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```CHEM 312: Lecture 2
Nuclear Properties
 Modern Nuclear Chemistry:
Chapter 2 Nuclear Properties
 Nuclear and Radiochemistry:
Chapter 1 Introduction,
Chapter 2 Atomic Nuclei
• Nuclear properties
 Masses
 Binding energies
 Reaction energetics
Q value
 Nuclei have shapes
2-1
Nuclear Properties
• Systematic examination of measurable data to determine nuclear
properties
 masses
 matter distributions
• Size, shape, mass, and relative stability of nuclei follow patterns
that can be understood and interpreted with models
 average size and stability of a nucleus described by average
nucleon binding in a macroscopic model
 detailed energy levels and decay properties evaluated with a
quantum mechanical or microscopic model
Simple example: 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
Simple property dictates nucleus behavior. Number of protons and
neutron important
2-2
Which are the 4 stable odd-odd nuclei?
2-3
Data from Mass
• Evaluation of Mass Excess
• Difference between actual mass of nucleus and expected
mass from atomic number
 By definition 12C = 12 amu
 If mass excess negative, then isotope has more
binding energy the 12C
• Mass excess==M-A
 M is nuclear mass, A is mass number
 Unit is MeV (energy)
 Convert with E=mc2
• 24Na example
 23.990962782 amu
 23.990962782-24 = -0.009037218 amu
 1 amu = 931.5 MeV
 -0.009037218 amu x (931.5 MeV/1 amu)
 -8.41817 MeV= Mass excess= for 24Na
2-4
Masses and Q value
• Atomic masses

From 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 mass difference between parent and daughter

Mass excess values can be used to find Q (in MeV)
• β- decay Q value
AZA(Z+1)+ + β- +n + Q

 Consider β- mass to be part of A(Z+1) atomic mass (neglect
binding)
 Q= AZ-A(Z+1)
14C14N+ + β- +n + Q

 Energy =Q= mass 14C – mass 14N
* Use Q values
(http://www.nndc.bnl.gov/wallet/wccurrent.html)
2-5
 Q=3.0198-2.8634=0.156 MeV
Q value
•
•
Positron Decay
AZA(Z-1)- + β+ +n + Q


Have 2 extra electrons to consider
 β+ (positron) and additional atomic electron from Z-1 daughter
* Each electron mass is 0.511 MeV, 1.022 MeV total from the
electrons

Q=AZ – (A(Z-1)- + 1.022) MeV
90Nb90Zr- + β+ +n + Q


Q= 90Nb – ( 90Zr + 1.022) MeV

Q=-82.6632-(-88.7742+1.022) MeV=5.089 MeV
Electron Capture (EC)

Electron comes from parent orbital
 Parent can be designated as cation to represent this behavior
AZ+ + e- A(Z-1) + n + Q


Q=AZ – A(Z-1)
207Bi207Pb +n + Q


Q= 207Bi –  207Pb MeV

Q= -20.0553- -22.4527 MeV=2.3947 MeV
2-6
Q value
• Alpha Decay
 AZ(A-4)(Z-2) + 4He + Q
 241Am237Np + 4He + Q
Use mass excess or Q
value calculator to
determine Q value
 Q=241Am-( 237Np+4He)
 Q = 52.937-(44.874 + 2.425)
 Q = 5.638 MeV
 Alpha decay energy for
241Am is 5.48 and 5.44 MeV
2-7
Q value determination
• For a general reaction
 Treat Energy (Q) as part of the equation
 Solve for Q
• 56Fe+4He59Co+1H+Q
 Q= [M56Fe+M4He-(M59Co+M1H)]c2
* M represents mass of isotope
 Q=-3.241 MeV (from Q value calculator)
• Mass excess and Q value data can be found in a number
of sources
 Table of the Isotopes
 Q value calculator
 http://www.nndc.bnl.gov/qcalc/
 Atomic masses of isotopes
 http://physics.nist.gov/cgibin/Compositions/stand_alone.pl
2-8
Q value calculation examples
•
•
Find Q value for the Beta decay of 24Na
24Na24Mg+ +b- + n +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 + n +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
2-9
Terms from Energy
•
Binding energy

Difference between mass of nucleus
and constituent nucleons
 Energy released if nucleons
formed nucleus

Nuclear mass not equal to sum of
constituent nucleons
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
 Some mass converted into energy
that binds nucleus
 Measures relative stability
Binding Energy of an even-A nucleus is generally higher than adjacent odd-A
nuclei
Exothermic fusion of H atoms to form He from 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)

Responsible for the abnormally high natural abundances of these
2-10
elements

Elements up to Fe formed in stellar fusion
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 from thermal
neutron?
2-11
Binding-Energy Calculation: Development
of simple nuclear model
• Volume of nuclei are nearly proportional to number of nucleons present
 Nuclear matter is incompressible
 Basis of equation for nuclear radius
• Total binding energies of nuclei are nearly proportional to 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 and how they
may arrange
 Developed from mass data
 http://en.wikipedia.org/wiki/Semi-empirical_mass_formula
2-12
Liquid-Drop Binding Energy:
2
2



 N -Z 
 N -Z  
2/3
2 -1/ 3
2 -1
EB  c1 A1 - k 
  - c2 A 1 - k 
  - c3 Z A + c4 Z A + 
 A  
 A  


• 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
2-13
Liquid drop model
2
2



 N -Z 
 N -Z  
2/3
2 -1/ 3
2 -1
EB  c1 A1 - k 
  - c2 A 1 - k 
  - c3 Z A + c4 Z A + 
 A  
 A  


• 2nd Term: Surface Energy
 Nucleons at surface of nucleus have unsaturated forces
 decreasing importance with increasing nuclear size
• 3rd and 4th Terms: 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
•  term: Pairing Energy
 binding energies for a given A depend on whether N and Z are even or
odd
even-even nuclei, where =11/A1/2, are the most stable
 two like particles tend to complete an energy level by pairing opposite
spins
 Neutron and proton pairs
2-14
•
Certain values of Z and
N exhibit unusual
stability

2, 8, 20, 28, 50, 82,
and 126
•
Evidence from different
data

masses,

binding energies,

elemental and
isotopic
abundances
Concept of closed shells
in nuclei

Similar to
electron closed
shell
Demonstrates limitation
in liquid drop model
Magic numbers
demonstrated in shell
model
•
•
•

Nuclear
structure and
model lectures
Magic Numbers: Data
comparison
2-15
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 b-stable
nuclide
 nearest the
minimum of the
parabola
2-16
Friedlander & Kennedy, p.47
Even A mass parabola
• For even A there are usually two or three possible b-stable isobars
 Stable nuclei tend to be even-even nuclei
Even number of protons, even number of neutron for
these cases
2-17
R=roA1/3
• Nuclear volumes are nearly 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 for equation above
• Nuclear radii can mean different things
 nuclear force field
 distribution of charges
 nuclear mass distribution
2-18
•
•
The radius of the nuclear force field must be less than
the distance of closest approach (do)

d = distance from center of nucleus

T’ =  particle’s kinetic energy

T =  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
2Ze 2
T' T do
2 Ze 2
do 
T
2-19
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
2-20
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
2-21
Nuclear
potentials
•
•
Scattering experimental
data have has approximate
agreement the Square-Well
potential
Woods-Saxon equation
better fit
Vo
V
1 + e( r - R ) / A

Vo=potential at center
of nucleus

A=constant~0.5 fm

R=distance from
center at which
V=0.5Vo (for halfpotential radii)

or V=0.9Vo and
V=0.1Vo for a dropoff from 90 to 10% of
the full potential
• ro~1.35 to 1.6 fm for SquareWell
• ro~1.25 fm for Woods-Saxon
• ro~2.2 fm for Woods-Saxon with
drop-off from 90 to 10%
• Nuclear skin thickness 2-22
Nuclear Skin
Nucleus Fraction of nucleons in the “skin”
12C
0.90
24Mg
0.79
56Fe
0.65
107Ag
0.55
139Ba
0.51
208Pb
0.46
238U
0.44
 (r ) 
o
[( re - Re ) / ae ]
1+ e
2-23
Spin
• Nuclei possess angular momenta Ih/2
 I is an integral or half-integral number known as nuclear
spin
For electrons, generally distinguish between electron
spin and orbital angular momentum
• Protons and neutrons have I=1/2
• Nucleons in nucleus contribute orbital angular momentum
(integral multiple of h/2 ) 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 state
2-24
Magnetic Moments
• Nuclei with nonzero angular momenta have
magnetic moments
 From spin of protons and neutrons
• Bme/Mp is unit of nuclear magnetic moments
 nuclear magneton
• Measured magnetic moments tend to differ
from calculated values
 Proton and neutron not simple structures
2-25
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
2-26
Parity
•
•
•
System wave function sign change if 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
2-27
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
 Used in description of nucleus
2-28
Study Questions
• What do binding energetics 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
• What is the density of nuclear material?
• Describe nuclear spin, parity, and magnetic
2-29
moment
Question
• Comment in blog
• Respond to PDF Quiz 2
2-30
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