SFB C4 06/06 3/15 Dissipation per cycle depending on frequency
SFB C4 06/06 4/15 Mechanical losses For small loss angles: relaxation strength frequency of acoustic wave relaxation time Single anelastic process (single relaxation time): Maximum at
SFB C4 06/06 5/15 Mechanical losses relaxation time relaxation constant activation energy Boltzmann constant Especially for stress induced transitions between states of minimal energy:
SFB C4 06/06 6/15 Dissipation due to stress induced hopping of alkali-ions in -quartz W. P. Mason in Physical Acoustics, edited by W. P. Mason (Academic Press Inc., New York, 1965), vol. 3B, p. 247. O Si
SFB C4 06/06 7/15 Q-Measurement on crystalline quartz 3“ 12 mm measured frequency: 11565 Hz calculated frequency : 10793 Hz @ 300 K mode shape: MeasurementMeasurement + Fit defects phonons 910 -7 s 3.4 meV 510 -13 s 53 meV 110 -14 s 163 meV 410 -13 s 194 meV Temperature [K] Damping Q -1
SFB C4 06/06 8/15 3“ 12 mm measured frequency : 11565 Hz 17115 Hz 61720 Hz calculated frequency: 10793 Hz 16987 Hz 61121 Hz @ 300 K Q-Measurement on crystalline quartz Temperature [K] Damping Q -1
SFB C4 06/06 9/15 Mechanical losses in solids External lossesExternal losses –e.g. suspension losses, residual gas damping… Internal lossesInternal 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
SFB C4 06/06 10/15 Dissipation caused by interaction of acoustic wave with thermal phonons Akhieser damping Landau-Rumer damping Temperature [K] Damping Q -1
SFB C4 06/06 11/15 Dissipation caused by interaction of acoustic wave with thermal phonons Akhieser damping Landau-Rumer 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. Interaction of acoustic waves with individual thermal phonons.
SFB C4 06/06 12/15 Crystalline silicon less than 10 14 doping atoms per cm 3 : 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 W. P. Mason in Physical Acoustics, edited by W. P. Mason (Academic Press Inc., New York, 1966), vol. 4A, pp. 299.
SFB C4 06/06 13/15 n-doped silicon Origin of dissipation: Movement / transition of conduction electrons between minima of energy in k-space Minima occur along the six 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
SFB C4 06/06 14/15 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.
SFB C4 06/06 15/15 How to determine the mechanical loss factor? total damping sum of internal anelastic processes background damping (suspension, residual gas damping…) directionality (anisotropy!) of and