Created: Apr, 2003 Update: January 28, 2008
Sun. Dec. 7, 2003:  I made calculations of <u2> where the region of averaging is a hollow sphere with inner radius R1 and outer radius R2. For each plot, the white curve averages <u2> from R1 = 0 to R2 = Rp and is therefore the same as <u2>p. All of these cases are for (SPH,0) modes.
   Dec 3 2003: I made calculations of < u2 >p.
Fig. 1203(a)
The yellow curve is < u2 >p The "area under the curve" for a pseudomode is supposed to be Mp+m / Mp . See equation (B2) in the core-shell paper. Purple tickmarks simply denote peak locations. The actual sum of < u2 >p values (blue and red parts together) under the first peak is 0.892 Mp+m/Mp
Fig. 1203(b)
The yellow curve is < u2 >p The "area under the curve" for a pseudomode is supposed to be Mp+m / Mp . See equation (B2) in the core-shell paper. Purple tickmarks simply denote peak locations. The actual sum of < u2 >p values (blue and red parts together) under the first peak is 1.061 Mp+m/Mp
Fig. 1203(c)
The yellow curve is < u2 >p The "area under the curve" for a pseudomode is supposed to be Mp+m / Mp . See equation (B2) in the core-shell paper. Purple tickmarks simply denote peak locations. The actual sum of < u2 >p values (blue and red parts together) under the first peak is 0.847 Mp+m/Mp
Fig. 1203(d)
The yellow curve is < u2 >p The "area under the curve" for a pseudomode is supposed to be Mp+m / Mp . See equation (B2) in the core-shell paper. Purple tickmarks simply denote peak locations. The actual sum of < u2 >p values (blue and red parts together) under the second peak is 1.038 Mp+m/Mp
Fig. 1203(e)
The yellow curve is < u2 >p The "area under the curve" for a pseudomode is supposed to be Mp+m / Mp . See equation (B2) in the core-shell paper.
Fig. 1203(f)
The yellow curve is < u2 >p The "area under the curve" for a pseudomode is supposed to be Mp+m / Mp . See equation (B2) in the core-shell paper. This matches Figure 1 in the "audi" paper.
sph0u2p6.cpp - THz scale
Ag in BaO-P2O3: Matrix boundary conditions


Whether the outer matrix surface has free or fixed boundary conditions hardly affects the frequency dependence of amplitudes inside the nanoparticle. This serves as a test of the macroscopic limit of matrix size being reached.


   In every case the radius of the nanoparticle is Rp = 4.9 nm and the radius of the matrix Rm is 64 times larger. The boundary conditions at the outer surface of the matrix can be chosen in two ways:
(1) free surface
(2) fixed surface
   For all plots, the horizontal scale goes up to 30 cm-1. The effect of changing the matrix boundary conditions is to shift the frequencies of the individual modes, but the overall dependence of amplitude on frequency is unchanged.

(SPH,l=0)
Free surface
Fixed outer surface

(TOR,l=1)
Free surface
Fixed outer surface

(SPH,l=2) - longitudinal part
Free surface
ags264.4

ags264e.gif
Fixed outer surface
ags264h.4

ags264he.gif

(SPH,l=2) - transverse part
Free surface
ags264.4

ags264.gif
Fixed outer surface
ags264h.4

ags264h.gif

Important programs: (Borland Turbo C++ for DOS)
scp85b.cpp - generates ".3" file of ν and A-F
- modes can be TOR or SPH, but for m=0 oonly
- matrix outer surface boundary can be ffree or fixed
 scp86a.cpp - reads ".3" file, normalizes and writes to ".4" file
- efficient integration using multiple 11D integrals
 scp87b.cpp - reads ".4" file and finds inner product of any two modes
- efficient integration using multiple 11D integrals
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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