Continuum Phonon Approach to Embedded Nanoparticle Raman Spectra
1. Motivation for Calculation
(This section can be skipped if you want to get directly to the details.)
The underlying motivation of all the analysis that follows is to
calculate the low frequency Raman scattering spectrum of a nanoparticle
embedded in a glass matrix. The results reported here fall substantially
short of that goal, however.
I proceed from the simplified assumption that low frequency
Raman spectra of nanoparticles are caused by the mechanical vibrations
of the nanoparticles. This approach completely ignores realistic features
of what the electrons in the nanoparticle are doing.
Raman scattering from a microscopic object (atom,
molecule, nanoparticle) is associated with low frequency
variation of the 3×3 polarizability tensor α(t). This tensor refers to a
single nanoparticle. The dipole moment of the nanoparticle is p(t). The
electric field (from a laser) induces a time dependent dipole. Radiation
from the dipole leads to Rayleigh scattering and Raman scattering.
One idealization made is that only the electric field at a single point needs
to be considered. This field is E(t).
pi(t) = αij(t) Ej(t)
The assumption that I make is that the variations of the
polarizability tensor depend solely on mechanical vibration of the
nanoparticle as well as the associated vibration of the glass matrix in its
vicinity. The details of what this connection should be are not clear to
me. Therefore, I begin with a much simplified approach, which is that
the variation of the polarizability depends on surface displacement of
the nanoparticle only. While this assumption is surely inadequate, it
provides a starting point of something specific to calculate.
The surface displacement is a vector valued function
with units of metres that is defined everywhere on the spherical surface
of the nanoparticle.
Another important issue is the physical origin of the
vibrations. I take the attitude that these vibrations correspond to
equilibrium thermal fluctuations of a solid object in thermal equilibrium
at temperature T. I ignore quantum mechanical effects and treat
the system as a classical elastic continuum at temperature T. For a
nanoparticle at room temperature this is in fact justifiable for the
portion of the Raman spectrum below about 50 cm-1. For such
low frequency vibrations, the quantum of energy is small compared
to kBT, their mean energy. This assumption has completely broken
down by the time optical phonon modes are reached, but it seems
likely to be adequate for the vibrations associated with LOFIRS.
It may be questioned whether the nanoparticle is
really in thermal equilibrium. In particular, if the incident laser beam
is "driving" the mechanical vibrations then equilibrium is not the
correct picture. Also, the temperature distribution might be
significantly nonuniform. But these possibilities are ignored here.
Also ignored are:
The foregoing comments are meant to provide a
motivation for the calculation that follows. No rigorous
connection with Raman spectra is being claimed here.
2. Definition of Calculation
(For convenience, all definitions and assumptions are
stated in this section.)
The "matrix" region surrounds the inner "nanoparticle" region.
The matrix is a sphere with outer radius Rm.
The radius of the nanoparticle is Rp. The matrix and
nanoparticle are concentric. The limit is taken that Rm
approaches ∞.
Both material coordinates and real space coordinates are
used, and it is important to keep clear which is which. Let r be
material coordinate position. R(r,t) is the actual position
(at time t) of the material point whose equlibrium position is r. The
displacement field u is defined by
u(r,t) = R(r,t) - r.
Material parameters:
ρm = density of matrix (kg/m3)
ρp = density of nanoparticle (kg/m3)
clm = longitudinal speed of sound in matrix
ctm = transverse speed of sound in matrix
clp = longitudinal speed of sound in nanoparticle
ctp = transverse speed of sound in nanoparticle
The boundary conditions at the outer surface of the
matrix are zero displacement. (Note, however, that if the boundary
conditions were those of a free surface, the final result of this
calculation will be the same.)
The boundary conditions that apply at the boundary
of the matrix and the nanoparticle are the usual ones:
1. continuity of the displacement field
2. continuity of three components of the stress field
The limit of small vibrations is assumed so that all
behavior is linear.
The system is in equilbrium at temperature T, so that
each mechanical degree of freedom has average mechanical energy
kBT, where
kB = 1.32×10-23 J/K
(i.e. Boltzmann's constant).
3. Vibrational Mode Classification
Because of the spherical symmetry of the problem,
quantum numbers l and m can be introduced to
classify eigenfunctions of vibration. l takes non-negative
integer values. m varies from -l to +l, but
will not be mentioned again, since it just corresponds to a degeneracy
index, and is unrelated to the final results. The l=0 case has
full spherical symmetry and is called the "breathing" mode.
Also important is the polarization of the modes. As is
always the case in the vibration of a solid, there are three polarizations
for a travelling wave: (See md30.htm for details.)
(1) longitudinal spheroidal
(2) transverse torsional
(3) transverse spheroidal
4. General Form of Displacement Field u
Our purpose is to calculate the displacement field
u inside the nanoparticle for a given value of l.
The general form of the displacement field is as follows. There
are nine parts of the wave, each with a complex valued amplitude
coefficient:
- incoming wave in the matrix (long sphh)
- incoming wave in the matrix (tran torr)
- incoming wave in the matrix (tran sphh)
- outgoing wave in the matrix (long sphh)
- outgoing wave in the matrix (tran torr)
- outgoing wave in the matrix (tran sphh)
- non-singular standing wave in the nannoparticle (long sph)
- non-singular standing wave in the nannoparticle (tran tor)
- non-singular standing wave in the nannoparticle (tran sph)
Because the nanoparticle is assumed to be much smaller
than the matrix, the incoming wave's mean squared amplitude can
be determined from the requirement that its average energy is kBT.
These amplitudes were calculated earlier in
md30.htm. These are expressed in terms of |A|2
for a range of phonon frequencies centered at ω (in rad/s) of width dω.
These nine complex coefficients are related by the
boundary conditions. However, there is never a 9×9 system of
equations to solve. It is always simpler than that. There are
four "versions" of the calculation:
Version A: l = 0, "breathing mode"
Version B: l > 0, "spheroidal", longitudinal incident wave
Version C: l > 0, "spheroidal", transverse incident wave
Version D: l > 0, "torsional"
Version A: l = 0, "breathing mode"
The displacement field has the form, with complex amplitudes A, B and C:
- incoming wave in the matrix (long sphh) A
- non-singular standing wave in the nannoparticle (long sph) B
- outgoing wave in the matrix (long sphh) C
From md30.htm: (setting l = 0 in this case)
|A|2 =
dω kBT (2 l + 1) /
(π2 ρm clm
ω2)
|A| is the magnitude of complex number A. This, together with
the boundary conditions at the surface of the nanoparticle, determines B and C,
up to a factor of eiφ, since the complex phase of A is unknown.
Check on correctness: By energy conservation, |A| = |C|.
Version B: l > 0, "spheroidal", longitudinal incident
The displacement field has the form, with complex amplitudes A, B, C, D and E:
- incoming wave in the matrix (long sphh) A
- non-singular standing wave in the nannoparticle (long sph) B
- non-singular standing wave in the nannoparticle (trans sph) C
- outgoing wave in the matrix (long sphh) D
- outgoing wave in the matrix (trans spph) E
From md30.htm:
|A|2 =
dω kBT (2 l + 1) /
(π2 ρm clm
ω2)
This, together with the boundary conditions at the surface
of the nanoparticle, determines B, C, D and E.
Check on correctness: By energy conservation,
|A|2 = |D|2 + (coef???) |E|2.
Version C: l > 0, "spheroidal", transverse incident
The displacement field has the form, with complex amplitudes A, B, C, D and E:
- incoming wave in the matrix (trans spph) A
- non-singular standing wave in the nannoparticle (long sph) B
- non-singular standing wave in the nannoparticle (trans sph) C
- outgoing wave in the matrix (long sphh) D
- outgoing wave in the matrix (trans spph) E
From md30.htm:
|A|2 =
dω kBT (2 l + 1) /
(l(l+1) π2 ρm ctm ω2)
This, together with the boundary conditions at the surface
of the nanoparticle, determines B, C, D and E.
Check on correctness: By energy conservation,
|A|2 = (coef???) |D|2 + |E|2.
Version D: l > 0, "torsional"
The displacement field has the form, with complex amplitudes A, B and C:
- incoming wave in the matrix (trans toor) A
- non-singular standing wave in the nannoparticle (trans tor) B
- outgoing wave in the matrix (trans toor) C
From md30.htm:
|A|2 =
dω kBT (2 l + 1) /
(l(l+1) π2 μm ctm)
This, together with the boundary conditions at the surface
of the nanoparticle, determines B and C.
Check on correctness: By energy conservation, |A| = |C|.
Finding Surface Displacement
Once the amplitudes A, B, and C (or possibly D and E as well)
are known, the full form of the displacement field u inside and outside
the nanoparticle can be easily determined. The simplest thing to do with these
amplitudes is to find the amplitude of the surface displacement. This
displacement has r, θ and φ components. When plotting, it is
more appropriate to plot the square of each component surface displacement
since Raman scattering amplitude would be proportional to the square of
the displacement.
Equivalence to Complex Frequency Approach
The above described approach allows the position and
width of peaks to be determined. It requires a "leap of faith" to suppose
that these peaks correspond to peaks in a Raman scattering spectrum.
However, what can be verified is to compare these peaks to those
obtained using the Complex Frequency Approach for the vibrational
frequencies of a nanoparticle embedded in a glass matrix. There are
two bases of comparison for each peak:
It is convenient to take the results of version C and
version D and combine them (simply adding the square displacements).
It is not possible to make a single plot of displacements
versus ω because for each l there is a different
θ dependence of displacement. Thus, a plot if needed for each
value of l and for both for the spheroidal and torsional cases.
(Much more would need to be known about details of the physical mechanism for Raman
scattering in order to make a single plot of intensity versus ω.)
Table I: Basis for comparison
Continuum
Complex Frequency
1.
center frequency of the peak
Re(ω)
2.
HWHM of the peak
Im(ω)
For the case of CdS in SiO2 the two approaches
give almost equivalent results (see md45.htm).
However for the case of Si in SiO2 the width of the peaks is
wide enough that the two approaches are not identical.
Advantages of This Approach
If only peak frequencies and widths are desired, then
it is more straightforward simply to use the complex frequency approach.
However, this approach has some possible advantages:
- The conceptual connection with the coontinuum of phonons in the
matrix is made (a point which is obscured in the complex frequency
approach).
- Information is available about the ammplitude of surface vibration
associated with each peak, and this is given in metres, so there is a
basis for comparing amplitudes of different peaks to see which are
expected to be big and which should be smaller in the Raman spectrum.
Daniel Murray
Associate Professor
Math, Stats & Physics Unit
University of British Columbia - Okanagan
Kelowna, BC, Canada
daniel "dot" murray "at" ubc "dot" ca