CRYO/Q
Relaxation mechanisms in solids
Anja Zimmer
Friedrich-Schiller-University
Jena, Germany
3rd ILIAS-GW Meeting, London
October 27th 2006
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Dissipation
stress
strain
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Dissipation per cycle depending on frequency
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Mechanical losses
Single anelastic process
(single relaxation time):
tan     
2  f  
1   2  f   
2
For small loss angles:
  tan     
    T,f 
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Maximum at
2 f   1
2  f  
1   2  f   
2
   T

relaxation strength
f
frequency of
acoustic wave

relaxation time
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Mechanical losses
  0  e
Ea
k BT
Free en ergy
Especially for stress induced
transitions between states of
minimal energy:
Ea
Distance
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
relaxation time
0
relaxation constant
Ea
activation energy
kB
Boltzmann constant
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Dissipation due to stress induced hopping of
alkali-ions in a-quartz
Si
O
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W. P. Mason in Physical Acoustics, edited by W. P. Mason
(Academic Press Inc., New York, 1965), vol. 3B, p. 247.
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 3“  12 mm
Q-Measurement on crystalline quartz
Measurement + Fit
-4
10
-5
10
-1
Damping
Damping QQ-1
-6
10
910-7 s
3.4 meV
510-13 s
53 meV
110-14 s
163 meV
410-13 s
194 meV
phonons
defects
-7
10
-8
10
-9
10
-10
10
0
20
40
60
80
100
120
140
160
180
200
Temperature
Temperature[K]
[K]
mode shape:
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measured frequency:
11565 Hz
calculated frequency : 10793 Hz
@ 300 K
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 3“  12 mm
Q-Measurement on crystalline quartz
-5
Damping Q-1 -1
Damping Q
10
-6
10
-7
10
-8
10
0
20
40
60
80
100
120
140
160
180
200
Temperature [K]
Temperature
[K]
measured frequency :
11565 Hz
17115 Hz
61720 Hz
calculated frequency:
10793 Hz
16987 Hz
61121 Hz
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@ 300 K
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Mechanical losses in solids
• External losses
– e.g. suspension losses, residual gas damping…
• Internal losses
– „ideal“ solid:
• thermo-elastic damping
• interaction of acoustic waves with thermal phonons
of the solid
• interaction of acoustic waves with electrons of the
solid
– „real“ solid:
• additional damping caused by defect induced losses
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Dissipation caused by interaction of acoustic wave
with thermal phonons
Damping Q-1
-6
10
-7
10
0
20
40
60
80
100
120
Temperature [K]
2  fac  th  1
Landau-Rumer
damping
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2  fac  th  1
Akhieser
damping
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Dissipation caused by interaction of acoustic wave
with thermal phonons
2 f ac  th  1
Landau-Rumer
damping
• Interaction of
acoustic waves with
individual thermal
phonons.
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2 f ac  th  1
Akhieser
damping
• Perturbation of equilibrium
of thermal phonon
distribution.
• Reestablishment affords
increase in entropy and such
leads to a partly absorbation
and attenuation of the
acoustic wave.
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Crystalline silicon
• less than 1014 doping atoms per cm3:
mechanical losses are dominated by interactions of
acoustical waves with thermal phonons of the
crystal
• higher doping concentrations:
dissipation by interactions with additional electrons
and holes respectively increases
SFB C4 06/06
W. P. Mason in Physical Acoustics, edited by W. P. Mason
(Academic Press Inc., New York, 1966), vol. 4A, pp. 299.
12/15
n-doped silicon
• Origin of dissipation:
Movement / transition of conduction electrons between
minima of energy in k-space
• Minima occur along the six
<100> directions
• Unstressed crystal:
Minima possess same energies
and numbers of occupation
• Stress along crystal axis rises energy
of parallel located minima and lowers
that of perpendicular ones
• Consequence:
Reestablishing equilibrium by flow of electrons from minima of
higher energy to lower ones
• delay = intervalley relaxation time is origin of an effective
energy transfer from the acoustic wave to the thermal bath
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p-doped silicon
• Energy surfaces of the valence band become reshaped by
stresses induced by acoustic waves.
• It is assumed that a flow of holes from regions of higher
energy to that of lower energy of the same surface occurs and
not between surfaces.
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How to determine the mechanical loss factor?
total damping
background damping
(suspension, residual gas
damping…)
sum of internal
anelastic processes
1
Q1  f 0 ,T    Qrel,i
 f0 ,T   Qbg1  f0 ,T 
i
Q
1
 f0 , T    i  f0 , T 
i
2  f 0  i
1   2  f 0  i 
2
1
 Q bg
 f0 , T     f0 , T 
f0  f0  T      T 
directionality (anisotropy!) of
i
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and
i
  f 0,n     f 
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