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 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 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 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 12 C = 12 amu If mass excess negative, then isotope has more binding energy the 12 C Mass excess= =M-A M is nuclear mass, A is mass number Unit is MeV (energy) Convert with E=mc 2 24 Na 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 24 Na 2-4
Masses and Q value Atomic masses From nuclei and electrons Nuclear mass can be found from atomic mass m 0 is electron rest mass, B e (Z) is the total binding energy of all the electrons B e (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 A Z A (Z+1) + + β - +ν + Q Consider β - mass to be part of A (Z+1) atomic mass (neglect binding) Q= A Z- A (Z+1) 14 C 14 N + + β - +ν + Q Energy =Q= mass 14 C mass 14 N * Use Q values (http://www.nndc.bnl.gov/wallet/wccurrent.html) Q=3.0198-2.8634=0.156 MeV 2-5
Q value Positron Decay A Z A (Z-1) - + β + +ν + 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= A Z ( A (Z-1) - + 1.022) MeV 90 Nb 90 Zr - + β + +ν + Q Q= 90 Nb ( 90 Zr + 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 A Z + + e - A (Z-1) + ν + Q Q= A Z A (Z-1) 207 Bi 207 Pb +ν + Q Q= 207 Bi 207 Pb MeV Q= -20.0553- -22.4527 MeV=2.3947 MeV 2-6
Alpha Decay Q value A Z (A-4) (Z-2) + 4 He + Q 241 Am 237 Np + 4 He + Q Use mass excess or Q value calculator to determine Q value Q= 241 Am-( 237 Np+ 4 He) Q = 52.937-(44.874 + 2.425) Q = 5.638 MeV Alpha decay energy for 241 Am 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 56 Fe+ 4 He 59 Co+ 1 H+Q Q= [M 56 Fe+M 4 He-(M 59 Co+M 1 H)]c 2 * 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/cgibin/compositions/stand_alone.pl 2-8
Q value calculation examples Find Q value for the Beta decay of 24 Na 24 Na 24 Mg + +β + ν +Q Q= 24 Na- 24 Mg M ( 24 Na)-M( 24 Mg) 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 22 Na 22 Na + + e - 22 Ne + ν +Q Q= 22 Na - 22 Ne M ( 22 Na)-M( 22 Ne) 21.994436425-21.991385113 0.003051 amu 2.842297 MeV From mass excess -5.181 - -8.024 2.843 MeV 2-9
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 B tot (A,Z)=[ZM( 1 H)+(A-Z)M(n)-M(A,Z)]c 2 average binding energy per nucleon B ave (A,Z)= B tot (A,Z)/A Some mass converted into energy that binds nucleus Measures relative stability Terms from Energy 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 4 He 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 2-10 Elements up to Fe formed in stellar fusion
Mass Based Energetics Calculations Why does 235 U undergo neutron induced fission for thermal energies while 238 U doesn t? Generalized energy equation A Z + n A+1 Z + Q For 235 U Q=(40.914+8.071)-42.441 Q=6.544 MeV For 238 U Q=(47.304+8.071)-50.569 Q=4.806 MeV For 233 U Q=(36.913+8.071)-38.141 Q=6.843 MeV Fission requires around 5-6 MeV Does 233 U 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
E B = c Liquid-Drop Binding Energy: 2 2 + N Z 2/3 N Z 2 1/3 2 A 1 k c2 A k c3z A c4z A A A 1 1 1 +δ c 1 =15.677 MeV, c 2 =18.56 MeV, c 3 =0.717 MeV, c 4 =1.211 MeV, k=1.79 and δ=11/a 1/2 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
E B = c Liquid drop model 2 2 + N Z 2/3 N Z 2 1/3 2 A 1 k c2 A k c3z A c4z A A A 1 1 1 2nd Term: Surface Energy Nucleons at surface of nucleus have unsaturated forces decreasing importance with increasing nuclear size 3rd and 4 th 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 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/a 1/2, are the most stable two like particles tend to complete an energy level by pairing opposite spins Neutron and proton pairs 2-14 +δ
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 β-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 β-stable isobars Stable nuclei tend to be even-even nuclei Even number of protons, even number of neutron for these cases 2-17
Nuclear Shapes: Radii R=r o A 1/3 Nuclear volumes are nearly proportional to nuclear masses nuclei have approximately same density nuclei are not densely packed with nucleons Density varies r o ~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
Nuclear Force Radii The radius of the nuclear force field must be less than the distance of closest approach (d o ) d = distance from center of nucleus T = α particle s kinetic energy T = α particle s initial kinetic energy d o = distance of closest approach in a head on collision when T =0 d o ~10-20 fm for Cu and 30-60 fm for U T ' = 2Ze T 2 d o = T 2Ze d o 2 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 R e (half-density radius)~1.07 fm d e ( 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 V = 1+ e o V ( r R) / A V o =potential at center of nucleus A=constant~0.5 fm R=distance from center at which V=0.5V o (for halfpotential radii) or V=0.9V o and V=0.1V o for a dropoff from 90 to 10% of the full potential r o ~1.35 to 1.6 fm for Square- Well r o ~1.25 fm for Woods-Saxon with half-potential radii, r o ~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 12 C 0.90 24 Mg 0.79 56 Fe 0.65 107 Ag 0.55 139 Ba 0.51 208 Pb 0.46 238 U 0.44 ρ ( r) = 1+ e ρ [( r e o R e )/ a e ] 2-23
Spin Nuclei possess angular momenta Ih/2π 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 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 state 2-24
Magnetic Moments Nuclei with nonzero angular momenta have magnetic moments From spin of protons and neutrons µ B m e /M p 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(a 2 -c 2 ), R 2 = (1/2)(a 2 + c 2 )= (r o A 1/3 ) 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 moment 2-29
Pop Quiz Answer PDF quiz on Lecture 2 page Send in when complete Provide comments in blog http://rfssunlv.blogspot.com/ 2-30