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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