2. First Excitation Potential of Hg

2.1 Theory

The Franck-Hertz experiment is one of the classic demonstrations of the quantization of atomic energy levels. Electrons emitted by the filament are accelerated through mercury vapor. When the accelerating voltage reaches V₁, the electrons have just enough energy to excite the mercury atoms from the ground state to the lowest excited state. Thus, many of the electrons lose their energy and cannot reach the collector; this is signaled by an abrupt drop in collector current as the accelerating voltage is increased past V₁.

Figure 1: Qualitative trend of collector current vs. accelerating voltage.

As the accelerating voltage is increased beyond V₁, electrons which have been brought to rest as a result of exciting mercury atoms are again accelerated until they can produce another excitation. Thus, a second peak occurs at V₂, where ideally one expects V₂ - V₁ = V₁. However, contact potentials in the system displace the first peak at V₁ from its expected value, and V₁ itself is not a good measure of the excitation potential. The sequence of electron acceleration and atomic excitation may continue as the accelerating voltage is further increased so that a series of peaks may be observed.

Location of several peaks gives somewhat independent determinations of the excitation energy. However, analysis of the data requires some thought. If you simply subtract the voltages at which successive peaks occur and average the differences, you effectively discard all of the data except that for the first and last peak! A least squares fit of the data to V_n = a + n b should be made. V_n is the voltage of the n’th (n = 1, 2, ...) peak, b is the first excitation potential, and a is influenced by the contact potentials. The least squares fit section summarizes the least squares fitting procedure.

Breakdown of the gas in the tube will occur when the accelerating voltage becomes too high. Onset of breakdown may be delayed by increase in vapor pressure (higher oven temperature) or decrease in number of electrons (lower filament current).

2.2 Apparatus & Equipment

Apparatus

Figure 2: Frank-Hertz apparatus and electrical connections.

Equipment

1. Franck-Hertz tube, Klinger KA6040, filled with Hg
2. Oven, maintain temperature in range 180 - 210 °C
3. Variac, to regulate oven temperature
4. Thermometer, to 200+ °C
5. Power Supply & Amplifier, Klinger KA6045
6. Furnishes filament current, If, adjustable, not to exceed 0.3 A
7. Furnishes accelerating voltage, V_A, adjustable by 10-turn pot., 0 - 70 V
8. Maintains collector (M) about 1 volt negative relative to accel. electrode (A) Amplifier for detection of collector current, I_c; null and sensitivity adjustment
9. DC Digital Voltmeter, to measure accelerating voltage, V_A
10. AC Ammeter, to measure filament current, I_f
11. DC Voltmeter, to monitor collector current, I_c (monitors output of Klinger amplifier)

2.3 Precautions

Personal
The oven becomes hot; do not touch it. It may be heating when you arrive.

Apparatus
Do not let oven temperature exceed 210 °C. Filament current, If, should not exceed 0.3 A.

2.4 Procedure

Figure 3: Experiment Control Panel

Open the Frank-Hertz.vi LabView program. Set the apparatus switch on the control panel to the Hg experiment.

Control Panel Information and Picoammeter Setup
The Frank-Hertz.vi LabView program controls the power supply, and records the current generated as measured by the picoammeter. The control panel has parameters for the voltage increment, starting voltage, ending voltage and scan rate. When the parameters are set and the program run button is pressed the power supply will be run from the starting voltage to the ending voltage in the increments specified. The scan rate determines how quickly the increment will be added. The voltage increment and scan rate should be varied to investigate the visible effect on the "quality" of the data collected.

Before taking data, the picoammeter must be zeroed (by pressing the zero check button) and the range must be setup. The meter has several range settings. These settings effect how the picoammeter reports the measured current to its display as well as to the computer (in the form of a proportional output voltage). There are three nano-amp range settings that allow for a maximum reading of either 2nA, 20nA or 200nA. The lower the range setting the better the resolution the meter can provide in that range. When the picoammeter is set to a given range, the switch on the control panel must be set accordingly so that the current is reported accuratly. The picoammeter display will read "OVERLOAD" if the measured current is above the range setting. Hovever, as long as the data curve remains smooth on the computer dislay, the meter is still reporting the current accuratly.

In addition, there should be a potential difference between the anode and the loop to increase the collection of electrons. The picoammeter can provide this voltage. Press the voltage source "Operate" button, then set the voltage to the desired value.

Experimental measurements.
For this experiment the potential difference between the anode and the loop should be about +0.500V.

Carefully locate peaks in the current and record the corresponding voltages. Absolute values of the current are of no importance; only the voltages corresponding to "breaks" in the current are of interest. Obtain one good set of data which shows many peaks.

The quality and number of peaks that you can observe is sensitive to oven temperature and filament current. Lower temperature facilitates observation of lower order peaks, but breakdown may occur before higher order peaks can be observed. Reduction in filament current delays breakdown and facilitates observation of higher order peaks.

Suggested trial conditions for Klinger KA6040 Hg tube:

Temperature 200°C (Variac = 62), filament current I_f = 0.25A
Temperature 170°C (Variac = 55), filament current I_f = 0.23A

By trial and error, you may improve upon the suggested starting values. Oven temperature and filament current must remain nearly constant as a complete set of peak voltages is recorded. As the accelerating voltage is increased, distinct peaks may not be found beyond a certain point, and finally the tube will break down.

Present your results as a plot of accelerating voltage for the maxima in collector current versus n, where n is the order of the peak (n = 1 for V about 7 volts, n = 2 for V about 12 volts, etc.).

Determine the first excitation potential of mercury. The slope of a "best fit" straight line drawn through the data is a satisfactory procedure. Alternatively, you may use the least squares fitting procedure outlined in the next section, alternatively your spreadsheet software or calculator may have a least squares fitting routine. Make the plot even if you do not use it to determine the first excitation potential.

Your report should include a discussion of the historical role played by the Franck-Hertz experiment in the development of quantum theory.

2.5 Calculating a Least Squares Fit by Hand

Observable Data Sets (n,vn) e.g. n vn 1 2.0 2 2.6 3 2.9 4 3.7 etc.

Note that v_n is an observed value at n, while V_n is calculated from the "best fit" equation: V_n = a + bn.

Choose a,b to minimize: E ≡ Σ (V_n - v_n)² (all sums are over n)

E = a² Σ 1 + b² Σ n² + 2ab Σ n - 2a Σ v_n - 2b Σ nv_n + Σ v_n²

Set (δE/δa)_b = 0:

Set (δE/δb)_a = 0:

Substitute your data sets, (n₁, v_n for all n), into these two equations. Solve the two equations for a and b.

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