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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.
4
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
5
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
6
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
7
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.
8
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
9
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.
10
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 ____________________
11
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.
12
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
13
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
14
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?