revision: phys490 advanced nuclear physics

25/03/2019 E.S. Paul: PHYS490 Advanced Nuclear Physics Revision 1

Revision: PHYS490 Advanced Nuclear Physics

Building Blocks and Energy Scales

Depending on energy and length scales, different constituents may be considered as the building blocks of the atomic nucleus

3/25/2019 PHYS490 : Advanced Nuclear Physics : E.S. Paul 2


The Strong Force

The strong force is fundamentally an interaction between quarks

It is really a residual colour force mediated by the exchange of gluons


Properties of the N-N Force

The force is spin dependent

The force is charge symmetric

The force is (nearly) charge independent

The force has a non-central component

The force depends on the relative velocity or momentum of the nucleons

The force has a repulsive core

‘Exchange model’: force mediated by pion exchange


Shell Model – Mean Field

Assumption – ignore detailed two-body interactions Each particle moves in a state independent of the other

particles The Mean Field is the average smoothed-out interaction

with all the other particles An individual nucleon only experiences a central force

A nucleon in the Mean Field of N-1 nucleons

N nucleons in a nucleus


(Wrong) Magic Numbers


Simple Nuclear Shapes The general shape of a nucleus can

be expressed in terms of spherical harmonics Yλμ(θ,φ) - the λ = 2 term describes quadrupole deformation

The parameters β and γ describe the shape:

β is related to the distortion

γ is related to the lengths of

the principal axes

Quadrupole β and γ Parameters


prolate

x = y < z

Axially symmetric shapes

prolate

prolate

oblate

oblate

oblate

x = y > z

x > y = z

x = z < y

x = z > y

x < y = z

Triaxial shapes : x ≠ y ≠ z

γ = n 60°

γ ≠ n 60°

60°

0°

-60°


Nilsson Model

In order to introduce nuclear deformation Nilsson modified the harmonic oscillator potential to become anisotropic:

V = ½m[ω12x2 + ω2

2y2 + ω32z2]

Oscillations are now along the deformation axis (ω3) or perpendicular to it (ω1 and ω2)

He also added a spin-orbit force


Nilsson Single-Particle Diagrams

Z

N


Nilsson Labels The energy levels are labelled by the asymptotic quantum

numbers: Ωπ [N n3 Λ]

‘N’: N = n1 + n2 + n3 is the oscillator quantum number

‘n3’: n3 is the z-axis (symmetry axis) component of N

‘Λ’: Λ = ℓz is the projection of ℓ onto the z-axis

‘Ω’: Ω = Λ + Σ is the projection of j = ℓ + s onto the z-axis

‘π’: π = (-1)N = (-1)ℓ is the parity of the state


The Λ, Σ, Ω Quantum Numbers

The z-axis (deformation axis) is the quantisation axis

Spin projections: Ω = Λ + Σ = Λ ± ½


Nuclear Moments of Inertia Nuclear moments of

inertia are much lower than rigid-body values – a consequence of nuclear pairing

They are also much larger than those of a pure superfluid

Short-range pairing correlations between nucleons introduce a degree of superfluidity to the nucleus (cf correlated pairs of electrons in superconductors)


Backbending The nuclear moment of inertia

increases with rotational frequency

Around spin 12ħ a dramatic rise occurs

The characteristic ‘S’ shape is called a backbend (158Er)

A more gradual increase is called an upbend (174Hf)

Fictive Coriolis and centrifugal forces in the nucleus’ rotating frame of reference overcome the pairing force between a specific pair of nucleons: Now I = R + J


Band Termination

neutron backbend

proton backbend

Gamma Ray Energy


Band Termination in 158Er At termination 158Er can be thought of as an inert 146Gd

core plus 4 protons and 8 neutrons which generate a total spin 46ħ

The configuration of the 12 valence particles is:

π(h11/2)4 ν(i13/2)2 (h9/2)3 (f7/2)3

The terminating spin value of 46 is generated as:

(11/2+9/2+7/2+5/2) + (13/2+11/2) + (9/2+7/2+5/2) + (7/2+5/2+3/2)

Parity: (-1)4 (+1)2 (-1)3 (-1)3 = +1


High K (Iz) Bands

If we have many unpaired nucleons outside the closed shell then alignment with the x-axis becomes difficult because the valence nucleons lie closer to the z-axis, i.e. they have high Ω values

The sum K of these projections onto the deformation (z) axis is now a good quantum number

K = Iz = Σjz = ΣΩ


K Forbidden Transitions It is difficult for rotational bands with high K values to

decay to bands with smaller K since the nucleus has to change the orientation of its angular momentum.

For example, the Kπ = 8- band head in 178Hf is isomeric with a lifetime of 4 s. This is much longer than the lifetimes of the rotational states built on it.

The Kπ = 8- band head is formed by breaking a pair of protons and placing them in the ‘Nilsson configurations’:

Ω [N n3 Λ] = 7/2 [4 0 4] and 9/2 [5 1 4] In this case: K = 7/2 + 9/2 = 8 and π = (-1)N(1).(-1)N(2) = -1


K Isomers in 178Hf A low lying state with

spin I = 16 and K = 16 in 178Hf is isomeric with a half life of 31 years !

It is yrast (lowest state for a given spin) and is ‘trapped’ since it must change K by 8 units in its decay


Limits of Nuclear Existence

Proton Dripline

Neutron Dripline

Known Nuclei

Fission Limit

Terra Incognita

Stable Nuclei

Segre Chart

Proton Rich: Proton Radioactivity

The half-lives of proton radioactivity are sensitive to the orbital angular momentum of specific states

A centrifugal barrier occurs in the potential proportional to the orbital angular momentum


ℓ = 0 Coulomb Barrier

ℓ = 5 Coulomb Barrier plus Centrifugal Barrier


Neutron Rich: Nuclear Haloes

The spatial extent of 11Li with 3 protons is similar to that of 208Pb with 82 protons !

11Li is modelled as a core of 9Li plus two valence neutrons


Borromean System

Halo nuclei have provided insight into a new topology with a Borromean property

The two-body subsystems of the stable three-body system 11Li (9Li + n + n) are themselves unstable ! bound

unbound


New Magic Numbers


Super Heavy Elements (SHE)

Neutrons

Protons

Roentgenium

Darmstadtium

Meitnerium

Hassium

Bohrium SHE

Quantal shell effects stabilise energy

Up to Z = 112 results confirmed

Dubna: Z = 114, 116, 118

Berkeley: Z = 118 ‘discovered’ then retracted

Copernicium (2010)


Mesoscopic Systems

‘Mesoscopic’ systems contain large, yet finite, numbers of constituents, e.g. atomic nuclei, metallic clusters


Nanostructures and Femtostructures

‘Nanostructures’: intense research is ongoing for quantum systems that confine a number of electrons within a nanometre-size scale (10-9 m), e.g. grains, droplets, quantum dots

Nuclei are femtostructures (10-15 m)

All these systems share common phenomena but on very different energy scales:

nuclear MeV; molecular eV; solid-state meV


Supershell Structures

Metallic clusters also exhibit a supershell structure

The basic shell structure is enveloped by a long wavelength oscillation (beat pattern)

Nuclei become unstable well before the first half-period of the long wavelength oscillation is seen


Collision Kinematics The Q value is:

[ (MA + Ma) - (MB + Mb) ] c2

Exothermic (Q > 0) reactions give off energy – kinetic energy of reaction products

Endothermic (Q < 0) reactions require an input of energy to occur. By considering the kinetic energy available in the centre-of-mass frame, the threshold energy is:

Ta > |Q| [ (Ma + MA) / MA ]


The Compound Nucleus Consider the reactions:

a + A C* a + A*

b + B*

γ + C*

The incident particle a enters the nucleus A and suffers collisions with the constituent nucleons, until it has lost its incident energy, and becomes an indistinguishable part of the excited compound nucleus C*

The compound nucleus ‘forgets’ how it was formed and its subsequent decay is independent of its formation: “Bohr’s Hypothesis of Independence”


Origin of the Elements Big Bang: 1H, 2H, 3He, 4He, 7Li (Z = 3) Thermonuclear fusion in a rapidly expanding mixture of

protons and neutrons Interstellar Gas: Li, Be, B (Z = 5) Spallation and fusion reactions between cosmic rays and

ambient nuclei Stars: Successive energy-releasing fusion or ‘burning’ of light elements Low (< 8 M

): Li, C, N, F (Z = 9)

Massive (> 8 M

): Li, B, C, to Fe (Z = 26) (maximum BE)


Hydrogen Burning: The pp Chain

Basically, four protons (hydrogen nuclei) are transformed into one alpha particle (helium nucleus: 2 protons + 2 neutrons)

This process liberates 27 MeV of energy, since helium is more bound than hydrogen


Hydrogen Burning: CNO Cycle

The CNO (carbon-nitrogen-oxygen) cycle converts H (hydrogen) into He (helium) by a sequence of reactions involving C, N and O isotopes and releasing energy in the process.

It occurs in stars with masses › 1.5 M


Helium Burning: Triple Chain


Creation of Elements Beyond Fe

Fusion of elements up to Fe (Z=26) releases energy, the nuclear binding energy

To produce elements heavier than Fe via nuclear fusion requires an input of energy – the binding energy decreases for heavy nuclei

So, how are elements heavier than iron formed ?

Supernovae explosions, explosive nucleosynthesis, proton and neutron capture reactions

25/03/2019 E.S. Paul: PHYS490 Advanced Nuclear Physics Revision 36

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revision: phys490 advanced nuclear physics

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