Soluble model for Xray scattering from CDWs with dislocations

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Soluble model for X-ray scattering from CDWs
with dislocations

• N. Kirova1 and S. Brazovskii2
• 1LPS, Université Paris-Sud, Orsay, France
• 2LPTMS, Université Paris-Sud, Orsay, France

Concern Nonsymmetric profiles of the scattering
intensity. Limitations Scattering only from
centers of dislocation dipoles. Direct
realizations surface structures of dislocation
lines.
ECRYS-2005
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Dislocations What, where and why
Dipole of the dislocation and its
image stretches the surface layers by one atomic
period.
Dislocation
Realisation in usual crystals array of
dislocations submerge below the crystal surface
to release the tangential stress from the
epitaxial covering. Experiment neutron
scattering, ILL-Grenoble.
Analogous regime for formation of dislocations
in CDWs Field effect at the side junction. The
penetrating electric potential is equivalent to
the stretching tension. Theories Brazovski and
Matveenko, Hayashi, Miller. Experiments Delft,
IRE - Moscow
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Current conversion normal ? sliding in
CDWs. Dislocation as a leading edge of new
periods. Experiments Cornell, Grenoble Synchrotro
n space resolved studies
Individual surface dislocation in CDWs Speckles
experiments Lebolloch' and Ravy
Related topic asymmetric X-ray profiles from
CDWs with defects. Friedel phase shift as a non
quantized version of dislocation loops
Ravy, Pouget et al
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Examples of nonsymmetric peaks in X-ray
scattering intensity profiles
Extrinsic defects - impurities
Intrinsic defects, stress of CDW by The external
electric field
P. Monceau et all NbSe3
Also Cornell
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X-ray scattering by imperfect CDW crystals
CDW uu0cos(q0r?) Scattering intensity by
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Origin of the peak splitting
Far from deformed regions no strain, fconst,
I(q) max at qQ-Q00
Within deformed region (impurity, dislocation)
dfa,
In 1D for random Poisson distribution
For dislocation a2p , hence no shift, no
broadening
We need 3D effects
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Single dislocation line near the crystal surface
Single dislocation line centered at (X,Y)
x
Dislocation
Kx,y CDW elastic modulii. No normal stress at
the surface
Deformations are given by the dipole the actual
DL at (X,Y) and its image at (X,-Y).
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Random array of parallel dislocations
l mean spacing betweendislocation lines
Result of calculations
Io Bessel function
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Regime of unlesolved peaks Yltltl Shallow and/or
rare dislocations
Bleaching (1-n)
I(q)
nexp(-qY)
q
1/l
1/Y
z
Exp(-qY) form factor for dislocation line
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Regime of two resolved peaks lltltYDeep and/or
concentrated dislocations
Gaussian peak at the streched CDW position
Broad region of the split off peak - Gauss in q1/2
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Conclusion

• X-ray scattering from the crystal surface in
  presence of aligned array of dislocations
  positioned in a characteristic depth Y below it
  with the mean spacing l along the surface
  strong dependence on the aspect ratio n 2p Y/l
• ngtgt1
• Bare peak at q0 exists for qgt0 only within an
  interval qlt1/(nY),
• its height is exponentially reduced exp (-n )
• Scattering peak is shifted by d q n /Y
  corresponding to a periodicity of the stretched
  (squeezed for d qlt0) surface layer.
• Its shape is Gaussian with the width w (n /Y)1/2
  
• nltlt1
• No additional peaks can be resolved. Bare peak at
  q0 becomes non-symmetric acquiring a width wn
  /Y in direction qgt0. This slope shape exp
  (-qY/n).
• More general cases can be studied on the bases of
  this model
• distributions of the depth Y and of lines angles
  correlations in lines positions, etc.
• The exact result can be used to verify the
  numerical procedures.

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Thanks!

• Welcome Reception !!