Origin of the incommensurate phase of quartz: I. Inelastic neutron scattering study of the high temperature β phase of quartz

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Origin of the incommensurate phase of quartz: I.

Inelastic neutron scattering study of the high temperature β phase of quartz

G. Dolino, B. Berge, M. Vallade, F. Moussa

To cite this version:

G. Dolino, B. Berge, M. Vallade, F. Moussa. Origin of the incommensurate phase of quartz: I.

Inelastic neutron scattering study of the high temperature β phase of quartz. Journal de Physique I,

EDP Sciences, 1992, 2 (7), pp.1461-1480. �10.1051/jp1:1992222�. �jpa-00246633�

Classification

Physics

Abstracts

63.20D 64.70R

Origin ^of the incommensurate phase of quartz

_:

^{I. Inelastic}

neutron scattering study ^of the high temperature p phase ^of

quartz

G. Dolino

(I),

B.

Berge (I),

M. Vallade

(I)

and F. Moussa

(2)

(1) Laboratoire de

Spectromdtrie Physique

(*), Universit6

Joseph

Fourier (Grenoble I), BP 87, 38402 Saint-Maxtin-d'Hdres Cedex, France

(2) Laboratoire L£on Brillouin (CEA-CNRS), CEN

Saclay,

91191 Gif-sur-Yvette Cedex, France

(Received 25 November J99J,

accepted

Jl March J992)

Rdsl~md. L'existence de la

phase

incommensurable du quaxtz est attribu£e h _une interaction

entre le

gradient

du mode _mou

optique

de la transition a-p et un mode

acoustique

transverse.

Pour vdrifier _ce modme, des mesures de diffusion

indlastique

des neutrons, de haute rdsolution,

ont dt£ faites. Un mode _mou r£solu _en centre de _{zone a,} pour la

premidre

fois, dt£ observ£ _vers I THz h 250 K, dans la

phase

p du quartz, confirmant le caractdre

displacif

de cette transition.

Le long de [£ 0 0], _une forte interaction est observde _{entre ce} mode _mou et la branche

acoustique

_{ayant une} ddformation de cisaillement a~. L'arnollissement des deux branches mixtes, rdsultant de cette interaction, _a dtd suivi _en fonction de la

temp£rature.

Pr~s de la transition, _un minimum

apparat

_sur la branche basse

ft£quence, qui

d£croit continuement

jusqu'h

0 pour

£

= 0,035 h la transition incommensurable. En raison d'un arnortissement

important,

la branche molle est surarnortie

prks

de la transition, _cc

qui produit

_un

pic quas161astique.

Le

long

de [f f 0], oh le mode _{mou est}

coup16

avec le mode

acoustique longitudinal,

_aucun minimum n'est

observ£. Ces rdsultats sont _en bon accord _avec les

pr£dictions

du modkle de couplage avec un

gradient, d6velopp6

dans l'axticle suivant.

Abstract The

origin

of the incommensurate

phase

of quaxtz ^{is attributed} to a

gradient

interaction between the

optical

soft mode of the a-p transition of quaxtz ^and a transverse acoustic mode. To _test this model

high

resolution inelastic neutron

scattering

studies of the lattice

dynamics

of quartz have been

performed.

For the first time, a resolved _zone center soft mode has been observed in the p

phase

of quartz at I THz _at 250 K,

confirming

the

displacive

character of this transition.

Along

[f 0 0] _a strong interaction has been observed between this soft mode and the acoustic branch With a~ ^{shear strain. The}

softening

of the two mixed branches

produced by

this interaction has been followed

by decreasing

temperature. ^{Near the transition} a

dip

_appears in the lower

frequency

branch, Which goes

continuously

to 0 _near f

= 0.035 at the incommensurate

phase

transition. Due _{to a} large

damping,

the soft branch is

overdarnped

_near the transition leading to a

quasielastic

peak.

Along

[f f 0] Where the soft mode is

coupled

With the

longitudinal

acoustic mode, _no

dip

is observed in the lower

frequency

mode. These results _are in

good

_agreement with the

predictions

of the

gradient

interaction model discussed in the

following

paper.

(*) URA 08, associ£e _au CNRS.

1. Introduction.

The relevance of the soft mode

(SM) concept

^{for the}

understanding

of structural

phase

transitions has been

supported by

_numerous

investigations using light scattering [I]

or

inelastic neutron

scattering [2].

In

general,

lattice instabilities related _to SM _{can occur} not

only

at

high

_symmetry

points

of the Brillouin _zone, but also _at

arbitrary positions

inside the Brillouin _zone,

leading

to the existence of incommensurate

(inc) phases,

_as shown

by

the classical

example

of

K~SeO~ [3].

The results of inelastic neutron

scattering

studies of inc

phases

have been

recently

reviewed

[4], dealing mainly

with the measurements of

phasons,

the

specific

^excitation of inc

phases.

This paper is _a

report

on the result of

high

resolution inelastic neutron

scattering

measurements

performed

in the

high

temperature

fl phase

of quartz ^{in order} to elucidate the

origin

of the inc

phase

discovered in this material

[5, 6]

and its relations to the mechanism of the classical

a-p

transition

It is _now clear that the

properties

of the low temperature a

phase

of quartz are well

described

by

the Landau

theory

of lst order

transition,

with _an order

parameter

_1~

corresponding

to the rotation of

SiO~

tetrahedra

[7, 8].

However the mechanism of the transition is _not

completely

understood _: in

general

the

a-p

transition is considered _to be a

displacive

_{one, as a} SM has been observed

by

Raman

spectroscopy

in the _a

phase [9]

and

by

inelastic _neutron

scattering

in the

fl phase [10].

However _as in this _neutron

experiment only

_an

overdamped

excitation _was

observed,

the

displacive

nature of the transition has been

discussed,

and indeed _a recent neutron determination of the _structure of the

fl phase

_was in favor of _a disordered structure

[I Ii.

In the theoretical work of

Aslanyan

and

Levanyuk [12]

the

origin

of the inc

phase

of quartz was attributed to a

gradient coupling

term,

(u~ _u~~) ^~~

2

u~

~~

,

between the

&x &y

strains u,~ and the

spatial

derivatives of the order parameter ~. This interaction

produces

an

anisotropic coupling

between the SM associated to _1~ and the acoustic modes associated _to u,~, ^{which can lead to the existence of an inc}

phase.

This paper describes the results of

high

resolution inelastic neutron

scattering experiments (already presented

in _a short conference report

[13]).

In the

following

_paper

[14]

the

phenomenological gradient coupling theory

will be discussed _as well _{as a more}

microscopic

model of

nearly rigid Si04

tetrahedra. Before

presenting

our measurements, we will first recall

briefly

the

properties

of the inc

phase

of

quartz

^and the results of

previous

inelastic

neutron

scattering

studies of this

crystal.

The inc

phase

of quartz ^{has been the}

subject

of several review papers

[8, 15, 16]

_: this

phase

which exists around 850 K in _{a narrow} temperature ^{interval of 1.5 K in between the usual} a

and

p phases

is characterized

by

the

superposition

of 3 _waves at 120°

producing

a 3 q structure. Even in this small

temperature

range,

physical properties

show

large

variations. For

example

the modulation wavevector q which is close to the

[I

0

0]

direction of the

reciprocal lattice,

decreases from 0.033 at T~ ^to 0.022 _at T~ (T~ ^and T~ are the transition

temperatures

from the

p phase

to the inc

phase,

and from the inc

phase

to the _a

phase, respectively).

Recent studies of the inc

phase

have been

mainly

^{concemed with} various _non linear effects

produced by

the interaction of modulation _waves with mobile

point

defects

[17, 18]

and

by

studies of the transition around T~ where _a I q

phase (with

_a

single

modulation

wave)

_was observed in _an interval of _a few 0.01K

[19, 20].

As the

long history

of the SM of quartz ^has

recently

been reviewed

[8],

we will

only present

the results of

previous

neutron measurements of the

dispersion

curves of quartz ⁱⁿ a and

p

phases.

In

1967,

Elcombe

published

measurements of the

dispersion

_curves of the 7 lower

frequency

^modes

along [0

0

ii

at room

temperature [21].

More detailed measurements of the

[0

0

ii

acoustic modes _were later

published by

Joffrin _et al.

[22]

in _a

study

of acoustical

activity,

while the

dispersion

curves of the 6 lower

frequency

modes

along [f

0

0]

and

[I f 0]

_were measured

by

Domer et al.

[23].

In

1970,

Axe and Shirane

published

the first _neutron measurement related to the

a-p

transition

[10]

: in the

p phase, they

observed _{a zone} center

SM,

which remained

overdamped

even at 214 K above the transition

temperature.

^{A fit}

by

_a

damped

harmonic oscillator gave ^a

mean field behavior for the

frequency

_wo.

Along [f

0

0],

_a flat low

frequency

branch _was observed further studies of this low

frequency

branch and of its interaction with transverse acoustic modes _were

performed

in the _a

phase by

Bauer et al.

[24]

and

by Boysen

et al.

[25].

(In

this last paper, one measurement was also

performed

in the

p phase,

at 13 K above the transition

temperature.)

In all these measurements no feature

anticipating

the existence of the inc

phase

_was

observed. The existence of _a

premonitory quasielastic scattering

around

(0.035

0

0)

_was first observed in 1985 _at the ILL

by

^several of the

present

authors

[26].

In later works

[27]

performed

at the KFA

(Jolich)

and LLB

(Saclay),

also with thermal neutrons, the

dispersion

curves of the 4 lower

frequency

branches _were measured at 100K above the transition

temperature, showing

the strong

anisotropy predicted by

the

gradient coupling

model between

[f

0

0]

and

[f f 0].

However it is

only by performing

the

present high

resolution

measurements with cold neutrons that _a

complete

verification of the

gradient coupling

model has

finally

been obtained.

2.

Experimental

results.

After _a

description

^of _our

experimental conditions,

_we will first

give

a

general presentation

of the

dispersion

curves of the

p phase

^of quartz _; then we will present our results, first _near the

zone center and then

along [f

0

0]

and

[f f 0].

2.I EXPERIMENTAL coNDiTioNs.-The present

experiment

_was

performed

_at the

Laboratoire Ldon BRllouin _on the

triple

axis spectrometer ^{4Fl installed} on a cold _source. The energy of the incident beam _was selected

by

two

(002) pyrolitic graphite

monochromators

separated by

_an horizontal collimator

Ho (the

first monochromator _was

vertically bent).

Additional horizontal collimators

Hi

and

H~

_were

placed respectively

before and after the

sample

and the energy of the scattered beam _was obtained with _a flat

graphite analyzer.

While

some measurements of the

high

_energy

(E

_~ l THz

dispersion

_{curves were}

performed

with

an incident beam of wave-vector k~ = 2.662

l~

_with

a

graphite filter,

most of the present

measurements _were

performed

at

kj=1.55h~~

with _a

beryllium

filter. For the

higher

resolution

conditions,

we used collimator

divergences

of 25' for

Ho

and of 40' for

Hi

and

H~.

To further increase the resolution _some measurements _{were even}

performed

with

k,

=

1.38

h~

_{A crucial}

improvement

for

high

resolution measurements _near the _{zone center}

was to _use vertical collimators

Vi

and

V~ (of 60'divergence)

before and after the

sample.

The main effect of these vertical collimators _{was to} reduce the vertical

divergence

of the beam and to cut the

high

_energy tail of acoustic modes. Measured

phonon

_{groups were} fitted _to the

response function

g(w)~ [(w2- w()2+ (yw)2]~

of _a

damped

harmonic oscillator of

frequency

_wo and of

damping

_y, convoluted to the instrumental resolution

function, using

the standard LLB program written

by

Hennion

[28].

Two fumaces _were used _: for

higher temperatures,

up to 250 K the standard LLB fumace gave a temperature

stability

of _± 2 K. For _{more accurate} measurements near the

transition,

we utilized _a

laboratory

built fumace which gave a temperature

stability

of _± 0.05 K in the temperature range between

T;

₌ 850 K and 900 K. Two

samples

of natural quartz were used _:

the

larger

_one for the

high temperature

measurement was a

cylinder

with axis

parallel

to

OZ,

of 30 _mm diameter and 50 _mm

height.

The smaller _one for

higher

_accuracy measurements close to the transition temperature was a cube of 20 _mm sides. Both

samples

were

positioned

with the Z axis

vertical,

_so that measurements were

performed

in the

(0

0

1) plane.

With

k,

= 1.55

h~ only

a few

points

of the

reciprocal

_space could be

reached,

_among them the

largest

scattered

intensity

was measured in the

(I 0)

zone.

2.2 DISPERSION _{CURVES lN THE}

P

PHASE OF QUARTZ. The 6 lower

frequency dispersion

curves of

fi~quartz

at 250 K in the

(0

0

1) plane

_are shown in

figure

I. In the

following

_we

will consider

only

the soft mode

(SM)

and the 3 acoustic

branches,

the

longitudinal

_one

(labeled LA)

and the _{two transverse ones}

(TAT

and

TA~). (Near

the _zone center, the shear strain of

TAT

is in the

(0

0

1) plane

and its

slope

^is

proportional

to the square ^root of the elastic constant C~~; the shear strain of

TA~

is

perpendicular

to the

(0

0

1) plane

and its

slope

is

proportional

to ^~~l). As _a consequence of the

hexagonal

_symmetry of the

p phase,

the acoustic branches have _an

isotropic dispersion

_near the _zone center.

(We

recall that in the

reciprocal

coordinates used in this paper, ^{the wavector modulus} q =

f along [f

0

0]

and q =

If _{along [f f 0]).}

The 4 lower

frequency

modes have different

couplings along [f f 0]

and

[f

0

0]

_:

along [f f 0]

there is _a

gradient coupling

between the SM and the

LA,

both of

Ai symmetry

while

the 2 TA modes of

A~ symmetry

are not

coupled

with the SM.

Along [f

0

0],

the SM of

3~ symmetry

^is

coupled

with the 2 TA modes of

3~

symmetry, ^{but there is} a

strong gradient

interaction

only

with

TAj,

while the

coupling

with

TA~

is of

higher

order in q ; the LA of

3j

_symmetry is not

coupled

with the SM. Due _to the strong

gradient coupling,

_{a gap appears}

f fi r

/

/ ^,

LA ,'

',

' ITA,

,' ^,+-

2 /~"~

_T~

',' ', 2

,"' ₁₀₍₀₁ ^",

1'

/

M' ',

" (SM/TA ,'~ ^{~ ~~~~}

",

,"

,,~'~~'~

^s~

,/

' r §001

jjooj

~

o.~ o-s o.3 o

Fig.

I. -LOW

frequency dispersion

curves at 1250 K, in the p

phase

of quaxtz,

along

[£ 0 0] and [£ f 0]. (The insert shows _a

paxtial

view of the hexagonal Brillouin zone.) The full lines

correspond

to

symmetric modes while the dashed lines

correspond

to antisymmetric modes. (Below 3 THz, these lines

correspond

to the results obtained by the present measurements and presented with _more details in the

following pictures

above 3 THz, the lines

give only

_an indication of the

topological

behavior

expected

from symmetry considerations.)

in the

crossing region

between the _two

coupled

branches

(labelled (SM/TAi)~

and

(SM/TAj)_

for the

high

and low energy branches

respectively).

The

dispersion

_curves of

figure

I _at 250 K _are rather similar _to those obtained

previously

at 950 K

[27]

and shows the

same strong

anisotropy

between

[I

0

0]

and

[f

f

0]. However,

in the present

experiment

we were able _to observe _{a new}

important feature,

_a resolved _{zone center} SM at I THz.

2.3 ZONE CENTER SOFT MODE.

2.3, I Measurements at q = 0.

Only

_an

overdamped

_zone center SM has been observed _so

far in the

p phase

of quartz

[10].

With the

high

resolution conditions with

k,

₌ 1.55

h~

~, ^a resolved SM is _now observed at

(I 0)

_as shown

by

the _curve

(a)

^of

figure 2,

measured _at 250 K

(I,e.

400 K above the transition

temperature

_{T~ =} ⁸⁵⁰

K).

The main

difficulty

in this

measurement was the strong ^low

frequency

contamination

coming

from the acoustic branches

collected within the finite volume of the resolution

ellipsoid.

The _use of vertical collimators

was crucial in order to reduce this

quasielastic peak appearing

below 0.3 THz _;

although

_very

intense

(~

10~

n/mn),

the

Bragg peak

itself does not disturb the

experiment

^{because its width}

is

only

0.01THz

(FWHM).

Intensity

(n/~mn)

>"~ _i

THz~

~

i

200 '

~ 100

~ T

800

' T(K)

0 ^'

Id

₉₀₀

'

,"

'

~

l 0 0 ^'

0

,

"'-

-l 950

I '

+

~ ^~

50

,,~/

⁺

j

⁺⁺

,,_ +_+ ⁺

b)

₁₀₅₀

0 ^~~

~ ^{+ +}

',~,---~---~~

^~~~~~

o SM uJ

0 0.5 _1.5 ITHz)

Fig.

2. Plots of the (I 0) neutron groups of the soft mode (SM) measured at various temperatures in the p

phase

of quaxtz. ^{The full} lines are the results of the fit to the response function of

damped

harmonic oscillators, With

frequencies

_wo, indicated

by

vertical _arrows. The dashed lines show the

respective

contributions of the SM and of the

background.

The insert shows the linear _mean field behavior of

WI

as a function of the temperature.

The full line of

figure

2a shows the fit of the SM to the response function of _a

damped

harmonic oscillator. Several measurements

performed

at

1250K, give

_{a mean}

frequency

wo = ± 0.03 THz and _{a mean}

damping

_y

=

0.75 _± 0,15 THz. Measurements

performed

with k~ =

2.662

h~

_at

_{(I 0), (3}

₀

_{0), (2 0)}

_and

₍₂

₂

₀₎

_{do not show}

a resolved

peak

but

only

_a shoulder around I THz.

With

decreasing

temperature, ^{the maximum} of the SM is hidden

by

the

parasitic quasielastic peak,

but the results of the fit

given

in column

(a)

of table I show that the SM

becomes

overdamped

below 900 K. At 860 K the fit with 2 free

parameters (w

o and

y)

is not

meaningful

because the response function becomes very close to a Lorentzian of width

w)y

_then

we fixed _y

= 0.5 THz

by extrapolation

from

higher temperature

^values. The inset of

figure

2 shows the linear variation of

ml

_{as a} function of the temperature. ^The

points

between 860 and 050 K _{are on a}

straight line,

which

extrapolates

to 0 at

To

₌ 847

K,

I-e- 3 K below

T;.

This value of T~

To

₌ 3 K is smaller than the value of 10 K obtained

by

Axe and

Shirane from _neutron measurements of the

overdamped

SM

[10]

and

by

Bachheimer and

Dolino,

from the variation of the order parameter ^{in the} a

phase [29].

2.3.2 Measurements in the

vicini~y of

the r

point.

For small q

values,

the

gradient coupling

of the SM with acoustic modes is _so small that the modes _can be considered _as

uncoupled.

Measurements

performed

around

(110)

^for _{q =} ^0.02 at 250

K,

with

angles increasing by

30°

from

if I 0)

to

ii f 0) give

results similar to those measured at

(1.02 0),

shown

by

curve

(a)

^of

figure

3 _: the SM is

again

observed _near I

THz,

but _on the low

frequency

sides

one observes _{now a} resolved

peak produced by

the TA modes around 0.22THz with _a

shoulder

produced by

^the LA mode at 0.33 THz. With _a

decreasing temperature,

the SM

frequency

decreases while the acoustic

peaks

remain _at the _same

frequencies

_as shown

by

curve

(b)

for T

= 950 K. At lower temperatures ^{the SM becomes}

overdamped

and appears as a

quasielastic peak. Then,

_{one can}

improve

the resolution

by using

neutrons of lower energy

with k~ =

1.38

h~

as shown

by

_curves

(c), (d),

and

(e)

of

figure

3. These _curves measured at

(10.987 0),

^{show the}

growth

of the

overdamped

SM below the acoustic modes. The SM

frequencies

fitted _{to an}

overdamped

harmonic oscillator with _a fixed _y

=

0.5 THz

give

_a

mean field linear variation for

ml,

_as shown in the inset of

figure 3,

which

extrapolates

to 0 _at

To

₌

T,

7 K

(with

_a lowest measured value of _wo

= 0.09 THz _at T

= T~ + 1.5

K).

The value of

To

obtained here is lower than that determined _at the _{zone center ;} this is

probably

as a

consequence ^{of the} uncertainties in

extracting frequencies

from

overdamped

modes. In conclusion the results obtained either at the _{zone center or} in its

vicinity

_are in

agreement

with

a mean field behavior of the

SM, although

_more

complex

^behavior with _a central

peak

_very

close _to

Ti,

_cannot be

completely

excluded due to the limited resolution of _our data. Within the present mean field

analysis

the

frequency

of the _zone center SM has _a finite

frequency

wo= 0.08 ±0.03 THz _at T~. A linear

extrapolation

for

ml _gives

_{a zero}

_frequency

_at

a

temperature

To

with T~

To

= 5 _± 2 K.

2.4 MEASUREMENTS _ALONG

Ii

0

0].

2.4, I

[f

0

0] dispersion

_{curves at} 250 K. The

expected coupling

between the SM and the

TAj

is most

clearly

shown

by

the

dispersion

of the

purely

transverse

(I

+q ^I -q

0)

modes _at 1250 K. The

phonon

_groups

corresponding

to the 2 mixed modes

(SM/TAi)~

and

(SM/TAj)_

measured for different values of the wavevector q are shown in

figure

4a.

The

high frequency (SM/TAi)~

branch starts at I THz with the horizontal

dispersion

of _an

optical

mode but with

increasing

_q, the

(SM/TAi)~

branch

progressively

takes _a steeper

slope

close to the linear

dispersion

characteristic of _an acoustic mode behavior. On the low energy

side,

_near the _zone center, there is _a

single peak corresponding

to the unresolved TA

~~ ^~

lt/

~

/ ^mn

~~j

0.01 To 500

'

0 10 20

0 T= 85%K

500

,,

'>,,~

0

'1.,,_ Id)

_{8 66}

500 ^'

0

fi[~~ II'

8 8

1000'

500

jb)

q50

o

t

soo

'

la) 12so

$,

_*

_~f

SM LU

0 TA LA 0.5 (THz)

Fig.

3.- Temperature ^{variations of} neutron groups measured in the

vicinity

of the (I 0) zone

center,

showing

the SM

softening

I) _curves (a), (b), measured at (1.020 0) With

k,

₌ 1.55

h~

~, ii)

curves (c), (d), (e), measured at (1 0.987 0) with _k~

=

1.38

A~

' Vertical _arrows show the SM

frequency

wo,

given by

^the fit to

damped

harmonic oscillators. The TA mode shows small temperature variations,

remaining

at wo =

0.19 THz. The dashed lines show the contribution of the SM_ and of the TA modes.

The insert shows the linear _mean field behavior of WI near T~ = 850 K (with _y

= 0.5 THz).

modes. For q =

0,15,

_a

splitting

of this

peak

_appears, ^which is _more

clearly

visible at q =

0.20. The lower

frequency

branch

already

observed

by

Axe and Shirane

[10]

is identified

as the

(SM/TAi)-

branch. The medium

frequency

branch

going

to 2.05 THz _at the _zone

boundary

is the

TAz

_one. The interaction of the SM and of the

TAT produces

a gap of

0.7 THz around q = 0.05 between the

(SM/TAi)~

and the

(SM/TAj)_.

The

[I

0

0] dispersion

curves can also be measured in _{a more}

longitudinal configuration along (I

_{+ q}

10)

_as shown in

figure

4b. The

anticrossing

of the

(SM/TAi)~

and of the

(SM/TAi)-

^is no

longer observed,

_as the

intensity

of the

(SM/TAi)+

decreases

quickly

with

increasing

_q. One observes

clearly

the LA mode _at 0.72 THz for q = 0.04. For

higher

values of q, the

dispersion

_curve of the LA is very close to that of the

(SM/TAi)~

observed in

figure

4a. One _can also note _an

interchange

of the relative intensities of the

(SM/TAi)_

and of the

TAz occuring

between

(I

+ q

0)

and

(I

+ q I q

0).

This is

mainly

due to the

strong anisotropy

of the _structure factor of the

(SM/TAj)_,

shown in

figure

5 of the

following

_paper.

The

[I

0

0] dispersion

_curves obtained from these measurements at 250 K _are

plotted

in

figure

4c. While the dashed lines _are

only

a

guide

for the eye, the full lines _were calculated

1(a.u.) (SM/TA~)_

~~?

_{~~~ ~~}

,

0,1 '

' '

'

,

j ^,

/

' f

' ^' / ^{' '}

q =0

uJ

j~)

^{2 3 0}

jb)

~lTHzl _is

_N

_{/TA,i~ ,,}

~

LA

, _,

/ ,

/ ,,~

, ,

/ ,'

,

q

0 o-1 0.2 0.3

(C)

Fig.

4. Measurements of the low

frequency

[£ 0 0]

dispersion

curves at 250 K in the p

phase

of quaxtz : a) _{neutron groups} measured

along

(I _{+ q} I q 0) for different values of q. The full lines _are

given by

_a fit to

damped

harnlonic oscillator behavior; b) neutron groups measured

along

(I

+q10)

for different values of q; c)

plot

of the

dispersion

_curves: (+) (I _+q I -q 0)

measurements ; (x) (I _{+ q} 0) measurements. The full lines, calculated from the

gradient coupling

model, show the behavior of the 2 mixed branches _{for q «} 0.13. For

larger

values of _q, the dashed lines

are only a guide for the eye.

from the

gradient coupling

^model introduced

by Aslanyan

and

Levanyuk [12].

For q w 0.13 there is a rather

good

agreement ^{with the}

experimental results,

while for

larger

_q this

model becomes

inadequate

in order to describe _our

experimental

results.

The

gradient coupling model, developed

in the

following

_paper, considers _an

expansion

in q of the SM

frequency w(~(q)

= A

(T To)

+

gq~

+

hqi _coupled by

_a term

aq~

to _an acoustic branch

ml

=

~"q(

_where

only

^the

damping

_y of the SM is introduced. The values of the P

parameters

^{used in} the calculations _are

given

in table I of the

following

_paper. The

profile

of the measured

phonon

groups shown in

figures

4a and b _were fitted _to the _sum of the response

functions of

uncoupled damped

harmonic oscillators and not to the calculated response function of the

coupled

modes. This

procedure,

^which enables _us to use a standard

computer

program and _to reduce the number of

parameters,

is

probably acceptable,

as no clear

evidence of interference effects were visible in the measured

phonon

_groups.

2.4.2

Temperature

variation

ofthe [f

0

0] dispersion

_curves.

Although

the

larger

variations

are observed for q w 0.2 _on the

(SM/TAj)~

and

(SMIAI)_ branches,

_some

softening

is also measured at q =

0.5 _: between 250 K and 850

K,

the

frequency

of the

TAz

decreases from 2.05 _to 1.9 THz while that of the

(SM/TAi)_

decreases from 1.05 _to 0.85 THz.

Systematic

measurements of the temperature ^{variations of the 2 mixed branches} were

performed

for

various values of q from 0.02 _to 0,I. Here _we will

present

^{the results obtained} at

q =

0.035,

which

correspond closely

to the initial satellite

positions

of the inc

phase

at

T,.

Due to the

anisotropy

of the intensities of the different modes shown in

figure

^{4 the}

variations of the 2 branches _were measured at different

positions

ⁱⁿ

reciprocal

_{space :} ^the

softening

of the

(SM/TAi)~,

shown in

figure 5a,

_was measured at

(1.035

0.965

0),

while the

softening

of the

(SM/TAi)-,

shown in

figure 5b,

_was measured _at

(1.03510).

Between 1250 K and T~ =

850

K,

the

(SM/TAi)~ frequency

decreases from wo = I to 0.6 THz. At T~ this mode shows _a linear acoustic

dispersion.

On the other

hand,

the lower

frequency peak, corresponding

to the

TAz,

remains at _a constant

frequency

_{wo =} 0.31±0.02THz with _a

small

damping

_y w0.07THz. In the _same way ^the

high frequency peak

of

figure 5b,

1(n/~mn) (nNmn)

TA~

,

(SM/TA,)+

T(K) 2

+ LA

'

900

+

+ ~

+

+

j ^{~~~ ~}

fl ^{+ +}

>

o uJ

°

[al

^{~~~~~ °}

lbl

Fig.

5.

Temperature

variations of the [f 0 0] low

frequency

_{neutron groups} in the p

phase

of qualtz measured :

a)

at

(1.035

0.965

0), showing

the

softening

of the

(SM/fAi )~, b)

at

(1.035 0), showing

the

softening

of the (SM/rAi)-.

Table I. Variations with temperature

of

the

frequency

_wo and

of

the

damping

_y

(both

in

THz) of damped

harmonic oscillators

fitted

_to

soft

mode

phonon

_groups in the

fl phase of

quartz.

(*The

values _at T~ were

extrapolated from higher

temperature measurements ;

(x)

indicates _a parameter with _a

fixed

value

x).

(a) ~b) (c)

SM

(SM/TAi)+ (SM/TAi)_

q =

0 q =

0.035 _q

=

0.035

T

(K)

_w

o y wo Y wo ^Y

250 ± 0.03 0.75 _± 0,15 1.01 ± 0.04 0.75 _± 0.13 _± ± 0.005

050 0.62 0.58 0.80 0.53 0.308 0.05

950 0.44 0.56 0.71 0.42 0.262 0.14

900 0.32 0.49 0.63 0.39 0.238 0.25

880 0.25 0.47 0.59 0.34 0,197 0.25

860 0.20

(0.5)

0.58 0.39 0,133

(0.3)

T,

= o,08

(0.5)

0.60 0.40 0

a) Zone center soft mode (SM) measured at

(I

0).

b)

High frequency

mixed branch measured _at (1.035 0.965 0).

c) Low

frequency

mixed branch measured at (1.035 0).

corresponding

to the LA

mode,

remains at a constant

frequency

_wo

= 0.61± 0.04 THz but with a

larger damping

_y ~0.4THz. The

(SM/TAi)_

shows _a

complete softening

from wo =

0.34 THz at 250 K _to 0 at

T;

while its

damping

increases from _y

= 0.05 to 0.3 THz _so

that this mode becomes

overdamped

around 870 K. The

phonon

_groups of

figure

5 _were fitted

by damped

harmonic oscillators. The temperature variations of the parameters ^{obtained for} the 2 mixed branches _are

plotted

in

figure

6 and _are

given

in columns

(b)

and

(c)

of table I.

The full lines of

figure

6

correspond again

to the

gradient coupling

model : the

frequencies

show _an

anticrossing

behavior which agrees rather well with the

experimental points.

There is also _a

good _agreement

for the variations of the

damping

of the 2 modes _: while at

high temperature only

the

(SM/TAi)+

has _a

large damping,

_near

T;

^the 2 modes have similar

dampings. Finally

_we

emphasize

that the

anisotropy

of the

intensity

observed _at 900 K for the

(SMffAI)_ corresponds

to the

anisotropy

^of satellite intensities in the inc

phase

_:

_strong

satellites _are observed at

(I

_±qo

0)

and _at

(I

I _± qo

0)

while

only

weak satellites _are observed at

(I

± qo I ± qo

0).

Between

T,

and

T;

₊ 50

K,

_more detailed measurements _were

performed

with the _use of _our

laboratory

built fumace with

high

resolution collimations. The

(SM/TAj)~

mode observed at

(1.035

0.965

0)

shows little variation in this

temperature

range.

On the other

hand,

_as shown in

figure 7a,

at

(1.035 0)

a

complete softening

of the

(SM/TAi)_

is observed: _at

T,+43K

^the

(SM/TAi)_

is yet

underdamped

^while at T~ ⁺ 21.5 K it

just

becomes

overdamped.

At lower temperatures a

strong quasielastic peak

uJ(THz)

I ₊

$(THz)

LA

+

~ ,

~l)-

+

x

o

800 _Tj loco 1200 T(K)

ja) ¹⁶⁾

Fig. 6.

-Temperature

variations of the parameters ^of

damped

harmonic oscillators fitted _to the [f 0 0]

phonon

_{groups :} a)

frequencies

_wo the

frequencies

of the LA and of the

TA2

modes _are nearly constant, while the

frequencies

of the _two mixed modes

(SMfTAI)+

and (SMfTAI)- show _a

typical anticrossing

behavior, in

good

agreement ^with the results of the

gradient coupling

model, shown

by

the full lines _; b)

dampings

_y; at

high

temperatures, ^all the

damping

is with the

(SMfTAI)+,

while _near T~, the 2 mixed modes have similar

dampings.

(+)

(1.035

0.965 0) measurements (x)

(1.035

0)

measurements.

grows, similar to that

already

observed at

(1

0.987

0)

in

figure

3. In this

overdamped regime

the

profile

of the

quasielastic peak

is almost Lorentzian with _a width r

=

wj/y.

The

temperature

^{variation of}

wj/y

near

T;,

is

plotted

in

figure 7b, showing

also _a linear _mean field

behavior,

close to the

prediction

of the

gradient coupling

model

given by

the full line. The

corresponding

numerical

parameters

are

given

in table II.

Although

the agreement ^between model calculations and

experiment

is rather

good,

_some differences appears for the lower

~lC) T-Tj(K)

~~ ^{I~IO THz}

11.~

~~

~

+

~

2500 16) 21,5

+

la)

o tu

T-Tj

j(~

^{~~ ~~~~~ ° ~°}

lb)

^{~° ~}

Fig.

7. a)

Temperature

variations of the

(SMfTAI)_

neutron groups, measured at

(1.035

0) in the p

yhase

^{of quaxtz, for different} temperatures close to T;. b)

Corresponding

temperature ^{variations of}

~ °

near

T,

(w

o and _{y are} the values of the

frequency

and of the

damping

of haIn1onic oscillators fitted to Y

the neutron groups of

Fig.

7a). The full line

corresponds

to the _mean field behavior of the

gradient

coupling

model.

Table II. Variations with temperature

of frequency

_wo,

damping

_y and width r

(all

in

THz) for

the

(SM/TAj)_

mixed mode at q = 0.035 _:

a)

Calculation with the

gradient

interaction

model

of

next paper. In the

overdamped region

below 871.5

K,

the width

of

the

quasielastic peak

is r

=

wj/y. _b) Experimental

results obtained _at

(1.035 0) for

the width r

of

the

overdamped soft

mode.

a)

Model

b) Experimental

T(K)

_{wo y} r=- r

893 0.223 0.193

880 0,193 0.222 0,155

871.5 0,167 0.238 0.l17 0.l10

866 0,146 0.251 0.085 0.073

861.4 0,122 0.260 0.057 0.060

860 0,l17 0.263 0.052 0.059

856.3 0.093 0.27 0.032 0.032

854.2 0.076 0.275 0.021 0.026

851.2 0.041 0.28 0.006 0.012

T,

₌ 850 0 0.28 0

temperature points

of

figure

7b where the measured

frequency

is

larger

than

expected

for _a

mean field behavior. This is

probably

_a consequence of the

inadequacy

of the standard

fitting procedure

around the conical

dispersion point occuring

at

T,.

Without information _on the

dispersion along [0

0

ii

^it is difficult _to

improve

_upon this situation.

Many

measurements similar _to those

presented

for q = 0.035 have been

performed

for different

temperatures

^and

different values of _q,

giving

the full

temperature

^{behavior of the}

dispersion

_curves in the

p phase

of quartz. ^{When the}

(SM/TAi)-

becomes

overdamped

around

870K,

it has _a flat

minimum around q = 0.05. With _a

decreasing temperature,

this minimum of the

dispersion

curve shifts to q =

0.035,

where the

frequency

_{goes to} 0 _at

T;.

Some

experimental points

_are

shown in

figures

2 and 6 of paper ^II ; we have fixed _y

= 0.3 THz for the entire

overdamped regime

in order to have less

dispersion

in the values of _wo.

2.4.3 Measurements

of

the

quasielastic scattering.

It is

interesting

to

study

the variation of the

quasielastic scattering

in the

overdamped regime

which is observed _near

T,

for

0wqw

0.07. The results of such measurements with low resolution collimation _at

k~

=1.55h~~

are shown in

figure 8,

where the intensities of the

quasielastic scattering

measured

along (I

+q

0)

_are

plotted

_{on a}

log

scale as a function of q for various

temperatures near

T;.

^{The lowest}

intensity

_curve, measured in the _a

phase

at

T~-

n/mn) I(n/mn)

i

_T-Ti(K)

-0.9

,-0,3

_~

/~ lj

0 2

T-Ti

/

₀

2b

lo

~,

o,os

q

Fig. 8.

Temperature

variations of the

experimental quasielastic scattering intensity (in log

scale) of the

overdarnped (SM/rAj)_

measured along (I + q 0) _{near T~.} The insert shows the temperature

variation (in linear scale) of the peak

intensity

close to

T,.

I.4K,

shows the

Bragg peak

and the residual

intensity

from the acoustic modes which decreases with

increasing

_q. Similar _{curves are} obtained _at

high temperature

^{in the}

p phase.

At

T;+43K

_a shoulder

extending

from q= 0.02 to q= 0.07 is

distinctly

observed

corresponding

to the low

frequency

tail of the

damped (SM/TAi)-.

For lower

temperature

this

quasielastic intensity

increases _as the

(SM/TAi)-

becomes

overdamped

around

T;

+ 20 K. At about

T;

+ II K _a broad maximum appears around q =

0.04

corresponding

_to

the existence of _a minimum in the

(SM/TAi)_

branches. With further

cooling

the maximum of this

quasielastic peak

becomes _more resolved and at

T;

it

changes continuously

into the

narrow elastic satellites of the inc

phase.

The inset of

figure

8 shows the variation of the

maximum

intensity

of the

quasielastic scattering

as a function of the

temperature.

^The transition temperature T~ is determined

by

^the start of the

sharp

linear increase of satellite intensities in the inc

phase.

The accuracy of this determination increases with better

collimations.

(Recently

ⁱⁿ

high

resolution

experiments using

neutron

[19]

and

synchrotron

radiation

[20]

it has been

possible

to observe the existence of _a

lq

^inc

phase

ⁱⁿ a small range ^of

a few hundreths of K in between the

p phase

and the usual

3q

inc

phase).

The

intjgrated ^intensity

^{I of a}

^damped

^{harmonic oscillator of}

^frequency

^{wo is}

^{given by}

1~

~'~)~

where

F;~~

^{is the inelastic} structure factor of this mode. The

gradient coupling

wo

model shows that _near

T;,

the structure factor of the

(SM/TAi)-

branch does not

change

much _{as a} function of q around 0.035.

Therefore,

the observation of _a maximum in the

quasielastic intensity

is _a direct consequence of the existence of _a

dip

in the

(SM/TAj)_

branch.

Some measurements _were also

performed

^{in the} inc

phase: although

the

increasing

intensity

of satellite

peaks

makes the detection of the diffuse

scattering

_more

difficult,

_a

quasielastic scattering along [f

0

0],

^is

always

observed. This

corresponds

to the additive

effects of

phason

and

amplitudon

excitations in the inc

phase.

As both modes remain

overdamped,

it _seems

nearly impossible

to separate ^their contribution. We have

only

noted _a small

broadening

of this

quasielastic scattering

in

going

_{near T~,} which

might correspond

to the

increasing frequency

of the

amplitudon.

To _resume, the results obtained

along [f

0

0]

_are in rather

good _agreement

with the

predictions

of the

gradient coupling

model _: at

high temperatures

^{there is} a

strong coupling

between the SM and the

TAT, Producing

two mixed branches

(SM/rAi)+

and

(SMfrAI)-.

With

decreasing

temperature, ^{the lower}

frequency

branch shows _a

dip

around q=

0.035 which _at

T, gives

rise

continuously

to the elastic satellites of the inc

phase.

2.5 MEASUREMENTS _ALONG

Ii f 0].

2.5,I

Dispersion

_curves at 250 K.

Along [f

£

0],

the _two

symmetric

modes

(SM

and

LA)

_are

coupled by

_a

gradient interaction,

while the 2

antisymmetric

modes

(TAj

and

TA~)

are not

coupled

with the SM.

Figures

9a and b show the _neutron groups measured

respectively along (1

+ 2

f

I

I 0)

and

(I

+ f I ₊

I 0)

for different values of q =

/

_{f. In}

figure 9a,

the

high frequency

mixed mode

(SM/LA)~

is

clearly

observed _: it starts at I THz _at q =

0 and for

larger

_{q it} takes _an acoustic character with _a linear

dispersion.

In

figure

9b this mode is observed

only

at small q < 0.04. The observation of the low

frequency (SM/LA)_

is

more difficult because its

frequency

is rather close to that of the TA modes:

along

(1

₊ 2

f

I f

0)

the _most intense low

frequency

mode is the

TA2 going

to 2.3 THz at the

zone

boundary.

The

(SM/LA)_

_appears first around q =

0.10 _{as a} shoulder _on the

high frequency side,

which becomes _a resolved mode for q = 0.13.

Along (I

₊

f

I + I

0)

the

interpretation

of the

experimental spectra

^is even more difficult _{: as} this direction is

along

a

symmetry ^axis of order two, one

expects

to observe

only symmetric

modes I,e, the 2 mixed

branches

resulting

from the interaction of the SM and of the LA. However for

q m 0.15, 3 low

frequency peaks

are observed _: in addition _to the

expected (SM/LA)_,

one

observes the 2 TA modes. The lower

frequency

one

corresponds nicely

to the

TA2 already

observed

along (1+

2

I

I

I 0).

The

highest frequency

_one, noted

TA(,

is found _{at a}

higher frequency

than the real

TAj

measured in other _zones

(in particular along (3 f

2

( 0)

where it is very

strong [26]).

Our

interpretation

is that the observation of TA modes

along (I

₊

I

I ₊

I 0)

is due to the finite size of the resolution

ellipsoid (in particular

in the vertical

[0

0

ii direction)

and that the

upward

shift of the

TAf

is due to the strong

anisotropy

of this mode.

The

[f I 0] dispersion

curves obtained from these measurements are

plotted

in

figure

9c the 2 full lines show the

prediction

of the

gradient coupling

model for the mixed

(SM/LA)±

modes,

calculated with the _same

parameters

as for

[I

⁰

0]

the

only

new

input

is the

slope

^of

the LA

mode, corresponding

to the elastic constant C

ii. As the

slope

^{of the LA} is

larger

than the

slope

of the

TAj,

the

anticrossing

of the 2 mixed modes _occurs around q = 0.06

producing

a smaller gap of 0.2THz. For _q ~0.13 the

gradient

model is _not able to

give

a

good

representation

and _a dashed line is drawn _{as a}

guide

for the eye. In the

anticrossing region.

the

expected dispersion

of the

(SM/LA)_

is masked

by

the presence of the

TA(.

If the identification of the

coupled

modes is exact for small and

large

_q, the

dispersion

of the

(SM/LA)_

in the

anticrossing region

cannot be very different from the theoretical

dispersion

curve shown in

figure

9c.

(Note

that for _{q <} 0.I, the

highest frequency peak

^of

figure

9b must

correspond

to the

superposition

of the

(SM/LA)_

and of the

upshifted TAj*).