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e

|e|Z

q

Coulomb Fields of Finite Charge Distributionsarbitrary nuclear charge distribution with normalization

Coulomb interaction

Expansion of

for |x|«1:

«1

Nuclear Deformations

e

|e|Z

q

Monopoleℓ = 0

Dipole ℓ = 1

Quadrupole ℓ =2

Multipole Expansion of Coulomb InteractionPoint Charges

Nuclear Deformations

Nuclear distribution

A Quantal Symmetry

symmetric nuclear shape symmetric

invariance of Hamiltonian against space inversion

both even or odd

Nuclear Deformations

n even p = +1n odd p = -1

If strong nuclear interactions parity conserving

Restrictions on Nuclear Field

Expt: No nucleus with non-zero electrostatic dipole moment

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

Nuclear Deformations

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

Neutron Electric Dipole Moment

?

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

Measure NMR HF splitting for

Transition energiesDw=4dnE

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

Nuclear Deformations

Experimental Results for dn

dn experimental sensitivity

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

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

Nuclear Deformations

q’

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

Sphere:

Nuclear Deformations

Q0 measures deviation from spherical shape.

z

z

z

z

b

a

Collective and s.p. Deformationscollectivedeformationdisc

collectivedeformationcigar

single particlearound core

singleholearound core

Q0>0 “prolate” Q0<0 “oblate”

Planar single-particle orbit:

Nuclear Deformations

Ellipsoidprincipal axes a, b

Deformation parameter d

y’

q

x’

Spectroscopic Quadrupole Momentz

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

intrinsic

What is measured in Lab system?

finite rotation through q

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’

Nuclear Deformations

Angular Momentum and Q

Qz =maximum measurable maximum spin (I) alignment

Legendre polyn. complete basis set

z

I

Nuclear Deformations

I couples with I to L =2

I

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

Vector Coupling of Spins

I≠0:

mI=I

q

Any orientation

quadratic dependence of Qz on mI

Nuclear Deformations

“The” quadrupole moment

F+DF

E+DE

F

E

Electric Multipole InteractionsInhomogeneous external electric field exerts a torque on deformed nucleus. orientation-dependent energy WQExamples: crystal lattice, fly-by of heavy ions

Axial symmetry of field assumed:

Taylor expansion of scalar potential U:

no mixed dervs.

Nuclear Deformations

monopole WIS

dipole

0

quadrupole WQ

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

No external charge

axial symm

Electric Quadrupole Interactions=Uzz

Nuclear Deformations

Field gradient x spectroscopic quadrupole moment mI2

mI=±2

excitedstate

I=2

mI=±1

mI=0

E2

isomer shift

I=0

mI=0

ground stateUzz=0 Uzz≠0

dN/dEg

Uzz=0

Eg

Quadrupole Hyper-Fine SplittingUse 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 g emission/absorption lines

- Estimates: atomic energies ~ eVatomic size ~ 10-8cmpotential gradient Uz ~ 108V/cmfield gradient Uzz ~ 1016V/cm2Q0 ~ 10-24 e cm2
- WQ~ 10-8 eV
small !

Nuclear Deformations

. .. .

I=8

I=6

I=4

I=2

Experimental Methods for Quadrupole Moments- Small “hf” splitting WQ of nuclear and atomic levels in Uzz≠0
- splitting of X-ray/ g 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 g spectroscopy

scattering experiments Uzz(t)

Nuclear spectroscopy of collective rotations model for moment of inertia

Nuclear Deformations

I=0

b

a

Collective Rotationsb : deformation parameter

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

Nuclear Deformations

Wood et al.,Heyde

Systematics of Electric Quadrupole Moments

Mostly prolate (Q>0) heavy nuclei

Q(167Er) =30R2

odd-Nodd-Z

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

Prolate

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

Nuclear Deformations

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

Oblate

8 20 28 50 82 126

Q0 Systematics

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

Nuclear Deformations

Møller, Nix, Myers, Swiatecki, LBL 1993

Nuclear Deformations

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