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RFSS: Lecture 2 Nuclear Properties

RFSS: 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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RFSS: Lecture 2 Nuclear Properties

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  1. RFSS: 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

  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 160 53 49 4 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 • 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

  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=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

  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 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

  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 = 52.937-(44.874 + 2.425) • Q = 5.638 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=-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

  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) • 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

  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=(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?

  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 • http://en.wikipedia.org/wiki/Semi-empirical_mass_formula

  13. 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

  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 • 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 • 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. 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

  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 30-60 fm for U http://hyperphysics.phy-astr.gsu.edu/hbase/rutsca.html#c1

  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 • 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. 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

  24. 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

  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. 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

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