Determination of h/e from the Photoelectric EffectBrian Lim and P. McEuenCornell University, Ithaca, NY 14853, USA(received 9 December 2005)By measuring the current generated in a photomultiplier tube (PMT), due to exposure from various frequencies the mercury arc, the photoelectric effect was observed. By varying the applied voltage and studying the subsequent I-V curves of the PMT, the stopping voltages of the generated photoelectrons are estimated, using linear approximations. Comparing the stopping voltages against frequency, h/e = 3.4±0.3×10−15Vs is determined within 18% from the accepted value of4.136×10−15Vs. Alternative methods to estimate stopping voltages are also briefly investigated and evaluated.1. IntroductionOne of the major break throughs by Einstein in his pivotal 1905 papers, the photoelectric effect1 helped to solidify the notion of light as quanta. Drawing from Planck's quantum hypothesis for blackbody radiation, that light has energy quantaE=hf, Einstein extended it to explain how the interaction of light with electrons of a metal give rise to photoelectrons. His photoelectric theory explains why and how the velocity of the electrons, and thus the measured voltage, varies with frequency and not light intensity. The latter affects the current of the electrons in this quantum theory, which is much contrary to the wave nature of light much accepted since Young's double slit experiment. First demonstrated successfully by Milikan2, the photoelectric experiment has become a staple in advanced undergraduate physics labs, using more advanced equipment, with the primary goal 1of determining the ratio of Planck's constant, h, against the electron charge, e, i.e. h/e, while revealing the intricacies involved in obtaining the results.2. Description of ApparatusThe experimental set-up (Figure 2.1) comprises (a) a mercury lamp (with a General Electric H100-A4/T 100W mercury bulb), to emit photons of specific frequencies, (b) a Hamamatsu photomultiplier tube 1P28 to generate photoelectrons, and (c) a disc, placed in front of the PMT with 5 filters: yellow, green, blue, violet, and ultraviolet. With these filters in front of the mercury source, the respective wavelengths, 5780A, 5461A, 4358A, 4047A, 3663A, are obtained. Current is measured with the Keithley 480 Picoammeter, and voltage with the Fluke 77 Multimeter. A black cloth is draped over the apparatus to reduce background light exposure.When conducting the experiment, it is best to have all the maximum forward currents equal to give the same normalization factor. Assuming that the magnitude of the current is directly proportional to the light intensity, the position of the mercury lamp, with respect to the PMT, is 2Figure 2.1: Apparatus set-up. The mercury source intensity can be adjusted by moving the lamp nearer/farther from the detector, or changing the angle of light incidence.+ –AV(a)(b)(c)adjusted to provide equal intensity. This is to compensate for the nonlinearity of the intensity of each frequency of the mercury spectrum (Figure 2.2), and the nonlinear spectral response of the PMT3. Unfortunately, this adjustment was done crudely by merely shifting the retort stand that supports the lamp until the current reading from the PMT is equal for the maximum forward currents.Furthermore, this calibration has to be repeated during the course of the experiment, due to a persistent drift of the maximum forward current. For the yellow and violet fequencies, with the weakest detection rates, the most profound drifting is observed, as the lamp is situated closest to filters and detector, causing their significant heating. The drifting is reduced a little when the lamp is brought farther away from the detector, but this compromises the requirement of maintaining a constant normalization factor of maximum forward current for all frequencies. Therefore, only the yellow frequency has a low current intensity, while the other frequencies are equally high. However, even with these measures, the drift is noticeable and causes the most significant of errors for this experiment.3. TheoryThe photoelectric effect involves the absorption of energy from a photon,E=h f, by an electron, such that the latter has sufficient energy to escape the metal surface of the cathode. Depending on the original energy of the electron in the metal, part of the photon energy goes into the work 3Figure 2.2: An example of the nonlinear intensity distribution of the mercury spectrum. Src: http://ocw.mit.edu/NR/rdonlyres/Physics/8-13-14Fall-2004-Spring-2005/A0478CD2-2136-4FA4-8EDC-85065AB6C2C1/0/jlexp002.pdffunction, while the remainder contributes to its kinetic energy. Thus derives the famous equation by Einstein:eV =12mev2=h f −, (3.1)where the energy of the electron can also be represented as eV, the electron charge multiplied by its voltage. Figure 3.1 illustrates the generation of photoelectrons from illumination.Of the electrons that escape, there is a maximum velocity which can be determined in terms of voltage. By applying a potential difference across the cathode and anode, opposing the electron path, the flow of photoelectrons can be made to stop. The stopping voltage, at which this just happens, can be used to calculate the ratio h/e, by the equationVs=hef −me. (3.2)The distribution of energy of the electrons in the metal surface can be explained using quantum mechanics and Fermi statistics4. Being Fermi particles, electrons follow the Fermi-Dirac distribution5n k =1eε k−/ kBT1which describes the average number of electrons distributed by their wavenumber, k, where ε(k) is the 4Figure 3.1: Circuit schematic of apparatusVAcathodeanodeelectron energy, μ is the chemical potential at that state, kB is the Boltzmann constant, and T is the temperature. In terms of energy, the density of states in 3D for a free particle isD ε ~ε6, giving the distribution of energy asn ε~∫0εεeε −/ kBT1dε.(3.3)The integrand in Equation 3.3 is plotted in Figure 3.2. Since ε ~ V and f ~ I, the I-V curve should follow Equation 3.3, and its derivative dI /dVvs. V should resemble Figure 3.3, within the validity of the assumptions.Figure 3.4 shows an energy level view7 of how the energy of an absorbed photon shifts the energy of a free electron in an Fermi sea (metal surface, for this case). Part of the energy hf is used to raise the electron energy above the Fermi energy, EF,
View Full Document
Unlocking...