Page 63 - Book Hosokawa Nanoparticle Technology Handbook

FUNDAMENTALS CH. 1 BASIC PROPERTIES AND MEASURING METHODS OF NANOPARTICLES
When the frequency for (
)=0 is described as
,
L
the following equation.

2 L 0 ()

2 T () (1.11.14)

This equation is well known as Lyddane–Sachs– 17.97 nm
Teller (LST) relation. In the case of lead titanate, the
phonon mode is not the only one. Therefore, the LST
relation should be rewritten as follows:

0
()
2
2
2
1LO 2LO 3LO (1.11.15)

()
2
2
2
1TO 2TO 3TO 10.55 nm
The subscripts indicate the order of the phonon mode Intensity (CPS) E (1TO)
from the lower frequency. A 1 (1TO)
In LST relation, the intrinsic dielectric constant (0) E(1LO) S.O.
should be calculated using the phonon mode fre-
quency which is determined by Raman analysis. Q.S. 9.45 nm
However, the phonon mode using Raman scattering is E(2TO) Silent
complicatedly overlapped. Therefore, all the phonon
modes need to be divided from the Raman spectrum
to determine the exact phonon mode frequencies. The A 1 (2TO)
observed Raman spectra were fitted by the damped E(2LO)+A 1 (2LO) E(3TO) A 1 (3TO)
harmonic oscillators to determine the accurate E(3LO) A 1 (3LO)
phonon mode positions. Then the Rayleigh scattering
was divided by the Debye relaxation equation. The
equation for fitting is as follows:
0 200 400 600 800
⎛ 1 ⎞ Raman shift (cm )
-1

I() ⎜ ⎝ e hkT 1 1 ⎟ ⎠

⎛ F
2Γ F

3 ⎞ Figure 1.11.2
⎜ r r 2 ∑ ii i ⎟ Change in Raman spectra of lead titanate with particle size

) 4Γ

⎟
⎜ ⎝
r i (

2 2 2 i 2 i 2 ⎠ and assignment of the phonon mode.
2
2
i
(1.11.16)
optical phonon mode (LO mode) and the transverse
where the former term is the Bose–Einstein factor
and,
, and F are the mode frequency, damping optical phonon mode (TO mode) are observed in E
i
i
i
factor and oscillator strength, respectively. Fig. 1.11.2 and A mode. In the case of lead titanate nanoparticles,
1
shows the Raman spectra for the PT nanoparticles 12 modes (3(A E)(LO) 3(A E)(TO)) generated
1
with different sizes and the fitting results by the above from three T 1u mode and B E mode (silent mode)
equation, together with the assignment for all the generated from T 2u mode are observed by Raman
phonon modes. The Raman spectra were measured scattering. In general, TO mode frequencies are lower
using triple monochrometer system in the range from than that of LO mode, and E mode frequencies are
10 to 900 cm 1 at room temperature. The 488nm line lower than that of A mode. The separation and the
of an argon ion laser with 100 mW power was used as assignment of the phonon mode from Raman spectra
the excitation source. In addition, all the slit width have been done according to the above procedure.
was adjusted at 100 m.
At room temperature, the crystal symmetry of lead 1.11.4 Measurement of the dielectric constant of
titanate is reported to be tetragonal. In this case, the nanoparticles
double-degenerated E mode ([100] direction) and
the non-degenerated A1 mode ([001] direction) are Fig. 1.11.3 shows the particle size dependence of the
observed, since the space group of the tetragonal per- dielectric constant of PT nanoparticles, which were
1
ovskite structure is C . In addition, the longitudinal calculated from the mode frequencies, determined
4v
40