Page 135 - Bio Engineering Approaches to Cancer Diagnosis and Treatment

6.1 Predicting the optical properties 133

where ψ and ξ′ are the Bessel function. M is the ratio between dielectric function ξ ′ n
ψn
n
n
2 πnR
of the nanoparticles and surrounded medium and X is the size perimeter: =x m . x=2πnmRy
Based on Drude-Lorentz model [2]: y
ω 2 ωΓ
2
ε = ε − p + i p , (6.11)
2
2
2
2
2
2
B ω + Γ 2 ωω +( 2 Γ ) ε=εB−wp2w +Γ +iwp2Γw(w +Γ ),
where ε and ε are the dielectric of the nanoparticles and bulk material, respectively. ε εB
B
Γ=Γ +νFR
Γ = Γ + νF is the damping term (R is the radius of the particle and νF is the Fermi ν F 0
0
R 4 πne 2
velocity) and ω = e is the plasma frequency of the free electron gas (m is me 2
wp=4πnee εOme
e
p
ε m e
O
the mass of the electron, n is free electron density, e is electron charge and ε is the ne εO
O
e
vacuum permittivity).
To calculate the amount of scattering, first, need to calculate absorption, extinc-
tion and scattering efficiencies. Based on Mie scattering coefficients:
∞
+
2
Q = 2 ∑( n 1 )R a( + b ), (6.12)
2
ext x 2 n =1 e n n Qext=2x ∑n=1∞2n+1Re(an+bn),
2 ∞
2
Q sca = ∑( n2 +1 ) a ( 2 n + b ), (6.13)
n
x 2 n =1 Qsca=2x ∑n=1∞2n+1(an2+bn2),
2
Q abs = Q ext −Q sca (6.14) Qabs=Qext−Qsca
ABSCAT and MATLAB program can assist to solve Eqs. (6.12)–(6.14) [2, 3].
6.1.2 Discrete dipole approximation
Discrete dipole approximation model was first introduced by Purcell and Pennypacker in
1973. This model provides the approximation for absorption, extinction, and scattering
cross-section. In this method, each particle defines as a mass composed of finite number
of elements with dipole interactions. Maxwell’s equations are converted to a simple
algebraic equation using Draine and Flatau methods [4]. For each dipole, the dipole
moment of the external electric field and internal electric field causing by the neighbor-
ing dipoles are calculated. The electric field of one dipole is calculated as follow:
iwd
− c  ω 2 
( ⋅rp r ˆ
r ˆ
E dipole = e  r ˆ × p ×+  1 − i ω   3 ˆ ) −  p , (6.15)



2
2 2
2
4 πε 0  cd  d 3 cd 2   Edipole=e−iwdc4πε w c drˆ×p×rˆ+1d −iwcd 3rˆ⋅prˆ−p,
3
0
where d and are distance to the sampling point and unit vector of which h is taken rˆ rˆ
r ˆ
r ˆ
from dipole to the location at which electric field is collected, respectively. P is the
vector related to the induce dipole moment,
P i = α E local , (6.16) Pi=αiElocal,
i
where α is the polarizability of the materials and E local is the electric field corre- αi
i
sponding to the incident wave, E inci , , and the effect of the electric field of the other Einc,i
dipoles [4]: