Last updated: January 23, 2008
Sound Speed Directional Dependence in Hexagonal
Crystals: CdS, CdSe, Al2O3, InN, BN, AlN and GaN
The directional dependence of the speed of sound in a hexagonal crystal is calculated from the bulk elastic constants. Minimum, maximum and average speeds of sound are found for CdS, CdSe, Al2O3, InN, BN, AlN and GaN. Longitudinal sound speed shows very little anisotropy, while the transverse speed of sound is much more anisotropic.

   Using the C++ programs hex2.cpp and dsprel6f.cpp, I found the minimum, maximum and average speeds of sound, transverse and longitudinal in a number of hexagonal crystals. The average is taken by uniformly randomly sampling different directions and taking a simple average. The transverse speed of sound has two branches.
   A hexagonal crystal has five independent elastic constants: C11, C12, C13, C33 and C44. Note that C66 = (C11 - C12)/2.

 Table I. Speed of sound in selected hexagonal crystals
material transverse (m/s) longitudinal (m/s)Notes:
min avg max (max-min)
-----------
min		 min avg max (max-min)
-----------
min
CdS 1763 1870 2069 17% 4234 4289 4409 4.1%sqrt(C33/ρ)=4409
CdSe 1493 1607 1825 22% 3501 3559 3750 7.1%sqrt(C33/ρ)=3750
Al2O3 6097 6438 6927 14% 10679 10913 11185 4.7% 
InN * 2598 2981 3285 26% 5391 5876 6308 17% *using elastic parameters theoretically calculated by K. Kim, W. R. L. Lambrecht and B. Segall, "Elastic constants and related properties of tetrahedrally bonded BN, AlN, GaN and InN", Phys. Rev. B 53 (1996) pages 16310-16326 ::
InN** 1211 1942 2513 108% 4897 5077 5282 7.9%**using experimental elastic parameters at 300K
BN 10548 11013 11693 10.9% 16360 16653 17570 7.4% 
AlN 6220 6419 6818 9.6% 10747 10986 11266 4.8% 
GaN 4132 4427 4839 17% 7596 7781 8044 5.9% 
ZnO 2778 2789 2817 1.4% 6011 6027 6084 1.2% 

   Exact checks on correctness can be applied. If C33 > C11, then the maximum longitudinal speed of sound is sqrt(C33/ρ).
   There is an interesting general pattern on the extent of anisotropy. In every case, transverse sound velocity shows more anisotropic variation than the longitudinal speed of sound. The anisotropy of the longitudinal speed of sound is always very small, taking the values 4.7%, 4.8%, 5.9%, 7.1%, 7.4% and 7.9%. Therefore, at least for these hexagonal crystals, the propagation of longitudinal waves can reasonably be approximated as isotropic.

References:

K. Kim, W. R. L. Lambrecht and B. Segall "Elastic constants and related properties of tetrahedrally bonded BN, AlN, GaN and InN" Phys. Rev. B 53, June 15, 1996, pages 16310-16326. ::

A. Tanaka. S. Onari and T. Arai, "Low-frequency Raman scattering from CdS microcrystals embedded in a germanium dioxide glass matrix" Phys. Rev. B 47 (1993) 1237-1243.

G. Cerullo, S. De Silvestri and U. Banin "Size-dependent dynamics of coherent acoustic phonons in nanocrystal quantum dots" Phys. Rev. B volume 60 (July 15, 1999) pdf ::

A. M. Alcalde, G. E. Marques, G. Weber. T. L. Reinecke "Electron acoustic phonon scattering rates in II-IV quantum dots: contribution of the macroscopic deformation potential" Solid State Communications 116 (2000) 247-252 pdf ::

V. N. Soloviev, A. Eichhofer, D. Fenske, U. Banin "Size-dependent optical spectroscopy of a homologous series of CdSe cluster molecules" J. Am. Chem. Soc. 123, 2354-2364 (2001) pdf ::

Eran Rabani "An interatomic pair potential for cadmium selenide" Journal of Chemical Physics vol. 116, no. 1 (2002) pdf ::

D. B. Murray "Molecular Dynamic Simulation of an Elastic Solid with Hexagonal Symmetry" link to article

D. B. Murray "Eight Point Force Molecular Dynamical Estimates of Vibrational Frequencies of an Isotropic Elastic Sphere" 2002 link to article

Sapphire Al2O3
C11=496
C12=164
C13=115
C33=498
C44=148
B = 240 GPa
http://www.tydex.ru/materials/materials2/sapphire.html

??? CdSe density 5.67 g/cc. !!!???
http://ncsr.csci-va.com/materials/cdse.asp
CdSe (Wurtzite) 5.81 g/cc

Article on CdSe modelled using pair potential
Table III gives calc and experi. elastic constants for wurtzite, zinc blende and rocksalt forms
CdSe Experimental values (Wurtzite = hexagonal):
C11 = 74.6 GPa
C12 = 46.1 GPa
C13 = 39.4 GPa
C33 = 81.7 GPa
C44 = 13.0 GPa
C66 = 14.3 GPa
B = 53.4 GPa
from: Eran Rabani "An interatomic pair potential for cadmium selenide" Journal of Chemical Physics vol. 116, no. 1 (2002) pdf ::

http://www.tydex.ru/materials/materials2/sapphire.html
mater. C11 C12 C13 C33 C44 density structure
Al2O3 496 164 115 498 148 3.98 g/cc hexagonal ("sapphire")

CdS properties:
http://www.efunda.com/materials/piezo/material_data/matdata_output.cfm?Material_ID=CdS
Piezo Data: CdS
Hexagonal
??? density 5684 kg/m^3. ???? seems wrong
Compliance SE 10-12 m2/N: (inverse of 6x6 stiffness C matrix ! )
20.69 -9.99 -5.81 0 0 0
-9.99 20.69 -5.81 0 0 0
-5.81 -5.81 16.97 0 0 0
0 0 0 66.49 0 0
0 0 0 0 66.49 0
0 0 0 0 0 61.36
http://www.issp.ac.ru/lpcbc/DANDP/cds_adv.html
Density 4.825 g/cc (agrees) ???!!!!
Young's modulus 45 GPa
http://www.vidrine.com/iropmat3.htm
Gives density as 4.82 g/cc. (agrees)

CdS hexagonal
C11=90.7 GPa
C12=58.1 GPa
C33=93.8 GPa
C13=51.0 GPa
C44=15.04 GPa
Source: Michael Conry "Notes on Wave propagation in anisotropic elastic solids"
pdf ::

Wurtzite InN. Elastic constants at 300 K.
C11 190 � 7 GPa Sheleg & Savastenko (1979); see also: Wright (1997), Kim et al. (1996)
C12 104 � 3 GPa
C13 121 � 7 GPa
C33 182 � 6 GPa
C44 10 � 1 GPa
http://www.ioffe.rssi.ru/SVA/NSM/Semicond/InN/mechanic.html

Wurtzite AlN. Elastic constants at 300 K.
C11 410 � 10 GPa McNeil (1993) see also Wright (1997)
C12 149 � 10 GPa
C13 99 � 4 GPa
C33 389 � 10 GPa
C44 125 � 5 GPa
http://www.ioffe.rssi.ru/SVA/NSM/Semicond/AlN/mechanic.html

Daniel Murray
Associate Professor
Physics
University of British Columbia Okanagan
Kelowna, BC, Canada
Email: daniel "dot" murray "at" ubc "dot" ca

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