1 jelly model look at homogeneous crystal; we inserted a point charge in this model inhomogeneous...

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1 Jelly model r Look at homogeneous crystal; We inserted a point charge in this model Inhomogeneous crystal Answer 1: = + Prob1: point charge field at position ? r ext r r ) ( r ind Poisson equ. : 2 4 ext ext r r 2 4 r r ext ind r r r 0 0 0 ext ind r r r 0 0 0 homogeneous crystal Screening effect + Point charge

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Page 1: 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

1

Jelly model

r

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

Answer 1: = +

Prob1: point charge field at position ?r

ext r r

)(rind

Poisson equ. : 2 4ext extr r

2 4r r

ext indr r r

0

0

0

ext

ind

r

r

r

0

0

0

homogeneous crystal

Screening effect

+ Point charge

Page 2: 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

2

Page 3: 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

3

Many body electrons systems

Page 4: 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

4

contents

Screening Effect :

Thomas-Fermi method

Lind hard method

Page 5: 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

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.

Page 6: 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

6

D free charges

E total charges = free charges + induced charges

Linear relation between E & D:

r ext r

D r E r

E is decreased in the presence of dielectric

D r d r r r E r ���������������������������������������������������� ����

Screening effect

ext r d r r r r ���������������������������������������������������� ����

nonlocal

Page 7: 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

7

1 extq qq

ext q q q

i q r ext i q re r d r d rd r e r r r ���������������������������������������������������� ����

Screening effect

ext i q rq d rd r e r r r ���������������������������������������������������� ����

r r R d r d R ���������������������������������������������������� ���� i q R rext q d Rd r e R r

�������������������������� ������������������������������������������������������ ����

i q R i q rd Re R d r e r ��������������������������������������� �����������������������������������������������������������

q q

?ind r

a.

= 0

Potential is Zero

r

Basic Principle in Hartree Model

Page 8: 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

8

Screening effect

Answer a:

0 ; 0 0ext indr r r

ind r d r r r r ���������������������������������������������������� ����

0 ; 0 0ext indq r r

Jelly model without free charges

Jelly model with free charges

2

2

4

4

ext extr r

r r

2

2

4

4

ext extq q q

q q q

2 4ext extq q q q q 4 ind q

2 4extq q q q q

2

2 4

extq qq

q q

Page 9: 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

9

2

2 2

4 41

q X q X qq

q q

�������������� 1 extq q

q

&

Screening effect

2

41

ind qq

q q

��������������

ind q

X qq

)(q

= ?

Solution : 1- Tomas-Fermi method (classic)

2- Lindhard method (quantum mechanical)

Hartree method for Jelly model

02

2

2ion elU U E

m

Page 10: 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

10

2

2

2 i i i ir e r r rm

ext indr r r

Screening effect

Tomas-Fermi dielectric constant

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

Semi classic

2 2

2

kk e r

m ����������������������������

343

322 f

L

kN

343

32

8fkN

nV

Quantum mechanicalclassical

]||[)( 02 nr ii

ind

&

Homogeneous gas

)]([4 3

kfKd

n

Inhomogeneous gas

Page 11: 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

11

1

1)]([

])([

KTk

e

kf

0indn r r e n r n

3

3 2 2

1

4exp 1

2

d kn r

ke r

m

0 3 2 2

1

4exp 1

2

d kn

km

��������������

e r

0 0ind r e n e r n

Tomas-Fermi dielectric constant

Screening effect

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

r

0n r n e r

Basic Tomas-Fermi Relation

is chemical potential

Page 12: 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

12

Tomas-Fermi methodScreening effect

0 0ind r e n e r n

As is small we can expand it: r

1 extq qq

Tomas-Fermi dielectric constant

1q

In cavity 00 k

2 02

41

nq e

q

02ind nr e r

0

0 0ind n

r e n e r n

2 0ind nq e r

ind q X q q 2 0nX q e Cte

2021

kq

q

04 2

002

kn

e

Now we have screening effect on Tomas-Fermi method

Page 13: 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

13

Screening effect

Tomas-Fermi method

Jelly model

Charge distribution

r

Prob2: charge distribution field at position ?r

Answer 2: ext extr q q r

2 4r r 2 4ext extr r

1 extq qq

202

1k

qq

ext Qr

r

2 2sin sinext i q r Qd r r dr d d q r dr d d er

2cos cos

0 0 0 0 0

sin 2 siniqr iqr

r r

Q rdr e d d Q rdr e d

ext i q r Qq d r er

)(qext =

cos siniqr u iqr d du 0

2ext u

r

dvq Q rdr e

iqr

Page 14: 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

14

cos

00 0

1 12 2iqr iqr iqr

r

Q dr e Q dr e eiq iq

20

4 4sin

Q Qqr dr

q q

2

2 2 20 02

4 / 4

1

ext q Q q Qq

k q kqq

3 2 20

4

2i q rd q Q

r eq k

2 sindq q dq d d

2 2 2

cos3 2 2 2 2

0 00

4 1sin

22iqr iqrQ q Q q

r dq e d dq eq k q k iqr

200

siniqr k rq

Q q qr Qe dq e

r q k r

����������������������������

Screening effectTomas-Fermi dielectric constant

Fourier Transform

Page 15: 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

15

ext Qr

r

k rQr e

r

���������������������������� potential shows the screening effect

Screening effectTomas-Fermi dielectric constant

ext r

r r

Potential energy

ext r

r r

Potential energy

Page 16: 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

16

Screening effectTomas-Fermi dielectric constant

We plot potentials in reciprocal lattice:

Without screening effect

Potential energy of one electron

Considering screening effect

ext Qr

r

20

2

4)(

kq

Qq

q

q

Page 17: 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

17

Screening effectLind hard method

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

;

22

F

kk k

n r

0

1 1i k q r

i k rk

q k k q

q ee

T TV V

������������������������������������������

����������������������������

q r

2 2 22

2 2k

kT

m m

0 0 0 01k k k k k

k k k k

V

2

2

2 k k ke r r rm

ext indr r r

0ext r n

])([)( 0nrnrind

&

First perturbation term

Page 18: 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

19

Screening effectScreening effect on Ion-Ion Interaction

Ion-Ion interaction :Direct interaction: coulomb inter.

Indirect interaction: screening inter.

1( )

2total ij

T i ji j

R R

���������������������������� 1

( )2

totalb i j screen

i j

R R

����������������������������

rdrrVRR bjiSC

rqiaa errdq

i

rqi ieqS rdeqqVqSqS rqqiab

q qSC

2

1

jj

rqi

q

bb eqVrV .

rqi

q

eqr

qqSq a

qVqSqV abb &

Atomic form factorStructure factor

Page 19: 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

20

Screening effectScreening effect on Ion-Ion Interaction

rdeqqVqSqS rqqiab

q qSC

2

1

rqiaa errdq

qq aa

rr aa

rdr *

( ) ( )S q S q

qqVqSqSRR abjiSC *

2

1

i

rqi ieqS

i

rqi ieqS * q ji

rqirqiiSC

ji eeRR,2

1 qqV ab

jiq

rrqiabSC

jieqqV,,2

1

Page 20: 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

21

Screening effectScreening effect on Ion-Ion Interaction

( ) ( )q q

i q rq e r d r

* *i q rq e r d r

k k q

4 1

Fk k k k q

qn q

V T T

22 2

8 1

Fk k

q mn q

V k k q

2 22

1 1

2 cosF Fk k k k

Iq kqk k q

22

3 3 20 0 0

sin

2 cos22

Fk

k

dk V dI k dk d

q kq

V

Potential is a real quantity

1 1

k q k q q k k qT T T T

2 2

2k

kT

m

&

Page 21: 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

22

Lind hard method

cos sinz d dz

12

3 2 20 1

22 4 22

2

FkV dz V kdk dz

I k dkqq kqz q zk

20

1 2ln

8 2

Fkq k

kdkq q k

2 2k dkx dx

q q

2

2

11 2 21ln

4 2 4 12 2

FF F

F F

qqkk k

Iqq

k k

2

22 2

118 21ln

4 2 112

FF F

F F

qqm r kk k

n qqV q

k k

( )2 f

q

k

ind q n q

2 2

2

2ind F

F

mk qq q

V k

2

4 2ind

F

qq q

k

ind q q

Screening effectScreening effect on Ion-Ion Interaction

Is q independent on Tomas-Fermi

&

Page 22: 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

23

( )2 f

q

k

q1

1/2

1

Screening effectScreening effect on Ion-Ion Interaction

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

( )Lin r

r

Page 23: 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

24

Lind hard dielectric constantScreening effect

22

44ind ind ind indr r q q

q

2

4 2ind

F

qq q

k

Lind hard relation

2

21

2 F

qq

q k

ext indq q q

2

2 2 F

qq q q q

q k

Lind hard model

0

( )lim ( ) Tomas Fermi

q

qlindherdq

0

lim 12q

F

q

k

Page 24: 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

25

Screening effectLind hard dielectric constant

( )Lin r

r

Tomas-Fermi model

Lind hard model

It has Bessel behavior & we can

see Friedel Oscillation

It is classical yukawa potential by considering

screening effect

Page 25: 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

26

Screening effectLind hard method

2 2 23 24

q qk k

f fdk

g ek qm

��������������

2 2

1

exp 12

kfkm

0 0 0 01k k k k k

k k k

V

20 ind

k kr e f r r r

/ 2 / 223 24 /

k q k qindf fd k

q ek q m

��������������

����������������������������

ind q q q

Page 26: 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

27

12

20

4b a

ZeG q V q q

V q

i q RSC b a

q

R V q q e ������������� �

������������������������������������������

2

20

4 i q RSC

q

ZeR G q e

V q

������������� �

����������������������������

34q

d qV

T b SCR R R ������������������������������������������

2 2

int int

2 sin

direct eraction Indirect eraction

Ze Ze qRG q dq

R R qR

Screening effectLind hard method

tot

R

Attraction &

repulsion

Page 27: 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

28

Screening effectScreening effect on e-e interaction

2e

r r ������������� � 2

k r ree

r r

������������������������������������������

������������� �

2

220

4 e

k k k

����������������������������

Fourier Transpose

Conclusion :

Screening effect

2

2

4 e

k k

����������������������������

Screening effect

e-e interaction becomes weak

Page 28: 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

29

Thanks for your attention

Page 29: 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

30