RFSS and NFSS: Lecture 2
Nuclear Properties
• Readings:
 Modern Nuclear Chemistry: Chapter 2
Nuclear Properties
 Nuclear and Radiochemistry: Chapter 1
Introduction, Chapter 2 Atomic Nuclei
 Nuclear Forensics Analysis, Chapter 2
• Nuclear properties
 Masses
 Binding energies
 Reaction energetics
 Nuclear shapes
2-1
Nuclear Properties
• Systematic examination of general nuclear properties
 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
Number of stable nuclei based on neutron and proton number
N
Z
Number
even
even
160
odd
even
53
even
odd
49
odd
odd
4
2-2
Mass excess
• Difference between actual mass and atomic
number
 Expression of binding energy relative to 12C
By definition 12C = 12 amu
* If mass excess negative, then isotope
has more binding energy the 12C
• Mass excess=∆=M-A
 24Na
 23.990962782 amu
 23.990962782-24 = -0.009037218 amu
• 1 amu = 931.5 MeV
2-3
• -8.41817 = Mass excess=∆
Masses
• Atomic masses

Nuclei and electrons
• Nuclear mass can be found from atomic 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
• Energy (Q) from difference in mass between parent and daughter
• Consider beta decay of 14C
14C14N+ + β- +ν + Q

 Energy = mass 14C – mass 14N
• Positron decay
• Electron Capture
2-4
Masses
• Electron Capture
 Electron comes from parent
orbital
 Parent designated as cation to
represent this behavior
• Alpha Decay
 241Am237Np + 4He + Q
 Use mass excess or Q value
calculator to determine Q
value
 Q = 52.937 – (44.874 + 2.425)
 Q = 5.638 MeV
 Alpha decay energy for 241Am is
5.48 and 5.44 MeV
2-5
Masses
• For a general reaction
 Treat Energy (Q) as part of the equation
 56Fe+4He59Co+1H+Q
 Q= [M56Fe+M4He-(M59Co+M1H)]c2
* M represents mass of isotope
2-6
For coursework please note if 1.022 MeV added to positron Q value
Q value calculation
•
•
•
Find Q value for the Beta decay of 24Na
24Na24Mg+ +β− + ν +Q


Q= 24Na-24Mg

M (24Na)-M(24Mg)
 23.990962782-23.985041699
 0.005921 amu
* 5.5154 MeV

From mass excess
 -8.417 - -13.933
 5.516 MeV
Q value for the EC of 22Na
22Na+ + e- 22Ne + ν +Q


Q= 22Na - 22Ne

M (22Na)-M(22Ne)

21.994436425-21.991385113

0.003051 amu
 2.842297 MeV

From mass excess
 -5.181 - -8.024
 2.843 MeV
Can also use Q-value calculator

http://www.nndc.bnl.gov/qcalc/
2-7
Terms
•
Binding energy

Difference between mass of nucleus
and constituent nucleons
 Energy released if nucleons
formed nucleus
Btot (A,Z)=[ZM(1H)+(A-Z)M(n)-M(A,Z)]c2

average binding energy per nucleon
 Bave(A,Z)= Btot (A,Z)/A
 Measures relative stability
•
Binding Energy of an even-A nucleus is generally higher than values for adjacent odd-A
nuclei
•
Exothermic nature of the fusion of H atoms to form He from the very large binding
energy of 4He
•
Energy released from fission of the heaviest nuclei is large

•
Nuclei near the middle of the periodic table have higher binding energies per
nucleon
Maximum in the nuclear stability curve in the iron-nickel region (A~56 through 59)

2-8
Thought to be responsible for the abnormally high natural abundances of these
elements
Mass Based Energetics Calculations
• Why does 235U undergo neutron
induced fission for thermal
energies while 238U doesn’t?
• Generalized energy equation
AZ + n A+1Z + Q

• For 235U

Q=(40.914+8.071)-42.441

Q=6.544 MeV
• For 238U

Q=(47.304+8.071)-50.569

Q=4.806 MeV
• For 233U

Q=(36.913+8.071)-38.141

Q=6.843 MeV
• Fission requires around 5-6 MeV

Does 233U fission from
thermal neutron
2-9
Binding-Energy Calculation
• Volume of nuclei are nearly proportional to the number of
nucleons present
 Nuclear matter is incompressible
 Basis of equation for nuclear radius
• Total binding energies of nuclei are nearly proportional to
the numbers of nucleons present
 saturation character
Nucleon in a nucleus can apparently interact with
only a small number of other nucleons
Those nucleons on the surface will have different
interactions
• Basis of liquid-drop model of nucleus
 Considers number of neutrons and protons in nucleus
2-10
Liquid-Drop Binding Energy:
2
2



 N −Z  
 N −Z 
2 −1
2 −1 / 3
2/3
EB = c1 A1 − k 
  − c3 Z A + c4 Z A + δ
  − c2 A 1 − k 
 A  
 A  


• c1=15.677 MeV, c2=18.56 MeV, c3=0.717 MeV, c4=1.211 MeV,
k=1.79 and δ=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
2-11
Liquid drop model
2
2



 N −Z  
 N −Z 
2 −1
2 −1 / 3
2/3
EB = c1 A1 − k 
  − c3 Z A + c4 Z A + δ
  − c2 A 1 − k 
 A  
 A  


• 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 protons
lowers binding energy
 4th term represents correction term for charge distribution
with diffuse boundary
• δ term: Pairing Energy
 binding energies for a given A depend on whether N and Z
are even or odd
even-even nuclei, where δ=11/A1/2, are the stablest
 two like particles tend to complete an energy level by
pairing opposite spins
2-12
Mass Parabolas
• Method of
demonstrating
stability for given
mass constructed
from binding energy
 Values given in
difference, can
use energy
difference
• For odd A there is
only one β-stable
nuclide
 nearest the
minimum of the
parabola
2-13
Friedlander & Kennedy, p.47
Even A mass parabola
• For even A there are usually two or three possible β-stable isobars
 Stable tend to be even-even nuclei
Even number of protons, even number of neutron for
these cases
2-14
Magic Numbers
•
Certain values of N
and Z--2, 8, 20, 28,
50, 82, and 126 -exhibit unusual
stability
• Evidence from
masses, binding
energies, elemental
and isotopic
abundances,
numbers of species
with given N or Z,
and α-particle
energies
• Concept of closed
shells in nuclei
• Demonstrates
limitation in liquid
drop model
2-15
Nuclear Shapes: Radii
R=roA1/3
• Nuclear volumes are about proportional to nuclear
masses
 nuclei have approximately same density
• nuclei are not densely packed with nucleons
 Density varies
• ro~1.1 to 1.6 fm
• Nuclear radii can mean different things
 nuclear force field
 distribution of charges
 nuclear mass distribution
2-16
Nuclear Force Radii
•
•
The radius of the nuclear force field must be less than
the distance of closest approach (do)

d = distance from center of nucleus

T’ = α particle’s kinetic energy

T = α particle’s initial kinetic energy

do = distance of closest approach in a head on
collision when T’=0
do~10-20 fm for Cu and 30-60 fm for U
2 Ze 2
T'= T −
do
2 Ze 2
do =
T
2-17
http://hyperphysics.phy-astr.gsu.edu/hbase/rutsca.html#c1
Measurement of Nuclear Radii
• Any positively charged particle can be used to
probe the distance
 nuclear (attractive) forces become significant
relative to the Coulombic (repulsive force)
• Neutrons can be used but require high energy
 neutrons are not subject to Coulomb forces
high energy needed for de Broglie
wavelengths small compared to nuclear
dimensions
 At high energies, nuclei become transparent to
neutrons
Small cross sections
2-18
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 methods that
determine nuclear force radii
• Re (half-density radius)~1.07 fm
• de (“skin thickness”)~2.4 fm
2-19
Nuclear potentials
• Scattering experiments has approximate
agreement the Square-Well potential
 Woods-Saxon equation better fit
Vo
V=
(r −R) / A
1+ e

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
2-20
Square-Well and Woods-Saxon Potentials
• 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%, skin thickness, of the full potential
2-21
Nuclear Skin
• charge density give information on protons distribution in
nuclei
 no experimental techniques exist for determining total
nucleon distribution
generally assumed that neutrons and protons are
distributed in same way
 nuclear-potential radii are about 0.2 fm larger than radii
of the charge distributions
Nucleus Fraction of nucleons in the “skin”
ρo
12C
(
)
r
=
ρ
0.90
[( r − R ) / a ]
1
+
e
24Mg
0.79
56Fe
0.65
107Ag
0.55
139Ba
0.51
208Pb
0.46
2-22
238U
0.44
e
e
e
Spin
• Nuclei possess angular momenta Ih/2π
 I is an integral or half-integral number known as the
nuclear spin
For electrons, generally distinguish between electron
spin and orbital angular momentum
• Protons and neutrons have I=1/2
• Nucleons in the nucleus contribute orbital angular
momentum (integral multiple of h/2 π) and their intrinsic
spins (1/2)
 Protons and neutrons can fill shell (shell model)
 Shells have orbital angular momentum like electron
orbitals (s,p,d,f,g,h,i,….)
 spin of even-A nucleus is zero or integral
 spin of odd-A nucleus is half-integral
• All nuclei of even A and even Z have I=0 in ground state2-23
Magnetic Moments
• Nuclei with nonzero angular momenta have magnetic
moments
 From spin of protons and neutrons
• µBme/Mp is used as the unit of nuclear magnetic moments
and called a nuclear magneton
• 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
• Measured magnetic moments tend to differ from calculated
values
 Proton and neutron not simple structures
Neutron has charge distribution
* Negative (from negative mesons) near edge
2-24
Methods of measurements
•
Hyperfine structure in atomic spectra
•
Atomic Beam method

Element beam split into 2I+1 components in magnetic field
•
Resonance techniques
•

2I+1 different orientations
Quadrupole Moments: q=(2/5)Z(a2-c2), R2 = (1/2)(a2 + c2)= (roA1/3)2

Data in barns, can solve for a and c
•
Only nuclei with I>1/2 have quadrupole moments

Non-spherical nuclei

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
2-25
Parity
•
•
•
System wave function sign change if the sign of the space coordinates change

system has odd or even parity
Parity is conserved
even+odd=odd, even+even=even, odd+odd=odd
 allowed 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 parity
 states with odd l have odd parity
2-26
Topic review
• Understand role of nuclear mass in
reactions
 Use mass defect to determine energetics
 Binding energies, mass parabola, models
• Determine Q values
• How are nuclear shapes described and
determined
 Potentials
 Nucleon distribution
• Quantum mechanical terms
2-27
Study Questions
• What do binding energetic predict about
abundance and energy release?
• Determine and compare the alpha decay Q
values for 2 even and 2 odd Np isotopes.
Compare to a similar set of Pu isotopes.
• What are some descriptions of nuclear shape?
• Construct a mass parabola for A=117 and
A=50
• Describe nuclear spin, parity, and magnetic
moment
2-28
Pop Quiz
• Using the appropriate mass excess calculate the
following Q values for 212Bi. Show the reaction
 β− decay
 β+ decay
 EC
 Alpha decay
• Which decay modes are likely?
2-29