LOCALIZED LATTICE INSTABILITY RELATED TO THE NUCLEATION PROBLEM OF MARTENSITE

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LOCALIZED LATTICE INSTABILITY RELATED TO THE NUCLEATION PROBLEM OF MARTENSITE

G. Guénin, P. Gobin

To cite this version:

G. Guénin, P. Gobin. LOCALIZED LATTICE INSTABILITY RELATED TO THE NUCLEATION PROBLEM OF MARTENSITE. Journal de Physique Colloques, 1982, 43 (C4), pp.C4-57-C4-73.

�10.1051/jphyscol:1982404�. �jpa-00221948�

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JOURNAL DE PHYSIQUE

CoZZoque C4, suppze'ment au n o 12, Tome 43, de'cembre 1982 page C4-57

LOCALIZED LATTICE INSTABILITY RELATED TO THE NUCLEATION PROBLEM

OF

MARTENSITE

G. ~ u g n i n and P. F. Gobin

Groupe drEtudes de MMe'aZZurgie Physique e t de Physique des M a t e ' r i m , I . N . S . A., B e t . 502, 69621 ViZZeurbanne Ceder, France

(Accepted 9 August 1982)

A b s t r a c t . The g e n e r a l i z e d s o f t mode i s n o t v e r y c o n v e n i e n t t o ex- p l a i n t h e m a r t e n s i t e n u c l e a t i o n p r o c e s s . I t must b e s u b s t i t u t e d by a l o c a l i z e d s o f t mode approach. The s t a r t i n g p o i n t f o r t h i s approach i s t h e s e n s i t i v i t y of t h e mechanical s t a b i l i t y t o t h e s t r a i n s . I t i s f i r s t d e m o n s t r a t e d t h a t c a l c u l a t i o n s u s i n g second and t h i r d o t h e r e l a s t i c c o n s t a n t s p r o v i d e a good approximation t o t h e s t r a i n producing t h e i n s t a b i l i t y . The e f f e c t s o f some p a r t i c u l a r s t r a i n such a s

"Bain s t r a i n " o r {110} <1T0> i n 6 phase a l l o y s a r e reviewed.

The s o f t zones around some s p e c i f i c l a t t i c e d e f e c t s l i k e d i s l o c a t i o n s and f r e e s u r f a c e s a r e t h e n a n a l y z e d . I t i s shown t h a t l a r g e l a t t i c e v i b r a t i o n s a r e l o c a l i z e d i n t h e s e s o f t zones. An a t t e m p t i s made t o e x p l a i n t h e r o l e o f t h e s o f t zones and a s s o c i a t e l a r g e v i b r a t i o n t o t h e n u c l e a t i o n p r o c e s s .

I n t r o d u c t i o n . Second o r d e r t r a n s i t i o n s a r e c h a r a c t e r i z e d by a c o n t i - nuous change from one phase t o t h e o t h e r a s shown i n f i g u r e 1. An obvious symmetry r e l a t i o n i s observed between t h e p a r e n t phase and t h e p r o d u c t , f o r example c u b i c and t e t x a g o n a l r e s p e c t i v e l y , and a n o r d e r p a r a m e t e r c a n b e e a s i l y d e f i n e d . With d e c r e a s i n g t e m p e r a t u r e and on r e a c h i n g T t h e r e i s a c o n t i n u o u s change i n t h e whole c r y s t a l from t h e h i g h synunegry phase t o t h e new lower symmetry phase. The p a r a m e t e r s a r e c l o s e t o t h o s e of t h e h i g h symmetry phase. T h e r e f o r e , t h e r e i s no n u c l e a t i o n problem a s d i f f e r e n t p h a s e s do n o t c o e x i s t . I n c a s e t h e t r a n s i t i o n i s due t o a mechanical i n s t a b i l i t y c o r r e s p o n d i n g t o a homogeneous s t r a i n , one of t h e e l a s t i c c o n s t a n t v a n i s h e s a t t e m p e r a t u r e TC The g e n e r a l s o f t mode c o n c e p t h a s been a p p l i e d s u c c e s f u l l y t o d e s c r i b e such second o r d e r phase t r a n s i t i o n s (1)

,

( 2 ) . According t o t h i s approach, t h e a p p e a r a n c e o f t h e new phase c a n b e c o n s i d e r e d a s t h e f r e e - z i n g o f t h e l o n g wavelength v i b r a t i o n s c o r r e s p o n d i n g t o t h e e l a s t i c c o n s t a n t which v a n i s h e s . T r a n s f o r m a t i o n s , o c c u r r i n g i n s e v e r a l a l l o y s such a s V3Si, Nb3Sn, In-T1 a r e o f t e n c l a s s i f i e d a s m a r t e n s i t e . They a r e d e s c r i b e d a s weakly f i r s t o r d e r t r a n s f o r m a t i o n s and t h i s b e h a v i o u r c a n b e understood a s a n e x t r a p o l a t i o n o f t h e scheme d e t a i l e d f o r second o r d e r t r a n s i t o n s . ( f i g . 2 ) . However, a s t h e t r a n s f o r m a t i o n s i s f i r s t o r d e r , i n t h e neighbourhood of TC two p h a s e s c o e x i s t w i t h f i n i t e d i f f e r e n c e s i n t h e i r l a t t i c e , and t h u s a n u c l e a t i o n problem e x i s t s . The a p p e a r a n c e o f o n e phase i n s i d e t h e o t h e r , t h e r e f o r e , i m p l i e s t h e e x i s t e n c e o f b o t h a s t r a i n energy due t o t h e f i n i t e homogeneous t r a n s - f o r m a t i o n s t r a i n and a n i n t e r f a c e energy. The s t r a i n energy term depends on t h e t r a n s f o r m a t i o n s t r a i n a m p l i t u d e and on t h e c o r r e s p o n d i n g

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982404

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C4-58 JOURNAL DE PHYSIQUE

elastic constant (which almost vanishes at the transformation point.) The interface energy term is more difficult to derive, but as a first approximation it can be considered as depending on the transformation strain amplitude (if the low temperature phase is assumed to be cohe- rent).

Finally in classical martensitic transformations, the difference in lattice parameters between the parent and product phases are large.

Changes in the lattice parameter of the product subsequently, after the transformation neither is observed. An order parameter is general- ly difficult or impossible to define. +

Thus there is a nucleation problem, the transformation being clearly first order with well defined coexisting phases. A "soft mode" is observed in a number of alloys exhibiting martensitic transformation, in particular thermoelastic transformations, and this will have an influence on the nucleation. The energy term associated with the transformation strain depens on the elastic constants and may become very small if the soft elastic constant corresponds to the transformation strain. Moreover, the large amplitude vibrations corresponding to the soft constant can be considered as attempts to firm a nucleus. For alloys which exhibit thermoelastic martensitic transformations

(Bphase alloys, ordered Fe3Pt, Ti-Ni) one of the elastic constants is low and decreases with temperature. (figure 4). However, the elastic constant vanishes not at the transformation temperature but

a

much lower temperature. Consequently when the nucleation process is assumed to be homogeneous the influence of this low elastic constant in

the nucleation process will be very limited. Nevertheless, the elastic behaviour is greatly affected in the neighbourhood of lattice defects as will be demonstrated in this paper.

General presentation. The idea that the mechanical stability of the lattice should be considerably modified by a homogeneous strain was first pointed out by Clapp

( 3 ) .

The starting point is the expression of the free energy per unit volume as a function of homogeneous strains

:

The crystal is mechanically stable when subjected to small homogeneous strains, if the free energy increases for whatever strain tensor 1 ~ ~ 1 :

Mathematically, the six eigenvalues of the Fij matrix must be positive.

When the cryStal is straln free the derivatives F,, , of the free ener-

L J

gy are taken in the neighbourhood

= 0

and are therefore identical to the elastic constants.

The stability conditions, in the case of cubic crystals, for example, are

:

---

* An order parameter can be considered in the case of f.c.c.

+ -h

b.c.c.

transformation based on the Bain strain amplitude, but intermediate values of this parameter lead to a structure which is not f.c.c. nor b.c.c. structure but a.b.c.t. structure. A gap is thus necessarily encountered when changing one phase into the other.

Therefore such a parameter is not a "strict0 sensu" parameter.

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I f t h e c r y s t a l i s s u b j e c t e d t o a f i n i t e b i a s s t r a i n t h e F i j v a l u e s become ( i f t h e e x p a n s i o n i s l i m i t e d t o t h i r d o r d e r t e r m s )

The F i j c a n b e c o n s i d e r e d a s t h e e l a s t i c c o n s t a n t s o f t h e s t r a i n e d l a t t i c e .

Clapp ( 3 ) h a s shown, i n p a r t i c u l a r , t h a t t h e Bain s t r a i n ( c l = E * =

-

2 E

,

E 3 = E r 4,516 ⁼^{0 )} ^{i s}e s p e c i a l l y e f f e c t i v e i n making some b . c . c . l a t t i c e s m e c h a n i c a l l y u n s t a b l e (Na, L i , BCu-Zn).

F o r a few p e r c e n t E > 0 ( e x t e n s i o n on a x i s 3 and c o n t r a c t i o n s on a x i s 1 and 2) t h e e l a s t i c c o n s t a n t r e l a t i v e t o t h e Bain s t r a i n i t s e l f

6 2~

v a n i s h e s (

-

⁼0 ) b u t f o r t h e same magnitude E < 0 t h e e l a s t i c con-

6 E L

s t a n t r e l a t e d t o a (110)

I:l-l~j

s h e a r ( A E ₁=

+

g, ₂ A E ₂= k

-,

a ₂ ^{A E}3,4,5,6=O) v a n i s h e s . The s t a b i l i t y e x i s t e n c e t h e n e x a c t l y c o r r e s p o n d s t o t h e c o n d i t i o n s t h a t t h e s e s i m p l e e l a s t i c c o n s t a n t s must b e p o s i t i v e . I n t h e more g e n e r a l c a s e where any s t r a i n i s a p p l i e d which c o m p l e t e l y de- s t r o y s t h e c r y s t a l symmetry, t h e s t a b i l i t y c r i t e r i a w i l l n o t e x a c t l y c o r r e s p o n d t o s i m p l e e l a s t i c c o n s t a n t s and t h e u n s t a b l e d e f o r m a t i o n mode i s n o t simple.

S i n c e t h e c r i t i c a l s t r a i n a m p l i t u d e i s s m a l l ( s e v e r a l p e r c e n t s ) and s i n c e s u c h a m p l i t u d e s a r e e n c o u n t e r e d i n t h e v i c i n i t y o f l a t t i c e d e f e c t s ( p o i n t , l i n e and s u r f a c e d e f e c t s ) , a l o c a l i z e d i n s t a b i l i t y should be c o n s i d e r e d . The i n s t a b i l i t y c o r r e s p o n d s t o a p a r t i c u l a r homogeneous s t r a i n mode. I f t h i s p a r t i c u l a r mode i s c l o s e t o t h e t r a n s f o r m a t i o n s t r a i n , o r a p a r t o f t h e t r a n s f o r m a t i o n s t r a i n , t h e n it i s e v i d e n t t h a t t h e u n s t a b l e zone p r e s e n t i n t h e v i c i n i t y o f a l a t t i c e d e f e c t w i l l have a f a v o u r a b l e i n f l u e n c e on t h e n u c l e a t i o n . Indeed a n u c l e u s may d e v e l o p i n such a r e g i o n w i t h o u t g e n e r a t i n g any o r v e r y s m a l l e n e r g y . On t h e o t h e r hand l a r g e a m p l i t u d e v i b r a t i o n s w i l l t a k e p l a c e i n t h e u n s t a b l e zone ond may b e c o n s i d e r e d a s a t t e m p t s t r a n s - form.

On t h e v a l i d i t y o f t h e t h i r d o r d e r a p p r o x i m a t i o n f o r t h e s t r a i n g t a b i l i t y c r i t e r i a . The f r e e energy expansion w r i t t e n above h a s been l i m i t e d t o t h i r d o r d e r t e r m s t o e v a l u a t e t h e mechanical s t a b i l i t y of t h e homogeneously s t r a i n e d l a t t i c e . I n o r d e r t o t e s t whether i n c l u - s i o n o f h i g h e r o r d e r t e r m s i n t h e expansion would g i v e a v e r y d i f f e - r e n t r e s u l t , a Landau formalism, r e c e n t l y used by Olson and Cohen (18) may b e f o l l o w e d . According t o t h e formatism,

where r l i s a n o r d e r p a r a m e t e r bound t o t h e homogeneous t r a n s f o r m a t i o n s t r a i n

rl = 0 : untransformed p a r e n t phase and r ~ = 1 : m a r t e n s i t e

E q u a t i o n ( 5 ) i s a p a r t i c u l a r c a s e of e q u a t i o n (1) because it c o r - r e s p o n d s t o a p a r t i c u l a r form o f t h e s t r a i n t e n s o r ~ E ~ ] . Q = 1 c o r r e s -

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C4-60 JOURNAL DE PHYSIQUE

ponds to the homogeneous transformation strain, where value E~ is around 0.2 to 0.3 for classical martensic transformations.

C 4 A E 2 - s E 3 t - E

Equation (5) becomes: AF = ---2. ₄ (6)

&o ^E0 0

For the temperature T = To where the energies of the parent phase and martensite are equal, Olsen and Cohen (18) proposed B = 2A and C = A.

With these conditions : 6 2 ~ f A

(-1 6E2 E-0

-

⁼^{2 - 7} which is the second order elastic constant E 0 of the B phase.

6 L ~ f A

-

- ₌27 which is the second order elastic constant 6 E 2 E-E

o E of the martensite

0

6 3 ~ f 12A

-)

-

=

-

is the third order elastic constant of the

&&3 E-0

E 0 B phase

These values correspond reasonably to observed elastic constants.

Indeed, the same elastic constants are encountered for the parent phase and the martensite. Moreover, the third order elastic constant is negative and about one order of magnitude higher (-) 6 as is the

E usual behaviour observed in metals and alloys. ⁰

If the parent phase is homogeneously strained along the particular direction associated with the transformation, mechanical instability will occur (18) when

6 'Af

-

⁼⁰^{and qs}⁼ ⁽¹

^-

^{1 /J3)}

^/

²⁼0.2113 and 62r1

If the Landau expansion is restricted upto third order terms, the mechanical instability is obtained when

This value, experimentally obtainable, is only 20% lower than the value obtained if the fourth order terms over included. Therefore, the nls value gives a good description of the sensitivity of the mechanical stability to the strain. This result, obtained using the simple phenomenologic approach, is well confirmed when comparing the results of Clapp (3) to those of Mac Donald (19) concerning the stability of the sodium subjected to a "Bain" strain. Using the calculation consisting up to third order terms described above, Clapp pre- dicted an instability c l = E~

-

0.015 and c3 = 0.032 from which we

-

can deduce a tetragonality c/a = 1.0477. A gellium model for alkaline metals based on atomic bonds evaluations and not limited to the third order terms led Mac Donald (19) to a mechanical instability as a

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function of similar strains for

=

1.0535. It can be seen that the two results are in good agreement and limiting to third order terms underestimates the critical strain only by an amount of 10% (Probably this difference is smaller than the experimental and calculation errors).

The foregoing shows that an expansion up to third order terms is quite sufficient to evaluate the effect of homogeneous strains on the mechanical stability. It must be reminded that the Landau formation used above has only a demonstrative value; it assumes that the appli- ed strain leads to an instability for the same type of strain. In fact, as already seen en Clappts paper (3) and as will be shown later, a strain in a given direction in the strain space can lead to instability corresponding to another strain direction.

Existence of localized soft modes around specific lattice defects.

To realise around known defects (dislocation, surfaces) the Bain strain considered by Clapp (3) appears to be difficult. However, shear strains (dislocations) or uniaxial displacements (surface) maybe more easy to encounter in a crystal. The effect of such strains on

the phase stability has been studied for B phase alloys

(5)

(10).

Soft zones around dislocations.

Among the possible shears in b.c.c. metals or alloys the (110}

<lTO> shear is of particular importance. This shear corresponds to the elastic constant

C ' = -

1 (Cll - C12) and is closely related to the homogeneous lattice shear alloys (see appendix 1). Finally this shear can be considered as constituting half the "Bain" strain (see appendix 2) which is very effective for the stability. The mechanical stabili-

ty

as a function of such a strain ha-s been studied. (4) (5). The cal- culations were performed for (011) @ill shear. The primary critical stability criterion is expressed by

:

where a

=

C

111 - '112

and =

'112 - '123 and

E

is the amplitude of the shear ( E ~

=

- -

^E

€ 2

- 7'

1,4,5,6

= O)'

This expression can be simplified, if we suppose that

Cll - ^C12

< <

Cll or C12 as is the case for phase alloys in the vicinity of the martensitic transformation.

Expression (7) then becomes

:

The critical strain amplitude

E - =

2J3 C1l - C12

leading to the in-

stability is only of the order of a few percent in the case of a

Cu-Zn-A1 alloy (6). Since the symmetry of the lattice is destroyed

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C4-62 JOURNAL DE PHYSIQUE

by t h e b i a s s t r a i n , t h e s t a b i l i t y c r i t e r i o n ( 8 ) d o e s n o t c o r r e s p o n d , a s e x p l a i n e d b e f o r e , t o a s i m p l e e x p r e s s i o n o f e l a s t i c c o n s t a n t s . However, it c a n b e shown ( 5 ) ( 6 ) t h a t t h i s c r i t e r i o n i s v e r y c l o s e t o t h e c o n d i t i o n t h a t t h e C ' t y p e e l a s t i c c o n s t a n t must be p o s i t i v e f o r t h e systems L101)

[lxl]

o r (110)

Dig.

The a p p l i e d s h e a r s t r a i n on t h e (011)

D l q

system i n d u c e s :

-

A r e s t o r i n g f o r c e unchanged f o r t h e s h e a r systems (011)

@ill

o r (011)

piq

(same systems a s t h e a p p l i e d s h e a r s t r a i n ) . - C I S =

7

1 ( C l l

-

C12)

-

^A r e s t o r i n f o r c e on t h e s h e a r systems (011) [ 0 i q on

(011)

@la

t h a t remains u n a l t e r e d w i t h r e s p e c t t o t h a t i n a s t r a i n f r e e c r y s t a l and i s g i v e n by C I S = 1

( C l l

-

C12)

-

An i n c r e a s e d r e s t o r i n g f o r c e on t h e s h e a r systems (110)

PioJ

o r (110) E l 0 1 where C r s =

-

1 2 ⁽ ^{C l l}

-

C12)

+

(2d

-

a ) . Such s t r a i n s w i t h s e v e r a l p e r c e n t a m p l i t u d e s a r e p r e s e n t i n t h e B phase a l l o y s . The d i s l o c a t i o n which " a p r i o r i " produce a l a r g e ( 1 1 0 ) < l i O > s h e a r s t r a i n a r e t h o s e which l i e on (110) p l a n e s w i t h

B u r g e r s v e c t o r <lTO>. I n 6 phase a l l o y s w i t h l a r g e e l a s t i c a n i s o t r o - py, e x i s t e n c e o f s u c h d i s l o c a t i o n s i s a t h e o r e t i c a l p o s s i b i l i t y ( 7 ) and even i f r a r e have been e x p e r i m e n t a l l y observed ( 8 ) ( 9 ) . A de- t a i l e d c a l c u l a t i o n a p p l i e d t o t h e c a s e o f Cu-Zn-A1 ( 5 ) ( 6 ) shows

( f i g u r e 5 ) t h a t a screw d i s l o c a t i o n w i t h B u r g e r s v e c t o r

= a. L01j.J (ao b e i n g t h e l a t t i c e p a r a m e t e r of t h e b . c . c . l a t t i c e ) g e n e r a t e s a wide domain o f s e v e r a l nanometers where t h e c r i t i c a l s h e a r s t r a i n i s a t t a i n e d . Even l a r g e r u n s t a b l e zones o f t h e c r y s t a l c a n be found when s u b m i t t e d t o t h e i n f l u e n c e of s e v e r a l d i s l o c a t i o n s ( 5 ) .

S o f t zones n e a r f r e e s u r f a c e

Clapp ( 3 ) a l r e a d y c o n s i d e r e d t h a t t h e f r e e s u r f a c e should be f a v o u r a - b l e t o a l o c a l i z e d s o f t mode b u t d i d n o t p r o v i d e any d e t a i l e d c a l - c u l a t i o n . I n a n o t h e r paper a t t h i s Conference (10) it h a s been demon- s t r a t e d how such a s o f t mode may be produced and p o s s i b l y e x p l a i n t h e e x i s t e n c e o f s u r f a c e m a r t e n s i t e . The s t r a i n c o n s i d e r e d a t t h e s u r - f a c e i s a s i m p l e d i s p l a c e m e n t normal t o t h e s u r f a c e . I f w e c o n s i d e r any s u r f a c e , t h e symmetry i s broken and t h e g e n e r a l c r i t e r i a f o r t h e s t a b i l i t y w i l l b e c o m p l i c a t e d and d i f f i c u l t t o i n t e r p r e t . However, a s s e e n i n t h e p a r t i c u l a r c a s e o f t h e s h e a r s t r a i n (110) <1i0> t h e primary c r i t i c a l c r i t e r i o n i s v e r y c l o s e t o t h e c o n d i t i o n C V s > 0. On t h e o t h e r hand, t h i s r e s t o r i n g f o r c e (appendix 1) i s of fundamental importance f o r t h e t r a n s f o r m a t i o n i n

B

phase a l l o y s . T h e r e f o r e , it i s o f i n t e r e s t t o s t u d y t h e v a l u e s of t e C I S t y p e r e s t o r i n g f o r c e s a s a f u n c t i o n o f t h e s u r f a c e s t r a i n a m p l i t u d e . When t h e s t r a i n i s z e r o a l l (110)

<la>

s h e a r systems a r e e q u i v a l e n t w i t h C f s = 1 ( C l l

-

C12) a s t h e r e s t o r i n g f o r c e . When a f i n i t e s t r a i n i s p r e s e n t , it i s e a s i l y shown t h a t t h r e e d i f f e r e n t r e s t o r i n g f o r c e s a r e e n c o u n t e r e d :

f o r t h e systems (110)

I>?o]

o r (1x0) l>10] which c a n be s h o r t e n e d t o t h e n o t a t i o n 110

(8)

- 1

- 7 ( F 2 2

+

F33

-

2F23) (11)

011

I n t h e s e e x p r e s s i o n s Fij = (-6 ~ . 6 c . 6 2 ~ ) E k a r e c a l c u l a t e d f o r p a r t i -

1 3

c u l a r v a l u e s of t h e s u r f a c e s t r a i n t e n s o r . I n t h e c a s e o f a s u r f a c e s t r a i n d i r e c t e d normal t o and outward o f t h e m a t e r i a l , a l l t h e C r s r e s t o r i n g f o r c e s d e c r e a s e w i t h i n c r e a s i n g s t r a i n a m p l i t u d e ; t h i s i s p h y s i c a l l y r e a s o n a b l e b e c a u s e a l l t h e i n t e r a t o m i c d i s t a n c e s i n c r e a s e . However t h e most a f f e c t e d r e s t o r i n g f o r c e s d e c r e a s e w i t h i n c r e a s i n g s t r a i n a m p l i t u d e ; t h i s i s p h y s i c a l l y r e a s o n a b l e b e c a u s e a l l t h e i n t e r a t o m i c d i s t a n c e s i n c r e a s e . However t h e most a f f e c t e d r e s t o r i n g f o r c e c o r r e s p o n d s t o t h e (110) p l a n e which i s t h e c l o s e s t t o t h e f r e e s u r f a c e . F o r example i f t h e f r e e s u r f a c e i s a (110) s u r f a c e , t h e c o r - r e s p o n d i n g C I S r e s t o r i n g f o r c e w i l l v a n i s h f o r a 3.5% s t r a i n ampli- t u d e i n t h e c a s e of Cu-Zn-Al, whereas t h e o t h e r r e s t o r i n g f o r c e s C ' s l O l and C ' s O l l w i l l v a n i s h o n l y f o r a 6.2% s t r a i n a m p l i t u d e . Few e x p e r i m e n t a l r e s u l t s a r e a v a i l a b l e f o r t r u e d i s p l a c e m e n t s a t f r e e s u r - f a c e b u t c a l c u l a t i o n s (20) seem t o show t h a t s u r f a c e d i s p l a c e m e n t s a r e p a r t i c u l a r l y l a r g e i n t h e c a s e o f b.c.c. c r y s t a l s and e x t e n d t o seve- r a l p l a n e s i n s i d e t h e c r y s t a l . T h e r e f o r e , it i s q u i t e p r o b a b l e , t h a t c l o s e t o t h e f r e e s u r f a c e t h e e l a s t i c p r o p e r t i e s a r e h i g h l y m o d i f i e d l e a d i n g t o s u r f a c e m a r t e n s i t e n u c l e a t i o n .

Consequences of l a t t i c e v i b r a t i o n s i n t h e s o f t zones. I t i s e v i - d e n t t h a t d r a s t i c s o f t e n i n g o f t h e r e s t o r i n g f o r c e s i n a l i m i t e d r e - g i o n of t h e c r y s t a l l e a d s t o s t r o n g changes i n t h e v i b r a t i o n s o f such a r e g i o n . U n f o r t u n a t e l y , a t t h i s moment t h i s problem h a s n o t r e c e i v e d a complete t r e a t m e n t . Clapp (11) h a s used r e c e n t r e s u l t s on a monato- mic l a t t i c e w i t h a p o i n t d e f e c t having weaker f o r c e c o n s t a n t s t h a n t h e o t h e r atoms ( 1 2 ) . A q u a s i l o c a l i z e d mode e x i s t s around t h e d e f e c t i n t h i s c a s e w i t h r e s o n a n t f r e q u e n c y wo lower t h a n t h e a c o u s t i c f r e q u e n c y The r e s o n a n t e f f e c t c a n be u n d e r s t o o d a s f o l l o w s : t h e q u a s i l o c a l i z e d mode around t h e d e f e c t w i t h f r e q u e n c y wo i s e x c i t e d by t h e normal l a t -

t i c e v i b r a t i o n s o f t h e same f r e q u e n c y . These normal l a t t i c e v i b r a - t i o n s a c t a s a d r i v i n g f o r c e e x t e r n a l t o t h e d e f e c t . A l a r g e e n e r g y t r a n s f e r c a n be produced between t h e e x t e r n a l f o r c e (normal v i b r a t i o n of t h e l a t t i c e ) and t h e d e f e c t which w i l l g e n e r a t e v e r y l a r g e v i b r a - t i o n s . The f o l l o w i n g r e s u l t i s o b t a i n e d f o r weak c o u p l i n g ( y 2 1 ) .

where wL = 2Jf/m i s t h e maximum-frequency o f t h e a c o u s t i c band.

f i s t h e f o r c e c o n s t a n t o f t h e p e r f e c t l a t t i c e f ' i s t h e f o r c e c o n s t a n t around t h e d e f e c t ( f ' << f )

f

-

f '

y = -

f

The maximum v i b r a t i o n a m p l i t u d e o f t h e atom which c o n s t i t u t e s t h e d e f e c t i s g i v e n by:

(9)

JOURNAL DE PHYSIQUE

where uo i s t h e a m p l i t u d e o f t h e weakly bound atom and A. t h e a m p l i t u - d e of t h e normal v i b r a t i o n a t f r e q u e n c y wo. I t i s t o be n o t i c e d t h a t a v e r y l a r g e v i b r a t i o n a m p l i t u d e w i l l o c c u r i f f ' << f . The disadvan- t a g e o f t h i s approach i s t h a t t h e d e f e c t c o n s i d e r e d i s a p o i n t d e f e c t whereas we a r e more i n t e r e s t e d i n s o f t zones where many atoms a r e i n - v o l v e d . I n a o n e d i m e n s i o n a l c r y s t a l it i s , however, p o s s i b l e t o c a l - c u l a t e t h e e f f e c t o f a s o f t zone having a n e l a s t i c c o n s t a n t C Z i n a c r y s t a l o f e l a s t i c c o n s t a n t C O

-

( 5 ) ( 6 ) (appendix 3 ) . I n o r d e r f o r a v i b r a t i o n mode t o e x i s t i n such a zone s e v e r a l l i m i t i n g c o n d i t i o n s must b e s a t i s f i e d i n r e l a t i o n t o normal modes o f t h e same f r e q u e n c y i n t h e s u r r o u n d i n g p e r f e c t c r y s t a l . I t f o l l o w s t h a t :

where AZ i s t h e a m p l i t u d e of t h e v i b r a t i o n i n t h e s o f t zone

A. i s t h e a m p l i t u d e o f t h e v i b r a t i o n of same f r e q u e n c y i n t h e p e r f e c t c r y s t a l . The v i b r a t i o n energy d e n s i t y i s enhanced a f a c t o r Co/CZ i n t h e s o f t zone. These r e s u l t s a r e q u i t e analogous t o t h o s e c o n c e r n i n g t h e i s o l a t e d atom (11): t h e v i b r a t i o n a m p l i t u d e i s enhanced by a f a c t o r (Co/Cz)1/2 and a l a r g e energy t r a n s f e r o c c u r s between t h e v i b r a t i o n s o f t h e p e r f e c t l a t t i c e and t h o s e of t h e modified l a t - t i c e . The r e s t o r i n g f o r c e C Z b e i n g v e r y s m a l l , it a p p e a r s t h a t t h e v i b r a t i o n s w i l l become v e r y l a r g e and c o r r e s p o n d t o v e r y l a r g e energy t r a n s f e r towards t h e s o f t zone. Of c o u r s e , s i n c e t h e s o f t zone h a s a v e r y s m a l l volume compared t o t h e volume of t h e p e r f e c t c r y s t a l t h i s t r a n s f e r w i l l have l i t t l e e f f e c t on t h e normal v i b r a t i o n s o f t h e l a t - t i c e f a r from t h e s o f t zone. These embryonic c a l c u l a t i o n s show t h a t it i s l i k e l y t h a t t h e s o f t zones e x i s t i n g i n t h e 6 phase a l l o y s i n d u c e v e r y l a r g e a m p l i t u d e v i b r a t i o n s whose e n e r g y d e n s i t y i s i n c r e a s e d r e - l a t i v e t o t h e p e r f e c t l a t t i c e .

The r o l e o f s o f t zones i n t h e n u c l e a t i o n o f m a r t e n s i t e . A q u a l i - t a t i v e d e s c r i p t i o n o f t h e m a r t e n s i t e n u c l e a t i o n i n t h e s o f t zones h a s been p r o v i d e d by Clapp (11) : The f o r c e c o n s t a n t s which have s o f t e n e d i n t h e s t r a i n e d r e g i o n s o f t h e l a t t i c e a r e supposed t o be s t r o n g l y t e m p e r a t u r e s e n s i t i v e n e a r t h e t r a n s f o r m a t i o n t e m p e r a t u r e and w i l l ge- n e r a t e q u a s i l o c a l i z e d low f r e q u e n c y modes t h a t w i l l c o u p l e d i r e c t l y w i t h b u l k l a t t i c e modes. I n c r e a s i n g l y l a r g e a m p l i t u d e s o f t h e s e v i - b r a t i o n a l modes w i l l o c c u r , and t h e s p a t i a l e n v e l o p e of t h e r e s o n a n t mode w i l l expand c o n t i n u a l l y , a s t h e r e s o n a n t f r e q u e n c y d e c r e a s e s i n approaching t h e t r a n s f o r m a t i o n t e m p e r a t u r e . A t some p o i n t i n t h e s p a t i a l expansion t h e l a t t i c e w i l l become s u f f i c i e n t l y u n s t a b l e w i t h r e s p e c t t o t h e i n c r e a s i n g s t a t i c and dynamic s t r a i n of a l o c a l i z e d r e g i o n such t h a t a t r a n s f o r m e d zone w i l l d e v e l o p and expand outwards r a p i d l y t h u s forming a n u c l e u s . T h i s i n t e r p r e t a t i o n i m p l i e s r a p i d changes i n e l a s t i c p r o p e r t i e s above Ms which d o e s n o t a p p e a r t o be t h e c a s e ( 2 ) .

Thus a n a l t e r n a t i v e e x p l a n a t i o n o f t h e r o l e o f s o f t zones i n c l a s s i c a l n u c l e a t i o n i s hereby proposed ( 1 3 ) : I n t h e s o f t zones t h e

(10)

weak e l a s t i c c o n s t a n t i s t h e one which c o r r e s p o n d s t o t h e l a t t i c e s t r a i n . A s a c o n s e q u e n c e r t h e s t r a i n energy term d u e t o t h e m a r t e n s i t e f o r m a t i o n i n such a zone w i l l be z e r o o r v e r y s m a l l : t h e o n l y r e s i s - t i v e term w i l l be t h e i n t e r f a c i a l term. A c r i t i c a l n u c l e u s much s m a l l e r t h a n f o r t h e p e r f e c t l a t t i c e w i l l t h e r e f o r e r e s u l t (reduced c r i t i c a l s i z e ) . I f t h e s o f t zone i s s m a l l e r t h a n t h e reduced c r i t i c a l s i z e e v e r y a t t e m p t t o n u c l e a t e w i l l f a i l . I f on t h e c o n t r a r y , t h e s o f t zone i s l a r g e r t h a n t h e reduced c r i t i c a l s i z e , n u c l e a t i o n t a k e s p l a c e . The a t t e m p t s t o form a n u c l e u s a r e p r o v i d e d by t h e h i g h ampli- t u d e l o n g wave v i b r a t i o n s l o c a l i z e d i n t h e s o f t r e g i o n . Indeed, t h e maximum s h e a r s t r a i n a t t a i n e d d u r i n g t h e v i b r a t i o n p e r i o d c a n b e con- s i d e r e d a s a c o o p e r a t i v e a t t e m p t t o n u c l e a t e . The l a t t i c e s t r a i n i s p r o g r e s s i v e l y produced o v e r a l a r g e volume c o r r e s p o n d i n g t o t h e long wave l e n g t h v i b r a t i o n . Consequently, a g r a d u a l t r a n s i t i o n form un- t r a n s f o r m e d t o t r a n s f o r m e d r e g i o n t a k e s p l a c e . The i n t e r f a c e i s f i r s t d i f f u s e , t h e n becomes s h a r p e r w i t h s t r a i n a m p l i t u d e ( f i g u r e 6 ) . T h i s s i t u a t i o n i s favoured by t h e f a c t t h a t d u e t o t h e low v a l u e of t h e e l a s t i c c o n s t a n t , t h e energy of i n t e r m e d i a t e s t a t e s o f t h e s t r a i n must b e s m a l l . I t might be o b j e c t e d t h a t t h e s h e a r s t r a i n s i n v o l v e d i n t h i s p r o c e s s a r e n o t e x a c t l y homogeneous s h e a r s t r a i n s , w i t h t h e wave l e n g t h b e i n g o n l y s e v e r a l i n t e r a t o m i c d i s t a n c e s . However, s i n c e t h e phonon branch c o r r e s p o n d i n g t o C ' d e c r e a s e s i n energy w i t h d e c r e a s i n g t e m p e r a t u r e s i m i l a r t o C ' (14) ( 1 5 ) , it i s l i k e l y t h a t t h i s phonon branch w i l l a l s o behave l i k e C ' a s a f u n c t i o n of t h e s t r a i n . I t i s t h u s r e a s o n a b l e t o suppose t h a t i n t h e B phase a l l o y s t h e l o n g wave s o f t e n i n g ( s o f t e n i n g of C ' ) d u e t o t h e d e f e c t s t r a i n e x t e n d s t o s m a l l e r wave l e n g t h s .

The b e h a v i o u r of h i g h a m p l i t u d e s h e a r waves d e r i v e d by computer s i m u l a t i o n i s provided i n a r t i c l e s by Suzuki and Wuttig (16) ( 1 7 ) . These a u t h o r s suppose t h a t s h e a r o s c i l l a t i o n s a r e p r e s e n t which a r e governed by two p o t e n t i a l s . One i s an i n t e r a c t i o n p o t e n t i a l c o n s t i t u - t e d by two minima, one f o r E = 0 ( p a r e n t p h a s e ) and t h e o t h e r f o r

E = 6 ( m a r t e n s i t e ) . The o t h e r p o t e n t i a l depends on t h e s t r a i n g r a - d i e n t which i n f a c t c o r r e s p o n d s t o an i n t e r f a c i a l e n e r g y term. The s i m u l a t i o n l e a d s t o embryo f o r m a t i o n i f t h e s t r a i n a m p l i t u d e i s l a r - g e r t h a n a c r i t i c a l a m p l i t u d e . I t must, however, be remarked t h a t t h i s c r i t i c a l a m p l i t u d e i s v e r y l a r g e and i n f a c t o f t h e same o r d e r o f magnitude a s 6 . A d e c r e a s e i n t h i s c r i t i c a l v a l u e i s observed i f t h e

s t r a i n g r a d i e n t term i s n e g a t i v e , b u t t h i s h y p o t h e s i s i s n o t p h y s i - c a l l y r e a l i s t i c s i n c e it c o r r e s p o n d s t o a n e g a t i v e i n t e r f a c e e n e r g y .

Conclusion. G e n e r a l l y i n a l l o y s undergoing t h e r m o e l a s t i c marten- s i t i c t r a n s f o r m a t i o n s , a v e r y weak e l a s t i c c o n s t a n t i s p r e s e n t which i s r e l a t e d t o t h e t r a n s f o r m a t i o n s t r a i n p r o c e s s . S o f t e n i n g of t h i s c o n s t a n t i n t h e b u l k a l o n e c a n n o t e x p l a i n t h e m a r t e n s i t e n u c l e a t i o n . However, i n t h e v i c i n i t y of t h e l a t t i c e d e f e c t s , due t o t h e anharmo- n i c i t y , r e g i o n s a r e p r e s e n t where t h i s e l a s t i c c o n s t a n t i s s t r o n g l y lowered which g e n e r a t e s v e r y l a r g e v i b r a t i o n a m p l i t u d e s . The conse- quence o f a l l t h e s e f a c t o r s on t h e e x a c t mechanism o f n u c l e a t i o n i s n o t e a s y t o d e r i v e . I t i s proposed h e r e t h a t n u c l e a t i o n o c c u r s when t h e reduced c r i t i c a l n u c l e a t i o n s i z e , which d e c r e a s e s w i t h d e c r e a s i n g t e m p e r a t u r e s , i s s m a l l e r t h a n t h e u n s t a b l e zone. The reduced c r i t i c a l s i z e i s o b t a i n e d from a c l a s s i c a l n u c l e a t i o n scheme w i t h a chemical d r i v i n g f o r c e i n c r e a s i n g w i t h d e c r e a s i n g t e m p e r a t u r e and a n i n t e r f a c e r e s i s t i v e term. The s t r a i n e n e r g y term i s supposed t o be z e r o o r v e r y s m a l l due t o t h e i n s t a b i l i t y . The a t t e m p t s t o form t h e n u c l e u s a r e p r o v i d e d by t h e l o n g wave h i g h a m p l i t u d e v i b r a t i o n s p r e s e n t i n t h e u n s t a b l e zones.

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Figure 1- Schematic of the beha- Figure 2- Schematic of the beha- Figure 3- schematic of the behaviour near a second order homo- viour near a weakly first order ,iour near a first order classi- geneous strain phase transition homogeneous strain phase transi- cal martensitic transition. tion. P : lattice parameters (example of the cubic to tetragonal transition) Cii: elastic constants relates to the homogeneous strain of the transition G : free energy ; HTP : high temperature phase; LTP : low temperature phase Mf-Af : temperature domain of the two phases.

(12)

(13)

Figure

5-

Iso-strain curves

z13

and

E~~

corresponding to a screw dislocation lying on

x 3 //

Loll] .

^xl

is parallel to

~ I O O ]

and

x2

is parallel t

l .

The curves are normalized by the factor

, where c is the strain amplitude.

Figure

6-

Schematic nucleation process a) Classical

:

the nucleus with the whole martensite strain extends in the radial (r) and thickness

(e)

directions. b) Non classical model proposed

:

the strain gradual-

ly takes the amplitude of the martensite strain in the whole soft

region during half a vibration period. The interface, therefore,

appears gradually at the boundary of the soft region.

(14)

Appendix 1

A good i d e a o f t h e atomic d i s p l a c e m e n t s i n t h e B+9R t r a n s f o r m a - t i o n i s provided by a hard s p h e r e model proposed by A h l e r s (21) and used by Olson and Cohen ( 2 2 ) . I n t h i s model a s h e a r on ( O 1 l ) g p l a n e s i n

@ T i )

d i r e c t i o n t r a n s f o r m s t h e ( i 0 1 ) p l a n e s i n t o f .c.c. t y p e c l o s e packed p l a n e . T h i s s h e a r i s a s s o c i a t e d ' w i t h a c o n t r a c t i o n i n t h e

@1q6

d i r e c t i o n d u e t o t h e hard s p h e r e s . The s t a c k i n g of t h e c l o s e packed p l a n e s d o e s n o t s t i l l c o r r e s p o n d t o t h e 9 R s e - u e n c e . A h l e r s h a s shown t h a t two s m a l l d i s p l a c e m e n t s c l o s e t o ~ 1 0 1 7 ~ d i r e c t i o n on two c o n s e c u t i v e c l o s e packed p l a n e s f o l l o w e d by a d i s p l a c e m e n t i n t h e o p p o s i t e d i r e c t i o n on t h e t h i r d p l a n e l e a d s t o t h e 9 R s t r u c t u r e . I t

i s n o t i c e a b l e t h a t t h e n e t d i s p l a c e m e n t a f t e r t h r e e p l a n e s i s z e r o . These d i s p l a c e m e n t s c a n t h e r e f o r e be c o n s i d e r e d a s s h u f f l e s . The hard

s p h e r e model a l s o l e a d s t o d i s p l a c e m e n t s t h a t a r e a s s o c i a t e d w i t h a c o n t r a c t i o n p e r p e n d i c u l a r t o t h e c l o s e packed p l a n e s . The mecanism i s t h e r e f o r e completed by a n i s o t r o p i c d i l a t a t i o n t o compensate t h e con- t r a c t i o n s .

Following a somewhat d i f f e r e n t approach, o n e of t h e p r e s e n t a u t h o r s ( 6 ) h a s a r r i v e d a t a mechanism c l o s e t o t h e o n e d e s c r i b e d above : t h e b.c.c. t o 9R t r a n s f o r m a t i o n r e d u c e s t o a homogeneous l a t - t i c e s h e a r on t h e (Oil), p l a n e i n t h e r0111, d i r e c t i o n , f o l l o w e d by a - _-_IJ s h u f f l i n g on t h e ( i 0 l ) g i n t h e

[lollB

d i r e c t i o n w i t h a wave l e n g t h

X

= 3d where d i s t h e s p a c i n g o f t h e c l o s e packed p l a n e s o f b . c . c . . Small second o r d e r d i s p l a c e m e n t s have been n e g l e c t e d i n t h i s d e s c r i p t i o n . T h i s g e o m e t r i c a l d e s c r i p t i o n i s i n t e r e s t i n g from a p h y s i c a l p o i n t o f view a s it g i v e s t h e p r o p e r o r i e n t a t i o n r e l a t i o n - s h i p s :

The mechanism, moreover, shows t h a t t h e homogeneous l a t t i c e s t r a i n i s a l m o s t ( a s second o r d e r d i s p l a c e m e n t s have been n e g l e c t e d ) a (011)

@ill

s h e a r ; t h i s c o r r e s p o n d s t o t h e s h e a r e l a s t i c c o n s t a n t , C ' , which i s of i n t e r e s t t o t h e p r e s e n t r e p o s t . The model a l s o p r e - d i c t s a h a b i t p l a n e c l o s e t o ( 0 1 1 ) 8 p l a n e o f s h e a r and t h e b a s a l p l a n e o f m a r t e n s i t e ( i 0 1 ) ( p l a n e of s h u f f l i n g ) i s a p p r o x i m a t e l y 60 d e g r e e s from t h e p l a n e of s h e a r i n a c c o r d a n c e w i t h t h e o b s e r v e d r e s u l t s ( 2 8 ) .

For t h e 6 t o 2H t r a n s f o r m a t i o n , t h e second o p e r a t i o n which g i v e s t h e c o r r e c t s t a c k i n g i n v o l v e s a homogeneous s h e a r (701) [101] B p l u s a s h u f f l i n g on t h e same system.

O t h e r models have been proposed, i n p a r t i c u l a r t h e B u r g e r s model (23) which assumes t h a t t h e c o n v e r s i o n o f ( 1 1 0 ) , p l a n e s i n t o c l o s e packed o n e s i s a c h i e v e d by homogeneous (112) [ l ? ~ ] s h e a r . T h i s s h e a r i s r e l a t e d t o t h e s o c a l l e d s p e c i a l mode ( 2 9 ) . I n f a c t , such a model g i v e s much d i s t o r d e d " c l o s e packed p l a n e s " when compared t o t h e model d e s c r i b e d e a r l i e r . Moreover, i n t h e c a s e o f 8+9R t r a n s f o r m a t i o n t h e second inhomogeneous s h e a r s on t h e (110) [ l l ~ ] ~ system i n c l u d e shuf- f l i n g p l u s homogeneous s t r a i n . T h e r e f o r e , t h e t o t a l t r a n s f o r m a t i o n

(15)

JOURNAL DE PHYSIQUE

i n v o l v e s homogeneous s h e a r s on (172)

[lii]

and (110) I l i 0 1 systems p l u s s h u f f l i n g on (110)

[ l i ~ ] .

T h i s s i t u a t i o n seems t o b e less p r o b a b l e t h a n t h e above o n e , e s p e c i a l l y i f we c o n s i d e r t h a t t h e e l a s t i c c o n s t a n t C 1 , o f t h e former model i s 2.5 t i m e s lower t h a n t h e Cs c o r r e s p o n d i n g t o t h e f i r s t s h e a r o f t h e B u r g e r s model.

I n c o n c l u s i p n t h e ( 1 1 0 ) < 1 i 0 > t y p e s h e a r c o r r e s p o n d i n g t o t h e C ' e l a s t i c c o n s t a n t i s o f s p e c i a l i m p o r t a n c e f o r t h e B+9R o r 6-2H marten- s i t i c t r a n s f o r m a t i o n s .

Appendix 2

E q u i v a l e n c e of two { 110}<1i0> t y p e s h e a r s t r a i n s w i t h a c o n s t a n t volume Bain t y p e s t r a i n when t h e s t r a i n i s s m a l l :

Assume a f i r s t s h e a r on (110) p l a n e i n t h e 0 d i r e c t i o n . The d i s p l a ~ e m e n t u ~ i s p r o p o r t i o n a l t o t h e d i s t a n c e from t h e o r i g i n .

d =

o%.o?

where

03

i s t h e u n i t v e c t o r normal t o t h e (110) p l a n e

where k i s t h e a m p l i t u d e o f t h e s h e a r s t r a i n

With a second s h e a r o f t h e same amount on (101) p l a n e i n t h e [lo?] d i r e c t i o n

The t o t a l d i s p l a c e m e n t i s g i v e n by :

(16)

The s t r a i n s a r e g i v e n by

The s t r a i n s e i j i n v o l v e p u r e s t r a i n s c i j p l u s r o t a t i o n s

T h e r e f o r e t h e two ( 1 1 0 ) < 1 i 0 > s h e a r s t r a i n s a r e e q u i v a l e n t t o a con- s t a n t volume t e t r a g o n a l s t r a i n p l u s a r i g i d body r o t a t i o n . The Bain t y p e s t r a i n i s v e r y c l o s e t o such a t e t r a g o n a l s t r a i n , t h e r e f o r e a s i n g l e { 1 1 0 ) < l i 0 > s h e a r s t r a i n c a n b e c o n s i d e r e d a s " h a l f " a Bain t y p e s t r a i n . I n t h i s c a l c u l a t i o n , t h e second o r d e r t e r m s have been n e g l e c t e d . I t i s t h e r e f o r e l i m i t e d t o s m a l l a m p l i t u d e s t r a i n s . However a complete c a l c u l a t i o n ( 2 4 ) shows t h a t even f o r t h e whole Bain s t r a i n ( k 2 0.24) t h e a p p r o x i m a t i o n i s q u i t e good.

Appendix 3

C o n s i d e r a f i r s t atom c h a i n w i t h (No+2) atoms, t h e ends b e i n g f i x e d ; t h e atoms i n t e r a c t o n l y w i t h n e a r e s t n e i g h b o r s . The frequen- c i e s o f t h e normal modes of t h i s c h a i n a r e g i v e n by (25) :

s i s mode number and wo

-

2(ko/m) where ko i s t h e f o r c e c o n s t a n t and m i s t h e atomic m a s s .

The maximum v i b r a t i o n a m p l i t u d e o f t h e atoms i s :

- r sII

A,, - As s i n

-

_No

₊

₁

r : 0 + N o

+

^1,r i s t h e number o f t h e atoms i n t h e c h a i n .

A second i d e n t i c a l c h a i n w i t h ( N Z

+

2) atoms c a n now b e c o n s i d e - r e d whose ends a r e a l s o f i x e d , t h e o n l y d i f f e r e n c e b e i n g t h e f o r c e c o n s t a n t k Z . Supposing t h e second c h a i n h a s a normal mode ( u ) whose f r e q u e n c y i s i d e n t i c a l t o t h e o n e o f a normal mode (s) o f t h e f i r s t c h a i n :

do s i n

s n

- url

2(No

+

1) - 'Z Sin 2 ( N Z

+

1)

(17)

The two chains may now be joinded by coincidence of the atom No + 1 of the first chain with the atom

0

of the second one. The vibration modes will not be affected (the mode will become a common mode) if the common atom is fixed: the net force on this atom must be

zero whatever is the time t. This condition gives

:

ko As sin - sn ^un

^{- 0}

No

+

+ kZ BU sin - _NZ ₊ ₁

^-

₍₁₅₎

BU is the amplitude of the mode

u

in the second chain.

When s and u are much smaller than No + ^{1 and NZ} +

1

which is the long wave approximation, Eqs (14) and (15) give

:

N o + 1

U

ko 1/2

- ^x ^-

⁼ ^(-)

and - Bu - -- - ko

^X

- NZ

⁺

s NZ +

1

s kZ As kZ N o + l X i i Consequently

:

Evaluation of the energy

:

The energy of the first chain is given by

:

The energy of the second chain is

:

Ez

= 1

m w2

8:

N 1) Considering mean energy per atom which is proportional to the energy density

:

-

1 2 2 .

^F:

^{= - m u} 1 2 2

Eoa

-

- m

w

Eza -

^B^: -

ko

2

As za 2 BU and - -

- - -

Eoa As kz

The elastic constant of each chain is proportional to the force

constant k. Hence, in conclusion, the amplitude of common vibration

mode is enhanced by a factor (Co/CZ)1/2 in the soft region and the

energy density by a factor Co/CZ. This result can be obtained by

other methods, in particular by the usual reflection and transmission

conditions for ultrasonic waves at an interface between two solids of

different elastic constants.

(18)

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-

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