Parametric crystal optics of nonmagnetic ferroics
Proceeding from the symmetry principles of crystal physics and thermodynamics we analyze parametrical optical phenomena induced by external fields of different kind and structural phase transitions in ferroelectrics and ferroelastics. Special attention is paid to the phenomena of spatial dispe...
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irk-123456789-1189362017-06-02T03:02:27Z Parametric crystal optics of nonmagnetic ferroics Vlokh, O.G. Proceeding from the symmetry principles of crystal physics and thermodynamics we analyze parametrical optical phenomena induced by external fields of different kind and structural phase transitions in ferroelectrics and ferroelastics. Special attention is paid to the phenomena of spatial dispersion (electro- and piezogyration). A phenomenological approach to the description of these phenomena is illustrated by the most expressive experimental results obtained for the following crystals: KH₂₍₁₋x₎ D₂x PO₄ (KDP, DKDP), K₂H₂AsO₄ (CDA), RbH₂PO₄ (RDP), Pb₅(Ge₍₁₋x₎ Six)₃ O₁₁, Pb₃ (PO₄ )₂ , K₂ Cd₂ (SO₄ )₃ , (NH₃ CH₂ COOH)₃H₂ SO₄ (TGS), and NaKC₄H₄O₆ 4H₂O (RS). Apart from new effects in the crystals of the A₂BX₄ group with incommensurately modulated structure, in particular, crystals [N(CH₃)₄]₂ ZnCl₄ , [N(CH₃)₄ ]₂ FeCl₄ , K₂ ZnCl₄ , K₂ ZnCl₄ , Co²⁺ are considered. Виходячи з симетрійних принципів кристалофізики і термодинаміки, проведено аналіз параметричних оптичних явищ, індукованих зовнішніми полями різного типу, структурними фазовими переходами в сегнетоелектриках і сегнетоеластиках. Особлива увага звертається на явища просторової дисперсії (електро- і п’єзогірація).Феномено-логічний підхід до опису цих явищ ілюструється найбільш виразними експериментальними результатами, одержаними для кристалів:KH₂₍₁₋x₎ D₂x PO₄ (KDP, DKDP), K₂H₂AsO₄ (CDA), RbH₂PO₄ (RDP), Pb₅(Ge₍₁₋x₎ Six)₃ O₁₁, Pb₃ (PO₄ )₂ , K₂ Cd₂ (SO₄ )₃ , а також для (NH₃ CH₂ COOH)₃H₂ SO₄ (TGS), NaKC₄H₄O₆ 4H₂O (RS). Окремо розглядаються нові ефекти в кристалах групи A₂Bx₄ неспівмірно модульованоюструктурою, зокрема у кристалах [N(CH₃)₄]₂ ZnCl₄ , [N(CH₃)₄ ]₂ FeCl₄ , K₂ ZnCl₄ , K₂ ZnCl₄ , Co²⁺. 1998 Article Parametric crystal optics of nonmagnetic ferroics / O.G. Vlokh // Condensed Matter Physics. — 1998. — Т. 1, № 2(14). — С. 339-356. — Бібліогр.: 50 назв. — англ. 1607-324X DOI:10.5488/CMP.1.2.339 PACS: 42.30.Lr, 42.70.a http://dspace.nbuv.gov.ua/handle/123456789/118936 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
Proceeding from the symmetry principles of crystal physics and thermodynamics
we analyze parametrical optical phenomena induced by external
fields of different kind and structural phase transitions in ferroelectrics and
ferroelastics. Special attention is paid to the phenomena of spatial dispersion
(electro- and piezogyration). A phenomenological approach to the description
of these phenomena is illustrated by the most expressive experimental
results obtained for the following crystals: KH₂₍₁₋x₎ D₂x PO₄ (KDP,
DKDP), K₂H₂AsO₄ (CDA), RbH₂PO₄ (RDP), Pb₅(Ge₍₁₋x₎ Six)₃ O₁₁,
Pb₃ (PO₄ )₂ , K₂ Cd₂ (SO₄ )₃ , (NH₃ CH₂ COOH)₃H₂ SO₄ (TGS), and
NaKC₄H₄O₆ 4H₂O (RS). Apart from new effects in the crystals of the
A₂BX₄ group with incommensurately modulated structure, in particular,
crystals [N(CH₃)₄]₂ ZnCl₄ , [N(CH₃)₄ ]₂ FeCl₄ , K₂ ZnCl₄ , K₂ ZnCl₄ ,
Co²⁺ are considered. |
| format |
Article |
| author |
Vlokh, O.G. |
| spellingShingle |
Vlokh, O.G. Parametric crystal optics of nonmagnetic ferroics Condensed Matter Physics |
| author_facet |
Vlokh, O.G. |
| author_sort |
Vlokh, O.G. |
| title |
Parametric crystal optics of nonmagnetic ferroics |
| title_short |
Parametric crystal optics of nonmagnetic ferroics |
| title_full |
Parametric crystal optics of nonmagnetic ferroics |
| title_fullStr |
Parametric crystal optics of nonmagnetic ferroics |
| title_full_unstemmed |
Parametric crystal optics of nonmagnetic ferroics |
| title_sort |
parametric crystal optics of nonmagnetic ferroics |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
1998 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118936 |
| citation_txt |
Parametric crystal optics of nonmagnetic ferroics / O.G. Vlokh // Condensed Matter Physics. — 1998. — Т. 1, № 2(14). — С. 339-356. — Бібліогр.: 50 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT vlokhog parametriccrystalopticsofnonmagneticferroics |
| first_indexed |
2025-07-08T14:56:07Z |
| last_indexed |
2025-07-08T14:56:07Z |
| _version_ |
1837091068782313472 |
| fulltext |
Condensed Matter Physics, 1998, Vol. 1, No 2(14), p. 339–356
Parametric crystal optics of
nonmagnetic ferroics
O.G.Vlokh
Institute for Physical Optics
23 Drahomanov St., UA–290005 Lviv, Ukraine
Received May 6, 1998
Proceeding from the symmetry principles of crystal physics and thermody-
namics we analyze parametrical optical phenomena induced by external
fields of different kind and structural phase transitions in ferroelectrics and
ferroelastics. Special attention is paid to the phenomena of spatial disper-
sion (electro- and piezogyration). A phenomenological approach to the de-
scription of these phenomena is illustrated by the most expressive experi-
mental results obtained for the following crystals: KH 2(1−x) D 2x PO 4 (KDP,
DKDP), K 2 H 2 AsO 4 (CDA), RbH 2 PO 4 (RDP), Pb 5 (Ge (1−x) Si x) 3 O 11,
Pb 3 (PO 4 ) 2 , K 2 Cd 2 (SO 4 ) 3 , (NH 3 CH 2 COOH) 3 H 2 SO 4 (TGS), and
NaKC 4 H 4 O 6 4H 2 O (RS). Apart from new effects in the crystals of the
A 2 BX 4 group with incommensurately modulated structure, in particular,
crystals [N(CH 3 ) 4 ] 2 ZnCl 4 , [N(CH 3 ) 4 ] 2 FeCl 4 , K 2 ZnCl 4 , K 2 ZnCl 4 ,
Co 2+ are considered.
Key words: crystal optics, ferroelectrics, optical activity, ferroics, phase
transitions
PACS: 42.30.Lr, 42.70.a
1. General remarks
1.1. Parametric crystal optics as a particular case of nonlinear optics
Parametric crystal optics is a particular case of nonlinear optics of high in-
tensity light beams under the conditions that one of the interacting waves is of a
low intensity, and frequencies of the others tend to zero. Nonlinear and parametric
optics have a similar phenomenological description and symmetric aspects.
Such a conclusion follows from the expression that describes time averaged free
energy of the unit volume [1,2]:
F = k1EE∗ + k2E0E
∗
0k3AafA
∗
af + . . .
+χ1EEE∗ + χ2EE0E
∗ + χ3EH0E
∗ + . . .
c© O.G.Vlokh 339
O.G.Vlokh
+Θ1EEEE∗ +Θ2EE0E0E
∗ + . . .
The formulas determining polarization P, magnetization M and mechanical
stress X can be deduced differentiating the above expression with respect to cor-
responding variables (E∗, H∗, A∗). The terms with the third χ and the forth Θ
rank tensors correspond to nonlinear optics. For example, the expression with χ1
describes the phenomena of wave generation with differential (subtractive) and
summable frequencies. In particular, the phenomena of optical detection and sec-
ond harmonic generation are described by the formula:
P ω±ω = χω±ω
1 EωEω
The expression with χ2 corresponds to the linear electrooptic effect.
1.2. Some tensor relations, symmetric aspects
It is known [3, 4] that optical properties of crystals are described by the material
equations taking into account a spatial dispersion [2]:
Ei = a′ijDj = (aij + iγijlkl)Dj,
where γijl = eijkgkl is an antisymmetric polar tensor, gkl is an axial gyration tensor
dual to the above one, eijk is a Levi-Civita tensor, kl is a wave vector, aij is a tensor
of the optical polarization constants without consideration of spatial dispersion (aij
is a reciprocal tensor with respect to εij tensor).
The parametric optical effects are determined by expanding aij or εij tensors
and γijl or gkl tensors into power series with respect to different fields, namely:
1. aij = a0ij + rijkEk +RijklEkEl+... - electrooptics [5, 6]; (1)
2. gij = g0ij + γijkEk + βijklEkEl+... - electrogyration [2, 7-10]; (2)
3. aij = a0ij + ρijklσkl+... - elastooptics [11]; (3)
4. gij = g0ij + ξijklσkl+... - elastogyration [2]; (4)
5. aij = a0ij + qijklpσklEp - elasto-electrooptics;
6. gij = g0ij +QijklpσklEp - elasto-electrogyration.
In inhomogeneous fields, gradient parametric optical effects are possible [12],
for example, a torsional-gyration effect [13]:
∆gij = ηijklm
∂σkl
∂xm
.
The restrictions as to the form of polar (aij, rijk, Rijkl, ρijkl, qijklp) and axial
(g0ij, γijk, βijkl, ξijkl, Qijklp) tensors are due to the point symmetry of crystals:
rijk = g0ij = βijkl = ξijkl = qijklp = 0 - in the crystals having inversion symmetry;
aij , Rijkl, γijk, ρijkl, Qijklp 6= 0 - in acentrically and centrally symmetric crystals.
The possibility for appearing the mentioned parametric optical effects agrees
with the following principles of the point symmetry:
1. The Curie principle – a superposition of the elements of the symmetry of
crystals and suitable fields.
340
Parametric crystal optics of nonmagnetic ferroics
2. The Neumann principle – obedience of the symmetry of the crystal to the
symmetry of the property.
The effects taking place in the magnetic field are additionally restricted not
only by the symmetry of the crystal but also by that of the kinetic coefficients in
non-dissipative media [14]:
1. The Onsager principle:
aij(H) = aji(−H).
2. The principle of Hermitian tensor:
a′ij(H) = a′ji(H) - for the real part of the tensor,
a′′ij(H) = −a′′ji(H) - for the imaginary part of the tensor.
Because of these features the following magneto-optic effects are possible in
the non-dissipative nonmagnetic crystals:
1. Magneto-optic change of optical birefringence (Kotton-Moutton):
∆a′ij = FijklHkHl. (5)
2. Magneto-optic activity with the symmetry ∞/m (Faraday), arises sponta-
neously in crystals with the magnetic ordering:
∆a′′ij = fijkHk ≡ ieijlϕijHk. (6)
3. Irreversible addition to the phase velocity of light [15]:
∆a′ij = δijklHkKl. (7)
4. Crossing effects of an optical activity of Faraday-type with the symmetry
∞/m [16]:
∆a′ij = ieijk(δlkmHkEm + ηlkmnHkEmEn). (8)
5. Magnetogyration effect as an analog of the electrogyration effect with the
symmetry of the axial tensor ∞2 is possible [2] only in dissipative crystals or the
crystals with magnetic ordering:
∆gij = δijkHk. (9)
1.3. Waves in gyrotropic crystals
In considering parametric optical effects in different directions of anisotropic
crystals, the superposition of an ordinary (or frequency) and a spatial dispersion
has to be taken into account. As a result, two waves with an orthogonal elliptic
polarization propagate in the crystal. The ellipticity of the waves is [2]:
k =
2g33
(a22 − a11)±
√
(a22 − a11)2 + 4g233n
2
,
where a11, a22, g33 are the tensor components in the physically caused laboratory
coordinate system.
341
O.G.Vlokh
Figure 1. Dependences of polarization
ellipse azimuth on phase differences at
different parameters k [17]: 1 - k=1; 2 -
k=0.6; 3 - k=0.414; 4 - k=0.4; 5 - k=0.2.
The polarization state of a wave
emerging off a plane-parallel plate un-
der the condition of a linearly polarized
incoming wave in one of the principal
planes of the crystal is determined by
the azimuth of the polarization ellipse
[2,17]:
tan 2χ =
2k(1 + k2) sinΓ
(1− k2)2 + 4k2 cos Γ
,
where Γ = 2π
λ
∆nd is a phase retarda-
tion of interfering waves, d is the sam-
ple thickness, ∆n = n2 − n1 is a linear
birefringence.
In figure 1 the dependences of angle
χ on Γ under different values of k are
shown.
Parametric effects through the
changes of aij and gij cause the chan-
ges of Γ and k, and, hence, χ chan-
ges. Their separation is a rather com-
plicated problem. Some results given below are a bright illustration of the current
progress in the field of theoretical foundations and experimental possibilities of
high-accurate polarimetry equipped with PC and lasers.
2. Parametric effects in proper ferroelectrics and ferroelastics
2.1. Electro- and piezooptics
Taking into account the facts that at phase transitions the parametric effects
arise spontaneously and their character is described by the behaviour of the cor-
responding order parameters, it is convenient to rewrite formulas (1)-(3) in terms
of polarization P and mechanical stress X:
∆aij = r∗ijkPk + R∗
ijklPkPl, ∆aij = πijklXkl, (10)
where r∗ijk = rijk/kij, R
∗
ijkl = Rijkl/k
2
ij, πijkl = ρijkl/Cklrs, kij and Cklrs are tensors of
dielectric susceptibility and elasticity, respectively. Thus, temperature dependences
of r and ρ for the induced effects in the region of phase transitions are determined
by k and C anomalies, i.e. these dependences are governed by the Curie-Weiss law
and the law of “two”. R∗, r∗ and π coefficients at free strain and constant electric
induction remain practically temperature independent. This is illustrated on the
examples of KDP (42m → mm2) and Pb3PO4)2 (3m → 2m) crystals (figures 2,
3). That is why the behaviour of parametric effects reflects the character and the
type of the transition as well as the influence of isotopic substitution H → D.
342
Parametric crystal optics of nonmagnetic ferroics
Figure 2. The temperature dependences
of the electrooptic coefficients r63 (1,2,3,4)
and r∗63 (1′) of KH2(1−x)D2xPO4 crystals
[18]. x: 1, 1′ - 0; 2 - 0.50; 3 - 0.82; 4 - 0.93.
Figure 3. The temperature dependences
of the elastooptic ρ′ (1) and piezooptic π′
(2) coefficients of Pb3(PO4)2 ferroelastics
[19]. ρ′ = n3
0(ρ11−ρ12), π′ = n3
0(π11−π12).
Spontaneous effects are determined by the symmetry of the initial paraelectric
phase. For example, in a TGS crystal (transition 2/m → 2) (figure 4) and in
Pb5Ge3O11 (6 → 3) (figure 5) the spontaneous electrooptic effect is quadratic,
though in the latter the paraphase is acentric (r∗ij3 = 0).
The spontaneous piezooptic effect is illustrated (figure 6) on the example of
K2Cd2(SO4)3 crystals (transition 23 → 222), where the first-order phase transition
is accompanied by a hysteresis of refractive indices. The behaviour of the refractive
indices in the ferroelastic phase is similar to the behaviour of the spontaneous stress
(strain). With repeating transitions, a mutual replacement of crystallographic axes
can take place in the ferroelastic phase, since the spontaneous stress can occur
along any of the principal directions with equal probability. Thus, the refractive
index change is described by either π12 or −π13 component.
2.2. Electrogyration
In analogy with electrooptic effects, relation (2) can be written in the form:
∆gij = γ∗
ijkPk + β∗
ijklPkPl, where γ∗
ijk = γijk/kij, β∗
ijkl = βijkl/k
2
ij.
The description of electrogyration under ferroelectric phase transitions is also
given on the basis of the paraphase symmetry. The consideration concerns only
the crystals having optical activity in the ferroelectric phase (pyroelectric classes
1, 2, 3, 4, 6, m, mm2) and the crystals having the so-called “weak optical activity”
with the symmetry ∞ mm (planar classes 3m, 4mm, 6mm).
343
O.G.Vlokh
Figure 4. Dependences of the birefrin-
gence of TGS crystals on temperature for
λ=550 nm [20]: 1 - nx−ny, 2 - nz−ny, 3 -
nx−nz. On the insert: dependences of the
spontaneous birefringence of TGS crystals
on the square meaning of spontaneous po-
larization: 1,2,3 - in the z, x and y axes
directions, respectively.
Figure 5. The temperature dependences
of the main refraction indices (1 - ne,
2 - n0) and birefringence (3) in the
Pb5Ge3O11 ferroics [21]. On the insert –
the dependences of the increasing refrac-
tion indices ∆ne (1′), ∆n0 (2′) and bire-
fringence (3′) on the square meaning of the
spontaneous polarization.
Figure 6. Refraction indices tempera-
ture dependences in K2Cd2(SO4)3 crys-
tals for λ=546 nm [22]. 1 - nz, 2 - ny,
3 - nx; a and b correspond to the inter-
changing nz and ny indices.
From this standpoint all crystals are
divided into two groups [2,23]:
1. Gyroelectrics: these are ferro-
electrics with a centrosymmetric para-
electric phase (they are optically in-
active) and the crystals of inversion
classes 6, 4, where g33 = 0, but
γ33 6= 0. Only linear spontaneous elec-
trogyration can occur in these crystals,
the opposite domains are enantiomor-
phous, the repolarization is accompa-
ined by the electrogyration hysteresis.
The temperature dependence of spon-
taneous electrogyration is similar to the
behaviour of Ps:
GS ∼ PS ∼ (Tc − T )1/2.
344
Parametric crystal optics of nonmagnetic ferroics
The coefficient of linearly induced electrogyration obeys the Curie-Weiss law,
γ ∼ (Tc − T )−1, and the law of “two”.
2. Hypergyroelectrics: these are acentric crystals (they are optically active) in
the paraelectric phase. The transitions are possible from 222, 32, 422, 622, 432, 23,
42m classes to 2, 3, 4, 6, mm2 classes. Spontaneous electrogyration is quadratic:
Gs ∼ P 2
s ∼ (Tc − T ).
Figure 7. Dependence of the light
polarization plane specific rotation
in TGS crystals on temperature (1)
and spontaneous polarization (2)
[24].
The opposite domains are non-enantio-
morphous; the optical activity does not chan-
ge under repolarization. The induced electro-
gyration is linear, γ ∼ (Tc − T )1/2 in the fer-
roelectric phase and quadratic in the parae-
lectric phase.
The properties inherent to gyroelectrics
were first revealed [24] in TGS
NH3(CH2COOH)3H2SO4 crystals (transition
2/m → 2). Figures 7, 8, 9 confirm the con-
clusions of item 1.
Lead germanate crystals Pb5Ge3O11
(transition 6 → 3) and solid solutions on
their basis give an excellent example of gyro-
electrics. At arising spontaneous polarization
and an external field application along z-axis
the electrogyration effect is not accompanied
by the electrooptic effect. Spontaneous elec-
trogyration (figure 10) has a linear character:
Figure 8. The dependences of induced
electrogyration coefficient γ12(o) and its
reciprocal γ−1
12 (•) value (T−Tc) for (100)
direction of TGS [25].
ρ =
π
λn0
γ∗
33Ps ∼ (Tc − T )1/2.
The repolarization is followed by an
electrogyration loop (figure 11) that,
unlike a quadratic electrooptic loop,
is linear. The induced electrogyration
(figure 12) is described as follows:
∆ρ
E
∼
1
4
(Tc − T )−1 at T < Tc,
∆ρ
E
∼
1
2
(Tc − T )−1 at T > Tc.
The properties of hypergyroelect-
rics can be featured on the ex-
ample of the Rochelle salt (RS)
(NaKC4H4O64H2O) (transition 222 → 2) or Ca2Sr(C2H5CO2)6 (transition 422 →
4) crystals. As it is shown in figure 13, the phase transition in the
345
O.G.Vlokh
Figure 9. The temperature dependences
of gyration component g11(o) and g33(•)
moduli of a single TGS. Insert - g11 vs
Ps(∆) and g33 vs Ps(△) relations in FE
phase [26].
Figure 10. Dependences of the polariza-
tion plane specific rotation on tempera-
ture (a), on (Tc − T )1/2 value (b) and
on spontaneous polarization (c) in crys-
tals Pb5(Ge1−xSix)3O11 [9,10,27]. x: 1 - 0;
2, 1’ - 0,03; 3, 2’ - 0,05; 4, 3’, 1” - 0,10;
5, 4’, 2” - 0,20; 6 - 0,40.
Figure 11. Hystereses of optocal activ-
ity (a) and birefringence increase (b) at
T = 200C in crystals Pb5(Ge1−xSix)3O11
[10,25]. x: 1, 1’ - 0; 2, 2’ - 0,40.
Figure 12. Temperature dependences
of the induced electrogyration ∆ρ/E
and value (∆ρ/E)−1 in crystals
Pb5(Ge1−xSix)3O11 [10,25]. x: 1, 1’ -
0; 2, 2 ’ - 0,10; 3, 3’ - 0,20.
Figure 13. Temperature dependences of
gyration gij tensor components in RS crys-
tals. λ=633 nm, 1 - g11, 2 - g22, 3 - g33.
On the insert - the dependence of increas-
ing ∆g11 on the square meaning of spon-
taneous polarization P 2
s [28,21,3,4].
346
Parametric crystal optics of nonmagnetic ferroics
RS crystal is accompanied by kinks in the g11-component temperature dependence,
in this case:
∆g11 = β∗
11P
2
x .
The induced effect is quadratic in the paraelectric phase and linear in the
ferroelectric phase (figure 14). Besides, rotation of the gyration surface is pro-
portional to Ps. Electrogyration properties under isotopic substitution H → D
change essentially; this is completely consistent with a pseudoproper character of
the ferroelectric transition (figure 15).
Figure 14. The dependence of the increas-
ing ∆g11 of RS crystals on the square
meaning of electric field strength E2 in
prototype phases [28]. 1 – T=300 K, 2 –
T=250 K. On the inserts: a – the temper-
ature dependences of square electrogyra-
tion effect coefficient β11 in a prototype
phase, λ=633 nm; b – the dependence of
gyration component g13 on electric field
E2 in a ferroelectric phase. T=305 K,
λ=633 nm.
Figure 15. Temperature dependences of
gyration gij tensor components in DRS
crystals [23]. λ=633 nm, 1 - g11, 2 - g22, 3
- g33.
2.3. Piezogyration
Piezogyration properties as well as piezooptical ones are of interest as to phase
transitions first of all for ferroelastic crystals. In this case expression (4) is written
in the following form:
∆gij = τijklXkl, τijkl = ξijrs/Cklrs.
Spontaneous piezogyration has the features of spontaneous quadratic electrogy-
ration: the initial paraelectric phase has to be acentrical; these crystals are hyper-
gyroelastics [30]. However, under deformation in crystals belonging to nonactive
347
O.G.Vlokh
Figure 16. The dependence of the opti-
cal activity increase δρ and birefringence
δ(∆n) on spontaneous deformations X1,
X3 in quarz crystals [31].
inverse classes and planar classes with
weak activity, the rise of optical activ-
ity is possible; the above classes are
gyroelastic ones. The following ferroe-
lastic transitions correspond to such
classes: 43m → 42m, 43m → 222,
4mm → mm2, 4mm → 2, 6mm →
mm2, 43m → 3m. The latter transition
is accompanied by the rise of weak op-
tical activity (piezogyration) with the
symmetry ∞mm. Under other tran-
sitions, an enantiomorphous domain
structure develops and switching is fol-
lowed by a complicated hysteresis.
An example of the hypergyro-
electrics is K2Cd2(SO4)3 crystals (tran-
sition 23 → 222) and SiO2 (quartz)
crystals (transition 622 → 32) [31]. Fig-
ure 16 presents some of the above re-
ported conclusions if a wide interval of
the incommensurate phase in quartz is
ignored.
In ferroelectrics-ferroelastics of axial classes the invariant relations exist be-
tween the coefficients describing electro-, piezooptics and electro-, piezogyration
effects in a paraphase and under spontaneous effects [32]: γijk/rijk = ξijkl/ρijkl
under the condition that aij, gij , χm, Ek are transformed by the same irreducible
representation.
2.4. Magneto-optics
The analysis of magneto-optic effects under ferroelectric phase transitions is
made on the example of Pb5Ge3O11 crystals (figure 17).
Figure 17. Temperature dependences of
total magnetooptical activity in the poly-
domain (1) and monodomain states (2, 2′ -
according to the opposite signs of sponta-
neous polarization, excluding spontaneous
electrogyration) of Pb5Ge3O11:Nd
3+ crys-
tals and their approximation (3, 4 – linear
and square effects regarding to Ps, respec-
tively) across spontaneous electrogyration
ρs ∼ Ps [9]. H=13.3kE, λ=514.5nm.
348
Parametric crystal optics of nonmagnetic ferroics
As it is seen from the figures (multidomain Pb5Ge3O11 samples), a parabolic de-
pendence of the magneto-optic rotation power is observed on the background of the
Faraday effect. This dependence is explained by the second term in formula (8), i.e.
the term corresponding to quadratic in P magnetopolarization activity. However,
in single domain samples, a sign of optical activity changes under repolarization,
i.e., it is bilinear with respect to P and H. The observed effect cannot be treated
as a spontaneous magneto-polarization activity (the first term in (8)), since in the
initial phase δ333 = 0. Moreover, the induced effect proportional to EH is absent
in the paraelectric phase; it arises only in the ferroelectric phase, where δ333 6= 0.
Hence, it is possible to assume [34] that in single domain crystals Pb5Ge3O11 there
exist two effects: a Faraday-type magneto-polarization activity induced by electric
and magnetic fields and a magneto-gyration effect under the condition of dissi-
pation and accompanied by a spontaneous polarization: ∆gij = γijklPkHl - in the
paraphase δ3333 6= 0.
3. Pseudoproper ferroelectrics and crystals with the incom-
mensurate structure
3.1. Electrogyration as a tool to study pseudoproper ferroelectrics
Let now proceed to clarify this thesis on the example of quadratic electrogyra-
tion in the KDP-type crystals. It is common knowledge that the phase transition
in these crystals is induced by the proton ordering on hydrogen bonds. While such
an ordering is not an immediate cause of the spontaneous polarization it, neverthe-
less, is defined as the order parameter being of the same transformation properties
as Ps.
The pseudoproper features of these crystals are apparent from the behaviour of
the tensor component g11 under the phase transition (figure 18) and β13 coefficient
of quadratic electrogyration in the paraphase (figure 19).
In particular, the temperature anomalies of β13 coefficient in KDP, DKDP and
CDA crystals are accompained by a change of the sign; they seem not to be referred
to the anomalies of k33. Coefficient β∗
13 = β13/k
2
33 depends on the temperature as
well. The explanation of the above peculiarities has come about via introducing the
following terms, besides polarization Ps, in the expression for the thermodynamical
potential: θ3 is the order parameter; Xi is mechanical stress; Tθ is the transition
temperature of the proton system under the condition that it does not interact
with the lattice.
Then one has [35]:
A = A0 +
1
2
β(T − T0)θ
2
3 +
1
2
γθ44 +
1
6
δθ63 +
1
2
ωP 2
3 −
−
1
2
SijXiXj −
1
2
b3iXiP3 −Q3iXiP
2
3 −
1
2
h3iXiθ3 −
−R3iXiθ
2
3 −W3iXiP3θ3 + fP3q3.
349
O.G.Vlokh
From where
∆g11 = β∗Q
13 P
2
3 + β∗R
13 θ
2
3 + β∗W13P3θ3 ∼= (β∗Q
13 χ
2
P + β∗R
13 χ
2
θ + β∗W
13 χPχθ)E
2
3 (11)
hence,
β∗
13 = β∗Q
13 − β∗W
13 (−f/β)(T − T ′
0)(T − Tθ)
−2,
where
T ′
0 = Tθ − (β∗R
13 /β
∗W
13 )(−f/β).
The approximation of temperature dependence of β∗
13 coefficient by this for-
mula agrees well with the experimental data (figure 19) for KDP, DKDP and CDA
crystals. The pseudoproper character of the ferroelectric transition is conditioned
by the coupling strength between the order parameter and the spontaneous po-
larization P . The value of ∆T = Tc − Tθ = f 2/βω approximately equals ∆T -value
derived from the proton-lattice interaction, where ∆T = F 2/kΩ2 (F is the interac-
tion constant, k is the Boltzmann constant, Ω0 is the frequency of the proton-lattice
interaction). The substitution (H → D) results in a strong shift of Tc and Tθ, but
does not affect the ∆T=55 K value, i.e. the strength of the proton-lattice interac-
tion. This interaction is essentially dependent on the anion substitution of PO3−
4
by AsO3−
4 . Then ∆T=65 K. The substitution of cations K+ by Rb+ brings about
such a pronounced increase of the proton-lattice strength that the electrogyration
effect in KDP crystals does not differ from the respective properties of proper
ferroelectrics.
Figure 18. Temperature dependences of
the gyration g11 tensor component in crys-
tals KDP (1), DKDP (2), CDA (3) and
RDP (4) [2, 36].
Figure 19. Temperature dependences of
the quadratic electrogyration coefficient
β13 (1, 2, 3) and β∗
13 (1’, 2’) of crystals
DKDP (1, 1’), CDA (2, 2’) and RDP (3)
[2, 36].
3.2. Parametric optical effects in crystals with the incommensurate struc-
ture
The effects of parametric crystal optics are especially sensitive to the incom-
mensurate structures. In particular, the character of the optical birefringence
350
Parametric crystal optics of nonmagnetic ferroics
changes clearly demonstrates the global hysteresis and its jump-like behaviour
along with the particular “parallelogram” - type cycles, non-smooth relaxation
processes (the kinetics), thermo-optical memory, etc. [37]. These effects are ex-
plained (without consideration of the zone-model) by adding the gradient terms,
a two-component gradient Lifshitz invariant among others, to the thermodynamic
potential. The invariant is as follows [38]:
ξ
∂η
∂x
− η
∂ξ
∂x
,
where η = ρ sinϕ , ξ = ρ cosϕ , ρ and ϕ are an amplitude and a phase of the order
parameter, respectively. Changing Ti to Tc requires transition from the plane-
wave model to the model of phase solitons and the model of commensurate and
incommensurate phase coexistence.
On this basis and taking into account the fact that the linear effects caused
by spontaneous polarization and spontaneous deformation are absent in the mod-
ulated incommensurate phase, the temperature changes of the birefringence are
described by the relationship [39, 40]: δ(∆n) ∼= αρ2+νρ2dϕ/dz - (α, ν - constants)
or in the plane-wave model δ(∆n) ∼= αρ2 ∼= (Ti − T )2β ∼= I, with a critical index β
and intensity of the incommensurate X-ray reflex I.
However, the above approach is not aimed to explain the nontrivial effects
in the incommensurate phase. An essential role is played by the interaction of
the defects (impurities) and phase solitons. This interaction depends on the ratio
between the mobilities of the defects Vd and solitons Vs [41, 42]. So, the three
boundary cases can arise:
1. Vs ≫ Vd – the attachement (pinning) of the incommensurate structure to
the defects (impurities);
2. Vs ≪ Vd – the attachement (pinning) of the defects (impurities) to the
incommensurate structures coupled with the creation of a “soliton density wave”.
3. Vs
∼= Vd – a “viscous” interaction of the incommensurate structure with the
defects (impurities).
Let us illustrate the conditions for these three cases realization on the example
of the optical birefringence study in the A2BX4 group crystals containing both
incommensurate ferroelectrics (phase transition mmm → 2mm) and ferroelastics
(mmm → 2/m).
1. An example of the global hysteresis in the [N(CH3)4]2ZnCl4 crystal, the “pa-
ralelogram”-type cycle in the K2ZnCl4:Co
2+ crystal and the optical birefringence
relaxation in [N(CH3)4]2FeCl4 are presented in figures 20, 21.
These effects manifest themselves under the condition of Vs ≫ Vd, i.e., the
condition of the incommensurate structure pinning on the defects or the constant
soliton density at the transition between different temperature regimes.
During the relaxation process because of the soliton nucleation, the crystal
passes through some intermediate metastable states that differ in the soliton den-
sity. The states are separated by a free-energy barrier. The gently sloping parts
of the birefringence kinetics dependence are related to the fixed soliton density.
351
O.G.Vlokh
As to the peaks, they are coupled with the transitions from one soliton density
to another (a sharp change takes place in the orderparameter phase, the wave
vector k).
Figure 20. Temperature dependen-
ces of the birefringence δ(∆n)b in
[N(CH3)4]2ZnCl4:1(•) - dT/dt=1.1 K/h;
2(o) - 0.11 K/h. Insert - partial cycles
of thermal hysteresis of the birefringence
δ(∆n)a in K2ZnCl4:Co
2+ [37].
Figure 21. Temperature dependence
δ(∆n)c of [N(CH3)4]2FeCl4 crystals. The
velocity of approaching to Tst=270,36 K
in the regime of heating is dT/dt=5,7 K/h
[37].
Figure 22. Temperature dependence of
the birefringence δ(∆n)a in K2ZnCl4
crystals kept for t=26h at Tst=510,5 K
[37,43]. Inserts: a - the anomaly form at
Tst−∆T ; b - the anomaly shift on global
hysteresis branches.
2. The case of Vs ≪ Vd is depicted
in figure 22. It is realized in the ther-
mooptic memory effect. The effect con-
sists in ordering the defects in the mod-
ulated structure field at the stabilized
temperature. Retracing this tempera-
ture point along the same branch with-
out going into the paraelectric phase
is followed by the appearance of an
anomaly. The anomaly on the global
hysteresis is shifted by the hysteresis
width along the temperature scale. The
anomaly is also observed at the temper-
ature at which the modulation period is
doubled.
The temperature and temporal
changes of the optical birefringence in
the incommensurate phase depend on
amplitude ρ and phase ϕ of the order parameter. The smooth dependence gov-
erned by ρ is superimposed on the anomaly part coupled with the ϕ behaviour.
352
Parametric crystal optics of nonmagnetic ferroics
Figure 23. Temperature dependence
of δ(∆n)c for [N(CH3)4]2FeCl4 crys-
tals at the temperature variation rate
dT/dt=1500 mK/h (2); 60 mK/h (3)
[37].
3. The case of Vs
∼= Vd “viscous” in-
teraction. The velocity of the solitons
(or DC) movement is of the same order
as the velocity of the defects (impuri-
ties) diffusion. Then, the temperature
curve of the birefringence under the
condition of decreasing velocity of the
temperature variation in the course of
experiment is greater, if followed by the
appearance of the steps (tooths) (figure
23). The rate of new DC nucleation and
their mobility decrease because of the
increase of the friction force. The dif-
ference of δ(∆n) values at two neigh-
bouring minimum points is a charac-
teristic of the increase in the order pa-
rameter amplitude ρ between the two
metastable states for k-localization.
Enhancing the effects can be
achieved with X-ray irradiation or the
influence of hydrostatic pressure, the
application of an external electric field,
mechanical stress or their gradients. At
the same time the above factors can be
accompanied by vanishing the peaks (phase factor) and appearing the steps on
the δ(∆n) temperature dependences. The steps correspond to the high-order com-
mensurations between which smooth transitions take place.
4. Summary
From the brief review of the effects of parametric crystallooptics in the phe-
nomenological approach which is based on the investigations conducted by the
author and his fellow-workers it may seem that the main problem is solved. How-
ever, it is not so, and not all the possible effects based on the phenomenological
approach have been determined. This, first of all, concerns magnetic crystals and
cross and gradient effects. The microscopic theory of these effects has been insuf-
ficiently worked out, especially their determination on the structural level. The
exclusions are papers [44-47], which are devoted to ferroics with the hydrogenium
(deuterium) bond, particularly the KDP group crystals and electrogyration in the
PbMoO4 and Pb5Ge3O11 type crystals [48-50].
From this approach the prognosis in searching for highly effective crystals,
practically useful in various elements of quantum and optoelectronics is left open.
Besides, on the examples of crystals with a non-corresponding structure the phys-
ical reality in parametric crystallooptics is broader than it appears on the basis of
353
O.G.Vlokh
the symmetrical aspects and phenomenological approach. In this respect a thor-
ough research in this field is far from being completed.
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Параметрична кристалооптика немагнітних
фероїків
О.Г.Влох
Інститут фізичної оптики,
290005 м. Львів-5, вул. Драгоманова, 23
Отримано 6 травня 1998 р.
Виходячи з симетрійних принципів кристалофізики і термодинаміки,
проведено аналіз параметричних оптичних явищ, індукованих зов-
нішніми полями різного типу, структурними фазовими переходами в
сегнетоелектриках і сегнетоеластиках. Особлива увага звертається
на явища просторової дисперсії (електро- і п’єзогірація). Феномено-
логічний підхід до опису цих явищ ілюструється найбільш виразни-
ми експериментальними результатами, одержаними для кристалів:
KH 2(1−x) D 2x PO 4 (KDP, DKDP), K 2 H 2 AsO 4 (CDA), RbH 2 PO 4 (RDP),
Pb 5 (Ge (1−x) Si x ) 3 O 11 , Pb 3 (PO 4 ) 2 , K 2 Cd 2 (SO 4 ) 3 , а також для
(NH 3 CH 2 COOH) 3 H 2 SO 4 (TGS), NaKC 4 H 4 O 6 4H 2 O (RS). Окремо
розглядаються нові ефекти в кристалах групи A 2 BX 4 з неспівмірно
модульованою структурою, зокрема у кристалах [N(CH 3 ) 4 ] 2 ZnCl 4 ,
[N(CH 3 ) 4 ] 2 FeCl 4 , K 2 ZnCl 4 , K 2 ZnCl 4 Co 2+ .
Ключові слова: кристалооптика, сегнетоелектрики, оптична
активність, фероїки, фазові переходи
PACS: 42.30.Lr, 42.70.a
356
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