1. CHEM 312: Lecture 2 Nuclear Properties
Readings:
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. 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 even odd even odd
Z even even odd odd
Number
Simple property dictates nucleus behavior. Number of protons and neutron important

3. Which are the 4 stable odd-odd nuclei?

4. 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
amu
= amu
1 amu = MeV
amu x (931.5 MeV/1 amu)
MeV= Mass excess= for 24Na

5. 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=D AZ-DA(Z+1)
14C14N+ + β- +n + Q
Energy =Q= mass 14C – mass 14N
Use Q values (
Q= =0.156 MeV

6. 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 MeV, MeV total from the electrons
Q=DAZ – (DA(Z-1) ) MeV
90Nb90Zr- + β+ +n + Q
Q=D 90Nb – (D 90Zr ) MeV
Q= ( ) 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=DAZ – DA(Z-1)
207Bi207Pb +n + Q
Q=D 207Bi – D 207Pb MeV
Q= MeV= MeV

7. 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=D241Am-(D 237Np+D4He)
Q = ( )
Q = MeV
Alpha decay energy for 241Am is 5.48 and 5.44 MeV

8. 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= 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
Atomic masses of isotopes

9. 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)
amu
MeV
From mass excess
5.516 MeV
Q value for the EC of 22Na
22Na+ + e- 22Ne + n +Q
Q= 22Na - 22Ne
M (22Na)-M(22Ne)
amu
MeV
2.843 MeV

10. 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 elements
Elements up to Fe formed in stellar fusion

11. 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=( )
Q=6.544 MeV
For 238U
Q=( )
Q=4.806 MeV
For 233U
Q=( )
Q=6.843 MeV
Fission requires around 5-6 MeV
Does 233U from thermal neutron?

12. 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

13. Liquid-Drop Binding Energy:
c1= 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

14. Liquid drop model 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

15. Magic Numbers: Data comparison
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

16. Mass Parabolas nearest the minimum of the parabola
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 -stable nuclide
nearest the minimum of the parabola
Friedlander & Kennedy, p.47

17. Even A mass parabola Stable nuclei tend to be even-even nuclei
For even A there are usually two or three possible -stable isobars
Stable nuclei tend to be even-even nuclei
Even number of protons, even number of neutron for these cases

18. R=roA1/3 Nuclear Shapes: Radii
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

19. 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’ =  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 fm for U

20. 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

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 methods that determine nuclear force radii
Re (half-density radius)~1.07 fm
de (“skin thickness”)~2.4 fm

22. Nuclear potentials ro~1.35 to 1.6 fm for Square-Well
Scattering experimental data have has approximate agreement the Square-Well potential
Woods-Saxon equation better fit
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%
Nuclear skin thickness

23. Nuclear Skin Nucleus Fraction of nucleons in the “skin” 12C 0.90
24Mg
56Fe
107Ag
139Ba
208Pb
238U

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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 moment

30. Question
Comment in blog
Respond to PDF Quiz 2