The Franck-Hertz Experiment

Kristin M. Beck1, ∗ and Jacob Mainzer1, †

1Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627

We experimentally reproduce the key elements of the historic Franck-Hertz experiment. Electronsare accelerated through monatomic mercury vapor in an electric field. Inelastic scattering betweenthe electrons and target atoms resulted in quantized excitation. We measured the excitation value tobe 4.93 ± 0.06eV. A qualitative framework for understanding the temperature and electron-densitydependence of the signal is presented. We also discuss the parameters under which a blue-whiteplasma is observed for our experimental geometry.

At the turn of the twentieth century, the physics com-munity had just begun to tackle the problem of quanti-zation as observed in the Millikan oil drop experiment,atomic spectra and the photoelectric effect. The Franck-Hertz experiment, first conducted in 1914, is a historicexperiment that showed quantized internal energy exci-tation in atoms. The experiment provided confirmationfor Bohr’s theory of the hydrogen atom [1].

In the Franck-Hertz experiment, electrons are accel-erated by an electric field through a monatomic vapor.The electrons traveling through the vapor collide withthe atoms. These collisions are completely elastic un-less the electron has more energy than the first excitedstate of the atom. In such an elastic collision, the energytransfered to the atom by the electron is found from basicnonrelativistic kinetics to be

∆K =4mM

(m+M)2K0 (1)

where m is the mass of the electron, M is the mass of theatom and K0 is the initial electron energy. For collisionswith atoms, the coefficient 4mM

(m+M)2 is small as M � m.In this scenario, ∆K ≈ 4 m

MK0 and the energy that theelectron loses (by transferring it to the atom) in a singleelastic collision is negligible.

When the energy of the incident electrons is greaterthan the energy necessary to excite the atom, inelasticcollisions can take the atom to one of its low-lying excitedstates. The electron loses this energy. After the collision,subsequent acceleration of the electron may bring it onceagain into an energy regime where it can inelasticallycollide with the atoms in the vapor.

To measure the energy of electrons after they havetraveled some distance in the atomic vapor, an electricfield that decelerates the electrons is introduced betweena grid and an anode [Fig 1]. The potential is chosen sothat electrons with relatively high kinetic energies willpass between the grid and anode, resulting in a measur-able current Iag between the anode and the grid. Lessenergetic electrons will pass through the grid and then re-turn to it instead of reaching the anode. These electronsdo not contribute to Iag. Nicoletopoulos has pointedout that using this grid technique for measuring the cur-rent is intrusive and effects the experimental results [2].

FIG. 1: Experimental setup and wiring. In the commercially-available Franck-Hertz experiment apparatus produced byKlinger S.A. Corporation, a glass tube with a small quan-tity of liquid mercury is heated so that some of the mercurygoes into the gas phase. Electrons are liberated within thistube by a heated filament (F) and accelerated through the gasby an applied voltage (Vacc) between the cathode (H) and thegrid (G). A retarding electric field produced by a 1.5V batteryis applied between the grid and the anode (A). The resultingcurrent flow between the grid and anode (Iag) is measuredby a precision electrometer (model 610B produced by Keith-ley Instruments). Vacc and Vf are controlled through customelectronics interfacing with a computer through a data acqui-sition card produced by National Instruments. A LabViewprogram was used to vary Vacc and record Iag.

Nonetheless, this historical method is sufficient to observethe desired quantization.

The physical picture presented above makes many ide-alistic assumptions. For example, it ignores the elas-tic collisions, electron-atom interactions in the region ofspace between the grid and anode, and velocity distri-butions of atoms and electrons. Refinements to this ex-planation of the experiment taking into account modernkinetic theory are described in Refs [3–5].

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FIG. 2: Relevant energy levels of atomic mercury (Hg). Directexcitation of the 3P0, 3P1, 3P2 and 1P1 states is possible fromthe 1S0 ground state through inelastic collisions with elec-trons. Transitions from the 3P0 and 3P2 to the 1S0 groundstate are forbidden. Also pictured is the 2.27eV transition3S1 →3 P2, which is probably responsible for green light ob-served during the experiment. Quoted energies are from [6].

In order to conduct the experiment, we used a commer-cially available Franck-Hertz apparatus from the KlingerS.A. Corp. In this apparatus, mercury vapor was heatedto temperatures between 130 and 180◦C in a glass tube.Electrons liberated from a heated filament by means ofan applied voltage, Vf , were accelerated through the mer-cury vapor by an applied electric field [Fig 1].

Mercury was used for this experiment because its va-por is monatomic. Monatomic vapors are preferred overmolecular gases such as hydrogen for this experiment.Molecular gases have closely-spaced rovibrational statesthat effectively wash out the quantized states in the exci-tation spectrum. Furthermore, mercury’s vapor densityis variable with temperature, an easily controlled experi-mental variable. Treating mercury vapor as an ideal gas,the molar density n in mol

m3 of mercury is given by

n =133.3224× 10−3204/(T+273.15)+8.009 Pa

R(T + 273.15)(2)

where T is the temperature in degrees Celsius and R isthe gas constant, kNA [1]. For the temperatures used inthis experiment, we had mercury vapor densities between0.04583 and 0.3071 mol

m3 . Also, it is valid in a first-orderapproximation to ignore the energy transferred from elec-trons to mercury atoms in elastic collisions as the coef-ficient 4mM

(m+M)2 that appears in Eq. 1 (which describesthe energy an electron transfers to an atom in an elasticcollision) is approximately 1.094× 10−5.

The lowest lying states of mercury that can be accessedby inelastic collisions are the transitions from the 1S0

FIG. 3: Image of the mercury tube during experimentation.Note the three bands that appear in the areas where greenlight is observed. Based on the color, this light is most likelythe 546.0750nm emission line 3S1 →3 P2 of mercury [7].

ground state to the 3P0, 3P1, 3P2 and 1P1 states [Fig 2].All of these transitions are equally likely to be excited;however, this experimental setup favors the 1S0 →3 P1

4.89eV transition [8].The preference for the 1S0 →3 P1 transition can also

be explained as follows. In our idealized description, elec-tron energy increases linearly in the electric field betweenthe filament and the anode until it reaches an energy(which is proportional to distance) at which it can ex-cite the mercury atoms. The 3P0 state is the first to beexcited as it has the lowest lying energy. However, the3P0 state is metastable and does not decay quickly. Thevapor is moving slowly enough that the majority of themercury atoms at the proper distance for excitation tothe 3P0 state are transferred to this state. With the satu-ration at this distance scale, electrons travel through thevapor unaffected by elastic collisions with these excitedatoms until they have enough energy to excite the 3P1

state. The transition 3P1 →1 S0 is allowed, so the tran-sition rate is fast enough that the atoms at the properdistance do not become saturated with excited atoms.So, the observed current spikes can be accounted for by1S0 →3 P1 transitions. Not all electrons lose energy inthese transitions, so some will excite the higher energytransitions to 3P2 and 1P1. Transitions at these otherenergies (i.e. to the 3P0, 3P2 and 1P1 states) account forsome of the smoothing in the observed current signal.

While Vacc is changed, green light is observed comingfrom the mercury chamber. Based on our knowledge ofthe mercury spectrum, this light is most likely coming

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from the decay of the 3S1 to the 3P2 state. This reason-ably strong transition involves the emission of a 2.27eVphoton. The model described is based on the assump-tion that atoms remain in a ground state configuration;however, to get to the 3S1 state, the atoms must be inhigher energy state. It is interesting to note that, for thefirst few excitations, a banded structure can be observedin this light [Fig 3]. These bands can be explained bythe space-dependence of the electron energies. As dis-cussed above, electron energy is roughly proportional todistance. After traveling the proper distance to excite the3S1 state, an electron can inelastically collide with a lo-cal mercury atom, exciting the 3S1 state. The electron’senergy is reduced by this inelastic collision. The electronthen again must travel some distance in the electric fieldto increase its energy to a level at which it can again ex-cite a mercury atom. This rough distance-dependence ofelectron energy sets up bands of atoms that are excitedto the 3S1 state. These atoms can decay to the 3P2 state,emitting the observed green light.

For the conducted experiment, we measured the peri-odicity of the electron energy at the grid as follows. UsingLabView, anode current Iag was recorded while Vacc wasincreased from 0.0 to 45.0V in increments of 0.05V witha scan rate of 10 scans per second. For each run, Iag

and Vacc were stored in a text file. Each file was thenanalyzed using Microsoft Excel 2003, looking at the de-pendence of Iag on Vacc [Fig 4]. Electrons that reachedthe anode were registered as negative current. Becauseelectrons were decelerated between the grid and anode,this current is a measure of the electron energy. We havereason to believe that the observed -2 nA offset visible inFig 4 and all other data sets is a nonphysical product ofour data acquisition system.

The excitation potential for the favored 1S0 →3 P1

transition was determined by multiplying the elementaryelectron charge with the measured voltage difference (forVacc) between the current peaks in a run. Data pointsrepresenting the current peaks were selected by visuallyinspection of the graphs produced by Microsoft Excel.For each run, the voltage difference between adjacentcurrent peaks was calculated. The mean and standarddeviation of these measurements were calculated on arun-by-run basis for all runs, including runs with varyingoven temperatures and filament voltages. The final ex-citation potential was calculated by taking the weightedaverage of these measured values.

The excitation potential was measured to be 4.93 ±0.06 eV. This value agrees with the expected value of4.86 eV corresponding to the 1S0 →3 P1 transition [7].

This determination of the excitation potential of mer-cury also gives us an estimate for Planck’s constant, h.We know

hc/λ = eV0 (3)

where λ is the wavelength of the emitted light, c is the

speed of light, e is the charge of an electron and V0 is theexcitation potential for the 1S0 →3 P1 transition.

Solving equation 3 for h and using the known spectralvalue of for the wavelength of the first excited state ofmercury, 253.6521nm [7], we find an experimental valueof h to be 6.68 ± 0.08 ×10−34J · s. This agrees with theaccepted value of 6.626 068 96± 33× 10−34J · s [9].

We measured the excitation potential in a range oftemperatures (10◦C increments between 130 and 180◦C)and electron densities (as determined by Vf , which wasvaried between 4.0 and 6.3V). Seven runs were performedat 180◦C with different filament voltages. Below a fil-ament voltage of 5V, the tube current was not largeenough to observe over the background noise. For Vf

above 5V, the data showed clear current peaks. Highervalues of Vf resulted in progressively better-defined andlarger-amplitude current peaks. This increased dataquality was a result of the the increased tube current.Altering the filament voltage did not result in any sta-tistically significant change to the calculated excitationpotential.

Six runs were performed at temperatures between130◦C to 180◦C. These runs were all taken with a Vf of6.3V. Altering the temperature did not result in any sta-tistically significant change to the calculated excitation.However, data analysis showed that, for higher temper-atures, the current peaks began at a slightly lower volt-age. The maximum offset observed was 0.8V. This phe-nomenon was a result of the initial kinetic energy dis-tribution of the electrons. For higher temperature, thekinetic energy distribution is greater than that in a lowtemperature environment. An electron with a large ini-tial kinetic energy required slightly less acceleration to

FIG. 4: Data from a run taken at 180◦C with Vf at 6.3V. Forthis run, the observed excitation potential was calculated tobe 4.89 ± 0.26eV. The arrows indicate the calculated value forthe excitation potential from all 6 data sets, 4.93 ± 0.06eV,showing qualitative matching of the calculated value with thistypical run.

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FIG. 5: A blue-white plasma formed at the filament (bot-tom of glass tube). This plasma was observed at a varietyof values for temperature, Vacc and Vf . The image shownabove was taken at a temperature of 140◦C with Vacc=25Vand Vf =6.2V.

reach the lowest lying excitation energies for mercury.An interesting effect was observed for particular sets of

Vf , Vacc and temperature. While the accelerating volt-age was ramped up, a white-blue plasma formed nearthe filament in the tube. This plasma grew as the accel-erating voltage was increased. When the plasma finallyattached itself to the filament, the current readings nolonger moved in a sinusoidal pattern and the glowingdischarge in the expected range for mercury excitationstopped [Fig 5]. We call the value of Vacc at which theplasma attached itself to the filament the onset voltage,Vonset.Vonset was observed to be dependent on both Vf and

the temperature of the tube. While this phenomenonwas not studied exhaustively, there seemed to be a nega-tive, linear dependence on the temperature and a positivelinear dependence on Vf (which, in turn, was roughly re-lated to the electron density in the tube) [Fig 6]. Furtherstudy is warranted to characterize and understand thisphenomenon.

In summary, experimental results of the historicFranck-Hertz experiment were reproduced, giving valuesin agreement with the expected values within one stan-dard deviation. We explored the parameter space of tem-perature and filament voltage and determined that themeasured excitation voltage was independent of theseparameters. Finally, we described an interesting phe-nomenon in which a blue-white plasma was formed and

attached itself to the filament under highly repeatableconditions.

This work was conducted during Advanced Lab(PHY243W) at the University of Rochester.

∗ Electronic address: kristin.beck@rochester.edu† Electronic address: jacob.mainzer@rochester.edu

[1] UR Advanced Laboratory Manual, The Franck-HertzExperiment [Online] http://www.pas.rochester.edu/

~AdvLab/2-Frank-Hertz/Lab02%20Hertz.pdf

[2] P. Nicoletopoulos, Phys. Rev. E 78, 026403 (2008)[3] R.E. Robson, et.al., J. Phys. B 33, 507 (2000).[4] G.F. Hanne, Am. J. Phys. 56 (8) (1988).[5] D.R.A. McMahon, Am. J. Phys. 51 (12) (1983).[6] E.B. Saloman, J. Phys. Chem. Ref. Data 35, 4 (2006)[7] Yu. Ralchenko, Kramida, A.E., Reader, J., and

NIST ASD Team, NIST Atomic Spectra Database (version3.1.5), [Online]. http://physics.nist.gov/asd3 (2008)

[8] A. C. Melissinos, Experiments in Modern Physics Aca-demic Press (1966).

[9] P.J. Mohr, B.N. Taylor, D.B. Newell, Rev. Mod. Phys.80, 633 (2008)

FIG. 6: The above graphs show how Vonset, the value of Vacc

at which the blue-white plasma attaches itself to the filament,varies with temperature (a) and Vf (b). These graphs sug-gest that the relationship between Vonset and temperature isnegative and linear, while the relationship between Vonset andVf (and thus the electron density in the tube) is positive andlinear.