Do spatially chaotic configurations possess glass-like properties?

Journal of Non-Crystalline Solids 75 (1985) 129-134 North-Holland, Amsterdam

DO SPATIALLY C H A O T I C C O N F I G U R A T I O N S

129

POSSESS G L A S S - L I K E PROPERTIES?

Relf S C H I L L I N G and Peter R E I C H E R T Institute of Physics,

U n i v e r s i t y of Basel,

Switzerland

For a chain with anharmenic and c o m p e t i n g interactions we prove the e x i s t e n c e of an infinite number of metastable chaotic 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 which e x h i b i t glass-like behaviour. The pair d i s t r i b u t i o n function shows more or less p r o n o u n c e d nearest-, n e x t - n e a r e s t etc. neighbour peaks and the absence of long-range order. The e x i s t e n c e of two-levelsystems is e s t a b l i s h e d and we compute exactly the a s s o c i a t e d energies and p o t e n t i a l barriers. We d e m o n s t r a t e that the c o r r e s p o n d i n g density of states is not c o n s t a n t but instead exhibits a scaling b e h a v i o u r w h i c h leads to a p o w e r - l a w b e h a v i o u r for the specific heat, with a fractional exponent smaller than one.

i.

INTRODUCTION A m o r p h o u s or glassy structures p o s s e s s rather d i f f e r e n t p r o p e r t i e s

c r y s t a l l i n e ones.

First of all, the atoms are a r r a n g e d irregularly,

short-range but no long-range order. b e h a v i o u r of phonons,

electrons

from exhibiting

The disorder strongly influences the

(localization),

etc., and p r o b a b l y also

implies the existence of a new type of low energy excitations. systems then allow one to explain the unusual

These two-level

low-temperature b e h a v i o u r of

a m o r p h o u s solids. Several concepts have been d e v e l o p e d to construct models for the structure and the p h y s i c a l p r o p e r t i e s of glass. The oldest one, o r i g i n a l l y due to 1 Z a c h a r i a s e n , is the r a n d o m network d e s c r i b i n g c o v a l e n t - b o n d e d glasses. Metallic glasses are better d e s c r i b e d by dense r a n d o m p a c k i n g of hard spheres 2'3 " In the last few years new concepts have been proposed.

Without being com-

plete, we m e n t i o n the curved-space d e s c r i p t i o n by Kleman and Sadoc 4 and the r e l a t e d work by N e l s o n 5.

Quite a d i f f e r e n t approach to glasses is the gauge

theory of amorphous structures by Rivier and Duffy 6, which is based on topological p r o p e r t i e s of r a n d o m networks. The q u e s t i o n as to whether the irregular a r r a n g e m e n t of the atoms in a glass can be c o n s i d e r e d as a m e t a s t a b l e e q u i l i b r i u m c o n f i g u r a t i o n which m i n i m i z e s an interaction p o t e n t i a l

is of basic importance and is still open.

There is, of

course, n u m e r i c a l evidence 7'8 for a small number of p a r t i c l e s that this may be true indeed.

We believe that the recent p r o g r e s s

in the studies of n o n l i n e a r

systems, e.g. d y n a m i c a l systems with temporal chaotic behaviour,

0022-3093/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

may p r o v i d e

R. Schilling, P. Reichert / Spatially chaotic configurations

130

us with a m i c r o s c o p i c e x p l a n a t i o n of the existence of a m o r p h o u s structures as spatial chaos.

This has been the m o t i v a t i o n of the work p r e s e n t e d here.

2. MODEL R e c e n t l y we have studied 9'I0 a t r a n s l a t i o n a l l y

invariant chain with

anharmonic and c o m p e t i n g interactions given by the n e a r e s t n e i g h b o u r potential:

C1 Vl ({Vn}) = -~ ~{ (Vn_a+_a o (Vn))2_ (c-a+-a_o (vn) )2} ,CI>0 which is p i e c e w i s e parabolic,

(la)

and a n e x t - n e a r e s t n e i g h b o u r p o t e n t i a l which is

chosen to be of the form:

C2 v2({v}) = T ~ (Vn+Vn+i-b)z ' c2~°

(ibl

where v n = Un+1-u n is the distance between a t o m n and n+l and a± = (a2±al)/2. Two p o s s i b l e shapes for V 1 and V 2 are p r e s e n t e d in Fig. trates the significance of al,a2,b and c. sgn

(Vn-C) c h a r a c t e r i z e s

p o t e n t i a l Vp(V)

i which also illus-

The important q u a n t i t y

the n e a r e s t n e i g h b o u r bonds.

o(v n) =

Fig. 2 represents a palr

which shares common features with Vl, V 2 and for which the

results we have found for the V 1 - V 2- model remain true.

~,~ ".,:~2

v2b %

¥p r¥I

V2

V=0

¥

\

v21 FIGURE 1 N e a r e s t and n e x t - n e a r e s t n e i g h b o u r potential: (a) d o u b l e - w e l l (b) singlewell p o t e n t i a l

FIGURE 2 Pair p o t e n t i a l V . V and V denote p i 2 the part of the nearest and nextnearest neighbour potential

Using V 1 and V 2 we get for 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 the n o n l i n e a r difference equation:

R. Schilling, P. Reichert /Spatially chaotic configurations

131

(2a)

2yV n + V n + 1 + V n _ 1 = ¢(v n)

where

~(Vn) = 2 [ b + ( v - l ) a + ] + 2 ( v - l ) a _ o ( v n)

Letting n > ~ ( l - / l - ~

±~2~)

exists a range for

I+CI/2C 2 •

then one can prove 9'I0 that for

(al,a2,b,c)

such that all solutions of

d e t e r m i n e d by all infinite sequences

Vn(O)

, y=

= A + B E i

(2b)

In I < i/3 there

(2) are u n i q u e l y

~ = {o n} , on = +i :

~[il

n-i

'

(3)

and their energy is given by an Ising-like model:

E(o) = E o -

H Z o + J n n n

q

n-m

o

(4)

o

nm

where the c o n s t a n t s A,B,E

,H and J depend only on ~ and (al,a2,b,c). The o result we have found above has the following p h y s i c a l meaning: if I~[ < i/3 and

(al,a2,b,c)

{Vn }

in a suitable range, then any arbitrary initial configuration

will relax to 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

action of V 1 and V2, where

on

= sgn

(vn

c)

{Vn (~)}

determine the r e l a x e d c o n f i g u r a t i o n uniquely.

Furthermore,

that all the c o n f i g u r a t i o n s

if

is always

3.

lql

(3) are metastable

Inl

CHAOTIC C O N F I G U R A T I O N S

o

In the following it

o we call p-normal,

(random)

either of p e r i o d one or

follows from

of +i and -I with p r o b a b i l i t y p and o n and

(3) by choosing

(l-p), respectively.

0 m are u n c o r r e l a t e d for n ~ m.

there also exist other chaotic configurations, o n = ±I

< i.

(T = OK)

A certain class of chaotic c o n f i g u r a t i o n s

r a n d o m sequences Such

it is easy to prove

< I/3.

The g r o u n d - s t a t e of our model is always periodic, two.

under the

i.e. the initial Ising v a r i a b l e s

e.g. take

Of course

o n = I for n ~ km and

for n = km for a fixed integer k and all integers m.

these r e p r e s e n t m i c r o c r y s t a l l i n e

structures with grain size p r o p o r t i o n a l to k.

In any case it should become clear that q u e n c h e d bond disorder.

o

characterizes the type of

For our p u r p o s e s we assume

o

to be p-normal.

What are the statistical p r o p e r t i e s of such chaotic configurations? of all, we have found that the j - n e a r e s t - n e i g h b o u r not d i s t r i b u t e d continuously.

However,

Using

distances

First

(j=l, 2 . . . . ) are

(3), we have d e t e r m i n e d the pair

13 2

R. Schilling, P. Reichert / Spatially chao tic configurations

distribution resolution

function 6.

G(6)(r)

which is represented

There are more or less p r o n o u n c e d

neighbour peaks,

indicating

the existence

in Fig.

3 with finite

nearest-,

next-nearest

of short-range

we have p r o v e d that the pair distribution

function

converges

irrational),

which means that there is no long-range

demonstrates

that the peaks themselves

6(8(0

-'"'1

....

i ....

I ....

order.

to one

order.

can be resolved

etc.

For r + (if A/B is

Furthermore,

Fig.

3

into finer peaks.

s(~&)

I''"-

3. 2.

5.

I.

O.

0. O.

2.

4.

6.

FIGURE

o.

rlA

for

6 = 0.1A,

structure

Structure

consists

4.

of two parabolic

TWO-LEVEL

with k n) .

chaotic

figuration

{Vn(a) }

"neighbouring" O' = On n { V n (J)}

(asymmetry)

equilibrium

otherwise

and

is obtained

configurations.

3

Apart from the

constant,

feature

of two-level

Starting

{Vn(O') } (i-l,i)

from

also (because

arises because

V1

systems

= oi4J ~ qn n= 1

As

o

, we consider

{7':[-1= -u i-l, C'~ =-O i

are such bonds for which

e~_ 1 = - a [

.

just by moving a few particles

the chain.

= E(a) - E (o')

(TLS).

number of

from one special con--

given by

where

{Vn(O)}

of the tunneling-model II'12,

%(09

from Fig.

HEAT

with frozen bond disorder

configurations

A~(O)

following

2, the chain may be in any of an infinite

over a barrier and then relaxing justification

4

2 n /A (for n --> i) appear

This special

In this section we will prove the existence

metastable

FIGURE

k-A/2tt

8.

parts.

SYSTEMS AND SPECIFIC

we have seen in section

6.

the mean lattice

extra peaks at k'n = (n-½) 2 ~ /A and at k n = ~n 4, k'n coincides

factor

4.

factor S (k) is given in Fig. 4.

expected peaks related to A = A + (p-I/2)B,

p = i/2 in Fig.

2.

3

Pair distribution function p = i/2 and q = 0.2

The corresponding

8.

Thus we have found a microscopic i.e. of the TLS.

follows (a i+n

from

- ~i-l-n )

Their energy

(4) : (5 )

R. Schilling, P. Reichert / Spatially chaotic configurations

133

The c o r r e s p o n d i n g p o t e n t i a l b a r r i e r s V i ( o ) can also be c a l c u l a t e d exactly 2 2 7 2 We have found that for all i, q and 0 they s a t i s f Y ~ l a - < Vi(o) < ~ i a_

i0

i.e. there exists a m i n i m u m barrier height Vmin, which is of the same order as the m a x i m u m value Vma x. C h a n g i n g more than two cross a barrier,

O-variables,

which means

p a r t i c l e s have to

more

one obtains a h i e r a r c h y of TLS, which are all localized if

nz

(o

- o')

n

= 0

n

A s s u m i n g that the p a r t i c l e s are oxygen atoms and that V 1 = a

= I~ (i: t u n n e l i n g distance),

= 0.1eV and max we have e s t i m a t e d that the q u a n t u m cor-

rections can be n e g l e c t e d for energies d e n s i t y of states n ( £ ) result p r e s e n t e d in Fig. JIq[3( ~ = 3).

larger than

~10-5eV.

is given by the d i s t r i b u t i o n of

A i.

5 for an energy resolution of order

Therefore,

the

This yields the JIql2(~

= 2) and

From this figure it becomes obvious that the spectrum of the

TLS is self-similar.

From this we get the scaling property:

n(lqls ) =

(p2 +

(l_p)2)inl-ln(c)

For the spectral d i m e n s i o n d d e f i n e d by n( £ ) = £n(p 2 +

~

8

(6)

d-i

we get from

(6)

(l-p)2)/£nlql

-I

0

I

e_

n3(~)I °fl° 11111.n. -I

FIGURE

Density of states n ~ ( e )

In p a r t i c u l a r n ( E ) refs.

ii and 12.

5

for V = 2,3, p = i/3 and

is not constant,

even for small energies,

Therefore the T L S - s p e c i f i c heat c(T)

given by a p o w e r - l a w c(T) ~ T ~

q

= i/4

in contrast to

is not linear but is still

if we assume that all TLS

(the most simple ones,

R. Schilling, P. Reichert /Spatial~ chaotic configurations

134

where only two contribute.

~-variables

are changed)

in a t e m p e r a t u r e range 0.1K - IK will

This is the case if the time scale

T for a m e a s u r e m e n t

is much

larger than i sec, because Vmi n sets a lower b o u n d for the tunneling-rate, which is 5.

F > 1 sec -I for the p a r a m e t e r s we have chosen.

CONCLUSIONS We have d e m o n s t r a t e d that anharmonic and c o m p e t i n g interactions,

frustration effects,

can lead to locally stable c o n f i g u r a t i o n s

which induce

(T=OK) with glass-

like properties. In p a r t i c u l a r the pair d i s t r i b u t i o n

function shows the e x i s t e n c e of short-

range and the absence of long-range order. Our model also p r o v i d e s a m i c r o s c o p i c systems.

justification of the two-level

The d e n s i t y of states, w h i c h is not constant,

b e h a v i o u r for the low-temperature

leads to a p o w e r - l a w

specific heat of the TLS with a fractional

e x p o n e n t d < I, which depends only on h and p. e x p e r i m e n t a l l y 13 for t h r e e - d i m e n s i o n a l

Such exponents are o b s e r v e d

glasses.

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