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  1. Gross Properties of Nuclei Sizes Nuclear Sizes and Matter Distributions

  2. Absorption of projectiles intersecting nuclear cross section area s jbeam current areal densityA area illuminated by beamL = 6.022 1023/mol Loschmidt # NT # target nuclei illuminated by beamMT target molar weightrT target densitydx target thickness [s]=1barn = 10-24cm2 Absorption Probability and Cross Section Mass absorption coefficient m: dN = -Nm dx Transmitted Incoming area illuminated by beam Illuminated area A dx Nucleus cross section area s Target Thin target, thickness dx« 1/m Nuclear Sizes elementary absorption cross section area per nucleus

  3. Cntr Amp/Disc Size Information from Nuclear Scattering (Pu-Be) n Source Basic exptl. setup with n source: Count Target in/target out Target n Detector Electronics DAQ d from small accelerator (Ed100 keV): T(d,n)3He En 14 MeV Experiment (approx. analysis) Nuclear Sizes J.B.A. England, Techn.Nucl. Str. Meas., Halsted, New York,1974 Equilibrium matter density r0

  4. q d Interaction Radii a scattering 16O scattering Nuclear Sizes 12C scattering D.D. Kerlee et al., PR 107, 1343 (1957) Distance of closest approach  scatter angle q P.R. Christensen et al., NPA207, 33 (1973)

  5. r(r), y(r) point nucleus r(r) r excitedground nuclear state Finite Size Effect in Muonic X-Rays Weak effect for e-atoms Enhanced in m-atoms (Replace e  m )Negative muon:m- e- mm = 207me Bohr orbits, am = ae/207 107 times stronger fields • mX-ray energies 100 keV– 6MeV • Isomeric/isotopic shifts DEis VCoul(r) En r 3d2p 1s DEis(2p) Finite size Nuclear Sizes DEis(1s) Point Nucleus

  6. Energy/keV Charge Radii from Muonic Atoms 2p3/2 1s1/2 2p1/2 1s1/2 Nuclear Sizes Engfer et al., Atomic Nucl. Data Tables 14, 509 (1974) Sensitive to isotopic, isomeric, chemical effects E.B. Shera et al., PRC14, 731 (1976)

  7. q/2 q/2 b a Electron Diffraction/Elastic Scattering Incoming electron plane wave = approximation to particle wave packet Detector l Center of nucleus: r=0 l Nuclear Sizes Different path lengths  phase differences jof elementary waves relative to center of nucleus  Interference patterns

  8. q/2 q/2 General Form of (e,e’) Scatter Cross Section Scattering angle q measures momentum transfer Amplitude f0 Phase Factorx No protons Amplitude Phase Factor Nuclear Sizes Determine amplitude factor f0 from classical PN approximation

  9. Scattering Form Factor Point nucleus (PN) has Rutherford cross section: a=b, jn=0  determine scaling factor Z protons Finite nucleus: integrate over space where proton wave function are non-zero Strength of Coulomb interaction same for each proton Nuclear Sizes Scatter cross section for finite size nucleus = cross section for point-nucleus x form factor F of charge distribution

  10. Relativistic (e, e’) Mott Cross Section In typical nuclear applications, electron kinetic energies K » mec2 (extreme) relativistic domain (b =v/c) check non-relativistic limit e- = good probe for objects on fm scale Nuclear Sizes Obtained in 1. order quantum mechanical perturbation theory, neglects nuclear recoil momentum.

  11. Elastic (e, e’) Scattering Data ds/dW diffraction patterns1st. minimum q(q)4.5/R 3-arm electron spectrometer (Univ. Mainz) X 10 Nuclear Sizes X 0.1 R. Hofstadter, Electron Scattering and Nuclear Structure, Benjamin, 1963 J.B. Bellicard et al., PRL 19,527 (1967)

  12. r r R Fourier Transform of Charge Distribution Form factor F contains entire information about charge distribution R Generic Fourier transform of f: 4.4a C Nuclear Sizes Fermi distribution r, half-density radius C diffuseness a C is different from the radius of equivalent sharp sphere Req

  13. Nuclear Charge Form Factor Form factor for Coulomb scattering = Fourier transform of charge distribution. Nuclear Sizes

  14. r r R Model-Independent Analysis of Scattering Interpretation in terms of radial moments of charge distributionExpansion: =1 Nuclear Sizes mean-square radius of charge distribution Equivalent sharp radius of any r(r):

  15. t=4.4a Nuclear Charge Distributions (e,e’) Rz(H) = (0.85-0.87) fmRz(He)= 1.67 fm C: Half-density radiusa: Surface diffusenesst: Surface thickness Leptodermous: t « C Holodermous : t ~ C Nuclear Sizes Density of 4He is 2 x r0 ! R. Hofstadter, Ann. Rev. Nucl. Sci. 7, 231 (1957)

  16. Charge Radius Systematics Note: Slightly different fit line, if not forced through zero. Nuclear Sizes r0(charge) decreases for heavy nuclei like Z/A  for all nuclei: r0(mass) = 0.17 fm-3 = const.  1014 g/cm3 (r0=1.07 fm)

  17. Mass and Charge Distributions Parameters of Fermi Distribution Charge density distribution: Mass density distribution: except for small surface increase in n density (“neutron skin”) Constant central density for heavy nuclides, not the very light (Li, Be, B,..) Nuclear Sizes

  18. Leptodermous Distributions R.W. Hasse & W.D. Myers, Geometrical relationships of macroscopic nuclear physics, Springer V., New York, 1988 Fermi Distribution (a C) C = Central radiusR = Equivalent sharp radiusQ = Equivalent rms radiusb= Surface width Leptodermous Expansion in (b/R)n Nuclear Sizes

  19. Additional Topics: Neutron Skin & Halo Nuclei Nuclear Sizes

  20. Studies with Secondary Beams Produce a secondary beam of projectiles from interactions of intense primary beam with “production” target  projectiles rare/unstable isotopes, induce scattering and reactions in “p” target Nuclear Sizes Tanihata et al., RIKEN-AF-NP-233 (1996)

  21. “Interaction Radii for Exotic Nuclei Derive sR =sTotal- selastic sR =:p[RI(P)+RI(T)]2 Nuclear Sizes KoxParameterization: Interaction Radius =(N-Z)/2 Tanihata et al., RIKEN-AF-NP-168 (1995)

  22. n 9Li 11Li n “Halo” Nuclei From p scattering on 11Li  extended mass distribution (“halo”). Valence-neutron correlations in 11Li: r1 = r2 = 5 fm, r12 = 7 fm Parameterization: tn 6He - 8He mass density distributions Experiment: dashed, Theory (fit):solid Nuclear Sizes Korshenninikov et al., RIKEN-AF-NP-233, 1996

  23. n n 8He Neutron Skin of Exotic (n-Rich) Nuclei Which n Orbits? Qrms(4He) = (1.57±0.05)fm Qrms(6He) = (2.48±0.03)fm Qrms(8He) = (2.52±0.03)fmV(8He) = 4.1 x V(4He) ! rmsmatter radii Tanihata et al., PLB 289,261 (1992) Thick n-skin for light n-rich nuclei: tn≈ 0.9 fm (6He, 8He) DRrms =Rnrms- Rprms Relativistic mean field calculations: tn eFPlausible because of weaker nuclear force 133Cs78 stable, normal n-skin thickness, tn~0.1fm181Cs126 unstable, significant n-skin, tn~ 2 fm Can one actually make 181Cs, role of outer n?? Nuclear Sizes Are there p-halos ?  Not yet known. D.H. Hirata et al., PRC 44, 1467(1991)

  24. End of Nuclear Sizes Nuclear Sizes