Copyright by the American Physical Society. Wang, Yu U., Sep 2006. “Diffraction theory of nanotwin superlattices with low
symmetry phase,” PHYSICAL REVIEW B 74(10): 104109. DOI: 10.1103/PhysRevB.74.104109.
PHYSICAL REVIEW B 74, 104109 共2006兲
Diffraction theory of nanotwin superlattices with low symmetry phase
Yu U. Wang*
Department of Materials Science and Engineering, Virginia Tech, Blacksburg, Virginia 24061, USA
共Received 6 June 2006; revised manuscript received 7 August 2006; published 18 September 2006兲
A nanotwin diffraction theory is developed. It predicts an adaptive diffraction phenomenon, where the Bragg
reflection peaks are determined by coherent superposition of scattered waves from individual twin-related
nanocrystals and adaptively shift along the twin peak splitting vectors in response to a change in the twin
variant volume fraction. Application of this theory to tetragonal phase explains the intrinsic lattice parameter
relationships of monoclinic M C phase recently discovered in ferroelectric Pb关共Mg1/3Nb2/3兲1−xTix兴O3 and
Pb关共Zn1/3Nb2/3兲1−xTix兴O3.
DOI: 10.1103/PhysRevB.74.104109
PACS number共s兲: 61.10.Dp, 61.12.Bt, 61.14.Dc
Ferroelastic crystals usually consist of structural twins to
accommodate the spontaneous lattice distortion and minimize the elastic strain energy.1 The twin-related domains are
crystallographically equivalent structural variants of different
orientations of the same low-symmetry phase 共e.g., tetragonal, rhombohedral, etc兲. They often self-assemble into highly
regular 共or “ordered”兲 patterns.2 When the twin boundary
energy density is small, the twin microstructure can miniaturize to form nanotwin superlattice,3 as occurred in martensitic crystals.4 Since the nanodomain size is much smaller
than the coherence length of diffraction radiation, scattered
waves from individual nanodomains coherently superimpose
in diffraction. Therefore, the relative phase angles of the
scattered waves must be considered. The purpose of this paper is to develop a nanotwin diffraction theory and report an
adaptive Bragg reflection phenomenon, which explains the
experimental observations in martensitic crystals4 and more
recently in ferroelectric Pb关共Mg1/3Nb2/3兲1−xTix兴O3 and
Pb关共Zn1/3Nb2/3兲1−xTix兴O3.5–8 Similar adaptive diffraction
phenomena should also occur in other nanoscale superlattices of low symmetry phases.
Nanotwins are believed to form in tetragonal ferroelectric
Pb关共Mg1/3Nb2/3兲1−xTix兴O3 and Pb关共Zn1/3Nb2/3兲1−xTix兴O3,9,10
where a new intermediate monoclinic M C phase5–8 is found
to be related to the conventional tetragonal phase through
three intrinsic lattice parameter relationships.10 The tetragonal nanotwin hypothesis explains these intrinsic relationships
based on a nanodomain averaging effect in low-resolution
diffraction used to measure the lattice parameters,9,10 because
significant reflection peak broadening is expected for diffractions of individual nanodomains. However, single-crystal
mesh scans of high-resolution x-ray and neutron diffractions
well resolve a characteristic fine triplet peak structure around
the 共200兲 Bragg reflection spot, which is assumed to consist
of 共200兲 twin peaks from two a domains and a 共020兲 single
peak from one b domain of the monoclinic M C phase.5–8
Therefore, if the observed monoclinic M C phase is indeed
tetragonal nanotwins, this diffraction phenomenon needs a
rigorous theoretical explanation. Application of this developed adaptive diffraction theory to tetragonal nanotwins
solves this issue.
In this paper, we consider a one-dimensional nanotwin
superlattice with a bilayer structural basis, since the size of
each twin variant domain along the layer plane is much
1098-0121/2006/74共10兲/104109共4兲
greater than the nanoscale layer thickness. The structure factor of the nanotwin superlattice can be written as
ñbasis共k兲 = ñ1共k兲 ⴱ ˜␪1共k兲 + ñ2共k兲 ⴱ ˜␪2共k兲,
共1兲
where a tilde ⬃ above a function indicates its Fourier transform, an asterisk ⴱ represents convolution operation, n1共r兲
and n2共r兲 are the electron density distributions in an ideal
共infinite兲 crystal with the same crystal structure and
orientation of respective twin layers of the bilayer basis,
and ␪1共r兲 and ␪2共r兲 are the shape functions describing twin
layers of the bilayer basis, which assumes one inside
each respective layer and zero outside. The function
nbasis共r兲 = n1共r兲␪1共r兲 + n2共r兲␪2共r兲 describes the electron density distribution in the bilayer basis. Using the structure factors ñ01共k兲 and ñ02共k兲 of respective twin layer crystals, where
n01共r兲 and n02共r兲 describe the electron density distribution in
the basis of respective twin layer crystals, the Fourier transform ñ1共k兲 and ñ2共k兲 become
ñ1共k兲 =
ñ2共k兲 =
兺 V−1ñ01共k兲␦共k − K共1兲兲,
K共1兲
兺 V−1ñ02共k兲␦共k − K共2兲兲,
K共2兲
共2兲
where V is the primitive cell volume of the crystal, ␦共k兲 is
Dirac delta function in three-dimensional reciprocal space,
and K共1兲 and K共2兲 collectively represent all reciprocal lattice
sites of respective twin layer crystals.
The nanotwin superlattice is formed by periodically repeating the bilayer basis through a specific translation vector
L, which is the primitive lattice translation of the onedimensional nanotwin superlattice. The electron density distribution in the superlattice is n共r兲 = 兺snbasis共r − sL兲, where
the summation is over integer s running from −⬁ to ⬁, i.e.,
assuming that the total thickness of nanotwin superlattice is
much greater than the thickness of its bilayer basis. The Fourier transform of function n共r兲 is
ñ共k兲 = ñbasis共k兲 兺 e−isk·L .
共3兲
s
Substituting Eqs. 共1兲 and 共2兲 into Eq. 共3兲 yields
104109-1
©2006 The American Physical Society
PHYSICAL REVIEW B 74, 104109 共2006兲
YU U. WANG
ñ共k兲 = V−1
冋兺
K共1兲
+
ñ01共K共1兲兲˜␪1共k − K共1兲兲
册
兺 ñ02共K共2兲兲˜␪2共k − K共2兲兲 兺s e−isk·L .
K共2兲
ñ共k兲 =
Â−T = Î − ␥␶ 丢 s,
共5兲
共6兲
which shows that the peak splits along the twin plane normal
direction.
For convenience, we select twin plane normal direction as
x axis and twinning shear direction as z axis to setup our
rectangular Cartesian coordinate system. Thus, ␶ = 共1 , 0 , 0兲,
共1兲
共1兲
共2兲
共1兲
共2兲
共1兲
s = 共0 , 0 , 1兲, and K共2兲
x = Kx − ␥Kz , K y = K y , Kz = Kz . The
shape functions ␪1共r兲 and ␪2共r兲 are window function of
height 1 in region x 苸 关−T1 , 0兴 and x 苸 关0 , T2兴, respectively,
where T1 and T2 are the thicknesses of respective twin layers
of the bilayer basis, and T = T1 + T2 = L · ␶ is the bilayer basis
thickness. The Fourier transform ˜␪1共k兲 and ˜␪2共k兲 are
˜␪ 共k兲 = f 共k 兲␦共k 兲␦共k 兲,
1
1 x
y
z
˜␪ 共k兲 = f 共k 兲␦共k 兲␦共k 兲, 共7兲
2
2 x
y
z
where ␦共␬兲 is a one-dimensional Dirac delta function, and
f 1共 ␬ 兲 =
2
␬T1 i共␬T /2兲
sin
e 1 ,
2
␬
f 2共 ␬ 兲 =
共9兲
where
where Î is identity matrix, ␥ is the magnitude of twinning
shear strain, the symbol 丢 represents vector dyadic product,
␶ is a unit vector normal to the twin plane, and s is a unit
vector along the twinning shear direction 共␶ and s are perpendicular to each other, and twinning deformation does not
change unit cell volume兲. The Bragg reflection peak splitting
is characterized by the vector difference between the corresponding twin pair of fundamental sites K共1兲 and K共2兲,
⌬K共1兲 = K共2兲 − K共1兲 = − 共␥s · K共1兲兲␶ ,
共1兲
⫻␦共ky − K共1兲
y 兲␦共kz − Kz 兲,
共4兲
The crystal lattices and reciprocal lattices of twin layer
crystals are uniquely related by the twinning deformation
matrix Â:1,11 r共2兲 = Âr共1兲, K共2兲 = Â−TK共1兲, where r共1兲 and r共2兲
are lattice site position vectors before and after deformation,
respectively, K共1兲 and K共2兲 denote the corresponding twin
pairs of reciprocal lattice sites 共i.e., Bragg reflection peak
splitting兲 due to twinning deformation, and the symbol −T
represents matrix inverse transpose operation. The matrix Â
is a special case of the invariant plane deformation matrix
and has the following form:1,11
 = Î + ␥s 丢 ␶,
1
共s兲
兺 兺 g共␬共s兲,K共1兲兲␦共kx − K共1兲
x −␬ 兲
VT K共1兲 ␬共s兲
2
␬T2 −i共␬T /2兲
2 ,
sin
e
2
␬
共8兲
which describe the broadening of fundamental peaks K共1兲
and K共2兲, respectively, due to nanoscale layer thickness and
their phase angles. The peak broadening is in the twin plane
normal direction along the twin peak splitting vector ⌬K共1兲.
Substituting Eq. 共7兲 into Eq. 共4兲 yields
g共␬共s兲,K共1兲兲 = ñ01共K共1兲兲f 1共␬共s兲兲 + ñ02共K共1兲 + ⌬K共1兲兲
⫻f 2共␬共s兲 + ␥s · K共1兲兲,
共10兲
and ␬共s兲 collectively represents the solutions of the following
equation for all possible integer values s,
␬共s兲 =
2␲s − L · K共1兲
.
L·␶
共11兲
Equation 共9兲 shows that the Bragg reflection peaks of nan共1兲
共1兲
共s兲
otwin superlattice are located at k共s兲 = 共K共1兲
x + ␬ , K y , Kz 兲 in
reciprocal space, which are distributed along the lines normal
to twin plane and passing both K共1兲 and K共2兲. In fact, k共s兲 are
the reciprocal superlattice sites of the nanotwins, whose reflection intensities are determined by the function
g共␬共s兲 , K共1兲兲 given in Eq. 共10兲. Note that the superlattice site
distribution, twin peak splitting, and peak broadening are all
along the same lines normal to the twin plane.
It is worth noting that for nanotwin superlattices of low
symmetry phases, the superlattice sites k共s兲 generally do not
coincide with the fundamental reflection sites K共1兲 and K共2兲
of the twin crystals, i.e., the values ␬共s兲 given in Eq. 共11兲 do
not reduce to 0 or −␥s · K共1兲. Therefore, the fundamental reflection peaks of the twin crystals at K共1兲 and K共2兲 will disappear, and new reflection peaks will appear at the reciprocal
superlattice sites k共s兲. This feature leads to an adaptive diffraction phenomenon, where the Bragg reflection peaks are
determined by coherent superposition of scattered waves
from individual twin-related nanodomains and, as will be
shown later, these peaks adaptively move along the twin
peak splitting vectors ⌬K共1兲 in response to a change in the
twin variant volume fraction. As a result, the lattice parameters measured from k共s兲 are different from those of the constituent crystals instead are a specific combination of them.
It is also noteworthy that only a few superlattice peaks
k共s兲 in the immediate vicinity of the fundamental sites K共1兲
and K共2兲 could possibly have visible intensities. According to
Eq. 共8兲, the peak broadening in ⌬K共1兲 direction has a typical
full-width broadness of 4␲ / T1 and 4␲ / T2 for K共1兲 and K共2兲,
respectively. As follows from Eq. 共11兲, the neighboring superlattice sites are spaced by a distance 2␲ / T in ⌬K共1兲 direction. Therefore, approximately 2T / T1 and 2T / T2 superlattice
sites around each respective K共1兲 and K共2兲 could produce
visible reflection peaks, depending on the magnitudes and
relative phase angle of the two terms in Eq. 共10兲. As shown
later for 共H00兲 reflection of tetragonal nanotwins, when
Hm␥ ⬍ 2, where m is the number of 共101兲 atomic layers in
bilayer basis, a single strong superlattice peak appears between the twin pair K共1兲 and K共2兲 and adaptively shifts along
⌬K共1兲 determined by the twin variant volume fraction. When
m becomes significantly larger, less broadened peaks concentrate on individual K共1兲 and K共2兲 and consequently lose overlapping, leading to a transition to a conventional diffraction
104109-2
PHYSICAL REVIEW B 74, 104109 共2006兲
DIFFRACTION THEORY OF NANOTWIN¼
the bilayer basis, m = m1 + m2, the twin layer thickness
T1 = m1at cos ␣, T2 = m2at cos ␣, T = mat cos ␣, and ␬共s兲
= 2␲共s − hm1 − hm2 cos 2␣ + 2lm2 sin2 ␣兲共mat cos ␣兲−1. Using
these results in Eq. 共10兲 determines the Bragg reflection peak
intensities at k共s兲.
In the following we consider the reciprocal superlattice
peaks around the fundamental 共H00兲 reflection spot 共i.e.,
h = H, k = l = 0兲 of tetragonal Pb关共Mg1/3Nb2/3兲1−xTix兴O3 and
Pb关共Zn1/3Nb2/3兲1−xTix兴O3, where the fine structure of the
共200兲 peak has been used to identify the new monoclinic M C
phase.5–8 The ferroelastic strain and twin peak splitting of
these materials are small, 共ct − at兲 / at ⬇ ␥ / 2 ⬃ 0.01,
⌬K共1兲 / K共1兲 ⬃ 0.01. While the structure factors ñ01共k兲 and
ñ02共k兲 can be determined from the atomic form factors, given
the pseudocubic symmetry of the crystal unit cell and the
small twin peak splitting, we assume for simplicity that
ñ02共K共1兲 + ⌬K共1兲兲 ⬇ ñ01共K共1兲兲. The superlattice reflection peak
intensity around 共H00兲 spot is proportional to 兩⌿兩2,
␺共s兲 =
FIG. 1. 共Color online兲 共a兲 Schematic of tetragonal nanotwin superlattice. 共101兲 twin planes are indicated by dashed lines. A tetragonal unit cell is highlighted by gray shadow in respective twin
variants. The primitive superlattice translation vector is L. The volume fractions of the first and second twin variants are ␻ and 1 − ␻,
respectively. The bilayer basis thickness is T. 共b兲 Adaptive diffraction phenomenon, where the Bragg reflection peak k共s兲 moves between the vanished fundamental peaks K共1兲 and K共2兲 of twin-related
nanodomains in response to a change in ␻.
pattern 共meanwhile, domain size with large m may also exceed the coherence length of diffraction radiation兲.
Now we apply this adaptive diffraction theory to the tetragonal nanotwin superlattice, as illustrated in Fig. 1共a兲. The
twin plane in cubic notation system is 共101兲. The twinning
shear strain is ␥ = 2 cot 2␣, where ␣ = tan−1共at / ct兲, and at
and ct are tetragonal lattice parameters. In the coordinate
system defined by ␶ 共x axis兲 and s 共z axis兲, the fundamental
site K共1兲 with index 共hkl兲 is K共1兲 = 2␲共h cos ␣ / at
− l sin ␣ / ct , k / at , h sin ␣ / at + l cos ␣ / ct兲, the twin peak
splitting
vector
⌬K共1兲 = −2␲共2 cot 2␣共h sin ␣ / at
+ l cos ␣ / ct兲 , 0 , 0兲, the primitive superlattice translation vector L = 关mat cos ␣ , 0 , 共m1 − m2兲at sin ␣兴, m1 and m2 are the
numbers of 共101兲 atomic layers in respective twin variants of
2
2
sin ␸1ei␸1 +
sin ␸2e−i␸2 ,
␬1
␬2
共12兲
where ␬1共s兲 = 2␲共s − Hm1 − Hm2 cos2 ␣兲共mat cos ␣兲−1, ␬2共s兲
= 2␲共s − Hm1 + Hm1 cos 2␣兲共mat cos ␣兲−1, ␸1共s兲 = ␲共s − Hm1
− Hm2 cos 2␣兲m1 / m, ␸2共s兲 = ␲共s − Hm1 + Hm1 cos 2␣兲m2 / m,
and the relative phase angle is ⌬␸共s兲 = ␸1 + ␸2 = ␲共s − Hm1兲.
Using cos 2␣ ⬇ ␥ / 2, it is ready to reveal that when
Hm␥ ⬍ 2, the superlattice peak k共s兲 corresponding to s0
= Hm1 has the highest intensity and is the k共s兲 closest to both
K共1兲 and K共2兲 and also the only k共s兲 located between them.12
For the 共200兲 spot, the condition Hm␥ ⬍ 2 gives m ⬍ 50, i.e.,
the bilayer basis is thinner than 14 nm 共with at ⬇ 0.4 nm兲.
The superlattice reflection peak corresponding to s0 = Hm1 is
the one that is observed in experiments 共for example, when
m1 = m2 = 10, the superlattice peaks corresponding to
s = s0 ± 1 and ±2 have respective intensities of 0.0163 and
0.0001 with respect to that of s = s0兲. The position of this
observed superlattice peak is k共s兲 = K共1兲 + 共1 − ␻兲⌬K共1兲, as illustrated in Fig. 1共b兲, where ␻ = m1 / m is the twin variant
volume fraction. Therefore, this superlattice reflection peak
k共s兲 adaptively shifts between K共1兲 and K共2兲 along the twin
peak splitting vector ⌬K共1兲 according to the twin variant volume fraction ␻. The lattice parameter measured from this k共s兲
around the 共H00兲 spot is a ⬇ ␻at + 共1 − ␻兲ct. The same adaptive behavior also occurs around the 共00L兲 spot, which gives
a lattice parameter c ⬇ ␻ct + 共1 − ␻兲at. The lattice parameter
measured from the 共0K0兲 spot is b = at, because the nanotwin
superlattice has homogeneous b1 = b2 lattice translation in the
y-axis direction, and the 共0K0兲 peak does not exhibit such an
adaptive reflection behavior. It is noted that these lattice parameter relationships are the same as obtained from the crystallographic analysis of tetragonal twins by volume fraction
averaging.9,10 Therefore, the adaptive diffraction theory validates the previously used twin-domain averaging
procedure.9,10 It is also noted that each set of tetragonal nanotwin superlattice diffracts incident waves just like one
monoclinic M C domain. When multiple sets of nanotwins of
different 兵101其 twin planes are present, the effect is just the
same as the presence of multiple M C domains of different
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PHYSICAL REVIEW B 74, 104109 共2006兲
YU U. WANG
FIG. 2. Triplet peak structure around the 共200兲 Bragg reflection
spot consisting of two 共200兲 superlattice peaks from nanotwins of
共101̄兲 and 共101兲 twin planes, respectively, and one 共020兲 superlattice peak from nanotwins of 共01± 1兲 twin planes. The vanished
fundamental peaks are not shown. Nanotwins of 共01± 1兲 twin
planes 共not shown兲 produce the same 共020兲 peak 共highlighted by
dotted line兲 as that from 关001兴-oriented tetragonal domain and give
the lattice parameter b = at.
orientations, which gives rise to the observed characteristic
triplet structure of the 共H00兲 reflection peak,5–8 which has
led to an interpretation of M C phase. As illustrated in Fig. 2,
the triplet peak structure around the 共200兲 Bragg reflection
spot consists of two 共200兲 superlattice peaks from nanotwins
of 共101̄兲 and 共101兲 twin planes, respectively, and one 共020兲
superlattice peak from nanotwins of 共011兲 and/or 共011̄兲 twin
planes. The 共200兲 superlattice peaks give the lattice parameter a ⬇ ␻at + 共1 − ␻兲ct. The nanotwins of 共011兲 and 共011̄兲
twin planes 共not shown in Fig. 2兲 produce the same 共020兲
peak 共highlighted by dotted line兲 as that produced by 关001兴oriented tetragonal domain and give the lattice parameter
ACKNOWLEDGMENT
Financial support from Virginia Tech through a startup
fund is acknowledged.
U. Wang, Phys. Rev. B 73, 014113 共2006兲.
M. Jin and G. J. Weng, Acta Mater. 50, 2967 共2002兲.
12 In fact, s = Hm gives the largest magnitudes for both terms in
0
1
Eq. 共12兲 with constructive superposition 共⌬␸ = 0兲, while the next
superlattice peaks correspond to s = s0 ± 1 with ⌬␸ = ± ␲ 共destructive superposition兲 and are located outside of the twin peak splitting vector ⌬K共1兲, thus having very small intensities. The peaks
further away have almost vanishing intensities because of rapidly decreased magnitudes of both terms in Eq. 共12兲.
13 D. E. Cox, B. Noheda, G. Shirane, Y. Uesu, K. Fujishiro, and Y.
Yamada, Appl. Phys. Lett. 79, 400 共2001兲.
14 D. La-Orauttapong, B. Noheda, Z. G. Ye, P. M. Gehring, J. Toulouse, D. E. Cox, and G. Shirane, Phys. Rev. B 65, 144101
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10 Y.
*Email: [email protected]
1 A.
b = at. This analysis also explains the superlattice peaks observed in powder diffraction.13–16 Recently, nanotwins of tetragonal phase with domain size about 10 nm has been directly observed by transmission electron microscopy 共TEM兲
in Pb关共Mg1/3Nb2/3兲1−xTix兴O3.17
It is worth noting that deviations from a perfect nanotwin
superlattice through fluctuations in m1 and m2, such as variations in the bilayer basis thickness and twin volume fraction,
will produce peak broadening. As in our ongoing research,
such an effect is under investigation by using a computational approach to explicitly account for the imperfections in
nanotwin superlattices. Nevertheless, the calculation of perfect nanotwin superlattices presented in this paper predicts
the lattice parameters and their intrinsic relationships, which
are in agreement with the diffraction experiments.5–8,13–16
In summary, we develop a diffraction theory of nanotwin
superlattices of low symmetry phases. This theory predicts
an adaptive diffraction phenomenon, where the Bragg reflection peaks are located at reciprocal superlattice sites in the
immediate vicinity of fundamental spots of the constituent
crystals and adaptively shift along the twin peak splitting
vectors in response to a change in twin variant volume fraction. We apply this theory to tetragonal nanotwin superlattice
and show that diffraction perceives tetragonal nanotwins
as monoclinic M C phase, whose lattice parameters are
intrinsically related to that of the tetragonal phase. It
explains the recent experimental observations in
Pb关共Mg1/3Nb2/3兲1−xTix兴O3 and Pb关共Zn1/3Nb2/3兲1−xTix兴O3.
Similar adaptive diffraction phenomena should occur in
other nanoscale superlattices of low symmetry phases.
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