X-ray diffraction of incommensurate structures in the soliton regime

~

0038-I098/86 $3.00 + .00 Pergamon Press Ltd.

Solid State Communications, Vol.58,No.2, pp.105-109, 1986. Printed in Great Britain.

X-RAY D I F F R A C T I O N OF I N C O M M E N S U R A T E S T R U C T U R E S IN THE SOLITON REGIME* J.M.

D e p a r t a m e n t o de Ffsica,

P ~ r e z - M a t o and G. M a d a r i a g a

F a c u l t a d de Ciencias, U n i v e r s i d a d del Pais Vasco, Apdo 644, Bilbao, Spain

(Received

10 July

1985

by E.F.

Bertaut)

In order to e l u c i d a t e the s e n s i t i v i t y of X-ray d i f f r a c t i o n to the structural d i f f e r e n c e s in an i n c o m m e n s u r a t e structure between the s i n u s o i d a l and soliton regimes, a general e x p r e s s i o n for the structure factor of a d i s p l a c i v e incomm e n s u r a t e structure in an ideal soliton regime is o b t a i n e d in terms of the so called "atomic s c a t t e r i n g m o d u l a t i o n factors", w h i c h reduce the e x p r e s s i o n to a form analogous to that of a c o m m e n s u r a t e structure. C o n s e q u e n t l y an approximate r e l a t i o n s h i p between the intensities of p a r t i c u l a r sets of satellites is d e t e r m i n e d and c o m p a r e d w i t h that e x p e c t e d in the s i n u s o i d a l regime. It is also shown that an ideal soliton regime gives place to a p a r t i c u l a r e x t i n c t i o n rule, and the c o r r e s p o n d i n g average structure has the atomic p o s i t i o n s s p l i t t e d like a d i s o r d e r e d structure. These results allow to e x p l a i n some r e c e n t l y r e p o r t e d m e a s u r e m e n t s in R b 2 Z n C I 4.

I. I n t r o d u c t i o n I n c o m m e n s u r a t e phases have been p r e d i c t e d to have a c h a r a c t e r i s t i c d e v e l o p m e n t as the t e m p e r a t u r e decreases, so that, right after the n o r m a l - i n c o m m e n s u r a t e t r a n s i t i o n the m o d u l a t e d d i s t o r t i o n is sinusoidal, c o r r e s p o n d i n g to the o r d e r p a r a m e t e r distortion, and as the t e m p e r a t u r e is further lowered coupled modes (higher harmonics) become s p o n t a n e o u s and superpose c r e a t i n g what is usually called soliton regime. In this regime the d i s t o r t i o n is p r a c t i c a l l y c o m m e n s u r a t e in q u a s i - m a c r o s c o p i c regions, that c o r r e s p o n d to the look-in phase, s e p a r a t e d by a p e r i o d i c array of narrow " d i s c o m m e n s u r a t i o n " regions. The lock-in phase t r a n s i t i o n takes place w h e n the d i s t a n c e s between these latter become m a c r o s c o p i c , so that their c o r r e l a t i o n disappears. E a c h c o m m e n s u r a t e region c o r r e s p o n d s then to a m a c r o s c o p i c domain of the lock-in phase. I-7 This soliton regime has been o b s e r v e d in d i f f e r e n t c o m p o m d s b y i ~ e v e r a l experim e n t a l techniques: NMR 8, EPRI~ d i e l e c t ric m e a s u r e m e n t s 14-16. However, structural analysis by m e a n s of X - r a y or neutron d i f f r a c t i o n have been until now unsuccessful to detect this regime. On the c o n t r a r ~ all the i n v e s t i g a t e d i n c o m m e n s u r a t e structures have been r e f i n e d c o n s i d e r i n g only a few h a r m o n i c s in the distortionl7-2~ It is usual for instance to c o n s i d e r harmonics up to the o r d e r of the highest *. W o r k s p o n s o r e d by the CAICYT

(Spain)

order distinct satellite r e f l e c t i o n observed. This order is however usually rather low in the whole stability range of i n c o m m e n s u r a t e phases, indicating, p r o v i d ed the m e n t i o n e d c r i t e r i u m is valid, that the soliton regime is far from being realized in most compounds. It is the aim of this paper to elucidate in a general context the sensitivity of a d i f f r a c t i o n e x p e r i m e n t to the structural d i f f e r e n c e s between a sinusoidal and an i d e a l i z e d soliton regime. The general features of the d i f f r a c t i o n d i a g r a m for b o t h regimes are d e t e r m i n e d and c o m p a r e d using the concept of atomic s c a t t e r i n g m o d u l a t i o n factors 23. For s i m p l i c i t y only o n e - d i m e n s i o n a l m o d u l a t e d distortions are considered. It will be shown that the soliton regime is c h a r a c t e r i z e d in the d i f f r a c t i o n pattern not by the a p p e a r a n c e of distinct higher order satellite reflections, but by the s u p e r p o s i t i o n on the lower order satellite p r o f i l e s of higher order ones, w h o s e d i f f r a c t i o n vectors are all to c o i n c i d e in the lock-in phase. An app r o x i m a t e law for their relative intensities can be o b t a i n e d and c o m p a r e d with that e x p e c t e d in the s i n u s o i d a l regime. An ideal soliton regime is shown also to produce a c h a r a c t e r i s t i c e x t i n c t i o n rule. This a n a l y s i s allows to c o n c l u d e that some results in a diffuse X - r a y scattering i n v e s t i g a t i o n of R2ZnC14 24, w h i c h were not u n d e r s t o o d at the moment of their p u b l i c a t i o n (see ref. 24), are a clear e v i d e n c e of the fact that the m e n t i o n e d c o m p o u n d a p p r o a c h e ~ as temper105

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X-RAY D I F F R A C T I O N

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ature decreases, the ideal soliton regime considered in this work. 2. Structure Factor An incommensurate displacive phase can be described by a commensurate atomic configuration usually called basic structure, w h i c h acts as a reference and is often taken to correspond to a high temperature normal phase, and the atomic displacements with respect to the basic structure ~(~,T), for each atom ~ (~=I,..~ in the basic unit cell and each cell T. Such displacement field, in the case of a one-dimensional modulated distortion, can be expressed as u(~,T) = --

--

/~ u u exp (i2~nk.T) n=0,±1,... - - n

(1)

STRUCTURES

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2

cluded, so that it must be considered in [~. These atomic positions correspond to what can be c o n s i d e r e d a theoretical "average" structure, t h a ~ as we will see below, does not always coincide with the average structure usually defined empirically in diffraction studies. 17 In the sinusoidal regime only zero and first harmonics (n=0,±1) in (5) are non negligible, while in the soliton regime higher harmonics will be relevant. Therefore the main general differences in the diffraction pattern of both regimes will arise from the differences on the corresponding atomic scattering modulation factors b ~ ( H ) . I n order to elucidate them t h e precise form of the functions u~(v) in an idealized soliton regime should be determined. 3. Soliton Regime

where k is the incommensurate m o d u l a t i o n wave vector. The atomic positions in the incommensurate structure are then given by

E ( P ' ~ )= _E~ + ~ + ~ ( U ' ~ )

(2)

where ~ is the atomic position of atom in unit cell of the basic structure. The magnitudes ~ represent the complex vectorial amplitudes of the different harmonics present in the distortion. The reflections in the diffraction pattern of such a structure are associated to diffraction vectors H, such that = G + h4 k

(3)

where G is any reciprocal lattice vector of the basic structure and h 4 is any integer. The corresponding structure factor for these diffraction vectors can be expressed in a compact form by introducing the so called "atomic s~wttering modulation factors" defined as L 3 1 b ~(H)= [ dv exp{2~H.u ~(v)+h4v} 20 where the continuous vectorial u~(v) is given by u~(v)=

~

(4)

function

_~ exp(i2~nv)

(5)

T~e structure factor can then be written in the form23: _~

f~(H)bP(H)exp(i2zH.~ ~)

u(~,T)= .J/jc~ (T)exp(i2~n'kc.T)

--

--

--n'

(7)

n'

The commensurate wave vector k_c is chosen to be the m o d u l a t i o n wave vector at the lock-in phase, and will be parallel and close to k. The sum extends to any integer n', such that n'kc is not equivalent to ~c under the basic reciprocal lattice translations (n'k ~ kc+G) ° The commen--C surate value of ~c assures that the sum (7) is finite, in contrast to (I), where the number of terms is in principle infinite, The relationship between both approaches is given by ~,

(T)= / ~ , ) u c~ e x p){ i 2 ~_n l k _- k _

.T}

(8)

where the sum is realized for the integers such that nk is equivalent to n'k_. If k and ~c do n ~ differ much, the amplitudes ~n,(T) will vary slowly w i t h T, having an approximate period along ~he wave vector direction of 1/Ik-k~ I. On the other side, we can--de~ine the following functions: -n' (v)= > j ~ e x p ( i 2 z n v ) n(n') Wu

n=±I,±2...

F(H)=

The structural distortion u(~,T) in (I) can also be given in terms of--a superposition of commensurate distortion waves w i t h T - d e p e n d e n t amplitudes:

(9)

so that

(6)

~=1...s where f~(H) are the atomic scattering factors and r ~ denotes the atomic positions in the--basic structure modified by the displacement c o r r e s p o n d i n g to the zero-harmonic amplitude ( ~ + ~ ) . Note that in (5) this amplitude has been ex-

cUn, (T)= W~ n, ( ( k - k c) .T)

(10)

The incommensurability of k assures that all the values taken by the functions W~, are relevant in order to describe the discrete set of amplitudes c~, (T). If p is the smallest integer such that P~c belongs to the basic reciprocal lat-

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x-RAY DIFFRACTION OF INCOMMENSURATE STRUCTURES

2

it is straightforward

from

(9) that

I07

satellites whose order is a multiple of r:

W~,(v+!) P

W~,(v)exp(i2nn'/p)

=

(11)

The relevant interval in v to describe the amplitudes ~ , ( T ) is then reduced to

(0,l/p). The functions W~,(v) determine those appearing in the expression for the atomic scattering m o d u l a t i o n factors and defined in (5):

Z

U~(V) =

_n,(V)

(12)

"

n'

•

U ~(v)=u~

if

r

-

the sum in (14) depends explicitly only on the value of i. Consider then two diffraction vectors and H', such that H'= H+np(k -_k) --

(18)

--C

where n is a small integer. The relationship between their satellite indices is then: h i = h4-nP

(19)

As ~c and k are supposed to be very close, both diffraction vectors will not differ much, and superpose at the lock-in phase. Furthermore if m p i s a multiple of r, both satellite indices, h 4 and h~, correspond to the same 1 index,-so that the values taken by the last factor in (14) is to be very similar for both d i f f r a c t i o n vectors. Hence the relative intensity of the c o r r e s p o n d i n g difraction peaks can be approximated by:

(13)

h4 Sin (3h4/r)

x

(~h4/r)

4. Sinusoidal Regime It is intersting to compare (20) with an analogous p r e d i c t i o n for the sinusoidal regime. In this case, the modulation functions u~(v) can be written in the form: U~(V)= u~exp(i2~v)

x{ 1 r

(17)

(20) r

From (4), the c o r r e s p o n d i n g atomic scattering m o d u l a t i o n factor is then

r

, 0~ 1
~-I < V <

--3

b ~ (H) = e

(16)

This extinction rule results from the step character of the m o d u l a t i o n functions and is not related w i t h those explained by symmetry arguments 27. The first two factors in (14) do not depend on the particular atom considered, so it is common to the whole structure factor (see eq. (6)). On the other side, if we write the satellite index h 4 in the form: h 4 = mr+l

Accordingly, the real vector functions u~(v) will also approach in the soliton regime a step configuration. We can take as an ideal soliton regime the limiting case, in w h i c h the discommensurations have negligible w i d t ~ so that the functions ~ ( v ) can be considered perfect step functions. This situation would correspond to a series of commensurate quasi-macroscopic domains distributed p e r i o d i c a l l y and separated by infinitely narrow domain walls. The m a x i m u m length of the steps in u_~(v) is restricted by (11) to be I/p. In fact, it can be proved ~ that, if rotational symmetry is also c o n s i d e r e d the steps have an equal lenth of 1/p.~ where t is the quotient between the orders of the point groups of the basic structure and the loc k - i n phase, so that the number of steps coincides w i t h the number of possible different domains in the lock-in pnas e26 . We have therefore in an ideal soliton regime that the function u~(v) is given by r (r=p.t) vectors {u~}, j=l...r, such J that --

I(H)=0 if h4=mr

exp (i2~ (H. _u~+h4 J/r) ) } (14)

~

+ c.c.

(21)

and the atomic scattering modulation factors become:

j=1,r

- i h 4 ( ~ + ~) bW (H) = e

w h i c h for the main reflections

reduces

J_h~4~ IH.u~ I)

(22)

to where

b u (H)= -?1

~

exp(i2~H.u~.) -- -]

(15)

~=

arg

(H.u~)

(23)

j=1,r The first inmediate conclusion, when (14) and (6) are c o n s i d e r e d , is that the diffracted intensity will be zero for

A c c o r d i n g to (6), we can consider in a statistical sense the relative intensities of two diffraction vectors H and H', which being very close have satellite --

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X-RAY DIFFRACTION

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indices ~ and h~ respectively, to be related : I(H')

J2h~(X) -

I(H)

(24)

~ h4 (x)

where x is taken of the typical order of m a g n i t u d e for 4~IH.u~ I . 5. The c o m p o u n d Rb2ZnCl 4 The above results can be c o m p a r e d w i t h some r e p o r t e d m e a s u r e m e n t s for Rb2ZnCI 4 24 This compound, similarly to other members" of the same family, has an o r t h o r h o m b i c basic structure (space group Pmcn), followed at 303K by an incor~nensurate phase w i t h k = ( ( 1 / 3 ) - 6 ) c * (6>0), w h i c h develops final~y at 190K Tnto a lock-in phase w i t h k =(I/3)c* and space group P21cn, so that T~ this case the index r d e f i n e d above is 6. If we denote by (hl,h~,h3,h4) , h i integers, a general d i f f r a c t i o n vector H= (hl,h2,h3)+h4k, the set of d i f f r a c t i o n vectors s a t i s f y i n g (18) are in this case of the form: (hl,h2,h3÷n,h4 -3n) In ref. 24 the d i f f r a c t i o n p e a k s corr e s p o n d i n g to the following d i f f r a c t i o n vectors were studied: (6,0,1,-3)

1=3

(6,0,2,-6)

i=0

(6,0,2,=4)

1=5

1=2

We have grouped t h e m a c c o r d i n g to their index i. For the last two groups, the o b s e r v e d relative intensities at 193K can be compared w i t h the approximate predictions for the sinusoidal and the soliton regimes given in (24) and (20) respectively. We have (with x=l in (24)): Exper. I_I/I 5 : 3xi04 I_7/I 5 :

0.2

I_4/I 2 : 5xi0 -3

Soliton 25

A more s i g n i f i c a n t fact is the t e m p e r ature d e p e n d e n c e of these intensities, w h i c h was shown in Figure 8 of the mentioned r e f e r e n c e and was not understood. A clear i n t e r p r e t a t i o n of this dependence arises from the table above: the differences in the increase rates of the different d i f f r a c t i o n peaks intensities is such, that these latter separate even further from the sinusoidal schema, while they rapidly a p p r o a c h the intensities d i s t r i b ution a s s o c i a t e d to the ideal soliton regime. The lock-in phase however takes place w h e n it is still far from being realized. An important evidence, w h i c h supports this i n t e r p r e t a t i o ~ is that the sixth order satellite (6,0,2,-6) could not be observed, despite the fact that satellites of similar and even higher order, like those listed above, were detected. This suggests that e x t i n c t i o n rule (16) p r e d i c t e d for an ideal soliton regime is satisfied. Finally it should be noted that standard X-ray analysis can o v e r l o o k the fine structure of the d i f f r a c t i o n peaks pro-files, failing to identify higher order satellite peaks, w h i c h will nearly superpose in the soliton regime w i t h the m a i n r e f l e c t i o n s or lower order satellites. This can result in a w r o n g a s s u m p t i o n of sinusoidal or nearly sinusoidal regime for the i n c o m m e n s u r a t e structure.

6. A v e r a g e Structure

(6,0,I,-I),(6,0,-1,5),(6,0,3,-7) (6,0,0,2),

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STRUCTURES

Sinusoidal 3x106

0.51

3.6xi0 -3

0.25

4.6xi0 -4

A l t h o u g h no definite c o n c l u s i o n can be o b t a i n e d from the tables above, it can be seen that the orders of m a g n i t u d e of the o b s e r v e d r e l a t i v e intensities differ w i t h respect to those p r e d i c t e d for both the s i n u s o i d a l and the ideal soliton regimes, except in the case of the peaks w i t h h 4 = ~ and 5 whose relative intensity agrees w i t h the soliton model. In the rest, the e x p e r i m e n t a l value is intermediate those given for the two ideal regimes.

A final point should be said about the so called average structure. T h ~ latter is usually e m p i r i c a l l y defined "" as that r e s u l t i n g from c o n s i d e r i n g in the diffraction p a t t e r n of the i n c o m m e n s u r a t e structure as o b s e r v e d o n l y the m a i n reflections (h4=0). It should c o r r e s p o n d to a c o m m e n s u r a t e configuration, where the m o d u l a t i o n on the atomic positions is averaged, and its effect on the d i f f r a c t ion is simulated by a fictitious temperature factor. Indeed the atomic scattering m o d u l a t i o n factors for the main reflections in the sinusoidal regime, a c c o r d i n g to (22), are:

J0 (47

I)

(25)

w h i c h are real and, for small values of the amplitudes u~ can be a p p r o x i m a t e d by b~

= 1-472

12

(26

The form of this e x p r e s s i o n coincides w i t h the first o r d e r e x p a n s i o n of the t e m p e r a ture factor usually c o n s i d e r e d in X-ray analysis, and t h e r e f o r e in this Situation it can be assimilated, w h e n c o n s i d e r e d in e x p r e s s i o n (6), to a t e m p e r a t u r e factor, if o n l y main r e f l e c t i o n s are being considered. Hence the average structure w i l l c o r r e s p o n d to the atomic positions r ~ in (6), w h i c h result from the addition-of the

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X-RAY DIFFRACTION OF INCOMMENSURATE STRUCTURES

zero harmonic to the basic structure positions. However, it should be taken into account that a c c o r d i n g to (15), if the structure is closer to the soliton regime than to the s i n u s o i d a l one, the atomic s c a t t e r i n g m o d u l a t i o n factors are not n e c e s s a r i l y real and their functional form w i l l a p p r e c i a b l e differ from that of a t e m p e r a t u r e factor. In this s i t u a t i o n the search of an average structure should p r o b a b l y produce i n c o n s i s t e n -

109

cies on the c a l c u l a t e d t e m p e r a t u r e factor, except if the atomic p o s i t i o n s are splitted. In fact, if we c o n s i d e r exp r e s s i o n (6) r e s t r i c t e d to the m a i n r e f l e c t i o n s w i t h the form (15) for the atomic s c a t t e r i n g m o d u l a t i o n factors in the ideal soliton regime, it can be i n t e r p r e t e d as the structure factor of a c o m m e n s u r a t e s t r u c t u r e w i t h the basic lattice and each atom s p l i t t e d into r positions, r~+ u~, j=l,...,r.

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