Phase Pinning by EPR Probe in Biphenyl Doped with Naphthalene

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Naphthalene

A. Veron, J. Emery

To cite this version:

A. Veron, J. Emery. Phase Pinning by EPR Probe in Biphenyl Doped with Naphthalene. Journal de

Physique I, EDP Sciences, 1997, 7 (8), pp.977-1001. �10.1051/jp1:1997199�. �jpa-00247379�

Phase Pinning by EPR Probe in Biphenyl Doped with

Naphthalene

A. Veron

(*)

and J.

Emery

Laboratoire de Physique de

I'#tat

_{Condens4 (**1,}

avenue Olivier Messiaen,

72085 le Mans Cedex 9, France

(Received

22 January1997, received in final form I April 1997, accepted 21 April 1997)

PACS.61.72.-y Defects and impurities iii crystals; microstructure PACS.64.70.Kb Solid-solid transitiou8

PACS.64.70.Rh Commensurate-incommeu8urate tran8itions

Abstract. The naphthalene Electronic Paramagnetic Re80nance probe used to study the

phase transitions in biphenyl does not account for the plane _wave modulation of the incommen- surate phase II. Its EPR spectra yields _a phase distribution which looks like _a "multi-soliton regime". Moreover, the splitting between the edge singularities is not symmetrical, in contra-

diction with _a linear _one. This behaviour is not exhibited by the phenanthrene EPR probe

wbose spectra account well enough for the plane _wave modulation with _a linear coupling to the order parameter. To analyse the defect behaviour of the probe, _we first introduce _a coupling between the probe and the modulation _wave and determine the phase distribution within the

phenomenological Landau theory. Furthermore, _a calculation based _on intermolecular interac- tion is performed in order to describe the microscopic origin of this distribution and is applied

to naphthalene and phenanthrene molecular probes.

1. Introduction

In _an incommensurate system, ^the

phase

correlation radius of the modulation is "infinite" _even far from the

phase

transition, ^but the defects which

couple

to the modulation break this

long

range order [I] ^and can

impose

a

phase

value which minimize their energy. The

experimental

data

concerning

the influence of defects _on the

properties

of incommensurate

phase

refer

mostly

to the multi-soliton

regime [1-3].

Generally,

the Electron

Paramagnetic

Resonance

(EPR) probes

introduced diluted in _a dia-

magnetic

material _are chosen in such _{a way} that

they

do not

modify

the

crystalline

cell: the relevant parameters are the

charge

and the ionic radius. Such _a situation is encountered in

Rb2ZnC14 doped

with Mn2+ _which substitutes Zn2+ [4]. ^{In another} case, the

paramagnetic

ion fits the size of the substituted ion but needs _a

charge compensation

_as in

ThBr4

_or ThC14 where

Gd~+

substitutes Th~+ with _a

charge

compensation on the Br~ _or Cl~ site

[5,6].

^{In all}

these _cases, the defect is _{a non} extensive one,

and,

in the incommensurate

phase,

these EPR

probes

account for the modulation.

(*) Author for correspondence

(e-mail: veronfllola.univ-lemans.fr)

(** UPRES-A 6087

© Le8

#dition8

_{de Physique 1997}

The theoretical

study

of EPR lines in _an incommensurate

phase

is

complicated. Nevertheless,

Blinc [7j gave a

simplified

method to reconstruct the _resonance spectra. ^{At the} transition, each

high

temperature line

gives

rise _{to a} distribution that is the _sum of local lines whose width and

position depend

_on the local

phase

of the modulation:

1(r)=k.r+io (i)

where

#

takes all the values between 0 and 21r module 21r.

Therefore, I(H),

the

intensity

of the incommensurate line for _a

magnetic

field

H,

is

given by:

j+~ jH ^H(#)

^~

~~~~

-n

~~~~~

t(#) ^~~

where the function

j

^~

~~~

t(4 (~)

describes the local line shape. It is _an individual line

resulting

from _an EPR

probe

which

resonates for _a

magnetic

field

H(#). I(#)

_represents its

width,

and

#

is the local

phase

of

the modulation.

P(#)

is the distribution of the

phases

# which accounts for the

regime

of the modulation:

P(#)

is constant in the

plane

_wave

regime

^and is

given by

the sine-Gordon

equation ⁱⁿ the multi-soliton

regime

_[7]. ^In all _cases the

phase

values between 0 and 21r _are taken into account and the EPR spectra account for the

phase

distribution.

In the

microscopic

description, a local line position is defined

by:

H(4)

= Ho +

hilt(4)

₊

h~it~(4); i1(4)

₌ ^A

cos(4

+

40). (4)

It is _{as a} function _{up to} second order of the local

amplitude tt(#)

= A

cos(#

₊

#o)

^{of the order}

parameter. Ho is the line

position

in the

high

temperature

phase

and A is the order parameter

amplitude.

For _a molecular

crystal

such _as

biphenyl,

the EPR

probes

_are molecules with

large

extensions which

nearly modify

half of _a cell.

Therefore,

_we expect them to act on the

phase

^{of the}

modulation _as defects would.

Figure

la shows _an EPR spectrum ^of

phenanthrene-hio

in the incommensurate

phase

II of

biphenyl.

^This spectrum exhibits the features of _a

plane

_wave modulation [8].

Figure

16 shows the EPR spectrum ^of

naphthalene-d8

in the _same conditions:

this spectrum ^is

typical

of _a multi-soliton

regime

_[7] while the modulation is _a

plane

_{wave one [9].}

Hence,

the two molecular

probes

do not exhibit the _same

behaviour,

and the

naphthalene

_one

yields

_a defect character.

This paper devoted to this

problem

is outlined _as follows. In Section 2, _we recall the _proper- ties of the

biphenyl crystal,

the

experimental

method and the characteristics of the EPR

probes

in the normal

phase.

In Section 3, _we present the

experimental

results for

phenanthrene

and

naphthalene

in the incommensurate

phase II,

and _we discuss the

particular

behaviour of the

naphthalene

used _as the EPR probe. In Section 4, we determine the

phase

distribution of the EPR

probe

and

apply

it to simulate the

experimental

spectra. ^In the last part, we describe the

microscopic origin

of the

probe

behaviour.

2. Characteristic Features of

Biphenyl

and EPR Probes

2.I. BIPHENYL PROPERTIES. The

biphenyl

molecular

crystal

with the chemical formula

Ci2Hio consisting

of two

phenyl rings

connected

by

_a

single

^C-C

bond,

has been the

object

of extensive studies

during

^{these last few} years. At _room temperature, ^{in the} normal phase

al bl

i ~~

~ ~

~ ~

3490 3560 3630 4225 4275 4325

Magnct,c fidd (Gauss) Magnct,c fidd (Gauss)

Fig. 1. Typical lines in the incommensurate pha8e II of bipheuyl:

a)

with the phenanthrene-hio probe;

b)

with the uaphthalene-die probe.

r~

fi ^b

a

Fig. 2. Crystal structure of biphenyl iii the high temperature phase ^I: _{a =} 8.12

1,

_b

= 5.63

1,

c = 9.51

I,

_fl

= 95.1°, from reference [12].

(named phase I),

the

crystal

symmetry is

P21la

with two molecules in the unit cell

(Fig. 2).

When it is cooled from the

high

temperature

phase,

the

crystal

exhibits two structural

phase

transitions toward _some incommensurate states at TI

= 40 K and TII _# 17 K

[9,10].

These transitions _are characterized

by

_a modulated twist of the two

phenyl rings

around the

long

molecular axis.

At TI ₌ 40

K,

_a second order

phase

transition _occurs and the

crystal

becomes incommensu-

rate. The order parameter has four components

(n

₌

4) [9-11]

and the _wave vector star has

four _arms:

~

~j

^{~ j} _{~~ ~}

(i

db)~~

_~

~

~j

^~~ j ~~~ ~

(i db)~~

~'~~ " ~~ ~~ ~~~

2 ^{~"~ ~ ~} 2

for this

complicated

phase

(named phase II).

It is worth

noting

that the

high

order satellites

were not observed in this

phase,

which _means that the modulation is _a

plane

_{wave one}

[12,13].

After _a first order

phase

transition which _occurs at

TII

" 17

K,

the

crystal

remains

(down

to

very low

temperature)

in _a second incommensurate

phase (namely phase III)

whose modulation

wave vector is:

+ q5 #

~~ ~~~b*

(6)

2 and exhibits _some satellites _{up to} the third order.

These

phase

transitions _were

extensively

studied

by

electronic

absorption

and emission spec- tra in pure

biphenyl-d12 (14],

neutron

scattering [9,10,12,13], X-ray scattering [15,16]

and

Raman

scattering [17,18].

Local

investigations

_were also

performed

such _as Nuclear

Magnetic

Resonance

(NMR) [19,20]

and Electronic

Paramagnetic

Resonance

[21-24]. Nevertheless,

^the first

investigations by

EPR

[21,22]

_were

interpreted

without any reference to the incommensu-

rate characteristics of the states.

There _are two ways ^to consider _{an n}

= 4 order parameter:

. a

superposition

of two

displacement fields,

^each _one associated with the _wave vectors

+q~i

and +q~2

(2q

modulated

system);

. a bi-domain _structure

where,

in each

domain,

the

displacement

is associated with the _wave vectors +q~i _or

+q~2 (lq

modulated

system).

Several _neutron

scattering [10,11]

and the Raman

scattering

[25] ^{studies showed the}

lq

bi- domain structure of the incommensurate

phase

II. A second class of

studies,

which

gives

_more direct evidence of _this

behaviour,

is concerned with local

techniques:

NMR [26]

experiments performed

_on

hydrogen

nuclei and EPR studies

[23,

24] ^without any ^stress. The _more direct

proof

of the bi-domain _{structure was}

performed by comparing

an EPR

study

without and under stresses [27]. ^This

experiment

showed _a

change

in the spectrum when stress _was

applied,

demonstrating

the

disappearance

of _one domain.

2.2. EXPERIMENTAL METHOD. In order to

perform

_an EPR

experiment

_on

diamagnetic

biphenyl

_we had _to include _some

probes

in the _pure

biphenyl

in order to simulate conditions in the

pioneer

work [28]. ^The

naphthalene (or phenanthrene)

diluted molecules substitute the

biphenyl

_one with _a concentration

equal

to 0.5%. The

biphenyl crystal

was grown

by lowering

the melt

through

_a temperature

gradient (Bridgman method).

The

products, biphenyl, naph-

thalene and

phenanthrene,

_were

purchased

^from the Aldrich _company and _were

purified

before

use.

The

ground

state in both

naphthalene

and

phenanthrene

is not

paramagnetic. Therefore,

the EPR

experiments

_were

performed

in the lowest

triplet

state of these molecules [29]. ^The

particular

device to used to

perform

these

experiments

is described elsewhere

[24, 30],

EPR measurements were

performed

_{on an} X-band

spectrometer (9.5 GHz) equipped

^with a cryostat.

The

crystal

_was cooled down

by

helium gas

flowing

in _a Dewar inside the cavity. The precise temperature ^{of the}

sample

_may differ because of the temperature

gradient

of the flow.

2.3. PROPERTIES oF THE EPR PROBE. In the normal

phase,

two systems of

lifes

are

observed [8, 24] ^which

correspond

to the two

inequivalent naphthalene (or phenanthrene)

Table I. Orientation

of

the

naphthalene f2$j (Na)

and

phenanthrene f23j (Ph)

spin Hamil- tonian and

of

the

biphenyl f31j (Bi)

^molecttlar azis

(8M,

4lM

)

in the

crystal frame

_a,

b,

c*.

8Na

4~Na

8Bi

4~Bi

8ph

_4~Ph

Oz

axis 18° 7° 17° -5° 16° 0°

oy

axis 95° -63° 101° -58° 98° -63°

Oz

axis 108° 38° 103° 35° 105° 29°

molecules in the cell. Three lines for each molecule _are

typical

for _a S

= I spin: ^the Ams ₌ 2 transition in low field and the two

Ams

= I transitions in

high

field. The

anisotropy

_curves which describe the line

positions

_verstts the direction of the static

magnetic

^field in the three

crystallographic planes, give

the

spin

Hamiltonian parameters ^defined

by:

m=2

7lo ₌

flS§H

+

£ _{Bfof (7)}

m=-2

and the orientation of the EPR

probes

in the

biphenyl

host

crystal.

In the molecular axis system ^the

spin

Hamiltonian parameters _are:

B(

= 340 x 10~~

cm~~; B]

= -150 _x 10~~ cm~~ _{for the}

naphthalene

_[24]

probe

and

B(

= 354 _x

10~~ cm~~; B]

= -484 _x

10~~ cm~~

for the

phenanthrene

_{one [8].}

The

§

tensor is

isotropic

with the free electron value _g

= 2.0023. The molecular _{axes are}

given

in Table I and _can be

compared

with those of the

biphenyl

molecule in the _pure

crystal [31].

We _{can see} that the molecular

probes nearly

take the _same orientation _as the

biphenyl

_one.

Nevertheless,

it is

worthy

to note that these molecules differ in their sizes since the molecular widths are1

=

2.01,

_2.8

I,

_3.0

1

_in

biphenyl, naphthalene, phenanthrene, respectively

and the molecular

lengths

are L

= 7.0

1,

_3.9

1,

_7.0

1 respectively (see Fig. 3).

We _{can see} that the width of the

phenanthrene

and

naphthalene

molecule _are

higher

than that of the

biphenyl molecule,

which must

probably

hinder the rotation of the

probe

^around its

long

axis.

Biphenyl

and

phenanthrene

have the _same

length

^while

naphthalene

is shorter. This geometry _may favour _a rotation around _an axis

perpendicular

to the

long

axis of the molecular

probe.

3.

Experimental

Results in the Incommensurate Phase II

3. I.

PHENANTHRENE-hio

SPECTRUM PARTICULARITIES

[8].

The behaviour of the

phenan-

threne-hio Probe

is _a classical _one [8]. ^If we consider _one line of its EPR spectra ^{shown in}

Figure

4, we can see that this

high

temperature ^line

splits

at the

phase

transition into two

symmetrical singularities.

This behaviour

corresponds

to _a

typical

linear

splitting

^described

by

the hi _Parameter

(in Eq. (4)). Moreover,

the

singularity

intensities _are

symmetrical,

which

means that the

phase

distribution

P(#)

is uniform _as

expected

for _a

plane

_wave

regime.

^The

line

splitting

^follows the law

(TI -T)fl

with

fl

=

0.4I,

which

corresponds

to the order parameter

~ £".;

."

~ x

--

y ~

a) b)

<

^~n

i~

~

C)

Fig. 3. Size comparison of different aromatic molecules. The naphthalene molecule is shorter than the biphenyl _one while the phenanthrene _one 18 longer:

a)

biphenyl: Lb

= 7.0

I,

₁₆

= 2.0

ii

b)

phenanthrene: Ln

= 7.0

I,

in

= 3.0

ii

_c) uaphthaleue: Lp ₌ 3.9

i~

in

= 2.8 I.

behaviour for _a universal class _n

= 4, d

= 3 [32].

Finally,

it _was shown that this

probe

^does

not allow to differentiate the two domains. If _we remember the

plane

_wave character of the modulation in

phase

II of

biphenyl,

then it appears that the

phenanthrene-hio

EPR

probe

behaviour accounts well

enough

for this modulation.

The parameters

hi

and h2 in

equation (4) originate

from the modulated

spin

Hamiltonian parameters [8,

23, 24, 27]. Indeed,

in the incommensurate

phase

the

spin

Hamiltonian reads:

i~(i)

= i~o +

ii~(« (8)

m=2 m=2

~~(~) ~j ~l~i~(~)°i~~ ~j ~2~i~(~)°i~ (~)

m=-2 m=-2

where 7lo is the

high

temperature

phase spin

Hamiltonian and

A7l(#)

the modulated

spin

Hamiltonian

developed

_up to second order of the

displacement

field

experienced by

the EPR

probe.

The main results obtained _are the

following:

. the

comparison

between the X band spectra _[8] and those obtained in low field

experiments

[21, 22] ^{have shown that the main} parameters in the modulated spin Hamiltonian

AYi(#)

_are

~l,2~' (~), ~l,2~2

^~

(~), ~l,2~2

^~

(~l'

. the EPR

probe

does not account for _a twist

displacement,

but accounts for _a rotation dis-

placement;

T=40

3490 3560 3630

Magnetic

field

(Gauss)

Fig. 4. Phase transition of biphenyl shown with the phenanthrene-hio molecular probe behaviour in the

(b, c)

plane.

. symmetry considerations [23] ^{have shown that} a

large

second order term in the X band spectra ^{leads to} a more

importint _splitting

_than the _one observed in _zero field

experiments [21, 22]. Thus,

^the

splitting

is

essentially

_a first order function of the local

displacement field,

in agreement ^{with the critical} exponent value

fl

=

0.41,

I.e the local line

position H(#)

in equation

(4)

must contain _a

predominant hi cos(#)

term.

Furthermore,

the

investigation

of the incommensurate

splitting

_as a function of the static

magnetic

^field orientation [8] allows

us to determine the

spin

Hamiltonian:

A7l(#)

=

~j _{AiBf cos(# #?)Of (10)}

m=-2,-1,1

with the

following

numerical values at T

= 20 K [8]:

AiBp~

= 20 x 10~4

cm~~, AiBp~

= 0 _x

10~4 cm~~, AIB(

=13 _x

10~4

cm~~

it

=

#

in equation

(10),

and with the temperature

dependent amplitude ABf(T)

=

Af(Ti T)°'4~,

in agreement ^with the

experimental splitting.

T=42 5K

~~

~

~l~

~O

6K #

M+ ~O

~

~l$

~fl

~

2370 2405 2440

Magnetic

field

(Gauss)

2380 2410 2440

A4agnetic

field

(Gauss)

Fig. 5. Phase transition of biphenyl shown with the naphthalene-d8 molecular probe in the

(a, b)

plane.

This modulated spin Hamiltonian describes _a rotation with _an

angle amplitude

equal to 1.3°

around _an axis _u. This axis whose components ⁱⁿ the molecular axis system are

tt~ =

0.00,

_try

=

0.43,

_{ttz =} -0.90 is

perpendicular

to the

long

molecular axis which is

parallel

to the twist axis

(see Fig. 3).

Moreover,

the rotation

amplitude

of the molecular EPR

probe

is _a linear function of the twist

amplitude

of the

biphenyl

molecule. In this _case, the EPR

probe

cannot account for the

biphenyl

movement, but it does not act _on the

phase

of the modulation which remains _a

plane

wave one.

3.2.

NAPHTHALENE-d8

SPECTRUM PARTICULARITIES

[23,

24,

27]. Figure

5 shows the be- haviour of the

naphthalene-d8

line _{as a} function of temperature; the static

magnetic

field is in the

(a, b) plane.

This behaviour is similar for all the

lines,

^for the two

crystallographically

different molecular

probes

in the cell and in all the

planes.

^{When the} temperature is low- ered from the

high

temperature

phase,

each line exhibits different

changes

at TI ₌ 40 K and TII ₌ 17 K due to the

phase

transitions which _occur in the

biphenyl crystal.

In

phase

II _we observe three _or four

singularities

which characterize _an incommensurate

spectrum [23].

^{The EPR}

experiments

under stresses show that each spectrum is the super-

position

of two spectra [27] one spectrum per domain. In the

high

temperature

phase,

there

are two

crystallographically equivalent

sites.

Therefore,

in

phase

II _a site is the

image

of _an- other _one

by applying

the symmetry

plane

lost at the

phase

transition. In this

transformation,

the orientation of the

biphenyl

molecule

(or

the

naphthalene one)

is

changed. Thus,

in

phase

II _we have _two types ^{of molecules with the} same orientation but

belonging

to different domains

transforming

themselves

by

the symmetry

plane.

As the

displacement

fields _on the two types of molecules _{are not}

exactly

the _same, it

gives

rise to two different incommensurate

superposed

lines.

By applying

stresses which break the symmetry, one domain becomes

privileged

and

only

_one incommensurate line remains [27].

At the second phase transition, two

singularities

decrease while the two other increase in such _{a way} that the total

intensity

is

preserved

_[23].

During

this

phase

transition the

singu- larity

positions remain

nearly unchanged,

and the intensity

dissymmetry

of the two

remaining

singularities

in

phase

III is conserved. In the

phase

transition temperature _range, the interme- diate spectra can be simulated with _a superposition ^of a spectrum of

phase

II and _a spectrum of

phase III;

the

proportion being

temperature

dependent

_[23]. This result accounts for the

phase

coexistence which is

typical

for _a first order

phase

transition.

It has also been shown [24]

by

symmetry consideration and thanks to the modulated

spin

Hamiltonian that the

naphthalene probe

does _{not account} for the twist in

phase

III but turns round _an axis _u whose components in the molecular _{axes are tt~}

=

0.00,

_{try =}

0.85,

_tt=

= -0.52 and with _an

amplitude

equal to 2.5°. This movement is described

by

the modulated

spin

Hamiltonian of equation

(10)

with the numerical values at T

= 15 K [24]

AiBp~

= -7 _x

10~~ cm~~, AiBp~

= 0 x

10~~ cm~~, AIB(

= -87 x

10~~ cm~~

3.3. DIscussIoN. In

phase

II

(Fig. 5), taking

into account the domain effect _on the

spectra, ^each spectrum associated _{to one} domain is characterized by two

dissymmetrical edge singttlarities

_[23]. Such spectra could be

analysed

in _one of several _ways, well known for _an

incommensurate system:

. the _more

simple

incommensurate spectrum is _one with _a linear

splitting (I.e.

hi

#

0 and h2 ₌ 0 in

Eq. (4))

where _two

singularities symmetrically

_grow on both sides of the

high

temperature ^line position ^{and have the} same

intensity.

This

hypothesis

is

clearly

in contradic- tion with the

experimental

results which show _a

dissymmetrical splitting

_[24]. Furthermore, it would be _{necessary to} introduce _{a non} uniform

phase" density

to account for the

singularity

intensities while the modulation is _a

plane

_wave

[12].

. a

dissymmetrical splitting

with

symmetrical singularities

is

typical

of _a pure second order term in

expression (4).

It may be in agreement ^with the fact that the

naphthalene

^molecular

probe

does not account for the twist

[24],

^and

consequently

it may mean that the local _am-

plitude

of rotation is

proportional

to the _square of the local twist.

But,

_as in the first _case, it is necessary to introduce _{a non} uniform

phase density

to account for the

dissymmetrical

intensities,

whose

origin

must to clarified.

.

Dissymmetrical singularity

intensities _can be attributed to the presence of first and second order terms in equation

(4).

Such _a reconstruction with relation

(2)

is sketched in

Figure

6.

However,

these parameters cannot account for the

dissymmetrical splitting

which would be due to a

predominant

second order term h2

larger

than the first order _one. Moreover, the

intensity dissymmetry

obtained in this

hypothesis

is insufficient

(see Fig. 6).

When h2 is weak

compared

with

hi,

the spectrum ^exhibits _a small

dissymmetry

which increases when the ratio

h2/hi increases;

then when h2 _" hi the

dissymmetry

stops to increase but the intense

singularity splits

into two

singularities

^when h2 is still

increasing. Therefore,

_we still must

additionally

introduce _a

phase

distribution whose

origin

must also be clarified.

There _{are no} other classical

possibilities

to account for the spectra: in all the _{cases a}

phase

distribution is necessary while the modulation is sinusoidal

[12,13]

^and some

large

^{second order}

terms _are also necessary in the two last

hypotheses. Symmetry

considerations show

[23,

24]

that _a

large

second order term in X band spectra leads to a more important

splitting

than that

a)

2370 2405 2440

AIagnetic

field

(Gauss)

Fig. 6. Reconstruction of _a naphthalene-d8 incommensurate spectrum ^of phase III with _a line

coiitainiug first and second order terms

(experimental:

dashed line, theoretical: solid

line): a)

hi ₌ 17.0 Gau8s, h2 ₌ 5.0 Gauss; b) hi

= 15.8 Gauss, h2

= 8.0 Gau8s; c) hi ₌ 13.5 Gau8s, h2 =10.5 Gauss.

observed in _zero field

experiments. Thus,

^the

splitting

must

essentially

be _a first order function of the local

displacement field,

I-e- the local line

position H(#)

must contain a

predominant

hi

cos(#)

term.

However,

^the

splitting

between

edge singularities

as a function of temperature

follows _a law

(TI T)fl

with

fl

= 0Al [23] ^which

corresponds

to an order parameter behaviour for _a universal class _n

=

4,

d

= 3 [32]. ^{This result} suggests ^{that the}

amplitude

of the rotation

linearly depends

_on the twist

amplitude.

These remarks _argue in favour of _a first order

splitting. But,

in this _case, it is not

possible

to have any

dissymmetrical singularity

intensities and any

dissymmetrical singularity splitting.

The

single soltttion,

to accottnt

simttltaneottsly for

the

first

order

splitting

and the

dissymme- tries,

is to introdttce _a

special phase

distribtttion which

forbids

_some

displacement amplitttdes

in the relation

~(#)

=

Acos(#

₊

#o).

We _see that this reqttirement _is

eqttivalent

to

forbid

_some

phase

valttes.

One

might

think that the

naphthalene

molecular

probes pin

the

phase

of the modulation _{as a} classical defect does.

However,

this

hypothesis

does not support ^{the results of the}

phenanthrene

which is _a

longer

molecule than

naphthalene

and would have to act more

seriously

on the

phase.

According

to this

remark, naphthalene

has a very particular behaviour.

The influence of defects _on structural

phase

transition _was

extensively

studied these last

years

[1-3].

A

particular

_{case concerns} the incommensurate system.

By

their action _on the

modulation, the defects _are classified into two classes: fixed defects and mobile defects.

A defect of the first class fixes the

phase

of the modulation _{at a}

preferential

value which

corresponds

to a minimum value of its energy. Two

limiting

cases arise:

.

high coupling:

each defect imposes its

phase

_on its site;

. low

coupling:

the defect concentration is _so

important

that the modulation cannot

adapt

itself to each

defect,

and _a

phase

distortion appears in the

region

of

high

defect

density.

The defects of the second class

originate

the memory effect due to the diffusion of the interaction between

punctual

defects and the modulation _wave. In this interaction the defects fit the local

phase

of the modulation

by moving (displacement

or

orientation). Memory effects,

due to the competition ^between the

density

defect _wave and the

locking potential,

_are observed in the

vicinity

of the

locking

transition.

Although

_no

locking

transition exists in _our case, we

think the

naphthalene probe

could be considered _{as a} mobile

defect,

because this molecule fits the modulation

(rotation

of the molecular

probes). Nevertheless,

_our measurement system

does _not allow _us to evidence _any

hysteresis phenomena.

To account for this

particular phase distribution,

_we consider two models of

phase density

in the

following

section.

4. Determination of the Phase

Density P(~fi)

In _a multi-soliton

regime,

the

phase #(r)

is _no

longer

_a linear function of the _space variable

r.

However,

it is

equivalent

to _a

plane

_wave

regime #(r)

= k

r but with _{a non} uniform

phase density P(#).

If the

probe exactly

accounts for the

modulation,

it

experiences

all the

phases

with _a

phase

distribution

given by P(@.

But in _{a more}

general

case, where the

probes

behave _as

defects,

_one could

imagine

^{that the}

probes privilege

_some

particular phases

of the modulation in such _a way that the phases

experienced by

the

probe

_are distributed with _a

phase

distribution

P'(@

different of

P(@.

So _even if the modulation is _a

plane

_{wave one,}

the probe could _account for _an effective "multi-soliton"

regime

which would result from the interaction between the

probe

and the modulation. If _we

emphasize

that in _an EPR

experiment

we measure what

happens exactly

_on the defect

itself,

then the observed

phase

distribution

should be the

P'(@

_one defined above.

In the

following section,

we will show how _a sinusoidal modulation leads to _{a non} uniform

phase

distribution _on the defects.

4.I. DISTRIBUTION _{oF THE} PHASE EXPERIENCED _{BY THE} DEFECTS. When the EPR

probe

behaves _{as a} defect, its

displacement

field is _no

longer

the modulation _one.

However,

it should

depend

on it

through

the movement of the

neighbouring

molecules which

approximately

follow the modulation

displacement

^field. A reasonable _way to determine the

phase

distribution

P'(4l)

is to _assume that the defect

displacement

field depends _on the

displacement

field

phases

of the

neighbouring biphenyl

molecules. We transfer to the defect

phase

the

displacement

field anomalies, that _means the defect

phase

distribution is _no

longer

the _one of the _pure

biphenyl.

In the

following

model, _we lbok at the

crystal overall,

while in the model

developed

in the next

chapter

_we calculate the lobal field

displacement

of the molecular

probe

for _a

given

value of the

displacement

field of the

neighbouring biphenyl

molecules.

Then,

the

probe

local

phase

is

subjected

to two kinds of energy:

. an elastic _energy which

always

tends to

bring

back the _wave to a

plane

_{wave one;}

. a defect _{energy as a} result of the molecular

packing

of the molecular

probe.

The free _energy of the system ⁱⁿ a continuous model is [2]:

F =

/ _jk ())

+

~lvqlx _x>)~lI)

₊

_I(x >)fix))ldx ₍₁₁₎

~

~

where the first _term in the

integrand

represents ^{the elastic} energy of the modulation and the second term the free _energy of the defects

linearly coupled

to the two components _J~ and

(

of the local order parameter. ^The

phase #

is defined

by:

J~ ⁼ A _cos

#

and

(

₌ ^A sin

# (12)

where A is the

amplitude

of the order parameter, _xj represents ^the

position

of the

jth

defect

along

the modulation.

If _{we assume V~} and

l~

to be delta-functions

(assumption

of local

defect),

I-e-

V~(x)

=

V/d(x)

^and

V, ix)

=

I~°d(z),

^{then the free} energy becomes:

where

#j

₌

#(x~).

We minimize the free _energy in two steps:

.

firstly,

_we fix the

phase

_{#~ on} the defect and _we minimize the first term, the solution reads [33]:

iii)

₌ ^{~~~~ ~}

#j

+ ^~ ~~

#j+i

for xj ^< x < x~+1

(14)

xj+i xj xj+i xj

so that the

phase #

varies

linearly

between the

jth

and the

ii

₊

I)th defects;

.

secondly

_we insert this result in the free _energy and _we minimize it in relation with the variables

#~,

^{then the free} energy reads:

and

fter

tion,

we in:

l~j)) ^{j j ~-)}

^{~ "~}

^~~'

^~~~~

After the first term of

equation (17)

is divided

by

the factor (x~

xj-i),

_we

recognize

in the continuous limit the second derivative of the

phase #.

Now _we

replace,

in the second member of

equation (17),

the additional factor

I/(xj xj-i) by I/ld

where Id represents ^the _average distance between two consecutive defects. It is then obvious that this factor is the concentration cd of defects in the

crystal. Therefore, making

the continuous

approximation, equation (16)

writes:

~~

~ ^~~~~

^~

^{~~~~ ~}

~~'~

_~~~~

where

iii)

now represents ^{the continuous}

spatial

variations of the

defect phase.

We

recognize

the

equation

which describes the soliton

regime

[7]. ^{It is} now

possible

to find the

phase

distribution

by integrating equation (18),

since _we have the relation

P(#)

_c~

dx/d#.

First

integration gives:

Ill1

^~

^{II l}

^~ = 2KiC°S

<

_C°S <o ₁₁

9)

or

((~)

^~ = 2Kcos

# (2K

_cos

#o Ko)

with Ko ₌

(((~)

^~

(20)

x x ~

Equation (18)

is the _{same as} the _one

describing

the

pendulum motion,

two cases which

depend

_on the initial conditions _can be considered:

. The first

corresponds

to the usual multi-soliton

regime

and is

analogous

to the

turning pendulum.

It _occurs when:

2Kcos#o Ko

< -2K.

(21)

In this _case all the

phases

between 0.2 _1r modttlo 21r _are realized and the

phase density

is

given by:

~~~~

_~

~(

^~

_~/Ko ^Kco~#o

+

2Kcos#

^~

i

~~~~

. The second

corresponds

to the

oscillating pendulum

and _occurs when:

2K < 2K _cos

#o Ko

_< 2K.

(23)

In this _{case, some}

phases

_are forbidden because 2K _cos

# (2K

_cos

#o Ko)

must be _a positive

quantity. Thus,

the

phase density

can be written _as:

P(#)

for

#

_E]

ii, Ill (24a)

~/2K(cos # cos11

and

P(w

= 0 for

i ii ii, Iii (24b)

where

ii

is defined

by cos(#1)

₌

~~~°~~(~ ^~°

^{and is named "cut}

phase".

2

Pi

dj

Although

the

phase density diverges

at

+#i,

the

integral ~~i _{~/2K(cos #}

_converges

cos

ii

toward _an

elliptic

function in such _{a way} that _{we can} normalize the distribution. We will _see in the next section how this distribution accounts for the

experimental

results.

4.2. COMPARISON _{WITH THE} EXPERIMENTAL RESULTS. The second

phase

distribution is

more suitable to account for the

experimental

results of

naphthalene. Indeed,

_we recall that the

intensity

^{of the} line _{as a} function of the

magnetic

^field H is

given by equation (2). Therefore,

if _we introduce the

previous phase density,

_we obtain:

+~i

(H H(#)) d#

~~~~ /-~i

_~(fb)

_~/2K(C°~ (25)

l~ ~°~ ~~~

with

H(#)

₌ Ho + hi _cos

#

for _a first order

splitting.

The

singularities

of such _a system are situated _{at :}

Ho

+ hi which

corresponds

to an extremum of

H(#)

and

(26)

Ho +

hi

_cos

ii

because the

phase

distribution

diverges

at

+11

Table II.

E~perimental

resttlts: A and B chain

properties f23j.

Parameters Distribution

P(#)

Rotation parameters

Rotation axis:

AB(

= -87 _x 10~~ cm~~ Cut

phase

= 100° _tt~

= 0.00

Chain A

ABp~

₌ 0 uy = 0.85

AB]

= -6 x

10~~

cm~~ d

= 0.001 _uz

= -0.52

Rotation

amplitude:

a = 2.5°

Rotation axis:

AB(

= 37 _x 10~4 cm~~ _Cut

phase

= 140° _tt~

= 0.00

Chain B

ABp~

= 0 try = 0.37

AB]

= -12 _x 10~~ cm~~ d

= 0.001 _ttz

= -0.93

Rotation

amplitude:

a = 2.5°

Moreover,

the second

singularity

will be _very intense because it is associated with the cut

phase.

Let _us remark that

if11

*

1r/2,

then the intense

singularity

will appear near

Ho,

the

high

temperature line

position

in agreement ^{with the}

experimental

results.

We _are able _{now to} simulate the spectra with _a modulated

spin

Hamiltonian

containing only

first order terms, ^this

procedure

_was described elsewhere [23] ^{where it} was also demonstrated

that this modulated Hamiltonian has the form:

A7l(#)

=

(ABp~Op~

+

ABp~Op~

₊

AB(O() _{cos(#). (27)}

We recall that _we must define two

spin Hamiltonians,

_one for each chain

[23],

and _we

precise

that the

phase density

used in this simulation is not

exactly

the

previous

_one but _a similar _one where _a parameter d has been added to monitor the

intensity

of the

singularities:

P14)

-

~/j

₊

~~~l

~~~ ~~

128)

We think the parameter d accounts for the competition between the lattice modulation and the defect modulation _wave which would have to appear in _{a more} realistic model. We have verified that this

phase density gives

the correct

splitting

for any direction of the

magnetic field,

in

particular,

a

splitting

toward

high

_or low field in agreement with the

experimental

result. It _was necessary ^to

verify

this point to confirm the coherence of the model with the

experimental

results. The

spin

Hamiltonian parameters allow _us to determine the movement of the

probe [23j.

^{All the} results _are shown in Table II and _some simulated spectra are shown in

Figures

7 and 8.

In the last section, we show the

origin

of this

particular phase

distribution for

naphthalene.

5.

Origin

of the Phase Distribution

The _purpose of this section is _to deduce the behaviour of the molecular

probe

from the inter- actions between the molecules in the

crystal.

When the

overlap

of electron clouds is small,

T=50K

's

~~

G/ j

~

~

~

~ ~

~

T=20K T=20K

2375 2415 2455 2630 2670 2710

Magnetic field

(Gauss) Magnetic

field

(Gauss

_j

Fig.

7

Fig.

8

Fig. 7. Reconstruction of uaphthaleue-d8 incommensurate spectra of phase II, in the (a,

b)

plane,

with _a phase distribution containing a cut phase

(experimental:

da8hed line, theoretical: solid

line).

Fig. 8. Reconstruction of naphthalene-d8 incommensurate spectra of phase II, in the

la, c)

plane, with _a pha8e d18tribution containing a cut phase

(experimental:

da8hed line, theoretical: solid

line).

the intermolecular _{energy may} be

separated

into three terms: the interaction energy of the static molecular

multipole

_moments, the

dispersion

_energy and the

repulsive

_energy due to the clouds

overlap

_[34]. When there is _a centre of symmetry, ^{the first} contribution is

mainly

_a

quadrupole

one. The repulsive _energy takes into account the interactions between individual

atoms of the two

molecules,

U = Vi

(H-H)

+

V2(C-H)

_{+ V3}

(C-C) (29)

where the three

potentials

_on the

right-hand

side

apply

to intermolecular

hydrogen-hydrogen, hydrogen-carbon

and carbon-carbon contacts,

respectively.

The

dispersion, quadrupole-quadrupole

and

hydrogen-hydrogen repulsion energies

have been calculated for the

equilibrium

structure of

crystalline naphthalene

for several

displaced

struc- tures which _are small rotations of the molecules about their symmetry axes [34]. ^{The authors} show that the

dispersion

and

quadrupole

interaction _{are not} minimized at the

equilibrium

struc- ture while the

hydrogen-hydrogen repulsions

have _a strong minimum.

Consequently,

the _attrac- tive

dispersion

and

quadrupole

interaction determine the

crystal

cohesion while the

hydrogen- hydrogen repulsion

determines the orientations of the molecules.

Although

this work [34]

was

performed

for

naphthalene,

it is reasonable to use this result for molecules

belonging

7

5 '

, ^Probe 6

' '

'

3 4

8

Fig. 9. This figure shows the bipheuyl molecules

(solid

line) in the plane of the probe

(dashed line)

taken into account to calculate the total potential _energy of the probe molecule. The equivalent

molecule in the plane just above and just below _are also taken into account in this calculation.

to the _same

family,

such _as

biphenyl

_or

phenanthrene. Besides,

_we

point

out that this conclu- sion has been used

by

_some authors for several aromatic molecules

[35-37].

Our

approach

to the

problem

of the molecular EPR

probe

behaviour in the incommensurate

phase

of

biphenyl

is based _on the assumption that it is the intermolecular repulsive interactions which _are of

primary importance

_[34]. The orientation of the molecular

probe

is assumed to be

given by

the minimum of the total

repulsive

_energy of the molecular

probe

in the field of the

neighbouring biphenyl

molecules.

Only

the first and second

neighbours

are taken into account since the other molecules have _no contact with the

probe (see Fig. 9).

The total

repulsive

energy is _{a sum over} pairs of atoms H-H

(it

is assumed that the

hydrogen

contacts

give

the

largest

contribution to the

repulsion

_{energy [34]}

).

The

potential

function for H-H

repulsion

is

represented

_{>by an}

exponential

V(r)

₌ A

exp(-cr) (30)

where _r is the distance between the two atoms, c a constant which characterizes the hardness of the

potential

and A is _an

arbitrary

constant. The _{constant c} is found to be 2,I

l~~

_for

naphthalene,

lA

i~~

_for

phenanthrene

and 2

i~~

_for

_biphenyl.

Figures 10a,

b and _c show the total

potential

_energy of _one Tnolecule

(biphenyl, phenanthrene

and

naphthalene respectively)

in the

high

temperature

phase

_{as a} function of its orientation.

The molecule is rotated about _one of its molecular _axes

(x,

_{y or}

z)

and the

angle

is zero at the

equilibrium position.

We _can notice _a

particular

^{behaviour of}

nap.hthalene

when this molecule

a)

~

. .

~

.

.

~s

. I

.

.

(

.

~' .

~ 40

"

.

~

~

. m

if

~

. .

g

.

.

U~

Ui

m

~

m

io 5

0 5

io

io Angle