Lattice dynamics of a monoclinic CsH₂PO₄ crystal
The phonon dispersion relations in a CsH₂PO₄ crystal in a paraelectric phase by means of a rigid molecular-ion model are calculated. The phonon spectra in the quasi-harmonic approximation at various values of temperature and hydrostatic pressure are obtained. The condensation of Au external phonon...
Збережено в:
| Дата: | 1999 |
|---|---|
| Автори: | , , , , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут фізики конденсованих систем НАН України
1999
|
| Назва видання: | Condensed Matter Physics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/119928 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Lattice dynamics of a monoclinic CsH₂PO₄ crystal / Ya.I. Shchur, R.R. Levitskii, O.G. Vlokh, A.V. Kityk, Y.M. Vysochansky, A.A. Grabar // Condensed Matter Physics. — 1999. — Т. 2, № 1(17). — С. 93-108. — Бібліогр.: 38 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-119928 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1199282017-06-11T03:03:51Z Lattice dynamics of a monoclinic CsH₂PO₄ crystal Shchur, Ya.I. Levitskii, R.R. Vlokh, O.G. Kityk, A.V. Vysochansky, Y.M. Grabar, A.A. The phonon dispersion relations in a CsH₂PO₄ crystal in a paraelectric phase by means of a rigid molecular-ion model are calculated. The phonon spectra in the quasi-harmonic approximation at various values of temperature and hydrostatic pressure are obtained. The condensation of Au external phonon branch in the centre and at the boundary of the Brillouin zone is obtained. В рамках моделі жорстких іонів обчислюються фононні дисперсійні співвідношення в параелектричній фазі кристала CsH₂PO₄. Фононні спектри отримані в квазігармонічному наближенні для різних значень температури і гідростатичного тиску. Виявлено конденсацію зовнішньої фононної гілки Au в центрі і на границі зони Бріллюена. 1999 Article Lattice dynamics of a monoclinic CsH₂PO₄ crystal / Ya.I. Shchur, R.R. Levitskii, O.G. Vlokh, A.V. Kityk, Y.M. Vysochansky, A.A. Grabar // Condensed Matter Physics. — 1999. — Т. 2, № 1(17). — С. 93-108. — Бібліогр.: 38 назв. — англ. 1607-324X DOI:10.5488/CMP.2.1.93 PACS: 63.20.-e, 63.20.Dj http://dspace.nbuv.gov.ua/handle/123456789/119928 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The phonon dispersion relations in a CsH₂PO₄ crystal in a paraelectric
phase by means of a rigid molecular-ion model are calculated. The phonon
spectra in the quasi-harmonic approximation at various values of temperature and hydrostatic pressure are obtained. The condensation of Au external phonon branch in the centre and at the boundary of the Brillouin zone
is obtained. |
| format |
Article |
| author |
Shchur, Ya.I. Levitskii, R.R. Vlokh, O.G. Kityk, A.V. Vysochansky, Y.M. Grabar, A.A. |
| spellingShingle |
Shchur, Ya.I. Levitskii, R.R. Vlokh, O.G. Kityk, A.V. Vysochansky, Y.M. Grabar, A.A. Lattice dynamics of a monoclinic CsH₂PO₄ crystal Condensed Matter Physics |
| author_facet |
Shchur, Ya.I. Levitskii, R.R. Vlokh, O.G. Kityk, A.V. Vysochansky, Y.M. Grabar, A.A. |
| author_sort |
Shchur, Ya.I. |
| title |
Lattice dynamics of a monoclinic CsH₂PO₄ crystal |
| title_short |
Lattice dynamics of a monoclinic CsH₂PO₄ crystal |
| title_full |
Lattice dynamics of a monoclinic CsH₂PO₄ crystal |
| title_fullStr |
Lattice dynamics of a monoclinic CsH₂PO₄ crystal |
| title_full_unstemmed |
Lattice dynamics of a monoclinic CsH₂PO₄ crystal |
| title_sort |
lattice dynamics of a monoclinic csh₂po₄ crystal |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
1999 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119928 |
| citation_txt |
Lattice dynamics of a monoclinic CsH₂PO₄ crystal / Ya.I. Shchur, R.R. Levitskii, O.G. Vlokh, A.V. Kityk, Y.M. Vysochansky, A.A. Grabar // Condensed Matter Physics. — 1999. — Т. 2, № 1(17). — С. 93-108. — Бібліогр.: 38 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT shchuryai latticedynamicsofamonocliniccsh2po4crystal AT levitskiirr latticedynamicsofamonocliniccsh2po4crystal AT vlokhog latticedynamicsofamonocliniccsh2po4crystal AT kitykav latticedynamicsofamonocliniccsh2po4crystal AT vysochanskyym latticedynamicsofamonocliniccsh2po4crystal AT grabaraa latticedynamicsofamonocliniccsh2po4crystal |
| first_indexed |
2025-07-08T16:55:31Z |
| last_indexed |
2025-07-08T16:55:31Z |
| _version_ |
1837098582132391936 |
| fulltext |
Condensed Matter Physics, 1999, Vol. 2, No. 1(17), p. 93–108
Lattice dynamics of a monoclinic
CsH2PO4 crystal
Ya.I.Shchur 1 , R.R.Levitskii 2 , O.G.Vlokh 1 , A.V.Kityk 1 ,
Y.M.Vysochansky 3 , A.A.Grabar 3
1 Institute of Physical Optics,
23 Dragomanov Str., 290005 Lviv, Ukraine
2 Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 290011 Lviv, Ukraine
3 Uzhgorod State University,
46 Pidhirna Str., 294000 Uzhgorod, Ukraine
Received November 19, 1997
The phonon dispersion relations in a CsH2PO4 crystal in a paraelectric
phase by means of a rigid molecular-ion model are calculated. The phonon
spectra in the quasi-harmonic approximation at various values of tempera-
ture and hydrostatic pressure are obtained. The condensation of Au exter-
nal phonon branch in the centre and at the boundary of the Brillouin zone
is obtained.
Key words: lattice dynamics, soft mode, structural phase transition,
phonon spectrum
PACS: 63.20.-e, 63.20.Dj
1. Introduction
Caesium dihydrogen phosphate CsH2PO4 (CDP) belongs to a group of mono-
clinic phosphates of the KH2PO4-type, like KD2PO4 (DKDP), RbD2PO4 (DRDP),
TlH2PO4 (TDP), RbHPO4 (LHP). Owing to their interesting properties, the CDP
crystal and its deuterated analogue CsD2PO4 (DCDP) have been intensively stud-
ied by X-ray diffraction [11, 21, 34], neutron scattering [9, 12, 22, 33], dielectric
[5,7, 31] acoustic [1, 14, 24], Raman scattering [8, 13, 20, 38], hyper-Raman scat-
tering [28], IR absorption [30] and optic [35] measurements. However, a lattice
dynamics of the CDP crystal in the region of phase transitions (PTs) and a triple
point have been investigated insufficiently.
In this paper a theoretical treatment of the lattice dynamics of CDP within
the framework of a rigid molecular-ion model in the quasi-harmonic approximation
c© Ya.I.Shchur, R.R.Levitskii, O.G.Vlokh, A.V.Kityk, Y.M.Vysochansky, A.A.Grabar 93
Ya.I.Shchur et al.
in paraelectric (PE) phase under the influence of hydrostatic pressure and tem-
perature is presented. We restrict ourselves to the consideration of low-frequency
external phonon modes which gives the most complete information about the na-
ture of PTs. Moreover, in monoclinic phosphates of the CDP-type the PTs are
often accompanied by unit cell multiplication (e.g., in DRDP [10, 29] and TDP
[2, 26] crystals). Therefore, the use of a CDP crystal as a model object for the
investigation of structural PTs seems to the quite reasonable.
2. Structure and symmetry
In a high-temperature PE phase the CDP crystal has a space group P21/m
with two molecules per unit cell (a = 7.906A, b = 6.372A, c = 4.883A, β = 107.73o
[21] ). The CDP crystal structure is depicted in figure 1 (according to [9]). The
peculiarity of this structure is the presence of two types of hydrogen bonds O–
H. . .O. The shorter bonds (R = 2.47 Å) link the PO4 groups into zig-zag-like
chains running along the b axis. In the PE phase the protons are disordered on
those bonds. The longer bonds (R = 2.54 Å), which are directed approximately
along the c axis, cross-link the chains to form (b, c) layers. The protons on the longer
bonds are ordered on one of the two possible equilibrium sites at all temperatures.
Figure 1. The P-T phase diagram of a CsH2PO4 crystal in the PE phase (ac-
cording to [20]).
At Tc = 156 K and atmospheric pressure, a CDP crystal undergoes the PT
into the PE phase with the space group P21 and the spontaneous polarization
Ps‖b (Z = 2, a = 7.87 Å, b = 6.32 Å, c = 4.89 Å, β = 108.3o at T = 83 K [12]). In
the PE phase a proton ordering on the shorter hydrogen bonds occurs.
By means of the dielectric [32] and neutron [25] investigation, it was shown that
at some values of hydrostatic pressure and temperature, the PT into the antifer-
94
Lattice dynamics of a monoclinic CsH2PO4 crystal
Figure 2. The structure of a CsH2PO4 crystal in the PE phase (according to [9]).
roelectric (AFE) phase takes place in a CDP crystal. The pressure-temperature
phase diagram of a CDP crystal is depicted in figure 2 [21]. The neutron scattering
study [25] has revealed that the PT into the AFE phase is accompanied by a unit
cell doubling along the a axis (Z = 4, a = 15.625Å, b = 6.254Å, c = 4.886Å,
β = 108.08o). The antiparallel ordering of protons on the neighbouring chains
running along the b axis takes place. The atom coordinates determined by [25]
allowed the authors to identify the symmetry of the AFE phase both with the P21
space group and with the P21/a one. Due to the Raman spectra studies of the
DCDP crystal at T = 83 K and P = 800 MPa [20] a conclusion was drawn that
the symmetry of the AFE phase is P21/a. Besides, as it follows from the general
symmetry consideration (Curie principle [3]), if there is an inversion centre in the
PE phase, it must persist in the AFE phase, too. We shall assume that in the AFE
phase the P21/a symmetry is realized. For the lattice dynamics consideration, an
orthonormal set of axes X, Y and Z was used. In the PE phase, the principal pa-
rameters of the monoclinic lattice mP with respect to the orthonormal axes are
the following:
a = (0, τy, 0), b = (0, 0, τz), c = (τ cosα,−τ sinα, 0), (1)
where τ = c; τy, τz are the corresponding principal parameters along the Y and
Z axes; α = 17.73o. At the same time, the reciprocal lattice parameters can be
written as
b1 = (2π tanα/τy , 2π/τy, 0), b2 = (0, 0, 2π/τz), (2)
b3 = (2π/(τ cosα), 0, 0).
95
Ya.I.Shchur et al.
3. Group-theoretical classification of lattice vibrations
A general group theory analysis of the normal modes of a CDP crystal in the
PE phase for 14 wavevectors q which correspond to all the symmetrical points
and lines of the Brillouin zone (BZ) of the monoclinic lattice mP was made. A
symmetrized form of the dynamical matrix D(q), the matrices of irreducible mul-
tiplier representations (IMRs) T (q, h) (here h is an element of the wavevector point
group G0(q)) and the symmetry vectors E(q) for wavevectors qi(i = 1, 14) were
obtained. However, in this work we present the results for the most interesting
from the physical point of view cases (q7, q1, q12, q13, q3, q11) which will be used in
numerical calculations. Kovalev’s notation [15] was used for the identification of
wavevectors, elements of the wavevector point groups G0(q), symmetrical points
and lines of the BZ.
We carried out the lattice dynamics study of a CDP crystal within the rigid
molecular-ion approximation [36] where the H2PO4 groups are treated as rigid and
nondeformable. Hence, there are two Cs+ ions (k = 1, 2) and two (H2PO4)
− ions
(k = 3, 4) in the unit cell of a CDP crystal. In this case, there are 18 external
branches in the phonon spectrum of a CDP crystal.
The symmetry vectors in the centre of the BZ (q7 = 0) corresponding to the
translational displacements and librations of the structural units of the crystal
are given in table 1. They are used for diagonalization of the dynamical matrix
and phonon frequencies classification according to IMRs. The point groups of the
wavevectors (q7, q1, q12, q13, q3, q11) and normal modes classification by the IMRs
of these groups are shown in table 2. The IMRs of the above mentioned groups
are given in table 3.
4. Model
The crystal potential energy Φ in the rigid molecular-ion approximation is
written as [6]
Φ =
1
2
∑
l,k
k⊂K
∑
l′,k′
k′⊂K′
1
4πε0
ZKkZK′k′e
2
r(lKk, l′K ′k′)
+ a exp
{
−
br(lKk, l′K ′k′)
R(k) + R(k′)
}
, (3)
where a = 1822 eV, b = 12.364; l, l′ are unit cell indices; K, K ′ are the indices
of atomic or molecular vibrating units in the unit cell; k, k′ are the indices of
atoms in molecular units; k ⊂ K means that atom k belongs to molecule K; e
is an electron charge; ZKk and RKk are effective charge and radii parameters of
Kk atom, respectively; r is a distance between Kk and K ′k′ atoms. The first and
second terms of expression (3) represent the long-range Coulomb and the short-
range Born-Mayer-type repulsive energy, respectively. Moreover, we assumed that
the total potential energy is a sum of the pair interactions of the atoms of different
molecules. Interactions between the atoms within a molecule have been neglected.
The unknown parameters ZKk and RKk are determined from the lattice equi-
librium conditions with respect to any macroscopic internal strains [4] and also
96
Lattice dynamics of a monoclinic CsH2PO4 crystal
Table 1. Symmetry vectors for CsH2PO4 in the PE phase in the centre of the
BZ (q7 = 0). Designations x, y and z refer to the translations along the X, Y and
Z axes; x̃, ỹ and z̃ refer to the librations about X, Y and Z axes.
Representation Symmetry vectors
Ag x(1)− x(2), y(1)− y(2), x(3) − x(4), y(3)− y(4),
z̃(3) + z̃(4)
Bg z(1) − z(2), z(3) − z(4), x̃(3) + x̃(4), ỹ(3) + ỹ(4)
Au z(1) + z(2), z(3) + z(4), x̃(3) − x̃(4), ỹ(3)− ỹ(4)
Bu x(1) + x(2), y(1) + y(2), x(3) + x(4), y(3) + y(4),
z̃(3) − z̃(4)
Table 2. Point groups of wavevectors and classification of normal modes for
CsH2PO4 (−1
2
< µ1, µ2, µ3 <
1
2
) in the PE phase.
Point group Elements of
Wavevector G0(q) of G0(q) Classification
wavevector
q7 = 0 2/m h1, h3, h25, h27 5Ag + 4Bg + 4Au + 5Bu
q1 = µ1b1 + µ3b3 m h1, h27 10A′ + 8A′′
q12 =
1
2
b3
2/m h1, h3, h25, h27 5Ag + 4Bg + 4Au + 5Bu
q13 =
1
2
b1
q3 = µ2b2 2 h1, h3 9A+ 9B
q11 =
1
2
b2 2/m h1, h3, h25, h27 9E
Table 3. Irreducible multiplier representations of some point groups G0(q) of
wavevectors for the P21/m space group.
G0(q7), G0(q12), G0(q13)
h1 h3 h25 h27
representation
Ag 1 1 1 1
Bg 1 -1 -1 1
Au 1 1 -1 -1
Bu 1 -1 1 -1
G0(q1), G0(q3) G0(q1) : h1 h27
representation G0(q3) h1 h3
A′ A 1 1
A′′ B 1 -1
G0(q11) h1 h3 h27 h25
representation
E 1 0 1 0 0 1 0 1
0 1 0 -1 -1 0 1 0
97
Ya.I.Shchur et al.
with respect to the condition of molecule electroneutrality. Thus, the constraints
for the crystal potential are
∂Φ
∂U i
α(lK)
∣
∣
∣
∣
0
= 0,
∂Φ
∂Sαβ
= 0,
∑
K,k
ZKk = 0, (4)
Figure 3. The phonon disper-
sion relations in a CsH2PO4
crystal in the PE phase at
room temperature and atmo-
spheric pressure along the fol-
lowing directions: a) b1, b) b2,
c) b3.
where U i
α(lK) is a displacement of molecule K
in the unit cell l; α is the Cartesian components
X, Y and Z; i refers to the translation (t) or
rotation (r); Sαβ is a macroscopic strain.
At the same time, we suppose that a pri-
ori the quantum particle proton located on the
hydrogen bond O–H. . .O cannot be taken into
account directly in the quasi-harmonic approx-
imation. Therefore, the influence of the protons
is taken into consideration indirectly by means
of unequal values of the effective charges ZKk
and radii RKk of oxygen ions in H2PO4 groups.
With the help of ab initio calculations within
the extended Huckel method it was shown [27]
that the charges and radii of oxygen ions in the
H2PO4 group depend on proton localization on
the O–H. . .O bond in one of the minima of a
double-well type potential in which the proton
moves.
Proceeding from the aforesaid, we have used
the following values of the model parameters
Z(Cs) = 1.18, Z(P) = 0.34, Z(O1) = −0.39,
Z(O2) = −0.41, Z(O3) = Z(O4) = −0.36;
R(Cs) = 2.83, R(P) = 1.0, R(O1) = 1.36,
R(O2) = 1.32, R(O3)R(O4) = 1.44. The charge
Z(O1) is smaller than Z(O2) because the pro-
tons on the bonds O1–H. . .O2 are localized near
the oxygen ions O1 at all temperatures. The
charges Z(O3), Z(O4) and radii R(O3), R(O4)
are equal, respectively, because the protons on
the bonds O3–H. . .O4 tunnel between the two
possible off-centre equivalent positions in the
PE phase.
Therefore, in reality the lattice dynamics of
a crystal which consists of the Cs+ and (PO4)
−
ions with indirect consideration of the proton
influence is studied.
98
Lattice dynamics of a monoclinic CsH2PO4 crystal
5. Results and discussion
5.1. Phonon dispersion relations in CDP
The lattice dynamics calculation of the CDP crystal is carried out by means of
the program DISPR [6] which has been modified by us in order to use the group-
theory information to a greater extent. Figure 3 shows the calculated phonon
dispersion relations along the directions b1, b3 (q7 = 0 → q1 = µ1b1 + µ3b3 →
q13 =
1
2
b1 or q12 =
1
2
b3) and (q7 = 0 → q3 = µ2b2 → q11 =
1
2
b2). The compatibility
relations between the IMRs along these directions are presented in table 4.
Table 4. Compatibility relation between the irreducible multiplier representations
along the b1, b2 and b3 directions in CsH2PO4 in the PE phase.
q7 = 0 q1 = µ1b1 q13 =
1
2
b1
q7 = 0 q3 = µ2b2 q11 =
1
2
b2
q7 = 0 q1 = µ3b3 q12 =
1
2
b3
Ag Ag Ag
Bg A′ Bg Bg A
E
Au A′′ Au Au B
Bu Bu Bu
The accuracy of the calculation was controlled at different stages by the corre-
spondence of the calculated dynamical matrixD(q) to the one determined theoreti-
cally by means of general symmetry requirements [18]. As one can see from figure 3,
at the boundary of the BZ at point q11 =
1
2
b2, the phonon modes become two-fold
degenerate. This is a consequence of two-dimensionallity of the irreducible multi-
plier representation E of the group G0(q11). Note, that the time-reverse symmetry
does not require any additional degeneracy for all the considered cases (q7, q1, q12,
q13, q3, q11).
The disagreement in frequencies while approaching the centre of the BZ from
different directions can be explained by LO – TO splitting. Only polar modes
which transform according to the IMRs Au and Bu possess LO – TO splitting. It
follows from the eigenvectors analysis that the vibrations propagating along the
b1 and b3 directions are purely transverse (Au symmetry) or quasi-transverse (Bu
symmetry). At the same time, the modes propagating along the b2 direction are
purely longitudinal (Au symmetry) or purely transverse (Bu symmetry). Therefore,
at b2 → 0, the modes of Au symmetry have larger frequencies as compared with
those when b1 → 0 or b3 → 0 (ωLO > ωTO). On the contrary, the frequencies of the
Bu symmetry at b2 → 0 are smaller than the frequencies at b1 → 0 and b3 → 0,
because the modes along the b1 and b3 directions are quasi-transverse and have
some longitudinal components.
99
Ya.I.Shchur et al.
Table 5. Comparison of the experimental and theoretical values of external modes
frequencies in the centre of the BZ in CsH2PO4.
experiment (cm−1)
representation Raman IR calculation (cm−1)
Ag 43 40
49 45
75 88
118 122
219 172
Bg 45 44
61 59
110 106
234 157
Au acoustic acoustic
38 39, TO
61, LO
74 72, TO
82, LO
100 139, TO
161, LO
Bu acoustic acoustic
acoustic acoustic
97, QTO∗, q = (0.001, 0, 0)
76 96, TO, q = (0, 0.001, 0)
106, QTO, q = (0, 0, 0.001)
122, QTO, q = (0.001, 0, 0)
106 108, TO, q = (0, 0.001, 0)
112, QLO, q = (0, 0, 0.001)
149, QTO, q = (0.001, 0, 0)
146 123, TO, q = (0, 0.001, 0)
150, QLO, q = (0, 0, 0.001)
∗QTO, quasi-transverse optical mode
OLO, quasi-longitudinal optical mode
100
Lattice dynamics of a monoclinic CsH2PO4 crystal
Comparison of the calculated phonon frequencies in the centre of the BZ
(q7 = 0) with those obtained from Raman and IR investigations [19] is presented
in table 5. For most of the calculated frequencies, a good agreement with the
corresponding experimental results is obtained.
Table 6. The components Sij of
the elastic compliance matrix in
CsH2PO4 at T = 293 K (according
to Prewer et al. (1985))
ij Sij(GPa)−1 ij Sij(GPa)−1
11 1.82 12 -0.219
22 0.103 13 -1.17
33 0.772 15 0.249
44 0.133 23 0.138
55 0.450 25 -0.150
66 0.117 35 -0.181
46 0.033
The calculation of the external phonon
dispersion relation along the b1 direction
attracts special interest since the PT into
the AFE phase occurs in a CDP crystal
at some values of temperature and hy-
drostatic pressure. Usually, such a PT is
related to the external mode condensa-
tion at the BZ boundary at point q13 =
1
2
b1. With the help of group theory con-
sideration [17], one can show that the
IMR Au is responsible for the AFE PT in
a CDP crystal with the symmetry change
P21/m → P21/α. The same IMR Au is
responsible for the PT into the FE phase
(space group P21). This PT is caused by
the external mode condensation in the
centre of the BZ.
The phonon spectra calculation of a CDP crystal at different values of temper-
ature and hydrostatic pressure was carried out. Moreover, we remained within the
framework of the quasi-harmonic approximation at specific values of T and P . The
influence of interatomic anharmonicity on lattice dynamics is taken into considera-
tion indirectly through the change of the lattice parameters a, b and c, which were
determined from the experimental data of thermal expansion and ultrasonic mea-
surements. We assumed that the temperature and hydrostatic pressure influence
the lattice parameters only (without a change of fractional atomic coordinates in
a unit cell) according to the next linear laws:
a = (1 −KaP )aT , b = (1 −KbP )bT , c = (1 −KcP )cT , (5)
where Ka, Kb and Kc are linear compressibility components along the a, b and c
axes, respectively, at the applied hydrostatic pressure P ; aT , bT and cT are the
lattice parameters at temperature T .
The linear compressibility Klmn of a monoclinic crystal with the P21/m sym-
metry along the given direction [l, m, n] in the Cartesian coordinate system X, Y
and Z is written as [23]
Klmn = (S11 + S12 + S13)l
2 + (S12 + S22 + S23)m
2 (6)
+ (S13 + S23 + S33) n
2 + (S16 + S26 + S36)lm,
here Sij are components of the elastic compliance matrix which is inverse to the
elastic constant matrix Cij. The values of Sij components for the CDP crystal,
101
Ya.I.Shchur et al.
determined at room temperature by the ultrasonic waves velocities measurements
[24], are presented in table 6.
Thus, the components of the CDP crystal linear compressibility along the crys-
tallographic a, b and c axes have the following values: Ka = 0.022, Kb = −0.260
and Kc = 0.390. At the same time, we assume that the linear compressibility is
independent of both the temperature and hydrostatic pressure.
To determine the thermal dependence of the lattice parameters a, b and c, the
dilatometric investigations of monodomain specimen have been carried out. The
linear thermal expansion (∆1/1) observed along the a∗, b and c directions as a
function of temperature, where a∗ ⊥ (b, c), is shown in figure 4. As one can see,
∆1/1 along the b and c axes are essentially larger than ∆1/1 along a∗ direction.
This can indicate a quasi-layer nature of the CDP crystal along the (b, c) plane.
In other words, interactions between the ions within the same layer are to a great
extent larger than interactions between the ions from different layers. The perfect
cleavage that occurs along the (b, c) plane confirms a relative weakness of interlayer
forces. Therewith, the linear thermal expansion along the c axis increases with the
temperature decrease, i.e. the CDP crystal expands along this axis at cooling.
The thermal expansion coefficient a along c axis becomes negative. This unusual
behaviour of the coefficient a is considered in detail in [37].
5.2. Phonon dispersion relations in a CDP crystal at differe nt values of tem-
perature and hydrostatic pressure
Figure 4. The linear thermal
expansion along the orthonor-
mal a∗, b, and c axes of a
CsH2PO4 crystal.
Using the data of thermal expansion (figure 4)
and linear compressibility (table 6) of a CDP crys-
tal, with the help of expression (5) it is easy to
obtain the lattice parameters at various values of
T and P . The calculated dispersion relations for
external phonon modes of the B symmetry (G0q1)
along the b1 direction at different values of T and
P are presented in figure 5. As one can see, at
increasing the hydrostatic pressure and decreas-
ing the temperature, a lowering of most of the
phonon branches is observed. At T = 130 K and
P = 241 MPa, the lower optic phonon branch of
the Au symmetry falls to zero in the centre of the
BZ (at room temperature and atmospheric pres-
sure the value of this optic mode is 39 cm−1. This
implies that in a CDP crystal the PT into the PE
phase must occur.
At the same time there occur such displacements of structural units of the
crystal to new equilibrium sites which define the structure of the PE phase. These
displacements, determined from the analysis of the eigenvector corresponding to
the soft phonon mode Au, are schematically presented in figure 6. As follows from
this figure, the Cs+ and (PO4)
− ions shift along the b axis in opposite directions
102
Lattice dynamics of a monoclinic CsH2PO4 crystal
Figure 5. The dispersion relations of phonon modes of the B symmetry along
the b1 direction in a CsH2PO4 crystal at different values of temperature and
hydrostatic pressure:
a) T = 150 K, P = 200 MPa; b) T = 135 K, P = 230 MPa;
c) T = 130 K, P = 241 MPa; d) T = 121 K, P = 286 MPa.
103
Ya.I.Shchur et al.
with simultaneous PO4 group rotation in the ac-plane. The calculated displace-
ments of structural units both translational and rotational are in good qualitative
agreement, with neutron scattering data [12].
Figure 6. Schematic representation of ions displacements corresponding to the
ferroactive Au soft phonon mode determined from the eigenvector analysis.
Variation of lattice parameters with temperature and hydrostatic pressure is
accompanied by a change of effective charges and radii. For preserving the equilib-
rium conditions (4) it is necessary to change the model parameters to the fol-
lowing values Z(Cs) = 1.21, Z(P) = 0.33, Z(O1) = −0.40, Z(O2) = −0.42,
Z(O3) = −0.36; R(Cs) = 2.85, R(P) = 1.0, R(O1) = 1.35, R(O2) = 1.31,
R(O3) = R(O4) = 1.44 at T = 130 K and P = 241 MPa.
In figure 7a, the dispersion relation of the soft phonon mode Au in the plane (b1,
b3) at T = 130 K and P = 241 MPa is depicted. As one can see from the P − T
phase diagram (figure 2), the PT into the FE phase takes place at T = 130 K
and P = 251 MPa, i.e. the calculated values of temperature T = 130 K and
hydrostatic pressure P = 241 MPa are in good agreement with the experimental
data. For other values of T , there is no good coincidence between the experimental
and the calculated values of P at which the PT into the FE phase occurs. It
should be noted that the calculated frequencies of external phonon modes are
more sensitive to the influence of hydrostatic pressure on the lattice parameters
than to the temperature influence. So, at T = 121 K and P = 286 MPa (an
experimental value of hydrostatic pressure obtained from the P−T phase diagram
for T = 121 K is P = 370 MPa), the lowering to zero of the same lower optic
104
Lattice dynamics of a monoclinic CsH2PO4 crystal
phonon branch of the Au symmetry, active at PT into the FE phase, takes place
at the BZ boundary at point q13 =
1
2
b1 (figure 5). In other words, the PT into the
AFE phase, accompanied by the unit cell doubling along the a axis, must occur in
a CDP crystal at some values of T and P . Here the displacements of the crystal
structural units to new equilibrium sites of the AFE phase are similar to those
presented in figure 6, only they are opposite in the neighbouring cells. In figure 7
the calculated dispersion relation of the soft phonon mode Au in the plane (b1, b3)
at T = 121 K, P = 286 MPa (i.e. at the PT into the AFE phase) is presented.
a) b)
Figure 7. The dispersion relation of the Au soft phonon mode in the (b1, b3)
plane at a) T = 130 K and P = 241 MPa; b) T = 121 K and P = 286 MPa.
6. Conclusions
In this paper we have reported the results of lattice-dynamical calculations
of a CDP crystal based on the rigid molecular-ion model which includes the
Coulomb and short-range interactions. The protons on hydrogen bonds in the
quasi-harmonic approximation have not been taken into account immediately but
indirectly by means of choice of the unequal values of effective charges and radii of
oxygen ions in H2PO4 groups. This approach provides a reasonable explanation of
the observed Raman and IR data for a CDP crystal in the external modes region
where the effects of internal vibrations are not expected to be felt. In the case when
anharmonic particles of the protons were taken into consideration immediately as
hard atoms (e.g., Cs, P, or O) there was a significant aggravation of lattice equi-
librium conditions and appearance of unphysical results (the phonon frequencies
with imaginary values).
The lattice dynamics calculation of a CDP crystal at different values of tem-
perature and hydrostatic pressure was carried out. Thereat, we supposed that in
the quasi-harmonic approximation the influence of T and P would be displayed
only through the change of the lattice parameters a, b and c. For this purpose,
105
Ya.I.Shchur et al.
dilatometric investigations of a CDP crystal were performed. Using the ultrasonic
waves velocities measurements [24], the values of linear compressibility along the
a, b and c axes were obtained. This enabled us to calculate the phonon spectra
of CDP at various values of T and P . At decreasing temperature and increasing
hydrostatic pressure the lowering of most of the phonon branches was observed. At
T = 130 K and P = 241 MPa, the lower optic phonon branch of the Au symmetry
falls to zero in the centre of the BZ (q7 = 0). At T = 121 K and P = 286 MPa,
the same lower optic phonon branch tends to zero at the boundary of the BZ at
point q13 =
1
2
b1. In other words, the obtained results qualitatively show that the
PT in a CDP crystal takes place either into the FE phase (condensation of the Au
mode in the BZ centre) or into the AFE phase (condensation of the Au mode at
the BZ boundary) depending on a correlation of interatomic forces which depend
on the temperature and hydrostatic pressure. It confirms once more an extremely
important role of a proton subsystem in phase transitions in the compounds of
this type.
References
1. Anisimova V.N., Yakushkin E.D, Majszchyk J. Acoustic and dielectric properties of
quasi-one-dimensional ferroelectric CsH2PO4 in an external electric field. // Journ.
Mater. Science Lett., 1993, vol. 12, No. 3, p. 588–590.
2. Arai M., Yagi T., Sakai A., Komukae M., Osaka T., Makita Y. Elastic instability in the
ferroelastic phase transition of TlH2PO4 studied by Brillouin scattering. // J. Phys.
Soc. Japan, 1990, vol. 59, No. 2, p. 1285–1292.
3. Blinc R., Žekš B. Ferroelektrics and antiferroelectrics. Moskow, Mir, 1975 (in Russian).
4. Boyer L.L., Hardy J.R. Static equilibrium conditions for rigid-ion crystals. // Phys.
Rev. B., 1973, vol. 7, p. 2886–2888.
5. Brandt N.B., Chookov S.G., Kulbachinskii V.A., Smirnov P.S., Strukov B.A. // Fiz.
Tverd. Tela, 1986, vol. 28, p. 3159 (in Russian).
6. Chaplot S.L. A computer program for external modes in complex ionic crystals (the
rigid molecular-ion model). Report N BARC 972 Bhabha Atomic Research Centre
(Bombay), 1978.
7. Deguchi K., Nakamura E., Okaue E., Aramaki N. Effects of deuteration on the dielec-
tric properties of ferroelectric CsH2PO4. II. Dynamic dielectric properties. // J. Phys.
Soc. Japan, 1982, vol. 51, p. 3575–3582.
8. Fillaux F., Marchon B., Novak A. Shape of Raman OH stretching band and proton
dynamics in CsH2PO4. // Chem. Phys., 1984, vol. 86, p. 127–136.
9. Frazer B.C., Semmingsen D., Ellenson W.D., Shirane G. One-dimensional ordering
in ferroelectric CsD2PO4 and CsH2PO4 as studied with neutron scattering. // Phys.
Rev., 1979, vol. 20, p. 2745–2755.
10. Hagiwara T., Itoh K., Nakamura E., Komukae M., Makita Y. Structure of mono-
clinic rubidium dideuteriumphosphate, RbD2PO4, in the intermediate phase. // Acta
Cryst., 1984, vol. C40, p. 718–720.
11. Itoh K., Hagiwaga T., Nakamura E. Order-disorder type phase transition in ferro-
electric CsD2PO4 studied by X-ray structure analysis. // J. Phys. Soc. Japan, 1983,
vol. 52, p. 2626–2629.
106
Lattice dynamics of a monoclinic CsH2PO4 crystal
12. Iwata Y., Koyano N., Shibuya I. A neutron diffraction study of the ferroelectric tran-
sition of CsH2PO4. // J. Phys. Soc. Japan, 1980, vol. 49, p. 304–307.
13. Kasahara M., Aoki M., Tatsuzaki I. Raman study of the phase transition in CsH2PO4
and CsD2PO4. // Ferroelectrics, 1984, vol. 55, p. 47–50.
14. Kityk A.V., Shchur Ya.I., Lutsiv-Shumskii L.P., Vlokh O.G. On the acoustic properties
of CsD2PO4 and RbD2PO4 crystals near phase transitions under hydrostatic pressure.
// J. Phys.: Condens. Matter, 1994, vol. 6, p. 699–712.
15. Kovalev O.V. Irreducible Representations of the Space Groups. New York, Gordon
and Breach Science Publishers, 1965.
16. Lifshits I.M. About thermal properties of chain-type and layer-type structures at low
temperatures. // Zhurn. Eksper. Teoret. Fiz., 1952, vol. 22, p. 475–486.
17. Liubarskii G.Ya. The Application of Group Theory in Physics, New York, Pergamon
Press, 1960.
18. Maradudin A.A., Vosko S.H. Symmetry properties of the normal vibrations of a crys-
tal. // Rev. Mod. Phys., 1968, vol. 40, p. 1–76.
19. Marchon E. Etude vibrationelle des transitions de phase d’un ferroelectrique pseudo-
unidimensionnel CsH2PO4, Paris, These de doctorat d’etat, 1983.
20. Marchon B., Novak A. Antiferroelectric fluctuations in CsH2PO4 and Raman spec-
troscopy. // Ferroelectrics, 1984, vol. 55, p. 55–58.
21. Matsunaga H., Itoh K., Nakamura E. X-ray structural study of ferroelectric cesium
dihydrogen phosphate at room temperature. // J. Phys. Soc. Japan, 1980, vol. 48,
p. 2011–2014.
22. Nelmes R.J., Choudhary R.N.P. Structural studies of the monoclinic dihydrogen phos-
phates: a neutron-diffraction study of paraelectric CsH2PO4. // Sol. Stat. Commun.,
1978, vol. 26, p. 823–826.
23. Nye J.F. Physical Properties of Crystals, Oxford, Clarendon, 1967
24. Prawer S., Smith T.F., Finlayson. The room temperature elastic behaviour of
CsH2PO4. // Aust. Journ. Phys., 1985, vol. 38, p. 63–83.
25. Schuele P.J. , Thomas R. A structural study of the high-pressure antiferroelectric
phase of CsH2PO4. // Japan Journ. Appl. Phys., 1985, vol. 24, p. 935–937.
26. Seliger J., Zagar V., Blinc R., Schmidt V.H. O17 study of the antiferroelectric phase
transition in TlH2PO4. // J. Chem. Phys., 1988, vol. 88, p. 3260–3262.
27. Stetsiv R. Electron states and optical effects in the KH2PO4-type crystals with hy-
drogen bonds. Thesis on Doctor of Philosophy degree, Lviv, 1993.
28. Shin S., Ishida A., Yamakami T., Fujimura T., Ishigame M. Central mode in hyber-
Raman spectra of the quasi-one-dimensional hydrogen-bonded ferroelectric CsH2PO4.
// Phys. Rev. B, 1987, vol. 35, p. 4455–4461.
29. Suzuki S., Arai K., Sumita M., Makita Y. X-ray diffraction study of monoclinic
RbD2PO4. // J. Phys. Soc. Japan, 1983, vol. 7, p. 2394–2400.
30. Ti S.S., Rumble S., Ninio F. Infrared spectra of paraelectric and ferroelectric
CsH2PO4. // Sol. Stat. Commun., 1982, vol. 44, p. 129–135.
31. Yasuda N., Kawai J. Dielectric dispersion associated with the de-electric-field-enforced
ferroelectric phase transition in the pressure-induced antiferroelectric CsH2PO4. //
Phys. Rev. B, 1990, vol. 42, p. 4893–4896.
32. Yasuda N., Okamoto M., Shimuzu H., Fujimoto S., Yoshino K. , Inuishi Y. Pressure-
induced antiferroelectricity in ferroelectric CsH2PO4. // Phys. Rev. Lett., 1978,
vol. 41, p. 1311–1313.
107
Ya.I.Shchur et al.
33. Youngblood R., Frazer B.C., Eckert J., Shirane G. Neutron scattering study of pres-
sure dependence of short-range order in CsD2PO4. // Phys. Rev. B, 1980, vol. 22,
p. 228–235.
34. Uesu Y., Kobayashi J. Crystal structure and ferroelectricity of cesium dihydrogen
phosphate CsH2PO4. // Phys. Stat. Sol. (a), 1976, vol. 34, p. 475–481.
35. Ushio S. Temperature variations of optical indicatrix and birefringence in ferroelectric
CsH2PO4. // J. Phys. Soc. Japan, 1984, vol. 53, p. 3242–3249.
36. Venkataraman G., Sahni V.C. External vibration in complex crystals. // Rev. Mod.
Phys., 1970, vol. 42, p. 409–470.
37. Vlokh O.G. Shchur Ya.I., Hirnyk I.S., Klymiv I.M. About thermal expansion of
RbD2PO4,CsH2PO4 and CsD2PO4. // Fiz. Tverd. Tela, 1994, vol. 36, p. 2890–2895
(in Russian).
38. Wada M., Sawada A., Ishibashi Y. Some high-temperature properties and the Raman
scattering spectra of CsH2PO4. // J. Phys. Soc. Japan, 1979, vol. 47, p. 1571–1547.
Граткова динаміка моноклінного кристала CsH2PO4
Я.І.Щур 1 , Р.Р.Левицький 2 , О.Г.Влох 1 , А.В.Кітик 1 ,
Й.М.Височанський 3 , А.А.Грабар 3
1 Інститут фізичної оптики,
290005 Львів, вул. Драгоманова, 23
2 Інститут фізики конденсованих систем НАН Укpаїни,
290011 Львів, вул. Свєнціцького, 1
3 Ужгородський державний університет,
294000 Ужгород, вул. Підгірна, 46
Отримано 19 листопада 1997 р.
В рамках моделі жорстких іонів обчислюються фононні дисперсійні
співвідношення в параелектричній фазі кристала CsH2PO4. Фононні
спектри отримані в квазігармонічному наближенні для різних значень
температури і гідростатичного тиску. Виявлено конденсацію зовніш-
ньої фононної гілки Au в центрі і на границі зони Бріллюена.
Ключові слова: динаміка гратки, м’яка мода, структурний фазовий
перехід, фононний спектр
PACS: 63.20.-e, 63.20.Dj
108
|