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Long-range order and diffraction of X-ray waves on multilayered crystalline films
M. Feigelman, V. Pokrovsky
To cite this version:
M. Feigelman, V. Pokrovsky. Long-range order and diffraction of X-ray waves on multilayered crys- talline films. Journal de Physique, 1981, 42 (1), pp.125-131. �10.1051/jphys:01981004201012500�.
�jpa-00208980�
Long-range order and diffraction of X-ray waves
on multilayered crystalline films
M. Feigelman and V. Pokrovsky
L. D. Landau Institute for Theoretical Physics, U.S.S.R. Academy of Sciences, Moscow, U.S.S.R.
(Reçu le 1 S avril 1980, révisé le Il août, accepté le 12 septembre 1980)
Résumé.
2014On calcule les facteurs de structure en diffraction des rayons X pour des films minces de smectique B.
On prédit un important effet de taille, associé à l’épaisseur du film, sur la forme non lorentzienne du facteur de structure incohérent.
Abstract.
2014Structure factors for the coherent and diffuse scattering of X-ray waves on thin films of smectic B
are calculated. A strong size effect with respect to the width of the film as well as a non-Lorentzian form for the diffuse scattering structure factor are predicted.
Classification
Physics Abstracts
71.30
-78.80
1. Introduction.
-The transition from a two- dimensional crystal to a three-dimensional crystal
with an increase in the number of layers has been
studied experimentally by Moncton and Pindak in [1].
They have also studied the X-ray diffraction on thin films of a liquid crystal of butyloxybenzilidene octylanilene (40.8). The latter is smectic B in a certain temperature range. The crystal was prepared as a freely suspended film consisting of a certain number
of monomolecular layers. The number of layers varied
from 4 up to the value exceeding 100.
The problem of the development of long-range order
due to an increase in the number of layers is of theore- tical interest as well. It is acknowledged (Mermin [2], Berezinsky [3]) that a two-dimensional crystal has
no long-range configurational order. The coherent
scattering amplitude in an infinite sample equals
zero whereas the diffuse scattering amplitude has peaks at momentum transfers close to certain vectors of the reciprocal lattice. The number of such peaks
and the character of singularities are dependent on temperature (Jancovici [4], Reatto and Chester [5]h
Mikeshka and Schmidt [6], Imry and Gunther [7]).
On the other hand, in a three-dimensional crystal
the coherent scattering amplitude is different from
zero and in an ideal infinite crystal has a £5-like cha- racter whereas diffuse scattering has peaks of the
Lorentz form in the vicinity of each Bragg vector.
In the present work we investigate theoretically
how the transition from a two-dimensional to a
three-dimensional picture of diffraction takes place.
To bring the theory closer to the experimental situation, certain real features of smectic B are intro- duced into it (a small modulus of the interplane shift).
Throughout this paper the temperature regime
under consideration is assumed to be sufficiently
far from the A-B phase transition point so the influence of vortices can be neglected.
2. Scattering and fluctuations in a multilayered crystal.
-From the very beginning it will be our
assumption that the number of layers N in the crystal
is large. The crystal will be described as a continuous elastic medium, isotropic in the plane (x, y) which corresponds to hexagonal symmetry. The elastic energy of the medium is described by the Hamilto-
nian :
Here ulk
-components of the deformation tensor ; A., Jl, v, y and É - elastic constants ; the Greek indices a, fi acquire the values 1, 2. Expressing the Hamiltonian (1) in terms of the displacement vector Ui, we obtain :
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01981004201012500
126
We have replaced uZ by v to stress the plane symmetry and 8 + 2 y by e. The quantities y, E are assumed to be
the same order and small in comparison with other elastic constants. This means that we regard as small all
the forces which resist the shift of the planes with respect to one another. Let us finally come to the Fourier
representation :
The finite width of the film is taken into account in (3) since the z component of the wave vector acquires a dis-
crete set of values xd
=::!:: 2 ;n; n N
=0, l, ..., N 2 where d-distance between the planes. From the expression
for the elastic energy (3) we may get the fluctuational averages :
Note that equations (4)-(6) are valid as for approxima-
tions in continuous media in general, only in the region of small k, K. Elsewheré k will be regarded as small, yet, the essential values of K in certain cases
appear to the order 1/d. In these cases we shall replace
the quantity x2 by its analogue for the discrete system of planes (21d’) (1 - cos Kd). In the denominator of
(4)-(6) we have neglected the quantities proportional
to e, since their contribution is always small in comL
parison with other terms.
1Equations (5), (6) should be specified. Since y is small, the next term of the expansion over k2 should
be taken into account. Hence, instead of (5), (6) we
shall obtain
The coefficient c has the order of magnitude of Jla2
since it is physically associated with the bend of the
plane but not with the interaction between different
planes.
3. Cohérent scattering. - The cross-section for the elastic X-ray scattering off a crystal is proportional
to the formfactor S(q) equal to :
where q
-momentum transfer, ra
-radius-vector of a particle with the vector number a. The vector ra
can be represented in the form r.
=a + ua. Then (7)
takes the form :
The first term of (8) corresponds to the cohérent,
and the second term to the diffuse, scattering. In this
section we shall restrict ourselves to the coherent
scattering formfactor
Let us calculate the Debye-Waller factor exp(- W ).
As is known, in the harmonic approximation
W = ’ (qu) )2. We shall be interested only in values
q equal to reciprocal lattice vectors :
where gl, g2
-basic vectors of a hexagonal flat lattice, g3
-vector perpendicular to the plane,
111, n, 1
-integers. Let us first consider the case when
the vector q lies in the plane (1 = 0). In this case we
have :
Let us separate the term with x
=0 from the sum over K in (9), and replace the remaining sum by the integral
over K :
The first term on the right-hand side of (10) corresponds to « two-dimensional » fluctuations of the crystalline
film as a whole. With logarithmic accuracy the contribution of two-dimensional fluctuations can easily be estimat- ed as
where R - linear dimension of the system.
The quantities 1"
=/L and fi
=JlL where L
=Nd - width of the film, play the role of two-dimensional Lame coefficients. At a fixed number of layers N and R - oo the Debye-Waller factor exp( - 2 W) ten4s to
zero according to the law :
The contribution of three-dimensional fluctuations to the quantity ( Ua ufl > is equal to :
At N - oo we get the Debye-Waller factor for a three-dimensional system :
In (14) we have neglected the difference of the quantity from unity. Note that the value of
exp( - Wq) for the minimum value of q is a parameter of the crystalline long-range order occurring due to the
weak interplane coupling. A similar result has been obtained by Berezinsky and Blank [7] and by Pokrovsky
and Uimin [8], for magnetic systems and by Mineev [9] for layered crystals.
In the general case when the reciprocal lattice vector q has a z-component, we add to the quantity 2 W
of (9) a term of the form
’where
Note that the contribution of the mixed terms ua v* ) to the quantity W is equal to zero by virtue of symmetry.
The final expression for the Debye-Waller factor is of the form
where Y/q, wfl’ (q are defmed in (12) and (15).
128
At 1 = 0 size effects in the Debye-Waller factor may be observed at a number of layers
For the samples of smectic B dealt with in [1] this quantity is roughly equal to 10-15. At 1 # 0 size effects in the
Debye-Waller factor may be observed at sufficiently larger N :
Under the same experimental conditions N2 is of order 300. Thus we predict a strong dimensional depenr
dence for the coherent scattering in the given regions N (cf. (17), (18)).
At sufficiently large q the coherent scattering cross-section in its maximum gets compatible with the diffuse
scattering cross-section. We can then observe only the Bragg peaks for which the following condition is fulfilled 1:
At 1 = 0 and N 11 5, the most important term on the left-hand side is the first. Therefore, the condition under
which the Bragg peaks are observed is :
At 1 :0 0 the terms involving 1 are important. Then the condition is of the form :
With an increase in N, the number of the observed peaks increases up to
4. Diffuse scattering.
-In the vicinity of the Bragg peaks the diffuse scattering in a two-dimensional sys-
tem has singularities (cf. Introduction). Let us see what happens to them in a multilayered system. Let q lie close to any vector of the inverse lattice b. A small difference q - b will as usual be denoted as q, and the components of this vector lying in the plane of the layer and normal to it, by k and x respectively. It follows from (8) that the
diffuse scattering cross-section is proportional to the Fourier-component of the respective formfactor :
At large distances the harmonic approximation is valid :
where 4 j
=1, 2, 3 and
The mean values of ( ui(P) uj(- p) ) have been calculated in section 2 (Eqs. (4), (5’), (6’)). Let us first treat the simplest situation when b3
=0. The components of the correlation function Gp(r) may be calculated by the
method employed in section 3 for the Debye-Waller factor :
where
where p and z are the components of the vector. The quantity G,(,’) should be estimated with the logarithmic
accuracy :
. 1The contribution of three-dimensional fluctuations to the correlator G,,,P can conveniently be represented in the
form of the difference (cf. (22)). The mean values of the form ua(o) up(O) > have been calculated in (13). The
correlation function ( ua(r) up(O) > can be represented in the form
where va
=pa/p and
(see [11], p. 741).
In the two most interesting limits we have
C-Euler constant,
These asymptotics correspond to the following behaviour of ( u,,,(r) uo(O) )(3) in various asymptotic regions :
where l5k,l-Kronecker symbol. Equation (30) has a simple physical interpretation : at small distances fluctuations of parallel displacements in various planes are independent. Therefore in each plane the correlation functions
are identical to those of a two-dimensional crystal.
130
The behaviour of the formfactor Si(q) depends on the parameter ka , of the asymptotic form (30) of the correlation function. We get :
1, we can make use
Note that in (32) there is no dependence on K. This is natural since (32) corresponds to the region of incoherent two-dimensional fluctuations. In another asymptotic region ka fô « 1 the exponent in (21) may be considered
small, therefore the exponential function may be replaced by its argument. It actually means that the quantity Si(q) is a linear combination of the correlation functions ( u«(q) up(- q) ) :
.
