Disordered structure of the cubic phase of quinuclidine at 295 K

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Disordered structure of the cubic phase of quinuclidine at 295 K

R. Fourme

To cite this version:

R. Fourme. Disordered structure of the cubic phase of quinuclidine at 295 K. Journal de Physique,

1979, 40 (6), pp.557-561. �10.1051/jphys:01979004006055700�. �jpa-00209139�

Disordered structure of the cubic phase ^of quinuclidine ^{at 295 K}

R. Fourme

Laboratoire de Physicochimie Structurale,

Université Paris XII, ^avenue du Général-de-Gaulle, ⁹⁴⁰⁰⁰ Créteil, France

(Reçu le 27 octobre 1978, accepté ^le 20 février 1979)

Résumé.

Le spectre de diffraction X de monocristaux de quinuclidine ^{a été} enregistré ^{à 295 K et} interprété.

La maille élémentaire est cubique ^à faces centrées avec un paramètre a

8,913 Å, le groupe d’espace étant Fm3m.

Partant d’un squelette moléculaire rigide, plusieurs types de désordre réorientationnel ont été soumis à l’analyse :

un modèle où les molécules toument de manière isotrope ^{autour de leur centre de masse et un autre à réorien-}

tations gênées ^entre plusieurs orientations équiprobables. Dans le second cas, ^on trouve un bon accord entre les facteurs de structure expérimentaux ^{et calculés} (facteur ^résiduel pondéré ^R’

^4,5 %); l’empilement ^molé-

culaire a été trouvé par combinaison d’une méthode de Monte Carlo ^{avec un} affinement par moindres carrés

en groupe rigide. ^Il n’y ^a pas coincidence entre l’axe ternaire d’une molécule et l’un des axes ternaires du cristal :

chaque ^molécule peut occuper 24 positions discernables.

Abstract.

Single-crystal X-ray diffraction data were collected and interpreted ^{for the} plastic phase ^of quinu-

clidine at 295 K. The unit-cell is face-centred cubic with a

8.913 Å, space group Fm3m. Assuming ^a rigid ^mole-

cular skeleton, ^several types of orientational disorder were investigated : isotropic reorientations of the mole- cules about their centre of gravity and hindered reorientations between equally weighted orientations. For the second model, ^{there is a fair} agreement ^between observed and calculated structure amplitudes (residual ^{R’ =4.5} %);

the packing has been found by ^a Monte Carlo method coupled ^{with a} rigid-group least-squares refinement. The threefold molecular and crystal ^{axis do not} coincide and there are 24 possible orientations for each molecule.

Classification

Physics ^Abstracts

61.10

1. Introduction.

Quinuclidine, N(CH2CH2)3CH, formally ^{known as} 1-azabicyclo (2.2.2) octane, is ^a

globular, cage-like molecule with C3v symmetry.

Thermodynamic properties ^are characteristic of a

so-called plastic crystal [1] : quinuclidine undergoes

a solid-solid phase transition at 196.00 K with an

entropy ^increment (6.34 e.u.) ^much larger ^{than that}

for the subsequent melting ^{at 430 K} (3.4 e.u.) [2].

In the room temperature phase (phase I), ^a study using X-ray diffraction methods has been reported : powder diffraction patterns ^{were indexed on the basis} of a cubic face-centred cell with a

8.95 + 0.01 Â

at room temperature [3]. ^There is residual entropy ⁱⁿ the low-temperature phase II, about Rln2 e.u. [4].

To account for transitional entropy ^{increment in} plastic crystals, Guthrie and McCullough [5] ^have suggested ^a simple model based on a superposition

of molecular symmetry ^{elements with} subgroups ^of

the lattice site symmetry. Following ^this idea, ^Westrum

et al. [2] ^{assumed that the} quinuclidine ^molecules

occupy Oh ^lattice sites ; then, only ^the C3v subgroup provides ^{a natural set of} symmetry elements for the cage (the ^{threefold molecular and} crystal ^{axis are}

then coincident) ; using the four threefold lattice axis and the head-for-tail distinguishability of the mole- cule, ^sixteen possible orientations should be _appa- rent for the disordered crystal ¹ phase. ^The plastic

phase ^{of a similar} compound, triethylenediamine N(CH2CH2)3N (TEDA) ^{has been} investigated by X-ray diffraction [6] ; ^{it was} suggested that molecules at each lattice site undergo hindered reorientations between eight equally weighted orientations all centred

on the site with coincident threefold molecular and

crystal axis ; owing ^{to the nature and limited amount}

of the experimental ^data (a powder pattern ^{with a} few lines), ^{this is} only ^{a tentative} interpretation.

The present analysis, ^{based on} single crystal X-ray diffraction, ^was undertaken to provide ^{a structural}

basis to related investigations ^{on the} dynamics ^of

disorder in the plastic phase ^of quinuclidine [7, 8].

2. Expérimental.

Quinuclidine ^{is a} highly ^vola-

tile and hygroscopic compound. ^Good single crystals

with dimensions _up to five millimeters were grown

by slow sublimation in Lindemann glass capillaries (00.5 mm) sealed to an evacuated vessel : the material

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01979004006055700

558

was the resultant of several previous sublimations to

increase the purity ^{of the} laboratory grade quinucli-

dine sample. ^A capillary, ^{sealed at both} ends, ^was mounted on the goniometer ^{head of a} Syntex P21

four-circle diffractometer equipped ^{with a} molyb-

denum X-ray ^{tube and a} graphite monochromator.

Crystal ^{data are :} quinuclidine ; cubic face-centred cell

Z

4. F(OOO)

^{248. The} non-extinction rules are :

with no additional extinctions.

Those conditions are valid for space groups F23, Fm3, F432, F43m ^{and Fm3m.} Groups belonging ^to

the Laue _group m3 (F23 ^and Fm3) ^{were discarded}

because reflexions hkl and khl were not found signi- ficantly different. The space-group Fm3m ^was kept

on the basis of arguments ^{which are discussed here-} after.

All the reflexions in a half-sphere ^of resolution were

measured _up to 2 0

50° using ^{0-20 scans with}

variable speeds ranging ^{from 0.5} deg . min -1 ^{1 to}

57.3 deg. min-1. ^{Two standard} reflexions, ^measured

every 20 reflexions did not show any significant

variation. The intensity of each reflexion was corrected for the Lorentz and polarization effects and for the variation of irradiated volume since the crystal ^was

a long cylinder. ^The absorption correction was negli- gible.

One scale factor and an overall temperature ^factor (B

⁸ A 2) ^were determined by ^a Wilson statistics calculation. Finally, ^the intensities of equivalent

reflexions were averaged ^{and a} complete ^{set of 49} independent reflexions, ^{all with a} positive ^{net count,}

was obtained ; among them, ¹³ reflexions had inten- sities I 2 u(l) ^{but were} also included in calculations.

Low-angle ¹¹¹ and 200 reflexions are very intense and might ^be expected ^to suffer from secondary

extinction. Calculations for structure determination

were performed with 47 reflexions and residual fac- tors were recalculated including ¹¹¹ and 200 reflexions.

3. Structure détermination.

The microwave _spec-

trum of quinuclidine ^{at room} temperature ^{has been}

previously investigated [9] ; ^the progression ^observed

was ascribed to a torsional oscillation of the molecular skeleton. The potential ^{function was shown to have}

two symmetrical ^{minima at} 0

± 10. 70 (0 ^is equal ^to the dihedral angle between the NC1 1 C2 ^and Cl C2 C7 planes) ; ^we have calculated that this tor- sional oscillation displaces C 1 ^and C2 ^atoms ± ^0.11 A apart ^{from the} plane ^defined by NC1 C2 C7 ^{at the}

untwisted conformation. (At ^room temperature ^the

amplitude ^of 0 ^reaches ± ²²⁰ (see ^{figure 2 of ref.} [9]).

This yields ^an angle amplitude of ± ^{1 D° for} any ^vector

bound to C1 or C2, ^{or to the} hydrogen ^{atoms bound}

to them, and situated in a plane perpendicular ^{to the} C3 molecular axis. Should this skeletal torsion be present ^{in the} single crystal phase ^{it would not} modify

the overall molecular symmetry and its effect would be much smaller than that of the lattice thermal motion (the ^overall temperature factor, B

8 Â2, corresponds ^{to a root} mean-square displacement ^of

~ 0.3 Â). We then concluded that it is valid to use

for crystallographic calculations a rigid ^molecular

model constrained to C3v symmetry ^{and a} single ^iso- tropic temperature factor ; this model was built from

compilation of bond distances and bond angles ⁱⁿ

similar compounds ^and specially ^{TEDA at room tem-}

perature ⁱⁿ phase ^II [10].

As a result of the molecular librations, ^bond lengths

obtained from electron density calculations are shorter than their actual values [11] ; ^{so, our model was built}

from uncorrected interatomic distances and angles.

The C-H distances were fixed at 1.0 Á, ^as usually

found in X-ray analysis (Fig. 1).

Fig. 1.

Assumed molecular skeleton of quinuclidine ^and labelling

of atoms.

Several models for the disorder were examined.

The function minimized during refinements was

W(j ^{Fo I -} 1 Fe /)21w I Fo 1’. ^The weight w ^{is calcu-}

lated as w = 1/(a + Fo 1 ⁺ c Fo 12) ^{according to}

[12] ^{with a}

^{12 and c}

^0.04; this scheme down-

weights the contribution of weakest reflexions. The

weighted ^{residual R’} is defined as the _square root of the function. The crystallographic residual R is defined

as usual : R = L Il Fo 1 - Fc 1 III ^{1 Fo 1.}

3.1 MODEL WITH FREE ROTATION.

In this model, molecules freely ^rotate about their centres of mass

which are fixed at the face-centred cubic lattice points

in space group Fm3m. The scattering function for a

spherically symmetric ^atom rotating ^{at radius r from}

a point (xyz) ^is

where s

2 sin 0/À ^and (j, il, Ç) ^are coordinates in

reciprocal space [13]. ^Two parameters only ^were

refined, the scale factor and one overall isotropic

temperature ^{factor. The} value for R’ was 0.15

(R

0.16) ^{with B}

^{12 Â 2.}

3.2 FRENKEL MODEL.

Instead of a free rotation,

it can be assumed that the molecules occupy ^a discrete number of orientations [14]. Those orientations are

related by space group operations. ^If n 1 and _n2 are

the total numbers of discemable orientations in phase ¹

and II, ^the transitional entropy (6.34 e.u.) suggests that nlln2 ²⁴ (Rln24

^6.32 e.u.). ^{It has been}

claimed that _n2

2 on the basis of the residual entro- py in the low temperature phase [4], ^{so that one} gets n, ^48. These considerations led us to assume, as a starting point, that the molecules were in general positions ^of space group Fm3m with their centres of

mass fixed at special positions ^of multiplicity ^{4. The}

two space groups F43m and F432 where discarded because the maximum number of discemable orien- tations would be only ^24. The three translational parameters of the model were fixed by constraining

the centre of mass at the origin ^{of the cell and the}

three rotational parameters ^were searched with the program Pythie. ^The algorithm ^{of this} program associates a Monte Carlo generation of molecular parameters ^{to a} rigid-group refinement, ^{with an} increasing ^set of low-angle reflexions. Each generation

is called a trial and the procedure is controlled by

the evolution of a weighted ^residual [15]. ^{The scale}

factor and the thermal parameter ^{were fixed at the} values determined by the statistics on structure ampli-

tudes. Convergence ^was obtained for a number of

trials, all solutions belonging ^{to the same set of sym-}

metry related orientations. A further rigid-group ^refi-

Fig. ^2.

Fo-synthesis ; projection ^{on the} (100) plane. ^This figure

is printed ^with 20 grey levels on a linear scale adjusted ^{so that}

white corresponds ^to the minimum electron density ^{and black cor-} responds ^to the maximum density.

Fig. ^3.

(Fo - Fc)-synthesis : projection ^{on the} (100) plane. ^Same

comment as in figure 2 ; the maximum electron density ^{on this}

map is

1/28 ^{of the} corresponding ^{one in the} Fo-map.

Table 1.

Observed and calculated ^structure factors

on the absolute scale : a) for the model with molecules

in general positions ; b) for ^{the more} constrained model.

560

Table II.

Structural data for ^{the cubic} phase of quinuclidine. ^1.

Atomic coordinates :

coordinates CA)

in an internal orthogonal frame;

fractional coordinates in the unit-cell ( x 10’) (hydrogen ^{atoms are labelled} according ^to the bonded carbon atom). ^Il.

^Eulerian angles ⁱⁿ degrees (with e.s.d.) for the molecular frame, defined ^{as in} reference [16]. el

^57.9 (0.4) ; 02

^34.7 (2.2) ; 03

^20.1 (3.1). ^III.

^Overall temperature fac-

tor (with e.s.d.) ^7.6 (0.2) A2.

nement was performed ^{with the} program Orion [16].

The scale factor, ^the isotropic temperature ^{factor and} three orientation parameters ^were adjusted ; ^several weighting ^{schemes were} tried, giving closely ^related results ; the results that have been retained were

obtained with weights calculated as above ; ^{at the end} of this refinement, R

^{0.059 and} R’1

^0.045.

As this point, ^the centroid coordinates were also allowed to refine. The fit was not improved signifi- cantly and the initial position ^was kept. ^A projection

of the structure on the (100) plane ^was calculated ; ^a diference-map ^{did not show} residual electron density higher than about 0.2 ^e. A - 3 (Figs. ^{2 and} 3). ^Observed

and calculated structure factors are listed in table la and structural data in table II.

The angle between the molecular axis in _any orien- tation and the nearest diagonal of the unit-cell is 10.40. We have specially checked this point ^{in the} following way : keeping the centroid at the cell origin,

the threefold axis of the molecule was taken as always lying along ^a threefold axis of the unit cell. This can

be easily ^{done with} program Orion since the refe-

rence frame S2, in which refinement is performed, ^can

be rotated at will [16]. The molecule has then a single degree ^of freedom, rotation about the threefold

molecular axis. This angular parameter ^{was modified}

by increments of 50 and the residual factor R was

calculated for each value. The two conformations with the lowest R-values were refined, using ^three adjustable parameters : ^the angular parameter, ^the scale factor and the isotropic temperature factor. The best result is R

0.099, R2

^0.085 (table Ib). ^The

ratio 9t

R’IR’ is 1.888. According ^{to the tests of}

Hamilton [17] ^{on 9t, the more} constrained model can

be rejected ^{at a level of} significance better than 0.005.

So, ^{there is no} coincident threefold molecular and

crystal axis, contrary ^{to the} assumption ^{of Westrum}

et al. [2].

4. Discussion.

The time and _space averaged ^struc-

ture of plastic quinuclidine ^is reasonably well fitted

by ^a simple model in which there are 48 symmetry related discernable orientations randomly occupied

about each lattice site. The local pattern ^{of disorder}

cannot be inferred from those results alone. It should be noted that one given ^{molecule cannot} occupy all 48 orientations but only ²⁴ because it is impossible

to invert a rigid non-centric body ; neighbouring ^mole-

cules _may belong ^{either to the same set} of orientations

or to the set related by inversion, ^so that the _average

structure is centrosymmetric. ^In addition, ^molecular reorientations about the molecular threefold axis

exchanging symmetry ^{related atoms have no effects}

on the diffraction pattern ^{but are detected} by ^NMR

because the nuclear spins provide ^a labelling ^{for the}

atoms ; ^{for each} labelled molecule, ^{there are}

24 x 3

72 discernable orientations. Evidence for such reorientations is given ^elsewhere [7, 8]. ^Corre-

lations between molecules cannot be deduced from the Bragg diffraction data; ^diffuse X-ray scattering might provide useful information and a preliminary investigation ^has been undertaken [18].

The low-temperature phase ^of quinuclidine ^is hexagonal ^{with a}

^{6.1 A and c}

^{10.0 A} [3]. ^The

molecular packing ^is probably ^{similar to that in}

TEDA II which has been solved from single crystal

neutron diffraction data [10] ^{and was found to be an}

ordered structure of untwisted molecules. The residual entropy ^{in the} low-temperature phase ^of quinuclidine might be related to the head-for-tail distinguishability

in this molecule, which does not exist in TEDA.

Most calculations have been performed ^{with a}

local version of the X-Ray system [19]. Other programs

are cited in the text. Fourier-maps ^{were drawn with}

a program written by ^A. Rimsky. ^Useful discussions with and communication of their results on quinu-

clidine by ^{C. Brot} and B. Lassier-Govers are gratefully acknowledged.

References

[1] TIMMERMANS, J., ^{J. Chim.} Phys. ³⁵ (1938) ^331.

[2] WESTRUM, ^E. F., WONG, ^{W. K. and} MORA WETZ, E., ^J. Phys.

Chem. 74 (1970) ^2542.

[3] BRUESCH, P., Spectrochem. ^{Acta 22} (1966) ^861.

[4] AMZEL, ^L. M., CUCARELLA, ^{M. C. M. and} BECKA, L. N.,

J. Phys. ^{Chem. 75} (1971) ^1073.

[5] GUTHRIE, G. B. and MCCULLOUGH, ^J. P., ^J. Phys. ^Chem.

Solids 18 (1961) ^53.

[6] NIMMO, ^{J. K. and} LUCAS, ^B. W., ^Acta Cryst. ^{B 32} (1976) ^597.

[7] BROT, C., LASSIER-GOVERS, B., LECHNER, ^{R. E. and} VOLINO, F., this issue of this journal.

[8] VIRLET, ^{J. and} BROT, C., this issue of this journal.

[9] HIROTA, E. and SUENAGA, S., ^{J. Mol.} Spectrosc. ⁴² (1972) ^127.

[10] NIMMO, ^{J. K. and} LUCAS, ^B. W., ^Acta Cryst. ^{B 32} (1976) ^348.

[11] CRUICKSHANK, D. W. J., Acta Cryst. ⁹ (1956) ^754.

[12] CRUICKSHANK, D. W. J., ⁱⁿ Computing Methods and the Phase Problem in X-Ray Crystal Analysis (Pergamon Press) 1961, p. ^45.

[13] JAMES, ^R. W., ^{in The} Optical Principles of ^the Diffraction of X-Rays (G. ^{Bell and} Sons) 1962, p. 230.

[14] FRENKEL, J., ^Acta Physico-Chimica U.S.S.R. 3 (1935) ^23.

[15] ANDRÉ, D., FOURME, R. and RENAUD, M., Acta Cryst. ^{A 28} (1972) 458.

[16] ANDRÉ, D., FOURME, ^{R. and} RENAUD, M., ^Acta Cryst. ^{B 27} (1971) 2371.

[17] HAMILTON, ^W. C., ^Acta Cryst. ¹⁸ (1965) ^502.

[18] LEVELUT, ^A. M., Private communication (1978).

[19] STEWART, ^J. M., KUNDELL, F. A. and BALDWIN, ^J. C., ^The

X-Ray ⁷⁰ system, University ^of Maryland, College Park,

Maryland (1970).