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PHONON DISPERSION CURVES OF ORDERED PHASES OF T.B.B.A.
J. Benattar, A. Levelut, L. Liebert, F. Moussa
To cite this version:
J. Benattar, A. Levelut, L. Liebert, F. Moussa. PHONON DISPERSION CURVES OF OR- DERED PHASES OF T.B.B.A.. Journal de Physique Colloques, 1979, 40 (C3), pp.C3-115-C3-119.
�10.1051/jphyscol:1979324�. �jpa-00218720�
PHONON DISPERSION CURVES OF ORDERED PHASES OF T.B.B.A.
Abstract. — The lattice dynamics of a deuterated single crystal of T.B.B.A. have been measured by mean of inelastic neutron scattering.
The acoustic phonon propagation, specially the transverse mode propagating perpendicularly to the planes of molecules, has been observed in the crystalline phase and in both smectic B and E phases. While the energy of the longitudinal modes varies just a little in the different phases, the energy of the transverse mode, near the zone boundary, increases in the smectic phases.
An attempt of analysis of the phonon width in the smectic phases has been undertaken.
1. Introduction. — Terephtal-bis-butyl-aniline (T.B.B.A.) exhibits nine phases. We have been interested in three of them :
— at room temperature, it has a crystalline struc- ture,
— at 115 °C, it undergoes a transition into a smectic B phase,
— on cooling down, to 85 °C, it becomes a smectic E like structure [1, 2, 3].
The three phases have a three-dimensional order with a layered crystal, but in both smectic phases, the fluctuations of the atoms around their mean positions are of great amplitude (1 or 2 A) and only a few Bragg spots are measurable as a consequence of a very large Debye-Waller factor. In the smectic B phase, the molecules undergo a reorientational motion around their long axis and thus the array is pseudo- hexagonal in each layer, while in the smectic E phase a freezing of this reorientational motion leads to a herring-bone packing in each layer. Nevertheless large vibrations and distortions of the molecules take place. Let us point out that in the crystalline phase, though stable at room temperature, there is
still a large thermal motion of the terminal aliphatic chains of the molecules.
Previous neutron inelastic scattering measurements have been done on non-deuterated single crystal [4, 5].
In spite of a large incoherent scattering, it has been possible to determine the beginning of the acoustic branches of phonons in these three phases.
However, the transverse mode propagating per- pendicularly to the layers of molecules has been observed only in the solid phase.
For the smectic phases, it is interesting to know whether well defined transverse phonons can propa- gate perpendicularly to the layers. De Gennes and Sarma [6] have predicted for the smectic B phase, a vanishing shear constant as a consequence of a melting of the terminal aliphatic chains.
So neutron coherent inelastic scattering measure- ments have been made on a nearly fully deuterated single crystal.
2. Experiments. — These measurements were carried out on a cold source triple-axis spectrometer at the reactor EL. 3 in Saclay. To analyse energies less than 0.3 THz, we have used incident neutron beam of wave-length larger than 4 A provided by a double monochromator of pyrolytic graphite. Higher- (*) Associe au C.N.R.S.
JOURNAL DE PHYSIQUE Colloque C3, supplément au n° 4, Tome 40, Avril 1979, page C3-115
J. J. BENATTAR, A. M. LEVELUT and L. LIEBERT Laboratoire de Physique des Solides (*),
Université Paris-Sud, Bâtiment 510, 91405 Orsay Cedex, France F. MOUSSA
Laboratoire Léon-Brillouin, CEN Saclay, BP N° 2, 91190 Gif sur Yvette, France
Résumé. — La dynamique de réseau d'un monocristal deutérié de T.B.B.A. a pu être étudiée au moyen de la diffusion inélastique de neutrons. La propagation de phonons acoustiques, en par- ticulier du mode transverse se propageant perpendiculairement aux plans des molécules, a pu être mise en évidence dans la phase cristalline et dans les phases smectiques B et E. Alors que l'énergie des modes longitudinaux varie peu d'une phase à l'autre, l'énergie du mode transverse s'accroît, au voisinage du bord de zone, dans les phases smectiques. Un essai d'analyse de la largeur des phonons dans les phases smectiques a été entrepris.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979324
C3-116 J. J. BENATTAR, A. M. LEVELUT, L. LIEBERT AND F. MOUSSA
order contamination was suppressed by the use of a cooled Be filter. To analyse energies larger than 0.3 THz, we have worked with incident neutrons the
SmE
a
wave-length of which was 3
A.
Second order conta- mination being suppressed (at 83%)
by a pyrolytic graphite monochro~ator set for Bragg reflexion with 1.5 as the wave-length.he
scattering plane was the-plane (a*, e*t (Fig. 1).This plane is parallel to the long axis of the molecules of T:B.B.A. in the three phases. c* is perpendicular to the layers. Most of the experiments have been made in the Brillouin zone centered on the 400 reflexion of the solid phase, or the 200 reflexion of the smectic phases. This allows one to follow a longitudinal acoustic mode propagating in the a* direction and a transverse mode, in the c* direction.
2.1 ACOUSTIC LONGITUDINAL MODES PARALLEL TO
a*. - The results, for the three phases, are reported on figure 2. We can see that the dispersion curve is quite the same for the crystalline phase and for the smectic E phase, except for the width of the mode which is larger than the resolution of the spectrometer in the smectic E phase. In the smectic B phase, the initial slope of the curve is lower than in both other phases, the phonons are damped and we are not able to detect them at energies higher than 0.55 THz, because of a lack of intensity.
FIG. 1. - a ) Schematic description of the structures of the 3 studied phases by their projections on the (a, c) and (a, b) planes ; notice the difference of scale between the two projections. In the case of the (a, c) projection, molecules are represented by full lines
when located at y = 0 and dotted lines for y = 112.
Unit cell parameters, density and space group for the three studied phases of T.B.B.A.
a b c Space Multi-
Phase A A
a P
p group plicity- - - - - - - - Crystal 17.57 5.75 53.2 115.50 1.08 A2/a 8 SmE 10.38 5.24 28.31 123.60 1.02 Pa 2 SmB 10.15 5.18 28.6 119.0 101 C2/m 2 b) Reciprocal lattice cross-section in the scattering plane for the crystalline phase. In the smectic phases, Bragg points are noted
- +
FIG. 2. - Longitudinal phonon dispersion curves for the 3 phases :
& : crystalline phase ; : smectic B phase ;O : smectic E phase ; B.Z. : Boundary of the zone. Transverse phonon dispersion curve
for the solid phase only (+).
An other interesting feature of these measurements of the longitudinal modes in the crystalline phase, is the presence of a further mode measurable only near the zone boundary (Fig. 3). One can see on that energy scan, at room temperature, two not-well- defined wide maxima, the second one, for reasons of continuity, is the longitudinal acoustic phonon; on heating the sample.to 80 OC, it remains in the crystal- line phase and the intensity of the first maximum grows much more than expected from the Bose population factor :
exp
[ k ]
kF3 T - I201,200,201, they have nearly the same positions as in the crys- talline phase.
PHONON DISPERSION CURVES O F ORDERED PHASES O F T.B.B.A. C3-117
FIG. 3. - Constant Q-scans in the crystalline phase at the point : 2 R
Q, = 4.5 x - with a, = 17.238 A
a x
+
: T = T,,, ; 0 : T = 80 OC. These scans correspond to an annihilation of phonons. The full and dotted lines are guides foreyes.
which leads to a ratio of 1.2, &at is obviously too small.
It was very difficult to follow the dispersion of that mode and further measurements are needed.
We think either the rotation of the methyl groups at the ends of the molecule of T.B.B.A. or the librations the whole molecule around its own axis c, are perhaps responsible of that extra mode, but it has to be confirmed.
2 . 2 TRANSVERSE ACOUSTIC MODES PARALLEL TO c*.
With the chosen geometrical configuration of the neutron scattering process, the polarization of the measured phonons is around the a* direction, so these modes are usually called transverse though their polarization is not perpendicular to the wave- vector.
In the crystalline phase, this dispersion curve had already been measured 141. With the deuterated crystal, we have been to follow it in both smectic phases.
Following a constant-q scan, we observed a large inelastic scattering peaked on v = 0 and varying with the q direction. That intensity is specially high in the c* direction and acoustical phonons are no
longer measurable along such scans. So, the dispersion curves have been determined only from constant- energy scans. However, near the zone boundary, the dispersion curve usually becomes flat and a constant energy scan is not a good method to deter- mine it. So, we do not know exactly what is the true value of the zone boundary energy, but it seems sure that it is higher than in the crystalline phase (about 0.2 THz instead of 0.14 THz).
FIG. 4. -Transverse phonon dispersion curves for the three phases : A : crystalline phase ; : smectic B phase ;O : smectic E
phase.
The experimental results are reported on figure 4.
The energy of the phonons increases when the crystal becomes smectic. We have also observed a broadening of the neutron groups in the smectic phases.
In the solid phase, we have also measured transverse modes propagating in the a* direction, with a c*
parallel polarization (undulations of the layers) but this mode cannot be seen in the smectic phases, because of the lack of intensity in the convenient Brillouin zone (Fig. 2).
3. Analysis of results and discussion. - As it is evident the phonons have a width in energy in both smectic phases, we have tried to calculate this width from the measured neutron intensities. We have taken as a scattering function for the phonons :
T(v,) is the width in energy, v, is the dispersion law for the phonons. To calculate the scattered intensity one needs to know the dispersion law in the whole
C3-118 J. J. BENATTAR, A. M. LEVELUT, L. LIEBERT AND F. MOUSSA
reciprocal space around the analysed point. The simplest law is chosen :
q,, being the component of q along c* and q, being the component perpendicular to c*.
The scattering vector Q being related to q by : Q = z
+
q with z = 4a*in the crystalline phase and z = 2 a* in the smectic phases.
The measured intensity of the scattered neutrons at the point v,, Q O at which the triple-axis spectrometer is set, is given by
,.
R(v - v,, Q - Q , ) is the resolution function of the triple-axis spectrometer which is known from the apparatus parameters.
We have fitted the calculated intensities to the measured ones with two parameters : the width T(v,) and c, which is a kind of phase velocity, for each scan; c' remains constant in the fit, having been determined separately.
It is clear that on6 cannot perform the four-dimen-
sional integration. One is led to make severe sim- plifying hypotheses to have only a one-dimensional integration. So the precision of the values of the fitting parameters is about 25
%.
3.1 LONGITUDINAL MODES. - The values of the fitting parameters for this mode are reported on figure 5.
In the domain of energies less than 0.4 THz, which is the maximum for which one has been able to measure the phonons in the smectic B phase, the values of the phase velocity for both smectic phases are neighbouring, within the precision of the deter- mination of the fitting parameters, and they are about
30
%
lower than the velocity in the solid phase.For higher energies, the velocity in the smectic E phase seems to increase and to go to the solid phase value.
The width
r
increases in both smectic phases with increasing energy, but, while it seems to follow ar
= Dv2 law in the smectic E phase, except for the highest energy values, the behaviour of the para- meterr,
in the smectic B phase, seems to be linear with the energy. In this phase, the quality of the scans at energies larger than 0.4 THz, did not allow us to analyse them.3 . 2 TRANSVERSE MODES. - The values of c and
r
are repoxted on figure 6 .FIG. 5. - Fitting parameters
r
and c for longitudinal phonons in smectic B and E phases. Full lines are guides for eyes. The dotted line is a parabola. The dash and dotted line is a straight line :I3 : smectic B phase; 0 : smectic E phase; w : crystalline phase (only c is fitted for these points).
FIG. 6. - Fitting parameters
r
and c for phonons in smectic B and E phases. The ultra-sound measurements are reported for both smectic phases (at v--
0). Full lines are guides for eyes. The dotted line is a parabola : : smectic B phase; 0 : smectic E phase;A : crystalline phase (only c is fitted for these points).
PHONON DISPERSION CURVES OF ORDERED PHASES OF T.B.B.A. C3-119
It has been difficult to analyse phonons with energy greater than 0.15 THz because in a constant energy scan, near the zone boundary, the measured intensity has a double origin due to the phonons being broad : the neutrons are scattered by phonons in the [200]
f
zone and in the [20?] zones, according to the direction of the scan. As we do not know the dynamical struc- ture factor and the Debye-Waller factor, it is not easy to estimate in a given measured intensity what belongs to a proper phonon in the zone where the scan is done and what belongs to the tail of the sym- metrical phonon in the following zone.
The main features of our results are :
- the widths of the phonons are of the same order of magnitude for both smectic phases ;
- in the range of low energies ( v < 0.1 THz) the fitting parameter c takes rather the same value for the three phases. For higher energies, to have a good fit, one has to take greater values of the phase velocity c in the smectic E phase. It is obvious on figure 4 that for a given wave-vector, the energy of the phonon increases in the smectic phases. The origin of that is not understood, it looks like some coupling with a mode which should appear in these phases.
We have also reported in figure 6 the transverse velocity measured with ultrasound waves in both smectic phases [7]. These measurements have been made at 110 MHz, that is three order of magnitude lower than ours. Therefore, as they correspond to a mean value of the transverse velocity over several propagating directions in the smectic layers, one
cannot merely compare them with the neutron measurements.
Nevertheless, it is clear that transverse modes can propagate perpendicularly to the layers in the smectic phases of T.B.B.A., and the shear modulus does not vanish.
4. Conclusion. - From these measurements, we can deduce some conclusions :
- Both longitudinal and transverse phonons can propagate in the three ordered phases of T.B.B.A.
Particularly the transverse mode which propagates perpendicularly to the layers.
- The longitudinal mode does not change very much from the crystalline phase to the smectic phases.
- On the other hand, the transverse mode has an energy and a width which are greater in the smectic phases than in the crystalline one.
- Beside the phonon scattering, large quasi- elastic scattering is observed in the smectic phases.
These neutron scattering experiments have been done in the (a*, c*) plane of the single crystal.
Further measurements are needed, in other direc- tions, to study the anisotropy of the propagation of phonons, and to confirm the extra mode, specially its behaviour with temperature. Furthermore, a theoretical model of the interaction between the molecules in a given layer, and between the layers themselves, is necessary to interpret quantitatively the measured phonon dispersion curves.
References
[I] DOUCET, J., MORNON, J. P., CIIEVALLER, R., LIFCHTIZ, A., [4] DOUCET, J., LAMBERT, M., LEVELUT, A. M., PORQUET, P., Acta Crystallogr. B 33 (1977) 1701. DORNW, B., J. Physique 39 (1978) 173.
[2] DOUCET, J., LEVELUT, A. M., LAMBERT, M., Mol. Crist. Liq. [5] Bid., Neutron Inelastic Scattering I (1977) 549, IAEA.
Crist. 24 (1973) 317. [6] DE GENNES, P. G., SARMA, G., Phys. Lett. A 38 (1972) 219.
[3] DOUCET, J., LEVELUT, A. M., LAMBERT, M., Phys. Rev. Lett. [7] UNAL, H., Thesis 1978.
32 (1974) 301.
