The Nuclear Atom

1
The Nuclear Atom

• Atomic Spectra Emission of radiation by atoms
  in a flame or electrical discharge and absorption
  of radiation by atoms.
• Rutherford Scattering Scattering of alpha
  particles from atoms showed that the positive
  charge is concentrated in an atomic nucleus.
• Bohr Model of the Hydrogen Atom The angular
  momentum of electrons in an atom is quantized in
  units of Plancks constant divided by 2p.
• X-Ray Spectra Emission from atomic transition
  deep within the atom.
• Franck-Hertz Experiment Quantization of atomic
  energy levels exhibited by scattering as opposed
  to radiation.

2
Atomic Spectra

• Early in the 19th century, Fraunhofer saw
  dark bands on the solar spectrum.
• In 1885, Balmer observed hydrogen spectrum
  and saw colored lines.
• Found empirical formula for discrete wavelengths
  of lines.
• Formula generalized by Rydberg for all
  one-electron atoms.

3
Atomic Spectra

• (1) Continuous spectrum from an incandescent
  light bulb.
• (2) Absorption-line spectrum (schematic) of suns
  most prominent lines H, Ca, Fe, Na.
• Emission-line spectra of Na, H, Ca, Hg, Ne.

4
Atomic Spectra Demo
5
Nuclear Atomic Model

• Atomic Spectra
• Lower-energy optical absorption/emission lines
  from materials indicate quantized electron energy
  levels.
• Bohr model predicts energy transitions for
  one-electron atoms.

6
Hydrogen Energy Levels
E? 0 eV
Energy
E1 -13.6 eV
Lyman
7
Rydberg Formula

• Rydberg constant R ? 1.097 107 m-1
• nfinal 1 (Lyman), 2 (Balmer), 3 (Paschen)

• Example for n 2 to 1 transition

8
Rutherford Scattering Nuclear Model

• Rutherford scattering probes the atom. (Hit it
  with something!)
• Beam of a particles (He2) strikes a thin gold
  metal foil.
• Atoms in the foil scatter the alpha particles
  through various scattering angles q that are
  detected with a scintillation screen.
• 180º scattering can occur (back scattering),
  indicating a hard core interaction between the a
  particles and atoms in the foil.
• Nuclear model by Rutherford explains large
  scattering angles.

9
Rutherford Scattering

• Atomic Model must include
• 10-10 m diameter, electrons, neutral atom.
• Model 1 - Thomsons Plum Pudding Model.
• Model 2 - Rutherfords Hard Core Nuclear
  Model. ? Nuclear Model proven correct by
  Rutherfords experiment.

Nuclear Model
Plum Pudding Model
q too small
large q possible
10
Rutherford Scattering

• Scattering occurs due to Coulomb repulsion
  between incoming positively charged a particles
  and atomic nuclei in metal foil.
• b impact parameter (distance of closest
  approach, ?b gives ?q)
• s pb2 scattering cross section

11
Rutherford Scattering

• Scattering fraction f fraction of
  particles scattered through angles gt q for given
  b.

• Radius of closest approach rd
• Derive using conservation of kinetic and
  potential energy.

12
Rutherford Scattering Nuclear Size

• Rutherford Scattering can be used to determine
  nuclear size.
• At low energies, the incoming ? particle will
  scatter without penetrating the nucleus.
• At higher energies, the ? particle will penetrate
  the nucleus, and the number of observed large
  angle scattering events will be reduced.
• Data for aluminum shows a nuclear size of ? 10
  fm.

13
Rutherford Scattering Fraction f Problem

• A gold foil (Z 79, n 5.91028 atoms/m3) of
  thickness 2 mm is used in a Rutherford experiment
  to scatter a particles with energy 7 MeV. Find
  the fraction f of particles scattered at angles
  q gt 10.
• First, find the impact parameter b for q 10
  and then solve for f.

14
Rutherford Backscattering Spectroscopy (RBS)

• Rutherford Backscattering is widely used to
  evaluate thin film samples. Provides elemental
  composition and depth profiling.
• Backscattered beam is energy analyzed, energy of
  scattered ions depends on
• element (energy loss depends on momentum transfer
  to the target atom)
• angle
• location in solid - as the ions travel through
  the material, they lose energy.
• Large beam size (1 mm) - poor lateral resolution
• depth resolution ? 20 Å
• weak signal for low atomic number elements
• poor mass resolution for high atomic number
  elements

15
RBS of Si crystal

• For backscattering at the sample surface, the
  only energy loss is due to momentum transfer to
  the target atom. Si surface 1.1 MeV
• Scattering from atoms below the surface includes
  energy losses as beam moves through bulk sample.
  Total energy loss corresponds to depth of
  scattering atom.

16
RBS of thin film on Si Crystal

• RBS of two TaSi films of different compositions
  on Si substrates. 2.2 MeV incident He ion
  beam.
• The high energy peak arises by scattering from Ta
  in the TaSi film layer. The peak at lower energy
  is from Si.
• For backscattering at the sample surface, the
  only energy loss is due to momentum transfer to
  the target atom. Ta surface 2.1 MeV , Si
  surface 1.3 MeV

Height --gt concentration Width --gt layer
thickness Energy --gt element and depth

17
Bohrs Model of the Atom
18
Bohrs Model of the Atom

• Problem Classical model of the electron orbiting
  the nucleus is unstable. Why?
• Electron experiences centripetal acceleration.
• Accelerated electron emits radiation.
• Radiation leads to energy loss.
• Electron eventually crashes into nucleus.
• Solution In 1913, Bohr proposed the quantized
  model of the H atom to predict the observed
  spectrum.
• Electrons can only occupy orbitals with specific
  quantized energy levels.
• Bohr model will ultimately be replaced with
  quantum mechanics.

19
Bohrs Model of the Atom

• Bohrs model puts electrons into quantized
  orbits.
• When radiation is absorbed, electron is promoted
  to higher energy orbit.

• When electron drops to lower energy orbit,
  radiation is emitted.

20
Quantization of L and f

• Bohr proposed two quantum postulates
• Postulate 1 Electrons exist in stationary
  orbits (no radiation) with quantized angular
  momentum.

• Postulate 2 Atom radiates with quantized
  frequency f (energy E) when electron makes a
  transition between two energy states.

hc 1240 eV nm
21
Quantization of r and E

• Quantized angular momentum L leads to quantized
  radii and energies for an electron in a hydrogen
  atom or any ionized, one-electron atom.

• Derivation uses the following

22
Rydberg formula

• Energy transitions yield general Rydberg
  formula.
• Applicable to ionized atoms of nuclear charge Z
  with only one electron.

23
Periodic Table
24
Unknown Transition Problem

• If the wavelength of a transition in the Balmer
  series for a He atom is 121 nm, then find the
  corresponding transition, i.e. initial and final
  n values.

25
Series Limit Problem

• Find the shortest wavelength that can be emitted
  by the Li2 ion.
• The shortest l (or highest energy) transition
  occurs for the highest initial state (ni ?) to
  the lowest final state (nf 1). Remember that
  Z3 for lithium.

26
X-Ray Spectra

• In 1913 Moseley measured characteristic x-ray
  spectra of 40 elements (energy keV).
• Observed a series of x-ray energy levels called
  K, L, M, etc.
• Analogous to optical series for hydrogen (Lyman,
  Balmer, Paschen)
• X-rays vs. optical light
• Higher-energy x-ray transitions for heavier
  elements.
• Lower-energy optical transitions for lighter
  elements.
• Moseley Plot gives equation with similarities to
  Rydberg equation.

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X-ray Spectra Stylized Diagram of Atomic Levels
Ma for n 4 to 3
n 4
La for n 3 to 2
n 3
n 2
Ka for n 2 to 1
a for n1 to n b for n2 to n g for n3 to n
28
X-ray Spectra Moseley Plot
K Series n 2,3,... to n 1

• Derived from Bohrs formula with Z-1 effective
  charge instead of Z due to shielding of nucleus.

L Series n 3,4,... to n 2
29
X-ray Spectra Unknown Z Problem

• If the wavelength of the Ka x-ray line (n  2
  to 1 transition) for an unknown element is
  l  0.08 nm, find the element number Z.

30
The Crab Nebula at different wavelengths

• Remnant of a supernovae explosion seen on
  Earth in 1054 AD 6000 light years from Earth. At
  the center of the bright nebula is a rapidly
  spinning neutron star, or pulsar, that emits
  pulses at 30 Hz.

31
Franck-Hertz Experiment

• In 1914 Franck and Hertz directly measured the
  energy quantization of atoms via the inelastic
  scattering of electrons.
• Electron IN and Electron OUT (same electron)
• Summary of Experiment
• Measure current of electron beam (I) vs.
  accelerating grid voltage (V) inside a glass tube
  filled with Hg gas (5 eV transition).

32
Franck-Hertz Experiment

• Higher Voltages

Lower Voltages
33
Franck-Hertz Experiment Modern Physics Lab
Franck-Hertz Tube
Acceleration Voltage
Electron Beam
Collector
Voltage Sensor 1 Acceleration Voltage
Voltage Sensor 2 Collector Current
34
Franck-Hertz Experiment I-V Data

• 4V e reaches collector.
• EK 4 eV Observe maximum current I.
• 5V e excites (1) Hg atom.
• Promote Hg e- to excited state.
• EK 5eV - 5eV (to excite Hg) 0e does not
  reach collector.
• Observe minimum current I.
• 6V e excites 1 Hg atom reaches collector.
• Ek 6eV - 5eV 1eVe- barely reaches collector.
• Current starts to rise again.
• 10V e excites 2 Hg atoms.
• Ek 10eV - 2(5eV) 0e does not reach
  collector.
• Observe 2nd minimum in current I.
• Continue with same logic to explain multiple
  minima in IV curve.