Nuclear and Particle
Physics
Lecture 7
Main points of lecture 6
Liquid Drop model – nucleus viewed as being similar to a
droplet of incompressible fluid
(A − 2Z)2
Z2
B(A, Z) = avA − asA − aC 1 / 3 − aa
+ δ (A,Z)
A
A
Pairing
Coulomb
Volume
term
term
term
Asymmetry
Surface
term
term
2/3
Parameters are extracted from fits to experimental data
Model gives good description of changes in binding energy per
nucleon, and therefore M(A,Z), for all nuclei.
Allows predictions of minimum stable isobars, energy released in
fission/fusion …etc
Dr Daniel Watts
3rd Year Junior Honours
Course
Monday January 30th
Notes
Notes
Experimental evidence of shell effects in atomic physics
Experimental evidence of shell effects in the nucleus
Separation energies
Isotopic abundances
Ionization energies
Nucleon number
Atomic radius (nm)
Atomic radii
Z
Neutron capture cross section
Notes
Notes
SHELL MODEL
Experimental evidence of shell effects in the nucleus
Stable Isotopes/ Isotones
ANALOGY:
atomic electron configuration
⇒ high stability at closed-shell structures
(e.g. noble gases)
⇒ chemical properties determined by
valence electrons
ASSUMPTIONS:
ordered structure within nucleus
nucleons move independently in potential well
allowed energy states determined by V(r)
use V(r) in Schrödinger eqn. ⇒ predict quantised energy levels
IDEA:
fill in states according to n and l quantum numbers
try to reproduce nuclear MAGIC NUMBERS
Z, N = 2, 8, 20, 28, 50, 82, (126)
Energy of 1st excited state of even-even nuclei
Question: How can you get independent nucleon motion in a
densely packed nucleus? How do nucleons travel far enough
between collisions to have definite spatial orbits?
Explanation: For nucleons to collide they must be quantum states
available for scattered nucleon(s) to go into – many collisions are
blocked in the nucleus as the possible states which the scattered
nucleons could enter are already filled (Pauli blocking).
Therefore nucleons generally orbit the nucleus as though they
were transparent to each other!
Notes
Notes
What form for the potential?
Short range of the nuclear
force → expect potential to
be similar shape to the
proton (charge) distribution
NB. Proton distribution ~
same size as matter (protan
and neutron) distribution
e.g. 208Pb: 82 prot 126 neut.
∴ neutrons more densely
packed!
Exclusion principle
(applies independently to neutron and protons)
Shell model
+
spin-orbit interaction L·S
(Mayer, 1949)
Energy splitting ∆E ~ L·S
ENERGY of nucleon DECREASES when L // S
J = L + S has MAXIMUM possible value when L // S
higher J ⇒ lower energy
Use: n, l, j and mj as good quantum numbers
Exclusion principle ⇒ 2j + 1 possible values of mj
NOTE: labelling the state with j includes the effect of spin
⇓
occupancy of each level:
2(2l +1)
ml
ms
infinite
square well potential
harmonic
oscillator potential
quantum energy states
of potential well
+ angular momentum effects
spin-orbit
splitting
multiplicity
occupation of state = 2j+1
closed shells
indicated by
“magic numbers”
of nucleons
Only smallest magic numbers are reproduced – even with a realistic
potential which reflects the measured charge distribution
PUZZLING PROBLEM
Notes
Differences between atomic and nuclear shell models
1.
Spin-orbit interaction much stronger in nuclei than in atoms
2.
Its sign is opposite to the atomic case
3.
Nuclear spin-orbit interaction is NOT of MAGNETIC origin,
but rather a PROPERTY of NUCLEAR FORCE
4.
Atoms have either LS or JJ coupling
Nuclei almost all have JJ coupling as spin-orbit interaction
is much stronger in nuclei than in atoms
n = radial behaviour of wave function
radial node quantum number
l = angular behaviour of wave function
orbital quantum number
N.B. n is NOT the same as in atomic physics ( nprinc = nrad + l )!
⇓
there are no restrictions on the value of l for any given n !
(see for example Eisberg & Resnick, p.536-541)