Jelly model

1. Screening effect Look at homogeneous crystal; We inserted a point charge in this model Inhomogeneous crystal Answer 1: = + Jelly model Poisson equ. : + Point charge Prob1: point charge field at position ? homogeneous crystal

3. Many body electrons systems

4. contents Screening Effect : Thomas-Fermi method Lind hard method

5. Screening effect We consider all other charges presence in medium on our calculations, as this causes some reductions in our quantities then we recall it as a screen.

6. Screening effect D free charges E total charges = free charges + induced charges Linear relation between E & D: E is decreased in the presence of dielectric nonlocal

7. Screening effect a. = 0 Potential is Zero Basic Principle in Hartree Model

8. Jelly model without free charges Screening effect Answer a: Jelly model with free charges

9. 0 Hartree method for Jelly model Screening effect & = ? 1- Tomas-Fermi method (classic) Solution : 2- Lindhard method (quantum mechanical)

10. Tomas-Fermi dielectric constant Screening effect 1- Tomas- Fermi method: our hypothesis is slow varying Semi classic classical Quantum mechanical Homogeneous gas & Inhomogeneous gas

11. Tomas-Fermi dielectric constant Screening effect is chemical potential In presence of free charges, density become a function of r Basic Tomas-Fermi Relation

12. As is small we can expand it: Tomas-Fermi dielectric constant r 2 ( ) k e = + 0 q 1 2 q In cavity Now we have screening effect on Tomas-Fermi method Tomas-Fermi method Screening effect

13. Prob2: charge distribution field at position ? Tomas-Fermi method Screening effect Answer 2: Jelly model Charge distribution =

14. Tomas-Fermi dielectric constant Screening effect Fourier Transform

15. Potential energy Potential energy potential shows the screening effect Tomas-Fermi dielectric constant Screening effect

16. Without screening effect Considering screening effect Potential energy of one electron Tomas-Fermi dielectric constant Screening effect We plot potentials in reciprocal lattice:

17. & First perturbation term Lind hard method Screening effect 2- Lind hard method : First principle; we can use time independent perturbation theory (quantum model) ;

18. j j Screening effect on Ion-Ion Interaction Screening effect Direct interaction: coulomb inter. Ion-Ion interaction : Indirect interaction: screening inter. & Atomic form factor Structure factor

19. Screening effect on Ion-Ion Interaction Screening effect

20. Screening effect on Ion-Ion Interaction Screening effect Potential is a real quantity &

21. Is q independent on Tomas-Fermi Screening effect on Ion-Ion Interaction Screening effect & Lind hard method

22. 1 1/2 q 1 Screening effect on Ion-Ion Interaction Screening effect When q 0 Lind Hard model tends to Tomas-Fermi model

23. Lind hard dielectric constant Screening effect Lind hard relation Lind hard model

24. Lind hard dielectric constant Screening effect It has Bessel behavior & we can see Friedel Oscillation Lind hard model It is classical yukawa potential by considering screening effect Tomas-Fermi model

25. Lind hard method Screening effect

26. R Lind hard method Screening effect Attraction & repulsion

27. Screening effect Screening effect Screening effect on e-e interaction Screening effect Fourier Transpose e-e interaction becomes weak Conclusion :

28. Thanks for your attention Thanks for your attention