Soluble model for X-ray scattering from CDWs with dislocations

1. 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

2. Image dislocation (X,-Y) z (X,Y) y b 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

3. source drain v~I v=0 Formation of new planes in the electronic crystal Elimination of additional planes Current conversion: normal  sliding in CDWs. Dislocation as a leading edge of new periods. Experiments: Cornell, Grenoble Synchrotron 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

4. S. Ravy et all: Blue bronze K0.3(Mo0.972V0.028)O3 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

5. X-ray scattering by imperfect CDW crystals CDW: u=u0cos(q0r+) Scattering intensity by

6. Origin of the peak splitting Far from deformed regions – no strain, φ=const, I(q) max at q=Q-Q0=0 Within deformed region (impurity, dislocation) dφ=a, In 1D for random Poisson distribution For dislocation: a=2p , hence: no shift, no broadening We need 3D effects

7. Image dislocation (X,-Y) z (X,Y) y b 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).

8. Random array of parallel dislocations l – mean spacing betweendislocation lines Result of calculations: Io –Bessel function

9. Regime of unlesolved peaks Y<<l :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

10. I(q) exp(-n) reduction w q 2p/l Regime of two resolved peaks l<<Y:Deep and/or concentrated dislocations Gaussian peak at the streched CDW position Broad region of the split off peak - Gauss in q1/2

11. 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 • n>>1: Bare peak at q=0 exists for q>0 only within an interval q<1/(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 q<0) surface layer. Its shape is Gaussian with the width w= (n /Y)1/2 • n<<1 No additional peaks can be resolved. Bare peak at q=0 becomes non-symmetric acquiring a width w=n /Y in direction q>0. 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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