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  1. Gross Properties of Nuclei Sizes Nuclear Sizes

  2. Absorption upon intersection of nuclear cross section area s jbeam current areal densityA area illuminated by beamL = 6.022 1023/mol Loschmidt# NT # target nuclei in 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 Illuminated area A dx Nucleus cross section area sT Thin target, thickness x  Target/Absorber 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 12C scattering D.D. Kerlee et al., PR 107, 1343 (1957) Nuclear Sizes Distance of closest approach  scatter angle q P.R. Christensen et al., NPA207, 33 (1973)

  5. b a Elastic Electron Scattering Simplified treatment: Incoming plane wave= approximation to particle wave packet. Independent nucleons, impulse approximation. Detector l l Elementary scattering waves arrive at detector with phase differences  calculate relative to nuclear center Nuclear Sizes probability amplitude for proton n

  6. Momentum Transfer and Scatter Angle in (e,e’) For given energy, scattering angle q determines momentum transfer. Controllable by experiment (E, q). <bra* | ket> Carry out integration over r, rn’ except rn Nuclear Sizes =f0 x density of proton n at rn

  7. Separation of Variables Point nucleus (PN): 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 nucleus = cross section for point-nucleus x form factor F of charge distribution

  8. Mott Cross Section for Electron Scattering 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.

  9. 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)

  10. 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

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

  12. 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):

  13. 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)

  14. 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)

  15. r(r) point nucleus r(r) r Excited ground nuclear state Muonic X-Rays Effect exists for also for e-atoms but is weaker than for muonsNegative muon:m- e- mm = 207me Replace electron by muon  “muonic atom” Bohr orbits, am = ae/207 107 times stronger fields • X-ray energies 100keV–6 MeV • Isomeric/isotopic shifts DEis VCoul(r) En r 3d2p 1s DEis(2p) Finite size Nuclear Sizes DEis(1s) Point Nucleus

  16. Energy/keV Charge Radii from Muonic Atoms 2p3/2 1s1/2 2p1/2 1s1/2 2p3/2 1s1/2 Nuclear Sizes 2p1/2 1s1/2 Engfer et al., Atomic Nucl. Data Tables 14, 509 (1974) R(A) sensitive to isotopic, isomeric, chemical effects. Adding n dilutes rZ(r) E.B. Shera et al., PRC14, 731 (1976)

  17. Mass and Charge Distributions Parameters of Fermi Distribution Charge density: Mass density distribution scales: except for small surface increase in n density (“neutron skin”) Constant central density for all nuclides, except 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. 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)

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

  21. 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

  22. n n 4He 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) ! rms matter 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 eF Plausible 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)

  23. End of Nuclear Sizes Nuclear Sizes