Determining Planck's Constant Using LEDs

In this paper a value for Planck’s constant is measured. The value found with this experiment is h = (5.6 ± 0.5)×10−34 J·s which within 2σ of the accepted value of h = 6.626×10−34 J·s, this constitutes reasonable agreement with theory for the purposes of this experiment. Introduction My problem is that I don’t know the value of Planck’s constant and I don’t want to take anyone else’s word for it. In order to determine Planck’s constant, the energy of a photon with a known frequency must be measured. In the context of light emitting diodes (LEDs), the band gap of the LED serves to give an approximate energy measurement for the photons emitted. Using a spectrometer, the wavelength of the emitted photons can be determined for improved experimental results. This procedure was first described in 1974 by O’Connor and O’Connor [1] as a way to measure Planck’s constant using the basic functionality of LEDs. As recently as 2008, Zhou and Cloninger described a method for using computers to improve data collection and analysis for such an experiment with the appropriate software [2]. The experiment described is useful for illustrating the operating characteristics of LEDs to undergraduate students. Band gap theory and it’s implications are presented in a straightforward manner through making use of the same type of threshold energy concept presented by Einstein in his 1905 paper explaining the photoelectric effect [3]. The methods presented in this paper can also be used to provide moderately experienced undergraduate students with a basic introduction to quantum mechanical concepts as they apply to interactions between electromagnetic radiation and solids through the measurement of one of the fundamental constants of nature. Experimental Design Max Planck first described the quantization of electromagnetic radiation emitted by a black-body as a solution to the Ultraviolet Catastrophe predicted by the classical RayleighJeans formulation describing radiation of a black-body [3]. In Planck’s formulation, the energy of the emitted radiation is proportional to the frequency via E = hf , (1) where E is the energy of the radiation, f is the frequency of the emitted radiation and h is Planck’s constant. Einstein took this argument one step farther by hypothesizing that all emitted electromagnetic radiation is quantized depending on the electron energy orbital structure of the material doing the emitting. The development of solid state physics not only verified this prediction but also determined the method by which semi-conductors take advantage of such a threshold emission behavior to either emit light or pass electrons across the band gap from the valence band to the conduction band in a semi-conducting material. For this paper, Equation 1 must be modified slightly since it is easier to make a voltage measurement than it is to make an energy measurement in relation to the light emitted by LEDs. When this is done we obtain E = eV0, (2) where E is the energy of the emitted photons, e is the charge of the electron and V 0 is the turn-on voltage of the LED. Figure 1 shows a diagram of the circuit used to perform this experiment. A power supply is connected to a switch which, when off, charges a 3 mF capacitor bank and, when on, disconnects the circuit from the power supply and discharges the capacitor bank through a current limiting resistor and an LED. By measuring the discharge curve of the capacitor bank using the cursors on an oscilloscope and recording the voltage and time at which the LED turns off during the capacitor bank discharge, V 0 for the LED is measured. Setting the right hand sides of both Equations 1 and 2 equal to each other we can solve for Planck’s constant and obtain h = eV0 λ c , (3) where we recognize that λ/c is just the inverse of the frequency of the emitted photons, h is Planck’s constant, e is the charge of the electron and V 0 is the turn-off voltage of the LED [2]. Since we measure λ and V 0, and we know c and e are fundamental constants we find a value for Planck’s constant. Instead of using Equation 3 however, the data can also be fit to a linear function described by Equation 1 using Matlab to give the undergraduate experimenter the option to practice data fitting and proper error analysis. Analysis To find Planck’s constant first the circuit must be wired properly according to Figure 1 or Figure 2. After this is done the leads of an oscilloscope are placed across the capacitor bank. By charging the capacitors with the power supply and then discharging the capacitors across the current limiting resistor and LED to ground, the discharge curve is used to determine the turn-off voltage of the LED according to an exponential fit described by y = Ae −t B + C, (4) where y represents the voltage of the capacitor bank with respect to time, V(t), B is the time constant of the circuit τ which is equal to RC for this resistor-capacitor circuit and C is the turn-off voltage of the LED. By choosing the values of the resistor and capacitor such that the decay of the capacitor bank over a period of a few time constants is observed on the oscilloscope, the trace is stopped at the time the LED turns off. After measuring the voltage and time data points of the exponential decay in charge experienced by the capacitor bank, Matlab is used to fit the data. Figures 3 through 6 show the decay curves of the capacitors across the different colored LEDs. The final data point for each figure is the turn-off voltage for the different LEDs. The reason different colors are used is because the threshold energy of the band gap is different for each diode since energy is inversely proportional to wavelength of emitted photons. This diversity of band gaps provides an opportunity to determine the best value possible for Planck’s constant. Figure 7 shows the linear fit used to determine Planck’s constant. The experimental value is h = (5.6 ± 0.5)×10−34 J·s which is almost within 2σ uncertainty of the accepted value of h = 6.626×10−34 J·s constituting reasonable agreement between the data collected and the accepted value for Planck’s constant. Conclusion This paper describes an experiment through which beginning undergraduate students can become familiarized with the basic concepts of quantum mechanics while in the context of an introductory electronics course. For this experiment, the cursor functionality of an oscilloscope was used to measure the exponential decay curves of the capacitor bank across the different LEDs. Such a method opens the experiment up to systematic error because of the lack of resolution on the screen of the o-scope. For improved results, students could write and use a Labview interface with a Labjack device that outputs the data collected to some type of spreadsheet data file. Better agreement with the accepted value for Planck’s constant should be attainable with such a set up. References 1. O’Connor, Leah R. O’Connor, Patrick J. ”Measuring Planck’s Constant Using a Light Emitting Diode. ” The Physics Teacher. 12. (1974): 423-425. 2. Cloninger, T. Zhou, F. ”Computer-Based Experiment for Determining Planck’s Constant Using LED’s” The Physics Teacher. 46. (2008): 413-415. 3. Dubson, Micheal A. , Taylor, John R. , Zafiratos, Chris D. (2003) Modern Physics for Scientists and Engineers. Prentice Hall. Figure 1: Shown is a diagram of the circuit used to perform this experiment. The three capacitors in parallel add up to have a capacitance of 3 milli-Farads, the current limiting resistor placed in series with the LED ensures that the diode won’t burn out. Figure 2: Shown is a picture of the circuit used to perform this experiment. The three capacitors in parallel add up to have a capacitance of 3 mF, the current limiting resistor placed in series with the LED ensures that the diode won’t burn out. Figure 3: Shown is the Matlab fit for the exponential decay of the capacitor bank across the red LED. Figure 4: Shown is the Matlab fit for the exponential decay of the capacitor bank across the yellow LED. Figure 5: Shown is the Matlab fit for the exponential decay of the capacitor bank across the green LED. Figure 6: Shown is the Matlab fit for the exponential decay of the capacitor bank across the blue LED. Figure 7: Shown is the Matlab fit for the linear relationship between energy and wavelength described by Equation 1, where the slope of the fit is Planck’s constant.