Longitudinal relaxation of mechanically free KH₂PO₄ type crystals. Piezoelectric resonance and sound attenuation
Within the framework of proton model with taking into account the piezoelectric interaction with the shear strain ε₆, a dynamic dielectric response of KD₂PO₄ type ferroelectrics is considered. Experimentally observed phenomena of crystal clamping by high frequency electric eld, piezoelectric reso...
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| Cite this: | Longitudinal relaxation of mechanically free KH₂PO₄ type crystals. Piezoelectric resonance and sound attenuation / R.R. Levitskii, I.R. Zachek, A.P. Moina, A.S. Vdovych // Condensed Matter Physics. — 2008. — Т. 11, № 3(55). — С. 555-570. — Бібліогр.: 44 назв. — англ. |
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Levitskii, R.R. Zachek, I.R. Moina, A.P. Vdovych, A.S. 2017-06-06T17:51:56Z 2017-06-06T17:51:56Z 2008 Longitudinal relaxation of mechanically free KH₂PO₄ type crystals. Piezoelectric resonance and sound attenuation / R.R. Levitskii, I.R. Zachek, A.P. Moina, A.S. Vdovych // Condensed Matter Physics. — 2008. — Т. 11, № 3(55). — С. 555-570. — Бібліогр.: 44 назв. — англ. 1607-324X PACS: 77.22.Ch, 77.22.Gm, 77.65.-j, 77.84.Fa, 77.65.Fs DOI:10.5488/CMP.11.3.555 https://nasplib.isofts.kiev.ua/handle/123456789/119382 Within the framework of proton model with taking into account the piezoelectric interaction with the shear strain ε₆, a dynamic dielectric response of KD₂PO₄ type ferroelectrics is considered. Experimentally observed phenomena of crystal clamping by high frequency electric eld, piezoelectric resonance and microwave dispersion are described. Ultrasound velocity and attenuation are calculated, peculiarities of their temperature dependence at the Curie points are described. Existence of a cut-off frequency in the frequency dependence of attenuation is predicted. В рамках протонної моделi з врахуванням п’єзоелектричної взаємодiї зi зсувною деформацiєю ε₆ розглянуто динамiчний дiелектричний вiдгук сегнетоелектрикiв типу KD₂PO₄. Враховано динамiку п’єзоелектричної деформацiї. Явно описано явища затискання кристалу високочастотним електричним полем, п’єзоелектричного резонансу i НВЧ дисперсiї, що спостерiгаються на експериментi. Розраховано коефiцiєнт поглинання звуку. Описано особливостi коефiцiєнта поглинання в околi точок переходу. Передбачено наявнiсть обрiзаючої частоти у частотнiй залежностi коефiцiєнта поглинання звуку. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Longitudinal relaxation of mechanically free KH₂PO₄ type crystals. Piezoelectric resonance and sound attenuation Поздовжня релаксацiя механiчно вiльних кристалiв типу KH₂PO₄. П’єзоелектричний резонанс та поглинання звуку Article published earlier |
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| title |
Longitudinal relaxation of mechanically free KH₂PO₄ type crystals. Piezoelectric resonance and sound attenuation |
| spellingShingle |
Longitudinal relaxation of mechanically free KH₂PO₄ type crystals. Piezoelectric resonance and sound attenuation Levitskii, R.R. Zachek, I.R. Moina, A.P. Vdovych, A.S. |
| title_short |
Longitudinal relaxation of mechanically free KH₂PO₄ type crystals. Piezoelectric resonance and sound attenuation |
| title_full |
Longitudinal relaxation of mechanically free KH₂PO₄ type crystals. Piezoelectric resonance and sound attenuation |
| title_fullStr |
Longitudinal relaxation of mechanically free KH₂PO₄ type crystals. Piezoelectric resonance and sound attenuation |
| title_full_unstemmed |
Longitudinal relaxation of mechanically free KH₂PO₄ type crystals. Piezoelectric resonance and sound attenuation |
| title_sort |
longitudinal relaxation of mechanically free kh₂po₄ type crystals. piezoelectric resonance and sound attenuation |
| author |
Levitskii, R.R. Zachek, I.R. Moina, A.P. Vdovych, A.S. |
| author_facet |
Levitskii, R.R. Zachek, I.R. Moina, A.P. Vdovych, A.S. |
| publishDate |
2008 |
| language |
English |
| container_title |
Condensed Matter Physics |
| publisher |
Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Поздовжня релаксацiя механiчно вiльних кристалiв типу KH₂PO₄. П’єзоелектричний резонанс та поглинання звуку |
| description |
Within the framework of proton model with taking into account the piezoelectric interaction with the shear
strain ε₆, a dynamic dielectric response of KD₂PO₄ type ferroelectrics is considered. Experimentally observed
phenomena of crystal clamping by high frequency electric eld, piezoelectric resonance and microwave dispersion
are described. Ultrasound velocity and attenuation are calculated, peculiarities of their temperature
dependence at the Curie points are described. Existence of a cut-off frequency in the frequency dependence
of attenuation is predicted.
В рамках протонної моделi з врахуванням п’єзоелектричної взаємодiї зi зсувною деформацiєю ε₆ розглянуто динамiчний дiелектричний вiдгук сегнетоелектрикiв типу KD₂PO₄. Враховано динамiку п’єзоелектричної деформацiї. Явно описано явища затискання кристалу високочастотним електричним полем, п’єзоелектричного резонансу i НВЧ дисперсiї, що спостерiгаються на експериментi. Розраховано коефiцiєнт поглинання звуку. Описано особливостi коефiцiєнта поглинання в околi точок переходу. Передбачено наявнiсть обрiзаючої частоти у частотнiй залежностi коефiцiєнта поглинання звуку.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/119382 |
| citation_txt |
Longitudinal relaxation of mechanically free KH₂PO₄ type crystals. Piezoelectric resonance and sound attenuation / R.R. Levitskii, I.R. Zachek, A.P. Moina, A.S. Vdovych // Condensed Matter Physics. — 2008. — Т. 11, № 3(55). — С. 555-570. — Бібліогр.: 44 назв. — англ. |
| work_keys_str_mv |
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| first_indexed |
2025-11-26T00:10:45Z |
| last_indexed |
2025-11-26T00:10:45Z |
| _version_ |
1850595825567137792 |
| fulltext |
Condensed Matter Physics 2008, Vol. 11, No 3(55), pp. 555–570
Longitudinal relaxation of mechanically free KH2PO4
type crystals. Piezoelectric resonance and sound
attenuation
R.R.Levitskii1, I.R.Zachek2, A.P.Moina1, A.S.Vdovych1
1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii Str.,
79011 Lviv, Ukraine
2 Lviv Polytechnic National University, 12 Bandera Str., 79013 Lviv, Ukraine
Received June 5, 2008, in final form July 9, 2008
Within the framework of proton model with taking into account the piezoelectric interaction with the shear
strain ε6, a dynamic dielectric response of KD2PO4 type ferroelectrics is considered. Experimentally observed
phenomena of crystal clamping by high frequency electric field, piezoelectric resonance and microwave dis-
persion are described. Ultrasound velocity and attenuation are calculated, peculiarities of their temperature
dependence at the Curie points are described. Existence of a cut-off frequency in the frequency dependence
of attenuation is predicted.
Key words: ferroelectrics, piezoelectric resonance
PACS: 77.22.Ch, 77.22.Gm, 77.65.-j, 77.84.Fa, 77.65.Fs
1. Introduction
Ferroelectric compounds of the KH2PO4 family have been studied for nearly 70 years. Extensive
experimental data have been accumulated; several models of the phase transitions in these crystals
have been proposed and explored (see [1–14]). At the end of 1960-s, attention of theoretical and
experimental studies of the ferroelectric compounds has shifted to the problem of the dynamic
phenomena. Investigation of the low-frequency dispersion of their dielectric permittivity provides
an important information about mechanisms of the phase transitions or dielectric response of the
crystals. Despite the success in constructing a microscopic theory of the KH2PO4 family crystals
mostly phenomenological models were used in interpreting the experimental data as regards the
dynamic characteristics. Such models provide no information about the microscopic nature of the
dielectric dispersion and do not allow one to adequately describe the effect of different factors
on its temperature and frequency dependencies. This problem has not been solved by the Green
functions method or Bloch kinetic equation method [8,9,11,12] either.
Most theoretical works on the relaxation phenomena in KH2PO4 family ferroelectrics is based
on stochastic Glauber model [16]. For the first time this approach was used to describe relaxational
phenomena in KD2PO4 in [17], where the main features of longitudinal relaxation were explored
within the four-particle cluster approximation in the paraelectric phase in KD2PO4. In that work
the long-range interactions between deuterons was not taken into account, and the theoretical re-
sults were not compared with experiment. Later [18–22] a more sophisticated model of the KD2PO4
type ferroelectrics and ND4D2PO4 type antiferroelectrics was proposed, in which longitudinal dy-
namic properties of these crystals were calculated within the four-particle cluster approximation
for short-range interactions and mean field approximation for long-range interactions. It was shown
[12,15,23–25] that the proposed [18–22] theory permits a satisfactory description of longitudinal
relaxation in the KH2PO4 type ferroelectrics. An attempt to develop a more consistent theory of
the KH2PO4 family ferroelectrics within the four-particle cluster approximation with taking into
account the tunneling (Ω) was made in [26–28]. The model was not sophisticated enough to com-
c© R.R.Levitskii, I.R.Zachek, A.P.Moina, A.S.Vdovych 555
R.R.Levitskii et al.
pare its results with experimental data for the dynamic characteristics of the crystals. At the same
time, in these works the fact of suppression of the dynamic characteristics of KH2PO4 type crystals
by short-range correlations was established for the first time. Instead of the tunneling frequency, an
effective parameter Ω̃ (Ω̃ � Ω) renormalized by short-range interactions was obtained. It should
be noted that the established [26–28] suppression explains, most likely, the Debye character of
dispersion of the dielectric permittivity observed in these crystals.
It should be stressed that the ferroelectrics of the KH2PO4 family are also piezoelectric. The
piezoelectric coupling is revealed particularly when the crystals are subjected to external electric
fields and mechanical stresses of certain symmetry. At the ferroelectric phase transition in the
KH2PO4 type crystals, a spontaneous strain arises which changes their tetragonal symmetry.
Studies of the piezoelectric coupling effect on the phase transition and on physical properties
of the KH2PO4 type ferroelectrics were initiated in [29], where the Slater model [30] was modified
by taking into account the splitting of the lowest ferroelectric level by the strain ε6.
Fundamental are the results obtained in [31,32], where the proton ordering model was modified
for the first time by taking into account the deformational molecular field related to the strain ε6
as well as the splitting of “lateral” deuteron configurations. In [33,34], with taking into account all
possible splittings of configurational energies (“upper/lower”, “lateral”, and single-ionized config-
urations) due to the strain ε6, the phase transition and the effect of the stress σ6 [33] and field E3
[34] on the physical characteristics of K(H0,12D0,88)2PO4 crystals were explored. Tunneling was
not taken into account. A satisfactory agreement of theoretical results and experimental data was
obtained.
Physical characteristics of the KH2PO4 family crystals with taking into account piezoelectric
coupling and tunneling are explored theoretically in [35].
So far the model consideration of longitudinal dielectric characteristics of the KD2PO4 type fer-
roelectrics has been restricted to the static limit and to the microwave region [15,18–25]. Attempts
to explore the piezoelectric resonance phenomenon within a model that does not take into account
the piezoelectric coupling are pointless. Conventional proton ordering model of the KD2PO4 type
ferroelectrics does not permit one to describe the effects related to the difference between the free
and clamped regimes or the phenomenon of crystal clamping by high-frequency electric field.
It seems natural to calculate, within the proposed in [33] proton ordering model with piezoelec-
tric coupling, the dynamic characteristics of the KD2PO4 type ferroelectrics in a wide frequency
range from 103 Hz to 1012 Hz, including the piezoelectric resonance region. In [36] within the
framework of the four-particle cluster approximation for the proton ordering model, the thermo-
dynamic and longitudinal dielectric, piezoelectric, and elastic characteristics of the KD2PO4 have
been calculated. It was shown that at the proper choice of the model parameters a good quantitative
description of available experimental data by the proposed theory is obtained.
In this work, following the approach developed in [37], we shall calculate the longitudinal
dynamic dielectric characteristics of the KD2PO4 type ferroelectrics and study their temperature
and frequency dependences. We shall investigate the effect of crystal clamping by longitudinal
high-frequency external electric field. Expressions for the sound velocity and attenuation for a
certain propagation direction will be obtained; their temperature and frequency dependences will
be explored.
2. System of equations for time-dependent distribution functions
of deuterons
We shall consider a system of deuterons moving on O–D. . .O bonds in deuterated ferroelectric of
the KD2PO4 type. A primitive Bravais cell of these crystals consists of two PO4 groups (tetrahedra)
with four hydrogen bonds attached to one of them (“A” type tetrahedra), while the hydrogen
bond attached to the other tetrahedron (“B” type) belongs to the four nearest structure elements
surrounding it (figure 1).
The Hamiltonian of the deuteron subsystem with taking into account short-range and long-
range interactions in the presence of mechanical stress σ6 = σxy and electric field E3 along the
556
Longitudinal relaxation of mechanically free KH2PO4 type crystals
1
2
1
1
1
2
2
2
2
3
1
4
a
b
γ
Figure 1. Primitive Bravais cell of the KD2PO4 type crystal. One of the possible ferroelectric
configurations is shown.
crystallographic axis c, which induce contributions to polarization P3 and strain ε6 of the crystal,
consists of the “seed” and pseudospin parts [33,35]:
Ĥ = NH(0) + Ĥs , (2.1)
where N is the number of primitive cells. The “seed” energy of the primitive cell corresponds to the
heavy ion subsystem and does not depend explicitly on hydrogen arrangement. It is expressed via
the strain ε6 and electric field E3 and includes the elastic, piezoelectric, and electric counterparts
H(0) = v̄
(
1
2
cE0
66 ε
2
6 − e036E3ε6 −
1
2
χε0
33E
2
3
)
, (2.2)
where v̄ = v
kB
, v is the primitive cell volume; cE0
66 , e036, χ
ε0
33 are the so-called “seed” elastic constant,
coefficient of the piezoelectric stress, and dielectric susceptibility, respectively. These quantities
determine the temperature behavior of the corresponding observable characteristics far from the
transition point Tc.
The pseudospin part of the Hamiltonian is
Ĥ = N
[
H0 + 2νc(η
(1))2
]
+
∑
q
Ĥ
(4)
q6 ,
where
Ĥ
(4)
q6 = −
4
∑
f=1
z6
β
σqf
2
+
ε6
4
(−δs6 + 2δ16)
4
∑
f=1
σqf
2
− ε6(δs6+2δ16)
(σq1
2
σq2
2
σq3
2
+
σq1
2
σq2
2
σq4
2
+
σq1
2
σq3
2
σq4
2
+
σq2
2
σq3
2
σq4
2
)
+ (V +δa6ε6)
(σq1
2
σq2
2
+
σq3
2
σq4
2
)
+(V −δa6ε6)
(σq2
2
σq3
2
+
σq4
2
σq1
2
)
+ U
(σq1
2
σq3
2
+
σq2
2
σq4
2
)
+ Φ
σq1
2
σq2
2
σq3
2
σq4
2
, (2.3)
and
z6 = β(−∆c + 2νcη
(1) − 2ψ6ε6 + µ3E3),
(
β =
1
kBT
)
.
The dynamic properties of the KD2PO4 type ferroelectrics will be explored within the pro-
posed dynamic model [21], based on the stochastic Glauber model [16] ideas. When calculating the
557
R.R.Levitskii et al.
dynamic characteristics, we shall restrict our consideration to the four-particle cluster approxima-
tion, providing a successful description of thermodynamic characteristics of the crystals [8–10,15].
Using the approach, developed in [18–21], we obtain the following system of equations for the
time-dependent deuteron distribution functions:
−α d
dt
〈
∏
f
σqf
〉
=
∑
f ′
〈
∏
f
σqf
[
1 − σqf ′ tanh
1
2
βε
z
qf ′(t)
]
〉
, (2.4)
where ε
z
qf ′(t) is the local field acting on the f ′th deuteron in the qth cell, which can be found from
the Hamiltonian (2.3).
The expressions tanh
βεz
qf′
(t)
2 are presented in the following form
tanh
βε
z
q1
2
= P z
6 σq3 +Qz
61σq2 +Qz
62σq4 +Rz
6σq2σq3σq4
+Mz
61σq2σq3 +Mz
62σq3σq4 +Nz
6σq2σq4 + Lz
6 ,
tanh
βε
z
q2
2
= P z
6 σq4 +Qz
61σq1 +Qz
62σq3 +Rz
6σq1σq3σq4
+Mz
61σq4σq2 +Mz
62σq3σq4 +Nz
6σq1σq3 + Lz
6 ,
tanh
βε
z
q3
2
= P z
6 σq1 +Qz
61σq4 +Qz
62σq2 +Rz
6σq1σq2σq4
+Mz
61σq1σq4 +Mz
62σq1σq2 +Nz
6σq2σq4 + Lz
6 ,
tanh
βε
z
q4
2
= P z
6 σq2 +Qz
61σq3 +Qz
62σq1 +Rz
6σq1σq2σq3
+Mz
61σq2σq3 +Mz
62σq1σq2 +Nz
6σq1σq2 + Lz
6 . (2.5)
Since σqf = ±1, we find for the coefficients P6, . . . , L6:
P z
6 =
1
8
(lz1 − lz2 + nz
1 − nz
2 +mz
1 −mz
2 +mz
3 −mz
4),
Qz
61 =
1
8
(lz1 − lz2 − nz
1 + nz
2 +mz
1 +mz
2 −mz
3 −mz
4),
Qz
62 =
1
8
(lz1 − lz2 − nz
1 + nz
2 −mz
1 −mz
2 +mz
3 +mz
4),
Rz
6 =
1
8
(lz1 − lz2 + nz
1 − nz
2 −mz
1 +mz
2 −mz
3 +mz
4),
Nz
6 =
1
8
(lz1 + lz2 + nz
1 + nz
2 −mz
1 −mz
2 −mz
3 −mz
4),
Mz
61 =
1
8
(lz1 + lz2 − nz
1 − nz
2 +mz
1 −mz
2 −mz
3 +mz
4),
Mz
62 =
1
8
(lz1 + lz2 − nz
1 − nz
2 −mz
1 +mz
2 +mz
3 −mz
4),
Lz
6 =
1
8
(lz1 + lz2 + nz
1 + nz
2 +mz
1 +mz
2 +mz
3 +mz
4), (2.6)
where we use the following notations
lz1
2
= tanh
β
2
[
±w + (δs6 + δ16)ε6 +
zz
6
β
]
, nz
1
2
= tanh
β
2
[
±(w − w1) − δ16ε6 +
zz
6
β
]
,
mz
1
4
= tanh
β
2
[
±(ε− δa6ε6 − w) − δ16ε6 +
zz
6
β
]
, mz
2
3
= tanh
β
2
[
∓(ε+ δa6ε6 − w) − δ16ε6 +
zz
6
β
]
.
Taking into account the symmetry of distribution functions
η(1)z = 〈σq1〉 = 〈σq2〉 = 〈σq3〉 = 〈σq4〉,
η(3)z = 〈σq1σq2σq3〉 = 〈σq1σq3σq4〉 = 〈σq1σq2σq4〉 = 〈σq2σq3σq4〉,
η
(2)z
1 = 〈σq2σq3〉=〈σq1σq4〉, η
(2)z
2 =〈σq1σq2〉=〈σq3σq4〉, η
(2)z
3 =〈σq1σq3〉=〈σq2σq4〉, (2.7)
558
Longitudinal relaxation of mechanically free KH2PO4 type crystals
from (2.4) with taking into account (2.5) and (2.7), we can obtain a closed system of equations for
the time-dependent single-particle, pair, and three-particle deuteron distribution functions in the
KD2PO4 type crystals
α
d
dt
η(1)z
η(3)z
η
(2)z
1
η
(2)z
2
η
(2)z
3
=
c̄11 c̄12 c̄13 c̄14 c̄15
c̄21 c̄22 c̄23 c̄24 c̄25
c̄31 c̄32 c̄33 c̄34 c̄35
c̄41 c̄42 c̄43 c̄44 c̄45
c̄51 c̄52 c̄53 c̄54 c̄55
η(1)z
η(3)z
η
(2)z
1
η
(2)z
2
η
(2)z
3
+
c̄1
c̄2
c̄3
c̄4
c̄5
. (2.8)
Here we use the following notations
c̄11 = −(1 − P z
6 −Qz
61 −Qz
62), c̄12 = Rz
6, c̄13 = Mz
61, c̄14 = Mz
62, c̄15 = Nz
6 , c̄1 = Lz
6 ,
c̄21 = (2P z
6 + 2Qz
61 + 2Qz
62 + 3R6), c̄22 = −(3 − P z
6 −Qz
61 −Qz
62), c̄23 = (Nz
6 +Mz
62 + Lz
6),
c̄24 = (Nz
6 +Mz
61 + Lz
6), c̄25 = (Mz
61 +Mz
62 + Lz
6), c̄2 = (Nz
6 +Mz
61 +Mz
62),
c̄31 = 2(Nz
6 +Mz
62 + Lz
6), c̄32 = 2Mz
61, c̄33 = −2(1 −Rz
6), c̄34 = 2P z
6 , c̄35 = 2Qz
61, c̄3 = 2Qz
62 ,
c̄41 = 2(Nz
6 +Mz
61 + Lz
6), c̄42 = 2Mz
62, c̄43 = 2P z
6 , c̄44 = −2(1 −Rz
6), c̄45 = 2Qz
62, c̄4 = 2Qz
61 ,
c̄51 = 2(Mz
61 +Mz
62 + Lz
6), c̄52 = 2Nz
6 , c̄53 = 2Qz
61, c̄54 = 2Qz
62, c̄55 = −2(1 −Rz
6), c̄5 = 2P z
6 .
(2.9)
In the single-particle approximation from (2.4) we obtain the following equation for distribution
functions
α
d
dt
η(1)z = −η(1)z + tanh
1
2
z̄z
6 . (2.10)
3. Dynamic characteristics of a mechanically free K(H1−xDx)2PO4 crystal.
Piezoelectric resonance
In this section we shall consider vibrations of a thin l× l square plate of a KD2PO4 crystal, cut
in the [001] plane, induced by time-dependent electric field E3t = E3e
iωt. This field, in addition
to the shear strain ε6, also induces diagonal components of the strain tensor εi. For the sake of
simplicity, we shall neglect the diagonal strains.
Dynamics of deformational processes in KD2PO4 will be described using classical Newtonian
equations of motion of an elementary volume
ρ
∂2ui
∂t2
=
∑
k
∂σik
∂xk
, (3.1)
where ρ is the cell volume, ui are the displacements of an elementary volume along the axis xi,
σik is the mechanical stress. The shear strain ε6 is determined by the displacements ux = u1 and
uy = uz, that is
ε6 = εxy =
∂u1
∂y
+
∂u2
∂x
.
In our case a shear strain σxy = σ6 is different from zero and [36]
σ6 =cE0
66 ε6−e036E3+
4ψ6
v
mz
Dz
6
+
2δa6
v̄Dz
6
Ma6−
2δs6
vDz
6
Ms6+
2δ16
vDz
6
M16 , (3.2)
where
Ma6 = aa6 −
a
a6
, Ms6 = sinh(2zz
6 + βδs6ε6), M16 = 4b sinh(zz
6 − βδ16ε6).
559
R.R.Levitskii et al.
Thus,
ρ
∂2u1
∂t2
= cE0
66
∂ε6
∂y
+
4ψ6
v
∂η(1)
∂y
+
2δa6
v
∂
∂y
(
Ma6
D6
)
− 2δs6
v
∂
∂y
(
Ms6
D6
)
+
2δ16
v
∂
∂y
(
M16
D6
)
, (3.3)
ρ
∂2u2
∂t2
= cE0
66
∂ε6
∂x
+
4ψ6
v
∂η(1)
∂x
+
2δa6
v
∂
∂x
(
Ma6
D6
)
− 2δs6
v
∂
∂x
(
Ms6
D6
)
+
2δ16
v
∂
∂x
(
M16
D6
)
.
At small deviations from the equilibrium we can separate in the systems (2.6) and (2.8) the
static and time-dependent parts, presenting the distribution functions, effective displacement fields
u1, u2, and the strain ε6 as sums of equilibrium values and of fluctuational deviations
η(1) = η̃(1) + η
(1)
t , η(3) = η̃(3) + η
(3)
t , η
(2)
i = η̃
(2)
i + η
(2)
t , (i = 1, 2, 3),
ε6 = ε̃6 + ε6t, u1,2 = ũ1,2 + u1,2t, (3.4)
zz
6 = z̃6 + z6t = −β∆̃ + 2βνcη̃
(1) − 2βψ6ε6 − β∆t + 2βνcη
(1)
t − 2βψ6ε6t + βµ3E3t .
We substitute the expressions (3.4) into the system of equations (2.8), (2.10), expand into a
series over the time dependent terms the coefficients c̄ij , c̄i, limit oneself to linear approximation.
We exclude parameter ∆t and obtain the system of equations for fluctuating parts of distribution
functions:
d
dt
η
(1)
t
η
(3)
t
η
(2)
1t
η
(2)
2t
η
(2)
3t
=
c11 c12 c13 c14 c15
c21 c22 c23 c24 c25
c31 c32 c33 c34 c35
c41 c42 c43 c44 c45
c51 c52 c53 c54 c55
η
(1)
t
η
(3)
t
η
(2)
1t
η
(2)
2t
η
(2)
3t
− 1
2
βµ3E3t
c1
c2
c3
c4
c5
+βψ6ε6t
c1
c2
c3
c4
c5
− βδs6ε6t
c1s
c3s
c21s
c22s
c23s
+ βδa6ε6t
c1a
c3a
c21a
c22a
c23a
− βδ16ε6t
c̃11
c̃21
c̃31
c̃41
c̃51
. (3.5)
The expressions for coefficients of the system (3.5) are given in [36].
Taking into account (3.4), equations for the displacements (3.3) can be rewritten as follows:
ρ
∂2u1t
∂t2
= c16
∂ε6t
∂y
+ c26
∂η
(1)
t
∂y
, ρ
∂2u2t
∂t2
= c16
∂ε6t
∂x
+ c26
∂η
(1)
t
∂x
, (3.6)
where we use the following notations
c16 = cE0
66 +
4βψ6
vD6
f6 −
2β
vD6
[
δ2s6 cosh(2z̃6 + βδs6ε̃6) + δ2164b cosh(z̃6 − βδ16ε̃6)
+δ2a62a coshβδa6ε̃6
]
+
2β
vD2
6
(−δs6Ms6 + δ16M16 + δa6Ma6)
2,
c26 =
4
v
(
ψ6 −
ϕ
η
6
D6
f6
)
,
We shall look for solutions of the system (3.5) and (3.6) in the form of harmonic waves
η
(1)
t = η(1)(6, x, y)eiωt, η
(3)
t = η(3)(6, x, y)eiωt,
η
(2)
1t = η
(2)
1 (6, x, y)eiωt, η
(2)
2t = η
(2)
2 (6, x, y)eiωt,
η
(2)
3t = η
(2)
3 (6, x, y)eiωt, ε6t = ε6(x, y)e
iωt,
u1t = u1(y)e
iωt, u2t = u2(x)e
iωt. (3.7)
560
Longitudinal relaxation of mechanically free KH2PO4 type crystals
Finally, solving the system (3.5) with taking into account (3.7), we find that
η(1)(6, x, y) =
βµ3
2
F (1)(αω)E3 +
[
−βψ6F
(1)(αω)
+βδs6F
(1)
s (αω) − βδa6F
(1)
a (αω) + βδ16F
(1)
1 (αω)
]
ε6(x, y), (3.8)
where we use the notations
F (1)(αω) =
p(4)(iαω)4 + p(3)(iαω)3 + p(2)(iαω)2 + p(1)(iαω) + p(0)
(iαω)5 + p4(iαω)4 + p3(iαω)3 + p2(iαω)2 + p1(iαω) + p0
,
F (1)
s (αω) =
p
(4)
s (iαω)4 + p
(3)
s (iαω)3 + p
(2)
s (iαω)2 + p
(1)
s (iαω) + p
(0)
s
(iαω)5 + p4(iαω)4 + p3(iαω)3 + p2(iαω)2 + p1(iαω) + p0
,
F (1)
a (αω) =
p
(4)
a (iαω)4 + p
(3)
a (iαω)3 + p
(2)
a (iαω)2 + p
(1)
a (iαω) + p
(0)
a
(iαω)5 + p4(iαω)4 + p3(iαω)3 + p2(iαω)2 + p1(iαω) + p0
,
F
(1)
1 (αω) =
p
(4)
1 (iαω)4 + p
(3)
1 (iαω)3 + p
(2)
1 (iαω)2 + p
(1)
1 (iαω) + p
(0)
1
(iαω)5 + p4(iαω)4 + p3(iαω)3 + p2(iαω)2 + p1(iαω) + p0
. (3.9)
Expressions for p4, . . . , p0, p
(4), . . . , p(0), p
(4)
s , . . . , p
(0)
s , p
(4)
a , . . . , p
(0)
a , p
(4)
1 , . . . , p
(0)
1 in (3.9) are
given in [36].
Taking into account the relations (3.6) and (3.7), we obtain the following wave equations for
u1, u2:
cE66(αω)
∂2u1
∂y2
+ ρω2u1 = 0, cE66(αω)
∂2u2
∂x2
+ ρω2u2 = 0, (3.10)
where
cE66(αω) = cE0
66 +
4βψ6
vD6
f6 +
2β
vD2
6
(−δs6Ms6 + δ16M16 + δa6Ma6)
2
+
4βψ6
v
[
−ψ6F
(1)(αω) + δs6F
(1)
s (αω) + δ16F
(1)
1 (αω) − δa6F
(1)
a (αω)
]
− 4ϕ3f6
vD6
β
[
−ψ6F
(1)(αω) + δs6F
(1)
s (αω) + δ16F
(1)
1 (αω) − δa6F
(1)
a (αω)
]
− 2β
vD6
[
δ2s6 cosh(2z̃ + βδs6ε̃6) + 4bδ216 cosh(z̃ − βδ16ε̃6) + δ2a62a coshβδa6ε̃
2
6
]
.
Equation (3.10) can be written as
∂2u1
∂y2
+ k2
6u1 = 0,
∂2u2
∂x2
+ k2
6u2 = 0, (3.11)
where k6 is the wavenumber
k6 =
ω
√
ρ
√
cE66(αω)
. (3.12)
We shall look for the solutions of (3.11) in the form
u1 = A1 cos k6y +B1 sin k6y, u2 = A2 cos k6x+B2 sin k6x.
As a result
ε6(x, y) = k6
[
−(A1 cos k6y +A2 cos k6x) + (B1 sin k6y +B2 sin k6x)
]
. (3.13)
The boundary conditions are set as follows:
ε6(0, 0) = ε6(l, l) = ε6(0, l) = ε6(l, 0) = ε0. (3.14)
561
R.R.Levitskii et al.
The values of ε0 are determined from (3.2), using the relation (3.10)
ε0 =
e36(αω)
cE66(αω)
E3 . (3.15)
where
e36(αω) = e036 +
βµ3
v
[
−ψ6F
(1)(αω) + δs6F
(1)
s (αω) + δ16F
(1)
1 (αω) − δa6F
(1)
a (αω)
]
. (3.16)
Taking into account the boundary conditions (3.14), we find from (3.13)
ε6(x, y) =
ε0
2
[
−cosk6l − 1
sin k6l
(sin k6y + sin k6x) + (cos k6y + cos k6x)
]
. (3.17)
Using the expression, relating polarization P3 to the order parameter η(1) and strain ε6, as well
as relation (3.8), we find that
P3(x, y, t) = P3(x, y)e
iωt, (3.18)
where
P3(x, y) = e36(αω)ε6(x, y) + χε
33(αω)E3 ,
and
χ33(αω) = χε0
33 +
βµ2
3
2v
F (1)(αω).
Now we can calculate the dynamic dielectric susceptibility of a free crystal χσ
33(αω)
χσ
33(αω) =
1
l2
∂
∂E3
l
∫
0
l
∫
0
P3(x, y)dxdy. (3.19)
Taking into account (3.17), we find that
1
l2
l
∫
0
l
∫
0
dxdyε6(x, y) =
2ε0
k6
tan
k6l
2
=
ε0
R(ω)
, (3.20)
where
1
R(ω)
=
2
k6l
tan
k6l
2
.
With (3.18) and (3.20) from (3.19) we obtain that
χσ
33(αω) = χε
33(αω) +
1
R(ω)
e236(αω)
cE66(αω)
. (3.21)
4. Attenuation and velocity of ultrasound in KD2PO4 crystals
Pulsed ultrasonics provide a useful method for the investigation of crystal behavior. The ultra-
sound wavelength is usually much smaller than the sample dimensions. Therefore, the dynamical
variables, such as elementary displacements and order parameter, depend only on the spatial co-
ordinate which is the direction of sound propagation.
If thin bars of the crystal are cut along [001], then we shall consider a transverse sound wave
propagating along the bar and polarized along [010]. Among ∂ui
∂xj
the only nonzero derivative is
562
Longitudinal relaxation of mechanically free KH2PO4 type crystals
∂u2
∂x
; therefore, instead of the systems (3.5) and (3.6), we can write
d
dt
η
(1)
t
η
(3)
t
η
(2)
1t
η
(2)
2t
η
(2)
3t
=
c11 c12 c13 c14 c15
c21 c22 c23 c24 c25
c31 c32 c33 c34 c35
c41 c42 c43 c44 c45
c51 c52 c53 c54 c55
η
(1)
t
η
(3)
t
η
(2)
1t
η
(2)
2t
η
(2)
3t
+ βψ6ε6t
c1
c2
c3
c4
c5
−βδs6ε6t
c1s
c3s
c21s
c22s
c23s
+ βδa6ε6t
c1a
c3a
c21a
c22a
c23a
− βδ16ε6t
c̃11
c̃21
c̃31
c̃41
c̃51
,
ρ
∂2u2t
∂t2
= c16
∂ε6t
∂x
+ c26
∂η
(1)
t
∂x
. (4.1)
Solving the system (4.1), we obtain the same wavenumber as found above
k6 =
ω
√
ρ
√
cE66(αω)
. (4.2)
Using (4.2), we can find the sound wave velocity
v66(ω) =
ω
Re|k6|
= Re
√
cE66(αω)√
ρ
(4.3)
and the contribution of the pseudospin subsystem to the sound attenuation
α6(ω) = α60 − Im|k6| = α60 − Im
∣
∣
∣
∣
∣
ω
√
ρ
√
cE66(αω)
∣
∣
∣
∣
∣
, (4.4)
where α60 is a constant frequency and temperature independent contribution of the other mecha-
nisms to the experimentally observable attenuation.
5. Discussion
Let us evaluate the above found dynamic characteristics of mechanically free K(H1−xDx)2PO4
crystals, cut as l × l square plates (l = 1 mm) in the [0,0,1] plane. It should be noted that the
developed theory is valid, strictly speaking, for highly deuterated KD2PO4 crystals, only. However,
the tunneling in undeuterated crystals is weakened due to short-range interactions [26], which is
indicated by the experimentally established relaxational character of ε∗33(ν, T ) dispersion see [15,23–
25] in KH2PO4. Therefore, proton tunneling can be neglected as well as the obtained expressions
used for undeuterated KH2PO4. In numerical calculations we shall use the values of the model
parameters determined in describing the static and dynamic permittivity of a mechanical free
crystal [36] and given in table 1.
Unfortunately, no experimental data are available to perform a quantitative comparison of the
theoretically obtained temperature and frequency dependences of the dynamic characteristics of
mechanically free crystals in the piezoelectric resonance region.
In figures 2 and 3 we plot the frequency dependences of the real and imaginary parts of the
dynamic dielectric permittivity of mechanically free KH2PO4, KD2PO4 crystals in the paraelectric
phase at different temperatures ∆T . For the MH2XO4 crystals, a resonance dispersion takes place
in the frequency range, 3 ·105–3 ·108 Hz, and for KD2PO4 in the 5 ·105–5 ·108 Hz range. At ω → 0
563
R.R.Levitskii et al.
Table 1. The optimal set of the model parameters for the K(H1−xDx)2PO4 crystals.
x Tc T0
ε
kB
w
kB
ν3(0)
kB
µ3−, 10−18 µ3+, 10−18 χ0
33
(K) (K) (K) (K) (K) (esu · cm) (esu · cm)
0.00 122.5 122.5 56.00 422.0 17.91 1.46 1.71 0.73
0.81 205.6 204.8 85.82 781.5 33.44 1.76 2.02 0.42
1.00 220.1 219.0 93.05 868.6 35.76 1.84 2.10 0.34
x ψ6
kB
δs6
kB
δa6
kB
δ16
kB
c0
66 · 10−10 e0
36
(K) (K) (K) (K) (dyn/cm2) (esu/cm2)
0.00 -150.00 82.00 -500.00 -400.00 7.10 1000.00
0.81 -200.00 52.73 -957.39 -400.00 6.45 1914.77
1.00 -138.64 45.64 -1068.18 -400.00 6.30 2136.36
P
−
R
−
P+ R+
(s) ( s
K
) (s) ( s
K
)
0.35 0.0100 0.43 0.0160
1.95 0.0082 4.19 0.0001
2.84 0.0077 4.54 0.0349
we obtain a static dielectric permittivity of a free crystal. Taking into account the dispersion (3.12),
we find the equation for the resonance frequencies
ωn =
π(2n+ 1)
l
√
cE66
ρ
,
where taking into account the fact that in the 5 ·105–5 ·108 Hz frequency range cE66(ω) is practically
frequency independent. The resonance frequencies are inversely proportional to sample dimensions.
106 108 1010 1012
0
200
400
600
800
106 108 1010 1012
10−3
100
103
106 108 1010 1012
0
100
200
300
400
500
600
106 108 1010 1012
10−3
100
106 108 1010 1012
0
20
40
60
80
100
106 108 1010 1012
10−3
100
ε’
33
ε’’
33
∆T, K
ε’
33
ε’’
33
ε’
33
ε’’
33
1
1
2
2
3 3
ν, Hz ν, Hz
ν, Hz ν, Hz
ν, Hz ν, Hz
Figure 2. Frequency dependences of the real and imaginary parts of the dynamic dielectric
permittivity of a mechanically free and clamped KH2PO4 crystal at different ∆T , K: 1 – 5,
[38], [39]; 2 – 10; 3 – 50 [38], [39].
564
Longitudinal relaxation of mechanically free KH2PO4 type crystals
106 108 1010 1012
0
200
400
600
800
106 108 1010 1012
10−3
100
103
106 108 1010 1012
0
100
200
300
400
500
600
106 108 1010 1012
10−3
100
103
106 108 1010 1012
0
20
40
60
80
100
106 108 1010 1012
10−3
100
ε’
33
ε’’
33
∆T, K
ε’
33
ε’’
33
ε’
33
ε’’
33
1
1
2
2
3
3
ν, Hz ν, Hz
ν, Hz ν, Hz
ν, Hz ν, Hz
Figure 3. Frequency dependences of the real and imaginary parts of the dynamic dielectric
permittivity of a mechanically free and clamped KD2PO4 crystal at different ∆T , K: 1 – 5, 2 –
10, 3 – 50. , , [40].
The dashed lines in figures 2–3 describe the low-frequency permittivity of a clamped crystal.
With increasing frequency and temperature ∆T , the amplitudes of the resonance peaks decrease.
With increasing temperature ∆T the last resonance peak shifts to higher frequencies. Analogous
multi-peak resonance dispersion is also observed in the ferroelectric phase. Above the resonance
frequency, the crystal is clamped by the high-frequency field; above 109 Hz the clamped permittivity
has a relaxational dispersion. The theoretical frequency curves ε′33(ω) and ε′′33(ω) well accord with
experimental data. The resonance dispersion in KH2PO4 is schematically presented in [43] for the
ferroelectric phase at 104 − 106 Hz.
In figure 4 we show the temperature curves of the real part ε′33(ω, T ) of the free dielectric
permittivity of KH2PO4 at different frequencies. Below the frequency of the first resonance peak,
the temperature variation of ε∗33(ω, T ) essentially coincides with that of the static permittivity of
a free crystal. Near the resonance frequencies, the sharp peaks in the temperature curve of permit-
tivity appear, the number of which increases with an increase of frequency, whereas the magnitudes
decrease. Upon further increase of frequency, numerous resonance peaks of small amplitude arise
around the curve of clamped permittivity. At even higher frequencies the peaks disappear. The
resonance peaks for the real and imaginary parts of the permittivity for a given frequency are
observed at the same temperature ∆T . The character of piezoelectric resonance curves is different
for different crystals considered here.
In figure 5 we plot the calculated temperature dependences of the sound attenuation α6 for
KH2PO4, K(H0,195D0,805)2PO4 at different frequencies along with the experimental points α6(T )
taken from [41,42]. A good quantitative description of experimental data is obtained, especially
565
R.R.Levitskii et al.
−10 −5 0 5 10 15
−2000
−1000
0
1000
2000
−10 −5 0 5 10 15
10−3
100
103
−10 −5 0 5 10 15
−1000
−500
0
500
1000
1500
2000
−10 −5 0 5 10 15
10−3
100
103
−10 −5 0 5 10 15
0
200
400
600
800
1000
−10 −5 0 5 10 15
10−3
100
103
ε’
33
ε’’
33
∆T, K
∆T, K ∆T, K
ε’
33
∆T, K
ε’’
33
∆T, K
ε’
33
∆T, K
ε’’
33
∆T, K
1 1
2
2
3
3
Figure 4. Temperature dependences of the real and imaginary parts of the dynamic dielectric
permittivity of a mechanically free KH2PO4 crystal at different frequencies ν, MHz: 1 – 3, 2 –
10, 3 – 30.
for K(H0.195D0.805)2PO4.
Near the transition temperature Tc, a sharp increase of attenuation is obtained. In the ferro-
electric phase, the attenuation decreases much faster when moving away from the transition than
in the paraelectric phase.
The dependence of attenuation α6 on the square of frequency ω2 for KH2PO4,
K(H0,195D0,805)2PO4 crystals at different temperatures ∆T is shown in figure 6. As one can see,
attenuation varies proportionally to the square of frequency. The closer the temperature is to the
Curie point, the larger is the rate of this variation.
With increasing frequency, starting from ∼ 107 Hz, the theoretical attenuation sharply increases
and then reaches a saturation at about 1010 Hz (figure 7). In the paraelectric phase, the cut-off
frequency decreases with approaching the transition temperature Tc. The cut-off frequency also
decreases with increasing deuteration x. Such high values of attenuation at saturation, in fact,
mean the absence of sound propagation.
In figure 8 we depicted the calculated temperature dependence of sound velocity v66 for KH2PO4,
K(H0,195D0,805)2PO4.
566
Longitudinal relaxation of mechanically free KH2PO4 type crystals
−5 0 5 10 15 20 25
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
−5 0 5 10 15 20 25
0
1
2
3
4
5
α
6
, cm−1
∆T, K ∆T, K
1
2
3
2
3
4
5
4
1
α
6
, cm−1
a) b)
Figure 5. Temperature dependences of sound attenuation for KH2PO4 (a), at different frequen-
cies ν, 106 Hz: 1 – 10, [41], 2 – 30, [41], 3 – 50, [41], 4 – 70, [41], 5 – 90,+[41] and
K(H0.195D0.805)2PO4 (b) at ν, 106 Hz: 1 – 5, [42], 2 – 15, [42], 3 – 25, [42], 4 – 45, [42].
0 1 2 3 4 5 6 7
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 1 2 3 4 5 6 7
0
1
2
3
4
5
6
7
8
9
10
x1016 ω2, c−2x1016 ω2, c−2
α
6
, cm−1 α
6
, cm−1
1
2
3
4
1
2
3
4
a) b)
Figure 6. Frequency dependences of attenuation for KH2PO4 (a), K(H0.195D0.805)2PO4 (b) at
different temperatures ∆T ,K: 1 – 2, 2 – 5, 3 – 10, 4 – 50. , , – [42].
106 108 1010 1012
10−1
100
101
102
103
104
105
106
106 108 1010 1012
10−2
10−1
100
101
102
103
104
105
ν, Hz
α
6
, cm−1 α
6
, cm−1
ν, Hz
4
3
2
1
1
2
3
4
a) b)
Figure 7. Frequency dependences of attenuation for KH2PO4 (a), K(H0.195D0.805)2PO4 (b) at
different temperatures ∆T ,K: 1 – 2, 2 – 5, 3 – 10, 4 – 50.
567
R.R.Levitskii et al.
−20 0 20 40 60 80 100
0
2
4
6
8
10
12
14
16
18
x 104
−20 0 20 40 60 80 100
0
2
4
6
8
10
12
14
16
18
x 104v
66
, cm/c
∆T, K ∆T, K
v
66
, cm/c
a) b)
Figure 8. Temperature dependences of sound velocity for KH2PO4 (a), K(H0.195D0.805)2PO4
(b).
It has a minimum at T = Tc. Below 1010 Hz, the magnitude of cE66(ω) is frequency independent;
therefore, we calculated the velocity using (8.3) and experimental data for cTE
66 ([44] for KH2PO4
and [42] for K(H0.195D0.805)2PO4) and crystal density ρ. The results are shown in figure 8 by
symbols ◦.
At the frequency of microwave dispersion of permittivity, a sharp increase of sound velocity v66
should be observed, after which the frequency curve of velocity saturates at about ∼ 1011 Hz. The
saturation values of sound velocity are temperature independent.
6. Conclusions
Within the proton ordering model with taking into account the shear strain ε6 we explored a
dynamic response of the KD2PO4 type crystals to an external harmonic electric field E3. Dynamics
of the pseudospin subsystem is described within the stochastic Glauber approach. Dynamics of the
strain ε6 is obtained from the Newtonian equations of motion of an elementary volume, with
taking into account the relations between the order parameter of the pseudospin subsystem and
the strain in the static limit. Expressions for the longitudinal dynamic dielectric permittivity of
KD2PO4 crystals are obtained. Evolution of dynamic permittivity with increasing frequency via
the piezoelectric resonances (105–108 Hz) to the clamped crystal with the microwave (109–1012 Hz)
relaxational dispersion is obtained. Temperature and frequency dependences of sound attenuation
α6 and velocity v66 in K(H1−xDx)2PO4 are obtained. An anomalous increase of attenuation in
the phase transition region and the temperature curve of sound velocity are described. A presence
of cut-off frequency for sound propagation (an anomalous increase of attenuation at frequencies
corresponding to the microwave dispersion of the dielectric permittivity) is predicted.
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Поздовжня релаксацiя механiчно вiльних кристалiв типу
KH2PO4. П’єзоелектричний резонанс та поглинання звуку
Р.Р.Левицький1 , I.Р.Зачек2, А.С.Моїна1, А.С.Вдович1
1 Iнститут фiзики конденсованих систем НАН України, 79011 Львiв, вул. Свєнцiцького 1
2 Нацiональний унiверситет “Львiвська полiтехнiка”, 79013 Львiв, вул. С. Бандери 12
Отримано 5 червня 2008 р., в остаточному виглядi – 9 липня 2008 р.
В рамках протонної моделi з врахуванням п’єзоелектричної взаємодiї зi зсувною деформацiєю ε6
розглянуто динамiчний дiелектричний вiдгук сегнетоелектрикiв типу KD2PO4. Враховано динамiку
п’єзоелектричної деформацiї. Явно описано явища затискання кристалу високочастотним електри-
чним полем, п’єзоелектричного резонансу i НВЧ дисперсiї, що спостерiгаються на експериментi.
Розраховано коефiцiєнт поглинання звуку. Описано особливостi коефiцiєнта поглинання в околi то-
чок переходу. Передбачено наявнiсть обрiзаючої частоти у частотнiй залежностi коефiцiєнта погли-
нання звуку.
Ключовi слова: сегнетоелектрики, п’єзоелектричний резонанс
PACS: 77.22.Ch, 77.22.Gm, 77.65.-j, 77.84.Fa, 77.65.Fs
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