Nanotechnostructured Catalysts TiO Nanoparticles for Water Purification - Nanotechnologies for Water Environment Applications - page 54

band (Brus, 1984, 1986; Lin et al., 2006). Consequently, the effective bandgap is
reduced without shifts of E V and E C . Continuingly decreasing the particle size from a
few tenths of a nanometer to near its Bohr radius around 1.5 to 5 nm (Pan et al., 2005))
induces the quantum size effect due to drastic changes in its electronic properties.
According to the effective mass approximation theory, the three dimensional
approximation can be expressed as:

* *

⎡

22

⎤

! υ +

U(r) F(r) = E F(r) (Eq. 3.16)

⎢ ⎥

n

2m ∗

⎣ ⎦

,where F(r * is the envelope of wave; E n is the discrete energy level, (g-cm-s -2 ); m * is the
effective mass of electron (g); ! is the reduced Plank's constant, (g-cm-s -2 ); and U(r) is
the pseudopotential (g-cm-s -2 ). Assuming the electron is confined in a spherical radius
(R , M) of the semiconductor, the potential is:

0if0 r R ≤≤

= ⎨ ∞

U(r) (Eq. 3.17)

if r ≥ R

⎩

Using the above approximations with proper boundary conditions, the solution
for the discrete energy levels in the nanocrystalline gives (Paeker and Grimmeiss, 1991):

π

22

!

E = ; n = 1, 2, 3...... (Eq. 3.18)

n

2m * R

2

Efros and Rosen (1998, 2000) have rearranged the equation by considering the
quantum confinement in the nanocrystalline. They express the first lowest excitation
state as:

2

⎡⎤

2 2

! π e

E(R) =+ E − 1.786 = 0.248E

(Eq. 3.19)

⎢⎥ g Ry

2R μ ε R

⎣⎦

, where E g is the bandgap of the bulk semiconductor, (eV); μ is the reduced effective
mass (g); is the dielectric constant of the semiconductor, (C 2 dyne -1 cm -2 ); R is the
radius of the spherical semiconductor (cm); and e is the charge of electron (g 0.5 cm 1.5 s -1 ).
E Ry is the Rydberg energy, (s -1 ), the exciton binding energy of an e - -h + pair, which is
defined as:

e

2

2 μ

e

4

E = =

!

(Eq. 3.20)

Ry 22

2 εα ε

B