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Nuclear Force, Structure and Models

- Readings:
- Nuclear and Radiochemistry: Chapter 10 (Nuclear Models)
- Modern Nuclear Chemistry: Chapter 5 (Nuclear Forces) and Chapter 6 (Nuclear Structure)
- Characterization of strong force
- Charge Independence
- Introduce isospin
- Nuclear Potentials
- Simple Shell Model (Focus of lecture)
- Particle Model and collective models
- Nucleus as a Fermi Gas

Nuclear Force

- For structure, reactions and decay of nuclei
- electromagnetic, strong and weak interactions are utilized
- Fundamental forces exhibit exchange character
- operate through virtual exchange of particles that act as force carriers
- Particles consistent with Heisenberg Uncertainty Principle
- Exist for a short time and cannot be detected
- Can be used to determine distance
- If photon
- Mass particle

Exchange particles

- Exchange particles for different forces
- graviton for gravitational
- pion for the strong interaction between nucleons
- photon for the electromagnetic force
- W and Z bosons for the weak interaction
- For an exchange particle of zero mass (the photon), range of the force is essentially infinite
- For strong interaction between nucleons, the range is less than 1.4 fm
- mexchange 140 MeV/c2
- for weak interaction, the W and Z bosons have masses m 90 GeV/c2
- R 10-3fm
- For strong force in nucleus pion is particle

Strong Force

- Nuclear in nature due to short range
- Range of a nucleon
- Nuclear force is strongly attractive and forms a dense nucleus
- Nuclear force has a repulsive core
- Below a distance (0.5 fm) nuclear force becomes repulsive
- force between two nucleons has two components
- spherically symmetric central force
- asymmetric tensor force
- Spin dependent force between nucleons
- Consider 2H
- Proton and neutron
- Parallel spin 3S
- Can be in excited state, 3D
- Antiparellel is unbound 1S

Charge Independent Force

- Strong force not effected by charge
- np , nn, pp interactions the same
- Electromagnetic force differs
- Strong force examined by:
- Nucleon-nucleon scattering
- Mirror nuclei
- Isobars with number of p in one nuclei equals number of n in other
- Similar energy for net nuclear binding energy
- Normalize influence of Coulomb Energy
- Proton and neutron two states of same particle

Isospin

- Isospin is conserved in processes involving the strong interaction
- Isospin forms basis for selection rules for nuclear reactions and nuclear decay processes
- Property of nucleon
- Analogy to angular momentum
- T=1/2 for a nucleon
- +1/2 for proton, -1/2 for nucleon

Isospin

- For a system with isospin T
- 2T+1 members of multiplet
- Similar to 2S+1
- T3=(Z-N)/2
- T3 is third isospin component
- Consider A=14 isobars
- 14C, 14N, 14O
- 14C and 14O are mirror nuclei
- 14C has 6 protons and 8 neutrons
- 14Ohas 8 protons and 6 neutrons
- T3=±1
- 3 isospin states
- -1, 0, +1
- Energy similar for states
- 14N T3=0
- Defines isospin states for nuclei, helps define strong force properties

Nuclear Potential Characteristics

- Particles in a potential well
- Nuclear forces describe potential
- Small well
- Well stabilizes nucleons
- Free neutrons decay, in well no decay
- Mixture of nucleons stable
- 2 protons (2He) unstable
- 2 neutrons unstable
- A=3
- Mixture of n and p stable (3 protons unstable)
- Nuclear force acts between nucleons in uniform way
- Protons have additional Columbic repulsion that destabilize proton-rich nuclei
- Very neutron-rich nuclei are also unstable
- Light, symmetric nuclei (Z=N) are favored
- Nuclear force depends on the spin alignment of nucleons
- Potential energy of two nucleons shows similarity to chemical bond potential-energy function

Nuclear Potential Characteristics

- Lacks spherical symmetry
- Central potential and tensor (set of vectors) interaction
- Potential has finite range
- Large and repulsive at small distances
- At larger distances, potential approaches zero in exponential fashion
- Not square well
- Depends on quantum state of system
- potential-energy curve that describes stretching of chemical bond depends on electronic state of molecule
- Potential energy of two nucleons shows similarity to potential-energy function of chemical bond
- Lacks spherical symmetry
- Central potential and tensor (set of vectors) interaction
- Potential has finite range
- Large and repulsive at small distances
- At larger distances, potential approaches zero in exponential fashion instead of resembling square well
- Depends on quantum state of system
- potential-energy curve that describes stretching of chemical bond depends on electronic state of molecule

Exchange character between nucleons

Similar to electron exchange between bonded atoms in chemical bond

Can be described by semiempirical formulas

Can describe scattering of one nucleon by another up to energies of several hundred MeV

Charge Symmetry and Charge Independence

Neutron and proton have differing magnetic moments

different potential energies due to magnetic interaction

Isospin used to describe difference between proton and neutron

Observation that difference in properties of pair of mirror nuclei can be accounted for by differing Coulomb interactions in the two nuclei

nuclear part of proton-proton interaction in given quantum state identical to that of two neutrons in same quantum state

Mirror nuclei

N and Z number exchanged

13C and 13N

Nuclear PotentialShell Model

- Interactions among nucleons in nucleus replaced by potential-energy well within which each particle moves freely
- Concerned with detail properties of the quantum states
- Properties determined by shape of potential energy well
- Experimental Evidence
- ground-state spin of 0 for all nuclei with even neutron and proton number
- Magic number for nuclei
- Systematics of ground-state spins of odd-mass-number nuclei
- Dependence of magnetic moments of nuclei upon their spins
- Properties of ground states of odd-mass-number nuclei to first approximation considered those of odd nucleon alone
- All other nucleons provide potential-energy field that determines the single-particle quantum states

Shell Model

- Model nucleus as a spherical rigid container
- square-well potential
- potential energy assumed to be zero when particle is inside the walls
- Particle does not escape
- Energy levels in figure
- Harmonic oscillator potential
- parabolic shape
- steep sides that continue upwards
- useful only for the low-lying energy levels
- equally spaced energy levels
- Potential does not “saturate”
- not suitable for large nuclei
- Change from harmonic oscillator to square well lowers potential energy near edge of nucleus
- Enhances stability of states near edge of nucleus
- States with largest angular momentum most stabilized

Shell Model

- Shell filling
- States defined by n and l
- 1s, 1p, 1d, …
- States with same 2n+l degenerate with same parity
- 2s = 2*2+0=4
- 1d = 2*1+2 =4
- 1g=2*1+4=6
- 2d=2*2+2=6
- 3s=2*3+0=6
- Spin-Orbit Interaction
- Addition of spin orbit term causes energy level separation according to total angular momentum (j=ℓ+s)
- p=1; j=1/2 and 3/2
- split into fourfold degenerate 1p3/2 and twofold degenerate 1p1/2 states
- g=7/2 and 9/2
- states with parallel coupling and larger total angular momentum values are favored
- closed shells 28, 50, 82, and 126 because of the splitting of the 1f, 1g, 1h, and 1i
- Each principal quantum number level is a shell of orbitals
- Energy gap between shell the same

Odd-A Nuclei

In odd A nucleus of all but one of the nucleons considered to have their angular momenta paired off

forming even-even core

single odd nucleon moves essentially independently in this core

net angular momentum of entire nucleus determined by quantum state of single odd nucleon

Configuration Interaction

For nuclides with unpaired nucleons number half way between magic numbers nuclei the single-particle model is oversimplification

Contribution from other nucleons in potential well

Odd-Odd Nuclei

one odd proton and one odd neutron each producing effect on the nuclear moments

No universal rule can be given to predict resultant ground state

Filling ShellsLevel Order

level order given is to be applied independently to neutrons and protons

proton levels increasingly higher than neutron levels as Z increases

Coulomb repulsion effect

order given within each shell essentially schematic and may not represent exact order of filling

Ground States of Nuclei

filled shells spherically symmetric and have no spin or orbital angular momentum and no magnetic moment

ground states of all even-even nuclei have zero spin and even parity

increased binding energy of nucleon in nuclei with even number of like nucleons

Filling ShellsFilling Shells

- lowest level is 1s1/2,
- s since ℓ=0, j=ℓ+s=1/2
- level has only 2ℓ+1=1 m-value
- hold only 2 protons in the proton well and two neutrons in the neutron well
- next levels are 1p3/2 and 1p1/2 pair
- N=1 ħ
- 4He exact filling of both N=0 harmonic oscillator shells for neutrons and protons
- expected to have an enhanced stability
- Consider shell filling when the N=0 ħ and N=1 ħ shells filled
- eight protons and eight neutrons
- 16O should be especially stable
- other shell closures occur at 20, 28, 50, 82, and 126 nucleons
- unusually large numbers of isotopes and isotones due to enhanced stability
- A few stable nuclei have both closed neutron and proton shells
- very strongly bound (relative to their neighbors)
- 4He, 16O, 40Ca, 48Ca, and 208Pb
- doubly closed shell nuclei have been synthesized outside stable range
- 56Ni, 100Sn and l32Sn (unstable)

Filling Example

- Consider the isotope 7Li
- 3 protons and 4 neutrons
- 2 protons in 1s1/2, 1 proton in 1p3/2
- 2 neutrons in 1s1/2, 2 neutrons in 1p3/2
- spin and angular momentum be based on unpaired proton
- spin should be 3/2
- nuclear parity should be negative
- parity of a p-state (odd l value, l=1)
- Excited state for 7Li?
- Proton from 1p3/2 to 1p1/2
- Breaking paired nucleons requires significant energy, neutrons remain paired
- Bound excited state corresponds to promotion of proton
- 1p1/2 corresponds to 1/2-

Filling Example

- Consider 57Ni
- 28 protons, 29 neutrons
- Protons fill to 1f7/2, all paired
- Single neutron in 2p3/2
- 3/2– spin and parity
- Excited state of 57Ni
- From2p3/2 to 1f5/2

Filling Levels

- consider 13C
- 7th neutron is unpaired
- p ½ state
- ½-
- 51V unpaired nucleon is 23rd proton, f 7/2
- 7/2-
- Not always so straight forward
- examine 137Ba
- 81st neutron is unpaired, h 11/2
- spin 11/2-
- measured as 3/2+
- high spin does not appear as ground
- Deformation impacts level filling

For configurations in which odd proton and odd neutron are both particles in their respective unfilled subshells, coupling rules are:

if Nordheim number N (=j1+j2+ l1+ l2) is even, then I=j1-j2

if N is odd, I=j1j2

Parity from sum of l states

Even positive parity

odd negative parity

prediction for configurations in which there is combination of particles and holes is I=j1+j2-1

Shell Filling: Spin and parityShell Model Example

- Consider 38Cl
- 17 protons (unpaired p in 1d3/2)
- l=2 (d state) and j=3/2
- 21 neutrons (unpaired n in 1f7/2)
- l=3 (f state) and j=7/2
- N= 2+3/2+3+7/2 = 10
- Even; I=j1-j2
- Spin = 7/2-3/2=2
- Parity from l (3+2)=5 (odd), negative parity
- 2-
- Consider 26Al (13 each p and n)
- Hole in 1d5/2, each j = 5/2, each l=2
- N=5/2+5/2+2+2=9
- N=odd; I=j1j2
- I = 0 or 5 (5 actual value)
- Parity 2+2=4, even, +
- 5+

Particle Model: Collective Motion in Nuclei

- Effects of interactions not included in shell-model description
- pairing force
- lack of spherically symmetric potential
- Nonspherical Potential
- intrinsic state
- most stable distribution of nucleons among available single-particle states
- since energy require for deformation is finite, nuclei oscillate about their equilibrium shapes
- Deformities 150 <A<190 and A<220
- vibrational levels
- nuclei with stable nonspherical shape have distinguishable orientations in space
- rotational levels
- polarization of even-even core by motion of odd nucleon

Prolate: polar axis greater than equatorial diameter

Oblate: polar axis shorter than diameter of equatorial circle

Shell change with deformation

- Energy of a single nucleon in a deformed potential as a function of deformation ε.
- diagram pertains to either Z < 20 or N < 20. Each state can accept two nucleons

f7/2 deformation

Nilsson Diagram

- 50<N<82
- 137Ba
- 81st neutron is unpaired, spin 11/2
- measured as 3/2+
- Deformation parameter should show 3/2 d
- 3/2+
- 1st excited state ½+
- Oblate nuclei

Magnetic Moment

- Magnetic moment depends on nuclear structure
- measure of the response of a nucleus to an external magnetic field
- composed from
- net effect of the motion of the protons
- plus the intrinsic spins of the protons and neutrons
- Nuclei with non-zero spin have magnetic moments
- Protons and neutrons have magnetic moments
- for neutrons positive charge in center, negative charge on periphery

Experimental Evidence of Spin and Magnetic Moments

- Hyperfine structure in atomic spectra
- nuclear and electronic magnetic interaction
- splitting in structure

Experimental Evidence

- Atom beam experiments
- orientation to a magnetic field
- beam split into 2I+1 components
- Nuclear Magnetic Absorption
- 2I+1 orientations
- ß and g decay experiments
- orientation of gamma
- Nuclear Reactions
- energy of reaction
- Often magnetic moments are less than those expected for single particles
- indicate that the nuclear wave function is not completely dominated by one particle.
- large amount of variation in magnetic moments indicates the complexity of the nuclear structure
- Nucleon pairing has limitations in explaining properties

The magnetic moments of the odd-proton (A) and of the odd-neutron nuclei plotted as a function of the nuclear spin, j. The data generally fall inside the limits and are better reproduced as 60% of the limits.

Emphasizes free-particle character of nuclear motion

Treat average behavior of the large number of nucleons on a statistical basis

Treats the nucleus as a fluid of fermions

Confines the nucleons to a fixed spherical shape with a central potential

nucleons are assumed to be all equivalent and independent

Nucleus taken to be composed of a degenerate Fermi gas of neutrons and protons confined within a volume defined by the nuclear potential

degenerate gas since all particles are in lowest possible states within the Pauli principle

the gas can be characterized by the kinetic energy of the highest state

two identical nucleons can occupy same state, each with opposed spins

Fermi Gas ModelFermi Gas Model

Potential energy well derived from the Fermi gas model. The highest filled energy levels reach up to the Fermi level of approximately 28 MeV. The nucleons are bound by approximately 8 MeV.

V = nuclear volume, p is momentum

Rearrange to find kinetic energy (e) from

p=(2Me)1/2

M is neutron mass

Fermi gas model is useful high energy reaction where nucleons are excited into the continuum

The number of states is

Fermi Gas ModelReview: Summary and Comparisons of Nuclear Models

- Fundamental model is independent-particle model
- difficulty lies in nucleon-nucleon interactions not included in effective potentials
- residual interactions
- Extreme single-particle model
- uses residual interactions to cause even number of identical nucleons with same n, l, and j quantum numbers to couple to net angular momentum of zero
- configuration mixing neglected
- pairing of nucleons takes account of short-range correlations in nucleon motions

Intrinsic States

extreme single-particle model predicts ground-state spins and parities of nearly all odd-mass nuclei

also useful in describing excited states of nuclei, particularly near closed shells

Rotational States

as still more nucleons or holes are added beyond closed shells, configuration mixing that results from residual interactions causes permanent spheroidal deformation of nucleus

excited states better described as rotational states

Collective Nonrotational States

quadrupolar fluctuations about nondeformed shape

backbending: change of slope in plot of level energy versus spin

SummaryInteracting-Boson Model

model for even-even nuclei that can handle intermediate nuclei

treats nucleus as if it were composed of nucleon pairs

whereas both proton and neutron are fermions, nucleon pair considered boson

assumes only valence nucleons, paired to l=0 or 2, contribute to low-lying states

SummaryQuestions

- What is a nuclear potential
- What are the concepts behind the following:
- Shell model
- Particle model
- Fermi model
- How do nuclear shapes relate to quadrupole moments

Pop Quiz

- Using the shell model determine the spin and parity of the following
- 13C
- 99Tc
- 238U
- 96Nb
- 242Am
- Compare your results with the actual data. Which isotopes are non-spherical based on the results?

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