1

Determining the Critical Potentials for Helium: The Franck-Hertz Experiment

Trent H. Stein, Michele L. Stover, and David A. Dixon

Department of Chemistry, The University of Alabama, Shelby Hall, Tuscaloosa, AL, 35487-0336

Introduction:

The model for the atom developed over a number of years on the basis of key experiments and

insights. In 1897, J. J. Thomson showed that cathode rays are what we now call electrons and

measured the charge to mass ratio of the electron by using crossed electric and magnetic fields.

He showed that the mass of the electron was small, more than 1000 times smaller than the

hydrogen atom. For this work, Thomson received the Nobel Prize in Physics in 1906. E.

Rutherford shot alpha particles at a gold foil and showed that only a few back-scattered. This led

to the nuclear model of the atom in 1911. Rutherford had already won the Nobel Prize in

Chemistry in 1908 for his studies of the disintegration of the elements and the chemistry of

radioactive processes. Building off of this work, Niels Bohr introduced his model of the

hydrogen atom in 1913 with the energy of the states of the atom quantized. He used classical

mechanics and electrostatics with the key idea of quantizing the angular momentum. He

predicted that the electrons in atoms can only exist in certain bound states (energy levels).

In 1914, J. Franck and G. Hertz confirmed the Bohr model for atoms that electrons only occupy

discrete quantized energy levels and made the first non-optical measurement of the quantum

nature of atoms. The experiment involved sending a beam of electrons though mercury vapor

and observing the loss of kinetic energy when an electron struck a mercury atom and excited it

Sketch of Franck-Hertz Apparatus

2

from its lowest energy state to a higher one. This occurred at 4.9 eV and all electrons with at

least this amount of energy would lose only 4.9 eV showing the quantum nature of atoms. There

were already hints of this in the solar spectrum and in the emission of light from atoms heated up

in a Bunsen burner but this was the first proof of this. In a second paper in May 1914, Franck and

Hertz then showed that the light emitted from a collision of the electrons with mercury atoms

was exactly at wavelengths corresponding to 4.9 eV which showed the relationship between

wavelength and energy as well as that between absorption and emission from excited atomic

states. Remember that they did not have today’s light sources so they used an electron whose

energy they could precisely control as the excitation source. Franck and Hertz were awarded the

Nobel Prize in Physics in 1925 for “their discovery of the laws governing the impact of an

electron upon an atom”.

Bohr’s derivation

Potential Energy (P.E.) of two charges = q1q2/4πε0r with ε0 = 8.854 x 10-12

C2J

-1

So for a nucleus with charge Z interacting with 1 electron, (r = electron-proton distance, e =

charge on the electron, h = Planck’s constant, F = force, L = angular momentum)

P.E. = -Ze2/4πε0r

Total E = ½(mev2) - Ze

2/4πε0r

F = mea and F(coulomb) = dV/dr = Ze2/4πε0r

2

For uniform centrifugal motion: a = v2/r so we can write

F = Ze2/4πε0r

2 = mev

2/r (1)

Bohr hypothesizes that only discrete levels are present for an electron orbiting the nucleus so

quantize angular momentum so

L = mevr = nh/2π with n a positive integer (fit n intervals into 2π = 360º, a circle) n = 1, 2, 3, …..

L = mevr = nh/2π (2)

This condition was later reinterpreted by de Broglie to imply that the electron existed in a

standing wave pattern with n full wavelengths along the orbit.

We now have 2 equations in 2 unknowns: V and r

Solve (2) for v = nh/2πmer and substitute into (1)

Ze2/4πε0r

2 = men

2h

2/r(2πmer)

2 = n

2h

2/r(2π)

2mer

3

rn = n2h

24πε0/4π

2Ze

2me = n

2h

2ε0/πZe

2me = (n

2/Z)a0

3

With a0 = Bohr radius = h2ε0/πe

2me = 0.529 Å

Then vn = Ze2/2ε0nh

Substitute back into total E expression to get:

En = -Z2e

4me/8n

2h

2ε0

2 = -2.18 x 10

-18 J (Z

2/n

2) = -13.61 eV (Z

2/n

2)

or we can write

En = -Z2e

2/2a0n

2

where we express the energy in units of Rydbergs with 1 RH = -2.18 x 10-18

J = -13.61 eV = 0.5

a.u. (atomic units)

What is the experiment?

One implication of Bohr’s assumptions is that the energy can only take on certain discrete

values. In particular, if the electron’s lowest possible energy is –Eo, then the other available

energies for the H atom are:

o o o oE E E E, , , ,...

4 9 16 25

(3)

are the available energy levels for the electron. The value for Eo in the Bohr model is 13.6eV for

the H atom as noted above.

It is difficult to measure the energy of an electron orbiting an atom directly. One can eject the

electron if a light source with enough energy is available but detecting the photoelectrons is hard

and one has to have the atoms in the gas phase. One can measure the absorption of light but

many atoms do not readily absorb in the visible. One has to get the atoms in the gas phase and

have an intense light source plus be able to measure the light absorption. One can measure the

energy emitted in the form of light, when an electron drops from a higher energy (higher n) state

to one with lower energy (lower n). Conservation of energy says that the amount of energy

emitted in such a transition is simply

Eemitted Eatom = Ef Ei (4)

Eo

1

n f

2 1

n i

2

, (5)

where ni and nf are integers corresponding to the electron’s initial and final states, respectively.

Because of hydrogen’s simple atomic structure (a nucleus plus one electron), Bohr’s model

applies directly.

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The helium transitions you will be investigating are different from the hydrogen spectrum in at

least two important respects:

1. Helium is a two-electron atom. Each electron must therefore interact not only with the

nucleus, but also with the other electron. This renders the energy spectrum of helium much

more complicated than that of hydrogen. The energy levels cannot be calculated accurately

without a complete quantum-mechanical treatment, which is quite difficult to do even though

there are only two electrons. (One has to accurately predict the interaction of two electron

probability densities as the electrons are not really point charges.)

2. You will be inducing transitions in the helium with incident electrons, rather than looking at

spontaneous transitions that emit photons as with hydrogen. This means that the selection

rule (∆l = ±1) that applied to the spontaneous transitions may no longer apply because that

rule is a consequence of the fact that photons have 1 unit of spin angular momentum and

angular momentum is conserved.

An energy level diagram for helium is shown in Figure 1. The singlet states are on the left and

the triplet states are on the right. Note that "singlet" states are those in which the two helium

electrons have opposite spin, while "triplet" states are those whose electrons have the same spin.

Figure 1. Helium energy level diagram showing the electron configuration. Spectroscopic

term and energy above the ground state of the first few energy levels of helium. The

horizontal dashed line indicates the ionization potential.

1s53s

1s43s

1s33s

1s23s

1s53p

1s43p

1s33p

1s23p

1s33d

1s43d

1s53d

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Experiment:

Basic idea: In a tube that has been evacuated and then filled with helium, free electrons are

accelerated by a voltage VA to form a divergent beam passing through a space at a constant

potential. To prevent the walls of the tube from becoming charged, the inner surface is coated

with a conducting material and connected to the anode A (see Figure 2). In the tube, there is a

ring-shaped collector electrode R, through which the divergent beam can pass without touching

it, even though the ring is at a slightly higher potential.

A small current IR, with a value in the order of picoamperes (10-12

amps), is measured at the

collector ring, and is found to depend on the accelerating voltage VA. It shows characteristic

maxima, which are caused by the fact that the electrons can undergo inelastic collisions with

helium atoms during their passage through the tube and excite the He atom into electronically

excites states.

The kinetic energy E of an electron is as follows: E = e • VA

where e is the elementary electron charge. If this energy corresponds exactly to a critical

potential of the helium atom (an excited state), all of the kinetic energy may be transferred to the

helium atom. In this instance the electron can then be attracted and collected by the collector

ring, thus contributing to an increased collector current IR. As the accelerating voltage is

increased, successively higher levels of the helium atom can be excited, until finally the kinetic

energy of the electron is enough to ionize the helium atom. As the accelerating voltage is

increased further, the collector current shows a steady increase.

Figure 2. Schematic diagram of critical potential tube

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Safety Instructions:

The Critical Potential tube is a hot cathode tube. Treat them carefully.

Do not subject the tube or leads to any tension or mechanical stresses.

If voltage or current is too high, or the cathode is at the wrong temperature, it can

lead to the tube becoming destroyed. Do not exceed the stated operating parameters.

Equiment:

Critical Potential Tube (helium)

Battery Unit (with AA battery)

Grounded Shield

Tube Holder

Control Unit (with charger)

DC Power Supply

7 Experiment Leads (connectors)

Multimeter

LabQuest 2 (with charger)

2 Differential Voltage Probes

USB connector

DC Power

Supply

Figure 3. Franck-Hertz Apparatus

LabQuest

Battery

Unit

Grounded

Shield

Tube

Holder

Control

Unit

Critical

Potential

Tube

Differential

Voltage

Probes

Multi-

meter

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Procedure Step A: Using the LabQuest

1) The Franck-Hertz apparatus should have already been set up by your TA. It should look

like Figure 3.

2) In this experiment, you will only be adjusting the voltage and current on the power

supply and the minimum and maximum accelerating voltage on the control unit.

3) You will NOT be adjusting any connectors, chargers, or the tube itself. Follow the

instructions carefully to avoid damage to the equipment. The pins on the tube are very

fragile and the tube is under vacuum.

4) Turn on the LabQuest. The screen should display 2 different potentials (red and blue).

5) To the right of the screen, you will see a small grey box with mode, rate,

and duration (see figure to the right). Use the stylus (located on the back of

the LabQuest) to tap on the box.

6) Set the Mode to Time based, the Rate to 10000 samples/s, and the Duration to 0.1 s.

(NOTE: The interval setting sets itself after the rate is set). Then tap OK.

7) You should now be back at the screen with the two potentials. Make sure that your rate

and duration have changed. (NOTE: If they did not, repeat step 6.)

8) In the top right corner, tap on the box with the graph .

9) To collect a set of data, press the play button or tap on the green arrow at the

bottom left of the screen.

10) After you collect your data, save it by tapping the file cabinet in the upper right

corner.

11) To switch between runs, tap on the button directly to the left of the file cabinet that says

run # . Then tap on the particular run you would like to see.

Procedure Step B: Setting the Accelerating Voltage (VA)

1) Twist the dial on the multimeter clockwise to 200m in the VDC section. The display

should turn on and read 0.00 mV (NOTE: There is a 200 (V) setting and a 200m (V)

setting.)

2) Since the leads on the multimeter have fixed pin tips, you will have to hold them in place

to make sure they are getting good metal-to-metal connection to take measurements.

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3) To adjust the maximum VA, connect the

COM input (black lead) on the

multimeter to the ground ( ) for the

output and the Voltage input (red lead)

on the multimeter to #3 for the output on

the control unit (see Figure 4).

4) Make sure that the leads are touching the

metal sides of the probe holes and slowly

turn the knob #3 clockwise or counter

clockwise to increase or decrease the

maximum VA respectively (see figure 4).

5) Adjust the maximum VA to be somewhere between 20 and 30 mV.

6) To adjust the minimum VA, leave the COM input on the multimeter in the ground ( )

for the output and move the Voltage input on the multimeter from #3 to #4 for the output

on the control unit (see Figure 4).

7) Hold the leads in place and slowly turn the knob #4 to adjust the minimum VA to be

somewhere between 10 and 20 mV (see figure 4). Note: Do not set the minimum and

maximum VAs equal to each other.

Procedure Step C: Data Collection

1) On the power supply, make sure that all 4 of the knobs (labeled current and voltage) are

turned off (to their counter-clockwise limit). Do not force the knobs, they will stop

turning at this limit. Then, press the power button on the power supply.

2) The top two knobs are the coarse (on the right) and fine (on the left) knobs for current

whereas the bottom two knobs are the coarse and fine knobs for voltage (NOTE: the fine

and coarse knobs are used for small and large adjustments respectively.)

3) You should see two values displayed on the power supply. Both should be approximately

zero. Voltage is on the left and current is on the right.

4) On your power supply, in the upper right corner of the screen, you will see a red CC. This

stands for constant current.

5) Slowly turn the coarse knob for the current clockwise until the red CC disappears and a

green CV appears at the bottom of the screen. This means you are in constant voltage.

6) Next, slowly turn the coarse knob for the voltage clockwise until the green CV disappears

and the red CC reappears. (NOTE: If your voltage gets to 4 V and the red light has not

Figure 4. Control Unit

knob

knob

V

(Max)

V

(Min)

COM

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turned on, STOP increasing the voltage. Raise your hand, and your TA will come check

your apparatus.)

7) Repeat steps 5 and 6 until your voltage is ~ 3 to 4.5 Volts and current is ~ 1 to 1.3 Amps.

8) At this point, your bulb should be lit up (see Figure 3). (NOTE: Do not move the bulb or

the stand to do this.)

9) On your LabQuest, tap play and collect your first

data set which corresponds to the first ionization

energy. Your data should look similar to Figure

5. (NOTE: Unlike Figure 5, your time is set to

0.1 s (which is twice as long as Figure 5), so you

will either see pieces of or a complete duplicate

set of curves. Ignore these.)

10) Does your data look like a smooth single line

(Figure 5) or does it jagged and distorted?

11) Slightly adjust your minimum and maximum VA on the control unit and your current and

voltage on the power supply within the ranges provided in this step.

a. Minimum VA: 10 – 20 mV

b. Maximum VA: 20 – 30 mV

c. Voltage: 3 – 4.5 V

d. Current: 1 – 1.3 Amps

12) Save your first run on the LabQuest. Then press

13) Continue to make adjustments until you get data

that looks like Figure 5. (NOTE: You do not have

to save all of the bad runs).

14) Save the “best” run on the LabQuest. Record which

run is your best run on the next page under the ‘1st

ionization’ data section. Also record the minimum

and maximum VAs from the control unit along with

the voltage and current from the power supply.

(NOTE: Don’t forget units)

15) It is also possible to get two complete sets of

curves in a single acceleration that correspond to

the 1st and 2

nd ionization energies (see figure 6).

16) To achieve this result, adjust the minimum and maximum VA on the control unit and the

current and voltage on the power supply within the ranges provided in this step.

a. Minimum VA: 10 – 20 mV

b. Maximum VA: 35 – 45 mV

c. Voltage: 2.0 – 3.0 V

d. Current: 0.9 – 1.0 Amps

Figure 5. 1st ionization

Figure 6. 1st and 2

nd ionization

play again.

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17) Tap play on the LabQuest. Your data should look similar to Figure 6. (NOTE:

Remember you are running for twice as long as the figure, so you will see a duplicate.)

18) Slightly adjust your minimum and maximum VA on the control unit and your current and

voltage on the power supply until you get a smooth single line that looks like figure 6

(see step 16 for the ranges). NOTE: You do not have to save all of the bad runs

19) Save the “best” run on the LabQuest and record which run is your best on this page under

the ‘1st and 2

nd ionization’ data section. Also record the minimum and maximum VAs

from the control unit along with the voltage and current from the power supply. (NOTE:

Don’t forget the units)

20) On the LabQuest, tap File (in the upper left-hand corner of the screen) and then Save.

Choose a name for your data collection and tap Save/OK. (NOTE: This will save all of

the runs that you have collected regardless of which one is currently on the screen).

21) Unplug the two differential voltage probes from the LabQuest.

22) Slowly adjust the voltage and current down on the power supply until the knobs are

turned off (to their counter-clockwise limit). Then press the power button.

23) At this point, you are done with your apparatus. Inform your TA and move on to the Data

Analysis Section.

Data:

1st ionization (step 14)

Best Run Number _________________

Minimum VA ______________ Maximum VA ______________

Voltage ___________________ Current ____________________

1st and 2

nd ionization (step 19)

Best Run Number _________________

Minimum VA ______________ Maximum VA ______________

Voltage ___________________ Current ____________________

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Data Analysis:

1. Use the USB cord provided to connect the LabQuest to the computer in the lab. If the

computer begins to install software/drivers/etc., wait until the installation is complete

before moving on to the next step.

2. Open Logger Pro. Click on File, go the LabQuest Browser, and click Open.

3. Click the file name you choose in step 20 in the previous section and click Open. (NOTE:

A prompt may appear. If it does, click continue without data collection.)

4. To the right of the screen, you should see your graphs superimposed on one another. To

the left, you should see a table with all of your data. Your data will be separated into runs

that contain 3 columns each (see figure below).

5. Use the scroll at the bottom of the screen to see your best run for your 1st ionization data

set on the previous page (NOTE: You may have to expand the table window to see all 3

columns for the run at once).

6. Click . The entire column should be selected. While holding down the Shift

key, click the other two column titles ( and ). At this point, all of the

data for the run should be selected.

7. Copy and paste the data into an Excel spread sheet. Insert a line above the data and add

back your column titles as they did not transfer over with the data.

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8. Follow steps 5 – 7 to copy the data from your best run for the 1st and 2

nd ionization data

set on the previous page into the spreadsheet. (NOTE: Use a blank column or a line to

separate your two runs on your Excel sheet)

9. Save your spreadsheet and email a copy of it to you and your lab partner.

10. From this point forward, the rest of the lab report can be completed outside of lab. If you

need help using Microsoft Excel, ask your TA before you leave lab. This lab requires

you use Excel to do multiple graphs and calculations.

Attach your four graphs to this report

a) Give each of your graphs a title, label your axes (including units), and label your lines if

you have more than one on a single graph

b) Graphs should be scatter plots with smooth lines and markers (see figure to right)

11. Plot a graph of Time (x-axis) vs. Potential (y-axis) for each of your two runs. NOTE:

Each graph should have 2 lines as you have two different potentials (see figures below).

Also you may need to manually adjust the ranges on the axes to better see the data.

12. At this point, you are done with your 1st and 2

nd ionization data set (step 19).

13. Copy and paste all 3 columns of your data from the 1st ionization data set (step 14) onto a

second tab in your Excel spreadsheet.

14. Delete the potential column that does not have the curves.

15. Additionally delete any partial data you may have. You only need one set of ionization

curves (see figure on next page).

-0.5

0

0.5

1

1.5

2

0 0.05 0.1

Po

ten

tia

l (V

)

Time (s)

1st ionization

-1

0

1

2

3

0 0.05 0.1

Po

ten

tia

l (V

)

Time (s)

1st and 2nd ionization

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16. Use the equation to calculate the collector current in

picoamps (pA). (NOTE: You will need to do this for all of your voltages. You should

have around 400.)

17. Plot a graph of Time (x-axis) vs. IR

(y-axis). (see figure to the right)

18. Identify the time of the tallest peak.

You can do this by hovering the

mouse over the peak. Record this

time in your spreadsheet as t1. (see

figure to the right) NOTE: Make sure

you record the time of the peak and

NOT the current of the peak.

19. Identify the time of the ionization

threshold, the point where the line

begins to increase before it drops to

zero. Record this time in your spreadsheet as t2. (see figure above) NOTE: You may want

to make your graph larger and/or adjust the values of the x-axis to make this easier to see.

20. Use the equation

to calculate the energy in eV at each of

your times, t (you should have about 400 of them). (NOTE: t1 and t2 are constants you

determined in steps 18 and 19)

21. Plot a graph of IR (y-axis)

vs. Energy (x-axis). (see

figure to the right)

22. Use the chart on page 4 to

identify the energy levels

of each peak.

23. Label each peak. (NOTE:

Use the terms to the left of

the energies in the chart,

minus the 1s, as your

labels. Example: 4p or 33s)

0

400

800

1200

1600

-0.005 0.005 0.015 0.025 0.035 0.045

I R (

pA

)

Time (s)

Time vs IR

t2

t1

0

200

400

600

800

1000

1200

1400

17.5 19.5 21.5 23.5 25.5

I R (

pA

)

E (eV)

IR vs E

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Questions:

1) How do these measurements support the ideas of quantum mechanics?

2) What are some possible sources of error in this experiment?

3) How does this experiment compare with spectroscopy of the hydrogen atom?