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High-Order Diffraction Satellites and Temperature Variation of the Modulation in the Incommensurate
Phase of Rb2ZnCl4
I. Aramburu, G. Madariaga, D. Grebille, J. Pérez-Mato, T. Breczewski
To cite this version:
I. Aramburu, G. Madariaga, D. Grebille, J. Pérez-Mato, T. Breczewski. High-Order Diffraction Satel- lites and Temperature Variation of the Modulation in the Incommensurate Phase of Rb2ZnCl4. Jour- nal de Physique I, EDP Sciences, 1997, 7 (2), pp.371-383. �10.1051/jp1:1997150�. �jpa-00247333�
J. Phys. I £Fonce 7 (1997) 371-383 FEBRUARY 1997, PAGE 371
High-Order Dilfraction Satellites and Tenlperature Variation of
the Modulation in the Inconlmensurate Phase of Rb2ZnC14
1. Aramburu (~>*), G. Madariaga (~), D. Grebille (~), J-M- PArez-Mato (~) and T. Breczewski (~)
(~ Departamento Fisica Aplicada I, Escuela T4cnica Superior de Ingenieros Industriales y de
Telecomunicac16n, Universidad del Pais Vasco, Alameda de Urquijo sIn. 48013 Bilbao, Spain (~) Departamento Fisica de la Materia Condensada, Facultad de Ciencias,
Universidad del Pais Vasco, Apdo. 644, 48080 Bilbao, Spain (~) Laboratoire de Chimie-Physique du Sulide (**), #cafe Centrale,
92295 Chitenay-Malabry Cedex, France
(~) Departamento Fisica Aplicada II, Facultad de Ciencias, Universidad del Pais Vasco, Apdo. 644, 48080 Bilbao, Spain
(Receiied 23 July 1996, received iii final form 7 October 1996, accepted 21 October 1996)
PACS.61.44.Fw Incommensurate crystals
PACS.64.70.lth Commensurate-incommensurate transitions PACS.61.10.-I X-ray diffraction and scattering
Abstract. The intensity of a set of main reflections and satellites up to seventh order has been measured as a function of temperature within the incommensurate phase of ltb2ZnC14.
The observed temperature dependence of the diffraction peaks can be explained by a variation of the static structural modulation through the changes in 3 parameters: the amplitude of the
primary mode, the soliton density that describes its anharmonicity, and the amplitude of a third harmonic modulation. The eigenvectors of the primary mode and the third harmonic used for the static modulation have been derived from previously published structural data on the incommensurate and the ferroelectric structure. The soliton density behaviour, determined for the first time from X-ray diffraction data, agrees with results derived from other techniques.
1. Introduction
The diffraction pattern of an incommensurate (IC) phase is characterized by the appearance of satellite reflections that cannot be indexed with simple rational numbers in the lattice corre-
sponding to the main reflections. In IC phases with a one-dimensional displacive modulation, the appearance of these satellites is related to the freezing of a soft mode of wave-vector qi in
the structure. Therefore, near the transition temperature (Ti) to the high symmetry phase,
satellites will be located at +qI from the main reflections (first order satellites). As temperature is lowered several changes take place in the structural distortion: the period of the modulation varies, and the amplitude of the first harmonic (the order parameter of the transition) usually
(*) Author for correspondence (e-mail: wmbarleitllg.ehu.es) (**) UA 453
© Les (ditions de Physique 1997
increases. Besides, according to certain phenomenological models [1,2j, higher order harmonics
are expected to progressi,>ely condense in some compounds in order to produce the continuous appearance of the lock-in domains. This temperature dependence of the structural modulation provokes changes in the intensity and location of the first order satellites, and the appear-
ance of new satellites located at the wave-vectors +nql around the main reflections (n~~-order satellites).
From a phenomenological point of view [3], a typical compound where this type of struc- tural variation with temperature should be observed is Rb2ZnC14. This material undergoes
at Ti " 303 K a continuous transition from a Normal phase (N) (Pmcn, Z
= 4) to a one-
dimensional modulated IC phase with a wavevector qi " 3 d(T) c*, due to the freezing
of a 22 soft mode. On cooling, at TL " 191 K, d vanishes and the IC phase locks into a ferroelectric commensurate phase (F) (P21cn, Z
= 12), with a triplicated cell along c* and spontaneous polarization along a*. The detection of high-order satellites [4,5] in the X-ray
diffraction pattern of this compound seems to confirm the presence of high order harmonics in the structural modulation. Besides, measurements carried out with other techniques [6,7j also support this hypothesis. However, up to the moment, only the first harmonic of the structural
modulation has been determined [8]. The aim of the present work is to characterize all the relevant distortions present in the structural modulation of Rb2ZnC14, and to determine their temperature variation from a careful measurement and analysis of the temperature dependence
of high order satellite intensities.
The intensity of a set of main reflections and satellites up to the fl~ order has been measured all along the IC phase of Rb2 ZnC14. Their temperature variation can be explained introducing
a structural model where the static modulation is described as a superposition of 22 (harmonics 1, 5, 7, and 23 (harmonic 3) distortions. In this model, the temperature dependence of the structural modulation is characterized by the temperature variation of 3 parameters: the order parameter (the amplitude of the 22 distortion), the soliton density [I] that describes its anhar- monicity, and the amplitude of the 23 distortion (the polar mode). The adjusted temperature
variation of these three parameters is able to fairly explain the intensities measured.
2. Experimental
The crystals were grown isothermally at 310 K by the dynamic method from acid aqueous so-
lutions (pH < 2) which contained a stoichiometric ratio of rubidium chloride and zinc chloride.
The product of synthesis was purified by recrystalization from distilled water. The obtained colourless single crystals were large, transparent, and of good optical quality. The chemical composition of the crystal was confirmed by atomic spectrometry (Rb) and chemical analysis (Zn). The crystal used in the experiment had a volume 5 x 2 x 0.5 mm3 with the a-axis
normal to the large face (setting Pmcn). X-ray diffraction experiments were performed on the
high resolution two axis goniometer of the Laboratoire de Chimie-Physique du Solide of the
(cole Centrale de Paris, using the Cu-Ko radiation of
a rotating anode generator of18 kW, monochromated with the (400) reflection of a post-sample InP single crystal. The angles 29 and ~J were accurately measured by means of incremental photoelectric encoders (accuracy
2 x 10~~°). The sample was mounted in a cryostat, and cooled by nitrogen gaseous conduc- tion with a thermal stability of about 0.05 K. The measuring accuracy was better than 0.I K.
~~easurements were performed in reflection mode using two types of step scans (9 29 and
~J 9) for each reflection and at each temperature. It was also possible to map the diffraction intensity distribution in selected areas of reciprocal lattice planes in order to locate the peak intensity maxima.
N°2 MODULATION IN ltb2ZnC14 VERSUS TEMPERATURE 373
(601-1) 4
o
o o
o
~ O
~ #
3
~ O
~
(603-7)
~
.~ (6002)
,5~
)
g
~~~ '~~
fl
(602-4)
o (degrees)
fi (60.15)
(603-7)
i~
i
ot~ ~~a
~. o.
~o
o.
~
~
oo o.~ '
~.
o
Fig. 1. Intensity distribution around (601-1) at T
= 198 K. It can be observed that even at 6 K above the lock-in transition the peaks are clearly solved (d ie 0.018). The inset shows the profile of the 7~~ order satellite in the (w 9) scan at T
= 195 K. The continuous line represents the fit of the
experimental points to a Gaussian curve with an asymmetric background.
The misfit parameter d was obtained from the rotation angles of the sample for the (6000), (6040) and (60-11) reflections. Unlike the results obtained with samples grown by other
methods [9], no pinning of d over the IC phase was observed. The lock-in transition was clearly
detected (d = 0) and took place at TL " 192 K. The temperature dependence of the intensity
of main reflections and satellites of different orders was measured bet~N.een 286 K and 195 K.
Together with the satellite reflections reported in [4j other main reflections and satellites of l~~
and 2~d order up to
a total of 11 reflections were measured. Although the weakest reflections
were only detected near TL (it T
= 195 K), their peaks were clearly solved (see Fig. I). Except for the (6040) reflection, the scans (~J 9, 9 29) approximately corresponded to scans along
c* and a*, respectively.
The reflection profiles were fitted to Gaussian curves with symmetric background except
at 195 K (d m 0.015), where the overlapping along the (~J 9) direction of higher order satellites with main reflections and first order satellites recommended the use of an asymmetric background. The values of the peak intensity and background thus obtained were corrected for
Lorentz-polarization and absorption. Unlike the results obtained in other studies [5,9-12], the width of the reflections remained practically constant, except at 195 K where the satellites of 2~d, 3~d and 5~~ orders showed a remarkable broadening in their (~J 9) scan. In the particular
5
286K 210K
4 ' (6000) '
' (6040)
,
? ~
W~ 3 '
b~ ~~~'~~~
©
~4 2 (601i)
(602-2)
1 (601-3)
(600-2)
, (60-15)
0
20 40 60 80 100
TrT
Fig. 2. Temperature dependence of the integrated intensities for the measured reflections in a log- log plot (T m 303 K). In the case of the main reflections and the ((60-11 ), (6011 ), (600-2)) satellites,
the temperature dependence of the intensity clearly changes at 210 K. The error bars are inside the
symbols in the main reflections and first order satellites.
case of the l~~ order satellite (60-11) no appreciable broadening between 286 K and 195 K was
observed in both scans, in clear disagreement with the results indicated in [12]. In the case of the 3~~ order satellite (601-3) the broadening of around 25% observed at 195 K was clearly less than the previously reported at the same temperature in [5j (r- 80%). Besides, this broadening
took place in a very reduced temperature range above TL (/hT < 8 K), in contrast with the continuous change with temperature indicated in the same reference (/hT m 20 K). The
origin of this broadening is not clear and, according to the continuous temperature dependence
obtained for the misfit parameter d(T)~ it seems not to be related with the pinning by defects of the modulation wave, in contrast with some proposals [13]. The integrated intensities were calculated as the product of the peak intensity and the full width at half maximum (FWHM) in the (~J 9) scan. The variance a~(I) for the experimental integrated intensities was calculated from counting statistics [14]. Finally, the weights (I la()F)o)) of the observed structure factors
were calculated. As the background was typically less than 10% of the peak intensity for all reflections between 286 K and 200 K, their weights were essentially equal in that temperature range.
The temperature dependence of the integrated intensities is represented in Figure 2 in a
log-log plot. A simple way to quantitatively characterize this temperature dependence is by
means of effective exponents (fle~) defined as: I(t)
-J t~fleff, where t e Tj T. The determined (2fle~) are listed in table I, together with the intensities relative to the first order satellite
(60-11) at 200 K. In the case of the ((60-11), (6011), (600-2)) satellites, the value of the fle~ clearly changes at 210 K. For that reflections, the average fle~ obtained for the whole
temperature range is also indicated. It can be observed that, except in the case of the fifth order satellite, the values reported by Andrews and Mashiyama [4] coincide with the average fl~~ obtained, while their relati,>e intensities are similar to the ones of Table I. In contrast, the
too high value reported by Ehses [5] for the fle~ of the third order satellite (-J 5.I) is perhaps
a consequence of the big broadening of the satellite observed over a large temperature range in that measurement.
N°2 MODULATION IN Rb2ZnC14 VERSUS TEMPERATURE 375
Table I. Effective exponents of the satellite reflections measured and intensities relative to the first order satellite (60-ii ) in comparison with the ones determined by Andrews &
Mashiyama (ABM) (Ii.
Satellite In/I~o-11 2fle~
200 K ATOM (193 K) Range 2 fle~ Average ATOM
60-11 1. 1. T > 210 K 0.65 + 0.07 0.70 + 0.03
0.68 + 0.05
T < 210 K 2.82 + 0.03 0.92 + 0.05
6011 2A x 10~~ T > 210 K 0.62 + 0.04
0.66 + 0.04 T < 210 K 2.6 + 0.2
600-2 1-1 x 10~~ 10~~ T > 210 K 2.24 + 0.09
2.6 + 0.2 2.7 + 0.1
T < 210 K 5.72 + 0.08
602-2 2.8 x 10~~ T < 251 K 1.9 + 0.2 1.9 + 0.2
402-2 2.6 x 10~~ T < 251 K 1.61+ 0.08 1.61+ 0.08
601-3 2.5 x 10~~ 3 x 10~~ T < 251 K 3A + 0.2 3A + 0.2 3.6 + 0.1
60-15 3.7 x 10~~ 3
x 10~~ T < 220 K 9.2 + 0.7 9.2 + 0.7 5.8 +1
3. Description of the Structural Modulation
In order to explain the observed temperature variation of the diffraction intensities reported above, a realistic model of the static distortion in the IC phase and its temperature variation
are required. This model can be constructed taking as main basis the sinusoidal structural modulation determined at 210 K and reported in [8].
The IC phase of Rb2ZnC14 is due to the freezing of a 22 soft mode. Near Ti the static distortion of this symmetry is essentially harmonic, as determined in [8]. But, as temperature is further
lowered, it is expected [3] that new harmonics of this symmetry arise in order to produce the continuous appearance of the six domains that characterize the lock-in phase. According to the superspace group transformations [lsj, the harmonics of 22 symmetry will be the ones of order
(1, 6m +1 (m E Z+ ). The progressive appearance of these harmonics in the 22 distortion can be described from a phenomenological point of view in the framework of the Landau theory of phase transitions [16]. The usual approach to describe this anharmonic distortion is to use the
lock-in phase as a reference [17], so that the order parameter is spatially inhomogeneous in the IC phase. It has been demonstrated [18] that in this approach, within the constant amplitude
approximation (which is rather well satisfied [19,20]), the atomic displacements corresponding
to the 22 distortion can be described by the atomic modulation functions:
u((u)
= piu(~ cos (2~9(ns,u) + ~fi(~ + qL r~ (1)
where
~J labels the atoms, a = z,y,z and u is the internal coordinate (u = qi .I, being I the cell index and qI the wave-vector of the modulation). qL is the value of the wave-vector in the lock-in phase and r" the average position of the atom ~J in the average unit cell. pi and
9(ns,u) represent the amplitude and the inhomogeneous phase of the order parameter, while
(u(~,~fi(~) are the amplitude and phase of the first harmonic in the modulation at a given temperature, describing the eigenvector of the 22 frozen mode. By definition pi
" I at that
temperature and will increase at lower temperatures, while the mode eigenvector is expected
to remain essentially constant. The other term varying with temperature in (I) is 9(ns,u).
Starting from general arguments [18j or through an specific free energy expansion [3j it can be demonstrated that, in the case of Rb2ZnC14, 9(ns,u) satisfies the following sine-Gordon
equation [21]:
~~)~
= ~~~~~~ (l k~cos~(39)) (2)
du ~
being K(k) the complete elliptic integral of the first kind. The parameter k can be related with the soliton density [1,3], ns:
~
~~ 2K(k) ~~~
that measures the anharmonicity degree of the modulation. Choosing 9(0) e 0, the function
9(u) is perfectly determined at each temperature by the value of ns and equation (2). For temperatures near Ti, ns m I and 9(u)
= u, so that the distortion is described only by a
first harmonic (sinusoidal limit). As temperature is lowered, pi increases and ns decreases, acquiring 9(u) a progressive stepped shape with as many steps as the number of domains present in the lock-in phase (6 in this case). Finally, at T
= TL, ns = 0 and 9(u) will be
constituted by 6 perfect steps (solitonic limit). Thus, if the values of (u(~,~fi(~) are known at a certain temperature from
a structural determination as that reported in [8], equation (I)
allows to describe the temperature variation of the 22 structural distortion in the IC phase by
means of the changes in only two global parameters, pi and ns. This Landau-type model of the
temperature variation of the 22 structural distortion is further supported by the experimental
fact that the eigenvector of the distortion is rather constant even after the lock-in transition
(see appendix).
However, the 22 distortion is not the only distortion present in the IC phase. In fact, the
polarization P~ in the F phase, of symmetry B2u, suggests a significant presence in the IC distortion of a 3~~ harmonic of 23 symmetry (the one that becomes B2u when qi is locked into
qL). The atomic displacements in the IC phase can, therefore, be approximately described by
the modulations:
u((u) = pin(~ cos (2~9(ns,v) + ~fi(~ + qL r")
+p3u(~ cos (2~3u + ~l(~ + 3qL r~) (4)
where we are neglecting other secondary modes of smaller amplitude (harmonics (2, 6m + 2 (m E Z+)) of E4 symmetry (see Tab. II in the appendix) and the eventual soliton config-
uration of the third harmonic. Within this model, once known the values of (u[~,~fi(~) and
N°2 MODULATION IN Rb2ZnC14 VERSUS TEMPERATURE 377
Table II. Amplitudes (x10~) of the symmetry modes that describe the ferroelectric phase of Rb2ZnC14 as a distortion of its basic structure. The basic cell parameters have been taken as
units of length. It can be observed that the main contributions correspond to modes of 22 and B2u symmetries.
Aig(Y) Zj~~(z) Ej~~(z) Z(~~(z) Z(~~(z) Z(~~(y) Z(~~(y)
Zn -250 -123 -65 -205 x 585 x -37 103 42 -21
Rb (1) -320 -217 301 844 x I 705 x I 79 98 -251 -31
Rb (~) 320 -80 -72 98 x I l104 x I 17 -49 30 12
Cl Ii) -420 457 -29 -442 x 2643 x I -8 83 166 23
Cl (~) 610 -247 850 -3379 x I 543 x I 91 68 -44 -21
Cl 13) -630 -555 -627 1279 x I 233 x I -65 75 29 -12
Alg(Z) Z~~~(Z) ~'~~(Z) ~/~IV) ~~~~(#) ~~~~(~)
Cl (3) 188 120 532 -867 x I -684 x x I 86 x I 7 28
(u(~,~fi(~) at a certain temperature, only three global free parameters: pi, p3, ns characterize the structural modulation at each temperature.
As the first harmonic of the structural modulation in the IC phase of Rb2ZnC14 had been
previously determined [8] at 210 K, the magnitudes (u(~,~fi(~) are known. But the amplitudes and phases corresponding to the third harmonic of the modulation (u(~,~fi(~ have never been determined. An approximate estimation of them can be made from the knowledge of the B2u distortion in the F phase, when this phase is analyzed as a distortion of the normal phase (N). This is done in the appendix and the values obtained used in the subsequent analysis of the diffraction intensities. A comparison between the intensities of the set of reflections
measured as a function of temperature with those predicted by the model allows to estimate the temperature variation of the three model parameters. In the next section the details of this determination are given.
4. Temperature Dependence of the Modulation
For each set of parameters (pi,p3,ns) and each temperature, the moduli of the theoretical structure factors )F)c for the reflections measured were calculated according to the following procedure:
the cell parameters and the wave-vector were the ones experimentally determined at each temperature;
the atomic diplacements were assumed as given by equation (4) with the values of the
(u(~,~fi(~) taken from Hedoux et al. [8], and (u(~,~fi(~) as estimated in the appendix;
as non-modulated anisotropic thermal parameters fl( were considered those experimentally determined in [8], corrected at each temperature according to a linear law;
equations (2, 3) were used to obtain the phase configuration 9(ns, u) for the relevant ns;
for every symmetry independent atom, the amplitudes of a Fourier expansion of expression
(4) in terms of the variable t = u + qL r" were introduced in the program REMOS (22j in
order to generate the structure factors.
