Optical Properties of Gold Nanoparticles
with Matlab
PHY??? Intro to Nanotechnology
Dr. Zohdy, Dr. Martins, Evan J. Dawley
Introduction
Particles with dimensions on the order of a billionth of a meter are called nanoparticles. When
nanoparticles of certain metals such as gold are exposed to electromagnetic radiation, the
nanoparticles may absorb or scatter the radiation unlike macroscopic metals. The electrons on
the surface of the nanoparticles oscillate at a particular frequency. As a result, the optical
response surrounding that frequency is much stronger. A typical measure of optical response
is known as extinction efficiency (absorption plus scattering vs. wavelength). The goal of this
lab is to examine the extinction efficiency for different sizes of gold nanoparticles. At what
frequencies do gold nanoparticles exhibit the largest absorption and scattering?
The equation for extinction efficiency is given by,
Qext 

2
 kr 
2
  2n  1 Re a
n 1
n
 bn 
where:
k
2
(1)

r  radius of nanoparticle
an , bn  extinction coefficients
The extinction coefficients are computed through the following:
1
an 
bn 
m 2 jn  mx   xjn  x     jn  x   mxjn  mx  
m 2 jn  mx   xhn(1)  x     hn(1)  x   mxjn  mx  
 jn  mx   xjn  x    jn  x   mxjn  mx  
 jn  mx   xhn(1)  x    hn(1)  x   mxjn  mx  
where:
m  complex refractive index relative to medium

1
2
(2), (3)
(ratio of permeability of sphere to medium)
x  kr
jn  x   spherical bessel function
hn(1)  x   jn  x  +iyn  x  (spherical henkel function)
Matlab has built in traditional Bessel and Henkel functions which are not identical to the
spherical Bessel and Henkel functions. The traditional and spherical kinds are related by
jn  x  
yn  x  

2x

2x
J n  0.5  x 
(4), (5)
Yn  0.5  x 
J and Y are Matlab’s Bessel function of the first and second kinds, respectively. For n = 0, the
spherical functions are
sin x
x
 cos x
y0  x  
x
j0  x  
(6), (7)
The derivatives in Equations (2) and (3) can be expressed as
 xjn  x    xjn 1  x   njn  x 
 xh  x    xhn(1)1  x   nhn(1)  x 
(8), (9)
(1)
n
2
cn 
dn 
1 jn  x   xhn(1)  x    1hn(1)  x   xjn  x  
1 jn  mx   xhn(1)  x    hn(1)  x   mxjn  mx  
1mjn  x   xhn(1)  x    1mhn(1)  x   xjn  x  
m 2 jn  mx   xhn(1)  x    1hn(1)  x   mxjn  mx  
Matlab Assignment
Using the expressions in the introduction, plot the extinction efficiency from wavelengths 350
nm to 1000 nm (in increments of 0.01 nm) for gold nanoparticles of size 5 nm, 10 nm, 20 nm,
and 50 nm in a vacuum (i.e. the permeability ratio is 1). A data file entitled gold.dat will be
provided to you. The file contains the real and imaginary parts of the refractive index for each
wavelength from 350 nm to 1000 nm. You can load the file directly into a matrix using the
command, load gold.dat.
Note: Equation (1) contains an infinite series with index n. However, the series can be
truncated after a certain number of elements because the remaining numbers in the series are
negligible. For each wavelength the maximum index is given by
nmax  x  4 x1/3  2
(10)
When you plot the extinction efficiency, plot all cases on a single plot. In your m-file, this
plotting process is simple if you can create a matrix of size 4 x m (where m is the number of
wavelengths). Make sure you label your axes and title your plot. Find the wavelengths at
which the nanoparticles have an extinction efficiency peak.
Hint: It may be helpful to access a particular row or column of a matrix. Using a “:” in the
correction position will do the trick. For example, access all of the column data for row 1 by the
expressions A(1,:) or access all of the row data for column 2 through A(:,2).
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