Interactions between substitutional and orientational orders. The phase transitions in enantiomeric and racemic crystals of TMHP. III. A thermostatistical approach of the two phase transitions in the racemic solid solution of TMHP

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Interactions between substitutional and orientational orders. The phase transitions in enantiomeric and racemic crystals of TMHP. III. A thermostatistical approach of the two phase transitions in the racemic

solid solution of TMHP

J. Lajzerowicz, J. Lajzerowicz-Bonneteau, B. Suchod

To cite this version:

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

tational orders. The phase transitions in enantiomeric and racemic crystals of TMHP. III. A thermo-

statistical approach of the two phase transitions in the racemic solid solution of TMHP. Journal de

Physique I, EDP Sciences, 1991, 1 (4), pp.573-583. �10.1051/jp1:1991153�. �jpa-00246352�

Classification

Physics

Abstracts

61.50K 64.70K

Interactions _between substitutional and orientational orders.

The phase transitions in enantiomeric and racemic crystals of TMHP.

III. A thermostatistical approach of the _two phase transitions in the racemic solid solution of TMHP

J.

Lajzerowicz,

J.

Lajzerowicz-Bonneteau

and B. Suchod

Laboratoire de

Spectromdtrie physique.

Universit6

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)

Rksumk. Dans la

phase

solution solide haute

temp6rature

du

t6tram6thyl

2,2,5,5

hydroxy

3

pyrrolidine ~TMHP),

on a sur

chaque

site molbculaire soit _une moldcule droite soit _une molbcule

gauche (d6sordre

de

chiralit6) chaque

mo16cule pouvant

prendre

deux orientations diff6rentes

(d6sordre orientationnel).

_{Il y a} donc quatre ^stats

possibles

_sur

chaque

site. Une telle situation _se d6crit I l'aide de trois

paramdtres

d'ordre. L'«

6nergie

libre _»

(type Landau)

du

systdme prfisente

un terrre du troisidme ordre qui

couple

_ces trois

paramdtres.

Nous _avons 6valu6

num6riquement (m6thode

de

Kit#igorodski)

les

bnergies

d'interaction intermolbculaire _en fonction des

paramdtres

d'ordre. Ceci _{nous a}

permis

de

prbvoir,

conformbment _aux rbsultats

expbrimentaux,

_que le

paramdtre

^d'ordre d'orientation condense le

premier

^entrainant I

plus

^basse

tempbrature, grfice

aux tenures du troisidme ordre la mise _en ordre des chiralit6s. Los

changements

de

syrn6trie

trouvds et les

temp6ratures

de transition calcu16es _{sont en} accord avec les observations.

Abstract. In the

high

temperature ^racemic solid solution of

tetramethyl

2,2,5,5

hydroxy

3

pyrrolidine (TMHP),

there is on each molecular site either a

right

molecule, _{or a} left molecule

(chirality disorder)

each molecule

having

two

possible

orientations

(orientational disorder).

There _are then four

possible

states on each site. Such _a situation is described

using

three order parameters. ^The « Landau free _{energy »} of the system ^contains a third order _term that

couples

these three parameters. ^We have

numerically

evaluated

(Kitiigorodski

method) the intermolecu- lar interaction

energies

_{as a} function of the order parameters. ^{This allows} us to

predict

from the

experimental

results that the orientational order parameter condensates the first, and, because of the third order term, leads to

ordering

of the chiralities at lower temperature. The symmetry

changes

found and the calculated transition temperatures are in agreement ^{with the} observations.

1. Inhoducfion.

In the two

preceding

articles

[1, 2]

we described the structural

phase

transitions observed on

enantiomeric and racemic

crystals

of

tetramethyl 2,2,5,5 hydroxy

3

pyrrolidine,

TMHP

(Fig. I).

The

high temperature phase

of enantiomeric

crystals (I.e. crystals

formed

by

molecules of the _same

chirality,

either

right-handed

D _or left,handed _L

molecules)

exhibits

574 JOURNAL DE

PHYSIQUE

I M 4

Q9

C₄ C3

Cl C8

Cl₁ C5 c₂ Cl

~

OS

Fig. I. The molecule of TMHP.

Numbering

of the atoms.

orientational disorder of the molecules _on their sites _; the _two orientations _are deduced from

one another

by

_a

binary

_symmetry axis

(for example

Dl and

D2).

At low

temperature

(transition temperature

³⁰⁵

K)

the

crystals

_are ordered

ill.

Racemic

crystals

_are solid solutions at

high temperatures.

^On a molecular site there is either

a D molecule _{or a} L molecule. In

addition,

these molecules

display

_as for enantiomeric

crystals

an orientational disorder.

Globally

_{on a}

given

site _{one can} find either Dl _or D2 _or Ll _or L2. The local site symmetry ^{is m2m}

[3].

^{This double disorder of} orientation and of

chirality

substitution decreases with temperature.

X-ray

diffraction studies enable _us to

give

a

description

of the evolution of the

crystalline

structure with the temperature

[2].

This evolution is

complex

and it is

interesting

to go a step further with thermostatistical

considerations. Vfhat _are the

pertinent

order

parameters?

What _are the interactions involved ?..

We have

already

used _a thermostatistical

approach

to

study

two dimensional order in racemic solid solution

crystals [4]

and the

ferro-paraelectric

order-disorder

phase change

of _a

molecular

compound [5].

In this field _a

thermodynamic study

_on

camphor crystal

_may also be mentioned

involving positional

and orientational disorder of the constituent enantiomers in the

crystalline

state

[10].

It is this statistical

approach

of the

ordering

and of the

phase

transitions encountered in the TMHP racemic

crystals

that _we

report

here. We evaluate

successively

the intermolecular

potential

_energy

(empirical calculations),

the

entropy

^{and the free} energy. We shall be able then to

predict

and

explain

the

phenomena.

2. Initial

bypothesis.

We shall describe the

high temperature

structure of racemic

crystals

and

give

_some

characteristics of the low temperature

phase.

These results _were obtained

by X-ray

diffraction

[2].

At

high temperatures

the

crystals

_are

orthorhombic, having

_{space group} Cmcrn with 4 molecules in the cell _{a x} b _{x c.} The molecular sites _are numbered

1, 2,

3 and 4

(Fig. 31)

_;

and 2 _{are on}

layer

called 0

(z 0).

3 and 4 _on

layer

I

(z

~

l/2).

Sites I and 2

(respectively

3 and

4)

_are deduced

by

a translation

(1/2, 1/2, 0) (C

Bravais

lattice). Layers

0 and I _are

deduced

by

_a two fold axis

2j [001]

at the

origin.

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) (Fig. 2). Consequently

_a

given

atom of the molecule

on site I of the cell _{m can} have the coordinates _{; xyz,} iyz, xy2 or fyi. ^This can be summarized

by

_{a;~ x, y,}

p;~

z with _a~~

= ± I and

p;~

= ± l.

We choose for Dl a;~ =

p;~

₌ l and denote

Di by (+

,

+

'),

_a,~

=

l, p;~

= l for

Ll,

with Ll denoted

by (-

,

+

),

and then D2 is

(-

,

and L2 is

(+

,

).

Each molecular site is _a thus 4 state

system

:

(+

,

+

), (-, ), (+

,

)

and

(-

,

+

).

It is

important

to note that

1~~ ^{= a}j

p~ represents

the

chirality

of the

j-th

molecule

independently

of site I and of the cell

y

2 ^x

i j

Fig.

2. Molecular disorder of the site of symmetry ^m2m with _mean

occupation

Dl

(25 fb) D2(25 fb) Ll(25

fb) L2(25 fb).

I) Projection

on the

plane

_{yz :} ( DlLl

(---)

D2L2.

II) Projection

_on

the

plane

_xy

( )

DlL2

(---)

D2Ll.

O

~

l -1 3 -3

b 2 4

1

fit

^~ x 3 -3

Layer

0(2zo)

Layer

1(2=1/2)

8

~~ ~~ ~~,

22 21 ₄₂ 41 42' 41'

2a 11 31 31'

2i 22'

~~2'~

-2b-

2=0 z=1/2 2« -1/2

~

Ii1

.020

all(a31+a31')

+0.027 al1("21+"22+"21'+"22') _+0.098 ~ll(fl32+~32')

-0.011 )11(~21+~22+~21'+~22') _+0.l10 all(U4l+U42+U4l'+U42')

Layer 0 +0.044 RI1(fl21+fl22+~21'+~22') LaYC" -@,014 II1(~41+~42+~41'+~42')

zY0 -0.046 all("12+"12')

zy ±1j2 +0.043 all(~41-~41'-~42+~42')

+0.100 )11(~12+~12') -0.043 $11(a41-"41'-"42+"42')

Fig.

3. I)

Numbering

of the four molecular sites of the cell _ax b _{x c.} II)

Numbering

of the 14 molecular sites j nearest to site 11. The 8 molecular sites of the low temperature

phase:

cell

2ax2b _xc, lattice C, are 11, 12, 21, 22

(layer

0, z~0) and 31, 32, 41, 42

(layer

I,

z

l/2). III)

The different terms of the intermolecular potential _energy

(kcal/mole)

for the 14 molecular

pairs

I

I-j.

m l~j = + I

corresponds

to _a D and

1~~ ⁼ to _a L molecule. a~,

pj

^and

1~~ ^are the three order

parameters

necessary ^to describe this four _states

system [13]. They belong

to three

irreducible

representations

of the local site

symmetry

^{of the}

high temperature phase.

In the

high

temperature disordered

phase

the _mean values for _one site I _are

(a;~)

=

0, (p;~)

=

0, (1~;~)

=

0,

I

representing

_any of the sites

1, 2,

3 _or 4. In the low

temperature phase

these

spatial

_mean values _{are no}

longer

_zero.

We _assume

that,

whatever the

temperature,

^the conformation of the molecules

(bond

lengths

and

angles,

dihedral

angles)

remains the _same and the

position

and orientation

parameters

of the molecule _on their site also remain the _{same ;} but the four states

previously

described will _no

longer

be

possible

_{on a}

given

site.

Analysis

of certain

X-ray

diffraction intensities

justifies

these

hypotheses [2].

For the low

temperature phase

_{we assume} also that the cell parameters are 2 _{a x} 2 b _{x c} and that the Bravais lattice is C. These

again

are results of _an

X-ray

diffraction

investigation,

mentioned in reference

[2].

^{We label the}

eight

molecular sites in this C cell

according

to

figure

3II.

Starting

from the

high

temperature

phase

and

using only

these

hypotheses

_we

intend to examine how the

occupation

of these different sites varies with temperature. ^It

would be

interesting

to be able to deduce the temperature

dependence

of different

surstructure

peak

intensities

([2], Fig. 3).

3. Intermolecular

potential

_energy

@dgh temperature phase).

3.I DIRECT CALCULATION. TO calculate the intermolecular

potential

_{energy One} must

consider the

following energies [6-7]

_:

1)

^Van der Waals _energy

2)

Electrostatic _energy

(mostly dipolar interactions,

the

dipoles being

localized _on the NO

group)

3) Hydrogen bonding

_energy.

(The

molecules deduced

by

the translation b _are related

by hydrogen

bond between the _group

NO(6)

of _one molecule and

O'(9)H

of the

following (Fig. I)).

As _{we are} interested in energy differences _we do _not take into _account

energies

2 and 3. In effect in the structure, whatever be the molecule

occupying

the site I, the NO group is almost

parallel

to the y axis and the atomic coordinates of atoms N and O do not

depend

_on

a, and

p;.

We then _assume that both the electrostatic _energy and the energy of

hydrogen

bonds _{vary very} little

during

the

ordering (between

two

neighbouring

molecules I and

j

^the intermolecular distances

O(9)...

O

(61'

varies between

2,68h

_and

2,73h

_{and the}

angles defining

the geometry ^of the bond vary

by

less than 5° when _a;, a~,

p;, p~

^{take the values} + I _or I. We

already

had to deal with _a similar situation

[4]).

For the Van der Waals interaction _energy between two atoms m, and

nj separated by

r~~ we use the

expression arjj~

_{+ b exp}

(-

_cr~ with the _{constants a,} b and _c

given by G1glio [8].

We used the program PACK of

Goldberg

and Schrnueli

[9].

The summation radius is 8

J~.

Our stated

potential

_energy includes

only

Van der Waals interactions

(we neglect

interactions

between N-O and O' atoms

coupled by hydrogen bonds).

For two molecules located at I and

j

the most

general form

of intermolecular

potential

energy is _:

U,y=A+Ba,+Ca~+Dp,+Ep~+Fa;aj+Gp;p~+Ha;p~+Ia~p;+Ja,p;

+Kajp j+La;a~p;+Ma;a~pj+Na;p;pj+Pa~p;p~+Qa;ajp;p~ (1) (this

is due to the values ± I for the variables _a and

p).

To calculate these different coefficients

A,

B... for each

pair (ij ),

it is

only

_{necessary to put}

in I and

j altematively

the different molecules

Dl, D2, Ll,

L2. One _can do this for all

pairs (I, j ),

where I is

fixed,

for

example

I

= I I and

j

^variable : this will

mainly

involves 14

pairs (I I, j ) (Fig. 2II).

The maximum value found for a

pair

is 4.2

kcal/mole

for Dl in I I and Ll in 21. The energy calculated for

pairs, beyond

number

14,

_are less than 0.01

kcal/mole (we

must

remember that these values have little absolute

significance

_:

only

variations _are

meaningful).

Actually,

with _no

hypothesis

about the cell _or

symmetry (I.e.

invariance under translation

or

symmetry operations) grouping

of _some

pairs

_{seems necessary.} For

example

the _energy of the two

pairs

11-31 and 11-3l'consists of

only

six _{terms :}

~ ₊

~(fl31 fl31')

+ ~"

lI("31

₊

"31')

+

l~fl1)(fl31

+

fl31')

₊

+

l~i"11 "31(fill fl3I)

_"

II

"31'(fill #31')1

₊ l~"

Ii

fill ("31fl3I

+

"3I'fl3I')

To obtain the total intermolecular

potential

_energy it is necessary ^to add these

(I I, j ) pair

contributions and then

repeat

the

operation

after

changing

the

origin

molecule II.

In the final

expression

linear terms like

B(p~j p~j,) disappear,

and if _we retain

only

terms greater ^than

0,01kcal/mole,

12 _terms remain _one is constant and eleven

depend

_on the

different parameters

(Fig. 3III).

The

only

terms that arise in the

expression

of the intermolecular

potential

_energy are terms

quadratic

in _a,

p,

_~

(remember

that _a;

p;

=

~;).

This is _a consequence of the

hypothesis

that

only

the

parameters

_a,

p

_{or ~} can vary from _one site to another rather than the

geometrical _parameters

of conformation and

position.

In

particular,

this eliminates third order _terms like a;

p~

1~~ that involve three

sites, representing

for

example

the

dependence

of the

chirality

_1~~ of the molecule in site k _on the

parameters

_a; and

p~

^{of molecules in sites}

I and

j.

3.2 INTERACTION ENERGY IN FOURIER SPACE. In the

following

_We Shall _use the

primitive

cell A

=

(a

₊ b

)/2

_; B

=

(a b)/2

_; C

= c

(Fig. 41)

^{instead of the cell} a,

b,

_c ^which is face- centered C. This

primitive

cell contains two

molecules,

_one on site I

(Fig. 31, layer 0,

z

0),

the other _on site 3

(layer I,

_z

l/2).

Each site I at

position

_r; in the

crystal being

characterized

by

the three parameters a;,

p;,

_{1~,, we} can define for the

crystal

three order parameters

(these

_are

densities)

a(r)

=

jja, 8(r-r;) P(r)

"

£ P< 8(r r;) (2)

1~(r)

=

jjl~, 8(r-r;).

y x

b

B A

a

y x

~tX-tX-tX~

~~

~~~'~~~~~

Fig.

4.

I)

The

high

temperature

elementary

cell A _x B _x C.

II)

Condensation of the _a parameter at the first transition.

The Fourier transform of these order parameters is calculated _; for

example

for

a :

«(q)=~£«;exP2iarq.r;

i

where

q(qyqyqz) represents

_a

reciprocal

_space vector

(not

_a

reciprocal

lattice

vector).

In

fact,

_as there _are 2 molecules _per cell

(layers

0 and

I),

_we have to consider six order

parameters

:

ao(q), a~(q), po(q), p~(q), l~o(q), 1~~(q).

^As has

already

been concluded from

X-rays

studies

[2],

_no modification _occurs in the _z direction _on

lowering

the temperature.

Hence the vectors q will be limited to

q(qx,

_qY,

0)

of the

(X, Y) plane.

Since the energy is _a

quadratic

form of the order parameters it is easy to find its

expression

in q space from the data

given

in

figure

3. For

example

the terms in

a

(q)

in this interaction energy

expression

_{are :}

U

=

£ (A(q)I«o(q) «o(-

_{q + «}

i(q) «I(- q)I

+

K(q)I«o(q) «I(- q)1

₊

+ K*

(q)I«o(- q) «I(q)1)

with

A

(q )

=

[0.027(cos

2 orqy + cos 2 _orq

y)

0.046 _cos 2 _or

(qy

+ q

y)]

_x 2

K(q)

=

[0.020

+ 0.110

(exp

2 _I orq _y + exp 2

iorq y)]

x 2 ~~~

N

being

the number of cells of the

crystal.

The _sum

jj

is taken _over the Brillouin _zone. It is to be noted that _a

(q )

is related

directly

to

the _mean values of linear combinations of the _a, evaluated from the

crystallographic

investigation.

For

example ao(0, 0,

0 is the _mean value of

(aj

+

a~), a,(0, 0,

0

)

that of

(a~

₊

a~)

and

ao(0,1/2, 0)

that of

(aj a~)...

It is

noteworthy

that the coefficients _are

symmetrical

in qy, qy in this

expression.

4. Calculation of enhopy ^{and free} energy.

We consider in the

crystal

_a small volume AV

containing

_n

molecules,

where _n is small

compared

to

N,

the total number of molecules in the

crystil,

_but

_{large compared}

_to the number of molecules in the cell.

We have

already

mentioned

~paragraph2)

that each molecular site is _a 4 state

system

:

(+,

₊

), (+

,

)(-,

₊

)(-, ).

If _among the _n molecules considered _we call

n(+

,

+

),

n

(+

,

), n(-

,

+

)

and n

(-

,

the number of molecules in each state, ^then :

n(+,+ )+n(+,- )+n(-,+ )+n(-,-

_=n

n(+,+ )+n(+,- )-n(-,+ )-n(-,-

= an

n(+,+ )-n(+,- )+n(-,+ )-n(-,- ) =fin

~~~

n(+,+ )-n(+,- )-n(-,+ )+n(-,- )

_=l~n.

Hence

n(+,+ ) =n(I+«+p+1~)/4 n(+,- ) =n(I+a-p-1~)/4...

On each site I _one has

always

_~,

= a,

p;,

but tills relation is not valid for the local _mean values _a,

p,

_~, ^i-e- _~ ~

up.

In the

spirit

of the

Bragg

Williams

approximation,

the

following expression

can be written for the entropy

S

=

k

Log

^~"

(k

is the Boltzman constant

~

(+

,

+ )" ~

(+

,

)" ~

(~

,

+ )" ~

(~

,

l'

For small values of the order

parameters

^the

expansion

is

S=

-kn[1/2(a~+ p~+1~~) _apl~ +1/2(a~p~+P~I~~+ a~~~)+

+

1/12(a~

₊

P ~+

1~~) +

].

It is

important

to notice and this is _one

of

the main

points of

the present

investigation

the presence of the third order term

up1~.

This is due to the fact that the

product

of the three

representations

to which _a,

p

and

1~

belong

includes the

identity.

The

quadratic part

^{of the} entropy ^is

easily expanded

in Fourier space. The third order term will

give

rise to terms such _{as a}

(q~) p (qp)1~ (q~)

with _:

(translation invariance)

_q

~

+ q_p +

q~

= 0.

(5)

The free energy is obtained

by calculating

F

= U TS. This

expression

is _an

approximation because, firstly, only

the second order terms _are present ⁱⁿ

U,

and

secondly,

the

expansion

for S is valid

only

for small values of _a,

p

^and 1~.

5.

Analysis

of the results.

5. I HIGH TEMPERATURE PHASE TRANSITION. In _a first Stage _We limit _Our discussion to the

quadratic

_part of the free energy. Terms in _a,

p

and _{1~ are}

independent.

For

a it is easy to

diagonalize U~ (formula (3));

the lowest

eigenvalue

is

E(q)

=

A(q)- [K(q)K*(q)]"~

Its minimum value

E(q)~i~

_occurs at qy

=1/2

and qy

=1/2.

The order

parameter

^associated with the lowest

eigenvalue

is such that

ao(1/2, 1/2, 0)

= a,

(1/2, 1/2, 0).

The

corresponding phase

transition temperature ^{is such} that

kT~

=

E~~(q). Numerically T~

₌ ^{300 K. The} same discussion for

p

and

1~ shows that at

T~

^these

parameters

are far removed from their instabilities. Then at 300 K the

system undergoes

_a

phase

transition with

a ~

0,

while still

p

= 1~ = 0 and

q(a)

=

1/2,1/2,

0.

The structure is shown in

figure

4II. The cell volume doubles. A site

having

_a

positive

_mean

value of parameter a is _one

occupied preferentially by

Dl _or L2 the orientational

disorder, relating

for

example

Dl _to

D2,

is _no

longer

total.

Inside the

crystal

there will be

antiphase

domains with _{a ~} 0 and _{a <} 0.

The agreement is very

satisfactory

between these results and those obtained

by crystallogra- phy [2], namely ao(q)

=

aj(q),

^the same

description

of the order

parameters,

^{and close}

agreement

^for the transition temperature

(300

K

compared

with 325

K).

It is

tempting

to

apply

the energy coefficients used here to the transition observed in the enantiomeric

crystals (see appendix).

5.2 Low TEMPERATURE PHASE TRANSITION. What _are the consequences of the

a condensation ? To discuss this

point

_we ^shall

replace

_a in the free _energy

expression by

its

mean

value,

denoted

(

_a

)

Firstly

_we will examine

only

the

quadratic

terms in

p

and _1~..

(once again,

to be

rigorous

one should consider

po, pi,

_'lo and _l~i but since the

following generalities

remain valid _we shall consider

only

_a and

p, thereby simplifying

the

expressions).

One _can write _:

F=N/2jB(q~)p(qp)p(-qp)+c(q~)~(q~)~(-q~)+D(q, j«j)p(q~)~(q~)+

+D(-q, I«))P(-qp)7J(-q~)1. (6)

Five

points

_are worth

noting

_:

* The

symmetry change

induced

by

the

a condensation makes

p

^and _1~

belong

to the _same

representation

_;

* The coefficients

B(q)

and

C(q)

come

firstly

from the

quadratic part

^{of the initial} free

energy and of the entropy ^and

secondly

from

higher

order terms like

la)2 pi

la )~ p

^~

la )

is _no

longer

_a small

quantity

and these last terms cannot be

neglected

_{; we}

were not

able, however,

to evaluate

numerically

their contribution.

* The third order

entropic

term in

apl~, already mentioned,

is the _cause of the linear

coupling

between

p

^and _1~. The coefficient D is

directly proportional

to

la )

and _can be written

D(q, la )

=

D'(q)(a (q~))

its

sign changes

from _one domain

la )

to the other

la ).

As _we

neglected

third order terms

contributing

_to the Van der Waals interactions

~paragraph 3.I),

the numerical value of D is not

completely

reliable.

* Due _to the orientational

symmetry

of the

high temperature phase

the free energy is invariant _on

permuting

_qy and qy; for

example B(qy, qy)

₌

B(qy, qy).

* As

la )

is modulated in the _space

(q~

=

1/2, 1/2,

0

)

and _{q~ + q}

p + q

~ ⁼

0

(formula (5)) (except

for _a

reciprocal

lattice

translation)

then

qp

+ q~ =

(1/2,1/2, 0).

Writing

from

(6)

the

equilibrium

conditions for P ^and 1~, we obtain

equations

of the

following

form

BP (qp

# D' (tY

(-

_q~

))

_7~

(-

_q

~ +

higher

Orders

c _~

(q~

= D'

la (-

_q

~

)) p (-

_{q +}

higher

orders

These

equations

look like those

describing parametric amplification

and oscillations in electronics and in _non linear

optics,

where

(a) represents

the pump field. Above _a

given

threshold of this _pump field it will induce self oscillation in the

p

and ~ fields

[11].

The condition _{q~ +}

qp

+

q~

=

0 will

correspond

to the

phase matching

condition

(of

_course for

optical phenomena

_an additive condition between the

frequencies

is

needed).

This

type

^of mechanism for the

phase

transition _can thus be called

parametric.

From the

crystallographic analysis [2]

two

possible

solutions _were found q_p =

(0, 1/2,

0

),

_q

~ ⁼

(l/2, 0,

0

)

and _q

p ⁼

(1/2, 0,

0

),

_q

~ ⁼

(0, 1/2,

0

)

These modes

satisfy

the

previous

^condition

_qp

_{+ q}

~ ⁼

l/2, 1/2, 0) and, through

the fourth

point

^mentioned

above, they

contribute

equally

to the free energy. We limit _our discussion to these modes. Moreover in order to

simplify

the

writing

we shall _not mention anymore the qz coordinate

(qz

=

0).

The

quadratic

_part of the energy can thus _now be written _: F

=

N/2[Bp ~(0, 1/2)

₊

CI~ ~(l/2, ₀₎

₊ 2

Dp (0, 1/2)

1~

(1/2, 0)

₊

+

Bp ~(l/2, 0)

₊

CI~ ~(0, 1/2)

+ 2

Dp (1/2, 0)

1~

(0, 1/2)] (7)

These two

independent (at

this

level) quadratic

forms must be

diagonalhed

_:

(Gill')+

_{~~'(I +}

~fi~fi)+ _~fi(1)

with

1l'j

=

p (0, 1/2)

cos fJ _{+ ~}

l/2, 0)

sin 8

4ij

=

p (0, 1/2)

sin fJ _{+ ~}

(l/2, 0)

_cos fJ

the _same for 1l'~ and

4i~

^with

p (1/2, 0)

and _~

(0,1/2) G,

H

=

1/2 ((B

₊ C

)

_±

[(B

C )~ + 4 D ~]~'~)

tg28=2D/(C-B)

1l' and

4i,

the _new order parameters, are linear combinations of

p

and _1~. Since

fJ

depends

_on the

temperature,

^the

eigenvectors

_are not fixed

by

the symmetry.

1l'i

and ~Pj

belong

_to the _same

representation.

When G

=

0,

_a second

phase

transition _occurs.

If _we allow _{as we} have done in this article and in

agreement

^{with the}

experimental

results the solution

qp

=

(0, 1/2)

and

q~

=

(1/2, 0)

vith

po

=

pi

and l~o = 1~ j, and if _we

replace

la ) by

its observed value of almost 0A at 285 K

[2],

^then

taking

into _account the

quadratic

terms that _we have

already evaluated,

_we find

Tp,

~ ^~

275 K. This

temperature

^is very close _to the

experimental

_one

(285 K).

G

=

0 vill induce condensation of

1l'j and/or P~.

Indeed three different situations _can arise

according

to the

higher

order terms _:

a)1l'j~0 1l'~=0

b) Pi

=0

P~~0

c)1l'j=±0 1l'~~0.

Cases _a and b

correspond

to different ordered

phases

with the _same energy, while _{case c}

corresponds

to a situation of different energy.

Clearly

the condition of lowest energy ^must

prevail.

We _are not able to find

this, since,

_as

explained above,

_we do not have the _exact numerical values of the different coefficients.

The

interpretation

of the

X-ray

diffraction data

[2]

^led to solutions a and b. In the

crystal

zones would then exist in which

Pi

~ 0 and other zones in which 1l'~ ~ 0. Let _us consider _a

zone in which

Pi

~

0,

when the second

phase

transition _occurs and G becomes

negative.

~Pj ^{is also different from} zero. The

largest

terms of fourth order in the energy are those in

PI

and

1l'/~Pj

_near the

transition,

when

1l'j

^is

proportional

to

(G(~'~,

~Pj ^{will be}

proportional

to

1l'/

~

G ^~'~ The _same

reasoning

can be used for

regions

characterized

by

the parameters 1l'~ and ~P~.

Below this second transition

eight

different domains _can appear in the

crystal they

_are characterized

by

_{± a, ±}

(1l'j

vith 1l'~ ₌ ⁰ or ±

P~(

with

Pi

₌ ^0. As stated

already,

the existence of these _numerous domains may

possibly

be the _cause of the

peak

width observed for the surstructure reflections

[2].

We have thus described the low

temperature phase

transition

by

the

ordering

of the chiralities. This

ordering

_appears when _a

(the

orientational

parameter)

has condensed. It _can be said that _a

triggers

this transition.

6. Conclusion.

In this _{paper we}

present

a theoretical

analysis

of the two

phase

transitions that _occur

successively

when the

system

goes from the

high

temperature ^to the low

temperature phase.

This

analysis

accounts for the main features of the

phase

transitions.

In the

high temperature phase

the system ^has two types ^{of disorder} :

orientational,

and substitutional. To describe this situation _we have to introduce three order parameters with _a third order

coupling

between them. We _are

able,

to a certain extent, to evaluate

numerically

the

quadratic

part of the interaction _{as a} function of the order

parameters (knowing

the

structure of the HT

phase)

and to make

predictions

about the nature of'the order and the

phase

transition

temperatures

^~vith

relatively good

_success.

In

spite

of this rather

satisfactory general picture,

several

experimental

results mentionned in the

previous

article

[2]

_are still not

fully

understood _:

the

analysis

of the

experimental

results and theoretical considerations lead _us to conclude that the low

temperature phase

is

monoclinic,

but _no monoclinic

angular

deviation of the orthorhombic lattice _was detected. This may be due to the fact that the monoclinic deviation» is

proportional

to the square of the low temperature

phase

order

parameter

which is _{never very}

large (see below)

_;

we have _no clear

interpretation

for the

temperature dependence

of the vidths and intensities of the

superstructure spots (type II)

associated with the low

temperature phase.

The behaviour is reminiscent of critical

phenomena [12],

but

except

^{for certain}

pathological

cases

(low dimensionality

for

example)

it is difficult to conceive of such

large

effects _over such

a wide

temperature

^interval. We therefore

suggested [2]

that

they

_may be due to the

large degeneracy

of the system

(8 types

^of

domains)

with _numerous

possibilities

of domain

walls,

but this

explanation

is not

entirely satisfactory

either

the order

parameters

saturate at rather low values

(ca. 0.4-0.5).

We

suspect

that when the low temperature ^{transition takes}

place

_on

cooling,

the variation of _a

(the

orientational

parameter)

is

blocked,

in turn

blocking

the parameters

p

and

1~ associated with the other

phase

transition. Our theoretical

discussion,

limited to the

quadratic

term

only,

is of _course unable to describe this

phenomenon.

It is the

growth

of the order parameter a that

triggers

the low

temperature transition,

but it is conceivable that _a

positive higher

order term will induce _a

type

^of frustration which _can account for the observed behaviour.

Finally

it cannot be excluded that the chosen vectors q~,

qp, q~

are not the correct _ones.

May

be

they

have _a component

parallel

to _oz ? This could induce _a modulated structure,

mixing

in very definite way the twins _we

already

described. Extra reflections would then

appear: these _were not

observed,

but _{we may} not have looked for them

systematically

enough.

Acknowledgments.

We want to thank G. Comrnandeur for

synthetizing

the

compounds,

G. d'Assenza for his

help

in

X-Ray experiments

and E. Geissler for

rereading

and

correcting

the

English.

Appendix.

We have

already

mentioned several times

[1,2]

that the

high temperature

structures of racemic

crystals (solid solutions)

and enantiomer _{are very} close _: cell

parameters, position

and

orientation of the molecules _are similar. In these conditions it is

interesting

to use for the enantiomer the numerical calculations

presented

here for the solid solutions.

In the

expression

for the

potential

_energy U _one must set _a

=

p

and _1~

=

0,

and in that for

the

entropy, p

_{= ~ =} ^{0. The} coefficient of the _a ^~ term in the

expression

for the free energy is then

(Fig. 3III)

_:

A

(q)

₊ kT

= 0.040 ₊ 0.224

(cos

2 orqx + cos 2 orq

y)

₊ 0.054 _cos 2 _or

(qy

_{+ q}

y)

₊ kT.

An

instability

will appear for _a q ^vector that mininfizes A

(q)

_; one finds qy = q _y =

1/2.

The

corresponding

temperature

kT~

= minimum value of

A(q)

is 150 K.

In

spite

of _non identical structural

parameters,

the correct

ordering

of the low

temperature phase

is found the calculated transition

temperature, although

far from the

experimental

one

(305 K),

is of the

right

order of

magnitude.

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