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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.
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