On structural phase transitions in perfect crystals

MECHANICS RESEARCH COMMUNICATIONS 0093-6413/81/060341-08502.00/0

ON STRUCTURAL PHASE TRANSITIONS

Gareth P. Parry School of Mathematics,

Vol.

8(6), 341-348,1981. Printed in the U S A Copyright (c) Pergamon Press Ltd

IN PERFECT CRYSTALS

University of Bath, Bath, Avon,

England.

(Received 6 July 1981; accepted for print 23 September 1981)

Introduction

A prototype analysis of continuous phase transitions at b.c.t, configurations of perfect single crystals was presented in [i]. That calculation involves some simplifying, generally unjustified, features but it is good enough to demonstrate the limitations of Landau's theory of second order transitions. This work generalises [i] and shows that there are no unique isothermal elastic moduli defined at structural phase transitions. The implications of the results are discussed in the context of Landau's analysis.

Analysis

This is an extension of work initiated in [i] and [23, concerning phase transitions

in single crystals modelled by an internal variable theory.

The

calculations bear significantly upon Landau's theory [3] of second order phase transitions, which depends crucially upon the smoothness of a thermodynamic potential expressed in terms of order parameters. fying assumptions,

With some simpli-

it was shown in [11 that an internal variable theory with

a smooth potential generally induces sufficient roughness in the corresponding thermoelastic potential at a structural transition to disallow any expansion in terms of macroscopic order parameters.

So it is incorrect to use

Landau's method to gauge the behaviour of an order parameter measuring the macroscopic

strain in a structural phase transition via the conventional

thermoelastic picture. The main feature of this paper is the removal of some simplifying assumptions made in [i] and [21, although the work is still not quite general.

However,

the extension is significant enough to uncover new and unexpected properties of the isothermal

elastic moduli near the transition point. 341

342

G A R E T H P. PARRY

Briefly,

the p r o b l e m is to model a phase t r a n s i t i o n c o n t r o l l e d by temperature

in an u n s t r e s s e d b.c.t,

monatomic

tion and c o n v e n t i o n s of [i], background, lem.

single crystal.

We shall follow the nota-

to w h i c h the reader should also refer for the

basic g e o m e t r y and s y m m e t r y c o n s i d e r a t i o n s r ~ l e v a n t to the prob-

The b.c.t,

lattice may be thought of as two p a r a l l e l and c o n g r u e n t

simple lattices,

one of w h i c h has lattice vectors w h i c h are the edges of the

c o n v e n t i o n a l b.c.t, 'centre atoms'

unit cell and the other of w h i c h is c o m p o s e d of the

of the b.c.t,

lattice.

We e n v i s a g e a p h a s e t r a n s i t i o n where

this lattice of centre atoms shifts o f f - c e n t r e at some critical t e m p e r a t u r e 8 = @o"

W i t h the d i s p l a c e m e n t of a typical centre atom, or the

'shift', de-

noted by the v e c t o r ~ and the i n c r e m e n t in the C a u c h y Green tensor by ~, the Heln%holtzfree energy per unit mass

To ensure objectivity,

is a s s u m e d to take the form

we take the c o m p o n e n t s of ~ to be

nents on the b.c.t,

lattice vectors,

material directions

(see [i] for discussion).

In e q u i l i b r i u m c o n f i g u r a t i o n s ,

c o v a r i a n t compo-

w h i c h are p r e s u m e d to t r a n s f o r m as

~ and ~ assume values such as to render

s t a t i o n a r y at given @, and such c o n f i g u r a t i o n s are p r e s u m e d stable w h e n ~ is a b s o l u t e minimum.

~

(~,~,0)

Therefore,

at equilibrium,

= 0

(2)

and this g e n e r a l l y gives m a n y solutions of the form

= ~(~,e) w h e r e each ~

i

(n,@)

(3) is single v a l u e d . The f u n c t i o n

~(~,0)~ min ~ (~i(~,0),~,0) i

(4)

may then be r e g a r d e d as the t h e r m o e l a s t i c potential.

F r o m [i], the m o n a t o m i -

city of the crystal g u a r a n t e e s that ~ is even in ~, and t h i s ensures that z ~ is a s o l u t i o n of

(2).

Of course,

this is saying that the b.c.t,

f i g u r a t i o n is always an e q u i l i b r i u m c o n f i g u r a t i o n stable). det

There may be more than one s o l u t i o n to (~,~,0)

tion.

Henceforward,

(but it is not always (2) only w h e n

= O ,

w h e r e det is the d e t e r m i n a n t ,

con-

(5) so this e q u a t i o n d e t e r m i n e s

we assume that ~ =~,

~=~,

@ = 0

the c o n s t r a i n ~ i m p o s e d by the s y m m e t r y of the b.c.t,

the phase transi-

satisfies (5). W i t h o configuration, we find

PHASE

TRANSITIONS

that close to this solution

of

IN

PERFECT

CRYSTALS

(5) , to significant

343

order,

1 2 1--2 1 2 = ~a~ 1 + ~ 2 + ~ 7 3 + c171~ 2 + d (q1317173+Q2317273) + ~ie

2 2 + 1~ h ( ~ l 2+172)173 2+<~ °, (174+ 1 72 ) + i~ fii4 3 + ~ g171172

with Z. and Q l z3 a = a

+

o a

:

the components +

alQll

o

a3Q33

+ alN22

with

+

a2Q22

+

a

of ~ and ~ respectively,

(6)

+ a2qll

a3Q33

b : b O + b I(QII + D22 ) + b 2 Q 3 3 c = Coql2

,

with the a , b , 1

c,

1

components

to which

simplify

of b.c.t, parameters, plications

parameters,and

of ~ were disregarded

t h e present

matters

allow excess

d, e, f, g, h material

¢o

with

= ~(~,~,8).

1

The off diagonal extent

(7)

in what follows,

equalities

between

free energies, which which

With ~ given by

analysis

is an improvement

we shall

material

(6),

call upon

parameters,

on that work. 'genericity'

as in [i].

we are only allowed one equation

is that which derives follow

in [i], and this is the

in some more

from

(5).

symmetric

To

to disIn the class

between

material

(We shall disregard

configurations

com-

cf. [1].)

(2) becomes

a~ 1 + c172 + dQ1373 + e~ 1 + g17~2±z + h~l~3

= O

a--~2 + c171 + dQ23Z3 + e172 + gz1722 + h17217~ = 0 bZ 3 + d(Q1371 + Q 2 3 Z 2 ) + fZ~ + h(Z 1 + 7 2 ) Z 3 = O . At the putative

transition,

(5) holds,

(8)

so that either

a

or b O

The case where b

vanishes

may be dealt with quickly.

vanishes

By genericity,

O

there non-zero,

a

is

O

so that the first two equations

to lowest order,

ther~

O

and the third equation

reduces

of

(8) say that 171 = 172 = O,

to

~ 3 ( b + f~3 ) = O . There are three distinct

(9) solutions

for Z3, to wit

173 = O

½ 73 = + (-b/f) and these

solutions

By inspection,

, where bf
the critical

so that each path through

(iO)

as b ÷ O . point

is a relative

the critical

point

m i n i m u m when

f is positive,

is a path of relative

minima

344

GARETH

in t h a t c a s e .

P.

PARRY

Putting

(ii) and,

corresponding

to t h e

¢o = ¢ ( ~ , ~ , e )

,

1 b 2 + ¢o

¢~

=

4

solutions

, where

b < O,

that

~(~,@)

= ~o

, where

b > 0

,

~i

, where

b < O

.

I t is c l e a r

from

discontinuous

only

= O

a

in t h i s

order

path

the

passes

through

case,

so t h a t

o f ~3

' and

follows

[i]

isothermal

there

this

exactly.

2

only with

that considered the material

=

this

is a d e v i a t i o n

is l o s t

parameter

system

For

c .

e~ 1 + g~2 ) = 0

) = 0

with

three

of

(i)

~i = z2 = O

(ii)

~i

classes

f r o m n o w on.

of

the coupling

future

reference•

that f r o m b.c.t.

in t h e t h e r m o e l a s t i c

Differences

come

about

in

of

(8),

~

= 0 to 3

This

system

term which if c

differs

arises

vanishes,

from

through we have

O

,

(15)

,

solutions

given

by

,

= O

,

~2 = ± (-~/ e) ½

,

where

ae < O

42 = O

,

71 = i ( - a / e ) ½

,

where

ae < O

(iii) ~i = + ( N I / A ) ½ N 1 = ga-ea

are

(14)

i n [i] b y v i r t u e

~2(a+e~2+g~l

with

Notice

0

O

~l(a+

moduli

2 = 0 + g~l~2

az 2 + cn I + ez~ + g z 2 z l deal

elastic

b = O.

~ O b y g e n e r i c i t y , so t h a t b y t h e t h i r d o a n d t h e f i r s t t w o e q u a t i o n s r e d u c e to

--

shall

that

vanishes.

o

a ~ 1 + c~ 2 + e~

We

~o)

, b

o lowest

equilibrium

analysis

(13)

(¢i

nonvanishing

This

the case where a

form of

symmetry

in t h e

description.

When

the

if t h e

tetragonal

nature

(12)

f

it is c l e a r

has

(iO),

,

'

~2 = ±(N2/A) ½

N 2 = ga-

ea

,

' where

A = e 2 - g2

(16) NIA and N 1A>O ,

and where

,

the

+- s i g n s

are

PHASE TRANSITIONS

IN PERFECT

CRYSTALS

345

unrelated. Correspondingly,

define

,o = ~i =_--

~2

1 a2 __ 4 e 1 a2 4 e

,o

+

~O

ae < O ,

when

nu

,

when ae < O ,

~3 = - ~ 2 1 {e(N 12 + N2) + 2 g N I N 2 } This critical positive.

point

Note

~3_~2

(17)

+ ~o , when NIA and N 2 A > O "

is infinitesimally

stable provided

that e and A are both

the relations

: _! N2 4Ae 2 '

~3_~i

= ___l 4Ae

N2 1 '

(lS)

which would allow us to find ~, as in [i]. Returning reduce able

to the general

the system to a quartic,

solutions.

sense

'close'

sufficiently Firstly

small

notice

e ( ~ l)+ ~4 4

above,

solutions

which are in some

that is, solutions

where

c is

with a and a, or N 1 and N 2.

of

(14)

+~)+~

1

2+~o ,

2 g~l~2

this may be rewritten

either

(19)

as

(20)

+ 2c~in 2) + ~o

the trivial

(ii)

close

given by

solution

to ~i = O , ,

e(~

1 2 2+~o - 7 gITl~2

2

(i)

with ~

to establish

obtained

+ C ~ I~2 + ~

Close to the solutions

= ~c

but that the quartic does not yield manage-

in comparison

+~i ~n~

1 or as ~(a~ 12 + a ~

Zl

that it is easy to

that with ~3 = O, ~ is given by

and that by virtue

4

(14), one soon finds

So we are content to the solutions

* = ~ii 2

z2

(16) one determines

that

is unperturbed,

~2 = ±(-~ /el½

, with a < O

o = ~2 + 8c2'

, there

is a solution (21)

= ±(-V/e) ½ , and ~,~ given by O

2

e~ 2

O

e an 2

N1 Via

form of

(22)

2~N 1

(20) , one determines ~i

1 -~ a

1 a c2 +

4 e

2 N1

~o

f

when a < O

w

(23)

346

GARETH

to

lowest

are

close

order

in c.

to the

The

trivial

P.

PARRY

linearisation

branch.

fails

if a : O

There

is also

the

two

solutions

, in w h i c h

solution

case

we

corresponding

to

½ ~2 = O, 7[1 = t(-a/e) vanishes,

and

ing m a t e r i a l the

this

One

is a g e n e r i c

parameters.~

1/3

with

N

1

fails

if e i t h e r

involving = O

, then

N1 or N 2

no p r e s u m p t i o n we

find

regard-

a solution

of

given 1/3 ,

(-e~/A)

~c2/3 +

(in

given

+3

___

24)

@ -

,

25)

27

potential

*=

'

by g o 22

L

4 e

by

~2

~

1/3

~o

4/3

I

c

+

than

the

fact one

needs

more

dii) Finally,

close

to t h e

26)

third

leading class

of

terms

given

solutions,

in

(24)

putting

to d e t e r m i n e

@).

here

½

o 7TI

i (NI/A) ½

:

,

iT2

:

+-(N 2 / A )

,

]72

:

o ]72

(27)

,

finds o = ]71

]71

with

~

+

acc

,~ g i v e n

~^2 0 2 0 2 za ] 7 1 4 2 K =

with

corresponding

-i

,

the

that

the

terms

involving

c in the

form

c 2

+

~TN2 f ~NI 2 (NI,N 2)

(29)

quartic

fails

the c l a s s

solutions.

If we

(28)

have

linearisation

General

,

and determines

f is a h o m o g e n e o u s

(ii)

@c

gN2)/,

potential

c ½ ± ~ ( N I N 2) where

+

by

k ~ l (eN 1

if e i t h e r

N

in N 1 a n d N2 (involving or N

1

vanishes,

2

g a n d e).

in w h i c h

case

The

we are

closetD

features

restrict

attention

N1/N 2 < 6 , with

@

(29)

that

terms with

to deal

to a d o m a i n

where

N2/N 1 < @ ,

> i, w i t h

comparison tent

possibility,

in p a r t i c u l a r

o ]72 = ~2

'

~ and @here :

one

the

form ]71 : @c

and

of

(30)

c sufficiently which

represent

the d i f f e r e n c e s

only

with

this

small,

then

it is c l e a r

perturbations given

case,

in

where

(18).

from

(23),

in the p o t e n t i a l For b r e v i t y

the o r d e r i n g

given

we in

are

(26) small

shall [i]

be

still

and in con-

PHASE T R A N S I T I O N S

applies

IN P E R F E C T C R Y S T A L S

347

(with the new i n t e r p r e t a t i o n of the ~l).

Notice that ~ i _ ~o is g e n e r a l l y h o m o g e n e o u s of the second degree in N I , N 2 and c.

In c o n t r a s t to the case w h e r e c = O, w h e r e ~ i _ ~o is p o l y n o m i a l ,

the sec-

o n d p a r t i a l d e r i v a t i v e s of ~ w i t h r e s p e c t to ~ are then g e n e r a l l y not d e f i n e d at ~ = ~.

That is to say,

the i s o t h e r m a l elastic moduli are not d e f i n e d there,

a l t h o u g h they have an unique value on any r e a s o n a b l y d i r e c t p a t h into ~ = O and they are b o u n d e d in the domain s p e c i f i e d by

(30).

Specifically,

moduli

d e t e r m i n e d on e q u i l i b r i u m paths into the t r a n s i t i o n will appear to suffer finite d i s c o n t i n u i t y .

It is w o r t h r e m a r k i n g that the d e t e r m i n a t i o n of m o d u -

li via m e a s u r e m e n t s of wave v e l o c i t y t h e r e f o r e needs rather careful c o n s i d e r ~ tion, w h i c h we do not a t t e m p t in this l i m i t e d space.

(It is not even obvious

that p l a n e i n f i n i t e s i m a l waves are n o n - d i s p e r s i v e ) . Order p a r a m e t e r s So as to e m p h a s i z e the d i s t i n c t i o n b e t w e e n the b e h a v i o u r of order p a r a m e t e r s at v a r i o u s

'levels', we s u m m a r i z e L a n d a u ' s argument.

If the H e l m h o l t z

e n e r g y may be w r i t t e n as ~(z,@), w h e r e z is "some quantity.,

free

which determines

the e x t e n t to w h i c h the atomic c o n f i g u r a t i o n in the less s y m m e t r i c a l p h a s e d e p a r t s from the configuration in the m o r e s y m m e t r i c a l phase", the latter phase, then e x p a n d i n g in p o w e r s of z ~(z,8) = ~(0) + B ( 0 ) z + y ( 0 ) z 2 + 6 ( O ) z 3 + E ( O ) z ~ . . . .

and z = O in (31)

The e q u i l i b r i u m c o n f i g u r a t i o n s m u s t be m i n i m a w i t h r e s p e c t to v a r i a t i o n s of z, so that O = 8(0)

+2y(O)z

+ 3 6 ( 0 ) z 2 + 4 Z ( O ) z 3 ....

Since z = O m u s t be a s o l u t i o n of since this is a c o n t i n u o u s

(32), 8(0)

transition,

(32)

m u s t v a n i s h identically,

y(O ) = O.

Also,

and

~ is a m i n i m u m if

O

and only if 6(0 ) = O and Z(O )>O. O

z2 ~

T h e n from

(32), close to O ,

O

O

y'(0 ) o 6@ , 2Z (8)

(33)

O

where 60 is the i n c r e m e n t in O, and the dash d e n o t e s the derivative. there is more than one order parameter,

When

there is a c o m p a r a b l e r e s u l t to

(33)

with z 2 i n t e r p r e t e d as the sum of the squares of the o r d e r parameters. W i t h the p i c t u r e o b t a i n e d in this paper,

w i t h ~ g i v e n by

(4), we p r e s u m e that

the m a t e r i a l comes into the t r a n s i t i o n a l o n g ~ = O, where ~ = ~o, and leaves a l o n g a p a t h c o r r e s p o n d i n g to some n o n t r i v i a l ~ and a s s o c i a t e d p o t e n t i a l ~i,

i J O. ~i(~,O)

By v i r t u e of the a n a l y s i s in the last s e c t i o n = f(~,O)

+

r e g u l a r terms in ~

(depending on 0),

(34)

348

GARETH P. PARRY

where

is h o m o g e n e o u s of d e g r e e two in ~ , and is g e n e r a l l y not poly-

f(~,e)

nomial. Since the e q u i l i b r i u m p a t h is stress free, we have

o=-~ (~,@) ~f

= ~ is g e n e r a l l y not a s o l u t i o n of this equation,

The p a t h e = 8

o

,

(35)

+ regular terms •

but ~ = ~ at

so that ~f

w h e r e a(8)

(~

e ) +a(e

o

o

)

(36)

,

is the first of the r e g u l a r terms in

35).

By v i r t u e of the homo-

g e n e i t y of f, ~f

0

o)

=

(37) ,

Thus the r e g u l a r terms zn so that g(@ ) = O also. o A g a i n by h o m o g e n e i t y it follows that

~ ~6e where ~

is

Afortiori,

(35) b e g i n with ~'(@

o

) 68.

(38)

,

a constant

such

~f

that~(~,e

o) = a (eo)

any order p a r a m e t e r w h i c h is a linear f u n c t i o n of ~ is also pro-

p o r t i o n a l to 6e, w h i c h result should be c o n t r a s t e d w i t h L a n d a u ' s result is r e c o v e r e d if, for example,

(33).

we introduce the

'lost' order

p a r a m e t e r ~, w h i c h s a t i s f i e s

½ (in loose notation), order p a r a m e t e r s

by v i r t u e of

is p r o p o r t i o n a l

(16), so that the sum of the squares of the to 6@.

References [i]

G.P. P a r r y

On phase t r a n s i t i o n s

i n v o l v i n g internal

strain. Int. J.

Solids Structures 17, 361-378(1981) [23

G.P. P a r r y

on shear induced p h a s e t r a n s i t i o n s

in p e r f e c t crystals,

to appear in Int. J. Solids Structures. [3]

L.D. L a n d a u

On the t h e o r y of phase transitions.

In Collected Paper&

of L.D. Landau (Edited by D. Ter Haar) pp 193-216. P e r g a m o n Press and G o r d o n and Breach,

London

(1965) .