Soft modes and structural phase transitions - Indian Academy of

Bull. Mater. Sci., Vol. 1, Numbers 3 and 4, December 1979, pp. 129-170

(~ Printed in India.

Soft modes and structural phase transitions* G VENKATARAMAN Keactor Research Centre, Kalpakkam 603 102 MS received 19 June 1978 Abstract. This paper presents a survey of soft modes and their relationship to structural phase transitions. After imrodncing the concept of a soft mode, tk.e origin of softening is considered from a lattice-dynamical point. The Landau theory approac& to structural transitions is then discussed, followed by a generalisation of the soft-mgde cortcept through the use of the dynamic order-parameter susceptibility. The relationship of soft m,sdes to broken symmetry is also examined. Experimental results for several classes of crystals arc next presented, bringing out various features such as the co-operative lalm-Tellev effect. The survey concludes with a discussion of the central peak, touching upon both the experimental results and the theoretical speculations. !.

Introduction

I a m grateful for this opportunity to discuss soft m o d e s and structural phase transitions. Although I have never worked in this area, I used to keep abreast o f this subject in view o f m y interest in lattice dynamics, but lately i began to lose touch. The present assignment has enabled me to pick up the threads again and become aware o f the excitement sweeping the field. Historically speaking, R a m a n and Nedungadi (1940) seem to have been the first to observe a soft mode in a strt~ctural phase transition. This was not generally well-known in the past, and I became aware o f it when Prof. Chandrasekhar mentioned it at the Academy meeting in the year following Prof. Raman's death. It is gratifying that this fact has now got into a b o o k (Blinc and Zeks 1974), ensuring better recognition to Raman's pioneering contribution. The next important event in the history o f soft modes is undoubtedly the prediction made independently by Cochran (1959) and Anderson (1960) that phase transitions in certain ferroelectrics might result from lattice dynamical instability. Great excitement was aroused when soon after this, Cowley (1962) discovered in his neutron scattering experiments on SrTiOs that one o f the q = 0 optic modes showed a softening behaviour as the temperature was decreased. Unfortunately, there was no concomitant (ferroelectric) phase transition b u t a soft mode and a related phase transition were discovered in ~rTiOa a little later. The subject o f soft modes soon became wide open, a variety o f experimental techniques like N M R , E R R , light scattering, * Survey presented at the discussion meeting on phase transitions arranged by the Indian Academy of Sciences at Bangalore in June 1978. 129

130

G V enkatar aman

neutron scattering, etc. being applied to a study of the problem. In 1971, fresh excitement was aroused through the discovery of a central peak accompanying the soft mode, leading to renewed vigour on both the experimental and theoretical fronts. In view of all this, it is appropriate that we spend some time at this meeting to catch up with what has transpired in this field.

2.

General aspects of soft modes

Let us begin with the question, " w h a t is a soft mode9. ". Operationally, one may define the soft mode as a collective excitation whose frequency decreases anomaloously as the transition point is reached. On account of its historical association, the term soft mode always brings to mind a lattice dynamical mode but, as we shall see later, one could have other types o f soft modes as well. Further, softening could occur not only as one approaches the transition temperature from above but also from below. In what follows, we shall by and large restrict attention to lattice dynamical soft modes. Such modes trigger a lattice instability, leading to a structural phase transition either of a second or first order. Table 1 offers a brief classification of the structural changes that could enstte. In some cases, such changes are accompanied by the onset of either ferroelectricity or antiferroelectricity. I shall not be overly concerned with the ferroelectric aspects of these phase transitions since Prof. Srinivasan will be covering them in his presentation. Figure 1 shows a schematic of the different kinds of soft modes that have been observed so far. Of interest is the result for N'bsSn where a sizable chunk of a whole branch goes soft. Experience todate regarding the spectral character of the soft mode is summarised in figure 2 from which one finds that the mode can be either underdamped or overdamped (depending on the anharmonicity prevailing). Figure 3 gives a schematic view of the temperature dependence of the soft mode frequency. It is to be noted that whereas in a second-order transition, the soft Table 1. Classification of structural phase transitions with typical examples. im

Structural transitions ii

i

i

''' I Antiferrodistortire (no. of formula units in unit ceil changes)

Ferrodistortire (no change in no. of formula units in unit cells) I

1

displacive (A) (BaTiOa)

I

t

I

i

displacive (C) (SrTiOO

Order-disorder(B) O,U-h¢l) r

i

Z

Order-disorder (D) (NH~r) i

|

A. reference to Blinc and Zeks (1974) will show that in categoryA, only feiTod~trics seem to occur. In category C, both ferroelectriesand antiferroelectricsare poss~le 2. Birgeneau et al (1974)mention that there is a third group of transitions in which tb.ere is a strong coupling betweenorder-disorder and displacive types of variables Examples cited are KDP and PrAIO~.

Note : 1.

Soft modes and structural phase transitions

131

SrTi0 3

Bo Ti 0 3

TI

>T2 >I"3

t..0

q

q

Nb3Sn

K2SeO4 OJ

0J

Y2 q

q

Figure 1. Schematic drawings of the various kinds of soft modes observed so far. Notice the softening can cccur anywhere in the Britlouin zone. Examples of materials exhibiting such behaviour are also given.

(a)

W

(b)

Frequency Figure 2. Typical lineshapes of soft modes. (a) corresponds to underdamped, and (b) to overdamped modes.

132

G Venkataramat~ (o)

(1)

(3)/

0

~g

{b) H)

I I I I I

Tc Temp

3. Variation of soft mode frequency with temperature for (a) seoand-ord~r and (b) first-order transitions. The numbers m parenthesis denote degeneracy.

mode frequency actually vanishes at the transition point, in a first-order transition, the change of phase occurs before the mode frequency is able to go to zero. Attention is drawn to the fact that soft modes exist also below To. This is not unexpected because if the structural transition is due to the condensation of a soft mode that exists above Te then correspondingly there must be agencies below T, which seek to restore the symmetry of the high-temperature phase as temperature is increased from below To. Thus it is that as many soft modes appear below To as exist above T,. We next consider the order parameter associated with structural phase transitions, and their relation to the soft modes. The concept of the order parameter will be formally introduced later but for the moment let us discuss some qualitative aspects without being too fussy about the definition. The order parameter, as all of us know, is a measure of the order resulting from the phase transition. We shall denote it by the symbol g. In all structural phase transitions, the average value of g denoted by ( g ) is non-vanishing below To and vanishes above 2",. The change is continuous across the transition temperature for a so:end-order transition, and is discontinuous for a first-order transition (figure 4). It is important to realise that g itself is not a static quantity even though ( g ) is. Thus g can fluctuate, and in fact these fluctuations are derived by a suitable superposition of the atomic displacements associated with the soft mode vibrations. The fluctuations are so organised that ( g ) vanishes identically in the high-tempera-


133 (o)

E o o

(b) c~

Tc Zemp

4. Variation of the order parameter with temperature for (a) second-order and (b) first-order transitions.

ture phase but becomes non-vanishing in the low-temperature phase. The fluctuations become important near T0 and diminish as one moves away from T~ on either side. Correspondingly, the soft modes also become stiffer. 3.

Softening from a lattice dynamics point

Having noted the existence of soft modes, the next question obviovs~y is, "why do modes become soft ?'" This can be tackled in a variety o f ways and we shall start with the lattice dynamists' approach, Many years ago~ Born (see Born and Huang 1954) established that a crystal lattice will become unstable if one of its normal-mode frequencies becomes purely imaginary. Let us pursue this line of argument. Now the lattice dynamical Hamiltonian is usually written in the form (Venkataraman et al 1975)

where the Q's denote the normal coordinates and the P's the conjugate momenta. The Hamiltonian of equation (1) is purely harmonic and since anharmonic effects play a crucial role in the softening process, the above Hamiltonian obviously needs

t34

G Venkatararaan

to be enlarged. Assuming for simplicity only fourth-order anharmonic effects (a common practice), the above Hamiltonian may be expanded to an effective Hamiltonian of the form

ql

"'~ J"~"' (2) In the above, the usual fourth-order term has been replaced by one of the form ( QQ' ) Q" Q"' so as to define an effective quadratic potential for the phonons. It is possible in principle to diagonalise (2) to the form

H.n -- ~ ~

:-(q

E°2

(q

,

qJ

where & denotes the renormalised mode frequency, and is related to the bare frequency too by & ~ ( q ) __.,~zo(q)q_2 ~

"~-' ,"--t¢',,~ ~, V~,m(q, K) ~ Q ( / ) Q ~, m .)/"

(4)

Let (q,j) denote the soft mode, and we further suppose that the forces in the crystal are such as to make ~ (q/] negative. However, from experiments we know that ~2 for the soft mode must be positive. This is obviously possible only if the anharmonic effects represented by the second term on the r.h.s, of (4) overwhelm the negative contribution from coo ~. It now becomes conceivable that as temperature is lowered, the anharmonic contribution may not be able to counter the effects of tog sufficiently. In turn this would lead to ~ becoming negative, producing in its wake a crystal instability, as required by Born's theorem. Nature of course averts this disaster by bestowing a different structure, better conducive to lattice stability. An estimate for the transition temperature may be obtained from equation (4) (Blinc and Zeks 1974). On evaluating the ( . . . . ) term ontl-e r.h.s., one obtains

[1 (/)]( f l = k ~1'

× c o t h 2fl~

~=1

).

(5)

Since the renormalised frequency occurs on both sides, one has obviously a selfconsistency problem. An approximate solution could however be obtained by replacing ~ by coo on the r.h.s, leading to the result t¢ -i 1 x ~ ( ~ ) ~ too2(q.)+2 ~ ' V~,,(q, x) ~2og0(/ ) ] coth [ ~ f l t o 0 ( , ) ] . ltc

(6)


135

Notice the prime on the summation on r.h.s. This implies the exclusion of soft modes which is necessary as their bare frequencies are purely imaginary. Assuming now that o9~ < k~T, we can recast the above as

where

~ (soft mode) --~ a (T -- T.),

(7)

Z V.~sil"/

(8)

a =k B

coo ~ ,

lt¢

and

To =

[coo ~ (soft mode)/a],

-

(9)

denotes the transition temperature. It is worth emphasising that by their very nature, the soft modes are expected to make substantial contributions to anharmonic effects which, however, have been carefully excluded in (6) ! The above exercise is therefore not good for quantitative purposes although it serves well as a plausibility argument. Methods are of course available to accommodate the aaharmonic effects contributed by the soft modes but we shall not discuss them here. 4.

Landau's

theory

There is another way of looking at structural phase transitions, and that is via Landau's theory. Landau and Lifshitz (1959) discuss this at some length in their classic book on statistical physics--and many of you may already be familiar with it. So I shall content myself with presenting a bare outline, adequate for the present purpose. In a second-order phase transition, the change of state is continuous (i.e. order parameter changes continuously). Taking note of this, Landau assumed that the Gibb's free energy g (T, P, ~/) in the neighbourhood of the transition point should be expandable as a series in powers of a certain quantity g, the order parameter. To arrive at the nature of this expansion, Landau and Lifshitz consider the density p (x, y, z) which describes the probability distribution of atom positions in the crystal, p then must evidently reflect the symmetry group of the crystal which means that for T > T , p must be consistent with the symmetry group Go of the high-temperature phase. Likewise, for T < To, p must be consistent with the group Go of the low-temperature phase. This enables one to write P = Po + ~P,

(10)

where pc corresponds to the symmetry of Go a]ad p to the symmetry of G. It is clear that ~p has the same symmetry as p. Using group-theoretic arguments, Landau and Lifshitz then show that $p may be written =

2: i

c # , x (rx),

(ll)

where ~b~x are basis functions which transform according to the irreducible representation F x of Go. The index i runs over the set of such functions. Now the Gibb's free energy g will not only be a function o f temperature and pressure but also of p and hence of the coefficients C,. Introducing the notation •z :

~, CL C~ = 7,~k (so that Z 7[ = 1), f

f

(12)

136

G Venkataraman

the free energy expansion upto fourth order is written in the form* g = go + A~v ~ + V 4 Z¢g V,~f~ (7,).

(13)

Hero V is the order parameter informally introdt:ced earlier and now defined by (12). The quantity f ~ is an invariant of fourth order constructed from 7~. Such invariants occur when g is expanded in terms of the C,'s because g itself is invariant under the symmetry group of the crystal. The sum over a in (13) contains as many terms as there are individual invariants of corresponding order. The stable state of the crystal is found by minimising g with respect to ~ and h. The stability conditions are: ag/a~, = o, a * g/a~, ~ > o.

04)

Nothing has been said so far about the sign of the coefficients of the ~2 and terms. Landau and Lifshitz show that the coefficient of ~4 must always be positive; also it is not strongly temperature-dependent. A on the other hand varies rapidly with temperature and can have any sign. The state (~) = 0 (no order) is stable for A > 0 whereas when A < 0, (~) is non-vanishing**. Transition from high symmetry Go to low symmetry G accompanied by the appearance of an order parameter (~) thus occurs when A changes sign. Figure 5 shows a sketch o f g in varim:s situations. The appearance of the double well explains why (V) becomes non-vanishing below To. Several footnotes mr:st now be added to the above disct:ssion. Firstly, in the context of soft modes, the basis functions entering the expansion (11) will be eigenvectors of the soft mode of the high symmetry phase. If the wave-vector of the soft mode is q, then i runs over all the partners corresponding to the irreducible representation F~ (q) of the wave-vector group Go (q) and over similar sets corresponding to all other wave-vectors belonging to the star of q. T~

4O

A1" " E

E

¢ " 20 U

o,&

A__A

•

• m

-

AI'='A 2

O I 0

r

I

I

I

0.1

0.2

0.3

I

[kOk] Reduced wovevector (k)

Figure 22. Dispersion relations along [kok] for some of the low frequency excitons in PrA10= at 77 K. See also figure 21 (After Birgeneau et al 1974),

! 52

G Venkataraman

Pr At03, ~.3(k,O,k)

3°I

0 151K11) • 161K(2) z~ 170K(3) A 20OK[4 )

:~2.O

O~spersenc~ y 210K

e'

/ /

Ii, t'-

'/

"d~o ~ 0

[

0.04 Reducedwovevector(k) 0-02

Figm'e 23. [101l acoustic phonon of PrAIO, which couples to the Ax -+ B1 exeiton. Observe the softening of the acoustic branch similar to what happens in Nba~n (After Birgencau et al 1974).

7" 6.

Tb~ (MoO~)~

[ shall now move away from perovskites and look at another substance, namely Tb~(MoO4)3 which also has unusual properties. At 159°C it undergoes a tetragonal to orthorhombic transition, in the low temperature phase, the substance is ferroelectric, and coupled with the two possible polarisation states :E P,, are the two mechanical configurations (figure 24) described by opposite shears (zE u,~). Just as the polarisation can be switched by an applied field, one mechanical configuration can be switched into another by an applied stress (ferro-elasticity). Further, both ferroelectricity and ferroelasticity are so coupled that P, and u , change simultaneously. Neutron scattering experiments by Dorner et al (1972) revealed that the phonon triggering the transition was at the zone boundary as may be seen from figure 25. Now a soft mode at the zone boundary usually results in an antiferroelectric structure [e.g. as in (NDa)DzPO4] and cannot directly produce a spontaneous polarisation. However, the order parameter associated with the soft mode can couple with shear strain which in turn can produce polarisation by piezoelectric coupling. The free energy expansion must thus not only involve the order parameter ~ but also u~, and P,. The names secondary and tertiary order parameters have been suggested for the latter two. Other than those associated with the order parameter ~,, one has the additional equilibrium conditions Ogle,=, = o ;

~glaP, = o.

153

go['t modes and structural phase transitions

PE

Fff(+)

FE(-;

Figure 24. Sketch of the configuration of terbium molybdatc in the para and ferroelectric phases. The solid lines in the lower half describe the unit cells of the paraphase and the dashed lines that of the ferrophase, projected on to the x-y plane. Observe the opposite shear of the two FE configurations (After Dorner et al 1972). 10

4

0

_

ty!

200

I

_ I

400 Temp. ('~C)

600

800

Figure 25. Temperature dependence of the soft mode at the point M in terbium molybdate (After Dorner et al 1972).

By using these, the free energy expansion can be expressed in terms o f g alone, with suitably redefined expansion coeiticiems. Dorner et al (1972) have it in the form

g = ½ ,o~ ~,~-÷ ¼~4 X Ba f ~

(7,)

t~

÷ 1 / 6 ~ , z w . J ~. (~,3 ÷ o/

. . .

154

G Venkataraman

where o~r, B~ and Ira are suitably defined coefficients. Having eliminated the secondary and tertiary order parameters, the analysis can be carried out as usual. Among other things, knowing the values of the coefficients, the soft mode frequen cies in the two phases can be calculated [recall equation (15)]. Figure 26 shows a schematic plot of what can result. ~2

B2>O

w2

B2=O

~2= 2

I

(o2MO i

,

,

D

BI < Bz To, Qo = 0. In turn this causes the parameter C ( = e o n s t . ×BQ °) to vanish, producing a decoupling between x and y. Thus in the linear approximation, the central peak disappears above T,. However, Feder has shown that if a more general form for A F t h a n that in (42) is considered, i.e. one with quartic terms, then the central peak survives even above T,. Figure 39


165

SrTiO3 (figure 14). Along with fluctuations in Q, Feder considers also the fluctuations in entropy, and to deal with these he introduces the variables x = Ko (0., -- Q0), y -~ K, (£', -- S°),

(38)

where Oz is the most probable value for Qz (at a given time t) and QO is the equilibrium value. (QO denotes the order parameter and obviously, for T > To, it vanishes), g'~ and S o which refer to entropy are similarly defined and the K's are suitable constants. If A F (x, y) denotes the change in the free energy of the system in going from x = 0, y = 0 to x = x, y = y, then knowing AF, the correlation function Cn (t) for the order parameter fluctuations can be calculated as sketched below. First wc manipulate 6",,, (t) = (x (o) x (t)) into the form

where

C,, (t) = S dx' dy' po (x', y') x' (x (t))c,

(39)

po (x,y) = exp { -- f l ~ F ( x , y)},

(40)

denotes the equilibrium probability distribution for finding the system in the state (,x, y). Further, (x (t))o is the conditional average defined by

(x(t)), = S d x d y p ( x y t

[ x' y' t = 0 ) x ,

(41)

where p (x y t I x' y' t = 0) is the probability that starting from x' y' at t = 0, the system reaches the state x, y and time t = t. If AF, p and (x (t))0 are known then C,, (t) can be deduced. Feder first considers a linear model in which A F has the form

f l A F = ½ (o~ x 2 + Cxy + ½ 097,y2,

(42)

where co,, C, and o9, are appropriate parameters. are governed by the equations

The quantities (x (t))0 and (y (t)),

O O'--t(x (t)), = -- p, 09~ (x (t)}o - - / t , C(y (t)}o,

8t (y (t))o = -- la~ C (x (t)), -- & o9' (y (t)),,

(43)

where/t,,/a~ are appropriate mobilities. Under the as,~umption/to o9~ ( = ?,) & 09 ~ ( = 7u), i.e. the order parameter fluctuation relaxes much faster than entropy fluctuation, Feder shows that

C,,(op) = (

....

)o9 + 7;

?~ +

(44)

which has the same structure as sketched in figure 37 b. Both terms on the r.h.s. of (44) give peaks centred around co = 0 but the sharper one is due to the slower variable, i . e . y . As already noted, for T > To, Qo = 0. In turn this causes the parameter C ( = e o n s t . ×BQ °) to vanish, producing a decoupling between x and y. Thus in the linear approximation, the central peak disappears above T,. However, Feder has shown that if a more general form for A F t h a n that in (42) is considered, i.e. one with quartic terms, then the central peak survives even above T,. Figure 39

Soft m o d e s and strtwtural p h a s e transitions

167

Thomas (1974) too has examined the question of the central peak, and has given a detailed discussion of the situation when it arises due to thermo-distortive coupling (figure 40), i.e. coupling of the order parameter with heat fluctuations. Without coupling there are the two responses

5Q (q, co) = ;go (q, co) V~t (q, co), T S S (q, co) = p o (q, 09) (e¢~t) (q, co),

(47)

where Po~t is an external power source and p0 is the response of the thermal system. The other symbols have the same meaning as earlier. With coupling, the responses change to )~ and p defined by the equations

6Q = 7. (q, co) [ V ~ -k- (g/T) po Pox,], T S S = p (q, co) [g CQ Doq ~- Zo V~xt -k

Pearl.

(48)

In the above, CQ is the specific heat at constant order parameter and Do the bare heat diffusion coefficient and g the coupling. When solved, one obtains Doq 2 -- ico g (q, co) = go (q, co) D (q, co) q~ -- io9'

1

(49)

(50)

P (q' co) = D (q, co) q~ -- ico"

Thus the order-parameter response not only shows the soft mode character (con. tained essentially in ;(0) but also a central peak (coming from the diffasion term D (q, o~)). This is the way the Rayteigh peak arises in the scattering of light from liquids. Thermo-distortive coupling can arise only if the order parameter transforms according to identity representation of the symmetry group (since energy transforms that way). In this context it may be noted (Blinc and Zeks t974) that if the soft mode is non-degenerative above To, then below Tc it will split imo several modes one of which wilt transform according to the identity representation, making thermo-distortive coupling possible and hence also the central peak. Taking the case of SrTiO3, we see precisely this. On the other hand, the appearance of a peak

,xt

Figure 40. Block diagram skowing the thermal and lattice (soft mode) responses in the absence of and in the presence of thermo-distortive coupling.

168

G Venkataramat~

above To in SrTiOa must be visualised in terms of coupling to a variable transforming like Rz~.* " Microscopic" theories of the central peak have also been attempted (Cowley 1974 ; Cowley and Coombs I973 ; Wehner and Klein 1972; Klein 1974 ; Niklasson and Sjolander 1974). In these, one does not deal with phenomenological or macroscopic variables but rather starts from the phonon picture of the solid, and attempts to calculate the lattice dynamical response L (q, e)) defined by the relation

(v (q, co)) = L (q, a)) V~,~ (q, o~), where Volt is the external force field disturbing the lattice producing an average displacement pattern (U). The response function can be generally written as L (q, o9) = -- [097 -- co= (q) -- M (q, co)]-I,

(51)

[compare with (27)~, all the anharmonie effects being comprehensively included in the (complex) self-energy M ( q , o)). The exercise now reduces to one of calculating M which is usually done via phonon transport theory, i.e. by considering phonon population dynamics. According to Niklasson and Sjolander (I974)one may, from such an analysis, infe~ a stru.cmre for M (q, t) as shown in ftgt:re 41. Here one sees three distinct relaxation times related respectively to the inverse Debye frequency, the thermal phonon life-time r and the hydrodynamic relaxaion time [~-~(1/qZ)]. The long-lived component is the one which eventually produces the central peak. The essential point is that if properly done, phonon transport dynamics can produce a central peak without resort to phenomenological models. The latter however have the virtue of giving better insight. Summarising the situation with respect to the central peak, one may say that many aspects of it have been studied experimentally and established. On the theoretical side, it seems to be universally accepted that the peak is due to the existence of a long time scale in the fluctuations. The details of the physical origin of this long time scale are still not quite clear and to this extent, I would think the ball rests in the theoreticians's court. I must also draw attention to the fact that computer simulation (Schneider and Stoll 1976) has revealed the existence of evanescent, short-range ordered clusters close to T, in the high temperature phase. The appearance of the central peak above T, must obviously be related in some manner to these clusters but the study of the connection is still in an infant stage (Bruce and Schneider 1977). 9. What good are soft modes?

Before winding up, I should perhaps add a few remarks about the practical utility of soft modes. In these days of doing things relevant, such speculation is not out of place! Fortunately, my job has been made easy by Fleury (1973) who has addressed himself precisely to this problem. ,u

*The presence of thermo-distortive coupling would according to Thomas (1974), render g into a nonergodie variable (on account of the large number of degrees of freedom that then become accessible). Once w is nonergodic the considerations of Schneider et al (discussed earlier) no longer become applicable,


169

Time Figure 41. Qualitative behaviour of M (q, t) showing the three differentrelaxation regions. Fleury notes that the frequency of the q -----0 soft mode in Perovskite-type crystals (if it exists), can be varied by applying an electric field. Such phonons are IRactive and since the frequency can be varied, tuning applications are possible. One explicit suggestion is for a tuner based on stimulated Raman scattering off soft phonons in the presence of an applied field. Dv.ring stimulated Raman scattering, soft mode phonons are generated. Being IR active they can appear outside the crystal as far-infrared radiation which can then be tuned by varying the applied electric field. The low frequency character of soft modes suggests thermal applications also. As is well-known, low frequency phonons dorvlnate heat transport. By manipulating the soft-mode frequency electrically, the heat conductivity of the material could be altered rapidly. Thus, electric-field tunable thermal switches are conceivable. Perhaps the most important application may be in the realm of superconductivity. It is already known that the soft modes in A-15 compounds are in some way responsible for the high transition temperatures exhibited by these. It is conceivable that materials can be tailor-made to have the desirable soft-mode characteristics and hence also favourable superconducting properties. So, as Fleury says, '" it may come to pass that higher temperature superconductors will provide the best answer to the question what good are soft modes 2" In conclusion, I would like to acknowledge my indebtedness to malxy authors whose work I have freely drawn upon in preparing this talk.

References Anderson P W 1960 lzv. dkad. Nauk. p 290 Anderson P W 1963 Concepts in solids (New York: Benjamin) Aubry S 1975 J. Chem. Phys. 62 3217 Axe J D, Shirane G and Muller K A 1969 Phys. Rev. 183 820 Axe J D, Shapiro S Iv[, Shirane G and Kiste T 1974 in Anharmonic lattices, structural transitions and melting ed. T Riste (Leiden : Noordhoff) p 23 Birgeneau R J, Kjems J I~ Shirane G and Van Uitert L G 1974 Phya. Rev. B10 2512

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