Nuclear and Particle
Physics
Lecture 8
Main points of lecture 7
SEMF gives good description of masses and binding
energy of nuclei
But …See strong evidence for shell like
structure in the nucleus
Magic numbers – certain numbers of
protons or neutrons produce very
stable nuclei.
Due to shell closure (e.g. as seen for
noble gases in atomic physics)
From analogy with atomic physics try a theory where
nucleons move independently in a central potential
→ Shell model
Choose a realistic
potential
(reflecting
radial distribution
of
nucleons in the
nucleus)
Dr Daniel Watts
3rd Year Junior Honours
Course
Thursday February 2nd
Don’t reproduce magic numbers !!
Need to introduce spin-orbit coupling L.S
Notes
Notes
Experimentally determined Nuclear properties
Shell Model predictions
&
single-particle features
Nuclear states characterized by two quantum numbers:
J = total angular momentum (nuclear spin)
π = parity
Ground state properties:
positron decay
even Z, even N ⇒
J π = 0+
even Z, odd N
odd Z, even N
Jπnucleus = Jπ
⇒
STABLE
unpaired nucleon
remaining part of configuration (i.e. closed shell)
⇒ inert core with Jπ = 0+
valence
nucleon
17
8
1d5/2
1d5/2
1p1/2
1p3/2
1p1/2
1p3/2
1s1/2
neutrons
17
9
Examples
O
protons
1s1/2
neutrons
electron decay
F
protons
both ground states have: Jπ=5/2+
Two nuclei (1) and (2) with Z1=N2 and N1=Z2 are called
MIRROR NUCLEI
Relative ordering of ground-state + excited state very similar
⇒ p-p and n-n interactions are very similar (charge symmetry)
Difference in ground state energy due to additional electrostatic
potential energy in nucleus with higher Z (see tutorial question)
http://www.nndc.bnl.gov/wallet/wcccurrent.html
Notes
Notes
Energy level scheme for
17O
and
SHELL MODEL SUMMARY
17F
ordered structure within nucleus
nucleons move independently in potential well
allowed energy states determined by V(r)
V(r) =
Remember
− V0
⎞
⎛
1 + exp⎜⎜ r − R ⎟⎟
a ⎠
⎝
Woods-Saxon
Coulomb repulsion
adds to potential
Typical values:
V0 ~ 50 MeV
R ~ 1.25 A1/3
a ~ 0.5 fm
Neutron
potential well
Proton
potential well
But! Only smallest magic numbers are reproduced
introduce strong inverted spin-orbit interaction
Remember
V(r) = VWS + Vso(r)lxs
l = orbital angular momentum of individual nucleon
s = intrinsic spin of individual nucleon
j=l+s
17
8
O
17
9
F
⇒ j = l ± 1/2
(if l=0 then j=+1/2 only!)
energy splitting = ½(2l+1) =2
higher j ⇒ lower energy
Notes
Notes
Examples
Nuclide
Z
N
Shell Model
Observed Ground State
17O
8
9
l = 2; j = 5/2
J = 5/2; + parity
17F
9
8
l = 2; j = 5/2
J = 5/2; + parity
43Se
21
22
l = 3; j = 7/2
J = 7/2; - parity
209Pb
82
127
l = 4; j = 9/2
J = 9/2; + parity
209Bi
83
126
l = 5; j = 9/2
J = 9/2; - parity
Now let’s build a nucleus…
spuds if pug dish of pig
spdsfpgdshfpig
Exercise:
Determine spin and parity assignments for the ground states
5He, 7Li, 11C, 15N, 19O
of the following nuclei:
Woods-Saxon
potential
Woods-Saxon
potential
+
spin-orbit interaction
Notes
Notes
What about excited states in the shell model?
Magnetic moments arise from:
i) intrinsic magnetic moments of nucleons
ii) magnetic moment from oribital angular momentum of nucleons
Ground state properties
Given by unpaired nucleon with all nucleons in
lowest possible energy states.
Magnetic moments of even-even nuclei are zero.
Excited states
Shell model can explain the spins and parities
of some excited states through reconfiguration of
nucleons to give different valence nucleon
17
8
Shell model predictions of magnetic moments of nuclei
For nuclei with an odd neutron or proton shell model predicts two possible
values corresponding to the two possible orientations of the nucleon spin
with respect to the orbital vector for the valence nucleon
O
1/2-
1/2+
Jπ=5/2+
2s1/2
1d5/2
1p1/2
1p3/2
1s1/2
2s1/2
1d5/2
1p1/2
1p3/2
1s1/2
2s1/2
1d5/2
1p1/2
1p3/2
1s1/2
Note : Mirror nuclei show similar orderings and energies in their
excited states (e.g. 17O and 17F)
Measured values
tend to lie
between the shell
model predictions
Dipole moment more
sensitive test of model
than nuclear spins and
parities
Notes