Interactions between substitutional and orientational orders. The phase transitions in enantiomeric and racemic crystals of TMHP. II. X-ray structural investigations of the two phase transitions in the racemic solid solution of TMHP

(1)

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Interactions between substitutional and orientational orders. The phase transitions in enantiomeric and

racemic crystals of TMHP. II. X-ray structural

investigations of the two phase transitions in the racemic solid solution of TMHP

J. Lajzerowicz-Bonneteau, B. Suchod

To cite this version:

J. Lajzerowicz-Bonneteau, B. Suchod. Interactions between substitutional and orientational orders.

The phase transitions in enantiomeric and racemic crystals of TMHP. II. X-ray structural investigations

of the two phase transitions in the racemic solid solution of TMHP. Journal de Physique I, EDP

Sciences, 1991, 1 (4), pp.559-572. �10.1051/jp1:1991152�. �jpa-00246351�

(2)

Classification

Physics

Abstracts

61.50K 64.70K

Interactions between substitutional and orientational orders.

The phase transitions in enantiomeric and racemic crystals of TMHP.

II. X-ray structural investigations of the two phase transitions in the racemic solid solution of TMHP

J.

Lajzerowicz-Bonneteau

and B. Suchod

Laboratoire de

Spectromdtrie Physique,

Universitb

Joseph

Fourier, Grenoble I, B-P. 87, 38402 Saint-Martin-d'Hdres Cedex, France

(Received

25

September

1990,

accepted

in

final form

12 December

1990)

Rdsumd. Au-dessus de 330K les cristaux

racdmiques

du

tdtramdthyl

2,2,5,5

hydroxy

3

pyrrolidine (TMHP)

sont des solutions solides. Sur _un site donna _{il y} _a soit _une moldcule droite soit _une molbcule

gauche (dbsordre

de

chiralit6)

de

plus chaque

molbcule peut occuper deux

positions possibles (dbsordre d'orientation).

_{II y} _a donc quatre ^btats

possibles

_sur

chaque

site. On

a btudib, _par diffraction de rayons X, la mise _en ordre en

tempbrature

dbcroissante. Paramdtres de maille, intensitds diffractbes

(taches

de

Bragg

et diffdrentes taches de surstructure) et

largeurs

de raies ont dtd mesurds. II y a deux transitions de

phase (325

K et 285

K)

_; l'ordre I basse

tempbrature

est du type ^ordre alternd DLDL..., I 100 K, ii _est loin d'dtre achev6.

L'interpr6tation

n6cessite l'introduction de trois

paramdtres

d'ordre

coup16s.

Abstract. Above 330 K racemic

crystals

of

tetramethyl

2,2,5,5

hydroxy

3

pyrrolidine (TMHP)

are solid solutions. On _a

given

site there is either _a

right-handed

_or a left-handed molecule (chirality disorder) in addition each molecule _can occupy two

possible

positions

(orientational

disorder). There _are then four

possible

states _on each site. The

ordering

that arises with

decreasing

temperatures ^is

analysed using X-ray

diffraction cell parameters, diffracted

intensitiis

(main Bragg and different superstructure

reflections)

and reflection width have been measured

versus temperature. There are two

phase

transitions

(325K

and

285K)

_; the order at low

temperature îs ôf âltemate type DLDL..., _even at 100 K it is far from

complete.

Three

coupled

order parameters are

required

for the

interpretation.

1. Inhoducfion.

In the

preceding

article

[I]

_we described the _structure of the enantiomeric

crystals

of

tetramethyl 2,2,5,5 hydroxy

3

pyrolidine (CSNO~Hj~, TMHP).

We studied the orientational

order-disorder structural transition which _occurs in these

crystals

around 305 K _: in the

high temperature (HT) phase,

at a

given

^site the molecule _can take _two

possible

orientations deduced

by

_a 2 fold axis

parallel

to b. The

C222j high temperature

space group becomes

P2j2j2j

at low

temperatures.

^In both _cases there _are 4 molecules _per cell.

(3)

In the solid state there is

miscibility

between the

right

handed

(D)

and left handed

(L)

molecules for all the concentrations and the

resulting crystals

_are solid solution

crystals.

We have

published

the structure at 300 K of the racemic

(50

fb

D,

50 fb

L)

solid solution

crystals [2].

Its structure is _very similar to that of the enantiomers

nearly

identical cell

parameters,

space group Cmcm

(a

_supergroup of

C222j)

and similar

arrangement

^of the molecules. On _a

given site, however,

from _one cell to

another,

there is either _a D _or _a L molecule each with two

possible

orientations. These four

possibilities Dl, D2, Ll,

L2 _are related

by

the local site symmetry ^m2m.

We wondered how this structure would evolve at low temperatures.

Ordering

of the

molecular orientation _can be

predicted leading

to a solid solution

having only chirality

disorder. One _can

expect

also _an

ordering

of chiralities

leading

either _to _an

alternating

order DLDL

(racemate)

_or to

phase separation

between D and L molecules. Such situations have been found and

investigated

in different

types

of racemic solid solutions

[3, 4].

This paper is

organized

_as follows: in section 2 _we report the

experimental X-ray

diffraction

investigation,

in section 3 _we describe the

high

temperature ^structure ^and introduce the order parameters, ^and ⁱⁿ section 4 _we

analyse

and discuss the temperature

dependence

of the different structural results.

2.

Experiment.

Thermograms

of TMHP

crystals

were

performed.

Unlike the enantiomers

[I], they

do _not show any well defined

peak (Fig. I).

A very weak and broad shoulder

is, however,

observed

spreading

from 220 K to 320 K.

C L/G/K

o.5

o.4 ''~'~~

o-s

210 250 2@0 330~~~~

Fig. I. Thermogram.

Crystals

of racemic solid solution. Tj ^and T~ are defined in figure 3.

A

preliminary X-ray

diffraction

study

on a

single crystal

_was made _on films

(Weissenberg

and

Explorer Camera).

When the

temperature

^is

lowered, superstructure spots

appear : in

fact,

_some

already

exist at _room

temperature (in

_our

preliminary publication [2]

we did not take these spots ^into

account).

A _more

systematic

and

thorough study

_was then made _on _a four circle

X-ray

diffractometer.

At different

temperatures

we measured the cell parameters, ^certain diffraction intensities and the width of _some main and

superstructure

reflections. Data collection _was

performed

at 100 K with

comparatively long counting

times because _even _at that

temperature

intensities of

superstructures ^remain very weak. At low temperatures ^cell parameters are

2a, 2b,

c and the Bravais lattice is C.

In

figure

2 _we

plot

the temperature variation of the cell

parameters,

^which appears very

regular.

At several times tests _were made _on the cell parameters ^for

possible

monoclinic

symmetry

at low

temperature.

None _was detected.

The

temperature dependence

of the intensities of _a few reflections _are

displayed

_on

figure

3. Two very different types ^of behaviour _are observed. The first is

type

I reflections

with indices such _as h ₊ k

=

2 _n ₊ I with the cell _a,

b,

_c

(these

reflections

correspond

to the C

(4)

,_«~ -

__O~~"'

%

,' w" ,' ,,~' ,,., «,*

zg _o- _»~

,--~"

73

₁₂₃ 223

Fig.

I(hk)I

a--~>c--

i

I

T,=285 TIKJ

Fig. 3. ^- Different of

X-ray : D) main _Bragg eflections,

(+) type I perstructure (h

+

with the cell

2a,

_2b,

c).

extinctions of

the

igh^perature ; the second concerns type II

reflections ^hich

eedsthe2a, 2 b, _c

celland which

have

odd _h and k _(for

either

h

+

k = 4n or

h + k

=

4 n + 2). Two

phase ^ansitions are then xhibited : one at

uperstructure

of

(5)

**

+ o -~

l y*

I-DID

~

o.oos

+

,

f

~

'

~

a~

i

, , i i

a o o a

173 223 273

~~~

Fig.

4. Halfwidth _at half maximum of different

X-ray profiles

_versus temperature

(D)

main

Bragg

reflections, (+) typeI superstructure, (o) typeII superstructure.

Different behaviours of reflections _can be _seen also in their

X-ray intensity profile.

Several reflections _were scanned

along

the three axis

a, b,

c* in

figure

4 _are

plotted

the half width at half maximum

(HWHM)

of their

profile

_versus the temperature.

Along

the three scanned

directions,

the results _are very similar. For the main

Bragg

reflections and the

type

^I

superstructure, ^the ^HWHM ^around 0.001

h

_for _the _former _and _0.002

h~

_for _the _latter

shows _very little variation with

temperature.

We found _more _or less the _same results for the enantiomeric reflections

(the

instrumental diffractometer resolution _can be estimated _at

around

0.001h~~).

_The _behaviour _of _type _II reflections is very

strongly temperature

dependent.

Between 260 K and 170 K their HWHM varies from 0.015

h~

_to _0.005

h~

they

_are _very difficult to measure

properly

above 260 K.

It should be mentioned that the

techniques

used to measure the diffracted intensities is not

appropriate

for the measurement of

type

^II

superstructure

intensities

just

below T~. These

peaks

_are _very broad and weak and their intensities _are

certainly slightly

underestimated.

All these measurements were

reproduced

several times the

phenomena

_are reversible. It is noticeable that the evolution kinetics is very

slow, especially

at low temperature for type ^II intensities _: for

example

the

intensity

of the

(1, 5, 2) reflection,

measured at 230 K after _a

rapid

temperature

descent,

increases

by

10 fb in ten hours. The measurements

presented

here have been done

during

slow _ascent and descent of

temperature (about fourty

hours for the

complete range).

It should be

interesting

to

study

with _more details the kinetic

aspects

^{of the}

orderings

studied.

(6)

3.

IIigh temperature ~IIT)

structure order

parameters.

We recall the characteristics of the

high temperature

enantiomeric _structure

[I]:

cell parameters a =

10.028

(5) h,

_b

=

6.651

(4) h,

c ₌ 13.987

(6) h

_at _T

= 323

K,

_space _group

C222j,

four molecular sites numbered I and 2 for

layer

0

(z

0

)

and 3 and 4 for the

layer

I

(z

~

l/2) (Fig. 51).

On each molecular site

(symmetry 2)

there _are _two

possible

molecular site orientations Dl and D2

(Ll

and L2 for the other enantiomeric

crystal).

A

given

atom of the molecule D _on site I of cell _m has the coordinates _a;~ _x, _y, _a

;~ z with a,~ = ± l

(Dl

_or

D2).

Then in this

orientationally

^disordered

phase (a;~)

=

0. Below the

phase transition,

the

amplitude

of _a; is _no

longer

_zero and _goes to I _at low temperature

(domains

_appear with

tX = ±

1).

Y

~ j

)

2 4

-1-x 3-3

Layer 0

(z=0)

La,er I(z=1,2)

11-1 -l1 -32-31

22 21 42 41

fl 12-11-1

~-31-3

21 22 41 42

1-12-l~

31-32-31

Fig.

5.

Labelling

of the molecular sites _:

(I) high

temperature

phase

cell _a, b, _c 4 molecular sites (1, 2, 3,

4)

_;

(II)

low temperature

phase

cell 2a, 2b, _c C lattice, 8

independant

molecular sites.

In the HT structure of the racemic solid solution the characteristics _are _: cell parameters

a =

10.034

(8) h,

_b

=

6.672

(6) h,

c = 13.95

(1) h

at T

=

325

K,

_space _group

Cmcm,

four

molecules sites numbered

1, 2, 3,

and 4 _as before

(Fig. 51). (In

order to

simplify

the

X-ray

structure factor

expressions

and the

comparisons

with the enantiomeric structure we describe Cmcrn group with _a _z

=

I/4

translation of the

origin

with respect to the classical

description).

On each molecular site

(symmetry m2m)

there is either _a D molecule with 2

possible

orientations Dl and D2 _or _a L molecule also with 2

possible

orientations Ll and L2.

Consequently

_a

given

atom of the molecule _on site I of the cell _m _can have the coordinates _:

xyz,

iyz, xyf

or

$yf.

This _can be summarized

by

_aj~x, _y,

pi

~

z with _al

mm ± I and

pj~

₌ ^±I ^and ^for ^the ^three ^other ^sites

by

_a~~x,

y+1/2, p~~z

and -a~~x, -y,

p~

~

z +

1/2

and _a

~,,, .<, i +

1/2, p~,,,

= + 2

We choose for Dl a;~ =

p;~

= l and denote

Di by (+,

₊

),

while a,~ =

l,

P;m

₌ I for

Ll,

_so that Ll is

(-,

₊

)

likewise D2 is

(-, )

and L2

(+, ).

Each

molecular site is thus _a 4 _state system :

(+,

+

), (-, ), (+,

and

(-,

+

).

It _can be

seen that _~;~

=

a;~p;~ represents

the

chirality

of the I-th

molecule,

while ~,~ = + l

corresponds

to a

D,

and ~,~ =

l _to _a L molecule.

In the

high temperature

disordered

phase

the _mean value for _one site I is

«; =

(«<ml

= °

P;

=

lP;m)

= ° ~; =

(~;m)

=

o

I

representing

_any of the sites 1,

2,

3 _or 4

(Fig. 51).

In the low

temperature phase

the

spatial

mean values _a;,

p,,

_~; of the parameters _a;~,

p;~

and ~;~

(i

now

being

related to _one of the 8

independent

sites

II, 12, 21, 22, 31, 32,

41 and 42

(Fig. SII)

_are _no

longer

_zero.

a;,

p;,

and ~; are the order

parameters

_: we shall define this notion _more

explicitly

for the

thermodynamic investigation

of the transition in the

following

article.

(7)

4.

X-ray

^diwracfion results.

It is recalled that superstructure reflections that appear when the temperature ^decreases are

few and weak: _at 150K for 100 main reflections measured

(observed

HT reflections:

h and k _even, h ₊ k

=

4

n) only

30 reflections of

type

I

(h

and k _even, h ₊ k

= 4 _n ₊

2)

and

100 of

type

II

(h

and k

odd)

_are measurable. The maximum intensities in these three

following

ratio

250/1/2.

Such weak superstructure intensities

certainly imply

_a state that is still

poorly

ordered. With such

experimental

data _we _cannot

hope

to _use classical

crystallographic

structure resolutions at different temperatures. ^We ^will see that

information _on the order parameters can nevertheless be obtained and

interpreted.

4.I X-RAY STRUCTURE FACTOR EXPRESSIONS As A FUNCTION OF «,,

p,,

^and _1~,. ^I5et us

consider the low temperature

phase

cell parameters 2 _a,

2b,

_c, C Bravais lattice and the molecular sites mentioned in

figure

SII.

In cell _m, for the molecule _on site 11, ^with _p atoms of coordinates

aiix~, y~,

pi

j

z~,

^the

expression

for the structure factor is _:

Fjj~

=

jjf~exp20rj(hajjx~+ky~+fpjjz~)

p

=

jj f~

_exp 2

orjky~(cos

²

orhx~

+

ja

ii sin 2

orhx~) ^(cos

²

orfz~

+

jp

ii sin 2

orfz~)

P

" I~I + "

lI1~2

+

fl

II 1~3 ⁺ 'i

lI1~4

Fj, F~, F~

and

F~

are

complex trigonometric

functions of the coordinates

x~, y~,

z~

(it

is recalled that

1~ ii ⁼ ^a ii

p ii).

The coordinates of the atoms _on the 7 other molecular sites of the cell _m

being simple

transformations of

x~, y~,

z~

(sign changes

_or

translations)

the cell structure factor _can be

expressed

_as follows

Fm(hkf)

=

fi +jgi

+

Ai(«;m)f2 +jA2(«;m)

_g2 ₊

Bi(P;m)f3

₊

+jB2(P;m)g3+Di(7J;m)f4 +jD2(7J;m)g4

where

f,,

_g, _are

trigonometric

functions

depending

_on

^hkf

and atomic

coordinates,

A j,

A~, Bj...

_are linear functions of _a;~,

p;~,

_1~,~ which characterize the different sites of the

m cell. Of _course linear combinations differ

according

to

hkf parity categories

^of reflections _:

we

give

_more details in the

appendix.

~

For the whole

crystal (M cells)

the structure factors

F(hkf

=

jj F~

will

depend

_on the

Mm

i

mean values a; =

(a,~), p;

₌

(p;~)

^and _1~; ₌

(1~,~),

i

representing

the 8 molecular sites of the low temperature ^cell.

4.2 THE _MAIN BRAGG REFLECTIONS. The intensities of these reflections

undergo

almost

no

change

when the temperature ^decreases

(except

for _a usual small

increase), figure

3. With data collections at

293, 273,

173 and 100 K _we refined the structure with

only

these main reflections. We used molecular block refinements

(ORION

_program

[5])

^and ^the molecular model found in the enantiomeric refinement

[I].

Positional and orientational molecular

parameters

are

refined,

that is to say six

parameters

per molecular site. Like in the HT

structure the _space _group used is Cmcm and the cell is axbxc. These three refinements _are

very

satisfactory (R~

=

0.05

).

(8)

From these results _one _can conclude that _: the

hypothesis

of _a

rigid

molecule is _correct

the

positional

and orientational parameters vary very little with temperature: ^the

crystallographic

coordinates of the center of

gravity

of the molecules lie inside the

following

intervals uj :

(-

0.022

0.024),

_u~ _:

(0.333

_;

0.335)

and u~ :

(0.003 0.005)

with

tr

(u

0.0006 and for the orientation

angles

0

j

(87

_;

88], 0~.

(1

2], _0~ (80

_;

81]

with

tr(0)

~

0.3. These results _are very similar to those for the enantiomer structure

[I]

on average the sites

11, 12,

21 and 22

stemming

from site I _are

occupied by

Dl ₊ D2 ₊ Ll ₊ L2.

(The

_same for

31, 32,

41 and

42).

Thus the evolution of the structure with

temperature

^deals

only

with site occupancy, with still

everywhere

the four

possibilities Dl, D2, Ll,

and L2 _on each

layer.

These different

results _are very

important

and will be used

systematically

in the

study

of low

temperature phases

both for the

crystallographic study

and for the

analysis

of the intermolecular

energies.

It follows in

particular

that the molecular sites must remain _as far _as

possible crystallographically dependent

to

justify

the fact that the

geometrical parameters

^of these sites

are

practically

constant.

In the

expressions

for the structure factors of the main

reflections,

terms

depending

on the order

parameters

_a;,

p;,

_1~, must be

identically

_zero _; this leads to

(appendix)

_:

ZW,=0 ~ fl,=0 ~ 7j,=0

t ,

for

layer

0 _as well _as for

layer

I and whatever be the temperature. ^This ^confirms ^the

type

^I

superstructure

reflections?

They correspond

to the

extinctions of the C lattice in the _a _x b _x _c cell.

They

reflect the fact that sites I and 2

(or

3 and

4)

which _were deduced

by

_a

translation,

_no

longer

_possess translation symmetry.

They

were measured at 273 K and at 173 K. As said

before,

these intensities _are few and weak and their values vary little below 273 K

(Fig. 3).

We

compared

these

experimental

values with those calculated

according

to the results in the

appendix.

The best

agreement corresponds

to _zero contributions from terms

depending

on

p;

and _1~, parameters.

Taking

into account the results found in 4.2

also,

this leads to the

following

results _:

PI

_"

fl2

_"

fl3

_"

fl4

_" °

'II _" 'l2 " 'l3 "'l4 "

°

For the a,

parameters

we find

(Fig. 61)

:

aj(= a~)

m

a~(= a~)

m 0.40

(between

273 K and 173 K this numerical value increases very

slightly

^from 0.39 to

0.42).

As many intensities _are very weak it is _not _easy _to

identify X-ray intensity

extinctions but

according

to what _was done

previously

_we _can ^conclude ⁱⁿ ^favour ^of the space group Pbnm

(n°

62

Pnma)

with _aj

= a~

(the

_space _group

Pb2jm

with _aj ~ _a~ could also be suitable but

some molecular sites would become

inequivalent,

which does not _seem very

probable (see

3b),

and _moreover it would be necessary to assume that this transition is

accompanied by

the

loss of several symmetry

elements).

The symmetry ^of ^the ^molecular ^sites ^has gone from m2m

(high temperature)

to _m

(mirror plane perpendicular

to

z) (p;

= 1~; = 0 _a; _~

0).

(9)

To summarize _: when the temperature ^decreases ^the orientational disorder

(related

to the

a

parameter)

decreases _: _on one site the _mean _occupancy

Dl(25fb) D2(25 fb) Ll(25 fb) L2(25fb)

becomes _a

preferential

_occupancy, for

example Dl(35fb) L2(35fb) D2(lsfb)

Ll(lsfb)

which satisfies the condition _a= 0.40

p

_=1~ =0. It _can be said that the intermediate

phase

is _one with

chirality

disorder _a racemic solid solution in which _some orientational disorder

persists (the

value of _a;

being

less than

I).

It is

pointed

out that the structure of this intermediate

phase

is very similar with _a mirror

m in addition to the LT structure of the enantiomeric

crystals ~previous article).

The orientational disorder that

disappears

in both _cases is associated with the

disappearance

of the 2 fold axis of the molecular sites.

a~a

c~~a~

b[

^I

a-a a

al'-all'-all' Dl -L2 -Dl

fi)"' _'fill" D2' 'Ll

,'

,

/ ,

all -all -aj ^L2-Dl ^-L2

'fij" fij"/

_' /

,

~ _,/

,

, /

all -all -a§ Dl-L2-Dl

ii j v

Fig. 6.-(I)

The _a order parameter condenses first

(I

means

-a).

The cell remains _a, b, _c

(p~

= 1~, =

0).

The

ordering

is of antiferro type.

(II)

The low temperature

phase

in monoclinic cell a~,

b~,

_c~

(b~

=

c). (III)

The

analysis

of the

symmetries

and of the

X-ray

extinctions allows _a

description

of the structure

by

means of the order parameters a, p', p ", ~' and _~ " for the layer 0 (a,

pi, pi', ~(,

~i' for the layer I).

(IV~

Inspection of diffracted intensities leads to p

= p" and

~'

= ~^"

(.

The figure represents ^the

hypothesis

_a p'

= ~'

= l actually even at J00 K _one is far from the saturated

phase.

4.4 STRUCTURE OF THE LOW TEMPERATURE

(LT)

_PHASE.

for structure factors

(appendix)

for these

reflections,

^it ^is

necessary that terms in _a be null: this

corresponds

to the conditions _on parameters

a; found

previously (4,I),

no

discontinuity

_or

special

variation is observed in the intensities of

type

^I reflections

during

the second transition

(Fig. 3).

This _means that the

p;

and _1~; terms

appearing

in the

corresponding

structure factor

expressions

_are _zero.

(10)

The different conclusions deduced tkom these observations _are consistent. It follows that

the

type

II structure factors

depend only

_on

_parameters p

^and _1~ ^with

(Figs.

SII and

6III)

:

fill "~flI2~fl' fl2I ~~fl22~fl" fl31"~fl32"fli fl4I

"

~fl42"fli'

and the _same relations for _1~.

A

chirality

order thus _appears, with alternation

along

the _rows of molecules

parallel

to the y axis. More

precisely,

if for

example

the site I I is

preferentially occupied by Dl,

then site 12 is

preferentially occupied by L2,

and site 21 will be

occupied by

either Ll _or

D2,

and then 22 either

by

D2 _or Ll, The _same

applies

to

layer

I with the additional fact that if 11 is

preferentially occupied by

Dl then 31 will be

occupied by

Dl _or

L2,

since

a ii remains

equal

to a~j.

It is

important

to note that this

ordering requires

molecular

d%fusion

from site to site within the

crystal.

At this

stage

^of the

investigation, therefore,

the LT _structure

depends

_on

p', P", Pi, Pi'and ~', ~", ~i,

b,

_c, d of the

(h, k, f )

reflections

(see appendix).

4A.2

Symmetry of

the low temperature

phase.

^The evolution

previously

^described means

that the mirror symmetry ^{of the} ^sites

disappears.

^The

binary

_symmetry elements related to the y axis

(2j

axis and

glide planes)

also

disappear (Fig. 6III).

The

crystal

becomes monoclinic

(this

is true _even if

p

^'

=

p

^"

).

The

remaining binary

syrnmetry ^axis can be either in the _a direction _we would then have

a C monoclinic lattice with _a cell 2 _a _x 2 b _x _c _or in the _c

direction,

the lattice

being

then P with _a cell of half volume. The latter solution is consistent with the observed extinctions _: the

chosen cell will be monoclinic with a~ =

(a

₊

b);

_c~

=

(-

a

+b); b~

= c

(Fig. 6II),

the space group is

P2j/n,

there _are 8 molecules in the whole cell _: there _are then 2

independent

molecules. In fact _we observe _no monoclinic deformation of the cell parameters, ^which means

that the two molecular sites remain related

by

a strong

pseudo _symmetry (same

molecular

conformation, coupling

of the

position

and orientation

parameters).

^Monoclinic

syrnmetry

ⁱⁿ the diffracted intensities _was not observed

either,

but this could be

explained by

the existence of twins

(domains)

for which the diffraction reflections will

exactly superimpose.

The evolution of the symmetry ^from ^the ^HT

phase

enables _us to

predict

the existence of

eight

different twins if _one considers

only

_one molecular

layer (in

the

following paragraph

the

stacking

of the successive

layers

will be

seen).

These different

twinnings

_are

represented

in

figure

71. The

X-ray

intensities diffracted

by

the domains deduced

by

translation _are

identical,

while for the other domains

they

_are different.

4A.3 Structure

of

the LT

phase. Choosing

four

particular

arrangements we calculate

numerically

all the

trigonometric

terms that _occur in the

expression

for structure factors

(11)

f @

I I I I f

l

~ ~

"~' "~'

l I

---~---

~

l

~

aj' al'

@ @

,L2 ,L2,

~ ~ ^~ ~

L2-Dl-L2

L2-DljL2

'Ll

£

^'D2

_j

fiL2' ~L2'

jj ---(---_-__

,Dl

_~Dl

~ ~ ~( ~

Dl-L2-Dl Dl-L2-Dl

'D2

£

_'Ll

_~

~Dl'

'

~Dl'

Fig.

7. (I) A, B, C, D _: four

possible

domains (the four others _can be obtained

by changing

_a into

a).

A and C (B and

D)

_are deduced

by

translation and

give equal

diffracted intensities (the

following parallel

molecular

plane

is identical and described Avith the _same parameters a,

p).

(II) Same as I

assuming complete

order. For each domain the monoclinic cell is shown.

(relations (I))

if the

respective

volume of the different twins _were

known,

_We would be able to evaluate

p', p

"... from the observed diffraction intensities.

Examination of the _numerous intensities that _are null _or very weak

suggests

as a first step that

p'

=

pi

and

p

^" ₌

pi':

the 21 monoclinic axis _passes

through

the

origin

molecules of

rows

parallel

to the y~ axis

(orthorhombic z)

_are identical

(according

to _our

convention, they

have the _same

parameters

a,

p,1~)

_; then the successive molecular

planes

_are descrihed in the _same _way.

The

expressions

I for the structure factors then

simplify greatly

for

example

for reflection>

with h ₊ k

=

4 _n and

f

= 2 _n one has

4

j (p'

₊

p

^"

)

_cos 2 orhx _cos 2

orky

sin 2

orfz (1~'

₊

1~^"

)

sin 2 orhx sin 2

orky

sin 2

orfz

To

justify

the weak value of _some

intensities,

it is still _necessary _to make

p'( p "(

and

l'l'(

~

l'l"I.

The

expressions

of structure factors

simplified further;

for

example

for _a reflection

previously

mentioned _:

if

p'

=

p

^" then 1~' ₌ 1~'

,

giving

8

jp'

_cos 2 orhx _cos 2

orky

sin 2

orfz

if

p'

=

p

^" ^then

1~

'

= 1~

"

,

giving

8

j ~'

^sin 2 orhx sin 2

orky

sin 2

orfz

(12)

(The

intensities

depend only

_on

p~

_or

~'.)

With the two

possibilities p

^"

= ±

p'

_we ^find

again

the domains

previously mentioned,

A and B of

figure

71. Since monoclinic deformation is not

observed,

the intensities diffracted

by

these domains _are

superimposed,

and

they

_are _very different.

For the diffraction data collected at 100 K the best numerical agreement

corresponds

to

domains of

practically equal

volumes with

la

~0.42

(typel superstructures)

and

p' ~'

0.50

(type

II

superstructures).

For _a

particular site,

for

example,

this

gives

the _mean

occupation

Dl 60

iii,

L2 15

fb,

Ll 15

fb,

D2 10 fb _; order is still far from

complete

at 185 K below the transition It should be

emphasized

that _we have

only

_a small

quantity

of

experimental

data of low

quality (§ 4)

and the molecular block methods used _are still not very ^accurate. The

reliability

factor R

computed

with the LT diffraction data is about 0.25.

For easier visualisation of the different domains

obtained,

in

figures

6IV and 7II _we represent

perfect

order

(where

a

given

site is

always occupied by

the _same enantiomer with the _same

orientation),

I-e-

la

=

p

= ~ =

l.

5. Discussion.

The

study

of diffracted

X-ray

intensities enables _us to describe the _structure of racemic

crystals

of TMHP at different

temperatures.

The

high temperature phase (T

_~ 325

K)

has _a double molecular disorder both in orientation and substitution

(chirality).

The symmetry ^of the molecular sites is m2m. The

description

of the

progressive ordering requires

three

coupled

order parameters _a,

p,

_~ to describe the _mean

occupation

state of _a molecular site. The

a

parameter condenses first this

corresponds

to the

disappearance

of _one of the orientational disorders of the molecules _on their sites

(rotation by

_or around the y

axis).

At 285 K

chirality ordering

starts

(p

and 1~),

requiring

diffusion of the molecules from _one site to another.

Along

molecular _rows

parallel

to _ox and _oy _an

alternating

order DLDL... is

generated,

while the _rows

parallel

to Oz become homochiral. The low temperature

phase

is of racemate type.

In both enantiomer and racemic

crystals

there _are identical homochiral molecular

planes

in the abc axis these _are the

(110) planes (Figs.

61 and

IV).

In the first structure these

planes

are deduced

by

translation

[I],

for

example

_a. In the second structure these alternate

chirality planes

_are deduced

by glide

^mirror

planes,

for

example

_a mirror

(001)

with translation _a

[6, 7].

Can the

strong

difference in behaviour of the two

types

of superstructure

(diffraction

intensities

Fig.

3 and

peak

width

Fig. 4)

be understood in terms of the two kinds of order described ? For the type ^I intensities associated with the first

transition,

there is _no _reason that the

vanishing

of the rotation of the molecule around

Oy

should generate ^small

domains,

and

accordingly

the related

superstructure

intensities have widths at half

height

similar to those of the main

Bragg

reflections. It _can be noted that the width of this

peak

is

practically

the _same _as in the _case of enantiomeric

crystals [I].

For the type ^II intensities _one

hypothesis

is that of

ordering

of

chiralities, implying

that diffusion of the molecules from site to site

[8]

leads to the appearance of small size domains.

Consequently superstructure

reflections _are broad domain dimensions of 50 _to 150

h

are

measured between 250 and 150 K. These widths

practically

do not

change

below 150

K,

the diffusion kinetics

being

_very much slowed. Another

possible hypothesis

is that the intensities and widths of these

superstructure

^could ^be fluctuation effects

[9].

^The ^broad

temperature

range observed is not in favour of the

latter,

but nethertheless does _not rule it out

completely.

Numerous

questions

remain:

Why

does the orientational order parameter _a, ^which

approaches

0.40 _near 290K

(40°

below the

transition),

not increase further at low temperature ^? In the _case of enantiomeric

crystals

40° below the transition

ill,

the value of the orientational order parameter is about 0.9. How _can the

temperature dependence

of the

JOURNAL DE PHYSIQUE i, M4, AVRIL (Ml 25

(13)

type

^II

superstructure

intensities

(Fig. 3)

be

interpreted?

To what do these three order parameters

correspond precisely

?

In _an

attempt

to _answer these

questions

and go

beyond

the

descriptive

_stage

given by

the diffraction data _we have taken _a thermostatistical

approach

to the

problem.

This is the

topic

of the

following

article.

Appendix.

Different

expressions

of We

X-ray

structure factors

(2a,

2

b,

_c

cell)

The

numbering

of the 16 molecular sites of the C cell 2 _a _x 2 b _x _c is shown in

figure

6. Each site is caracterized

by

the

parameters

_a;,

p;

there _are

eight couples (a;, p ;)

per cell.

The coordinates of _an atom

~f)

of the molecule located at the site I I _are _a

ii x, y,

p

ii z

those of the _same atom of the molecule located at the site 12 _: _a

j~ x, y ⁺

1/2, p

j~ ^z those of

the _same atom of the molecule located at the site 31: a~jx, y,

p~iz+1/2.. (if

a ii ⁼

p

ii ⁼ a~i =

p~j

=

I,

the molecules _on sites I I and 31 _are Dl molecules

they

_are

_a

2 orhx

j

(- 1)~(cos

2 orhx ja

~~ sin 2 orh.<

)(cos

2

orfz

+

jp

4~ sin 2

orfz

₎

The

expression

in

cur(>

hracLei; _can be

e,paniicii

and ihc dilieiL,iii term,

regrouped, leading

to the

following

[era,;

Cj

cos 2 orhx _cos 2

orky

_cos 2

orfz

Aj

(a,

sin 2 orhx sin 2

orky

_cos 2

orfz B~(p, )

_cos 2 orhx sin 2

orky

sin 2

jDj

_J~, sin ~ 7r/1; ;in 2 ml, ;in ~

«f=.

(14)

The values of

Cj

and

C~

are 8 _or 0

according

to the different

categories (hkf)

of the diffraction reflections.

At

and

A~ represent

^linear combinations of the parameters a,

Bi, 82,

combinations of

parameters p;

_;

Cj, C~

n n +

f

=2

n + I

f=2

n

f=2

n + I

(a) (b) (c) (d)

Cj ^8'

0 0 0 0 0 0 0

C~

⁰ ⁸ ⁰ ⁰ ⁰ ⁰ ⁰ ⁰

Is'

I I I I I I I I

Ai, Bj, Di

e" I I I I I I I I

e~ I I I I I I I I

A~, B~, D~

e" I I I I I I I -1

e~ I I I I I I I I

As _an illustration for the main reflections

(h

and k _even and h ₊ k

= 4 _n,

f even)

this

general expression

contains _seven terms.

jj f18

^cos ² ^orhx ^cos ²

^orky

^cos ²

^orfz

^{sin 2} ^orhx ^{sin 2}

^orky

^cos ²

^orfz ^jj

^a ^; ⁺

all atomsof 8

the molccuic ii

_1~;

jj

_1~;

iaym1

(15)

+

j

sin 2 orhx _cos 2

orky

_cos 2

orfz ^iay«o jj

_a;

jj

_a;

layer1

cos 2 orhx sin 2

orky

sin 2

orfz ^laycr jj

_@,

jj

_@;

1.

o iaym

When _we consider the structure factors _not

just

for _one cell but for the whole

crystal

the

expression depends

of the _mean values

(a;) (p;)

and

(1~;),

I-e- the order parameters. ^In the text we call these _mean values _a;,

p;

and _1~;.

The numerical values of the

purely trigonometrical parts

of the structure factors _can be calculated _;

by choosing particular

distributions of molecules _on different sites _one _can

give

_a

zero value to several coefficients

A, B,

D and _as the atomic coordinates _are

known,

the calculation is

simple.

Comparison

between the

experimental

measured diffraction intensities and the

previously

mentioned calculations

fives

information _on the order

parameters

^of ^the structure.

References

Ill

SUCHOD B,, LAJZEROWICZ-BONNETEAU

J.,

J.

Phys.

I1

(1991)

553-558.

[2] ^CHION B., LAJzEROWICz-BONNETEAU J., COLLET A., JACQUES J., Acta Cryst. 832

(1976)

339.

[3) LAJZEROWICZ-BONNETEAU J., CHION B., LAJZEROWICZ J., J. Chem.

Phys.

74

(1981)

3500-3509.

[4] LAJzEROWICz-BONNETEAU J., LAJzEROWICz J. and BORDEAUX D.,

Phys.

Rev. 834

(1986)

6453-

6463.

[5] ANDRE A., FOURME R., RENAUD M., Acta Cryst. B 27

(1971)

2371.

[6] ^BENEDETTI E., PEDONE C., SIRIGU A., Acta

Cryst.

829

(1973)

730-733.

[7] ^BORDEAUX D., GAGNAIRE G., LAJzEROWICz-BONNETqAU J., Acta Cryst. C39

(1983)

466-470.

[8] This _process of molecular diffusion inside the

monocrystals

is _a local _process and is different from the long _range diffusion _process that _we observed in the phenomena of

chirality

segregation

that lead not to racemates but to

conglomerates

[4].

[9] BRUCE A. D., J. Phys. C14

(1981)

193-210.