The Proper Es of Nuclei Nucleons

The Properes of Nuclei

Nucleons

• The nucleus is made of neutrons and protons. • The nucleons have spin ½ and (individually) obey the Pauli exclusion principle.

Protons p 938.3 MeV 2.79µN

Neutrons n 939.6 MeV -1.91µN

1 Nuclides

• A nuclide is a nucleus with a speciﬁc number of neutrons and a speciﬁc number of protons.

• Z = # of protons N = # of neutrons • A = mass number (N+Z) X = chemical symbol

• Isotopes = same Z, diﬀerent N • Isotones = same N, diﬀerent Z Example: Carbon-14 or

Nuclear Properes

Forces, Masses and Shapes

2 Nuclear Forces

• The Strong Nucleon-Nucleon Force – Binds nucleons together – Short range (≈ 3 fm) – Aracve over most distances – Repulsive “hard” core – Spin dependent – Charge Independent (n-n, n-p, and p-p forces the same aer electrostac repulsion removed) – The potenal “well” that two nucleons experience is about 40 MeV deep and a few fermi wide.

n-p and p-p interacons

3 Nuclear Potenal

• Each proton or neutron in the nucleus feels an average force from the other nucleons. • This force can be modeled as a potenal well.

Size and Shape

• Describing the nucleus : – Charge density – Mass density – Force density • Most (but not all) spherical

1/3 -15 R = r0A r0 = 1.2 x 10 m =1.2 fm

4 Size and Shape

The surface thickness, t, is roughly constant across nuclei at approximately 2.3 fm.

Size and Shape

• Isomers (shape) • Most nuclei are spherical. • Some deformed nuclei in Z= 57 – 71 region

5 Determining the size of the nucleus

• The size of nuclei can be determined in a variety of ways – Electron Scaering – Mirror Nuclei Decays – Alpha Scaering – Aenuaon of neutron beams

Determining the size of the nucleus: Electron Scaering

• Stanford Linear Accelerator

500 MeV electrons, λ = 2.5 fm

Electrons scatter from nuclear charge density

6 Determining the size of the nucleus: Electron Scaering

• Diﬀracon paerns from high energy electrons scaered from 16O and 12C. • The minimum occurs at sinθ = 0.61 λ/R • R = (1.23 ±0.01)A1/3 fm

Determining the size of the nucleus: Electron Scaering

• Radius from electron scaering experiments.

• R = (1.23 ±0.01)A1/3 fm

7 Determining the size of the nucleus: Mirror Nuclei

• Mirror nuclei are isobars with the proton and neutron numbers reversed. • As the strong force is independent of charge the mirror nuclear should diﬀer only in terms of electrostac energy.

Determining the size of the nucleus: Mirror Nuclei • 15N and 15O are mirror nuclei. • The decay energy should be • The 15O nucleus can equal to the electrostac spontaneously emit a positron energy diﬀerence. and neutrino decaying to 15N.

• Analysis gives

R = 1.2 A1/3 fm

8 Determining the size of the nucleus: Alpha Scaering

Alpha particles are close enough to feel the strong nuclear force.

• To probe maer distribuon strongly interacng parcles are used.

Determining the size of the nucleus: Neutron Aenuaon • A beam of fast neutrons is aenuated by interacng via the strong force with the nuclei (i.e. neutrons are absorbed or scaered).

• R = 1.4 A1/3 fm

9 Mass of Nuclei

• Mass Spectrometers can separate isotopes by their mass.

Mass and Binding Energy

• When two or more objects come together under the inﬂuence of an aracve force, they become bound and the system loses mass.

This lost mass leaves the system as energy: E = mc2

10 Binding energies

• The nuclear binding energy is the diﬀerence between isolated neutrons and protons and the bound nucleus. • It is tradional to use atomic masses in the binding energy formula.

Binding Energies - Example

11 Curve of the Binding Energy

Binding Energies and the Liquid Drop Model

• The liquid drop model provides a theorecal framework to develop a general formula for binding energies: • Energies associated with – Volume – Surface area – Coulomb repulsion between protons

12 Semi-Empirical Binding Energy

Volume Electrostatic Pairing

Surface Area Symmetry

Paring:

Symmetry and Pairing

• The Pauli principle requires we ﬁll up the energy levels with one proton or neutron per state. • Fill up levels with n and p individually • Spin pairing • Proton potenal is slightly higher in energy

13 Chart of the Nuclides

• Stable Nuclei: – Light Nuclei: N ~ Z – Heavy Nuclei: N > Z

Pairing Interacon

• Stable Nuclei: – 266 stable nuclides – 159 – even Z, even N – 50 – odd Z, even N – 53 – even Z, odd N – 4 – odd Z, odd N

14 Neutron and Proton Separaon Energies

• Energy needed to remove one neutron from a nucleus

• Energy needed to remove one proton from a nucleus

Properes of Nuclei

Magnec and Electric

15 Magnec moment of electron

Nuclear Magnec Moments

• Proton: g = 5.5856912 +/- 0.0000022 • Neutron: g = -3.8260837 +/- 0.0000018

• Nuclei have magnec moments that are combinaons of intrinsic nucleon moments and “orbital magnet moments”

16 Electric Quadrupole Moments

• The charge distribuon of the nucleus can be represented as a sum in terms of mulpole moments.

Hyperﬁne Spling

• The magnec and electric properes of nuclei can be detected through the coupling of the EM ﬁelds of the nuclei with atomic electrons. • Magnec hyperﬁne spling arises from the dipole-dipole interacon between the nucleus and the atom.

21cm line of Hydrogen

17 Models of Nuclei

Models of Nuclei

• Independent Parcle Models – “Shell Models” – Based on an analogy to harmonic oscillators and square wells. – Assume nucleus has some stac average potenal made from the individual N-N interacons (mean ﬁeld approximaon.) • Collecve Models – Based on analogies to ﬂuid dynamics – Nucleus made of a “nuclear maer” – Allows non-spherical shapes and collecve moons

18 The Shell Model

• Model the nucleus by a potenal well of some shape. • Protons and neutrons ﬁll the well according to the Pauli Principle. • Harmonic oscillator? • Square well?

The Shell Model

• The large gaps between levels we associate with closed shells. • Notaon: nl • n counts the number of levels with a given angular momentum l

19 Wood-Saxon Potenal

• An intermediate form based on nuclear maer distribuon.

Shell Structure with Wood-Saxon

Spling diﬀers somewhat at higher levels.

20 Do Nuclei have shell structure?

• Binding energy of last neutron

Do Nuclei have shell structure?

• Neutron absorpon cross-secons

21 Do Nuclei have shell structure?

• Binding energy vs. formula

Do Nuclei have shell structure?

• Quadrupole Moments

22 Magic Numbers

• There are numbers of neutrons and protons that yield parcularly ghtly bound nuclei. • The magic numbers:

2, 8, 20, 28, 50, 82, 126

• 15N – magic neutron number • 58Ni – magic proton number • 40Ca – doubly magic

Magic Numbers

• 2, 8, 20, 28, 50, 82, 126

23 Spin Orbit Interacon

Goeppert-Mayer and Jensen Nobel Prize1963

Spin-Orbit Interacon

• The total angular momentum of a nucleon is labeled by j.

• As a single nucleon has s= ½ the possible j values are

24 Spin Orbit Interacon

• The expectaon value is

• And the jth level has a degeneracy of

Spin Orbit Interacon

• Consider the 1f level which has a degeneracy of 14.

• Possible j values are 5/2 and 7/2 thus the levels are 1f5/2 and 1f7/2 . • The degeneracy of these levels (2j+1) are 6 and 8 respecvely. • The spin-orbit potenal causes an energy diﬀerence proporonal to the diﬀerence in the expectaon value of l⋅s for each state.

• The energy spling increases with increasing orbital angular

momentum l. If fSO(r) is negave then the states with the higher j will be pushed down in energy.

25 Spin-Orbit Interacon

2, 8, 20, 28, 50, 82, 126, 184

Predicng Properes of Nuclei from the Shell Model • Ground State spin and parity • Magnec Dipole Moments • Electric Quadrupole Moments • Excited States

26 Ground State Properes

• The group state nuclear spin and parity is determined primarily by the valance nucleon. • Closed shells have j = 0 and + parity. • Like nucleons pair up to give j = 0. • The unpaired nucleon(s) outside the closed shell determine spin and parity. • By this model all even-even nuclei should have a ground state spin of 0. • Parity is determined by the orbital angular momentum of the valance nucleons: (-1)l

Example: Oxygen and Nitrogen

• 16O – Z=8, N=8 (doubly Magic) – ground state is 0+ • 17O – Z=8, N=9 (1 neutron outside closed shell) – ground state is 5/2+ • 15N – Z=7, N=8 (neutron shell closed, one unpaired proton) – ground state is 1/2-

27 Magnec Moments

• In the simplest model the magnec moment of the nucleus is determined by the valence nucleon(s). • The magnec moment is a combinaon of the spin magnec moment and the orbital magnec moment of the nucleon.

• But lz and sz do not have well deﬁned value in a system of well-deﬁned jz. • There are two possible j states:

Magnec Moments

• The comparison between the computed (lines) and measured values (dots) is shown. The lines are known as “Schmidt” lines.

odd Z odd N

28 Excited States

• We should be able to explain the spin and parity of the excited states of nuclei by using the shell model. • As energy is added to the nucleus the nucleons are promoted to higher levels in the shell model.

Excited states

(8)

1/2 - ?? 3/2 -

mirror nuclides Pairing energy increases with Independent particle shell model states

29 All have 1 unpaired

nucleon in f5/2 level. Multiple nucleon excited states

2p3/2

1f5/2

1 nucleon beyond 20

Collecve Models

• Nuclei made of strongly interacng nuclear maer. • Deformed (non-spherical nuclei). • Rotaonal, vibraonal moons of nuclei. • Liquid-drop Model – Binding Energies

30 Even-Even Nuclei Low Lying Energy Levels of 130Sn

Even-Even Nuclei

Energies of the lowest 2+ states of even-even nuclei

31 Nuclear Vibraons

• Constant density shape deformaons

quadrupole sextupole octopole phonons: 1 2 3

2+ 0+,2+,4+ 0+,2+,3+,4+,6++

Nuclear Rotaons

• Some nuclei have non-spherical shapes. These deformed nuclei can undergo rotaon yielding a set of rotaonal bands built on top of shell model states.