Complex Frequency (Decaying) Torsional Modes...

Updated: January 23, 2008
Complex Frequency (Decaying) Torsional Modes of an Si nanosphere Surrounded by an SiO₂ Matrix

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   C++ programs used:
scp77.c - graphical root search
scp77c.c - precision root finder (needs initial guess)

ν = 0.01 S v_lp / ( d c )
ν = complex valued frequency expressed in terms of "wavenumbers" (cm^-1)
S = complex dimensionless frequency parameter. S = Re(S) + i Im(S).
v_lp = longitudinal speed of sound in nanoparticle (m/s)
d = nanosphere diameter (m)
c = speed of light (m/s)

   The displacement field u(r,t) is assumed to have the form inside and outside the nanosphere (of radius R) as follows:
u(r,t) = e^iωt ∇ × (r B j_l(k_tp r) P_l(cos(θ)) )        (r ≤ R)
u(r,t) = e^iωt ∇ × (r C h⁽²⁾_l(k_tm r) P_l(cos(θ)) )       (r ≥ R)
where
B and C are complex amplitude constants
   One boundary condition is that u(r,t) is continuous at r = R. For toroidal modes, only u_φ is nonzero. The other boundary conditions is local force balance on the boundary (at r=R) between the nanoparticle and the matrix. In other words, the components of the (3×3) stress tensor σ_rr, σ_rθ, σ_rφ must all be continuous as the boundary r=R is crossed. For toroidal modes, only σ_rφ is nonzero.

Table 1. l = 1 Toroidal:
Re(S)	Im(S)	S (without matrix)
0.2800	0.2604	n/a
1.2738	0.1566	1.2710
2.0074	0.1551	2.0057
2.7187	0.1547	2.71
3.4220	0.1545	3.42
4.1221	0.1544	4.12

My impression is that the S=0.2800+0.2604i mode arises from the possibility of (nearly rigid) rotational oscillations. These would not be Raman active for a perfect sphere, but they could be Raman active for a non-spherical particle. Apart from that, the frequencies of the modes have not been shifted much by the presence of the matrix.

Table 2. l = 2 Toroidal:
Re(S)	Im(S)	S (without matrix)
0.1522	0.2809	n/a
0.6656	0.3318	0.5517
1.5757	0.1678	1.5738
2.3201	0.1601	2.3186
3.0380	0.1576	3.03
3.7459	0.1564	3.75

I don't understand the origin of the S = 0.1522+0.2809i mode. Apart from that, the frequencies of the modes have not been shifted much by the presence of the matrix.

Table 3. l = 3 Toroidal:
Re(S)	Im(S)	S (without matrix)
0.2867	0.3119	n/a
0.9720	0.3648	0.8523
1.8638	0.1785	1.8622
2.6213	0.1654	2.6196
3.3474	0.1608	3.34
4.0610	0.1586	4.06

   Once again, there is an additional low frequency mode (S=0.2867+0.3119i) that was not present in the absence of the matrix. These l = 3 toroidal modes are not Raman active for an isotropic elastic sphere in an isotropic, homogenous matrix. However, it seems possible that under actual conditions these modes might contribute to the Raman spectrum.
   The programs (scp77.c,scp77c.c) are not able to explore l ≥ 4 due to memory space limitations.

   Silicon does not have isotropic elastic properties. It is a cubic crystal. However, for simplicity it was approximated here as an isotropic material by using the speeds along the [100] crystal direction. In the silicon nanoparticle I used the following sound speeds:
v_lp = 8430 m/s
v_tp = 5840 m/s
ρ_p = 2.34 g/cc
In the SiO₂ matrix, I assumed:
v_lm = 5720 m/s
v_tm = 3750 m/s
ρ_m = 2.2 g/cc

   Alternative, but relatively common, dimensionless frequencies for nanosphere vibrations are ξ and η, where
ξ = ω R / v_lp
η = ω R / v_tp
where ω = angular frequency (rad/s)
R = radius of nanosphere (metres)
v_lp = longitudinal sound velocity inside nanoparticle (m/s)
v_tp = transverse sound velocity inside nanoparticle (m/s)
ω = η v_tp / R
S and η are related by:
S = η v_tp / ( v_lp π )
For the case of an Si nanoparticle, with the isotropic sound speeds I assumed,
S = η (5840 m/s) / (3.14159×8430 m/s) = 0.2205 η
η = 4.535 S

   Spherical hankel functions of the first kind (inward travelling waves) h⁽¹⁾_l(x) and second kind (outward travelling waves) h⁽²⁾_l(x) are:
h⁽¹⁾_l(x) = j_l(x) + i n_l(x)           (mathworld.wolfram.com)
h⁽²⁾_l(x) = j_l(x) - i n_l(x)
In particular, (noting that time dependence is e^iωt)

j₀(x) = sin(x) / x		standing wave - nonsingular at origin
n₀(x) = - cos(x) / x		standing wave - singular at origin
h⁽¹⁾₀(x) = (-i/x) e^ix		inward travelling wave
h⁽²⁾₀(x) = (i/x) e^-ix		outward travelling wave

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

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