Table Of Content^COJSMJCJM
mmmm$&
In ftS and Qua^ tab
W
Pierre Vladimir
Toledano Dmitnev
Wxt\<S;$&iwm<i
RECONSTRUCTIVE
PHASE TRANSITIONS
In Crystals and Quasicrystals
This page is intentionally left blank
RECONSTRUCTIVE
PHASE TRANSITIONS
In Crystals and Quasicrystals
Pierre Toledano
University of Amiens, France
Vladimir Dmitriev
University of Rostov on the Don, Russia
World Scientific
Singapore • New Jersey • London • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
P O Box 128, Farrer Road, Singapore 912805
USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Calaloging-in-Publkation Data
Tol&Jano, Pierre.
Reconstructive phase transitions : in crystals and quasicrystals /
Pierre Tol&lano, Vladimir Dmitriev.
p. cm.
Includes bibliographical references and index.
ISBN 9810223641. -- ISBN 9810228406 (pbk.)
1. Quasicrystals. 2. Crystals. 3. Phase transformations
(Statistical physics) I. Dmitriev, Vladimir. II. Title.
QC173.4.Q36T65 1996
530.4'14-dc20 96-9517
CIP
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 1996 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
photocopy is not required from the publisher.
This book is printed on acid-free paper
Printed in Singapore by Uto-Print
V
To Zoya To Olga
This page is intentionally left blank
vii
FOREWORD
Reconstructive phase transitions constitute the most fundamental and widespread
type of structural transitions existing in Nature, forming a large majority of the transitions
found in elemental crystals, alloys, minerals and complex fluids. Despite this fact they have
been persistently considered by theorists as the "uninteresting case" (P W Anderson,
Pierre Curie Colloquium, 1980). Such a conclusive statement is mainly based on the
presupposed opinion that the absence of group-subgroup relationship between the
symmetries of the phases surrounding a reconstructive transition forbids the definition of
an order-parameter, which is the primary step for a theoretical approach to a phase
transition
In some respect this pessimistic view results from the popularity of Landau's theory
of second-order transitions (the "interesting case") in which the author examines the
situation of continuous or almost continuous transitions between group-subgroup related
phases It also stems from the fact that models of phase transitions in statistical physics
such as the Ising model, lead to a lowering of symmetry between the high and low
temperature phases The Landau theory and the complementary theory of Critical
Phenomena offer the possibility of working out the universal properties of systems
undergoing phase transitions on the sole basis of symmetry considerations Along this line
some classes of structural transitions have been thoroughly investigated during the past
decades, such as ferroelectric and ferroelastic transitions, or transitions in thermotropic
liquid crystals
Reconstructive transitions have been left aside from the preceding theoretical
framework The aim of our book is to demonstrate that these transitions can indeed be the
subject of a full theoretical treatment providing a strong revision and a better
understanding of the Landau concepts, and most importantly on the condition that one
leaves the realm of pure symmetry for the world of concrete structures and effective
transition mechanisms
For a long time reconstructive transitions have been a center of interest for
metallurgists and crystallographers. They have investigated comprehensively the phase
diagrams involving such "transformations" and the corresponding displacive and ordering
processes They also established a nomenclature and proposed some classifications.
However, while the specific features of the various subclasses of reconstructive transitions
viii
have been clearly disclosed experimentally, a unifying theoretical picture could not emerge.
This is mainly because the nature and symmetry of the transition order-parameters have
not been well understood and related organically to the critical variables and to the
thermodynamic functions.
It is the purpose of the theory presented in this book to show the intimate
connections between the set of critical degrees of freedom (atomic displacements or
probability of occupancy of selected positions by the atoms), the order-parameters, the
phase diagrams, and the static and dynamical properties of the two main types - displacive
and ordering - of reconstructive transitions. On that goal the structures are expressed
under the form of lattice functions corresponding to a sum of stationary density waves
The increment 8\\i of the lattice function at the transition provides a natural definition and
the explicit form of the order-parameter as a function of the critical degrees of freedom
Thus, the order-parameter corresponds to the phase or to the amplitude of 5v|/ for,
respectively, displacive and ordering mechanisms
Striking differences appear in our theoretical description of displacive
reconstructive transitions with respect to the standard Landau approach to displacive
transitions Thus, the order-parameters are non-linear periodic functions of the critical
displacements instead of linear functions Therefore the thermodynamic potentials have the
meaning of truncated Fourier series, instead of Taylor expansions Their minimization,
which is performed with respect to the critical displacements, yield two sorts of stable
states: the Landau states which are group-subgroup related to the initial phase, and
additional states for which the group-subgroup relationship is lost
For ordering reconstructive transitions there exists, as in the current description of
continuous order-disorder transitions, a linear relationship between the order-parameters
and the critical variables. We show that such transitions generally occur between
differently ordered close packed structures which are derived from a disordered parent
structure for definite occupancies of the atomic sites and specific crystallogeometrical
requirements The fulfillment of these conditions creates new symmetry operations in the
ordered phases and prevent the possibility of a group-subgroup relationship between them
Accordingly, for both displacive and ordering reconstructive transitions the absence
of a group-subgroup relationship between the symmetries of the phases is the result of
specific values taken by the critical variables, i.e the phases correspond to limit states
This essential property is shown to have significant consequences on the static and
dynamical features of reconstructive transitions, as well as on the critical behavior of the
relevant physical quantities
This book is a monograph Its subject is however related to the wider field of
discontinuous phase transitions. We have therefore included a long introductory chapter
devoted to a unified presentation of the phenomenological theory of first-order phase
transitions In this chapter the results concerning the solutions of the radial problem in the
Landau theory, which have been essentially obtained by Gufan and his coworkers, are
detailed They provide a useful basis for working out the phase diagrams of reconstructive
phase transitions Along the same line, a presentation of the concepts of the theory of
Singularities is given. These concepts allow to place on a rigorous basis some empirical
ix
rules used for the construction of thermodynamic potentials and of the corresponding
phase diagrams.
In Chapter II the density wave theory of reconstructive phase transitions is
formulated and illustrated by a number of working examples The applicability of the
theory to real crystalline systems is shown in Chapters III and IV In these Chapters we
have preferentially, but not exclusively, considered reconstructive transitions in crystals of
the elements since the simplicity of their structures allows to show more clearly the
connection between the transition order-parameters and the critical displacements and
ordering of the atoms Only few examples of minerals have been discussed because their
transitions generally possess an implicit morphotropic character The description of
morphotropic transitions, also predominantly found in complex fluid systems, requires
further theoretical considerations which are beyond the subject of this book
The predictive character of the density wave approach to phase transitions is well
demonstrated in the examples of crystal-quasicrystal transitions examined in Chapter V.
The approach permits a determination of the atomic positions in the quasicrystal structure
providing their positions in the crystal structure of reference are known. The loss of the
crystal's translational symmetry and the onset of a non-crystallographic point-group
symmetry, which characterize a crystal-quasicrystal transition, are described in terms of
combined displacements and disordering of the atoms
This book materializes many years of a theoretical investigation of concrete
examples of reconstructive transitions in crystals and quasicrystals We feel indebted to the
colleagues and friends who collaborated with us during this period We are especially
grateful to Yuri Gufan and Serguei Rochal who took part to the most important steps in
our effort to insert the interpretation of reconstructive transitions into a general theoretical
framework We have also benefited from the collaboration of Vladimir Lorman and Bruno
Mettout We thank Victor Torgashev for many helpful discussions and Evgeni Kut'in who
allowed us to quote some of the unpublished results of his thesis In many occasions Jean-
Claude Toledano helped us to clarify the formulation of our theoretical approach At last,
we want to thank Vladimir Dvorak who offered us the possibility to write the last pages of
our book in the beautiful city of Prague.
P. T and V. D., March 1996
