4511_Lec_23Jan08

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Stability of Nuclei

• Nuclei are stable because the composite state is
  energetically advantageous compared to the
  disassembled parts
• Relative stability is measured by the binding
  energy per nucleon

Average energy needed to release a nucleon from
the nucleus

• Low-mass nuclei, A?20
• B/A bounces up and down, overall increasing
  steeply with A
• A?60
• B/A maximal at 8.7 MeV per nucleon for 56Fe
• Agt60
• B/A falls slowly to 7.6 MeV per nucleon for 238U

Stable nuclei!
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"Wallet Card (http//www.nndc.bnl.gov/wallet/wccu
rrent.html)
Nuclear Information

• Properties of all known isotopes of a given
  element
• Mass info, abundance, spin/parity,
  stability/lifetime, decay modes

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• Sizes and shapes reveal more properties of nuclei
  and their constituent nucleons
• Rutherford scattering gave rough picture
  (spherical, tens of fm)
• Subsequent experiments (higher energy, electrons
  instead of ?s, many different nuclei) revealed
  details
• Experimental improvements required theory
  upgrade. Electrons are relativistic and have
  spin 1/2

• Mott scattering
• e- with spin scattering from spinless nucleus
  use relativistic QM

• Falls off with ? faster than Rutherford
• Extreme relativistic case, typical for electrons

Big Effect
(Natural Units)
Rutherford happens at 180?
Mott does not
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Why not?

• es with ??1 have well-defined helicity
  (frame-indep. definition of spin)

• Kinematics of 180? scattering
• e with h -1 approaches along z-axis and is
  turned around
• Maintaining h -1 requires spin flip
• Conservation of momentum demands that something
  take the angular momentum. Spinless nucleus
  cant do it. Orbital angular momentum is ?
  motion and cant do it
• Scattering at 180? of electron from spinless
  nucleus is impossible

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Hofstadter Measuring Nuclear Sizes and Shapes

• Observation Mott formula works only for small ?
  or q2
• At big q2, e probes inside charge distribution,
  sees only part ? cross section smaller than
  predicted
• Need realistic charge distributions!

125 MeV es on Au foil
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Nuclear Form Factors
Function that parameterizes the disagreement of
the cross section with the Mott formula, i.e. the
departure of the charge distribution from
pointlike
? Cross-section measurements give info about
charge distribution. In practice, experiments
have limited range in ? or q2 and data must be
fitted to determine parameters
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eNucleus Scattering as Diffraction

• 400 MeV electrons on 12C
• Fall-off in cross section ?1/q4
• Dip in cross section is equivalent to
  diffraction minimum

Dashed Born approx. (plane wave on
sphere) Solid Exact
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• In practice
• Assume a candidate charge distribution
• Fourier transform it to get the corresponding
  form factor and differential cross section
• Fit to the scattering data.
• Repeat to optimize parameters of the charge
  distribution

52? min. for 12C gives a 2.5 fm

• Example Sharp-Edged Sphere

?(r)?0 for rlta
?(r)0 for rgta
Multiply by Mott to get d?/d?.
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• Better Guess Woods-Saxon Potential

d - fuzziness
(Soft-Edged Sphere)
a - radius

• Hofstadter collected and fitted data for
  e-nucleus scattering for beam energies 500 MeV to
  1 GeV
• Found W-S fits measured cross sections (form
  factors) for nuclei with A gt 40 very well not
  too badly below 40, except for very small nuclei
• Other forms gave better fits for nuclei below A
  40

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Charge Distribution Parts List
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Hofstadters Data
Charge density in nuclear core is same for all
Rate of fall-off (fuzziness) is quite consistent

• a ? A1/3 ? V ? A. Since M ? A, nuclear density
  M/V1 ? nuclear matter is incompressible, like a
  fluid

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Specific example

• Electron scattering from 40Ca, the usual form of
  calcium, and 48Ca, a neutron-rich isotope
• Scattering angle 15?-55?
• 7 orders of magnitude in d?/d?
• Three minima ? very good info about charge
  density
• 48Ca minima shifted to slightly smaller ? (q2) ?
  it is the larger nucleus

Multiplied by 0.01 to separate for comparison