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Sizes. Gross Properties of Nuclei. Nuclear Deformations and Electrostatic Moments. z. e. |e|Z. q. Coulomb Fields of Finite Charge Distributions. arbitrary nuclear charge distribution with normalization. Coulomb interaction. Expansion of. for |x|«1:. «1. z. e. |e|Z. q.

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sizes

Sizes

Gross Properties of Nuclei

Nuclear Deformations

and Electrostatic Moments

coulomb fields of finite charge distributions
z

e

|e|Z

q

Coulomb Fields of Finite Charge Distributions

arbitrary nuclear charge distribution with normalization

Coulomb interaction

Expansion of

for |x|«1:

«1

Nuclear Deformations

multipole expansion of coulomb interaction
z

e

|e|Z

q

Monopoleℓ = 0

Dipole ℓ = 1

Quadrupole ℓ =2

Multipole Expansion of Coulomb Interaction

Point Charges

Nuclear Deformations

Nuclear distribution

a quantal symmetry
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
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
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 d n
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

intrinsic quadrupole moment
z

q’

Intrinsic Quadrupole Moment

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

Sphere:

Nuclear Deformations

 Q0 measures deviation from spherical shape.

collective and s p deformations
z

z

z

z

z

b

a

Collective and s.p. Deformations

collectivedeformationdisc

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

spectroscopic quadrupole moment
z’

y’

q

x’

Spectroscopic Quadrupole Moment

z

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
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
zVector Coupling of Spins

I≠0:

mI=I

q

Any orientation

quadratic dependence of Qz on mI

Nuclear Deformations

“The” quadrupole moment

electric multipole interactions
z

F+DF

E+DE

F

E

Electric Multipole Interactions

Inhomogeneous 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

electric quadrupole interactions
Maxwell equs. 

No external charge

axial symm

Electric Quadrupole Interactions

=Uzz

Nuclear Deformations

Field gradient x spectroscopic quadrupole moment  mI2

quadrupole hyper fine splitting
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 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 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

experimental methods for quadrupole moments
. .. .

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

collective rotations
z

b

a

Collective Rotations

b : deformation parameter

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

Nuclear Deformations

Wood et al.,Heyde

systematics of electric quadrupole moments
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

q 0 systematics
Q0 Systematics

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

Nuclear Deformations

Møller, Nix, Myers, Swiatecki, LBL 1993

slide20
No more deformations!

Nuclear Deformations

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