Observation of a new memory effect in a modulated structure

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Observation of a new memory effect in a modulated

structure

J.P. Jamet, P. Lederer

To cite this version:

J.P. Jamet, P. Lederer. Observation of a new memory effect in a modulated structure. Journal

de Physique Lettres, Edp sciences, 1983, 44 (7), pp.257-264. �10.1051/jphyslet:01983004407025700�.

�jpa-00232190�

Observation

of

a new

_memory

effect in

a

modulated

structure

_(*)

J. P. Jamet and P. Lederer

Laboratoire de

_Physique

des Solides _(**), Université Paris-Sud, Bâtiment 510, 91405

_Orsay,

France

(Re!pu le 23

_septembre

_1982, r~vise les 15 decembre 1982 et _{20 janvier} 1983,

accepte

le

_{1 D fevrier}

₁₉₈₃₎

Résumé. 2014 Nous décrivons la

_première

observation d’un nouvel effet de mémoire dans une structure

modulée

_{(thiourée deutériée).}

Nous

_présentons

une

_{explication qualitative}

de cet effet basée sur

l’interaction de défauts ou

_{d’impuretés}

mobiles avec le

_paramètre

d’ordre de la modulation. Nous

montrons _que cet effet peut être utilisé pour obtenir les

_{lignes d’égal}

vecteur d’onde de modulation dans le

_plan

E, T du

_diagramme

de

_phase

_{par des} méthodes

_{thermodynamiques.}

Nous

_suggérons

que les

systèmes à

ODC comme

_{NbSe3 présentent}

eux aussi une modulation

_périodique

de

concen-tration de défauts, ce

_qui

rendrait _compte des effets non

_linéaires

en

_champ

_{électrique. (ODC :}

Onde

de Densité de

_Charge.)

Abstract. 2014 We _report the first observation of a new _memory effect in a modulated structure

(deute-rated

thiourea).

We _present a

qualitative explanation

of this effect in terms of mobile defects, or

impurities, interacting

with the modulation order _parameter. We show that this

effect

can be used to

obtain the lines of constant modulation wavevector in the E, T

_plane

of the

_{phase diagram, by}

thermo-dynamic

methods. We _suggest that a similar

periodic

modulation of defect concentration occurs in CDW _systems, like

_NbSe3,

and accounts for the non

_linearity

induced

_by

the electric field.

_{(CDW :}

Charge Density Wave.)

Classification

Physics Abstracts

61. 70 - 64. 70K - 66.30J

Thermal

_hysteresis

effects are

_present

in ferroelectrics and modulated structures and are known to exist in

_{dipole density}

wave

_systems

such as

SC(ND2)2

[1],

Rb2ZnCl4

[2],

TMAZnC14

[3],

BaNaNbs015

[4]

or

charge density

wave

_systems

such as

_{TTF-TCNQ [5]. They}

are observed

_by

using

neutron

_diffraction,

_specific

heat,

dielectric

_{susceptibility}

or

_linear

_{birefringence}

experi-ments. The effect of

_{impurities by doping}

has been

_investigated

for

_example

in

_{Rb2 - xKxZnC14}

_[6]

but there

is,

to our

knowledge,

no

_{experimental study concerning}

the effect of dislocations or

tunnelling

impurities

on the

_properties

of a modulated structure. In some

_systems

_[4],

the rather

large

thermal

_hysteresis

which is observed on the

_q-wavevector

evolution of the modulation

_plays

in favour of the influence of

_impurities

or defects on the modulated structure. The

_pinning

of the incommensurate modulation

_by

the defects has been

_investigated

from a theoretical

_point

of

view in the case of the

_{charge density}

waves

[7j.

(*) Le contenu de cet article a 6t6

_présenté

en

_partie

au

_Colloque

du C.N.R.S. : Parois et

_Domaines~RCP

608) Aussois,

janvier

1982.

(**)

Associé au C.N.R.S.

L-258 JOURNAL DE _{PHYSIQUE -} LETTRES

We

_present

here an effect observed in the.modulated

_phase

of thiourea which we call a _memory

effect and which can

_presumably

be attributed to an

_ordering

of defects or

_impurities

or other mobile entities

_by

the modulation order

_parameter;

we believe that this effect

which,

to our

knowledge,

has never been observed before is

tightly

related to the existence of

_hysteresis

in the

following

sense : if the

_effect

is

_extrinsic,

then

_hysteresis

is most

_likely

of extrinsic

_{origin (even}

though

different defects

_may~explain

one or

_other);

if the effect is

_intrinsic,

then

_hysteresis

is

probably

also intrinsic.

~---1. Linear

_{birefringence}

and dielectric

susceptibility experiments.

The

_{susceptibility}

measurements have been

_{performed by using}

a

_Sawyer-Tower

like circuit.

The reference sinusoidal

_signal

at 1.5 kHz is

_{given by}

a

_{high stability}

low distortion

_generator

and the detection uses a lock-in

amplifier coupled

to a X-Y recorder and a

_{digital voltmeter;}

no

_significant

difference has been found on the

_{susceptibility}

between 10 Hz and 600 kHz which

means that s = _~o. For the measured

sample (thickness :

0.62 mm, area : 6

_mm2)

the Noise to

Signal

Ratio

_(N.S.R.)

was 2 x

10-4,

while the kink

_amplitude,

in the case of

_figure

_1,

_represents

about 10 times the noise

_{amplitude (furthermore}

the absolute

_precision

on the

_{susceptibility}

measurement

is

_certainly

not better than

_10-’).

On the other

_hand,

the

_{birefringence}

measure-ments

on the same

_sample

used an

acousto-optical

modulator to

_improve

the

_signal

to noise

ratio

_{[8] ;}

the

_light

source is a stabilized HeNe Laser

_(~,

= 6 328

A) ;

the

optical phase

shift is

~ 15

_{radians/mm [9]}

in the modulated

phase (i.e.

between 190 K and 216

K)

and the noise

ampli-tude is

10- 3

_radian/mm;

as an

_example

the kink

_amplitude

is about 5 x

10- 3

_radian/mm

if

To

= 194

K,

but it is about

10- 3

radian/mm

at 202 K. We

_report

here the most accurate results

obtained

_{by susceptibility. Birefringence}

results will be

_reported

elsewhere.

In the modulated

_phase

of thiourea

₍₁₉₀

K T 216

_K,

in zero d.c. field and at

_atmospheric

pressure)

the dielectric

_{susceptibility}

and the

_{birefringence}

exhibit a

_large

thermal

_{hysteresis :}

Fig.

1. -

a)

Typical

temperature evolution as a function of time used to _{get the memory} effect. The _step located around 201 K

_during

the

_{cooling phase}

will _generate a kink

during

the

_{heating phase}

at a

_larger

temperature T ~ 202 K

_(due

to the

_{hysteresis effect).}

b) Experimental susceptibility

curves for a cleaved _sample with dimensions 2 x 3

mm2,

thickness 0.62 mm.

The dotted lines _represent the curve obtained

during

a continuous run and the solid line the curve obtained

Fig.

2. -

E, T

_phase

_diagram of

_{SC(ND2)2 ;}

the solid lines are from _[10]. The

_flame-shaped

curve is the

envelop

of the commensurate

_phase

_{with q} = b*/8 and the

triangles

_are obtained by

using

the memory effect; the dotted lines

_{are just}

a

_help

to _{the eyes,}

_{they join}

the

_points

obtained

_{by using}

the same _temperature

step

during

the

_cooling

_phase

and

_applying

various field values

_during

the

_heating

_phase which result’in

various kink

_positions.

The theoretical

_analysis

_given

in _[12] for

_{the q}

= b*/8

phase

assumed, in the absence

ofUmklapp

term, a theoretical line _(dashed dotted _line) which is also the middle line

_{of the q}

= b*/8

commen-surate

_phase;

the excellent agreement of the theoretical and

_experimental

curvatures is an excellent check of the _theory

_given

in

_[12].

it is null within

_experimental

_accuracy at and above the second order

_phase

transition

tempera-ture

_{(To N}

216

_K)

_separating

the

_paraelectric

and modulated

_phases

and increases

monotoni-cally

below

_To

down to the commensurate

_phase

transition

_(q

=

b*/9 )

at

T 9 ~

192 K. At 190

K,

a first order transition to the ferroelectric

_{phase (q}

=

0)

_occurs and below this

_temperature

the

hysteresis

effect is associated to _{the presence of ferroelectric domain walls. If} a non-zero d.c. field E is

_{applied parallel}

to the ferroelectric axis « a _», the situation is not

_drastically

modified

apart

from the onset of the field-induced commensurate

_{phase with q}

=

b* j8 [10].

Inside the

domains of

_stability

of the commensurate

_phases

_(q

=

b*/8

and q

=

b*/9),

the

hysteresis

effect

as measured

_{by birefringence}

or

susceptibility experiments

is different from zero.

2. Thermal

_cycle

and memory effect.

Standard measurements of the

_{susceptibility}

in the modulated

_phase

_(i.e.

between 190 K

and

216

K)

involve

_usually

a slow

(

3

mK/s)

and continuous

heating

rate from the low

tempera-ture to the

_high

_temperature

_{phase boundary}

and vice versa. As is well

_known,

the

_resulting

curves

exhibit a thermal

_hysteresis.

We have obtained an

_interesting

effect when the

_{heating (or cooling)}

procedure

is

_slightly

less

_simple,

as shown on

_figure

la.

_Starting

first to cool in zero external field

from above 216 K

_(with

a constant

_cooling

_rate

_{I dT/dt}

_I

= 3

mK/s)

_one allows the

_temperature

to stabilize at an

_arbitrary

value,

_say

_To

= 201 K

during

_a time interval

Ato -

600

s, then to

decrease

_again;

before

_{going through}

the first order transition

_separating

the modulated and

ferroelectric

_phases,

the

_temperature

is

_again

stabilized at

_{T 1}

= 191 K

(for

_a time interval which

experimentally

has not exceeded two

_days)

and then increased

_(dT/dt

= 3

mK/s) :

_one then

observes a small

_negative

kink on the

susceptibility

or

_{birefringence}

curve at a

_temperature

_slightly

above

_{To =}

201 K. The

_amplitude

of this kink is rather small

_{A/// ~}

10 - 3

but it is

_clearly

visible on

_figure

_lb,

in a

_{susceptibility experiment.}

One _may now use the same

cooling

procedure

as above but

_during

the

_{heating phase,}

a d.c.

sam-L-260 JOURNAL DE

_{PHYSIQUE -}

LETTRES

ple

does not become

_{ferroelectric)}

then a kink is still observed but at a

_higher

_temperature

than

with

_{Ej c}

= 0. The loci of these kinks _are

displayed

_on

figure

2.

3. Remarks.

a)

The

_{experimental procedure}

used in

_(la)

can be modified for several parameters such as

To, TI,

or the

_cooling

and

_heating

rates; the kink

_amplitude

is smaller when

_To

is increased and it is null at and above the modulated

_{paraelectric phase}

transition. If

_To

is inside the domain of

stability

of a commensurate

_{phase (q}

=

b*/8,

with

Ed.,. 0

0 for

_example)

then the domain of

stability

of this

_phase

_appears to be

_slightly

increased. If

_during

the

_cooling

_process, the

tempera-ture is allowed to stabilize several

_times,

then several kinks will be « written ». At last the kink

amplitude

seems to be

_{sample dependent}

and

_consequently

it is most

_probably

related to the

density

of

_impurities

or defects

_present

inside the

_system;

the

_dependence

of the kink

_amplitude

with the duration of the

_{temperature step}

at

_To,

has not

_yet

been

_{investigated quantitatively;}

all we can _say, at

_present

time,

is that a few minutes seem to be sufficient to somehow « saturate »

the memory effect.

b)

The memory effect has been observed for several

heating

or

_{cooling procedures,}

which are not

_reported

here. A

_{susceptibility}

kink not shown on

_figure

lb is also observed upon

_cooling

again

at the locus called «

_T-step

position

>r.

c)

We do not believe that the observation of the _memory effect could be an artefact due to

trivial causes such as : _space

_{charge (it}

is observed on short-circuited

_samples

in zero d.c.

_field)

or electrical contact

_{instability (the samples}

are

_gold-plated

under

vacuum).

4.

_{Interpretation.}

How can we

_interpret

such a

_{phenomenon ?}

When

_during

the

cooling

_{process, the}

_temperature

is allowed to stabilize at

_{To (at}

time

_to),

the modulation wavevector

_stops

_decreasing,

so that the

extrinsic defects

(dislocations

for

_example)

or

_impurities,

with

_{relatively high mobility}

can relax

in the

_periodic

field of the modulated

_dipolar

field. This can be understood on the basis of a

_simple

steric condition : in

_thiourea,

the modulation is a modulation of the orientation of electric

_dipoles

(which

are

_{planar molecules)}

with the b-axis

[11] ; impurities (for example

interstitial

molecules)

thus

_experience

a modulated interaction with the lattice :

where

_P(r)

is the local uniaxial

_{polarization,}

_ðrfri

is the Kronecker’s

_{symbol and r~}

is the

_impurity

position;

we have

_{developed Yr(r)}

to lowest order in

_{P(r). (Thus}

we do not treat

_properly

the low

_temperature

_case).

_Vo

is taken here to be

_equal

for all

_impurities,

for

_{simplicity. (In}

the CDW

case, it is

_probably

more

_appropriate

to assume

_{t~(r) ~}

_p(r),

where

_p(r)

is the

_{charge density.}

In other

_systems,

one _may have more

_{complicated impurity-modulation}

interactions

potentials).

As

a

result

of

~M,

at thermal

_equilibrium,

the

_impurity

concentration is also

_periodic

if the

impurity

mobility

is not zero, with the

reciprocal

modulation

periodicity

(2qo)-1.

When the

modulation is sinusoidal :

where

_bo

_1;

_bo

_depends

on the nature of the

_impurities,

_defects,...

which can interact with

the modulation

_{potential ;}

also it is reasonable to assume that

_bo

_goes to zero with the

_amplitude

of the

modulation, i.e.,

_{as a}

first

_{approximation :}

_{bo -}

_{Pqo ~2}

where

_P~

is the Fourier transform of

_P(r).

Notice that

C(r)

is the _average concentration in the

_{plane perpendicular}

to the

modula-tion axis at

_point

« r » ; in a

3d-system,

the concentration can be modulated as in the formula

When the modulation is

_strongly

anharmonic

_{(soliton limit),}

as is the case near the

commen-surate-incommensurate transition _{at q} =

b*/9,

the

impurity

concentration

profile

should also

exhibit a

_large

number of harmonics and a

_periodic

wall structure.

Thus the

_impurity

_gas can exhibit a

_periodic

modulation with wavevector

_{qo if a}

sufficient number of

_impurities

has a relaxation time T in the modulation

periodic

field,

such that

where

_Ato

= 600 _s in the

experiment

described above.

a)

When the

_temperature

T continues to decrease below

_To

= 201

K,

the modulation

wave-vector

_q(T)

varies with

T,

i.e.,

with time. As a result the

periodic impurity

_pattern

is submitted

to a time

_{dependent potential.}

The

_sign

of the

_{potential-gradient,}

i.e. the

force,

alternates in time

at a

given

impurity

site,

while its

intensity

increases with

decreasing

_temperature

due to the order

parameter

monotonous

_increase;

so if the rate of

_change

of the

_temperature

is

_sufficient,

_impurities

do not follow the time

_dependent

modulation,

and the

_periodic

concentration

_pattern

_keeps

the initial

_periodicity.

When the

_temperature

becomes

_{stationary again}

at

_T1

at time tl, the

impurities

experience

a

periodic potential

with a different

_{periodicity corresponding}

to the wave-vector _q1’

_usually

incommensurate with _qo.

At

_relatively

short times _{after tl,} i.e. _{t1 t ~ t1} + r, when the

_temperature

is stabilized at T =

T 1,

the concentration

pattern gets

the form :

where b,

is different from

_bo

because the order

_parameter

varies with

_temperature,

and where

b’ 0

bo.

This

_pattern

is out of

_equilibrium

as

_long

as

_{bo =~}

0,

but we

_expect

it to have a

long

life-time because the time _ir necessary to thermalize the

_impurities

_(i.e,

to erase the

_impurity

modula-tion with wavevector

_2qo)

involves

_tunnelling

of defects or

_{impurities through}

the

_potential

barriers due to the

_periodic

_modulation,

so that r, is much

larger

than

t( t,

>>

t).

Clearly

T, increases

with

_decreasing

_temperature

because the

_height

of the

_potential

barriers increases. Therefore at

time t such that

the

_system

exhibits a modulated concentration of

impurities

with fundamental Fourier compo-nents of wavevector

_2qo

and

_2q1

_(and,

_possibly

their harmonics if the

_dipolar

modulation is not

simply sinusoidal).

(Note

that all we can _say about _ir is that in our

_sample

_tr > 48

h).

b)

When the

_temperature

is

_increasing

from

_{T 1 (after}

a time shorter than

_’tr)’

there will be

an extra free energy

density

be an extra _{free energy}

_density

_6F,

when the wavevector

_q(T)

will

go

through

qo. Indeed :

Note that the

_{general expression}

for 5F is

_proportional

to sin 2~ where ø. is the

_phase

diffe-rence between the modulation and the

_{periodic impurity}

_{pattern. 4>}

is chosen so that

_{Yo 60}

0.

(There

exists also a term

_{b’ 0}

_{Co Po}

_Pq

_~,i2q

which we

_neglect

_here,

as discussed in

Appen-dix

_1.)

Because of this extra term, the

amplitude

of the modulation

_experiences

a small variation

when

_q(T)

=

L-262 JOURNAL DE PHYSIQUE - LETTRES

This

_change

is reflected in the uniform

_{susceptibility.}

We find

_(see

A

_{1) :}

with

y is a

_{phenomenological}

coefficient defined in A I.

~Pg -

1

_{bx .}

Note

that

2013~- = -

-

in zero

field,

see

appendix.

B

Pq

x

/

Thus the uniform

_{susceptibility}

exhibits a

_(negative)

kink at the

_temperature

_To(E)

defined

by

q [ To (E )]

=

qo.

Therefore,

if we « write » the modulation wavevector _{qo in the system} in

zero

field,

we can determine the lines where the modulation wavevector _qo

_stays

constant in the

_(E,

_T)

_plane.

The

_preliminary

results of our measurements are shown on

_figure

2 for three values of qo. The results are in a _very

good

_agreement

with neutron diffraction results at low d.c. field values

[16]

and at

_high

field values

[17].

c)

When the _system is heated at or above the modulated

_paraelectric

transition

_{( ~}

216 K

in zero d.c.

_field),

the

_periodic

field due to the modulation

vanishes,

so that the

_{periodic impurity}

concentration relaxes to a uniform one, in a time Ot ~ r. Thus if one cools the

_system

down

from

_high

_temperature,

one

_expects

no _memory

_effect,

as observed.

5. Conclusion.

We have described a new _memory effect observed in the modulated

phase

of thiourea and shown

that it allows to determine

iso-q

lines in the E-T

_plane

of the

_{phase diagram by susceptibility}

measurements. We have

_suggested

that this effect can be understood

_if,

in the

_crystal,

there exist defects or

_impurities

with a

_sufficiently

broad

_spectrum

of relaxation

_times,

and which interact

with the

_dipolar

modulation.

Likewise,

iso-q

lines may be determined

using

the same effect

by

birefringence

measurements.

Notice that the

_theory

outlined here is a weak

_coupling

one, in the sense that the

modulation-defect

_coupling

does not

_change

the

_equilibrium

value of the modulation wavevector. Situations

where a

_strong

_coupling

occurs should lead to a

quite

different behaviour.

A number of unanswered

_questions

must still be clarified : what are the nature and the relevant relaxation times of the

_impurities

or defects which allow to _{observe the memory} effect ? How does the

_intensity

_{of the memory effect}

_depend

on the

_cooling

or

_heating

rates or on the time

intervals between

temperature

stabilizations ? More

_{fundamentally,}

we

_expect

the memory

effect to shed some

_light

on the nature of the

_{global hysteresis}

in thiourea : is this

_hysteresis

intrinsic ? If it

is,

is it due to the Peierls

_potential

felt

_by

discommensuration walls

_[12],

or is it

due to the intrinsic defect lines

_{[13] ?}

or is this

_hysteresis

extrinsic

_(due

to the

_crystal

inhomo-geneities,

dislocations,

etc.).

Somewhat like the

_hysteresis

described

_by

Neel

_{[14] ?}

We feel that those

_questions,

which are

quite

basic in the

understanding

of three dimensional modulated

structures, and in

_particular

in the

_problem

of the

_{piecewise analyticity}

of the

_{thermodynamic}

functions

_[12]

may be attacked from the

point

of view of their influence on the memory effect.

Experimental

as well as theoretical work is in progress on these

_questions

and will be

_reported

elsewhere.

Finally,

we would like to

_suggest

that a

periodic

defect concentration is

_expected

to occur

in

_practically

all modulated _structures, with a wide variation in relaxation

_times,

orders of

magni-tude,

etc.,

depending

on the

specific

nature of each _system, on the nature of the mobile

_defects,

on the

_temperature

at which modulated structures

_order,

etc..

_NbSe3

is a

_particularly

_interesting

tempe-rature for CDW

formation,

as well as the

_periodic

noise

_spectrum

_[18]

can be understood

_provided

one assumes a

_{pinning potential}

of the CDW to the

_impurities.

A random distribution of fixed

impurities

cannot

_provide

the sinusoidal

_{potential -}

sin

_~,

where ~ is the CDW

_{phase [19].}

On the

_contrary,

if a sufficient amount of

_{impurities (or}

_defects,

or

else,

etc.)

is able to relax in the

_periodic

field of the

_modulation,

as

_happens

in

_thiourea,

the

_appropriate

sin ~

_potential

arises most

_naturally

from

_{commensurability}

between CDW and

_impurity

_modulation,

and

accounts for the essential

_experimental

features. In order to check this

_explanation,

one should

measure the threshold fields after

_{quenching, annealing}

and look for time

_dependent

effects

linked with defect diffusion.

After this _paper was

_completed,

we received a

preprint

on the

phase

diagram

of

_NaN02

measured

_by

neutron diffraction under electric field

_[20].

In this _parent

_system

of

thiourea,

the measured

_iso-q

lines are

_strikingly

similar to those shown in

_figure

2. As

_pointed

out in

refe-/~p (y.)B2

rence

_{[ 20 ],}

the

_{iso- q}

line curvature is well accounted for

_{by introducing}

a

_{term r~}

~(~) (201320132013 )

in

_equation

_(1.2),

_{with r~}

> 0. We consider the results of reference

_[20]

as a confirmation of the

memory effect described here.

Acknowledgments.

We are

_pleased

to thank Prof V.

_Janovec,

G. Montambaux and M. Barreto for

_stimulating

discussions and F.

_Denoyer

and D. Durand for

_showing

us their neutron data

_prior

to

publica-tion. J. Ferré is

_acknowledged

for continuous interest while this work was

completed.

We thank

Seb Doniach for

_helpful

remarks.

Appendix

I.

In the _presence of the modulated

_impurity

concentration the free energy of the modulated

phase

of thiourea is

_{[ 10] :}

with

In

_fact,

_(1.2)

is a

simplified

version of the free energy for

thiourea;

in order to describe the modulated

_part

of the

_{phase diagram}

in the

_high

field

_region,

one needs to take into account

a sixth order term

_{[12] ’"}

_DP;(r).

_Taking

D =

0,

_{as we} do

here,

allows _a clearer derivation

of the _memory

_effect

and is

_only

_acceptable

in thiourea at low fields

_(E

1

_kV/mm).

In

_(1.2),

_{Px(r )}

is the uniaxial

_polarization

and

_Ex

the

_applied

d.c. electrical

field;

Ao

=

( T -

To )/T,

B and _{y are}

_positive

constants and

_{a(T, E)}

is

_{negative [15].}

We consider that the

_phases

of the

system are described

_by

a uniform

_{polarization Po}

and a sinusoidal

_polarization

of wavevec-tor q. The

Umklapp

terms have been

_neglected.

Then :

where

L-264 JOURNAL DE PHYSIQUE - LETTRES

We shall

_neglect

the second term in

_(1.5)

for two reasons : first of

_all,

in most of the modulated

phase

P~ ~ ! I Pq

I Po,

and

_secondly,

there is no

way

to mark the

_system

at _{qo, and} test an echo

at 2 2n 2n

.

at

2qo

because 9b

~(T)

~.

Then

_Po

and

_Pq

are determined

_by

aF/aPo

= 0 and

aF/aPq

=

0,

i.e. :

In

_(1.6),

the terms of order

_P~

are

_negligible

in the modulated

_phase

at low and intermediate field values

_(E

1

_kV/mm

in deuterated

_thiourea).

At

last,

we have :

and at low fields :

in the limit

_{Ao -}

2

_{~ ~>}

2

_{bo Vo Co}

we obtain the result of the text. The latter is thus seen to

be a weak

_coupling

limit,

as

_emphasized

in the conclusion. References

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