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
Aage Bohr
Ben Mottelson
(1917 – 1986)
(1922 –
(1926 –
U.S.A
Denmark
USA / Denmark
Between 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
The electrostatic energy associated
with charge distribution (r) in electric
potential (r) is:
 3
eZ    (r ).d r
ELECTRIC CHARGE

 , d 3r
  3
U    (r ) (r )d r
  3
P    (r )r .d r
ELECTRIC DIPOLE

r

+
-

 2 2 3
Q    (r ) 3z  r .d r
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 Q
PROLATE
“Cigar”
Q0
Q0
OBLATE
“Dou-nut”
Q0
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]
The nuclear Quadrupole moment Q
Quadrupole Moment (Barns)
Q
Magic Numbers
The regions of sphericity
Limits of
manmade
nuclides
56
28
16
8
Ni28 (5.6d )
QUESTION –
will mankind
ever make
this double
magic?
O8
40
20
Ca 20
Quadrupole moment on the Shell model
Q0
An extra proton or neutron added to a
closed shell configuration makes an
OBLATE ellipsoid
Q0
A proton or neutron removed
from a closed shell configuration
make a PROLATE ellipsoid
Quadrupole moment on the Shell model
missing
Odd proton
extra


Q    (r )(3z 2  r 2 ).d 3r
  nlj
*

(r )(3z  r ) nlj (r )d r
2
2
3
2 j 1
 r2 
1( j  1)
QM
expression
missing
Odd neutron
extra
Advanced
treatment
For a uniformly charged
sphere:
3 2 3 2 2/3
 r  R  Ro A
5
5
2
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
Nuclide
19
9 10
F
Q
0.06b

Shell Nilsson Expt
0.05
5
2
1
2
1
2
3
2
3
2
3
2
3
2
21
10
Ne11 0.09b
0.09
5
2
23
11
Na12 0.14b
0.11
5
2
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 
and
J  I
1 2
J2
E  I 
2
2I
1 ˆ2
Schroding Eqn : J   E
2I
 2 J ( J  1)
EJ 
,  J ,m  YJ ,m ( ,  )
2I
The nucleus can also vibrate
Breathing mode –
Quadrupole Deformation
First observed in 1977 – very
high energy
Rotating wave with Sherical
Harmonic Wavefunction –
circulates – or vibrates the
nucleus.
Requires nuclear fluid
compressibility
Spin = 2, Parity = +
Nuclear vibrations are bosonic
106
46
Pd 60
The giant dipole resonance
p

n
PHOTON CROSS SECTION (mb)
High energy photon
E
E
14MeV

.c 
.3x10 23 Fs 1
 c
197 MeV .F
 2 x10 22 s 1

PHOTON ENERGY (MeV)
Nuclear magnetic moments
z
s
The magnetic moment of the nucleus
comes about because
j
l
(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.
s





 p  g sp  N ( s / )  5.586 N ( s / )

 n  g sn  N ( s / )  3.826 N ( s / )
proton
For the PROTON we must add the mag mom due to ORBITAL motion to get
the full mag. Mom.


g lp  N

l
g sp  N

s
g lp  1
For the NEUTRON we can write down a similar equation but define that gln=0
After time averaging rapid motion about j then

l (l  1)  34 
1
i  2  N ( g li  g si ) j  ( g li  g si )
j  1 

gln  0
See SEC 6.6
MAGNETIC MOMENT (nuc magnetons)
Magnetic dipole moments
JACKNIFE
ODD NEUTRON
STRETCH
NUCLEAR SPIN J
Shown are the
“Schmidt lines”
Magnetic dipole moments
MAGNETIC MOMENT (nuc magnetons)
Shown are the “Schmidt Lines” named after German physicist T. Schmidt
who discovered these lines empirically in 1937.
ODD PROTON
STRETCH
ODD NEUTRON
JACKNIFE
NUCLEAR SPIN J