HAL Id: jpa-00210502
https://hal.archives-ouvertes.fr/jpa-00210502
Submitted on 1 Jan 1987
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
A molecular dynamics simulation for an ODIC phase. - III. Analysis of the correlation functions
Ph. Buchet, R.M. Pick
To cite this version:
Ph. Buchet, R.M. Pick. A molecular dynamics simulation for an ODIC phase. - III. Analysis of the cor- relation functions. Journal de Physique, 1987, 48 (5), pp.821-835. �10.1051/jphys:01987004805082100�.
�jpa-00210502�
A molecular dynamics simulation for an ODIC phase.
III. Analysis of the correlation functions
Ph. Buchet and R. M. Pick
Département de Recherches Physiques (UA 71), Université Pierre et Marie Curie, 4, place Jussieu,
75005 Paris, France
(Requ le 6 octobre 1986, accepté le 13 janvier 1987
Résumé. - Une simulation numérique d’un modèle 2-d d’une phase plastique a été réalisée à trois
températures. Les diverses fonctions de corrélations relatives aux déplacements atomiques et aux orientations moléculaires calculées sont bien décrites, à toute température, par un modèle phénoménologique de couplage
linéaire déplacement-orientation. Les coefficients correspondants sont correctement prévus, à haute tempéra- ture, par la théorie microscopique de Michel et Naudts, mais d’importants désaccords apparaissent à plus basse température. Nous discutons l’origine de ce désaccord, ainsi que la signification physique des temps de relaxation calculés.
Abstract.
2014A 2-d numerical simulation of a rotator phase has been performed at three different temperatures. The various displacement and orientation correlation functions are correctly described, at all temperatures, by a phenomenological model of linear displacement-orientation coupling. The model
coefficients are correctly predicted at high temperature by the Michel-Naudts theory, but large discrepancies
appear at lower temperatures. The origin of these discrepancies as well as the meaning of the computed
relaxation times are finally discussed.
Classification
Physics Abstracts
63.50
-64.60C
1. Introduction.
Plastic crystals, also called ODIC, or rotator phases,
are thermodynamically stable states of matter, inter- mediate between the solid and the liquid states, which can exist in molecular, or iono-molecular systems. In such a phase, there exists a long range translational order, but an orientational disorder,
each molecule possessing several equivalent possible
orientations. The long-range translational order al- lows one to define a mean lattice for the position of
the centre-of-mass of each molecule, or atom, of the
crystal, each site of the lattice being characterized by
its coordinates and its symmetry. This symmetry
imposes, in turn, the number of equivalent orienta-
tional potential wells available to a given molecule.
The basic problem of the rotator phases is the understanding of the orientational dynamics of the
molecules : how long does a molecule remain in the
vicinity of the bottom of a potential well (residence
time T) ? How long does it take it to jump from one potential well to another ? Through which proces- ses ? What is the influence of the centre-of-mass
positions and motions on this dynamics ?
Various experimental techniques have been used
to try to elucidate those points : dielectric, NMR, or inelastic incoherent neutron scattering measure-
ments, IR and Raman line shape analyses, have
shown that the residence time may be as long as 10- 9 to 10- 8 s, but that, close to the melting point, it
decreases and is typically 10-12 to 10-13 s. Diffraction
experiments have revealed the existence of anomal-
ously large Debye-Waller factors, nearly tempera-
ture independent, making clear the existence of
important centre-of-mass displacements. Also, in
some particular cases, such as NaCN or KCN, the softening of an elastic constant in the vicinity of the ordered-plastic phase transition has dramatically
shown the existence of a collective coupling between
these centre-of-mass motions and the molecular orientations. A good review of various of these aspects can be found in reference [1].
Another approach to this problem has been, more recently, the use of large computers for undertaking
numerical simulations of the motions of the molecules once the interaction potential between the
molecules, or rather between the various atoms of these molecules has been defined. Since the pioneer
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01987004805082100
work of S. Pawley [2] on SF6, and of M. Klein et al. [3] on NaCN, several such calculations on
different plastic crystals have been performed and analysed, looking at the dynamical properties of
either individual molecules, or of collective variables
describing some specific properties of a given plastic crystal.
This paper is the third of a series devoted to the
analysis of numerical simulations of a model de-
signed to be a 2-d analog of the rotator phase of an
alkali cyanide. With the idea of carefully studying
the reorientation dynamics, this model was chosen to be as simple as possible. Each unit cell contains
two ions per cell, a positive Na and a negative CN.
The former interacts via Coulomb forces, with a symmetrical dumb-bell having its negative charge
located at its centre-of-mass. An additional repulsive Born-Mayer potential acts between each end of the
dumb-bell and the nearest neighbouring Na ions,
and ensures the dynamical stability of the rotator phase.
In the first paper of this series [4] (hereafter
referred to as I) the static properties of the model
were studied. It was shown that the system had a low temperature rhombic phase with one formula unit
per cell, the CN dumb-bell being located along the long diagonal of the cell. A molecular dynamics
simulation on a 10 x 10 lattice showed that a phase
transition occurred around 280 K towards a mean
square lattice rotator phase. In this phase, the
maximum probability of the dumb-bell orientation is obtained for () = 4 (modulo ; ), () being the angle
between the [1, 0] direction of the lattice and the dumb-bell. This means that the dumb-bell tends to be at the bottom of one of two equivalent potential
wells. It also means that, due to the repulsive Bom- Mayer forces, the two Na atoms aligned with the
dumb-bell are pushed outwards, while the two Na
atoms perpendicular to the bumd-bell can, and do
move inwards : in other words, most unit cells
closely resemble, at a given time, the low tempera-
ture rhombic cell.
The single particle rotational properties of the
bumd-bell were also studied in I. Taking sin 2 0 as a dynamical variable (a variable which takes the value
+ 1 at the bottom of one potential well and 2013 1 at the bottom of the other well) the one-particle
orientational dynamics was studied through the
correlation function sin 2 0 (0) sin 2 (J (t )) , at
320 K. It was found that its Fourier transform had
essentially a Lorentzian line shape at low frequency
but had a slower decrease for to > 30 cm- l. This was
interpreted as a reorientation process governed by a
residence time T, but with a finite transit time from
one potential well to the other. This residence (or relaxation) time was found to be approximately
1 pcs.
The librational properties of the dumb-bells were
also studied, at this temperature, in a local approxi- mation, through the self correlation function of the
dynamical variable cos 2 0 (t). The librations were
found to exhibit a broad spectrum centred at
co = 120 cm- l, with a HWHM of approximately
70 cm- 1. This width was shown to have, as sole origin, the static deformation of the rhombic cell,
which varies from cell to cell.
In the second paper of this series [5], (referred to
as II) the collective dynamics of the rotator phase
was investigated at the same temperature through
the study of the space-time Fourier transforms of
some correlation functions. The chosen variables
were the sodium and dumb-bell centre-of-mass dis-
placements, uNa(t ) and UCN(t) as well as the angular
variable sin 2 0 (t). First, it was shown that the phonon frequencies could be correctly predicted by
mean force constants, taking into account the
Coulomb forces and the mean value (see Sect. 2) of
the Bom Mayer potential. It was also shown that the phonon linewidths (HWHM), which may amount to 10 % to 15 % of the phonon frequencies, had, in
most cases, the same origin as the librational line widths, i.e. the local, static, disorder of the crystal.
This picture turned out to be mostly incorrect for the T.A. phonons propagating along the [1, 0] direction ;
in this particular direction, these phonons are strongly coupled to the sin 2 0 (t ) variable (see
Sect. 2 and 4). This leads to an apparent shift of the phonon frequencies, and an important apparent
increase of their linewidth.
The specific purpose of the third paper of this series is to thoroughly analyse the coupling between
the translational and the orientational variables
briefly discussed in II. This study will be performed by
a) extending the previous M.D. simulations to
higher temperature (520 K and 820 K), and b) analysing the corresponding response functions in the framework of phenomenological [6] and microscopic [7, 8] theories which have been proposed
to describe the molecular orientational dynamics of
these rotator phases. The present analysis, which has
been briefly summarized in [9], is the first one where
the various predictions of these theories are com- pared, at different temperatures, with the results of numerical simulations. This comparison will show that, while the phenomenological model of [6]
always allows for a correct description of the various correlation functions, the prediction of their line shape and relative intensity by the presently available microscopic theories is only good, for the model
under study, at high temperatures. This result has
led us to look for possible low-temperature improve-
ments of the theory. As we shall show at the end of
this paper, we obtained only moderate success in
that respect, and more efforts are clearly needed in
this direction.
In order to put into perspective the various results obtained here, this paper is organized as follows.
Section 2 presents a summary of the main tech-
niques and results of the microscopic theory of
Michel and Naudts [7] under a form appropriate to
the 2-d problem under study. It also points out its relationship with the phenomenological model of
Yamada et al. [6] as already noticed by Michel and
Courtens [10]. Section 3 presents a qualitative de- scription of our M.D. simulation results at 520 K and 820 K. They are numerically analysed with those
previously obtained at 320 K, in the framework of [6], in section 4. The comparison between the par- ameters obtained in that section and their predicted values, following [7], is performed at the beginning
of section 5. We, then, analyse the various approxi-
mations inherent to the microscopic theory we made
use of and show that several simple generalizations
of [7], including a suggestion of Michel and Rowe [8]
do not seriously improve, in our case, the agreement with the M.D. simulations. We finally discuss, in section 6, the problem of the relaxation time, and
show the internal consistency of our high tempera-
ture results, as well as the relationship between the present 320 K results and those published earlier [4, 5].
2. Summary of the linear coupling theory.
A microscopic theory of the dynamical behaviour of rotator phases has been proposed by K. Michel and
J. Naudts [7] in 1976 and later extended by Michel
and coworkers, the latest extension having been proposed in [8]. We shall, in this section, summarize
the main ideas contained in [7] and subsequent
papers, as well as the approximations related to the
linearized formulation we are going to use, in a
language appropriate to the specific case of our 2-d
model.
Let us define the position of the ions of our crystal (sodium and dumb-bell centres-of-mass) by R’
where L labels the lattice cell, and i the nature of the
ion (i = 1 for Na and i = 2 for CN). In view of the interactions present in our model, the potential
energy of the crystal may be written as
where OL is the angle between the dumb-bell of cell
L with the [1, 0] direction. (Note that (1) does not depend, simultaneously on two angular variables,
because the Bom Mayer forces, which are the only angular dependent forces, are sufficiently short
, range not to lead to interactions between neighbour- ing dumb-bells.)
Equation (1) is expanded in two different ways.
a) First R£ is expressed as RL + U£ where R°‘ is the position of the (i, L ) atom in the perfect square lattice
The expansion of (1) in successive powers of the
UL yields
where V 1, V2, - - - are the first, second,
...derivatives of the pair potential with respect to the relative
displacement UL - UL,, computed at the ionic equilibrium positions ; in this expansion, only the
Born Mayer contribution gives rise to angular depen-
dent contributions.
b) As for the 0 L dependence of V 0, V 1, and V2, these functions are projected on a basis formed by some complete orthonormal set of functions of
OL- It has been shown [11] that it was convenient to
choose this set in such a way that each function transforms, under the symmetry operations of both
the molecule point group and of the site of the dumb-bell centre-of-mass point group, as some ir- reducible representation of pach point group. The set of functions sin p’ 0 and cos p’ 0 agrees with such a definition. As each term of (3) is invariant under all the operations of the molecular point
group, the expansion of VO, Vl, V2,... contains, in fact, only terms with p’ even (p’ = 2 p ).
Both the UL and the whole set of angular variables
are dynamical variables of our problem, and one can
use thermal Green function techniques to obtain space-time correlation functions between these vari- ables. Combining this technique with a Mori-Zwan- zig [12] continuous fraction expansion, taking into
account, as the mostly relevant angular variables, those related to p = 1, and truncating the continuous
fractions at their lowest significant term, simple expressions for those correlation functions can be deduced. They turn out to be identical to those
which can be obtained from a free energy expression
built up following the following rules.
a) Writing the set of angular variables as C ’ (0
with
-the first term of the free energy expansion is
with
In practice, (5) is limited, as already mentioned, to p = p’ =1 which leads to a 5 m,m’ interaction constant; (5) is thus quadratic in sin 2 0 L and
cos 2 e L, separately.
b) The second term of the free energy is derived from the second term of (3) by expanding it on the
C2 p (9 ) basis, and again truncating the expansion after p = 1. This term linearly couples sin 2 OL (or
cos 2 0 L) with the displacements f U", I and, by
construction, the coupling coefficients do not depend
on the temperature.
c) The third term of the free energy is, similarly,
derived from the third term of (3) by expanding it on
the C m (0 ) basis, but retaining only the p = 0 term.
This gives rise to terms quadratic in (U£) with
coupling coefficients again independent of the tem-
perature.
The approximate free energy reads :
d) All the coefficients of (7) depending only on
RL - Ro,, (7) is space Fourier transformed and its
phonon part (related to the last term of (7)) diagonalized into normal modes. (7) then transforms into
where Qq is a normal mode coordinate, and
(8) appears as a harmonic free energy expressed with
the help of the set of independent variables C2’(O (q» and 6q.
e) The dynamical matrix related to this set of variable is obtained from the equations of motion :
These two equations make clear that, while the phonons obey a normal Newton law, the angular
variables are governed by diffusive equations charac-
terized by relaxation times r’ ; for each wave
vector, the preceding set of linear equations is
identical to a 6 x 6 dynamical matrix G-1(q, w ).
f) G (q, w ) being the inverse of this matrix, the various dynamical correlation functions between the 6 variables are obtained from the usual expression
Im ( kT co G(q, w )) . ) In particular, p if one neglects
the indirect coupling between sin 2 0 (q, Cù) and
cos 2 0 (q, w ), one obtains
The neglect of this indirect coupling is an approxi-
mation which is particularly valid for sin 2 0 (q, w )
because it turns out that
Consequently, except in subsection 5.3,
cos 2 0 (q, w ) will always be neglected so that, in (11) and (12) we shall restrict ourselves to the
m = 2 index.
Remark. - Some other coupling mechanisms
have also been .neglected in writing down (11) and (12). In principle, there is no reason for admitting
that the relaxation dynamics of C 2 ’(0 (q)) should not couple, at least this variable with -C"(O(q))
(m’ =A m ) so that one should write
In the present work we have assumed, for simplicity,
and in line with the neglect of indirect couplings just
stated above, a diagonal form for r I" (cf. (10b)).
This approximation turned out to be sufficient, as we
shall see in section 4, for a coherent description of
our numerical results.
Equations (11) and (12) are the correlation func-
tions, as obtained, in the lowest order approxi- mation, by K. Michel et al. in [7] ; the only unknown quantities in the approach presented here are the
relaxation times T2 q, which are simply a first order
approximation for the continuous fractions appear-
ing in the Zwanzig Mori treatment [12]. In fact, as
shown in [10], (7) or (11) and (12) are not completely
new results in the present context. Such expressions
had already been obtained by Yamada et al. [6] for
the case of disordered phases, in which a molecule
can have only two possible orientations (or positions). Making the approximation of instant jumps between one orientation and the other, Yama- , da et al. used a pseudospin variable, U L, to describe
the occupation probability of the two orientation
wells, labelled by + and -, writing
0 ’L is a discrete variable, which replaces sin 2 8 L in
the context of this section, and, on a phenomenologi-
cal basis, Yamada et al. proposed an expression for
the free energy identical to (8) except for the allowance of an interaction energy between spins
located on different sites, i.e. for a q dependence of the g coupling constant. They thus obtained the correlation functions (11) and (12).
The only important point to be mentioned is that,
in [6], equation (10b) basically describes the jump of
an individual pseudospin from one potential well to
the other, which means that the corresponding
relaxation (or residence) time has no wave vector dependence. Furthermore, the hypothesis of instant jumps implies the neglect of the reorientation time for a molecule, i.e. implies the existence of a long
residence time in a potential well. The typical
timescale is given by the libration of an individual molecule in its potential well. One thus expects that
equations such as (11) or (12) would be valid provided that :
where V L is a typical libration frequency. In our model, such a typical value is 4 THz, which gives
As indicated in section 1, the purpose of this paper is to analyse the validity of (11) and (12) for
the description of the correlation functions obtained in our M. D. simulations. Before proceding to this
analysis, we have to make precise which correlation functions were actually computed, and to roughly
describe their general features. This is the subject of
section 3.
3. Numerical computations and qualitative analysis.
The 2-d model of I, the properties of which have
been summarized in section 1, has been studied at 520 K and 820 K using the same numerical values for
the forces, masses and lattice parameters as at 320 K, on a 10 x 10 lattice cell sample. This repre- sents a constant volume computation, which makes
easier the comparison with the theory, and each
simulation was run for 2 x 104 time steps of 10- 14 s.
We first verified that, at 820 K, the crystal had not
melt. The self correlation function related to sin 2 () was then computed, and its Fourier transform is given, for the 820 K run, in figure 1. One sees that, contrary to what was obtained at 320 K (cf. I),
the low frequency part of this spectrum has not a Lorentzian lineshape, and the origin of this effect
will be discussed in section 6.3. The same correlation function computed at 520 K led to a curve inter-
mediate between the 320 K and the 820 K cases [13].
Fig. 1.
-Self correlation function
sin 2 ø (CL) ) * sin 2 ø (CL) ) > at T=820K.
The space and time Fourier transforms of the correlation functions for the sin 2 OL variable, the
transverse and the longitudinal displacements were
later computed for q along [1, 0] and q along [1,1]
for both temperatures ; typical spectra for both directions are given in figures 2 and 3. The phonon frequencies and linewidths deduced, by inspection,
from the maxima and from the HWHM of the
phonon response functions were essentially the same
as at 320 K. Nevertheless, important changes were
found for the TA, q along [1, 0] branch. The
Fig. 2.
-Dynamical correlation functions for sin 2 0 (q) (top) and for T.A. phonon (bottom) for q along [1, 0] at
T = 820 K (q is in a * unit).
apparent phonon frequencies were closer to their
mean field values (see Sect. 2 and [5]) and their
HWHM smaller. Furthermore, and mostly notice-
able for small q, the shape of the response functions
dramatically changed, at low frequency, between
320 K and 820 K, and that was also true for the sin 2 B (q) variable, as is shown in figure 4 : at 320 K,
the T.A. response function is dominated by the coupling to sin 2 0 (q) ; the central peak of the latter
is so important in the phonon response function that the T.A. phonon is barely visible at q = a . Con-
10
versely, at 820 K, the T.A. phonon dominates both response functions, and the situation is, again,
intermediate at 520 K.
The existence of this coupling between the centre- of-mass motion and sin 2 0(q) was predicted in II through the study of d2 q. It was shown that sin 2 0 (q) should couple to the transverse phonons
for q along [1, 0] (and mostly to the T.A. branch,
due to the shape of the dispersion curve) and to the longitudinal phonons for q along [1,1]. We need to
Fig. 3.
-Dynamical correlation functions for sin 2 0 (q) (top) and for the L.A. phonon (bottom) for q along [1, 1]
at T = 820 K (q is in*a* unit).
Fig. 4.
-Dynamical correlation functions for sin 2 0 (q )
(left) and for the T.A. phonon (right) for q = 10 a* 10 at
T = 320 K, 520 K and 820 K.
substantiate these qualitative findings and that previ-
ous theoretical analysis through a numerical analysis
of these correlation functions, which will be done in the next section.
4. Numerical analysis of the response functions.
4.1. INTRODUCTION AND METHODOLOGY. - The
present work is not the first one in which the results of a M.D. simulation of a rotator phase is analysed
with the help of the linear response theory. This
method has already been used by Lynden Bell
et al. [14] who made use, for such a study, of the
results of a 3-d simulation of the plastic phase of NaCN, performed in the vicinity of the transition
temperature to the ordered phase. They took into
account the seven wavevectors available in their simulation. Assuming the validity of the linear
coupling theory, they showed that they could deduce
the different parameters (phonon frequencies, trans-
lation-rotation coupling constants and rotation-rota- tion coupling constants) pertinent to their orienta- tional variables from the sole consideration of some
static correlation functions. With the help of these coefficients, they looked for the relaxation times which could best fit their time correlation functions,
and produced quite reasonable fits. They finally compared the translation-rotation coupling constants
obtained by this method with the values predicted by [7] and obtained a good agreement between the two sets of values.
Because (11) and (12) depend on frequency and
not on time, and also because we wish to test on the
same footing all the parameters entering these expressions, we have taken, in this study, a different approach. We have fitted, at the same time, the displacement-displacement and the orientation- orientation correlation functions (11) and (12), look- ing all at once for the best values of g2 2 d22 qg i w q and T 2 2 q (in short g (q ), dq and T (q )). Also, as
mentioned in the introduction, the phonons have a
« natural » linewidth related to the angular depen-
dence of the force constants. This linewidth can be estimated on theoretical grounds because one can
also compute the phonons of the low temperature rhombic phase (cf. I). One can then relate this
linewidth to the difference in phonon frequencies,
between two phonons propagating with wave vectors
ql and q2 which become equivalent (and equal to q)
,
in the high temperature phase (cf. II). We have
taken this linewidth into account by replacing in (11)
and (12) (c q by
where 7"q is the HWHM computed with the above recipe.
Practically speaking, the fit has been performed simultaneously on the two (or three) response func-
tions corresponding to the displacement(s)-dis- lacement(s) and sin 2 0 (q)* sin 2 0 (q) correlation functions, and, in order to increase the numerical accuracy, we have summed up the response functions
corresponding to all the wave vectors which belong
to the star of q. Once the best fits for all q parallel to
the same direction were obtained, we checked if
some parameters had a constant value within the
error bar. Whenever that was the case, we fixed them at their mean value, recomputed, for each q, the other parameters and verified that this procedure
did not induce a significant change in each reliability
factor.
4.2 NUMERICAL RESULTS.
-The analysis has been performed for the 820 K, 520 K, and 320 K runs for
q along [1, 0], and for the first two runs only for q
along [1,1]. The first important result is that (11)
and (12) can reproduce our correlation functions at each temperature, in spite of the considerable evolu- tion of their shape with temperature (cf. Fig. 4). A typical example of the quality of the fit is given in figure 5. The overall agreement is always good.
Nevertheless, on both curves, there exist additional oscillations, which are correlated between the two response functions. Such oscillations appear in all
Fig. 5.
-Comparison between the M.D. correlation functions (dashed line) and the numerical fit (solid line)
for q = 2 a , at T = 820 K, for sin 2 0 (q) (top) and for
10
the T.A. phonon (bottom).
our spectra, but are not correlated between different wavevectors, even if they belong to the same star.
They are thus neither a numerical noise, nor intrinsic properties of our system and they clearly deserve
further study.
Our numerical values are summarized in figure 6
for g (q) and in figure 7 for d (q) (1), for the five
Fig. 6.
-i (q) kT as a function of q, for q along [1, 0] (left)
and along [1,1] (right) at 820 K (x ), 520 K (D) and
320 K (0). (The curves are guides for the eye.)
Fig. 7. - d (q) (with To = 820 K) as a function of q, for
kTo ’
’ .q along [1, 0] (left) and along [1,1] (right) at 820 K (x ),
’
520 K ( ) and 320 K (0 ). The dashed curves (- - -)
are predictions of equations (14a) and b. (The other
curves are guides for the eye.)
(1) Due to the short range character of the Bom Mayer potential, there exist only two independent parameters in the coupling between sin 2 0 L and the displacements, and only one combination enters into the coupling with a longitudinal phonon propagating along a * + b *. Further-
more, as we have taken identical masses for the sodium and the dumb-bell, for q along [1,1], the dynamical matrix
is automatically diagonalized. We can thus express, for each q along [1,1] and for j being a longitudinal phonon,
d’, in term of one combination of the translation-rotation
coupling coefficients, named d(q), and of known par- ameters. It is this d(q) which is represented, in figure 7,
for q along [1,1] in the unit d (q ) : The situation is even
,, / k-To
simpler for q along [1, 0] where sin 2 (J (q) couples, in practice, only to the T.A. branch. This explains why the index j is omitted in the rest of the paper.
wavevectors and the two directions analysed. They
are displayed p in unit g (q) and d (q) (with
kT Jf (
To = 820 K), which appear as natural units in our
problem (see (11)). The smoothness of these curves
confirms that the spurious oscillations discussed above did not alter the quality of the fitting pro- cedure. The corresponding values for T (q) are given
in table I which shows that, except for T = 820 K and q along [1,1] T (q) depends only on the tempera-
ture.
Table I.
-Relaxation time i(q) (in pcs). First four
lines : values obtained in the fitting procedure. Last
line : predicted values following equation (30).
5. Discussion of the coupling parameters
5.1 COMPARISON WITH THEORY. - As the com-
puted correlation functions are correctly described by equations (11) and (12), at all temperatures, one
can compare the numerical values obtained in our
fits with those predicted by the Michel and Naudts
theory [7], which gives a microscopic basis to
Yamada’s ideas.
From a qualitative point of view, there exist two obvious differences between our numerical results and the theoretical predictipns : firstly, g (q) has a q
dependence, at least at low temperatures (see Fig. 6) while, following (6), no such dependence should
exist. Secondly, the numerical analysis of figure 7
shows that d (q) is roughly proportional to JkT,
instead of being temperature independent, as pre- dicted in section 2.
The quantitative analysis of our parameters can be performed with the help of the results obtained
in (II). From the appendix of that paper, we obtain
that
Table II. - Coefficient (X4(T) of Pe ff(O) (Eq. (19))
and coefficient a4 of the single particle potential. The
values of ai (in deg. K) are obtained using respec-
tively equation (18) (second line), 24 (third line) and 25b (last line).
Also, writing for the single particle potential (first
term of (3))
for large enough temperature, one gets, with the help of (6)
Using equations (14)-(18), we have computed
go and d (q ) and these results are given in table III for
f3.go and as dashed lines in figure 7. It is apparent that, at this temperature, there is a rather good agreement between our numerical results and the
microscopic theory for both quantities for the two
Table 1111.
-Orientation-orientation coupling cons-
tant. First line : values obtained by the fitting procedure for q - 0. Second, third and fourth lines : values
obtained with equation (17) using the corresponding
lines of table 11 for a’. Last line : self consistency
Bs correction, to be added to the three preceding lines.
directions studied. When the temperature decreases,
this agreement worsens in three different ways.
-
Following (17), (3go should increase with de-
creasing temperature, because a4 is negative, while it actually decreases ;
-
A dispersion of g (q) appears clearly at low temperature ;
-
d(q) decreases with temperature, instead of
being temperature independent.
Thus, if the microscopic linear theory always predicts a correct order of magnitude for all the
coefficients of the various response functions, and is
in quantitative agreement with the high temperature simulation, it is not able, in the form discussed in section 2, to describe correctly, for our M.D. compu-
tation, the thermal evolution of these quantities. We shall, in the next three paragraphs, both discuss
some implications of this result and look for possible improvements of the theory. The discussion of the relaxation time, T (q) also obtained in our fit, will be
the subject of section 6.
5.2 ANHARMONICITY VERSUS LOCALIZATION.
-The linear microscopic theory, as summarized in section 2, contains three different approximations.
- The use of the potential V °( 6 ) as a local
potential from which effective interaction constants
g’2p!{’p are derived.
- The restriction of the orientational variables to
p=1.
- The neglect of the anharmonic contributions in the third term of (3), and later in (7).
We shall postpone the influence of the first
approximation to paragraph 5.3 and first focus our
discussion on the last two.
The anharmonic contributions contained in the last term of (3) contain both displacement and
orientational variables. There may appear, in each term, an arbitrary number of displacement variables,
bust only one orientational variable, as (1) depends
on OL only. The influence of the anharmonic dis-
placement terms is a well known problem [15] of
lattice dynamics ; it leads to frequency shifts, and
line width contributions which are, at the lowest
perturbation order, proportional to kT. On the basis of the diagrammatic treatment of this effect, one
could expect that the inclusion of these anharmonic terms would lead to similar effects on the trans-
lation-orientation coupling constants d2’ q and to no
direct effects on g2P 2P. The role of the anharmonic contributions related to the influence of the orien- tational variables is more complex to analyse, as
these variables are also subjected to sum rules of the
form
and has not yet been fully studied. It should manifest itself, first of all, on the phonon frequencies, as these
terms directly enter into the expression of the force constants. Nevertheless, these frequencies have
been found to be hardly temperature sensitive.
Consequently, we expect the role of the anharmonic contributions to be mostly related to the displace-
ment anharmonicities, and thus to increase with temperature : the agreement between our compu- tation and the theory should thus worsen with
increasing temperature.
The neglect of the C!J:p ( (J) terms for p > 1 has
quite the opposit effect. The effective probability P eff( (J ) of finding a molecule with orientation 0 may be written as (cf. I)
a 4(T) can be obtained from our simulation at each temperature, and its value is given in table II. The
decrease of a4(T) with increasing temperature means that, as expected, the molecules are less localized at the bottom of their potential well at high
temperature, and thus spend less time in a given
orientation. When studying the dynamics of a molecule, one should then include more and more
orientational variables as the temperature decreases,
because of the greater localization of the molecules at low temperature. The neglect of C ’ (0 ) functions
for p > 1 is thus a high temperature approximation.
Our computations show that this second approxi-
mation is a more drastic one than the neglect of the
anharmonic contributions, a result that we had not
anticipated.
There is no easy remedy to this situation. One could think of simply adding, within the framework of (8), other Czp(Ø) variables, and the lowest func- tion which couples to sin 2 0 (t ) is sin 6 0 (t ). Never- theless, in equation (10b), one considers each func- tion C 2 p ( 6 (q, t ) ) as an independent-dynamical vari- able, which is linearly coupled to all the others by
the I.h.s. of this equation. This is incorrect, as, for
each CN, 0 (t) is the only dynamical variable.
We have tried to overcome this difficulty by considering that the variable sin 6 0 (q, t ) has a
faster dynamic than sin 2 ø (q, t) and follows the latter adiabaticaly. As shown in appendix A, once
sin 6 0 (q, w ) has been eliminated using this adiaba- tic approximation, the dynamical matrix coupling
the phonons j; q to sin 2 0 (q, w ) reads, in complete analogy with section 2
where, in the high temperature approximation
2 2 = go is still given by (17), but where
(wtfff has an expression given by equation (A.9),
and d6 q is, in agreement with section 2, derived from
the coefficient in sin 6 0 of the second term of (3).
Once can verify that d2 q and d6 q have exactly the
same analytical form. They differ only by a numerical
coefficient (D6 instead of D2), which is the same for
the two directions q along [1, 0] and q along [1,1].
Using our Bom Mayer potential we have obtained
D6/ D2 = - 6.4 x 10- 4. As the product of d 2J a4 is
positive, (21) predicts a decrease of the effective coupling parameter with decreasing temperature.
But at 320 K, Q a4 is only equal to - 0.23, so that the
correction is still too small by three orders of
magnitude to explain the bahaviour of our data. The
inclusion of C!/:p ( Ø) for p:::. 1 is thus a problem
which cannot be easily dealt with in the framework
of a mean field linear theory.
We have also looked for other possible explana-
tions for the thermal behaviour of d(q) such as the
influence of the Debye Waller factor, and of the orientational probability, Peff (0). None of them [13]
was able to explain our results.
5.3 INFLUENCE OF THE LOCAL POTENTIAL.
-The method through which g2 2 = go is derived in the
microscopic theory has been summarized in sec-
tion 2. It leads, in the high temperature limit, to (17). As a4 is negative, (17) predicts an increase of the theoretical value (3 go with decreasing tempera- ture, this ratio being always larger than 2.
This is opposite to our numerical results. 13 gexp (0
is always smaller that 2 and decreases with decreas-
ing temperature. This result is, in fact, coherent with table II, because a4(T), which characterized
Peff (0 ), is always negative. It means that one has always a smaller probability of finding a molecule
with 6 = 0 modulo ’T 2 4 than with 0 - ’T
modulo ’T 2 ). Consequently, if one writes :
the effective single particle potential Veff(O) has a
maximum for 0 = 0 and a minimum for 0 = 4 4 ’
contrary to V 0(0 ) : the strong orientation translation
coupling renormalizes so violently the single particle potential, that the orientational probability is rever-
sed from what can be predicted from yO( (J). Writing
and identifying (19) with (22) and (23) in a high
temperature expansion yields
One can expect to take into account, in an ad hoc
manner, the renormalization of the single particle potential by using a4(T) instead of a4 in (17). This gives a geff which is given, in the third line of table III under the form f3 geff, and is very close to f3 gexp (0) (first line) at 820 K and 520 K, and predicts, even at
320 K, a value in rather good agreement with the
experiment.
The effect of this translational-rotational coupling
has also been considered, on a theoretical basis, by
K. Michel and M. Rowe [8]. Starting from (8), one
can, in a static approximation, eliminate the phonons
and define an effective interaction energy between the orientational variables under the form
(Note that, for the sake of completeness, we have
used here also the cos 2 0 (q ) variables. ) The second
term of (24) represents, in real space, an interaction between the orientational variables which depends
on their relative distances RL - RL,, and contain,
inter alia, a RL = Ro, term. Considering this term as
an additional single particle potential, this adds to
V ° ( 9 ) a second term :
where BS and BC are positive quantities.
Michel and Rowe suggested that 9’2 ’2 should be computed with the help of (6b) using
V 0 ( (J ) + V 1 ( (J ) as the relevant single particle poten-
’
tial. This leads to the replacement, in (17), of
BS - BC
a 4 by a 4 + BS 2 Bc . . We have computed BS and
BC from the phonon frequencies and the expression
of the coupling coefficients and obtained
This quantity is positive, contrary to a4, and yields
the values J3 gKM which are reported in the fourth
line of table III and have also the correct tempera-
ture time dependence, though the agreement with
13 ge,,p (0) is slightly worse than in the case of J3 9 eft.
5.4 ORIENTATIONAL INTERACTION ENERGY AND SELF CONSISTENCY. - The third and fourth lines of table III seem to indicate that (17) gives results
consistent with the M.D. simulation once the single
orientational potential has been properly modified along the lines suggested in subsections 5.2 or 5.3. In
fact, in [8], Michel, Parlinski and Rowe suggested to incorporate into g2 2 a second correction which amounts to transform, in the case of the orientational variable sin 2 8, (17) into
’
where BS has been defined through (25a) and a4 stands for the coefficient of cos 4 6 in the single particle potential. In the original paper, [8], a4 was
DS _ DC
specifically equal to a4 + 2 and the correction
(26) was introduced to compensate for the use of the additional potential (25b). In fact, as shown in appendix B, this correction is necessary (neglecting again, for the sake of simplicity the orientational variable cos 2 6) to insure a consistent value of
sin2 2 6) written under the form
The correction appearing in (26) is, again, a high temperature approximation which supposes that, as
...
