The Shell Model of the Nucleus

5. Nuclear moments.

The Collective Model of the Nucleus

[Sec. 6.2, 6.4, 6.5, 6.6, 6.7 Dunlap]

The Bohr-Mottelson Collective Model of the Nucleus

James Rainwater

(1917 – 1986)

U.S.A

Aage Bohr

(1922 –

Denmark

Ben Mottelson

(1926 –

USA / DenmarkBetween about 1950 – 1955, Bohr and Mottelson followed the idea on collective nuclear motion suggested by Rainwater. All three received the 1975 Nobel Prize – “For their work on the connection of collective nuclear motion with single particle motion”

Electric nuclear moments

rdrrU 3)()(

The electrostatic energy associated with charge distribution (r) in electric potential (r) is:

r

rd3 ,

rdreZ 3.)( rdrrP

3.)( rdrzrQ 322 .3)(

ELECTRIC CHARGE ELECTRIC DIPOLE ELECTRIC QUADRUPOLE

+ -++ +

-

-

Electric and Magnetic nuclear moments

“For symmetry reasons the electric dipole moment of the nucleus (as well as all other static multipole moments with odd parity; for example – magnetic monopole, magnetic quadrupole or electric octupole) must VANISH.

WHY? Because the both the strong and the electric forces are invarient under the parity operation – which means we should never get different nuclear properties under space inversion

The nuclear Quadrupole moment QPROLATE “Cigar”

OBLATE “Dou-nut”

Unlike atoms – nuclei can easily distort from spherical state. Unlike atoms the potential the nucleons move in is formed by the nucleons themselves. [In an atom it is the potential coming from the nucleus that dominates]

0Q

0Q

0Q

The nuclear Quadrupole moment Q

Q

Magic Numbers

Qu

adr

upo

le M

om

en

t (B

arn

s)

The regions of sphericity

)6.5(Ni285628 d

816

8 O

204020 Ca

QUESTION – will mankind ever make this double magic?

Limits of manmade nuclides

Quadrupole moment on the Shell model

A proton or neutron removed from a closed shell configuration make a PROLATE ellipsoid

An extra proton or neutron added to a closed shell configuration makes an OBLATE ellipsoid

0Q 0Q

Quadrupole moment on the Shell model

Odd proton

Odd neutron

missing

extra

missing

extra

2

322*

322

)1(1

12-

)()3()(

).3)((

rj

j

rdrrzr

rdrzrQ

nljnlj

QM expression

Advanced treatment

For a uniformly charged sphere:

3/2222

5

3

5

3ARRr o

Deformation effects shell states Nilsson Model

In a nonspherical nucleus a definite direction is defined – the direction 3 along the axis. The total angular momentum j is projected along the 3rd axis to give a new quantum no. K. = ± j, ±(j-1), ± (j-2) etc. This phenomenon also splits the degeneracy previously seen for spherical (closed shell) nuclei

The Nilsson Shell Model Plot for Deformed Nuclei

23

23

25

122311

23

23

25

112110

21

21

25

10199

0.11 0.14b Na

0.09 0.09b Ne

0.05 0.06b F

Expt Nilsson Shell Q Nuclide

Irrotational collective rotation of the nucleus

Nuclear collective rotation occurs around an axis perpendicular to the symmetry axis “3”

The rotation is called irrotational because the nucleus is not quite solid – It is largely the “skin” of “outer shape” of the nucles that is rotating

Collective rotational motion

J

Consider energy of a rigid rotator with moment of inertia

),( ,2

)1( E

ˆ2I

1 :Eqn Schroding

2I

J

2

1

,,

2

J

2

22

mJmJ YI

JJ

EJ

IE

and IJ

The nucleus can also vibrate

Breathing mode –

First observed in 1977 – very high energy

Requires nuclear fluid compressibility

Quadrupole Deformation

Rotating wave with Sherical Harmonic Wavefunction – circulates – or vibrates the nucleus.

Spin = 2, Parity = +

Nuclear vibrations are bosonic

6010646 Pd

The giant dipole resonance

PHOTON ENERGY (MeV)

PH

OT

ON

CR

OS

S S

EC

TIO

N

(mb)

High energy photon122

123

102

103..197

14.

sx

FsxFMeV

MeVc

c

EE

p

n

Nuclear magnetic moments

l

j

proton

z The magnetic moment of the nucleus comes about because

(1) We have charged particles – protons moving around the center of the nucleus (i.e. p, d, f, g etc states)

(2) Both protons and neutrons have their own INTRINSIC magnetic moments.

)/(826.3)/(

)/(586.5)/(

ssg

ssg

NNsnn

NNspp

s

s

For the PROTON we must add the mag mom due to ORBITAL motion to get the full mag. Mom.

sl

NspNlp gg

For the NEUTRON we can write down a similar equation but define that gln=0

1lpg

0ln gAfter time averaging rapid motion about j then

1

)1()()( 4

3

21

j

llggjgg silisiliNi

See SEC 6.6

Magnetic dipole moments

ODD NEUTRON

MA

GN

ET

IC M

OM

EN

T (

nuc

mag

neto

ns)

NUCLEAR SPIN J

JACKNIFE

STRETCH

Shown are the “Schmidt lines”

Magnetic dipole moments

ODD NEUTRON

MA

GN

ET

IC M

OM

EN

T (

nuc

mag

neto

ns)

NUCLEAR SPIN J

ODD PROTON

JACKNIFE

STRETCH

Shown are the “Schmidt Lines” named after German physicist T. Schmidt who discovered these lines empirically in 1937.