phys490: nuclear physicsns.ph.liv.ac.uk/phys490/chapter12.pdf · constituents t = 0 matter ......

3/4/2020 PHYS490 : Advanced Nuclear Physics : E.S. Paul 1

PHYS490: Nuclear Physics

Advanced Nuclear Physics

Chapter 12



12. Mesoscopic Systems


Micro – Meso - Macro

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


Finite Fermionic Systems

The behaviour of micro particles (atoms, electrons, nuclei, nucleons and other elementary particles) can be described by quantum theory

Macroscopic bodies obey the laws of classical mechanics

These two ‘worlds’ largely differ from each other

In nature there is no sharp border between the micro and macro world and there are objects that exist in the intermediate range

The atomic nucleus, a finite fermionic system, is an example of such a mesoscopic system


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


NucleiHe-dropletsMetal clusters

N

complexity

Mesoscopic systems

E

Emergent phenomena:-Liquid-gas surface, droplet features -superconductivity / superfluidity-thermal phase transitions -shell structure, quantal shapes (liquid)-spatial orientation, rotational bands-rotational/magnetic response-quantum phase transitions

mac

rosc

opic

Quantum dots

Nanoparticles


Quantality Parameter

The ‘quantality’ parameter (Mottelson 1999), Λ = ħ2 / M a2 V0, measures the strength of the two-body attraction V0 expressed in units of the quantal kinetic energy associated with a localisation of a constituent particle of mass M within the distance a corresponding to the radius of the force at maximum attraction

For small Λ the quantal effect is small and the ground state of the many body system will be a configuration in which each particle finds a static optimal position with respect to its nearest neighbours (crystalline)


Nuclei as Quantum Liquids

If Λ is large enough the ground state may be a quantum liquid in which the individual particles are delocalised and the low-energy excitations have ‘infinite’ mean-free path

Constituents T = 0 matter3He Λ = 0.21 ‘liquid’ 4He Λ = 0.16 ‘liquid’H2 Λ = 0.07 ‘solid’ Ne Λ = 0.007 ‘solid’Nuclei Λ = 0.4 ‘liquid’


Fermi Liquid Droplets ‘Clusters’ are aggregates of

atoms or molecules with a well-defined size varying from a few constituents to several tens of thousands

Conduction electrons in clusters are approximately independent and free

Nucleons in nuclei also behave as delocalised and independent fermions

Hence analogies exist between these two systems


The Spherical Droplet

Both clusters and nuclei are characterised by a constant density in the interior and a relatively thin surface layer

The Liquid Drop Model can be used to calculate the binding energy of a charged droplet

The binding energy can be expanded in powers of A1/3 (i.e. radius) where A is the number of constituents


Spherical Droplet Energy The energy of a droplet may be expressed as:

ELD(N,Z) = fA + 4πσR2 + WZ + C Z2e2/R

= fA + bsurfA2/3 + WZ + bcoulZ

2A-1/3

Here R = r0A1/3 is the radius of the droplet, A the

number of atoms and Z is the net charge

The first term (fA) is the ‘volume energy’ which contains the binding energy per particle f of the bulk material

The second term (4πσR2)is the ‘surface energy’ where σis the coefficient of surface tension


Spherical Droplet Energy (cont)

The third term (WZ) contains the ‘work function’ W which is the energy required to remove one electron from the bulk metal

The fourth term (C Z2e2/R) represents the ‘Coulomb energy’ of the charged constituents

In nuclei the charge is evenly distributed because the symmetry energy (quantal effect) keeps the ratio of neutron to protons roughly constant: thus C=3/5

For a cluster charge tends to accumulate at the surfaceand C tends to 1/2 for a large cluster


Shell Structures A bunching together

of the energy levels of a particle in a two- or three-dimensional potential represents a shell structure

Metallic clusters show shell structuressimilar to nuclei

Clusters can contain more constituents than stable nuclei


Supershell Structures

Metallic clusters also exhibit a supershellstructure

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


Periodic Orbit Theory

Supershell structure from interfering periodic orbits


Loss of Spherical Symmetry Deformation occurs in subatomic and mesoscopic systems

with many degrees of freedom, e.g. nuclei, molecules, metallic clusters

The microscopic mechanism of ‘spontaneous symmetry breaking’ was first proposed by Jahn and Teller (1937) –for molecules

Nuclei with incomplete shells tend to deform because the level density near the Fermi surface is high (unstable) for a spherical shape

When the shape of the nucleus changes, nucleonic levels rearrange such that the level density is reduced (stable) –‘nuclear Jahn-Teller effect’


Shapes Of Clusters

Nuclei can easily deform because they consist of delocalised nucleons (liquid)

The presence of heavy discrete ions leads to a more varied response of clusters

Nevertheless, similar shapes are predicted for nuclei and clusters despite the very different nature of the interactions between the constituents


Differences Between Atomic Nuclei and Metallic Clusters

There is only one kind of nuclear matter

It has a single ‘equation of state’

However, all materials have their own equation of state

In a cluster, as in bulk matter, it is the constituents that determine the density and binding energy


Nuclear Molecules

Speculation about the existence of clusters in nuclei, such as alpha particles, has existed for a long time

Initially stimulated by the observation of alpha particle decay

Ikeda Diagram


Beryllium-12

A beryllium nucleus containing 8 neutrons and 4 protons has been found to arrange itself into a molecular-like structure, rather than a sphericalshape that some naïve theories might suggest

Beryllium-12 can be thought of as two alpha particles and four neutrons

M Freer et al. Phys. Rev. Lett. 82 (1999) 1383


Chain States: Nuclear Sausages

Cluster Model calculations for 12C show evidence for a ‘chain state’ consisting of three α particles in a row – axis ratio 3:1 (i.e. ‘hyperdeformed’)

Similarly calculations for 24Mg show evidence for a chain state consisting of six α particles in a row –axis ratio of 6:1 !


Bloch-Brink Cluster Model

Brink presented the light alpha conjugate nuclei as almost crystalline structures

These nuclei contain specific arrangements of the alpha clusters

Narrow resonances in 12C + 12C scattering data suggested larger clusters may occur

‘Nuclear Molecules’

Carbon Hoyle State

An excited 0+ state (7.6 MeV) in 12C, known as the Hoyle state and important in the creation of this abundant element, is thought to correspond to the triangular combination of three alpha particles

See Chapter 14!



Binary Cluster Model

It has been observed that measured quadrupolemoments of many superdeformed bands follow:

Qo ~ 2 Ro2[ Z A2/3 – Z1 A1

2/3 – Z2 A22/3]

This expression results from considering the states of the nucleus (Z, A) to be composed of two clusters (Zi, Ai) in relative motion

For example, a strongly deformed band has recently been found in 108Cd (Z = 48)

The predicted fragmentation for 108Cd is: 58Fe (Z = 26) + 50Ti (Z = 22)

Summary

Femtostructures

Finite quantum system

Physics derived from limited number of constituents

Nuclear molecules, cluster structure


Advanced Nuclear Physics

Edward Paul


phys490: nuclear physicsns.ph.liv.ac.uk/phys490/chapter12.pdf · constituents t = 0 matter ......

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