Large phase fluctuations near TI and phase pinning in incommensurate ThBr4 : Gd3+ through E.S.R. measurements

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Large phase fluctuations near TI and phase pinning in

incommensurate ThBr4 : Gd3+ through E.S.R.

measurements

J. Emery, S. Hubert, J.C. Fayet

To cite this version:

Large phase

fluctuations

near

_TI

and

phase

pinning

in

incommensurate

_{ThBr4 :}

Gd3+

_through

E.S.R.

measurements

J.

_Emery

_(1),

S. Hubert

₍₂₎

and J. C.

_Fayet

₍₁₎

(1)

Université du Maine, Laboratoire de

_{Spectroscopie}

du Solide (*), 72017 Le Mans Cedex, France

(2)

Institut de

_Physique

Nucléaire, B.P. n° _{1, 91406 Orsay Cedex,} France

(Re~u le

_{29 fevrier}

1984, revise le 2 mai, accepte le 14 mai ₁₉₈₄₎

Résumé. 2014 Récemment la coexistence de raies « comme normales » et incommensurables dans les spectres R.M.N. de

87Rb,

a été observée au-dessous de _T, dans

_Rb2ZnBr4,

et a été _{interprétée} en

termes d’ondes stationnaires de fluctuations de

_phase

entre des défauts

_épingleurs.

Nous

_analysons

un

effet

_identique

dans les _spectres R.P.E. de

_{ThBr4 :}

Gd3+. Nous excluons un effet

_extrinsèque

de la sonde E.P.R.

_qui

ne manifeste aucune tendance à accrocher la

_phase

et nous montrons _{que des} fluctua-tions de

_phase

_{inhomogènes peuvent} rendre _compte des observations. Une

_application

sommaire du modèle

_développé

_{par Blinc}

_indique

_{que le «}

_pinning

»

_{apparait probablement}

sur les dislocations. Abstract. 2014

Recently,

the coexistence of « normal-like » and incommensurate 87Rb N.M.R. lines have been observed in

_Rb2ZnBr4

below _TI, and has been

_interpreted

in terms of a

_standing

wave

regime

of

_phase

fluctuations between

_pinning

defects. We

_analyse

a similar effect in the E.P.R. _spectra

of ThBr4 :

Gd3+. We rule out the

_possibility

of an extrinsic

_pinning

effect due to the E.P.R.

_probe

and we show that

_{inhomogeneous phase}

fluctuations can account for the observation. A crude

appli-cation of the model

_{developed by}

_{Blinc indicates that the average distance between}

_pinning

centres is

compatible

with their

_being

dislocations. Classification

Physics Abstracts

61.16N - 64.70 - 73.60F

Introduction.

ThBr4 (/3) undergoes

a normal to incommensurate structural

_phase

transition at 95 K

_[1-6].

Neutron

_scattering

data

_{[3] give}

a

precise description

of the soft coordinates and of the associated

displacement

field below

_1,.

No lock-in transition is observed down to 10 K, and the modula-tion

_obeys

a

_plane

wave

_regime.

An

_important

and rare result

_[3]

is the

_unambiguous

observation of the

_phason

and

_amplitudon

modes below

_T,,

in accordance with

_theory.

Two

_types

of local measurements with the

_help

of extrinsic

_probes

have been realized. The luminescence _spectrum of

U4 +

_[6]

at 4 K would seem to indicate a

_{partial phase pinning by}

the

U 4 +

_impurities.

In a

previous

work we have shown the

_sensitivity

of the E.P.R.

G d3 + -n. n. n

Br

-vacancy to the transition

_[5].

(*) E.R.A. no 682.

L-694 JOURNAL DE _{PHYSIQUE -} LETTRES

In this Letter we focus our interest on :

i)

the evolution of the

E.P.R. lineshape

across the incommensurate

_phase

in order to

_investigate

the

_possibility

of

_{phase pinning by}

the

_probe.

ii)

a

particular

effect limited to the _{temperature range}

_{T, -}

2.7 T

_T]

which we

_interpret

in terms of

_large

_phase

fluctuations. A similar effect has been

_recently

observed

_by

87 Rb

N.M.R.

spectroscopy

[7]

in

_Rb2ZnBr4.

1. Theoretical

_background.

The static

_displacement

field below 95 K can be

_{represented by}

a

_{superposition}

of a rotation and of a twist of the Br- tetrahedra

_{[3] : surrounding}

the

Th4+

ions

r and t _represent normalized vectors for the rotation and for the

twist,

A is the

_amplitude

and

QJ

= kz ₊

00

is the

phase

of the

plane

_wave

modulation,

_z is the coordinate of the Br-

along

the incommensurate wave vector

k(O,

0,

k),

a is a

mixing

coefficient between the two

_components,

the

_phases

of which differ

by ~

_[3].

2

The static

_displacement

field induces a local shift of the E.P.R. lines of a

_{paramagnetic probe}

substituted for

Th4+.

A _power

_expansion

limited to second order

gives :

We have assumed a short range

probe.

In the critical

region

one has A oc

_{(T I -}

T)~.

The

parameters hi depend

on the E.P.R. transition and on the orientation of the

_magnetic

field;

they

are not

_dependent

on _temperature. Within the model

[8]

the static line

shape

is

given by :

and exhibits a continuous

_{absorption background}

with

singularities

centred at resonance fields such as

2013.

=

0. j (H - AH)

_represents

_a local line

shape

with _a finite line width

including

various kinds of

_broadening

irrelevant to the transition.

The static line

_shape

can be modified if the local extrinsic

_probe

behaves as a

_{phase pinning}

defect. Then some value of the

_phase

is

_locally

favoured,

and one should obtain a

_spurious

sin-gularity

at the

_{corresponding}

values of AH. Such a

singularity

does not mirror the modulation of the bulk.

Moreover a

_{simple theory}

of

_structurally

incommensurate

_phase

predicts

that the modulation wave can move

_{freely [7, 9]. Generally,}

discrete lattice effects or intrinsic defects lock the modula-tion to the

_underlying

lattice

_[7].

Just below

_T,,

the thermal energy should be able to overcome the

_pinning

_energy and restore a

_floating

incommensurate

_{phase [7],}

_inducing

_spatial

and

temporal

fluctuations of the

_{phase 0.}

In other words since the

_phason

mode has in

_principle

no _gap, one may

expect

substantial thermal fluctuations of the local

_{phase 0 :}

This line shift

_permits

us to define a characteristic

frequency :

and we _may consider two

_limiting

cases :

- The

phase

fluctuations are slow with _respect

_{to Ve}

and the E.P.R. measurement

_gives

an instantaneous response. The random

phase

fluctuations do not

_modify

the E.P.R. _spectrum and one observes two

_{edge singularities separated by}

- The

phase

fluctuations are fast and motional

_narrowing

occurs. A

simple

calculation,

gives :

We have assumed

_homogeneous

Gaussian fluctuations with mean

_square

0’

_).

It turns

out that the

_phase

fluctuations of mean _square

_amplitude

of

order -r

induce an

_{important spectral}

narrowing

of the

_{edge singularities}

and can lead to an

unresolved,

«

normal-like »,

single

line.

This effect is

_quite

similar to the enhancement of the

_Debye-Waller

factor of satellites reflections

by phase

fluctuations,

in

_{scattering experiments [10].}

2. E.P.R. line

_shape

below

_T,.

In

_{ThBr4 :}

Gd3+,

one observes several

_paramagnetic

centres :

Gd3+

substituted for

Th4+

at

D2d

sites, Gd3 +

associated to nearest

_neighbour

_(n.n)

_{Br- vacancy,}

Gd3 +

associated to next

nearest

_neighbour

_(n.n.n)

Br- _{vacancy. This last}

_point

defect is the most

_easily

tractable to

_study

the

_phase

transition

_[5]

_by

E.P.R. We shall consider the associated

_high

field E.P.R. line

corres-ponding

to

_Mg

= 5 H ~ , for

H in the

_{(001) plane}

at about 25° _away from

_{[100], [5].}

The concentration

of Gd3 +

is about 100 _{p.p.m., and the}

_doping

induces the same concentration of extrinsic Br- vacancies. It is worth

_noting

that the mean distance between

_paramagnetic

probes

is of the order of ~ 20 in lattice

_parameter

units.

Typical

line

_shapes

are

represented

in

figures

1,

2 and 3. Between

_{TI -}

2.8 K and

_{TI -}

15

K,

one observes two

_{edge singularities (Fig. 1),}

i.e. a

_typical

incommensurate line

_shape.

This

_type

of _spectra have been

_{previously analysed [5].}

The

_splitting

between the

_{singularities mainly}

arises from the rotational

_component

of the

_displacement

field and exhibits a critical behaviour

AH oc A oc

_{(TI -}

T)~

_{with #}

= 0.34 ± 0.02. At lower

_temperature,

three and four

singularities

become _apparent. Their

_origin

is doubtful since

_{phase pinning by}

the

_probe

_may alter the line

shape.

To

_clarify

this

_point,

we have used the theoretical model of section 1 to reconstruct the

experimental

lines. The

_computation

results indicate that the modification of the

_experimental

lines

_shape

are

_{satisfactorily}

_{explained by}

an

_increasing

contribution of second order terms

associated with an increase of the

_amplitude

of the modulation when the

_temperature

decreases. An excellent _agreement, between the theoretical model and the

_experimental

results is obtained in the

_temperature

_range

_{TI, TI -}

15

K,

with A oc

_{(T -}

T, )0.31

_[3]

and all

_h/s

constant

_(Fig.

_1).

Below,

a

_good

_agreement is obtained

(Fig. 2)

with the same values of the

~/s.

Nevertheless,

inho-mogeneous saturation

effects,

consistent with relaxation mechanisms induced

by

the

phason

L-696 JOURNAL DE _{PHYSIQUE -} LETTRES

Fig.

1. -

Experimental

spectrum at _{TI -} 7 K _{(dashed line)}

_compared

to the theoretical _{prediction (full}

line).

Fig.

2. -

Experimental

spectrum at _{T, -} 15 K (dashed

line),

and theoretical _spectrum

_{(full line).}

One notices the third

_singularity

attributed to a

cos~ ~

terms which become

_important

at lower temperatures.

effects

_{becoming important.}

It is worth

_noting

that the best fit of the theoretical line

_shape

to

the

_experimental

one is obtained

by considering

Gaussian local line

shape f (H - AH).

We conclude that the line

_shape

mirrors the

_plane

wave modulation

_regime,

the two _component

displacement

field and their

_phase

difference

of 7r .

Moreover the

_{probe, which}

is a

_point

defect,

with

_charge

misfit and

_{compensation by}

a near

vacancy,

does not exhibit _any

_tendency

towards

phase pinning.

For

_comparison,

the

U4 +

_impurity,

which fits the host

_[6],

would seem to induce some

_degree

of

_{phase pinning.}

We

_suggest

similar

_{investigations}

on various

_point

defects to

clear the mechanisms of

_phase

_pinning

which is

_generally

invoked and still rather obscure.

3. Evidence for

_{large phase}

fluctuations near

_TI.

Fig.

3. -

a)

Experimental

spectrum at

_{TI -}

35 K. _b) The theoretical one _{(not fitted),} which

_permits

us to

establish the

_validity

of the model. The

_singularity

2 arises from twist terms.

Fig. 4. -

Experimental

spectra near _T,. One notices at

_{TI -}

1.8 K a coexistence of « incommensurate »

singularities with a « normal like » line.

in the bulk which would smooth the

_{edge-singularities}

of the « incommensurate » line. This

L-698 JOURNAL DE _{PHYSIQUE -} LETTRES

We have studied the « normal » line above

_T,

_and

the « normal like » _{line below}

_{T I,}

_{by assuming}

a line

_{shape given by}

.

where N is a normalization constant and

_Ho

is the central resonance field.

This model has been set _up to include static contributions

_through

the Gaussian _parameter

h(J

and

_dynamical

contributions

through

the Lorentzian

_{parameter hL}

to the line width and the line

_shape.

At 110 K the _parameters are

_h~,

= 3.64 G and

hi.

= 5.11

G,

and _can be attributed

to

_{inhomogeneous}

_broadening

_{by superhyperfine}

interactions and to

_{homogeneous broadening}

by

non critical

fluctuations,

respectively.

The _parameters have been fitted to the

_experimental

line

_shapes.

The results are

_represented

in

_figure

5 where we have

_plotted

the Gaussian

_~h~

and the Lorentzian

_~hL

critical

broadening

of the E.P.R. line. One can

identify T) by

a

change

of the Gaussian

broadening

~h~

(Fig.

5).

This determination of

_Tt

_agrees with the

_temperature

determined from the critical behaviour of the

_{edge-singularity}

_spectrum

(Fig. 6).

Above

_T,,

the line

_shape

and the line width have a normal critical

_behaviour,

_frequently

observed

near structural transitions.

_Increasing

fluctuations and critical

slowing

down,

associated with

a soft mode

plus

a central

peak

account for it. At

_{TI -}

7 K the local line

_shapes

involved in the normal « incommensurate » _spectrum

_(Fig.

₁₎

are Gaussian and narrow. This indicates a

weak influence of fluctuations. The « normal like » line below

_T,

and the normal line above

_TB,

exhibit a continuous behaviour of their Lorentzian

_{broadening, figure}

6. This is an indication

that the « normal like » line _may arise from motional

_narrowing

of an « incommensurate »

Fig.

5. - Critical Gaussian

Fig. 6. -

Graphical determination of _{TI by}

_{extrapolation}

of the

_edge

_singularity spectrum.

spectrum

by large

fluctuations as indicated

by

the model

developed

in section 1. Indeed this model

_applies

to the _{temperature range}

_[T,,

_{TI - 2.7]}

in which the

_preponderant

term of the line shift is A cos

_4J.

Nevertheless one cannot

_explain

the coexistence of

_{edge singularities}

and of the central line

_{by homogeneous phase}

fluctuations. One has to assume that the

_phase

is static in some

regions

of the

crystal

4Jõ > ~

0 and exhibits

large

fluctuations in other ones

~/( ~ ~ ~ 2013.

N.M.R.

_{spectra of 87Rb}

have

_given

evidence for the same

_type

of

phenomenon [7]

in

_Rb2ZnBr4.

4.

_{Standing-wave}

model.

According

to the model

_{developed by}

Blinc

_{[7], pinning}

defects are involved in the

_{interpretation}

of the observed _spectra. Between these

_defects,

distant

_by

_I,

the

_phason

mode would lead to

standing

waves,

creating regions

of small and

large

fluctuations at the nodes and antinodes.

For

_large

I,

low

_frequencies

and therefore

_large

thermal fluctuations are allowed. In a crude

C

,{

description

one _may write : 1

~ :2013 = -r

for the lowest allowed

_frequency.

For a certain cut

2 v

off

_{length Ie,}

the

_amplitude

of fluctuation is too small for substantial motional

narrowing.

One may state the condition I =

Cp

_>

l

_{or v}

Cp

with

_Cp

the

_{phason velocity.}

y ₂

v ~

_{2 Ic}

p p y

On the other hand the

_frequency

has to be

_{large enough}

with _respect to the characteristic

E.P.R.

_{frequency Ve :}

At

_{TI -}

2.7

_K,

the « normal like » line

_disappears.

This means that the distance between

pinning

defects is too small to fulfil the motional

narrowing

conditions. With

_{Cp =}

2 x

1012

(in

units of lattice

_parameter

_{s-1 )}

_[3],

one finds I

_{= Ie}

= 8 x

103

_(in

units of lattice

_parameter).

This distance more

likely corresponds

to dislocations than to

_point

defects. The known

_point

defects induced

_by

the

Gd3 +

_doping

are much less

distant,

which

_implies

that the

_paramagnetic

L-700 JOURNAL DE _{PHYSIQUE -} LETTRES

Conclusion.

We have shown that the E.P.R.

_probe

used to

_investigate

the incommensurate

_phase,

does not

pin

the

_phase

of modulation and reflects

correctly

the

_plane

wave

_displacement

field. The

_probe

is not

_responsible

for the coexistence of « incommensurate » and « normal like » lines. This

interesting

effect can be

_analysed

in terms of

_{inhomogeneous}

_phase

fluctuations due to

_phase

pinning

defects. A _very

_simple

_application

of the

_standing

wave

_model,

_{previously developed by}

Blinc,

indicates an _average distance between

_pinning

defects

_compatible

with their

_being

dislo-cations.

Nevertheless a definite

interpretation

of this

effect,

also observed in

Rb2ZnBr4,

needs more

experimental

and theoretical

investigations.

References

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