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=== Quantization of bound electrons in atoms ===
{{main | Atomic theory | Bohr atom | Bohr-Sommerfeld model}}
Experiments with light and matter in the late 1800s uncovered a reproducible but puzzling regularity. When light was shown through purified gasses, certain frequencies (colors) did not pass. These dark absorption 'lines' followed a distinctive pattern: the gaps between the lines decreased steadily. By 1889, the [[Rydberg formula]] predicted the lines for hydrogen gas using only a constant number and the integers to index the lines.<ref name=Whittaker>{{Cite book |last=Whittaker |first=Edmund T. |title=A history of the theories of aether & electricity. 2: The modern theories, 1900 - 1926 |date=1989 |publisher=Dover Publ |isbn=978-0-486-26126-3 |edition=Repr |___location=New York}}</ref>{{rp|v1:376}} The origin of this regularity was unknown. Solving this mystery would become first major step toward quantum mechanics.
Throughout the 19th century evidence grew for the [[atomic theory|atomic]] nature of matter. With JJ Thomson's discovery of the electron in 1897, scientist began the search for a model of the interior of the atom. Thomson [[Plum pudding model|proposed]] negative electrons swimming in a pool of positive charge. Between 1908 and 1911, [[Rutherford model | Rutherford]] showed that the positive part was only 1/3000th of the diameter of the atom.<ref name=baggott/>{{rp|26}}
Models of "planetary" electrons orbiting a nuclear "Sun" were proposed, but cannot explain why the electron does not simply fall into the positive charge. in 1913 Neils Bohr and Ernest Rutherford connected the new atom models to the mystery of the Rydberg formula: the orbital radius of the electrons were constrained and the resulting energy differences matched the energy differences in the absorption lines. This meant that absorption and emission of light from atoms was energy quantized: only specific energies that matched the difference in orbital energy would be emitted or absorbed.<ref name=baggott/>{{rp|31}}
Trading one mystery – the regular pattern of the Rydberg formula – for another mystery – constraints on electron orbits – might not seem like a big advance, but the new atom model summarized many other experimental findings. The quantization of the photoelectric effect and now the quantization of the electron orbits set the stage for the final revolution.
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