By the end of this experiment, students are expected to be able to do the following:

Technical:

• Perform basic wiring, and use measurement tools like a multimeter and oscilloscope
• Use advanced scope techniques (like high frequency reject triggering, averaging, using cursors, and saving traces)
• Keep a lab notebook (with a focus on keeping track of multiple data sets and saved files; noting settings, procedure and calibration information; and making quick back-of-the-envelope calculations in-lab).

Data Collection:

• Calibrate from measured quantities (e.g. $V_{scope}$) to desired quantities (e.g. electron kinetic energy).
• Make choices about apparatus settings to obtain “optimal” data (e.g. setting voltage ramp limits, tuning filament current, and adjusting scope triggering/averaging)
• Estimate uncertainties from scope data (including considerations of skew due to features on a sloped background)

Analysis:

• Import and re-plot data files (with annotations)
• Fit linear data
• Use a calibration formula to convert quantities
• Propagate uncertainties to calculated quantities
• Compare values to literature

Students will gain exposure to and practice with the following topics:

Experimental topics covered:

• Measurable (and correctable) systematic biases
• Smearing of features due to resolution effects
• Techniques to enhance signal-to-noise

Physics topics covered:

• Atomic Physics: energy level structure of helium, parahelium vs. orthohelium, inelastic electron collisions, electronic excitation, selection rules, ionization, and wavelength-energy conversion
• Electricity and Magnetism: acceleration of electrons and charged He+ ions
• Electronics: circuit with multiple voltage sources, floating power supplies, voltage dividers, 60 Hz noise, and important grounding considerations
• Experiment Design: use of an oscilloscope to measure short time signals, grounding and noise considerations, interpretation of signals with multiple contributing physics effects, apparatus considerations (e.g. accelerating voltage choices, helium gas density in the tube, current saturation, signal amplification)
• Physics History: history of the development of atomic physics and quantum mechanics

In 1900, Max Planck attempted to establish a firm theoretical foundation for understanding the blackbody radiation spectrum. In order to do so, he reluctantly introduced the idea that electrons in the walls of a black body cavity emitted energy (in the form of light) that was quantized in multiples of some elementary unit. Despite this quantization of energy, Planck still believed that the light traveled outward in a continuous wave.

In 1905, in his study of the photoelectric effect, Albert Einstein found that more generally any traveling light wave (not just that emitted by blackbody radiation) is composed of these discrete chunks. Together, their work established that photons of light carry energy $E=hf = hc/\lambda$, where $f$ is the frequency of the light, $\lambda$ is the wavelength, $c$ is the speed of light, and $h$ is now called Planck's constant.

In 1913, Niels 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) still remained 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, 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). As a result, their initial interpretation of the results was incorrect.

With time, however, the correct interpretation emerged. 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 Physics.

In the version of the Franck-Hertz experiment presented here, electrons with charge $e$ are boiled off a hot cathode and accelerated by a potential difference, $V_a$. From classical physics, with no radical (quantum) assumptions, these electrons have energy $E=eV_a$. 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.

In our setup, we instead will 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;
  • Note that 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.

Figure 1: Geometry of the critical potential tube

The anode is held at a potential, $V_a$, which is more positive than the cathode. Thus, electrons generated at the cathode are accelerated to a kinetic energy of $K = eV_a$ 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, undeflected, 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 $V_a$ linearly with time. Thus, the electron energy will also increase with time. Collisions between electrons and helium atoms will occur, but as long as the electron 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 $V_a$.

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.

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, $E = hc/\lambda$.

For the hypothetical atom in the prelab, we ignored the spin of the electron. Here, it will be important to keep track of the spin because the orientation of one electron (relative to the other) adds an additional small separation in the energy levels.

• An electron has an intrinsic spin of s = 1/2.
• Neutral helium has two electrons. Therefore the total spin can be either $S = 0$ (if the spins point in opposite directions) or $S = 1$ (if the spins point in the same direction).
  • An atom where $S = 0$ (electrons in opposite directions) is called parahelium.
  • An atom where $S = 1$ (electrons in the same direction) is called orthohelium.
  • Because of the Pauli exclusion principle, when both electrons are in the ground state they must have oppositely directed spins; therefore, the ground state is parahelium.

In Table 1 below, we are using the expanded spectroscopic notation $n^{2S+1}L$, where $n$ is the principle quantum number, $S$ is the total spin, and $L$ is the angular momentum quantum number (such that S is L = 0, P is L = 1, and D is L = 2). In this notation, an excited state $n^3L$ is an orthohelium state and $n^1L$ is a parahelium state.

Multimeters

To begin, connect the two digital voltmeters that monitor the $V_{a,min}$ and $V_{a,max}$ values produced by the console.

• Plug the Hertz tube console into the wall outlet. (A light in the lower right corner will illuminate when the console is powered on).
• Connect one voltmeter to monitor the voltage at Output 3. To do so…
  • …Connect the “V” side of the voltmeter to Output 3.
  • …Connect the COM“ side of the voltmeter to any one of the ground jacks on the console.
  • …Turn the meter on to the DC voltage scale.
• Repeat with Output 4.

If everything is connected correctly, you should be reading voltage values from the two outputs in the range from 0 to 60 mV. Adjust the MAX and MIN control knobs and observe how the voltages change.

Oscilloscope

Next, connect the FAST Outputs 1 and 2 to oscilloscope Channels 1 and 2 in order to monitor the $V_a$ ramp voltage and the ring collector voltage, respectively. (We will not use the SLOW outputs on the console.)

• Turn on the oscilloscope.
• Using a BNC to double banana adapter, connect the red side of Channel 1 to FAST Output 1 and connect the black side of Channel 1 to any ground jack on the console.
• Repeat with Channel 2.

At the moment, Channel 2 should be a flat signal at 0 V (as the collector ring is not yet attached to the console). However, Channel 1 should display a sawtooth voltage signal with a frequency of 20 Hz (or period of 1/20 Hz = 50 ms).

• Trigger the scope on Channel 1 to achieve a stable trace.
  • Suggestion: Trigger on the falling slope of Channel 1 in order to catch the sawtooth cleanly at the start of each cycle. If you attempt to trigger on the rising slope, you may find that noise in the signal causes the trace to jiggle around from trigger event to trigger event. If you find that your trace is acting this way, use the “high frequency” reject setting (found in Trigger Menu: More: Coupling: HF Reject).
  • Adjust the Channel 1 voltage and time base to fill the screen with a single $V_a$ ramp.
• Adjust the $V_{a, min}$ and $V_{a,max}$ knobs on the console while observing the effect on the scope.

Figure 3: The wire with the in-line fuse (to be used as one connection from power supply to filament)

Now connect the tube to the console.

With the filament power supply OFF, connect the supply to the tube base.

• Connect the positive side of the power supply (red) to jack F3 on the back of the tube base.
  • Use the cable with the in-line fuse (shown in Fig. 3) to make this connection. This will protect the filament if it attempts to draw too much current.
• Connect the negative side of the power supply (black) to jack F4 on the back of the tube base.

Next, connect the anode and cathode to the $V_a$ ramp…

• Connect $V_A$ Output 1+ (in the box in the upper left corner of the console) to jack A1 on the back of the tube base. (This is the anode voltage.)
• Connect $V_A$ Output 1- to jack C5 on the back of the tube base. (This is the cathode voltage.)
• Connect the short green jumper wire between jack C5 and jack F4. (This ties the low voltage side of the filament circuit to the cathode voltage.)
  • NOTE: This connection is made internally on some of the tube bases already. This jumper guarantees the connection in case your tube base is one of the ones where the internal connection is missing.

In order to maintain the ring collector at +1.5 V relative to the anode, connect a AA battery between the anode and ground.

• Connect the negative side of the battery holder to the anode (either jack A1 on the tube base or $V_A$ Output 1+ on the console).
• Connect the positive side of the battery holder to any ground jack on the console.

Finally, connect the ring collector to the console.

• Connect the BNC cable from the top of the tube to INPUT 2 on the console (in the box in the lower left corner of the console).

With the ring current now connected to the console, you should observe a signal on Channel 2 of the oscilloscope. (Zoom in if the signal still appears flat.) The signal will likely be noisy (and may also have a periodic component with a frequency of 60 Hz) because the ring is collecting any stray electrons from the room which pass through the tube near the ring. In order to suppress this background noise, we can use a grounded shield which will intercept these electrons and direct them to ground (instead letting them pass through the tube to be picked up by the ring).

• Slide the carboard and metal shield around the tube, making sure that the tab slides into the grove along the tube base and the shield is pushed all the way back so that it fully covers the tube.
• Connect the wire from the shield to any ground jack on the console.

With the shield in place, the signal on the scope should now be much less noisy (though likely still not quite flat).

A light on the front of the power supply will indicate whether the power supply is in constant voltage (CV) or constant current (CC) mode.

• When the power supply is in CV mode, it will attempt to maintain the voltage set by the voltage control knob by adjusting the current provided (up to the maximum current allowed by the current control knob).
• When the power supply is in CC mode, it will attempt to maintain the current set by the current control knob by adjusting the voltage difference (up to the maximum voltage difference allowed by the voltage control knob).

If you have no need to limit a voltage or current, then you can turn one of the knobs up to maximum and control the power supply by the other knob. (Typically this means CV mode… so turning current to max and using the voltage knob to adjust.)

If, however, you wish to limit a voltage or current (for example to prevent a fuse from blowing or to protect a sensitive circuit), then you can turn the corresponding knob down to a lower lower level.

In this experiment – where we wish to make sure we never exceed about 1.2 A of current – we can set this limit by slowly turning up the voltage knob until we reach a reading of about 1.2 A. Then, we can slowly turn down the current knob until the light on the front of the supply changes from CV to CC. At that point, the supply is now limited to about 1.2 A and you should not be able to exceed that amount no matter what the voltage setting.

First, the flux of electrons emitted by the filament wire increases as the accelerating potential increases. The electron current density, $J$, (in units of current/area) is given by a formula known as Child's Law,

where $k$ is a proportionality constant that depends on the mass of the electron, the charge of the electron, and the separation distance between the anode and cathode.

Child's law is a consequence of the space charge limiting effect. The first electrons to be emitted from the filament when it is heated form a “cloud” around around the wire known as space charge. The negative electric potential created by this cloud makes it harder for future electrons to be ejected. However, as the potential between anode and cathode is increased, this repulsive effect is reduced, leading to the increase in the flux of electrons emitted.

Second, even though the trajectories of the accelerated electrons form a cone that does not intersect with the collection ring (see Fig. 1), it is always possible that an electron undergoes an elastic scatter off a helium atom and changes direction. As the flux of electrons increases, the number of electrons that are randomly scattered into the collection ring increases proportionally.

Therefore, since the electron flux increases as $V_A^{3/2}$, the background upon which the peaks appear also should increase as $V_A^{3/2}$.

As was the case in the prelab question, the first set of peaks corresponds to the minimum energies required to excite an electron from the ground state of helium up to one of the possible excited states. As there are many possible excited states, there are many possible excitation energy peaks… though just as in the prelab, some of these energies may be so closely spaced that we can't distinguish them as separate and they instead merge together into one peak.

We will spend more time on Day 2 investigating these individual dips, but for now let's focus just on the first dip (as the mark of the beginning of these possible excitations).

For the second set of peaks, the interpretation is a bit more complicated. Here, electrons have enough energy to excite two different atoms.

• Some of the electron energy excites atom 1 from ground to some excited state.
• The rest of the energy goes to atom 2 to excite it from ground to an excited state.
• The total energy the electron started with is the sum of those two excitation energies.
• Since there are multiple possible excitations for each atom ($1^1S \rightarrow 2^1S$, $1^1S \rightarrow 2^1P$, etc.), there will be an energy peak in the second set corresponding to each pair of transitions.
• The resulting set of peaks will therefore look similar to the first set of peaks, but you should notice that there are more peaks overall (and that the intensities of those peaks differ).

Given the complexity of this spectrum, we won't try to identify each peak. Again, for now let's focus just on the first dip (as the mark of the beginning of these possible two-atom excitations).