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

2

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

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

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

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

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

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

12 Tomas-Fermi method Screening effect As is small we can expand it: Tomas-Fermi dielectric constant In cavity  k q q   Now we have screening effect on Tomas-Fermi method

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

14 Screening effect Tomas-Fermi dielectric constant Fourier Transform

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

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

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

18 Screening effect

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

20 Screening effect Screening effect on Ion-Ion Interaction

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

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

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

24 Lind hard dielectric constant Screening effect Lind hard relation Lind hard model

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

26 Screening effect Lind hard method

27 Screening effect Lind hard method R Attraction & repulsion

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

29 Thanks for your attention

30