On considering the elastic constant table as the matrix of an operator ; consequences in ferroelasticity

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On considering the elastic constant table as the matrix of an operator ; consequences in ferroelasticity

A. Bulou

To cite this version:

A. Bulou. On considering the elastic constant table as the matrix of an operator ; consequences in ferroelasticity. Journal de Physique I, EDP Sciences, 1992, 2 (7), pp.1445-1460. �10.1051/jp1:1992220�.

�jpa-00246632�

Classification

Physics

Abstracts

64.70K 62.200 62.65

On considering the elastic _constant table

_as

the matrix of

_an

operator _; consequences in ferroelasticity

A. Bulou

Laboratoire de

Physique

de l'Etat Condens£

(*),

Facult£ des Sciences, Universit£ du Maine, 72017 Le Mans Cedex, France

(Received 24 December 1991,

accepted

in

final form

17 March

1992)

R4sum4.-Les

consdquences

de la convention associde h la contraction des indices des composantes ^du tenseur des constantes

61astiques

_C,~~i sont discutdes. On _{montre que, pour} Etre

manipul£

_comme la matrice d'un

op£rateur,

le tableau des _constantes

dlastiques

doit Eke dcrit

avec de nouvelles conventions ddfinies par :

C$~

₌ C,~~t ^si a, fl

= 1, 2, 3,

C$~

=

/C~~~t

_si

(a

= 1, 2, 3 _et fl = 4, 5, 6), ou (a ₌ 4, 5, 6 et fl

= 1, 2, 3) et

C$~

= 2C,~~t ^si a, fl ₌ 4, 5, 6.

Los

C$

~ sont les composantes d'un tableau 6 _x 6

[C"].

En

adoptant

_pour les d£formations e,~ la convention u~ = e,~ si I

= j et u~ ₌

/

e;~ si I *

j

_on conserve pour

I'£nergie

la for~ne

habituelle E

=

C$~

_{u~ u~.} On _montre que les valeurs

©~

per~nettant ^d'£ctire

I'£nergie £lastique

~ sous une for~ne

diagonale

E

=

£ ©~

[E ]~ (oh les E~ sont des ddfor~nations

sym£trisdes)

sort 2

~ _,

"

les valeurs propres du tableau [C"]

qui

peut ^{ainsi Eke}

qualifi£

de matrice des constantes

61astiques

les _« valeurs propres » du tableau des constantes

61astiques

d6fini _avec les conventions usuelles _ne permettent pas d'£crire la forrne

diagonale

correcte dans le _cas des groupes

quadratiques

4,

4,

_{4/m et dans} le _cas des groupes

rhombo£driques.

Ce rdsultat _a des

consdquences

dans l'dtude des transitions

ferrodlastiques

_propres ou

pseudo-propres,

ces transitions dtant associ£es _au ramollissement de l'un des

C~. Quelques

_arguments montrant l'int£rEt de l'utilisation, _pour d'autres types ^de tenseurs, de la convention de contraction

proposde

sont

pr£sent£s.

Abstract The consequences of the convention associated with the contraction of the indices of the elastic constant tensor components C,~~i are discussed. It is shown that, in order to be handled like the matrix of an operator, ^the elastic constant table must be written with _new conventions _:

C$~

_{= C,~~t} if _a, fl ₌ 1, 2, 3,

C$~

₌

/

C,~~i ^if (a

= 1, 2, 3 and fl ₌ 4, 5, 6) _or (a ₌ 4, 5, 6 and fl =1, 2, 3),

C$~

=

2C,~~t

^if a, fl ₌ 4, 5, 6. The

C$~

are the components ^of a 6 _x 6 table

[C "].

Using

for the strains

e,~ the convention u~ = e,~ if I

= j ^and u~ =

/

e,~ if I * j, the usual

for~n

E=jC$~u~u~

for the elastic _energy is

preserved.

It is shown that the ter~ns

C~

of the elastic energy written in _a

diagonal

for~n E ~

=

£ C~[E~]~

(where the E~ are the 2

« =1

(*) CNRS U-R-A- _n 807.

JOURNAL DE PHYSIQUE I T 2, N'7, JULY 1992 ⁵²

symmetry-adapted

strains) are the

eigenvalues

of the table [C "] that _can be called matrix of the elastic constants; the

«eigenvalues»

of the elastic constant table defined with the usual conventions do _not lead _to the

right diagonal

for~n in the _cases of the

tetragonal ~,

m

4,

4 groups and in the _cases of the

trigonal

_groups. This result has consequences in the studies of the proper and

pseudo-proper

ferroelastic

phase

transitions since

they

are associated with the

softening

of _one of the

C~.

Some arguments

showing

the

advantage

of _an extension of the conventions

proposed

in this paper to other kinds of tensors are

given.

I. Introduction.

The conditions for

stability

of _a

crystal

with

respect

to elastic deformations have been

described

by

Born and

Huang [I]. They

deduced

stability

criteria in terms of

specific

conditions between the elastic constants. These conditions became of

special

interest in the field of structural

phase

transitions characterized

by

_a deformation of the unit

cell,

the _so- called proper

(and pseudo~proper)

ferroelastic

phase

transitions

[2]

sometimes called elastic

phase

transitions. Boccara

[3]

showed that such transitions result from the violation of the

stability

criteria

given by

Bom

Huang

and deduced that

they

_are associated with the

vanishing

of _{some «}

eigenvalues

of the elastic constants matrix _».

Aubry

and Pick

[4]

further deduced that such transitions must be associated with the

softening

of sound

velocity propagating

in _a

particular

direction with _a

peculiar polarization. Hence,

these transitions _are of

special

interest since the critical behavior is

strongly

related to the

dimensionality

of the soft acoustic

subspace [5-9]. Fairly

few materials exhibit proper or

pseudo-proper

ferroelastic

phase

transitions

[10].

One of the best known

examples

is encountered in

BiV04

that has been studied in

great

detail

by

David

[11],

from both the

experimental

and theoretical

points

of views.

Although

the nice

approach given by

Boccara to determine the elastic constant combination that softens _{at a} ferroelastic

phase

transition works in _a number of _cases, it has

already

been

noted that in _some circumstances the result differs from the _exact combination deduced from the calculation of the extrema of the sound

velocity [12, 13].

In this article it is shown that in

order _to be handled like _a matrix

(and

to lead _to

meaningful eigenvalues),

the elastic constant table

requires

_new conventions for the definition of its components. ^These conventions have been

previously

introduced in reference

[10].

2. Strain energy

density.

A

homogeneous

deformation of _a

crystal

is described

by

the

components

of the strain tensor e,~. ^{The strain energy}

density

of the

crystal

_can then be

expressed

ⁱⁿ terms of these strain

components ^{and the isothermal elastic} constants

C,~~i

as

~

~

i I I I ~ljkf

E>j ~kf

(~)

~lj~li~lt~l

Since the strain tensor is

symmetrical

_e,~

= e~~

)

and since the elastic _{constants are}

symmetric

in the

ij

and

kf indices,

it is

possible

to _use the

Voigt

notation where _two indices

(ij)

are

replaced by

_one

(a

with the

following

convention _:

=

(i

₊

j)

~ *j.

_~~~

Using

this

notation,

the strain elastic energy

density

_can be written with _two summation indices

(instead

of

four) running

from _one to six.

However,

to do

this,

additional conventions must be used _to take into account the fact that, in summation

(I),

_some terms appear twice or

four times.

Actually

two conventions _are encountered _:

in the _{more common one}

[14],

the strain components

e~~ ^are

replaced by

_e~ such that

~"

~ij

^~~ ~j

~~~

e~ = 2

e~~ if *

j

while the elastic constants _are

simply given by C]p

=

C~~;i

^(~) ; then

(I)

reads _:

~

~

j _{I I ~~P}

~a ~P

(~)

~lfl~l

The

C(~

_are the elements of _a 6 _x 6 table

[C

~].

some authors

[15, 16]

define the elastic

compliance (or

elastic modulil

by s[

p ⁼ si~~i ^and

so the elastic constants _are

given

_as follows _:

C[~

= C~~~i ^{if a,}

p

=

1, 2,

3

if

(a

=

1, 2,

3 and

p

=

4, 5, 6)

C[~

= 2 C,~~y ^or ~~~

if

(a

=

4, 5,

6 and

fl

= 1, 2,

3)

C[~

=

4 C~~~i ^{if a,}

fl

=

4, 5,

6.

Equation (I)

then reads _:

E

"

l I ~SP

_{e« eP}

16)

«=,p=~

provided

^{the strain}

components

are such that _:

e~ = e~~ for I

=

j

and for I #

j. (7)

The

C[p

_are the elements of _a 6 _x 6 table

[C~].

As shown in section

3,

in the

general

_cases, neither the

[C

_~] table _nor the

[C

_~] table _can be

manipulated

like _an

ordinary

matrix.

However,

if the

following

conventions _are used _:

u~=e~~

^for

I=j

~ ⁽⁸⁾

u~ =

2 ej~ for I #

j

(I)

The elastic compliances are then defined

by

_:

s[~

_{= S,pit} if _a, fl ₌ 1, 2, 3 if (a

= 1, 2, 3 and fl ₌ 4, 5,

6)

S[~

₌ ²_{S,~it or}

if (a

= 4, 5, 6 and fl ₌ 1, 2, 3)

S[~

= 4S,~~i ^{if a,} fl

= 4, 5, 6.

C$~

=

C,j~i

^if _a,

p =1, 2,

3 if

(a

₌

1, 2,

3 and

p

=

4, 5, 6)

C$p

=

/ C;y~i

or

(9)

if

(a

=

4, 5,

6 and

p

=

1, 2, 3)

C$~

₌ 2

C;~ki

^if a,

p

=

4, 5,

6

equation (I)

reads _:

6 6

E

=

j £ £ C$~

_{u~ u~}

(10)

«=ip=i

and

then,

the elastic constant table

[C "]

exhibits _new

properties

and it _can be

manipulated

as an

ordinary

matrix. For _reasons that will appear

below,

this convention _can be called the

«

unitary

_» convention _or the _«

symmetrical

_» convention.

In the three above

conventions,

_{symmetry can} further be used to reduce the number of

independent

elastic constants as described

by Nye [14].

VECTORIAL _NOTATION. As

pointed

out

by

Boccara

[3],

formulations

(4) (as

well _as

(6)

and

(10))

can be written in _a shorter way if _one considers the _e~

(or

_{e~ or}

u~)

_as the

components

^of

a six-dimension vector, say

e) (or [e)

_or

[u)) using

the Dirac

notation,

and the elements

C(p (or C$p

or

C$p)

of the table _as the

components

^{of the matrix of} an

operator

C~

(or

C~ _or

C~)

in _a six dimension space. The strain energy

density

then reads :

E

=

(e[C~[ e) (II)

E

=

(e[C~[ e) (12)

E

=

(u[C~[ u) (13)

Although

this formal notation is

allowed,

the tables

[C

_~] and

[C~]

cannot be handled like

ordinary

matrices since the _« basis _» where

they

_are written is not normalized.

3. Action of the

symmetry operators

of the

point

_group.

Let _{us now} consider that the symmetry ^{of the}

crystal

is described

by

the

point

_group

G

=

(R)

with elements R.

3,I MATRiCES _OF TRANSFORMATiON OF THE COMPONENTS OF A vEcToR. To each

element R of G

corresponds

in the three dimensional vectorial space an

operator

^{R that} acts on the _vectors

(active sense).

This

operator

^is

fully

defined

by

its action _on the vectors of _an

orthonormalized basis vi

)

,

u~)

,

u~)

;

R[u,)

3

= =

£ V(R)j; _[u~) (14)

j=1

Then,

under _a

symmetry operator R,

_a vector

Iv)

such that _:

3

[V)

₌

£

_U,

[Vi) (15)

<=1

is transforJned into _{a vector}

iv')

:

Iv')

=

Rlv) (16)

with

components

v,'

given by

_:

3

vi

=

£

v

(R

_);~

vj ⁽¹⁷⁾

j

The V

(R

_);~ are the

components

^of a 3 _x 3 matrix

[V (R)]

and the _set of matrices for the different elements of G forms _a

representafiion

^r~ of the group. This

representation

is

unitary (V (R

_),~

= V

(R~ ~)j;)

^{since the matrices} are

representations

of

(symmetry) _operators

written

on an orthonormalized basis.

3.2 MATRICES _OF TRANSFORMATION OF THE COMPONENTS OF THE

(e)

_«VECTORS».

Under _a

symmetry

operator

R,

the

component

_e;~ ^{of the strain} tensor becomes

e(

^{such that} :

3 3

e/j

=

£ £

V

(R

_{);~ e~i V}

(R

~)i~

(18)

f _{i k i}

I-e- the second rank

symmetric

tensor transforms like _a matrix. Since the

[V (R )]

matrices _are

unitary, (18)

can be written _:

3 3

e[

₌

£ £

V

(R

_)~~ V

(R

_{)~i e~i}

(19)

f I k i

which is the form the _most

commonly

used.

A

symmetrical

tensor

(e~i

₌

ei~)

_can be characterized

by only

six components. ^{In this} case

equation (19)

also reads _:

~ ~ ~

ei

=

z

v

(R

_)~~ v

(R )j,

_e~~ +

z z _{(v (R ),}

~ v

(R )ji

+ v

(R ),i

^v

(R

_)j~

)

_e~i

(20)

This relation _can

altematively

be written :

3 3

e/j

=

£ £ T(R),~

_{~i e,I}

(21)

k= i f»k

where the terms

T(R)~j

_~i are

given by

:

~ ))' _~

=

~ _)~ ~ _)~

+ V

(R );I

V

(R

_)j~

~

#

~~~~

The

T(R);~,~i

_are elements of 6 _x 6 matrices

[T(R)].

It _can be shown that these matrices form _a

representation ^r~

of the group G I-e- for any elements

R~

^and

R~

of G _:

~T(Ra)iiT(Rb)1~ ~T(RaRb)1. (23)

The characters

X(R)~

_{of this}

representation

are those of the

symmetrized

_square of the

r~

representation (characters

_X

(R ))

:

~

(x (R )2

+ x

(R

2) x

(R ) (24)

= ~

The

[T(R)]

_are the fix 6 matrices

required

to calculate the transformation of the six

components

^of a

symmetrical

second rank tensor like the strain _tensor under the action of _an

operator

^R.

Using

the contracted notation defined

by (7), equation (21)

reads _:

6

ej

=

z

T

(R )j

p ep

(25)

«

where the

components T(R )[p

of the _two index matrices _are

easily

deduced from the four

component

ones

T(R)~j

~i

according

to

(22).

Vectorial notation. As mentioned in section

2,

the e~ can be considered _as the

components ^of a vector

[e)

of _a six-dimension space

spanned by

a basis

([ei),

[e~),

..,

[e~)).

Within this

framework, comparing (25)

with

(17),

it appears that the

[T (R )~]

matrices _can be identified with the matrices of _an

operator T(R

_{)~ and}

by analogy

with

(16)

_:

[e')

=

T(R)~ [e) (26)

3.3 MATRiCES _OF TRANSFORMATiON OF THE

e)

_« vEcToRs ». -The elastic strain energy

density

_can be also written in terms of the strains e~ defined

by (3).

The e~ can be connected with the e~

using

a 6 _x 6

diagonal

matrix

[A

with components _A~~ ^defined

by

:

A,~

= &~~ if I,

j

₌

1,

2 _or 3

A,~

=

/

&,~ if I,

j

=

4,

5 _or 6

~~~~

(where

&~~ ⁼

l if I

=

j

^and

&~~ ⁼

0 if I #

j). Using

the vectorial notation _as in

2,

the

components

^of

e)

_are related with the

components

of

[e) by

_:

18> =

A~ ie>

It _can be deduced that under _a symmetry operator the e~ are transformed into

e[ according

to _:

6

e[

₌

£ T(R)[p

_BP

(28)

a

or in short

e')

=

T

(R

)~ e

) ₍₂₉₎

where the

T(R )[p

_are the

components

^{of the matrices}

[T(R )~] given by

:

iT (R

_)~

i

= iA

21iT (R )ei iA

₂₁

(30)

The

[T(R)~]

_are 6 _x 6 matrices that forJn _a

representation ^r~~

of the group G,

equivalent

to

r~ _(or r~~, _they

have the _same characters.

3.4 UNITARY REPRESENTATION. In the

general

_case, neither l~~~ _nor l~~~ _are

unitary since,

as seen in table

I,

the matrices of transformation of

e)

and

[e)

associated with element R~ _are not

equal

to the

transposed

matrices of element R. On the other hand the matrices

corresponding

to the transformation of the components ^{of the} strain _« vector »

[u)

form _a

unitary representation ^r~~ They

_are related _to those relative to

is )

and

[e) by

:

iT(R)~i

=

iA-hi(R)~iiAi

=

iAiiT(R)eiiA-11. ₍₃₁₎

Table I. Form

of

the matrices

of transformation of

the components

of

the _« vectors _»

[e), e)

and

[u)

under the action

of

a

point

symetry operator ^R.

~~

^{2 V}

^(R

^{)ik V}

^(R

^)~(

jj~

j~~ej

~

~~

_~~~ ~ _~~~

(f

_{# k}

₎

V

(R )ik

V

(R

_)~k V

(R

_)~k ^V

(R

_{)~( +} V

(R

_{)~( V}

(R

_)~k

g

*

g

# I and

(I

_# k

~ ~~~ ~ ~

~~~

V

(R

_)~

k V

(R

_)~(

~~ ~~

_~~

2 V

(R ),

_k ^V

(R

_)~k V

(R ),k

^V

(R )( ^fi~R _);(

_V

_(R

)~ k

g

_#

I) g

#

I)

and

(I

_{# k}

(R

_)~~ v

(R

~~~

/

_V

_(R

);~ ^V

(R

_)~i

~~ ~~

_~~

/

_V

_(R

),~ ^V

(R

)~~ V

(R

_),~ V

(R

~ _~(R

),

_I V

(R

)~~

Q

# I

) Q

# I and

(f

_# k

)

From

(31)

and

(22),

it is

easily

deduced that _: if _a, p =1,

2,

3

T

(R )j

p ⁼ T

(R )j,,

_{,~ =} ^v

(R ),

_~ ^v

(R ),,

if _a

= 1,

2,

3 and p

= 4,

5,

6

T(R )1~

=

T

(R )i.

~i =

/

_v

_{(R ),~}

_v

_(R

)~i

(32)

if _a

=

4, 5,

6 and

p =1,

2, 3 T

(R )$

p ⁼ T

(R )(,

» =

/

_V

_(R

_{);~ V}

_(R

)y~

j +,

if a =

4, 5,

6 and

p =1, 2,

3

T

(R )$

p ⁼ T

(R )(,

_{w =} V

(R

_{);~ V}

(R )ji

+ V

(R );I

V

(R _)jk

j *,. t +i

The

unitary

of the

[T(R)~]

matrices follows from the

unitary

of the

[V (R)]

matrices.

3.5 INVARIANCE _{OF ENERGY UNDER THE ACTION OF THE SYMME~RY OPERATORS.} if _a

convenient _« basis _» is

chosen,

the elastic energy

density

_can be written in _a

diagonal

form _{as :}

6

E

=

£ ^C[E~]~ (33)

~

a =1

The

C _(having

the dimension of _an elastic

constant) only depend

_on the elastic constants

C;~~i

^and are thus invariant under any symmetry

operator

of the

point

_group

G;

the

E~ depend

_on the strains e~~. The elastic _energy

density being

invariant under all the

symmetry operators

^of

G,

the

E~

must transform

according

to its irreducible

representations, they

_are

symmetry-adapted

strains. In

particular,

if _some

E~ corresponds

to a 2 _or 3 order irreducible

representation,

the associated

C

value is

degenerate

2 _or 3 times

respectively.

Let _us consider the usual

expression

for the energy

given by (4)

based _on the

e)

strain.

[C

_~]

(like [C

_{~] and}

[C ~])

is invariant under any

symmetry

operator ^{of G while the}

e)

_are transformed into

e') according

to _:

je'j

=

T(R)~ jej

and

je'j

=

jej ni~ ₍₃₄₎

where stands for the

adjoint

_operator I-e- with the

transposed

matrix. The invariance of the energy then

requires

that _:

E

=

(e[

C~

[e)

=

(e[ T(R)~C~T(R)~ e) (35)

so that _:

C~

=

T(R

_{)~ C~}

T(R

)~

(36)

Since,

in the

general

_case,

T(R

)~ is not

equal

to

T(R

~)~, ^{the above relation does} not ensure

that the

«

operators

» C~ and

T(R

)~ commute I-e- it does _{not ensure} that

they

have _common

«eigenvectors

_{» :} the

«eigenvectors

_» of C~ _may not

belong

to the _«

symmetry adapted

strains _», invariant

subspaces

of

T(R

)~. ^{On the other hand}

according

to

(31)

it _can be deduced that _:

T

(R

)~ ₌ ^AT

(R

_{)~ A} and T

(R

)~

= A T

(R

)~ ^A

(37)

since

[A]

is

diagonal. Then,

since

[T(R )~]

_are

unitary, (36)

and

(37)

lead to :

AC ~ A

=

(T(R ~)~)

AC ~ A

(T(R

_)~

) (38)

The above relation _ensures that AC~A and

T(R)~

_commute, and _{so a common} basis of

eigenvectors

for these

«operators»

^{exists: the}

eigenvectors

of the «operator» C~

=

AC ^~ A transform like the symmetry

adapted

strains I-e- like the irreducible

representations

of G into which

r~~ (equivalent

to

r~~

_and

r~~

_reduces.

The

components

^{of the elastic} constant table

[C"]

_are deduced from the relation C~

= AC ^~ A which leads to the values

given by (9).

The

[C

~] table _can be

manipulated

like the matrix of _an operator ^that commutes with the symmetry operators ; in the

following,

it will be called _« the matrix of the elastic constants _».

It _must be noted that the conventions

(9)

are

required

to

easily

calculate the terms

C

of the

diagonal

form

(31)

_as the _«

eigenvalues

_» of a matrix. This convention is _not necessary to calculate the conditions for elastic

stability

_as described

by

Bom and

Huang [I]

and

given by Cowley [5].

3.6 EXAMPLE. The fact that the

«

eigenvectors

_» of the usual elastic constant table may ^not be invariant under the symmetry

operators

^{of the}

point

_group is

readily

illustrated in the _case of the

point

_{group C~~.} The

decomposition

of the

symmetrized

_square shows that the

symmetry-adapted

components ^{of the strain} tensor transform like _:

2Aje2E.

Then the _«

right

» elastic _constant matrix _must exhibit two

eigenvalues

with

degeneracy

two

associated with the _two E symmetry ^strain components. ^A

quick

look at the character tables

[17],

or calculation

using

a

projection operator (2)

shows that the E

symmetry

strains

correspond

to :

a

^b c

b _-a d

c d

~

I-e- to the _« vector _{» :}

18E) =a(181) 182)) +(184) +j185) +(186)

The

(conventional)

elastic _constant table

[C

_~] for this

trigonal

_symmetry reads

[18]

_:

Cii C12 C13 Cm

° °

C12 Cii C13 -C14

° °

~

ci~ ci~ c~~

0 0 0

~~

~

c14 c14

0 c<J4 ^{0 0}

0 0 0 0 c<J4

c56

0 0 0 0

c5~ c~~

with C

~~ ⁼

(C

ii

C12)/2

^and

C5~

=

C

j~. Its _«

eigenvalues

_» C ^~ _can be obtained

according

to :

C~18E)

=

al _EE)

and

they

_{appear as} the solutions of two second-order

equations

:

C^~~

(C

ii C

j2 +

C44)

C ^~ ₊

(C

ii

C12) C44 2(C14)~

= 0

@~- _(C66

+

C44)

C ^~ ₊

C~~ C44 (C5~)~

₌ 0.

These _two

equations

_are different and lead to four different _«

eigenvalues

_» while two, twice

degenerated,

_are

expected. If,

_on the other

hand,

_one considers the

[C~]

elastic constant

m_atrix,

one has _to

replace C44, Cm, Ci~

and

C5~ by C~j

=

2C44, C~j

=

2C~~, C(~

/ _C14

_{and C ]~} =

=

/

_{C (4}

" 2

C14 respectively.

In this _case the above two

equations

lead _{to a}

single

_{one :}

C

~~- _(C

ii C

i~ + 2

C44)

C ~ +

2(C

ii C

j~) C44 4(C14)~

= 0

so that the

expected degeneracy

conditions for the

eigenvalues

C~ _are satisfied.

4. Discussion.

4.I SOFT ELASTIC CONSTANTS ASSOCIATED _WITH FERROELASTIC PHASE TRANSITIONS.

Boccara

[3]

showed that those

phase

transitions such that the order

parameter

^is a

deformation of the unit cell

(the

so-called proper or

pseudo-proper

ferroelastic

phase

(2) The method is described in detail in [10] within the framework of the convention introduced in the present ^axticle.

transitions)

are associated with the

vanishing

of _some elastic constant or of _some combination of elastic _constants

C

associated with

strain(s) E~

that does

(do)

not

belong

_to the

totally symmetric

irreducible

representation

of the

point

_group,

However,

_as shown

above,

the

C

_are

eigenvalues

of the

[C

_~]

(given

in Tab.

II)

but not of the usual elastic constant table

[C~].

These

eigenvalues

of the

[C~]

_are shown in table III.

It appears that the _«

eigenvalues

» of the elastic constant table defined with the usual conventions _{are even}

qualitatively

different from the

C

in the

trigonal point

_groups and in the

tetragonal 4, 4,

4/m

point

_groups. In the latter _{case, a} ferroelastic

phase

transition _can be

second order and _so the

complete vanishing

of _a

C

is

expected;

such _a situation is

encountered in

BiV04,

_{a system}

widely

studied. The

present

paper shows

why discrepancies

were observed between the

expressions

of the soft elastic constant as deduced from the

Table II. Elastic _constant tables and elastic

compliance

tables _as

defined according

to relation

(9) for

the

dijfierent crystallographic point

_groups

(with

respect to the system

of

_axes

defined

in

[18]).

For the omitted

cubic,

orthorhombic and

tetragonal

4 _mm,

422, 12

_m, 4/mmm

classes,

these tables have the _same

form

_as with the usual convention.

Hexagonal

classes

cl cj~ cj~

0 0 0

cj~ cjj cj~

0 o 0

cj~ cj~ cj~

0 0 0

0 0 0

c~4

0 0

0 0 0 0

c~4

0

0 0 0 0 0

cl -cj~

Tetragonal

classes 4.

I, _4/m

c

j

_{c j~ c j~} 0 0 c j~

C12

C

li

C

13 ° ° C

16

C 13 C₁₃

C13

^{° ° °}

0 0 0

c~j

0 0

0 0 0 0 c_~4 0

C 16 C₁₆ ° ° ° C

t6

Monoclinic

point

_groups

C~l ~(2 ~~3

° °

~~6

~ (2 ~(2 ~ (3 ^{° °} ~ (6

~~3 ~(3 ~13

° °

~~6

0 0 0

C~4 C(5

0

0 0 0

C(5 C]5

0

C[6 C(6 C(6

0 0

Cj6

Table II

(continued).

Trigonal

classes

3, 3

~~l ~~2 ~(3 ~(4 C(5

°

~~2 ~~l ~(3 ~(4 C(5

°

C (3 ~ (3

~(3

^~ ~ ~

~(4 ~~4

°

~~

~

~~~~

cj~ -cj~

o ⁰

C~j /C(~

o o o

/ca /~a

_{~a ~a}

15 14 ll~ 12

Trigonal

classes 3 _m,

32, 3m

C~l ~(2 ~~3 _C~4

° ~

C~2 ~~l ~(3 ~C~4

° ~

~a₁₃ ~a _~a o o o

13 33

~j~ ~j~

~

C~j

0 0

o o o 0

C~j /C(~

o o o

o

/cj~ _{cj~ _cj~}

«

eigenvalues

_» of the elastic constant table

[I Ii

and the values deduced from the _extrema of the sound velocities

[12].

4.2 ELASTIC _{CONSTANT MATRIX AND ELASTIC COMPLIANCE MATRiX.} In addition _to the

fact that the conventions

presented

in

(8)

and

(9)

_{are necessary} to handle the elastic constant table like _an

ordinary matrix,

it has the

advantage

of

being

more «

symmetric

_». The relation

between the strains and the stresses :

3 3

«,~

=

£ £ C,j~i

e,I

(39)

t i f _i

reads with the usual conventions _:

6

"

a ^"

£

c

j

p ⁸p

(40)

fl I

where the BP ^are defined

by (3)

and where the «~ are defined

by

_a relation

equivalent

to

(7).

Using

the conventions defined

by (8), (40)

can be written with contracted indices _{as :}

6

r~ =

£ C$p

_Up

(41)

fl=i

where the conventions for the contraction of indices for the stress tensor

components

are

defined

exactly

like those for the strains I-e-

~~

"~J

^~°~ ~~

(42)

r~ =

/

«~~ for I #

j.

Table III.

Components C ofthe

elastic energy associated with the

non-totally symmetrical symmetry-adapted

strains. The _« vectors _»

defining

the

symmetry-adapted

strains _are

given

in the

first

column

(upper line) together

with

(lower lines)

the

corresponding

irreducible

representations (in parentheses) for

the

dijfierent point

groups ; in the monoclinic

point

_groups,

the

[u~)

axis is

parallel

_to the

binary

axis and/or

perpendicular

to the mirror

plane.

The

Care _given

in column 2 with the usual notation

for

the elastic constants

C~p

m

C[p (upper line)

and with the convention

given by (9) (lower line).

The

degeneracy ofthe $

is

given by

the

order

of

the associated irreducible

representations.

Cubic classes

Strain

C~

Ul)

_~

'~2)

~

'~31' '~I) '~2)

Cjj C12 T(E), Td(E), o(E)

~ ^~ ~ ~

Ii 12

Th(Eg), Oh(Eg) l~~~~d~~~)~ ~~~),

~$~

~h(Tg), Oh _(T2g)

^~

Hexagonal

classes

Strain

C~

('~l) '~21' '~6) ₎

C6(E2), C6v(E2), D6(E2)

^~

II

~12

~°3h(E'), 1~3h(E'), C~l

^C₍₂

~°6h(E2g), 1~6h(E2g) ( ~4) ~5) )

C6(Ei), C6~(Ej), D6(Ej)

²

C~

~3h(E" ), D3h(E" ), C~j

~6h(Elg), 1~6h(Elg)

Trigonal

classes

3,

3

Strain

f

I

j ~c

~~ c

~~ + 2

c~)

±

[(c

ii c

i~ 2

c~)~

₊

2

+

16(C

_{)~ +}

C/5)

_]~'~)

~~~ iii/i~lli/~'i~~

^{~ ~~~}

^{i (C ii}

^{C <~ +}

^C~i)

⁼

^{i(C ii}

^{C <~}

^C~i)~

⁺

~

~(~l~

_~

Table III

(continued).

Trigonal

classes 3 _m,

32, 3

m

Strain

C~

j _(C

ii C

i~ + 2

C~~)

±

i(C

ii C

i~ 2

C44)~

+

('~l) '~21' '~6)1'('~41' '~5))

₁₆

C3v(E ), D3(E), D3d(Eg)

~ ~

(C ii

C _{[2 +}

C$)

±

[(C ii

^C _[2

C~)

+ 2

2 1/2

~ ~ _~41

Tetragonal

classes 4

1,

4/m

Strain

f

(C

ii C

j~ + 2

C~~)

±

[(C

ii C

j~ 2

Cm)~

+

2

j~ j j~ j j~ j

⁺¹⁶

^C)~]~'~)

l 2 , 6

~4~~

~'

~4~~

~'

~~~~~~~ (~

_~l ~

~2 ~

~$6)

t

[(~

_~l ~ _~2

~$6)~

₊

+ 8 £~^a₁₆

jl/2j

( ~4), ~5) )

~

CM

C4(E), 54(E), C4h(Eg) C~j

Tetragonal

classes 4 _mm,

422,

42 _m,

4/mmm

Strain

~

'~l) '~2)

~4v(~l), ~4(Bl), D2d(Bl), ^{~~~ ~~~}

D

4h

(B1 )

^~

(l

_{~ (2}

C~~(B~),

~~~~),

D~~(B~),

^~

^~~~

D~~(B~~) ~~~

( ~4) ~5) ⁾

~ £~

C4~(E), D4(E), D2d(E),

^~

C~j

D4~ (Eg)

Table III

(continued).

Orthorhombic

point

_groups

Strain

~

U6)

~2v(1~2), D2(Bl),

^~

^~66

~2h(~ _lg) ~$6

(U5)

~2v(l~l ), ~2(1~2),

^~

j~~

~

(~ ~55

2h 2g

~~~~ 2

C44

C~~(B~), D~(B~),

~

~2h(1~3g) ~~

Monoclinic

point

_groups

Strain

f

1u4>,

_1u5>

1(c44

+

c55)

=

[(CM C55)~

+ 4

Cisi~'~i C2(B), C~(A"),

C~~(B ((Cj4+ C]5)

_±

[(Cj4- C]5)~

+

4C()l~'~)

~ 2

It follows that the

components

^{of the elastic}

compliance

table

[s~]

must be defined

exactly

like the elastic constant table components

[C "]

the form of the

[C

~]

given

in table II is also the

form of the

[s~].

Nye [14] already pointed

out that the existence of two conventions in

defining

the contracted form of the elastic constants

(C[~

and

C(p

in this

paper)

_can make it difficult to compare data

published by

different authors. The choice of _one notation rather than the other is

arbitrary.

More

recently

Sirotine and Chaskolskiia

[19] emphasized

that _a

«

symmetric

_»

convention would be

really

better but

they

did _{not use} it since the data

given

in the literature would have to be corrected. The

present

article

brings

an additional _reason for

choosing

such

a convention. The

advantage

of

using

_a

symmetric

notation is _not limited _to the field of

elasticity.

For the different _tensors

describing physical properties

it is

required

_to know the

specific

conventions for the contraction

(3). Using

the

«

symmetric

_»

convention,

_a

single

rule would have _to be defined I-e- any tensorial

quantity

Q~~__ ⁼ Q~~,_ with _two indices I and

j running

from I _to 3 would be transformed into _a

quantity

with _a

single

index Q~__

running

(3) ^The

necessity

of

introducing

_a

multipling

factor 2 in the contracted form of the

piezoelectric

stress coefficients is not always mentioned [18].

from I _to

6, provided

Q«,,

=

Q,j,,

^if a =

1, 2,

3

Q~

=

/ _Q,j

_if

a = 4,

5,

6 _;

the process would have _to be

applied

as many times _as there _are

pairs

of indices

(running

from I _to

3)

to be contracted.

5. Conclusion.

It has been shown

that,

in the

general

case, the standard elastic constant table _cannot be treated like _an

ordinary

matrix. On the other

hand,

_{a «}

symmetric

_» convention associated to the process of contraction of the indices leads _{to a} form such that the _«

eigenvalues

of the

elastic constant table» _are

meaningful.

The

«

symmetric»

convention makes it easy to transform the

expression

of the elastic energy into _a

diagonal

^{form. This is} useful in the

study

of the so-called ferroelastic

phase transitions,

_a

variety

of structural

phase

transitions characterized

by

the

vanishing

of _one of the components ^{of the}

diagonal

form for the elastic

energy. The results

presented

in the article

explain

the

origin

of the

discrepancies

encountered in the literature

conceming

the soft elastic _constant in the ferroelastic

BiV04.

Taking

into _account the above

considerations,

the

approach given by

Boccara

[3]

remains the

quickest

and best way to

predict

the soft elastic constants of _a ferroelastic

phase

transition.

The above

problem conceming

the elastic

properties pointed

out a more

general problem

related to the _use of _«

non-symmetric

_» conventions when

defining

_a contracted form of _a tensor. In this paper, it has been shown that the _use of _{a «}

symmetric

_» notation may

give

the contracted form of _a tensor additional useful

properties

_; there is _no doubt that _{a « non-}

symmetric

_» notation cannot do this. This

gives

_an additional _reason to

apply

a _«

symmetric

_» convention to any kind of

(symmetric

_or

antisymmetric)

_{tensor, a} notation that lies _{on a}

single

rule and _so, that would avoid the

knowledge

^of the various conventions characteristic of each of them.

Acknowledgments.

Thanks _are due to Prof. J. Nouet and to Prof. M. Rousseau for fruitful discussions.

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