Announcements Nuclear Sizes and Nuclear MassesRutherford ExperimentElectron Scattering by the Nuclear Charge DistributionSlide Number 5Nuclear Charge DensitySlide Number 7Quick Note on Muon InteractionsThe Distribution of Nuclear Matter in NucleiSlide Number 10The Masses and Binding Energies of Nuclei in Their Ground StatesMore about binding energyAtomic Mass UnitBinding energy characteristicsSlide Number 15The semi-empirical mass formulaSemi-empirical formula 2The -stability valleyWill the atom with odd A and Z < Zmin, --decay?+ decay for odd A and Z > Zmin?More about K-capture-stability valley for nuclei with even ASlide Number 23Even A,  stabilityThe masses of the -stable nucleiMonday, September 29, 20081Announcements • Link to lectures:• http://www.physics.drexel.edu/~jelena/Phy s476676Lectures/Lectures476676Post.ht ml• Contacted the bookstore. They will have the newest 6th edition of “Particles and Nuclei” book available soon.• Homework for chapter 4: 4.1, 4.4, 4.5, 4.8Monday, September 29, 20082Nuclear Sizes and Nuclear Masses• Nucleus is a bound assembly of protons and neutrons• Protons and neutrons are together called nucleons• Chemical element X• A total number of nucleons• Z total number of protons• A – Z, total number of neutrons, called the mass number of the nucleus 3517 Cl and 3717 Cl• Isotopes are nuclei which only differ in number of neutrons• Isobars are nuclei which differ in number of protons, but have the same mass number AXZAMonday, September 29, 20083Rutherford Experiment• Rutherford’s experiment: size of the nucleus ~10-15 m, while size of the atom ~10-10 m•Monday, September 29, 20084Electron Scattering by the Nuclear Charge Distribution• High precision data came in ~50’s. Nuclear charge distribution probed by elastic scattering of high energy electron beams• Electromagnetic interaction dominant• Details of the nuclear charge density can be probed with incident particles, whose de Broglie λ is smaller than nuclear charge density change.• Electron λ/2π ~ 1 fm, has p = 2 π ~/λ and energy E = (p2c2 + m2c4)1/2 ~ 200 MeV (relativistic)Monday, September 29, 20085Experiments measure differential cross- section dσ(E,θ)/dΩ,where θ the scattering angle and incident electron energy EExample: Fig. 4.1.Electrons scattered from nucleus have higher energy due to very small recoil of the nucleus compared to recoil of atomic electrons.ρch(r)=ρ0ch1+er−RaMonday, September 29, 20086Nuclear Charge Density• Density function: eρch (r)• In the case of spherically symmetric nuclei, e ρch (r) is only r dependant• Finding ρch (r) from dσ(E,θ)/dΩ is only partially possible (Fig. 4.2.)Monday, September 29, 20087• It is customary to assume shape of ρ(r) with a few parameters and then determine parameters by fitting to the scattering data.• A typical form:where one determines R and a, while ρ0ch is a normalization constant, when integrate over entire space to get total charge Z.Function ρ has no fundamental significance, but it conveniently describes an almost uniform charge distribution that extends to R and falls to 0 over well-defined surface region of thickness ~a.Example Fig. 4.3: while R increases with the size of the nucleus a stays pretty much the same.ρch(r)=ρ0ch1+er−RaMonday, September 29, 20088Quick Note on Muon Interactions• Muons are also used to probe nuclear charge distributions• Muon is often considered as a heavy electron (x 207 me ), same charge and unstabe with mean life of 2.2x10-6s.• However, due to the larger mass, muons have atomic orbits that are much smaller in radius (207 times) and lowest muonic states partially lie in the nucleus.• Energies of these states depend on nuclear charge distribution• How do you produce muonic atom?• When the beam of negative pions hits the target material, muons produced in the pion decay get captured in the outer atomic orbitals.• Just before these muons decay, many fall to lower orbits, emitting X-rays in the transitions• By measuring X-ray energies and comparing to predictions based on different R and a in the nuclear charge density distribution, one can determine values of R and a.• This method for finding R and a agrees well with results from electron scatteringMonday, September 29, 20089The Distribution of Nuclear Matter in Nuclei• Neutrons are neutral• From measured charged density in the nucleus we can get some idea about distribution of nuclear matter • Lets assume that protons are point like objects and neglect Coulomb forces. Then:– Proton number density ρp (r) ~ ρch (r) (nuclear charge density)– Strong nuclear force is short range and charge independent, so ratio of proton to neutron density is the same everywhere in the nucleus– ρn (r) / ρp (r) = N / Z– So, the total density of nucleons is ρ = ρn + ρp or ρ = (A / Z) ρchMonday, September 29, 200810– Example Fig. 4.4– At the center of nucleus ρ is roughly the same for all nuclei– It increases with A, but limits at ρ0 = 0.17 nucleons fm-3 for large A– ρ0 is “density of nuclear matter– Based on A = (4/3 π) R3 ρ0 , which implies – R = 1.12 A1/3 fmMonday, September 29, 200811The Masses and Binding Energies of Nuclei in Their Ground States• The spherical liquid drop model• Uniform density• Bound QM system in ground state (1 keV or more difference for excited states – x100 more than for atomic states)• B(Z, N) energy is needed to pull apart nucleus into Z protons and N neutrons• B(Z, N) is binding energy and is positiveMonday, September 29, 200812More about binding energy• Mnuc (Z, N) = Zmp + Nmn – B(Z< N) / c2• B(Z, N) is of the order of 1% of the rest mass energy mnuc c2.• Mass of ion is measured directly, and not the mass of the bare nucleus:• Ma (Z, N) = Z(mp + me ) + Nmn – B(Z,N)/c2 –belectrons /c2• Where belectrons is binding energy of atomic electrons (negligible ~20.8 Z7/3 eV)Monday, September 29, 200813Atomic Mass Unit• Atomic masses are known very accurately (mass spectrometers …) • The unit used is atomic mass unit which is defined as 1/12 of the mass of the neutral 12C• 1 amu = 931.49432 ± 0.00028 MeV/c2• Masses of radioactive atoms are determined by measuring the energy release in the decay using E = mc2Monday, September 29, 200814Binding energy characteristics• Table 4.2. shows binding energies, as well as binding energy per nucleon