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(Q-3)   Atomic Energy Levels


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Frequency ν and Wavelength λ

    Imagine a continuous electromagnetic wave with frequency ν emitted for an entire second. It covers a distance of c meters, where c meters/sec is the velocity of light, and that distance contains ν waves. Each wave therefore occupies a "wavelength"

λ = c / ν
so that
ν = c / λ

Obviously, the frequency ν is proportional to the quantity 1/λ appearing in Balmer's formula. Even though λ is more easily measured in the lab, Balmer's formula suggested that the frequency ν may be more meaningful physically. Later the frequency ν also appeared in Einstein formula E = hν. What did all that mean?

Energy Levels

Many people helped in interpreting that message, and the story told here is a great oversimplification. The foundation was Einstein's formula, discovered in 1905

E = hν

It suggested that an electromagnetic wave gave up its energy in definite portions ("photons"), and the size of such portions was proportional to the frequency ν of an electromagnetic wave (ν is the Greek letter "nu"; the letter "f" is also used sometimes).

    One of the early interpreters of the Balmer formula was a young Danish physicist named Niels Bohr, in 1913. Before that Bohr had been a distinguished member of Denmark's national soccer team--although, supposedly, the real star of the team was his brother Harald Bohr, who later distinguished himself in mathematics. Examining the Balmer formula and also the Ritz combination principle (further below), Bohr suggested that atoms could exist (even briefly) only in certain energy levels, and light was emitted only when an atom descended from some higher energy level to a lower one.

The hydrogen atom, for instance, had energy levels

– hcR /n2     (n = 1,2,3...)

    (Energies of bound electrons are all negative, their magnitude increasing with the strength of the binding. An electron with energy zero is just at the point of breaking free, while having a positive energy means it is unbound. Similarly, the energy of a planet or satellite bound by gravity is negative, rising to zero if it acquires escape velocity.)
When an electron descends from a higher energy (large n) to a lower one (smaller n), by Balmer's formula together with Einstein's, the energy of the photon emitted was exactly the amount of energy given up:

hν   =   hc/λ   =  hc R [1/n2   –   1/m2]         ( m>n,     n,m = 1,2,3...)


The Ritz Combination Principle

In 1908 Ritz found that Balmer's formula was just one prominent example of a more widespread phenomenon, in "line spectra" emitted by hot gas. If one formed differences between the values of (1/λ) in pairs of spectral lines of some atom, sometimes two different pairs gave the same difference. That meant of course that the differences in frequency (c/λ) also matched (c the speed of light).

    Suppose that one spectral line, of frequency ν1 resulted from a jump from energy A to energy B, and another line, of frequency ν2, originated in a jump from energy C to energy D. Then

h ν1 = A – B
h ν2 = C – D

In such atoms, it might perhaps be possible for transitions to also occur from A to C, and from B to D, causing the emission of photons with frequencies ν3 and ν4. In those cases

h ν3 = A – C
h ν4 = B – D

If such transitions are possible, then (as may be checked)

ν3 – ν4 = (1/h)[( A – C) – (B – D)]   =                                
                                          =   ν1 – ν2

As noted earlier, the wavelengths of spectral lines (and hence their frequencies) could be established with very high precision. If such equalities as the one written above occurred by blind chance, they should be quite rare. Ritz and others found matching differences (of various sorts) much more prevalent, and that supported the idea that atoms would exist (at least briefly) at different energy levels.

More on Atomic Energy levels

    If atoms are left undisturbed, they usually drop to the lowest available energy level and stay there, in their "ground state." Occasionally, however, they may also be pushed up to some higher energy ("become excited") e.g. by a collision with a fast atom or electron, one which got extra speed from an electric voltage or from some source of heat. An atom elevated to one of its higher "excited levels" soon falls back to a lower level ("undergoes a quantum jump"), emitting a photon whose energy corresponds to the difference between the levels. That need not be the ground state: the atom might descend to that state in several steps, emitting a photon at each step on the way.

    Usually such a return occurs very quickly--in nanoseconds, perhaps--but not always. The green and red colors of the polar aurora are emitted at well defined wavelengths, which for a long time could not be matched by anything observed in the lab. In the end they were traced to unusual excited levels of the oxygen atom. In dense oxygen, e.g. in laboratory light sources, the extra energy is quickly removed during collisions, but in the high atmosphere, where collisions are rare, the excited state can persist for 0.5-1 seconds, until a green or red photon is emitted and removes the energy. The long lifetime of these levels explains the gradual fading and brightening of the rays of which auroral "curtains" are made up, a visually fascinating spectacle, a bit similar to the brightening and fading of flames in the fireplace. Each ray is created by a beam of electrons guided along a magnetic field line.

    The Ritz combination principle allowed physicists to translate the jumble of observed spectral lines into a more orderly (and smaller) scheme of energy levels. Based on understanding based on the quantum theory of the atom (see further below), such levels can be sorted into meaningful families, and various questions can be addressed, e.g. why certain transitions exist while others seem "forbidden," even though they too are expected to release energy.

    One question was, why did levels sometimes split into two or more narrowly separated levels. For instance, the yellow light of sodium actually contains two closely spaced wavelengths.

    A related effect was the splitting of a single spectral wavelength into several closely spaced ones, when light was emitted from the region of a strong magnetic field. This is the Zeeman effect, discovered 1896 by the Dutchman Pieter Zeeman, and the separation of the wavelengths often gives an indication of the strength of the magnetic field in the source region of the light. It was the Zeeman splitting of spectral lines emitted from sunspots that led George Ellery Hale in 1908 to realize that sunspots were in fact strongly magnetized, to about 1500 gauss (0.15 Tesla).

    Radioactive nuclei emit gamma rays (γ rays), photons with energies about a million times higher than those of atoms. It took a while to measure their wavelength with any precision, but by about 1949 it was realized that they too displayed the Ritz principle. That confirmed the long-standing suspicion that nuclei too had energy levels, and that the associated transitions were the source of gamma rays.

Questions from Users: ***   Energy levels: plus or minus?

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Author and Curator:   Dr. David P. Stern

Last updated: 13 February 2005
Re-formatted 27 March 2006