Updated: July 23, 2003
Vibrational Basis Functions for a Nanoparticle in a Finite Spherical Matrix




1. Basic Definitions:
   All of this refers to the vibrational modes of a spherical nanoparticle of radius Rp and density ρp inside a spherical glass matrix of radius Rm and density ρm. The outer surface of the glass matrix is free. But the resultant light scattering properties of the nanoparticle will be just the same if the outer surface of the glass matrix is assumed to be fixed.
   It is important to make a careful distinction between material coordinates (R) and space coordinates (r). In material coordinates, the position of a given atom j always has the same value Rj. However, the position of atom j in space varies with time as rj(t) due to vibrations of the material. If the entire system is in equilibrium with zero stress then Rj = rj for every atom j.
   Deviations from equilibrium are represented by the displacement field u. The displacement field (in metres) within the nanoparticle and matrix is given by u(R,t), where boldface is used to indicate vector quantities.
r = R + u
u = r - R
   Vibrations of the system cause density fluctuations. Density can be referenced to space coordinates so that dm = ρs.c.(r) d3r. But density can also be referenced to material coordinates so that dm = ρ(R) d3R. Note that ρ(R) = ρp at any material point R inside the nanoparticle. Mechanical vibrations do not cause ρ(R) to change. At all material points in the glass matrix ρ(R) = ρm.
   We are interested in normal modes of frequency ω. All quantities discussed here are real valued. Time dependence goes as sin(ωt), so that for an eigenfunction, u(r,t) = uj(r) sin(ωjt), where j is an index used to label eigenfunctions and eigenfrequencies. ω is in radians per second. Frequencies in wavenumbers (cm-1) are denoted by ν.

Definition of the six (real valued) amplitudes:
A - SPH LONG MATRIX, 1st kind Bessel
B - SPH LONG MATRIX, 2st kind Bessel
C - SPH TRAN MATRIX, 1st kind Bessel
D - SPH TRAN MATRIX, 2st kind Bessel
E - SPH LONG PARTIC, 1st kind Bessel
F - SPH TRAN PARTIC, 1st kind Bessel

Important C++ programs:
scp81p4.cpp - 6×6 determinant zeroes to find modes, A-F, write to '1:' .txt file
scp83a2.cpp - read in 1: or 2: .txt file, and verify that <ui|uj> = δij
scp84a3.cpp - auto MC normalizer. Reads 1: file (such as 81a.txt) and generates 2: file (such as 81a2.txt)

Si 5 nm SiO2 10 nm: 81a2.txt   (This is a '2:' file)

2. Orthogonality of Normal Modes:
   The velocity field v is defined in terms of the displacement field u by:
v(R,t) =  ∂
----
∂t 		u(R,t)
[2.1]
   Consider a single normal mode {plmn} with angular frequency ωplmn whose velocity field is
vplmn(R) sin(ωplmn t).
[2.2]
   The velocity field due to a general disturbance of the nanoparticle is
vtot(R,t) = Σ xplmn vplmn(R) sin(ωplmn t) + Σ yplmn vplmn(R) cos(ωplmn t)
[2.3]
   The total kinetic energy of the system is
Ktot = (1/2) ∫ ρ(R) vtot · vtot d3R
[2.4]
   Next, consider the kinetic energy due to a single normal mode:
Kplmn(t) = (1/2) ∫ ρ(R) || xplmn vplmn sin(ωt) + yplmn vplmn cos(ωt) ||2 d3R
[2.5]
   The key assumption is that the total kinetic energy is the sum of the kinetic energy associated with individual normal modes:
Ktot = Σ Kplmn(t)
[2.6]
   This last relation holds for any state of the system, that is, for any possible set of values of the coefficients
{xplmn ; yplmn }
   It is useful to consider the special situation where all of the coefficients are zero with the exception of xp'l'm'n' = 1 and xp"l"m"n" = 1, where {p'l'm'n'} and {p"l"m"n"} are distinct (not all the same). In that case,
Kp'l'm'n' = (1/2) sin(ωp'l'm'n' t)2 ∫ ρ(R) || vp'l'm'n' (R) ||2 d3R
[2.7]
Kp"l"m"n" = (1/2) sin(ωp"l"m"n" t)2 ∫ ρ(R) || vp"l"m"n" (R) ||2 d3R
[2.8]
vtot = sin(ωp'l'm'n' t) vp'l'm'n' (R) + sin(ωp"l"m"n" t) vp"l"m"n" (R)
[2.9]
and since at every time Ktot must equal Kp'l'm'n' (t) + Kp"l"m"n" (t) therefore
0 = sin(ωp'l'm'n' t) sin(ωp"l"m"n" t) ∫ ρ(R) vp'l'm'n' (R) · vp"l"m"n" (R) d3R
[2.10]
so that
0 = ∫ ρ(R) vp'l'm'n' (R) · vp"l"m"n" (R) d3R
[2.11]
It follows also that
0 = ∫ ρ(R) up'l'm'n' (R) · up"l"m"n" (R) d3R
[2.12]
   This relationship can be used when selecting an inner product on the vector space of displacement fields. Suppose that u1 and u2 are two functions of R. Let their inner product (denoted as < 1 | 2 >) be defined by:
< 1 | 2 > =   ∫ ρ(R) u1 · u2 d3R
------------------------
     ∫ ρ(R) d3R
[2.13]
   Thus chosen, displacement fields of normal modes of vibration are orthogonal with respect to this inner product. The next natural step is to normalize the displacement fields so that

< plmn | PLMN > = δpP δlL δmM δnN
[2.14]
   The choice of denominator in the definition of the inner product is for convenience only. It allows the normalized u to be dimensionless. The denominator is the total mass of the nanoparticle-matrix system, denoted as M:
M = ∫ ρ(R) d3R
[2.15]
   The normalization of displacement fields is:
∫ ρ(R) || uplmn ||2 d3R = M
[2.16]
   A general disturbance of the system is made up of a superposition of normal modes using the coefficients x and y:
utot(R,t) = Σ xplmn uplmn(R) sin(ωplmn t) + yplmn uplmn(R) cos(ωplmn t)
[2.17]
   Since utot has units of metres, xplmn and yplmn also have units of metres.
   The time averaged kinetic energy associated with a single mode {plmn} is

<< Kplmn >>time = (1/4) M ωplmn2 (xplmn2 + yplmn2)
[2.18]
   Adding on the potential energy associated with the strain field of the nanoparticle and matrix, the total energy associated with mode {plmn} is
Eplmn = (1/2) M ωplmn2 (xplmn2 + yplmn2)
[2.19]
   Assuming thermal equilibrium, and also assuming that hω is small, the average energy of a single normal mode is kBT where kB is Boltzmann's constant and T is the temperature in Kelvin degrees.


3. Checks on Correctness:
1. I checked for several examples that the basis functions really are orthogonal.
2. By plotting ur and uθ as a function of r, I verified that both functions are continuous at the nanoparticle-matrix boundary.
3. By plotting σrr and σrθ as a function of r, I verified that both components of stress really do approach zero as the outer surface of the matrix is approached.
4. I plotted the stress difference as a function of r near the nanoparticle-matrix boundary and verified that the difference really does approach zero at the boundary.

4. Projection Formalism
   Suppose that {uj} (where j=1,2,3...) is a normalized set of eigenfunctions such that <ui|ui> = δij. In particular, consider ui, and suppose that the actual displacement field at a given moment of time is u(r,t) = xi(t) ui, where x(t) is a real valued scalar function of time. Since the basis function are orthonormal,
xi(t) = <u(r,t)|ui>, where we are using the same inner product define above. xi(t) is the projection of the displacement field onto eigenfunction i. Let H represent the total energy (potential and kinetic) of the system. Then H = (1/2) M ωi (xi)2 + (1/2) M (dxi/dt)2. Here M is the total mass of the nanoparticle-matrix system. Thus, a single normal mode can be expressed in terms of a one dimensional harmonic oscillator.

5. Quantization
   Since a given eigenfunction corresponds to a one dimensional harmonic oscillator with coordinate xi, the motion can be quantized by introducing a complex wavefunction ψ(xi). This quantum system would have eigenfunctions with discrete energy levels spaced by hbar ω, in the usual way. To quantize the whole system we introduce a multivariable wavefunction ψ(x1,x2,x3,x4...). Because each coordinate corresponds to a classical normal mode, there are eigenfunctions of the variable-separated form: ψ(x1) ψ(x2) ψ(x3) ψ(x4) × ... The eigenstates of this system corespond to an integer ni for each classical normal mode, where ni can be 0, 1, 2, etc. The system eigenstates the correspond to integer series: (n1,n2,n3,n4,...). If the system is raised to temperature T at equilibrium, then the quantum state of the system is a mixture of eigenstates, with a Bose-Einstein distribution for each classical normal mode.

Phonon Creation Operators
   For a system with one mechanical degree of freedom whose Hamiltonian is:
H = p2/2m + (1/2) m ω2 x2,
the phonon creation operator is:
a+ = sqrt(mω/(2hbar)) x - i p / sqrt(2mhbarω)
and the phonon annihilation operator is:
a = sqrt(mω/(2hbar)) x + i p / sqrt(2mhbarω)
where x and p are understood to be linear operators rather than scalar variables. In particular [x,p] = xp-px = i hbar.
   In terms of these,
H = hbar ω ( a+a + 1/2 )
x = sqrt(hbar/(2mω)) (a + a+)
p = -i sqrt(m hbar omega/2) ( a - a+)
[ a,a+] = a a+ - a+ a = 1
Reference: heppc16.ucsd.edu

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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