The Franck-Hertz Experiment David Ward The College of Charleston Phys 370/Experimental Physics Spring 1997 Abstract One of the most important experiments supporting quantum theory is the Franck-Hertz experiment, which demonstrates the discrete energy values in which an atom may exist.The method employed here differs from the original experiment in that the emission spectra are not analyzed.Instead, the changes in an electron beam's current after traversing a partially evacuated tube with mercury (Hg) vapor present is analyzed as the electron beam's energy is increased.It is found that the Hg only absorbs energy in discrete amounts by graphing the electron beam current after traversing the length of the tube as a function of the accelerating potential.Distinct dips are seen in the current at different potentials.The distance between peaks from our data is 5.01 0.5 eV corresponding to an energy transition of Hg from the ground state to the 63P1 orbital. Introduction In the early part of the twentieth century the structure of the atom was studied in depth.In the process of developing and refining a theory of atomic structure, scientists developed a number of theories and techniques designed to explain and investigate experimental atomic phenomena. Scientists had developed a periodic table of the known elements without realizing the part that the structure of atoms played in placing an element in that table.It had been found that atoms are related to electromagnetic phenomena, since molecules could be dissociated into their component elements by applying a current to them.The differences among magnetic, insulating and conducting materials were suspected to be related to atomic structure.Heated elements emitted light when a current passed through them.The light emitted by elements is at specific frequencies; which indicated a characteristic internal structure specific to each element. In 1897, J.J. Thomson developed a technique to deflect charged particles through magnetic fields.This technique provided indications about the charge-to-mass ratio andsign ofthe charge of an electron.In 1909, Ernest Rutherford, Hans Geiger and ErnestMarsden developed scattering experiments to probe the internal charge and mass distribution of atoms.In 1915, Niels Bohr developed mathematical techniques for predicting atomic behavior that incorporated the new ideas of quantization and correlated to experimental analysis of the wavelength spectra of hydrogen. Making quantum theory fit the experimental data led to refinements in understanding the atomic structure.For example, Bohr's calculations of the wavelength of hydrogen's spectra were slightly different from the experimental results.Resolving the differences led to the understanding that the electron does not simply revolve about the hydrogen nucleus, but both revolve around their combined center of mass. Prior to the experiments performed by James Franck and Gustav Hertz, the quantum theory had been applied to atoms to describe radiation of photons only.Franck and Hertz decided to study electrons, atoms, and the transfer ofkinetic energy.Their experiment extended the application of quantum theory to electrons and their energy levels, and provided a new technique for studying atomic structure. Theory Via thermionic emission, electrons are discharged into a partially evacuated glass tube.The tube has a small amount of mercury in it that when heated sufficiently becomes a vapor.The vapor pressure is proportional to the temperature to which the mercury is heated.An accelerating potential is established inside the tube by applying a potential difference between the cathode (thermionic emitter) and a wire mesh anode that acts as an electron filter.The electron filter removes low energy electrons and permits higher energy electrons to pass through the filter into a region of retarding potential (~-1.5V).During the electron beam's trip through the tube, it occasionally encounters a mercury atom.In general, the electron will collide inelastically or elastically with the mercury atom dependent upon the energy of the electron.Other possibilities, such as ionization of the mercury atom or spin polarizing collisions if the electron energy is below ~10 eV, may also result.1The occurrence of ionization can be minimized by choosing the proper vapor pressure. A typical thermionic electron has an energy of about E ~ KbT, where T is the absolute temperature of the metal and Kb is the Boltzmann constant.For usual emitter temperatures (~3000 K) the energy of thermionic electrons is about 0.1 eV.The electrons that pass through the filter have an energy around the accelerating potential times the charge of an electron.If an electron traverses the distance between the filter and the anode collector unimpeded or interacts with Hg only by elastic collisions it will have an energy of eV electron volts minus the eV from the retarding potential, and will be absorbed at the anode collector.By colliding with mercury along the way an inelastic collision may result and the mercury may absorb up to but not exceeding eV electron volts.Quantum theory predicts that the mercury atom will only be able to absorb energy in discrete values.The smallest amount of energy that it can absorb is related to the orbital transition of an electron in the ground state to the next energy level.Usually, the atom will radiate this absorbed energy and return to the ground state via a photon of wavelength corresponding to the energy by the relation: Em - Eg = h ,(1) where Em - Eg is the energy difference between the excited and ground states, h is Planck's constant, and is the frequency of the photon related to its wavelength and the spectral line associated with this excitation by, = c / = Em- Eg. (2) The incident electron loses this energy Em- Eg in an elastic collision.Electrons that lose most of their energy to one or more elastic collisions will not have enough energy to overcome the retarding potential, and thus return to the filter where they are absorbed. The general results from the original Franck-Hertz experiment are:2 a)the frequency of any line in the emission spectrum is independent of the accelerating potential V, but, b)any particular line will only appear if the accelerating potential V islarger than some critical value that was different for different lines. They also showed that the "turn-on potential", V, for any line could be calculated in the following way. 1)Using the measured spectrum of Hg and the Einstein hypothesis regarding the source of the discrete spectrum, one constructs an energy spectrum for the mercury atom. 2)A particular line, gm, associated the transition Eg Em, would appear in the observed spectrum if and only if the potential V satisfied the inequality, eV Em - Eg.(3) The interpretation of this effect within the Einstein picture of radiative emission is that unless the electrons have sufficient energy to excite an atom from the ground state to the energy state Em then no photons associated with transitions from that energy state will be emitted.Since the emission spectrum could not be related to any kind of normal modes of vibration of the atomthe results strengthened the belief of quantum theory that the atom exists in only discrete energy levels. The original experiment differs from the one in this lab in that only the detection of a discrete energy absorption by the Hg atoms is measured.The emission spectrum could have been resolved but probably would not be very convincing since the possibility of this apparatus emitting at the desired wavelength due to something other than excitation radiation is strong.Referring to the energy level diagram in figure 1, it is clear that the first excited state of mercury corresponds to an energy level of 4.67 eV.This is not, however, the orbital observed in this experiment.When the energy of the electrons reach 4.67 eV excitation to the 63P0 state is possible, but only a few electrons give up 4.67 eV due to its small cross section with the result that the electrons have a large mean free path.The remaining electrons continue to gain energy where many excite Hg to the 63P1 state as it has a high cross section above 4.9 eV.Some electrons continue up to 5.4 eV or more and excite Hg to the 63P2 state.The tendency for a larger fraction of the electrons to gain enough energy to excite 63P1and 63P2 states increases with decreasing vapor pressure and also depends on the specific tube design.3In this lab the tube was heated to 179 degrees Celsius and resulted in a vapor pressure too high for significant excitations to higher energy levels.However, in the temperature range of ~150 degrees Celsius and a distance from the electron filter to the anode of about ~5 cm, higher excitation levels may be observed in the data.4 After the energy of the electrons increases above ~5 eV, the accelerating potential may be increased until on the average the electrons have sufficient energy to excite two Hg atoms.This trend continues until the accelerating potential is too high and ionization results.By plotting the accelerating potential (proportional to the energy by an amount of the charge of an electron e) versus the resulting currents at the anode collector, the excitation of Hg is represented by dips in the otherwise monotonic curve of current as a function of accelerating potential.The distance in electron volts between peaks represents the excitation energy of Hg.Apparatus and Procedure Essentially the apparatus is the same as described in the theory section.The tube is enclosed and heating coils in the bottom of the enclosure provide heat to the tube.A thermometer may be inserted into the enclosure from above to measure the temperature of the tube.The temperature is controlled by adjusting the current to the heating coils.Circuit diagrams are provided in figure 2 along with the operational parameter ranges. The temperature should be adjusted to ~180 degrees Celsius, allowing about 10 minutes for changes to take place and for everything to be close to thermal equilibrium.We settled for 179 degrees Celsius after about an hour of fine tuning the temperature.The filament should then be heated by applying ~6.3 V potential difference to the emitter.When the emitter glows red a small accelerating potential should be established between the emitter and the electron filter.After applying -1.5 V retarding potential to the anode collector, measure the current through the anode collector.The accelerating voltage is then increased in increments of ~0.25 V and the current through the anode collector is measured for each accelerating potential.The data is then analyzed as described in the theory. Analysis There are several questions that need to be resolved regarding the data.From the graph in figure 3, the data seems to fit the theory well; there are six dips in the range of 30 V, which is expected from the theory.However, since the data is not continuous and apparently has fluctuations beyond the instrumental error, what method should be employed to find the distance between the peaks in the graph, and what uncertainty was encountered in the measurement of the current other than that due to instrumental error?Furthermore, once this is established we see that the position of the first peak does not correspond to the average potential difference between peaks, and the average excitation energy seems to be higher than the expected 4.9 eV. Addressing methods of estimating the uncertainty in the measurements made with the ammeter, there was in general a bit of fluctuation in the measurements made.Usually the needle on the ammeter would settle on or around a value.The instrumental error in these measurements is only 0.1 pA, but because readings were not always stable some other uncertainty should be included in this.The instrumental error, really, changed when a different range was used, as it was in this experiment, but unfortunately no data regarding changes in the range of the ammeter were recorded.In the graph of the data there are small and large deviations from the smooth path expected, and these may be used to get an estimate for the uncertainty. Refer to figure 4, if before and after an apparent deviation, there seems to be a return to the smooth path, then the deviations may be ignored and a smooth line inserted where the deviations occurred.This line may then be estimated as linear with some slope m.By comparing each deviation with a corresponding point on the line, then the magnitude of the deviation can be found from the estimated linear equation.If these deviations are considered to be primarily due to error in reading a fluctuating or unstable needle on the ammeter, then the average of these deviations may be used as an estimate of the uncertainty.The uncertainty in the ammeter readings was calculated by this method. The average distance between the peaks, which is also the excitation energy, would simply be the distance between the tops of two consecutive peaks if the data were continuous, but since it is not continuous then the top of each peak must be estimated.This was estimated by finding two similar (usually exactly the same) current measurements near the peaks, but on opposite sides of the peaks, and estimating the top of the peak to be half the distance between these two values. For the first peak, there was an observed maximum of 75 pA, but to the left was a measurement of 73 pA, and to the right was a measurement of 74 pA.These last two values were used to estimate the accelerating potential at which the top of the peak occurs.This value was calculated to be 6.76 eV, which is just a bit more than the observed maximum at 6.74 eV.The same method was employed for each of the peaks, except for the last peak (sixth peak) which never started its descent in our observations, and so no peak could be estimated; this is also the point at which ionization discharge was observed.The values for these peaks and their distances between one another (the measured excitation energy) is contained in figure 3. Although the accelerating potential differences between peaks is used to determine the excitation energy, the uncertainty in this calculated value is more than justthe uncertainty in the accelerating potential since uncertainty in the measured currentsintroduces uncertainty in the location of a peak.The relationship between the uncertaintyin the current and the potential is described by the area of an ellipse as illustrated in figure 5.The method used to estimate the location of the peaks only produced the location in terms of the potential.Since the current at this potential is not known, then the equation of an ellipse of the form, V2/ a+ I2/ b = 1(4) where V and I are potential and current respectively and a and b are the uncertainties in the potential and current respectively, may be used to estimate the uncertainty in the differences in potential between peaks as a function of the uncertainty in both the current and the potential.Had the data been almost continuous such that the intervals of increasing potential were less than the uncertainty in the potential, then the uncertainty ofthe differences between peaks would have been within the uncertainty of the potential.The uncertainty in the excitation energy is approximated by the differential of the potential from equation 4 as follows:VdV= [ (a/b) I / (a-I2(a/b) )1/2] dI.(5) With this approximation the uncertainty in the excitation energy of Hg came out to be ~0.5 for every peak.There was variation in the values, but not enough to affect the significant digits. The problem of the occurrence of the first peak not corresponding to the average excitation energy is easily resolved in terms of the contact potential difference.Because of the loss and gain of energy by the electron beam due to the work functions of the cathode and anode, which have different work functions, there is a shift in the occurrence of the peaks.This does not affect the calculation of the average excitation energy because it was acquired by differences in the peak values, and since the effect due to the contact potential differences does not vary then the effect is subtracted out in the differences between peaks.The contact potential difference may be calculated by subtracting from the first peak the average potential difference between peaks.This leads to a contact potential of 1.75 V, or an energy difference in the work functions of 1.75 eV. The accepted value of the excitation energy in the Franck-Hertz experiment is 4.89 eV,5 but the values for the excitation energy acquired should not be expected to be this value.The vapor pressure of the mercury and its associated mean free path have a lot to do with this.The primary considerations in deviations from 4.9 eV are the vapor pressure of the mercury, which is temperature dependent, and the distance between the electron filter or accelerating grid and the collector anode.Hanne reports that the excitation energy for vapor pressure times filter collector distance of 4 mbar cm is 5.15 eV, for 20 mbar cm it is 4.9 eV, and for 100 mbar cm it is 4.8 eV.6No measurements of the filter collector distance were made or found in the notes provided with the apparatus, and although the temperature was recorded as 179 degrees Celsius, this is not enough to calculate the vapor pressure.It is, however,reassuring to know that deviations from the accepted value are expected for different configurations. There appeared to be a small repetitive variation in the data on the descending side of peak three, four, and five, however,the energy values associated in the differences between these variations did not correspond to any of the transition energies associated with the mercury atom. The variations are a bit larger than the others encountered which were assumed to be due to observational error from the fluctuations in the measurements.The occurrence of these larger variations seem to suggest that they are systematic.A very possible cause may be the regular change in the range of the ammeter, which later encountered technical problems and was sent to be repaired.However, no data was recorded regarding when the change in range took place, and since the ammeter has not been functional the areas of interest could not be investigated further.If this is not the cause of the apparently systematic variations, then they are probably just a random variation due to the same uncertainty as the other smaller variations.It should be noted that in favor of the latter argument, there were variations of the same size as the repetitive variations occurring at random.Conclusion The excitation energy associated with the absorption of energy by mercury atoms was found to be5.0 0.5 eV in this experiment.The value was obtained by taking the average of the energies from the graph in figure 3, and in particular the energies associated with the estimated peaks and not the measured ones were used.This differs from the accepted value of 4.89 eV by ~2.5 %, but from the literature by Hanne we see that this does not necessarily indicate that we did something wrong.Since the excitation energy observed may vary due to configuration differences, the higher value we obtained seems that much more reasonable.This does not mean that the discrete energy amount responsible for the excitation of Hg changes, but only that deviations from this value are observed systematically because of the method used to determine the excitation energy.The uncertainty in the excitation energy observed seems a little high, but the crude method of approximating the contribution to the uncertainty from the uncertainty in the measured current is better than assuming that the uncertainty in the current does not contribute to the uncertainty in the excitation energy.Regardless of the accuracy of the data, the graph in figure 3 which demonstrates the absorption of energy in discrete values is of far greater importance than the actual amount of energy that is required for excitation.If the focus of this lab was to obtain as accurately as possible this energy value, then a method more similar to the original experiment in addition to the method used here would have been employed. 1D.R.A. McMahon, "Elastic electron-atom collision effects in the Franck-Hertz experiment," Am. J. Phys. 51, 1086-1091 (1983). 2C. Garrod, Twentieth Century Physics, (Faculty Publishing, California, 1984). 3G.F. Hanne, "What really happens in the Franck-Hertz experiment with mercury?," Am. J. Phys. 56, 696-700 (1988). 4F.H. Liu, "Franck-Hertz experiment with higher excitation level measurement," Am. J. Phys. 55,366-369 (1987). 5C. Garrod, Twentieth Century Physics, (Faculty Publishing, California, 1984). 6G.F. Hanne, "What really happens in the Franck-Hertz experiment with mercury?," Am. J. Phys. 56, 696-700 (1988). Bibliography C. Garrod, Twentieth Century Physics, (Faculty Publishing, California, 1984). G.F. Hanne, "What really happens in the Franck-Hertz experiment with mercury?," Am. J. Phys. 56, 696-700 (1988). F.H. Liu, "Franck-Hertz experiment with higher excitation level measurement," Am. J. Phys. 55, 366-369 (1987). D.R.A. McMahon, "Elastic electron-atom collision effects in the Franck-Hertz experiment," Am.J. Phys. 51, 1086-1091 (1983). S.T. and A.R. Thornton, Modern Physics for Scientists and Engineers, (Saunders College Publishing, 1993).