Electric quadrupole Electric quadrupole momentmoment

Introduction to quadrupoleIntroduction to quadrupole

A general distribution of electric charge may be characterized by its net charge

its dipole moment, its quadrupole moment and higher order moments .

Radiation from a monopole sourceRadiation from a monopole source

Radiation from a dipole sourceRadiation from a dipole source

Radiation from a linear quadrupole Radiation from a linear quadrupole sourcesource

3 spatial regions3 spatial regions

�� 1)the near (static) zone1)the near (static) zone

�� 2)the intermediate (induction) zone2)the intermediate (induction) zone�� 2)the intermediate (induction) zone2)the intermediate (induction) zone

�� 3)the far (radiation) zone 3)the far (radiation) zone

Mass or charge elementMass or charge element

Electric potentialElectric potential

∫ ′−′′

=rr

vdrr

)(

4

1)(

0

ρπε

φ′− rr4 0πε

∫−= drEr .)(φ

=

′

−

′+=′−

′−=

−−−

−

cos212

1

2

211

1

θr

r

r

rrrr

rrR

+

′−

′+

′−

′−

...)cos2(8

3)cos2(

2

11

1 22

2

2

2

2

θθr

r

r

r

r

r

r

r

r

rr

Multipole potential in generalMultipole potential in general

∫ ∫ +′′′+′′= cos)(

1)(

11)( vdrrvdrrV θρρ

( )∫

∫ ∫

+′−′′

+′′′+′′=

...1cos32

1)(

1

cos)()(4

)(

223

20

vdrrr

vdrrr

vdrr

rV

θρ

θρρπε

′=′

′→′

zr

rrr

cos

)()()( *

θ

ψψρ

22)()( rrZZ −=′⇒′−→′ ψψ

Why the second term is zero?????

∫ ′−′′′=

′=′

vdrreQ

zr

)1(cos)(

cos

22 θρ

θ

rr ′>>

Potential expansion in general Potential expansion in general

)(1

)(1

4

1)(

4

1)(

00

+′′′⋅

+′′=′−

′′=

∫

∫ ∫

vdrrr

vdrrrr

vdrr

ρ

ρπε

ρπε

φ

rr

...)(32

1

)(4

1

23

,5

30

+′′′−′′

+′′′⋅

∑ ∫

∫

′

vdrrxxr

xx

vdrrr

r

ijji v

jiji ρδ

ρπε

r

Electric quadrupole fieldElectric quadrupole field

( )

)1cos3()(

)(3

22

2

′−′′′=

′′′−′′=

∫

∫

vdrreQ

vdrrxxQ ijjiij

θρ

ρδ

...030

...030

)1cos3()(

22

2

⇒<′−′⇒<⇒>′−′⇒>

′−′′′= ∫

rzQ

rzQ

vdrreQ θρ

Interaction of energy between Interaction of energy between nuclear nuclear

charge distribution & external charge distribution & external potentialpotential

∫ ∫

...)()(

)()()()(

2

22

0

+

′∂∂′′+

′∂

∂′′+

′′=′′′==

∫∫

∫ ∫

z

Vzrdv

z

Vzr

vdxVrvdrVrqE

extext

extvext

ρρ

ρρ

( )0

2

22

2

1cos

2

3

4

1

=

∂∂

−′=z

extQ z

VQeE θ

+==

)1(cos

II

mI

I

I Zθ

→∂

∂−

+−==⇒

→−′

)()12(

)1(3

4

1

)2

1cos

2

3(

2

22

2

Z

V

II

IImIeQE

Qe

extQ

θ

][ −−=⋅

⇒⋅++=→+=

⋅=⋅−=

)()()(1

2

)(

222

222

s

SLJSL

SLSLJSLJ

SLrfBE µ

][][

=

−=→

=

−=→+=

+−+−+>=⋅<

−−=⋅

00232.21

)1()1()1(2

1

)()()(2

2

s

Bss

l

Bll

slj

g

Sg

g

Lg

SSLLJJSL

SLJSL

hh

h

µµµµµµµ

SpinSpin--orbit coupling effectorbit coupling effect

→−=

−≠

jg

Jg

landaeBlJ

BlJ

µµ

µµh

)1(2

)1()1()1(

)1(2

)1()1()1(

++++−+

++

+−+++=

JJ

SSLLJJg

JJ

SSLLJJgg

s

lJ

J

S

L

Jµ Lµ

Sµ

Lets return to nucleusLets return to nucleus

==

∝⋅−=

e

JB

BE

e

eIM

hµµµ

µ

][ )1()1()1(2

1

2

2,

2

222

+−+−+>=⋅<

+⋅+=+=

==

JJIIFFJI

JJIIF

JIF

mp

eIg NNII

h

hµµµ

Sodium Sodium

=⇒

==

→→=2

1

2

10

3)2

3(

2

1223 Js

lSINa

==

→+⋅⋅⋅⋅⋅⋅⋅−=+−=

=

2

1

2

1

2

3

2

1

2

3....

222 2

F

FJIJIF

s

SPLITTING OF ENERGY LEVELs

=+⋅⋅⋅−=→=

=+⋅⋅⋅⋅−=→==→

2,113131

2

3,

2

1

2

11

2

11

2

1,132

Fj

jslP

=−⋅⋅⋅−=→=

=+⋅⋅⋅−=→=→+=

3,2,1,02

1

2

3

2

3

2

3

2

3

2,12

1

2

3

2

1

2

3

2

1

Fj

FjJIF

[ ]

][ )1()1()2)(1(

)()1(

12

2

+=+−++

=>⋅<−>⋅<

=−+=∆

⋅=

+h

h

FAFFFFA

JIJIA

FEFEE

JIAE

FF

][

181.3612

4.164.1601

)1()1()2)(1(2

=⇒=∆⇒=→=

=⇒=∆⇒=→=

+=+−++

hh

hh

h

AMHZ

EFF

AMHZ

EFF

FAFFFFA

[ ]24

2

2

2

1

)12()12(2

)1()1(23

3

I

JJII

JJIIJIJI

z

VeQEQ

=

−−

++−>⋅<+>⋅<

∂∂=

−−hh

2

1

2

1 ,0

2

12 PSE

J

Iif Q ⇒=→

=

=

z

VeQBBAE

P

∂∂=+=∆

→

→ 2

2

23

2

32

3

BAE

BAEz

−=∆−=∆

∂

→

→

01

12

2

2

The coupling of Q (a property of the nucleus) with an EFG

(a property of a sample) is (a property of a sample) is called the quadrupole interaction

Determining parameters in 2 Determining parameters in 2 isotopesisotopes

1)(IA

Iµ

=2

1

2

1

)(

)(

I

IA

A

Iµ=

The effect on the nuclear energy levels for a transition, The effect on the nuclear energy levels for a transition, such as in Fe or Sn, for an asymmetric charge such as in Fe or Sn, for an asymmetric charge

distribution. The magnitude of quadrupole splitting is distribution. The magnitude of quadrupole splitting is shown shown

Presure field produced bz a Presure field produced bz a longitudinal quadrupolelongitudinal quadrupole