Post on 03-Dec-2014
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