Local order and lattice dynamics in the ordered phases of T.B.B.A. (Terephthal-Bis-Butyl-Aniline)

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Local order and lattice dynamics in the ordered phases of T.B.B.A. (Terephthal-Bis-Butyl-Aniline)

A.M. Levelut, F. Moussa, J. Doucet, J.J. Benattar, M. Lambert, B. Dorner

To cite this version:

A.M. Levelut, F. Moussa, J. Doucet, J.J. Benattar, M. Lambert, et al.. Local order and lattice

dynamics in the ordered phases of T.B.B.A. (Terephthal-Bis-Butyl-Aniline). Journal de Physique,

1981, 42 (12), pp.1651-1663. �10.1051/jphys:0198100420120165100�. �jpa-00209363�

Local order and lattice dynamics in the ordered phases of T.B.B.A.

(Terephthal-Bis=Butyl-Aniline)

A. M. Levelut (*), ^{F. Moussa} (**), ^{J. Doucet} (*), ^{J. J.} Benattar (**), ^M. Lambert (*) (**) ^{and B. Dorner} (***)

(*) Laboratoire de Physique ^{des Solides} (+),

Université Paris-Sud, ^Bâtiment 510, ⁹¹⁴⁰⁵ Orsay, ^France

(**) Laboratoire Léon-Brillouin, ^C.E.N. Saclay, ^{91191 Gif sur Yvette} Cedex, ^France (***) ^Institut Laüe-Langevin, ¹⁵⁶ X, 38042 Grenoble Cedex, France

(Reçu ^{le 18 mai} 1981, accepté ^{le 26 août} 1981)

Résumé.

Nous faisons le point ^{sur la structure et la} dynamique de réseau des phases smectiques ^{ordonnées du}

T.B.B.A. L’étude de la dynamique ^{de réseau au} moyen de la diffusion cohérente inélastique ^{des neutrons confirme}

les résultats de diffraction des rayons X : les phases smectiques Bc ^et Ec présentent ^{un ordre} conventionnel tridi- mensionnel. Toutefois les molécules effectuent des mouvements de grande amplitude qui ^rendent difficile l’obser- vation des modes collectifs dans certaines directions. Nous avons repris, ^{avec les} neutrons, l’étude, ^{faite aux} rayons X, de l’ordre local en chevron présent ^{dans la} phase smectique Bc ^{du T.B.B.A. Grâce à} l’analyse ^en énergie, ^{les mesures} neutroniques ^{donnent un meilleur ordre de} grandeur des dimensions des différents domaines ordonnés en chevron.

Il semble que l’angle d’inclinaison de la molécule n’induise qu’une ^faible anisotropie ^{dans un} type ^{donné de} domaine,

et donc que ^ce soit le couplage ^{entre couches} smectiques qui ^{favorise la} croissance d’un type ^{de domaine} privilégié près ^{de la} transition SmBc ~ SmEc. ^De plus ^{nous avons} pu ^{mettre en} évidence un temps de vie des domaines qui

est à peine supérieur ^au temps ^de relaxation des mouvements individuels de rotation des molécules.

Abstract.

This paper reviews the main results ^on the structure and the dynamics ^{of the ordered smectic} phases

of T.B.B.A. The study ^{of lattice} dynamics by ^{means of neutron coherent inelastic} scattering ^{bears out the} X-ray

measurements : the smectic Bc ^and Ec phases ^{show a} conventional three-dimensional order, ^however the molecules may perform ^{motions with} large amplitude, this makes the observation of collective modes rather difficult. Moreover

we have restarted with neutron techniques ^{the work done with} X-rays ^{on the local} herring-bone ^{order in the smectic} Bc phase ^of T.B.B.A. Because of the energy analysis, ^the neutron measurements give ^{better measure of the sizes of}

different types of domains. It seems that the molecule tilt angle ^induces only ^{a little} anisotropy ^{in a} given domain,

and thus it is the coupling between smectic layers ^which favours the growth ^{of a} preferential ^{domain near the}

transition SmBc ~ SmEc. ^{Moreover a lifetime} of the domains has been measured, and appears ^to be slightly greater than the relaxation time for individual rotational motions of the molecules.

Classification

Physics ^Abstracts

61.10

61.30

63.20

64.70E

The ordered smectic phases ^{can be} classified in four

families ; ^{all these} phases ^{exhibit a mean} three-dimen- sional order, ^but fluctuations around the equilibrium positions ^{located on the} crystalline ^{lattice sites are} large compared ^{with those} existing ^{in an usual} crystal-

line phase. ^{The four} phases BA, Bc, EA, Ec (1) ^{are made}

of periodically ^stacked layers ; they ^{differ from one}

another by ^{the kind of order inside a} layer ^and by

the possible ^tilt angle of the molecular axis with respect

to the normal to the layers.

T.B.B.A. undergoes ^{at 113.5 OC a} transition from a

stable crystalline phase ^{to a smectic} Bc phase. ^The

(+) ^{Associé au C.N.R.S.} (L.A. ^no 2).

(’) ^These phases ^are often called B instead of BA, ^{H or G instead} ofBc ^{and E instead} of EA.

molecules, ^{the tilt} angle of which is about 30o, ^are arranged ^{on a} quasi hexagonal lattice. ^The Bc phase

remains in a super-cooled ^{state and at 84 OC it turns}

into a smectic Ec phase where the order inside a layer

is different. Studies performed ^{on the structure and}

on the dynamics ^{of both smectic} phases, ^{have shown}

the following ^{results :}

-

the X-ray diffraction measurements [1], yield

information about the mean structure and the short range order, ^{but cannot} separate elastic from inelastic processes ;

-

the incoherent inelastic neutron scattering ^mea-

surements [2] ^{or the nuclear} magnetic ^resonance

measurements on different nuclei [3, 4] which inform about the _proper motions of a single ^{molecule, but}

are quite insensitive to the correlated motions of the molecules.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198100420120165100

Inelastic coherent neutron scattering experiments performed ^on fully deuterated single ^domain crystals

of T.B.B.A. complete ^the results obtained from the above mentioned methods and allow us to study ^the

collective motions. Previous experiments ^{on non}

deuterated samples ^{gave us} interesting but incom-

plete ^results [5] because of the strong ^incoherent background. ^{Here, the main results} obtained from the inelastic coherent neutron scattering ^on fully

deuterated samples ^are reported. ^These measurements have been performed ^{on the} crystalline phase ^and

on the Bc ^and Ec ^smectic phases. They ^are compared

with the results obtained by ^other techniques. ^First

of all, ^{let us} recall what is known about these ordered

phases ^{of T.B.B.A.}

1. Structure and dynamics ^{of T.B.B.A.}

^The X-ray scattering patterns of both smectic phases ^show

that the mean order of the centres of mass of the mole- cules is three-dimensional. Table 1 gives ^{the lattice} parameters ^and the space group of the elementary

Table I.

Space group, lattice parameters of ^the crystalline ^{and smectic} phases of ^T.B.B.A.

cells of the studied phases ; ^we observe reflections

showing ^{that the molecules are} arranged ⁱⁿ equidis-

tant layers (001), reflections typical of the order in the

layer (h, k, 0) ^and reflections (h, k, 1) ; ^{all the} Bragg peaks are narrow. The observed reflections (h, k, 1)

allow us to conclude that there is no translational disorder in the layer plane ^{from one} layer ^{to the next.}

The intensity decreases much faster with h, k, ^{1 increase}

than in the crystalline ^{case because the} fluctuations of the atoms about their equilibrium positions ^are large (> ² Á) ^even though ^{the mean} positions ^remain perfectly ^{defined at} points ^{of a} three-dimensional lattice. Some of the motions which give ^{rise to fluc-}

tuations have been observed by ^different techniques.

In the smectic Bc phase, ^molecules undergo ^orien-

tational jumps ^{around their} long axis, ^which gives

an almost hexagonal structure. The incoherent quasi-

elastic neutron scattering [2] shows this orientational motion and the relaxation time is about 10- Il s.

The rotation of the molecules around their long ^axis

is not completely [4] isotropic and the orientations of

neighbouring molecules show some correlation. The

phenyl rings ^of neighbouring molecules, ^{in a} given

layer, ^are arranged ^{in a} herring-bone ^like packing,

the local rectangular ^{structure has a} cell twice as large

as the hexagonal ^cell [6]. ^{Diffuse zone} boundary spots

are associated with this short _range order and ^are visible on X-ray diagrams, in ^the reciprocal plane [hko] perpendicular ^{to the molecular} axis. There are three possible orientations of the local rectangular

cells (Fig. lb) ^obtained by ^{rotation of 2} 03C0/3 ^around

the molecular axis. In each local cell, ^{there are mole-} cules with two different orientations (modulo 03C0).

Four equivalent ^diffuse spots ^are associated with each orientation of the local cell. On the whole, ¹² diffuse spots ^{located on a circle centered at the} origin

of the reciprocal ^{space are observed} (Fig.1a).

Fig. ^la.

(a*, b*) reciprocal plane of the smectic phase Bc : ⁰

Diffuse spot ^of type ^{1. 0} Diffuse spot ^of type ^{2. 0 Diffuse} spot ^of type 3. 2022 Bragg peak.(llPzone ^of scattering. ^The type ¹ diffuse spots become Bragg peaks ⁱⁿ the smectic Ec phase.

Fig. ^lb.

^{The three} orientations of local herring-bone ^domains

and the symmetrical ^{ones with} respect ^to the b axis. Only ^{the domains}

of type ¹ change ^with temperature.

The X-ray diagram ^is compatible ^{with the} picture

that the molecules jump ^{into six} positions (modulo n).

In the smectic Bc phase, ^{there is a tilt} angle ^between

the molecular axis and the normal to the layer plane ( ~ 300 in the case of T.B.B.A.), ^{so the six} positions

are not equivalent since the direction of the molecule is not an axis of symmetry. Among ^{the twelve observed}

diffuse spots, ^{four of them are connected with the} rectangular ^cell the small side of which is identical to the binary ^{axis of the} primitive ^cell describing ^the

monoclinic structure of the SmBc phase ; ^the spots

are narrower and more intense than the others and the difference is more and more important ^{as the} SmBc SmEc transition is approached.

The anisotropy ^of molecular rotations has been estimated by ^{Blinc et al.} [4] by ^{means of} quadrupole

resonance measurements. The authors determined the anisotropy ^{of the} rotation of the -CH = NC bonds of the molécule ; they ^{estimated the} probabilities Pi, P2, P3 of the molecule in the three positions (modulo n). They ^{find, near the} SmBc - SmEc transition :

In the SmEc phase, ^{the molecular rotations are}

frozen. Therefore the incoherent quasi-elastic ^neutron scattering [7] ^{and the} quadrupole ^resonance [4] ^show

that the molecule is able to perform ^a jump ^{of 1800.}

The SmEc phase ^has layers ^{with a} long range order with a herring ^bone structure. The four diffuse spots mentioned above become Bragg spots ^{in the} SmEc phase, while the others do not fully disappear ^{at the} transition, but remain visible until two degrees ^below

the transition temperature. ^{From a certain} point ^of view, this transition looks like the transition which

occurs in the plastic crystals.

Moreover, ^{two other} types of disorder Lave been established in these smectic phases :

-

from X-ray diagrams, ^{one can} conclude in favour of nearly longitudinal displacements ^{of the} molecules, quite well correlated along ^{one array.}

These correlated motions are a proof of the existence of interlayer ^bonds.

-

from nuclear magnetic ^resonance measurements

[3], ^{one concludes that the} molecule does not seem to be rigid. ^{The results are well} explained by ^variations

of the conformation ^« trans »-« cis » ^« trans » obtained

by the rotation of ^one outside phenyl ^{around the}

C-C bond of the terephthaldehyde group (Fig. 2).

The crystalline phase is better ordered and is quite

different from the smectic phases ^because reflections with high ^{indices are} observed. That means the Debye-

Waller factor is the same as that of an usual crystal [8].

The molecules have a « cis » conformation but the

homologous phenyl rings ^{of the molecules are} nearly parallel, ^{the structure} is indeed quite different from the herring-bone packing of the smectic phases (Fig. 3).

However some disorder remains since the terminal

Fig. ^2.

Different conformations of the molecule of T.B.B.A. in the smectic phases.

Fig. ^3.

Projection ^{of the} crystalline phase ^{structure on the} (a, b) plane along ^{the c direction.}

methyl groups have motions with large amplitude [2].

A study ^{of the structure shows that one of the} methyl

groups performs ^a nearly complete rotation, ^{the last}

C-C bond describing ^{a cone while the other} methyl

group oscillates in a plane [8].

The smectic phases appear ^to have a three-dimen- sional periodic ^{mean order so, there should be acous-}

tical phonon ^{branches as in the solid} phase. ^However,

there are numerous molecular motions and _many of them are correlated, ^at least, ^{on short} range and these motions should be likely coupled ^{with the acoustical} phonons.

Let us note that a. low frequency optical ^mode (19 cm-1) ^{has been ob§erved in Raman} spectroscopy [9, 10, 11], ^{but the authors do not} agree about the

phases where this mode can be seen.

All the results we have just ^{discussed concern}

T.B.B.A. which is ^one of the most studied mesomorphic compounds. ^{We have tried to} complete ^our knowledge

of this system by measurements of the collective modes, by ^means of inelastic coherent neutron scat-

tering, ^{in both ordered smectic} phases and in the

crystalline phase.

2. Experiments.

^{We have used a} nearly fully

deuterated compound ^where only ^the hydrogen ^atoms

located in a position of the methine groups ^were

partly substituted (40 %). ^The deuterated terephthal- dehyde ^{has been} provided by ^Merck, Sharp ^and Dohme, while the butylaniline has been deuterated

by ^{L. Liebert} (Orsay); the latter deuteration has been controlled by high resolution nuclear magnetic

resonance. The powder of deuterated T.B.B.A. obtained

by the reaction of both compounds ^{in alcohol has}

been dissolved in deuterated solvent. Large single crystals ^{have been} grown from this solution by ^Pro-

fessor J. P. Chapelle (Orsay) ^the weight ^{of which}

varies from 0.4 _g to 1 _g. The mosaic spread ^{was about}

2°.

The neutron scattering measurements have been carried out on two triple axis-spectrometers :

-

IN2 ^at the Institut Laüe-Langevin, ^{in Grenoble.}

The used incident wavelengths ^{were 2.35 Á and 4 Á}

with a graphite filter and a beryllium ^filter respectively

to reduce higher ^order contamination.

-

H 1 ^at Laboratoire Léon-Brillouin in Saclay.

This spectrometer ^{is set on a cold source. The wave-} length ^{of the neutrons} has been varied between 3 Á

and 5.4 Á. A beryllium filter has been used too and

a pyrolytic graphite ^{set in} reflection position ^for higher-order wavelengths ^{has been used for wave-} length smaller than 4 A.

Recently, ^{we have} improved ^the precision ^{of our}

measurements by using ^a very high resolution triple-

axis spectrometer, ^IN 12, ^at the Institut Laüe-Langevin.

It is installed on a neutron guide ^{from the cold source}

with a « eut-off » for wavelengths less than 2 A. Fixed incident neutron wavelengths ⁱⁿ the range 4 A-5 Á

have been obtained with a vertically ^curved pyrolytic graphite monochromator; ^the analyser ^{was a flat} pyrolytic graphite crystal. The collimation before the monochromator is determined by ^{the neutron} guide. The horizontal collimations were 30’-30’-30’

after the monochromator and before and after the

analyser, respectively. ^A beryllium ^{filter was used}

too. Only ^{some of these new results are mentioned} hère. The whole of them will be published ^elsewhere.

Two scattering planes ^were explored ⁱⁿ reciprocal

space :

-

The plane ^defined by (a*, c*) which contains the molecular long ^{axis c in} the smectic phases, ^and

which is normal to’the layers ^{in the three} phases (smectic ^and crystalline) (Fig. ⁴ a).

-

The plane ^defined by (a*, b*), ^{normal to the}

axis c and which contains the binary axis b for the three phases.

Our investigations ^were concerned with the follow-

ing points :

-

the phonons propagating along ^{a* and} b*,

-

the study of the local ordering ^{related to the} SmBc ~ SmEc transition,

Fig. ^4a.

Reciprocal plane (a*, c*) ^{of the solid} phase ^{of T.B.B.A.}

Fig. ^4b.

The three acoustic modes and two optic modes propa-

gating along ^a in the solid phase of T.B.B.A.

-

the study ^of phonons propagating along ^{the c*}

direction, ^i.e. perpendicular ^{to the} layers ^{in the} crys- talline phase. These excitations have ^a peculiar ^beha-

viour in the smectic phases, ^so they ^{will be treated} separately.

3. Phonons in the solid phase, comparison ^{with the}

smectic phases.

^The study ^of non-deuterated single crystals [5] had allowed us to determine the beginning

of several acoustic branches; ^the large ^incoherent background prevented ^{us from} measuring phonons

far from the zone centre. With deuterated crystals ^we

have been able to measure phonons ⁱⁿ nearly ^the

whole Brillouin Zone [13]. ^{One has to note that all}

Table II.

Sound velocities in ^m. s-1 measured by ^neutron experiments ^{values put in brackets} correspond ^{to a}

mean velocity of transverse modes propagating along ^c and measured by ultrasonic techniques.

(*) Values measured by ultra sound techniques ^{at 110 MHz} by Unal and Bacri.

the studied phases ^are monoclinic (point group 2/m).

In almost all experiments, ^{one can observe several} (2 ^or 3) ^neutron groups connected ^to the scattering by phonons ^{of a} given propagating ^{vector q but with}

different polarizations.

In the crystalline phase ^the measurements have been performed mainly around three intense Bragg points ^{in two} perpendicular planes ^{of the} reciprocal

space :

-

in the plane (a*, c*) : [4, ^0, 0] ^and [2, 0, 10] ^these

two vectors being nearly perpendicular,

-

in the plane (a*, b*) : [4, 0, 0] ^and [0, 2, 0].

In the smectic phases, only ^the Bragg point [4, ^0, 0]

has an homologous point ⁱⁿ [2, 0, 0] and there ’are

no more intense Bragg points ^{in the} c* direction while in the plane (a*, b*) ^{one can scan in the} neighbour-

hood of [1, 1, 0]. ^{Table II} gives ^the propagation ^direc-

tions and the rough directions of polarization ^{of the}

observed acoustic ^{modes, and} gives the measured sound velocities too.

In figure ^{4b we} report ^the dispersion ^{curves of} pho-

nons propagating along a* and in figure ^{5 those}

which propagate along b* both in the crystalline phase.

All the dispersion ^{curves have been} determined _up

to the zone boundary except ^{for the} nearly longitu-

dinal mode parallel ^to b*. The value of _{the energy} of this mode is rather high ^{at the zone} boundary

and the lack of scattered intensity prevented ^{us from} measuring ^the dispersion ^{near the zone} boundary.

We did not explore ^the reciprocal plane (b*, c*)

because of the lack of intense Bragg peaks ^{in the}

smectic phases. ^The nearly longitudinal ^mode pro-

pagating perpendicularly ^{to the} layers ^{has been}

measured only ^{in the} crystalline phase, because the

Bragg peak [2,0,10] disappears in the smectic phases.

Here, ^we shall give only ^{an outline on the} phonons propagating along c*, ^the polarization of which is

Fig. ^5.

^The three acoustic modes propagating along ^{b* in the}

solid phase of T.B.B.A.

roughly along ^{a*. A more elaborate} discussion will be published ^elsewhere.

3.1 PHONONS PROPAGATING ALONG THE a* DIREC- TION.

As this direction is not a symmetrical ^one,

in a given geometry ^{of the} scattering process, the three

acoustic modes theoretically ^{can be observed at the}

same time (Fig. 4b).

Figure ^{6a shows} conditions where ^« transverse » modes are observed simultaneously. Moreover, owing

to the fact that the space group is ^non symmorphic,

the modes are degenerate ^{at the zone} boundary, ^i.e.

some optical branches and acoustic branches cross

the zone boundary ^{at the same values of} energy with

a slope different from ^zero.

Fig. ^6.

a) Example ^{of the} simultaneous observation of two

« transverse » modes propagating along ^a (Âo

^4.1 Â) ^{at the} point

M in figure 4a). b) Shift of the energy of the mode with ^a q compo- nent along ^{c at the} point ^{N in} figure 4a).

For the sake of clarity ^we have chosen ^to represent the dispersion branches in an extended zone scheme

as they have been measured. The dynamic ^structure

factor of the acoustic modes vanishes near the [5, 0, 0]

reciprocal lattice point while it does not for the optical

modes in the ^same conditions. The lowest dispersion

curve (Fig. 4b) corresponds ^to modes with polarization

vectors roughly parallel ^{to the c axis. In a} simple picture, ^{one can} say that the molecules oscillate along

their long axis while remaining parallel ^{to one another}

and one can relate these motions to the scattered

intensity ^{observed on} X-ray diagrams ^from scattering

in a plane perpendicular ^{to the c axis.} Figure ^{6b shows}

the shift of the _energy of the mode (0.13 ^{THz in} figure 6a) ^{when the} propagating ^vector q has a componént along ^{c* while the} component along ^a* being nearly

the same. We observe a rather important increase of the _energy which means that the propagation ^{is easier} along ^the a* direction than anywhere ^else.

Furthermore, ^{in a} geometry ^{of the} triple axis spec- trometer which favours the observation of the nearly longitudinal ^a* mode, ^we have measured two modes.

One is an acoustic mode and the other, ^measurable only for 0.3 a* _q 0.7 a*, ^{has an} energy lower than the _energy of the « longitudinal » ^{mode but} higher

than the ^« transverse » ones. For the same ^above- mentioned reasons, the dispersion ^{curve of this extra}

mode crosses the zone boundary ^{with a finite} slope.

The dependence ^{of the extra mode on the} temperature is shown in figure ^{7. The} intensity rises with the ^tem- perature while the _energy of the mode decreases

slightly.

At the transition to the SmBc phase ^{this mode}

Fig. ^7.

^The temperature behaviour of the optic mode of low energy in the solid phase (Âo

^2.36 À).

vanishes, so, it seems perhaps ^to be related to the librations of the molecules around their long ^axis

which become nearly free rotations at the transition.

In the crystalline phase ^{indeed, the} phenyl ring planes

are nearly parallel ^{to the} (b, c) plane ^{so, the} displace-

ments induced by librations of the phenyl groups

are roughly parallel ^{to the a} direction. Moreover,

the temperature behaviour of this mode can be

compared ^{to the} measurements performed by Pynn,

Otnes and Riste [12] ^{on, P.A.A.} powder. They ^have

observed an intensity ^{near the} Bragg peaks, increasing

with the temperature, in the solid phase. According

to these authors, ^this increase, ^{which is} large ⁱⁿ planes perpendicular ^to the molecular long axis, ^{is due to the} phenyl ^libration amplitude which rises with the tem-

perature. However, the T.B.B.A. structure is not a

Fig. ^8.

Comparison ^of dispersion ^curves of ^{the «} longitudînal »

mode propagating along a* for the three phases : ^{A solid} phase.

0 smectic Ec phase. D smectic Bc phase. ^{The values of} q ^are given

in Å-1.

simple ^{one and} many molecular motions can occur, so connecting motions and modes _may be hazardous.

In the a* direction of propagation, in both smectic

phases, ^{we have} just ^{been able to measure «} longi-

tudinal » acoustic phonons (Fig. 8). ^A strong quasi-

elastic scattered intensity ^much higher ^{than the} phonon ^{one made the} energy ^scans difficult to per- form (Fig. 9) (cf. § 3. 3). ^{So almost all} scans were

q-scans. Moreover, ^{the neutron} groups ^were wider than the apparatus resolution and the intensity ^was

low.

Fig. ^9a.

Energy-scan (annihilation) ^at Q

[1.725, 0, 0] ^{in the}

smectic Bc phase (03BB0

³ Â). ^The dotted line is the estimated quasi- elastic-scattering, the full line is a guide ^for the eyes.

As for the intensity ^of X-rays scattered in planes perpendicular ^{to c axis,} which has been seen in the smectic phases, ^{we have not been able to detect it}

with elastic neutron scattering. ^It is clear that the lowest ^« transverse » mode is ^no longer ^measurable

because the intensity ^{of the} Bragg peak [2, ^{0, 10]}

vanishes in the smectic phases.

3.2 PHONONS PROPAGATING ALONG THE b* (OR b)

DIRECTION.

In the direction of the binary ^{axis of}

the primitive cell there is a symmetric representation containing ^the quasi longitudinal acoustic mode and

an antisymmetric representation containing ^{the two} quasi transverse modes. This does not restrict the

amplitude ^{of an atom at a} general position ^{to be} parallel ^or perpendicular ^to q.

Nevertheless there exists a selection rule for a

binary axis, ^for Q // q // b ^where only the modes of the even representation ^{are visible.}

Fig. ^9b.

^Neutron elastic-scattering ^{in the} (a*, c*) plane ^{in the} SmEc phase. ^{Note the} spreading ^{of the} intensity along ^{the c direction}

around the Bragg peak.

Fig. ^9c.

^{Elastic scan} along ^{the a*} direction in the smectic Bc phase showing the mosaic spread ^of [2, 0, 0] ^and [2, 0, 1] Bragg reflexions, ^{in addition to a} large ^scattered intensity. ^In the insert an

energy ^scan at the point [2.2, ^0, 0.5] proves ^{that this} intensity ^is mostly quasi-elastic.

Therefore, ^we have been able to follow these modes,

in the geometry Q

^{t +} q with T

[0, 2, 0] ^and

q

[0, 03BE, 0] (Fig. 5). ^{In the} geometry :

we have observed two neutrons groups. One can

interpret this latter result in the following ^two ways :

-

as the polarization ^{vectors are not} along sym- metrical directions, ^we have measured both ^« trans-

verse phonons » (let ^{us note that} it should have been

possible theoretically ^{to measure} three modes all

together). The second neutron group ^was only ^detect-

ed for _q > 0.25 b* and an energy ^v > 0.4 THz;

moreover, ^{as we} have not scanned in the reciprocal plane ^defined by (c, b), ^we cannot assert that this mode is an acoustic phonon,

-

this second neutron group may be the ^extra

phonon already ^{observed in the a direction and}

related to the librations of the molecules. No definite conclusions can be drawn at present.

3. 3 PHONONS PROPAGATING IN THE C* DIRECTION.

In the solid phase. ^we have observed well defined

phonons. ^{Much attention was} paid ^{to the} dispersion

curve of transverse modes which corresponds ^{to shear}

oscillations of the layers ^with respect ^{to each other.}

In a recent investigation which will be published elsewhere, ^{we observed a} TO branch in the solid

phase in addition to the TA branch. The two modes have a strong ^{inelastic structure factor in} the neigh-

bourhoo.d of the reciprocal lattice points (400), (401),

(401), (402).

We have completed ^this study ^of phonons ^{in the}

solid phase by ^an investigation ^of optical ^{modes at}

zone centres. Figure ^{10 shows an} energy-scan ^at the

[2, 2, 0] ^{zone centre. The} peak ^is badly defined, ^{but a} rough maximum appears ^at about 0.5 THz. It could be the mode measured in Raman spectroscopy (20 ^cm-1

^0.6 THz) [9,10,11].

Fig. ^10.

Energy-scan ^{at the} [2, 2, 0] ^{zone centre} in the solid phase.

The solid line is a guide ^{for the} eyes.

In the smectic Bc and smectic Etc phases, it is difficult to ascertain whether ^or not true acoustic transverse

oscillating ^{motions can} propagate (no ^{mode is}

observed in the optic range). ^When looking ^{at the}

constant-energy ^scans (Fig. lla) ^{it seems that the}

behaviour is the ^{same as} in the solid phase ^{since well}

defined peaks ^{are visible} and, ^{for a} given energy v, we observe the same maximum in the intensity ^at

± _q in two different Brillouin zones centred on

[2, 0, 0] ^and [2, 0, 1]. ^{We did not observe} anything

around [2, 0, 1], ^{which is} quite normal because the elastic structure factor is _very low at this point.

On the contrary, ^when looking ^{at the} constant-Q

scans (Fig. 11 b) ^no transverse oscillating ^motion

could be clearly detected. Furthermore a strong intensity ^{around v}

⁰ appears ^near the [20 l] ^reci- procal ^{row. The} origin ^{of this} intensity ^{which has}

elastic and quasi-elastic contributions will be analysed

in a further publication. ^{Let us} just mention that this

intensity, the q dependence of which is similar to that of the inelastic scattering factor in the solid phase ^is

due to some relaXing correlated motions. These results are to be compared ^with measurements on a

high resolution X-ray spectrometer ^{on smectic B} liquid crystal ^films [ 14]. ^{D. E.} Moncton and R. Pindak

point ^{out that in addition to} the coherent Bragg scattering owing ^to the three-dimensional long ^range

order in a bulk smectic B, ^{there is} also ^a strong ^diffuse scattered intensity along ^the [20 IJ axis which is likely

due to the same relaxing mechanism. At least we must underline that this quasi-elastic intensity ^remains

almost unchanged ^{at the smectic} Bc ~ ^smectic Ec

transition.

As far as the interpretation ^of peaks ^{in constant}

energy ^scans [13] is concerned it is _very dangerous ^to plot such maxima in (q - 0)) space without analysis

of constant Q ^{scans in the same} region. Recently ^we performed by ^constant Q ^{scans a vast} study ^{of the} concerned-region ⁱⁿ reciprocal space.

The data ^are not yet completely analysed. ^All

that can be said for the ^moment is that only quasi-

elastic scattering ^{has been observed. This means either}

that the transverse phonons ^{in c direction are over-} damped in the smectic Bc phase ^or that there exists another relaxing motion with a similar structure factor which provides ^a huge intensity ^{such that} eventually existing phonons ^{cannot be} separated.

We did not measure systematically in the smectic

phases all the modes detected in the crystalline phase.

Indeed ^some have been already determined on the non-deuterated crystal. On the other hand we have

mostly ^{focused our attention on the} scattering

connected with the rotation of molecules in the smectic Bc phase.

4. SmBc - SmEc transition.

As mentioned above, the local order in the SmBc phase ^{which has}

been identified as a herring-bone ^structure is charac- terized by ^{12 diffuse} spots ^{in the} (a*, b*) plane, ^located

at [ ± ^{2, ±} 1, 0], [ ± 5/2, ± 1/2, 0] ^and [ ± 1, ± 3/2, 0]

points ^on X-ray diagrams [6] (Fig. 12), corresponding

to three orientations of domains with the local herring-

bone order, respectively type I, type ^{II and} type ^III.

The spreading of this diffuse intensity ^{out of the} (a*,

b*) plane ^extends along ^{the c* direction at} any ^tem-

Fig. 11.

a) Representation ^{in the} (Q-hv) space of the scans corresponding ^to the transverse modes propagating along c* in the smectic Etc phase. ^The dotted line is the energy-scan ^at Q

[2, 0, 0.5]. b) Energy.scan ^{at the zone} boundary : ^{A at} Q : [4, 0, -f] in the solid phase. ^{0 at} Q : [2, 0, 0.5] in the smectic Bc phase. The full and broken lines are guides for the eyes.

perature ^{which means the} herring-bone ^{domains are} quite uncorrelated from layer ^to layer.

At high temperature, the twelve spots ^are equivalent but, ^on cooling down, ^the spots with half integer

indices remain the same while the four spots ^with integer indices become narrower and more intense.

Finally ^{at the the} temperature of the transition

SmBc ~ SmEc, they ^become Bragg spots.

We undertook neutron experiments ^to separate the inelastic part ^{from the elastic} part ^{in the} scattering (which ^is impossible ^with X-ray techniques) ^{and to} get ^more quantitative ^{results on the} size of different domains in the layers, ^{i.e. on} the coherence length ^of

each domain. Figure 12 shows the intensity maps in the plane (a*, b*) ^{in the} SmBc phase. The map a) ^is obtained from an X-ray negative photograph (wave- length ^{of 1.54} Á) by oscillating crystal techniques.

The _map b) ^{results from neutron elastic} scattering

measurements (wavelength ^{of 4} Â). ^{On the latter, the}

diffuse spots ^{stand out} against ^the background ^better

than on the former, this is due to the definition in energy window of the scattered neutrons. However, the inelastic broadening of the scattered intensity ^on

the diffuse spots ^{was smaller than the} resolution of the instrument IN2. Therefore the q dependence ^{of the}

energy integrated intensity has been measured with this instrument.

4.1 q-SCANS ON THE DIFFUSE SPOTS.

Such Scans are shown in figure 13. We have performed ^these

scans in order to measure the in-plane spread ^{of the}

diffuse spots ^{and to} determine the size or coherence

length ^{of the} herring-bone domains connected with a

given ^diffuse spot. The mosaic spread ^{of the} sample

was about 20 in the SmEc phase. ^We have calculated the line shape arising ^{from a} broadening ^{of a} b-function Bragg peak ^{for such a} sample. The width of the peak given by ^the triple-axis spectrometer ^{can be} neglected ^{and one can evaluate the minimum size of}

domains directly ^{from the} experimental measurements.

Let us also note that the experimental resolution in the c* direction was about 0.1 times the q ^{width in this}

direction.

These values are reported in table III. The anisotropy

does not change very much with temperature ^and remains weak. At high temperature ^{near the} melting point (113 OC), ^{the sizes of the three} types ^{of domains}

are similar. When decreasing ^the temperature ^down

to the transition smectic Bic --+ ^smectic Ec, ^{the size of}

the domains of type ¹ increases while those of types ^II

Fig. ^12.

Intensity maps in the (a*, b*) plane in the smectic Bc phase : a) X-ray scattering ; b) ^neutron scattering ; c) theoretical calculated intensity from reference [16].

Table III.

Coherence lengths ^{and maximum} intensities of ^the diffuse spots ^{in the smectic} Bc phase.

and III remain nearly ^constant (only ^{the domains 1}

and II have been studied, ^{but from our} X-ray ^{data and}

from a symmetry argument ^{we can state that the} behaviours of the domains of types II and III are

identical). ^This type ^of pretransitional anisotropic

diffuse scattering ^is usually ^described by [15] : ^{The vector} q has the centre of the diffuse spot ^{as the}

Fig. 13.

Q-scans ^near the diffuse spots ^at different temperatures, in the smectic Bc phase. a) Reciprocal (a*, b*) plane ^{with the} pathes

followed in the Q-scans : ^{4 :} [3-03BE, 03BE, 0] ; ^{4 ’ :} [2 03BE,03BE, 0]; ^{4 "} [3/2 ^{+ 2} 03BE, 03BE, 0]. All the full lines are guides ^{for the} eyes. b) Q-scans along ^d.

c) Q-scans along J ’. The broken line is a t5-function Bragg peak measured with the triple ^axis spectrometer. d) Q-scans along ^L1".

origin, Çx’ 03BEy, ^{Çz are} the correlation lengths ^{in three} orthogonal directions. Their variation with tempera-

ture is usually j§ ^oc (T - T 0)-1 ^{where To is the}

phase transition temperature ^{of a} second order phase transitiom. j ^{is a} parameter ^{which we define as}

At q

^{0 the} intensity ^should vary ^as ç2. ^From

table III it can be seen that the intensity at q

⁰ varies like çî ^{as well as like} 03BE22 ^within experimental

error (çz ^{has not been} measured). ^{The data at} only

three temperatures ^{are not} comiplete enough ^{to obtain} by extrapolation ^a To which should come out to be below the actual temperature Tc ^of the first order

phase transformation.

Now to account for the different behaviour between the two kinds of domains (type ^{1 and} types ^{II and} III),

we can say that the tilt angle (300 ^with respect ^{to the} layer normal) ^{acts as an} éxternal field favouring type ¹ domains. Apparently ^{the influence} of this external field is _very weak in the disordered phase ^{as all three} types of domains can be observed. In the ordered smectic Ec phase this field manifests itself in the fact that only type ¹ domains condense out.

The ordering ^{mechanism is very} probably ^stetic -hindrance, ^{as can} be found from structure data in

comparing the size of a particular ^{molecule and the}

distances to its neighbours. Steric hindrance is the basis of ^a model calculation by Descamps ^{and Cou-}

Ion [16]. Within this model a particular ^{molecule can} perform rotational jumps between six equivalent positions. ^The probability ^of occupying ^a given position ^is calculated depending ^{on the} momentary orientations of the neighbouring molecules. To account for steric hindrance, ^the probability ^{that the} phenyl rings ^{of two} neighbouring ^{molecules are on the same} line, ^is put ^{to zero.} This model produces ^{short range} herring-bone ^{order. An} intensity map calculated from this model, ^see figure 12c, corresponds ^to the expe- rimental results in figure 12b. The main difference is

essentially ^{due to the} huge Debye-Waller ^{factor which} strongly reduces the experimental intensities at large angles.

It is a puzzling observation that the measured Bragg intensity ^at (2, 1, 0) ^{in the smectic} Etc phase ^{is of the}

same order of magnitude ^{as the} intensity ^{in the centre}

of the diffuse spot ^{in the sniectic} Bc phase ^{near the}

phase transition. It is astonishing that below a first

order phase transition the increase in intensity ^{is not}

larger, especially ^{since in the smectic} Bc phase only

the intensity ^{in the} centre ^{of the} diffuse spot ^was considered and not the integral ^{of the diffuse} intensity

in the reciprocal space.

4.2 ENERGY ANALYSIS OF THE DIFFUSE SPOTS.

Despite ^{the short} range order due to steric hindrance,

the individual molecules perform rotational diffusion around their long ^axis. High resolution inelastic

neutron scattering experiments [7] ^{found a relaxation}

time _ÏM for this diffusion of about 10- II s. From the neutron data it could not be decided whether ^a particu-

lar molecule performs continuous rotational diffusion

or a jump diffusion between different orientations. On the other hand NQR ^results [4] ^{show that} the molecules

are almost equally distributed over six orientations,

the 0° and 1800 orientations having ^a slightly higher probability ^{than the} four other orientations.

In order to improve ^the dynamical model of the smectic Bc phase ^{it seems} interesting ^to compare the relaxation time of the rotational diffusion of one

molecule _iM to the domain life time _zp. For that purpose, ^we have performed high resolution measure- ments on the diffuse spots with the three-axes spectro-

meter IN12 at the ILL. We used k,

^{1.25 Â-’}

and with the other experimental conditions given above, the energy width of a vanadium scan was

0.0118 THz.

Four energy ^{scans were} performed ^and analysed :

at the two centres of the diffuse spots (2, 1, 0) ^and (5/2, 1/2, 0), ^{at the} valley between them at (2.25,

Fig. 14.

Energy scans on a diffuse spot ^and between diffuse spots, at 114 oC, in the smectic Bc phase : (0) Q

2,1, ^0. The full line represents the best fit with i

1.4 x 10 - 11 ^s. (A) Q

2.25, 0.75,0. The broken line represents the best fit with r

0.9 x 10-11 s.

(The ^{scan at} 5/2, 1/2, ⁰ being nearly ^{the same as the scan at} 2, 1, ^{0 is}

not shown.)

0.75, 0) ^{and at a} point (1.5, ^0.5, 0) ^far away from the diffuse scattering (Fig. 14). ^{For lack of} time, ^these experiments ^were only performed ^{at one} temperature : 114 °C. For the four _scans, an inelastic broadening

was detected. The measured intensities were fitted to a sum of two functions : one Gaussian (width

0.011 8 THz) accounting for the elastic incoherent

scattering and for all quasi-elastic scattering ^which might ^be present with energy widths smaller than the resolution and one Lorentzian to describe the visible

quasi-elastic scattering.

The Lorentzian ^was folded with a Gaussian of the resolution width. Both functions were modified cor-

responding ^{to the} varying transmission of the instru- ment [17]. ^The fit included as well two parameters ^for the background (one ^{constant and one for a linear} slope). ^{The data were distinct} enough ^{so that all} parameters ^{could be varied} simultaneously ^{and the}

width of the Gaussian when left free came out very close to the vanadium width.

We observe that the intensity ^described by ^the

Gaussian is almost the same in all ^cases. Thus, ^we

conclude that the gaussian describes that part ^{of the} incoherent scattering ^which appears elastic within the _energy resolution. We further conclude that the diffuse intensity ^is entirely ^defined by ^the intensity

contained in the Lorentzian. In the centres of the diffuse spots ^the heights ^of the Lorentzian are compa- rable to the heights of the Gaussian. The width of the Lorentzian curves at the centre of each spot ^{is the}

same within experimental ^{error and is 2 r = 0.022 THz.}

This corresponds ^{to :}

In the valley the Lorentzian has half the height ^of

that at the centre and corresponds ^to TD

At (1.5, 0.5, 0) ^the intensity of the Lorentzian is about 15 % ^{of the one at the centre} and the width corres-

ponds again to ^{0.9 x 10-11 ± 0.2 x lO-11 s.}

Hervet et al. [2] have measured a time 03C41 of the order of 10-11 s which is a characteristic decay ^{time for an}

individual molecule which is independent ^{of the}

existence of positional correlations between neigh- bouring molécules ; ^the corresponding residence time iM for a sixfold jump model is then _rm

L 1/2 [18].

From coherent inelastic scattering ^{at well localized} points ^{in the} reciprocal space, ^we observe a charac- teristic time _03C4D of the orientational correlations

between neighbouring molecules : a « domain » life time. The significant result is that _{03C4D ~} 1.4 x 10-11 s

is larger ^than iM. This should always ^{be the case but}

the fact that the two times are not very different from

each other leaves room for speculative ideas about the

mechanism of the molecular motions. We surmise that

the motion of an individual molecule is not completely

independent from the motions of its neighbours.