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The Frank-Hertz Experiment

March 25, 2014

Instructor: Prof. Robert Kramer

Dane Mettam

Lab Partners:

1. Abstract:

2. Introduction/Background:

3. Equipment Desc & Procedure:

4. Data:

5. Analysis:

6. Conclusion

Overall Lab Grade:

/ 100

/ 100

Robert Maldonado, Xuan Chen

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1. Abstract

The Franck-Hertz effect is replicated in order to verify the claim that energy is

transferred in quantized packets. Energy loss of free electrons to inelastic collisions with

mercury is investigated. A discrete quantization interval is investigated by observing the

current as electrons collide with gaseous mercury in a vacuum diode. An accelerating

voltage is applied to a grid and when the electrons collide inelastically with the mercury

atoms, the current is dropped. This is done multiple times to determine the set voltage

required to excite mercury. Mercury was observed to absorb 4.93±0.53eV and produced a

249.6nm wavelength. Current levels dropped each time a mercury atom was excited and

thus produced a dramatic decrease in current at the observed eV intervals. This

experiment proved without a doubt that inelastic collisions absorb the same amount of

energy at every collision, all the time.

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2. Introduction/Background

It was found that energy levels are emitted and absorbed in discrete packages

called photons. From the Bohr model the equation E=hf (equation 1) was obtained which says

that the energy of any beam of light is dependent upon the frequency of the light. This

was proven by the photoelectric effect. However, these effects still were not observed

with electrons on atoms. In 1914, J. Frank and G. Hertz conducted an experiment to test

the Bohr model and observe excitation of atoms with discrete energy levels and compare

with the excited atoms spectrum.

In the replication of the Franck-

Hertz experiment, electrons are

thermionically emitted in an oven from

a cathode by an acelerating voltage

through a grid to another grid and to an

anode. In between the two grids is

mercury vapor that the electrons collide

with. Two types of collisions occur

between the electron and mercury atom, elastic and inelastic collisions. In the elastic

collisions, the electron bounces off of the atom like a bouncy ball on a hard floor. Some

energy is lost form the moving electron to the stationary atom but not much since the

atom is so much larger. However, if the electrons are provided with enough energy with

the accelerating voltage then it will collide inelastically. An in elastic collision will take

nearly all of the energy from the electron and transfer it to the atom. The atom now in an

excited state will eventually stabalize by emitting the energy it gained in the collision as

light with the same energy. The emission of light returns the atom to its ground state. The

light emitted can hit the cathode to release further electrons as in the photoelectric effect.

The electron after the collision, continues on to the second grid with the loss of energy. A

small retarding voltage of 1.5 Volts on the grid is enough to prevent the electron from

reaching the anode. All of the other electrons that had elastic collisions still have more

than enough energy left over to go through the retarding voltage to the anode where it is

read as a small current. The graph shows the electrons position in a vacuum tube with the

described set up as a function of position.

Graph 1: Set up for an observation of a single excitation

Image 1: Diagram of inside the Franck-Hertz Tube

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Equation 3

There are two ways to measure the energy

levels absorbed and emitted by the atoms. The first

way is to detect the light it produces and measure its

wavelenth. This meathod is difficult and

impractical. The other meathod of measuring the

quantized energy levels is by measuring the current

with applied voltage. In this meathod, the current

will drop when the electrons collide inelastically. If

the electrons are given enough energy when initially

accelerated, they can have multiple collisions before they no longer have enough to

overcome the retarding voltage. Inelastic collisions will drastically reduce the current

and is seen on graph 2. If energy is indeed quantized, then the mercury atoms will take

the same amount of energy to exite it in every collision. This will be a dip in current at

regular voltage intervals. This regular interval will be the change in voltage (ΔV). The

observed ΔV is the same energy that the mercury atom absorbs. The emission of light

that it produces will be of the same magnitude. This energy should be the same as a line

in its spectrum of light. If ΔV is consistent and it corilates with a specific frequency of

light produced from mercury, then the theory that energy is quantized is validated. Also,

it will verify that atoms emit and absorb only in discrete amounts.

To verify this with the experiment, a larger voltage is applied to the electron so

that it can have multiple collisions. With mercury, ΔV is supposed to be 4.86eV which

corrilates to a wavelength of 253.7nm. To calculate the ΔV into the wavelenth, a simple

calculation is done. The voltage is divided form the constant 1237.8x10−9. 1.2378−6

𝛥𝑉=λ

As the temperature increases of the oven increases, the cathode also increases in

temperture which in turn increases the average energy of conduction electrons within the

cathode. As the electrons gain kinetic energy, they will eventually overcome the work

function of the cathode. So while the heat aids in the emission of electrons and vaporizes

the mercury, it also heats the anode which has its own work function. The work function

difference between the anode and cathode can be calculated with the first dip in current

versus the average. First an average of all of the ΔV is taken. The first ΔV should be

higher than the average and subtracted from each other. This gives the work function (ф).

𝛥𝑉̅̅ ̅̅ − 𝛥𝑉1 = ф

It is imperative that the temperture within the tube remains at a constant

temperature for the success of the experiment. The reason for this is that

mercury’s vapor pressure will vare greatly with temperature. Differences with the

temperature will create differences with the vapor pressure and will affect the overall out

come. If more heat is added then the vapor pressure will be higher. With a higher vapor

pressure, more elections will collide (N)which will inturn create more collisions per

volume (n). Increased collisions create flatter peaks in when analyzing current and

voltage. A lower temperature will allow more electrons to flow in to the anode and could

damage the tube. To calculate the change in pressure, the Clausius-Clapeyron equation is

Graph 2: Electron current as a

function of acceleration voltage

Equation 2

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used. The equation when graphed gives a linear description of temperature (T) to pressure

(P). There are three constants with in it. ∆𝐻𝑣𝑎𝑝 which is the enthalpy of vaporization, R

which is the universal gas constant, and C which is dependent upon the type of material

used and gives the y-intercept.

𝑙𝑛 (𝑃𝑓

𝑃𝑖) = −

∆𝐻𝑣𝑎𝑝

𝑅(

1

𝑇𝑓+

1

𝑇𝑖) + 𝐶

The electrons, when they collide with the mercury atom doesn’t just keep going

straight, it scatters in different directions. Very little kinetic energy from collisions is lost

from elastic collisions of the electron on the atom. The atom will deflect the electron into

a different direction. The electron will collide multiple times throughout the tube and will

average out a mean free path. Normally the mean free path of a particle is represented

with λ but since that is already being used for the wavelength in this report, the mean free

path will now be represented with ( ). The scattering is affected by the cross section (σ)

of the mercury atom. The number density of mercury is the same number as the number

of collisions per unit volume (n) which can also be refered to as atoms/𝑐𝑚3. The mean

free path can be calculated with equation 5. To determine the probability of whether any

given electron will scatter while moving throught the vapor can be calculated with

equation 6. The cross section can be estimated with the experiment if desired and used to

find the thickness ( )of the vapor or the probability of scattering.

=1

√2𝑛𝜎 𝑃(𝑠𝑐𝑎𝑡𝑡𝑒𝑟) = 𝜎𝑛

Equation 4

Equation 5 Equation 6

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3. Equipment Description and Procedure

1. Connect everything together and heat

up the mercury to 180±10̊C. Do not

exceed 200̊C. Wait 20 minutes for

mercury to heat up.

2. Set the ammeter or picometer to

measure positive and negative current.

3. With accelerating voltage at 0 volts,

increase the filament current until the

tube has an orange glow that is dim.

4. As filament current is increased; observe the anode current.

5. Ensure that the tube is not glowing blue and or white throughout the experiment.

If it does then lower the accelerating voltage by at least 10 volts.

6. Increase accelerating voltage to about 60 volts. Increase retarding voltage to 1.5

volts.

7. Record the current as a function of accelerating voltages. Repeat this step for

accelerating voltages of 50 volts, 40 volts, and 30 volts.

8. Record and calculate all data.

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4. Data

The chart below shows all of the data collected from the picometer. Each time the

current dropped down to a point, that point was subtracted from the next lowest point to

create a potential difference. Each difference is roughly the same and represents the

quatization energy of electrons colliding with mercury atoms inelastically. All voltage

changes are summed up then divided by 21 (the number of potential differences) to

obtain the mean value. The value of each difference was then subtracted from the mean

value then squared to get all of the values in the right most column. These values were

then summed together and divided by 20 (number of trials minus one). This gave the

variance. The square root was then taken to give the standard deviation. The reported

value is the mean plus or minus the standard deviation or 4.93±0.53eV.

Mean Quantized Voltage and Standard Deviation

Low-V High-V ΔV mean Σ(ΔV-V)^2

60V

7.09 12.11 5.02 4.929524 0.008185907

12.11 16.88 4.77 4.929524 0.025447907

16.88 21.98 5.10 4.929524 0.029062067

21.98 27.15 5.17 4.929524 0.057828707

27.15 32.00 4.85 4.929524 0.006324067

32.00 36.87 4.87 4.929524 0.003543107

36.87 41.75 4.88 4.929524 0.002452627

41.75 46.55 4.80 4.929524 0.016776467

50V

7.11 10.75 3.64 4.929524 1.662872147

10.72 15.75 5.03 4.929524 0.010095427

15.74 20.41 4.67 4.929524 0.067352707

20.41 25.64 5.23 4.929524 0.090285827

25.64 30.42 4.78 4.929524 0.022357427

30.42 35.19 4.77 4.929524 0.025447907

40V

8.65 13.64 4.99 4.929524 0.003657347

13.64 18.37 4.73 4.929524 0.039809827

18.37 23.54 5.17 4.929524 0.057828707

23.54 28.12 4.58 4.929524 0.122167027

30V

5.02 11.51 6.49 4.929524 2.435085347

11.51 17.20 5.69 4.929524 0.578323747

17.20 21.49 4.29 4.929524 0.408990947

Sum=103.5 Sum=5.673895238

Mean=4.930 Variance=0.283694762

Standard Deviation=0.532630042

Reported Value=4.93±0.53eV

Chart 1: Mean quantized voltage and standard deviation

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The chart on the previous page

came from graphs such as the one

right. The graphs are the current as

a function of the accelerating voltages.

Two points are chosen as shown and

The difference is automatically

calculated. The error of each

measurement is approximated to how

certain the person moving the cursor

onto the right spot feels that it is

center. Unlike the picture to the right,

the points chosen were actually at the lower points where the current bottoms out. The

difference between each bottom is about the same.

Each bottom signifies that a mercury atom has absorbed that specific amount of

energy. It absorbs the same amount at each bottom and does not change regardless of

accelerating voltage.

The wavelength of the light

produced form the excited mercury is

calculated to be 249.6nm which is consistent

with the spectrum of mercury that has a

significant line at 253.7nm. The actual

energy that mercury needs to create this

wavelength is 4.86eV which is within the

error. The percent error of the energy

needed is 1.44% and the percent error for

the wavelength calculation is 1.62%. In the

image to the right, it is clearly seen that

mercury has a spike of light at around 254nm which is within the error of calculation.

The equation to convert the voltage to wavelength is below.

𝜆 =1237.8𝑥10−9

𝑒𝑉

The anode and cathode each has a different work function as is calculated using

the first ΔV in a set of ΔV for a given accelerating voltage and is subtracted from the

mean value. This gives the potential between the two probes.It only affects the first

calculation. This equation is given on page 3, equation 3. ф=1.56±0.53V.

Graph 3: Current as function of voltage with 60V accelerating voltage

Image 7: Spectrum wavelengths of Hg radiation

Equation 7

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5. Analysis

The results of the experiment had very consistent results that signify a very low

probability for error. This shows with a percent error of 1.44% in the accelerating

voltage. The 4.93±0.53eV is within the error of 4.86eV. The mercury atom absorbs the

4.93±0.53eV of energy from the collisions and emits it in light quanta of equal value with

a frequency of 1.20x1015Hz and a wavelength of 249.6±54.6nm.

There were a few sources of error. Locating the right spot on each bottomed out

point since it was not perfectly pointed was probably the biggest source of error. Each of

those spots was varied a bit and not exact. Not many points were measured outside the

standard deviation, 80.95% of the ΔV were within 1 sigma of the error and 90.48% were

within 2 sigma. Another source of error was the heat of the oven. It was given in the lab

instructions that the temperature was supposed to be 180̊C but we kept it at 190±1̊C.

There was no adjustment in the calculations or in data collection to compensate for this.

The last source of error was the work function and potential between the anode and

cathode. The only initial voltage drop that makes sense to find the work function off of

was the final set when the accelerating voltage was set at 30 volts. This first value

should be higher than the rest but for whatever reason at 40V, 50V, and 60V; this was not

seen. The work function was calculated using only that number (6.49V) and the mean

voltage (ΔV).

One thing I was never able to figure out was what the constant 1237.8x10−9 was

from in determining the wavelength. This constant was found in Melissinos, Experiments

in Modern Physics but never derived. In short the accuracy of this constant cannot be

determined but it is a correct translation for electron volts to wavelength and provided me

with a very convenient way of determining the emitted spectrum frequency of mercury.

The error of the voltage was calculated by taking the mean of all of the ΔV. 𝛥𝑉̅̅ ̅̅

was then subtracted from each individual ΔV and then squared. The sum of all of these

was taken then divided by the number of trials minus 1 (20). The square root of this was

then taken to be 𝛿±0.53eV. This calculation is show below.

𝛿 = √∑(𝛥𝑉̅̅ ̅̅ −𝛥𝑉𝑖)2

𝑛−1

Equation 8

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6. Conclusion

This experiment was extremely successful and it showed that energy is absorbed

and emitted in discrete packages. The energy lost form inelastic collisions matched the

energy needed for mercury to emit a 249.6nm photon. Mercury was observed absorb

4.93±0.53eV to produce the 249.6nm wavelength. This emission matches up with what is

seen from the spectrum of mercury on page 7. In this investigation of quantization of

mercury vapor in a vacuum diode, the current levels dropped each time a mercury atom

became excited from the inelastic collisions and this occurred at regular intervals of

4.93±0.53V. This experiment proved without a doubt that inelastic collisions absorb the

same amount of energy at every collision, all the time. It also verified the results first

seen by Franck and Hertz in 1914.

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7. Assigned Questions

1. The mean free path of electrons propagating in a gas is given by 𝜆−1=σn where σ

is the total cross section summed for all processes, elastic and inelastic, and n is

the number density of the mercury. The number density can be found using the

tables for mercury vapor pressure in the CRC handbook. Assume that the elastic

scattering cross section is the area of a disk with the radius of a mercury atom.

Find the mean free path and 𝑡1/2 for the vapor at 180̊C. Compare this to the size

of the Franck-Hertz tube, and explain what is different in the case of elastic

scattering.

I could not find the vapor pressure of mercury or the size of the exact size of the tube.

Elastic scattering will move the electron in a random direction with almost all of its

momentum intact while inelastic collisions will eliminate most of its momentum and

energy.

2. Why is the retarding voltage needed to observe the Franck-Hertz effect?

The retarding voltage exists to ensure that electrons that scatter inelastically to not reach

the anode. That way, only elastic collisions are detected by the anode so that when a

sharp drop in the current is observed, it is the retarding voltage eliminating the electrons

that excite the mercury atoms. When it excites the atoms it loses most of its energy and

cannot move past it.

3. Should you use the positions of the peaks or of the valleys to determine the

excitation energy? Or both? Explain.

The valley points should be used when determining the excitation energy since it is a bit

sharper than the peaks. This will allow for a more precise measurement.

4. Why are some peaks and valleys smeared out rather than sharp?

The peaks and valleys are not pointed because of the energy dependence of the excitation

cross section and the fact that the thermionically emitted electrons are not distributed

evenly. Basically, the electrons are not evenly emitted from the cathode and the collisions

don’t exactly occur evenly at all point within the vacuum tube.

5. How would increasing the temperature affect your observations? Would there be

a higher or lower background current? Sharper of less sharp peaks? More peaks?

Increasing the temperature would increase the pressure of the mercury within the tube.

Increased pressure is calculated with the Clausius-Clapeyron equation. This would create

more collisions (N) and therefore more collisions per volume (n). Since each collision

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produces ultraviolet light, the increased collisions would cause significantly more

electrons to be emitted via the photo electric effect. Overall, this will lower the current

that actually gets to the anode and mask continuous discharge. By the time you would

notice it being continuous, it would already be damaged. Additionally, there would be an

increase in inelastic electrons moving back towards the cathode because of the retarding

voltage. The number of peaks would be unaffected but they would be flatter.

6. Look up the spectrum for mercury to see if there is a spectral line (or lines) whose

wavelength (energy) corresponds to the energy differences you measured for

mercury.

There is a very prominent spectral line at 253.7nm that corresponds to the accepted

electron voltage of 4.86eV. A 249.6nm wavelength was calculated from the observed

4.93±0.53eV.

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8. References Davidson, MW. Fundamentals of Mercury Arc Lamps. N.d. Chart. Zeiss Microscopy and Digital

Imaging, Tallahassee, Florida. Web. 8 Apr 2014. <http://zeiss-

campus.magnet.fsu.edu/articles/lightsources/mercuryarc.html>.

Franck-Hertz experiment with mercury. Diss. Leybold, Germany: Web.

Lab handout: PHYS 34300 Modern Physics; Robert Kramer, spring 2014 34300 lab.

Kramer, R. (2014). Modern physics notes. Informally published manuscript, Department of

Chemistry and Physics, Purdue University Calumet, Hammond, IN, Retrieved from

https://mycourses.purduecal.edu/webapps/portal/frameset.jsp?url=/webapps/blackboard/exec

ute/launcher?type=Course&id=_97498_1

Melissinos, A., & Napolitano, J. (2003). The franck-hertz experiment In J. Hayhurst

(Ed.), Experiments in Modern Physics. San Diego: Academic Press. doi:www.academicpress.com

Mettam, Dane. 2014. Photograph. n.p. Web. 25 Mar 2014.