Orientational disorder in plastic molecular crystals - III. — Infrared and Raman spectroscopy of internal modes

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Orientational disorder in plastic molecular crystals - III.

- Infrared and Raman spectroscopy of internal modes

M. Yvinec, R.M. Pick

To cite this version:

M. Yvinec, R.M. Pick. Orientational disorder in plastic molecular crystals - III. - Infrared and Raman spectroscopy of internal modes. Journal de Physique, 1983, 44 (2), pp.169-183.

�10.1051/jphys:01983004402016900�. �jpa-00209583�

Orientational disorder in plastic ^molecular crystals

III.

Infrared and Raman spectroscopy ^{of internal modes}

M. Yvinec and R. M. Pick

Département ^de Recherches Physiques (*), Université P. et M. Curie, 4, place Jussieu, 75230 Paris Cedex 05, ^France

(Reçu ^le 23 juin 1982, accepté le 14 octobre 1982)

Résumé.

La spectroscopie ^{Raman et} infrarouge des modes internes dans les cristaux moléculaires à désordre d’orientation est analysée ^{dans le} formalisme, précédemment développé [5, 6], des fonctions de base adaptées

aux

symétries ^{du site et} de la molécule. Dans certains

la contribution rotationnelle

profils ^{Raman et infra-}

rouge des modes internes peut être déconvoluée de la largeur purement vibrationnelle

d’autres effets dus à des

couplages ^{variés. La} partie purement rotationnelle du profil spectral ^{est ici} analysée ^à l’aide d’un petit ^nombre

de fonctions d’autocorrélations rotationnelles indépendantes ^du point ^de

^{de la} symétrie. ^{On montre de} plus

que l’intensité intégrée des raies internes permet de déterminer expérimentalement ^les premiers coefficients indé-

pendants de la densité de probabilité d’orientation des molécules.

Abstract.

The formalism of symmetry adapted functions for molecular orientations introduced previously [5, 6]

is here applied ^{to the} analysis of the Raman and infrared spectroscopy of internal modes in orientationally ^disor-

dered molecular crystals. ^In

^favorable cases, the contribution to the Raman and infrared lineshapes arising

from the rotational dynamics of the molecules

be disentangled from other contributions arising ^{from the}

vibrational lifetime

various coupling ^{effects. Here,} the rotational lineshapes

analysed ^{in terms of} independent, symmetry adapted rotational self-correlation functions. Furthermore, it is shown that the integrated intensity ^of

internal modes provides

measurement of the first symmetry independent coefficients in the development ^{of the}

orientational probability density ^function.

Classification

Physics ^Abstracts

02.20

63.50

78.30

1. Introduction.

The initials ODIC (Orientational

Disorder In Crystals) ^are presently widely ^{used to}

name those high temperature phases of molecular

crystals in which the centres of mass of the molecules still form a regular crystal lattice while their orienta- tions display ^some degree of disorder. This structural disorder is usually ^{related to a} complex dynamics ⁱⁿ

which the molecules can perform. large amplitude

reorientations as well as librational motions around

some preferred orientations. Recently many experi-

mental techniques ^{such as} NMR, X-ray ^{or neutron} scattering, optical infrared and Raman spectroscopy have been used to probe the rotational dynamics ^of

the molecules in the plastic phases [1, 3]. Here, ^we shall focus on the Raman and infrared spectroscopy of internal molecular modes. Indeed, ^{in ODIC} phases,

the internal molecular vibrations often give ^{rise to}

broadened infrared and Raman lineshapes ^{which are}

at least partly ^{due to} the rotational freedom of the molecules [2, 4]. ^This paper merely ^{intends to use the}

formalism of symmetry adapted ^functions (SAF), ^lar- gely developed ^{in two} previous papers [5, 6] ^to analyse

the rotational information which ^can be extracted from the Raman and infrared lineshapes of internal modes.

In fact, ^this paper is mainly the continuation of those two previous papers [5, 6] which hereafter will be referred to as (I) ^and (II) respectively. ^In (I), ^{we intro-}

duced a canonical basis for the functions of the mole- cular orientations. This basis is made of symmetry adapted ^functions (SAF) which transform according

to some irreducible representations of both the site group and the molecular symmetry group. In (I) ^this

basis was used to give ^a complete ^and unique ^deve- lopment of the static orientational probability density function, PO(Q), (in short, p.d.f) which takes into account, a priori, ^the simplifications arising ^{from the}

molecular and the site symmetries. ^In (II), ^{this for-}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01983004402016900

malism was extended to describe the rotational dyna-

mics of molecules in terms of independent symmetry

adapted rotational correlation functions. This was

used to show how experimental ^data arising ^from

neutron scattering experiments ^{could be} analysed.

In this last _paper of the series, ^{we intend to} apply ^the

same method to Raman and infrared spectroscopy.

This implies ^to express Raman and infrared line-

shapes ^{in terms of} appropriate rotational correlation functions. This approach ^was first introduced by

Gordon [7] ^and since, it has been widely extended in the framework of molecular liquids [8, 9]. ^{The SAF}

formalism allows to extend those previous ^{results to}

the case of anisotropic crystalline ^sites. In summary, it allows to make precise ^{which ones of the} independent

rotational correlation functions are measured through

each spectral component of each internal molecular mode depending ^on both the site symmetry ^{and the} symmetry of the internal molecular mode.

The plan ^{of this} paper is as follows : part ^{2 is devoted}

to generalities ^{about Raman and infrared} spectroscopy of internal modes in molecular crystals. ^{It intends to}

stress the basic assumptions ^{which are to be} made, if one wishes to extract rotational informations from

lineshapes ^which generally ^involve many other effects such as vibrational relaxation, vibration-rotation

coupling, ^Coriolis coupling, ^{etc. Part 3} applies ^the

formalism of the symmetry adapted ^{functions to the} expression of the rotational correlation functions relevant for Raman and infrared scattering. ^In part 4,

we show that some information about the statistical distribution of molecular orientations can be obtained from the measurement of integrated intensity ^of

Raman internal lines. At last, ^we analyse ⁱⁿ part ⁵ the rotational information coming ^{out from Raman}

and infrared lineshapes.

2. Generalities on Raman and infrared spectroscopy of internal modes. The various approximations.

We shall not, of course, develop ^{here the} general theory of interaction between light ^{and matter but} simply ^start from the basic formulae which _express the Raman scattered intensity and the coefficient of infrared absorption ^{in terms of} appropriate ^correla-

tion functions.

The first one relates the Raman intensity ^{to the}

fluctuations of the macroscopic polarizability ^{of the}

material sample :

In (2 .1 a) :

R((o) is the ratio of the scattered intensity ^{with a} frequency ^{shift W} = Wd - Wi ^to the incident intensity,

kd ^{is the} length of the scattered wave vector, ei and _ed are the polarization ^{vectors of} respectively

the incident light ^{and the} analysed ^scattered light,

E(t) is the total polarizability ^{tensor of the} scattering

volume at time t,

and the symbol > stands for an average ^{over a} thermal equilibrium distribution of the target ^states.

The second basic formula relates the infrared absorp-

tion to the fluctuations of the dipole ^{moment :}

with

IR(w) is the fraction of incident _energy absorbed per unit thickness of the sample,

w and _ei are the frequency ^{and the} polarization

vector of the incident light,

11(t) is the total dipole ^moment in the volume V of the sample ^{at the time t.}

In the case of molecular crystals, ^{the total} polariza- bility ^{tensor and} dipole ^{moment of the} sample ^{can be}

written as a sum of molecular contributions

FormVIae (2 .1 ) ^and (2.2) ^{are still} quite general ^{in the}

sense that they provide ^{the Raman and infrared} spec- tral intensity ^{over the whole} frequency range and must,

therefore, include the whole time dependence ^{of the} polarizability ^tensor EL(t) ^and dipole ^moment IIL(t).

More precisely, ^these quantities depend ^{on the one}

hand on the external (translational ^and reorientational) dynamics of the molecule which gives ^{rise to the low} frequency ^{part of the spectra} (either purely ^rotational

or including ^a « collision induced » translational con-

tribution). ^{On the other} hand, the molecular polariza- bility ^and dipole ^moment reflect the deformation of the molecule through its various internal vibrational modes. Each internal mode j’ gives ^rise (if ^{this mode} is, in the ODIC phase, sufficiently decoupled ^{from all}

the other internal modes) ^{to a} spectral ^{line, the inten-} sity ^and lineshape of which is the main subject ^{of our} study ^here. Usually the internal lines are well apart from the low frequency part ^{of the} spectra ^{and we} shall assume that there is no overlapping ^between spectral ^lines arising from different internal modes.

With such an hypothesis, ^{the Raman} and infrared

lineshapes corresponding ^{to a} given ^internal mode j’

arise from contributions ^to the polarizability ^tensor

and dipole ^{moment which can} be written :

where QLj’,,’(t) ^are the normal coordinates related to

the mode j’ of the molecule L,

n’ labels the different partner coordinates if j’ ^{is a} degenerate ^{mode and} ELj’,,’(t) and - ^{are the}

derivative tensors :

Inserting (2. 3) ^into (2. 2) ^and (2 .1 ) ^we finally get ^the basic formulae which give ^{the Raman} and infrared

lineshapes ^{related to a} molecular internal mode :

At this stage, it is clear that Raman and infrared lineshapes depend ^on complicated correlation functions which involve both the dynamics of the internal mode and the external dynamics of the molecules. In fact, ^{the exter-}

nal dynamics of the molecules _appears twice in the formulae (2 , 5a) ^and (2.5b) : once, indirectly, through ^its

influence on the vibrational dynamics (the time evolution of the internal coordinate QLj’n,(t) ^{may depend on}

the relative orientation and position of the Lth molecule with respect ^{to its} neighbours) ^{and once} directly, through ^its coupling ^to the ^detection process (the derivative tensors ELj’,,’(t) ^and "Lj’,,’(t) ^{depend on the mole-}

cular orientations and positions). ^{To extract some} information about the rotational dynamics of the molecules

we have to disentangle ^which contribution to the lineshape ^{is due to} the rotational motions and which one

arises from non rotational effects such as purely vibrational relaxation or inhomogeneous broadening. ^We

are thus lead to make two main approximations.

1) ^{The first one concerns} the detection mechanism and is known as the Kastler Rousset hypothesis. ^It

assumes that, in the molecular _axes, the individual Raman and infrared tensors ELj,,,, ^and °Lj’n’ have definite components (for instance cartesian component . ^and 7r’,) depending only ^{on the} ( j’, n’) considered vibra- tion. In particular, ^these components ^are unaffected by ^the dynamics of all the surrounding molecules and are

time independent. Thus, ^the components ^{of these tensors in the} crystal ^axes depend ^{on time} only through ^the

molecular reorientational motion : as usual, the orientation of the molecule L at time t is described through

the rotation OL(t) ^which brings ^the crystal ^axes (x, y, z) ⁱⁿ coincidence with the molecular fixed ^axes (xL, yL, zL).

In the crystal axes, the cartesian components (E Lj’n’(t) "b ^and II"Lj’n’(t)) of the derivative polarizability ^and dipole

moment derive from the molecular fixed axis components (e Jlfl ^and nj:n’) ^{through the following rotational}

relation :

...-.nJ. f." ...-.nJ. .. v

where M(OL(t)) is the usual rotation matrix associated with the three Euler angles corresponding ^{to the rota-}

tion QL ^{in the} crystal ^axes.

2) The second hypothesis, ^{which we} shall call the decoupling approximation, ^{concerns the} dynamics ^of

the crystal. ^{It assumes} that the rotational motions and the vibrational dynamics ^are statistically independent

so that the correlation functions appearing ⁱⁿ (2. 5) ^can be written as a product ^{of a} purely rotational corre-

lation function by ^a vibrational one which leads to

This approximation is, ^of course, ^a very crude one and it is perhaps ^{useful to make a little more} precise ^what

it really implies ^{for the} dynamics of the molecules. In a molecular crystal, ^{it is} usually reasonable to consider

that the dynamics ^of the j’ internal mode of a given ^molecule couples only ^{with the same} mode of the neigh-

bouring molecules.’ Thus, ^{for a} given orientational configuration, ^{we can write a} potential ^{energy for each set}

{ QLj’.’ } ^{of normal} coordinates j’ ^{under the} following ^form

Wj’ ^{is the} frequency ^{of the} mode j’ ^{for an} isolated molecule and the coupling ^term V L iZ({ q }), which, a priori, depends ^on the orientation of all the molecules, may include a static part ^{and a} fluctuating ^one.

The decoupling approximation ^assumes that the fluctuations of the coupling ^{terms are} unimportant ^either

because their amplitude ^is negligible (in ^{the weak} coupling approximation) ^{or because} they ^{are too fast} compared

to the correlation time of vibrational coordinates (in ^{the case} of motional narrowing).

In part 4, ^{we shall see} that these two approximations (namely the Kastler Roussel hypothesis ^{and the} decoupling approximation) ^are sufficient to derive ^some information about the static aspect of the orientational disorder (measurement of the first coefficients of PO(Q)) ^{from the} integrated spectral intensity of internal modes.

In part 5, ^we shall furthermore assume that there is no coupling between the internal modes of different molecules, ^i.e.

in that _case, the decoupling approximation simply ^assumes that the local molecular field VLL(T ^S2 }) ^{does not} depend ^{much on} the molecular orientations, ^so that the vibrational correlation function can be written as

Up to now, ^we have neglected the effect of Coriolis coupling ^which may be important ^{if the} mode j’ ^is

in the vector representation of the molecular _group. Up ^{to first order,} Coriolis forces induce a coupling ^between

the (n, n’) degenerate coordinates of the mode j’, ^{which is} proportional ^{to the} angular velocity of the molecule.

As it will be clear later, ^{the Raman} and infrared detection _process decouples the normal coordinate n and n’

and thus, Coriolis effect can simply be taken into account in the vibrational correlation function ovib

Before discussing ^these points, ^{it is useful to} simplify expressions ^{(2.6) and (2.7).} This is the _purpose of part ^3.

3. Application ^of group theory ^to the Raman and infrared spectroscopy of molecular crystals.

^{It is well-}

known that Raman and infrared experiments yield ^different spectra according ^{to the} polarization of incident and scattered light. ^{If we} introduce cartesian components ^{relative to the} crystal ^{axes, formulae} (2.1) giving

the Raman and infrared intensity ^can be rewritten as

Of course, all the cartesian components Rab,cd(W) ^and IRab(w) ^{of the Raman} and infrared tensors are not inde-

pendent and the number of different spectra ^{which can be} obtained depends ^{on the} symmetry group of the

crystal. ^Our purpose here is to show that in the ^case of molecular crystals, ^the group theory analysis ^{can be}

made at the same time for both the site and the molecular symmetries. ^This approach ^{allows to} specify ^the

different independent informations resulting ^{from Raman} and infrared lineshapes corresponding ^to each internal molecular mode. In the following, ^{we shall} assume, for simplicity, ^{that we are} dealing ^{with a molecular} crystal

with one molecule _per unit cell so that there is no distinction to be ^made between the site symmetry group and the crystal symmetry group.

a) First, ^it is convenient to define the irreducible components ^{of a tensor with} respect ^{to the} crystal ^and

molecular symmetry group. Let ^{us assume} that we have a tensorial object ^T defined for instance by its cartesian

components Tab in the crystal ^{axes. It is} well-known that this tensor can be split ^{into its} spherical components

T7 ^which, under the effect of a rotation of the axis system, transforms according ^to the irreducible representation Di of the rotation _group. This splitting ^{involves a} projection operator i such that

(see ^notes (1) ^and (’)).

Now, through ^a unitary transformation a applied ^{to the} spherical components TJJ’, ^{we can define new}

tensorial components Tf which transform according ^to the irreducible representations ^{of the} crystal group

This unitary transformation a has been used in (I) ^to define, ^{from the} spherical ^harmonics YJJ’, ^{the site} symmetry harmonics Yf. As in this _paper, the Greek letter A is a shortened notation for a composite ^index

r is the name of an irreducible representation ^{of the} group S,

11 labels the independent ^r subspaces ^{if this} representation is included more than once in the I manifold,

p numbers the different partners ^{for a} degenerate representation.

Thus, ^from (3. 5) ^and (3. 3) ^{we can write the} decomposition ^{of the tensor T into} irreducible tensorial _compo- nents of the crystal group under the form

In the same way, if t is a molecular ^tensor with cartesian components, fA’b’ in the molecular axis, ^{we can}

define its spherical components t"

and, through ^the unitary transformation introduced in (I), ^{we can} define the irreducible components t1’ ^with

respect ^to the molecular symmetry group

In full analogy ^with (3. 6a), the Greek letter h’ is a composite index relative to the molecular _group

From (3. 5b) ^and (3. 3b) ^the decomposition ^{of t} into molecular irreducible tensorial components ^writes

b) ^The analysis ^{of Raman} and infrared spectra ^into independent components ^with respect ^{to the} crystal

group is now straightforward. Indeed, ^the crystal polarizability ^{tensor E and the} dipole ^{moment II can be} split

into crystal irreducible tensorial components El (1

0, ¹

2) ^and IIIa (I

1), which from (3 .1 ) ^and (3 . 2) ^{leads to}

(1) ^We follow, here, the definition of irreducible spherical

tensorial components given by ^Rose [10]. ^This implies ^{that if}

tensorial object ^{T has the} spherical component TML ⁱⁿ

given (x, y, z) ^axis system, ^the spherical component T ML

of T in

axis system (x’, y’, z’) obtained from the old

one (x, y, z) through the rotation Q

(a, fl, y) (defined ⁱⁿ (x, y, z))

where D"’(0)

D7m’(ex, f3, y)

^the Wigner ^function

given e.g. by ^Rose [10].

(1) The formula (3. 3) should in fact include

auxiliary

index to label the independent ^{sets of} spherical components which belong ^{to the same I} representation of the rotation group.

In this article, however,

only dealing with dipole

moments and polarizability ^{tensors which}

first rank tensor and second rank symmetric ^tensors respectively. ^A

first rank tensor projects only

the manifold I

1,

and

second rank symmetric ^{tensor has two sets of} spherical

components,

^{for 1}

^{0 and}

^{for 1}

^2.

for Raman intensity (with h, 12

^0, 2) ^{and to}

for infrared absorption (with /1

12

I).

The correlation functions of irreducible ..tensorial components appearing ^{on the} right hand side of (3. ^9a

and b) ^{have to} reflect the symmetry ^{of the} crystal, ^{so that} they ^{must be all zero} except when, ^{the two} implied components À1

(F 1, p 1, p 1) and ).2

(T 2, /"21 P2) ^are contravariant which means that they ^are equivalent

partners (p 1 = ^{p2) in} conjugate representation (T

F*, 2 ^or T

F2 ^{if the} representations ^are real), ^{in which}

case the result is independent ^{of the} degeneracy ^index (pl

p2)

Thus, ^one independent spectral component arises for each possible association of two conjugate representations

of the crystal group.

Let us take, ^for instance, the well-known example ^{of a cubic} crystal (symmetry group Oh). ^The isotropic part (I

⁰ component) ^{of the} polarizability ^{tensor is of course in the} A1g representation of Oh, ^{while the} anisotropic

part (1 = 2 component) splits ^{into the two} representations Eg ⁺ F2g. This, altogether, give ^three independent

Raman spectra RA , RE ^and RF2.’ ^{In the case of a liquid, the spherical} components (1

^{0 and I}

2) ^are already

irreducible components, ^which yield only ^two independent ^Raman spectra :

c) ^{If now we turn} towards the molecular aspect ^{of the} crystal, ^{we are led to make} explicit ^the crystal ^irre-

ducible components (ELj’Il"t(t) ^and 17 Lj’?i’,l ’(t)) of the individual molecular derivative tensors E Lj’?i’ ^and "Lj’Il"

This can be done easily by ^first going through ^the spherical components ^{of these tensors in the} crystals ^axis

Then, making ^use of the Kastler Rousset approximation, ^{we can} derive the spherical components ^{in the} crystal

axes from the components in molecular axes as a function of the rotation QL(t) which characterize the orienta- tion of the L molecule with respect ^{to the} crystal ^axes

In the molecular _axes, the spherical components ^are related to the molecular components through :

Thus, using the definition given ⁱⁿ (I) ^{for the} canonical basis of symmetry adapted function for molecular orien- tations

we obtain from (3.13, 14, 15)

At last, the molecular irreducible components Gj’n’ ,r ^and 1tj’n’ ,r ^are quite simple. Indeed, ^{for a} given vibrational

mode j’, ^the derivative tensors Ej’n’ ^and 1tj’n’ ^have only irreducible components ^which belong ^{to the} representation 0393’j ^{of the} mode j’ in ^{the molecular} group. Thus, ^{if a}

(r’, 11’, p’) ^{we can write}

The constants g j, and 7r ’l characterize the Raman and infrared activity ^{of the} vibrational mode j’.

(03BC’)

For a given j’ mode, the number of infrared constants

nr(/1;) ^{is equal to} the number of r J, representations ^includ-

ed in the I

1 tensorial subspace (if ^{zero, the} mode j’ ^is simply inactive for infrared absorption). ^{In the same} way,

,

the number of Raman constants eil ^{is equal to the number of r;.} representations present ^{in the 1}

^{0 and} 1

2 subspaces.

For instance, ^{table III of} paper (I) shows that for a tetrahedral (Td) molecule, ^{the Raman} active modes are

the modes of symmetry A1, ^{E and} F2. The derivative polarizability ^{for an} Al ^{mode has one} isotropic component (I

0), ^{while the tensors} corresponding ^{to the mode E or} F2 ^{have one} anisotropic (I

2) component. Only ^the F2

modes are infrared active through ^{one I}

¹ component. ^{In the case of a} C3v ^{molecule, the} A, ^{mode has two}

Raman constants, ^one for 1

0 and one for 1

2, ^{while the E modes are Raman active} through ^two independent

l

2 constants. The A1 ^{and E modes are} also infrared active with one constant each.

Finally, inserting (3.16) ^into (3.9) ^and making ^{use of the} decoupling approximation (3.7), ^we obtain the

independent ^Raman and infrared spectral components ^{under the} following ^{form :}

In the formalism, the rotational contribution to the Raman and infrared lineshapes of internal mode _appears in terms of correlations between functions of the molecular orientations, ð ’V" (Q L( t»), ^{which are just the} symmetry

adapted functions introduced in (I). This formulation is going ^{to be used} throughout ^{the next two} parts ^where it _appears to be _very useful especially ^{in the case of} high ^{molecular and site} symmetry.

4. The integrated intensity of Raman internal line : measurement of the p.df. P o(Q).

One of the most

important ^{features in the} description ^of plastic phases is the statistical distribution of molecular orientations.

It can range between two extreme cases : a) ^an isotropic distribution of molecular orientations and b) ^{a distri-}

bution of the molecules _among a finite set of possible orientations. As usual, ^{we call} p.d.f. (probability density function) the function PO(Q) which describes the probability density, ^{for a molecule, to} have the orientation Q.

In (I), ^{we used} the canonical basis { AÅ/’(Q) }, ^{adapted to} the site and molecular symmetry, ^to write down an

explicit development ^{of the} p.d.f. P o(Q)

This development allows, a priori, ^the symmetry of the molecule as well as the site symmetry ^to be taken into

account and yields ^{a reduced set} of relevant and non redundant coefficients (A ’0’6), the values of which characte-

rize the amount of anisotropy present ⁱⁿ the statistical distribution of molecular orientations.

Let us simply recall that those coefficients are divided into two classes :

(i) Coefficients of the first kind : A for which the indices h§ ^and AO+ belong ^{to the} identity represen- tation (To , ^and To+) ^of respectively the site and the molecular _group.

(ii) Coefficients of the second kind : A "0-’0- ; they ^exist only ^when both the site (S) and molecular (A)

groups contain improper rotations, ^{in which case the} indices Ào ^and Ao belong respectively ^{to the second} representation (r 0- ^and to ) of S and fl which induces the identity representation ^{of the} subgroups Sr (of S)

and Ar (of m) which contain only pure rotations.

The _purpose of this part ^{is to show how the} measurement of the integrated intensity ^{of Raman} internal lines

provides ^a numerical estimate of some of the first kind coefficients in the development (4.1) ^of PO(O). Indeed,

if we neglect the variation of the scattered wave number, kd, ^{over the} frequency range of a given ^internal line,

we obtain, by integrating (3.18a)

Now, ^{we shall assume} that the instantaneous correlation between internal modes of different molecules vanishes so that

where n(wj’) ^{is the Bose} factor for the mean frequency (oil ^{of the mode j’ in the plastic phase. The relation (4. 3)}

is obviously ^{valid if there is no} coupling between the internal modes of different molecules. However, ^this appro- ximation can be justified ^{in much more} general ^cases. Indeed, ^{since the} dephasing processes arising ^{from the}

fluctuations of the coupling potential ^{written in} (2. 8) ^and (2. 9) play ^{no role in} the correlation values at t

0, the relation (4.3) ^{is a valid} approximation ^as long ^{as the} dispersion of the vibrational frequencies _wj, arising

from the coupling ^{is not too} large ^{so that the Bose factor} n(wj’) ^{can be} considered as a constant over the whole Wj’ frequency range.

where ,1’1

(r JI, ^{,u i, n’)} A2

(r JI, ^{11;, n’)}

N is the number of scattering ^molecules.

In 4. 4 the thermal _average on the right hand side involves only ^one molecule and thus depends only ^{on the} p.d.£ Po(Q)

Introducing ^the development (4.1) ^of PO(Q) ^into (4. 5), ^we get

where the projection coefficients ^{can be} expressed ^{in terms} of the usual Clebsh-

Gordan coefficients (lm ) 1 11 12 m1 m2)

Finally, inserting (4. 6) ^into (4. 4), ^we obtain the integrated intensity ^{of each} independent spectral component of an internal mode as a linear combination of the coefficient A ’l" ^of PO(Q)

(4. 9) ^{is still a} very complicated expression, ^{but it can be} simplified ^{with the} help ^{of the two} following ^{remarks :} First, the indices À’1 ^and À2 ^{refer to the same} representation rJ, of the molecular _group; this implies ^that,

in the generalized ^Clebsh coefficient

1’ refers to the identity representation ^{of the}

molecular _group. Therefore, ⁱⁿ any ^case, the integrated ^Raman intensity depends only ^on the coefficient of the first kind A ).o+fó+ ^of _Po(S2).

Second, ^{there is a more} important selection rule. Indeed, ^the polarizability ^{tensors are second rank} symmetric

tensors which implies ^{that the two indices} 11 ^and 12 ⁱⁿ (4. 9) ^are restricted to the value 0 or 2. As the product ^{of the}

symmetry adapted ^functions A).l’l(Q) A).;2(Q) ^{projects only on the} subset of function {A).’;" (Q) ^with ) li - 12 I ^I h ⁺ /2 }, ^formula (4.9) ^involves only the coefficients A ).,;.’ ^of Po(S2) ^{with 1 4. Therefore,}

Raman scattering ^can provide information only ^on the coefficients of PO(Q) of the first kind with 1 _4.

The above treatment applies ^{as well to the} integrated ^infrared absorption ^{lines. However the} last selection rule implies that infrared absorption ^involves only the coefficients A ).0; ).ó+ ^of PO(Q) ^{with 1} 2, ^{which is not}

very interesting ^{since in} many ^cases this subset includes only ^the first coefficient Ao, ^the value of which is fixed

by the normalization of PO(Q) :

In principle, ^{in the} general ^case, (4.9) ^{is not} very useful because neither the constants

eil Y’ characterizing

the Raman activity ^{of the} mode j’ ^nor their relative values are known. However, in many cases, two ^{or more} 03BC’

independent spectra depend ^{on the same constants} Ej, ( i) which allows to eliminate them. Let _us, for instance consider the example of tetrahedral (TJ molecules in a cubic (0h) site. We know from (I) that, ^{in this} case, the

p.d.f. p oCQ) ^has only ^{one non} trivial coefficient A4 ^{for 1 :( 4 :}

Each vibrational mode E or F2 ^{of the} Td molecule is Raman active through ^{a traceless Raman tensor deter-}

mined by ^{one constant} e(l

2) ^which gives ^{rise to two} independent spectral components REg ^and RF2g in ^the

crystal group. For each mode, ^{the constant} (1

2) is eliminated by forming ^{the ratio}

g g

If we take the usual definition of the spectral components RE ^and RF2g, ^{and apply (4.8) and (4.9) to this}

case we obtain :

for E vibrational modes and

for F2 vibrational modes.

This method has recently ^{been used to} determine, ^{in the} plastic phase ^of potassium perchlorate, KCI04,

the unique coefficient A A,gA ^{as a} function of the temperature [11]. ^It has also be used, ^{at room} temperature, in adamantane C 1 oH 16 [ 12], ^{where it was} found that the value obtained for this unique coefficient was- the same

for various internal modes and also agreed with its value determined by ^neutron scattering [ 13].

In the case of a linear molecule, ^the p.d.f. ^is only ^a function of the polar and azimuthal angles (0, CP) ^{of the C°°}

molecular axis and can be developed ^{on the site} symmetry ^harmonics Si (cf. appendix A)

with

In the case of a site with cubic symmetry, ^{the first non trivial term} arises for 1

4, and the above technique applied ^{to the Z +} stretching ^{mode of a} linear molecule gives [ 14] :

This technique ^{has been used to measure} the a4 coefficient for the high-temperature phase ^{of KCN} [ 15] ^{and the}

obtained value is again ⁱⁿ agreement ^{with the} corresponding neutron measurements [ 16].

5. The rotational contribution to Raman and infrared lineshapes.

^As already ^{stated in} part 2, ⁱⁿ general,

the Raman and infrared detection _processes involve, the vibrational and orientational dynamics of the molecules in a complicated ^mixed way. However, things ^become simpler if, besides the Kastler Rousset hypothesis ^{and the} decoupling approximations, ^{we can further assume} that there is no coupling between the internal modes of different molecules (3). ^Indeed, under these conditions, inserting (2.11) ^into (3.18a) ^{we find the} independent

Raman spectra under the form

where

with

ø j!b(t) ^{being defined by (2.11).}

Of _course, an equivalent formula holds for infrared lineshapes, ^{if we} replace ^{the Raman constants} by the infrared ones In other words, ^{the Raman} and infrared lineshapes

appear ^{as a} convolution product ^{between a} vibrational contribution and a purely rotational one.

Owing ^{to the} decoupling hypothesis between the internal modes of different molecules, the rotational

bandshapes (4.3) ^are given ^{in terms of the}

self-correlation functions At/Àí*(QL(O») 12 .(QL(t)) > ^which

involve only ^{one molecule.} Thus, ^Raman scattering and infrared absorption ^{act as} incoherent detection _pro- cesses, which, like incoherent neutron scattering, only reflect the individual aspect of the reorientational dyna-

mics. _{In paper} (II), this individual rotational dynamics ^has been described through ^the s.o.d.f (self-orientational density function) P(Ql Q2’ t) which is the joint probability ^{for a molecule to} have the orientation Q, ^{at a time}

(3) Note that this approximation ^is stronger ^{than that} leading ^to (4. 3). Indeed, ^the latter, which refers to

coupling

at t = 0, implies ^the neglecting of the time fluctuations of

the second term of the r.h.s. of (2.8). ^The approximation

used here neglects this second term

whole.

tl and the orientation Q2 ^{at the later} time t2

t

^{+ t} (cf. (II)). ^{Of course,} this function can be developed ^on

the canonical SAF basis (4) :

and it is obvious to identify ^the coefficients B of this development with the self-orientational correlation func- tions appearing ⁱⁿ (5.3); ^{indeed :}

Furthermore, ^{it was shown in} (II) that, owing ^to the site and molecular symmetry, ^the development (5.4)

includes only ^a small number of non trivial and independent coefficients, ^which implies that there is also a small number of independent symmetry adapted self-correlation functions. To make things ^more precise ^{we recall}

here the conclusions obtained in (II) :

If we set

(with ^{the usual} meaning ^{of these} notations),

(i) ^One independent coefficient of the first kind arises for each possible association of mutually contravariant

representations for both the site _group and the molecular _group

(ii) Furthermore, when both the site and the. molecular _groups include improper rotations, coefficients of the second kind arise from the product of« anticontravariant » representations ^{for both} groups :

In (5.3), the molecular indices À; and A’ refer to the same real representation rj, ^{of the mode j’. Therefore,}

each independent component J j’t1lf: ^{of the} rotational spectrum ^{is a} linear combination of the Fourier transform

self-correlation coefficients BFF" ^{(W) which belongs to the correct symmetry} of the molecu- lar and site _{groups :}

(4) ^We have found convenient to take, ⁱⁿ (5.4),

^defini-

tion slightly different from the

used in (II), ^where

,&A2A(Q 12 )* ^{appears instead of} A-12 12’(Q2 ). ^As

^{result, the}

selection rules given ⁱⁿ (5.7) ^and (5.8)

^not absolutely

identical to those given in that paper.

where Vj’ ^{is the} degeneracy ^of the j’ vibrational mode.

This result is _very similar to the result obtained in (II)

where the incoherent part ^{of neutron} scattering ^data

was analysed ^{with the same} formalism. However,

Raman scattering (resp. ^infrared absorption) ^measures

only ^the independent self-correlation functions for

11 ^and 12 equal ^{to 0 or 2} (resp. 11

12

1), ^while

neutron scattering implies ^no selection rules on the I values (5, 6).

Under the previous approximations (Kastler

Rousset i.e. rigid infrared and Raman tensors, decou-

pling approximation and absence of vibrational

coupling between various molecules), the usefulness of (5.1) ^for measuring the various correlations func- tions BÀI;í Àj;2(t) contained in a given ^band profile

is limited by ^our knowledge of the vibrational contri- bution C,ib(W) (Eq. (5 . 2)). ^{In the case of} long enough

residence times, ^a sufficiently good approximation might ^{be to assume} that the main vibrational contri- bution consists in an inhomogeneous broadening.

The latter can be estimated from the band shape (or

the site splitting) in the low temperature ^ordered

phase,. ^This proved ^{to be a} reasonable hypothesis ⁱⁿ

the case of the plastic phase ^of neopentane [17] ^for

instance. A limiting ^{case of this} approximation ^{is of}

course C%?(m)

^{6((o -} w j’) ^{which implies the neglect}

of both inhomogeneous broadening and vibrational relaxation effects.

Assuming ^the possibility of the deconvolution process implied by (5. 1), ^the previous analysis ^of

infrared and Raman data is especially useful when the linear combination (5.9) ^{reduces to one} single ^term.

Then each independent spectral component, ^once deconvoluted from the vibrational linewidth, ^is just proportional ^{to one} independent self-correlation function. This is for example ^{the case when} Td ^mole-

(5) ^{It is to} be noted that the independent rotational self- correlation function B03BBl103BB03BB 03BB03BB2(t)

involves the selection rules

i

03B4n1n2 ^and ðn’ln2’

Thus, the Raman and infrared detection processes involve each normal component ^{n’ of}

degenerate ^mode indepen- dently ^{from its} partners

^{when these} partners ^coordi-

nates

dynamically coupled through e.g. Coriolis effect.

(6) ^{The above} analysis (5. 7) ^is valid for any type ^{of site} symmetry and, ⁱⁿ particular is in full agreement ^{with the} results given by ^Steele [9] for the rotational lineshapes ⁱⁿ

molecular liquids. Indeed, ^{in the}

^of

liquid ^{the site} indices Àl and À2

just ^the spherical indices ml and m2 and if

keep dealing ^{with the} spherical components ^of the molecular derivative tensors

get ^the isotropic (1

0)

and anisotropic (l

^{1 (IR), 1}

² (Raman)) components ^of spectral lineshapes ^{under the} following ^{form :}

where 6Q,(t)

Q,(O) - 1 Q,(t) is the reorientation under- gone by the molecule at time t.

cules reside in an Oh site. From table I in (II), ^{we know}

that there are four independent correlation functions for 11

12

^2, namely : B + EgE B-, B+ F2gE ^and B+22g22. ^{The two} independent spectral components

(Eg ^and F2g) ^{of a} vibrational mode of symmetry ^E

are respectively proportional ^to B I EgE ^and B + 22gi,

while the two components ^{of an} F2 vibrational mode

give B + EgF2 ^and B+ 22g22· ^A precise analysis ^{of these}

four independent ^functions may be a first step ^{for the}

building of realistic reorientation models in plastic crystals.

The selectiveness of Raman and infrared scattering

may be particularly ^{useful in} some cases where it allows to clearly distinguish the contributions arising

from different kinds of molecular motions. Let us for

example consider the stretching (E +) mode of the linear CN- ions in NaCN [14] (cf. appendix A). ^This

mode gives rise, in the cubic site of the CN- ion,

to three independent spectra A 1 g, Eg, F2g. ^The A 1 g

spectrum derives from the (1

0) component ^{of the} Raman tensor and is independent of rotational motion ; ^{it thus} gives ^an estimate of the effect of vibrational relaxation of the Raman lineshapes. ^The

E spectrum ^{measures the} correlation S Eg, 1 (0)

S2."(t) > of the function

which, ^{in the case of} NaCN, appears ^to be extremal for the six preferred [100] orientations of the CN- ions.

Thus, ^the Eg ^{spectrum does not see the small ampli-}

tude librations of CN- ions around the [100] ^orien-

tations and can be used to evaluate the dynamics ^{of the} large amplitude reorientations between preferred

orientations. On the other hand, ^the F2g ^spectrum

involves both the reorientation and librational mo-

tions of the CN- ions, and if their residence time in a

preferred orientation is long enough, ^{it is} dominated,

at large enough frequency, by the librations.

6. Concluding ^remarks.

^The object ^{of this} paper has been to show how it is possible, through ^infrared

and Raman spectroscopy of the internal modes of the molecules of an ODIC single crystal, ^to obtain infor- mation on the orientation of these molecules. More

precisely, ^we have discussed how some independent

coefficients on the p.d.f. P oCQ) ^{could be} derived from the integrated intensity ^{of these} modes, ^and how,

in a similar _way, some independent coefficients of the s.o.d.f. P(ol, Q2, t) could be obtained from the study

of the band profiles. ^We have shown that the obtention of this information relies on a certain number of

approximations : the Kastler Rousset hypothesis ^of

Raman and infrared tensors rigidly ^{fixed to the mole-}

cules is one of them. More important is the role played by ^the coupling between the vibrations of different

molecules, and the influence of the molecular field

on the frequency of the internal modes of a single

molecule; ^{we have} given the conditions under which

some coefficients of P(Q) ^and P(Qll Q2, t) ^{could be} obtained, and shown that the restrictions in the latter

case are more stringent than in the former. The problem

of extracting rotational information, ^{in the case when}

the vibrational coupling ^between neighbouring ^mole-

cules cannot be neglected, ^{has not been discussed} here,

and has not yet ^{received a} proprer ^answer in plastic crystals, ^{even when the} vibrational-rotational cou-

pling ^is neglected. Further studies are clearly ^needed

to clarify ^this important aspect ^{of the} problem. Finally,

in all this _paper, the population life time TB (18) ^of

the mode j’ has been considered to be infinite; ^more precisely ^we have assumed that Arm T

> 1, where Arm is the HWHM due to both the rotational and the

inhomogeneous broadening ^caused by ^{the local}

molecular field. This should not be an important restriction, ^as this condition seems to be _very frequently

fulfilled.

The same type ^of analysis ^{as the one} presented ⁱⁿ

this _paper had been done in (II) ^{for neutron} scattering data, and it would be easily ^{extended to the} analysis

of data arising from various other experimental techniques. ^{In each case} (see (II)), simplifying assumptions ^{will be} necessary in order to extract the

same information, ^{but these} approximations ^differ

from one technique ^to the other. Thus the simulta-

neous use of various techniques ^is necessary in order to ascertain the validity of the obtained results.

Appendix ^A.

^THE CASE OF LINEAR MOLECULES IN A PLASTIC CRYSTAL.

The symmetry group of a linear molecule is either D ooh ^or Coov depending ^{on the}

existence of an inversion center. The irreducible

representations ^{of these} groups ^are given [19] through

the following character tables I and II.

Table I.

Character table for ^the Coov group.

From (I) ^and (II), ^{we know} that, ^{in the case of a} linear molecule, the SAF writes

where £

(T, U, p) ^{refers to the site} symmetry.

(In ^{the case of a} liquid, ^{the SAF are} just ^the Wigner

function D ’T’ (Q).)

The development of the orientational p.d.f., Po(Q)

then takes the form :

where (0, 0) ^{are the} polar ^{and azimuthal} angles ^{of the} Coo molecular axis (cf. (II), appendix B) ^{while the} development ^{of the} s.o.d.f., P(Oll Q2, t) ^is

The selection rule arising ^{from the} Coo ^molecular symmetry implies ^that ml + m2

^0, which insures that

P(Ql’ Q2’ t) ^{does not} depend ^{on each} angle q1 and _q2 but only ^{on the} difference _cp

_q2.

In particular, ^{in the case of a molecular} liquid ^{we find :}

with C1 ’(t)

Bm,m’ - m I m’ (t).

As can be seen from A. 3 (and ^A. 4b), P(Q l’ Q2’ t) ^{does not} depend, a priori, only ^{on the} polar and azimuthal

angles of the molecular axis at time 0 and _t, but on the full three Euler angles, ^as it involves AÀ;"(Q) ^{with m # 0}

and Dmm’(Q). ^{We shall} find, ⁱⁿ fact, ^that through ^{Raman and infrared} spectroscopy, correlation functions involv-

ing m

^{0 and «m}

^{1 » are} detectable : in this specific case, light scattering ^and light absorption by ^internal

modes turn out to be somewhat superior ^to incoherent neutron spectroscopy ^{which can} give information on m

0 correlation functions.

One can show [ 18] ^{that the} internal vibrational modes are either stretching modes, (which belong ^{to the Z +} representation ^of Coov, ^{or the} Lg+ ^{and Eu} representations ^of D ooh) ^or bending ^modes (belonging ^{to the} n(Coov),

Tig ^or 1ru(Dooh) representations), ^The stretching ^{modes Z + and} Eg ^{are Raman} active with Raman tensors which

project on eg ^and E2O ; ^{the Z + and} Lu+ ^{are infrared} active, with infrared components p?. ^{But for the bending} modes,

the 7T and ng modes have anisotropic ^Raman components E2 while the 7r and _1ru modes have infrared components

Ill.

Table II.

Character table for ^the D ooh group.

Direct applications ^of (5.9) and of its infrared counter part then show that the stretching ^modes provide

information on the m

0 coefficients of (A. 3) ^and (A. 4b) ^{while the} bending modes allow us to detect the m

1 ones. Consequently ^the stretching ^{modes do not} give ^us informations on the third Euler angle, ^which

is normal, ^{as a} stretching mode leaves the molecule with the Coo symmetry. ^{On the other} hand, ^the bending ^modes

are sensitive to the rotational dynamics ^{of the} bending plane, ^{which means that} the third Euler angle ^{is a relevant} quantity ^{which can} be measured.

Let us take the example ^{of a linear molecule} sitting ^{on a cubic site. A} stretching ^mode gives ^{rise, in Raman} scattering, ^{to an} Alg(l

^{0) spectrum, and to two 1}

^{2 spectra,} Eg ^{and a} F2g, both sensitive to the reorientation processes. Their lineshapes ^are respectively proportional ^to the Fourier transform of the following correlation functions

where, for instance

If the stretching mode is infrared active, ^{there is} only ^one independent absorption spectrum, ^of symmetry Flu, proportional ^to the Fourier transform of

where

A Raman active bending mode, ^{on the other} hand, ^will give ^{rise to} only ^{two 1}

² spectra, respectively proportional ^to the Fourier transform of

while an infrared active one gives ^{rise to one I}

¹ absorption spectrum proportional ^{to the} Fourier transform of