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Incommensurate phase of quartz : III. Study of the coexistence state between the incommensurate and the α-phases by neutron scattering and electron microscopy
G. Dolino, J.P. Bachheimer, B. Berge, C.M.E. Zeyen, G. van Tendeloo, J. van Landuyt, S. Amelinckx
To cite this version:
G. Dolino, J.P. Bachheimer, B. Berge, C.M.E. Zeyen, G. van Tendeloo, et al.. Incommensurate phase of quartz : III. Study of the coexistence state between the incommensurate and the α-phases by neutron scattering and electron microscopy. Journal de Physique, 1984, 45 (5), pp.901-912.
�10.1051/jphys:01984004505090100�. �jpa-00209824�
Incommensurate phase of quartz : III. Study of the coexistence state between the incommensurate and the 03B1-phases by neutron scattering
and electron microscopy
G. Dolino (1), J. P. Bachheimer (1), B. Berge (1), C. M. E. Zeyen (2), G. Van Tendeloo (3),
J. Van Landuyt (3) and S. Amelinckx (3) (4)
(1) Laboratoire de Spectrométrie Physique (LA n° 8, CNRS), Université scientifique et médicale de Grenoble, BP 68, 38402 St-Martin d’Hères, France
(2) Institut Laue-Langevin, BP 156X 38042 Grenoble, France
(3) University of Antwerp (RUCA), Groenenborgerlaan 171, 2020 Antwerp, Belgium (Reçu le 16 novembre 1983, accepté le 25 junvier 1984)
Résumé. 2014 Une étude détaillée de la transition d’accrochage de la phase incommensurable du quartz a été faite par deux techniques complémentaires : la diffusion élastique des neutrons et la microscopie électronique. Les expé-
riences avec les neutrons ont été faites sur des échantillons massifs (quelques cm3) avec une mesure simultanée de la dilatation thermique. La transition entre la phase incommensurable et la phase 03B1 est du premier ordre, et, en général,
une interface unique traverse l’échantillon. Une décroissance continue du vecteur d’onde de la modulation a été observée au voisinage de cette interface. Une image directe de cet état de coexistence est obtenue par la micro-
scopie électronique, mais seulement sur des échantillons très minces (quelques 100 nm), ce qui permet d’observer de quelle façon la période de la modulation incommensurable augmente de 10 nm à une taille macroscopique, par l’intermédiaire de « dislocations » du réseau de la modulation incommensurable. (Des résultats analogues obtenus
sur AlPO4 sont aussi présentés). Les résultats de ces deux techniques expérimentales sont bien corrélés et une
discussion de ces résultats par rapport aux expériences précédentes, surtout de diffusion de la lumière, est présentée.
Abstract 2014 A detailed study of the lock-in transition of the incommensurate phase of quartz has been made by
two complementary techniques : elastic neutron diffraction and electron microscopy. The neutron experiment was
made on bulk samples (a few cm3) with a simultaneous thermal expansion measurement. The transition from the incommensurate phase to the 03B1-phase is 1st order and in general a single interface sweeps the sample. The modula-
tion wavevector was observed to decrease continuously in the vicinity of this interface. Electron microscopy allows
to obtain a direct space picture of this coexistence state on very thin samples (a few 100 nm) and to observe how the incommensurate modulation period increases from 10 nm to macroscopic size with the help of « dislocations » in the incommensurate networks of columnar Dauphiné twin domains. (Analogous results obtained on AlPO4 are
also presented). The results of these two experimental techniques are well correlated and are discussed in relation to
previous results on the quartz transition mainly from light scattering studies.
Classification
Physics Abstracts
64.70K
1. Introduction.
In recent years the study of incommensurate phases
arose considerable interest and numerous examples
of such phases are known in different types of materials such as magnetic systems, conductors and insulating
dielectrics. In the first paper of this series [1] we
have presented results obtained by neutron dif-
fraction showing that quartz is one more material
(4) Also S.C.K./C.E.N., 2400 Mol, Belgium
to be added to this list Indeed within a small tempe-
rature range of about 1.3 K between the usual a
and P phases, an incommensurate phase has been
shown to exist. Further results on the variation of the elastic constants in the vicinity of these transition temperatures, obtained by Brillouin spectroscopy,
were described in [2] with a discussion of the possible
mechanism involved, following a proposal by Asla-
nian and Levanyuk [3]. In the present paper we
investigate more specifically the transition from the incommensurate (INC) phase to the low tempe-
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01984004505090100
rature commensurate a-phase using two different techniques : elastic neutron scattering and electron
microscopy.
A large amount of work has recently dealt with the understanding of incommensurate modulations and the nature of phase transitions from the com- mensurate to the incommensurate state. The more
generally accepted picture is the following [4] : Upon cooling from a normal commensurate high temperature phase there is a second order transition at a temperature Ti into an incommensurably modu-
lated state, within which some order parameter presents a sinusoidal plane wave modulation 11(r) =
110 exp i(qo. r + 4». Both the modulation amplitude
110 and the incommensurate wave vector qo vary
continuously with temperature in the incommen- surate phase, while the phase 0 has an indeterminate value leading to the existence of a gapless excitation : the phason. Upon further cooling there, generally
is a lock-in transition to a commensurate state, preceded by a loss of the purely sinusoidal character of the modulation and the appearance of higher
harmonics with wave vectors 2 qo, 3 qo... Thus commensurate regions appear in the crystal, separated by narrower regions with a rapid variation of the
phase 0, McMillan’s discommensurations [5]. The
incommensurate modulation then corresponds to
a spatial average over these regions. This state is
also called the multisoliton regime, as the discom-
mensurations correspond to the soliton solution of the Sine-Gordon equation in a continuous approxi-
mation where the amplitude is constant. This state
is similar to some kind of microdomain structure.
One can then picture the lock-in transition as being
due to the growth of the commensurate regions with
the gradual disappearance of the discommensurations
leading to a homogeneous commensurate phase or
more likely to an ordinary domain structure of macroscopic size. The density of the discommen- surations can be considered as an order parameter for this lock-in transition. Recently direct obser-
vations of such discommensurations were made by
electron microscopy in 2 H-TaSe2 and their evolution
during the lock-in transition has been followed [6, 7].
There is however some uncertainty as to the thermo-
dynamic description of such transitions [4] : the simplest theoretical treatment, based on exponential
interaction between discommensurations, leads to
a continuous decrease of the density of disco-
mensurations (or of qo) as [log (T - Tr ,)] -’ where Tc is the lock-in transition temperature. Experi- mentally however, most lock-in transitions are found to occur with a jump of qo from a finite value to zero,
a behaviour which has been described by ’more sophisticated theories introducing a coupling with
other physical variables such as elastic strains [8].
In the case of quartz the transition from the high temperature fl-phase to the INC-phase appears to be of second order, with some premonitory effects
on the fl-phase side, such as the softening of elastic
constants [2]. On the other hand the transition from the INC-phase to the commensurate low tempe-
rature a-phase is first order as clearly shown by the
behaviour of thermal parameters, and the existence of a thermal hysteresis of about 1.2 K between heating
and cooling [9, 10].
However, several experiments have produced evi-
dence for the systematic presence of heterogeneous
structures during this first order transition (pre- viously believed to occur between the classical (x and
#-phase). In 1956 intense light scattering, interpreted
as critical opalescence, was discovered by Yakolev
et ale in the close vicinity of the a-fl transition [11]. The
existence of extensive microdomains of the Dauphine
twin type was conjectured by R. A. Young to inter- pret his X-ray observations [12]. Shapiro and Cum-
mins have shown that the light scattering is indeed
elastic [13], hence it must be produced by static
centres, which could well be the microdomains
suggested by Young. Indeed an inhomogeneous
interface was observed directly by some of the present authors, using electron microscopy [14]. In the vicinity
of the transition the a-phase breaks up into arrays of columnar triangular domains, with a mesh size of a few tens of nm, which decreases in moving to
the high temperature side, suggesting that the fl-phase
may be a spatial or a time average of a Dauphine
twin structure. A recent neutron structure deter- mination in the #-phase at 863 K supports this
« disorder » model without however being able to
discriminate between a static or a dynamic’
disorder [15]. Subsequent optical measurements (light scattering and optical microscopy) by two of us
have shown that for massive samples, two different regions can be observed between the a and
fl-phases [16, 17]. At first, the previous experiments
were interpreted as some complex (and ill under-
stood) interfacial structure between the a and the
#-phase. More recently a different interpretation
has been given by Aslanian and Levanyuk [3] :
to explain the systematic occurrence of these struc- tures in the 10-’ to 10- 8 m range as shown by
electron microscopy [14] and in the 10 - 6 to 10-4 m
range as deduced from light scattering [17,18], they pro-
posed the existence of an incommensurate phase.
This suggestion has recently been fully verified by
neutron scattering [1, 9]. It is now clear that the
heterogeneous structures described above are related to the occurrence of this INC-phase. In particular
it now appears that the electron microscopic obser-
vations of [14] were the first direct-space pictures
of an incommensurate phase.
To gain more information on this lock-in transi- tion we have conducted new experiments dealing specifically with this problem made by two comple- mentary techniques : neutron diffraction and electron
microscopy. While neutron diffraction can be made
on bulk samples of a few cm’, with quite accurate
temperature control, it only yields information in
reciprocal space (part 2). On the other hand, a direct picture of the sample is obtained with electron micro- scopy but only on very thin specimens of about 10-’ m
thickness and in large temperature gradients (part 3).
Some results on AIP04, an isomorphous substance
with quartz will also be presented. Despite the quite
different nature and conditions of these two series of experiments their results are rather consistent
as discussed in part 4.
2. Neutron scattering.
2 .1 EXPERIMENTAL CONDITIONS. - The neutron scat-
tering data were collected on the D10 triple axis spectrometer at the Institut Laue-Langevin. The sample was a cylinder of natural quartz of good optical quality (diameter 12 mm, height 30 mm).
Its optical axis, parallel to the Z-axis, was vertical
so that neutron scattering was measured in the
horizontal XY plane. The sample was heated in
a standard ILL furnace with thermocouple regu- lation but it was supported by a special device allowing
a better control of temperature stability and uni- formity, and a simultaneous measurement of the
sample’s thermal expansion in the Z direction. Tem- perature measurements were made with a 50 Q
platinum resistor, and an a.c. bridge giving a relative uncertainty of 2 x 10-2 K, while the vertical gradient
in the sample controlled by a differential thermo-
couple and an extra heater was reduced to 0.1 K/cm.
The dilatometer was made of silica glass rods trans- mitting the sample’s thermal expansion to a sensor
which was kept at room temperature, with a relative resolution of Al_,/l. L-- 10 - 6.
The spectrometer was operated in the three axis mode with a PG 002 monochromator, selecting a
neutron beam of wavelength 0.243 nm and with
a PG 004 analyser set to zero energy transfer. Thus the energy resolution was 0.1 THz or better and the momentum resolution about 0.3 and 0.2 nm -1 along
and transverse to the momentum transfer vector.
Among the Bragg peaks within our experimental
range (Q , 45 nm-’) in the XY plane the (030)
reflection produces the most intense satellites (this peak lying along a symmetry axis of the fl-phase, only 4 satellites are observed, whilst around most
of the Bragg peaks, 6 satellites are present) [1]. The
coordinates will be given with respect to the reci- procal lattice of quartz.
2.2 EXPERIMENTAL RESULTS. - We shall first describe
a complete temperature cycle obtained upon heating
and then cooling the sample. The thermal expansion
of quartz during a similar temperature cycle has
been presented in [19]. The results of the simultaneous
measurements of the thermal expansion and of neutron scattering are presented in figure la and lb respecti-
Fig. 1. - a) Variation of the thermal expansion A/z of the quartz sample along the.OZ axis, during a complete tempera-
ture cycle through the transition (arbitrary units). b) Tempe-
rature variation of the satellite position qo (+) and width Ws (x) upon cooling obtained from neutron scattering.
The inset shows the corresponding satellite shapes for a 3 0 > scan at the temperature indicated by an asterisk *.
In figure 1 a and 1 b, solid curves correspond to homogenous phases, while dashed curves indicate heterogenous states
were high and low temperature phases coexist
vely. In the a-phase the thermal expansion is roughly
linear within the temperature interval of a few degrees plotted. Nucleation of the high temperature phase
occurs at a temperature Th. Some superheating
is necessary to overcome the nucleation barrier of this 1 st order transition; the sample cools down
somewhat after nucleation. Since the small platinum
resistor is closely coupled to the sample this tempe-
rature drop is readily observed. Subsequently the
temperature increases again, following the furnace
temperature and the transition to the high tempe-
rature phase is completed. Upon cooling, the incom-
mensurate phase reappears below T;, and satellites
are observed in the neutron scattering pattern, as is shown in the inset of figure I b for a 3 0 > scan.
In figure 1 b are plotted the temperature variations of two satellite parameters : qo the reciprocal dis-
tance from a satellite to the closest Bragg peak, cor- responding to the modulus of the modulation wave vector and WS the satellite width. Already at Ti + 2 K
the highest temperature in figure 1, weak scattering humps, are observed but with a faint intensity and
a large width, probably of inelastic origin. At Ti
the intensity begins to increase sharply whilst the
width narrows drastically as shown in figure 1 so
that the humps transform into well defined satellite
peaks. The variation of qo, from 0.034 to 0.030 a*
occurs in a range a little larger (by 0.001 a*) than
in [1 ].
Upon further cooling the nucleation of the a-phase
occurs at at temperature Tc, and it is followed by a
small transient temperature rise. The transition to the a-phase is slow enough to allow neutron measure-
ments during the coexistence of INC and a-phases.
(In Fig. 1 the one phase state corresponds to full lines, while the coexistence state corresponds to
dashed lines.) The satellites are observed to move
towards the Bragg peak while their width increases
so that just before the end of the transition, only
one broadened peak is observed When the transition is completed, only the central narrow a-phase Bragg peak remains.
We have made a more detailed observation of this coexistence state by changing the temperature in very small steps. To present these results more
clearly, we shall first describe the probable mor- phology of the two phases during their coexistence in the sample, which is shown schematically in figure 2 (although no direct observation was made in the present experiment, this morphology can be
deduced from numerous previous observations on
massive samples, obtained by X-ray topography [20]
or optical microscopy [16, 17]). Nucleation of the
a-phase is systematically observed in the colder part of the sample, in the present case the upper part,
as there is a small vertical temperature gradient of
about 0.1 K/cm, estimated from the temperature
range of the coexistence state. Upon further cooling
the a-phase expands and an interfacial region pro-
gressively sweeps the central scattering volume of the
sample as shown in figure 2. If we neglect elastic
strains resulting from the difference in specific volumes
of the two phases the total length of the sample is
a rough measure of the position of the interface.
To get a better spatial resolution the neutron beam
height was reduced from 8 mm, as used for the results of figure 1 to 4 mm.
The results obtained in this way are condensed into figure 3; the central curve gives the thermal
expansion during this new cycle whilst the surrounding diagrams show satellite scattering patterns along iden tical ç 3 0 &) scans at several characteristic stages. We shall begin our description in the fl-phase
at the higher temperatures of figure 3 (T; + 1 K).
On curve (a) two weak humps are observed in the thermal diffuse scattering. Upon cooling below Ti
the appearance of the incommensurate phase is
marked by the growth of sharper and more intense
satellite peaks (curves b and c). As in paper [1] these
curves were fitted by 4 Gaussians one for each satellite,
the third one for the Bragg peak, and the fourth one for
Fig. 2. - Schematic representation of the coexistence of the a-phase with the incommensurate phase in the sample used for the neutron scattering experiments (lengths are in mm).
the broad background The resulting fitted curve is in good agreement with the experimental points, but for
some parameters, particularly the width of the broad
humps in the p-phase some uncertainty subsists. The temperature variation of qo, the distance from the satellite to the Bragg peak, and Ws, the satellite width,
are shown by the full curves in figure 4a. After nuclea- tion of the a-phase at Tc, the temperature was lowered
by small steps so that several scans (curves d to h) could
be made during this coexistence state which could be made to last 1 hour and a half. Just after nucleation
there is a decrease of the measured satellite intensity,
but their positions and widths are not changed (curves
d and c are very similar). This stems from the fact that the scattering volume, which is in the middle of the
sample, is not much perturbed by the nucleation of the
a-phase at the upper edge. While the temperature decreases further, the interface moves down and a change in the satellite shape is observed : qo decreases whilst the width WS increases (curve e) so that the
satellites meet the Bragg peak (curves f and g). In the
last steps before the transition to the a-phase is completed, there is only a structureless broadening of
the bottom of the Bragg peak (curve h). The corres- ponding variations of qo and Ws as a function of temperature in the coexistence state are shown by
dashed curves in figure 4a.
However, since this transition occurs at practically
constant temperature we have also plotted in figure 4b (left diagram) the variation of qo and WS as a function
of the interface position, given by the thermal expan- sion of the sample in the coexistence state. This clearly
shows that a strong decrease of qo occurs in the vicinity
of the interface. When the interface comes nearer to the lower crystal boundary, its motion accelerates under
Fig. 3. - The central curve shows the variation of the thermal expansion Al., (arbitrary units), while the surrounding curves
show the shapes of the satellites for ; 3 0 ) scans at different stages. The full lines correspond to homogeneous phases, the
dashed lines to phase coexistence. The dot-dashed line corresponds to a 2nd temperature cycle made without going fully to
the a-phase. In the surrounding diagrams the Bragg peak is shown reduced 20 times.
Fig. 4. - a) Variation of the satellite position qo (+) and width Ws (x) as a function of temperature (corresponding to Fig. 3). b) Gives the variation of qo and Ws upon cooling plotted as a function of the thermal expansion ð-Iz given in figure 3.
c) The same but upon heating (corresponding to the dot-dashed curve of Fig. 3).
the action of elastic image forces, and the sample turns completely into the a-phase, with only the Bragg peak remaining in the diffraction pattern (curve i).
During a further temperature cycle we have interrupted
the cooling before completion of the transition into the a-phase, and reheated again. The corresponding
thermal expansion curve is shown by the dot-dashed
line of figure 3. During this reheating the satellites
moved away from the Bragg peak (curve j to m). The
variation of qo and WS during this temperature rise are also plotted as a function of temperature in figure 4a
and as a function of the thermal expansion in figure 4c (right diagram). Just before the end of the coexistence state (curve m) the satellites are clearly separated but
