1. Gross Properties of Nuclei
Sizes
Gross Properties of Nuclei
Nuclear Sizes

2. Absorption Probability and Cross Section
Incoming
Transmitted
Mass absorption coefficient m: dN = -Nm dx
Illuminated area A dx
Nucleus cross section area s
Target
Thin target, thickness x 
Absorption upon intersection of nuclear cross section area s
j beam current areal density A area illuminated by beam L = /mol Loschmidt# NT # target nuclei in beam MT target molar weight rT target density dx target thickness
[s]=1barn = 10-24cm2
Nuclear Sizes
elementary absorption cross section area per nucleus
W. Udo Schröder, 2008

3. Size Information from Nuclear Scattering
(Pu-Be) n Source
Amp/Disc
Cntr
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
W. Udo Schröder, 2008

4. Interaction Radii q d Distance of closest approach  scatter angle q
a scattering
16O scattering
12C scattering
D.D. Kerlee et al., PR 107, 1343 (1957)
Nuclear Sizes q d
Distance of closest approach  scatter angle q
P.R. Christensen et al., NPA207, 33 (1973)
W. Udo Schröder, 2008

5. Elastic Electron Scattering
Incoming plane wave= approximation to particle wave packet
Impulse Approximation for interaction:
Detector a b
phase difference of elementary waves relative to center of nucleus l
Center of nucleus: r=0 l
Nuclear Sizes
probability amplitude for proton n
W. Udo Schröder, 2008

6. Momentum Transfer and Scatter Angle in (e,e)
q/2
Scattering angle q determines momentum transfer
<bra* | ket>
Nuclear Sizes
f0 x density of proton n at rn
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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
W. Udo Schröder, 2008

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.
W. Udo Schröder, 2008

9. Elastic (e,e) Scattering Data
ds/dW  diffraction patterns 1st. 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)
W. Udo Schröder, 2008

10. Fourier Transform of Charge Distribution
Form factor F contains entire information about charge distribution R r R
Generic Fourier transform of f:
4.4a
Nuclear Sizes C
Fermi distribution r, half-density radius C diffuseness a
C is different from the radius of equivalent sharp sphere Req
W. Udo Schröder, 2008

11. Nuclear Charge Form Factor
Form factor for Coulomb scattering = Fourier transform of charge distribution.
r-Distribution
Function r(r)
Form Factor
q-Distribution
Point 1
constant
Homogeneous sharp sphere
r0 for r  R =0 for r >R
oscillatory
Exponential
exponential
Gaussian
Nuclear Sizes
W. Udo Schröder, 2008

12. Model-Independent Analysis of Scattering
Interpretation in terms of radial moments of charge distribution Expansion: =1
mean-square radius of charge distribution
Nuclear Sizes r R
Equivalent sharp radius of any r(r):
W. Udo Schröder, 2008

13. Nuclear Charge Distributions (e,e)
Rz(H) = ( ) fm Rz(He)= 1.67 fm
t=4.4a
C: Half-density radius a: Surface diffuseness t: 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)
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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)
W. Udo Schröder, 2008

15. Muonic X-Rays r(r) r X-ray energies 100keV–6 MeV
Effect exists for also for e-atoms but is weaker than for muons Negative muon: m-  e- mm = 207me
Replace electron by muon  “muonic atom”
Bohr orbits, am = ae/207
107 times stronger fields
r(r) r
X-ray energies 100keV–6 MeV
Isomeric/isotopic shifts DEis
3d2p 1s
DEis(1s) r
VCoul(r) En
Point Nucleus
point nucleus
Excited ground nuclear state
DEis(2p)
Finite size
Nuclear Sizes
W. Udo Schröder, 2008

16. Charge Radii from Muonic Atoms
Energy/keV
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)
W. Udo Schröder, 2008

17. Mass and Charge Distributions
Parameters of Fermi Distribution
Charge density:
Mass density distribution:
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
W. Udo Schröder, 2008

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 radius R = Equivalent sharp radius Q = Equivalent rms radius b = Surface width
Leptodermous Expansion in (b/R)n
Nuclear Sizes
W. Udo Schröder, 2008

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)
W. Udo Schröder, 2008

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)
W. Udo Schröder, 2008

21. “Halo” Nuclei
From p scattering on 11Li  extended mass distribution (“halo”). Valence-neutron correlations in 11Li: r1 = r2 = 5 fm, r12 = 7 fm
9Li n
11Li
Parameterization: tn
6He - 8He mass density distributions
Experiment: dashed, Theory (fit):solid
Nuclear Sizes
Korshenninikov et al., RIKEN-AF-NP-233, 1996
W. Udo Schröder, 2008

22. Neutron Skin of Exotic (n-Rich) Nuclei
8He n
Which n Orbits?
Qrms (4He) = (1.57±0.05)fm
Qrms (6He) = (2.48±0.03)fm
Qrms (8He) = (2.52±0.03)fm V(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 ~0.1fm 181Cs126 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)
W. Udo Schröder, 2008

23. End of
Nuclear Sizes
Nuclear Sizes
W. Udo Schröder, 2008