Sizes

W. Udo Schröder, 2007 Nuclear Sizes 2 Absorption Probability and Cross Section Absorption upon intersection of nuclear cross section area  j beam current areal density A area illuminated by beam L = /mol Loschmidt# N T # target nuclei in beam M T target molar weight  T target density dx target thickness []=1barn = cm 2 Target dx Incoming Transmitted Mass absorption coefficient dN = -Ndx Thin target, thickness x  elementary absorption cross section area per nucleus Illuminated area A Nucleus cross section area 

W. Udo Schröder, 2007 Nuclear Sizes 3 Target n Detector Electronics DAQ (Pu-Be) n Source Size Information from Nuclear Scattering Basic exptl. setup with n source: Count Target in/target out d from small accelerator (E d  100 keV): T(d,n) 3 He  E n  14 MeV J.B.A. England, Techn.Nucl. Str. Meas., Halsted, New York,1974 Amp/ Disc Cntr Experiment (approx. analysis)  Equilibrium matter density  0

W. Udo Schröder, 2007 Nuclear Sizes 4 Interaction Radii  scattering 16 O scattering 12 C scattering P.R. Christensen et al., NPA207, 33 (1973) D.D. Kerlee et al., PR 107, 1343 (1957)  d Distance of closest approach  scatter angle 

W. Udo Schröder, 2007 Nuclear Sizes 5 Electron Scattering a b Detector phase difference of elementary waves Impulse Approximation: Whole is sum of parts, no interactions among parts. Incoming plane wave= approximation to particle wave packet Center of nucleus r=0 probability amplitude for proton n

W. Udo Schröder, 2007 Nuclear Sizes 6 Momentum Transfer and Scatter Angle in (e,e)  Scattering angle determines momentum transfer

W. Udo Schröder, 2007 Nuclear Sizes 7 Separation of Variables Point nucleus: a=b,  n =0  Scatter cross section for finite nucleus = cross section for point-nucleus x form factor F of charge distribution

W. Udo Schröder, 2007 Nuclear Sizes 8 Mott Cross Section for Electron Scattering In typical nuclear applications, electron kinetic energies K » m e c 2  (extreme) relativistic domain e - = good probe for objects on fm scale Obtained in 1. order qu. m. perturbation theory, neglects nuclear recoil momentum.

W. Udo Schröder, 2007 Nuclear Sizes 9 Elastic (e,e) Scattering Data R. Hofstadter, Electron Scatt. and Nucl. Struct., Benjamin, 1963 J.B. Bellicard et al., PRL 19,527 (1967) X 10 X arm electron spectrometer (Univ. Mainz) d/d  diffraction patterns d/d  diffraction patterns 1 st. minimum q()  4.5/R

W. Udo Schröder, 2007 Nuclear Sizes 10 Fourier Transform of Charge Distribution  r R Generic Fourier transform of f: Form factor F contains entire information about charge distribution Fermi distribution , half-density radius C diffuseness a R C 4.4a C is different from the radius of equivalent sharp sphere

W. Udo Schröder, 2007 Nuclear Sizes 11 Nuclear Charge Form Factor Form factor for Coulomb scattering = Fourier transform of charge distribution.r-Distribution Function (r) Form Factor -Distribution Point1constant Homogeneous sharp sphere  0 for r  R =0 for r >R oscillatory Exponentialexponential Gaussian

W. Udo Schröder, 2007 Nuclear Sizes 12 Model-Independent Analysis of Scattering mean-square radius of charge distribution  r R Equivalent sharp radius of any (r): Interpretation in terms of radial moments of charge distribution Expansion: =1

W. Udo Schröder, 2007 Nuclear Sizes 13 Nuclear Charge Distributions (e,e) R. Hofstadter, Ann. Rev. Nucl. Sci. 7, 231 (1957) t=4.4a C: Half-density radius a: Surface diffuseness t: Surface thickness Leptodermous: t « C Holodermous : t ~ C

W. Udo Schröder, 2007 Nuclear Sizes 14 Charge Radius Systematics  0 (charge) decreases for heavy nuclei like Z/A  for all nuclei:  0 (mass) = 0.17 fm -3 = const.  g/cm 3

W. Udo Schröder, 2007 Nuclear Sizes 15 Muonic X-Rays Negative muon:  -  e - m  = 207m e Replace electron by muon  “muonic atom” Bohr orbits, a  = a e / times stronger fields (r) r 1)X-ray energies 100keV–6 MeV 2)Isomeric/isotopic shifts E is 3d 2p 1s E is (1s) E is (2p) r V Coul (r) E n point nucleus ground excited nuclear state

W. Udo Schröder, 2007 Nuclear Sizes 16 Charge Radii from Muonic Atoms Engfer et al., Atomic Nucl. Data Tables 14, 509 (1974) Energy/keV E.B. Shera et al., PRC14, 731 (1976) 2p 3/2  1s 1/2 2p 1/2  1s 1/2 Sensitive to isotopic, isomeric, chemical effects

W. Udo Schröder, 2007 Nuclear Sizes 17 Mass density distribution: except for small surface increase in n density (“neutron skin”) Mass and Charge Distributions Charge density: Constant central density for all nuclides, except very light (Li, Be, C,..)

W. Udo Schröder, 2007 Nuclear Sizes 18 Leptodermous Distributions C = Central radius R = Equivalent sharp radius Q = Equivalent rms radius b = Surface width R.W. Hasse & W.D. Myers, Geometrical relationships of macroscopic nuclear physics, Springer V., New York, 1988 leptodermous

W. Udo Schröder, 2007 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 Tanihata et al., RIKEN-AF-NP-233 (1996)

W. Udo Schröder, 2007 Nuclear Sizes 20 “Interaction Radii for Exotic Nuclei Derive  R = T -  el,  R =:[R I (p)+R I (T)] 2 Tanihata et al., RIKEN-AF-NP-168 (1995) =( N-Z)/2 Kox Parameter ization:

W. Udo Schröder, 2007 Nuclear Sizes 21 “Halo” Nuclei From p scattering on 11 Li  extended mass distribution (“halo”). Valence-neutron correlations in 11 Li: r 1 = r 2 = 5fm, r 12 = 7 fm 6 He - 8 He mass density distributions Experiment: dashed, Theory:solid 9 Li n n 11 Li Korshenninikov et al., RIKEN-AF-NP-233, 1996 tntn

W. Udo Schröder, 2007 Nuclear Sizes 22 Neutron Skin of Exotic (n-Rich) Nuclei 8 He n n Q rms ( 4 He) = (1.57±0.05)fm Q rms ( 6 He) = (2.48±0.03)fm Q rms ( 8 He) = (2.52±0.03)fm V( 8 He) = 4.1 x V( 4 He) ! matter radii D.H. Hirata et al., PRC 44, 1467(1991) Thick n-skin for light n-rich nuclei: t n ≈ 0.9 fm ( 6 He, 8 He) Relativistic mean field calculations: t n   F 133 Cs 78 stable, insignificant n-skin, t n ~0.1fm 181 Cs 126 unstable, significant n-skin, t n ~ 2fm Can one make 181 Cs ?? p-halos ? Coulomb barrier keeps p together, expansion could reduce it Tanihata et al., PLB 289,261 (1992)

W. Udo Schröder, 2007 Nuclear Sizes 23