Nuclear and ParticlePhysics

3rd Year Junior HonoursCourse

Monday January 30th

Dr Daniel Watts

Lecture 7

Main points of lecture 6

3/13/2

AZaAaAa)Z,A(B

2

Csv −−= + δ (A,Z)A

)Z2A(a2

a−−

Liquid Drop model – nucleus viewed as being similar to a droplet of incompressible fluid

Volumeterm Asymmetry

termSurface

term

Coulombterm

Pairingterm

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 infission/fusion …etc

Notes Notes

Experimental evidence of shell effects in atomic physics

Atomic radii

Ionization energies

Ato

mic

rad

ius

(nm

)

Z

Separation energies Isotopic abundances

Neutron capture cross section

Nucleon number

Experimental evidence of shell effects in the nucleus

Notes Notes

Experimental evidence of shell effects in the nucleus

Stable Isotopes/ Isotones

Energy of 1st excited state of even-even nuclei

ASSUMPTIONS: ordered structure within nucleusnucleons move independently in potential wellallowed energy states determined by V(r)

SHELL MODEL

ANALOGY: atomic electron configuration⇒ high stability at closed-shell structures

(e.g. noble gases)⇒ chemical properties determined by

valence electrons

use V(r) in Schrödinger eqn. ⇒ predict quantised energy levels

IDEA: fill in states according to n and l quantum numberstry to reproduce nuclear MAGIC NUMBERS

Z, N = 2, 8, 20, 28, 50, 82, (126)

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

infinite square well potential

harmonic oscillator potential

Only smallest magic numbers are reproduced – even with a realisticpotential which reflects the measured charge distribution

Exclusion principle(applies independently to neutron and protons)

⇓

occupancy of each level: 2(2l +1)

ms

ml

What form for the potential?

PUZZLING PROBLEM

Short range of the nuclear force → expect potential to be similar shape to the proton (charge) distribution

NB. Proton distribution ~ same size as matter (protanand neutron) distribution e.g. 208Pb: 82 prot 126 neut. ∴ neutrons more densely packed!

(Mayer, 1949)

multiplicity

occupation of state = 2j+1

Shell model +

spin-orbit interaction L·S

Energy splitting ∆E ~ L·S

ENERGY of nucleon DECREASES when L // S

J = L + S has MAXIMUM possible value when L // Shigher 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

closed shellsindicated by

“magic numbers”of nucleons

quantum energy states of potential well

+ angular momentum effects

spin-orbit splitting

Notes

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

Differences between atomic and nuclear shell models

n = radial behaviour of wave functionradial node quantum number

l = angular behaviour of wave functionorbital quantum number

(see for example Eisberg & Resnick, p.536-541)

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 !