Miscellaneous exercises

The following exercises deal with several late 19th and early 20th century experiments which first determined crucial properties of matter. The probability of meeting an exercise of this kind in any written test is small but nonvanishing...

1. In a Franck-Hertz experiment the energy threshold for Hg absorption is 4.9 eV. By observing the spectrum emitted by the excited Hg vapor by a spectrometer, an isolated peak is observed at 2536 Å.

   1) Assuming that the energy of the emitted radiation is equivalent to the energy absorbed by the Hg atoms at threshold, estimate the value of the Planck constant.
   RESULT: h=E λ /c=6.640×10^-34 J s.

   2) Assuming that the experiment is realized in a cylindrically symmetric geometry, give a rough description of the space distribution of the radiating atoms, as a function of the applied accelerating voltage.

2. Factory-made plastic spheres are available of standard diameters 0.5, 1, 1.5 and 2 µm, and density ρ=300 kg/m³. Suppose you decide to use them in place of oil drops to make a Millikan-type experiment to determine the elementary charge. Knowing that the arc-discharge electric field in air is of the order 1 MV/m, choose the best suited spheres to buy. Compare with the wavelength of the illuminating light.
   RESULT: d ~ (6 q E/ π ρ g )^1/3 ~ 2 µm
3. Atoms are disposed in a simple cubic array of lattice spacing a=0.91 Å. Consider Bragg diffraction of (110) diagonal planes.

   1) find the largest wavelength for which a first-order diffracted maximum can occur.
   RESULT: λ_max = &sqrt;2 a = 1.287 Å.

   2) if electrons of kinetic energy 300 eV are used as diffracting probe, at what angles θ from the (110) plane the 1st and 2nd order maxima occur?
   RESULT: θ₁ = 0.582 rad = 33.38°; no second-order diffraction (sin(θ₂)>1).

Comments and debugging are welcome!