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Calculated Vibrational Frequencies of a
Wurtzite Crystal Structure CdS Sphere
Vibrational frequencies of a Wurtzite crystal structure Cadmium Sulfide sphere are calculated based on the five bulk linear elastic constants. Comparison with polarized Raman scattering experimental data shows good agreement.

   The elastic properties of a crystal are not isotropic. Unlike polycrystalline materials such as metals, or disordered materials such as ceramics, glasses and plastics, a single crystal does not have a Young's modulus and Poisson ratio. Rather, the stress varies depending on the axis along which strain is applied.
   There has been much interest in the observation and effects of low frequency phonon modes of spherical nanoparticles, and in particular nanoparticles made of Cadmium Sulfide (CdS), including experimental studies [Tanaka, Onari & Arai 1993] and theoretical analysis [Ovsyuk and Novikov 1996] [Alcalde, Marques, Weber and Reinecke 2000 pdf ::].
   In order to estimate the vibrational frequencies of CdS spheres, all this past work has begun with the approximation that CdS is an isotropic, homogeneous, continuous, linear elastic material and referred to the original solution [Lamb 1881] as well as to more recent references incorporating restatements of the analysis [Nishiguchi and Sakuma 1981] [Tamura, Higeta and Ichinokawa 1982] [Fujii, Nagareda, Hayashi and Yamamoto 1991]. In practice, actual nanoparticles of CdS deviate from this ideal in a number of ways: (1) diameter of the spheres varies within a sample; (2) they are not perfectly spherical; (3) they are not homogeneous, particularly at their surface [Tamura, Higeta and Ichinokawa 1982]; (4) they are often embedded in a glass or plastic matrix which damps and shifts the frequency of their vibration [Tamura, Higeta and Ichinokawa 1982] [Ovsyuk and Novikov 1996] [Alcalde, Marques, Weber and Reinecke 2000 pdf ::]; (5) the speed of sound is reduced at very high frequencies; (6) the elastic properties of the bulk material are not isotropic. It is this last issue, bulk material anisotropy, that is addressed here.
   Cadmium Sulfide can exist in at least two structures. The first is the cubic zinc sulfide structure, reminiscent of a diamond crystal [Kittel, 4th ed., figure 30, page 31] with a cubic unit cell of edge length 0.582 nm [Kittel, 4th ed., page 31]. The second is the hexagonal zinc sulfide structure, usually called the Wurtzite structure [Kittel, 4th ed., figure 31, page 32] with unit cell dimensions a = 0.413 nm and c = 0.675 nm [Kittel, 4th ed., page 33]. My impression is that nanoparticles of CdS, once suitably annealed, are predominately of the Wurtzite structure. All of the following analysis assumes this.
   Computer programs (with listings written in C++) are available to accurately compute the vibrational frequencies of an anisotropic elastic sphere accurate to within the order of 1%. They are available for materials with cubic [Murray 2002] (C++ listing: cc3mod.cpp) and hexagonal [Murray 2002] (C++ listing: hc2mod.cpp) symmetry.
   Calculations have been made for spheres of CdSe, Y2O3, Si, InAs, NaCl, CsBr and RbI.
   A separate computer program hex2.cpp is used to determined the five force constants starting from the five elastic constants of the material. The lattice used is hexagonal with lattice constants a = 1 and c = 0.93. The ratio c/a must be chosen so as to avoid having some negative force constants. The c/a ratio chosen for the lattice in the calculation is unrelated to the c/a ratio for the actual crystal structure, since only the bulk, low frequency, linear elastic constants are being simulated.
   Five different force constants are required in order to represent the general elasticity of a material with hexagonal symmetry. Such a material has five independent elastic constants, C11, C12, C13, C33 and C44. Other elastic constants are dependent on these through: C11=C22, C33, C12=C21, C13=C31=C23=C32, C44=C55, C66=(C11-C12)/2. Finally, constants Cij and Cji with i=1,2,3 and j=4,5,6 are all 0.

Table I. Force constants for lattice simulation
of hexagonal Cadmium Sulfide
(Using C++ program hex2.cpp.)
constant units value
a (metres) 1
c (metres) 0.93
ρ (g/cc) 4.825
C11 (GPa) 90.7
C12 (GPa) 58.1
C13 (GPa) 51.0
C33 (GPa) 93.8
C44 (GPa) 15.04
C66 (GPa) 16.3
k6pt (N/m3) 5.581e9
ksp1 (N/m) 2.567e10
k3pt (N/m3) 1.134e10
ksp3 (N/m) 1.968e10
ksp4 (N/m) 8.706e9

Figure 1. Fourier transforms of lattice simulations of CdS spheres are shown. The initial disturbance of the lattice is as shown at lower left. For each plot, the horizontal axis is η = ω R/(1765 m/s). The vertical axis is 1/R, with infinite R at the top. R at the bottom is 3. The maximum radius and number of atoms, N, used is shown at the lower right. (hc2mod.cpp cdssph0a.gif...)

Table II. Vibrational frequencies (ω in rad/s)
of a CdS (Wurtzite form) nanosphere of radius R
mode
type 		angular momentum quantum number
l 		η ω R frequency, ν,
for 10 nm diam.
sphere (cm-1)
sph0 6.98 12320 11.09
sph1 3.85 6795 6.11
sph2 2.66 4695 4.22
sph2 2.77 4889 4.40
sph2 3.18 5613 5.05
sph2 5.20 9178 8.26
sph2 5.40 9531 8.58
sph2 5.60 9884 8.89
sph2 6.00 10590 9.53
sph3 3.75 6619 5.96
sph3 4.05 7148 6.43
sph3 4.32 7625 6.86
sph3 6.80 12002 10.80
sph4 5.40 9531 8.58
sph5 6.40 11296 10.16
tor1 5.55 9796 8.81
tor1 6.20 10943 9.85
tor1 6.50 11472 10.32
tor2 2.55 4501 4.05
tor2 2.60 4589 4.13
tor2 2.75 4854 4.37
tor3 3.90 6884 6.19
tor3 4.20 7413 6.67
tor4 5.30 9355 8.42
tor4 5.60 9884 8.89
tor5 6.50 11472 10.32
tor5 7.00 12355 11.12

Figure 2.Calculated vibrational spectrum for selected modes of a 10 nm diameter CdS (Wurtzite crystal structure) nanoparticle
(hc2fmod.cpp cds.frq readfrq3.cpp cds.scr s2g.cpp cds.gif)

Figure 3.Calculated vibrational spectrum for selected modes of CdS (Wurtzite crystal structure) nanoparticles of various diameters as observed in the experiment of Tanaka, Onari and Arai.
(hc2fmod.cpp cds.frq readfrq3.cpp cds.scr s2g.cpp cds.gif)

Figure 4.  Raman active spectrum for selected modes of
CdS (Wurtzite crystal structure) nanoparticles of diameters 7.5 nm
as observed in the 1993 experiment of Tanaka, Onari and Arai.
(hc2fmod.cpp cds.frq readfrq3.cpp cds.scr s2g.cpp cds.gif)


Comparison with Experimental Data

   One earlier experiment [Tanaka, Onari & Arai 1993] performed Raman scattering on annealed CdS microcrystals embedded in a germanium dioxide glass matrix. Selection rules associated with Raman scattering restrict the modes that can be observed. They felt that they were able to view the n=0 spheroidal mode from the polarized (HH) Raman spectrum which exhibits a single peak. This peak is somewhat broad, which they attribute to the range of particle diameters. The frequency of the peak is roughly inversely proportional to particle diameter. The largest particle size which showed the peak was 16.1 nm diameter. The location of the peak center is 7.4 cm-1. This corresponds to ω = 1.394×1012 rad/s. This makes ω R = 11221. The above theoretical analysis predicts the l=0 spheroidal peak at ω R = 12320, which is 10% higher. There are two effects that would be expected to lower the experimentally seen frequency: (1) the small size of the particle means that the phonon wavelength is not infinitely large relative to the unit cell size (0.4 to 0.7 nm), making phonon dispersion possibly relevant; (2) coupling to the surrounding germanium dioxide glass matrix will dampen and somewhat lower the frequency of the vibrations. A simple calculation based on an isotropic elastic sphere surrounded by a fluid matrix [Murray 2002] predicts a frequency reduction of about 7%. Another important effect to consider is that the rapid damping of the oscillations will lead to lifetime broadening.
   The same experiment [Tanaka, Onari & Arai 1993] observed a second Raman peak at a lower frequency by observing the depolarized (HV) spectrum. This is not as clear, and was only visible for the smallest particle size (7.5 nm), making inverse radius dependence impossible to confirm. They argued that the l=2 spheroidal peak could cause this scattering. For the 7.5 nm particles, the broad peak is located roughly at around 8 cm-1. This corresponds to ω = 1.56×1012 rad/s and ω R = 5860. Theory showed that there is not a single peak in that region, but a group of at least twelve peaks. This is illustrated in Figure 3. Somebody has said [Ovsyuk & Novikov 1996, section III A] that only spheroidal peaks with even l and torsional peaks with odd l are Raman active. It has also been pointed out that the quadropolar modes have the greatest Raman activity [Portales et al. 2002 pdf ::]. Figure 4 shows the Raman active peaks only. Looking at Figure 4, there does in fact seem to be a group of peaks whose center is at approximately 8 cm-1.
References:

H. Lamb, "On the vibrations of an elastic sphere" Proc. London Math. Soc. 13, 189 (1881-1882).

N. Nishiguchi and T. Sakuma, "Vibrational spectrum and specific heat of fine particles" Solid State Comm. 38 (1981) 1073-1077.

A. Tamura, K. Higeta and T. Ichinokawa, "Lattice vibrations and specific heat of a small particle" J. Phys. C: Solid State Phys. 15 (1982) 4975-4991.

M. Fujii, T. Nagareda, S. Hayashi and K. Yamamoto, "Low-frequency Raman scattering from small silver particles embedded in SiO2 thin films" Phys. Rev. B 44 (1991) 6243-6248.

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.

N. N. Ovsyuk and V. N. Novikov, "Influence of a glass matrix on acoustic phonons confined in microcrystals" Phys. Rev. B 53 (1996) 3113-3118.

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 ::

H. Portales, L. Saviot, E. Duval, M. Gaudry, E. Cottancin, M. Pellarin, J. Lermé and M. Broyer "Resonant Raman Scattering by Quadrupolar Vibrations of Ni-Ag Core-shell nanoparticles" (Preprint Mar 22 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

D. B. Murray "Breathing Mode Vibrations of an Isotropic Elastic Sphere Surrounded by a Fluid Medium" 2002 link to article

D. B. Murray "Vibrational Frequency of Silicon Nanoparticles" 2002 link to article

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)

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


Daniel Murray
Associate Professor
Math, Stats & Physics Unit
University of British Columbia - Okanagan
Kelowna, BC, Canada
daniel "dot" murray "at" ubc "dot" ca

For a list of related articles click here.


W. Q. Chen, H. J. Ding, R. Q. Xu "Three-dimensional static analysis of multi-layered piezoelectric hollow spheres via the state space method" International Journal of Solids and Structures 38 (2001) 4921-4936 ::

Chen, W. Q. and Ding, H. J. (2001) Free vibrations of multi-layered spherically isotropic hollow spheres. International Journal of Mechanical Science, Vol. 43(3), 667-680. ::

H. Cohen, A. H. Shah and C. V. Ramakrishnan 1972 Acustica 26 329-333, "Free Vibrations of a Spherically Isotropic Hollow Sphere"

P. R. Heyliger and A. Jilani 1992 International Journal of Solids and Structures 29 2689-2708 "The Free Vibrations of Inhomogeneous Elastic Cylinders and Spheres"

Chen, W. Q., Cai, J. B., Ye, G. R. and Ding, H. J. (2000) "On eigenfrequencies of an anisotropic sphere". ASME Journal of Applied Mechanics, Vol. 67(2), 422-424.

Chen, W. Q. "Effect of radial inhomogeneity on natural frequencies of an anisotropic hollow sphere" Journal of Sound and Vibration, Vol. 226(4), 787-794.(1999) ::

Chen, W. Q. and Ding, H. J. (1997) "On free vibrations of an embedded anisotropic spherical shell" ASME Journal of Pressure Vessel Technology, Vol. 119(4), 481-487.

Ding, H. J. and Chen, W. Q. (1996) "Natural frequencies of an elastic spherically isotropic hollow sphere submerged in a compressible fluid medium". Journal of Sound and Vibration, Vol. 192(1), 173-198.

W. Q. Chen and L. Z. Wang, "Free Vibrations of Functionally Graded Piezoceramic Hollow Spheres with Radial Polarization" Journal of Sound and Vibration vol. 251, pages 103-114 (2002) ::

W. Q. Chen, "Vibration Theory of Non-Homogeneous Spherically Isotropic Piezoelastic Bodies" Journal of Sound and Vibration" vol 236 pages 833-860 (2000) ::

Palko JW, Kriven WM, Sinogeikin SV, Bass JD, Sayir A "Elastic constants of yttria (Y2O3) monocrystals to high temperatures"
JOURNAL OF APPLIED PHYSICS 89 (12): 7791-7796 JUN 15 2001
http://hercules.geology.uiuc.edu/~stas/Abstracts/JAP01_89_7791.htm
Room temp: C11=223.7(0.6) C12=112.4(1.1) C44=74.6(0.7) GPa K = 149.5(1.0) GS=66.3(0.8) [Voigt-Reuss-Hill average]

Formula for Voigt and Reuss for cubic crystal (in french; has error in C' definition)
http://www.ens-lyon.fr/LST/WEBHP/elasticite/node2.html
GV = (2C'+3C44)/5 = Voigt Bound
GR = 5C'C44/(2C44+3C') = Reuss Bound
GS=(GV+GR)/2 = Voigt-Reuss-Hill average
C'=(C11-C12)/2
Also, for a cubic crystal, the VRH bulk modulus is (C11+2C12)/3

"Gas-phase-condensed Y2O3:Eu3+ nanoparticles of < 10 nm contain multiple phases: monoclinic Y2O3:Eu3+, cubic Y2O3:Eu3+, monoclinic Eu2O3, and a disordered phase. Larger particle sizes predominantly form in the metastable monoclinic phase" "Annealing as-prepared 4-nm Y2O3:Eu3+ at 500-900 C produces cubic-phase Y2O3:Eu3+ and removes all other phases. " link

Noninertial mechanism for electronic energy relaxation in nanocrystals Ho-Soon Yang,* Michael R. Geller, and W. M. Dennis Department of Physics and Astronomy, University of Georgia, Athens, Georgia 30602-2451 pdf ::

H. S. Yang, M. R. Geller Dennis, "New Mechanism for Electronic Energy Relaxation in nanocrystals" Aug 15 2000 preprint

H. S. Yang, M. R. Geller, W. M. Dennis, "Noninertial Mechanism for electronic energy relaxation in nanocrystals" Phys. Rev. B vol. 62 (2000) 9398. ::

T. Takagahara, J. Lumin. 70, 129-143 (1996)
"Electron-phonon interactions in semiconductor nanocrystals"

List of recent publications by Takagahara group (in Japanese)
http://www.hiei.kit.ac.jp/~takaghra/

http://www.chem.vt.edu/chem-dept/tissue/nano/

"While the elastic modes of spherical nanoparticles can be calculated analytically, this is not the case for nonspherical nanoparticles." "Visscher et al. ...mode structure for ellipsoidal nanoparticles..." ::

W. M. Visscher, A. Migliori, T. M. Bell and R. A. Reinhart, "The normal modes of free vibration of inhomogenous and anisotropic elastic objects" J. Acoust. Soc. 90 (1991) 2154-2161.

M. K�gl, L. Gaul A 3-D Boundary Element Method for Dynamic Analysis of Anisotropic Elastic Solids Computer Modeling in Engineering & Sciences, 1(4) (2000), pp. 27-43 abstract

U. Banin, G. Cerullo, et al. "Quantum confinement and ultrafast dephasing dynamics in InP " Phys Rev. B vol 55, March 1997 ::

Has general axis formula for Young's modulus!
1/E = (C11+C12) +...
http://www.crystran.co.uk/optics.htm
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 ::

CdS colloidal quantum dots:
S. A. Empedocles, D. J. Norris, M. G. Bawendi, Phys. Rev. Lett 77, 3873 (1996).
   Another experiment [Cerullo, De Silvestri & Banin 1999 pdf ::, see figure 6] used femtosecond pump-probe spectroscopy to look at a CdSe nanocrystal with diameter 3.6 nm in a polymer film. Excitation of the breathing (n=0 spheroidal) mode led to damped oscillations with frequency 21 cm-1 and damping time 1.5 ps. This frequency corresponds to ω = 3.96×1012 and ω R = 7120. But this is for CdSe and not for CdS.

http://www.spie.org/web/abstracts/2300/2362.html Paper #: 2362-33 Femtosecond pump-probe studies of CdSe microcrystals embedded in a germanium dioxide glass matrix, pp.304-311 Author(s): Toshihiro Arai, Univ. of Tsukuba, Tsukuba, Ibaraki, Japan; Akinori Tanaka, Univ. of Tsukuba, Sendai, Japan; Kiyoto Matsuishi, Univ. of Tsukuba, Tsukuba, Ibaraki, Japan; Seinosuke Onari, Univ. of Tsukuba, Tsukuba, Japan; Yoshihiro Maruyama, Hamamatsu Photonics KK, Tsukuba, Ibaraki, Japan; Mitsuru Ishikawa, Hamamatsu Photonics KK, Tsukuba, Ibaraki, Japan.
Abstract: We report here results of experiments on the transient absorption spectra of CdSe microcrystals embedded in a germanium dioxide glass matrix. The transient absorption spectra were measured by means of femtosecond pump-probe method. In the sample with the smaller microcrystal size within the category of the strong confinement regime (the individual particles confinement regime), the absorption bleachings due to the carrier dynamics in the quasi-zero- dimensional quantum confined states were observed. In the sample with the larger microcrystal size within the category of the weak confinement regime (the exciton confinement regime), the absorption bleachings due to the hot carrier in the 3D electronic states were studied at various excitation intensities.!31
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