Origin of the incommensurate phase of quartz : II. Interpretation of inelastic neutron scattering data

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Origin of the incommensurate phase of quartz : II.

Interpretation of inelastic neutron scattering data

M. Vallade, B. Berge, G. Dolino

To cite this version:

M. Vallade, B. Berge, G. Dolino. Origin of the incommensurate phase of quartz : II. Interpretation of inelastic neutron scattering data. Journal de Physique I, EDP Sciences, 1992, 2 (7), pp.1481-1495.

�10.1051/jp1:1992223�. �jpa-00246634�

Classification

Physics

Abstracts

63.20D 64.70R

Origin of the incommensurate phase of quartz

:

II. Interpretation of inelastic neutron scattering data

M.

Vallade,

B.

Berge

and G. Dolino

Universitd

Joseph

Fourier (Grenoble I), Laboratoire de

Spectrom£trie Physique

(*), BP 87, 38402 Saint-Maxtin-d'Hdres Cedex, France

(Received 25 November J99J, accepted in final form ii March J992)

Rksl~md. Los rdsultats de

l'investigation

_par diffusion

indlastique

des neutrons des modes de basse

frdquence

^du quartz p, ddcrits dans l'article

pr6cddent

[Ii, sont

interpr6tds

h l'aide de deux

approches

diff6rentes _: I) _un modkle

ph6nom6nologique,

directement issu d'un

ddveloppement

du type

Landau-Ginzburg

de

l'6nergie

libre _{; cc} modkle n'est valable que pour la

panic

du spectre relatif _aux

phonons

de grande

longueur

d'onde, mais il perrnet ^d'6tablir une connexion aisle _avec les donndes

therrnodynamiques

it) _un modme

microscopique

de

dynamique

de r6seau, qui est

une extension du modme de Grimm-Domer (modkle h t6trakdres

rigides)

on montre que les

principales caract6ristiques

du spectre ^des

phonons

de basse

ft6quence,

et en

particulier

l'amollissement d'un mode

3~

h _un vecteur d'onde incommensurable

prks

du centre de _zone, peut Etre

compris

_{par une}

analyse

des mouvements de t6trakdres

Si04

_Presque

rigides.

Abstract. The results of _an inelastic neutron

scattering investigation

of the

low-frequency

modes of p quartz, ^{described in the}

preceding

_paper [I], _are

interpreted using

_two different

approaches:

I) _a

phenomenological

model

directly

derived from _a

Landau-Ginzburg

_type

expansion

of the free energy ; this model is

only

relevant for the

long-wavelength

_part of the

phonon

spectrum ^but it allows an easy connection with

therrnodynamical

data _; it) _a

microscopic

lattice

dynamical

model, which is _an extension of the Grimm-Domer model _; it is shown that the main

properties

of the

low-frequency phonon

spectrum and, in

particular,

the

softening

of _a

3~

mode _{at an} incommensurate _wave vector close to the zone,center, can be understood

by analysing

the motions of

nearly rigid

Si04 tetrahedra.

I. Introduction.

The various

physical

mechanisms

leading

to the _occurrence of incommensurate

(inc) phases

in dielectrics have been _an active field of research

during

the recent years

[2].

From _a

phenomenological point

of

view,

inc

phase

transitions in dielectrics have been related to the presence ^{of the so-called} « Lifshitz invariant _»

j3] (which

_can

only

be found when the order

(*) Assoc16 _au C-N-R-S-

parameter ^{has several}

components)

_or to _{a «}

Lifshitz,type

invariant»

[4] (when

the order parameter ^is

coupled

to another

thermodynan~ic variable).

On the other hand the

microscopic

mechanisms

leading

to the existence of _an inc

phase

have been

clearly

elucitaded

only

in _a restricted number of materials _: for

biphenyl,

the

competition

between intramolecular and intermolecular interactions has been invoked to

explain

the _occurrence of _a modulated _structure

[5].

^In

NaN02

the inc

phase

is

thought

to

result from the

coupling

of

large amplitude

reorientations of

N02

_groups with the

translational motion of Na

[6]. Microscopic

models have also been

proposed

for _some other systems e,g.,

K~Se04 17], propylammonium

tetrachlorometallates

[8].

The relative

simplicity

of the quartz structure makes attractive the

study

of the mechanism

leading

to the inc

phase existing

_near the a-p ^{transition in this material}

[9]. According

to the

phenomenological theory

of

Aslanyan

et al.

[10, iii

the _occurrence of _an inc

phase

in quartz type

crystals

would result from _{a «}

gradient coupling

_» between the elastic strains and the

gradient

of the order parameter _1~ ^{of the}

a-p

structural transition. This kind of

coupling corresponds,

^{in the soft-mode}

picture,

to an interaction between _an

optical

mode and _an acoustic mode close _to the Brillouin zone-center as shown in reference

[12].

A

striking

illustration of the

physical meaning

of this kind of

coupling

in

quartz-type crystals

is

provided by

the so-called _«

exaggerated gradient

method _» of Heine and Mc-Connell

[13].

The order parameter _1~ ^{of the}

a-p phase

transition

corresponds mainly

to altemate rotations of

Si04

tetrahedra around _axes

lying

in the basal

plane

of the

hexagonal

lattice. It

corresponds

to a

zone center

optical

mode. In the _«

phase

_{1~ can} take _two

opposite

values

(according

to the _two

possible signs

of rotation of each

tetrahedron), leading

to

Dauphind twinning.

A very thin domain wall between these twins

(as represented

in

Fig, I) corresponds

to _a strong

(and probably quite exaggerated) gradient

of _1~ but _{one can}

easily

_see that deformations of the unit cell

(-I.e. strains-)

_{are present} in the wall.

Furthermore,

the

comparison

between

figures

la and 16

clearly

shows

that, according

to the direction of the

gradient

of

1~, the strains _are either pure shear

(VI~ along

the

Oy

type

directions)

or

purely compressive

strains

(VI~ along

the Ox type

directions).

In

spite

of their

qualitative

character these

pictures

_are of _some

help

to

understand the mechanism

leading

to the incommensurate

phase

in quartz: ^{the actual}

gradient

of

1~ which is present ⁱⁿ a

long-wavelength

inc

modulation, although

much smaller than that described in

figure I,

is associated with strains of different

kinds, according

to its

direction. One is then

logically

^lead to ascribe the

origin

of the inc

phase

to an

interplay

between _two modes of atomic

displacements

:

optical,like

^and acoustic-like. A controverse took

place

some years ago about the relative

weight

of the _two modes

[14].

In _a recent

preliminary work,

Gouhara _et al.

[15] reported

_{on an}

X-ray

determination of the inc structure

and

they

confirmed that the modulated atomic

displacements correspond

to a

mixing

of

acoustic and

optical

_components

(with

_a

large

contribution of the acoustic

one).

As the inc

phase

arises from the

high

temperature p

phase through

_a 2nd order

transition,

the

dynamical

critical fluctuations above the

p-inc

transition

point T,,

must also involve _a

mixing

of acoustic and soft

optical phonons.

This

point

_was

already

evidenced in

preliminary

inelastic _neutron

scattering experiments [16]

and _{a more} detailed

investigation,

described in the

preceding

paper

[Ii, brings

_a

stronger support

to this

point

of view. The purpose of this paper is _an attempt to

give

a

quantitative interpretation

of the temperature

dependence

of the

low-energy phonon

branches in the

p phase

and thus _to propose ^a mechanism

leading

to the inc

phase.

The

analysis

will be made _{at two} different levels _:

.

First,

_as the inc _{wave vector} is

only

a small fraction of the

reciprocal

lattice unit _vector

(qo>0.03a*),

_a continuous-medium

approximation

is relevant. We consider _a

simple

phenomenological theory

which follows the

original

model of reference

[10] by considering

only

_a soft

optical

mode

(SM)

and acoustic

phonon modes,

with small _{wave vectors.} The main

x x

a)

b)

Fig, ^I.

-Exaggerated

gradient

picture of

a

Dauphind

_twin wall in

The

figure

_the

arrangement

of the

comersharing Si04 tetrahedra,rojected on the (0

0

^{) lane :} a) wall

erpendicular to the Oy axis.

The

radient of the order

parameter

(v,u~ )

is associated with _a

unit

cell. b)

Domain wall erpendicular to

the

Ox axis. The gradient (v~u~) is associated

with

a

compressive strain

of

the unit cell. In figure la the two _twins have been

translated

along

order

deformed.)

interest of

this

approach

_is

to

provide an easy connection

macroscopic ^operties,

Sect. 2).

. _In a second step, an ttempt

is

ade

to

interpret the

of the

modes over

the

_whole

Brillouin zone, by considering

a

icroscopic model. _As a

starting point

for this

study, _we use the

Grimm-Domer model

[17,

18] which _assumes that

frequency ^ptical modes are those hich leave the Si04 trahedra

The

method

used to find _{these modes is}

described in

ection 3.

A

imple

Bom,von rman

force

constants

and a uasi-harmonic

approximation)

is hen ^oposed in

section

emarks.

2. henomenological _radient

coupling model.

As usual

in the

_soft-mode

theory of

phase

transitions, the _phonon requencies

within the

pproximation,

using

an effective ^otential energywhich

has

_the

same

form

as the andau-Ginzburg free

energy.

_In

the

_high emperature

p phase

of

has already

been _{shown that} the potential

energy

corresponding

to

the long _velength

soft

optical _phonon

mode

and

to

In this

expression ~(q)

is the

amplitude

of the

uncoupled soft-optical

mode with _{a wave} vector q, and

wj(q, _T)

is the

corresponding frequency.

Close to the Brillouin _zone center we

assume that _:

wj(q, _T)

m A

(T To)

+

gq~

₊

hq~

U~~~(q), UT~~(q)

^and

U~~(q)

are the

amplitude

of the two transverse acoustic modes

(polarized respectively

in the

(0 01) plane

and

perpendicular

to this

plane)

and the

longitudinal

acoustic mode. The

uncoupled

acoustic mode

frequencies

_{are :}

W(Ai(~)

_~

j~ _{~~ W(A2(~)}

"

~ _{~~, W/A(~)}

~

)~ ~~

(p

=

volumic _mass,

C~~ ⁼ elastic

constants).

The bilinear

coupling

between

1~

(q)

and

U~A (q)

_or

_UTA~(q),

in

equation (I) corresponds

to the

Lifshitz-type

invariant introduced

by Aslanyan

and

Levanyuk j10].

This _term exhibits

anisotropy (since

^it

depends

_on the

angle

_~b between q and the twofold Ox axis of the

hexagonal p phase).

One _can note that for

4

= 0

(mod ar/3)),

_~

(q )

is

only coupled

with the LA mode, whereas for

4

=

(ar/2) (mod (ar/3)),

_1~

(q)

is

only coupled

with the

TAT

mode.

This agrees with the _«

exaggerated gradient picture

_» of

figure I,

since q is

along

the

gradient of1~.

A bilinear

coupling

between

1~(q)

and

UT~~(q)

^{is also allowed}

^by

^{symmetry but it is}

proportional

to

q~ and,

for small q, it _can be

neglected.

^{A similar}

description

^of the interaction between _a soft

optical

^mode and _an acoustic mode has

already

been used

by

Axe et al, to

explain

^{the anomalous}

dispersion

of acoustic branch in

KTaO~ j19].

In order _to describe the finite linewidth of the

phonon

_{groups, a}

damping

_y has _to be introduced for the soft

optical

mode

(the damping

of acoustic

phonons

is

proportional

to

q~

and it _can be

neglected

in the small _wave vector

limit).

The

coupled

mode

complex frequencies

_are then obtained from the secular

equation

_:

ml

w y w ~) aq^~ cos 3

#

_aq^~ sin 3

#

det

aq~cos

3

# (w/~ w~

₀

= 0.

(2)

aq~

sin 3

#

0

(w(A w~)

For q

along

the 3 lines

( ii

0

0)

directions of the

reciprocal lattice)

_cos

(3 #)

₌ ^{0 and}

equation (2)

reduces to _a 4th order

polynomial equation

for _w.

Explicit

solutions _can

only

be

obtained in the limit _y

= 0 _:

ml _(q)

=

iwl(q)

₊

_WlAi(q)i

±

~/iwl(q) _WlAi(q)i~

+ 4

a~q~j ⁽³⁾

The lower branch

w_(q)

exhibits _a minimum

equal

to 0 at the

p,inc phase

transition

temperature

^T ₌

T;

for _an inc modulation wave-vector q = qo.

Ti

and qo are related

by

the relations _:

A

(T~ To )

₌ 1/2

£

g

q(

=

hq( ₍₄₎

66

Since h is assumed to be _~

0,

_an inc

phase

_can

only

_occur when g _<

(pa~/C~~).

When _y is

non-vanishing, equation (2)

must be solved

numerically.

^{The solutions behave}

essentially

in the _same way as in the _y

= 0 _case except ^{that both}

frequencies

_{w~ are now}

complex,

the mode

coupling producing

_a transfer of

damping

from the soft-mode to the acoustic mode, as observed in Brillouin

scattering experiments [12].

Most of the

input parameters

^of this

model,

have been deduced

by fitting

^the inelastic

neutron data of

[Ii

obtained at the

highest temperature (T= Ti +400K)

in the range

[q[

_< 0.I a*

along

the 3 direction.

(This

limited range of wave-vector _was chosen rather

arbitrarily,

but it appears ^to be _an upper limit of

validity

of the q

expansion

used for the

uncoupled frequencies).

The coefficient h

(proportional

to

(Tj To) according

to

(4))

was

much _more

sensitively

deterJnined from the value of qo and from the spectrum ^at T

T,.

The elastic constants C

ii and

Cm

were taken from ultrasonic data

[201.

The numerical

values of the

parameters

are summarized in table1

[211.

The

dispersion

_curves

along

3((f00))

directions and A directions

((f to))

for the

temperature

_range

(T;

_< T

< T

; + 400

K)

were then calculated with this set of parameters. ^{The fit} was

improved by allowing

_a linear

temperature dependence

of the

damping

coefficient _y

(from

0.5 THz _at

Ti,

to 0.7 THz at

Ti

+ 400

K).

A

general picture

of the

dispersion

_curves is shown in

figure

2.

The main conclusions which _can be drawn from _a

comparison

of calculated and

experimental

_{curves are} the

following

_:

I)

The overall agreement is correct. In

particular

the

strong anisotropy

^of the soft modes and of the

TAj

and LA modes in the basal

plane

is

fairly

well

reproduced

with _a minimum of

adjustable parameters. Neglecting

the

coupling

between the soft

optical

mode with the

TA~

mode is

justified

since the

dispersion

_curve of this mode is almost

isotropic

and exhibits

no anomalous

dispersion.

ii)

The main

discrepancy

concems the value of the

parameter Ti To (calculated

to be

0.7K whereas the

experimental

value is

5±2K). Any _attempt

to

adjust Tj- To

to its

experimental

value results in _a

strong

^{increase of} the value of the

parameter

h

(according

to

Eq. (4))

which leads to _a curvature of the

dispersion

_curve at T

= T~

by

far _too steep ^for qo < q < 0, I. This

discrepancy

_may arise from _some

inadequacy

of the _mean field

theory

and

(or)

of the

quasi,harmonic-approximation.

Let _us recall that _an anomalous temperature

dependence

of _some elastic constants and of the thermal

expansion

above

Ti [221

shows the

important

role

played by

fluctuations combined with strong anharmonic

couplings.

Unfortu-

nately, owing

to the limited resolution and _to the

difficulty

of _a reliable deconvolution of _our neutron

scattering data,

it _seems

meaningless

at the present ^{time to introduce additional}

fitting parameters

ⁱⁿ the model

(such

_as non-classical critical

exponents).

On another

hand,

it is noticeable and somewhat

puzzling that,

in

equations (4),

the terms

pa~/Cm

and g

nearly

Table I. Parameters

of

the

phenomenological

model

of

section 2

(with JFequencies

in THz and _wave vectors in units

of

the

reciprocal lattice).

~~~

= 350

(C11

= 166.5 _x

llfN/m~)

P

~~~

=

120

(Cm

=

57.075 _x

llf N/m~)

P A

= 0.0025 _g

= 140

a =131.2 h =1600

y =

0.5(T/To) To

= 850 K

T; To

= 0.74 K qo =

0.033 _a *

/

~ 2

o

0

0.00 .02 .04 .06 ,08 .10

a) b)

Fig.

2.

-Dispersion

_curves of the

interacting phonon

branches

according

to the

phenomenological

model of section 2. The lines _are calculated with the _set of parameters ^{of table I and the}

points

are

experimental

values

(Ref. [I]).

al Wave-vector

along (I

0

0)

direction

(3

_or

Oy).

b) Wave-vector

along (I

² I

0)

direction (A _or Ox). (~ D) T T~ ₌ ^0.75 K, (. . .) T T, ₌ 12 K, (... 6)

T T, ₌ 50 K, (~ m) T T, = 100 K, (- o) T T~ = 400 K.

cancel each

other,

I.e. the _constant g is close to the threshold value for which _an inc

phase

_can

occur.

(This

is related _to the small range of

temperature (1.5 K)

_over which the inc

phase

exists,

and to the small value of the inc _wave vector

qo).

This close

cancellation,

which appears

as a pure coincidence in the

phenomenological theory,

_can be better understood in the

light

of the

microscopic

model described in the next section.

3.

Rigid

tetrahedra mode

analysis.

In this section _we present an

attempt

to derive

simple physical

_arguments which _can shed

some

light

_on the existence of _an inc

phase

in

quartz, by considering

the

peculiar

structures of this material. In

principle

the _occurrence of the inc

phase

should be found

by solving

the full

dynamical problem (27 phonon branches) using

_an

adequate

force _constant model. Several models of this kind have been

proposed [23]

most of them _are relative _to a-quartz ^but some

of them also _concem

p-quartz [24, 25].

It is not

surprising

however that such models do not

predict

the inc

instability,

because the characteristic

dip

_on the lowest

phonon

branch is

expected

to be _a

tiny

detail in the full

phonon spectrum,

^{and its} existence is related _{to a} subtle

balance between weak force constants. In

addition,

_even if _a model _were able _to exhibit this

dip,

it would

probably

obscure the

physics

under

heavy

^numerical calculations. Therefore it

seems to be _more advised to concentrate _our attention _on the lowest energy

phonon

modes.

As

already

often

noted, quartz

^is built of rather

rigid Si04

units which share _{a common comer}

(see Fig. 3).

The covalent character of the SiO

bonding explains

this

rigidity

and the

relatively

open ^structure of

p

quartz.

Megaw [26] analysed

the

a-p

transition in

quartz

as the result of rotations of

rigid Si04

tetrahedra. This idea _was then

developed by

Grimm _et al.

[17]

who tried _to relate the thermal

expansion

_near the transition _to these rotations. The _same

j,

5 6

I.o

7

y3

Fig.

3.

-Projection

of the unit cell of

p-quartz

_on the basal

plane.

(o) _oxygen atoms, (o) silicon atoms.

assumption

of

rigid

tetrahedra _was later used

by Boysen

_et al.

[181

to

analyse

the low

frequency part

^{of the}

phonon

spectrum ^of a and

p quartz

^that

they

have measured

by

inelastic neutron

scattering.

The main

assumption underlying

^their

analysis

is that intertetrahedron force constants are smaller than intratetrahedron _ones. Therefore the lowest energy

phonon

modes _must

correspond

to

nearly rigid

motions of

Si04 units,

if any motion of this kind is allowed

by

the structure. As _{a matter} of

fact,

these

particular

modes

merely correspond

to the

« extemal modes _»

widely

used in the context of lattice

dynamics

of molecular

crystals.

In the

case of covalent framework

materials, however,

_severe

topological

constraints _are

imposed

_on

the motion of

rigid

molecules

by

the fact that

they

share _{a comer} and the existence of

« extemal modes _» cannot be taken _as

granted

in the

general

_case. The

analysis

of

Boysen

et al.

[18] only

^{concemed the existence} of

rigid

tetrahedron modes

(RTM)

at some

high

symmetry

points

in the Brillouin-zone

(r

and

M).

In _a

preceding

_paper

[16]

_{we gave} the results of _{a more}

general analysis

valid for any

point

inside the Brillouin _zone. Since _no detail

was

given

in that paper

conceming

the method used to derive these

RTM,

and since this method is

thought

to be of _some

general

interest to

predict

the existence of soft modes in framework

crystals,

_we

give

ⁱⁿ the

following

_{a more}

complete description

of the

procedure.

Let _us consider _a 3-dimensional framework of N identical _comer

sharing

molecular units with

p comers shared. The total number of

degrees

of freedom is 6N

(3

translations

T~~ ^{and 3 rotations}

R~~ (per

unit

I).

The three

displacements

of each atom shared

by

two units

(I

and

j)

_can be

expressed

_{as a} function of both the

(T~,

_R~ ^{and the}

(T~, Rj).

There _are then 3 relations

linking

the motions of _two connected units _: these

compatibility

relations will be

called «constraints». The total number of constraints in the system ^is

clearly ~~~

2

(neglecting

surface

effects)

since

(Np/2)

atoms _are shared. For small atomic

displacements

these ~

)P

constraints lead to _a

system

^{of ~}

~P

homogeneous

linear

equations

with 6N 2

unknown

(T~~

and

R~j).

For tetrahedral units

(as

in

quartz)

_p

= 4 _so that the number

of equations

is the _{same as} the number

ofunknown.

This _case is

particularly interesting

since _non-

trivial solutions

only

exist under

special

conditions.

Using

a harmonic

approximation,

the

equation

of motion _can be written

using

Fourier components

T~;(q)

and

R~;(q)

where the index I denotes _now the molecular units

belonging

to a

given

unit cell

(I

= I to

n)

and q is a

vector of the lst Brillouin _zone

(BZ).

For each q there _are then ~ ~P

equations

with fin 2

complex

unknown

T,(q)

and

R,(q).

Let _us consider the

p

_quartz structure

(with

the

assumption

of

perfect Si04

tetrahedra for the sake of

simplicity).

There _are three

(Si04)

units per unit cell

(see Fig. 3).

The 18 constraints

correspond

to the

displacements

of 6 oxygen

atoms O~ = 4 to

9) (see

tab.

II). They correspond

to _a

system

of 18

homogeneous equations

in which the _{wave vector} q appears

through

the

phase

^factors

e;.

_{For q} at a

general position

ⁱⁿ

the BZ there _{are no} non-trivial

solutions,

but solutions exist when q lies in the

(0

0

1) plane

or

along

the

[0

0

Ii

directions

(and

also

along particular

lines _on the BZ

edge).

In

solving

the

system

^of

equations,

^{it is} convenient to use

symmetry adapted

combinations of

T,(q)

and R~

(q)

which transform

according

to irreducible

representations

of the small _group of the _wave

Table II. Atomic

displacement

_{components u~~}

(in

^the

(x,

_y,

z) reference JFame)

_as a

jknction of

the translation and rotation components

of

^the

Si04

units,

for rigid

tetrahedron modes

ofwave

vector q R

(

e

)

is the matrix

corresponding

to a rotation

ofangle

e around the _z- axis.

T~,

and

R~;

_are translations and rotations components ^{in the} local

(x;,

_y;,

z,) reference frame (see Fig.

3

).

The _set

of18 equations

^relative to the oxygen

displacements

determine the

possible

RTM with _{wave vector q}

(coordinates

11~D.

Si ui

~

=

Ti

j2arj~

~

u2«"R~p ~

2p

j4arj~

"3«"R«fl $

3fl

~

U4a ~

(Tl

~

~l

X Pa

)a

"

°3 ~afl ~) _(T3

_~

_~3

X

Pd)fl

115 _a ^" R

«fl

~)

_{1'~2 +}

_~2

_X

Pb )fl "

°3 ~afl ~) _(~3

₊

_~3

_X

_Pc)fl

"8

a ^"

~ap ~) _(T3

~

~3

_X

Pa)p

_"

°2 ~afl ~/

_1'~2

~

~2

X

Pd)p

"9

a

"

(Tl

_~

~l

_X

Pb)a

"

°2 ~ap ~) _(T2

_~

_~2

X

Pc)fl

91 = exp(2 ^ar(7~

-

)) ; 9j

=

xp(2 ar(7~ <))

°2

_"

exp(2 iar(- -

_~

+

_<))

; 93

" XP(217r(I

vector q, so that the system

splits

into

independent sub,systems corresponding

to each

representation.

The various solutions _are summarized in table III. In this table is also indicated the character of the RTM when q goes ^{to zero.} Three different cases are found _:

I)

^{the RTM behaves} as a pure translational

displacement

^and is called acoustic

(A), it)

it behaves _{as a} pure

optical

mode

(O) (center

of _mass

invariant), iii)

it is _a

mixing

of acoustic and

optical components (A+O).

At the BZ center

l~

it is clear that the 3 uniforJn translations leave the

Si04

units undeforJned _so that

they

are RTM

(labelled Qx(r~), Q~(r~)

and

Q~(r~)).

There exists _an additional solution

Q(r~)

^which

corresponds

to the

order

parameter

_1~ ^{of the}

a-p transition,

since it involves altemate rotations of the

Si04

around the

Ox-type

directions.

Compatibility

relations can be found

by looking

at the

q - 0 limit

along

various directions of the

reciprocal

_space. For q

along [0

0

fl (A line),

one

has 3 RTM

belonging

to different

representations

_{A~, A~, A~} and _:

Q(A4)

-

Qx(r6)

+

joy(J~6)

Q (A6)

-

Qx(r6) ioy(J~6) Q (A2)

-

Q (r3)

The former _{two are} associated with the TA modes

polarized

in the

(0

0

1) plane

and the latter is the soft

optical

mode.

Table III.

Rigid

tetrahedron modes with _{wave vectors}

along

the

3,

A and A directions.

T,,

and

R,;

_are the

(unnormalised)

translation and rotation components relative _to the I-th

SiO~

unit

(see Fig.

3

).

The

corresponding

atomic

displacements

_can be calculated

using

table II. In the last line is indicated the character

of

each RTM when _{q goes to zero}

(O

=

optic,

TA

=

transverse

acoustic).

_«

=

art

_{; y =} ^"

< y'

=

"

((

I

)

_;

y"

=

"

(<

+

1).

3 3 3

~l zg oo) A(oon

Txi ^'2 cos a a. cos a

Ty1 °

~~2m $11 4cosZy)slay T~~ I(d3+1)sin _a

Rxi 2 _{cos a}

4COS~Y'l~coSy

RyI ⁰ ~

xe2i" "(4C°S~tI)slay

R~j ^{2i sin} a

~ sinix Tx2

as Q(62) Q(62)

~'~

~~~~ ~'~

~ y ~

Tz2 U

l(4

^{cosla-I) sina}

Ry2 xe""

x ~lu sina

x

e~"

Rz2

C°S" ~~~

xe-2iu

xe.2iu e~4iy

Ry3 -Ry2 xe-2>a _x e-4.y

Rz3 0

q ~ o ~6) lox+ ;Qy) ~6)

When q is

along

_a

general

direction in the

(001) plane, only

one RTM is found and it behaves _{as a} pure TA mode

polarized along [0

0

Ii (the

mode labeled

TA~

in Sect.

2).

No

additional RTM is found in the

particular

_case when q lies

along ii I 0) (A directions),

but

one extra RTM is

actually

found when q lies

along ii

0

0) (3 directions).

This last

mode,

noted

Qa(3~),

_{appears to} be the most

interesting

_one. As _a matter of fact it is the

only

RTM which exhibits _{a «} mixed _» character

(A

₊

O)

_:

Qa(32)

- 0.6

Q (1~3)

0.8

Qx(1~6) (The

modes have been normalized

using

_:

£

_m;

[u~~(q)[~

=

l.)

i, a

This

singular

behaviour deserves _some comments, ^{since actual}

phonon

modes _are

always

pure acoustic _or

optical

modes when q goes to zero. This _«

hybrid

_» mode is _a consequence of the infinite force constants which _are

implicitly

assumed in the

rigid Si04

tetrahedron

hypothesis.

Modes which _are not RTM have _an infinite

frequency.

This is the _case, in

particular,

for the acoustic modes

(at

_q #

0)

which _{are not}

RTM,

such _as the LA mode and the

TAT

mode. The

corresponding dispersion

branches have _an infinite

slope

at q # 0 for

perfectly rigid

tetrahedra. The

crossing

of this acoustic

phonon

branch with _an

optical

mode

occurs

right

_at the r

point,

and

hybridization

of acoustic and

optical

modes takes

place

at

vanishingly

small q.

(With

_a more realistic

assumption

of finite intra tetrahedron force constants, the

crossing

would _{occur at} finite _wave vectors and the

singular

behaviour would be removed _see below Sect. 4

-).

It is

interesting

to note that this

type

^{of behaviour is}

quite

reminiscent of the TO

phonon-photon

mode interaction in the

polariton region

_:

assuming

_an

infinite

light velocity

results in _{a «} mixed

photon-phonon

_» excitation at q =

0 which leads _to the LO-To

splitting.

This

singular

behaviour

disappears

when _a finite

light velocity

is taken

into _account.

An additional

interesting

feature of the

Q~(3~)

mode is found when _one considers the _mean

change

Ad of the

(Si-Si)

distance for this mode. As shown in

figure

4 Ad vanishes _at r and _at the BZ

edge point

M and it is

quite

small

along

the whole 3 line for this

particular

mode.

Since the Si-O-Si force _constant is known to be the

strongest

among the intertetrahedron force _constants

[2, 5]

_{one can expect} the

Q~(3~)

mode to be flat and soft

along

the whole 3 line.

Let _us summarize what kind of

qualitative

conclusions

conceming

the

phonon dispersion

curves of

p quartz

can be inferred from _our

assumption

of

rigid

tetrahedra _:

I)

the existence of three

low-lying

branches

(two

TA and _one

optical)

_{for q}

along [0

0

fl (A line)

it)

for q

along if

0

0) (3 lines),

a very soft and flat

branch, corresponding

to a strong

hybridization

of _an

optical

and _an acoustic

(TAj)

components near

l~

and _a

moderately

low

frequency (TA~)

branch _;

iii)

for other directions of q in the

(0

0

1) plane,

_a

moderately

soft

(TA~)

branch

roughly isotropic

as

long

as q lies in this

plane.

A

comparison

with

experimental

data

[Ii

shows that these

predictions

are

essentially

correct.

FurtherJnore,

inelastic neutron

scattering

structure factors _can be calculated for the various

RTM. The results

conceming

the zone-center soft-mode

Q(r~)

and the soft-branch

Q~(3~)

_are shown in

figure

5. It is

striking

that the

largest

^calculated structure factors _are found

along

the _« fourth

hexagon

_», ⁱⁿ

agreement

^{with the}

large

scattered intensities observed

along

these directions

[27].

The RTM

analysis provides

_us with _a

qualitative explanation

of the mechanism

leading

to an inc

phase

_; the

Q~(3~)

mode possesses the two

key

features which _are

required

for the

Ad

z ' /

2c '

' / /

~ //

A

r z

Fig.

4.- Mean relative change Ad of the Si-Si distance for the various

rigid

tetrahedra modes considered in the model of section 3. Ad is defined

by

Ad

=

i<J) jj

_[(1~~ -1~~),r~~]~)~'~ ^where _1~~ ^{is the} (norrnalised)

displacement

of the I-th silicon in the mode and

r~~ the unit vector

along

the

equilibrium

Si- Si direction. (Ad for the shear mode

Qc(3~)

is also

represented

using a dashed line.)

) ) £ L

IQ, ~ .-.--"l«<i<11]ibjjjjjl«>"-"..-zJ.-

~ JL

Q Q S J L

L Q Q 7

$ 4

Z

L /

IQ'

L

IO.

ii, oi 12,o) _13, oi 14, o) 15,o)

Fig.

5. Inelastic structure factors calculated with the atomic

displacements

of the

Q~(3~)

mode (Tab. III). The size of the bars is

proportional

to the squared modulus of the _structure factors. The circles _are relative _to the zone-center optic-mode Q

(r~).

Maximum

scattering intensity

is found along the «4th

hexagon»

in agreement ^with

experimental

observations. (The

figure corresponds

to the f

= 0 strate of the

reciprocal space.)

occurrence of _a modulated

phase, according

to our

phenomenological approach

_{: a} flat soft

dispersion

branch

along

3

(g small)

and _a

large SM-TAT hybridization

at small q

(gradient coupling

coefficient _a

large).

^The ratio _r of the

optical

mode

component

to the TA mode

component

^is r= 0.75 in _our model and it _can be

compared

with the ratio of the

corresponding

static

displacements

in the inc

phase recently

determined

by

Gouhara _et al.

[15]

(r

_<

0,15).

It is noticeable that all the above conclusions have been deduced from

purely geometrical considerations,

without any

adjustable parameters.

^{In order} to be _a bit _more

quantitative

_one has to choose _a

specific

^force constant model.

4. Lattice

dynamical

model of fl quartz ^within the

nearly rigid

tetrahedra

approximation.

To follow _on the lines of the

analysis

of the

preceding section,

_we shall

attempt

a calculation of the

dispersion

_curves of

p

_quartz

by assuming

almost undeformable

Si04

units. As _{we are}

mainly

interested in the lowest

frequency phonon branches,

_we do not

try

to

diagonalize

the full 27 _x 27

dynamical

matrix but _{we assume} that it _can be truncated _{to a} 4 _x 4 sub-matrix

corresponding

to the lowest energy modes

(acoustic

and soft

modes).

Such _an

approximation

is better _near the Brillouin _zone center

(where

these modes have

frequencies

much lower than other

modes)

than _near the _zone

edge

where

they

are

relatively

close to the hard modes. Our

aim is to

keep

_a maximum

physical transparency by introducing

the minimum number of

fitting parameters.

^The set of modes chosen to write the 4 _x 4

dynamical

sub-matrix includes the RTM

complemented by

non-RTM

purely

^acoustic

components.

^For q

along

A the 3

acoustic modes and the soft mode

belong

to different irreducible

representations.

The two TA modes and the

optical

mode _are then the RTM of table III _;

only

_one LA

component

^should have to be

added,

but since it

belongs

to the irreducible

representation Ai

^{it does} not

couple

with the

previous

modes and it is of _no interest for _our purpose. For q

along 3,

_one LA component

(with

_symmetry

3j)

and _one

TAT component (with symmetry 3~)

should be added to the _two RTM

Q~(3~)

and

Q~(3~).

One must note that there is _a

large

arbitrariness in the choice of the non-RTM modes since

only

their behaviour close to q = 0 is well defined. A reasonable

guide

in

building

these modes is _to select those which

keep

the Si-O bond

length invariant,

since the

corresponding

force constant is

by

far the strongest one

[25].

Unfortu-

nately

this

prescription

is not sufficient _to

uniquely

define them. One has

found, however,

that the

shape

of the

dispersion

curves does not

depend

very much _{on a}

particular choice,

at least in the

vicinity

of the r

point (which

is the _most

important

to discuss the inc

instability).

The force constant model _we have chosen, is of the

simple

Bom,Von Karman central force

type,

as that

recently

used

by

Bethke _et al.

[25]

to fit the whole

dispersion

_curves of

p

quartz.

One considers the interactions between 5 different kinds of atom

pairs

:

(Si-O), (O-O)i, (Si- Si), (O-O)~

and

(O-O)~.

The former two are

intratetrahedral, nearest-neighbour

interactions.

The other three _are intertetrahedral interactions

((O-O)2 corresponds

to interactions

between oxygen atom

pairs

like

(04-06)

and

(04-O~)

both

separated by

_a distance

~ _~

(O-O)~

to interactions between atom

pairs separated by

^~ where _a is

~

~

^~

/~

~

/~

the lattice

parameter).

There _are then 10 force constants

(5 longitudinal (L

and 5 transverse

(T)

force

constants).

According

to our

approximation,

L

(Si-O)

is considered to be infinite. The other two strong force _constants L

(O-O )i

^and L

(Si-Si)

which _are best determined from

fitting

the hard modes have been fixed _to values close to those found

by

Bethke et al.

[25],

The

equilibrium

condition

imposes

one relation between the 5 _transverse force constants so that 6

independent

force constants _are left _as

adjustable parameters.

^The

dispersion

_curves for _q

along

^{3 have} been

fitted _to determine these parameters. ^{As shown in}

figure

6 a shallow

dip

_can be obtained _near qo =

0.035 a*

along

the lowest 3 branch.

(In

Ref.

[25]

this

dip

could

only

be

reproduced by introducing

_an ad hoc

phenomenological coupling

between the SM and

TAT modes.)

In order to

reproduce

the

temperature dependence

of the

modes,

it _was found sufficient to allow for _a

small linear

change

of the force constants

(mainly T(Si-Si) (cf.

Tab.

IV)).

As

expected,

the

TA~

mode is found to be

only slightly coupled

with the other

3~ modes,

_so that the

dip

close to

T;

is

mainly

due _to the interaction of the RTM

Q~(3~)

with the pure acoustic component

Q~(3~),

_as assumed in the

phenomenological theory. Furthermore,

the

eigenvector

of the

dynamical

matrix relative to the lowest branch is

quite

close to the RTM _« mixed _» mode

Qa,

for any q

larger

than 0.005 _a * In

particular

at the inc _wave vector the

eigenvector Q

is _:

Q

=

[0.96 Qa,

0.27

Qcl

/

~ o II

~ i

/~ _--, _, O/

~ i

/~

/

, ^{/ /}

y/ ^/~

' /

~

/~

' /

"

O o

u

~ u

0 ^{° °}

A r Z 0.00 .02 .04 .06 .08

a) b)

Fig.

6.

Dispersion

_curves of the lowest energy

phonons

of p quartz

along

the 3 and the A directions of the

reciprocal

_space. The lines _are the results of the

« nearly

rigid

tetrahedron model

» with the force

constants of table IV. The

points

_are

experimental

values (Ref. [I]). (~ D) T

T,

=

0.75 K, (... 6)

T T~ = 50 K, (- O) T T,

= 400 K. a) Dispersion _curves along the whole Brillouin _zone. b)

Enlarged

view of the

vicinity

of the incommensurate wave-vector

along

the 3 direction.

Table IV. Force constants

ofthe

«

nearly rigid

tetrahedron

approximation

_»

ofsection

4. L and T

refer

to the

longitudinal

and _transverse

force

constants

of

the Born-Von-Karman model.

t ₌ T

T;.

The values _are in

N/m.

(si-o) (o,o)i (si,si) (o-o)~ (o,o)~

L _oo

(17.487

0.006

t) (48.7

+ 0.0164

t)

2.0 5.96

T 14.737

(-

1.375 0.006

t) (-

1.31 + 0.0164

t)

1.61 1.70