐๐ก๐๐๐๐๐๐
ฮ๐ธ๐ ๐ก๐๐๐๐
โข Dilution factor =๐๐๐๐
๐๐๐๐ก๐๐๐๐๐
Thermoelastic damping
โข Due to a non-zero coefficient of thermal expansion, there is temperature gradients created when a substance expands and contract
โข These leads to heat flow, leading to additional energy dissipation
โข Famously high for sapphire at room temperature
โข Does not affect torsion pendulums, at least to first order due to lack of volume change
Thermoelastic damping
Thermoelastic Q for sapphire fibres at room temperature
Surface loss
โข Literature suggests that surface loss dominates sample loss for fibresโข Thinner fibres generally leading to lower Q
โข Emphasis on fibre handling in high Q papersโข โThe measured Q was strongly dependent on
handling, with a pristine flame-polished surface yielding a Q 3-4 times higher than a surface which had been knocked several times against a copper tubeโ- S. Penn et al. 2001
โข Surface treatments are essential in getting high Q
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Fused silica flexural mode
A. Gretarsson and G. Harry 1999
Surface loss
โข Why do surfaces have such low Q?
โข Surfaces generally defer greatly in properties from the bulk. The surface would be polycrystalline in nature, covered with impurities, and made up of randomly orientated crystallites.
โข Dislocation density would also be higher near the surface of any machined surface
Surface loss model
โข Assume surface layer has a different (constant) Q from bulk
โข Assume surface layer has a fixed thickness (for a specific sample)
โข Assume elastic constants are identical between bulk and surface
โข Assume energy stored in surface layer << energy stored in bulk (๐ธ๐ ๐ข๐๐๐๐๐ โช ๐ธ๐๐ข๐๐)
โข1
๐๐๐๐=
1
2๐
ฮ๐ธ
๐ธ=
1
2๐
ฮ๐ธ๐๐ข๐๐+ฮ๐ธ๐ ๐ข๐๐๐๐๐
๐ธ๐๐ข๐๐+๐ธ๐ ๐ข๐๐๐๐๐โ
1
2๐
ฮ๐ธ๐๐ข๐๐+ฮ๐ธ๐ ๐ข๐๐๐๐๐
๐ธ๐๐ข๐๐
Surface loss
โข How do we get ๐ธ๐ ๐ข๐๐๐๐๐
๐ธ๐๐ข๐๐?
Surface loss
โข Potential energy per unit volume due to strain is 1
2๐0๐
2
โข ๐0is Young modulusโข ๐ is strain
โข Potential energy per unit volume due to torsion is 1
2๐๐พ2
โข ๐ is shear modulusโข ๐พ is strain angle
โข๐ธ๐ ๐ข๐๐๐๐๐
๐ธ๐๐ข๐๐โ
โ ๐ ๐พ2๐๐
๐ ๐พ2๐๐(Assume fixed thickness h
and same shear modulus for damaged surface and bulk)
Surface loss for round torsion fibre
โข1
๐๐๐๐=
1
๐๐๐ข๐๐+
1
๐๐ ๐ข๐๐๐๐๐
โ ๐ ๐พ2๐๐
๐ ๐พ2๐๐=
1
๐๐๐ข๐๐+
1
๐๐ ๐ข๐๐๐๐๐โ8
๐ท
โข ๐๐ ๐ข๐๐๐๐๐ dominates at small diametersFused silica torsion mode
K. Numata et al. 2007
Data range of K. Numata et al. 2001
Extrapolation
C-axis Orbe Pioneer fibre
Graph with 1
๐๐๐๐ก๐๐๐๐๐= 0
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
10 100 1000 10000
Q
ratio of strain energies (1/m)
Q with surface loss model
A-axis Orbe Pioneer fibre
Silicon flexural mode
R. Nawrodt et al. 2013
Fused silica flexural mode
A. Gretarsson and G. Harry 1999
Surface quality
โข Tensile strength is very dependent on surface quality30Systems with small dissipation, V.B. Braginsky, V.P. Mitrofanov, V.I. Panov, 1985, taken from Libowitz 1976
Suspension thermal noise in torsion pendulums (off resonance, ๐ โซ ๐0)
โข ๐2 ๐๐ is the mean square of the angular displacement within the frequency interval ๐๐
Length of fibre
Q factorMoment of inertia of the torsion pendulum
Angular frequency
Fibre diameter
Shear modulusTemperature
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Surface quality
โข Some quantification here:โข Assuming a safety factor s
โข And replacing D
โข We see that improving surface quality is key to low thermal noise
Suspension thermal noise for torsion pendulum
Tensile strength
Safety factorIndependent of scale, but not of shape
Length of fibre
32
Conclusion
โข Q is a dimensionless parameter that characterises dissipation of a system
โข This can be used to calculate the thermal noise spectrum
โข For mechanical Q, surface losses play a big role in thin materials