Verifying the Quantization Atomic Energy States
By Michael Munschy and Shishir Kotra
Section 1: Abstract
The quantized nature of atomic energy states was verified by a procedure similar to the original
experiment by Franck and Hertz in 1914. A current of electrons was accelerated through a range of
kinetic energies and allowed to collide with a heated sample of mercury vapor. The current passing
through the sample was measured, and revealed that at regularly spaced intervals of kinetic energy, the
mercury atoms were excited into higher energy states by inelastic collisions with the accelerated
electrons. The regular, discretized occurrences of these absorption events are evidence of the
quantization of atomic energy states.
Section 2: Introduction
The Franck-Hertz experiment was a Nobel Prize winning experiment which lead to the discovery
of the quantized nature of energy changes within an atom. By measuring the current passing through
the Mercury vapor, it was observed that at regular voltage intervals there was a sudden drop in current
and a spontaneous emission of light from the mercury molecule. The wavelengths of this emitted light
matched that of known spectral lines for mercury, and became one of the first observations of the
quantized nature of energy changes in atomic systems. Our experiment will seek to verify the results
obtained by Franck and Hertz in 1914. We will use an apparatus which is conceptually similar to that
used by Franck and Hertz, with the added benefit of modern oscilloscopes and computer technology.
A sample of mercury in an evacuated tube will be heated to various temperatures, and an
electron current will be accelerated through a range of potentials. A combination of multi-meters,
oscilloscopes, and computer software will monitor the interaction of the electrons and the mercury
vapor by measuring the electric current flowing through the sample.
Section 3: Theory
The energy levels of electron transfers within the electron shells of atoms have been well
established, and will provide the primary check against our experimental data. The energy levels for
electron transitions in the first few electron shells of mercury are shown in Figure 1 below. When an
atom of mercury absorbs energy, an electron can be excited into a higher energy level. The electron
quickly discards this extra energy as a photon with a well defined wavelength, and returns to its original
energy.
Figure 1: Energy levels of some of the
electron transitions possible in a mercury
atom. An electron is excited from the
ground state into a higher energy level,
then spontaneously decays back to the
ground state via the emission of a photon.
In our experiment, the source of additional energy for the mercury atoms comes from collisions
between the electrons within a mercury vapor cloud. An electron traveling through the mercury vapor
will frequently collide with the molecules in the gas. The mean free path of the electron is the average
distance an electron is able to move before colliding with one of the molecules in the mercury vapor,
and can be calculated from the following equation:
𝜆=
1
(1)
√2𝜋𝜂𝑅02
In this equation, 𝜆 is the mean free path length, 𝑅0 = 1.5x10−10 m is the radius of a mercury atom, and
𝜂 is the number density of the atoms in the mercury vapor. The number density can be found with an
application of the ideal gas law:
𝑃𝑉 = 𝑁𝑘𝐵 𝑇
(2)
Where 𝑃 is the vapor pressure in pascals, 𝑉 is the volume of the sample, 𝑁 is the number of mercury
atoms in the vapor, 𝑘𝐵 is the Boltzmann constant, and 𝑇 is the temperature in Kelvin. The number of
𝑁
atoms per unit volume is the number density, and solving (2) in terms of 𝑉 gives an expression for the
number density:
𝜂=
𝑁
𝑃
=
𝑉 𝑘𝐵 𝑇
(3)
Equation (3) shows that the number density is directly proportional to the ratio of the gas pressure and
temperature. Substituting (3) into equation (2) reveals that as the number density increases, the mean
free path for an electron traveling through the vapor will decrease. This has a direct consequence on the
energy involved in the collisions between electrons and the mercury vapor. As an electron is subjected
to an electric field it gains kinetic energy, as described by the following equation:
𝐾 = 𝑞ℯ 𝐸̅ 𝜆
(4)
were 𝜆 is the mean free path determined by (1), 𝑞ℯ is the fundamental charge of an electron, and 𝐸̅ is
the electric field created by the potential difference applied to the sample. Equation (4) reveals that for
longer mean free paths and, by extension, smaller number densities, the average kinetic energy of the
electrons when they collide with mercury atoms will increase.
In general, electrons will collide with mercury atoms elastically, losing very little kinetic energy in
the process. These electrons will proceed through the sample to be collected and measured as a
current. If an electron has a kinetic energy close to an energy level of mercury, then the electron will
undergo and inelastic collision, imparting a large portion of its kinetic energy into the electrons of the
mercury atom and exciting them into a higher energy level. This will cause the colliding electron to be
collected at a greatly reduced kinetic energy. If these lower energy electrons can be “filtered out” of the
measured current, then the current becomes a tool for observing when these inelastic collisions are
taking place. This is the method used in our experimental apparatus.
Section 4: Experimental Method
The primary apparatus used in our experiment was a Franck-Hertz tube and heating oven
connected to a Keithley electrometer. Figure 2 shows a schematic view of the mercury tube in the oven.
The filament is heated by applying approximately 5.5 Volts to it. Once heated, the filament will emit
electrons which can be detected by the electrometer as a current. An accelerating voltage, 𝑉𝐴 is applied
between the anode and cathode. This voltage creates the electric field in equation (4), and is directly
proportional to the kinetic energy gained by the electrons. Between the grid anode and the
electrometer, a retarding voltage of 1.5 volts is applied in opposition to the accelerating voltage. If an
electron does not have sufficient kinetic energy to overcome this retarding potential, then it will be
absorbed by the grid anode and not register as part of the current measured by the electrometer. This
effectively acts as a filter for low-energy electrons. After imparting a large portion of their kinetic energy
to the mercury atoms during inelastic collisions, the electrons will not have enough energy to overcome
the retarding voltage, and the electrometer will show a drop in the current being detected. This allows
us to observe which acceleration voltages produce the inelastic collisions characteristic of the excitation
of the mercury atoms.
Figure 2: Schematic view of the Frank-Hertz tube. A voltage is applied
to the filament to heat it up and cause it to emit electrons. An
accelerating voltage is applied between the cathode and anode to
create an electric field. Finally, and electrometer measures the current
of electrons passing through the mercury vapor in the tube.
In supplement to the main apparatus, two Keithley multimeters allow for precise measurements
during data collection. A thermocouple is connected to one of the multimeters to monitor the
temperature within the oven, while another multimeter is used to measure the acceleration voltage
applied between the cathode and anode. To control this voltage, a customized power supply is used
with a potentiometer which can very minutely adjust the accelerating voltage. To collect the data
produced by the multimeters and electrometer, the devices are connected to a computer and used as
inputs for the Labview software package. The software rapidly samples each of the three devices and
produces graphs of temperature versus time, and current versus acceleration voltage. Once data
collection is finished, the program exports the data set into a text file which can be uploaded into the
Kaleidagraph software for additional analysis.
Prior to collecting data, the oven was allowed to warm up to a specified temperature. Due to
the nature of the heating element, the temperature in the oven tends to fluctuate by a few degrees over
time. Once the oven reached the desired temperature, the power supply was turned on and data
collection began. The acceleration voltage was slowly increased from 0V to 30V while the electrometer
measured the resulting currents. Once data collection was completed the results were exported and the
apparatus was reset to a new temperature. In total, five different temperatures were tested to study
what effect temperature may have on the process.
Section 5: Data and Analysis
A total of five temperatures were tested in the manner described in Section 4, and are
represented in Figures 3 through 7. During the experiment we encountered some issues with data
collection, the most damaging of which was an increasing frequency of system freezes. This lead to
some of the data sets having large gaps in the recorded data. Since the electron absorption events
happen within a narrow range of voltages, it is suspected that some of the data sets do not represent
the “whole story”. These concerns will be addressed on a case-by case basis. We’ll begin by looking at
the two cleanest data sets.
Figure 3 shows a graph of the current measured by the electrometer as a function of the
acceleration voltage, which was collected at a mean oven temperature of 120.07 ± 2.29C. In the graph
we see that the first visible occurrence of inelastic collisions occurs at 8.39 ± 0.01 V, which is marked by
the sudden decrease in measured current at lower voltages. At this point, most of the electrons passing
through the mercury sample are undergoing inelastic collisions, imparting most of their kinetic energy
into exciting the electrons of the mercury atoms. This causes a majority of the electrons to not have
enough kinetic energy left to overcome the stopping voltage at the grid anode, and prevents those
electrons from contributing to the measured current. As the voltage is increased beyond this point, the
electrons are able to regain some energy after inelastic collisions, which causes more and more
electrons to make it to the electrometer. Once the electrons gain enough kinetic energy they are again
able to undergo an inelastic collision with mercury, which manifests as another minima at 13.71 ± 0.01
V. The change in voltage between these minima is 5.32 ± 0.01 V, which corresponds to the energy
expected from the 3𝑃2 → 1𝑆0 transition shown in Figure 1. The next absoption event happens at 18.87
± 0.01 V, after an increase of 5.16 ± 0.01 V. The final occurrence of inelastic collisions is observed at
24.54 ± 0.01 V, after an increase of 5.67 ± 0.01 V. An average ∆𝑉 of 5.38 ± 0.01 V correlates to the
expected 3𝑃2 → 1𝑆0 transition energy of 5.43 eV.
8.39𝑉
13.71𝑉
18.87𝑉
24.54𝑉
Figure 3: Current versus temperature for data collected at an oven
temperature of 120.07 ± 2.29 C. Noted are the regularly spaced
minima in the measured current.
Figure 4 displays the data collected at an oven temperature of 140.66 ± 2.11 C, and is a graph
of the measured current versus the applied voltage….
Figure 5 shows the data collected in our third “representative set”. The data was collected at an
average over temperature of 151.31 ± 0.62 C….
Figure 6 is a graph of the data collected at an oven temperature of 102.15 ± 2.70 C. The key
feature of this graph is noticeably widened peak in the current, and the relative lack of the regularly
spaced absorption events visible in Figures 3 and 4. It was during this data run that the first of the
equipment problems began….
Figure 7 shows a graph of the data collected at an oven temperature of 171.26 ± 2.53 C. In
contrast to the key feature of Figure 5, this data set shows a very rapidly varying current. We
hypothesized that this might be due to a rapid fluctuation in the temperature, which would have a
dramatic effect on the mean free path and thus affect the kinetic energy of the electrons. Figure 8
shows the result of temperature comparison between the 171 C and 140 C data runs. There does not
appear to be a dramatic difference in the temperature fluctuation, which rules out our hypothesis. A
Figure 8: A comparison of the temperature fluctuations during data collection at 140 C (left) and
171 C (right). The relative similarity suggests that temperature variation is not the cause for the
erratic current fluctuations seen in Figure7.
more likely explanation is that at the higher temperature …
Section 6: Conclusion
Our data reveals the regular occurrence of inelastic collisions between the accelerated electrons
and the mercury atoms in the Franck-Hertz tube. The fact that these inelastic collisions only occur at
specific collision energies verifies that electron transitions within the mercury atoms can only occur for
very specific energy gains. This means that the energy transitions in atoms are quantized, and agrees
with the results discovered in the original experiment by Franck and Hertz.
13.63𝑉
18.23𝑉
23.43𝑉
28.75𝑉
Figure 4: Current versus temperature for data collected at an oven
temperature of 140.66 ± 2.11 C. Noted are the regularly spaced
minima in the measured current.
13.54𝑉
18.92𝑉
23.75𝑉
28.75𝑉
Figure 5: Current versus temperature for data collected at an oven
temperature of 151.31 ± 0.62 C. Noted are the regularly spaced
minima in the measured current.
Figure 6: Current versus temperature for data collected at an oven
temperature of 102.15 ± 2.70 C. A key feature of this graph is the
very broad peak between 8V and 22V.
Figure 7: Current versus temperature for data collected at an oven
temperature of 171.26 ± 2.53 C. The defining characteristic of this
graph is the very rapid variance in the measured current.