BACKGROUND

Figure 1: Energy level diagram of the hydrogen atom.

You are probably familiar with the model for an isolated atom that consists of a positively charged nucleus about which electrons are distributed in successive orbits. Niels Bohr introduced this planetary model of the atom in 1913 to account for the wavelengths present in the atomic spectrum from hydrogen gas in which each atom has one electron. He also postulated that only those orbits occur for which the angular momentum of the electron is an integral multiple of h/2, i.e. n h/2, where n is an integer and h is Planck's constant. Using this non-classical postulate together with some relations from classical physics, Bohr showed that the energies of the electrons occupying these allowed orbital states had a simple dependence on the integer n, namely:

, n=1,2,3,...

(1)

where me is the mass of the electron and is the permittivity. Note that these energies are negative, so that the lowest energy occurs when n =1. Instead of expressing the energy in joules, it is sometimes convenient to use the energy unit of electron volts (eV), where 1 eV is the energy gained by an electron on passing through a positive potential difference of 1 volt. If En is in joules, we divide by the conversion factor 1.6 x 10-19 J/eV. In units of eV, we get

(2)

for the energy levels of the hydrogen atoms determined from Eq. (2). Note that the inverse square dependence of the energy levels produces a relatively large energy difference between the levels associated with small n, whereas the higher levels are very close to each other. In order for the electrons to remain in their planetary orbits, they must continually change direction. Therefore, they are constantly under the influence of a force and this means they are always being accelerated. In classical physics, a charged particle radiates energy when it is accelerated. Thus a classical planetary electron would soon radiate away its energy and no longer remain in orbit. This led Bohr to his second non-classical postulate which states that an electron remaining in one of the allowed orbital states does not radiate energy, and that radiation is emitted only when an electron goes from a state of higher energy (n2) to a state of lower energy (n1), where n2 > n1. The energy of the quantum of radiation emitted, hv, is equal to the difference in energy of the two states. The second Bohr postulate then states: hv = En2 - En1, where v is the frequency of the emitted radiation and En is in joules. If we substitute the expression of En from Equ. (1),

It is usual to write this expression in terms of the wave number

(3)

where R = Rydberg constant = = 1.097 x 107 m-1

The wavelengths of some of the hydrogen spectral lines as calculated from Equ. (3) are shown in Fig. 1, and they agree with experimental observations, confirming the predictions of the Bohr model with its non-classical postulates. Bohr's picture of electrons in discrete states with transitions among those states producing radiation whose frequency is determined by the energy differences between states can be derived from the quantum mechanics which replaced classical mechanics when dealing with structures as small as atoms. It seems reasonable from the Bohr model that just as electrons may make transitions down from allowed higher energy states to lower ones, they may be excited up into higher energy states by absorbing precisely the amount of energy representing difference between the lower and higher states. James Franck and Gustav Hertz showed that this was, indeed, the case in a series of experiments reported in 1913, the same year that Bohr presented his model. In its lowest energy, with n =1, the atom is said to be in its ground state. From Fig. 1, we see that raising the electron in a hydrogen atom from the ground state (n=1 ) to its next highest state (n=2) requires an energy absorption of (13.6--3.39) = 10.21 eV. This energy may be provided in several ways. The atom may absorb a photon, as in the photoelectric effect experiment. Alternatively, if we heat the gas, the mean kinetic energy of an atom E = 3/2 kT, where k = Boltzmann's constant = 1.38 x 10-23 J x K-1 = 8.6 x 10-5 eV x K-1

This thermal energy would be converted, at a collision, into excitation energy. We may calculate the gas temperature required to provide 10.21 eV of thermal excitation energy

10.21 = 3/2 kT, T = 8 x 104 K

Obviously, such a high temperature isn't practical for laboratory experiments. A more convenient way is to accelerate a beam of electrons in an electric field and permit these electrons to collide with the electrons in the gas atoms. In this way energy may be transferred from the electron beam to the gas atoms. If the energy of the electrons in the impinging beam is less than the separation of the ground state from the first excited state, then no energy is transferred and the collisions are termed elastic. If the beam energy is equal to or greater than the separation of the lowest states, then energy is absorbed equal to the energy separation of the states and the collision is termed inelastic. The impinging electrons may be left with some of their initial energy if that energy is greater than the energy separation of the two levels. Franck and Hertz used a beam of accelerated electrons to measure the energy required to lift electrons in the ground state of a gas of mercury atoms to the first excited state. A schematic diagram of their apparatus is shown in Fig. (2).

Figure 2: Franck-Hertz Tube

The sealed tube contains a small amount of mercury which is vaporized by heating the tube with a surrounding furnace. The tube contains a cathode which is heated to produce electrons, a perforated anode maintained at a positive voltage Va with respect to the cathode, and a collector electrode held at a small voltage Vc that is negative with respect to the anode. A meter, A, measures the collector current. Fig. 3 shows the collector current as a function of anode voltage.

Figure 3: Oscilloscope traces of Franck-Hertz signals.

(a) Trace for lower temperatures; (b) Trace for higher temperatures

Electrons emitted by the cathode are accelerated by voltage Va as they move toward the anode. If eVa is less than the energy E(2,1) = E(n=2) - E(n=1), only elastic collisions will occur between the electrons and the mercury atoms. The accelerated electrons will pass through the perforated anode and will be measured at the collector. When the electron energies are equal to E(2,1 ) the collector current begins the drop because some electrons make inelastic collisions with mercury atoms and cannot overcome the potential barrier Vc. The current rises as Va is increased since electrons can now make inelastic collisions before reaching the anode and still gain enough energy to reach the collector. As Va is increased, successive minima in collector current occur with a spacing of V = . They occur because an electron can now gain enough energy to make an inelastic collision well before reaching the anode. This electron can gain additional energy to make a second inelastic collision with a second minimum in the collector current. This process will repeat as Va is increased, producing a series of equally spaced minima in the measured collector current as shown in Fig. 3. From the spacing of the minima we can calculate the energy difference between the ground state and first excited state. This permits us to determine the frequency or wavelength of the radiation emitted by a mercury atom when it drops from the excited to the ground state.

Why did Franck and Hertz (and now you) use mercury vapor instead of hydrogen gas? The reason is that hydrogen atoms combine in pairs to form hydrogen molecules. Therefore, some of the energy lost in inelastic collisions of the electrons with a hydrogen gas would result from separating the hydrogen molecules into atoms and this would complicate the analysis of the measured collector current. A mercury gas consists of single atoms. Mercury atoms have 80 electrons (and protons) instead of the one for hydrogen. A more detailed quantum mechanical analyses is required to account for the energies of all these electrons. However, for the purposes of the Franck-Hertz experiment, it is sufficient to note that 78 of these electrons are bound much more strongly to the nucleus than the remaining two. Therefore inelastic collisions of the mercury atoms with the accelerated electrons only excite these more weakly bound "outer" electrons. Therefore the collector current will measure the difference between the ground and first excited state of these "outer" electrons.