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Opeoluwa Matthews

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Opeoluwa Matthews

Advanced Lab Individual Experiment

The Frank-Hertz Experiment

Thursday, March 3, 2011

The Frank-Hertz Experiment: Demonstrating the Quantization of Kinetic Energy

Transfer in Electron-Atomic Collisions

Abstract:

The purpose of this experiment was to show that the transfers of energy to atomic electrons of

mercury through collisions occurs for discreet energies and are consistent with the quantummechanically allowable energy transitions for mercury. This was demonstrated by accelerating electrons

through increasing potentials in a Frank-Hertz tube and observing the corresponding currents at the end

of the tube.

Introduction:

This experiment replicates the Frank-Hertz experiment conducted by German physicists James

Frank and Gustav Ludwig Hertz in 1914. It succeeded the Bohr-Heisenberg Model of the atom

introduced in 1913, which sought to explain the spectral emission lines of the hydrogen atom. Bohr

proposed that the change in energy levels were related to the frequency of the radiations by E=hv . It

was eventually shown that that absorption of photons occurred for only discrete frequencies of light and

thus discrete energies hv .

The Frank-Hertz experiment tests the generalization of Bohrs findings. It tests whether or not

kinetic energy transfers, like photons, occurred also in discrete energies. To test this, Frank and Hertz

bombarded gases with electrons in a chamber and tested whether the electrons lost their kinetic energy

at discrete energies. Their findings won them the Nobel Prize in Physics in 1925.

Experimental Procedure:

Experimental Setup:

The apparatuses required for the Frank-Hertz Experiment include Frank-Hertz tube containing

mercury, a copper filament, a heating grill, a variable-potential Anode and a variable-potential Collector.

These can be seen in Figure 1 below.



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Figure 1. Schematic of the Frank-Hertz Experiment.

As can be seen in the schematic above, the mercury atoms are vaporized within the Frank-Hertz tube.

Electrons are then emitted by thermionic emission from the filament. The Collector at the end of the

tube, connected to a programmable DC power supply, accelerates the electrons through the Frank-Hertz

tube at varying voltages. The electrons are then able to collide with the mercury atoms and move

towards the Collector.

If the electrons are at the right energies to transit the mercury atoms to a higher energy level, the

collision will be inelastic. But if the electrons are not at the discreet energy levels required, the collision

is simply elastic and the electrons retain their energy.

Note that after the collision of the electrons with a mercury atom, the Collector could still accelerate

them to higher energies. For this reason, there is a Grid Anode at the end of the Frank-Hertz tube that

provides a retarding potential to ensure that those electrons that have had inelastic collisions but have

still been accelerated after the collision will be drawn to the Grid Anode and not make it to the

Collector.



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At the Collector is a picoammeter to measure the current at the end of the tube. The current indicates

how many electrons are losing their kinetic energies to inelastic collisions.

Conducting the Experiment:

The Frank-Hertz tube was placed at a certain temperature, typically between 150 and 200oC. The Grid

Anode was placed at a retarding potential, typically between 1.5 and 2 volts. Then the Collectors

accelerating voltage was varied from around 2 volts to 30 volts at increments of .1 volts, with the

current at the collector recorded at each increment. Then, a graph of current against voltage was

plotted to show the collision patterns (if any) of the electrons.

Results and Interpretation:

Below will be a series of graphs detailing what experiments were performed. The error bars will

be in black.

Figure 2 below is a graph of current at the Collector against voltage taken with Annode Grid

retarding voltage of 1.5v.

0

5 10-10

1 10

-9

1.5 10-9

2 10-9

2.5 10-9

3 10-9

3.5 10-9

4 10

-9

0 5 10 15 20 25

1_5v__150-155

c u r r e n t ( n A )

volts(v)



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Figure 2. Graph of current at collector versus voltage. Results were recorded for .1 volt increments of the

Collectors potential ranging from 2 volts to 23 volts. The Annode Grid was set at a retarding potential of 1.5volts.

The chamber was at a temperature between 150 and 155oC.

In the above graph, the average distance between the peaks was found to be 4.96±.07v.

In Figure 3. Below, measurements are recorded for Annode Grid retarding voltage of 1.5v, like figure 1,

but the temperature of the chamber is raised.

-2 10-10

0

2 10-10

4 10-10

6 10-10

8 10-10

1 10-9

1.2 10-9

0 5 10 15 20 25 30

1_5v__170-180

C u r r e n t ( n A )

Volts(v)



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Here, the average distance between peaks was found to be 5.07 ±.8 volts.

Comparing Figure 2 and 3, it is interesting to see how raising the temperature reduces the overall

current at the Collector. Figure 4 is Figure 2 and 3 plotted on the same axis to demonstrate this more

clearly.

-1 10-9

0

1 10-9

2 10-9

3 10-9

4 10-9

0 5 10 15 20 25 30

1_5v__170-180

1_5v__170-180

1_5v__150-155

C u r r e

n t ( n A )

Volts(v)

Figure 4. Graph of current versus voltage for 1.5v Annode Grid, 150-155oC chamber temperature

plotted on the same axis with graph of current versus voltage for 1.5v Annode Grid and 170-180oC.



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The lowering of the overall current is expected when the temperature is raised because the mercury

atoms have more kinetic energy with higher temperatures, thus allowing for more collisions of the

electrons at lower electron energies.

Figure 5 below is a graph of current versus voltage for 1.7v Annod Grid and 150-155oC chamber

temperature.

0

5 10-10

1 10-9

1.5 10-9

2 10-9

2.5 10-9

3 10-9

3.5 10-9

0 5 10 15 20 25

1_7v__150-155


volts(v)




Here, the average distance between peaks was found to be 5.19 ± .08v. Again, comparing Figure 5 withAnnode Grid at 1.7v with Figure 1 with Annode Grid at 1.5v, both at the same chamber temperature, we

can see how a change in the Annode Grid voltage can affect the current v voltage curve. Figure 6 below

plots the graphs of both above delineated configurations on the same axis.



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0

5 10-10

1 10-9

1.5 10-9

2 10-9

2.5 10-9

3 10-9

3.5 10-9

4 10-9

0 5 10 15 20 25

1_7v__150-155

1_7v__150-155

1_5v__150-155


volts(v)

Figure 6. Graph of current versus voltage for 1.5v Annode Grid, 150-155oC chamber temperature

plotted on the same axis with graph of current versus voltage for 1.7v Annode Grid and 150-155oC.

As expected, we see that increasing the Annode Grid voltage lowers the overall current at the Collector.

This is because a higher retarding voltage at the Annode Grid means less electrons have the energy to

make it to the Collector.

In Figure 7 below, measurements are taken along the inside of a dip to detect smaller energy transitions.

Hence measurements are taken from 16 to 17volts in .01volt increments for 1.5v Grid voltage and 170-

180oC chamber temperature. Hence it is similar to zooming into Figure 3 between 16 and 17 volts.



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1 10-10

1.5 10-10

2 10-10

2.5 10-10

3 10-10

3.5 10-10

15.8 16 16.2 16.4 16.6 16.8 17 17.2

zoomed in 16-17v 1_5v__170-180

C u r r e n t ( n A )

Volts(v)

Figure 7. Graph of Collector current versus voltage taken between 16 and 17 volts in .01volt increments.

Annode Grid retarding voltage is 1.5 volts and chamber temperature is 170-180oC.

As can be seen, there is also oscillation going on in the small interval between 16 and 17 volts. Theaverage distance between peaks was found to be 0.20 ± .01v.

Figure 8 below, like Figure 7, takes measurements at .01volt increments but between 19.5 and 20.5

volts.



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4 -

-

-

-

¡

.-

¡

.4 -

¡

.-

¡

. -

¡

.4 19. 19. . .4 20.

z i 19.5-2 .5v 1_5v__170-180

C u r

r

t (

)

lts( )

Figure 8. Graph of Collector current versus voltage taken between 16 and 17 volts in .01volt increments.

Annode Grid retarding voltage is 1.5 volts and chamber temperature is 170-180oC.

Here, since the peaks are all but prominent and the errors are so large that the peaks could be slid up

and down to produce a linear curve, it seems unreasonable to calculate the distance between peaks.



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Error Analysis:

Current vs Voltage Graphs:

Current:

All the error expected in this area is related to the equipment. For experiments with 0.1 volt increments,

the error due to the machine was measured by taking 2 volt increments within the range of the

measurements taken and calculating 10 times each what the value of the current is. The same was done

for the .01v-increment measurements, except the errors were measured at .1volt increments, instead.

The standard deviation was found and each value was then mapped to the error regime that

corresponds to the mark of the last regime and the standard deviation calculated for its maximum value.

Of course, a computer algorithm, in Python, was used to make this mapping as it would be impossible

within the time frame of the experiment to do it manually. A python algorithm was also used to clean

up the data as copied from the logger pro interface, which would again be impossible if done manually.

Voltage:

All the error expected is related to the fact that when a command from the computer is sent to the

variable-potential collector to change to a certain potential, the value is always off by a small amount.

That amount was found to be typically a constant 0.16% of the command that was sent. That value was

used as the error.

Conclusion:

The average distance between peaks for the normal 0.1v-increment sweeps was found to be 5.07

±0.07eV (as it is electrons we are dealing with). This value is 3.5% off the expected value of the

transition of Mercury from the 6s6s1S0 ground state to the 6s6p

3P1 excited state, which is 4.9eV.

By quantum mechanics, the above stated transition is by far the most probable transition given the

accelerating potentials used. Hence, at higher accelerating potentials, the peaks simply correspond to

when electrons have integer multiples of the energy for transition of Mercury from the 6s6s1S0 ground

state to the 6s6p3P1 excited state. So, what happens in the chamber is they are able to have multiple

inelastic collisions until they have no more kinetic energy.

For the smaller voltage increments, an oscillation was spotted, with an average peak width of 0.20 ±

.01v. Nonetheless, after significant searches, a transition with energy as low as 0.20eV was not found.

Nonetheless, the clear sign of periodicity in the current-voltage graphs show there is a good chance that

such a transition can be found.

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