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Atoms and Nuclei PA 322 Lecture 12 Unit 3: Introduction to nuclear properties (continued) (Reminder: http://www.star.le.ac.uk/~nrt3/322) Topics • Basic nuclear properties – full list: • mass, charge, mass & charge distribution, radius, abundances of isotopes (if nuclide stable), decay modes, half-lives (if unstable), reaction modes, spin magnetic dipole moment, electric quadrupole moment … – covered in last lecture: size, distribution of mass and charge – covered in this lecture • mass and abundances • binding energies • stability PA322 Lecture 12 2 Mass and abundance of nuclides • Key measurements are: – mass of nuclide – mass number of nuclide = A = N + Z – relative abundances of different isotopes [same element (same Z), but different A] – NB: nuclide mass ≠ mass number • Atomic weights of elements from simple chemical techniques and knowledge of Avogadro’s Number – but these reflect only average over abundances of different isotopes of same element – accuracy limited • Atomic numbers from, e.g., X-ray spectroscopy (wavelengths of Kα lines ∝ 1/Z2) PA322 Lecture 12 3 detector Nuclide masses can be determined with very high precision using mass spectrometer E ion source PA322 velocity selector Lecture 12 momentum selector 4 Mass spectrometer Determination of nuclide masses and abundances • • • Ions of the material to be tested are produced by discharge or electron bombardment Ions enter “velocity selector” with range of velocities v – orthogonal electric and magnetic fields E and B1 – only ions with v = vsel emerge: where forces due to electric and magnetic fields exactly cancel (other ions blocked). detector E ion source velocity selector momentum selector in practice B1 and B2 can be the same magnetic field Ions with single velocity vsel enter momentum selector – uniform magnetic field B2 separates ions by momentum and hence mass (since single value of velocity): produces trajectories with different radii r – final position of ion in detector thus determines nuclide mass PA322 Lecture 12 5 Mass spectrometer Determination of nuclide masses and abundances • Velocity selection – force from electric field – force from magnetic field FE = q E FM = q (v x B1) – for orthogonal fields, forces cancel for vsel = E/B1 • Momentum selection – motion perpendicular to B field: mvsel = qB2 r mv2/r = qvB r = (mvsel )/(qB2) – nuclide mass is thus uniquely determined by r – mass number directly obtained from mass: nearest integer with mass expressed in u (1/12 12C mass) – mass spectrometer separates isotopes with different nuclide mass PA322 Lecture 12 6 Mass spectrometer Determination of nuclide masses and abundances 36Kr Atomic weight = 83.798 • • • • PA322 Measurements made with respect to 12C Absolute accuracy ~10-6 Relative accuracy ~10-8 by comparing materials with similar mass effective atomic weight of element = ave. over all isotopes Lecture 12 7 Ball-park estimate of energy of nucleons • Nucleons to be confined to the atomic nucleus, i.e. with r ≤ Rnucleus • Uncertainty principle: p is nucleon momentum ℏ ≲ Δp Δx thus implies ℏ / Rnucleus ≲ Δp putting p ~ Δp and E = p2/2mp E = ½ mpv2 = ½ (mpv)2 /mp this gives E ~ ℏ2/ (2mpR2) for R ~ 2 fm get E ~ 5 MeV • Simple estimate of (kinetic) energy of nucleon is E ~ 5 MeV • Compare to energy of outer electrons in atom E ~ 1-2 eV PA322 Lecture 12 8 Origin of binding energy • Binding energy due to fact that nucleons are in potential well, ie. have –ve potential energy – energy needs to be supplied to remove nucleons net potential due to nuclear force (attractive) and Coulomb force (repulsive between protons) nuclear force at short range > Coulomb force V(r) -B r – But also accounts for kinetic energy. PA322 Lecture 12 9 Binding energy • Nuclide masses < sum of parts (if they were equal determining accurate masses would not be very interesting) • Mass deficit due to binding energy of nucleus (mass energy of deficit is the binding energy) • Mass energy of nuclide with atomic number Z and mass number A: (mA – Zme) is quantity measured in mass spectrometer but mA is quantity usually tabulated can be ignored, typically ~10-6 of mc2 mc2 = mA c2 – Z mec2 + Be atomic mass energy mass energy of electrons electronic binding energy A Z • Binding energy of a nuclide X is difference between its mass energy and its constituent nucleons, ie. B = [ Z mp + N mn – { m(AX) – Zme } ] c2 m = m(AX) A nucleons PA322 mass energy of nuclide Lecture 12 10 Binding energy • Binding energy can be rewritten in terms of hydrogen atom masses: B = [ Z m(1H) + N mn – m(AX)] c2 • Binding energy concept sometimes expressed as mass defect Δ where Δ = (mA – A) c2 • mass defect is directly related to binding energy, but with offset • Example 16O: mass m = 15.994915 u (from Table in Krane) A Binding energy B = [8 mp + 8 mn – {15.994915 – 8 me}] B = [8.05821176 + 8.06932008 - 15.994915 + 0.00438864] u B = 0.1370 u = 127.62 MeV B/A = 7.98 MeV/nucleon [Mass defect Δ = 15.994915 u – 16 = -4.737 MeV] 1u = 931.50 MeV PA322 Lecture 12 11 Binding energy per nucleon as function of A PA322 Lecture 12 12 Binding energy dependence with mass number • Dependence of binding energy per nucleon with mass number A – strong increase at low A – more gentle decline at high A – approx constant for most A at value of B/A~ 8 MeV – maximum binding energy per nucleon at A ~ 60 (close to Fe) • Implications – combining two nuclei with low A increases B and thus releases energy: FUSION! (~4 MeV per nucleon) – splitting nucleus with high A increases B and thus releases energy: FISSION! (~ 1 MeV per nucleon) – processes do not occur spontaneously (fortunately!), require activation energy PA322 Lecture 12 13 neutron rich stable nuclides are shaded black, unstable (radioactive) nuclides are shaded grey PA322 Lecture 12 14 Stability of nuclides • Overall pattern: – stable nuclides • for small Z, N~Z and A~2Z • for large Z, N>Z • most stable nuclei have even Z and even N • significant number with even-odd and odd-even configurations • only a very few odd-odd nuclei are stable (e.g.21H 147 N ) – unstable nuclides are typically those furthest from line of stability • exhibit radioactivity, transformed into different nuclides PA322 Lecture 12 A N Z number stable even even odd even odd 166 8 odd even odd odd even 57 53 15 Note, axes flipped compared to previous plot. PA322 Lecture 12 16 PA322 Lecture 12 17 Stability of nuclides • Reasons for stability/instability – must be related to properties of nuclear and Coulomb forces (and balance between them) – can be explored explicitly in terms of contributions to total binding energy (e.g. the semi-empirical mass model which predicts B(A) ) – for small Z, N~Z and A~2Z • similar numbers of protons and neutrons must be preferred configuration – for large Z, N>Z • Coulomb repulsion increases with Z; additional neutrons needed to dilute repulsive force – most stable nuclei have even Z and even N – significant number with even-odd and odd-even configurations – only a very few odd-odd nuclei are stable • related to properties of nuclear force: – binding energies must be Beven-even > Bodd-even ~ Beven-odd > Bodd-odd PA322 Lecture 12 18