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The chain structure of BaTiO3 and KNbO3
Solid State Communications, Vol. 6, pp. 715- 719, 1968. Pergamon Press. Printed in Great Britain
THE CHAIN STHUCTURE OF BaT1O3 AND KNbO3 R. Comes, M. Lambert and A. Guinier Service de Physique des Solides associê au C. N. R. S. Faculté des Sciences, Orsay, Essonne, France (Received 11 July 1968) Among the different phases of BaT1O3 and KNbQ5 only the rhombohedral form is completely ordered. The orthorhombic, tetragonal and cubic phases are only partially ordered: the structure as determined by classical diffraction data is an average structure. A description is given for the Ti and Nb displacements in the orthorhombic form. This form is built with ordered chains directed along [0101if the polarisation is directed along [101]. 3 from BaTiQ~single DIFFUSE scattering crystals of X-rays’ has 2already and electrons been observed by several authors. We have recently shown, on the basis of this diffuse scattering, that the cubic, tetragonal and orthorhombic phases of BaTiO~ and the isomorphous phases of KNbQ~are partially disordered. Consequently the structures of these phases as determined by classical Bragg reflection data are average structures.4 In the rhombohedral phase no diffuse scattering is observed, which means that the rhombohedral phase is the only perfectly ordered structure. The BaTi(~,(or KNbq~) units in the rhombohedral phase are all directed along the same [111) direction, which is the triad axis; in other words, when we consider the Ti or Nb atoms which are displaced from the centro-symmetric position, these atoms are all displaced along the same [1111direction. The three higher symmetry phases appear to be the result of several possible orientations of the BaTiQ~ (or KNbQ 3) units along the different <111> dIrections. According to the number of allowed orientations, that is to say the number of different <111> directions along which the Ti (or Nb) atoms are displaced, the average resulting symmetry is orthorhombic (2 directions), tetragonal (4 directions) or cubic (the 8 <111> directions).
of the observed diffuse scattering. This will be done below in the simplest case of the orthorhombic phase. The average structure of the orthorhombic phase is most easily pictured as derived from the original cube by stretching It along a face diagonal and compressing it ~Jong the other; these diagonals are the orthorhombic axes, In the following we will use the term orthorhombic but we always refer to the cubic or pseudo-cubic axes. In all the three higher symmetry phases the diffuse scattering is restricted to [100) relplanes directions between the orientations of two successive BaTiO3 (or KNbO3) units, or in other words between the displacements in two successive cells of the Ti or Nb atoms. The result of this is a chain structure; The chains scatter partly independently from one another giving this particular intensity distribution. In particular, If we consider the orthorhombic phase when the polarisation is directed along [101], we observe diffuse scattering in the (010) relplanes; no diffuse scattering is found
It is possible to give a more detailed description of these average structures if we take into account the distribution in reciprocal space 715
716
THE CHAIN STRUCTURE OF BaTiO3 AND KNbO3
________
~
~
: -
~J
______ -
.
Vol. 6, No. 10
y direction or [010], and N3 in the z direction or [001] (Fig. 3). The crystal is built up with chains parallel to [0103; along one of these chains all the atoms have the same displacement with respect to the average position: in the first chain C~([010] chain of Fig. 2) the atoms are displaced by +dy with respect to the average position, in the second chain C2 ([010] chain of Fig. 2) by -dy with respect to the average position (the displacements -4-dy and -dy are shown on the average structure of Fig. 2). In addition there is no correlation between two different chains, which means that all the chains of the type C1 or C2 are distributed at random on the nodes of the xoz plane. The diffracted amplitudes of the chains are respectively: Sin N2rrs.b 5~1-~.~S.g
A1 =fexp(-2i~’s.dy)
FIG. I
Sin N3rrs.b A2
Diffuse streak pattern from a KNbO3 single crystal in the orthorhombic phase. The crystal is not perfectly single domain, some very weak diffuse scattering may be seen in the (100) relpianes (vertically on the figure). Radiation: monochromatic Mo K a Exposure 2 hr. in the (100) and (001) relpianes (Fig. 1). The distribution of the intensity can be explained if we build the orthorhombic phase with two types of rhombohedral chains directed along [0103. In the first chain all the Ti (or Nb) atoms are displaced in the [111] direction (position 1 in the [010] chain of Fig. 2). In the second chain all the Ti are displaced in the [111] direction (position 2 in the [010] chain of Fig. 2). The average resulting structure gives an average displacement along [101] as determined by classical structure analysis of the orthorhombic phase. It is easy to show by means of a short calculation,~ that such a chain structure for the orthorhombic phase allows the prediction of the main features of the diagram of Fig. 1. For simplicity we will consider a crystal with only one atom per unit cell, let us say the Ti (or Nb) atom of Fig. 2. The crystal has N1 unit cells in the direction x or [100], N2 in the
=
f exp (+2i yr5~dy) Sin irs. b 2
where ~ b f
=
p
=
lattice parameter along the [010] direction scattering factor of the unique atom.
=
To calculate the total diffracted amplitude we must sum the contributions of all these chains; the simplest method is to introduce identical C0 chains which give a diffracted amplitude A0: —
Sin N2 irs.b A0
=
~
A1
-i- ~
A3
= ~
I
—
Sin rrs.b
[exp (-2ius. dy)
+
exp (+2irrs. dy)
I
we can write: A1
=
A0
+
(A1
-
A0)
A2
=
A0
+
(A2
-
A0)
and the total diffracted amplitude is:
~ n1, n2
A0
+
L~ (A n1, n~
~
-
113
A0)
Vol. 6, No. 10
THE CHAIN STRUCTURE OF BaTiO3 AND KNbO3
I;
I,
2~
2~
2~
_
‘—
2~
7
.
~,
7
_
FIG. 2 The Ti or Nb displacements In the [010] and [0 10] chains of the average orthorhombic ~structure.
p19] CHAIN
position I Tiin orNbAtoms
~oio
[ooij
~oi~CHAIN 2 Tiinorposition Nb Atoms
~~RISA1IO
AVER A GE SIR UCT URE
ORTHORHOMBIC PHASE CHAINS EXIST ONLY ALONG
Lolol
717
718
ThE CHAIN STRUCTURE OF BaTIO3 AND KNbO3
Vol. 6, No. 10
From this expression we can conclude: (1) That the diffuse scattering is restricted to
Cr~,n3
(010) relpianes, as is the case in the diaN2b
/ /1 ‘.
/
/ / / / /
gram of Fig. 1. (2) The scattered intensity must be zero in the (010) reiplane passln~through the origin of reciprocal space (s~= 0 gives sin r~2s. = 0), this isalsoverifiedinFig. 1.
I 47/
f/I/I / / / / / / /7 / / / I / / / / / / I~
(3)
thofu N2b Is related juattiie onedimenslonalcase of the more
e—n1a N1 a
The simplified model we have used for the calculation gives a good qualitative account of the main features of the diagram of FIg. 1. For this calculation, as well as for the description related to Fig. 2, we have only considered one atom (Ti or Nb) per unit cell. The case of KNbQ3 Is not drastically different from this simplified one atom model. The Nb atom which has the greatest displacement, also has a large scattering factor as compared to those of the 0 atoms and K atoms. As a consequence of this, the diffuse scattering in the relpianes is relaticely uniform for a KNbO3 single crystal as may be seen In Fig. 1.
FIG. 3 The chain Cn1, ~ of the simplifled model used for the calculation and its coordinates in the xoz plane. The diffracted amplitude of the chain Cn ~ is An ~ 1,
1,
where n1 and 113 correspond to the coordinates of the nodes in the xoz plane (Fig. 3); A~1,~ is at random either equal to A2 or A2.
For BaTiO3 the Ti atop~has a scattering factor comparable to that of the 0 atoms and much smaller than the scattering factor of Ba. As a consequence the diffuse scattering in 4 which corresponds to the an structure factor of a one given relpiane shows important modulation,
The total diffracted intensity is then: —
/
—
L~S) — L1~sJ+ J. 2I~S)
—
r -c--’ +
I 2.
~
2 ‘ç’
)
~
~..—/
n n3 n’ ~ 1, ~
1r
(A
n1, n3
-A0)I
j
2 it0
-~ ~
2. ~,
(A* ~
-M)I
L. 2, .i L -j The first term I~(s) gives the intensity which is restricted to the nodes of the reciprocal lattice and corresponds to the usual Bragg reflections. ~‘
The second term 1 2(s) gives the diffuse scattering due to the disorder. From the definitions of A0, A1, and A2 it is easy to see that this term gives: 2N ~ 2 . — — sin I 22TTs.dy 2(s)—N1N3f sln Sifl2flS.E —
unit cell of a chain. A detailed analysis of the intensity distribution in the reiplanes then gives information on the displacements of the atoms relative to one another. The preliminary calculations which have been performed show that the unit cell In the higher symmetry phases has a 6 cell prostructure similar to the rhombohedral posed by R.F. Kay and P. Vousden. We have descrthed some details of the orthorhombic phase. It Is possible to generalise the chain model for the tetragonal and cubic phases. The experimental diagrams show that in the tetragonal phase there are two chain directlons [1001 and [010], and in the cubic phase three [100.~, phases [010] and [001 J; but in chain these directions higher symmetry the chain themselves become partly disordered.
Vol. 6, No. 10
ThE CHAIN STRUCTURE OF BaTIQ3 AND KNbO3
719
References 1.
LAMBERT M., QUITTET A. M. and TAUPIN C., Communication to the 6th International Congress of the I. U. C. Rome 1963. Acta crystallogr. 16, 82-31 (1963). LAMBERT M., Communication to the French Society of Crystallography, Rennes 1965. Bull. Soc. Scient. Bretagne 39, 93 (1964).
2.
HARADA J. and HONJO G., J. Phys. Soc. Japan 22, 45 (1967).
3.
HONJO G., KODERA S. and KITAMURA N., J. Phys. Soc. Japan
4.
COMES R., LAMBERT M. and GUINIER A., (1968).
5.
GUINIER A., X-ray Diffraction in Crystals, Imperfect Crystals and Amorphous Bodies, W.H. Freeman, London (1963).
6.
KAY H. F. and VOUSDEN P.,
19, 351 (1964).
C. r. hebd. Séanc. Acad. Sci., ParIs 266, 959
Phil. Mag. Series 7, 40,
1019 (1949).
Parmi les différentes phases de BaTiQ3 et de KNbO3 seule la forme rhomboedrlque est parfaitement ordonnêe. Les phases orthorhombique, tétragonale et cubique ne sont que partiellement ordonnêes: les structures dêtermlnées par les méthodes classiques ne sont que des structures moyennes. On décrit id la structure de in phase orthorhombique. Cette phase est constituée de chaines ordonnèes dirigées suivant [0103 si in polarisation est dirigêe suivant [lOll.
THE CHAIN STHUCTURE OF BaT1O3 AND KNbO3 R. Comes, M. Lambert and A. Guinier Service de Physique des Solides associê au C. N. R. S. Faculté des Sciences, Orsay, Essonne, France (Received 11 July 1968) Among the different phases of BaT1O3 and KNbQ5 only the rhombohedral form is completely ordered. The orthorhombic, tetragonal and cubic phases are only partially ordered: the structure as determined by classical diffraction data is an average structure. A description is given for the Ti and Nb displacements in the orthorhombic form. This form is built with ordered chains directed along [0101if the polarisation is directed along [101]. 3 from BaTiQ~single DIFFUSE scattering crystals of X-rays’ has 2already and electrons been observed by several authors. We have recently shown, on the basis of this diffuse scattering, that the cubic, tetragonal and orthorhombic phases of BaTiO~ and the isomorphous phases of KNbQ~are partially disordered. Consequently the structures of these phases as determined by classical Bragg reflection data are average structures.4 In the rhombohedral phase no diffuse scattering is observed, which means that the rhombohedral phase is the only perfectly ordered structure. The BaTi(~,(or KNbq~) units in the rhombohedral phase are all directed along the same [111) direction, which is the triad axis; in other words, when we consider the Ti or Nb atoms which are displaced from the centro-symmetric position, these atoms are all displaced along the same [1111direction. The three higher symmetry phases appear to be the result of several possible orientations of the BaTiQ~ (or KNbQ 3) units along the different <111> dIrections. According to the number of allowed orientations, that is to say the number of different <111> directions along which the Ti (or Nb) atoms are displaced, the average resulting symmetry is orthorhombic (2 directions), tetragonal (4 directions) or cubic (the 8 <111> directions).
of the observed diffuse scattering. This will be done below in the simplest case of the orthorhombic phase. The average structure of the orthorhombic phase is most easily pictured as derived from the original cube by stretching It along a face diagonal and compressing it ~Jong the other; these diagonals are the orthorhombic axes, In the following we will use the term orthorhombic but we always refer to the cubic or pseudo-cubic axes. In all the three higher symmetry phases the diffuse scattering is restricted to [100) relplanes
It is possible to give a more detailed description of these average structures if we take into account the distribution in reciprocal space 715
716
THE CHAIN STRUCTURE OF BaTiO3 AND KNbO3
________
~
~
: -
~J
______ -
.
Vol. 6, No. 10
y direction or [010], and N3 in the z direction or [001] (Fig. 3). The crystal is built up with chains parallel to [0103; along one of these chains all the atoms have the same displacement with respect to the average position: in the first chain C~([010] chain of Fig. 2) the atoms are displaced by +dy with respect to the average position, in the second chain C2 ([010] chain of Fig. 2) by -dy with respect to the average position (the displacements -4-dy and -dy are shown on the average structure of Fig. 2). In addition there is no correlation between two different chains, which means that all the chains of the type C1 or C2 are distributed at random on the nodes of the xoz plane. The diffracted amplitudes of the chains are respectively: Sin N2rrs.b 5~1-~.~S.g
A1 =fexp(-2i~’s.dy)
FIG. I
Sin N3rrs.b A2
Diffuse streak pattern from a KNbO3 single crystal in the orthorhombic phase. The crystal is not perfectly single domain, some very weak diffuse scattering may be seen in the (100) relpianes (vertically on the figure). Radiation: monochromatic Mo K a Exposure 2 hr. in the (100) and (001) relpianes (Fig. 1). The distribution of the intensity can be explained if we build the orthorhombic phase with two types of rhombohedral chains directed along [0103. In the first chain all the Ti (or Nb) atoms are displaced in the [111] direction (position 1 in the [010] chain of Fig. 2). In the second chain all the Ti are displaced in the [111] direction (position 2 in the [010] chain of Fig. 2). The average resulting structure gives an average displacement along [101] as determined by classical structure analysis of the orthorhombic phase. It is easy to show by means of a short calculation,~ that such a chain structure for the orthorhombic phase allows the prediction of the main features of the diagram of Fig. 1. For simplicity we will consider a crystal with only one atom per unit cell, let us say the Ti (or Nb) atom of Fig. 2. The crystal has N1 unit cells in the direction x or [100], N2 in the
=
f exp (+2i yr5~dy) Sin irs. b 2
where ~ b f
=
p
=
lattice parameter along the [010] direction scattering factor of the unique atom.
=
To calculate the total diffracted amplitude we must sum the contributions of all these chains; the simplest method is to introduce identical C0 chains which give a diffracted amplitude A0: —
Sin N2 irs.b A0
=
~
A1
-i- ~
A3
= ~
I
—
Sin rrs.b
[exp (-2ius. dy)
+
exp (+2irrs. dy)
I
we can write: A1
=
A0
+
(A1
-
A0)
A2
=
A0
+
(A2
-
A0)
and the total diffracted amplitude is:
~ n1, n2
A0
+
L~ (A n1, n~
~
-
113
A0)
Vol. 6, No. 10
THE CHAIN STRUCTURE OF BaTiO3 AND KNbO3
I;
I,
2~
2~
2~
_
‘—
2~
7
.
~,
7
_
FIG. 2 The Ti or Nb displacements In the [010] and [0 10] chains of the average orthorhombic ~structure.
p19] CHAIN
position I Tiin orNbAtoms
~oio
[ooij
~oi~CHAIN 2 Tiinorposition Nb Atoms
~~RISA1IO
AVER A GE SIR UCT URE
ORTHORHOMBIC PHASE CHAINS EXIST ONLY ALONG
Lolol
717
718
ThE CHAIN STRUCTURE OF BaTIO3 AND KNbO3
Vol. 6, No. 10
From this expression we can conclude: (1) That the diffuse scattering is restricted to
Cr~,n3
(010) relpianes, as is the case in the diaN2b
/ /1 ‘.
/
/ / / / /
gram of Fig. 1. (2) The scattered intensity must be zero in the (010) reiplane passln~through the origin of reciprocal space (s~= 0 gives sin r~2s. = 0), this isalsoverifiedinFig. 1.
I 47/
f/I/I / / / / / / /7 / / / I / / / / / / I~
(3)
thofu N2b Is related juattiie onedimenslonalcase of the more
e—n1a N1 a
The simplified model we have used for the calculation gives a good qualitative account of the main features of the diagram of FIg. 1. For this calculation, as well as for the description related to Fig. 2, we have only considered one atom (Ti or Nb) per unit cell. The case of KNbQ3 Is not drastically different from this simplified one atom model. The Nb atom which has the greatest displacement, also has a large scattering factor as compared to those of the 0 atoms and K atoms. As a consequence of this, the diffuse scattering in the relpianes is relaticely uniform for a KNbO3 single crystal as may be seen In Fig. 1.
FIG. 3 The chain Cn1, ~ of the simplifled model used for the calculation and its coordinates in the xoz plane. The diffracted amplitude of the chain Cn ~ is An ~ 1,
1,
where n1 and 113 correspond to the coordinates of the nodes in the xoz plane (Fig. 3); A~1,~ is at random either equal to A2 or A2.
For BaTiO3 the Ti atop~has a scattering factor comparable to that of the 0 atoms and much smaller than the scattering factor of Ba. As a consequence the diffuse scattering in 4 which corresponds to the an structure factor of a one given relpiane shows important modulation,
The total diffracted intensity is then: —
/
—
L~S) — L1~sJ+ J. 2I~S)
—
r -c--’ +
I 2.
~
2 ‘ç’
)
~
~..—/
n n3 n’ ~ 1, ~
1r
(A
n1, n3
-A0)I
j
2 it0
-~ ~
2. ~,
(A* ~
-M)I
L. 2, .i L -j The first term I~(s) gives the intensity which is restricted to the nodes of the reciprocal lattice and corresponds to the usual Bragg reflections. ~‘
The second term 1 2(s) gives the diffuse scattering due to the disorder. From the definitions of A0, A1, and A2 it is easy to see that this term gives: 2N ~ 2 . — — sin I 22TTs.dy 2(s)—N1N3f sln Sifl2flS.E —
unit cell of a chain. A detailed analysis of the intensity distribution in the reiplanes then gives information on the displacements of the atoms relative to one another. The preliminary calculations which have been performed show that the unit cell In the higher symmetry phases has a 6 cell prostructure similar to the rhombohedral posed by R.F. Kay and P. Vousden. We have descrthed some details of the orthorhombic phase. It Is possible to generalise the chain model for the tetragonal and cubic phases. The experimental diagrams show that in the tetragonal phase there are two chain directlons [1001 and [010], and in the cubic phase three [100.~, phases [010] and [001 J; but in chain these directions higher symmetry the chain themselves become partly disordered.
Vol. 6, No. 10
ThE CHAIN STRUCTURE OF BaTIQ3 AND KNbO3
719
References 1.
LAMBERT M., QUITTET A. M. and TAUPIN C., Communication to the 6th International Congress of the I. U. C. Rome 1963. Acta crystallogr. 16, 82-31 (1963). LAMBERT M., Communication to the French Society of Crystallography, Rennes 1965. Bull. Soc. Scient. Bretagne 39, 93 (1964).
2.
HARADA J. and HONJO G., J. Phys. Soc. Japan 22, 45 (1967).
3.
HONJO G., KODERA S. and KITAMURA N., J. Phys. Soc. Japan
4.
COMES R., LAMBERT M. and GUINIER A., (1968).
5.
GUINIER A., X-ray Diffraction in Crystals, Imperfect Crystals and Amorphous Bodies, W.H. Freeman, London (1963).
6.
KAY H. F. and VOUSDEN P.,
19, 351 (1964).
C. r. hebd. Séanc. Acad. Sci., ParIs 266, 959
Phil. Mag. Series 7, 40,
1019 (1949).
Parmi les différentes phases de BaTiQ3 et de KNbO3 seule la forme rhomboedrlque est parfaitement ordonnêe. Les phases orthorhombique, tétragonale et cubique ne sont que partiellement ordonnêes: les structures dêtermlnées par les méthodes classiques ne sont que des structures moyennes. On décrit id la structure de in phase orthorhombique. Cette phase est constituée de chaines ordonnèes dirigées suivant [0103 si in polarisation est dirigêe suivant [lOll.