Jelly model

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# Jelly model - PowerPoint PPT Presentation

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.

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## PowerPoint Slideshow about 'Jelly model' - summer

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Presentation Transcript

Screening effect

Look at homogeneous crystal; We inserted a point charge in this model Inhomogeneous crystal

Jelly model

Poisson equ. :

+

Point charge

Prob1: point charge field at position ?

homogeneous crystal

contents

Screening Effect :

Thomas-Fermi method

Lind hard method

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.

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

Screening effect

a.

= 0

Potential is Zero

Basic Principle in Hartree Model

Jelly model without free charges

Screening effect

Jelly model with free charges

0

Hartree method for Jelly model

Screening effect

&

= ?

1- Tomas-Fermi method (classic)

Solution :

2- Lindhard method (quantum mechanical)

Tomas-Fermi dielectric constant

Screening effect

1- Tomas- Fermi method: our hypothesis is slow varying

Semi classic

classical

Quantum mechanical

Homogeneous gas

&

Inhomogeneous gas

Tomas-Fermi dielectric constant

Screening effect

is chemical potential

In presence of free charges, density become a function of r

Basic Tomas-Fermi Relation

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

Prob2: charge distribution field at position ?

Tomas-Fermi method

Screening effect

Jelly model

Charge distribution

=

Tomas-Fermi dielectric constant

Screening effect

Fourier Transform

Potential energy

Potential energy

potential shows the screening effect

Tomas-Fermi dielectric constant

Screening effect

Without screening effect

Considering screening effect

Potential energy of one electron

Tomas-Fermi dielectric constant

Screening effect

We plot potentials in reciprocal lattice:

&

First perturbation term

Lind hard method

Screening effect

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

;

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

Screening effect on Ion-Ion Interaction

Screening effect

Potential is a real quantity

&

Is q independent on Tomas-Fermi

Screening effect on Ion-Ion Interaction

Screening effect

&

Lind hard method

1

1/2

q

1

Screening effect on Ion-Ion Interaction

Screening effect

When q 0 Lind Hard model tends to Tomas-Fermi model

Lind hard dielectric constant

Screening effect

Lind hard relation

Lind hard model

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

Lind hard method

Screening effect

R

Lind hard method

Screening effect

Attraction & repulsion

Screening effect

Screening effect

Screening effect on e-e interaction

Screening effect

Fourier Transpose

e-e interaction becomes weak

Conclusion :