Bohr Model In addition to the atomic line spectra of single electron atoms, there were other successes of the Bohr Model X-ray spectra Frank-Hertz experiment
X-ray Spectra Recall the x-ray spectra shown a few lectures ago
X-ray Spectra Moseley found experimentally that the wavelengths of characteristic x-ray lines of elements followed a regular pattern f Kα 3cR Z 4 ( ) A similar formula described the L-series x-rays f Lα 5cR Z 36 ( ) 7.4 3
X-ray Spectra 4
X-ray Spectra 5
X-ray Spectra Moseley s law can be easily understood in terms of the Bohr model If a K (n) shell electron is ejected, an electron in the L (n) shell will feel an effective charge of Z- (Z from the nucleus electron remaining in the K shell) We have then for the n to n transition λ f K α ( Z ) c λ R 3cR 4 ( Z ) Exactly agreeing with Moseley s law 6
X-ray Spectra Applying the Bohr model to the L-series f L crzeff α 3 5cR 36 Now there are two electrons in the K shell and several in the L shell thus we might expect a (Z--several) dependence The data show Z eff (Z - 7.4) Aside, based on the regular patterns in his data he showed The periodic table should be ordered by Z not A Elements with Z43,6, and 75 were missing Z eff 7
Franck-Hertz Experiment 8
Franck-Hertz Experiment 9
Franck-Hertz Experiment 0
Franck-Hertz Experiment The data show The current increases with increasing voltage up to V4.9V followed by a sudden drop in current This is interpreted as a significant fraction of electrons with this energy exciting the Hg atoms and hence losing their kinetic energy We would expect to see a spectral line associated with de-excitation of λ hc ev.4 4 0 ev 4.9eV A 53nm
Franck-Hertz Experiment As the voltage is further increased there is again an increase in current up to V9.8V followed by a sharp decrease This is interpreted as the electron possessing enough kinetic energy to generate two successive excitations from the Hg ground state to the first excited state Excitations from the ground state to the second excited state are possible but less probable The observation of discrete energy levels was an important confirmation of the Bohr model
Frank-Hertz Experiment 3
Correspondence Principle There were difficulties in reconciling the new physics in the Bohr model and classical physics When does an accelerated charge radiate? Bohr developed a principle to try to bridge the gap The predictions of quantum theory must agree with the predictions of classical physics in the limit where the quantum numbers n become large A selection rule holds true over the entire range of quantum numbers n (both small and large n) 4
Correspondence Principle Consider the frequency of emitted radiation by atomic electrons Classical f classical using r n we find f ω π n a 0 classical V πr n h kme 4 me 4ε h 0 π 3 n ke mr 3 / 5
6 Correspondence Principle Quantum ( ) ( ) ( ) classical Bohr Bohr Bohr Bohr f n h me f me E hn E hn ne f n n n h E f n n h E f + + + 3 3 0 4 0 4 0 3 0 4 0 0 0 4 4 substituting for which for n large becomes ε πε h
Wilson-Sommerfeld Quantization Quantization appears to play an important role in this new physics Planck, Einstein invoked energy quantization Bohr invoked angular momentum quantization Wilson and Sommerfeld developed a general rule for the quantization of periodic systems 7
Wilson-Sommerfeld Quantization Pdq nh where P is some component of momentum and q is the corresponding coordinate and means integrate over one cycle 8
Wilson-Sommerfeld Quantization For a particle moving in a central field (like the Coulomb field) P L q φ Then Pdq Ldφ nh L dφ nh L nh π nh Which is just the Bohr condition 9
Wilson-Sommerfeld Quantization For a particle undergoing simple harmonic motion P p X q x d x Newton' s law m kx dt has solution x Asinωt then and dx p x ωacosωtdt m dx dt mωacosωt 0
Wilson-Sommerfeld Quantization Continuing on E E E ω E Pdq π ka cos p nhω π x dx cos ωtdt θdθ nhf mω A nh mω A E π ω nh ωtdt Which is just the Planck condition cos nh Recalling from a simple harmonic oscillator 0