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Phase transitions in C2O4H NH4 1/2 H2O : a light scattering study of the high pressure phase
J.L. Godet, M. Krauzman, R.M. Pick, H. Poulet, N. Toupry
To cite this version:
J.L. Godet, M. Krauzman, R.M. Pick, H. Poulet, N. Toupry. Phase transitions in C2O4H NH4
1/2 H2O : a light scattering study of the high pressure phase. Journal de Physique, 1989, 50 (13),
pp.1711-1728. �10.1051/jphys:0198900500130171100�. �jpa-00211026�
Phase transitions in C2O4H NH4 1/2 H2O : a light scattering study
of the high pressure phase
J. L. Godet, M. Krauzman, R. M. Pick, H. Poulet and N. Toupry
D.R.P. (LA 71), Université P. et M. Curie, 75252 Paris cedex 05, France (Reçu le 6 décembre 1988, révisé le 14 mars 1989, accepté le 16 mars 1989)
Résumé. 2014 AHO présente, à la pression ordinaire, une transition de phase ferroélastique du
deuxième ordre à 145 K (160 K pour AHOD). Une nouvelle phase (phase III) apparaît sous pression, que nous avons identifiée par les propriétés spécifiques de son spectre Raman. Celui-ci
se caractérise, en particulier, par deux modes de basse énergie, dont la fréquence varie fortement
avec la pression. La phase III est incommensurable, avec un vecteur d’onde q0//c*, dont le
module dépend de la pression [6]. Ces modes de basse fréquence sont deux phonons acoustiques,
rendus actifs par l’incommensurabilité, et nous proposons que la phase III soit, comme la phase II, due à une mise en ordre des ions ammoniums : leur orientation serait couplée à la troisième branche acoustique, ce couplage se faisant à q0 en phase III, et en centre de zone en phase II. Ce
mécanisme explique le comportement d’un pic central détecté en symétrie B2g.
Abstract. 2014 AHO exhibits, at normal pressure, a ferroelastic, second order phase transition at 145 K (160 K for AHOD). A new phase (phase III) has been identified under pressure, by its
characteristic Raman features. One specific aspect is the existence of two low energy modes, the frequency of which is highly pressure dependent. Phase III has been shown in [6] to be
incommensurate with a wave vector q0//c*, q0 strongly varying with pressure. These low
frequency modes are acoustic phonons belonging to two branches activated by the incommen-
surability, and we propose that phase III would be, as phase II, produced by the ordering of
ammonium ions linearly coupled to the third acoustic branch, this coupling taking place at
q0 for phase III and at the zone center for phase II. This mechanism explains the behaviour of a
central peak detected in the B2g geometry.
Classification
Physics Abstracts
33.20F
-64.60C - 64.70K
1. Introduction.
This paper is one of a series devoted to the study of the phase transitions which take place in
C2O4H NH4, 1/2 H20, ammonium hydrogen oxalate, here after called AHO. At room
2 y
temperature, AHO crystallizes in the space group Pmnb (D’6) [1,2] with Z
=8. In this
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500130171100
structure (phase I), the C204H- ions form a single set, each ion being in a general position,
while both the water molecules and the NH4 ions lie in a mirror plane. This implies the
existence of two distinct ammonium sets and X ray studies have shown that, for one of them,
this mirror plane has only a statistical character, each ion taking at random one of two
orientations symmetric with respect to this mirror plane. At normal pressure, AHO
undergoes, at Tc = 146 K, a second order, ferroelastic, equitranslational phase transition [3]
2
which leads to a Pl 21 n 1 (C5 2h) structure (phase II). The first paper of this series [4] (here after
n
referred to as I) was devoted to a Brillouin scattering study of the ferroelastic character of that transition. It was shown that the C55 elastic constant droped towards zero in the vicinity of Tc, and the analysis of C55(T) suggested that this decrease resulted from the influence of another degree of freedom linearly coupled to the elastic deformation e5.
This point was further studied through a series of Raman scattering experiments in [5] (here
after referred to as II). A strong central peak was detected in the B2g (ac) geometry of phase I,
in agreement with the prediction of I. With the help of the structural data of phases 1 and II,
this relaxational mode was identified with the B2g component of the pseudo-spin variable
which describes the disorder of one set of NH’ ions in phase I. The analysis of the line shape
and intensity of this peak as a function of temperature made clear that the NH’ ordering was
the driving mechanism of the transition, but that the very large coupling with the
e5 deformation shifted the temperature transition by more than 100 K. Consequently, the
03C9
=0 intensity of this central peak reached a maximal but finite value at Tc; also, its
linewidth decreased with decreasing temperature, down to a finite value, at Tc. Nevertheless,
this linewidth did not increase again just below Tc, but rather 15 K below Tc, an effect which
was explained in II as the result of the strong deformation of the lattice in phase II.
The present paper is devoted to the determination and the study, by Raman scattering, of
the phase diagram of AHO (and of the fully deuteriated analog C204D ND4, 1 2 D20, AHOD) between 78 K and room temperature, and for pressure up to 8 kbar. A new phase, called phase III, has been discovered, phase which exists only above the pressure (Py ) and below
the temperature ( TY ) of a triple point, the coordinates of which are different for AHO and AHOD. Phase III has been later identified as an incommensurate phase [6] and the analysis
of our data suggests a possible mechanism for its existence. Consequently, the present paper is divided as follows.
Section 2 gives a description of the phase diagram and of the first or second order character of the three transitions involved.
Section 3 is devoted to the three main experimental evidences which have made possible
the construction of the phase diagram.
Finally our results are discussed in section 4. We first recall (paragraph 4.1) how the
behaviour of new, low frequency modes, detected in phase III has been explained [6] by the
fact that phase III was incommensurate, with a large variation with pressure of the incommensurate wave vector. We then propose (paragraph 4-2) a model for the phase 1-111
transition which takes into account the ordering of the NH’ ions and their possible coupling
with an acoustic phonon. We show that such a mechanism can explain the thermal behaviour
of the central peak at various pressures (paragraph 4.3) and also (paragraph 4.4) that the
behaviour of an internal mode of the oxalate ion, at 851 cm- 1, which has been instrumental in
drawing the phase diagram, can be explained by our present knowledge of the structure of the
three phases.
A short summary with concluding remarks represent the last section of this paper.
2. Discussion of the phase diagram.
2.1 EXPERIMENTAL METHODS. - The phase diagram has been obtained with the help of a
pressure cell, described elsewhere [7], which allowed us to record Raman spectra between 78 K and 300 K for pressure below 8 kbar. The main point to be noticed is that polarization
measurements are possible with this cell, which is an important tool for the discussion of the recorded spectra. The a axis of the crystal was set vertically in the pressure cell so that, using
the two existing windows, the four following spectra b (aa ) c, b (ab ) c, b (ca ) c and
b (cb ) c have been recorded.
The transitions were detected through the appearance (or disappearance) of certain modes, by their typical changes in lineshape and/or intensity, by the splitting of some modes or by
their polarization. A close examination of the various spectra has shown that two spectral regions, namely the 830-900 cm-1, and the low frequency region (w « 100 cm-1 ) were the
most sensitive to the two transition lines we shall discuss, though some information obtained in other spectral regions, (especially 100-250 cm-1) was also instrumental in studying the
AHO and AHOD phase diagrams.
2.2 PRESENTATION OF THE PHASE DIAGRAM. - Anticipating on the description and the
discussion of the experimental evidences which lead to its construction, we present, in figure 1, the phase diagram of AHO in the pressure temperature domain covered in the present study. In the vicinity of 2 kbar and 135 K (2.6 kbar and 148 K for AHOD) (cf. Fig. 2)
it exists a triple point called Y, above which a new phase III has been discovered. The transition between phases 1 and III is second order (or possibly weakly first order) while, by
Fig. 1.
-Phase diagram of AHO.
still lowering the temperature (or the pressure), phase II is reached again through a transition
the first order character of which increases with increasing pressure. Below the triple point,
we have not detected signs of curvature of the transition line between phase 1 and phase II and
below 1 kbar, this transition is clearly second order.
The scarcity of measurement between 1 kbar and 3 kbar (1 kbar and 4 kbar for AHOD) has
not allowed us, either to give precise values for the pressure, Py, and the temperature, Ty, of the triple point, or to ascertain the fact that the phase I-II transition is always second
order. This is why, in figures 1 and 2, the vicinity of Y is represented by dotted lines which have been drawn only by extrapolation of the higher and lower pressure results. As shown in
Appendix A, the slope of the transition line between phases 1 and II agrees with a straight
forwards extension of the Landau theory of the ferroelastic transition as discussed in 1 :
indeed, as the principal deformations ei (i
=1 ... 3 ) couple linearly to the pressure and
quadratically to the order parameter Q, pressure modifies, through the induced deformations,
the term in Q 2 of the free energy, and thus the critical temperature. From the various coefficients determined in I, one predicts (see Eq.(A4) and numerical values given below) :
in agreement with the experiment. This means that, along this transition line, the application
of pressure does not substantially modidy the interactions governing the phase I-phase II
transition.
Fig. 2. - Phase diagram of AHOD.
Finally, from the extrapolation of the higher pressure results, the two transition lines 1-11 and II-III are found to have the same tangent at the Y point, as is predicted in a Landau theory with two competing order parameters [8]. On the other hand, extrapolating from the
low pressure side yields no change of slope between the 1-11 and the 1-111 transition lines, a
result which is not predicted by the theory but too uncertain to be further discussed.
3. Expérimental evidences of the phase transitions.
Four types of evidence have been used to build up the phase diagrams presented in figures 1
and 2. They are given below in the successive order of their importance concerning the
characterization of the nature of the transition.
a) The most outstanding observation is certainly the appearance of new lines in the low
frequency region 10-40 cm-1.
The other pieces of evidence are :
b) the thermal evolution of the central peak along isobaric lines ;
c) the change in the frequency splitting with Pressure and/or Temperature of a Raman
active internal mode located around 851 cm-1 ;
d) the sudden change in the intensity of some of the external modes ; such variation of the mode observed at = 72 cm-1 in the b (cb ) c geometry was used to determine the II-III transition line of obvious (cf. Fig. 3) first order character.
Fig. 3. - The low frequency Raman spectra, in the (cb ) geometry, at 80 K in phase II and phase III, close to the transition boundary.
3.1 THE LOW FREQUENCY MODES. - Above Py and below Ty, two new low frequency
modes have been recorded. They may be taken as the signature of existence of phase III, as they are recorded only in this phase. Their thermal and pressure behaviour is rather unusual
and, if properly interpreted, is source of much information about the new phase as will be
discussed in section 4.
a) Within experimental error (± 2 cm-1 ) the frequency of the 2 modes is independent of temperature even at 8 kbar in AHO where the temperature range of existence of phase III is
as large as - 50 K.
Fig. 4. - Phase III low-frequency Raman modes versus pressure (regardless of temperature). Note that
the upper curve can be deduced from the lower one by an affinity of ratio 1.8.
b) Conversely, their frequency is highly pressure dependent and the frequency of each of
them increases by a factor approximately equal to 2 from 4 kbar to 8 kbar in AHO (cf. Fig. 4).
Nevertheless the ratio between these two frequencies is constant, within experimental error,
in the whole pressure range.
c) Each of these modes has a definite polarization. The high frequency mode is recorded in the (ac ) scattering geometry (with a weak leakage in the (bc ) geometry, essentially due to inhomogeneous stresses on the sapphire windows of the pressure cell) while the low frequency
one is seen in the (ac ) geometry (with some (aa ) leakage).
d) At constant pressure, the intensity of these two modes is proportional to Tc (P T,
where 7c(P) is the phase 1-111 transition temperature at a given pressure. These modes
suddenly disappear at the phase III-II transition.
e) The frequency of these two modes does not vary upon deuteriation, at a given pressure, a
presumably accidental result which explains why only the set of curves related to AHO is given in figure 4.
f) Except for the central mode (see 3.2), no mode of lower frequency could be recorded in
our experiments in any of the studied scattered geometries, even at 8 kbar and 80 K.
3.2 THE THERMAL BEHAVIOUR OF THE CENTRAL PEAK. - The central peak has been studied
in the (ac ) geometry between - 10 cm-1 and 90 cm-1 and analysed with the same technics as reported in II at normal pressure. Typical linewidth and intensity results are given in figures 5
and 6, respectively for 6 kbar and 8 kbar, i.e. above Py. These figures may be characterized
by two distinct features. The central peak linewidth is pratically insensitive to the phase I- phase III transition, but, at 6 kbar where the II-III phase transition is accessible with our
pressure cell, there is an abrupt decrease of the linewidth below the transition. On the
contrary, the cv
=0 intensity of the central peak has a small but distinct accident along the 1-
Fig. 5.
-Width and maximum intensity of the central peak for P
=6 kbar.
Fig. 6.
-Width and maximum intensity of the central peak for P
=8 kbar.
III phase transition line, this intensity decreasing below TI-III before increasing strongly again.
When the phase III-II transition can be reached, this intensity abruptly decreases at the
transition (Fig. 5). An explanation of these two features will be given in paragraph 4.3.
In other scattering geometries, stray light prevented us to draw any conclusion concerning
the existence of a weaker central peak.
3.3 THE PRESSURE-TEMPERATURE BEHAVIOUR OF THE 851 cm-I INTERNAL MODE. - At normal pressure, in our Raman study of the phase 1-11 transition, we detected in phase 1 a
very narrow line (HWHM 3 cm-1), recorded at the same frequency (851 cm-1 ) in the four different scattering geometries which correspond to the Ag, Blg, B2g, B3g representations.
Below the phase transition, this mode splits, in each scattering geometry, into two components, symmetrical with respect to 851 cm-1, each component having the same intensity and the splitting being identical in all the four geometries. This splitting,
8 w, varies with temperature as :
As it was shown in 1 and II that the order parameter follows a Landau mean field theory :
this splitting is thus proportional to the order parameter in phase II. The detection of this
splitting has been used as the most direct signature of a phase transition and the main source
of information for the precise drawing of the 1-11 and 1-111 phase boundaries.
Below 3 kbar the same results as at normal pressure are obtained. Above 3 kbar the
splitting still exists and follows the same law as given by (2) but with f2 (P ) and Tc (P) taking the values given in table I. Nevertheless this splitting dw seems to be slightly
different in the (ab ) scattering geometry on the one hand and the (ac ) geometry on the other
hand (Fig. 7). Such a small asymmetry is also apparent on the linewidths which slightly differ
Table 1.
-Values of il (P ) and Tc (P) (Eq. (2)) relative to the two pressures 6 kbar and
8 kbar. Note that (P ) differ for the two polarizations (ab ) and (ac).
Fig. 7. - Variation with temperature of the splitting d W (T) for the 851 cm-1 line, at various pressures.
in the two polarizations. Both asymmetries disappear at the phase III-phase II transition
where 800 changes abruptly as is shown in figure 7. This abrupt change is one, but not the most characteristic sign of the first order character of this third transition line.
4. Discussion.
4.1 THE LOW FREQUENCY MODES AND THE INCOMMENSURATE CHARACTER OF PHASE III.
A possible interpretation of the exceptional behaviour of the two low frequency modes
detected in phase III, and, in particular, the fact that their frequencies vary with pressure, but that their ratio is independent on pressure (or temperature) was given in [9]..
It was proposed that phase III would be an incommensurate phase, characterized by a wave
vector qo which remains parallel to a given direction qo, but with a modulus varying strongly
with pressure and not with temperature. This interpretation was confirmed by a neutron
elastic experiment [6] which showed that qo was parallel to the cristallographic axis
The two low frequencies modes of figure 4 were then found to be two acoustic phonons with
wave vector qo, which are detected in the incommensurate phase through their coupling with
the frozen-in modulation. Using the elastic data of I, these acoustic branches were identified
as the LA mode (upper branch) and the TA mode, b polarized (lower branch of Fig. 4).
With the help of this information, of the value of 8 measured in [6] at 5.4 kbar and of the
frequencies reported in figure 4, one can conversly transform the frequency measurements of
figure 4 into a direct measure of the value of 6 through the relation :
.were v is either the LA or the TA, b polarized, sound velocity. Using this technique, one finds
that AHOD is the incommensurate system with the largest variation of 5, the latter changing
from 6 - 0.12 in the vicinity of Py to 5 ’- 0.25 at 8 kbar, this last value being confirmed by the
diamond anvil results of Bosio et al. [10].
This represents a variation of 8 twice as large, at least, as in thiourea [11], which exhibited up to now the largest reported variation of the incommensurate wave vector. Furthermore, though we have used only a limited set of pressure values, the curves of figure 4 appear quite
continuous and no sign of a partial lock-in has been detected up to now. d (P ) is thus either
continuous, or displays a harmless staircase behaviour, which may well extend far above the à
=0.25 value found at 8 kbar.
4.2 POSSIBLE NATURE OF THE ORDER PARAMETER. - The polarization measurements described in paragraph 3.1 may be used to discuss the symmetry of the mode which freezes at the I-III phase transition. As shown in [12], the symmetry of the hard modes detected by
Raman scattering in an incommensurate phase are obtained through the following rule. If
T s and Tu are respectively the irreducible representations of the frozen mode and of the hard
mode, in the high temperature phase, in the group of qo, the product TS Th is a representation
of the group of qo for 1 qo 1
-0. The hard mode will be detected in the Raman active
representations of the high temperature phase which are induced by T 7-h, considering the
group of qo as a subgroup of D2h. In the group of c *, which has C2v symmetry, the LA phonon belongs to the T 1 representation and the TA, b polarized, phonon to the T3 representation (Kovalev notations [13]). The only representation TS compatible with our polarization results
is T s - T 4 in which case :
in agreement with our Raman results. The representation of the mode which freezes-in is thus
unambiguously determined by our experiments, and one may be tempted to conjecture on the
nature of this mode.
In fact, T4 is the representation of the TA, a polarized, mode propagating along
c*, and, furthermore, this mode induces, in the b --> 0 limit, the e5 deformation which
characterized the phase 1-11 transition below 3 kbar. In other words, in the à - 0 limit,
T4 induces the B2g representation, which contains, as recalled in the introduction, both the
e5 deformation and the relaxation mode studied in II. These modes which are coupled at
q
=0 will remain so along c *, so that the same scenario which took place at q
=0 at low
pressure, could take place at some value 8c * above Py. A potential ordering of the NH’ ions at Sc * could strongly couple to the TA, a polarized, phonon, leading to a freezing
of this acoustic mode at the phase 1-111 transition. Though this mechanism is, presently, only hypothetical, it is supported by two experimental observations.
-
First, in this model, the TA, a polarized, mode gives rise to the amplitudon mode below
the phase transition. This mode should thus be visible in the A1g representation, with a temperature dependent frequency. Nevertheless its frequency which should be proportional
to ( Tc - T)112 in a Landau theory, could remain low in the whole part of the phase diagram we
have studied, and it could have also a weak intensity in the (aa ) geometry. The absence of any mode of lower frequency than those of figure 4 thus does not contradict the present hypothesis.
-
Second, in the neutron experiment [6], the satellite intensities are large ( 1/4 of the
neighbouring Bragg peak intensity at the lowest temperature used at 5.4 kbar) while they are
very weak in the X ray experiment. As the deuterium atoms have a large neutron cross section, while the hydrogen atoms are more or less invisible by X ray scattering, these satellite intensities indicate that the hydrogen, or deuterium, atoms play the most important role in the phase transition. This agrees with our hypothesis that the NH’ ion ordering is, also in phase III, the driving mechanism of the phase transition. Finally, the Raman results on the central peak recorded in the B2g geometry also agree with an ordering of the NH’ ions at the phase 1-111 transition, as we shall see in the next section.
4.3 THE CENTRAL PEAK BEHAVIOUR. - In 11, we analysed the shape and intensity of the
central peak which appears in the B2g geometry in term of the ordering of a set of NH’ ions. Our present results show that, above Py, the growth of an intermediate phase III
between phases 1 and II results, first, in a small decrease of the central peak intensity just
below the 1-III border line, followed by a second increase of this intensity when the temperature still decreases in phase III. Meanwhile, the linewidth decreases continuously
from phase 1 to phase III, with a higher slope in phase III than in phase I. The ordering of the NH’ ions at qo and - qo can explain such a result if the phase 1-111 transition is second order,
as indicated both by the behaviour of the 851 cm-1 line splitting in phase III (cf. paragraph 3.3) and the thermal behaviour of the incommensurate satellites detected in [6].
Such an effect is most clearly explained if we make use of the transition mechanism
proposed in 4.2, neglecting, to start with, the spin-acoustic phonon coupling. Generalizing (Al), if Q (q ) is the Fourier transform of the pseudo-spin variable which describes the two
possible orientations of the disordered NH’ ion (1) and q is a vector along the
c * direction, the relevant part of the free energy may be written as :
(1) More precisely, Q (q) is the Fourier Transform of the B2, linear combination of the four pseudo-
spin variables contained in the unit cell.
where the first and third terms of equation (6) describe the entropy, plus harmonic energy contribution to the free energy while the second and fourth terms contain the lowest relevant anharmonic contributions. In this model, the.critical temperature is given by kTc
=J(qo),
below which one has (using the equilibrium condition aF
=0) : aQ (qo)
Furthermore, assuming a relaxational dynamics for the q component of the pseudo-spin i.e.
where T is some individual relaxation time, yields, for the pseudo-spin susceptibility :
Above T,,, where (Q(qo))
=0, equation (9) leads to :
while, below Tc, where (IQ(Qo)12) is given by equation (7), one obtains :
As the B2g central peak intensity is given by :
where Rac is the Raman tensor associated with the pseudo-spin reorientation process, one expects that equation (10a, b) would explain the results presented in figures 5 and 6.
Indeed, above Tc, equation (10a) represents a Lorentzian profile, the W
=0 intensity of
which increases with decreasing temperature, while its linewidth decreases. Below
T,,, as J (qo ) > J (0 ), the first term of the denominator of equation (10b) may have two different behaviours : either it will increase again, if b (qo, 0) - b (qo, qo), or it will still decrease if 2 b ( qo, 0) J (0 ) leading to an increase of the w
=0 intensity, and a narrowmg
b (q0’ qo ) J(qo)
of the lineshape. This is what appears in figures 6 and 7, except for the small decrease of
intensity just below Tc. This additional decrease can be easily understood : it is related to the
fact that equation (11) is not correct below T,, because we have treated the free energy
equation (6) as if no relationship existed between the various Q (q ), while the pseudo-spins
variables are subjected to the constraint :
Close to T,, this constraint may be approximately taken care off, in the spherical approximation, which treats every vector q different from qo and - qo on the same footing, by replacing XRPA(q, w) by y (q, w ) where following Brout [14] :
In this expression ( ) Q(qo) represents a mean value computed on the spin system, in the presence of the order characterized by Q (qo) and Q (- qo), but neglecting any interaction between the spins. One then obtains :
One then obtains :
Replacing in equation (11) XRPA(O, w ) by X (0, cv ) given by equation (13), one sees that
below Tc, the co
=0 intensity is renormalized by the term (16a) which is a quantity constantly decreasing with temperature, while no such effect exists above T,, as (Q(qo)
=0. The
renormalization affects the intensity of the signal, but not its linewidth, in agreement with figures 5 and 6. These figures can then be explained by the interplay of two effects :
- just below T,, the reduction of the spin fluctuations through (16a) decreases the
susceptibility X (0, lù ) ;
.-
at lower temperatures, admitting a low enough value of b (qo, O)lb (qo, qo) the susceptibility diverges again.
The first effect does not affect the linewidth, which thus decreases below Tc as well as above, but with different slopes, the first term of the denominator being different in equations (10a) and (lOb).
If the model of the phase transition described in 4.2 applies, one must also take into account the pseudo-spin-T.A, a polarized, acoustic phonon coupling. As shown in Appendix B, such a
coupling does not affect the above given arguments, merely changing, below Tc, for each
q, J (q ) into an effective interaction renormalized by this coupling.
4.4 THE 851 cm-1 MODE. - As indicated in paragraph 3.3, the behaviour of this internal mode of the oxalate ion has been the most useful tool for drawing the phase diagrams of figures 1 and 2 ; this mode, which appears at the same frequency in the four different
scattering geometries in phase I, splits, for each of them, into two components of equal
intensities in phase II, the splitting being the same for each geometry and proportional to the
order parameter. The same kind of splitting appears in phase III, with an abrupt jump at the strongly first order phase III-II transition. Such a behaviour can be easily understood, if one
admits that their frequency is sensitive to the local environment of each molecule (local field effect) but that there is no dynamical coupling between these modes for the eight molecules
which belong to the unit cell, i. e. that there is no phase relationship between the vibrations of two molecules, so that one deals with an incoherent scattering phenomenon.
In phase I, where the 8 molecules form a single family, each molecule has the same local environment and thus the same frequency. The vibrational density of states is thus a
5 function at 851 cm-1.
- In phase II, where the space group symmetry is only P1 - 1 21 the oxalate molecules form
n
two distinct sets, and one expects two sets of frequencies representing two densities of states, with equal intensity for each polarization. In order to be more quantitative, let us assume that
the molecular local field (which modifies the intemal mode frequency from its free ion value)
is a function of the relative distances between one molecule and each of its neighbours. Below
the phase 1-11 transition, the position, and orientation, of every molecule is modified, to first order, by a quantity proportional to the order parameter : one thus expects a frequency change proportional to this order parameter for each set. Finally, if an operation of the D2h group sends one oxalate ion into another one, the frequency change for the second ion is the same as that of the first ion multiplied by the character of this operation for the irreducible
representation of the order parameter : the two sets of molecules in phase II correspond respectively to the + 1 and - 1 characters of this representation so that the two frequency
shifts are of opposite signs, and the splitting symmetrical.
- The phase III situation is somewhat more complex and has not yet been thoroughly analysed in the context of intemal mode frequencies. The modulation of the crystal is caused by the two modes which have frozen at qo and - qo. If i
=1, ..8 labels the eight oxalate
molecules in the phase 1 unit cell, the molecular field modulation in phase III, and thus, the frequency change for molecule i, can clearly be written, using the same argument as above :
As qo is incommensurate, when L spans the whole crystal, equation (17) generates, as shown in [12], a density of vibrational states centered at wo
=851 cm-1 of the form :
where àw ; i is, again, proportional to the modulus of the order parameter.
A priori, the 8 oxalate molecules could thus generate eight different densities of states but,
due to symmetry arguments, these densities are all identical. Indeed, the group of qo is C2,, which is a group of order 4, so that four molecules are related one to another by the symmetry operations of this group. Generalizing the case discussed for phase II, if
i is sent in j by an operation g of C’ , if tg is the fractional translation associated with
g, and X (g ) the character of the little representation (here T4) under which the order
parameter transforms, then, in the notations of equation (17) :
The four molecules related by the group of qo thus generate identical densities of states.
Finally, the inversion operation has to be included as qo and - qo coexist in the incommensurate distortion. This operation sends one set of four molecules into the second set. One easily shows that if this operation sends the molecule i of cell RL into the molecule i ’ of cell RL’, the change of frequency for molecule (i , RL ) is the opposite of that of molecule
(i ’, RL, ). Thus, in AHO, the modulation generates the same density of states for the eight
molecules. When broadened by the linewidth of 851 cm- 1 line in phase I, this density of states
is in fact quite similar to that given by the normal splitting of phase II.
The present argument thus predicts, in phase III, a behaviour totally similar to that of phase II, with two lines of the same intensity, a symmetrical splitting proportional to (Tc - T)1/2 ,
and a jump in this splitting at the phase III-II transition, which agrees with the results shown in figure 7 and table I. Nevertheless, this does not predict a different splitting for different
polarizations, a weak, but rather well established, effect, for which we cannot, presently,
offer a coherent explanation. It may be related to a dynamical aspect (development of a weak
intermolecular interaction under pressure) or to a detection mechanism effect (the frozen
-in mode may have also some librational component, which would modulate the molecular
orientation, and thus affect with different weights all the points of the density of states), but
none of these explanations seems presently very satisfactory.
5. Summary and final remarks.
The Raman study of AHO and AHOD as a function of temperature and pressure has allowed
us to draw the phase diagram of this crystal down to 78 K and up to 8 kbar and to discover the existence of a new phase III. The phase 1-111 transition is presumably second order, and the discovery of two low frequency lines existing only in phase III, the frequencies of which were strongly pressure dependent, has been instrumental in proposing that this phase is
incommensurate. Once ascertained by elastic neutron measurements [6] which showed that the incommensurate wave vector is parallel to c *, our Raman data are able to give a direct
measurement of the length of the incommensurate wave vector by light scattering, a type of results never obtained previously. Our polarization measurements of the two low frequency lines, as well as the behaviour of the central peak detected in the B2g(ac) geometry are
consistent with a model in which the mechanisms of the 1-11 and 1-111 transitions would be the same, namely the coupling between an ammonium ordering relaxational mode and the TA ,
a polarized, phonon propagating along c *. This coupling would take place at q
=0 for phase II, and at d (P ) c *, (where d (P ) is an increasing function of P with 6 (P ) > 0.12) for phase
III. Above the pressure of the triple point where the three phases coexist, a strongly first
order phase 11-111 transition always take place with decreasing temperature, and phase II can
be considered as the lock-in phase of phase III.
Our Raman data also indicate that, in phase III, the ordering of the ammonium ions is far from complete, as evidenced by the behaviour of the q
=0 relaxational mode in this phase.
Finally, our study of the 851 cm- 1 internal mode has revealed the existence of a specific density of states related to edge singularities. Such effects have been detected and analysed in great details in the NMR studies of incommensurate phases [15] but not yet reported in the
case of Raman studies of such systems.
The present study is far from complete. In particular, it will be neçessary to obtain more
information on the phase diagram in the vicinity of the triple point. Also the amplitudon
mode related to the incommensurate phase has not been detected and it will be important to
find out whether this is e.g. due to the narrow temperature range in which the study has been
done, or if this amplitude mode is visible in other [ (bb ) or (cc ) ]scattering geometries. In this
respect, light and neutron scattering experiments are quite complementary, and the use of the
latter technique can be quite usefull in the search of this amplitude mode, and will be necessary for verifying the symmetry of the mode which freezes-in at Tc. Finally, the free
energy of Appendix B has been written in an ad hoc manner ; it has neither been justified on a microscopic basis, nor does it contain any mechanism leading to a variation of qo with pressure. In this respect, it may be worthwhile noting that AHO is not the only case where the
interaction of a NH’ ion with a rigid planar system leads to an incommensurate phase. The
same situation happens in (C3H7NH3)2MnCI4 [16], where the three hydrogens of the NH3 group are bonded, in a disordered manner with the Cl atoms of the bidimensional
MnCl4 structure. Nevertheless the situation is rather different as, in the latter case, the modulation is within the planar structure, while it is perpendicular to it for AHO : the type of frustration leading to incommensurability might thus be not similar. Further theoretical as
well as experimental studies are thus needed to understand the very special behaviour of
AHO and AHOD.
Acknowledgment.
We thank Professor J.P. Mathieu who has carefully grown the AHO and AHOD single crystals used in the present experiments.
Appendix A.
In order to study a phase transition under pressure, it is necessary to make use of a Landau type expansion of the free enthalpy per unit cell :
where F is the free energy per unit cell, P the pressure and ei i
=1, 3 are the principal
deformations.
Following equation (2) of II, the relevant part of the free energy reads :
In this expression CS represents the bare elastic constant related to the es shear, and 8 i the coupling constants between ei and the square of the order parameter Q.
At a given pressure P, the equilibrium conditions are given by :
which leads, neglecting the terms in Q 3 and eliminating e5 and ei, to :
The transition temperature is thus :
so that :
03B4i
-r- has been evaluated in 1 and is negative for the three values of i, so that (A4) leads to a negative shift of the transition temperature. More precisely using the numerical results tabulated in 1 and II one has :
This yields :
Appendix B.
In the presence of a linear coupling between the pseudo-spin Q (q ) with an (acoustic) mode,
with normal coordinate Qa(q), the relevant part of the free energy of the system may be
expressed as :
where :
-
F is the free energy of the pseudo-spin system (Eq. (6)),
-
y (q ) is the coupling constant between the pseudo-spin and the acoustic phonon,
-
