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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.

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chem 312 lecture 2 nuclear properties
CHEM 312: Lecture 2Nuclear 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
nuclear properties
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 160 53 49 4

Simple property dictates nucleus behavior. Number of protons and neutron important

data from mass
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
masses and q value
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=DAZ-DA(Z+1)
    • 14C14N+ + β- +n + Q
      • Energy =Q= mass 14C – mass 14N
        • Use Q values (http://www.nndc.bnl.gov/wallet/wccurrent.html)
      • Q=3.0198-2.8634=0.156 MeV
q value
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=DAZ – (DA(Z-1)- + 1.022)MeV
    • 90Nb90Zr-+ β++n + Q
    • Q=D 90Nb – (D 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=DAZ – DA(Z-1)
    • 207Bi207Pb +n + Q
    • Q=D 207Bi– D207Pb MeV
    • Q= -20.0553- -22.4527 MeV=2.3947 MeV
q value1
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 = 52.937-(44.874 + 2.425)
    • Q = 5.638 MeV
    • Alpha decay energy for 241Am is 5.48 and 5.44 MeV
q value determination
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/qcalcr.jsp
    • Atomic masses of isotopes
      • http://physics.nist.gov/cgi-bin/Compositions/stand_alone.pl
q value calculation examples
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
terms from energy
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
mass based energetics calculations
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?
binding energy calculation development of simple nuclear model
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
liquid drop binding energy
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
liquid drop model
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
magic numbers data comparison
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
mass parabolas
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 -stable nuclide
    • nearest the minimum of the parabola

Friedlander & Kennedy, p.47

even a mass parabola
Even A mass parabola
  • 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
nuclear shapes radii
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

R=roA1/3

nuclear force radii
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 30-60 fm for U

http://hyperphysics.phy-astr.gsu.edu/hbase/rutsca.html#c1

measurement of nuclear radii
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
electron scattering
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
nuclear potentials
Nuclear potentials
  • 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
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

Nuclear Skin
slide24
Spin
  • Nuclei possess angular momentaIh/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
magnetic moments
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
methods of measurements
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
parity
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
topic review
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
study questions
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
pop quiz
Pop Quiz
  • Using the appropriate mass excess data calculate the following Q values for 212Bi. Show the reaction
    • b-decay
    • b+decay
    • EC
    • Alpha decay
  • Which decay modes are likely?
  • Send answers as e-mail or bring to next class meeting
  • Provide comments in blog when complete