In 1900, Max Planck attempted to establish a firm theoretical foundation for understanding the blackbody radiation spectrum. In order to do so, as a purely formal assumption, he reluctantly introduced the idea that electrons in the walls of a black body cavity emitted energy, quantized in multiples of an elementary unit. He also believed that, once emitted, the light radiated outward in a continuous stream, like water waves.

In 1905, in his study of the photoelectric effect, Albert Einstein focused on the emission and absorption of light, rather than its transmission. He extended the idea that light is generally emitted and absorbed in chunks, later called photons, having energy E = hf.

In 1913, Neils Bohr introduced his model of the hydrogen atom, which included the concept of discrete (quantized) energy states. In Bohr’s model, transitions from one energy state to another would help explain the discrete lines observed in the emission spectra of the elements. However, when measuring spectral lines with a diffraction grating spectrometer (assuming a wave nature of light), one is measuring wavelengths, not energies. Therefore, the relationship between energy and wavelength (or frequency) was only theoretical, not tested by experiment.

In 1914, James Franck and Gustav Hertz demonstrated that atoms in a gas may absorb energy due to collisions with electrons and that this transfer of energy always occurs in discrete, measurable amounts. At the time of their work, however, Franck and Hertz were apparently unaware of Bohr's findings, and the only mechanism known to them for such energy transfer was ionization, i.e., the complete removal of an electron from the atom. It is an interesting historical note that Franck and Hertz misinterpreted the discrete energy losses they observed to be due to ionization rather than to excitation of the atomic energy levels postulated by Bohr.

With subsequent reinterpretation, their work constituted an important experimental verification of quantum physics, confirming Bohr's postulate.

If atoms absorb energy from the electrons in discrete and measurable chunks, and the wavelength of the subsequently emitted light can be measured, then the relation between the energy and wavelength of photons can be determined. Thus, the Franck-Hertz experiment is a crucial link which lends credibility to the Bohr atomic model and the Planck-Einstein quantum hypothesis. For this work, Franck and Hertz were awarded the 1925 Nobel Prize.

In this version of the Franck-Hertz experiment, electrons with charge e are boiled off a hot cathode and accelerated by a potential difference Va. From classical physics, with no radical (quantum) assumptions, these electrons have energy eVa. These electrons pass through helium gas (mercury gas in the original apparatus). If the electrons have sufficient energy to excite the helium atoms, the electrons lose energy and are attracted to a collector ring, where they are detected as a current. Thus, we expect certain energies to cause increases in current. We shall see!

The original Franck-Hertz apparatus was a vacuum tube, containing a drop of mercury in equilibrium with mercury vapor. The mercury vapor pressure was controlled by an oven.

For simplicity, we shall use a vacuum tube containing low-pressure helium gas. The geometry of the modified Franck-Hertz tube is shown in Fig. 1. This apparatus eliminates the need for a temperature-controlling oven. Mounted within the vacuum tube are the following:

• a tungsten filament, used to heat and boil electrons off a cathode;
• a cathode, the source of electrons;
• an anode, used to accelerate the electrons and to collect those electrons not captured by the collecting ring. The anode consists of two parts which are electrically connected – a metal cylinder near the cathode, and the inner surface of the glass vacuum tube which is coated with an electrically conductive material;
• A collecting ring, used to collect electrons which have lost energy to the helium atoms.

{ ${/download/attachments/165678716/fig_1.png?version=1&modificationDate=1507753183000&api=v2}$\\

Figure 1: Geometry of the modified Franck-Hertz tube.

The anode is held at a potential Va more positive than the cathode. Thus, electrons generated at the cathode are accelerated to a kinetic energy of eVa by the time they reach the anode can. In the absence of any collector ring potential and with no helium atoms present, the electrons would continue, un-deflected, to the inside of the glass tube. However, since the collector ring is held at a potential 1.5 V more positive than its surroundings, the electrons are attracted toward the collector ring as they pass it. The aperture of the anode cylinder is such that these electrons would miss the collector ring.  

In our experiment, we will increase Va linearly with time. Thus, the electron kinetic energy will also increase with time. Collisions between electrons and helium atoms will occur, but as long as the electron kinetic energy is less than an atomic excitation energy, these collisions will not result in significant energy absorption by the atom. Electrons scattered in this manner (elastically) are relatively unlikely to reach the collector ring, since they will have energies nearly equal e_Va._

QUESTION 1:  How do we know that elastically scattered electrons will have energy close to e_Va_?

An electron whose energy is reduced to less than 1.5 eV by exciting a helium atom will be attracted to the collector ring, and will contribute to a measured negative current. On the other hand, electrons not exciting atoms will miss the ring and continue to the anode on the inner surface of the tube.

The helium pressure and the geometry of the tube play a linked role in the design of the apparatus. The mean free path is defined as the average distance an electron must travel between collisions with helium atoms. Thus, the mean free path is determined by the He pressure.

In this tube, the helium pressure has been set so that there is a small probability of collision in the region between the cathode and the cylindrical anode (where the electrons are being accelerated), but a large probability for a collision thereafter (where the electrons are traveling at constant speed). Thus, when the collisions take place, the electron energy is well known.

The Hertz tube console (see Fig. 2) is used to control the accelerating potential Va and read the current from the collector ring. Va is supplied as a sawtooth ramp of frequency 20 Hz. The minimum and maximum voltages of the ramp can be set from 0 V to 60 V, using the MAX and MIN controls on the right hand side of the console.

• Output 1 provides an un-calibrated voltage, proportional to _V_a.
• Output 2 provides an un-calibrated voltage, proportional to the collector ring current.
• Output 3 provides Va(MIN)/1000.
• Output 4 provides Va(MAX)/1000.

A separate DC power supply provides current to the filament and a 1.5 V battery supplies a small potential to attract electrons to the collecting ring. A digital scope is used to view and measure the current in the collector ring and the accelerating voltage, Va.

As electrons leave or enter a metal surface, they are subject to electric field gradients due to the charges present in the metals. The potential barrier caused by these field gradients is called the work function of the metal. Work functions differ for different metals. In our tube, electrons leave the cathode and enter the anode. Thus, the electrons will experience field gradients at both interfaces. The net effect of these gradients is called contact potential difference (CPD). The CPD will effectively increase or reduce the accelerating voltage, depending upon their signs and magnitudes. Since the CPD is not of prime interest in this experiment, we will not attempt to measure it. You will find that multiple absorptions for the same atomic transitions will occur. Measuring differences between the accelerating potentials will eliminate the need to determine the CPD.  This idea will become clear later in the procedure.

Table 1 gives the wavelengths emitted by transitions from the lower excited states to the ground state of helium. It is important to note that these wavelengths were measured spectroscopically. Only later were they converted to energies using the Planck relation, E = hc/λ, where h is the Planck constant, c is the speed of light, and λ is the wavelength of each spectral line.

NOTE: Recall that, at the time of the Franck-Hertz experiment in 1914, the Planck relation was only an assumption, used to “fudge” the black-body theory. An independent method to measure energies was needed to verify the Planck relation. You are here to measure those energies!

Table 1: Energy levels and resonance energies of helium.

(Originally from Tables of Spectral Lines… Stringanov and Sventitskii; quoted in N.Taylor et.al., “Energy levels in helium and neon atoms by an electron-impact method”, Am. J. Phys. 49(3), March 1981.)

CAUTION:  The critical potentials tube may be easily destroyed!

• Never exceed a filament voltage of 3.3 V.
• Before making any connections make sure that the filament power supply is turned off and the Hertz tube console is unplugged.

Begin by connecting by connecting the filament power supply to the tube base as shown in Fig. 2(a). The Filament Power Supply should be off while making the connections.

{ ${/download/attachments/165678716/Hertz_crit_wiring_1.png?version=3&modificationDate=1510005899000&api=v2}$ Figure 2(a): Tube filament connections

Next connect the accelerating voltage and the ring collector as shown in Fig. 2(b). The Hertz Tube Console should be unplugged while making these connections.

{ ${/download/attachments/165678716/Hertz_crit_wiring_2.png?version=1&modificationDate=1510006141000&api=v2}$ Figure 2(b): Accelerating voltage and ring collector connections

Finally, make the connections to the digital multimeters and the oscilloscope as shown in Fig. 2©.

{ ${/download/attachments/165678716/Hertz_crit_wiring_3.png?version=1&modificationDate=1510006771000&api=v2}$ Figure 2©: Scope and DMM connections.

When completed the wiring should match that shown in Fig. 2(d).

{ ${/download/attachments/165678716/Hertz_crit_wiring_full.png?version=1&modificationDate=1510006975000&api=v2}$ Figure 2(d): Completely wired up Franck-Hertz critical potential tube

Plug the Hertz tube console into the AC outlet. On the Hertz tube console, set Va(MIN) to its minimum value, and set Va(MAX) to its maximum value by turning their control knobs. You should see their voltages on the respective meters. There is a 1/1000 voltage divider on outputs 3 and 4, so that a 1 mV reading on the DMM corresponds to 1 V between the relevant electrode and ground.

Press the autoset button on the scope to find a trace. Select the following settings:

• Trigger menu:
  • Source = Channel 1
  • rising slope
  • coupling =  HF reject (to stabilize the trace)
• Channel 1 menu:
  • DC coupling
  • probe = x1

You should see the periodic ramp voltage on channel 1. Adjust the channel 1 gain and time base to fill the screen with a single Va ramp. Take a moment to adjust Va(MIN) and Va(MAX) while observing the effect on the scope. Measure Va(MAX) on the socpe. To do so, press the cursor button and set cursor type to voltage. LEDs should light under the vertical position knobs. In this mode, the vertical position knobs move the voltage cursors (horizontal, dotted lines) up and down. Move cursor 1 to measure Va(MAX). 

QUESTION 2: How does this scope measurement of Va(MAX) compare with its meter reading?

We will need to know the electron energy at points where we see evidence of atomic excitations (current dips). Therefore, we must relate the voltage we measure on the scope to the actual accelerating voltage _Va _in the tube. To do so, set Va(MIN) to 0 V and Va(MAX) to its maximum value. Adjust the voltage and time settings on the scope so that one full sweep of the Va ramp fills the screen.

QUESTION 3:  Use the readings of Va(MIN)/1000 and Va(MAX)/1000 from the DMMs to determine the relationship between the voltage measured on the scope, to Va. Estimate the uncertainty in your voltage calibration.

Re-set Va(MIN) to 10 V and Va(MAX) to 50 V. Slowly turn up the filament voltage while observing the signal from the collector ring on channel 2 of the scope.  Make sure that the filament voltage does not exceed 3.3 V!  As the filament voltage is increased you should observe that the signal from the collector ring increases negatively. 

QUESTION 4: Why does increasing the filament voltage increase the (negative) current collected at the ring?

Adjust the time and voltage scales on both channels so that the signals fill the display as much as possible. The signal from the collector ring is noisy because it consists of a relatively small number of electrons being collected at the ring.

If the observed signal is already smooth, the scope can simply sample once and display the result. On the other hand, if the signal is noisy, the scope can take several samples, average them, and then display the smoother, average values. Of course, averaging takes some time, so the averaging mode is not well suited if you need to change conditions during the sampling time.

You should notice that there are two or three groups of small dips superimposed on the negatively increasing collector ring signal. These dips represent increases in the (negative) collector ring current caused by electrons’ exciting He atoms and thus, losing energy. The different groups of dips show the same set of atomic transitions being excited at multiple accelerating voltages. (See Fig. 3).

{ ${/download/attachments/165678716/fig_3.png?version=1&modificationDate=1507753183000&api=v2}$\\

Figure 3: The relationship between electron kinetic energy and collector current.

Referring to Fig. 3, point (a) shows electrons with KE = 0. Point (b) shows electrons whose KE has just reached an excitation energy of helium. These electrons lose their energy to the helium and return to KE = 0.  Point © shows electrons having lost their KE once before, returning to the helium excitation energy, and losing energy once more.

QUESTION 5:  Why do you observe the same transitions multiple times as Va increases?

The observation of multiple excitations of the same transition gives us a way to determine the contact potential difference (CPD) between the cathode and the collecting ring. The CPD will shift the positions of the dips associated with the atomic excitations. (See Fig. 3). However, when the same transition is observed multiple times, the potential difference between corresponding dips is a measure of the true energy Δ_E_ of the atomic transition. Knowing this, the CPD can be deduced. Since the CPD is not relevant to our experiment, we will not measure it, but instead measure the differences in accelerating voltage to determine the HELIUM transition energies.

 Set the scope to average over 64 sweeps and make a careful sketch of the voltage and collector current in your lab notebook.

Adjust Va(MIN) and Va(MAX) so as to zoom in on the group of dips at the lowest Va on the scope. You may have to adjust the scope trigger level to keep the dips centered on the display. Adjust the filament voltage, without exceeding 3.3 V, to maximize the number of dips visible in this group. Adjust the voltage scale on channel 2 so that the dips fill as much of the display as possible. Make sure that the scope is set to average over 64 sweeps and capture a spectrum of dips and the accelerating voltage ramp.

Use the voltage cursor to measure the ramp voltagesat each dip. Use your calibration to convert these voltages to values of Va. 

Table 1 gives the transition energies from several excited states of helium to the ground state. Note that these energies were not measured directly, but derived from measurements of wavelength. Many of these transitions are very close to one another and will not be resolvable as separate dips.

QUESTION 6:  Measure the full width at half maximum (FWHM) of your dips to estimate the energy resolution of the apparatus.

QUESTION 7:  Compare the energy differences between the dips in your spectrum with the transition energies shown in Table 1.

QUESTION 8:  Taking into account the energy resolution of the apparatus, identify the transitions, or groups of transitions, associated with each peak.

QUESTION 9:  How well does the Planck relation,  E = hc/λ, hold up under this test?