Raman spectroscopic study of LiH2 PO4

Article abstract of DOI:10.1016/j.ssc.2007.12.011

The dielectric constant of polycrystalline LiH₂PO₄ has been measured between 297 and 17 K. No marked changes were observed over this range, indicating that the room-temperature orthorhombic phase persisted up to 17 K. Raman spectra of polycrystalline LiH₂PO₄ were also measured at 297, 200, and 70 K in the frequency shift region of 15-4000?cm^-1 with Raman-active vibrational modes naively assigned to low-frequency (0-300?cm^-1) external and high-frequency (300-4000?cm^-1) internal modes. In addition to the internal modes of the PO₄ tetrahedra, the internal modes of the LiO₄ tetrahedra spectroscopically manifested themselves between 390-500?cm^-1. This frequency range overlaps those of ν₂ (PO₄) and ν₄ (PO₄). The LiH₂PO₄O-H vibrational frequencies were in good agreement with crystallographic reports that there are two types of hydrogen bonds: intermediate (long bonds) and strong (short bonds).

Full text of DOI:10.1016/j.ssc.2007.12.011

Solid State Communications 145 (2008) 487–492
www.elsevier.com/locate/ssc
Raman spectroscopic study of LiH2PO4
Kwang-Sei Leea,∗, Jae-Hyeon Kob, Joonhee Moona, Sookyoung Leea, Minhyon Jeona
a
Department of Nano Systems Engineering, Center for Nano Manufacturing, Inje University, Gimhae 621-749, Gyeongnam, Republic of Korea
b
Department of Physics, Hallym University, 39 Hallymdaehak-gil, Chuncheon 200-702, Republic of Korea
Received 24 November 2007; accepted 13 December 2007 by D.J. Lockwood
Available online 23 December 2007
Abstract
The dielectric constant of polycrystalline LiH PO has been measured between 297 and 17 K. No marked changes were observed over this
2
4
range, indicating that the room-temperature orthorhombic phase persisted up to 17 K. Raman spectra of polycrystalline LiH PO were also
2
4
−1
measured at 297, 200, and 70 K in the frequency shift region of 15–4000 cm with Raman-active vibrational modes naively assigned to low-
frequency (0–300 cm ) external and high-frequency (300–4000 cm ) internal modes. In addition to the internal modes of the PO tetrahedra,
the internal modes of the LiO tetrahedra spectroscopically manifested themselves between 390–500 cm . This frequency range overlaps those
−1
−1
4
−1
4
of ν (PO ) and ν (PO ). The LiH PO O–H vibrational frequencies were in good agreement with crystallographic reports that there are two
2
4
4
4
2
4
types of hydrogen bonds: intermediate (long bonds) and strong (short bonds).
c
ꢀ 2007 Elsevier Ltd. All rights reserved.
PACS: 63.20.Dj; 77.80.-e; 77.84.Fa; 78.30.-j
Keywords: A. Insulators; C. Crystal structure and symmetry; D. Phonons; E. Inelastic light scattering
1. Introduction
though electrical conductivity has been demonstrated to be
predominantly protonic [3–12].
MX2RO4-type (M = K, Rb, NH4, Cs, Tl; X = H, D; R = P,
As) crystals undergo ferroelectric or antiferroelectric phase
transitions at low temperatures [1,2]. They are also known to
exhibit high-temperature phase transitions (HTPT), as reported
in this class of materials by many investigators. However,
the experimental investigations of HTPT, e.g., of the phase
transformation temperature (Tp) and the metastability of HTPT,
are strongly dependent on the measurement conditions [3].
Lee raised questions about whether the polymorphic phase
transition actually exists and suggested that the term “HTPT”
should be replaced by “onset of partial polymerization at
reaction sites at the surface of solids” [3]. While some
researchers support this viewpoint [4–8], others maintain
that they are structural phase transitions [9–12]. Therefore,
the high-temperature superprotonic phase behavior remains a
controversial subject with an unclear microscopic nature, even
In contrast to tetragonal KH2PO4, RbH2PO4, and NH4H2PO4
(space group I42d–D21d2) [1–3], monoclinic CsH2PO4 (space
group P21/m–C22h) [3,13–16], and monoclinic TlH2PO4
(space group P21/a–C25h) [17–19] crystals, relatively little
work has been done on the low- and high-temperature behaviors
of lithium dihydrogen phosphate (LiH2PO4). Although the
dielectric constant of LiH2PO4 has been measured between
300 and 80 K, no discontinuous changes corresponding to a
phase transition were observed [20]. LiH2PO4 crystallizes in
the orthorhombic system with space group Pna21–C29v with
˚
a = 6.253, b = 7.656, c = 6.881 A, Z = 4 (Fig. 1), and a
factor group of mm2–C2v [21,22]; a point group that can show
piezoelectricity and pyroelectricity, and both weak piezoelectric
and pyroelectric effects have been observed [21,23]. As shown
in Fig. 2, two types of hydrogen bonds have been reported.
In the figure, one hydrogen atom, H1, is in an asymmetric
position along the [100] axis [linked by the O(3) · · · O(4, 4)III]
and the other hydrogen atom, H2, is in a general position and is
involved in an asymmetric bond along the [001] axis [linked
by the O(4) · · · O(2, 2)I]. These connect the PO4 tetrahedra,
∗
Corresponding author. Tel.: +82 553203207; fax: +82 553341577.
E-mail addresses: kwangsei28@hanmail.net, kslee@physics.inje.ac.kr
(K.-S. Lee).
c
0038-1098/$ - see front matter ꢀ 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ssc.2007.12.011

488
K.-S. Lee et al. / Solid State Communications 145 (2008) 487–492
Fig. 1. Projections of the structure of LiH PO onto the (001) and (100) planes. Only the oxygen atoms are shown. Dashed lines represent hydrogen bonds (after
2
4
Catti and Ivaldi, Ref. [22]).
be attached and to perform Raman scattering studies. The
powder pellet dielectric constant was measured between 297
and 17 K using an impedance analyzer (HP 4192A). Raman
˚
excitation was induced using an argon-ion laser 5145 A line
focused through a cylindrical lens to avoid local heating of the
sample. The 45◦-scattered light at the polycrystalline surface
was dispersed by a double-grating monochromator (Spex 1403)
connected to a GaAs photomultiplier tube (Hamamatsu R943-
02). The spectral resolution was 2 cm−1. The experimental
details were the same as in Ref. [24].
Fig. 2. Two types of hydrogen bonds at room temperature in LiH PO :
3. Results and discussion
2
4
III
I
O(3)–H · · · O(4, 4) along the [100] axis and O(4)–H · · · O(2, 2) along the
1
2
[001] axis (after Catti and Ivaldi, Ref. [22]).
The dielectric constant of the LiH2PO4 pellet was measured
from 297 to 17 K with cooling at a rate of 0.2 K min−1
.
building up a three-dimensional framework. In addition, LiO4
coordination tetrahedra are linked by vertices and form [100]
isolated chains.
As shown in Fig. 3, at 87 kHz and 297 K the dielectric
constant was 5.94. Haussu¨hl reported the LiH2PO4 single
crystal dielectric constant at 293 K between 10 and 100 kHz
along the three crystallographic axes as ε11 = 6.02, ε22 = 4.63,
and ε33 = 7.26 [23]. For a powder pellet composed of randomly
oriented polycrystals, the dielectric constant is averaged as
In a preliminary study examining both dielectric constants
and Raman spectra of LiH2PO4 in the 15–1400 cm−1 frequency
range, Lee et al. found no evidence of a dielectric anomaly
from room temperature down to 17 K [24]. Here, we report our
reinvestigation of the dielectric constant measurement between
297 and 17 K looking for a possible low-temperature phase
transition. We also extended the Raman scattering study at
297, 200, and 70 K between 15–4000 cm−1 to obtain a more
confident assignment of the external vibrations; spectroscopic
evidence for the vibrational modes of LiO4 in addition to the
PO4 internal vibrations; and O–H vibrations.
ε = 1 (ε11 + ε22 + ε33) = 5.97, in good agreement with
3
our measured value of 5.94. As shown in Fig. 3, there was no
dielectric anomaly in the given temperature range, indicating
that the room-temperature orthorhombic phase persists up
to 17 K. However, measurement of the dielectric constant
upon heating above room temperature revealed the onset of a
thermal transformation near 451 K. Therefore, the behavior of
LiH2PO4 at high temperatures supplies new information about
high-temperature transformation and protonic conductivity in
MX2RO4-type crystals, which have been reported in a separate
paper [25].
2. Experiments
LiH2PO4 was synthesized by the reaction Li2CO3
+
The Raman spectral density, I (ω), is proportional to the
2H3PO4 → 2LiH2PO4 + H2O + CO2, and then small, good
optical quality single crystals were grown by slow evaporation
from the aqueous solution at about 310 K. These crystals were
ground to form a disk-shaped pellet to which electrodes could
imaginary part of the generalized susceptibility, χ00(ω)
:
I (ω) ∝ [n(ω) + 1]χ00(ω) where the Bose–Einstein thermal
hω
¯
factor, n(ω) = (ekB − 1)−1 (hω is the excitation quanta
T
¯

K.-S. Lee et al. / Solid State Communications 145 (2008) 487–492
489
Fig. 3. Temperature dependence of the polycrystalline LiH PO dielectric
constant.
2
4
Fig. 4. Polycrystalline LiH PO Raman spectra at different temperatures in the
15–4000 cm frequency range.
2
4
−1
in the sample, kB, Boltzmann constant, and T , the sample
temperature). Fig. 4 shows the LiH2PO4 Raman spectra in
the frequency range of 15–4000 cm−1 at 297, 200, and
70 K. The Raman signals ride on the fluorescent background,
and produce very different spectra than tetragonal KH2PO4,
RbH2PO4, and NH4H2PO4 (space group I42d–D21d2) [1,2,9],
monoclinic CsH2PO4 (space group P21/m–C22h) [13–15], and
monoclinic TlH2PO4 (space group P21/a–C25h) [9,17–19].
Even if the crystal structures and symmetries of KH2PO4,
CsH2PO4, and TlH2PO4 are different, their crystallographic
structures are essentially composed of M+ (M = K, Cs,
Tl) cations and H2PO4− anions. Raman spectra of KH2PO4,
CsH2PO4, and TlH2PO4 show almost the same features in
the 300–1200 cm−1 frequency range [1,2,9,13–15,17–19].
Therefore, these vibrational modes have been assigned as
the internal vibrations of a PO34− ion modified slightly by
the surrounding crystalline field. Very different spectra have
the crystal, leaving 3 × 32 − 3 = 93 as the maximum number
of optical modes. Four irreducible representations, A1(z), A2,
B1(x), and B2(y), are allowed in its corresponding factor group,
mm2–C2v [26]. The symmetry species A1(z), A2, B1(x), and
B2(y) are all Raman-active, while those of species A1(z),
B1(x), and B2(y) are only active in the infrared spectra.
These spectra may be interpreted by roughly dividing them
into four parts on the basis of whether a given mode is
associated with collective protonic motions or relaxational
motions of the PO4 and/or LiO4, external lattice vibrations
between PO4 and LiO4 (including translations and rotations),
internal modes of PO4 and LiO4, or vibrations of the hydrogen
bonds (stretching and bending). In general, these may be
expected to occur in the order of increasing frequency. As the
spectral range above 15 cm−1 does not cover the collective
protonic or relaxational motions of PO4 and/or LiO4, we
discuss the remaining three regions: external (lattice) vibrations
between PO4 and LiO4 (including translations and rotations),
internal modes of PO4 and LiO4, and vibrations of the hydrogen
bonds.
been observed in the low-frequency range of 0–300 cm−1
,
which are related to the lattice vibrations originating from
the relative motions between M+ cations and H2PO4− anions.
The vibrational modes of LiH2PO4 at 297 K were tentatively
assigned according to this scheme in our previous paper [24].
Raman-active vibrational modes were assumed to consist of
3.1. External vibrations
low- (0–300 cm−1) and high-frequency (300–4000 cm−1
)
Consider the group of n nonequivalent points contained in
the primitive unit cell. Subtracting the three pure translations
(acoustic vibrations) leaves one with 3n − 3 optical modes.
The object is to classify these vibrations as external or internal,
where the LiH2PO4 internal vibrations arise from PO4 and
LiO4 motion, and the external vibrations, commonly known as
lattice vibrations, result from the relative motion between the
groups. The optical phonons are divided into translational and
rotational (or librational) types, which, in the limit of vanishing
forces among the groups, correspond to pure translations and
pure rotations. As mentioned above, the LiO4 tetrahedron
shares oxygen atoms with its four neighboring PO4 tetrahedra,
so that the decoupling of the motions of PO4 and LiO4 is
not as simple as in MH2PO4, which is composed of M+ and
H2PO−4 . In addition, the hindered rotational modes of PO4
modes. However, this criterion seems problematic, as the
crystal structure of LiH2PO4 is not composed of Li+ cations
and H2PO−4 anions, but of LiO4 and H2PO−4 , which share
oxygen atoms. Upon cooling a crystal, most lines become
narrower with increasing intensity due to the temperature-
dependent Bose–Einstein thermal factor n(ω), as shown in
Fig. 4. As some spectral lines, or bands, shift or split in the
low-temperature spectra, modes can be assigned with more
confidence in the same phase in the 70 K spectra than in the
297 K spectra.
The crystallographic unit cell (Pna21–C9 ), which is also
a primitive cell, contains four formula units2(vZ = 4) [21,22].
LiH2PO4 has eight atoms and therefore 8 × 4 = 32 atoms
in its primitive cell. The degrees of freedom are derived from
the 3 translational and 3 rotational motions. Three of these
degrees of freedom are responsible for the acoustic modes of
and LiO4 can be excited. The low-frequency (15–300 cm−1
region in Fig. 5 is assigned to external modes with a naive
)

490
K.-S. Lee et al. / Solid State Communications 145 (2008) 487–492
Fig. 5. Low-frequency Raman spectra of polycrystalline LiH PO at different
2
4
Fig. 6. High-frequency Raman spectra of polycrystalline LiH PO at different
2
4
−1
temperatures in the 15–400 cm range, showing mainly the external vibrations
as well as part of the internal vibrations of PO and LiO .
−1
temperatures in the 300–1300 cm range, showing the internal vibrations of
PO and LiO and some parts of the in-plane δ (O–H) and out-of-plane γ
4
4
4
4
(O–H) bending modes of hydrogen vibrations.
interpretation given in Table 1. As shown in Fig. 5, at
70 K in the frequency range of 200–300 cm−1, the intensity
of the weak shoulders, or bands, increases with decreasing
temperature. Consequently, distinguishing between external
and internal modes is not intuitively obvious. Recently, Shchur
calculated the lattice dynamics of CsH2PO4 [16]; a similar
lattice-dynamical calculation should be helpful in analyzing the
Raman spectra of LiH2PO4.
(Li2WO4); and 464, 432, 391 cm−1 (Li2MoO4) [28]. Similarly,
the lines or shoulders in the 390–500 cm−1 frequency range
of LiH2PO4 of Fig. 6 may be assigned as the internal modes
of LiO4. This frequency range overlaps with those of ν2 (PO4)
and ν4 (PO4), and may complicate the assignment of internal
modes of LiH2PO4. As the LiO4 tetrahedron shares oxygen
atoms with its four PO4 tetrahedra neighbors (Fig. 1), the
decoupling between the motions of the two tetrahedra is not
clear-cut. Using the relationship between crystallographic point
groups and their subgroups [29], the site symmetry of PO4 of
LiH2PO4 is anticipated to be 1 − C1 as is the site symmetry of
the LiO4 of LiH2PO4. In contrast, local site symmetries for PO4
in KH2PO4, CsH2PO4, and TlH2PO4 have been reported as 4−
S4 [1,2,9], m − Cs [13–15], and 1 − C1 [9,17–19], respectively.
In addition to the fundamentals of PO4 and LiO4, overtones
and combinations of PO4 and LiO4 can appear in the Raman
spectra, and it is therefore necessary to take overtones and
combinations into consideration for definite mode assignments
(Table 1). To definitely identify the LiO4 tetrahedron modes,
the 7Li–6Li isotopic shift should be measured.
3.2. Internal vibrations of PO4 and LiO4
The free tetrahedral phosphate anion PO4, with 43m–Td
symmetry, has nine vibrational normal modes, all of which are
Raman-active: the non-degenerate and completely symmetrical
A1 mode (stretching mode ν1), the doubly degenerate E modes
(bending modes ν2), and two triply degenerate F2 modes
(stretching modes ν3 and bending modes ν4). The values for
the frequencies of these modes in the solution reported in
the literature are ν1 ≈ 980 cm−1, ν2 ≈ 363 cm−1, ν3
cubic local site symmetry of the PO34− ion in the crystal lattice,
the molecular vibrations of the PO34−ion are modified slightly
by the surrounding crystalline field. The lines, shoulders, and
bands of the internal vibrations of the PO34−ion in Fig. 6 will
≈
1082 cm−1, and ν4 ≈ 515 cm−1 [27]. Examining the non-
3.3. O–H vibration
be assigned by referring to the frequencies of the PO34−modes
in the solution (Table 1). In addition to the PO4 tetrahedron,
LiH2PO4 also has a LiO4 tetrahedron [21,22]. This is a peculiar
structural aspect of LiH2PO4 compared to other MX2RO4-type
crystals (M = K, Rb, NH4, Cs, Tl; X = H, D; R = P, As)
without MO4 tetrahedra. Therefore, the LiO4 internal modes
are able to excite LiH2PO4. The oxo-anions form a very large
group, and the central atom may be a member of any group
in the periodic table. The only species containing a group I
element is the LiO4 tetrahedron, which exists in Li2CO3, the
spinel LiFeCr4O8, Li2MoO4, and Li2WO4 all of which have
the phenacite (Be2SiO4) structure. Bands assigned to the LiO4
Due to a certain anharmonicity of the hydrogen bond, the
hydrogen vibrations are expected to be very complex. As
the low-frequency modes, or bands, are often referred to as
collective proton modes, and the high-frequency complex broad
bands above 1500 cm−1, as O–H stretching vibrations, the
hydrogen modes are expected to be exhibited through the whole
Raman spectra range (0–4000 cm−1). According to Novak,
O–H · · ·O hydrogen bonding in solids is classified as strong,
intermediate, or weak [30]. There is an empirical correlation
between the stretching ν (O–H) frequency and the R(O · · · O)
˚
˚
˚
distances of 2.40–2.60 A, 2.60–2.70 A, and >2.70 A, for
the strong, intermediate, and weak bonds, respectively. As
the distance becomes longer, the bond weakens and the ν
(O–H) frequency increases to 700–2700 cm−1 for the strong,
7
tetrahedron on the basis of Li–6Li shifts occur at 497, 435,
416, 397 cm−1 (Li2CO3); 461 cm−1 (spinel); 471, 452 cm−1

K.-S. Lee et al. / Solid State Communications 145 (2008) 487–492
491
Table 1
Raman frequencies (cm ) tentatively assigned to external and internal
−1
vibrations of polycrystalline LiH PO at 70 K
2
4
Frequency Possible assignments
Frequency
Possible assignments
−1
−1
(cm
)
(cm
)
39 m
48 w
61 w
68 w
76 m
82 w
90 m
94 sh
451 w
471 m
491 m
496 sh
516 s
520 s
543 w
548 m
ν
4
ν
4
ν
4
ν
4
ν
4
ν
4
ν
4
ν
4
ν
4
(PO ) or LiO
4
4
4
4
4
(PO ) or LiO
4
(PO ) or LiO
4
(PO ) or LiO
4
(PO )
4
(PO )
4
(PO )
4
(PO )
4
104 w
109 w
113 w
119 m
123 sh
129 w
134 w
141 w
145 w
153 w
160 w
169 m
178 w
190 w
198 w
222 w
233 w
250 w
268 w
278 w
287 w
293 w
558 s, sh
(PO )
4
565 s
ν
(PO )
4
4
726 m
739 m
749 m
855 m
overtone of PO or LiO
overtone of PO or LiO
4
overtone of PO or LiO
overtone of PO or LiO
4
4
4
4
4
Lattice vibrations
4
4
892 vs
928 s
958 w, sh
975 w, sh
ν
ν
(PO )
4
1
(PO )
1
4
γ (O–H)
γ (O–H)
Fig. 7. High-frequency Raman spectra of polycrystalline LiH PO at different
2 4
temperatures in the 1200–4000 cm range, showing principally the internal
vibrations of O–H.
−1
(Table 1). The absence of ferroelectricity in LiH2PO4 seems
due to the structural asymmetry of hydrogen bonds. Although
the LiH2PO4 does not undergo a ferroelectric phase transition,
the general three bands (A, B, C bands) characteristics persist
but with complex splitting upon cooling, as shown in Fig. 7. The
interpretation of this spectral range should include the in-plane
δ (O–H) and out-of-plane γ (O–H) bending modes of hydrogen
vibrations as well as the stretching ν (O–H1) mode. The in-
plane deformation vibrations δ (O–H1) and δ (O–H2) give rise
to bands in the 1200–1300 cm−1 region, while the out-of-plane
γ (O–H1) and γ (O–H2) modes appear in the 900–1100 cm−1
region [30].
1053 s
ν
ν
ν
ν
ν
ν
ν
ν
(PO ) or γ (O–H)
3
3
3
3
3
3
3
3
4
1070 s, sh
1098 m
1120 w
1163 w
1187 w
1207 m
1222 m
(PO ) or γ (O–H)
4
(PO ) or γ (O–H)
4
(PO ) or γ (O–H)
4
(PO ) or γ (O–H)
4
(PO ) or γ (O–H)
4
(PO ) or γ (O–H)
4
309 w
323 w
334 s
363 s
380 s
394 m
402 s
411 w
421 m
436 w
ν
2
ν
2
ν
2
ν
2
ν
2
ν
2
ν
2
ν
2
ν
2
ν
2
(PO )
(PO ) or γ (O–H)
4
4
(PO )
4
(PO )
1282 w
1327 w
1469 sh, b
1655 m
1844 w, b
2069 w, b
2312 w, b
2770 w, b
3032 s
δ (O–H)
δ (O–H)
4
(PO )
4
(PO )
4
(PO ) or LiO
Cν (O–H )
2
4
4
4
4
4
4
(PO ) or LiO
4
(PO ) or LiO
4
(PO ) or LiO
Bν (O–H )
2
Aν (O–H )
ν (O–H )
4
4. Conclusions
(PO ) or LiO
4
2
1
Measurements of the dielectric constant of LiH2PO4
presented no evidence of a phase transition between 297
and 17 K. The low-frequency (15–300 cm−1) region in
Raman spectra was naively assigned to external modes. In the
frequency range from 200–300 cm−1, lowering the temperature
from 297 to 70 K increased the intensities of the weak
shoulders, or bands, precluding clear determination of external
versus internal modes. The vibrational modes appearing over
the 390–500 cm−1 range were assigned as the internal modes
of the LiO4 tetrahedra. This frequency range overlapped those
of ν2 (PO4) and ν4 (PO4). The O–H vibration frequencies
of LiH2PO4 were in good agreement with the findings of
crystallographic reports that there are two types of hydrogen
bonds: an intermediate (long bond) and a strong (short bond).
To obtain accurate measurements of the vibrational frequencies
of the symmetry species, and reasonable assignment of the
vibrational modes, spectroscopic studies of single crystals and
3080 s, sh
ν (O–H )
1
Intensity: - s: strong ; m: medium; w: weak ; v: very ; b: broad ; sh: shoulder.
2800–3100 cm−1 for the intermediate, and >3200 cm−1 for
the weak. According to this scheme, two types of LiH2PO4
hydrogen bonds can be described together, with the first,
an intermediate type, having a hydrogen atom, Hl, in an
asymmetric position along the [100] axis [R(O3–H · · · O4) =
˚
2.684 A] and the second, belonging to strong type having the
hydrogen atom, H2, in a general position and involved in an
asymmetric bond along the [001] axis [R(O4–H2 · · · O2) =
−1
˚
2.564 A]. The relatively sharp band near 3085 cm at 297
K in Fig. 7 is designated as a vibration of O–H1 belonging to
an intermediate hydrogen bond, while the three broad bands
near 2758, 2314, 1630 cm−1 at 297 K are designated as the
A, B, C bands of O–H2 and belong to strong hydrogen bonds

492
K.-S. Lee et al. / Solid State Communications 145 (2008) 487–492
7Li–6Li isotopic substitution should be made and a group-
theoretical analysis of the vibrational modes undertaken.
[11] S.M. Haile, C.R.I. Chisholm, K. Sasaki, D.A. Boysen, T. Uda, Faraday
Discuss. 134 (2007) 17.
[12] Y.-K. Taninouchi, T. Uda, Y. Awakura, A. Ikeda, S.M. Haile, J. Mater.
Chem. 17 (2007) 3182.
[13] B. Marchon, A. Novak, J. Chem. Phys. 78 (1983) 2105.
[14] S. Rumble, F. Ninio, S.S. Ti, Solid State Commun. 42 (1982) 767.
[15] F. Romain, A. Novak, J. Mol. Struct. 263 (1991) 69.
[16] Y. Shchur, Phys. Rev. B 74 (2006) 054301.
[17] P.V. Huong, M. Couzi, J.R. Vignalou, A. Tranquard, in: M. Balkanski,
R.C.C. Leite, S.P.S. Porto (Eds.), Proceedings of the Third, International
Conference on Light Scattering in Solids, Campinas, Brazil, Wiley,
New York, 1975, p. 845.
Acknowledgments
This work was supported by the 2005 Inje University
research grant and partially by the Korea Research Foundation
Grant funded by the Korean Government (MOEHRD, Basic
Research Promotion Fund) (KRF-2006-331-C00088).
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