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W. Udo Schröder, 2007

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Coulomb Fields of Finite Charge Distributions

|e|Z

e

z

r

r

r r r

r arbitrary nuclear charge distribution

with normalization 3 1d r r

Coulomb interaction

3( )e r

V r e dZ rr r

Expansion of 1

r r for r r

1 / 2 1 / 21 2 2 2

1 2

2 cos

11 ( 2cos )

r r r r r r rr

r rr r r

1 2 2

3

1 1 31 1

2 2 41 3 5

.....2 4 6

x x x

x

«1

for |x|«1:

221

0

1 11 cos 3cos 1 ...

21

(cos )r r r

r r Pr rr r r

20 1 1

11, cos cos , cos 3cos 1

2Legendre Polynomials P P P


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Multipole Expansion of Coulomb Interaction

|e|Z

e

z

r

r

r r r

3

2 2

2 23

3

3

3

2

3

2

2

1

cos

13

( )

...

c

.

o2

.

s 1

.

er r

e r d

e r

r r

er

V r e d rr r

e e Z

e e Z

e e Z

Z

Zd r r

Zd r r

Zd r r Q

r r

Monopoleℓ = 0

Dipole ℓ = 1 Quadrupole ℓ =2

Point Charges

Nuclear distribution


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A Quantal Symmetry

symmetric nuclear shape symmetric

invariance of Hamiltonian against space inversion

r

( ) (

1,

)

, 0

1E E E E

E

H r H r Parity Operator r r

H simultaneous eigen functions

evenH E

odd

3

:

( ) ( )(cos ) (cos )n nn n nel

Electrostatic multipole moments of r

M dr P r Pr r r

both even or odd

n even = +1n odd = -1

0elnM for n odd

If strong nuclear interactions parity conserving


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Restrictions on Nuclear Field

Expt: No nucleus with non-zero electrostatic dipole moment

Consequences for nuclear Hamiltonian (assume some average mean field Ui for each nucleon i):

2

11

2 22 2 2 2

2 2

11

1 11

ˆˆ ( ,...., ; ...)2

ˆ ˆ, 0

ˆˆ ˆ , 0( )

ˆ( ,...., ; ...), 0

( ,...., ; ...) ( , ....,

Ai

i Ai i

i i

A

i Ai

A

i A ii

pH U r r parity conserving

m

H

Since p px x

and U r r

U r r U r

1

11

; ...)

(| | , ....,| | ; ...)

A

Ai

A

i Ai

r

U r r

Average mean field for nucleons conserves inversion invariant, e.g., central potential


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Neutron Electric Dipole Moment

- + B 2

22

2n

n

n

n

n

nIntera

d

Bction d

B E

EB

E

d

qn =0, possible small dn ≠0. CP and P violation could explain matter/antimatter asymmetry

Measure NMR HF splitting for E B

Transition energies=4dnE

s

nd

B

E

B=0.1mG, tune with Bosc B. E = 1MV/m = 30Hz spin-flipof ultra-cold (kT~mK)EEkinkin=10=10-7-7eV, eV, =670Å =670Åneutrons in mgn.bottleguided in reflecting Ni tubes

PNPI (1996): dn < (2.6 ± 4.0 ±1.6)·10-26 e·cm

ILL-Sussex-RAL (1999): dn < (-1.0 ± 3.6)·10-26 e·cm

dn experimental sensitivity

From size of neutron (r0≈ 1.2fm): dn 10-15 e·m.So far, only upper limits for dn

Experimental Results for dn


7

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Intrinsic Quadrupole Moment

Consider axially symmetric nuclei (for simplicity), body-fixed system (’), z =z’ symmetry axis

’

z

r 3 2 2

0

2 20

( ) 3cos

3

1

cos Q

Q eZ d r

z

r

r r

r

z

Sphere: 2 2 2 2 2

0

3

0sph

r x y z z

Q

Q0 measures deviation from spherical shape.


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Collective and s.p. Deformations

Q0>0 “prolate” Q0<0 “oblate”

collectivedeformation

cigarsingle hole around core

zz z z

collectivedeformation

discsingle particle around core

z

b a

Planar single-particle orbit: 20 3spQ eZ z 2 2r eZ r

Ellipsoidprincipal axes a, b

2 2 20

; ; :2

2 45 5

coll

a b RR R b a

R

Q eZ b a eZ R

Deformation parameter


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0

Spectroscopic Quadrupole Moment

zbody-fixed {x’, y’, z’}, Lab {x, y, z}Symmetry axis z defined by the experiment

2 20 3Q eZ z r intrinsic

What is measured in Lab system?

2

0 2

2 20

13 ....... 3cos 1

cos( )

2

z

zQ eZ z r Q e

Q

Z

Q P

finite rotation through

Measured Q depends on orientation of deformed nucleus w/r to Lab symmetry axis. define Qz as the largest Q measurable.

How to control or determine orientation of nuclear Q?

Nuclear spin to symmetry axis, no quantal rotation about z’

z’

x’

y’


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Angular Momentum and Q

Qz =maximum measurable maximum spin (I) alignment

2 2 2 2

32

ˆˆ ˆ ˆ ˆ3 3

( ) (cos ) ( ) . .

z m II I I

I I I

I IQ eZ z r z r Q

m I m I

d r r P r nucl wave funct

Legendre polyn. complete

basis set

2

0( , ) ( , ) (cos )

I

I Id P

23 32 2

0( ) (cos ) ( ) (cos ) (cos )

I

z I IQ d r r P r d r P P

2

0

0 2 2 0, 1 2zQ for I I

for any Q

Spins too small to effect alignment of Q in the lab.

z I

I

I couples with I to L =2


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Vector Coupling of Spins

mI=I

z

1I I

I 2

01ˆ 3cos 12z

I I

I IQ Q Q

m I m I

I≠0:

1 cos cos1

II

m I I II I

0 (2 1

20 ,1 2)

10z for I

IQ Q

I

23 ( 1)

( )2 1

Iz I z I

m I IQ m Q m I

I I

Any orientation

quadratic dependence of Qz on mI

“The” quadrupole moment


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Electric Multipole Interactions

z

E+E

E

F+F

F

Inhomogeneous external electric field exerts a torque on deformed nucleus. orientation-dependent energy WQ

Examples: crystal lattice, fly-by of heavy ions

E U

Taylor expansion of scalar potential U:

2 2 2

2 2 20 2 2 20

0 0 0

1( ) ...

2U U U

U r U r U x y zx y z

0r center of nucleus

2 2

2, ,ij i

i j

U Ux x y z

x x x

Axial symmetry of field assumed:

3

2 2 22 2 2

0 2 2 200 0 0

( ) ( )

...2

W eZ d r r U r

eZ U U UeZU eZ U r x y z

x y z

monopole WIS

dipole

0 quadrupole WQ

no mixed dervs.

WIS: isomer shift, WQ: quadrupole hyper-fine splitting


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Electric Quadrupole Interactions

2 22 2 2

2 20 0

2QeZ U U

W x y zx z

Maxwell equs. 4 (0) 0U

2 2

2 20

2

200

12

U U U

zx y

2 2 2

2 2 20

U U U

x y z

No external charge

22 2 2

20

22 2 2

20

24

24

QeZ U

W x y zz

eZ Ur z z

z

2

20

4( )Q z I

eZ UmW Q

z

Field gradient x spectroscopic quadrupole moment mI

2

axial symm=Uzz


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Quadrupole Hyper-Fine Splitting

Use external electrostatic field, align Q by aligning nuclear spin I,Measure interaction energies WQ (I >1/2 ) Quadrupole hyper-fine splitting of nuclear or atomic energy levels

Slight “hf” splitting of nuclear and atomic levels in Uzz≠0

splitting of emission/absorption lines

dN/dE

E

Uzz=0

Estimates: atomic energies ~ eVatomic size ~ 10-8cmpotential gradient Uz ~ 108V/cmfield gradient Uzz ~ 1016V/cm2

Q0 ~ 10-24 e cm2

WQ ~ 10-8 eVsmall !

mI=±2

mI=±1

I=0

I=2

mI=0

mI=0E2

ground stateUzz=0 Uzz≠0

excitedstate

isomer shift


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Experimental Methods for Quadrupole Moments

Small “hf” splitting WQ of nuclear and atomic levels in Uzz≠0

splitting of X-ray/ emission/absorption lines

Measurable for atomic transitions with laser excitations

nuclear transitions with Mössbauer spectroscopy

muonic atoms:

107 times larger hf splittings WQ with X-ray and spectroscopy

scattering experiments Uzz(t)

Nuclear spectroscopy of collective rotations model for moment of inertia 2

0

2

1

2

(10 20)2

I

I IE

Q

keV

I=0I=2I=4

I=6

I=8

. .

. .


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Collective Rotations

20 0

4( , ) 1 ( , )

3 5R

R R YR

z

b a : deformation parameter

20

2

0

20

20

31 0.16

5

12

. . :

21 0.31

5:

98

I

rig

irr

eZR

Rotational and inversion

symmetry even I

E I I

rigid body mom o inertia

MR

hydro dynamical

MR

Q

Nuclei with large Q0 consistent w. collective rotations lanthanides, actinides

Wood et al.,Heyde

2

15 182

keV

;2

a bR R a b


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Systematics of Electric Quadrupole Moments

2

Q R

RZR

odd-Nodd-Z

Q(167Er) =30R2

Prolate

Oblate

8 20 28 50 82 126

Q<0 : e.g., extra particle around spherical core. pattern recognizable

1729 5163 123

8209

8 3, , ,O Cu Sb Bi

Q>0 : e.g., hole in spherical core pattern not obvious. If such nuclei exist, weak effect of hole for Q

27 55 115 176 16713 25 49 71 68, , , ,Al Mn In Lu Er

Tightly bound nuclei are spherical: “Magic” N or Z = 8, 20, 28, 50, 82, 126, …

Tightly bound nuclei are spherical: “Magic” N or Z = 8, 20, 28, 50, 82, 126, …

Mostly prolate (Q>0) heavy nuclei


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Q0 Systematics

Møller, Nix, Myers, Swiatecki, LBL 1993

Q0 large between magic N, Z numbersQ0≈0 close to magic numbers


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