Models of atomic nuclei
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Models of atomic nuclei. 1) Introduction 2) Drop model of nucleus 3) Nucleus as Fermi gas 4) Shell model of nucleus 5) Unified (collective) model of nuclei 6) Further models. Octup o le vibration s of nucleus. (taken from H-J. Wolesheima, GSI Darmstadt).

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Models of atomic nuclei

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Models of atomic nuclei

1) Introduction

2) Drop model of nucleus

3) Nucleus as Fermi gas

4) Shell model of nucleus

5) Unified (collective) model of nuclei

6) Further models

Octupole vibrations of nucleus.

(taken from H-J. Wolesheima,

GSI Darmstadt)

Extreme superdeformed states were

predicted on the base of models


Nucleus is quantum system of many nucleons interacting mainly by strong nuclear interaction. Theory of atomic nuclei must describe:

1) Structure of nucleus (distribution and properties of nuclear levels)

2) Mechanism of nuclear reactions (dynamical properties of nuclei)

Development of theory of nucleus needs overcome ofthree main problems:

1) We do not know accurate form of forces acting between nucleons at nucleus.

2)Equations describing motion of nucleons at nucleus are very complicated – problem

of mathematical description.

3)Nucleus has at the same time so many nucleons (description of motion of every its particle is not

possible) and so little (statistical description as macroscopic continuous matter is not possible).

Real theory of atomic nuclei is missing  only models exist.

Models replace nucleus by model reduced physical system.  reliable andsufficiently

simple description of some properties of nuclei.

Models of atomic nuclei can be divided:

A)According to magnitude of nucleon interaction:

Collective models(models with strong coupling) – description of properties of nucleus given by collective motion of nucleons

Singleparticle models (models of independent particles) – describe properties of nucleus given by individual nucleon motion in potential field created by all nucleons at nucleus.

Unified (collective) models– collective and singleparticle properties of nuclei together are reflected.

B)According to, how they describe interaction between nucleons:

Phenomenological models– mean potential of nucleus is used, its parameters are determined

from experiment.

Microscopic models– proceed from nucleon potential (phenomenological or

microscopic) and calculate interaction between nucleons at nucleus.

Semimicroscopic models – interaction between nucleons is separated to two parts: mean potential

of nucleus and residual nucleon interaction.

Volume energy

Surface energy

Coulomb energy

Asymmetry energy

Binding energy

Liquid drop model of atomic nuclei

Let us analyze of properties similar to liquid. Think of nucleus asdrop of incompressible liquidbonded together by saturated short range forces

Description of binding energy: B = B(A,Z)

We sum different contributions:B = B1 + B2 + B3 + B4 + B5

1) Volume(condensing) energy: released by fixed and saturated nucleon at nuclei:B1 = aVA

2) Surface energy: nucleons on surface → smaller number of partners → addition of negative member proportional to surfaceS = 4πR2 = 4πA2/3: B2 = -aSA2/3

3) Coulombenergy: repulsive force between protons decreases binding energy. Coulomb energy for uniformly charged sphere isE  Q2/R. For nucleusQ2 = Z2e2 a R = r0A1/3: B3 = -aCZ2A-1/3

4)Energy of asymmetry: neutron excess decreases binding energy

5) Pairenergy: binding energy for paired nucleons increases:

+ for even-even nuclei

B5 =0for nuclei with odd Awhere aPA-1/2

- for odd-odd nuclei

We sum single contributions and substitute to relation for mass:

M(A,Z) = Zmp+(A-Z)mn – B(A,Z)/c2

M(A,Z) = Zmp+(A-Z)mn–aVA+aSA2/3 + aCZ2A-1/3 + aA(Z-A/2)2A-1±δ

Weizsäcker semiempirical mass formula. Parameters are fitted

using measured masses of nuclei.

(aV = 15.85, aA = 92.9, aS = 18.34, aP = 11.5, aC = 0.71 all in MeV/c2)

Using Weizsäcker formula many regularities can be derived.

Behavior of stability curve:

During beta decay A is not changed. For nucleiwith odd Amasses is lying on parabola and onlyone stable isotopeexists. For nuclei with even A such parabolas are two due to pair contribution±δ → more stable isotopes exist. We find the most stable nucleus in the line of isobars:

reminder: M(A,Z) = Zmp+(A-Z)mn–aVA+aSA2/3 + aCZ2A-1/3 + aA(Z-A/2)2A-1±δ

We carry out derivation:mp – mn + 2Z0aCA-1/3+2aA(Z0-A/2)A-1 = 0

And we obtain:

Separation energy of neutron, proton and alpha particle:

Energy obtained by separation of nucleon α particle. Kinetic energy of produced α particle

after decay is:

Eα = [M(A,Z) – M(A-4,Z-2) – mα]c2

It is also negative value ofseparation energy.

γ vibrations



Ground state

The energy of stability border for different decays and particle emissions or fission can be obtained by Weizsäcker formula. Particle must come through Coulomb barrier before realization of decay or fission. It is problem for nuclei from fission.

Description of vibrations and rotations of nuclei can be obtained by drop model:

Nuclear matter is practically incompressible but production of surface vibrations is simple.

Quadruple – nucleus is changing from pressed to elongated ellipsoid. Octupole – deformation

has pear-shaped form. Energy of vibration state depends on frequency:

Ekv = nkvħω Eokt = noktħω

Where nkv, nokt are numbers of appropriate quanta. Quadruple quantum has spin J=2 and octupole has spin J=3. Special type of vibrations are independent oscillations of proton and neutron drops – giant dipole resonances.

For deformed drop possibility of rotation.

Description of rotational states:

where J – moment of inertia, spin I = 0,1,2, …

The Fermi-gas model

Nucleons arefermions(have spin 1/2).Because of Pauli exclusion principleonly one fermion can be in one state. States characterized by strictly given discrete values of energy and momentum exist at potential of nucleus. In the ground state all the lowest states allowed by Pauli principle are filled by nucleons. Such system of fermions is named asdegenerated fermion gas→ nucleons can not change their state (all near states are filled) → they can not collide and behave as noninteracting independent particles.

System of N fermions in volume V and with temperature T:

Probability of fermion occurrence in state with energy E:

wherekis Boltzman constant andEF – Fermi energy. We determineFermi momentumpF( nonrelativistic approximationEF = pF2/2m):

We introduce phase space:

extension of coordinate space by momentum space (6 – dimensional space). Space element is:

dV = dx·dy·dz → dV = d3r = r2sindr·d·d

If angular direction is not important, we integrate through all angles:

dV = 4π r2 dr

Analogously for momentum space element:

dVp =d3p = dpxdpydpz = 4π p2 dp

Phase space element: dVTOT = dV·dVp

From Heisenberg uncertainty principle:

ΔpxΔx ≥ ħ ΔpyΔy ≥ ħ ΔpzΔz ≥ ħ

VolumedVTOTof elementary cell in phase space ish3. numberdof elementary

cells with one particle with momentump  p+Δpis for volumeV:

Nucleons haves = 1/2  in every cellgs = (2s+1) = 2.ForT = 0:

p < pF 2 particles in cell

p > pF 0 particles in cell.

and then:

Fermi gas is degenerated for EF >> kT. ForEF << kT → classical gas and Maxwell distribution.

Nucleus is mixture of two degenerated fermion gases:

Z protons andNneutrons closed in volumeV = (4/3)R3 = (4/3)r03A. Fermi energyfor

neutrons and protons at nucleus:

in first approximation:mn mp =m, Z  N  A/2:

Deepness of potential well(binding of last nucleon isB/A):

V0 EF + B/A  37 MeV + 8 MeV  45 MeV

Further we calculate whole kinetic energy:

Hence forA = Z+Nnucleons:

Average kinetic energy per A(forZ A):

Shell model of nucleus

Assumption:primary interaction of single nucleon with force field created by all nucleons.

Nucleons are fermions  one in every state (filled gradually from the lowest energy).

Experimental evidence:

1) Nuclei with valueZ orN equal to 2, 8, 20, 28, 50, 82, 126 (magic numbers) are morestable

(isotope occurrence, number of stable isotopes, behavior of separation energy magnitude).

2) Nuclei with magic Z and Nhave zero quadrupol electric moments zero deformation.

3) The biggest number of stable isotopes is for even-even combination (156), the smallest for

odd-odd (5).

4) Shell model explains spin of nuclei.Even-even nucleus protons and neutrons are paired. Spin

and orbital angular momenta for pair arezeroed. Either proton or neutron is left over in odd

nuclei. Half-integral spin of this nucleon is summed with integral angular momentum of rest of

nucleus half-integral spinof nucleus. Proton and neutron are left over at odd-odd nuclei 

integral spinof nucleus.

Shell model:

1) All nucleons create potential, which we describe by 50 MeV deep square potential well with

rounded edges, potential of harmonic oscillator or even more realistic Woods-Saxon potential.

2) We solveSchrödinger equationfor nucleon in this potential well. We obtain stationary states

characterized byquantum numbers n, l, ml. Group of states with near energies createsshell.

Transition between shells  high energy. Transition inside shell  law energy.

3) Coulomb interaction must be included difference between proton and neutron states.

4)Spin-orbital interactionmust be included.Weak LS coupling for the lightest nuclei. Strong

jj coupling for heavy nuclei. Spin is oriented same as orbital angular momentum → nucleon is

attracted to nucleus stronger. Strongsplit of level with  orbital angular momentum l.

without spin-orbital


with spin-orbital



per level


per shell




Sequence of energy levels of nucleons given by shell model (not real scale) – source A. Beiser

Unified (collective) model of nucleus

The most part of nuclei has both single-particle and collective properties. Common description is possible by unified (collective) model:

Nucleus is divided to even-evenhard coreand some number ofouter valence nucleons.

Hard core need not be spherical (given by influence of valence nucleons) → possibility of rotation. We assume adiabatic approximation – ΔErot<<ΔEpart → collective motion proceed more slowly than single-particle → rotation does not disturb single-particle state → rotation bands over single-particle states: Erot = (ħ2/2J)[I(I+1)].

Often nonspherical deformed potential –Nilsson potential– is used.

Deformed potential → energy of single-particle level depends on magnitude of deformation.

Interaction between collective and single-particle states. → Mixture of different states.

Creation of complex system of single-particle, vibrational and rotational states (levels).

Unified model makes possible relatively good description of complicated system of energy levels at nucleus, state characteristics, transition probabilities, quadrupole moments of nuclei.

Further models of nuclei

Very strong influence of pairing. Nucleon pair can be so strongly bound, that we can look on it as on single particle (spins of two fermions are summed → boson). This is base ofinteracting boson model (IBA).

Strong pairing is also base of independent quasiparticle models.

Microscopic models use nucleon potential and solve equations of motion. Big problems with mathematical solution.

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