Neutron and infrared study of the dynamical behaviour of methane in NaA zeolite

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Neutron and infrared study of the dynamical behaviour of methane in NaA zeolite

E. Cohen de Lara, R. Kahn

To cite this version:

E. Cohen de Lara, R. Kahn. Neutron and infrared study of the dynamical behaviour of methane in NaA zeolite. Journal de Physique, 1981, 42 (7), pp.1029-1038. �10.1051/jphys:019810042070102900�.

�jpa-00209079�

1029

Neutron and infrared study ^{of the} dynamical ^{behaviour of methane}

in NaA zeolite ^*

E. Cohen de Lara

Service de Spectroscopie Moléculaire en Milieu Condensé, Laboratoire de Recherches Physiques (**),

Université Pierre-et-Marie-Curie, 4, place Jussieu, ^Tour 22, 75230 Paris cedex 05, ^France

and R. Kahn

Laboratoire Léon-Brillouin, ^{Orme des} Merisiers, ^{B.P. n°} 2, 91190 Gif sur Yvette, France

(Reçu le 18 décembre 1980, ^{révisé le} 20 fevrier 1981, accepté ^{le 24 mars} 1981)

Résumé.

L’étude par diffusion de neutrons et spectroscopie infrarouge ^{de la} dynamique ^{d’une molécule de}

méthane adsorbée dans la zéolithe synthétique ^NaA apporte plusieurs ^résultats intéressants : i) la molécule évolue

pendant ^un temps supérieur ^{à 10-9 s dans une même} cavité, ii) ^alors qu’à température ambiante elle peut occuper tout le volume de celle-ci, ^{au fur et à mesure} que la température s’abaisse, ^sa trajectoire ^{est limitée de} plus ^en plus à un déplacement ^{contre les} parois, ^{avec des} temps ^{de résidence} plus longs ^en certains sites. A 4 K elle est totalement

piégée ^{sur ces sites.} iii) ^Les temps ^de corrélation de ce mouvement (10-11 ^{s à 240} K) ainsi que les durées de vie de la vibration interne v1 conduisent à une énergie d’activation de 700 K.

Abstract.

The study by infrared and neutron spectroscopy ^{of the} dynamical ^{behaviour of one} methane molecule adsorbed in the synthetic zeolite ^NaA shows several interesting features : i) the molecule remains for a time longer

than 10-9 s in the same cavity, ii) ^{whereas at room} temperature ^{it moves} within the entire volume of the cavity,

when the temperature decreases its trajectory gets closer and closer to the walls with a longer trapping ^{time in}

definite sites. At 4 K the molecule is completely trapped ^{in these sites.} iii) ^From the correlation time of this motion

(10-11 ^{s at 240} K) ^{and the} life time of the _v1 internal vibration we obtain an activation energy of 700 K.

J. Physique ⁴² (1981) ^{1029-1038 JUILLET 1981,}

Classification

Physics ^Abstracts

68.45

1. Introduction.

Zeolites are porous alumino- silicate crystals presenting ^{in some} structural types nearly spherical ^cavities regularly distributed over

the crystal lattice. These cavities ^are connected

through pores ^or windows, whose diameters depend

on the nature of the zeolite (D. ^{W. Breck and R. M.}

Barrer [1]). Consequently ^{these materials offer} physi-

cists an interesting tool insofar as the cavities may be utilized to isolate one or several molecules, ^thus constituting ^a simple physical model. Furthermore

they ^{can be seen as} solid solvents since adsorbed molecules penetrate ^{into the} entire volume of the solid where they ^are uniformly distributed and sur-

rounded by ^{the atoms} constituting ^the cavity ^walls.

Our present purpose is ^to study ^the dynamical

behaviour of a methane molecule adsorbed in NaA type ^{zeolite. In this structure with chemical} composi-

tion Nal2(S’02AI02)12, ^{the mean} diameters of the cavities and access pores ^are respectively ^{11.4 and}

4.2 A. CH4, whose collisional diameter is of the order of 4 A, ^{can therefore} penetrate the cavities but its diffusion might ^be seriously hindered when crossing

the _pores. Inside each cavity ^certain degrees ^{of free-}

dom might ^vanish owing ^to the interaction with the surface or because of the available volume. This effect has been studied on methane (R. Stockmeyer

et al. [2], ^{P. Gamlen} [3]), ethylene (C. ^J. Wright,

C. Riekel [4], J. L. Carter et al. [5]), ^and nitrous oxide

(E. Cohen de Lara [6]).

This investigation ^{has been} approached experi- mentally using ^IR spectroscopy ^and neutron scatter-

ing methods and theoretically by calculation of the interaction of the molecule with the crystalline

structure. We shall give experimental results and the derived interpretations ^{but first we} shall review the

potential calculation model and some theoretical data.

1.1 ADSORPTION POTENTIAL ENERGY. - Iri phy-

sical adsorption ^{such as in the} NaA-CH¢ system, the forces involved are Van der Waals and electro-

66

(*) ^Neutron measurements performed ^{at the} Laue-Langevin

Institute.

(**) ^{Associe au C.N.R.S.}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019810042070102900

static forces : as we will see later the electrostatic field in cavities is strong enough ^to induce infrared forbidden bands. The potential interaction energy 0

of one CH4 ^{molecule inside a} cavity may be described

as follow : an electrostatic term QE (molecule ^{in the} cavity field) ^{and a} Lennard-Jones 6-12 potential ^for dispersion ^and repulsion (the ^{sum of} binary ^inter-

actions of the molecule with each ion of the crystal lattice). 4JE ^{stands on the one} hand for the energy due to molecular permanent ^moments (nil ^{in this}

case owing ^{to the fact that} CH4 ^{has neither a} dipole

nor a quadrupole) ^{and on the} other hand for the induction _energy due to the polarizability ^{cx; the}

field is calculated with the assumption that NaA is

purely ^{an ionic} crystal (E. Cohen de Lara et al. [7]).

Displacing ^{the centre} of the molecule within the whole volume of a cavity, ^{we are able to} give ^{a des-} cription of the interaction potential (Fig. 1) : ^it yields

various types ^of potential wells from which we cal- culate the potential barriers from one site to another.

Fig. ^1.

^{Potential curves of a} CH4 molecule in a NaA cavity. ^{In a} (r, 0, Q) reference frame centred in a cavity, ^we displace ^{.the centre of the}

molecule along different radii (0, 9). a) This ensemble of curves 0(r) is drawn in the symmetry plane ^{of the} cavity ^{where are} located the pecu- liar Na III cation and other Na II cations. Each curve corresponds ^to angles Q displaced by ^10°. b) ^{This curve} is the locus of the minima of 0. ^As we can see in a) ^and b) ^{there is a barrier to} jump ^{from the} potential ^well facing ^{Na III to} the Na II well.

Considering ^{there is a} peculiar ^{Na III} cation in NaA

zeolite, ^{we find the} deepest well where cjJrnin ^{is about}

6 kcal./mole (3 ^{200 K - 280} meV) - ^the experimen-

tal heat of adsorption ^{is 5.4} kcal./mole (D. ^{M. Ruth-}

ven, K. F. Loughlin [8])

^{and a second} type ^{of well} in front of Na II cations where Oj. ^{is about}

4.5 kcal./mole (2 ^{300 K - 200} meV). ^The height ^of

the barrier is 3 kcal./mole. The second derivative of the potential ^curves gives the order of magnitude ^of

the external frequencies, parallel ^and perpendicular

to the surface, vII = 2.5 x 1012 ^Hz, vl = 5 X lO+ 12 ^Hz.

These data will only guide ^{us in} interpreting ^our experimental results but our aim is to come back to the model and assumptions ^on which this potential

energy calculation depends.

2. Infrared experiments.

^Infrared spectroscopy

gives information on internal vibrations and rotation of the molecule as a whole, ^{and therefore on motions} performed ^{in a time range of 10-14 to 10-12 s.}

We shall not give details of the experimental ^tech-

nique ^{which was} described in a previous paper

(E. Cohen de Lara et al. [7]). ^{Let us recall} only ^that

NaA powder ^is compressed ^into pellets ^{of about}

100 _u.m thickness and outgassed ^at 400 OC and 10-’ torr vacuum for 16 hours in the infrared cell.

The zeolite is totally ^{free of water} after such an acti- vation treatment. Prior to analysing the variation of the vibrational bands with temperature (in ^the range 300 K to 200 K) ^we present the infrared data of CH4

adsorbed in NaA zeolite :

CH4 ^has four fundamental vibrations (G. Herzberg

[9]) : ^{in the gaseous} phase v3 (3 ⁰²⁰ cm-1) and V4 ( 1306 cm -1 ) ^{are infrared active,} while v 1 (2 914 cm -1 )

and _V2 (1526 cm-1) ^are only ^{Raman active. In the} adsorbed ^state beside the two allowed bands, ^the

forbidden band v, appears with approximately ^the

same intensity ^{as V3 and v4, due to the} dipolar ^moment

induced by ^{the field} existing inside the cavity. ^The stretching ^bands v 1 and _V3 are shifted towards lower

frequencies ^{as in the} liquid ^{and solid states.} Compa-

rison of the profiles ^shows that V 1 ^is obviously ^nar-

1031

rower than v3 and v4 : ^{since the} totally symmetrical

Raman band of a spherical ^{rotor is not} accompanied by rotational wings (G. Herzberg [9]) ^{we are able to}

infer that the v1, profile ^is strictly ^{due to} perturbations

of the vibrational frequency. ^In addition, the observed V1 band is located entirely below the gas phase ^fre-

quency ; owing ^to the fact that its intensity ^is propor- tional to the square of the field, this band does not appear when the molecule is at the centre of cavity (E ~ 0) ^{where its} frequency would be close to the gas frequency. ^{On the} contrary, ^the allowed v3 ^band

has a broad profile and the presence of wings ^leads

one to infer that CH4 ^{retains some} rotational free- dom inside the cavity.

The variation of these two bands with temperature is presented ⁱⁿ figure ^{2. Few} perturbations ^{are noticed}

on v3 ; its frequency ^{does not} change, ^{but the rota-}

Fig. ^2.

vie and _v3 infrared bands of CH4 adsorbed in NaA zeolite.

-.-.- 260 K;

230 K ; ^{210 K.}

tional wings slightly decrease with temperature. ^For

v 1 as seen in figure 3, ^the frequency shift increases when the temperature ^decreases (stronger interactions with the surface) ^{and at the same} time the width of

Fig. ^3.

Frequency ^{shift of} Vl vibration of CH4 ^{in NaA.}

the peak is reduced. Let us consider possible ^reasons

for this band width :

i) ^{one can} exclude the influence of the presence

probability of the molecule in the cavity ^{as in this}

case the profile ^{would be} asymmetric,

ii) ^a contribution is certainly given by ^the gradient

of the electric field acting ^on the molecule but in all

events this contribution should have _very little tem-

perature dependence,

iii) ^{we have neglected} the rotational effect, although rigorously, the electric field changes the orbital molecular symmetry ^and therefore the selection rule AJ

0.

Finally ^{we make the crude} assumption ^{that the}

width is only dependent ^on the life time _Ty of the vibration :

TV varies from 5 x 10-13 s for 273 K to 13 ^x 10-13 s for 210 K.

Now this band only exists when the molecule is close to the surface where the field is sufficiently

strong ^to induce the band. We may therefore consider that _iv is equal ^to the time the molecule remains on

the adsorption site. This time, iA (adsorption time) ^is

linked to the potential barrier that the molecule has to jump ^to leave the surface

,

Figure ⁴ presents ^{an Arrhenius} plot ^which gives ^a

value for Ao equal ^{to 700 K} (1.5 kcal./mole). ^{At this}

stage ^the physical significance ^of this number is not

clear; ^{it is difficult to} consider it as the activation energy for leaving ^the deepest well since the potential

map calculation gives ^{a value of 1 600 K for this} potential barrier. We shall discuss this point ^after presenting ^{the neutron} scattering ^results.

Fig. ^4.

^Arrhenius plot of the inverse of the width of _vl band.

3. Neutron scattering experiment.

^{Neutrons are} coupled with all the degrees of freedom of the mole- cule. Cold neutrons, with ^a typical incident energy of

10- 3 eV with a resolution of 10 %, ^see motions which

are performed ^{on a} time scale of 10- 10 to

lO-13 seconds.

3.1 EXPERIMENTAL. - One of the major experi-

mental problems with zeolites is their great affinity

for water. The samples ^were prepared outside the neutron spectrometer ^and sealed. Therefore the ^con- tainer has to fulfill two conditions : to be as transpa-

rent as possible ^{for neutrons} (low absorption

^no Bragg peaks

^no quasielastic scattering) ^{and to be} absolutely air-tight ^{at all} temperatures. ^{We chose to} work in cylindrical quartz containers (Q = 20 ^mm

-

height

^{80 mm - mass of} hydrated ^zeolite pow- der

15 g) ^since quartz ^{has a} coherent but quite

constant scattering spectrum ^for long wavelengths.

Zeolite was outgassed ^{at 400} OC for several days,

until the _pressure in the container was 10-’ ^torr.

For samples with adsorbed methane, ^{we introduced}

a known number of CH4 molecules in order to have about one molecule _per cavity

the distribution

over the cavities is certainly quite ^{uniform even at}

150 K the lowest temperature ^{of the} experiment [8]

and we sealed the quartz ^tube as near as possible ^to

the powder. With this method we reduced the dead

volume, ^{so that} the number of free molecules inside the container was low with regard ^to the number of adsorbed ones. The total transmission factor

(NaA ⁺ CH4 ⁱⁿ container) ^{was about 67} %. ^With

these conditions, ^we find that about one half of the scattered neutrons come from methane.

The neutron scattering experiment ^was performed

at the Laue-Langevin ^{Institute on the IN5 time of} flight spectrometer (R. ^E. Lechner et al. [10]) using

an incident wavelength ^{of 10 A} (elastic ^resolution

=

25 ueV FWHM). The energy spectrum ^{of scat-} tered neutrons was recorded at 6 angles ^simultan- eously, covering the elastic momentum transfer _range

0.15 Qo 1 A-’ .

For each temperature : 300, 270, 240, 200, 150, ⁴ K,

we took two spectra, ^one of zeolite with adsorbed methane and one of outgassed zeolite. Outside the

Bragg peaks ^{of NaA, the neutron} intensity ^is mainly

incoherent scattering and the contribution due to

CH4 ^{is the} difference of these two spectra, after ^cor- rections : we note that for a correct subtraction one

needs to make a correction of the bare zeolite _spec- trum. Indeed, whether there is or is not CH4 ⁱⁿ sample : i) ^{each atom of the zeolite does not see the}

same incident neutron flux; ii) when scattered by ^a

zeolite nucleus, ^{a neutron has a different} probability

of leaving ^the sample ^without being ^{absorbed or}

rescattered.

This correction is directly ^{related to} the methane

partial transmission factor and following ^Sear’s

paper [16] ^we have used for the first order transmis- sion factor H CH4 (ko, k), the result for a cylinder (equation ^9.2.3, p. 35) ^with ECH4

0.14 ± 0.03 cm-1.

This last value is calculated from the zero order transmission measurement. The energy distribution of the neutrons which have been scattered at least

once by ^a proton ^{of a} methane molecule is then :

(spectra CH4) =

= (spectra NaA+CH4 -H1(ko, k) (spectra NaA)) . Figure ^{5 shows an} example ^{for T}

^{240 K and three} scattering angles. On all the spectra, the elastic peak

width is equal ^to the resolution of the apparatus.

Fig. 5. - Time of flight ^neutron spectra ^scattered by ^methane

adsorbed in zeolite NaA (after corrections and subtraction of the NaA signal). ^The sharp peak ^{around zero} energy transfer has, ^for all angles, the width of the resolution function. Separation ^between

the inelastic and quasielastic ^{domains occurs} around 0.5 meV energy transfer.

3.2 DATA ANALYSIS.

We find that there is abso-

lutely ^no Q dependence ^on the width of the elastic

peak. ^{This means} that inside the time window of ^our measurements (t ^{10- 9} s) ^{there is no} long ^range

translation. This fact has greatly simplified ^{the ana-} lysis ^{and the} understanding ^{of the} motion of the methane : since we know that a molecule stays ^inside the same cavity during its interaction time with the neutron, ^{we shall consider N} independent ^closed

boxes (zeolite cavities) ^each containing ^one CH4

molecule.

The principal ^{term of the} scattering cross-section

is the incoherent scattering by protons ^{and in the}

Born approximation ^{for N} independent ^molecules

(W. ^{Marshall, S. Lovesey} [11]) :

1033

where Rj(t) ^{is the} position ^{vector of the} proton ^labelled j ^{at time t.}

The motion of one proton may be expressed ^as

the sum of four elementary ^{motions :}

1) the translation of the CH4 ^{centre of mass in a} cavity, rc ;

2) the vibration of the whole molecule with res-

pect ^{to the} cavity walls, ^{i.e. the} external vibration Uc ;

3) the rotation of the proton around the carbon i.e. the rotation of the molecule itself d;

4) ^its vibration with respect to c, i.e. the internal vibrations of the molecule v;

so

If we consider the time scale of these different motions :

i) ^we know these for the internal vibrations (10-14 s)

and for the rotation of the free molecule (10-12 s);

ii) the order of magnitude for the external vibra- tion is 10-13 s, and we may ^assume that the time scale for the translation of CH4 ^{in the} cavity ^{is about} 10- II s.

As usual, motions which differ so in the time scales

can be considered as uncorrelated and we can rewrite the scattering ^{function as a} convolution product ⁱⁿ

the Fourier _{space :}

Starting ^{from this} expression, ^{we shall now consider}

each term separately.

-

First, since the molecule is enclosed in a finite- sized box, ^we have for the translational motion

From this relation it is _easy to show in a very general

way that the scattering function is necessarily ^{of the}

form (V. ^{F. Turshin} [12]) :

Ao(Q) is the elastic incoherent structure factor (E.I.S.F.)

of this translational motion; ^{it comes} from the pro-

bability p(r) ^of finding ^{the centre} of the molecule in

an infinitesimal volume around a point ^r inside the

cavity

T(Q, w) ^{contains the} dynamical information about the trajectory ^and whatever the details of this motion

(diffusive

random walk

instantaneous jumps ^on

definite sites) (A. J. Dianoux et al. [13]), this function

can be approximated ^{at low} energy transfer (long times) by ^{a sum} of Lorentzian curves

An ^are coefficients which decrease rapidly ^{for low} Q

.

when n increases. Moreover the scattering ^function

satisfies the sum rule (W. ^{Marshall, S.} Lovesey [11])

which means that :

Since for our experimental conditions we are always

in the region ^of Q ¹ A-1, ^{we retain} only ^{the first}

Lorentzian term

-

Let us consider now the rotational motion of the proton around the carbon atom. We assume that :

i) methane is not oriented with respect ^{to the} surface : the four protons ^are equivalent ^{and we can} drop the j ^label.

ii) At least in the quasielastic region (hw ¹ meV, time intervals greater ^{than 5 x} lO-12 s) ^a proton ^has

a spherically isotropic rotational diffusion motion around the carbon (R. Stockmeyer et ^al. [2]).

The scattering function in that case is well known

(V. ^{F. Sears} [14]) :

JI spherical ^Bessel function ;

d radius of the sphere ^{i.e. C-H} length;

y-1 correlation time

Finally both vibrational motions are treated as

harmonic oscillators. The first excited state of internal vibration of CH4 (1 300 cm - 1) is, ^{on the one} hand,

very little thermally populated ^{and on the other} hand,

too high ^to be excited by incident neutrons ; so, in the energy range ^{we are} interested in, ^{we consider a} purely ^elastic scattering function and write it as a

Debye-Waller ^factor:

svib intQ’ ^w)

^{exp -} (Q2 V2 >] dw> (11)

For external vibration, energy levels are lower and there is certainly ^a frequency distribution 9((oo)

because of the slow motion of the whole molecule

inside the cavity. ^{We thus have to} consider the scat-

tering ^{of a neutron} by ^{a molecule} with and without

change ^{of state :}

where F(Q, cvo, a)) ^{is the} scattering ^{function of an}

harmonic oscillator with frequency roo (W. ^Marshall,

S. Lovesey [11] ^{and V. F. Turchin} [12]).

Introducing ^formulae (9), (10), (11), (12) ^{in rela-}

tion (3) ^and integrating ^{the result over the} spectrometer resolution function R(w, n) ^we get the number of neutrons scattered in the 2 0 direction and stored in the channel n ((o is the energy transfer and n is related to the time of flight of the scattered neutron). ^{In the} general ^{case the} complete equation ^{would be} very

complicated ^{with too} many parameters ^{to be useful} for a fitting procedure.

In our case fortunately ^{we can} distinguish quite unambiguously ^two parts ^{in the} spectrum (Fig. 5) :

a narrow quasielastic region ^I (AE ^0.5 meV) ^con- taining principally the motion with the longest

characteristic time i.e. translation in the box, ^{and a}

broad inelastic part ^II (AE > 1 meV) representing

the external vibration but also containing quasielastic scattering from rotational diffusion. Naturally ^this

last motion introduces intensity in the first narrow

region but since _{7 - 1} meV > A, ^{we can donsider} this term as being nearly ^{constant in} part ^I.

We have therefore analysed ^our data in the fol-

lowing way : in part ^{I the} intensity ⁱⁿ channel n is calculated by :

which relates the mean square vibrational amplitude

of the proton ^{to the} integrated intensity. ^{A least}

square fitting procedure ^{is then used to find the}

other parameters ^of (13) : normalization constant c, E.I.S.F. Ao(Q), correlation time A - I, ^{and two} para- meters for background BG(n).

In the inelastic part II, the number of physical parameters increases. Moreover in the _range of energy transfer below 5 meV ^our hypothesis ^of

uncorrelated external vibration and rotation is _wrong and the true scattering function has to be developed

in a quantum mechanical formalism. For these

reasons a real fitting procedure starting ^from equa- tions (3), (10), (11), (12) would be unrealistic and we

restrict our analysis ^{to find the} general shape ^and

the qualitative variation with temperature ^{of the} external frequency distribution g(wo), maintaining

the assumption of uncorrelated motions :

Taking ^for Srot ^{a sum} of three Lorentzian terms

(cf. (10)) ^{and for} g(cvo) ^a broad distribution and peaks,

we have extracted the parameters (amplitudes-posi-

tions and widths) giving approximately, ^{for each} temperature, ^the experimental Q ^{and w variation of}

the intensity.

4. Results.

Before giving ^the results, ^{we make}

a few remarks about the correction factors and the related uncertainties. We have already ^mentioned

how we took into account the influence of methane in the NaA intensity subtraction. As the scattering by NaA is almost purely elastic, ^the uncertainty ⁱⁿ

the H1(ko, k) coefficient leads to an uncertainty ⁱⁿ

the determination of the area under the elastic peak

of the methane, ^{i.e. on} its E.I.S.F.

Then in the difference spectrum, there remain

essentially ^{3 terms} arising ^{from :}

i) Neutrons which have been scattered by ^a proton and which have then reached the counter. This is the term we are looking for and with a good approxi-

mation it is equal ^to H1CH4+NaA)(ko’ k) ^x SincH4(Q, ro),

where H1 ^{can be} considered as a constant, ^at least in the quasielastic region.

ii) ^{Neutrons which are} first scattered by ^a proton, and then hit a nucleus of the zeolite (or the inverse

process). However the main part of the zeolite atte- nuation factor is due to absorption (Ào

¹⁰ A) ^and

very few of these neutrons come out. As we are not interested in absolute cross-section values, ^these

two terms are not important ^{in our data} analysis.

1035

iii) ^{Last but not} least, ^{neutrons which are scattered} twice by protons. Again following ^Sears [16], ^{we can}

estimate that the ratio of double to single scattering

is about 1/7. Because it is a rather complicated ^pro- blem, and because, ^{as we} shall see below, ^{it should}

not affect the main outline of our model, ^{we do not}

include this term in the scattering cross-section.

However, ^{from the} qualitative argument ^{that a}

non zero probability ^{of a} neutron to be scattered

elastically ^is restricted to a scattering in the forward

direction, we can see that most of the neutrons twice scattered will be found in the inelastic and the quasi-

elastic regions ^{of the} spectra.

4.1 GEOMETRY OF THE MOTION. - For reasons

mentioned before we consider that, ^{at least} during

the interaction time with a neutron, ^the molecule is enclosed in a roughly spherical ^{box, and} except ⁱⁿ front of one or perhaps several ions its interaction

potential with the walls depends mostly ^{on distance}

and very little ^on angular coordinates. Therefore in

a very simple ^{model, we assume that} CH4 ^is moving

more or less uniformly in the volume limited by ^two spheres ^{of radii} RI ^and R2. The E.I.S.F. for such ^a motion is presented ^{in the} appendix. For 300 K the best fit is obtained when R2

^3.5 A and Ri

⁰ (Fig. 6a), ^{which means that the centre of the molecule}

is distributed over all the volume of a 3.5 A radius

sphere. Notice that this value has ^a physical signifi-

cance : if we add to R2 ^{the CH} length (1.1 A) ^{and the}

hydrogen ^radius (0.6 A) ^we get ^the result that the

Fig. ^6.

Elastic incoherent structure factor of the methane in NaA at different temperatures. ^The curves are calculated with the _para- meters as written in the insert. For T

4 K (6b) ^the slight ^increase

is probably ^{due to a} failure of our rotational model.

whole molecule ^moves in a sphere ^{of 10.4 A diameter}

which is not very far from the NaA cavity ^diameter.

At lower temperatures (Fig. 6b) ^{there are two noti-}

ceable effects : i) ^{the curves} decrease faster at low Q .

and ii) a non zero minimum becomes visible for

Q ~ 0.9 A -1 below 240 K. In particular Ao(Q )

¹

for all Q ^{when T}

^{4 K. The first effect} is taken into account if R1 = ^{0 ; for} example the best fit for T

240 K gives R2

^3.5 A and RI

² A (Fig. 6c),

and for T 200 K, R2

Ri

^{3.5 A} (Fig. 6d)

which ^means that the molecule now remains close to the walls. The second behaviour seems to indicate

some trapping ^effect corresponding ^to sites where

CH4 ^{rests for} longer ^and longer ^{times as the} tempe-

rature decreases. To fit the experimental ^{curves we}

introduced a new physical parameter ^a = to t + I tl ^{ro + ri}

(Fig. 6b), T, being the time the molecule is trapped

and _To the time its moves along ^{the walls.}

We can discuss the validity of the model within the uncertainty ^{of the} experimental points. ^On figure ^{6, we show the} calculated uncertainties due to the statistical error in the counting ^{rate and on the}

transmission factor measurements. However we have noticed that the multiple scattering ^{effect leads to an}

excess of neutrons in the quasielastic peak ^{which on}

the one hand, ^affects predominantly ^the highest

values of E.I.S.F., ^{and on the other} hand, affects, ⁱⁿ

the same way, neighbouring values. For instance, ^the

decrease observed of the E.I.S.F. at Q

^{0.40 A}

between T

270 and T

200 K should be the

same after multiple scattering corrections. Thus, ^we

think that the introduction of the two parameters R 1

and a is valid even if the values obtained are only qualitative, especially ^{for a.} Indeed, this last _para- meter is sensitive to the discrepancies ^{in that} region

where the ^errors in the experimental points ^are important. ^{Moreover as our beam-time allocation}

was limited we have only ^a very few points ^{in this} region. ^As yet ^{it is} impossible ^to determine if there is one or several adsorption ^{sites or if the mean} trajectory changes below 200 K. Naturally ^{there are} probably other models of motion which could fit the data but this is, ^{in our} mind, ^the simplest ^model

which takes into account the temperature ^behaviour that ^we have measured.

4.2 CORRELATION TIME.

Figure ^{7 shows the}

results of the fitting procedure ^of part ^{I at 240 K} with equation (13). ^The experimental ^{data are} quite

well described with a simple Lorentzian function but

we find a slight increase of A with scattering angle.

We don’t think it is an indication of a Q dependence

of this parameter but think that it ^comes mostly

from higher ^{terms in} T(Q, w) ^{when the} Ai coefficients (i > 1) ^{are not} negligible ⁱⁿ equation (6). ^{The same} phenomenon ^{occurs for T}

^{150 K} (when ^the long-

range diffusion is certainly ^{too small to be} measured)

which supports ^our assumption. ^{Thus the corres-}

ponding motion is of diffusive type ^{with a} correlation

Fig. 7. - Neutron spectra ^versus energy transfer of methane in NaA at T

240 K. The points ^are the result of the subtraction

(after corrections) ^of (NaA ⁺ CH4) spectra ^{and NaA} spectra.

time _rc

2 n/A [15]. ^We have taken for A an ave-

rage of the values obtained with the fit of the lower

scattering angle spectra. Figure ^{8 shows} thatr,, obeys

an Arrhenius law (at ^{least in the} temperature range of the present experiment) ^{with an} activation _energy of about 700 K.

Fig. ^8.

^Arrhenius plot of the characteristic time for the transla- tional motion of the methane inside a NaA cavity.

We find the same value as that given by ^the v, IR band width. Insofar as the correlation time measured

by ^neutron scattering ^{refers to the motion of} CH4

in the whole cavity, ^{one can} imagine that this acti- vation energy is linked to the mean height ^{of the} potential ^{barrier which} keeps the molecule in the

vicinity of the walls.

4.3 EXTERNAL VIBRATIONS.

In this section, ^we shall analyse ^the rapid motion of the methane mole- cule. First, equation (14) shows that the intensity ^of

the quasielastic peak ^is proportional ^{to a} Debye-

Waller factor. In this term U2 > ^{is an} average value

over all the vibrations with respect ^{to the} walls, ^and V2 > ^is supposed ^{to be} negligible because it involves much higher frequencies. ^In the harmonic oscillator

approximation ^{we have}

g(wo) ^{is the} density ^{of the} probability ^for having ^an

oscillator with frequency coo. In figure ^{9 are} plotted

the variations with ^momentum transfer of the quantity

Fig. 9. - Elastic + quasielastic intensity (divided by ^{the form}

factor of the rotation J’(Qd)) for methane in NaA.

The slope ^{of these} straight ^lines gives ^the experimental

values of ( U2 > ^{at different} temperatures ^{and we note} that ( U2 > is reduced by ^a factor of 10 when _{T goes} from 240 to 150 K. Now, ^{when nw KT and if} g(wo) ^is independent ^of T, U2 > ^{should be} propor- tional to T, ^{which means that the} frequency ^distri-

bution varies with temperature. This result is in agreement ^{with the} analysis of the E.I.S.F. showing

how the motion in the cavity ^is restricted when T diminishes. More quantitative information is derived from the inelastic part ^{II. We} have fitted this part

at each temperature ^with equation (15) using ^for g(wo) ^{the sum of a broad} distribution centred around 10 meV and a narrow peak ^at about 6 meV. When the temperature changes it appears that the central

frequencies ^{remain constant} but the widths and the relative intensities of these bands _vary below 240 K.

The final result is shown in figure ^{10 for 2 extreme}

temperatures, together ^{with the} corresponding ^exter-

nal frequency distribution. Two results are clear :

i) ^{at low} temperature ^a relatively well-defined

peak becomes visible around ^v

1.5 x 1012 s-1,

1037

Fig. ^10.

^Inelastic scattering and related external frequency ^dis-

tribution of the methane in NaA. In the left part: 8 experimental,

-

fitted curve (see text),

vibrational contribution, ^{+ + + rota-} tional contribution; ^{In the} right part : g(v) calculated with the

same set of parameters ^as in the left part.

ii) ^it grows ^as the temperature decreases, ^{and this}

result is connected with the decrease of ( ^U2 ). ^We assign ^this peak ^to the vibration of the whole mole- cule trapped ^{in a} potential ^{well, and more} precisely

to the vibration parallel ^{to the surface} because this vibration is twofold degenerate.

Nevertheless we emphasize that these results on

external frequencies ^are semiquantitative. ^{A future} investigation ^will need, experiments ^{to look for the}

temperature dependence of the inelastic scattering

with thermal neutrons in an excitation creation mode, theory ^to develop ^{a better} approach ^{for the}

low frequency coupling between vibration and ^rota- tion.

5. Conclusion.

These measurements give ^a good

idea of the dynamical behaviour of methane adsorbed in zeolite NaA. Below 300 K the molecule remains in the same cavity ^{for times} longer ^{than 10- 9 s. At}

room temperature ^it occupies ^{more or less} uniformly

the whole volume of the cavity ^{and at} decreasing

temperatures, ^{its mean} trajectory ^gets closer and closer to the walls. At 200 K the molecule is always

as close as possible ^to the framework ions. The molecule also spends ^{more and more time in a defi-}

nite site (or may be several sites) probably ^{in front}

of the Na cations. This last effect becomes visible,

on a time scale of 10-10 _s, around 240 K and is the sole possibility ^{at 4 K. At the} present time, the results

on the E.I.S.F., ^the frequency shift of the _vi band,

the mean amplitude ^{and the} frequency distribution of the external vibration form a coherent picture ^and support this model. From another point ^{of view we}

notice that the width of the _v, vibrational band and the correlation time of the diffusive motion inside the cavity give ^{the same} activation _energy. However before claiming ^{that this} agreement ^{is not fortuitous} and that both techniques measure some mean poten- tial barrier between the wall and the centre of the

cavity, complementary experiments ^are necessary :

one needs ^to extend the infrared measurements into the low temperature region ^{and one needs a careful} analysis of the behaviour of the E.I.S.F. in the 0.5 Q ^{1.5 A -1} region.

Acknowledgments.

^{We wish to thank Mr Beaufils}

for his valuable help during ^{the neutron} scattering experiment.

Appendix.

^{In this} appendix ^we calculate the elastic incoherent ^structure factor Ao(Q) ^{for the}

model mentioned in the text. We recall that here, Ao(Q) depends only ^{on the centre of mass} coordinate.

1) The motion of CH4 is such that the probability p(r,;) ^of finding ^{the carbon atom within a volume}

defined by ^two spheres ^{of radii} R1 ^and R2 is constant, but is everywhere ^{else zero :}

2) We consider now that, ^{for 1} cycle, during ^the

time _To the motion is as described previously ^and during ri the molecule is trapped ^{in a} definite site RW

in the crystallographic ^cell.

In spherical coordinates, ^OZ being along ^the Q

vector

With

we have

Now, ^{as we have a} powder sample :

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