Hypersonic and dielectric studies of disordered single crystals, Rb1-x(ND4)xD2Aso4, Na1/2Bi1/2TiO3
and PbMg1/3Nb2/3O3, by brillouin scattering and dielectric measurements
by Chi-Shun Tu
A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in
Physics
Montana State University
© Copyright by Chi-Shun Tu (1994)
Abstract:
Temperature and frequency dependent measurements of dielectric permittivity, acoustic sound velocity
and damping have been carried out by using Brillouin light scattering and a capacitance and
conductance component analyzer on two different types of ferroelectrics, i.e. (i) FE-AFE mixed
deuteron glasses Rbl-x(ND4)xD2AsO4 (x=0, 0.10, 0.28), (ii) relaxor ferroelectrics Na1/2Bi1/2TiO3
(NBT) and PbMg1/3Nb2/3O3 (PMN).
In this study, three important results have been observed: (1)A broad and high-value maximum in
dielectric permittivity has been observed in relaxors NBT and PMN, which indicates that these
materials could be used for electrostrictive displacement transducers. (2) In FE-AFE mixed glasses
DRADA-x, a η2μ-type quadratic coupling, squared in order parameter and linear in strain, becomes the
main coupling contribution as ammonium ND4 concentration x increases from 0 to an intermediate
value. The results also confirm the presence of PE/FE phase coexistence in DRADA-0.10. (3) For both
FE-AFE mixed glasses and relaxor ferroelectrics, the order parameter(s) fluctuations, which are
generated by the local random fields originating from short-range randomly-placed cations, are the
main dynamic mechanisms for hypersonic and dielectric anomalies. In DRADA-0.10 and 0.28, these
local randomly-placed ions are ND4+ and Rb+ ions. In PMN, those randomly-placed cations are Mg+2
and Nb+5 which are randomly placed at B-site positions. In NBT, those randomly-placed cations are
Na+1 and Bi+3 which are placed randomly at A-site positions.
Two models, i.e. superparaelectric cluster and extrinsic bulk conductivity, also have been proposed to
explain the high-temperature and low-frequency dielectric anomaly in NBT. The elastic stiffness and
compliance constants are also calculated for PMN.
 
HYPERSONIC AND DIELECTRIC STUDIES OF DISORDERED SINGLE 
CRYSTALS, Rb1^ (ND4)xD2AsO4, Na172Bi172TTiO3 AND PbMg173Nb273O3, BY 
BRILLOUIN SCATTERING AND DIELECTRIC MEASUREMENTS
by
Chi-Shun Tu
A thesis submitted in partial fulfillment 
of the requirements for the degree
of
Doctor of Philosophy 
in
Physics
MONTANA STATE UNIVERSITY 
Bozeman, Montana
September, 1994
3 ) 3 1 %
- m
ii
APPROVAL
of a thesis submitted by
Chi-Shun Tu
This thesis has been read by each member of the thesis committee and has been 
found to be satisfactory regarding content, English usage, format, citations, bibliographic 
style, and consistency, and is ready for submission to the college of Graduate Studies.
Date Chairperson, Graduate Committee
Approved for the Major Department
Date Head, Major Department
Approved for the College of Graduate Studies
Date Graduate Dean
iii
STATEMENT OF PERMISSION TO USE
In presenting this thesis in partial fulfillment of the requirements for a doctoral ' 
degree at Montana State University, I agree that the Library shall make it available to 
borrowers under rules of the Library. I further agree that copying of this thesis is 
allowable only for scholarly purpose, consistent with "fair use as prescribed in the U.S. 
Copyright Law". Requests for extensive copying or reproduction of this thesis should be 
referred to University Microfilms International, 300 North Zeeb Road, Ann Arbor, 
Michigan 48106, to whom I have granted "the exclusive right to reproduce and distribute 
my dissertation for sale in and from microform or electronic format, along with the right 
to reproduce and distribute my abstract in any format in whole or in part."
Signature f l fa n
Date 3
iv
to my wife Rong-Mei (Ruth), because o f whom all things are possible, 
and to my son Stachus, in whom I  find the reasons
VVITA
The author was bom Chi-Shun Tu on January 21, 1962 in Kaohsiung, Taiwan. 
He is the fourth child of Rou-Nu Tu (deceased) and Hsiu-Hsia Tsai Tu and has three 
older sisters and one younger brother. He was married to Rong-Mei (Ruth) Chien in 
1988 and has one son, Stachus Igu Tu.
The author finished his undergraduate work with a B.S in physics at National 
Taiwan Normal University in 1985. After that time, he was in the Army for two full 
years to complete his obligation and then taught physics at high school for another two 
years. From Sept, of 1988 to June of 1990, he attended the University of Oregon in 
Eugene, Oregon, and graduated with a M.S. in physics. In September of 1990, he 
enrolled at Montana State University as a graduate teaching and research assistant until 
the fall of 1994, obtaining a Ph.D. in physics under Professor V. Hugo Schmidt.
vi
ACKNOWLEDGMENTS
I would like to sincerely thank my advisor Prof. V. Hugo Schmidt for his 
support, advice, and encouragement over the past four years. Whenever I became 
confused or frustrated by the experiments, he was there with the appropriate advice and 
help. Dr. Igor G. Siny and Greg Pastalan helped me to set up the experimental system 
and taught me Brillouin scattering skills. I give Igor and Greg my appreciation for 
sharing their ideas with me and for so much technical guidance that was given freely.
Thanks to Prof. G.F. Tuthill for endless theoretical teaching and discussion, to 
Bob Parker for his technical support on computer software and hardware issues, to Dr. 
Toby Howell who helped me to learn dielectric measurements, and to Norm Williams and 
Erik Anderson for their technical help and friendship.
My wife Rong-Mei (Ruth) deserves my deepest appreciation for her constant
*
support, encouragement and sacrifices. Stachus my son I rely upon to always bring a 
smile to my face and my heart.
My appreciation also goes to the National Science Foundation for the support I 
received under grant number DMR-9017429.
Finally, I would like to thank my parents, especially my mother. She gave us 
much help during the first six months since Stachus was bom and during writing the 
thesis. They may not understand what I am doing, but they understand me. Their 
encouragement and support played an important role in the completion of this thesis.
vii
TABLE OF CONTENTS
Page
I INTRODUCTION.....................;............ ..........................................................................I
Introduction to Disordered Crystals...............................................................  I
Relaxor Ferroelectrics (RF)..............................................................  3
Proton (or Deuteron) Glasses (PG)............................... ..................  7
Previous Knowledge for Crystals Discussed in This Work........................ . 11
' Na172Bi172TiO3 (NBT).........................................................................  11
PbMg173Nb273O3 (PMN)....................................................................  11
Rb1^ (ND4)xD2AsO4 (DRADA-X=O, 0.10 and 0.28)........... :............13
Purpose and Outline of The Present Work...................................................  13
2. EXPERIMENTAL PROCEDURES..........................................................................  15
Crystal Growth and Sample Preparation....................................................... 15
Dielectric Permittivity....................................................................................  16
Brillouin Light Scattering...............................................................................  17
General Description of Experiment...................................................  18
Characteristics of the Fabry-PeroLlnterferoriieter............................ 25
Pre-Alignment of Fabry-Perot Interferometer...................................  29.
Brillouin Spectrum Analysis................... ..........................................  30
3. THEORY....................................................................................................................  34
Dielectric........................................................................ ...............................  34
Dipolar Relaxation Equations............................................................  38
Superparaelectric Model (Nano-Scale Polar Cluster)!........................40
Equivalent Circuits of Dielectric Function.......,..........
Brillouin Light Scattering...........................................................
Spectrum and Principle of Brillouin Scattering...........
The Relation Between Velocity and Elastic Stiffness........................54
Selection Rules of Scattered Light....................................................  65
Static Coupling Theory of First-Order Phase Transition...................71
Debye Anharmonic Approximation...................................................  74
Dynamic Effects................................................................................ 75
VUl
TABLE OF CONTENTS-CONTINUED
Page
4. RESULTS AND DISCUSSION.................... ........................................................... 76
Rb1^ (ND4)xD2AsO4 (x=0, 0.10 and 0.28)............ ...................................... 76
Brillouin Back-Scattering Along LA[100] Phonon Direction...........76
N ai72Bi172TiO3 (NBT),........................................................ ...... .............. . 87
Frequency Dependent Dielectric Results............... ........ .................  87
Superparaelectric Cluster Model.............................. ............. 88
Surface-Modified Bulk Effect..............................................  94
Extrinsic Bulk Conductivity Model....................................... 95
Brillouin Backscattering Along LA[001] Phonon Direction.............108
PbMg173Nb273O3 (PMN)........................................................... ....... ,........... 116
Dielectric and Brillouin Scattering Results...................................... . 116
Determination of Elastic Stiffness Constants....................................  123
5. CONCLUSIONS......................................................................;..............................> 126
Comparison of Results..................................................................................  126
Similarities Among PMN, NBT and DRADA-0.10 and 0.28............126
Differences Among PMN, NBT and DRADA-0.10 and 0.28...........127
Applications...................................... ...... ...................................................... 128
Recommended Additional Work on This Problem.......................................  130
REFERENCES CITED 132
ix
LIST OF TABLES
Table Page
1. Parameters from the fits of Eq. (113) to measured values of frequency
shift at high temperature........................................................................................... 77
2. The parameters from fits of Eq. (34) to the real part of permittivity (e')
at five temperatures..................................................................................................  93
3. Elastic stiffness and compliance constants of PMN at several temperatures
from two scattering geometries 0S = 180° and 32.5±0.2°...................................... 125
4. Comparison of elastic stiffness and compliance constants at room
temperature................................................................................................................  125
XLIST OF FIGURES
I . Two-dimensional illustration of ordered and disordered states for both 
FE-AFE mixed glasses AxCNH^ ^ xH2BO4 [or AxCNDi^ xD2BO4] and 
complex relaxor ferroelectric systems (A^A2)BOg, A(BjB2)Og............................  2
Figure Page
2. A perovskite cubic unit cell. The point group usually is m3m for a 
paraelectric cubic ABOg perovskite-type structure. Solid lines indicate 
the BOg octahedron. A4 indicates one of three 4-fold rotation axes and 
4mm is the usual corresponding point group in the tetragonal ferroelectric 
phase. A3 indicates one of four 3-fold rotation axes and 3m is the usual
corresponding point group in the trigonal ferroelectric phase..............................  4
3. Mixed-ion arrangements of (a) ordered and (b) disordered cubic
perovskite ABOg structure.....................................................................................  6
4. A general schematic phase diagram vs. temperature for complex
relaxor ferroelectrics. "?" indicates one or two intermediate regions....................7
5. The structure of KDP at room temperature. Here just a few
hydrogen bonds are shown...................................................................................  7
6. Projection of KDP along c-axis in the ferroelectric phase (T<TC). The
solid line inside the square connects oxygen atoms at the same height.................  8
7. The structure of ADP projected along the c-axis. A NH4+ at height O 
is shown attached to nearby oxygens of PO4 tetrahedra. The heights
in unit of c of the phosphorus centers of the PO4 tetrahedral are indicated. 
The PO4 tetrahedron located above the ammonium, at height 1/2, and a 
number of protons, are not shown. The off-center motion of the ammonium 
ion is indicated by the arrow. This is one of the four possible directions of
this arrow in the (OOl) plane..................................................................................  9
8. Schematic representation of the antiferroelectiic phase
of ADP (or ADA)................................. ................................................................... 9
9. Schematic phase diagram of FE-AFE mixed glasses such as RADP.
PO: proton glass, PE/FE: PE and FE phase coexistence region.............................  10
Figure - Page
10. Dispersion curve of (o(k) of the longitudinal mode from monatomic 
linear chain. At any frequency below the cutoff frequency (Oc= IyJ K lM ,
two waves can propagate in both directions with opposite wave numbers...........  17
11. The determination of the Brillouin scattering angle 0 ............................................... 18
12. Symbols of scattering geometries, (a) Z(X5Y)X, (b)HV....................................... 19
13. Brillouin scattering set-up for small angles............................................................  21
14. Brillouin scattering set-up for backscattering.......................................... .............. 22.
15. The actual experimental connection between AD/DA converter
and instruments................................................................................ ......................  23
16. A typical Brillouin spectrum, d is the space between two mirrors of
Fabry-Perot interferometer. FSR means free spectral range............................. . 24
17. Actual positions of crystal, sensor and laser beam in the closed-cycle
helium refrigerator................. :...............................................................................  25
18. (a) A set-up for pre-alignment of the Fabry-Perot interferometer mirrors.
(b) The dashed line indicates the path along which the reflective mirror is
moved to check the radius of interference fringes................................................. 29
19. A typical transmitted interference pattern of a Fabry-Perot etalon........................ 30
20. (a) An illustrated Brillouin spectrum with larger FSR=Gt. R and B indicate
the Rayleigh and Brillouin peaks, respectively, (b) An illustrated Brillouin 
spectrum with smaller FSR=oc/2.................................................................... . 31
21. The Debye, Cole-Cole and Cole-Davison relaxation distribution functions...........40
xi
LIST OF FIGURES-CONTINUED
xii
LIST OF FIGURES-CONTINUED
22. Schematic representations of the properties of simple circuit combinations 
of ideal, frequency-independent elements of capacitance C, conductance G 
or resistance R, and inductance L, as shown in first column. The second 
column gives the complex impedance plot Z, the third the complex admittance 
plot Y and the fourth the complex capacitance C*. The fifth column gives 
the corresponding plots of log C'(co) (solid line) and log C(G)) (dotted line), 
against log to. Arrows indicate the sense of increasing frequency. Vertical
Figure Page
and horizontal axes are imaginary and real parts, respectively............................... 45
23. Light scattering diagram......................................................................................... 47
24. Momentum conservation of photons and phonon..................................................  51
25. Index ellipsoid. Two waves, polarized along D1 and D2 (mutually 
orthogonal), can propagate in the k direction (applicable to both
incident and scattered beams), with different velocities.......................................  53
26. Orientations of phonon propagation wave vector q /  /[100] and three
wave polarizations..................................................................................................  59
27. Orientations of phonon propagation wave vector q /  /[110] and three
wave polarizations...................................................................................................  61
28. Orientations of phonon propagation wave vector q /  /[cos6, sin 0 ,0] and
three wave polarizations.......................      62
29. (a) Elasticity vs. temperature for (i) Pryx linear coupling, (ii) Tq2IX 
quadratic coupling, solid line for either second-order or first-order 
and dotted line only for first-order and (iii) aqjx2 coupling, (b) The 
dynamic effects for a second-order q 2jx-type quadratic coupling
in typical pure ferroelectrics. v is phonon frequency shift and 
- 8 is phonon damping...............   73
30. Anti-Stokes components of the LA[100] Brillouin frequency shift for 
temperatures around the maximum value of half-width for DRDA. The
open circles are the measured data and solid lines are the Lorentz fits.................  81
Xin
LIST OF FIGURES-CONTINUED
31. Anti-Stokes components of the LA[100] Brillouin frequency shift for 
temperatures around the maximum value of half-width for DRADA-0.10.
The open circles are the measured data and solid lines are the Lorentz fits.......... 82
32. Anti-Stokes components of the LA[100] Brillouin frequency shift for 
temperatures around the maximum value of half-width for DRADA-0.28.
The open circles are the measured data and solid lines are the Lorentz fits...........83
Figure Page
33. Frequency shift and half-width vs. temperature of the LA[100] phonons 
for DRDA. The solid line for frequency shift is the Debye anharmonic 
calculation with parameters from Table I. The dashed line and solid circles 
for half-width are the qualitative estimates for the Landau-Khalatnikov and, 
pure lattice anharmonic contributions, respectively. The error bar indicates
the error range for the half-width experimental points.........................................  84
34. Frequency shift and half-width vs. T of the LA[100] phonons for 
DRADA-0.10. The solid line for frequency shift is the Debye anharmonic 
calculation with parameters from Table I . The solid line and solid circles 
for half-width are the estimates of fluctuations and pure lattice anharmonic 
contributions, respectively. The dashed line is a guide to the eye. The error
bar indicates the error range for the half-width experimental points......................  85
35. Frequency shift and half-width vs. temperature of the LA[100] phonons 
for DRADA-0.28. The solid line for frequency shift is the Debye 
anharmonic calculation with parameters from Table I. The dashed line 
is a guide to the eye. The error bar indicates the error range for
the half-width experimental points........................................................................  86
36. The temperature dependence of the real part of the dielectric relative 
permittivity in NBT at two frequencies upon heating. The A, B, and
C arrows indicate three anomalous regions...........................................................  94
37. Comparison of the dielectric anomaly in region A. The curves 3 and 4 
show only heating and cooling runs, respectively. The inset shows that
this anomaly is hidden by an increasing background at lower frequencies............  95
38. The temperature dependences of remanent polarization, Pr, and dielectric
permittivity (e') upon heating in region B where a ferroelectric state exists..........96
xiv
LIST OF FIGURES-CONTINUED
39. The real part of dielectric permittivity vs. temperature in region C at 
various frequencies upon heating. The solid line is the calculation of 
Eq. (41) with N=54 and dashed line is the calculation of Eq. (41) 
with N= 125..............................................................................................................  97
Figure Page
40. (a) The temperature dependence of the real part (e') of the dielectric
permittivity in NBT and PMN at 100 Hz upon cooling, (b) The 
temperature dependence of the imaginary part (e") of the dielectric 
permittivity in NBT and PMN at 100 Hz upon cooling..........................................  98
41. (a) a(=27tfe0e") vs. temperature at five different frequencies. The solid 
line is the fit of Eq. (121) for a c with parameters oo=2.12xl07 (Qm)"1 .
and w= 18860 K. (b) The imaginary part of dielectric permittivity,
£"p=Aa/(27tfe0), vs. temperature at five different frequencies. Aa=G-Gc............. 99
42. Complex representation of the NBT dielectric permittivity at four 
different temperatures. The numbers on the points indicate the 
frequencies in Hz. The dashed lines are regression lines...................... ............... 100
43. The strong thermal hysteresis behavior in region C at four different 
frequencies (10 kHz and 100 kHz in the left inset). Right inset: the 
difference of heating and cooling peak values shows a drop in the 
region of the ferroelastic phase transition at Tc;-820 K. The numbers 
on the points indicate the frequencies in kHz. The dashed line is
a regression line......................................................................................................  101
44. The temperature behavior of dc permittivity, B1, calculated from Eq. (41) 
with N=125. Inset: solid hexagons are from Eq. (41) with N=125, 
squares are the "measured" dc permittivity, solid circles are proportional 
to the tetragonal order parameter from neutron scattering, and triangles 
are the calculated dc permittivity as defined by the dc permittivity from 
Eq. (41) multiplied by the tetragonal order parameter. All curves are
adjusted to cross at 800 K........ .............................................................................  102
45. Relaxation times, Tpk and Trc, vs. 1000/T. The solid and dashed lines 
are fits of equation T=T0Cw r  with to=5.57x 10"18 s, W=25090 K for Tpk, 
and T0=3.54xlO-il s, W=30720 K for tRc......................................................................  103
XV
46. The real part of dielectric relative permittivity IO-4 e' vs. frequency (Hz) 
at five temperatures. The solid curves are fits of Eq. (34) with parameters
of Table 2 ..................................................................................:..................................... 104
47. (a) A actual connection of sample and electrods. (b) A series of two
parallel RC circuits. Z~‘ = —  + icoC, and Z ’1 = —  + ItoC2..................................  105
Ri R2
48. Plots of Eq. (124) with parameters: dbulk= l mm, dsur=10"4 mm,
Wbuik=ZOOOO K, Wsur=30000 K, e bulk=e'sur=100, and G0=IO7 (ohm-m)"1.............107
49. Brillouin backscattering spectra of NBT at different temperatures
near the damping maximum....................................................................................    112
50. Damping of the longitudinal acoustic phonons vs. temperature in the
vicinity of the cubic-tetragonal phase transition.......................................................  113
51. Temperature dependence of different characteristics in NBT: (a) Brillouin 
shift of the longitudinal acoustic phonons vs. T. (b) Half-width of
Brillouin components vs. T. (c) Intensity of the background vs. T ..................... 114
52. Intensity of the superlattice reflections from M and R points of the cubic 
Brillouin zone obtained in neutron scattering measurements vs. T to show 
the range of phase coexistence (shaded region). The phase sequence in 
NBT is shown at the top; Lparaelectric and paraelastic, ILferroelastic, 
III:antiferroelectric(?), IV: ferroelectric. The corresponding crystal
symmetries are shown in the middle. All data were obtained on heating runs...... 115
53. Anti-Stokes and Stokes components of the LA[001] Brillouin spectra 
for temperatures near the maximum damping value for Os=ISO0 in PMN.
The frequency interval between two the Rayleigh peaks is 17.035 GHz (FSR).... 119
54. (a) Frequency shift and .(b) Half-width vs. temperature of the LA[001] 
phonons for Os=ISO0. (c) The real part of dielectric permittivity vs. 
temperature at four different frequencies. The dashed line indicates
the ferroelectric phase transition temperature Tc~212 K........................................ 120
z
LIST OF FIGURES-CONTINUED
Figure Page
xvi
55. (a) Frequency shift and (b) Half-width vs. temperature of the LA[001] 
phonons for 0S=32.6°. (c) Frequency shift vs. temperature of the
TA[001] phonons. The dashed line is a guide to the eye...................................... 121
56. (a) Frequency shift and (b) Half-width vs. temperature of the LA[001] 
phonons for 0S=12.6°. (c) Frequency shift vs. temperature of the
TA[001] phonons. The dashed line is a guide to the eye...................................... 122
LIST OF FlGURES-CONTINUED
Figure Page
r
xvii
ABSTRACT
Temperature and frequency dependent measurements of dielectric permittivity, 
acoustic sound velocity and damping have been carried out by using Brillouin light 
scattering and a capacitance and conductance component analyzer on two different types 
of ferroelectrics, i.e. (i) FE-AFE mixed deuteron glasses Rbl x(ND4)xD2AsO4 (x=0, 
0.10, 0.28), (ii) relaxor ferroelectrics N ay2B iy2TiO3 (NBT) and PbMgy3Nb2ZgO3 
(PMN).
In this study, three important results have been observed: ( I ) A  broad and high- 
value maximum in dielectric permittivity has been observed in relaxors NBT and PMN, 
which indicates that these materials could be used for electrostrictive displacement 
transducers. (2) In FE-AFE mixed glasses DRADA-x, a r|^g-type quadratic coupling, 
squared in order parameter and linear in strain, becomes the main coupling contribution 
as ammonium ND4 concentration x increases from 0 to an intermediate value. The 
results also confirm the presence of PE/FE phase coexistence in DRADA-0.10. (3) For 
both FE-AFE mixed glasses and relaxor ferroelectrics, the order parameter(s) 
fluctuations, which are generated by the local random fields originating from short-range 
randomly-placed cations, are the main dynamic mechanisms for hypersonic and dielectric 
anomalies. In DRADA-0.10 and 0.28, these local randomly-placed ions are ND4"*" and 
Rb4" ions. In PMN, those randomly-placed cations are Mg+^ and Nb+^ which are 
randomly placed at B-site positions. In NBT, those randomly-placed cations are Na+ * 
and Bi+^ which are placed randomly at A-site positions.
Two models, i.e. superparaelectric cluster and extrinsic bulk conductivity, also 
have been proposed to explain the high-temperature and low-frequency dielectric 
anomaly in NBT. The elastic stiffness and compliance constants are also calculated for 
PMN.
V
ICHAPTER I 
INTRODUCTION 
Introduction to D isordered Crystals
If the composition in a crystal is not completely uniform (compositional 
heterogeneity) down to the unit cell scale or has cation fluctuations statistically or 
dynamically, then the crystal is usually called "disordered".1 In terms of symmetry, 
"disordered" means that there is no long-range translational symmetry inside the crystal. 
Among disordered materials, many experiments have concentrated on two different 
systems, i.e. ferroelectric (FE)-antiferroelectric (AFE) mixed crystals with the general 
formula A ^x(NH4)xH2BO4 [A=Rb, K, Cs and Tl, B=P and As]2"11 and complex relaxor 
ferroelectrics with formula (A1xAm1^ ) +2Bh4O3-2 or At i (BxBn^ x)+4O3-2.1,12,13 In these 
systems, local random fields originating from randomly-placed cations (or ions) play an 
important role for physical properties such as acoustic (sound velocity and damping) and 
dielectric anomalies. Since local random fields can suppress the long-range electric 
ordering and produce the order parameter fluctuations, a normal sharp phase transition 
usually is not observed in these materials. "Diffuse" (non-sharp peak) phase transition is 
the typical anomalous characteristic of these disordered materials. The two-dimensional 
distributions of ions (or cations) for both ordered and disordered states are given 
schematically in Fig. I.
2#  #
O  #
O  ®  O  
®  O  O  
O O O
Cj c )
#  O
#  #  
®  #  
#  #
O
O  O  
O  O
O O  
O O  #  
O O O  
O  O  
O  O
I J
0
O O O O O
#  O  ®  O  #
O
O O  ®
O  ®  O  ®  O
©  O  O  O  @
O  ®  O  #  O
#  O  #  O  #  
O O O O O
#  O  ®  O  ®
(A1A2)BO3, A(B1B2)O3, Ax(NH4) 1^ H 2BO4 and Ax(ND4) 1^ D 2BO4
Disordered State Ordered State
— A1(^)1 B 1(W) or A(w) rich region Uniform composition
-  A2(O), B2(O) or NH(D)4(O) rich region Full translational symmetry
Figure I. Two-dimensional illustrations of ordered and disordered states for both FE- 
AFE mixed glasses Ax(NH4)1^ H 2BO4 [or Ax(ND4)1^ D 2BO4] and complex relaxor 
ferroelectric systems (A1A2)BO3, A(B1B2)O3 .
For relaxor ferroelectrics, one can use thermal treatments such as annealing, 
quenching and ratio of cations to control ordering.1 In FE-AFE mixed proton glasses, 
the crystal ordering depends on ammonium concentration.2 7 From experimental results, 
some general differences have been found between ordered and disordered ferroelectrics: 
(I) Ordered (normal):
a) Sharp dielectric phase transition at Tc
b) Stable remanent polarization
3c) No strong dielectric frequency dependence at phase transition
d) Stable birefringence
e) No local random fields
f) Usually no intermediate phase between high- and low-symmetries 
(2) Disordered:
a) Diffuse dielectric transition which doesn't correspond to any specific 
change of symmetry (or phase)
b) No stable remanent polarization
c) Strong frequency dependence at diffuse transition
d) No stable birefringence
e) Strong local random fields .
f) Usually associated with a sequence of phase transitions, i.e. one (or 
more) intermediate phase occurred between high- and low-symmetries
In this study, five different single crystals have been measured, i.e. FE-AFE 
mixed deuteron glasses, RbxCND^) ^ _XD2AsO4 with x=0, 0.10 and 0.28 and complex 
relaxor ferroelectrics, Pb(MgiygNbgyg)Og (PMN) and NaiygBiygTiOg (NBT). The 
introductions to these two different systems are given below.
Relaxor Ferroelectrics (RF)
Perovskite is the name of the mineral calcium titanate (CaTiOg). Most of the 
useful piezoelectric (ferroelectric) crystals, such as barium titanate (BaTiOg), lead 
titanate (PbTiO3), lead zirconate titanate (PbZryxTixO3), lead lanthanum zirconate 
titanate (PLZT), sodium bismuth titanate NaiygBiygTiOg and lead magnesium niobate 
Pb(Mgy3Nbgyg)O3 etc., have perovskite-type structure, i.e. ABOg-type unit cell. Here 
"O" is oxygen, "A" represents a cation with a larger ionic radius, and "B" a cation with a
4smaller ionic radius. Fig. 2 shows a typical cubic ABO3 perovskite-type structure in the 
paraelectric phase and possible directions of distortion in the ferroelectric phase.14
Figure 2. A perovskite cubic unit cell. The point group usually is m3m for a paraelectric 
cubic ABOg perovskite-type structure. Solid lines indicate the BOg octahedron. A4 
indicates one of three 4-fold rotation axes and 4mm is the usual corresponding point 
group in the tetragonal ferroelectric phase. A3 indicates one of four 3-fold rotation axes 
and 3m is the usual corresponding point group in the trigonal ferroelectric phase.
Many of the complex relaxor ferroelectrics with perovskite-type structure are 
compounds with either A+2B44O3"2 or A+1B+50 3"2-type formula. In the perovskite 
family, there are many compounds with the formula A+3B+30 3"2, but among them no 
ferroelectrics have been discovered.13
The perovskite structure is essentially a three-dimensional network of BO6 
octahedra (see Fig. 2). It may also be regarded as a cubic close-packed arrangement of 
A and B ions with B ions filling the octahedral interstitial positions.13 The packing 
situation of this structure may be characterized by a tolerance factor t, which is defined 
by the following equations:13
5Ra + Rq -  t*J2(RB + R0) (I)
or
t _  Ra + r O
J2(Rb + Rq)
(2)
where Ra , Rb and R0 are the ionic radii of A, B and O ions, respectively. When t is 
equal to I, the packing is said to be ideal. When t is larger than I, there is too large a 
space available for the B ion, and therefore this ion can "move" inside its octahedron. In 
general, to form a stable perovskite structure, one requires that 0 .9< t< l.l. Besides the 
ionic radii, other factors, such as polarizability and character of bonds, must also be taken 
into account.13
The term "complex relaxor ferroelectiics" generally refers to the complex 
perovskites in which the charge classifications of (A'XA" 1.x)+2B+403_2 or A+2CtB1xB" 
x)+403"2 etc. are satisfied and unlike-valence cations belonging to a given site (A or B) 
are present in the correct ratio for charge balance, but are situated randomly on these 
cation sites. For example, sodium bismuth titanate N a ^B ii^TiOg has two unlike 
valence cations Na+ and Bi+3 distributed in the A site and lead magnesium niobate 
Pb(Mg173Nb2Zs)O3 has two unlike valence, cations Mg+2 and Nb+5 distributed in the B 
site,1,15,16 These randomly different cation charges give rise to random electric fields and 
cause random elastic distortion fields in microregions. These random fields tend to make 
the phase transitions "diffuse" instead of sharp as in normal ferroelectrics, and complicate 
the task of determining structure and state of electric order. The electric properties of 
relaxor ferroelectrics also depends on the cation ordering degree. The degree of ordering 
inside a crystal depend on several things such as method of crystal growth, cation 
valence, and thermal treatment (e.g. quenching and annealing). In certain compounds,
6such as P b S c^T a^O g  (PST) and P b ln ^N b ^O g  (PIN), the driving force for cation 
ordering is such that long-range coherence and corresponding normal FE (or AFE) 
behavior can be achieved through thermal annealing.15 An additional complication is the 
aging effect which can span a wide time scale in a nonlinear manner. Relaxor 
ferroelectrics usually exhibit a sequence of anomalies in the dielectric response as 
temperature changes.1,16 However, the dielectric maximum does not clearly mark a 
phase transition into a low-symmetry region.
It is important to note that B-site order in A+2(B'xB"i„x)+403"2 compounds 
could be either stoichiometric (ordered) or nonstoichiometric (disordered) as shown in 
Fig. 3.1,15
I
----  ABQ1----
w  * •  ^  -
+  >  h  H
L ±  l  
1 ' e - H
L
B' O  B" O  
(a) Stoichiometric (ordered)
B' or B" ®
(b) Nonstoichiometric (disordered)
Figure 3. Mixed-ion arrangements of (a) ordered and (b) disordered cubic perovskite 
ABO3 structure.
For example, B-site order in x=l/2 compounds, e.g. Pb(Scy3Taiz3)O3 and 
Pb(Mgy3W y3)O3, results in an average valence of +4 and thus is referred to as 
stoichiometric ordering.15 In contrast, nonstoichiometric ordering is found in x=l/3 
compounds, e.g. Pb(Mgy3Nb373)O3 and Pb(Cdy3Nb373)O3 where a tendency toward 1:1
7B'B" order is observed and as such the average charge value is lower than the expected 
valence of +4. The overall defect chemistry and average compensation mechanism of the 
nonstoichiometric ordering are not well understood at this time.
A sequence of phase transitions with one or two intermediate phase is a 
characteristic of complex relaxor ferroelectrics.1,16 A general schematic phase diagram is 
given in Fig. 4.
Low-symmetry High-symmetry
L.T. Temperature MT.
Figure 4. A general schematic phase diagram vs. temperature for complex relaxor 
ferroelectrics. "?" indicates one or two intermediate regions.
Proton (or Deuteroni Glasses (PG1
At room temperature, the hydrogen-bonded phosphates (MH2PO4) and 
arsenates (MH2AsO4) (M=Na, K, Rb, Cs, Ag, Tl and NH4) form a class of KDP-type 
isostructural crystals of tetragonal symmetry.17 The structure of KH2PO4 (KDP) at 
room temperature is given in Fig. 5.17
&  PO4
©  K
© H
C
Figure 5. The structure of KDP at room temperature. Here just a few hydrogen bonds 
are shown.
8When M is an alkali ion, the low-temperature state is ferroelectric. However, it should 
be noted that RDP can be grown in either a tetragonal or a monoclinic structure. Fig. 6 
is the c-axis projection of KDP structure in the ferroelectric phase (T<TC).17
' -1W ' h ^ - H  --W '
H , H
" h ^ ~ h - W 'hW h - -
, H
. ; . I
- - W ' hW h - W '
H H c e
Figure 6. Projection of KDP along c-axis in the ferroelectric phase (TcTc). The solid 
line inside the square connects oxygen atoms at the same height.
The KDP-type structure consists of two interpenetrating sublattices. One is a 
body-centered sublattice of PO4 tetrahedra and the other is a body-centered sublattice of 
K+ ions. The hydrogen atoms are dynamically fluctuating between off-center positions 
but are statistically distributed at the centers of the O H O  bonds in the paraelectric 
(PE) phase. They are in ordered off-center positions in the ferroelectric (FE) phase. In 
the FE phase the protons are either close to the "upper oxygens" or "lower oxygens" and 
the non-zero net polarization depending on domain type arises from the displacement of 
K+ and P5+ ions along the c-axis direction.
The ammonium compounds which exhibit a paraelectric (PE)-antiferroelectric 
(AFE) phase transition have NH4+ substituted for K+ position in the KDP-type unit cell. 
Two of the four "ammonium" protons form short bonds with the surrounding PO4 
tetrahedra as seen in Fig. 7.
9Figure 7. The structure of ADP projected along the c-axis. A NH4"1" at height 0 is shown 
attached to nearby oxygens of PO4 tetrahedra. The heights in unit of c of the phosphorus 
centers of the PO4 tetrahedral are indicated. The PO4 tetrahedron located above the 
ammonium, at height 1/2, and a number of protons, are not shown. The off-center 
motion of the ammonium ion is indicated by the arrow. This is one of the four possible 
directions of this arrow in the (OOl) plane.
In the antiferroelectric phase the positions of the "acid" protons are such that 
each PO4 tetrahedron has one close proton each at the upper and the lower sides, unlike 
in KDP where in the FE phase all the "acid" protons are attached either to the top or 
bottom of the PO4 tetrahedron. Such an arrangement of the "acid" protons results in 
local dipoles alternately pointing "up" and "down" along the a-axis (or b-axis) direction 
as shown in Fig. 8, and yielding zero net polarization.
Figure 8. Schematic representation of the antiferroelectric phase of ADP (or ADA)
10
It is important to note that the deuterated isomorphs show enhanced electric­
ordering temperatures (isotope effect).17 For instance, Tc is 146 K for Rbf^PO4 and Tc 
is 218 K for RbD2P04. Such isotope effects occur not only for pure ferroelectrics but 
also for FE-AFE mixed crystals.2
The FE-AFE mixed crystals A j.x(NH4)xH2B04 and A ^x(ND4)xD2BO4 usually 
exhibit a frequency dependent dielectric anomaly instead of a phase transition for certain 
intermediate values of x. In these systems, competition between the FE and the AFE 
orderings which are characterized by specific configurations of the acid protons, plays an 
important role for dynamic anomalies. The random distribution of the Rb (or K, Cs, Tl) 
and NH4 (or ND4) cations is the origin of the local random fields. At lower temperature 
the system eventually becomes nonergodic in which case the system is frozen completely 
on a practical time scale.8 This behavior is reminiscent of that of magnetic spin glasses, 
except that the random bias field caused by random Rb and NH4 neighbor placement 
biasing the O-H O bonds smears out the spin glass "transition". However, whether 
there is a real ergodic-nonergodic transition is still an open question. From the 
experimental results, it was found that small amounts of ammonium admixtures are 
sufficient to stabilize the tetragonal structure even at low temperature. A typical 
schematic phase diagram of proton (or deuteron) glass is shown in Fig. 9.7,18
x (NH4 concentration)
Figure 9. Schematic phase diagram of FE-AFE mixed glasses such as RADP. PO: 
proton glass, PE/FE: PE and FE phase coexistence region.
11
Previous Know ledge for Crystals D iscussed in This Thesis 
N a inB ijnTiO2 INBTl
Nai^B ii^TiOg undergoes a cubic-tetragonal-trigonal succession of structural 
phase transitions, ending in a ferroelectric state at room temperature. Different studies 
on NBT have been published in many papers,19'30 but its behavior is still far from clear. 
Most of the contradictions concern the number and location of different phases as well as 
the electrical state in each phase. According to x-ray23 and neutron measurements30 there 
are two structural phase transitions in NBT from the high-temperature cubic phase to a 
tetragonal and then a trigonal phase. The. tetragonal phase seems to be ferroelastic as 
shown by domain pattern visualization.24 However, the question of whether this phase is 
polar is still open. Some dielectric measurements indicate that an antiferroelectric phase 
exists between the tetragonal ferroelastic and trigonal ferroelectric phases. However, 
x-ray and neutron scattering studies have not provided any evidence to support this view. 
Neutron scattering indicates that the transition between tetragonal and trigonal phases is 
characterized by a region of phase coexistence. However, the nature of the high- 
frequency dielectric maximum near 640 K is still an open question.16
PbfMgi/sNbo/glOg (PMNT)
Dielectric results of PMN shows a frequency dependent "diffuse" phase 
transition associated with a broad maximum around 270 K.1 This anomaly is attributed 
to quenched random electric fields originating from charged compositional fluctuations.31 
Under a strong bias electric field (>4 kV/cm), an additional dielectric peak occurs at Tc-  
210 K below which the random fields are overcome by the external electric field.31
In the supposedly disordered PMN single crystal, a microstructure on a 
nanometric length scale with 1:1 cation ordering in B site was reported.32,33 Such 1:1 
ordering, when charge neutrality requires 1:2 stoichiometry, implies locally charged
12 -
regions causing fields which induce random polarization. Recent results show only a 5% 
tendency toward 1:1 ordering in these microregions.34
Three recent models for the relaxor ferroelectric crystal PMN have been 
proposed, namely the random field" model (RF),31 the dipolar glass model35 and the 
structural model.36 The first two models are based on the temperature and electric field 
responses of these microdomains. The last model, based on x-ray and neutron powder 
diffraction at 5 K, indicates an average cubic structure even at 5 K, but with some 20% 
of the material in non-cubic FE microdomains near 100 A size.
The RF model postulates random electric fields originating in a peculiarity of 
PMN's structure, i.e. local nonstoichiometric order causing local net charge, resulting in 
microdomains polarized along the 8 <111> directions, and assumes that the cubic 
anisotropy corresponding to these 8 directions is low enough so that the random fields 
can suppress the FE transition.
In the dipolar glass model for PMN, the "pseudospins" are nanometer-sized 
superparaelectric clusters which correspond to the microdomains in the random fields 
model. This model assumes that these superparaelectric clusters can flip at higher 
temperature, but freeze into a random configuration at lower temperature. However, the 
random field model seems to imply that microdomains have fixed orientations determined 
by local fields.
Recent results of the linear birefringence An«= <P//2>-<P_L2> in PMN under an
external electric field suggest an equilibrium phase transition temperature Tc in the 
absence of the random fields, 200 K<TC<234 K.31 In addition, by fitting the time 
dependence of the linear birefringence with two different exponents P below and above 
212 K, Westphal et al. propose that the Curie temperature of PMN is Tc~212 K. 
Viehland et al. also claim that the freezing temperature is Tf~217 K in PMN below which 
a dramatic change of the relaxation time distribution is observed and is interpreted as a
13
result of a long-range correlation of polar microdomains.31 Recent quasielastic-neutron 
scattering (QES) on PMN supports the above interpretations and shows that the 
correlation length is nearly temperature independent with a maximum value of 200 A 
below -217 K.35
Rb1^N D 1sLDoAsO1 ('DRADA-X=O. 0.10. and 0.281
Pure DRDA (x=0) single crystal exhibits a first-order PE<-»FE phase transition 
at Tc- 165 K as observed from spontaneous polarization measurement. The ND44" 
deuteron NMR spectra of DRADA-0.10 showed a gradual disappearance of the doublet 
near 131 K where the single broad linewidth grows to its full size,37 from which it is 
concluded that below 131 K the FE phase portion is greater than PE in the crystal and 
becomes the dominant ordering. This result is consistent with the presence of PEZFrE 
phase coexistence as evidenced by dielectric results which show that a gradual 
ferroelectric transition begins at Tm=146 K and is mostly completed at -120  K.7 The 
remaining PE material begins exhibiting dielectric dispersion characteristic of deuteron 
glass below a frequency-dependent temperature Tg, which is 60 K at 50 kHz. DRADA— 
0.28 has only a frequency dependent dielectric anomaly separating PE from deuteron 
glass behavior at Tg~65 K (f=0.1 MHz). Field-heated, field-cooled and zero-field-heated 
static permittivity revealed that below Te~38 K (nonergodic temperature) the system 
enters a nonergodic state in which for practical purposes the acid deuteron of the O- 
D O bond is frozen completely.8
Purpose and Outline o f The Present W ork
Both piezoelectric (strain linearly proportional to applied electric field) and 
electrostrictive (strain proportional to square of applied electric field) effects are 
important phenomena in ferroelectrics.13 These two intrinsic electric responses have
14
been used for applications in many ways such as nanometer-scale actuators and sensors. 
The main advantages of electrostrictive actuators include insignificant hysteresis and less 
creep and susceptibility aging (gradual change in response with time) which are the 
problems of piezoelectric actuators (or sensors). Bxillouin light scattering and dielectric 
methods are powerful tools to understand both piezoelectric and electrostrictive 
couplings. The elastic stiffness and piezoelectric stress constant can also be obtained 
from these measurements. The overall purpose of this study is to provide a basis for 
improving properties useful in applications and extending the useful temperature range 
upward and downward in disordered ferroelectrics.
In this study, FE-AFE mixed glasses Rbl x(ND4)xD2AsO4 (x=0, 0.10, and 
0.28) and complex perovskites N a ^B ii^TiOg and Pb(Mg1ZgM^yg)O2 have been 
measured by two different methods, i.e. Biillouin light scattering (Os=ISO0, 32.6° and 
12.6°) and dielectric permittivity (e*=e'-ie") between 0.02 and 300 kHz. The 
experimental procedure will be discussed in chapter 2.
15
CHAPTER 2
EXPERIMENTAL PROCEDURES 
Crystal Growth and Sample Preparation
Single crystals of R b^x(ND4)xD2AsO4 (x=0, 0.10, 0.28) for Brillouin 
scattering and dielectric permittivity measurements were grown here from aqueous 
solutions with certain ratios of RbD2AsO4 (DRDA) and (ND4)D2AsO4 (DADA) at 
room temperature. These starting materials were grown by following chemical reactions:
Rb2CO3 + As2O5 + D2O => 2RbD2As04 + CO2 
(ND4)2CO3 + As2O5 + 2D20  => 2(ND4)D2A s04 + CO2
These DRADA crystals were carefully polished to be rectangular with average size of 1.2 
x0.4x0.2 cm3. A general growth formula for this type of mixed crystals is given below.
KDP-type FE crystals + ADP-type AFE crystals => FE-AFE glasses
Ii Ii Ii
(Lx)-AH2BO4 + X-NH4H2BO4 => A 1 -x (N H 4 ) x H 2B 0 4
(Lx)-AD2BO4 + X-ND4D2BO4 A 1 -x (N D 4 ) xD 2 B ° 4
(Here A=Rb, K, Cs and Tl, B=P and As)
-
16
The solvent for deuterated crystals is heavy water. Such a "solution growth" is 
a two component system, i.e. solute and solvent, and so requires simultaneous control of 
two parameters (temperature and concentration x) during the crystallization process. 
The difficulty of crystal growth also depends on whether the cation sizes match each 
other. For instance, the rubidium and ammonium ions are a nearly perfect match in size, 
so high-quality and strain-free mixed single ciystals of Rb^x(NH4)xH2PO4 can be grow 
at all concentrations x.18
Relaxor ferroelectric single crystals NBT and PMN were grown by the 
Czochralski method in A. F. Ioffe Physical Technical Institute, with dimensions 5.57x 
6.03x5.74 (mm)3 and 5.37x4.77x2.82 (mm)3, respectively. The Czochralski growing 
method can be found in Ref. 38. For dielectric measurements, a silver paste (Ted Pella 
#16032) was used for the electrodes.
D ielectric Perm ittivity
For the audio frequency range (0.02-300 kHz), a Wayne-Kerr Model 6425 
Precision Component Analyzer with 4-lead connections was used for capacitance and 
conductance measurements. A Leybold RGD-210 closed cycle helium refrigerator was 
used for low temperatures (10-300 K) with a silicon diode temperature sensor put 
immediately below the sample holder. For high temperatures (300-900 K), an oven was 
used with a calibrated LakeShore PT-103 platinum resistance thermometer which was 
put just above the sample. The temperature accuracy is about ±0.1 K. All data were 
collected automatically with frequency swept in steps from 0.02 to 300 kHz. The 
measuring voltage was 0.5 V. The error of the dielectric relative permittivities (e' and e") 
is about 0.05%.
17
Brillouin Light Scattering
The light scattering caused by acoustic phonons is called Brillouin scattering, 
for which the frequency shift Av is between 0.01 and 5 cm '1. In Raman scattering, the 
frequency shift Av is usually caused by molecular vibrations or optical phonons and is 
generally larger than 5 cm '1. The elastic or quasielastic (Av=O) scattering processes 
which are caused by inhomogeneous distribution of the refractive index in a crystal are 
called Rayleigh scattering.
For visible light (usually from a high intensity laser beam), the photon wave 
vectors (of order IO5 cm '1), i.e. X-5000 A, are small compared with the Brillouin zone 
dimension (of order IO8 cm"1), thus information is provided by Brillouin scattering only 
about phonons in the immediate neighborhood of k =0. In order to estimate the order of 
magnitude of the acoustic phonon cutoff frequency (0C, a simple dispersion curve co(k) of 
the longitudinal mode from a monatomic linear chain with only nearest-neighbor 
interaction is considered in Fig. IO.39
Figure 10. Dispersion curve of a>(k) of the longitudinal mode from a monatomic linear 
chain. At any frequency below the cutoff frequency 0)C=2VK / M , two waves can 
propagate in both directions with opposite wave numbers.
For a monatomic linear chain lattice, the acoustic phonon frequency is 
to = 2 VkT m  I sin ka / 2 |.39 K is the force constant whose value depends on temperature
18
and different vibration modes. M is the mass of the atom. For small wave number, i.e. 
k a « l ,  we then have to = k a^ /K /M , i.e. a^/K / M =to/k=V0. V0 is the acoustic phonon 
velocity near the Brillouin zone center, i.e. k~0. When the wavelength is of the order of 
magnitude of the interatomic distance a (ka comparable to n), the frequency of acoustic 
phonons is bounded by the cutoff frequency, i.e. fc = VK / M / Tt =V0ZaTt. The V0 in
dielectric solids lies between IO3 and IO4 m/s. The interatomic distance is a few A. 
Taking V0=SxlO3 m/s and a=5 A, the cutoff frequency is fc=3.2xl012 GHz.
Brillouin light scattering can provide three types of basic information; (i) 
acoustic phonon frequency shift (sound velocity) from which the elastic and piezoelectric 
constants can be obtained, (ii) half-width at half height (damping or life time of phonon) 
and (iii) intensity of background which usually is related to the central component.
General Description of Experiment
The Brillouin spectra were obtained from several different scattering angles 
which include backscattering (6S=180°) and small angles (6S=12.6°, 32.6°). Scattering 
angle means the angle between incident and scattered wave vectors inside the crystal. An 
actual determination of scattering angle is given in Fig. 11.
8 : scattering angle 
n : refractive index
Figure 11. The determination of the Brillouin scattering angle 6.
19
The scattering geometry is usually expressed by a notation such as Z(X,Y)X for 
light scattering by a crystal. The notation stands for K1(Ep Es)Ks where E1 and Es are
polarization vectors of the incident ( I ) and scattered (S) beams, respectively. The first 
and last letters of Z(X,Y)X indicate that the propagation directions of the incident and 
scattered beams are parallel to the Z and X axes, respectively. The two letters in 
parentheses indicate that the polarization directions of the incident and scattered beams 
are parallel to the X and Y axes, respectively. Therefore, Z(X1Y)X corresponds to the 
experimental set-up illustrated in Fig. 12a. In case of isotropic materials (or in case the 
detailed specification of the scattering geometry is unnecessary or difficult), the geometry 
is expressed by two letters such as HV. "HV" means that E1 is parallel to the plane and 
Es is perpendicular to the plane, as illustrated in Fig. 12b. The plane is determined by K1 
and Ks.
A
A
Z
X
K
(a) (b)
Figure 12. Symbols of scattering geometries, (a) Z(X1Y)X, (b)HV.
20
In order to reduce the low-lying frequency mode of the Raman spectra, a 
narrow-band (I A) interference filter was used. AU samples were Uluminated by a Lexel 
Model 95-2 argon laser with 1=514.5 nm. Scattered Ught was analyzed by a Burleigh 
five-pass Fabry-Perot interferometer. The laser line broadening due to the jittering was 
claimed by the manufacturer to be about 10 MHz (half-width).
For small angle (0S<18O°) BriUouin scattering, the experimental set-up is shown 
in Fig. 13. The lens L1 serves to focus the incident beam into the sample which was set 
in the closed-cycle cryostat. The distance between L2 and the sample is the focus length 
of lens L2. The scattered beam from the sample is collected by L2 and then becomes 
paraUel. The lens L3 focuses the paraUel scattered Ught at the center of a 200 pm-radius 
pinhole, and the transmitted Ught is coUimated by the lens L3. The colUmated beam 
passes a narrow-band (I A) interference filter and then enters a Burleigh five-pass Fabry- 
Perot interferometer. The light from the Fabry-Perot interferometer is focused at the end 
of the telemeter with a 200 (im pinhole.
A Model R464 head-on photomultiplier (PMT) was used to coUect the 
transmitted Ught from the telemeter. A home-made high voltage power supply provides 
a maximum voltage of 1450 V for the cathode of the PMT. A home-made voltage 
amplifier was used to ampUfy the analog output from the PMT. The analog output from 
the ampUfier was fed into a home-made analog—^TTL transformer and then connected 
with a DeU 486 computer through a Computer Boards Inc. Model CIO-AD08 (8- 
channel) AD/DA converter. The BriUouin spectrum was displayed on the screen and 
stored as an ASCII file. A shutter was used to reduce the Rayleigh line intensity. If the 
, intensity of the Rayleigh line exceeds a certain value which was determined beforehand, 
the shutter is closed during the Rayleigh spectrum range. After a certain closing time 
which was also programmed preUminarily, it will be opened during the Brillouin spectrum
range.
Mirror
x Sample
Figure 13 Brillouin scattering set-up for small angles.
Mirror
Sample
Figure 14 Brillouin scattering set-up for backscattering.
23
Fig. 14 is the experimental set-up for backscattering in which a 5x5 mm2 prism was used 
to reflect the laser beam into L3.
A LABTECH NOTEBOOK software was modified with an AD/DA converter 
to interface the shutter, TTL transformer and ramp function generator. A detailed 
connection scheme between converter and instruments is given in Fig. 15.
RANGE
(4/-10V)
+IOVREF
LLGND
Ground
DIGITAL OUT 4 
DIGITAL OUT 3 
DIGITAL OUT 2 
(Shutter) d ig it a l  our I 
COUNTER 2 OUT 
COUNTER I OUT 
COUNTER I IN 
COUNTER OOUT 
(Amplifer) co unter  o in  
+12V PC BUS
19
18
17
16
15
14
13
12
11
10
9
8
"7
6
5
4
3
"2
I
Location of switch
37 CHO HIGH
36 CHi h ig h  (Ramp generator)
35 CH2 HIGH
'  34 CHS HIGH
33 CH4 HIGH
32 CHS HIGH
‘ 31 CH6 HIGH
‘ 30 CH7 HIGH
29 +5 VOLTS
28 DIG GND
27 DIGITAL IN 3
26 DIGITAL IN 2
25 DIGITAL IN I
24 INTERRUPT INPUT 
" 23 GATE 2
22 GATE I
21 GATEO
‘ 20 -12V PC BUS
Figure 15. The actual experimental connection between AD/DA converter and 
instruments.
Before acquiring a Brillouin spectrum, an adjusting program whose operation 
was observed on the computer screen was used to optimize the spectrum by optimizing 
the intensity of the Rayleigh lines. Acquisition of spectra is accomplished under 
computer control. The frequency interval of each scan is about 1.2 free spectral range
24
(FSR), within which two elastic peaks (Rayleigh lines) were included. Usually, each scan 
took 100 or 200 s depending on the intensity of the signal. A typical Brillouin spectrum 
is given in Fig. 16. The two maxima on both sides are Rayleigh peaks and the two 
smaller peaks between them are the Brillouin components.
Shutter closed| Shutter closedShutter open
Rayleigh lines -  
FSR= I/2d (cnr1)
4 0 0  -
Brillouin components
200  -
CHANNELS
Figure 16. A typical Brillouin spectrum, d is the space between two mirrors of the 
Fabry-Perot interferometer. FSR means free spectral range.
A Leybold RGD-210 closed-cycle refrigerator which can operate from 10 to 
475 K under vacuum (<10'3 torr) was used for temperature control with a LakeShore 
DRC-91C temperature controller. Fig. 17 shows the relative positions of sample, sensor 
and light beam in the closed-cycle helium refrigerator.
25
High-temperature stage 
with 25 ohm heater
Optical window
Laserbeam
Temperature sensor 
~  Optical sample holder
Figure 17. Actual positions of crystal, thermal sensor and laser beam in the closed-cycle 
helium refrigerator.
The temperature reading error was controlled to better than ±0.1 K and 
temperature was measured to ±0.01 K by using a calibrated silicon diode thermal sensor 
placed on the optical sample holder. For higher temperature (300-900 K), a closed oven 
with four windows was used with a calibrated LakeShore PT-103 platinum resistance 
thermometer and a manual controlled power supply. The temperature accuracy is about 
±0.1 K.
Characteristics of the Fabrv-Perot Interferometer40
The Fabry-Perot interferometer plays a key role in Brillouin light scattering. 
For better understanding of how the frequency shift and half-width were measured and 
determined, we briefly review its characteristics here.
The condition for constructive interference for a transmitted wave front is
2nd cos 6 =  mX (3)
26
where n is the refractive index of the medium between the two reflecting mirror surfaces, 
d is the mirror spacing, and 0 is the inclination of the normal of the mirror surface to the 
wave front direction, m  is the order of interference and X is the wavelength of the laser 
beam. In our case, the incident laser beam is perpendicular to the mirror surface (0=0). 
Since the medium between the two mirrors is air, the refractive index is n= l, Therefore, 
Eq. (3) becomes
If d decreases by X/2, a wave with the same wavelength is still able to pass the 
interferometer but the order of the interference lowers by one. Also, there will be 
another wave with different wavelength that may pass but with the same mth order 
interference. Hence we have
(4)
A t  I \d = (m  —1)— = m—  
2 2
(5)
8X =  X1 — X2 ~
1 2 2d
(6)
SX is called the free spectral range (FSR). In term of frequency v,
2d =
me
(7)v,
(m - l ) c
' V2
2d (8)
27
where c is the speed of light. From Eqs. (7) and (8), one can obtain
Av = V1- V 2 = —  (Hz)
1 2 2d
or
1
Av = — (cm '1) (Note: I Cnr1=SO GHz)
(9)
(10)
Here Av is the frequency interval between two nearest bright interference fringes (or 
Rayleigh peaks) and also called the free spectral range (FSR).
Assume the optical length d changes by X^/2a, where a  is not an integer. The 
wave front of wavelength X3 is allowed to pass this interferometer if the following 
formula is satisfied:
2(d — = (m — i)X3
where m  and i are integers. Then
. I T 1
Here we have used the relation, 2d=mX1. Then
* l ~ b
(H)
(12)
AX = X3 — X1 (13)
28
Normally m » i  and 2d» iX  (X-SxlO"5 cm), so
X2, ( i - - )_____a (14)2d a
or
Av — (i'-----) • FSR (15)
a
Here Av is the frequency shift from the original wave (Rayleigh line) and i is the shift 
order which is with respect to the Rayleigh line and is different from the order of 
interference. Experimental determination of shift order of Brillouin components will be 
discussed later in this chapter.
The Fabry-Perot interferometer which we used for experiments is a Burleigh 
Instr. Model RC-140. It uses matched sets of piezoelectric PZT transducers constructed 
from interferometrically matched PZT discs. This PZT material. offers linearity and 
hysteresis characteristics better than 1% and 0.5%, respectively. The RC-42 Ramp 
Function Generator drives the FP interferometer. The ramp generator controls the PZT 
to provide two functions, i.e. scanning and alignment.
The finesse is the measurement of the interferometer's ability to resolve closely 
spaced lines (close frequencies). The finesse (F1) for our system was about 40 (for single 
pass). For multi-pass system, the finesse is
F (16)
( 2 ' - l ) :
where p is the number of passes. In our five-pass system, the finesse is about 104. The 
contrast of multi-pass system is also improved significantly.
29
Pre-Ali gnment of Fabrv-Perot Interferometer
This is a rough parallel alignment procedure for Fabry-Perot mirrors and is 
critical for obtaining Brillouin spectrum. It should be completed before measurement. 
Before doing this pre-alignment, the ramp generator needs to be set at the following 
position: power "on", three biases "-one and half turn", ramp bias "-one quarter turn" 
and others "o ff. The cover of the Fabry-Perot interferometer must be removed and a 
piece of paper is placed in front of the front mirror as shown in Fig. 18a. A reflective 
mirror was used to reflect the interference pattern into the observer's eyes. How do you 
know that the two mirrors are on "rough" parallel position? The trick is to move the 
reflective mirror along the triangle indicated by dashed lines in Fig. 18b and adjust the 
two bigger knobs (labeled "L" and "R" in Fig. 18b) until the radius of the interference 
rings (shown in Fig. 19)40 is almost constant everywhere.
Figure 18. (a) A set-up for pre-alignment of the Fabry-Perot interferometer mirrors, (b) 
The dashed line indicates the path along which the reflective mirror is moved to check 
the radius of interference rings.
30
Figure 19. A typical transmitted interference pattern of a Fabry-Perot etalon. 
Brillouin Spectrum Analysis
Typical traces of the Brillouin spectrum have been shown in Fig. 16. The 
determination of phonon frequency shift and half-width can be done by selecting a 
suitable FSR, i.e. suitable Fabry-Perot spacing d (FSR= l/2d cm"1). One should avoid 
having the two Brillouin peaks close to each other so that they overlap, or separated too 
much so that they move into the "skirts" of the Rayleigh peaks. Since different materials 
have different values of frequency shift, a larger FSR (-2cm ^=60 GHz) usually is chosen 
first in order to have the Brillouin peaks appearing in first order with respect to the 
Rayleigh peak (see Fig. 20a). With such a large FSR, one can easily determine the actual 
phonon frequency shift range. However, a large FSR will reduce the resolution of 
phonon half-width. Thus, a smaller FSR is preferred in actual measurement, so that the
31
Brillouin peaks appear in the higher (second or third) order with respect to the Rayleigh 
peak (see Fig. 20b).
R R
' " " ..................................... Larger FSR= a ...............................................................
B (Stokes) B (Anti-Stokes)
r ........................... M  ------------
r  - ......................................  L
Figure 20. (a) An illustration of the Brillouin spectrum with larger FSR=Oi. R and B 
represent the Rayleigh and Brillouin components, respectively, (b) An illustration of the 
Brillouin spectrum with smaller FSR=cc/2.
The frequency shift can be determined by the following equation
Av — FSR 
M
(17)
where L is the total channel number between the Brillouin peak and Rayleigh line and M 
is the total channel number between the two nearest Rayleigh peaks. Here the Brillouin 
component has a half-width 8v (at half height) that arises from the finite lifetime of the 
phonons responsible for the scattering. A typical Lorentz profile equation which is a 
result of the assumption that phonons decay in time as e"5v t, can be used to fit the
spectrum, i.e.
32
i(x)=i ^ h r D (i8>
where I(x) is the spectrum intensity and x is the channel number. D is the background 
intensity and A is an arbitrary number. From fitting, the frequency shift Av and observed 
half-width Svobs can be determined as
Av = (N  — I H---- ) • FSR
M
(19)
OTJ
8V -  =  Itfsr (20)
where N is an integer (I, 2, 3 etc.). N=I means that the Biillouin peak appears in first 
order with respect to the Rayleigh peak, and so on.
For determination of natural-phonon half-width 5vph, the natural-phonon 
spectrum and the instrumental function were assumed to have a Lorentz distribution, and 
the broadening due to collection optics was assumed to have rectangular distribution. In 
this case, the natural-phonon half-width Svph is given by41
SVp l=  ( S v L - S v y ^ - S v ll,, (21)
where Svobs, Svinst, and Svang represent the observed, instrumental, and collection optical 
half-widths, respectively.
The Brillouin line broadening Svang due to finite collection angle 80ap is 
described by a well-known formula which to second order in 80 is42
(22)S v ,- . cot^ 89-P (Se. ) 1
Av 2 8
where 0 is the scattering angle and Av is the phonon frequency shift. In our case, 80ap 
is -0.057 (radians) obtained from measurements of fused quartz. According to Eq. (22), 
the broadening problem is severe in near-forward scattering (or small scattering angles) 
for which cot(0/2) is large. The half-width of the Rayleigh line in fused quartz was taken 
as the instrumental broadening Svins.
34
CHAPTERS
THEORY
Dielectric
The generally employed principle behind making measurements on a certain 
system is to subject it to a "force" and then examine how the system responds. Li the 
dielectric case, the "force" is an electric field. In order to reflect the intrinsic properties 
of the system, the applied field should be "suitably small". In other words, the applied, 
perturbation must not alter the nature of the system under study. This, in fact, is the 
essence of the linear response theory (LRT). Within this general framework two kinds of 
measurements may be performed: (i) response, in which the equilibrium value of some 
properties of the system under the influence of a time-independent field is measured, (ii) 
relaxation, in which a time-independent field that has been applied on the system for a 
long time is removed and the "free decay" of a system property such as polarization is 
investigated.
In dielectric materials, the intrinsic polarization is usually made up of three 
different components according to the nature of the charges displaced, i.e., electronic, 
ionic, and orientational (dipolar) polarizations. Only the dipolar contribution which 
comes from the reorientation of permanent dipoles, may lag the applied field and thus be 
responsible for dielectric dispersion at audio or radio frequencies. The other two
35
contributions (electronic and ionic) will become responsible for dispersion only if 
frequencies above IO12 Hz are applied.
Space charge (interfacial layers)43 polarization is an extrinsic effect and usually 
arises from the presence of electrons or ions capable of migrating over distances of 
macroscopic magnitude. Some of these charge carriers tend to become trapped and 
accumulate at lattice defects, impurity centers, voids, or at electrode surfaces and so 
distort the field and produce an apparent increase in the dielectric permittivity.43 
Interfacial polarization is of particular importance in heterogeneous or multiphase (or 
multilayer) materials. Due to the differences in the electrical conductivity of the phases 
present, charges move through the more conducting phases and build up on the surfaces 
that separate these from the more resistive phases.
For large fields, nonlinear properties of polarization P against the electric field E 
are the characteristic of ferro- and antiferroelectrics, and other dielectrics to a smaller 
extent. At constant temperature, the electric field E can be expressed by the Taylor 
series expansion as17
E = 4 p  + qP3 + £P5 (23)
%
Here, % is the static electric susceptibility, q and C1 are coefficients of the dielectric 
nonlinearity. If the applied electric field is small enough, however, the higher order 
nonlinear terms will not be evident. This is the case that we will be dealing with here. 
So, we will limit ourselves to the treatment of linear dielectric media, i.e. P = %E.
The dielectric permittivity tensor Eij is defined from the relationship between
electric field and electric displacement, i.e.
D (t) = E0 IE (I) + P  = B0EE(I) (2 4 )
36
i.e.
Di = E0EijEj - (25)
where E0 is the MKS constant.
In the simplest case, i.e. a harmonic electric field E = E0Cltot propagating in the 
material,.the electric displacement can be given by
D(CO5T).= D0 (T )ei(tot_S) (26)
i.e.
Doi (T )ei5c^ = E 0Eij ((O5T )Eoj (27)
This expression allows us to define the complex dielectric permittivity as
i.e.
Ey(CO5T )  =  Ey(CO5T ) - ^ ( C O 5T )  (28)
Ey(G)5T)
Doi (T) cos 8(G))
E0Eoj
; Eij(CO5T )
Doi(T)sinS(co)
eoE0j
(29)
The quantity Eij (Co) determines the component of D(t) with a phase difference
of ni l  with respect to E(t) and can be easily shown to be proportional to the dissipated 
power density (=C0£oe"Eo2/2) in the dielectric. For this reason, £"(co) is also called th e ' 
loss factor.
3 7
In the actual measurement, the capacitance C and conductance G were detected 
and then used in the following relationships to obtain complex dielectric permittivitty, i.e.
(30)
Eij(CD5T)
Cij(CD5T)
^ij(W)
Gij(CD5T)
CDCn
(31)
where C0 = e0A /  d is the geometrical capacitance, i.e., the capacitance that we would
obtain from the parallel plate capacitor when the sample is removed. A is the sample 
area which is contacted with electrode and d is the sample thickness. CO = 2%/(Hz) is the 
angular frequency.
The elements of dielectric tensor e- usually are anisotropic and depend on
crystal symmetry. From thermodynamic arguments, we know that the dielectric 
permittivity tensor is symmetric, i.e. Eij = Eji . Following are different dielectric tensors
which correspond to different crystal symmetries along some specific symmetry 
directions.14
Triclinic: H
*
8H*
£ 12
*
e I 2 *
e 22
*
8 13 *
e 23*
£ 13
*
e 23
*
£33
Gij =
Eli
*
e 12
0
812
*
822
0
0
0
*
£ 33
Monoblinic: (z along the 2-fold axis)
Orthorhombic: e;ij 0 4 :
0 0
e n  0
0
*
e 33
0
(three 2-fold axes are axes)
Trigonal
and
*
eIl 0 0
Tetragonal: */ i j  = 0 eli 0
0 0 4
and
Hexagonal
(z along the n-fold axis, n>2)
*
Ell 0 0
Cubic: Il
*6T 0 *Eli 0
0 0 *Eli
. A cubic crystal is an isotropic dielectric medium, i.e. the elements of the 
dielectric tensor are diagonal and equal, and do not depend on any reference axis.14
Dipolar Relaxation Equations
For most practical cases, the dielectric relaxation behavior of reorienting dipoles 
can be described quantitatively by the following equation:
E (CO1T )  =  E00 +
Ec — £„
[!  +  IiCOT(T)]1 a ]P
(32)
The different values of parameters a  and (3 are corresponding to different models. For
practical purposes, three most useful empirical models are given below.
39
(I) Debye equation: oc=0, P=I44
In this equation, there is a single relaxation time T. The real and imaginary parts 
of the dielectric permittivity can be separated as
S1 (O^T) =  S0 0 + - - 8 5  2 e2°° , Sn(O)1T)
• ! +  GTT (T)
COT(T)(Ss - S 00) 
! +  CO2T2(T)
(33)
The Cole-Cole plot of e" vs. s ' for various CO is a semicircle with its center at 
(E00+ ss) / 2  on the s ' axis.
(2) Cole-Cole equation: 0< a< l, P=I44
In this case, the inverse of the relaxation time T has a frequency distribution. 
The real and imaginary parts of dielectric permittivity can be given by
E1(COjT )  =  Ero+  I ( E s - S 00)
COS CCTt___________2__________
cosh (I -  a )x  + sin I  cot
Em(COjT )  =  I ( E s - S 00)
1+ s in h ( l - a ) x
cosh(l -  a )x  + sin I  cot
(34)
where x=ln(CQm/C0). In these equations, (OmT=I is the condition for the peak maximum 
in s". When plotting these equations in a Cole-Cole plot, we obtain a circular arc with its 
center lying below the real s' axis and intersecting the s' axis at angles of 7l(l-a)/2 from a 
vertical line through the center.
(3) Cole-Davison equation: oc=0,0<P<144
This description provides an asymmetric broadening (toward high frequencies) 
of a Debye peak by the introduction of the parameter p. In a Cole-Cole plot, this 
equation produces a right skewed arc.5,44
40
In term of frequency domain, these three different behaviors can be described as 
shown in Fig. 21.
----- CO
Debye
CO
Cole-Cole Cole-Davison
Figure 21. The Debye, Cole-Cole and Cole-Davison relaxation distribution functions.
Superparaelectric Model fnano-scale polar cluster!16
The behavior of dielectric permittivity in relaxor ferroelectrics usually is 
dependent on the size of polar cluster (or polar domain) which is induced by locally 
nonstoichiometric charged region. In the limit N— (N is the number of dipole in a 
cluster), we would have a long-range-order ferroelectric phase transition which usually 
occurs at lower temperature. At high temperature, i.e. N—>1, the real part of the 
dielectric permittivity e' should obey the Curie law e'= I+P^Z(E0VkT). Here p is the 
dipole moment of a unit cubic cell, V is the cell volume, E0 is the MKS constant, k is 
Boltzmann's constant and T is absolute temperature. At low temperature, if N remains 
finite (i.e. superparaelectric moment) instead of infinity, then the dc e' should obey the N- 
dipole-cluster Curie law Ez=Np2Z(E0VkT). In the following we will show a simple 
superparaelectric model in the DC limit.16
If we assume that each dipole interacts with the other dipoles in the same 
cluster, i.e. long-range interaction, then the Hamiltonian for one N-dipole cluster in an 
electric field E is
41
H = -
N
Z
i=l
pESi+JSiESj
j>i
N J
-P E Z S i - -  
i=i 1
ZSi ZS j-  ZSk
i=l j=l k=l
N j / N  Y  NJ 
^ pE E S i - 4  ES i (36)
where J represents a positive (ferroelectric) interaction which for mathematical and 
computational simplicity is assumed to exist among all dipoles in the N-dipole cluster, p 
is the dipole moment of an unit cubic cell. We define n+ as the number of dipole moment 
parallel to the field, i.e.
N = n+ + n_
E S i = m = n+ -  n_ = 2n+ —N 
i=l
(37)
Here we have assumed that each dipole has direction either along +z (T) or -z (T). "m" is 
the net dipole. Then, the partition function, Z, can be given by
z= X  (iL)eS m+SF(n,!"N)
n+=0 It+
(38)
Usually, pE/kT is much smaller than I, i.e.
= i + P i m
kT
(39).
4 2
Therefore, the mean aligned dipole excess and dc dielectric permittivity are given by
N! pE 2 (m2-N)j/2kTN
S
N
I  ---------— --------- .
n+=o n+ ! (N - n +)!
J/2kT
from Eq. (24), we have
(40)
e' (f  = 0 ,T ) = 1 +
p < m >
Ne0EV
(41)
where p<m>/NV is the cluster polarization. V is the unit cell volume ~(4A)3.
If (T-T0)/T0» 2 /N , where T0=NJZk, we can approximate the sum by an 
integral with limits extending to infinity, and replace the binomial coefficients by a 
gaussian, i.e.
!  ■ N l 
n+=o n + ! (N -n +)!
r = - _  (2 °+ -^
2Nt i i e 2N ^
(dn+ = ^ d m )
- ” £ p “
(42)
Here we use the relation
y—oo
Therefore, Eq. (40) becomes
(m)
lo“ e
[m2-Nj J/2kT—-
2Ndm
S j 0-H 1V =N ^ kT 1m2dm
( CO  2N 2kV
L e  dm
PE .( I
2kT 2N  2kT1 r1
pE N
4k  T - rL
(T0=NJZk) (43)
At higher temperature for which Eq. (43) is valid we obtain the Curie-Weiss 
law by substituting Eq. (43) into Eq. (41)
4e0 V k (T -T 0)
(44)
A dimensionless logarithmic derivative or "steepness parameter" S can be 
defined as
s _ T ( dT ) d(lnE ') 
e' d (lnT )
(45)
which is -I in the Curie-law regions at very low and very high temperatures and which
should have greatest magnitude near T0.
44
Equivalent Circuits of Dielectric Function43
Strong dispersive dielectric behavior at low frequencies had been noted for a 
long time in many dielectric materials. As one knows, the interfacial effect also shows a 
strong dielectric frequency dispersion and could be reproduced by an equivalent circuit 
of capacitance in series with parallel G-C (see Fig. 22f).43 This kind of inhomogeneous 
medium effect is already known under the general name of Maxwell-Wagner effect and it 
can be developed as a direct counterpart to the distributions of Debye-like relaxation 
behavior in a dipolar material. This model therefore, has all the superficial attraction, as 
well as all drawbacks of the distribution theories. Several equivalent circuits which 
represent several physical situations are given in Fig. 22. Following are some relations 
which might be helpful to understand Fig. 22.43
(for a series R-C circuit)
(for sample. A:area, d:thickness)
Y = G+mC
Z = T"1 =R +-----
(oC
F = (£"+Z£'X o -^ -  
d
C* =C+ i—  and 
(0
Brillouin Light Scattering
Spectrum and Principle of Brillouin Scattering
In this section we will present a theory of Brillouin light scattering from the 
viewpoint of classical electromagnetism. For practical interest, we will be limited to the 
case of the radiation zone (R » X » r ) ,  i.e., the wavelength of the emitted radiation X (~ 
5000 A) is much greater than the dimension r (~1 A) of the radiating source. R is the 
distance between observation point and origin. In this case, the vector potential A  can 
be given by45
Circuit Z Y X M  Comments
- r^o^wv-ll-
L R  11C
T - l t "
\/R
I___
tr
Z>
K
Resonance
—----------  "Leaky"
capacitor
Figure 22. Schematic representations of the properties of simple circuit combinations of 
ideal, frequency-independent elements of capacitance C, conductance G or resistance R, 
and inductance L, as shown in first column. The second column gives the complex 
impedance plot Z, the third the complex admittance plot Y and the fourth the complex 
capacitance C*. The fifth column gives the corresponding plots of log C'(to) (solid line) 
and log C"(co) (dotted line), against log CO. Arrows indicate the sense of increasing 
frequency. Vertical and horizontal axes are imaginary and real parts, respectively.
4 6
gikR oo z : v xm  __ m
A(R,C0) = —— X .- ----— JJ (r ,(0 )(n -r)  d3r (46)
cR m=0 m!
where ft is the unit vector of R. "r" is the position of the radiating source inside the 
medium. Since the terms of km(h - f )m go as (d/X ,)m « I, the dominant term is for 
m=0 (dipole radiation): Then the vector potential becomes
ikR
A(R, a>) = ——  j I  ( f , a))d3r 
cR
JkR
= - i k ------P(r,<D); [ J J ( r , a))d3r = - iro J rp ( r , a))d3r] (47)
cR
with
P = Jfp (f,(o )d3r (48)
P is the electric dipole moment. In the radiation zone, the fields are B = VxA and 
E =  Bxn,  i.e.
E(R,t) = l n x
-ito(t-—)
-Jfp (r,ti))e  c d r
^ x l h x 3jI P i
(49)
Here we use the relation E (R ,t)  = E(R,co)e 1C0t t' = t —
R - f
is the
c c
retarded time inside the medium. If the incident fight intensity is weak, then the local 
polarization would be written approximately by
47
where d  is the tensor of polarizability. Ein is the electric field of the incident wave 
within the medium. If it is a plane-wave field, then
EinCr,!) = E0ei(iE«?"tt,ot) (51)
where k0 = nco0 / c is the wave vector of the light wave and n is the refractive index of 
the medium. A light scattering diagram is shown in Fig. 23.
P =  O c E in (5 0 )
E
--->
------- >
------- >
---- >
Figure 23. Light scattering diagram.
To analyze the origin of the scattering, it is convenient to decompose a  into its 
time average part <ct> and the time-space fluctuations 5oc(r,t) produced by the thermal 
fluctuations in the medium. The thermal fluctuations cause off-diagonal components to 
appear in the polarizability tensor, so we have
p ( r ,t ' ) = [ <&>  + 5 d ( f , t ')] • E0ei(ko'F"“ 0°  (52)
dP
dt'
. d (5d)
-Ito0 < a  > -Ito0S a + — E0Ci(kO7-wOf)
dt'
(53)
4 8
d2P
<  OC > -O n2Soc -  2i(on-CO, 2 ^  ^
0 dt' d r
.g y ( k „ i - ro/ )  (54)
Since the characteristic frequencies of thermal fluctuations are small (<1013 Hz) 
compared to the light frequency in the optical region (~1015 Hz), we can regard Sa as a 
very weak function of time, i.e.
d2P [-(O02 < & > -CO02S a ]• g oe1^ o'r““ °t) = -co02P ( f ,t ') (55)
Substituting Eqs. (54) and (55) into Eq. (49) and carrying out the integration over the 
illuminated volume V at the retarded time t', we obtain
E rotal = J vE (R ,t)d3r = - ( ^ - )  - f i x fi x  Jv [< a  > + 8a ] • E0e
n|R-r|
„ _ i[k0-r-ro0(t— !---- 91
d r
v C y R
fi x jv [< a  > +8a] • (56)
Here we have used the relations
n; f i ^ -
R - r
(57)
The integral in Eq. (56) gives the superposition of phases of waves scattered from each
illuminated point in the crystal. The contribution to the integral from the <&> term is
zero except in the forward direction. This light oscillates at the same frequency as the
4 9
incident field and results only in elastic scattering. Scattering out of the incident direction 
results entirely from fluctuations 5OC in the polarizability.
We can decompose the fluctuations into their spatial Fourier components
5& (r ,t ')
_L 
2 n j
3
^2
q-r±to„(q)(t- HM -
E j5 a ( q ) e L
v- .
d3q (58)
where q  is the wave-vector of the fluctuation and to^(q) is the frequency of the 
fluctuation corresponding to this wave vector, i.e. ©^(q) = c|q|. The index fl denotes 
the possibility of a number of branches in the dispersion. In general, tog(q) can be 
complex to include a description of the damping of the fluctuation, to^(q) is double- 
valued with ±  for the degeneracy in the dispersion for positive and negative running 
waves.
p  Total 
c Scatt. n x Jv j8d(q) • g oei(ko?-<Oct)+iK'.(R-r)e
ItqTtmll (q)tT-
_nm/q)|R-r|^
d3rd3q
(59)
The exponential part of Eq. (59) can be rearranged as
i(k
e o
• f - m ot) + iK '- (R - f )  i [ q -T ttD fi(q )t +
i(k 0 • r - ©0t) + iK' ( R - r )  i [ q - r ± Wfl(q )t + k ^ • (R - f )]
i[k  • f  -  CO t] i [ q - f±  ©^ (q)!] i[(K '+ k  M R - f ) ]  
e °  °  e ^  e ^
_i[ko T -0ot] i[q r ± < y q ) t ]  ! [K -(R -J )]
![K -R -(G )0 + © ^(q))!] i[k o + q - K ] - f
(60)
-» Il COil —. —► —►
Here we have used k^ = — -  n and K = K' ± k ^ . Thus
E S  = nxLJ8«(q ). E ^ ^ - ^ ' V S - ^ W q
S X _J5a(q). E ^ ^ ^ ^ V q  • (X p  
q 71
¥ ? fix
h xJ56c(q)• E0ei[KR (“ 0+®^®)t]d3q - (271)18 (k0 + q -K )  
q
-C2TrH(^a) - Z i i x  nx56c(q)E0ei[K'R 
c R m L q=K-k0
(61)
We have used the delta function relation, i.e.
, ei[E„-«5-Rird3r=(2ii), 5(E(i +  g _ ^ (62)
Thus the wave vector which produces the scattering in the direction n  is that which 
satisfies the implicit equation
q = K - k 0 (63)
IkI = K = —( g)0 ±  g)u ) =
1 1  C ^ c
(6 4 )
51
where K  is the wave vector of the scattered photon and k0 is the wave vector of the 
incident photon, q  is the wave vector of the lattice fluctuation. Eqs. (63) and (64) 
represent the conservation of momentum among these three wave vectors, and 
conservation of energy, respectively. The plus and minus signs correspond to the. so 
called anti-Stokes and Stokes lines in the spectrum, respectively. Momentum 
conservation among photons and phonon is illustrated in Fig. 24.
q
Figure 24. Momentum conservation of photons and phonon.
In Eq. (64), there are three (QiaXq) for the same wave vector q , i.e. one 
longitudinal and two transverse modes. The f i x  f i x  boL(q) term gives the polarization 
of the scattered radiation. The same constraints can be derived by viewing the phonons 
as moving diffraction gratings (due to density fluctuation) with a velocity V = co /q , and 
the incident photons undergo Bragg reflection and Doppler shift.36
. From Eq. (63), we can calculate the magnitude of the wave vector q ,  i.e.
q2 = K2 + k o -2 K k ocos0 
x2 f  IMto0n S0 a S
T c
2^ -  2nillsf s(0o cose, (6 5 )
/
where Hi and ns are refractive indices for incident and scattered light respectively.
52
Because of co^«m0 (CDjlZa)0-IO'5), so we have CDs-OD0, i.e.
q = — (nf + Us - 2 n ins cos6)2 . 
c
(6 6 )
In an optically isotropic medium such as a cubic system, the index ellipsoid is a sphere 
[(xf + X2 + X3) /  e = 1] and Hi = ns = n .14 Then Eq. (66) becomes
q
q
2 n m O z I 1 -  2 HCD0 . 0— cos0 r  = ---------- s in—,
2 c 2
(67)
c 2
2 JtAv iCDIII£ (68)
.V  ’
where V is the phonon velocity (sound velocity) and Av is the phonon frequency (or 
Brillouin frequency shift). From Eqs. (67) and (68), we have
Av V 0 . 6—  = — 2nsin—. , (69)
v c  2
In an anisotropic medium, the index ellipsoid shown in Fig. 25 has important 
properties. The directions (polarizations) of the electric displacement vectors D1 and D2 
of two waves are along the semi-axes of the ellipse obtained from the cross section of the 
index ellipsoid inside the crystal with the equatorial plane which is perpendicular to the 
wave vector. The indices are equal to the lengths of the semi-axes of the ellipse.14
For optically biaxial crystals (triclinic, monoclinic and orthorhombic systems), 
the three principal axes of an ordinary ellipsoid are 2-fold rotation axes. An ordinary 
ellipsoid [xj Z n ^+ x ^ /n ^  + Xg Zn^ = I, H1 ^  n2 ^  n3] has two circular equatorial cross
sections which are symmetric with respect to two of the principal axes and which contain
53
the third axis. The directions orthogonal to these particular cross sections are the optical 
axes, i.e. only one index remains (the radius of the circle), waves with all polarization 
directions have the same velocity and there is no double refraction.
Figure 25. Index ellipsoid. Two waves, polarized along D1 and D2 (mutually 
orthogonal), can propagate in the k direction (applicable to both incident and scattered 
beams), with different velocities.
For optically uniaxial crystals (trigonal, tetragonal and hexagonal systems), one 
of the principal axes is a n-fold rotation axis An (n>2) and is the axis of revolution of the 
index ellipsoid. If the propagation direction is along An, the cross section of the index 
ellipsoid is a circular section, so the index is the same for any polarization direction. 
Thus the crystal will behave as an isotropic material for a wave propagating along An 
which is the optical axis. In this case, the equation for the index ellipsoid is
[(x  ^+ X2)/ nn + X3 /ng  = I] .14
In Brillouin scattering, one might worry about the problem of different indices 
and velocities inside the crystal caused by optical anisotropy. However, the difference 
between the principal indices is always small. The index ellipsoid is very close to a 
sphere. For instance, the indices of quartz are no=1.5442 and nc=1.5533.14
5 4
The Relation Between Sound Velocity and Elastic Stiffness14
Brillouin scattering is a tool for measurement of elastic constants (stiffness) and 
piezoelectric stress constants. The calculation presented here is a basic theory for a non­
piezoelectric material. The dynamic equation of lattice motion for a phonon can be 
written as14
d2ui _  CjTij 
P dt2 dxj
(70)
where U i is the displacement in the ith direction, p is the crystal density and Tij are 
elements of the stress tensor. One can use the generalized Hooke's law to form the 
relation between the stresses and the strains, i.e.14
Ty = CfklSkl (i,j,k,l=l,2,3 or x,y,z) (71)
and
s  ' (^ + iiL )
2 dx
(72)
where the Cfkl are the elastic stiffness constants (Young's Modulus) at constant electric 
field and Skl are the strain components. Substituting Eqs. (71) and (72) into Eq. (70) 
yields the phonon wave equations
(7 3 )
One can exchange "k" and "I" symbols, i.e.
Since elastic stiffness constants is a symmetry tensor,
CEijkl C
E
ijlk ’
and Eq. (73) becomes
d2ui
-ijkl
d2uk
dxjdxj
(74)
We seek plane-wave solutions of the phonon equation of motion in which the 
displacement vector can be written as
Ui = UioC1^  r Q)=27tAv (Brillouin shift) (75)
where uio is the displacement and i is the polarization index of the phonon. From Eqs. 
(68) and (74), we have c
’ PV2U1 = CftlIjI1Uk (76)
where V is the sound velocity. Ij and I1 are the components (projections) of the unit
phonon wave vector q=q! I q I along the j- and 1-axes. From now on, we will omit the 
superscript "E" for elastic stiffness constants, i.e. Cykl = Cyld.
For i= I , from Eq. (76) one has
56
pV U1-  C1111Z1Z1M1 + Cnx2IxI2Ul + C1113Z1Z3M1 
^C xx2xIxIxU2 + Cxx22IxI2U2 + Cxx23IxI3U2
^C 1x3xIlIxU3 + Cxx32IxI2U3 + C1133Z1Z3M3 
^C x2xxI2IxUx + Cx2x2I2I2Ux + Cx2l3I2I3Ux
^C x22xI2IxU2 + C1222I2I2U2 + Cx223I2I3U2 
r^Cx23xI2IxU3 + Cx232I2I2U3 + Cx233I2I3U3
^C x3xxI3IxUx +  Cx3x2I3I2Ux + C1313Z3Z3M1 
^C x32xI3IxU2 + Cx322I3I2U2 + Cx323I3I3U2 
^C x33xI3IxU3 + Cx332I3I2U3 + C1333Z3Z3M3 (77)
It is more convenient to use matrix notation than tensor notation, i.e.
Tensor notation: 11 22 33 23,32 13,31 12,21 
Matrix notation: 1 2 3 4 5 6
After changing tensor to matrix notation, Eq. (77) becomes
pV U1= C11Z1Z1M1 + C16Z1Z2M1 + C15Z1Z3M1 
w^Cx6IxIxU2 + Cx2IxI2U2 + Cx4IxI3U2
w^Cx5IxIxU3 + Cx4IxI2U3 + C13Z1Z3M3 
w^wC6xI2IxUx + C66I2I2Ux + C65I2I3Ux 
+C66I2IxU2 + C62I2I2U2 +  C64I2I3U2 
+C65I3IxU3 +  C64I2I2U3 + C63I2I3U3 
+C5xI3IxUx + C56I3I2Ux + C55I3I3Ux
+C56I3IxU2 + C52I3I2U2 + C54I3I3U2
+C55I5IxU3 + C54I3I2U3 + C53I3I3U3 (78)
The elastic stiffness constant tensor depends on the crystal symmetry and the 
reference frames. For orthorhombic (for all classes if three orthogonal diad axes are 
chosen as the reference frame), cubic (for all classes) and tetragonal (for classes 4mm,
57
42m, 422 and 4/mmm if the 4-fold rotation axis and two orthogonal diad axes are 
chosen as the reference frame), the elastic stiffness constant matrix can be expressed as14
T C 11 C 12 C 13 0  0  0  1
C u  C =  C =  0  0  0
C  _  C 13 C 23 C 33 0  0  0
u V -  0  0  0  C 44 0  0
0  0  0  0  C 55 0
0  0  0  0  0  C 66
The 6x6 coefficients Cij are symmetric with respect to the main diagonal. Thus, Eq. (78) 
becomes
p V  U 1- ( C 11Z1Z1 +  C 55Z3Z3 +  C 66Z2Z2 )Wj 
+ ( C 12ZiZ2 +  C 66Z2Z1 )u2
+ ( C b Z1Z3 +  C 55Z3Z1 ) « j  (79 )
Similarly, for 1=2 and 3, we have
P V 2U2 =  ( C 66I 1I 2 + C 12I 2I 1 ) Ul
+ ( C 66Z1Z1 +  C 22Z2Z2 +  C 44Z3Z3 )m2
+ ( C 23Z2Z3 +  C 44Z3Z2 )w3 ( 80 )
P V 2U3 =  ( C 55I 1I 3 + C 13I 3I 1 ) u ,
+ ( C 44I 2I 3 + C 23I 3I 2 ) u 2
+ ( C 44Z2Z2 +  C 33Z3Z3 +  C 55Z1Z1 )«3 ( 81 )
For non-trivial-solutions of this set of Eqs. (79), (80) and (81), the following determinant 
of the coefficient matrix must vanish for an arbitrary phonon wave direction q in the
58
crystal. This secular equation yields three eigenvalues which are associated with sound 
velocities of three phonon waves propagating in the same direction, and three mutually 
orthogonal eigenvectors which are the polarizations (displacement direction) of waves.
QM  +Csstt +C66I2I2 -pV2 
C66I1I2+C12I2I1 
C55I1I3 +C13I3I1
Q t t +Q6I2I1
C66I1I1+C22I2I2+C44I3I3-PV2 
C44I2I3 + C23I3I2
Ci3Iii3 +C55I3Ii 
C23I2I3 +C44I3I2 
C44I2I2+C33I3I3+C55I1I1-PV2
(82)
Four phonon propagation directions are discussed below:
(i) Phonon wave propagating along [100] direction (#//[100]), i.e. I1=I, 
I2=Is=O, then Eq. (82) becomes
C11-P V 2 . 0
0 C66-P V 2
0 0 C55-P V 2
(83)
The roots of this equation, i.e. the corresponding values of pV2, UreC11, C66 and C55. 
The unit eigenvector u = (UlsU2sU3) for the eigenvalue pV2 =C11 can be obtain by
0 C66-P V 2
^o 0
0
0
C55-PV2J W
(0 }
0
vOy
Thus
(uOm
U2
<U3,
— 0IoJ
(84)
Similarly for pV2 = C66 and pV2 = C55, the polarization vectors are
59
r U1)  f  CT
xu3y
(uOf o )
U2 0
^U3y Iv
for pV2 = C6
for pV2 = C5
Finally, we have
P lzL = C 11 % =  (1,0,0) longitudinal (LA) mode
p Vi = C66 nTl = (0,1,0) transverse (TA) mode
p V  = C55 nT2 = (0,0,1) transverse (TA) mode
The orientations of the phonon propagation wave vector and the three wave 
polarizations are given in Fig. 26.
z
n =[010]
n =[100]//
Figure 26. Orientations of phonon propagation wave vector q / /[100] and three wave 
polarizations.
(ii) Phonon propagating along [001] direction (^//[001]), i.e. I3=I, I1=I2=O.
Then Eq. (82) becomes
O (85)O
O
60
b
c « - p v =
0
0
0
C=-PV=
The eigenvalues pV2of this equation and the corresponding eigenvectors are
PX2 = C33 nL = (0 ,0 ,1 ) longitudinal (LA) mode
P Vn = C44 nTl — (0,1,0) transverse (TA) mode
P Xz — C55 fif2 — (1,0,0) transverse (TA) mode
(iii) Phonon propagating along [110] direction (^//[110]), i.e. I 1=I2= Ig=O.
Then Eq. (82) becomes
|(C „ + C « ,) -p V = 0
|(C «  +c„) ^ (C 11H-C11) - P V ' 0 =0 (86)
0 0 I ( C ssH-C14) - P V '
[(C11 + C66-2pV 2)(C66 +  C22-2 p  V2M C 12 + C66)2](C55 + C44- 2pV2)=0 (87)
In case of tetragonal symmetry with classes 4mm, 42m, .422, 4/mmm, one can use the 
following further symmetry relations
Thus Eq. (87) becomes
61
[(C11 + C66 -2 p  V2)2- (  C12 + C66)2] (2 C44 -2p  V2)=0 (88)
The eigenvalues pV2of this equation and the corresponding eigenvectors are
Ct2 + 2C66 
2 n L
LA
2
Ht2 — (0,0,1) TA
The orientations of the phonon propagation wave vector and the three wave 
polarizations are given in Fig. 27.
Figure 27. Orientations of phonon propagation wave vector q /  /[110] and three wave 
polarizations.
(v) Phonon propagation in the (001) plane (^//[h,k,0]), i.e. I 1=CosO, I2=SinO, 
I3=O (0 is the angle between phonon wave vector and [100] a-axis). Then Eq. (82) 
becomes
z
A n72 ” [ o o i ]
y
C11C O ^ C 66Sin 0 - p V  (Q2H-C66)CosOsinO 
(C12H-C66)CosOsinO C66Co^OH-C22Sin2O-PV2 
0 0 C44 sin2 0+C55 cos2 0 -  pV2
0
0 =0 (89)
62
In the case of tetragonal symmetry with classes 4mm, 42m, 422, 4/mmm, i.e. C11 = C22, 
C44 = C55, we have
C11 cos2 0 + C66 sin2 6 -  p V2 
(C12 + C 66) cos 0 sin 0 
O
(C12 + C 66) cos 0 sin 0 O
C66 cos2 0 + C11 sin2 0 -  p V2 O
O C44-P V 2
=O (90)
[(C11 cos2 0 + C66 sin2 0 -  pV2)2-(C l2 + C66)2 cos2 0 sin2 OKC44 -  pV2)=0 (91)
Thus, the eigenvalues pV2of this equation and the corresponding eigenvectors are
pV2 = C11 cos2 0 + C66 sin2 0 + (C12 +C 66IcosOsinO nL = (cos0,sin 0,0)
pV2, = C11 cos2 0 + C66 sin2 0 -  (C12 + C66) cosOsin 0 nT1 = (sin 0 , - cos0,0)
or ( -s in  0,cos0,O)
PVr2 =C 44 nT2 = (0 ,0 ,1 )
The orientations of the phonon propagation wave vector and the three wave 
polarizations are given in Fig. 28.
z
Ani =[001]
n
y
Figure 28. Orientations of phonon propagation wave vector q / /[cos0,sin 0,0] and 
three wave polarizations.
63
So far, the calculations presented above are only suitable for a non-piezoelectric 
material. For a phonon propagating in piezoelectric materials, Eq. (71) for the stress 
becomes14
Tj =C J iSh - (92)
where <j) is the electric potential. Then, Eq. (74) becomes
P ^ L  =  C j  ^ + e, ^
Sxjdx1 ail dxjdxa
(93)
However, the electric displacement
p -  ePHs H +  epaE (x -  ePH ^  ePa ^ (94)
must obey the Poisson equation SDp /  Sxp = 0 for an insulator, i.e.
S 2Uk
ePw S ^  " e
s S2<1)
^9x«Sxp
0 (95)
Therefore, with (3—>j for Eq. (95), the equation (93) of motion can be written as
P
/-iE S2Uk , e aije jH S2Uk _  zr ,E , eKije JH \  S Uk
'SM SxjSx1 -  ^  ^SxjSx1
(96)
6 4
So, the difference in the phonon equation of motion between non-piezoelectric and 
piezoelectric crystals is
c UkI- ^ r Ijkl -  C ijkl +  —
eja
(97)
where eaij is the piezoelectric stress constant and Ej01 is the dielectric permittivity at
constant strain (high frequency limit). Following is a simple case for a phonon 
propagating along the [100] direction (#//[100]) with tetragonal symmetry (class: 4mm) 
in a piezoelectric crystal, i.e.
" 0 0 0 0 ®113 0" ' 0 0 0 0 e15 0"
0 0 0 ®113 0 0 0 0 0 e15 0 0
_e311 e311 ®333 0 .0 0 _e3, e31 e33 0 0 0
then Eq. (83) becomes
o
o
c « -p v =
0
0
C55+ I l-P V 2
(98)
The eigenvalues pV2of this equation and the corresponding eigenvectors are ,
P ^  =  C11 fiL =  (1 ,0 ,0 )
P Vn =  C66 n-pi =  (0 ,1 ,0 )
+UIlZ n T2 =  (0»0>1)
LA mode 
TA mode
TA mode
65
Selection Rules of Scattered Light
In general, the propagation of light radiation in a crystal can be described 
completely in term of the dielectric impermeability tensor Py. By definition, we have14
:]; or E P =  I  (99)
where e is the dielectric tensor. According to the quantum theory of solids, the 
dielectric permittivity tensor depends on the distribution of electric charges in the crystal. 
The action of an external electric field results in a redistribution of the bond charges and 
a slight deformation of the ionic lattice. This result is reflected in a change of the 
dielectric impermeability tensor. This is the electro-optic effect. If we neglect higher- 
order term, then we have
Spy -  IykEk + PijrsTrs (100)
where Iyk and Pyrs are linear electro-optic and photo-elastic (elasto-optical) coefficients, 
respectively. In Brillouin scattering, we only consider the second term, namely
SPij =  PijrsTrs (101)
The photo-elastic coefficient tensor depends on the crystal symmetry. For orthorhombic 
(all classes), cubic (all classes) and tetragonal (classes: 4mm, 42m, 422 and 4/mmm) 
etc., the photo-elastic coefficients matrix can be expressed as
P11 P12 P13 O O O
P21 P22 P23 O O O
P31 P32 P33 0 0 0
0 0 0 P44 0 0
0 0 0 0 P55 0
0 0 0 0 0 P66
66
Following is a sample calculation showing how to obtain selection rules for 
different phonon modes. In this illustration we consider the case of phonon propagation 
along the [010] (b-axis) direction (q /  /  [010]):
For a longitudinal phonon polarized along [010], U1=U3=O, U2ocB1^ r , thus
- Tu l=O T12l=O T13l=O
T2Il =O T22lTK) T23l=O
T3Il=O T32l=O T33l=O
From Eq. (101), we have
Sp11 Pl2T22L SP12 = 0
OIi2
s-
Sp21 = o Sp22 = P22T22L
OIlefCO
SP31=O SP32=O SP33 = P:
Thus, in this case the fluctuation of the dielectric impermeability tensor P is
0 0 
SP22 0 
o sp33
The total dielectric impermeability tensor P is
Pij=Po+^
Pno+ SP11 0 ' 0
0 P22O+ SP22 0
0 0 Paso + SP33
67
Since e • (3 = I , the total dielectric tensor e turn out to be
8ij:
I
0
I
Q
Pno +  SP n
Q Q
P 220 +  S p 22
In AV
P 33O +  S P
I  S p 11 
no Pno
0
0
0
I  S p 22.
^22O Pz2O
0
0
0
I  S P 33
P33O P 33O
I
O0
I___ SPn o 0
Pno =5
O
 
• O
0  T 22 0
P 220
=J
O O
t—I 0 0 6f33
P 33O . L ' PL]
-Ejj0 + Seij
For our experimental situation, the polarization of the incident light is along 
[001]. Thus the fluctuation of the electric displacement is
SD = Se -E -SP33
P330
E
0
0
I
(102)
68
From Eq. (61), we know
Escatt o c fixCnxSD) oc [001]
Thus, the polarization of the scattered light is along the [001] direction. This means that 
for the longitudinal phonon mode the scattered light has the same polarization as the 
incident light.
For transverse phonons polarized along [100], U2=U3=O, U1Oce1^  r , ,
T117=O T12T*0 T13t=O
T217156O T22x=O T23t=O
T31t=O T32t=O T33t=O
and
P Il 0 fiP 12 =  2P66T12T fiP 13 =
fiP21 =  2P66T12T
0Il2# fiP 23 =
fiP31 = 0
OIlSi
28- fiP 33 =
Here we have used the symmetry relation, T12t=T21t. Thus, in this case the fluctuation of 
the dielectric impermeability tensor p is
0
5 Pij= %0
fiP 12
0
0
0
0
0
69
The total dielectric impermeability tensor P is
Plj=Po+ sP1
Pm SP„ O
sPm P „  O 
o o
Since e -p  = I ,  i.e.
&11 1^2 1^3 " Pno ' 6Pb 0  " 'I 0 0"
821 E22 E22 SPoi Pom 0 = 0 I 0
_e 31 ^32 3^3 _ O O f
5
I __
__ 0 0 I
(103)
Solving Eg. (101), we obtain
e i j - e ijo +  5 e ij~
I S p 12 0
Pirn P110P220
B P , I 0
PlloP220 P220
I
Q0 0
P330 .
0 Sp12
PuoPa2O
0
SP21
P110P220
0 0
0 0 0
The fluctuation of the electric displacement is
' 0 ' " 0 "
0 0
I 0
SD = Se-E (104)
70
Thus, there is no scattered light for this phonon mode.
. For transverse phonons polarized along [001], i.e. U2=U1=O, u3°=e^
and
T1im=O T12M=0 T13M=0
T2IM=0 T22M=0 T23M*0
T3IM=0 T32M5i0 T33M=0
S p 11 =  0 8 Pi2 =  Q 8 P 13 =  0
s p 2i =  0 8 P22 = ° 8 P23 =  2 P55T23M
SP31 =  0 8 P32 =  2 P55T23M 8 P33 =  0
Here we have used the symmetry relation, T23m=T32m. In the same way we find the 
fluctuation in the dielectric permittivity tensor
0
0
0 -
0
0
SPs:
0
%
P220P330
P220P330
0
Thus the fluctuation of the electric displacement is
SD = Se • E O
 o
'
. 
- _
_I
_  8 Pzs " o 'II P220P330 0
(105)
From Eq. (61), we know
• 71
Escatt 00 n x ( n x8D)°=[010]
thu s , the polarization of the scattered light is along the [010] direction. This means that 
for this transverse phonon mode the scattered light polarization is perpendicular to the 
incident light polarization.
In the same way we can obtain polarizations of scattered light for longitudinal 
(LA) and transverse (TA) modes for phonons propagating along different directions. In 
summary, one can find the following selection rules for Brillouin fight scattering: (i) The 
polarization of the scattered fight by the LA phonons in a crystal which has one of these 
symmetries; orthorhombic (all classes), cubic (all classes) and tetragonal (classes: 4mm, 
42m , 422 and 4/mmm), is the same as for the incident light, (ii) The polarization of the 
scattered light by one of the two TA modes is perpendicular to the polarization of. the 
incident light. The other TA mode does not produce scattered fight.
Static Coupling Theory of First-Order Phase Transition
In principle, the acoustic coupling contributions include both static and dynamic 
effects. Although the dynamic coupling always produces a negative contribution 
(softening) to the real part of the complex elastic stiffness change Ac*, the static effect 
can have either sign contribution, depending on whether the phase transition is of first or 
second order. According to the Landau free energy expansion with a single order 
parameter, the second-order transition gives a negative contribution in elastic stiffness 
change but the first-order transition can give either a positive (hardening) or negative 
contribution for both linear and quadratic couplings. Several first-order examples that 
show different elastic coupling behaviors can be found in Refs. 17 and 46.
The details of second-order coupling theory have been discussed elsewhere.47 
In the following, the first-order case, i.e. a4<0 and a6>0, will be discussed. In the spirit
72
of the Landau theory of phase transitions, the free energy, with one order parameter, can 
be expressed by
F (T |,  ( I , t )  =  F0 (T) +  J d 2 (T)I12 +  I U4T14 +  I U6T16 + . .  +  I  S 2JLl2 + I b 4JO.4 +• • •  +  Fc (T), JO)
(106)
'2  2
Fc (T)5(O) =  pTi|0 +  yn jo+ arijo +••• (107)
where r\ and jo designate the order parameter and the strain, respectively. The coupling 
terms Fc describes the interaction between strain and order parameter. P is the linear 
piezoelectric coefficient and 7 is the quadratic electrostrictive coefficient. The elasticity 
C=C00-AC is given by
yTl = Tls
(108)
where p is the sample density and V is the sound velocity. C0 0  is obtained in the high 
frequency limit. From Eqs. (106) and (108), we get
C=C0 0 -P 2X for linear coupling (109)
C=C00-^ y2Xns2 for quadratic coupling (HO)
where T)s is determined by (dF/dT))q^r|s=0. From Eqs. (106)-(1108), we can get the
following results for the low temperature ordered phase, i.e. T)s#0
73
X = ^ a 4Tis2 + 4 a6r|s) *
- a 4 ± V a 4 - 4 a 2a6
( 111)
( 112)
=> C=C - P2as
a4 -  4a2a6 ± 3a4y  a4 -  4a2a6
linear coupling
C=C00I
2-Ja4 - 4 a 2a6
quadratic coupling
Therefore, for a first-order transition, the elasticity C=C00-AC could be step-like up 
(positive contribution) or down (negative contribution) across the phase transition region 
for quadratic coupling. However, for a second-order transition, the elasticity always 
exhibits a downward step around Tc as temperature decreases for quadratic coupling.
Figure 29. (a) Elasticity vs. temperature for (i) pTi(i linear coupling, (ii) yr|2|i quadratic 
coupling, solid line for either second-order or first-order and dotted line only for first- 
order and (iii) a im 2 coupling, (b) The dynamic effects for a second-order r |2|X-type 
quadratic coupling in typical pure ferroelectrics. V is phonon frequency shift and 8 is 
phonon damping.
74
Fig. 29a gives the general behaviors of the elastic moduli for the three 
dominating lowest order coupling terms for both first- and second-order transitions. Fig. 
29b gives the dynamic effects for a second-order r |2(i-type quadratic coupling in typical 
pure ferroelectrics.48
Debye Anharmonic Approximation
In order to estimate the coupling effect in proton glass crystals, we will 
calculate the bare (uncoupled) phonon frequency, coa, by fitting the high temperature 
values (in the uncpupled region) of the measured frequency shift. The bare phonon 
frequency usually is defined for the phonon frequency in the absence of a phase 
transition. The temperature dependence of GDa can be described by the Debye 
anharmonic approximation as follows:10
GDa (T ,x) = GDa(0 ,x ) I - A (x)0 (x)F f  Q(X)VI T J. (113)
where 0 (x ) is the Debye temperature, x is the ammonium concentration, A(x) represents 
the amount of anharmonicity and F is the Debye function for internal energy,49
F ( f ) (114)
as tabulated, for example, by Abramowitz and Stegun.50
75
Dynamic Effects47
Two different damping mechanisms are usually responsible for acoustic 
anomalies, namely the Landau-Khalatnikov mechanism (relaxation portion) and order 
parameter fluctuations. A strong Landau-Khalatnikov mechanism contribution which is 
associated with a sharp damping maximum usually occurs for linear ri|i-type coupling 
(piezoelectric) material such as KDP family crystals. Here Tj is the order parameter and (I 
is the strain. On the other hand, the Landau-Khalatnikov mechanism also represents a 
relaxation characteristic of the order parameter to a strain.
The order parameter fluctuation damping which usually is associated with a 
broad maximum is a characteristic of a quadratic r |2{i-type coupling (electrostriction), 
squared in order parameter and linear in strain. For theoretical description, an approach 
which is based on the classical fluctuation-dissipation theorem can be used:
Bncmn(O) = < Sam(C)San(O) >dt (115)
It connects the dissipative (imaginary) part of the elastic modulus to the Fourier 
transform of a time-correlation function of internal stress fluctuation 5 am in a volume V. 
After several steps of calculation, the following result can be obtained:
Bncmn(CO) = ^  S  YijmYkln H  < Q1 ><  Qk > n r eiK,t < SQj(C)SQ*(0 ) > dt
-^kl IjlIc,!
. + C e itot <  SQi (I)SQj (I)SQk(O)SQ;(0 )  > dt} (116)
The sum of a Landau-Khalatnikov contribution in the first term and. a fourth-order 
correlation function describing the fluctuations is obtained.
76
CHAPTER 4
RESULTS AND DISCUSSION
Rbi .JN D 1LD2AsO,, Ix=O. 0.10. 0.28)
Brillouin Back-Scattering Along LAU001 Phonon Direction51
Actual LA[100] phonon spectra of the anti-Stokes Brillouin component are 
shown in Figs. 30-32 for x=0, 0.10 and 0.28, respectively. The data shown here are for 
several temperatures near the maximum value of half-width. The solid lines are fits of the 
Lorentz profile, from which the half-width Svobs and frequency shift Av can be obtained. 
The frequency shift for each compound indicates a positive coupling contribution with 
decreasing temperature. Comparing the half-width .for different x values, we notice that 
the average damping value increases as ammonium concentration increases. This 
dependence may result from the stronger local structural FE/AFE competition that can 
induce larger fluctuations, especially for intermediate x values such as 0.3<x<0.7 in 
RADP.3
The temperature dependences of the Brillouin shift and half-width for x=0 ,0.10 
and 0.28 are shown in Figs. 33-35, respectively. The solid lines in frequency shift are the 
calculations of Eq. (113) with parameters of Table I. These parameters A, Coa(T=O) and 
0  of Table I are from the fits of Eq. (113) to the high temperature (>270 K) measured 
values of the Brillouin shift. Here we assume that temperatures above 270 K are far 
above the coupled region since all phase transitions occur below 200 K. One finds that 0
77
and COa(0,x) tend to increase from the DRDA side to the DADA side, as expected in 
view of the higher frequency of the modes on the DADA-rich side. Also, the anharmonic 
factor A increases with ND4 content.
COa(T=O)
(GHz)
A(K-i) ' 0  (K)
DRDA 26.08 2.70x10-4 400
DRADA-0.10 26.37 2.76x10-4 480
DRADA-0.28 27.20 2.84x10-4 520
TABLE I. Parameters from the fits of Eq. (113) to measured values of frequency shift at 
high temperature.
Fig. 33 for x=0 (pure DRDA) shows a sharp damping peak at 164.8 K 
associated with an upward step in frequency which is expected, since the KH2PO4 (KDP) 
family has linear Tj(I-type piezoelectric coupling. Here Tj is the order parameter and p, is 
the strain. Such a sharp damping maximum for longitudinal acoustic phonons can be 
connected with the Landau-Khalatnikov relaxation-type mechanism. The dashed line in 
half-width of Fig. 33 is a qualitative estimate for this relaxation portion. We also try to 
estimate the pure lattice-anharmonic contribution by solid circles, which is the typical 
anharmonic half-width observed in Bdllouin scattering for dielectrics. For linear coupling 
with the assumption of a single relaxation time, the sound velocity and the attenuation a  
in the FE phase can be given by41
V j - V 02
I-KO2Tot"2
(117)
78
a CQ2V  1 V jj-V p
2 V3 ! + (O2Tof2
(118)
where t is the reduced temperature (Tc-T)ZTc, and T0 is the elementary individual-dipole 
relaxation time in the expression T=T0V1 for the relaxation time T. The velocities V00 and
V0 designate the high- and low-frequency limit velocities, respectively. From Eqs. (117) 
and (118) we can obtain the relation ’
T - T
COTq = — ----— . (119)
TC
By this relation we can calculate the elementary relaxation time T0. Here, 
Tm=164.8 K is the temperature at which the half-width is maximum and the FE transition 
temperature Tc is defined as the point at which the frequency shift curve is steepest. For 
the LA[100] phonons of DRDA, we obtain the following result:
Tc-T m-O.! K 
To~2.3xl0-14 s 
T -3 .8x10-12Z(Tc-T) s
The elementary dipole relaxation time T0 obtained from the LA[100] phonons 
of DRDA is short compared to that of other order-disorder ferroelectiics. For instance, 
T0 is 1 .2x l0 '13 sec for potassium dihydrogen phosphate (KDP) and 1 .3x l0 '12 sec for 
deuterated potassium dihydrogen phosphate (DKDP).41
79
Instead of a sharp peak, the damping data of Fig. 34 for x=0.10 exhibit a 
smooth growth from -200  down to -130 K, associated with a slowly rising anomaly in 
frequency shift above the Debye curve of Eq. (113). Such an increasing of damping 
which begins far above Tm= 146 K must be associated with the order parameter 
fluctuations. A qualitative estimate of this fluctuation contribution is given by the solid 
curve in Fig. 34 with maximum near 146 K. Beside this broad damping background, one 
can note that there is an additional peak appearing near 130 K which can be connected 
with the Landau-Khalatnikov maximum.
What are the origins of these two damping maxima in DRADA-0.10? The 
ND4+ deuteron NMR spectra of DRADA-0.10 showed a gradual disappearance of the 
doublet near 131 K where the single broad linewidth grows to its full size, from which it 
was concluded that below 131 K the FE phase portion is greater than FE in the crystal 
and becomes the dominant ordering.7 This result is consistent with the presence of 
PE/FE phase coexistence as evidenced by dielectric results which show that a gradual 
ferroelectric transition begins at Tm= 146 K and is mostly completed at -120 K. 
Furthermore, the real part of dielectric permittivity E11(T) shows deviation from the 
Curie-Weiss law below 160 K. This high-temperature dielectric anomaly below 160 K 
can be associated with the onset of short-range order due to partial freezing-in of the 
ND4 reorientations and implies a growth of local structural competition (between FE and 
AFE ordering). On the whole, one can expect that such FE/AFE ordering competition, 
which can suppress a long-range-order ferroelectric transition and generate the order 
parameter fluctuations, is responsible for both FE/PE phase coexistence and development 
of the broad damping peak centered at -146 K. Also, the rapid growth of FE ordering 
near 130 K is the origin of the Landau-Khalatnikov-Iike maximum in DRADA-0.10.
A even stronger damping temperature dependence which shows a growth from 
-300  down to -120  K (see Fig. 35) is observed for x=0.28, associated with a smoothly
80
rising frequency shift. This strong broad damping reveals that fluctuations are the 
dominant dynamic mechanism. As one knows, the dynamic fluctuation contribution for 
longitudinal acoustic phonons is a characteristic of a r |2|j,-type coupling, squared in order 
parameter and linear in strain.47 However, it is difficult to see a Landau-Khalatnikov 
maximum (associated with the Edwards-Anderson order parameter in this case) above 
such a strong fluctuation background. Taking into account earlier dielectric permittivity 
E11(T) results which showed a deviation from the Curie-Weiss law below 140 K, we 
conclude that neither FE nor AFE ordering but rather the local structural competition 
mechanism (between FE and AFE orderings) is the origin of this strong broad damping 
anomaly centered near 120 K. Since the fluctuations usually indicate a random force 
resulting from the substitutional disorder, the difference of damping behavior between 
DRADA-0.28 and DRADA-0.10 also implies that this random force is stronger in the 
x=0.28 crystal.
A main feature of the acoustic phonon spectra in DRADA (x=0,0.10, and 0.28) 
for LA[100] phonons is that the frequency shift shows a positive (hardening) instead of a 
negative coupling contribution (softening) at the phase transition. The sign of the 
coupling contribution may be related to the temperature responses of lattice parameters 
a(T) and c(T). A smaller lattice constant is usually associated with a stiffer interatomic 
force constant and implies a contraction Aa which leads to the LA[100] phonon 
hardening. However, the measurements we performed on pure DRDA for LA[001] Co- 
axis) phonons, not reported here, revealed a negative contribution at the FE transition 
temperature. This may be related to the expansion Ac observed in the related crystal 
KH2PO4 just below Tc.17
81
C
3
2c
3
O
U
LA[100]
x=0
165.0 K
164.9 K
164.8 K
164.6 K
164.4 K
Frequency (cm-1)
Figure 30. Anti-Stokes components of the LA[100] Brillouin frequency shift for
temperatures around the maximum value of half-width for DRDA. The open circles
are the measured data and solid lines are the Lorentz fits.
82
150 K
133 K
130 K
123 K
110 K
Frequency (cm-1)
Figure 31. Anti-Stokes components of the LA[100] Brillouin frequency shift for
temperatures around the maximum value of half-width for DRADA-0.10. The open
circles are the measured data and solid lines are the Lorentz fits.
83
136 K LA[100] 
x=0.28
125 K
120 K
115 K
100 K
Frequency (cm-1)
Figure 32. Anti-Stokes components of the LA[100] Brillouin frequency shift for
temperatures around the maximum value of half-width for DRADA-0.28. The open
circles are the measured data and solid lines are the Lorentz fits.
84
LA[100]
x=0
-  0.15 £
-  0.10
Temperature (K)
Figure 33. Frequency shift and half-width vs. temperature of the LA[100] phonons 
for DRDA. The solid line for frequency shift is the Debye anharmonic calculation 
with parameters from Table I. The dashed line and solid circles for half-width are 
the qualitative estimates for the Landau-Khalatnikov and pure lattice anharmonic 
contributions, respectively. The error bar indicates the error range for the half-width 
experimental points.
85
Temperature (K)
Figure 34. Frequency shift and half-width vs. T of the LA [100] phonons for 
DRADA-0.10. The solid line for frequency shift is the Debye anharmonic 
calculation with parameters from Table I. The solid line and solid circles for half­
width are the estimates of fluctuations and pure lattice anharmonic contributions, 
respectively. The dashed line is a guide to the eye. The error bar indicates the error 
range for the half-width experimental points.
86
LA[100]
x=0.28
N 28 —
£  27 -
3. 26 —
0.4
-  0.2
Temperature (K)
Figure 35. Frequency shift and half-width vs. temperature of the LA[100] phonons 
for DRADA-0.28. The solid line for frequency shift is the Debye anharmonic 
calculation with parameters from Table I. The dashed line is a guide to the eye. 
The error bar indicates the error range for the half-width experimental points.
H
al
f-w
id
th
 W
ph
 (
G
H
z)
87
N a1Z2B i1Z2TiQ3 (iNBTn)
Frequency Dependent Dielectric Results16
Fig., 36 reveals a new large anomaly at higher temperature region. In 
comparison with this anomaly, the "old" and well-known anomalies in the temperature 
regions indicated by arrows A and B in Fig. 36 are very small. The new anomaly in the C 
region was not seen previously in NBT, but similar anomalies have been observed in 
other perovskite crystals and ceramics. We will consider separately each of these three 
regions.
First, the anomaly in region A was found by Smolensky et al.12 At frequencies 
below 200 Hz we could hardly see this anomaly against the large background of 
dielectric permittivity. It appears as a pronounced shoulder in the region 0.2<f<l kHz 
(Fig. 37 inset). The two curves labeled 2 at 500 kHz were also measured on a NBT 
single crystal by Pronin et a/.20,21,25 These two sets of data agree well in the peak value 
and in the specific difference in behavior on the left side of the e ' peak in heating and 
cooling runs. Curve 3 (at 160 kHz) reproduces a recent result for an NBT single 
crystal,16,28 which is quite different from our data. Curve 4 in Fig. 37 represents very 
different behavior of NBT (f=200 kHz). We attribute these differences to sample quality 
and consider our NBT sample to be of good quality. Measurements in the frequency 
range 0.1-100 kHz show no evident dispersion for the peak position at T -640 K and are 
in agreement with most other results.16
Secondly, in region B a dielectric permittivity anomaly was found with 
frequency dispersion.16,12 Remanent polarization disappears completely above 550 K as 
shown in Fig. 38, so this dispersion can be attributed to domain wall displacement. Our 
measurements agree perfectly with those previous measurements and indirectly confirm 
the onset of ferroelectric behavior below 550 K.16
88
In region C, a huge frequency dependent peak for the real part e' of the 
dielectric permittivity was found in the higher temperature region (Figs. 36, 39 and 40a). 
A corresponding e" contribution also shows a round shoulder between 650 and -800 K 
(Fig. 40b). This hump is only seen clearly at frequencies below I kHz. The growth in e" 
was drastic enough to arouse suspicion about the correctness of the e' measurements. 
To check their correctness, we compared the dielectric measurements in NBT and in a 
sample of the well-known relaxor ferroelectric PbMgi/gNb^/gOg (PMN) under similar 
conditions. As shown in Fig. 40, at high temperatures above -800 K at nearly the same 
level of e" we did not find a pronounced peak in e' for PMN (Fig. 40a). We conclude 
that the huge peak of e' in NBT is not an artifact.
In the following, three possible models will be presented for these large low- 
frequency permittivity and loss anomalies.
Superparaelectric Cluster Model.16 In this model, we assume that the electrical response 
has two parts, i.e. the frequency-independent (probably ionic) conductivity response of 
free charges and the frequency-dependent polarization response of bound charges. Based 
on these assumptions, the permittivities can be written as
e
I H
e - ie (120)
a c a „ e - W / T (121)
Here the p and c subscripts refer to polarization and conductivity contributions, E0 is the 
MKS constant, f  is the measurement frequency, and we assume a thermally activated 
conductivity as in Eq. (121)., We obtained o 0=2.12x10^ S/m and W=18860 K by fitting 
the portion of the lowest-frequency (20 Hz) data in Fig. 41a which coincides with higher-
89
frequency data and thus represents the frequency-independent contribution from the 
conductivity. The remainder Ep" curves for various frequencies left after subtracting the 
conductivity contribution appear in Fig. 41b.
Figs. 39 and 41b can be viewed as the respective real and imaginary parts of 
the polarization contribution to the permittivity. Cole-Cole plots based on these data 
appear in Fig. 42 for temperatures of 600, 650, 700, and 750 K. AU these plots have the 
typical semicircular shape, with circle center somewhat below the real axis. 
Extrapolation of these circular arcs clockwise to the real axis provides estimated dc 
permittivity at 650 and 700 K, at which temperatures the polarization could not develop 
the full dc response at the lowest frequency of 20 Hz.
Fig. 43 compares permittivity peak location and magnitude for heating and 
cooling runs. The e' peaks occur at the same temperature upon heating and cooling at 
100 Hz and 10 kHz. For I kHz the heating run peak is at 813 K while the cooling run 
peak is at 800 K. We attribute this difference to a slight thermal hysteresis in the 
cubic/tetragonal ferrdelastic phase transition found at 820 K. This transition is the 
probable reason for the drop between 800 and 830 K of the difference [Jc-JhZ(Jh+Jc)I 
peak e' magnitudes upon heating (Jh) and cooling (Jc) shown for several frequencies in 
the upper right inset of Fig. 43.16
The dielectric behavior must be considered in conjunction with the NBT crystal 
structure, particularly as there appears to be a first-order ferroelastic cubic to tetragonal 
transition at Tc^=820 K. The order parameter (proportional to the square root of the 
tetragonal M-superlattice reflection intensity) is nonzero above Tc^, abruptly more than 
doubles at Tc^, and gradually doubles again as temperature drops to 640 K. Below this 
temperature there is a rapid drop of the M reflection and rapid rise of the R (trigonal) 
superlattice reflection, both accompanied by about 50 K thermal hysteresis and by 
coexistence of the M and R reflections.
90
The large dielectric peak at high temperature can be explained by assuming that 
these data he in the crossover region from independent dipole moments to 
superparaelectric moments of N dipoles. We will show that a superparaelectiic model 
reproduces the general shape of this dc envelope. The higher-temperature part of the 
crossover region, which we barely reached in our measurements, should obey a Cuiie- 
. Weiss law E7FCZ(T-T0). Most of our observed E7 envelope lies near T 0 where this E7 vs. 
T envelope becomes nearly a straight line.
With the help of Eqs. (40) and (41), we can calculate the dielectric permittivity 
in the crossover region. Fig. 39 shows the calculations for N=54 and N= 125, with dipole 
moment p adjusted so that at 800 K these curves cross the "dc" E7 curve (the curve along 
which low-frequency E7 curves coincide). The fit is better for N=125, but a larger N 
would fit still better.
To understand better where our data fit on the overall superparaelectric model 
permittivity curve, the logarithmic plot of permittivity against temperature in Fig. 44 is 
helpful. The calculated superparaelectric response for N= 125 dipoles in a cluster appears 
in Fig. 44. The -I slope regions at high temperature corresponding to independent 
dipoles and at low temperature corresponding to superparaelectric clusters are clearly 
evident. Our data, superimposed on this curve in the inset, appear only in the crossover 
region. The central portion of our data rises more steeply than the calculated curve but 
fits reasonably well, whereas there are large deviations of our data at the two ends. We 
now discuss how all these deviations are related to the structure, in particular to the 
tetragonal order parameter measured by Vakhrushev et al. obtained from the neutron M- 
reflection intensity.16,26
To model the effects of the transitions from cubic to tetragonal and tetragonal 
to trigonal phases, we assume that the number of clusters varies as the tetragonal order 
parameter. The tetragonal order parameter is assumed proportional to the square root of
91
the M-reflection intensity. Multiplying the superparaelectric cluster model permittivity of 
Eq. (41) by this temperature-dependent order parameter yields the calculated dc 
permittivity shown in the inset of Fig. 44.
Quite good agreement is seen in the inset of Fig. 44 between the calculated and 
"measured" dc permittivity. "Measured" is put in quotes because below 750 K the dc 
permittivity is found by a three-step process: First the conductivity contribution is
subtracted from e " to find the polarization contribution Ep", then Cole-Cole plots of Ep" 
vs. e'  are made, and finally the circular arcs of these plots are extrapolated clockwise, 
until they cross the e'  axis at the assumed value of dc e'. Confidence in this process is 
enhanced by the good fit of the data to circular arcs. Near 600 K the arcs become too 
short for accurate extrapolation. From 750 to 885 K the "measured" dc permittivity is 
taken as the "envelope" line in Fig. 39 along which low-frequency e' curves coincide.
The parameters predicted by the superparaelectric model can be compared with 
those for BaTiOg-family crystals to see if they are reasonable. Fitting the model with 
N= 125 to the dielectric data at 800 K we have, from the T vs. E1 data, e’= 2 .55x 104, 
which corresponds to a dipole moment p of 1.12x10 ^  (C m) per 0.4x0.4x0.4 (run)3 unit 
cell, giving a Curie-Weiss constant of C=LbxlO6 K. The closest comparison can be 
made with PbTiO3, which like BaTiO3 has a ferroelectric transition from a cubic to a 
tetragonal phase. For BaTiO3 this transition is only at 393 K, whereas in PbTiO3 it is at 
763 K which is much nearer the 820 K value reported by Vakhrushev et al. for NBT. 
PbTiO3 has Ps=0.75 (C nr2) giving p=4.47xl0"29 (C m) and has C=4. IxlO5 K. These 
values of p and C are about 3 times lower than for the above fit of the data to the 
superparaelectric model with N= 125. Taking another approach, we can fit the
superparaelectric model with the p value from PbTiO3 to the dc E1 value of 1.56x105 at 
640 K, obtained by extrapolating the Cole-Cole curve. At this temperature the 
permittivity is not increasing so fast with decreasing temperature, so it is reasonable to
92
assume that the permittivity is close to obey the low-temperature form, Ez=Np2Z(E0VkT). 
We then get N=391=(7.31)3 which correspond to a 2.92 ran cube for a superparaelectric 
cluster, whereas N=125=53 correspond to a 2.00 nm cube. We conclude that we can 
obtain a fairly good, fit with reasonable parameters if the superparaelectric moment 
correlation length is 2 to 3 nm.
The dynamic behavior of the large low-frequency peak is most strikingly 
illustrated by Fig. 45. Here we plot two experimental parameters against inverse 
temperature. One is Tpk, the inverse of the measurement angular frequency for which the 
dc e ' peak occurs at the given temperature. These peaks are quite sharp, so this Tpk is a 
good approximation to the dielectric relaxation time. The other parameter plotted is the 
"RC" relaxation time for the crystal, which as for any leaky dielectric is defined as
% c  = RC = E0Edc - R  = E0Edc — P = P E 0Edc (122)
d d A
where p is the inverse of the conductivity C displayed in Fig. 6a and determined as 
described in the associated text.
We see there is excellent agreement between these two x's over four decades of 
magnitude, and we emphasize that there are no adjustable parameters or model- 
dependent assumptions entering into these x's. We interpret this close agreement as 
showing that the mobile charge redistribution time xRC required for rearrangement of 
carriers around a given superparaelectric cluster is the relaxation time Tpk required for the 
cluster to reorient in the applied electric field. In other words, the cluster is "frozen" 
against polarization reversal until this charge redistribution can occur. The charge 
redistribution in turn is influenced by the cluster, as evidenced by the appearance of Ezdc 
in Eq. (122) for the charge redistribution time xRC. However, we have developed no 
quantitative model for this dynamic process.
93
The superparaelectric clusters undoubtedly have internal barriers to polarization 
reversal, as proposed by Cross. He estimated barriers of order 3000 K for larger clusters 
of volume (20 nm)3. This is an order of magnitude smaller than the energies 
corresponding to the slopes in Fig. 45, so the internal barriers do not appear to govern 
the dynamics in the NBT clusters.
The frequency dependence of e'(f,T) is illustrated for five temperatures in Fig. 
46. The crystal undergoes a relaxation process in the high temperature region for
frequencies below 50 kHz. The solid curves are fits to the Cole-Cole equation. The fits 
of Eq. (34) to the real part of the permittivity with E00 =1280 are presented in Fig. 46.
The parameters from fits at five temperatures are shown in Table 2 which indicates a 
millisecond relaxation time region between 700 and 800 K, which agrees with the data 
(Tpk and trc) as shown in Fig. 45.
650 K 700 K .750 K 800 K 850 K
fn-l(10-3 s) 384.7 33.99 8.65 1.12 0.30
a 0.104 0.122 0.158 0.103 0.195
576000 ■ 117300 74830 26290 10930
Table 2. The parameters from fits of Eq. (34) to the real part of permittivity (e') at five 
temperatures.
The parameter Edc(T) also shows good agreement with the data from extrapolation of 
Cole-Cole plots (Fig. 42), except at T=650 K. The relaxation time distribution parameter 
shows a drop (-35%) around 800 K as temperature increases, which may correspond to 
the paraelectric to ferroelastic phase transition.
94
200 Hz
100 kHz
TEMPERATURE (K)
Figure 36. The temperature dependence of the real part of the dielectric relative 
permittivity in NBT at two frequencies upon heating. The A, B, and C arrows 
indicate three anomalous regions.
1 -  This study
2 -  Pronin et ai
/ / 5 0 0
kHz N
/  3 -  Suchanicz
& Ptak
4 -  Zvirgzds et al.
400 500 600 700 800 900
TEMPERATURE (K)
Figure 37. Comparison of the dielectric anomaly in region A. The curves 3 and 4 
show only heating and cooling runs respectively. The inset shows that this anomaly 
is hidden by an increasing background at lower frequencies.
D
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Su ch an ic z  & Ptak
Sakata & 
Masuda
200 Hz
10
100 kHz
TEMPERATURE (K)
-  1.5
-  1.0
0.5
Figure 38. The temperature dependences of remanent polarization, Pr , and dielectric 
permittivity (e') upon heating in region B where a ferroelectric state exists.
D
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500 600 700 800 900
TEMPERATURE (K)
Figure 39. The real part of dielectric permittivity vs. temperature in region C at 
various frequencies upon heating. The solid line is the calculation of Eq. (41) with 
N=54 and dashed line is the calculation o f Eq. (41) with N= 125.
98
60  -
PMN -
TEMPERATURE (K)
Figure 40. (a) The temperature dependence of the real part (e') of the dielectric 
permittivity in NBT and PMN at 100 Hz upon cooling, (b) The temperature
dependence of the imaginary part (e") of the dielectric permittivity in NBT and PMN 
at 100 Hz upon cooling.
TEMPERATURE (K)
Figure 41. (a) o(=2jtfe0e") vs. temperature at five different frequencies. The solid
line is the fit of Eq (91) for CTc with parameters cto=2.12x107 (Qm)-1 and w= 18860
K- (b) The imaginary part of dielectric permittivity, £"p=ACT/(27tfE0), vs. temperature
at five different frequencies. Here Act=Ct -Ctc
IM
A
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X
V
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•  600
■ 650
▲ 700
x v  750 K
REAL PART OF DIELECTRIC PERMITTIVITY IO-4 £ '
Figure 42. Complex representation of the NBT dielectric permittivity at four 
different temperatures. The numbers on the points indicate the frequencies in Hz. 
The dashed lines are regression lines.
101
10 kHz
Z 3  CL%
T (K) r
IOOkHz
900 I :
100 Hz
700 800500 600
TEMPERATURE (K)
Figure 43. The strong thermal hysteresis behavior in region C at four 
different frequencies (10 kHz and 100 kHz in the left inset). Right inset: 
the difference of heating and cooling peak values shows a drop in the 
region of the ferroelastic phase transition at Tcl-820 K. The numbers 
on the points indicate the frequencies in kHz. The dashed line is a 
regression line.
102
CU
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g
S
Cd
S
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od
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NN
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TEMPERATURE (K)
Figure 44. The temperature behavior of dc permittivity, e’, calculated from Eq. (41) 
with N=125. Inset: solid hexagons are from Eq. (41) with N=125, squares are the 
"measured" dc permittivity, solid circles are proportional to the tetragonal order 
parameter from neutron scattering, and triangles are the calculated dc permittivity as 
defined by the dc permittivity from Eq. (41) multiplied by the tetragonal order 
parameter. All curves are adjusted to cross at 800 K.
R
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A
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A
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(s
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103
10OOZT(K)
Figure 45. Relaxation times, Tpk and t r c , vs . 1000/T. The solid and dashed lines are 
fits of equation T=T0Cw r  with to=5 .57x 1 0 18 s , W=25090 K for T . , and T =3.54x 
IO 21 s,W=30720 K fo r tRc. P
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7 0 0  K
7 5 0  K
8 0 0  K
8 5 0  K
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FREQUENCY (Hz)
Figure 46. The real part of dielectric relative permittivity IO"4 e' vs. frequency (Hz) 
at five temperatures. The solid curves are fits of Eq. (34) with parameters of Table
105
Surface-Modified Bulk Effect. A series of two parallel RC circuits which represent two 
surfaces and bulk (sample) can be used to describe a high-impedance surface layer 
problem. Fig. 47 is the circuit for this model.
r . . electrodes
sample
a
Bulk (sample) Surface layers
c I
____Il____ HIl Il
z I Z2
L a a a a J LAAAA__
R1 R2
b
Figure 47. (a) An actual connection of sample and electrodes.
I . „  . I
(b) A series of two
parallel RC circuits. Z1 = -----h itoC, and Z2 + icoG,
Following are equations which were used for this model.
Z  =  Zi +  Z 1
n - i d
I + ItoC1R1 I + ItoC2R2
Z ' - Z " (123)
D=I+ to2 (Cf Rf + Cf R f ) + M4C fcfR fR f
O-R1 + R2 + to R1R2 (C1R1 + C2R2)
d=to[C1Rf + C2Rf + to2 (C1 + C 2 ^ 1C2RfRf]
Pjdi
A A c0e-Wi/T
Ci
GpEjA
dj
ei
Cl
C0
or (i=l,2)
e' - i e (124)E0Ato Z
106
where dj and d2 are the thicknesses for sample and surface layer, respectively. We
assume that both R s are thermaly activated and that the C s are independent of 
temperature. Wi is the activation energy. Here we chose Ei=IOO for both bulk and
surface layer. For some parameters, this model reproduces a shoulder in e" similar to 
that seen in Fig. 41a as shown in Fig. 48. However, Fig. 48 shows only a plateau in e' 
and not the experimentally observed peak. The plateau level is independent of frequency 
whereas experimentally the peak amplitude increases rapidly with falling temperature. 
The failure to reproduce this characteristic peak behavior seems to rule out the possibility 
of the blocking surface interpretation.
Extrinsic Bulk Conductivity Model. In this model, a spatial power law for the 
distribution of barriers caused by crystal imperfection was assumed as
U(r) = B + B0( - i - ) Y (125)
R +  r
The large conductivity activation energy is associated with the high barrier for the carrier 
(ion or electron) to hop from one site to the next. The hopping conductivity obeys the 
equation G(T) = nvexp (-B  / kT)e2a^  /  kT . "ao "~4A is one lattice constant and n is the
carrier concentration. More detail of this model and analysis can be found in Ref. 52.
107
2x10*
£ '
10*
2x10® 
e"
5x10s
0 
2
1
tan 8
0
600 800 1000
Temperature (K)
f-too
200
1000 Hz .  „
*
•vV‘*
•  ■ A
# ■
■ a• M A
"‘" i iw ii i i i i i i i i ir " - -1****
. • y . ; c
I I I I I r I I I I I
Figure 48. Plots of Eq. (124) with parameters: dbulk= l mm, dsur=10"4 mm, 
Wbulk=20000 K, Wsu=SOOOO K, e'bulk=e'su =100, and Ct0=IO7 (ohm-m)"1.
108
Brillouin Backscattering Along LAFOOll Phonon Direction53,54
Fig. 49 illustrates qualitatively the changes in the Brillouin spectra obtained at 
different temperatures. The free spectral ranges of the interferometer are 30.20 GHz for 
all spectra at T>235 K and 34.60 GHz for the spectrum at the bottom of Fig. 49 at T<78 
K so that the Brillouin doublet in NBT appears in the second order with respect to the 
Rayleigh line. As seen from the spectra, the position and especially the half-width of the 
Brillouin components are strongly temperature dependent. Fig. 49 also shows that the 
background in the Brillouin spectra changes drastically and has the largest value at -650  
K which exceeds many times the peak intensity of the Brillouin doublet. Qualitatively, 
the same behavior was found from the central peak intensity as observed in Raman 
scattering. Strong background due to existence of a central peak in light scattered from 
NBT is the main origin of the spread in data point (Brillouin shift and half-width) in the 
vicinity of the anomalies.
Both the Brillouin shift and Brillouin half-width exhibit some anomalies in the 
vicinity of the first phase transition from a cubic to tetragonal phase at Tci~830 K. An 
increase of damping begins far above Tcl in the cubic paraphase (see Fig. 50) and can be 
connected with the contribution of order parameter fluctuations. The dashed curve in 
Fig. 50 is a qualitative estimate of fluctuation contribution. Beside the broad damping 
background, there is an additional peak in the tetragonal phase close to Tc l. This la tte r, 
peak can be attributed to the usual Landau-Khalatnikov mechanism but now it is shifted 
into the tetragonal phase due to frequency dispersion. Such a combination which 
includes both the Landau-Khalatnikov mechanism and order parameter fluctuations for 
longitudinal acoustic phonons implies that the Tj2U-Iype quadratic coupling, squared in 
order parameter and linear in strain, is the dominant mechanism in NBT.
Fig. 51 reveals a broad velocity dip, a broad damping peak and a strong 
background intensity. According to neutron scattering data, a first-order phase transition
109
occurs at Tci~820 K from a cubic paraphase to a tetragonal ferroelastic phase. The 
tetragonal distortion corresponds to an irreducible representation at the M point of the 
cubic phase Biillouin zone. The appearance, development and disappearance of the M- 
point superlattice reflections observed in neutron diffraction is illustrated in Fig. 52.
The next transition at TC2~593 K is to a trigonal phase. The trigonal structure 
is not a subgroup of the tetragonal structure, but its distortion corresponds to an 
irreducible representation at the R point of the cubic phase Brillouin zone. The 
tetragonal and trigonal phases coexist over a temperature range, and the point of 
maximum coexistence in the heating run is taken as TC2 in the phase diagram in Fig. 52. 
However, there is a sharp cusp at 640 K in the M-point neutron reflection intensity which 
signals the high-temperature onset of this transition. All the Biillouin scattering 
measurements were carried out upon heating.^
The neutron scattering data indicate that the phase transitions at Tci and TC2 
are initiated by some irreducible representations at the M and R points of the cubic phase 
Brillouin zone, respectively. It is well-known that in both these cases the anomalies for 
longitudinal acoustic phonons can be described by the Qrpu-type coupling. However, 
neither the step-like anomaly for sound velocity nor the Landau-Khalatnikov maximum 
for damping could be observed with certainty in the vicinity of Tc2- The velocity dip and 
damping peak look like reflections of each other in a horizontal mirror plane (Fig. 51a 
and 51b).
Fig.Slb shows considerable anomalous gradual growth in damping over a wide 
temperature range. Such behavior caused by fluctuations could originate in randomly- 
placed Na+ and Bi3+ cations. The usual Landau theory is a perturbation approximation 
and is not valid if large fluctuations exist. In NBT the separate relaxation portions of the 
acoustic anomaly which should be a sharp anomaly near the phase transition were not 
observed.
no
A "hybridization" may prevent us from observing relaxation contributions o f the 
usual form as seen in many ferroelectrics and ferroelastics. The "hybridization" may 
occur in NBT between two transitions, i.e. from the cubic phase to the tetragonal phase 
at Tci and to the trigonal phase at TC2. However, a linear superposition of two 
anomalies at the Tci and TC2 phase transitions cannot fit the data either. Both the 
velocity minimum and the damping maximum occur near T -650 K which is between 
Tci~820 K and TC2~593 K, This leads us to suggest a coupling of two different order 
parameter fluctuations. Each type of fluctuation contributes to acoustic anomalies with 
the same Tpu-type coupling between the fluctuating order parameter and strain.
As discussed above, the positions of this dielectric maximum at T=640 K and 
hypersonic damping maximum peak at T-650 K are probably initiated by competition of 
two coupled order parameters between two structural instabilities. The shift of damping 
maximum (measurements at about SxlO10 Hz) relative to the dielectric anomaly 
(measurements at IO5 Hz) is clearly seen. The shift may be due to the frequency 
dependence of fluctuations coupling electrostrictively to strain.
Fluctuations of the coupled order parameters or hybridized fluctuations 
manifest themselves in the spectra of high-frequency tight scattering also. A preliminary 
indication of increased central peak intensity between Tci and TC2 was obtained in the 
Raman scattering study. Fig. 51c shows a drastic background increase between Tci and 
TC2 compared to the level outside these transitions. This background corresponds to the 
peak intensity of the central component if there is only one broad central peak as 
suggested by the Raman spectra. The background increase when approaching TC2 from 
below and Tc i from above is comparable to the evolution of both the central peak in 
Raman scattering and overdamped soft modes detected in neutron scattering.
The shaded region in Fig. 52 shows the coexistence range of the R- and . M- 
point superlattice reflections, which corresponds to the trigonal and tetragonal phases.
I l l
The acoustic anomalies have no obvious relation to this -coexistence region centered near 
600 K because their extrema (at T-650 K) (Fig. 51a and 51b) are significantly shifted to 
higher temperatures. Although the neutron measurements can separate information from 
different points in the Biillouin zone so superlattice structures of different types (M and 
R) were studied independently (Fig. 52), the fight scattering spectra show an integral 
response.
A theoretical interpretation recently appeared for the polarization response of 
crystals with structural and ferroelectric instabilities. A simple model with two coupled 
structural and ferroelectric order parameters "shifts" the main dielectric anomaly from the 
ferroelectric transition to the structural one. One could not apply this model directly 
because NBT has two structural instabilities rather than one. However, the above "shift" 
makes it seem likely that those acoustic and dielectric responses of NBT are modified by 
the coupling between different order parameters.
IN
T
E
N
SI
T
Y
 
(p
u
ls
es
/s
ec
.)
112
600 - 
400  
1600 - 
1200  -
3500 -
2800 -
4800 - 
4200 -
1500
1000  -  
600 
400  
200  
100 4 
0
863 K
677 K
Vxrvxy1
504 K
246 K
100 200 300
CHANNELS
400
Figure 49. Brillouin backscattering spectra of NBT at different temperatures near 
the damping maximum.
Figure 50. Damping of the longitudinal acoustic phonons vs. temperature in the 
vicinity of the cubic-tetragonal phase transition.
114
O 200 400 600 800
Temperature (K)
Figure 51. Temperature dependence of different characteristics in NET: (a) Brillouin
shift of the longitudinal acoustic phonons vs. T; (b) Half-width of Brillouin
components at half maximum (or damping) vs. T; the instrumental broadening was
subtracted; (c) Intensity of the background in the Brillouin spectra vs. T.
Li
ne
w
id
th
 a
t H
al
f M
ax
im
um
 (
G
H
z)
115
P h a s e s :
TrigonalSymmetry: Tetragonal Cubic
Temperature (K)
Figure 52. Intensity of the superlattice reflections from M and R points of the cubic 
Biillouin zone obtained in neutron scattering measurements vs. T to show the range 
of phase coexistence (shaded region). The phase sequence in NBT is shown at the 
top; Lparaelectric and paraelastic, IIiferroelastic, III:antiferroelectric(?), IV: 
ferroelectric. The corresponding crystal symmetries are shown in the middle. AU 
data were obtained on heating runs.
116
PbMgy3Nb2Z3O3 CPMNl
Dielectric and Brillouin Scattering Results5^
Actual LA[001] phonon spectra of the anti-Stokes and Stokes Brillouin 
components for Qs=ISO0 are shown in Fig. 53. The data shown here are for several 
temperatures near the maximum value of half-width. Both frequency shift and half-width 
exhibit a strong temperature dependence over a wide temperature range. Fig. 53 also 
shows that the background in the Brillouin spectra changes drastically and has the larger 
values at T>~290 K. Since Brillouin spectra are sensitive to the central peak near the 
Brillouin zone center, this background anomaly implies an evolution of the central peak as 
evidenced by Raman scattering which indicates a strong central peak range between -300 
and -500 K. . However, central peak analysis is difficult because the peak is often 
superimposed by low-frequency Raman modes with complicated temperature dependence. 
Thus, a correlation between acoustic phonon damping and a central component in PMN is 
not clear yet.
The accumulated data for backscattering (0S=18O°) are shown in Fig. 54. The 
data indicate that a broad phonon softening starts at T>475, K and reaches a minimum 
near 290 K, associated with a broad maximum in damping. The velocity and damping 
temperature dependences look like reflections of each other. Such a growth of damping 
over a wide temperature range reveals that order parameter fluctuations are the dominant 
dynamic mechanism. . As one knows, the dynamic fluctuation contribution is a 
characteristic of a T|2|A-type quadratic coupling (electrostriction), squared in order 
parameter and linear in strain. However, a typical step-like frequency softening for T in ­
type coupling is not observed in this case. Beside this broad damping background, one 
can note that there is an additional damping peak near 212 K which can be connected with
117
the typical Landau-Khalatnikov maximum (relaxation-type). For comparison, dielectric 
measurements were also carried out at several frequencies and are presented in Fig. 54c.
What are the origins of these two damping peaks in PMN? The temperature 
dependence of the linear birefringence An®= <P//2>-<P±2> in PMN under an external
electric field suggests an equilibrium phase transition temperature Tc in the absence of 
random field, in the range 200 K<TC<234 K. In addition, by fitting the time dependence 
of the linear birefringence with two different exponents (3 below and above 212 K, 
Westphal et al. propose that the Curie temperature of PMN is Tc~212 K. Viehland et al. 
also claim that the freezing temperature is T^-217 K below which a dramatic change of 
the relaxation time distribution is observed which is interpreted as a result of a long-range 
correlation of polar microdomains. Recent quasielastic-neutron scattering (QES) on 
PMN supports the above interpretations and shows that the correlation length is nearly 
temperature independent with a maximum value of 200 A below -217  K. Since the 
Landau-Khalatnikov contribution is sensitive to the occurrence of long-range FE 
ordering, one can attribute this Landau-Khalatnikov-Iike maximum in PMN to the rapid 
growth of FE ordering at Tc~212 K.
The diffuse phase transition around 260 K is attributed to quenched random 
electric fields originating from charged compositional fluctuations at the B site of the 
ABOg structure. These random fields can suppress a long-range-order ferroelectric 
transition and generate dynamical order parameter fluctuations. Thus, one can expect that 
such random fields are responsible for the evolution of the broad damping background 
with a maximum near 290 K (see Fig. 54b) and modification of T|2|i-type frequency 
softening, i.e. a broad dip instead of a typical step-like anomaly. Similar acoustic 
anomalies, i.e. a broad damping maximum and a broad velocity dip which does not 
correspond to any specific phase transition or structural instability, are also observed in 
Na1Z2Bi1Z2TiO3 (NBT). The fluctuations in NBT are attributed to random fields
118
originating from short-range randomly-placed Na+ and Bi+3 cations at the A site of ABO3 
structure instead of Mg+2 and Nb+5 at the B site for PMN. Such acoustic similarity 
between PMN and NBT seems to imply the importance of short-range randomly-placed 
cations on complex ferroelectric relaxors.
The longitudinal LA[001] and transverse TA[001] phonon data for scattering 
angle 0S=32.6° are shown in Fig. 55. A phonon frequency softening is also observed at 
this angle for both LA and TA phonon modes, with a broad minimum at T-280 K. 
According to Eq. (22), the relative error in damping for the small-angle scattering is much 
larger than for the backscattering, thus the natural-phonon half-width Svph for small angles 
is not very meaningful. Our measurements are limited not only by the broadening of the 
collection angle but also by the broadening of the incident-laser light source and the 
resolution of the Brillouin spectrometer. Fig. 56 shows also a softening for 6S=13.0° and 
indicates a frequency minimum for both LA and TA modes at T -270 K. As one can see, 
due to dispersion the frequency minimum shifts to lower temperature as frequency (or 
scattering angle) decreases.
A broad phonon softening has been observed for both LA[001] and TA[001] 
modes starting at T>475 K followed by a hardening around 280 K. For 0S=18O°, a strong 
broad damping background is observed and is connected with order parameter 
fluctuations. An additional Landau-Khalatnikov maximum is also observed at Tc~212 K 
and implies a rapid evolution of long-range correlation of polar microdomains. Similar 
results are also obtained from our Brillouin scattering on Nay2Bi-^TiOg, i.e. a strong 
broad damping background and broad frequency softening. We conclude that the 
random-ion-caused fluctuations must play a important role for the similar acoustic 
anomalies in these complex ferroelectric relaxors.
IN
TE
N
S
IT
Y
 (p
ul
se
s/
se
c)
119
100 
400
200  -  
400 -
200  -  
500 -
400 -
600 -
400 -  
750 -
500 -
200
142 K
219 K
290 K
352 K
474 K
100 200 300 400
CHANNELS
Figure 53. Anti-Stokes and Stokes components of the LA[001] Brillouin spectra for 
temperatures near the maximum damping value for Gs=ISO0 in PMN. The frequency 
interval between the two Rayleigh peaks is 17.035 GHz (FSR).
120
LA[001]
45 —
44 -
-  1.0
— 0.5
L- 0 .0
15000 -
10
100 kHz10000  -
5000 —
100 200 300 400 500
Temperature (K)
Figure 54. (a) Frequency shift and (b) Half-width vs. temperature of the LA[001] 
phonons for 0S=18O°. (c) The real part of dielectric permittivity vs. temperature at 
four different frequencies. The dashed line indicates the ferroelectric phase 
transition temperature Tc~212 K.
H
al
f-w
id
th
 a
t H
al
f M
ax
im
um
 (G
H
z)
Fr
eq
ue
nc
y 
Sh
ift
 (G
H
z)
 
Fr
eq
ue
nc
y 
Sh
ift
 (G
H
z)
121
12.4
12.2
12.0
11.8
8.2 -  
8.0  —  
7 .8  -
(a) LA[001] o
o ° o
° o° '
° a?
<t> O O  .
Oq
% o o
°o
0 O o0
o
(C)
O0Ool
-  1.2 
-  0.8 
-  0 .4
O 0
TA(OOI)
O O
- O O O . 
z O
o° <f
°  ' 0vX0 0O Y 0
" l '} '  ° °
100 200 300 400 500
Temperature (K)
Figure 55. (a) Frequency shift and (b) Half-width vs. temperature of the LA(OOl) 
phonons for 0S=32.6°. (c) Frequency shift vs. temperature of the TA(OOl)
phonons. The dashed line is a guide to the eye.
H
al
f-w
id
th
 a
t H
al
f M
ax
im
um
 (G
Hz
)
122
N
1  
O
S
JZW
S'c0)3cr
2 
LL
4 .8  —
4 .6  —
(a) LA[001]
°  0  O  0O O  
O O O
° ° °  .  '
°0 O 0
°o °o
0 O
Oo
-  0 .3
0O
O O 0
% o o%° °°Oo o 0° 0 O —
0.2
0.1
O0 O °° O
N
r  3-2 H
-C
Cf)
5
C
CD3
S' 3.0 -
TA[001]
°0
° '  ' ' d  O «$P O 0 Z z ' 0
o° ' d p o  O , 4  « o °
O
II I i I i i i II I II i I I I I i n pi
100 200 300 400 500
Temperature (K)
Figure 56. (a) Frequency shift and (b) Half-width vs. temperature o f the LAfOOH
Z : :  TLd ihed6L  L L Z Z h e Z  -  —  -  TA,OOi1
H
al
f-w
id
th
 a
t H
al
f M
ax
im
um
 (G
H
z)
123
Determination of Elastic Stiffness Constants14,55
The high temperature phase in PMN is cubic with point group m3m whereas 
below 200 K a small trigonal distortion was observed by Shebanov from single crystal 
studies. Thus, we can consider cubic symmetry for calculation of elastic stiffness 
constants in PMN, i.e.
Cf Cf2 Cf2 0 0 0 '
Cf2 Cf1 Cf2 0 0 0
Ce= Cf2 Cf2 Cf, 0 0 0ij 0 0 0 Cf, 0 0
0 0 0 0 c : 0
0 0 0 0 0 CL
In our experimental situation, the phonon is along the [001] direction. From 
Eq. (85), one obtains
p v l  = Cf1 = = c3E3
PVi=PVL=C:= c : = c :
From Eq. (69),
V  = ^ V
2nsin(—)
2
where V is the phonon velocity, Av is the phonon frequency (or Brillouin frequency shift) 
and X0 is the wavelength of the incident light. Thus, one has the following relations
124
Cf, =<5 = <4 = PCt- ^ a-2n sin ( 0 / 2 ) ) 2 (126)
c : = , c ° = c ° = p ( „ x.°A> ^ j 2
2ns i n ( 0 / 2 ) '
(127)
where Avla and AVta are the longitudinal and transverse phonon frequency shifts, 
respectively. Since thermal strain ALZL0 is almost zero below 400 K, we will use the 
crystal density at room temperature, i.e. p=8.12xl03 kg/m3, for elastic stiffness 
calculation between 400 and 200 K. The temperature dependent refractive indices n(T) 
are obtained from Ref. 56. Furthermore, one can use the relation of the compliance 
constant tensor C? to obtain S^, i.e.
sCtpCpy -  ^ay  (128)
where 8™, is the unit tensor of dimension 6. Thus, we have
' /
C l C l c : 0 0 0 " fsf Sf2
C l C l Cf2 0 0 0 Sf2 sf
C l C l Cf1 0 0 0 Sf2 Sf2
0 0 0 Cf, Q 0 0 0
0 0 0 0 Cf, 0 0 0
Io 0 0 0 0 Ce'- ■ '4 4 / Io .0
Sf2 0 0 O^ f l 0 0 0 0 O ' )
Sf2 . 0 0 0 0 I 0 0 0 0
Sf1 0 0 0 0 0 I 0 0 0
0 sf, 0 0 0 0 0 I 0 0
0
0
0
0
sf,
0
O
<_
__
__
_ 0
Io
0
0
0
0
0
0
I
0
0
I j
Table 3 lists the elastic stiffness constants and compliance constants of PMN 
(with cubic point group m3m) at several temperatures. As seen in Table 3, both elastic 
stiffness constants Cj51 and Cb44 show a softening around 280 K as observed in the 
Brillouin phonon frequency shift. We attribute the slight difference in Cj51 from Os=ISO0 
and Os=32.5+0.2° to the error of determination for smaller angles.
125 ■
T(K ) n(T)
From Os=ISO0 
Cf1 (IO11NZm2)
From 0 
Cf (IO11NZm2) Cf,
,=32.5+0.2° 
(IO10NZm2) sf, (10’nm2ZN)
200 2.511 1.64 1.52 6.93 1.44
219 2.514 1.63 1.49 6.96 1.44
245 2.517 1.63 1.48 6.68 1.50
268 2.521 1.62 1.49 6.76 1.48
284 2.524 1.62 1.47 6.69 1.50
300 2.526 1.62 1.49 6.76 1.48
321 2.530 1.63 1.49 6.96 1.44
340 2.532 . 1.64 1.49 6.87 1.46
370 2.536 1.67 1.55 7.00 1.43
Table 3. Elastic stiffness and compliance constants of PMN at several temperatures from 
two scattering geometries 0s = 180° and 32.5+0.2°.
Table 4 compares the elastic stiffness constants at room temperature for several 
different ABO3-Iype ferroelectric crystals.
Cf (IO11NZm2) Present (PMN) BaTiO3 PbTiO3
Cf 1.62 2.11 2.35 ‘
Cf, . 0.68 0.56 0.65
sf, (IO-1V 2ZN) 1.48 1.78 1.54
Table 4. Comparison of elastic stiffness and compliance constants at room temperature.
One can note that the elastic stiffness constants C- of these pure perovskite-
type ferroelectric single crystals are larger than those in the complex crystal PMN.
126
CHAPTER 5 
CONCLUSIONS 
Comparison of Results
Beside some differences between the pure ferroelectric crystal DRDA and 
deuteron glasses DRADA-0.10 and 0.28 as indicated in chapter 4, several important 
similarities and differences have been found through this study between these disordered 
materials, i.e. deuteron glasses DRADA-0.10 and 0.28, and the relaxor ferroelectrics 
PMN and NBT.
Similarities Among PMN. NBT and DRADA-0.10 and 0.2851'55
(1) For all these crystals, DRADA-0.10 and 0.28, NBT and PMN, a broad 
maximum in dielectric permittivity has been observed. For both DRADA-0.28 and PMN, 
this broad dielectric peak value is frequency dependent.
(2) For DRADA-0.10 and 0.28, PMN and NBT, the temperature dependence of 
hypersonic damping shows a strong growth over a wide temperature range, which is 
attributed to order parameter(s) fluctuations. For DRADA-0.28, PMN and NBT, the 
damping peak occurs at higher temperature than the dielectric permittivity maximum. 
This shift of the damping maximum (measurements at IO10 Hz) relative to the dielectric 
anomaly (measurements at IO5 Hz) can be attributed to the frequency dispersion. 
However, these peaks do not correspond to any specific phase (or structural) transitions.
127
(3) For DRADA-0.10 and 0.28, PMN and NBT, the local random fields 
originating from randomly-placed cations (or ions) play an important role for both 
hypersonic and dielectric anomalies. In DRADA-0.10 and 0.28, these randomly-placed 
ions are ND4+ ions which tend to AFE ordering and Rb+ ions which tend to FE ordering. 
In PMN, those randomly-placed cations are Mg+2 and Nb+5 , randomly located at B-site 
positions. In NBT, those randomly-placed cations are Na+1, Bi+3 ions randomly located 
at A-site positions.
(4) For DRADA-0.10 and 0.28, NBT and PMN, a T|2|i-type quadratic 
(electrostrictive) coupling, squared in order parameter and linear in strain, is the main 
coupling contribution to acoustic anomalies.
(5) For DRADA-0.1, NBT and PMN, a sequence of phase transitions has been
observed.
(6) For both PMN and NBT, the damping peak is associated with a broad 
phonon softening in sound velocity with a minimum centered at the same temperature. 
These anomalies could not be explained by either the Landau-Khalatnikov maximum for 
damping or by a step-like coupling for sound velocity.
(7) Both DRADA-0.10 and PMN show an additional Landau-Khalatnikov peak 
which indicates the onset of long-range FE ordering.
Differences Among PMN. NBT and DRADA-0.10 and 0.2851'55
(I) The dielectric and hypersonic anomalies in NBT can be attributed to the 
fluctuations of two coupled order parameters which are initiated from two different 
structural instabilities and induce a intermediate transition between two different 
structures, i.e. tetragonal and trigonal symmetries. In PMN, the diffuse phase transition 
around 270 K is attributed to quenched random electric fields originating from charged 
compositional fluctuations at the B site of the ABO3 structure.31 For deuteron glasses
128
DRADA-0.10 and 0.28, the competition between FE ordering and AFE ordering is the 
origin for both the dielectric and hypersonic anomalies.
(2) In PMN the random electric fields have been attributed to charges in small 
regions in which according to transmission electron microscopy the Mg2+ and Nb5+ ions 
tend to order with a 1:1 ratio rather than the 1:2 ratio required for charge neutrality.32"34 
In NET, however, a 1:1 ratio for ordered Na+ and Bi3+ cations is stoichiometric and will 
not cause longer-range random electric fields. Instead, it is assumed that local charged 
regions in NBT originates from the random distribution of Na+ and Bi3+ cations.
(3) The dielectric permittivities of PMN and NBT are much larger than those in 
the deuteron glasses DRADA-0.10 and 0.28. The dielectric and hypersonic anomalies of 
both PMN and NBT occur at much higher temperatures than those in deuteron glasses 
DRADA-0.10 and 0.28.
(4) Both PMN and NBT exhibit a broad phonon softening along the [001] (or 
[100], or [010]) direction. However, both deuteron glasses DRADA-0.10 and 0.28 
show a phonon hardening along the [100] or [010] (a or b axis) direction.
Applications
Due to increasing interest in displacement transducers needed in different fields, 
especially optics, precision machinery and the microelectronics industry, recent 
developments in electromechanical actuators have been remarkable over the last few 
years. However, as is well documented in the literature, the true response of the 
piezoelectric materials deviates significantly from the ideal. There are four principal 
causes of nonideal behavior: hysteresis, aging effects, nonlinearity and creep. 
Electrostriction is a phenomenon which gives rise to a strain proportional to the square 
of the applied, electric field. It can be observed in all dielectrics: amorphous and
129
crystalline, regardless of symmetry. For centrosymmetric phases (such as m3m, 3m, 
4/mmm, 4/m etc.), the electrostrictive equation is:
Si = Q t i Pm=Qmieoter a ( E ) - I f B ^  = Q mi6^ L  (E )E^ (129)
and
( Q n Ql2 Ql2 O O O )
Q l2 Q n Ql2 O O O
Qmi =
Ql2 Ql2 Q n O O O
O O O Q44 O O
b O O O Q44 O
1 0 O O O O Q44,
Qmi are the elements of the electrostrictive tensor (using Voigt notation). Strains 
induced in a crystal due to the electrostrictive effect are, in general, very small. 
According to Eq. (129), in materials with a high dielectric permittivity (such as PMN, 
NBT etc.), however, it is possible to obtain electromechanical coupling as strong as that 
in the strongest piezoelectric materials. Several papers have proved from measurements 
on relaxor ferroelectrics that the electrostrictive properties of relaxor ferroelectrics offer 
several advantages over normal piezoelectric transducers: (I) large electrostrictive strains 
comparable to the best piezoelectric ceramics; (2) excellent positional reproducibility 
(insignificant hysteresis); (3) no poling is required, resulting in no aging effects; (4) very 
low thermal expansion coefficients (good thermal stability), (5) short response time.57'59 
Nomura et al. also found that the electrostrictive coefficients Qmi increase with the
degree of cation disorder.57 By using multilayer technology in an electrostrictive device, 
one can also increase the strain (displacement) significantly because the total 
displacement is proportional to the square of the number of layers, an important 
advantage over piezoelectric materials.57
I1 3 0
An electrostrictive displacement transducer can be used in the following fields: 
(I) optical devices: deformable mirror (to produce large deformations without 
hysteresis), Fabry-Perot resonator, etc. (2) surface physics: atomic force microscope 
(AFM), etc. (3) precision machinery and microelectronics. More actual examples can be 
found in Refs. 57 and 58.
Since the conductance of mixed FE-AFE crystals shows a strong temperature 
dependence at low temperature, they could be used to be a temperature sensor for low 
temperature after calibration. In a still lower temperature range, below the onset of the 
ferroelectric transition in proton glasses showing coexistence, the capacitance 
temperature dependence could be used for this purpose too.
Recommended Additional Work on This Problem
(I) In this study, only qualitative explanations are given for hypersonic 
anomalies in the relaxors NBT and PMN and deuteron glasses DRADA-0.10 and 0.28. 
Further theoretical models for these anomalies are important to understand these 
similarities and differences between these complicated disordered systems. (2) As 
mentioned in Ref. 16, the high-temperature (T>640 K) and low-frequency dielectric 
anomaly in NBT is still not clear. The question whether it is due to intrinsic properties, 
extrinsic bulk conductivity or an interfacial effect needs more experiments such as low- 
frequency dielectric measurements and reasonable ,physical models for its answer. (3) 
Phonon softening has been seen in the RADP-x system along the [110] phonon direction 
for both LA and TA modes, so one may expect to observe a similar softening behavior in 
DRADA-0, 0.10 and 0,28 along the [110] direction if,large clean crystals can be grown. 
(4) Further studies such as neutron scattering on DRADA-0.10 and 0.28 are critical for 
understanding the mechanism of ordering competition and the reason why those
131
hypersonic anomalies occur far above any phase transitions. (5) One could measure 
dielectric permittivity on NBT under a strong external electric field. One may see a 
nonlinear effect or an anomaly similar to that observed on PMN, i.e. an additional 
dielectric peak where the random fields are overcome by the external field. (6) More 
Brillouin scattering experiments along different phonon directions on NBT and DRADA- 
x are necessary for calculating elastic constants.
132
REFERENCES CITED
133
1. L.E. Cross, Ferroelectrics 76, 241 (1987).
2. V.H. Schmidt, S. Waplak, S. Hutton and P. Schnackenberg, Phys. Rev. B 30, 
2795 (1984).
3. E. Courtens, J. Phys. Lett. (Paris) 43, L199 (1982).
4. Z. Trybula, V.H. Schmidt, J.E. Drumheller, D.He and Z. Li, Phys. Rev. B 40, 
5289 (1989).
5. C.-S. Tu, V.H. Schmidt and AA.Saleh, Phys. Rev. B 48, 12483 (1993).
6. F.L. Howell, N.J. Pinto and V.H. Schmidt, Phys. Rev. B 46, 13762 (1992).
7. N.J. Pinto, Ph. D. Thesis, Montana State University, 1992.
8. N.J. Pinto, K. Ravindran and V.H. Schmidt, Phys. Rev. B 48, 3090 (1993).
9. E. Courtens, R. Vacher and Y. Dagorn, Phys. Rev. B 33, 7625 (1986).
10. E. Courtens, R. Vacher and Y. Dagorn, Phys. Rev. B 36, 318 (1987).
11. E. Courtens, F. Huard and R. Vacher, Phys. Rev. Lett. 55, 722 (1985).
12. GA . Smolensky, V.A. Isupov, AT. Agranovskaya, and N.N. Krainik, Sov. Phys. 
Solid State 2,2651(1961).
13. Yuhuan Xu, Ferroelectric Materials and Their Applications, (North-Holland 
press, New York, 1991).
14. E. Dieulesaint and D. Royer, Elastic Waves in Solids, (Wiley, New York, 1980),
15. CA . Randall, A.S. Bhalla, T.R. Shrout and L.E. Cross, J. Mater. Res. 5, 
829(1990)
16. C.-S. Tu, LG. Siny, and V.H. Schmidt, Phys. Rev. B 49, 11550 (1994). .
17. Landolt-Bornstein, New Series, Vol. 111/16b, edited by K.-H. Hellwege 
(Springer, Berlin, 1982).
18. U.T. Hochli, K. Knorr and A. Loidl, Advances in Physics 39, 539( 1990).
19. K. Sakata and Y. Masuda, Ferroelectrics 7, 347 (1974).
2 0 .1. P. Pronin, P.P. Symikov, V.A. Isupov, and GA . Smolensky, Pis'ma Zh. Tekhn. 
Sov. Tech. Phys. Lett. 5, 289 (1979).
2 1 .1. P. Pronin, P.P. Symikov, V.A. Isupov, V.M. Egorov, and N.V. Zaitseva, 
Ferroelectrics 25, 395 (1980).
134
22 .1.P. Pronin, N.N. Parfenova, N.V. Zaitseva, V.A. Isupov, and GA . Smolensky, 
Sov. Phys. Solid State 24, 1060 (1982).
23. J.A. Zvirgzds, P.P. Kapostins, J.V. Zvirgzde, and T.V. Kruzina, Ferroelectrics 
40, 75 (1982).
24. V.A. Isupov and T.V. Kruzina, Sov. Phys. Bull. Acad. Sci. USSR, phys. ser. 47, 
194 (1983)].
25. V.A. Isupov, I.P. Pronin, and T.V. Kruzina, Ferroelectrics Lett. 2, 205 (1984).
26. S.B. Vakhrushev, V.A. Isupov, B.E. Kvyatkovsky, N.M. Okuneva, I.P. Pronin, 
GA . Smolensky, and P.P. Symikov, Ferroelectrics 63, 153 (1985).
27. K. Roleder, I. Suchanicz, and A. Kania, Ferroelectrics 89, I (1989).
28. J. Suchanicz and W.S. Ptak, Ferroelectrics Lett. 12, 71 (1990).
29. J. Suchanicz and V.S. Ptak, Sov. Phys. Bull. Acad. Sci. USSR, phys. ser. 55, 
136(1991).
30. S.V. Vakhrushev, B.E. Kvyatkovsky, R.S. Malysheva, N.M. Okuneva, E.L. 
Plachenova, and P.P. Syrnikov, Sov. Phys. Crystallogr. 34, 89 (1989).
31. V. Westphal, W. Kleemann, and M.D. Glinchuk, Phys. Rev. Lett. 68, 847 
(1992).
32. A.D. Hilton, D.J. Barber, CA . Randall and T.R. Shrout, J. Mater. Sci. 25, 3461 
(1990).
33. N. de Mathan, E. Husson, G. Calvarin, J.R. Gavarri, A.W. Hewat and A. Morell, 
J. Phys.: Condens. Matter 3, 8159 (1991).
34. S.B. Vakhrushev, A A . Nabereznov, Y.P. Feng, S.K. Sinha, S.M. Shapiro, T. 
Egami, H.D. Rosenfeld and D.E. Moncton, Bull. Am. Phys. Soc. 39, 680 (1994).
35. D. Viehland, M. Wuttig, and L.E. Cross, Ferroelectrics 120,71 (1991).
36. E. Husson, M. Chubb and A. Morell, Mater. Res. Bull. 24,201 (1989).
37. N.J. Pinto, F.L. Howell and V.H. Schmidt, Phys. Rev. B 48, 5983 (1993).
38. J.J. Gilman, The Art and Science o f Growing Crystals, (John Wiley Sc Son Press, 
New York, 1963).
39. N.W. Ashcroft and N D. Mermin, Solid State Physics, (Saunders College, 1976).
40. G. Hernandez, Fabry-Perot Interferometers, (Cambridge, New York, 1986).
135
41. T. Hikita, P. Schnackenberg and V.H. Schmidt, Phys. Rev. B 31, 299 (1985).
42. H.G. Danielmeyer, I. Acoust. Soc. Am. 47, 151 (1970).
43. A.K. Jonscher, Dielectric Relaxation in Solids, (Chelsea Dielectrics Press, 
London, 1983).
44. V.V. Daniel, Dielectric Relaxation, (Academic Press, New York, 1967).
45. J.D. Jackson, Classical Electrodynamics, (Wiley Press, New York, 1962).
46. Landolt-Bomstein, New Series, Vol. I ll /16a, edited by K.-H. Hellwege 
(Springer, Berlin, 1982).
47. W. Rehwald, Adv. Phys. 22, 721 (1973).
48. J.J.L. Horikx,A.F.M. Arts, LI. Dijkhuis and H.W. de Wijn, Phys. Rev. B 39,
5726 (1989); LG. Siny, Feroelectrics 150, 119 (1993).
49. C. Kittel, Introduction to Solid State Physics, 5th Ed. (Wiley, New York, 1976).
50. Hankbook o f Mathematical Functions, 7th printing, edited by M. Abramowitz and 
LA. Stegun (Dover, New York, 1970), pp. 998.
51. C.S. Tu and V.H. Schmidt, "Temperature and concentration dependences of 
acoustic velocity and damping in Rbi_x(ND4)xD2As04 mixed crystals by 
Brillouin scattering", Phys. Rev. B 50, to appear I December 1994.
52. V.H. Schmidt and C -S Tu, "Dielectric and Brillouin scattering anomalies in an 
Nai/2Bi1/2Ti03 relaxor ferroelectric crystal", to appear in 9th Intemat. Symp. on 
Appl. of Ferroelectrics.
53. C S. Tu, LG. Siny and V.H. Schmidt, Ferroelectrics 152,403 (1994).
54. LG. Siny, C S. Tu and V.H. Schmidt, "Critical acoustic behavior of relaxor 
ferroelectric N ay2Biiz2TiO3 in the intertransition region", submitted to Phys.
Rev. B I.
55. C S. Tu, V.H. Schmidt and LG. Siny, Plan to submit to Phys. Rev. B I.
56. G. Bums and F.H. Dacol, Solid State Commun. 48, 853 (1983).
57. S. Nomura and K. Uchino, Ferroelectrics 4 1 ,117 (1982).
58. S.M. Hues, C.F. Draper, KR. Lee and R J. Colton, Rev. Sci. Instrum. 65,
1561 (1994).
59. J.V. Cieminski and H. Beige, J. Phys. D: Appl. Phys. 24, 1182 (1991).
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