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(Q-6) Expansion of the Bohr Model


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Energy levels of sodium and hydrogen

The Sodium Atom

    Following Bohr's early success with his model of the atom, he tried to extend it beyond hydrogen, in a joint effort with an older associate, the German theorist Arnold Sommerfeld. They noted that atoms like sodium were somewhat like hydrogen, since their chemical activities implied that just a single electron was involved. That suggested a single active electron orbiting far from the nucleus, while the rest of the electrons clustered up closer to the nucleus.

    The sodium atom itself is electrically neutral. Denote the electron charge by the letter e: if it has 11 electrons with total charge –11e, as suggested by data, then its nucleus must balance this with a charge +11e. Let 10 of the electrons cluster symmetrically around the nucleus: then at the location of the lone outer electron, their charge of –10e will approximately cancel the effects of 10e of the electric charge of the nucleus. The outer electron thus senses an electric force from just +e in the middle--the same as the electron in the hydrogen atom!

    As noted in the previous section, the quantity which was "quantized"--which could take only whole multiples of Planck's constant h--seemed to be the angular momentum, a vector associated with rotation around an axis, which defined the direction of that vector. If an electron has mass m, distance r from the axis of rotation, and its velocity around it is v, then its angular momentum is mrv.

    The observed energy levels of sodium have some similarities to those hydrogen (shown at the right edge of the drawing above, for comparison), although each level is now split into a whole family--the n=4 level, for instance, splits into (4s, 4p, 4d,4f) [the letters have historical reasons, of no interest here]. Bohr and Sommerfeld thought these represent orbits with different ellipticity, and therefore, different angular momenta. Presumably, this shifts the energies of the levels--though no one could calculate those shifts. The lines connecting energy levels represent "allowed" transitions, which can emit a photon. Obviously, only some of the many possible transitions occur in nature

Magnetism and Electron Spin

    Angular momentum of a rotating electric charge is also associated with magnetism: an electron orbiting a nucleus is equivalent (on the average) to an electric current flowing around its orbit, creating a magnetic field which at a distance resembles that of a small magnet at the center, perpendicular to the plane of the orbit. The angular momentum of the motion and the strength of that magnet are proportional. If the atom is placed in a magnetic field, the interaction may change the angular momentum and therefore the energy level--which may change up or down, depending on the direction of the "magnet."

    The result would be the further splitting of energy levels, into several closely spaced sub-levels. A "quantum jump" from there to a lower level could have one of several closely spaced energies, with the result that, instead of a single spectral line of a fixed frequency, we now see a "multiplet" of several closely spaced lines. That was observed in many atoms, though something in the theory did not seem to fit.

    Two young physicists, George Uhlenbeck and Sam Goudsmit seemed to find the reason in 1925: they suggested that maybe the electron itself was a spinning sphere, with a "spin" angular momentum and its own magnetism. By combining the spin with the orbital angular momentum in the way two vectors are combined, and inferring some experimentally derived quantum rules, the splitting could be understood. Rules for finding which energy levels could be linked and which could not were also derived. The magnetism of the spinning electron turned out to be exactly twice what was expected, something later explained by Dirac, using relativity. It was all a qualitative scheme, but its pattern generally agreed with data.

The Periodic Table

In chemical elements with many electrons (like sodium), the outermost electrons could be made to jump from one energy level to another. Electrons located deeper down, closer to the nucleus, were however also expected to have definite energy levels. Arranging those elements in the order of the weight of their atoms (obtained from their density as gases, or in other ways) brought out two features. One, each had one electron more than the one preceding it (something on which x-rays also gave evidence), and two, some patterns seemed to repeat, again and again.

    For instance, lithium, sodium, potassium--also rubidium and cesium--all had similar chemical behavior, consistent with a single active electron; the rest of the electrons formed in a tight spherical cluster centered on the nucleus. Such repeating patterns (quite a few exist) were first deduced in the 19th century by the Russian chemist Mendeleev ("Mendeleyev"), who used them to predict the existence of some yet-undiscovered elements.

    The electrons remaining in the tight central "cores" of lithium, sodium, potassium etc. seemed to represent quite stable arrangements. The elements just ahead of them on the list--helium, neon and argon--were "noble gases" which resisted any chemical combination, suggesting that all their electrons were in that core, in a bond so stable that no other atom could break into it.

    What determined the number of electrons in that "core"--and therefore, the number of elements one needed to pass before a new "period" was formed? Lithium was the 3rd element--so its core had 2 electrons. Sodium 11th--meaning 10 electrons in the core, 8 more than lithium, probably as a "second layer" on top of the one of lithium (and so forth). By then the arrangement of energy levels was known from spectra, each with a number of "allowed" states of angular momentum, and the idea was that they formed "shells" in which allowed energy levels were completely filled. The scheme indeed worked, but only after Pauli suggested that each level accommodated exactly two electrons of opposite spins. Among heavier elements, sometimes shells filled in an order different from the one expected (some "outer" ones ahead of "inner" ones), but again, the pattern seemed sensible.

    But only the pattern! One could perhaps classify the ordering of energy levels, but not predict their values. Nor could one predict the intensity of field lines--if an atom was at energy level "A", what were the relative probabilities of it jumping to "B" or to "C"? As one example--why was the transitions involving the double yellow line of sodium so prevalent that its light dominated all other emissions from that atom?

    And if the atom seemed to resemble a miniature planetary system, why did the frequencies of the photons it emitted seem unrelated to the orbital frequency calculated for such planetary motions? And why the arbitrary rules about angular momenta? The Bohr-Sommerfeld theory seemed to give plenty of hints, but no precise answers.

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Author and Curator:   Dr. David P. Stern
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Last updated: 16 May 2005
Reformatted 27 March 2006

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