The crystal structure of KCdF3

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The crystal structure of KCdF3

M. Hidaka, S. Hosogi

To cite this version:

M. Hidaka, S. Hosogi. The crystal structure of KCdF3. Journal de Physique, 1982, 43 (8), pp.1227-

1232. �10.1051/jphys:019820043080122700�. �jpa-00209499�

The crystal structure of KCdF3

M. Hidaka and S. Hosogi (*)

Department ^of Physics, Faculty ^of Science, Kyushu University 33, ^Fukuoka 812, Japan (Reçu ^{le 28} janvier 1982, accepté le 27 avril 1982)

Résumé.

La structure cristalline de KCdF3 ^est déterminée dans le cas de la troisième phase (ortho)rhombique

et affinée en utilisant une méthode de moindres carrés prenant ^en considération la structure en domaines. L’affine- ment utilise les intensités des réflexions de surstructure ponctuelles ^{X, M et R mesurées en} chambre de Weissenberg

en adoptant ^la technique ^{des films} superposés ^{et la radiation} filtrée MoK03B1. Le groupe d’espace ^de KCdF3 ^est

Pbnm-D162h (z ⁼ 4). ^{La structure est} bien décrite par les condensations successives des modes de phonon ^mou M3

et Z5 (semblables ^à Rx25 ⁺ Ry25). ^Le facteur R atteint 8 % lorsqu’on ^tient compte de l’intensité des reflexions X et M,

et 9,8 % ^{si l’on} inclut les réflexions ^R.

Abstract

The crystal ^{structure of} KCdF3 is determined in the third, orthorhombic phase ^{and refined} by using

a full matrix least-squares ^method by taking ^domain structures into consideration. Refinement is performed ^with superlattice-X, ^{M and R} point reflection intensity data collected on the multiple-film Weissenberg photographs

taken with filtered MoK03B1 radiation. Pbnm-D162h (z ⁼ 4) ^is assigned ^to the space group of KCdF3. ^{The R value}

reaches 8 % ^{for the} intensity data of the X and M point reflections. When the R point reflections are included in the

calculation, the R value is 9.8 %. ^{The structure is well described} by the successive condensations of M3

and Z5 (Rx25 ⁺ Ry25-like) soft-phonon ^modes.

Classification

Physics ^Abstracts

77.80

1. Introduction.

The structural analysis ^of crystals displaying structural phase transitions (SPT) ^{is of}

fundamental importance ^for studying the mechanism of the transition. Although ^{there are a} lot of investi-

gations concerning ^the transitions of the pseudo- perovskite compounds ABC3, ^{it is difficult to determine}

the structure in each phase because of the _appearance of poly-domains ^{in a} single crystal.

By ^{use of} X-ray and differential scanning calorimetry technique, ^the perovskite crystal KCdF3 ^{was found}

to have three transition points ^{at 485, 471 and}

243 K [1] : ^{The first two} transitions are caused by ^the

condensations of soft-phonon ^{modes at the cubic}

Brillouin zone boundary. ^At the third transition neither a change ^{in the} extinction rule for the diffraction pattern ^{nor a} change in the number of formula units is observed. The space groups of the four phases are, from the high temperature side, Pm3m, P4/mbm,

Pbnm and Pbn21 (1) respectively. The first and second transitions can respectively ^be explained ^{in terms of}

(*) Present address : International Institute for Advanced

Study ^of Social Information Science, Fujitsu Ltd, ^Numazu 410-03, Japan.

(1) ^Non centrosymmetric space group is assumed

the condensation of M3 ^and Rx. ⁺ R]s soft-phonon

modes at the M and R points in the cubic Brillouin

zone.

The condensation of soft-phonon ^{modes at the cubic}

Brillouin zone boundary ^is accompanied by ^the development ^of superlattice reflections, ^{which are}

related to the translational symmetry ^{of the lattice.}

Since they ^{do not exist in} the cubic phase, ^atomic displacements ^{due to the SPT} directly contribute to intensities of superlattice reflections, by ^{which the}

structure can be determined Some problems ^however

exist in the structure analysis of non-cubic perovskites :

First since ions mainly participating ^{in SPT are} usually light ^ones, intensities of superlattice reflections should be rather weak in the X-ray diffraction; second,

in low temperature phases, crystallographic ^domains

take place, and this makes diffraction patterns compli-

cated and consequently ^not easily interpreted.

The first problem is inevitable in the X-ray ^diffrac-

tion method, ^{but the neutron} diffraction method may be complementary. Unfortunately, absorption ^{of neu-}

trons by ^{Cd atoms is so} high ^{that neutron} diffraction with KCdF3 crystals ^{is not} practicable. ^The only

way which is practicable ⁱⁿ getting intensity ^{data for}

the structure analysis ^{is to use} X-ray diffraction at

temperatures ^{as low as} reasonable in each phase. ^This

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019820043080122700

1228

will make the intensities of relevant superlattice

reflections a maximum. It will be rather difficult ^to get intensity data for the phase II, because it _appears

only ^{in a narrow} temperature range. In the present

experiment, ^{the structure} analysis ^{was therefore}

undertaken only ^{for the} phase ^{III and was} qarried ^out

at room temperature.

The second problem ^{was solved} by ^Hidaka [2],

who determined the crystal ^{structure of} KMnF3

at 50 K taking the domain structure into account.

Furthermore, ^{the structure of} NaNiF3 [3] ^{at room} temperature, KMnF3 ^and SrTi03 [4] ^and KCaF3 [5]

were successively determined.

In this _paper, the procedure ^{of structure} analysis

of the phase ^{III of} KCdF3 containing six kinds of domains is presented and the results are discussed in terms of the successive condensation of soft -

phonon ^modes.

2. ExperimentaL

^The specimen used in the _pre- sent experiment ^{was of} cylindrical form, about 0.13 mm in diameter and 1.5 mm in length, ^{with the}

axis parallel ^{to the} [001]. The lattice constants were

determined by using high-angle reflections on oscilla- tion and Weissenberg photographs : a ⁼ 6.103,

b ⁼ 6.103 and c ⁼ 8.660 A, ^{which are} slightly ^different

from those given by ^Swanson, McMurdie, ^{Morris and}

Evans [6]. Although the values of a and b coincide within experimental ^error, the intensities of crystallo- graphically related reflections on Weissenberg photo- graphs clearly ^{show a two-fold} symmetry, instead of a

four-fold one, about the [001]. ^{For the structure ana-} lysis, Weissenberg photographs ^{with a} multiple-film arrangement ^were taken with filtered MoKa radiation for the reciprocal-lattice planes ^{from hk0 to hk5,}

where 185 independent superlattice reflections were

observed; ^the intensity ^{was measured} by using ^a

microdensitometer. The intensity ratio of the strongest

to weakest superlattice reflection is about 200 to 1.

Corrections were made for the Lorentz-polarization effect, ^for elongation ^of spots ^on the film and absorp- tion ; the linear absorption coefficient for MoKa is 80.0 cm-1.

3. Structure determination.

Since Weissenberg photographs have revealed the domain structure, ^we should treat the intensity of reflections taking ^the

influence of the domain structure into account. In the

specimen ^{used in the present} experiment, ^{one of the}

six kinds of domains was accidentally predominant;

we designate ^it the domain I-A. The indexing ^of

reflections on Weissenberg photographs ^{was made}

on the basis of this domain. In figure ¹ domains in the orthorhombic phase ^of KCdF3 ^are schematically

shown. Here ap, bp ^and cp ^{are the axes of a} pseudo- perovskite, ^{while the} orthorhombic cell has dimensions

-./2 ap x , ,/2- bp ^{x 2} cp. Although these six domains

are the same as those of NaNiF3 [3] ^and KCaF3 [5],

the axial angle y between ap ^and bp ^{is equal to 90° in}

Fig. ^1.

Crystallographic domains in the orthorhombic

phase ^of KCdF3. Domains labelled as I, II and III are speci-

fied by the direction of [001]p, while those labelled as A and B

are connected with each other by interchanging [100]p ^and [010]p.

KCdF3. Therefore lattice constants a and b take the

same value. Then a part of reflections coming ^{from the}

domains II and III completely overlap ^{with those}

from the domain I. This fact makes interpretation ^of

the diffraction pattern ^rather complicated.

Reflections on Weissenberg photographs ^{can be}

classified into five _groups with the indices as follows :

where n is an integer. Here, ^the group (i) corresponds

to fundamental-lattice reflections and the others to

superlattice reflections : i.e., ^the groups (ii), (iii) and (iv) respectively correspond ^{to the R, M and X} points

in the cubic Brillouin zone. For the _group (v), integral

indices can be assigned ^{if we assume} that the domain II

or III contributes to these reflections. As an example,

a zero-layer Weissenberg photograph ^{taken with}

filtered MoKa radiation is shown in figure ^{2, where}

reflections with the half-integer ^{index are visible.}

It is to be noted that the intensity ^{of the} group (v)

is much weaker than that of M and X reflections; ^this

indicates that the volume of the domain I is much

larger than that of the domain II and III. In general,

R points of the domain II and III overlap ^{with those}

of the domain I, ^{then the} intensity ^{at an R} point ^{on a} Weissenberg photograph ^is consequently ^{the sum of}

intensities from all the domains. On the other hand,

X and M reflections from the domain II and III appear at the position ^with half-integer ^indices.

Glazer [7] ^and Aleksandrov [8] have shown that

successive condensations of the soft-phonon ^modes

M3 ^and R25 ⁺ Ry 5 ^{lead to the} space group

Pbnm (D"). ^{Here we} quantitatively discuss the

Fig. ^2.

Zero-layer Weissenberg photograph ^of KCdF3

around the [001]p taken with filtered MoKa radiation.

Reflections with the half-integer ^{indices are} visible; ^{some of}

these reflections are labelled by ^circles.

volume ratio of the domains on the basis of this _space group. The space group Pbnm will be confirmed if the volume ratios of domains consistent with all the

intensity ^{data are} determined and if the structural parameters ^are determined with ^a reasonable reliability

factor.

In the space group Pbnm, ^the following ^conditions

should be observed :

It was found, however, that the conditions for the hOl and Okl ^were not fulfilled in the present ^case.

In figure ^3a the diffraction pattern along hOl and Okl

on the hkl photographs (I ⁼ odd) ^is schematically

shown. This figure suggests that there is a certain relation between the intensity of Okl and hOI.

Intensities of the following ^three pairs ^{of X} points

were measured : i.e., ⁰⁴¹ (88), ⁴⁰¹ (11); ⁰⁶³ (49),

603 (7); ⁰⁴⁵ (78), ⁴⁰⁵ (12), ^{where the} intensity ^is given

in the parenthesis. ^For these three pairs of reflections the intensity ^{ratio is} nearly ^{constant. This fact can be} explained ^{if we} regard hOl, ^{which is} prohibited ^{in Pbnm}

as h + 1 ⁼ 2 n + 1, ^{as the} Chl of the domain I. In

figures ^{3b and c the} diffraction patterns ^{from the} domains I-A and I-B are schematically ^shown respec-

tively. Superposition ^{of these} patterns gives ^the pat-

Fig. ^3.

^Schematic representation of diffraction patterns

along [100] ^and [O10] ^{on first} layer Weissenberg photograph.

a) ^Observed diffraction pattern. b) Diffraction pattern ^cau- sed only by ^{domain I-A, which is} predominant ^{in the} speci-

men. c) ^{That caused} by ^I-B.

tem shown in figure ^{3a. We can therefore assume}

that the space group of the phase ^{III of} KCdF3 ^is Pbnm; ^the co-ordinates of equivalent ^atomic positions

are given ^{in table I. On} the other hand, ^{from the} intensity ^of reflections with half-integer ^indices given below, ^{we can} determine the volume ratio for the domains I, ^{II and III.}

Table I.

Atomic positions ⁱⁿ KCdF3 (space group

Pbnm).

Since we see that the -12 ⁰ reflection is the 225 of the domain II and that the 110 ^{is the 221 of the domain III,}

the volume ratios of the domains I, ’II ^{and III are}

obtained to be 85, 9 ^and 6%, respectively. These ratios

were used as initial parameters ^{in the course of the}

refinement of the structure.

1230

4. Refinement of the structure.

A refinement of the structure was carried out by using the full matrix

least-squares method. The quantity minimized is

where i denotes i-th kind of domains, Ilk ^scale factor, F. ^and Fr the observed and calculated structure factors

respectively, and Vi the volume ratio for each domain;

the atomic scattering ^factors given in the International Tables for X-ray Crystallography ^are used. Unit

weight (Whkl ⁼ 1) ^is assumed for all the reflections.

Parameters refined are positional parameters, ^the

isotropic temperature factor for K and F atoms and the scale factors. The temperature factor of Cd atom cannot be determined, ^{since Cd atoms are at the} special position ^and consequently ^{there is no contri-}

bution of Cd atoms to superlattice reflections.

Since the quantity ^{X 2 is not a} linear function of the structural parameters, ^{it is} necessary ^to expand ^{X 2}

as a function of the parameters ^{and to use the method} of linear least-squares ^{to determine} optimum ^values

of the parameters. ^{To start the} calculation, approxi-

mate values of the parameters ^{should be} known;

if not, X 2 does not converge. The values of corres-

ponding parameters ^of NaNiF3 [3] ^{were used as}

initial values of the parameters. ^The key-shift ^method [9] ^was applied ^{at the} early stage ^{of the} refinement;

by this method the value of the R factors, ^{which is a} measure of goodness ^{of the} fit, becomes about 15 %

for each parameter. ^A plausible ^{set of} parameters ^thus determined was used as initial values for the ordinary least-squares refinement. Starting ^{from 20} %, ^{the value}

of the R factor decreased gradually ^and finally ^reached

8.0 % for the intensity data of the X and M point

reflections. When the intensity data of the R point

reflections were included in the calculation, ^{the R}

value was 9.8 %.

5. Results and discussion.

The final values of the structural parameters ^{of the} phase ^{III of} KCdF3

determined at room temperature ^are given ^{in table} II ; values on the left- and right-hand sides of the column

are determined by using the data of X and M reflections and those of X, ^{M and R} reflections, respectively.

An error for each parameter ^{value is} given ^{in paren-}

thesis. The difference between the corresponding parameter ^{values is not so} significant. Hereinafter we

describe the structure of KCdF3 by ^the parameter values determined from the intensity ^data excluding

those of R points. ^{Table III} lists the calculated and observed ^structure factors. The crystal ^{structure is}

shown in figure 4, ^{where arrows} indicate the displa-

cement of the ions from the positions in the cubic-

phase ^structure.

The structure of KCdF3 ^cannot be described by ^a

linear combination ^of displacements corresponding

to condensation of M3 ^and R’. 2 ⁺ R2 5 ^{modes, both}

Table II.

Structural parameters of ^the phase ^III of KCdF3 determined at room temperature.

of which consist in static rotational displacement ^of CdF6 octahedra. As the figure shows, ^{an additional}

shift of K atom is noticeable; the shift has a staggered arrangement along ^{the z axis. It} is therefore reasonable to connect the shift with an eigenvector of the irre- ducible representation ^{of the X} point ^{in the cubic}

Brillouin zone. It has been pointed ^{out that once a}

normal mode at the M point ^has condensed, ^{the M} point becomes the r point ^{in the} phase ^II [10, 11].

The R and X points in the cubic phase ^become equi-

valent in the phase II, ^both being ^Z point ^{of the} simple tetragonal ^{Brillouin zone.}

The R25 ^normal mode, ^{which is a} pure rotational mode of CdF6 octahedra around the four-fold axis in the cubic phase, is modified in the tetragonal phase

caused by the condensation of the M3 ^mode. Fujii, Hoshino, Yamada and Shirane [10] ^and Hirotsu, Harada, Iizumi and Gesi [11] ] ^{made a} normal-mode

analysis ^{for the Z} point ^{in the} tetragonal ^{lattice with}

the space group D 4h 5 ^{and the} eigenvectors ^{for each}

Fig. ^4.

Crystal ^{structure of} KCdF3 ^{at room} temperature.

Numerical figures ^are given ⁱⁿ A.

Table III.

Calculated (Fe) and observed (Fo ) structure factors, FcA ^and FcD ^are calculated structure for ^the

domains I-A and I-B respectively.

normal ^{mode were} determined According ^{to the}

notations of Fujii et al., triply degenerate R25 ^modes split ^into doubly degenerate ZS ^{modes and a non-} degenerate Zl mode, ^the Z5 mode, represents ^the rotational vibration of CdF6 octahedra around

pseudo-cubic [t00]p ^and [010]p ^{axes, while the} Zl ^mode

is accompanied by displacements ^{of both K and F 1}

atoms along the rotation axis [OOl]p’ ^{Since the succes-}

sive condensations of M3 ^and Z 1 ^{modes do not} change

the tetragonal symmetry ^{of the} lattice, Z5 mode should be taken for the condensation mode at the tetragonal-

to-orthorhombic phase transition.

The interatomic distances and bonding angles ⁱⁿ

CdF6 octahedron are shown in figure ^{5, from which}

1232

Fig. ^5.

Interatomic distances and bond angles ^{in the} CdF6 octahedron.

we find that the octahedra are rotated from the posi-

tion in the ideal perovskite structure; ^{at room tem-} perature the octahedra are rotated around [001]p

and [110]p ^{by as much as} 8.80 and 6.14 degree respec-

tively, while the distortion of ^an octahedron is not so

significant

The root-mean-squared displacement of the ion is related with the isotropic ^thermal parameter by

B ⁼ 8 n’ U2 ). ^{Therefore we} obtain that

Acknowledgments.

The authors wish to grate- fully thank Professor K. Horai for supplying crystals,

Professor A. Okazaki for various comments and critical reading ^{of the} manuscript ^{Thanks are also}

due to Dr. M. Ono for collaboration at an early stage of the work.

References

[1] HIDAKA, M., HOSOGI, S., ONO, ^{M. and} HORAI, K., Solid State Commun. 23 (1977) ^503.

[2] HIDAKA, M., ^J. Phys. ^Soc. Japan ³⁹ (1975) ^180.

[3] ^{HIDAKA, M. and} ONO, M., ^J. Phys. ^Soc. Japan ⁴³ (1977) 258.

[4] OKAZAKI, ^{A. and} ONO, M., ^J. Phys. ^Soc. Japan ⁴⁵ (1978) 206.

[5] ONO, M., INOUE, ^{K. and} HIDAKA, M., ^{to be} published.

[6] SWANSON, ^H. E., MCMURDIE, ^H. F., MORRIS, ^{M. C.}

and EVANS, ^E. H., 1970 ^Standard X-ray Diffraction

Powder Patterns, Natl. Bur. Std., Monograph ^25,

Sect. 8. through Landolt-Bornstein New Series

(Springer-Verlag Berlin, Heidelberg, ^New York) 1973, ^Vol. 7, p. 110.

[7] GLAZER, ^{A. M., Acta} Crystallogr. ^{B 28} (1972) ^3384.

[8] ALEKSANDROV, ^K. S., ^Sov. Phys. Crystallogr. ²¹ (1976)

133.

[9] ITO, T., ^Z. Kristallogr. ¹³⁷ (1973) ^399.

[10] FUJII, Y., HOSHINO, S., YAMADA, ^{Y. and} SHIRANE, G., Phys. ^{Rev. B 9} (1974) ^4549.

[11] HIROTSU, S., HARADA, J., IIZUMI, ^{M. and} GESI, K.,

J. Phys. ^Soc. Japan ³⁷ (1974) ^1374.