Ge - Germanium

Mechanical properties, elastic constants, lattice vibrations

Basic Parameters
Elastic Constants
Acoustic Wave Speeds
Phonon Frequencies

Basic Parameters

Bulk modulus7.5·1011dyn cm-2
Density5.323 g cm-3
Hardness on the Mohs scale6
Surface microhardness (using Knoop's pyramid test)780 kg mm-2
Cleavage plane{ 001 }

Elastic Constants at 300 K

C11 = 12.60·1011 dyn cm-2
C12 = 4.40·1011 dyn cm-2
C44 = 6.77·1011 dyn cm-2
Temperature dependences of elastic constants.
(Kardashev and Nikanorov [1985]).
Elastic constant C44 versus temperature
for pure (curve 1) and doped (curve 2. Nd=3.5·1019cm-3) samples.
(Bruner and Keyes [1961]).


For 300 K: 
Bulk modulus
(compressibility-1)
Bs=7.13·1011 dyn cm-2
Shear modulusC'=4.1·1011 dyn cm-2
[100] Young's modulusYo = 10.3·1011 dyn cm-2
[100] Poisson ratioσo= 0.26
Si, Ge. Young's modulus vs. direction in the (001) plane.
Wortman & Evans (1965)
Si, Ge. Poisson ratio vs. direction in the (001) plane.
Wortman & Evans (1965)

Acoustic Wave Speeds

Wave propagation
direction
Wave CharacterExpression for wave speedWave speed
[100]VL[C11/ρ]1/24.87 ·105 cm/s
VT[C44/ρ]1/23.57 ·105 cm/s
[110]Vl[(C11+C12+2C44)/2ρ]1/25.36 ·105 cm/s
Vt||Vt|| = VT = (C44/ρ)1/23.57 ·105 cm/s
Vt ort;[(C11-C12)/2ρ]1/22.77 ·105 cm/s
[111]Vl'[(C11+2C12+4C44)/3ρ]1/25.51 ·105 cm/s
Vt'[(C11-C12+C44)/3ρ]1/23.06 ·105 cm/s

Phonon Frequencies (in units of 1012 Hz)

phonon wavenumbers:  RemarksReferens
GeνLTO25')9.02 1012 HzT=300KNillsson & Nelin (1972)
 νTA(X3)2.385 1012 Hz  
 νLAO(X1)7.14 1012 Hz  
 νTO(X4 )8.17 1012 Hz  
 νTA (L3)1.87 1012 Hz  
 νLA(L2')6.63 1012 Hz  
 νLO(L1)7.27 1012 Hz  
 νTO(L3')8.55 1012 Hz  
Ge. Dispersion curves for acoustic and optical phonons.
Weber (1977)