PHY357 – Lecture 18

Nuclear Models

Liquid Drop model – brief review

Fermi Gas model

Shell model

Collective model

March 19, 2018

Nuclear Models

March 19, 2018

!  All models of nuclei are qualitative with tuned parameters !  Based on measurements + some intuition from other physical systems

!  One such analogue system is the atom !  Atomic structures were ‘understood’ only with quantum mechanics

!  With the Coulomb potential in Schrodinger Equation

!  Knowing the nature of the force (Coulomb potential) was not necessarily the first step in understanding the atomic system

!  We still can’t calculate based on the fundamental physics of the nuclear force !  Makes it challenging to pin down nuclear structure from first principles

!  Each of the models explains ‘some’ aspects of the nuclear structure

General Feature of Nuclear Models

!  General characteristics of all models !  Short range force – beyond a few fm nucleons are un-bound

!  They are essentially free to move around ‘inside’ the nucleus (r<R)

!  ‘hard core’ minimum radius, δ, where potential becomes infinite

!  More specific features !  The nuclear force is charge symmetric (pp, pn, nn all same)

!  Force appears to be spin dependent (spin 1 stronger than spin 0)

!  Force saturates beyond a few nucleon radii

!  Next-to-nearest neighbors don’t seem to interact

!  Can explain many of these features with pion-exchange

!  Nuclear analog of covalent bond (shared electron between atoms)

!  Neighboring nuclei ‘share quarks’ – must be same colour of quark

March 19, 2018

General Feature of Nuclear Models

!  General characteristics of all models !  Short range force – beyond a few fm nucleons are un-bound

!  They are essentially free to move around ‘inside’ the nucleus

!  ‘hard core’ minimum radius, δ, where potential becomes infinite

!  More specific features !  The nuclear force is charge symmetric (pp, pn, nn all same)

!  Force appears to be spin dependent (spin 1 stronger than spin 0)

!  Force saturates beyond a few nucleon radii

!  Next-to-nearest neighbors don’t seem to interact

!  Can explain many of these features with pion-exchange

!  Nuclear analog of covalent bond (shared electron between atoms)

!  Neighboring nuclei ‘share quarks’ – must be same colour of quark

March 19, 2018

Liquid Drop Model (review)

!  Based on physics of incompressible liquid

!  Include classical physics of liquid in model !  Surface tension

!  Nearest neighbor interactions only

!  Proton-proton Coulomb repulsion

!  Add some additional “Quantum” effects by hand !  Equal numbers of protons and neutrons are preferred

!  nn and pp pairs preferred

!  Taken together this leads to Semi-Empirical Mass Formula:

!  Can predict general features of nuclear fission from here

March 19, 2018

B.E = �a1A+ a2A2/3 + a3Z2

A1/3 + a4(N�Z)2

A ± a5A3/4

Fermi Gas Model

!  One of the first attempts to incorporate QM

!  Describe nucleus as proton/neutron gas !  Confined to the nuclear volume

!  Nucleons occupy energy levels inside well

!  Potential is spherically symmetric !  Radial size determines nuclear radius

!  Depth adjusted to get binding energy

!  Difference between p and n-wells

!  Changes with A to describe valley of stability

!  N=Z for low A

!  N= 1.7 Z for heavy nuclei

!  Coulomb repulsion also reduces proton binding

!  Fermi level adjusted to get typical BE/nucleon

March 19, 2018

Predictions of Fermi Gas Model

!  The nucleons are fermions !  Can be a maximum of 2 per energy state

!  Lowest energy states are most bound – filled first

!  The last filled state is at the “Fermi level” !  Determined by underlying nuclear binding force

!  Number of nucleons given by

!  Derivation given in text

!  For a typical nucleus (N=Z=A/2) we find !  pF ≈ 250 MeV/c (quite high momentum)

!  EF = pF2 /2mn = 35-40 MeV

!  Nucleons move around quite fast inside nucleus

March 19, 2018

nF

= 2Vtot

Vetat

= 21

(2⇡~)3 (4⇡

3)2A(r0pF )

3

state

Predictions of Fermi Gas Model March 19, 2018

!  Also explains binding energy dependence on excess neutrons

!  Sum the kinetic energy of neutrons and protons separately

Ekin (N,Z) = N(En) + Z(Ep) ≈ N (pnF )2 + Z (pp

F )2

!  Substitute expressions for Fermi momentum in terms of N,Z and A

!  Has a minimum for N=Z

!  Expand about N=Z for small Δ = (N-Z)/A

!  Results in

Ekin (N,Z) = E0 A ( 1 + 5/9 Δ2 + … )

!  First term is just a correction to first term of SEMF

!  Second term goes like (N-Z)2/A for N≠Z

!  Form used for N≠Z term in SEMF

!  Built up as the nuclear analog of the Atomic Shell model !  The latter describes the orbital structures of complex atoms

!  Textbook includes a quick review of salient features

!  Two electrons per orbital (spin up and spin down)

!  2l+1 orbitals for each angular momentum state (S, P, D, …)

!  Hyperfine splitting of J = L + S states

!  All of these have parallels in nuclear shell model

!  Predict the magic numbers for stable atoms (filled shells)

!  2, 10, 18, ...

!  He, Ne, Ar, ...

!  Least reactive, noble

gas nucleii

Shell Model

March 19, 2018

!  Hints nuclei also follow a shell model !  Nuclei with magic number

!  Have stronger binding energies

!  Have a larger number of isotopes/isotones

!  Neutron capture cross-section smaller than other nearby nuclei

!  Have very spherical charge distributions

!  Absence of quadrupole moment seen in other neighboring nuclei

!  All evidence of something ‘special’ about the configuration of these nuclei

Nuclear Shell Model

March 19, 2018

Ca Fe Sn Pb

!  Magic nuclei appear with !  N = 2, 8, 20, 28, 50, 82, 126

!  Z = 2, 8, 20, 28, 50, 82

!  Some even “doubly magic” !  Eg. 4He2 (N=Z=2)

!  Even more stable than others

!  Energy levels opposite come from a central potential of the form:

Nuclear Magic Numbers

March 19, 2018

Central Potentials/Shell Model

March 19, 2018

!  Important difference between atomic model and nuclear model !  No fixed central source of the potential

!  Like positive charge of nucleus in atomic model

!  Nucleons move in aggregated potential of all the other nucleons

!  The actual physics of the nuclear potential is not known

!  Central potential is a natural first guess !  Energy eigenstates will also be eigenstates of angular momentum

!  Central potential is rotationally invariant

!  Nuclear hamiltonian will commute with angular momentum

!  Can classify energy levels with quantum numbers like l and m

!  Like atomic solutions, wavefunctions can be classified by the number of nodes (n)

!  These are (usually) the boundaries between shells where probability to find nucleons vanishes

Specific Radial Potentials

!  To go further need an explicit form for V(r) !  Infinite square well

!  V(r) = -∞ (for r ≤ Rnucleus); V= 0 outside

!  Gives 2 (2l+1) states for each ‘n’

!  Explains magic numbers 2, 8, 18, 32, 50 (but not 20, 82, 126)

!  Simple harmonic oscillator potential

!  V(r) = ½ mω2r2

!  Leads to magic numbers 2, 8, 20, 40, 70 … (but not 18, 50, 82)

!  In general it is ‘easy’ to get the lower lying nucleii right but the heavy nucleii exhibit some other physics

March 19, 2018

Spin-Orbit Couplings

!  1940: Mayer and Jensen proposed !  Missing piece of the puzzle: spin-orbit

coupling between individual nucleons and the general ‘field’ of other nucleons in nuclei

!  This is like the fine-structure in atoms (when placed in a magnetic field)

!  Sign of the effect is opposite in nuclei ( j= l+ ½ more tightly bound)

!  Separates the two spin states in energy

!  More important for larger values of l

!  Thus more important for larger Z (or A) nuclei where L is larger

March 19, 2018

Vtot

= V (r)� f(r)~L · ~S

Predictions of the Shell Model

!  The shell model is successful predicting the structure of more nuclei !  Adjusting spin-orbit coupling ‘predicts’ the large binding energy gaps

!  Gets the nuclear magic numbers right as a consequence

!  Can predict the spin (and parity) of nuclei

!  Even-even nuclei all have L=0

!  nn or pp pairs combine to fill spin states progressively

!  No predictions of spin for odd-odd nuclei

!  There seems to be no ‘rule’ for how to combine spins of the ‘last’ proton with the ‘last’ neutron

!  For A=odd nuclei can get the spin state right most of the time

!  For example 178O

!  8 protons fill 3 lowest energy states ! L=0

!  8 neutrons do the same

!  Leaving one neutron in 1d5/2 state ! L=2, S=1/2 (JP = 5/2+)

March 19, 2018

Non-spherical Nucleii

!  The Shell model begins to shows some limitations for heavy nuclei !  Nuclei start to show non-spherical tendencies

!  Up to now we’ve only considered spherical potentials

!  Hence spherical nuclei

!  Non-spherical nuclei have un-predictable electric dipole moments

!  And significant electric quadrupole moments

!  Shell model predict ≈ 0 quadrupole moments for magic nuclei

!  And opposite signed quadrupole moments for neighbours (Z±1)

!  Large nuclei can have very large quadrupole moments

March 19, 2018

Collective Model

!  In an attempt to explain these return to liquid-drop model !  A. Bohr suggested a merger of shell model and drop model

!  Hard core of filled shells (and energy levels)

!  With “valence nucleons” free to move across the surface of the hard-core, like an extra molecule on the surface of a liquid drop

!  Introduces the non-sphericity necessary to explain many of the short-comings in the pure shell model.

!  The movement of the surface nucleons governed by a non-spherical potential

!  Have the possibility of introducing new energy levels

!  Vibrational and rotational

!  Needed to describe the behaviour of specific nuclei

March 19, 2018

Predictions of Collective Models

!  The Collective Model can predict !  Magnitude of large dipole moments + permanent quadrupole moments

!  Additional rotational and vibration energy levels

!  That have been seen and studied in exotic nuclei

!  A general reduction (with increasing A) in the energy level spacing of the even-even nuclei

!  Not predicted by the simple Shell model

!  While at the same time enhancing the spacing between closed shells at high A/Z

!  That are difficult for the shell model to predict

!  Like much of nuclear (and particle!) physics !  These models codify the experimental observations

!  Hint at an underlying structure

!  But, so far, haven’t actually led to a unique physical model

March 19, 2018

Summary

!  Nuclear Models are approximations to unknown nuclear binding !  Liquid drop model inspired by classical physics (+ quantum corrections)

!  Fermi-gas model inspired by quantum gas energy levels

!  Shell model is most sophisticated

!  Inspired by success of atomic orbital model in Chemistry

!  But limited by unknown underlying form of the potential

!  Some success describing

!  Nuclear magic numbers

!  Spin-parity of nuclei

!  Magnetic moments (not covered here)

!  Collective Model needed to describe physics/shape of exotic nuclei

March 19, 2018