Page 62 - Book Hosokawa Nanoparticle Technology Handbook

1.11 ELECTRICAL PROPERTIES FUNDAMENTALS
1.11.2 Novel characterization method for the dielectric If the Lorenz relation (E E P/3 ) is used as
0
loc
property the local electric field for the microscopic electric
polarization (P n
E ),
loc
The dielectric constant of nanoparticles can be esti-
mated by analyzing the phonon modes of Raman spec- ⎛ P ⎞
tra. It seems that we can estimate the intrinsic dielectric P n
⎜ ⎝ E 3 ⎠ ⎟ (1.11.6)
constant of nanoparticles by this method [3]. In this 0
method, the intrinsic dielectric constant is calculated
using Lyddane–Sachs–Teller relation (LST relation). When equation (1.11.6) is solved for P, the suscepti-
LST relation is described in the next section [4, 5]. bility P/E is obtained as below:
0
1.11.3 LST relation 1 P n
/ 0 (1.11.7)
E 1 ( − n
3 / 0 )
0
First, frequency dependence of the dielectric constant
for the ionic crystal ( ) is discussed in the long wave-

length limit region (wave number k 2 / ). The equa- and, insert it into 1 ,
tion of motion is described as follows when the n
electric field E(t) was applied to the crystal having a − 1 (1.11.8)
couple of a positive and a negative ion in the structure. + 2 3 0

Mu f u u ) qE t) The above equation is well known as Clausius–Mosotti
(
(
equation. This relates the dielectric constant to the elec-

Mu f u u ) qE t) (1.11.1) tronic polarizability, but only for crystal structures for
(
(
which the Lorentz local field is obtained. And, when
the polarizability
is replaced by the above equation,

where M and M are the mass of a positive and a the Clausius–Mosotti equation should be changed as

negative ion, respectively, u and u the displacement follows:
from the balanced position of the positive and the
negative ion, f the spring constant between the posi-
() 1 n ⎧ ⎪ q 2 ⎫ ⎪
tive and negative ion, q the electric charge for the ⎨
elec * 2
) ⎪ ⎬ (1.11.9)
2

ion. At 1/M* 1/M 1/M , u u u , the equa-
() 2 3 0 ⎩ ⎪ M (
⎭
0
tion can be arranged as follows:
In addition, in the case of low frequency (
0),
*
M u fu qE t() (1.11.2)

() 1 n ⎧ ⎪ q 2 ⎫ ⎪
⎨
⎬ (1.11.10)
If E(t) can be assumed as the sine wave of the angular
() 2 3 0 ⎩ ⎪ elec M *
⎪
2
0 ⎭
frequency, the displacement (u) is calculated as fol-
lows:
In contrast, in the case of high frequency which the
qE t() polarization cannot follow, the second part of the
u (1.11.3)
M (
2 0
2 ) right–hand side of the equation can be ignored,
*
therefore,
where
is the angular frequency in the system. The
() 1 n
(1.11.11)
0
dipole from the ion displacement is described as
() 2 3 0 elec
qu, therefore, the ionic polarizability (
) is
ion
described as follows:
Next, the Clausius–Mosotti equation was solved for
(
) using the above equations to give the following
u q 2

ion (1.11.4) equation:
2
2
*
E M (
−
)
0
() 0 ( )

( ) (1.11.12)

()

)
Then, the polarizability (
) is described as follows: 1 (/ T 2

=
q 2 (1.11.5) and
M (
2 0 −
2 )
elec
*
2

2 0 () 2 (1.11.13)
T
where
is the electronic polarizability. ()0 2
elec
39