2-1

Nuclear Properties

• Systematic examination of general nuclear properties at the general masses matter distributions

• Size, shape, mass, and relative stability of nuclei follow patterns that can be understood and interpreted with two models average size and stability of a nucleus can be

described by average binding of the nucleons in a macroscopic model

detailed energy levels and decay properties evaluated with a quantum mechanical or microscopic model

2-2

Masses

• Atomic masses Nuclei and electrons

• Nuclear mass m0 is electron rest mass, Be (Z) is the total binding energy of all

the electrons

Be(Z) is small compared to total mass

• Consider beta decay of 14C 14C14N+ + β- +antinuetrino + energy

Energy = mass 14C – mass 14N

• Positron decay

2-3

Masses

• For a general reaction

2-4

Terms

• Binding energy Difference between mass of nucleus and constituent nucleons

Energy released if nucleons formed nucleus

average binding energy per nucleon

* Measures relative stability Mass excess (in energy units)

* M(A,Z)-A Useful when A remains constant

2-5

Binding Energies

http://www.lbl.gov/abc/wallchart/chapters/02/3.html

2-6

• Binding Energy of an even-A nucleus is generally higher than the average of the values for the adjacent odd-A nuclei

this even-odd effect is more pronounced in graphing A vs. the binding energy from the addition of one more nucleon

• The very exothermic nature of the fusion of H atoms to form He--the process that gives rise to the sun’s radiant energy--follows from the very large binding energy of 4He

• Energy released from fission of the heaviest nuclei is large because nuclei near the middle of the periodic table have higher binding energies per nucleon

• The maximum in the nuclear stability curve in the iron-nickel region (A~56 through 59) is thought to be responsible for the abnormally high natural abundances of these elements

• Mass excess==M-A

Binding energy

2-7

Stable Nuclei

N even odd even oddZ even even odd oddNumber 160 53 49 4

• As Z increases the line of stability moves from N=Z to N/Z ~ 1.5 influence of the Coulomb force. For odd A nuclei only one stable isobar is found while for even A

nuclei no stable odd-odd nuclei

2-8

Terms

• Binding can be used to determine energetics for reaction using mass excess Energy need to separate neutron from 236U and 239U

2-9

2-10

Binding-Energy

• Volume of nuclei are nearly proportional to the number of nucleons presentnuclear matter is quite incompressible

• Total binding energies of nuclei are nearly proportional to the numbers of nucleons presentsaturation character

a nucleon in a nucleus can apparently interact with only a small number of other nucleons

*liquid-drop model of nucleus

2-11

Liquid-Drop Binding Energy:

• c1=15.677 MeV, c2=18.56 MeV, c3=0.717 MeV, c4=1.211 MeV, k=1.79 and =11/A1/2

• 1st Term: Volume Energy

dominant term

in first approximation, binding energy is proportional to the number of nucleons

(N-Z)2/A represents symmetry energy

binding E due to nuclear forces is greatest for the nucleus with equal numbers of neutrons and protons

124

3/123

23/2

2

2

1 11 AZcAZcA

ZNkAc

A

ZNkAcEB

2-12

• 2nd Term: Surface Energy Nucleons at surface of nucleus have unsaturated forces decreasing importance with increasing nuclear size

• 3rd and 4thTerms: Coulomb Energy 3rd term represents the electrostatic energy that arises from the

Coulomb repulsion between the protonslowers binding energy

4th term represents correction term for charge distribution with diffuse boundary

term: Pairing Energy binding energies for a given A depend on whether N and Z are even or

oddeven-even nuclei, where =11/A1/2, are the stablest

two like particles tend to complete an energy level by pairing opposite spins

2-13

Mass Parabolas

• For odd A there is only one -stable nuclidenearest the minimum of the parabola

• For even A there are usually two or three possible -stable isobarsall of the even-even type

Friedlander & Kennedy, p.47

2-14

Magic Numbers

• Certain values of N and Z--2, 8, 20, 28, 50, 82, and 126 --exhibit unusual stabilityevidence from masses, binding energies, elemental and

isotopic abundances, numbers of species with given N or Z, and -particle energies

accounted for by concept of closed shells in nuclei

Friedlander & Kennedy, p.49

2-15

Single-Particle Shell Model

• Collisions between nucleons in a nucleus are suppressed by the Pauli exclusion principleonly accounts for magic numbers 2-20

• Strong effect of spin-orbit interactions if orbital angular momentum (l) and spin of nucleon

interact in such a way that total angular momentum=l+1/2 lies at a lower energy level than that with l-1/2, large energy gaps occur above magic numbers 28-126

• Ground states of closed-shell nuclei have spin=0 and even parity

2-16

R=roA1/3

Nuclear Shapes: Radii

• Nuclear volumes are about proportional to nuclear masses, thus all nuclei have approximately the same density

• Although nuclear densities are high compared to ordinary matter, nuclei are not densely packed with nucleons

• ro~1.1 to 1.6 fm

• Nuclear radii can mean different things, whether they are defined by nuclear force field, distribution of charges, or nuclear mass distribution

2-17

Nuclear-Force RadiiThe radius of the nuclear force field must be less than the distance of closest approach (do)

T’=T-2Ze2/dd = distance from center of nucleus

T’ = particle’s kinetic energy

T = particle’s initial kinetic energy

do = distance of closest approach--reached in a head on collision when T’=0

T

Zedo

22

do~10-20 fm for Cu and 30-60 fm for U

2-18

Any positively charged particle subject to nuclear forces can be used to probe the distance from the center of a nucleus within which the nuclear (attractive) forces become significant relative to the Coulombic (repulsive force).

Since neutrons are not subject to Coulomb forces, one might expect neutron scattering and absorption experiments to be easier to interpret, however the neutrons must be of sufficiently high energy to have de Broglie wavelengths small compared to nuclear dimensions, but at high energies, nuclei become quite transparent to neutrons.

2-19

Square-Well and Woods-Saxon Potentials

Friedlander & Kennedy, p.32

2-20

ARro

e

VV

/)(1

Vo=potential at center of nucleus

a=constant~0.5 fm

R=distance from center at which V=0.5Vo (for half-potential radii) or V=0.9Vo and V=0.1Vo for a drop-off from 90 to 10% of the full potential

ro~1.35 to 1.6 fm for Square-Well, ro~1.25 fm for Woods-Saxon with half-potential radii, ro~2.2 fm for Woods-Saxon with drop-off from 90 to 10%--the “skin thickness”--of the full potential

Scattering experiments lead to only approximate agreement with the Square-Well potential; the Woods-Saxon equation fits the data better.

2-21

Electron Scattering

• Using moderate energies of electrons, data is compatible with nuclei being spheres of uniformly distributed charges

• High energy electrons yield more detailed information about the charge distribution (no longer uniformly charged spheres)

• Radii distinctly smaller than indicated by the methods that determine nuclear force radii

• Re (half-density radius)~1.07 fm

• de (“skin thickness”)~2.4 fm

2-22

Fermi Shape

]/)[(1)(

eee aRro

er

Friedlander & Kennedy, p.34

2-23

Nuclear Skin• Although charge density results give information on how protons are distributed

in the nuclei, no experimental techniques exist for determining the total nucleon distribution it is generally assumed that neutrons are distributed in roughly the same

way as protons nuclear-potential radii are about 0.2 fm larger than the radii of the

charge distributionsNucleus Fraction of nucleons in the “skin”12C 0.9024Mg 0.7956Fe 0.65107Ag 0.55139Ba 0.51208Pb 0.46238U 0.44

2-24

Spin

• Nuclei possess angular momenta Ih/2I is an integral or half-integral number known as the

nuclear spin• Protons and neutrons have I=1/2

• Nucleons in the nucleus, like electrons in an atom, contribute both orbital angular momentum (integral multiple of h/2 ) and their intrinsic spins (1/2)

• Therefore spin of even-A nucleus is zero or integral and spin of odd-A nucleus is half-integral

• All nuclei of even A and even Z have I=0 in ground state

2-25

Magnetic Moments• Nuclei with nonzero angular momenta have magnetic moments Bme/Mp is used as the unit of nuclear magnetic moments and called a

nuclear magneton

• Magnetic moment results from a distribution of charges in the neutron, with negative charge concentrated near the periphery and overbalancing the effect of an equal positive charge nearer the center

• Magnetic moments are often expressed in terms of gyromagnetic ratios g*I nuclear magnetons, where g is + or - depending upon

whether spin and magnetic moment are in the same direction

2-26

•Only nuclei with I1/2 have quadrupole moments

•Interactions of nuclear quadrupole moments with the electric fields produced by electrons in atoms and molecules give rise to abnormal hyperfine splittings in spectra

•Methods of measurement: optical spectroscopy, microwave spectroscopy, nuclear resonance absorption, and modified molecular-beam techniques

Methods of Measurement1) Hyperfine structure in atomic spectra

2) Atomic Beam method

split into 2I+1 components

3) Resonance techniques

2I+1 different orientations

Quadrupole Moments: q=(2/5)Z(a2-b2)

2-27

Statistics• If all the coordinates describing a particle in a system are

interchanged with those describing another particle in the system the absolute magnitude of the wave function representing the system must remaining the same, but it may change signFermi-Dirac (sign change)

each completely specified quantum state can be occupied by only one particle (Pauli exclusion principle)

Bose-Einstein (no sign change)no restrictions such as Pauli exclusion principle

apply• A nucleus will obey Bose or Fermi statistics, depending on

whether it contains an even or odd number of nucleons

2-28

Parity

• Depending on whether the system’s wave function changes sign when the signs of all the space coordinates are changed, a system has odd or even parity

• Parity is conserved

• even+odd=odd, even+even=even, odd+odd=oddallowed transitions in atoms occur only between an atomic

state of even and one of odd parity• Parity is connected with the angular-momentum quantum number l

states with even l have even paritystates with odd l have odd parity

2-29

Friedlander & Kennedy, p.39