Atoms and Their Constituents

• Prior to the 1890's those scientists who believed in the reality of atoms also believed them to be the most "elementary" particles of nature. They could be combined into molecules, but they could not be further subdivided.

• The discovery of radioactivity in 1895, followed shortly by studies showing that these phenomena involve a transmutation of elements, shook this belief in the atom as fundamental.

• Thompson's elucidation of cathode rays (1897) as beams of tiny charged particles suggested that these particles (electrons) come out of atoms, implying that atoms are not elementary after all.

• In 1909 Rutherford showed that the atom in fact consists of a heavy but very small nucleus carrying positive charge, surrounded by electrons.

• By about 1912 it was clear what atoms are made of. It was not clear how they manage to hold together in a stable configuration.

Rutherford's Model of the Atom

• The data from scattering of a particles (which are given off in radioactive decay of heavy elements such as radium) from a gold foil showed a surprising effect that occurred occasionally. Sometimes the incident a particle was deflected nearly completely backward.

• If the charge and mass in the gold atom is distributed more or less uniformly over its volume, the a particle should suffer a deflection of no more than a few degrees.

• Rutherford interpreted the data as showing that if fact most of the mass and (perhaps) all of the positive charge in the atom are concentrated in a very small volume. He named this massive positive object the nucleus.

• If the incident a particle collides essentially head on with this nucleus,  then significant deflection can occur.

• Rutherford showed that all of the cases of large angle deflection could be explained on his assumption.

• He thus proposed a model in which all of the atom's positive charge and nearly all of its mass are located in the nucleus. The electrons move around in the remaining space occupied by the atom, roughly like planets around the sun.

Difficulties With Rutherford's Model

• There was a serious problem with this "planetary" model of the atom: radiation.

• According to classical elctrodynamics an accelerated charge radiates, at a rate proportional to the square of the acceleration.

• An electron orbiting about the positive nucleus, held in place by the Coulomb force, will be continually accelerating as the direction of its motion changes. It should therefore continually radiate, losing energy and moving to a smaller orbit.

• A simple calculation shows that an electron starting from a circular orbit of the size of an atom will spiral into the nucles by radiating away its energy in a very short time, about 10^-8 s. This makes stability of the atom impossible.

Bohr's Theory of the Rutherford Atom

• In 1913 Bohr, who had worked for a time in Rutherford's lab and knew about the data and Rutherford's ideas, proposed a theoretical framework to apply to the atom.

• He had no explanation for the absence of the classically predicted radiation when the atom is in a stable state. He therefore asserted that classical electrodynamics does not apply at the atomic level.

• On the other hand, atoms do radiate sometimes — this is the principal source of our light. But this radiation is often confined to discrete frequencies, called spectral lines. Radiation at frequencies between these "lines" does not occur.

• Bohr asserted that this radiation consists of Einstein's photons, with frequency determined by their energy accourding to E = hf.

• The discreteness of the spectrum suggests that the energies available must be a discrete set. Bohr argued that this means the atom's own energy must be restricted to discrete "levels" and that the photon emitted when the atom moves from a higher level to a lower one will have energy E_i - E_f = hf.

• The allowed energy levels are those of stationary states, in which the atom can exist without radiating, at least for some time.

What are the Stationary States?

• Now Bohr needed a way to determine which states would be stationary.

• In this he was aided by the work of spectroscopists since the mid-19th century. They had catalogued the frequencies of the spectral lines of numerous atoms, especially hydrogen (which by 1913 was known to contain only one electron and thus be the simplest atom). The data on frequencies had been shown in many cases to be differences between a certain set of numbers, called "terms".

• In the 1880's Balmer had proposed a formula for the "terms" of hydrogen. It was this formula that Bohr used to find his theoretical framework.

• Bohr's papers contained several arguments to support his choice of stationary states. The one that had lasting significance was a new "quantum" postulate concerning angular momentum.

• He proposed the restriction that for the stationary orbits the angular momentum must be a multiple of h/2p, where h is Planck's constant.

• This was a radically new use of the quantum idea.

Details of the Stationary States for Hydrogen

• Bohr assumed that hydrogen consists of a single electron orbiting a nucleus with a large mass and charge +e. At first he considered only circular orbits.

• For such an orbit the total energy (non-relativistic) is

where r is the orbit radius, v its speed, m the electron mass, -e its charge, and k (in SI units) is 1/4pe₀.

• For circular motion the radial force is

so we can write the total energy as

• Now we apply Bohr's postulate about the angular momentum:

.

This marks the introduction into physics of the number n, the first of many quantum numbers.

• Solving for v from the radial force equation and doing a little algebra we find

• This formula gives the radii of the allowed orbits. Assuming that the smallest radius is not zero, we must start the sequence with n = 1. The the radii are multiples of the Bohr radius:

(Here we have introduced the standard symbol "h-bar" to stand for h/2p.) Numerically this radius is about 0.053 nm.
 
• The allowed radii are thus

• This then gives the allowed energies:

where






• Numerically we find E₁ = -13.6 eV.

The Spectrum of Hydrogen

• Using these formulas for the energy levels, Bohr could then predict the frequencies of light emitted when the atom moved from a higher level to a lower one. He simply used

• To reproduce the lines that Balmer had accounted for in his formula, one takes the final state to be n = 2. These frequencies are in the visible region. For example, with initial state n = 3 we find

Using h = 4.14 ¥ 10^-15 eV-s we find f = 4.59 ¥ 10¹⁴ Hz, which corresponds to a wavelength l = 656 nm, so this is the red line in the spectrum, called H_a  by the spectroscopists. Other lines observed in the Balmer series come from higher initial states.

• As technology improved so that spectra in the ultraviolet and infrared could be measured, other lines of hydrogen were compared to Bohr's formulas. They all conformed, within errors.

Weaknesses of Bohr's Theory

• The procedure proposed by Bohr was a hybrid, with one foot in classical physics (orbits, classical formulas for angular momentum, etc.) and one in quantum physics (photons, "quantized" angular momentum).

• Bohr could give no convincing reason why those states with quantized angular momentum were the stationary ones.

• Attempts to apply this procedure to atoms with more than one electron went nowhere. Nor could it be used to understand molecular structure.

• Nevertheless, it was a large step forward in our understanding of how different the laws at the microscopic level are from those of classical physics.

The Correspondence Principle

• To attempt to justify his angular momentum postulate, Bohr gave another argument which has outlived his theory.

• Consider the transition in an atom from a state with a large value of n to the next lower number, n-1. The emitted frequency is given by

(Here we have used w = 2pf.)

• Now let us calculate the angular frequency of the orbit, from the classical formula w = v/r. Using Bohr's formulas we find that






while



(The latter is obtained from the formula for kinetic energy.) Thus the angular frequency is



and so



But



so finally we have

• What does all this mean? What we have shown is that for large values of the quantum number n the orbital angular frequency is the same as the frequency emitted in a transition to the next lower level.

• But in classical physics a charge orbiting with a given orbital angular frequency will necessarily emit light with that same frequency. Thus Bohr's quantum description agrees with classical theory for large values of n.

• This would not have occurred if we had used a different rule concerning the angular momentum.

• Bohr took this situation to be an example of a general rule, which he called the Correspondence Principle:

The quantum description of nature and the classical description must merge into each other when systems of "classical" size are considered.

• In practice, situations with large quantum numbers are considered to be of classical size.

• The importance of this principle is that it serves a a guide when guessing the correct quantum description of a situation.

The Franck-Hertz Experiment

• In 1914 an experiment by Franck and Hertz demonstrated directly the existence of discrete energy levels in atoms.

• In essence this experiment, in which electrons were made to collide with atoms of Hg, looked at cases where the atom absorbs some of the kinetic energy of the electron by moving its own energy to a higher state. This loss of energy by the electron shows up as a decrease in current reaching a negative plate in the vacuum tube containing the Hg gas.

• As the initial energy of the electrons is increased, the current should rise. This was observed, but there were dips in the curve, indicating electron energies where the collisions with Hg atoms were inelastic, with the atom absorbing most of the electron energy.

• The fact that these dips were spaced apart showed that the atom is especially able to absorb electron energies of certain amounts, but not other amounts.

• This is easy to understand if it is assumed that the internal energy state of the atoms consists (as Bohr said) of discrete levels. The electron which has just the right energy to move the atom from one level to another will engage in the inelastic collision. Electrons with other energies will not, and their collisions will be elastic. (Since the atom is massive, an elastic collision leaves the electron energy essentially unchanged.)

• This simple experiment was very persuasive that Bohr's idea of discrete atomic energy levels is correct.